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![Chapter 1
Introduction
The New Code
Background
Rilles for the design of bridges have been the subject of
continuous amendment and development over the years,
amI a significant development took place in 1967. At that
time, a mee.ting was held to discuss the revision of British
Standard BS 153 I. t], on which many bridge design docu-
ments were based [2·1. It was suggested that a unified code
of practice should be written in terms of limit state design
which would cover steel, concrete and composite steel-
concrete bridges of any span. A number of sub-committees
were then formed to draft various sections of such a code;
the work of these sub-committees has culminated in Brit-
ish Standard 5400 which will, henceforth. be referred to in
this book as the Code.
The author understands that the Code Committee did not
intend to produce documents which would result in
significant changes in design practice but. rather, intended
that bridges designed to the Code would be broadly similar
to those designed to the then current documents. In addi-
tion the suh-committees concerned with the various ma-
terials and types of bridges had to produce documents which
would be compatible with each other.
In subsequent chapters, the background to the Code is
given in detail and suggestions made as to its interpretation
in praL'ticc. The remainder of this first chapter is concerned
with general aspects of the Code.
"*Code format
The Code consists of the ten parts listed in Table 1. I and.
at the time of writing, all except Parts 3 and 9 have been
published: drafts of these parts are available. Hence.
sufficient documents have been published to design con-
crete bridges. It sh(luld be noted that BS 5400 is both a
Code pf Practi<:e and a Specification. However. not all
aspects I of the design and construction of bridges are
covered; exceptions worthy of mention are the design of
"
Table I.IBS 5400 - the ten parts
Part Contents
I
2
:~
4
5
6
7
General statement
Specification for loads
Code of practice for design of steel hridges
Code of practice for design of concrete bridges
Code of practice f(lr design of composite bridges
Specification for materials and workmanship. steel
Specification for materials and workmanship. concrete.
reinforcement and prestressing tendons
Recommendations for m"terials and workmllnship.
concrete, reinforcement and prestressing tendons
9
10
Code of practice for bearings
Code of practice for fatigue
parapets and such constructional aspects as expansion
joints and waterproofing.
The contents of the individual parts are now sum-
marised.
Part 1
The philosophy of limit state design is presented and the
methods of analysis which may be adopted are stated in
general terms.
Part 2
Details are given of the loads to be considered for all types
of bridges. the partial safety factors to be applied to each
load and the load combinations to be adopted.
Part 3
Design rules for steel bridges are given but reference is not
made to Part 3 in this book. At the time of writing it is in
draft form.
Part 4
Design rules for reinforced. prestressed and '~()rnposile
(precast plus in-situ) concrete bridges are given in terms of
material properties, design criteria and methods of com-
pliance.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-7-2048.jpg)
![Part 5
~esign rules for steel-concrete composite bridges are
given and some of these are referred to in this book.
material from the Code both during the Code's drafting
stages and since publication.
Railway bridges
There should be fewer problems in implementing the Code
The. speci.fication of materials and workmanship in con. for railway 'bridges than for highway bridges, because Brit·
?~ctlon with structural steelwork are given, but reference,,1sh Rail have been using limit state design since 1974 [14].
IS not made to Part 6 in this book. ....;" ....~.~.w. ".,,"
Part 6
Part 7
The, speci.fication of materials and workmanship in con-
nectl~n "':Ith concrete, reinforcement and prestressing ten-
dons IS gIven.
Part 8
Recommendations are given for the application of Part 7.
Part 9
The design, lesting and specification of bridge bearings are
l'overed. At the time of writing Part 9 is in draft form but
some material on bearings is included in Part 2 as an
appendix which will eventually be superseded by Part 9.
Part 10
l.:o.ading~ for fatig.ue calculations and methods of assessing
f,lllgue life are gIven. Part 10 is concerned mainly with
s,tcel and steel-concrete composite bridges but some sec-
tlons are referred 10 in this book.
I~plementation of Code for concrete
bndges
Highway bridges
From the previous discussion, it can be seen that, if the
Code were to be adopted for concrete highway bridges: ,
I. Part 2. would replace the Department of Transport'~
Techmcal Memorandum BE 1/77 [3] and British
Standard BS 153 Part 3A r4 ] . '
2. Part 4 would replace the Department of Transport's
Technical Memoranda BE 1/73 [5J and BE 2/73 [6]
and Codes of Practice CP 114 [7], CP 115 [8] and
CP 11619]. '
.1, Parts 7 and 8 would replace the Department of Trans-
port's Specification for road and bridge works [10].
4. Part 9 would replace the Department of Transport's
Memoranda BE 1/76 [II] and 1M II [12].
. At the time of writing, the Department of Transport's
views on the implementation of the Code are summarised
in their Departmental Standard BD 1/78 [13]. This es-
sentially, states that the Code will, in due course', be
,~upp.lemented by Departmental design and specification
rCl.Jlllfcmcnts.ln addition, implementation is to be phased
o;('r an ~nst~ted. period of time, with an initial stage of
trial applicatIons of thc Code to selected schemes. How-
~:cr, ~t should be noted that several of the Department's
rcl'hnlcal Memoranda hnve been updated to incorporate
2
Development of design standards
for concrete structures
Before explaining the philosophy of limit state design it is
instructive to consider current design procedures for con-
crete bridges and to examine the trends that have taken
place in the development of codes of practice for concrete
, structures in general.
Current design procedures for concrete bridges are based
primarily on the requirements of a series of Technical
Memoranda issued by the Department of Transport (e.g.
BE 1173, BE 2/73 and BE 1/77); these in tllrn are based on
current Codes of Practice for. buildings (e.g. CP 114,
CP 11"5 and CP 116), with some important modifications
which reflect problems peculiar to bridges. '
Essentially, trial structures are analysed elastically to
determine maximum values of effects due to specified
working loads. Critical sections are then designed on a
,~modular ratio basis to ensure that certain specified stress
,;,Iimitations for both steel and concrete are not exceeded.
Thus, the approach is basically one of working loads and
permissible stresses, although there are also requirements
to check crack widths in reinforced concrete structures and
pto check the ultimate strength of prestressed concrete struc-
tures. This design pro~ess has three distinguishing
features - it is based on ~ Permissible working stress phil-
osophy, it assumes elastic material properties and it is
deterministic. Each of these, features will now be discussed
with reference to Table 1.2 which summarises the basic
requirements of the various structural concrete building
codes since 1934.
Permissible stresses
The permissible working stress design equation is:
stress due to working load ~ permissible working stress
where,
permissible working stress = material 'failure' stress
safety factor
Thus stresses are limited at the workinR load essentially to
provide an adequate margin of safety against/ai/lire. Such
a design approach was perfectly adequate whilst matedal
strengths were low and the safety factor high because the
pennissible working stresses were sufficiently low for ser-
viceability considerations (deflections and cracking) not to
be critical. In Ihis respect the most important consideration
is that of the permissible steel stress and Table 1.2 shows
Table 1.2 Summary of basic requirements from various Codes of Practice for structural concrete
Code Basis of analysis Steel stress Umltatlon Required load factor ' Additional design
and design (N/mml) requirements
OSIR Elastic analysis, with 140 for beams None for beams None
(1934) variable modular ratio 100 for columns 3.0 for columns
and permissible stresses f>0.45 fy
CP 114 As above, but m = 15 190 in tension For columns: Warning against
(1948) 140 in cQmpression 2.0 for steel excessive deflecti()Ds
f>0.50 fy 2.6 for concrete
····c
CP 114 Either elastic analysis 210 in tension 2.0 for steel Span/depth ratios
(1957) or load factor method 160 in compression 2.6 for concrete given for beams and slabs.
f>0.50 fy Warning against cracking
CP 115 Both elastic and ultimate Cracking avoided by 1.50 + 2.5L
(1959) load methods required limiting concrete tension
Warning against
or 2 (0 + L) excessive deflections
CP 114 Either elastic analysis 230 In tension 1.8 for steel More detailed span/depth
(as amended or load factor me.thod 170 in compression 2.3 for concrete ratios for deflection.
1965) f>0.55 £y Warning against cracking
CP 110 Limit state design No direct limit 1.6-1.8 for steel Detailed span/depths or
(1972) methods set, except by cracking 2.1-2.4 for concrete calculations for deflection.
and deflection
requirements
that the ratio of permissible steel stress to steel yield
strength has gradually increased over the years (i.e, the
safety factor has decreased). This, combined with the
introduction of high-strength reinforcement, has meant that
permissible steel stresses have risen to a level at which the
serviceability aspects of design have now to be considered
specifically. Table 1.2 shows that, as permissible steel
stresses have increased, more attention has been given in
building codes to deflection and cracking. In bridge
design, the consideration of the serviceability aspects of
design was reflected in the introduction of specific crack
control requirements in the Department of Transport
documents. It can thus be seen that the original simplicity
of the permissible working stress design philosophy has
been lost by the necessity to carry out further calculations
at the working load. Moreover, and of more concern, the
working stress designer is now in a position in which he is
using a design process in which the purposes of the various
criteria are far from self evident.
Elastic material behaviour,
It has long been recognised that steel and concrete exhibit
behaviour of a plastic nature at high stresses. Such
behaviour exposes undesirable features of working stress
design: beams designed on a working stress basis with
identical factors of safety applied to the stresses, but with
different steel percentages, have different factors of safety
against failure, and the capacity of an indeterminate struc·
ture to redistribute moments cannot be utilised if its plastic
properties are ignored. However, it was not until 1957,
with the introduction of the load factor method of design in
,CP 114, that the plastic properties of materials were
recognised, for all structural members, albeit disguised in a
workin~ stress format. The concept of considering the elas-
tic response of a structure at its working load and its plas-!
tic response at the ultimate load was first codified in the
Specific calculations for
crack width required
prestressed concrete code (CP 115) of 1959; and this code
may be regarded as the first British limit state design code.
Deterministic design
The existing design procedure is deterministic in that it is
implicitly assumed that it is possible to categorically state
that, under a specified loading condition, the stresses in
the materials, at certain points of the structure, will be of
uniquely calculable values. It is obvious that, due to the
inherent variabilities of both loads and material properties,
it is not possible to be deterministic and that a probabilistic
approach to design is necessary. Statistical methods were
introduced into CP 115 in .1959 to deal with the control of
concrete quality, but were not directly involved in the
design process.
Limit state design
The implication of the above developments is that it has
been necessary:
1. To consider more than one aspect of design (e.g.
strength, deflections and cracking).
2. To treat each of these aspects separately.
3. To consider the variable nature of loads and material
properties.
The latest building code, CP 110 [15], which introduced
limit state design in combination with characteristic values
and partial safety factors. was the culmination of these
trends and developments. CP 110 made it possible to treat
each aspect of design separately and logically, and to
recognise the inherent variability of both loads and material
properties in a more formal way. Although, with its intro-
duction into British design practice in CP 110 in 1972,
limit state design was considered as a revolutionary design
, 3](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-8-2048.jpg)
![Load
effect
Effect of 'If3 'IfL Ok
& 'If3 (effect of 'IfL Ok)
Effect of 'IfL Ok1----.(
'IfL Ok 'If3 '1ft. Ok Load
(a) Linear
Fig. 1.2(a),(b) Loads and load effects
important at the ultimate than the serviceability limit
state and thus a larger value .should be adopted for the
former.
It should be stated that the concept of using Y/3 can
create problems in design. The use of Y(3' applied as a
general multiplier to load effects to allow for analysis
accuracy, has been criticised by Beeby and Taylor [17].
They argue, from considerations of framed structures, that
this concept is not defensible on logical grounds, since:
I. For determinate structures there is no inaccuracy.
2. For many indeterminate structures, errors in analysis
are adequately covered by the capability of the struc-
ture to redistribute moment by virtue of its ductility
and hence Yp should be unity.
3. Parts of certain indeterminate structures (e.g. columns
in frames) have limited ductility and thus limited
scope for redistribution. This means large errors in
analysis can arise, and Yp should be much larger than
the suggested value of about 1.15 discussed in Chap-
ter 4.
4. There are structures where errors in analysis will lead
to moment requirements in an opposite sense to that
indicated in analysis. For example, consider the beam
of Fig. 1.3: at the support section it is logical to apply
Yp to the calculated bending moment, but at section
x-x the calculated moment is zero and Y/3 will have
zero effect. In addition, at section Y- Y, where a pro-
vision for a hogging moment is required, the appli-
cation of Yp. will merely increase the calculated sag-
ging moment. .
The above points were derived from considerations of
framed building structures, but are equally applicable to
bridge structures. In bridge design, the problems are
further complicated by the fact that, whereas in building
design complete spans are loaded, in bridge design posi-
tive or negative parts of influence lines are loaded: thus if
the influence line is not the 'true' line then the problem
discussed in paragraph 4 above is exacerbated. because the
designer is not even sure that he has the correct amount of
load on the bridge.
In view of these problems it seems sensible, in practice,
to look upon Yp. merely as a means of raising the global
load factor from Yo. Ym to an acceptably higher value of
6
Load
effect
'If3 (effect Of'lfL Ok) I---~
Effect of 'If3 'IfL Okl----If----7I~
Effect of 'IfL O"I------J'
'YfL Ok 'Yf3'/fL Ok Load
(b) Non-linear
- 'True' bending moments
--- Calculated bending moments
- - - 'If3 x calculated bending moments
X y
X
Fig. 1.3 Influence of Y/'J on continuous beams
Yp YfL Ym (Ym is defined in the next section). Indeed, in
early drafts of the Code, Yp was called Y1I (the gap factor)
and Henderson, Burt and Goodearl [18] have stated that
the latter was 'not statistical but intended to give a margin
of safety for the extreme circumstances where the lowest
strength may coincide with the most unlikely severity of
loading'. However. it could be argued that Yg was also
required for another reason. The YfL values are the same
for all bridges, and the. 1m values for a particular material,
which are discussed in, the next section, are the same,
irrespective of whether that material is used in a bridge of
steel, concrete or composite construction. An additional
requirement is that, for each type of construction, designs
in accordance with the Code and in accordance with the
existing documents should be similar. Hence,'-it is neces-
sary to introduce an additional partial safety factor (Y/I or
Yp.) which is a function of the type of construction (steel,
concrete or composite).
Thus Yf3 has had a rather confusing and debatable his-
tory!
Design strength of a material
At each limit state, design strengths are obtained from the
characteristic strengths by dividing by a partial safety fac-
tor (y",):
design strength = 'klym (1.4)
The partial safety factor. Ym' is a function of two other
partial safety factors:
Yml' which covers the possible reductions in the strength
of the materials in the'structure as a Whole as corn-
pared with the characteristic value deduced from the
control test specimens;
which covers possible weaknesses of the structure
Ym2' d . . th
arising from any cause other than the ~e uctlO~ .In e
strength of the materials allowed for to YmI' Includ-
ing manufacturing tolerances;
It is emphasised that individual values of Yml and Ym2
are not given in the Code but that values of Ym' for con-
crete bridges, are given in Part 4 of the Code; they are
discussed in Chapter 4. The values of Ym are dependent
, ...'.'"
upon:
. Material: concrete is a more variable material than
steel and thus has a greater Ym value.
2. Importance of limit state: greater values are used at
the ultimate than at the serviceability limit state.
because it is necessary to have a smaller probability of
the former being reached.
Design resistance of a structure or a
structural element
The design resistance of a structure at a particular limit
state is the maximum load that the structure can resist
without exceeding the design criteria appropriate to that
limit state. For example, the design resistance of a struc-
ture could be the load to cause collapse of the structure. or
to cause a crack width in excess of the allowable value at a
point on the structure.
Similarly the design resistance ofa structural element is
the maxim~m effect that the element can resist without
exceeding the design criteria. In the case of a beam. for
example, it could be the ultimate mome~t of resistance. or
the moment which causes a stress In excess of that
allowed.
The design resistance (R*) is obviously a function of the
characteristic strengths (tk) of the materials and of the par-
tial safety factors (Ym):
R* = function (/kly",) (1.5)
As an example. when considering the ultimate moment of
~esistance (Mil) of a beam
R* = Mil
(1.6)
and (see Chapter 5)
. ( at(f(Yrm)A.,)'
functlon ([k/y",) =(fvfy".,)A., d - (fc.,lY",c)b
(1.7)
where 'v,ICII =tk of steel and concrete. respecti~ely
y"••, Y"". =Ym of steel and concrete, respectively
A., =steel area
b =beam breadth
d =beam effective depth
at =concrete stress block parameter
. strengths. In such situations either values of R* or values
of the function of/k are given in the Code. An example is
the treatment of shear, which is discussed fully in Chapter
6: R* values for various values oflk are tabulated as allow-
able shear stresses.
Consequence factor
In addition to the partial safety factors YfL, Yp. and y~",
which are applied to the loads" load effects and material
properties. there is another partial safety factor (y,,) which
is mentioned in Part 1 of the Code.
Y" is a function of two other partial safety factors:
V"I. which allows for the nature of the structure and its
behaviour;
Y"z. which allows for the social and economic con-
sequences of failure.
logically. Y"l should be greater when failure occurs
suddenly. such as by shear or by buckling, than when it
occurs gradually. such as in a ductile flexural failure.
However, it is not necessary for a designer to consider Y,,1
when using the Code because, when necessary, it has been
Included in the derivation of the Ym values or of the func-
tions of Ilc used to obtain the design resistances R*.
Regarding Y"2, the consequence of failure of one large
bridge would be greater than that of Jne small bridge and
hence Y"2 should be larger for the former. However, th~
Code does not require a designer to apply Y"2 values: It
argues that the total consequences of failure are the same
whether the bridge is large or small, because a greater
number of smaller bridges are constructed. Thus, it is
assumed that. for the sum of the consequences, the risks
are broadly the same.
Hence, to summarise, neither Y"1 nor Y"2 need be con-
sidered when using the Code. '
Verification of structural adequacy
For a satisfactory design it is necessary to check that the
design resistance exceeds the design load effects:
R* ;r,S·
or function (/k, Ym) ;r, function (Qk' YfL' yp.)
(1.9)
(1.10)
This inequality simply means that adeq~ate l~a~
carrying capacity must be ensured ~t t~e ultimate h.mlt
state and that the various design cntena at the servIce-
ability limit state must be satisfied.
Summary
However, in some situations the design resistance is
calculated from
The main difference in the approach to concrete brid~e
design in the Code and in the current design do~uments IS
the concept of the partial safety fact?rs apphe~ ,'0 the
loads load effects and material properties. In addItion. as
is sh~wn in Chapter 4, some of the design criteria are dif-
ferent. However. concrete bridges designed to the Code
R* = [,function (jic))/y",
(1.8)
where Y"i is now a partial safety factor applied to the ~si~
tance (e.g. shear strength) appropriate to charactensttc
7](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-10-2048.jpg)
![should be very similar in proportions to those designed in
recent yearsj)ecause the design criteria and partial safetY
factors have been chosen to ensure that 'on average' this
will occur.
There is thus no short-term ~dvantage to be gained from
using the Code and, indeed, initially there will be the dis-
advantage of an increase in design time due to unfamili-
arity. Hopefully, the design time will decrease as designers
" .'
8
become familiar with the Code and can recognise the
critical'limit state for a particular design situation.
The advantage of the limit state format, as presented in
the Code, is that it does make it easier to incorporate new
data on loads, materials, methods of analysis and structural
behaviour as they become available. It is thus eminently
suitable for future development based on the results of
experience and research.
Chapter 2
Analysis
"* I .Genera reqUirements
The general requirements concerning methods of analysis
are set out in Part 1 of the Code, and more specific
requirements for concrete bridges are given in Part 4.
*"Serviceability limit state
Part 1 permits the use of linear elastic methods or non-
linear methods with appropriate allowances for loss of
stiffness due to cracking, creep, etc. The latter methods of
analYSis must be used where geometric changes signifi-
cantly modify the load effects; but such behaviour is
unlikely to occur at the serviceability limit state in a con-
crete bridge.
Although non-linear methods of analysis are available
for concrete bridge structures [19], they are more suited to
checking an existing structure, rather than to direct design;
this is because prior knowledge of the reinforcement at
each section is required in order to determine the stiff-
nesses. Thus the most likely application of such analyses is
that of <:hecking a structure at the serviceability limit state,
when it has already been designed by another method at
the ultimate limit state. Hence, it is anticipated that
analysis at the serviceability limit state, in accordance with
the Code, will be identical to current working load linear
elastic analysis.
Part 4 of the Code gives the following guidance on the
stiffnesses to be used in the analysis at the serviceability
limit state.
The flexural stiffness may be based upon:
1. The concrete section ignoring the presence of re-
inforcement.
2. The gross section including the reinforcement on a
,modular ratio basis.
3. The transformed section consisting of the concrete in
compression combined with the reinforcement on a
modular ratio basis./
However, whichever option is chosen, it should be used
consistently throughout the structure.
Axial, torsional and shearing stiffllesses may be based
upon the concrete section ignoring the presence of the re-
inforcement. The reinforcement can be ignored because
it is difficult to allow for it in a simple manner, and it is con-
sidered to be unlikely that severe cracking will OCClrr due
to these effects at the serviceability limit state.
Strictly, the moduli of elasticity and shear moduli to be
used in determining any of the stiffnesses should be those
appropriate to the mean strengths of the materials, because
when analysing a structure it is the overall response which
• is of interest. If there is a linear relationship between loads
and their effects, the values of the latter are determined by
the relative and not the absolute values of the stiffnesses.
Consequently, the same effects are calculated whether the
material properties are appropriate to the mean or charac-
teristic strengths of materials. Since the latter are used
throughout the Code, and not the mean strengths, the Code
permits them to be used for analysis. Values for the short
term elastic modulus of normal weight concrete are given
in a table in Part 4 of the Code, and Appendix A of Part 4
of the Code states that half of these values should be
adopted for analysis purposes at the serviceability limit
state. The tabulated values have been shown [20] to give
good agreement with experimental data. Poisson's ratio for
concrete is given as 0.2. The elastic modulus for rein-
forcement and prestressing steel is given as 200 kN/mmz,
except for alloy bars to BS 4486 [21] and 19-wire strand to
BS 4757 section 3 [22], in which case it is 175 kN/mm
z
.
It is also stated in Part 4 that shear lag effects may be
of importance in box sections and beam and slab decks
having large flange width-to-Iength ratios. In such cases the
designer is referred to the specialist literature, such as Roik
and Sedlacek [23], or to Part 5 of the Code, which deals
with steel-concrete composite bridges. Part 5 treats the
shear lag problem in terms of effective breadths, and gives
tables of an effective breadth parameter as a function of
the breadth-to-Iength ratio of the flange, the longitudinal
location of the section of interest, the type of loading (dis-
tributed or concentrated) and the support conditions. The
tables were based [24] on a parametric study of shear lag
in steel box girder bridges [25]. However, they are con-
sidered to be applicable to concrete flanges of composite
bridges [26] and, within the limitations of the effective
9](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-11-2048.jpg)
![I c'
, '
I
I
j ,:
should be very similar ;.
recent yearsl>ecause f'
factors have been c'
will occur.
There is thus
using the Coe'
advantage C'
arity. Hor
pplicable to concrete
ode permits the
s of analysis.
ed upon con-
~n-elastic dis-
_.,(S. Although such
_ -.If concrete bridge structure
-J and the HiUerborg strip method for
... .,clvisaged that the vast majority of structures
__,-.:;ofitlnue to be analysed elastically at the ultimate fimit
state. However, a simple plastic method could be used for
checking a structure at the ultimate limit state when it has
already been designed at the serviceabiIity limit state. Such
an approach would be most appropriate to prestressed con-
crete structures.
A design approach which is permitted in Part 4 of the
Code, and which is new to bridge design, although it is
well established in building codes, is redistribution of elas-
tic moments. This method is discussed later in this chap-
ter.
The stiffnesses to be adopted at the ultimate limit state
may be based upon nominal dimensions of the cross-
sections, and on the elastic moduli; or the stiffnesses may
be modified to allow for shear lag and cracking. As for
the serviceability limit state, whichever alternative is
selected, it should be used consistently throughout the
structure.
Part 4 of the Code also permits the designer to modify
elastic methods of analysis where experiment and experi-
ence have indicated that simplifications in the simulation
of the structure are possible. An example of such a sim-
plification would be an elastic analysis of a deck in which
the torsional stiffnesses are put equal to zero, although
they would be known to have definite values. Such a sim-
plification would result in a safe lower bound design, as
explained later in this chapter. and would avoid the com-
mon problems of interpreting and designing against the
torques and twisting moments output by the analysis.
However, the author is not aware of any experimental data
which, at present, justify such simplifications.
Elastic analysis at the ultimate limit state
The validity of basing a design against collapse upon an
elastic analysis has been questioned by a number of de-
signers, it being thought that this constitutes an anomaly. In
particular, for concrete structures, it is claimed that such
an approach cannot be correct because the elastic analysis
would generally be based upon stiffnesses calculated from
the uncracked section, whereas it is known that, at col-
lapse, the structure would be cracked. Although it is an-
ticipated that uncracked stiffnesses will usually be adopted
for analysis, it is emphasised that the use of cracked trans-
formed section stiffnesses are permitted.
In spite of the doubts that have been expressed, one
10
"*
should note that it is perfectly acceptable to use an elastic
analysis at the ultimate limit state and an anomaly does not
arise, even if uncracked stiffnesses are used. The basic
reason for this is that an elastic solution to a problem
satisfies' equilibrium everywhere and, if a structure is
designed in accordance with a set of stresses (or stress
resultants) which are in eqUilibrium and the yield stresses
(or stress resultants) are not exceeded anywhere, then a
safe lower bound design results. Clark has given a detailed,
explanation of this elsewhere [27].
It is emphasised that the elastic solution is merely one of
an infinity of possible equilibrium solutions. Reasons for
adopting the elastic solution based upon uncracked stiff-
nesses, rather than an inelastic solution, are:
I. Elastic solutions are readily available for most struc-
tures.
2. Prior knowledge of the reinforcement is not required.
3. Problems associated with the limited ductility of struc-
tural concrete are mitigated by the fact that all c~itical
sections tend to reach' yield simultaneously; thus stress
redistribution, which is dependent upon ductility, is
,minimised.
4. Reasonable service load behaviour is assured.
LocaI effects
When designing a bridge deck of box beam or beam and
slab construction, it is necessary to consider, in addition to
overall global effects, the local effects induced in the top
slab by wheel loads. Part 4 states that the local effects may
be calculated elastically, with due account taken of any
fixity existing between the slabs and webs. This conforms
with the current pr~ctice of assuming full fixity at the slab
and web junctions a:lld using either Pucher's influence sur-
faces [28] or West~igaard's equations [29].
As an alternativ~ to an elastic method at the ultimate
limit state, yield line theory, which is explained later, or
another plastic analysis may be used. The reference to
another plastic analysis was intended by the drafters to
permit the use of the Hillerborg strip method, which is also
explained later. However, this method is not readily ap-
.plicable to modern practice, which tends to omit transverse
diaphragms, except at !!Upports, with the result that top
slabs are, effectively, infinitely wide and supported on two
sides only.
In order to reduce the number of load positions to be
considered when combining global and local effects, it is
pemlitted to assume that the worst loading case for this
particular aspect of design occurs in the regions of sagging
moments of the structure as a whole. When making this
suggestion, the drafters had transverse sagging effects
primarily in mind because these are the dominant structural
effects in design terms. However, the worst loading case
for transverse hogging would occur in regions of global
and local hogging, such as over webs or beams in regions
of global transverse hogging; whereas the worse loading
case for longitudinal effects could be in regions of either
global compression or tension in the flange, in combi-
nation with the local longitudinal bending.
[-~-----
(a) Solid slab
[0000000000 I
(b) Voided slab
[0000.0_00_0.0 1
(c) Cellular slab
UUO'JO(d) Discrete boxes
I I I I I I I I I I(e) Beam and slab
Fig. 2.1(a)-(g) Bridge deck types
Types of bridge deck
General
Prior to discussing the available methods of analysis for
bridge decks, it is useful to consider the various types of
deck used in current practice, and to examine how these
can be divided for analysis purposes.
In' Fig. 2.1, the variolls cross-sections are shown dia-
grammatically. Those in Fig. 2.1 (a-e) are generally
analysed as two-dimensional infinitesimally thin structures,
and the effects of down-stand beams or webs are ignored;
whereas those in Fig. 2.1 (f and g) are generally analysed
as three-dimensional structures, and the behaviour of the
actual individual plates which make up the cross-section
considered.
The choice of a type of deck for a particular situation
obviously depends upon a great number of considerations,
such as span, site conditions, site location .and availability
of standard sections, materials and labour. These points
are referred to by a number of authors [30-33] and only
brief discussions of the various types of deck follow.
(f) Box girders
I(g) Widely spaced beam
and slab
Solid slabs
I
Solid slab bridges can be either cast in-situ, of reinforced
or prestressed construction, or can be of composite con-
struction, as shown in Fig. 2.2. In the latter case precast
prestressed beams, with bottom flanges, are placed ad-
jacent to each other and in-situ concrete placed between a~d
over the webs of the precast beams to form a composite
slab. The precast beams are often of a standardised form
[34,35].
Solid slabs are frequently the most economic form of
construction for spans up to about 12 m, for reinforced
concrete in-situ construction, and up to about 15 m, for
composite slabs using prestressed precast beams. The latter
are available for span ranges of 7-16 m [34] and 4-14 m
[35].
It is obviously valid to analyse either in-situ or com-
posite slabs as thin plates.
Voided slabs
For spans in excess of about 15 m the self weight effects
of solid slabs become prohibitive, and voids are introduced
11 '](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-12-2048.jpg)
![.,"1
~ ,
In-situ concrete
Precast beam
Fig. 2.2 Composite solid slab
Fig. 2.3 Continuous slab bridge
to reduce these effects. It is often necessary in continuous
slab bridges to make the centre span voided as shown in
Fig. 2.3, i~ order. ~o prevent uplift at the 'end supports
under certam condItions of loading. Voided slabs are gen-
erally used fol' spans of up to about 18 m and 25 m for
reinforced and post-tensioned construction respectively.
It should be mentioned that the cost of forming the voids
by means of polystyrene, heavy cardboard thin wood or
thin metal generally exceeds the cost of the concrete
replaced. Hence, economies arise only from the reduction
in the self weight effects, and, in the case of prestressed
construction, from the reduced area of concrete to be
!>tressed.
Voided slabs can be either cast in-situ, of reinforced or
~restressed construction, or can be of composite construc-
tion as shown in Fig. 2.4. The latter are constructed in a
similar manner to solid composite slabs, but void formers
are placed between the webs of the precast beams prior to
placing the in-situ concrete.
The presence of voids in a slab reduces the shear stiff-
ness of the slab in a direction perpendicular to the voids.
The implication of this is that it is not necessarily valid to
analyse such slabs by means of a conventional thin plate
an~lysi~ which ignores shearing deformations. Analyses
whIch ll1cJude the effects of shearing deformations are
available and are discussed later in this Chapter. However,
for the majority of practical voided slab cross-sections
these effects can be ignored.
Cellular slabs
Th~ i~troduction of rectangular, as opposed to circular,
vOIds 111 a slab obviously further reduces the self weight
effects; but causes greater shear flexibility of the slab, and
can result in construction problems for in-situ cellular slabs
due to the difficulty of placing the concrete beneath the
voids. The in-situ construction problems can be ove~come
by using precast prestressed beams in combi~ation with
in-situ concrete to form a composite cellular slab as shown
in Figs. 2.5 to 2.7. ,,'
The M-beam form of construction shown in Fig. 2.5 can
be used for spans in the range J 5 to 29 m [36], but has not
proved to be popular because of the two stages of in-situ
In-situ concrete
Precast beam
Fig. 1.4 Composite'v(}ided slab
Fig. 2.S Composite cellular slab using M-beams
In-situ concrete
Holes for
transverse pre-stress
Fig. 2.6 Composite cellular slab using box beams
'
In-situ concrete
Precast top hat beam
(a) Bridge cross section
(b) Beam detail
Fig. 2.7(a),(b) Composite cellular slab using top hat beams
[38]
In-situ concrete
Fig. 2.8 U-beam deck
concreting required on site and the necessity to thread
transverse reinforcement through holes at the bottom of the
webs of the precast beams [37].
There are a variety ot slundanl box beam seCIiOIlS
[34,35] (see Fig. 2.6) which can be used for spans in the
range 12 to 36 m, but the need for transverse prest.rcssing
tendons thrf)Ugh the dec.' creates site prohlE-lns,
The precast top hat, beam (see Fig. 2.7) which was
developed by G. Maunsell and P1II1ner [38] has the ndvan-
tage that no transverse rcinl'orcc'meJlt'or prestressing ten-
dons have to be threaded through the beam~.
In all forms or celhtlat slnb construction a considerahle
proportion of the cross,section i~ voided. It is generally
necessary to adopt either a thin plate analysis which allows
for the effects of shearing deformations, or ,to apply nn
appropriate modification to an Ilnalysis which ignores
them, as mentioned later in this chapter.
Discrete box beams
Discrete box beam decks can be constructed by casting an
in-situ top slab on precast prestressed U-beams [35,39] as
shown in Fig. 2.8. The advantage of such a form of con·
struction is that it is not necessary to thread transverse
reinforcement or prestressing tendons through the beams.
In addition, some benefit is gained from the torsional stiff-
ness of a closed box section, although this effect is not as
beneficial as it would be if the beams were connected
through the bottom, in addition to the top, flanges.
The structural behaviour of such decks is extremely
complex due to the fact that the cross-section consists of
alternate flexible top slabs and stiff boxes.,Strictly, such
decks should be analysed by methods which consider the
behaviour of the individual plates ,!hich make up the
cross-section but, in practice, they are often analysed by
means of a grillage representation.
Beam and slab
Beam and slab type of construction, consisting of precast
prestressed beams in combination with an in~situ top slab,
is frequently used for all spans; and precast beams are
available which can be used for span ranges of 12 to 36 m,
in the case of I-beams [34], and 15 to 29 m, in the case of
M-beams [36].
Beam and slab bridges are generally analysed as plane
grillages. This is not strictly correct because the neutral
surface is' a curved, rather than a plane, surface but, in
practice, it lis reasonable to consider it as a plan~.
Ox
x
o + iJO~ dx
It <1x
/ aOOy+ ...,..._Y- dy
dy
Fig. 2.9 Stress resultants acting on It plate element
Box girders
There is a wide range of box girder cross-sections and
,methods of construction. The latter include precast or in-
situ, reinforced or prestrellsed and the use of segmental
construction. Box girders are generally adopted for spans
in excess of about 30 m and a useful review of the various
structural forms has been carried out by Swann [40].
The structural behaviour of box girders and methods for
their analysis have been discussed by Maisel and Roll
[41].
Elastic methods of analysis
General
It is assumed that the reader is familiar with the, elastic
methods of analysis currently used in practice and only a
brief review of the various methods follows.
Orthotropic plate theory
An orthotropic plate is one which has different stiffnesses
in two orthogonal directions. Thus a voided slab is octho-
tropic, and a beam and slab deck, when analysed by means
of a plate analogy, is also orthotropic. It is emphasised that
bridge decks are generally orthotropic due to geometric
rather than material differences in two orthogonal direc-
tions.
If a plate element subjected to an intensity of loading of
q is considered in rectangular co-ordinates x,y which co-
incide with the directions of principal orthotropy, then the
bending moments per unit length (Mxo My), twisting
moment per unit length (Mxy) and shear forces per unit
length (Qx, Qy) which act on the element are shown in
Fig. 2.9.
13.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-13-2048.jpg)
![For equilibrium of the element it is required that [42]
'0
2
Mx_ 2 a2MXY + a2
My = -q (2.1)
'Ox2
ax 'Oy al
One should note that the equilibrium equation (2.1)
applies to any plate and is independent of the plate stiff-
nesses.
The constitutive relationships in terms of the plate dis-
placement (w) in the Z direction and the shear strains
(Yx, Yy) in the x and y directions respectively are [4~].
Mx =- Dx .2.. ( a.w _ Yx) - DJ ~ (~w-YY) .. (2.2)
ax, a,y 0)' oy
M D 2-.(aw - Y) - DJ 2-.(aw - Yx) (2.3)
y =- y ay ay y ax ax
M.<y = -Dxy [;y(~; -Yx) + ;x (~; -yy)J (2.4)
Qx =SxYx
Q.v = SyYy
(2.5)
(2.6)
where Dx, Dy are the flexural stiffnesses per unit length,
DI is the cross-flexural stiffness per unit length, Dxy is the
torsional stiffness per unit length and Sx. Sy are the shear
stiffnesses per unit length.
In conventional thin plate theory, it is assumed that Sx =
S =00 or Yx = Yy = 0, i.e. that the plate is stiff in shear: it
i; thelt possible to combine equations (2.2) to (2.4) to give
the following fourth order governing differential equation
for a shear stiff plate [42]
a4w ' a4w a4w_
Dx ax4 + 2(DI + 2Dxy) ax2al + Dy ay4 - q (2.7)
If the plate has finite values of the shear stiffnesses then
Ubove and Batdorf [44] have shown how it is possible to
obtain a sixth order. governing differential equation. How-
ever, as discussed later, it is reasonable for many bridge
decks to assume that one of the shear stiffnesses is infinite
and the other finite: it is then possible to obtain fairly
simple solutions to the goveming equations:
Series solutions
Many bridge decks are essentially prismatic rectangular
plates which are simply supported along two edges, and,
in such situations, it is possible to solve equation (2.7) by
making use of Fourier sine series for the displacement (w)
and the load (q) as follows (see Fig. 2.10)
w = ~ Ym(Y) sin m1l'x
m=1 L
(2.8)
q = ~ qm(y) sin !!!.!!!.
m= I
L
(2.9)
These expressions are chosen because they satisfy the
simply supported boundary conditions at x = 0 and L. Any
14
Fig. 2.10 Rectangular bridge deck
boundary conditions on the two other edges can be dealt !
Owith in the analysis. .
The series solution for bridge decks was originally due
to Guyon [45] and Massonnet [46], and was then
developed by Morice and Little [47] who, together with
Rowe, produced design charts which enable the calcula-
tions to be carried out by hand [48]. Cusens and Pama [49]
have published a more general treatment of the method
which extends its range of application and have also pre-
sented design charts for calculations by hand.
In addition to the above charts, computer programs exist
for performing series solutions such as the Department of
Transport's program ORTHOP [50].
It is emphasised that simple series solutions cannot be
obtained for non-prismatic decks in which the cross-
section varies longitudinally; nor for skew decks, because
it is not possible to satisfy the skew boundary conditions.
Although the series solutions are· for single span simply
supported decks, it is also possible to apply them to decks
which are continuous over discrete supports by using a
flexibility approach in which the di~crete supports are con-
sidered to be redundanci,es and zero displacement imposed
at each [49]. This apprdl).ch is used in the ORTHOP pro-
gram referred to above '
Series solutions for shear deformable
plates
If the bridge deck shown in Fig. 2.10 is considered to be a
voided or cellular slab, with the voids running in the span
direction x, then it is reasonable to consider the deck to be
shear stiff longitudinally (Sx = 00) but to be shear deform-
able transversely. In such a case it is possible to combine
equations (2.1) to (2.6) so that a series solution can be
obtained. This has been done by Morley [51] by represent-
ing the transverse shear, force by the following Fourier
sine series, in addition to using equations (2.8) and (2.9)
m= J
Qym sin m1l'x
L
(2.10)
This representation requires that, at the supports, Qy = O.
and the method is only applicable if there are rigid end
diap,hragms at the supports. Morley [51] presents design
charts which enable solutions to be obtained by hand, and
Elliott [52] has published a computer program which
solves the same problem. An alternative approach has been
presented by Cusens and Pama [49].
Folded plate method
The cellular slab, the discrete boxes or the box girders
shown in Fig. 2.1 can be considered to be composed of a
number of individual plates which span from abutment to
abutment and are joined along their edges to adjacent
plates. Such an assemblage of plates can be solved by the
folded plate method which was originally due to Goldberg
and Leve [53], and was subsequently developed by De
Fries-Skene and Scordelis [54] into the form in which it is
incorporated into the Department of Transport's computer
program MUPDI [55].
The folded plate method considers both in-plane and
bending effects in each plate and can thus deal with local
bending and distortional effects. It is a powerful method,
but the bridge must be right and have simple supports at
which there are diaphragms which can be considered to be
rigid in their own planes but flexible out of plane; in addi-
tion, the section must be prismatic because the solution is
obtained in terms of Fourier series. Continuous bridges can
be considered in the same way as that discussed previously
for the series solution of plates.
Finite elements
In the finite element approach, a structure is considered to
be divided into a number of elements which are connected
at specified nodal points. The method is the most versatile
of the available methods and, in principle, can solve
almost any problem of elastic bridge deck analysi~. The
reader is referred to one of the standard texts on fintte ele-
ment analysis for a full description of the method.
There are a great number of finite element programs
available which can handle a variety of structural forms. In
addition, there is a great variety of eleme~t shapes and
types: the latter include one-dimensional beam elements,
two-dimensional plane stress and plate bending elements,
and three-dimensional shell elements. The following
Department of Transport programs are readily available:
t. STRAND 2 [56] is for the analysis of reinforced and
prestressed concrete slabs and uses a triangular plate
bending element in combination with a triangular
plane stress element: in addition, beam elements,
which are assumed to have the same neutral axis as
that of the plate, can be used.
2. QUEST [57] is intended for the analysis of box gir-
ders and uses quadrilateral thin shell elements which
consider both bending and membrane stress resultants.
3. CASKET [58] is a general purpose finite element
program with facilities for plane stress, plate bending,
beam. plane truss, plane frame, space truss and space
frame elements, which are all compatible with each
other.
Analysis
Finite strips
The finite strip method is a particuhir type of finite element
analysis in which the elements consist of strips which run
the length of the structure and are connected along the strip
edges. The method is thus particularly suited to the
analysis of box girders and cellular slabs since they can be
naturally divided into strips.
The in-plane and out-of-plane Ciisplacements within a
strip are considered separately, and are represented by
Fourier series longitudinally and polynomials transversely.
Since Fourier series are used longitudinally, the method is
only applicable to right prismatic structures with simply
supported ends. However, intermediate supports can be
considered in the same way as that discussed previously
for the series solution of plates.
The finite strip method was originally developed by
Cheung [59] who adopted a third order polynomial for the
out-of-plane displacement function, and specified two
degrees of freedom (vertical displacement and rotation)
along the edges of each strip.
The in-plane displacement function is a first order
polynomial, which implies a linear distribution of in-plane
displacement across a strip, and there are two degrees' of
freedom (longitudinal and transverse displacements) along
the edges of each strip.
The use of a third order polynomial for the out-of-plane
displacement function results in discontinuities of trans-
verse moments at the strip edges, because only compati-
bility of deflection and slope is ensured. Hence, a large
number of strips is required in order to obtain an accurate
prediction of transverse moments. However, this can be
overcome by introducing a fifth order polynomial, which
ensures compatibility of curvature in addition to deflection
and slope. However, two additional 'degrees of freedom'
have to be introduced in order to determine the constants
of the polynomial; these 'degrees of freedom' are the cur-
vatures at the strip edges [60]. An alternative formulation.
which also uses a fifth order polynomial, involves the
introduction of an auxiliary nodal line in each strip and
only has the two degrees of freedom of deflection and slope
[49].
The auxiliary nodal line technique can also be adopted
for the in-plane effects, and a second order polynomial is
then used for the in-plane displacement function.
Grillage analysis
In a grillage analysis, the structure is idealised as a grillage
of interconnected beams. The beams are assigned flexural
and torsional stiffnesses appropriate to the part of the struc-
ture which they represent. A generalised slope-deflection
procedu(.e, or a matrix stiffness method. is then used to
calculate the vertical displacements and the rotations about
two horizontal axes at the joints. Hence the bending
moments, torques and shear forces of the grillage beams at
the joints can be determined.
Since the grillage method represents the structure by
means of beams, and cannot thus simulate the Poisson:s
15·](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-14-2048.jpg)
![ratio effects of continua, it should, strictly, be used only
for grillage structures. Nevertheless, the grillage method is
a very popular method of analysis among bridge engineers,
and it has been applied to the complete range of concrete
bridge structures [61]. When applied to voided or cellular
slabs or to box girders, a shear deformable grillage is fre-
quently used [61] in which shear stiffnesses, as well as
flexural and torsional stiffnesses, are assigned to the grill-
age members; and the slope-deflection equati~ns, or stiff-
ness matrices, modified accordingly. I
Guidance on the simulation of various tyJes of bridge
deck by a grillage is given by Hambly [61~ and West
[62].
Influence surfaces
A number of sets of influence surt'aces have been produced
in tabulated and graphical form for the analysis of isotropic
plates, and these are extremely useful in the preliminary
design stage of orthotropic, as well as isotropic, plates.
The influence surfaces have, g(lnerally, been derived
experimentally or by means of finite difference tech-
niques.
Influence surface va.1ues are available for rectangular
isotropic slabs [28], skew simply supported isotropic slabs
[63, 64, 65]. skew simply supported torsionless slabs [66J,
and skew continuous isotropic slabs [67, 68, 69].
Elastic stiffnesses
Plate analysis
When idealising a bridge deck as an orthotropic plate and
using a series solution, finite plate elements or finite strips,
the following stiffnesses are suggested for the various
types of deck. Whenever a composite section is being con-
sidered, due allowance should be made for any difference
between the elastic moduli of the two concretes by means
of the usual modular ratio approach. It is assumed
throughout that the longitudinal shear stiffness (Sx) is
infinite.
Solid slab
D,. = Dr = 12(I-v2)
Gh·1
D"y = 12
DI = vD)'
S,. = 00
(2.11)
(2.12)
(2.13)
(2.14)
where h is the slab thickness, E and v are the elastic
modulus and Poisson's ratio respectively of concrete, and
G is the shear modulus which is given by
E
G = 2(1 + v)
(2.15)
Second moment of area = I"
Fig. 2.11 Voided slab geometry
Voided slab
Elliott and Clark [70] have reported the results of finite
element analyses which were carried out to detemline the
flexural and torsional stiffnesses of voided slabs, and
which were checked experimentally. It was foueud th~~, for
a Poisson's ratio of 0.2, which is a reasonabl~ vahIliC to
adopt for concrete, the following equations for the~iff
nesses gave. values which agreed closely wi(h' those of
the analyses and the experiments.
Wi~h the notation of Fig. 2.11,
Dx = Elx
(2.16~
s(l- v2
)
Dy = ~~3 [1- 0.95 (~ r] (2.17)
D = - 1-084 -Gh
3
[ . (d r]~ 12 . h (2.18)
DI = vDy (2.19)
The author is not aware of any reliable published dat;!l
on the transverse shear stiffness of voided slabs. However,
an idea of the significal)c'e of transverse shear flexibility
can be obtained from theresults of a test on a model void-
ed slab bridge, with a depth of void to slab depth ratio of
0.786, reported by Elliott, Clark andSymmons [71]. The
experimental deflections· and strains, under simulated
highway loading, were compared with those predicted by
shear-stiff orthotropic plate analysis. It was found that the
observed load distribution was slightly inferior' to the
theoretical distribution in the uncracked state, bur wa~
similar after the bridge had been extensively cracked due
to longitudinal bending. In design terms, it thus seems
reasonable to ignore the effects of transverse shear flexi-
bility and to take the transverse shear stiffness 2S infinity.
Since the model slab had a void ratio which is very close
to the practical upper limit in prototype slabs, it is sug-
gested that the transverse shear stiffness caIn be taken as
infinity for all practical voided slabs. However, if it is
desired to specify a finite value of the transverse shear
stiffness, then approximate calculations carried Ot!l~ by
Elliott [72] suggest that, for practical void ratios, it is
about 15% of the value of a solid slab of the same overall
depth (h). The solid slab value is given by [73]
8,. =2Gh
6
(2.20)
Second moment
£&:£-g1+ s ~I
~-.~---I
Fig. 2.12 Cellular slab geometry
Hence, for a voided concrete slab
5 Gh
Sy =< 0.15 x '6 Gh ="8
Cellular slab
Second moment
of area =Iy
-9-j4 1 ~
(2.21)
Elliott [52] has suggested the following values for rela~
tively thin-walled cellular slabs having an area of vOi.d n~t
less than about .one-third of the gross area. The notation IS
illustrated in Fig. 2.12.
DI = v Dy
(2.22)
(2.23)
(2.24)
(2.25)
The transverse shear stiffness can be obtained by con-
sidering the distortion of a cell and ass~ming that points of
contraflexure occur in the flanges midway between the
webs as shown in Fig. 2.13 (as originally suggested by
Holmberg [74]). The shear stiffness can then be shown to
be
s = 24 Kw 12 + KI + K2
.I' S 12 + 4(KJ + K2) + KIK2
where
K Kw
I = Kfl
K KII'
2 = Kf2
Elw
KII• = h(l _ v2)
K - EI[1
fI - s(1- vi)
Elf?
Kj2 = $(1- v:Z)
Discrete boxes
With the notation of Fig. 2.14
D = Elx
x $
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(~.33)
(2.34)
If1
Fig, 2.13 Cellular distortion
Second
Second moment moment
~[Jl 0 U ~~ '~I+--.!..Z_~'
s
s
(e) Longitudinal (b) Transverse
Fig. 2.14(a),(b) Discrete box geometry
It is suggested that the torsi~nal stiffness be calculated by
assuming that the 'longitudinal torsional stiffness' is equal
to that of the shaded section shown in Fig. 2.14(a), and
the 'transverse torsional stiffness' is zero. If the sections of
top slab which do not form part of the box section are
ignored, the 'longitudinal torsional stiffness' is given by
the usual thin-walled box formula.
GI. a G (¢,) (2.35)
where Ao is the area within the median line of the box, t is
the box thickness at distance I from an origin, and the
integration path is the median line. Thus the 'longitudinal
torsional stiffness per unit width' is given by
GJx _ G (4A~ )s-s fdl
t
(2.36)
Since the 'transverse torsional stiffness' is taken to be
zero, the total torsional sHffness per unit width is also
given by equation (2.36). Clark [75] has demonstrated that
Dxy is one-quarter of the total torsional stiffness per unit
width calculated as above; hence
G (4A3)Dx)' = 4$ f~
(2.37)
Rather than calculat~ a specific value for the transverse
shear stiffness (Sv), it is suggested that the torsional stiff-
ness (D."v) be modified by applying the relevant reduction
factors given by Cusens and Pama [49].
Beam and slab
With the notation of Fig. 2.15
.7](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-15-2048.jpg)
![hI {
(a) longitudinal section
o
Second moment of area =Iy
......~"""'. rt(
S Torsional
j.-----!.y-----aJ~1S .. inertia =Jy
~~----~~y----_~~I~~-----S~y~--~. ~I
(b) Transverse section
Fig. 2.15(a),(b) Beam and slab geometry
Dx = Elx
s,.
D = Ely
y
Sy = 00
(2.38)
,(2.'39)
(2.40)
(2.41)
(2.42)
T~e ~orsional inertias (Jx and Jy) of the individual
longltudmal and, if present, transverse beams can be calcu-
lated by dividing the actual beams into a number (n) of
~om~nent rectangles as shown in Fig. 2.16. The torsional
I~ertla of the ith component rectangle of size b by h, .
given by I , IS
J, = k b~hi bl ~ h; )
J, = k b; M bl
~ h; (2.43)
The coefficient (k) is dependent upon the aspect ratio of
the rectangle, where the aspect ratio is always greater than
or equal to unity and is defined as b;lh; or h;lb; as appropri-
ate .. Valu~s o~ k are given in Table 2.1 [76]. The total
torsional me~tla. of a beam is then obtained by summing
those of the mdlvidual rectangles
n
.lr or Jv = l.: J;
;= I
Table 2.1 Torsional inertia constant for rectangles
Aspect ratio k
1.0
l.l
1.2
1.3
1.4
1.5
1.8
2,0
2.3
18
0.141
0.154
0.166
0.175
0.186
0.196
0.216
0.229
0.242
Aspect ratio
2.5
2.8
3.0
4.0
5.0
6.0
7.5
10.0
00
k
0.249
0.258
0.263
0.281
0.291
0.298
0.305
0.312
0.333
(2.44)
(a) Actual (b) Component rectangles
Fig. 2.16(a),(b) Torsional inertia of beam
,The I~tter calculation is an approximation and, in fact,
~nderestlmates the tru~ torsional inertia because the junc-
tion effects, where adJacen't rectangles join, are ignored.
~ack~on [77] has suggested a modification to allow for the
Junction eff~cts, .but it is not generally necessary to carry
out the modificatIOn for practical sections [62].
Grillage analysis
General
Guidance on the evaluation of elastic stiffnesses for vllri-
O~IS types of deck, for use with grillage analysis, has been
glv~n by Hambly [.61J and West [62]. In general, Ham-
bly ~ recom.mend.atlOns are modifications of those given
prevI?usl.y In t~IS chapter for plate analysis, whereas
~est s differ qUite considerably when calculating torsional
stlffnesses.
It. should.be noted that an orthotropic plate has a single
torsIOnal st.lffness (Dx~), whereas an orthogonal grillage
can have different torsIOnal stiffnesses (GC GC)' th
d' t' f . X> Y In e
Irec Ions 0 Its two sets of beams: Furthermore sl'nc
'11 . ' ' e a
gn .age ~annot simulate ttlr Poisson effect of a plate, Pois-
son s r.atto does not appear in the expressions for the flex-
ural stlffnesses, and a grillage stiffness equivalent to D
does nO.t occur: ~t is .again assumed that the longitudinall
shear stiffness IS Infimte.
. The following general recommendations are a combina-
tion of those of Hambly [61] and West [62] and of the
author's personal views. Reference should be made to
Hambly [61] for more detailed information concerning
edge beams and other special cases. The recommendations
are given in terms of inertias rather than stiffnesses
because this is the form in which the input to a grillag~
analysis program is generally required.
. As for orthotropic plate stiffnesses, discussed earlier,
differences between the elastic moduli of the concretes in a
composite section should be taken into account by the
modular ratio approach.
Solid slab
The inertias of an individual grillage beam should be
obtained from the following inertias per unit width by
multiplying them by the breadth of slab represented by the
grillage beam.
Longitudinal flexural
h3
transverse flexural = 12 (2.45)
h3
= 6(2.46)
8eam and slab
Longitudinal torsional = transverse torsional
Transverse shear area = (l(),
(2.47)
Voided slab
The inertias of an individual grillage beam should be
obtained from the following inertias per unit width by
multiplying by the breadth of slab represented by the gril-
lage beam: the notation is given in Fig. 2.11.
Longitudinal flexural = Ixs "
Transverse flexural == ~; [1- O.95(~r]
Longitudinal torsional =transverse torsional ==
rll-O.84(~n
h
Transverse shear area == 00 or '8
Cellular slab
(2.48)
(2.49)
(2.50)
(2.51)
The inertias of an' individual grill,age beam should be
obtained from the following inertias per unit width by
multiplying them by the breadth of slab represented by the
grillage beam: the notation is given in Figs. 2.12 and
2.13.
Longitudinal flexural
I
:::..!.
s
(2.52)
Transverse flexural == 1, (2.53)
Longitudinal torsional == transverse torsional =21" (2.54)
Transverse shear area =
24K... _____12__+_K~I~+__K;2____
where
K
_ K...
,--Kn
K~ ==~!':.
" Kf2
1...
K.. == h
If1
Kfl ==--s
S
Discrete boxes
12 + 4 (K, + K2) + K,K2
(2.55)
(2.56)
(2.57)
(2.58)
(2.59)
(2.60)
Various writers l39. 61. 621 have proposed different
methods of simulating a deck of discrete boxes by means
of a grilIage. It is not clear which is the most appropriate
simulation to adopt in a particular situation, but whichever
simulation is chosen the author would recommend calcu-
lating tile stiffnesses as suggested by the proponent of the
chosen simulation.
A longitudinal grillage beam can represent part of the top
slab plus either a single physical beam, or a number of
physical beams. If a longitudinal grillage beam represents
It physical beams. where II is not necessarily an integer.
then its inertias are given. with the notation of Fig. 2.15,
by
Flexural = nIx'
Torsional = n J.. + -T
(2.61)
(2.62)
(
S 11
3
)
A transverse grillage beam can represent either part of
the top slab. or part of the top slab with a transverse physi-
cal diaphragm. The shear area it taken to be infinity and
the flexural and torsional inertias to be, with the notation
of Fig. 2.15, and SR being the spacing of the tranverse
grillage beams.
Flexural = Sil 1"S,..
(
J h
3
)
Torsional = s/I i;.+6
Plastic method~ of analysis
Introduction
(2.63)
(2.64)
In this section, examples of plastic methods of analysis
which could he used in bridge design are given. Howev('1'.
as mentioned previously. it is unlikely that. with the poss-
ible exception of yield line theory for slabs. such methods
will be incorporated into design procedures in the nellr
future. Before discussing the xarious methods. it is neces-
sary to introduce some concepts used in the theory of phls-
dcity and limit analysis.
Limit analysis
It is useful at this stage to distinguish between the terms
limit analysis and limit state design. Limit analysis is il
means of assessing the ultimate collapse load of a strUl:-
ture. whereas limit state design is a design procedure
which aims to achieve both acceptable service load
behaviour and sufficient strength. Thus limit analysis can
be used for calculations at the ultimate limit state in a limit
state design procedure.
A concept within limit analysis is that it is often not
possible to calculate a unique value for the coIlapse load (If
a structure: this is alien to one' s experience with the theory
of elasticity. where a single value of the load. required Ii)
produce a specific stress, at a particular point in a strur'
ture, can be calculated. In limit analysis, all that it is gel1'
eraIly possible to state is that the coIlapse load is between
two values. known as upper and lower bounds to the col-
lapse load. For certain structures coincidental upper and
lower hounds can be obtained. and thus the unique value
of the coIlapse load can he determined. However, this is
not the general case and, for a vast number of commonly
14](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-16-2048.jpg)
![occurring structures, coincidental upper and lower bounds
have not been determined. It is thus necessary to consider
two distinct types of analysis within limit analysis;
namely, upper and lower bound methods.
An upper bound method is unsafe in that it provides a
value of the collapse load which is either greater than or
equal to the true collapse load. The procedure for calcu-
lating an upper bound to the collapse load can be thought
of in terms of proposing a valid collapse inechanismand
equating the int~rnal plastic work to the work done by the
external loads.
A lower bound method is safe in that it provides a value
ot' the collapse load which is either less than or equal to the
true collapse load. A lower bound to the collapse load is
the lond corresponding to any statically admissible stress
(or stress resultant) field which nowhere violates the yield
criterion. A statically admissible stress field is one which
everywhere satisfies the equilibrium equations for the
structure. The expression 'nowhere violates the yield
criterion' essentially means that the section strength at
each point of the structure should not be exceeded. It is
important to note that neither deformations in any form nor
stiffnesses are mentioned when considering lower
bound methods.
Lower bound methods
As has been indicated earlier, any elastic method of
analysis is a lower bound method, in terms of limit
analysis. because it satisfies equilibrium. There are other
lower bound methods available which employ inelastic
stress distributions; but these have been developed gener-
ally with buildings, rather than bridges, in mind. The
inelastic lower bourid design of bridges is complicated by
the more complex boundary conditions, and the fact that
bridges are designed for different types of moving concen-
trated loads. In the following, lower bound methods which
could be adopted for bridges are described.
Hillerborg strip method
One inelastic lower bound method, which is mentioned
specifical1y in the Code. is the Hillerborg strip method [78]
for slabs, in which the two dimensional slab problem is
reduced to one dimensional beam design in two directions.
This is achieved by the designer choosing to make Mxy = 0
throughout the slab. Thus the slab equilibrium equation
(2.1) reduces to
,iM a2
M
ax{ + ~ = -q (2.65)
It is further assumed that, at any point, the load intensity q
can be split into components exq in the x direction and
(1 - ex) q in the y direction, so that equation (2.65) can be
split into two equilibrium equations
a
2
M . )-~ = -exq
(2.66)
a2M~, = -(1 _ ex)q
ay2
20
Equations (2.66) are the equilibrium equations for beams
running in the x and y directions. The designer is free to
choose any value of ex that he wishes, but zero or unity is
frequently chosen.
The above simplicity of the strip method breaks down
when concentrated loads are considered, because it is
necessary to introduce one of the following: complex
moment fields including twisting moments [78]; strong
ban~~~.~ement [79]; or one of the approaches sug-
gested by Kemp [S0;81L.
A:further problem in applying the strip method to bridge
design is that difficulties arise with slabs supported only on
two opposite edges, as OCCurS with slab bridges and top
slabs in modem construction. where the tendency is to
omit transverse diaphragms.
In view of the above comments it is considered unlikely
that the method will be used in bridge deck design, but it
is possible that it could be used for abutment design as
discussed in Chapter 9.
Lower bound design of box girders
A lower bound approach to the design of box girders has
been suggested by Spence and Morley [82]. The basis of
the method is that, instead of designing to resist elastic
distortional and warping stresses. which have a peaky
longitudinal distribution, it is assumed that, in the vicinity
of an eccentric concentrated load, there is a transmission
zone, having a length a little greater than that of the load,
which is considered to act as a diaphragm and to transfer
the load into pure torsion, bending and shear in the
remainder of the beam. Thus, outside the transmission
zone, the beam walls are subjected to only in-plane stress
resultants.
The design procedure is to statically replace an eccentric
load by equivalent pairs of symmetric and anti-symmetric
loads over the webs. The anti-symmetric loads, W in
Fig. 2.17, result in a dJstortional couple. The length (L,) of
the transmission zone 'i,s then chosen by the designer and
equilibrium of the zone under the warping forces (wf and
ww) and corner moments (mJ considered. It follows [82]
that these can be calculated from
Wb
me = 8L,
2mc = wfb = wwc
(2.67)
(2,68)
The beam, as a whole, is then analysed, for bending and
torsion, as if a rigid diaphragm were positioned at the load:
this results in a set of in-plane forces in the walls of the
box. These in-plane forces are superimposed on the corner
moments and warping moments, and reinforcement
designed as described in Chapter 5.
1fM d"b .oment re Istn utlon
A lower bound method of analysis. which has been permit-
ted for building design for some time, is the redistribution
of elastic moments in indeterminate structures. The Code
permits this method to be used for prismatic beams,
w
w,
m~4-r__~
w:ft.
~4t_W...;_'~_-_-_-_-----+-.1:
me w, ~ b
(a) Wall moments and forces (bl Transmission zone
FIt.;. 1.17(3)· (~)Lower bound design of box girder [82]
Although the Code dnes not state the fact, the restriction to
pris!I1utic beams was intended to preclude redistribution of
moments in all structures and structural elements except
'small' beams, such as are used for the longitudinal beams
of beam and slab construction. This was because it was
thought that there was a lack of knowledge of redistri-
bution in deep members such as box girders.
The concept of moment redistribution can be illustrated
by considering an encastre beam of span L carrying an
ultimate uniformly distributed load of W. The elastic bend-
ing moment diagram, assuming zero self weight, is shown
in Fig, 2. 18(b), If the beam is designed to resist a hogging
moment at the supports of )'WLlt2, instead of WLl12, then
the beam will yield at the supports at a load of), Wand, in
order to carry the design load of W, the mid-span section
must be designed to resist a sagging moment of [(WLl24)
+ (WLlI2) -(),WLlI2)], as shown in Fig. 2. t8(c).
A similar argument would obtain if the mid-span section
were designed for a moment less than WLl24.
If the beam could be considered to exhibit true plastic
behaviour. with unlimited ductility, then any value of
),could be chosen by the designer. However, concrete has
limited ductility in terms of ultimate compressive strain,
and this limits the ductility of a beam in terms of rotation
capacity. As), decreases, the amount of rotation, after in-
itial yield, is increased. Thus, ), should not be so small that
the rotation capacity is exhausted. It should also be noted
that the rotation capacity required at collapse is a function
of the difference between the elastic moment and the
reduced design moment. Thus. it is convenient to think in
terms of this difference. and to definite the amount of
redistribution as J) = 1 - ), . An upper limit to /3has to be
imposed, when there is II reduction in moment, because of
the limited ductility discussed above.
The amount of redistribution permitted also has to be
limited for another reason: although a beam designed for a
certain amount of redistribution will develop adequate
strength. it could exhibit unsatisfactory serviceability limit
state behaviour. since, at this stage, the beam would
hehave essentially elastically, As the difference between
the ultimate elastic moment and the reduced design
moment increases. the behaviour at the serviceability limit
state. in terms of stiffness (and. thUS, cracking) deterior-
ates. Hence. the amount of redistribution must be
restricteD.
The t~()de states four conditions which must be fulfilled
when redistributing moments, and these are now discussed.
w
(cl Web
(al Loading
~ ..../1-WLl12
~
WLl24
(b) Elastic
/Elastic -WLt12
~RedlstrlbutCld .d-AWLi12 I WLi12-AWL:12
..... ~.",;
..... _--,"
WL/24 + WL/12 -. AWLi12
(c) Redistribution
~'~BMz1
A.... ,..~~ fWL'8
Redistributed - - - - _........ .r
(d) Overall equilibrium
/Elastic ultimate
~Elastic service ~ T
_. O.7WLi12 I':.> O.7WLt2~ .+_Redistribution ~ IWL,12
...... ~,.
RedistributeCi/........ - - - - - ....
(e) Serviceability conditions
··300kNm
~200kNm
2BOkNm
(f) Elastic ultimate moment envelope
Fig. 2.18(8)-(f) MOlllent redistribution
First. overaiI equilibrium must be maintained by keep-
ing the range of the bending moment diagram equal to the
'free' bending momcnt (see Fig. 2.18(d».
Second, if the beam is designed to resist the redistri-
21](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-17-2048.jpg)
![buted moments shown in Fig. 2.18(d), then, in the regions
A- B, sagging reinforcement would be provided. How-
ever,. these.· rc;gions would be subjected to hogging
moments af 'the -serviceability Iiinit state, where elastic
I:onditions obtain. The Code thus requires every section to
be designed to resist a moment of not less than 70%, for
reinforced concrete, nor 80%, for prestressed concrete, of
the moment, at that section, obtained from an elastic
moment envelope covering a1l combinations of ultimate
loads. For the single load case considered in Fig. 2.18(a),
this implies that the resistance moment at any section
should be not less than (for reinforced concrete) that
appropriate to the chain dotted line of Fig. 2.18(e). The
values of 70% and 80% originated in CP 110, where the
ratio of (Y!I_ Yf3) at the serviceability limit state to that at the
ultimate limit state is in the range 0.63 to 0.71. By provid-
ing reinforcement, or prestress, to resist 70 or 80% respec-
tively of the maximum elastic moments, it is ensured that
clastic behaviour obtains up to about 70 or 80% of the
ultimate load, i.e. at the service load. In the Code, the
service load is in the range 0.58 to 0.76 of the ultimate
load and, hence, the limits of 70 and 80% are reasonable.
Third, the Code requires that the moment at a particular
section may not be reduced by more than 30%, for re-
inforced concrete, nor 20%, for prestressed concrete, of
the maximum moment at any section. Thus, in Fig. 2.18(f),
the moment at any section may not be reduced by more
than 90 kNm for reinforced concrete. This seems illogi-
cal, since it is the reduction in moment, expressed as a
percentage of the moment at the section under conside,ra-
lion, which is important when considering limited duc-
tility. It is unclear why the CP 110 committee used the
moment at any section. However, this condition is always
over-ruled by the second condition which implies that the .
moment at a particular section may not be reduced by
more than 30 or 20% of the maximum moment at that
section. There is no limit to the amount that the moment at
a section can be increased, because this does not increase
the rotation requirement at that section, and the third con-
dition is intended to restrict the rotation which would occur
at collapse. Beeby [83J has suggested that the Code limits
were not derived from any particular test data, but were
thought to be reasonable. However, they can be shown to
be conservative by examining test data.
The fourth, and final; condition imposed by the Code is
that the neutral axis depth must not exceed 0.3 of the
effective depth, if the full allowahle reduction in moment
has been made. As the neutral axis depth is increased; the
IImount of permissible redistribution reduces linearly to
zero when the neutral axis depth is 0.6 of the effective
depth, for reinforced concrete, and 0.5, for prestressed
~·()ncrete. The reason for these limits is that the rotation at
failure, when crushing of the concrete occurs, is inversely
proportional to the neutral axis depth. Thus, if the concrete
l:rushing strain is independent of the strain gradient acros.s
the section, the rotation capacity and, hence, the permis-
sible redistribution increases as the neutral axis depth
decreases. Beeby [83] has demonstrated that the Code
limits on neutral axis dcpth are conservative.
A general point regarding moment redistrihution is that,
when this is carried out, the shear forces are also redistrib-
22
uted: the. author would suggest designing against the
greater of· the non-redistributed and redistributed shear
forces.
Moment redistribution has been described in some detail
because it is: a new concept in bridg~ design documents,
but it must be stated that it is difficult to conceive how it
can .be simply applied in practice to bridges. This is,.
because, in order to maintain eqJJilibrium by satisfying.
equation.(2.l), any l'~distribution of longitudinal moments
should be accompanied. by redistribution of transverse and- .
twisting momcmts. It would appear that redistribution in a
deck can only be achieved by applying an imaginary 'load-
ing' which causes redistribution. For example, longitudinal
support moments could be reduced, and.span moments
, increased~' by applying: an imagInary .'loading' consisting
of a displac.ement of the suppC)rt; the moments due to this
imaginary· 'loading" .wOIiid 'then be added to those of the
conventional loadfiigs., ' .. ":" ", ~ :
-......~ ""~ ... ,. ,
Upper boyn~ ·rnetho.ds'
Upper bound m~thods are more suitable for analysis (i.e.
calculating thel,ll~imate .strengtr of an existing structure)
than fqr~e.sig~.;ho~e.ye,~"as will ~e}een, it is possible to
use an upper.boun9 .method (yield line theory) to design
slab bridges ~n,d .t<ip,slat>s.. .
Introduction to. yield lin((]theory.
The read~r is r~ferre~to 'one of the specialist texts. e.g.
Jones and WoO(([~4),fdt'a·detai1ed ~xplanation of yield
line theory, since onlf sUfficient background to apply the
method is giver; in'the 'following.
The first stage in the Yi.tM line analysis of a slab is to
propose a valid collapse irtechanism consisting of lines,
along whichyieid· of the.. r~inforcemef!t takes place. and
rigid regions betwe~n the yield Hnes. A possible mecha-
nism for a slab bridge subjected to a point, load is shown in
Fig. 2.) 9; it can ·be seen tllat the geometry of the ykld line
pattern can be specified i!1 terms of the parameter <x. In
this pattern, only a single parameter is required to define
the geometry. but, in a more general case. there could be a
number ·of parameters; thus, in general, if there are 11
parametets, each will be designated <Xi where i takes the
values of one to/f.:, . ,.
A point on the slao is then given a unit deflection; the
deflection at any other point, and the rotations in the yield
lines, can be 'calculated in tenns of (Xi' In the slab bridge
example of Fig.. 2.19, it would be most convenient to con-
sider the point of application 'of the 10adP to have unit
deflection. '- .
Once the detlectioris at every point are known, the work
done by the external loads, in moving through their deflec-
tions, can be calculated from
E= l:[JJ pb d~.dyl (2.69)
. ,each rigid region
where p is the load per unit area on an element of slab of
side d.O(, dy and ·6,' which is a funCtion of OIi. is the deflec-
tion of the element. inc 'su'mmation in equation (2.69) is
/
ISagging
yield IIne5.----
/
~ Hogging
~/Yieldline
~
~
~
~
S
L
2"
L
~f
~-..- ..--.•..... ......_--.-.-....!.-......-...-.-.....-.----.--.--..-~
Fig. 1.19 Slab bridge mechanism
carried out for each rigid region between yield lines, and
the integration for each rigid region is carried out over Its
arclt. In practice, one is generally concerned with point
loads. line loads and uniformly distributed loads. For a
point load equation (2.69) reduces to the I.oad mult.ipli.ed
hy its deflection, and for a line load, or a Uniformly dlstnb·
uted load, it reduces to the total load multiplied by the
deflection of its centroid. These calculations are illustrated
in the examples at the end of this chapter.
Similarly, once the rotations in the yield lines are known,
the internal dissipation of energy in the yield lines can be
calculated from
D =l: 11m" 0" dl) (2.70)
each line
where 0", which is a function of «i, is the normal rotation
in a particular yield line,m" is the normal moment of resis-
tance per unit length of the yield line and I is the distance
along the yield line. The summation in equation (2.70) is
carried out for each yield nne, and the integration for each
yield line is carried out over its length. ..:
In general, a yield line will be crossed by a number of
sets of reinforcement, each at an angle «Pi to the normal to
the yield line. as shown in Fig. 2.20. If the moment of
resistance per unit length of the jth .set of reinforcement is
m·, in the direction of the reinforcement, then its contri-
b~tion to the moment of resistance normal to the yield line
ismj cos2 «PI' Hence. the total normal moment of resistance
is given by
mIl = l: mi cos
2
«P; (2.71)
Since «Pi are each functions of <X;. so also is mil'
In practice, it is often easier to calculate the dissipation
of energy in a yield line by considering the rotations (0,)
of thc yield line in the direction of each set of reinforce-
ment, and by considering the projections (I,) of the yield
line in tl}c directions normal to each set of reinforcement.
The dissip,ation of energy is then given by
1> =l: [l:fmj OJ d/j] (2.72)
each line
Vield Line
/
Fill. 1.10 Normal moment in yield line
The use of this equation is illustrated in the examples at
the end of this chapter,
The next stage in the analysis is to equate the external
work dOlle to the internal dissipation- of energy. and to
arrange the resulting equation as an expression for the
applied load (P in the example of Fig. 2,19) as a function
of «i' The minimum value of P for the proposed collapse
mechanism can then be found by differentiating the
expression for P with respect to each of «/. and
equating to zero. The resulting set of n simultaneous equa-
tions can be solved to give <X; and hence P.
It is emphasised that, although the resulting value of P
is the lowest value for the chosen mechanism, it is not
necessarily the lowest value that could be obtained for the
slab. This is because there could be an alternative mechan-
ism which would give a lower minimum value of P. It is
thus necessary to propose a number of different collapse
. mechanisms and to carry out the above calculations and
minimisation for each mechanism.
A major drawback of yield line theory is that the
engineer cannot be sure. even after he has examined a
number of mechanisms. that there is not another mechan-
ism which would give a lower value of the collapse load.
The engineer is thus dependent on his experience. or that
of others, when proposing collapse mechanisms: fortu-
nately, the critical mechanisms have been documented for
a number of practical situations.
.Yield line analysis of slab bridges
Yield line theory can be used for calculating the ultimate
strength of a slab bridge which has been designed by
another method. The various possible· critical collapse
mechanisms, and their equations, have been documented
by Jones (85) for the general case of a simply supported
skew slab subjected to either a uniformly distributed load
or a single point load. Granholm and Rowe (861 give guid-
ance on choosing the critical mechanism for a simply
supported skew slab bridge subjected to a uniformly distrib-
uted load plus a group of point loads (i.e. a vehicle), and
they also give the equations for such loadings. Clark l871
has extended these solut,ions to allow for different uniforn,
load intensities on various areas of the bridge and for the
application of a knife edge load. Although the above](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-18-2048.jpg)
![Concrete bridge design to BS 5400
(a) (b)
(d)
Fig. 2.21(a)-(e) Skew slab bridge mechanisms
authors give general equations for the various mechanisms
and loadings, it is very often simpler, in practice, to work
from first principles, as shown in the examples at the end
of thischapter~
The possible critical mechanisms which should ,be
examined when assessing the ultimate strength of a simply
supported skew slab bridge are those shown in Fig. 2.21.
In'the case of a continuous slab, similar mechanisms to
those of Fig. 2.21 would form but, in each case, there
would also be a hogging yield line at the interior support
as shown in Fig. 2.22(a) and (b). Alternatively, a local
mechanism could develop around the interior piers as
shown in Fig. 2.22(c). Although the!>e mechanisms are not
considered explicitly in the literature, similar mechanisms
are analysed in [841 and [85] and thus guidance is avail-
able in these texts.
Yield line des/gn of slab bridges
Although yield line theory is more suitable for analysis, it,
can be used for the design of slab bridges. When used for
desi'gn, as opposed to analysis, the calculation procedure is
very similar. A colIapse mechanism is proposed; the ex-
ternal work done calculated from equation (2.69), in terms
of the known Joads p; the internal dissipation of energy cal-
culated, frolll:. equation (2.72), in terms of the unknown'
moments of resistance mj; and the external work done
equated to the internal dissipation of energy. However,in-
steadofminimising the load with respectto the parameters (X;
which define the yield line pattern, as is done when imalys-
ing a slab, the moments of resistance mj are maximised
with respect to the parameters (Xi' Hence, in general, there
are, an infinite number of possible design solutions which
result in different. rcliltive values of the moments of resis-
tance. In practice, the designer cho().~es ratios for the'"
moments ofresistance, and it is usual to choose ratios which
do not depart too much from the ratios of the equivalent
elastic moments. Sometimes this is not necessary, because
24
(e).
(e)
the equations for one possible colIapse mechanism
invoives only one set of reinforcement. It is thus possible
to simpl.ify the design procedure because it is possible to
calculate mj for that set of reinforcement directly. The
values ofm; for the other sets of reinforcement can then be
calculated by considering alternative mechanisms. As an
example, if the transverse reinforcement is parallel to the
abutments, mechanisms (a) and (b) of Fig. 2.21 involve
only the longitudinal reinforcement. Hence, a suggested
design procedure for simply supported skew slab bridges is
to first provide suffici~nt longitudinal reinforcement to
prevent mechanisms (a) and (b) fornling under their
appropriate design load,S.The ~mount of transverse rein-
forcement required to 'prevent mechanisms (c) and (e)
forming can then be calculated by setting up equations
(2.69) and (2:72), equating them, and maximising the
unknown transverse moment of resistance with respect to
the parameters (Xi defining the colIapse mechanism. This
procedure is illustrated in Example 2.2 at the end of this
Chapter. .' ','
Clark [87] has designed model skew siab bridges by this
method and then tested them to failure: it was found that
the ra,tos 9f exp~rimental'to calculated 'ultimate load were
l.07 to 1,25 with 'a mean value of I:16 for six tests.
Gnmholm and Rowe [86] aiso tested' model skew slab
bridges and obtained ~aluesof the above ratio of 0.95 to
1.12 with a mean value of 1.08 for eleven tests. Thus,
although yield line theory is theoretically an upper bound
method. it can be seen, generally. to result in a safe esti-
mate of the ultimate load and, thus, to a safe design if used
as. a de~ig~l method. '
It should be emphasised that, 'because of the values of
the, p~rti~1 ;afcty facto~s that have been adopted in the
Code, i~ fill often b.e found that the calculated amount of
transverse reinforcement is less than the Code minima of
0.15% and 0.25% for high' yield and mild steel respec-
tively.. When this occurs, the latter values should ob-
viously be provided.
I
f
I~
/
I~
(a)
Fi~. 2.22(a)-(c) Continuous slab bridge mechanisms
(a) Top slab remote from
diaphragm
(e) Cantih:iv!ilr slab remote
from transverse beam
I
I
I
-r
I
I
I
I
I
I
I
I
I
I
I
I
I
Beem
or web
Fig. 2.2~(a)-(d) Top slab and cantilever slab mechanisms
(e)
I
I
I
I
I
I
I
I
I
I
I
I__J
f
f~
(b)
I I
I I
'I I
I I
I I
I I
I I
I I
I
I
I
I
I
I
I
I
IL__
------------~-----
Diaphragm
(b) Top slab near diaphragm
I
-I--Beam or web
I
I
I
I Freeedge~
I
I
I __ -
II __ ..-----
..---1
I~
1
I~
I~I
I
----------~--------
Transverse beam
(d) Cantilever slab near
transverse beam](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-19-2048.jpg)
![Torsional hinge
/
-, r--, r--, --, r--,t-
Il I II II II
II I I( II II
II I " II II
II I I~
Slab yield
lines
Longitudinal
beam
II I I~II I I)
II I I(
II I I)
Flexural
__~...-t hinge
II II Ie
II I IJ
II I Ie
II I P
II II I(
II II I'
II II I~
II II I
II II I~ II II_JL __ J L ____ JL __ Jl_
Support diaphragm
Fig. 2.24 Beam and slab mechanism [88]
Load in this
region
For a continuous slab, the design procedure is similar,
except that the ratio of longitudinal hogging to sagging
moment of resistance must be chosen by the designer. It is
suggested that this ratio should be chosen to be similar to
the ratio of the elasti~ support and span moments. The
longitudinal reinforcement can then be determine from
mechanism (a) of Fig. 2.22, and the transverse reinforce-
ment from mecahnisms (b) and (c).
Application to top slabs
Yield line theory can be used for the analysis and design of
top slabs in beam and slab, cellular slab and box girder
construction, and also for cantilever slabs. One should
~ote that the application to top slabs of the simple yield
hne theory outlined earlier is conservative because it con-
siders only flexural action, and the beneficial effects of
membrane action are ignored; as, indeed, they are also for
elastic design.
Possible collapse mechanisms for top slabs and can-
tilever slabs subjected to a uniformly distributed load plus
a point load are shown in Fig. 2.23. -
A possible design procedure is to calculate the amounts
of longitudinal and transverse reinforcement required to
resist the global bending effects, and then to determine the
additional amounts of reinforcement required to prevent
the mechanisms of Fig. 2.23 forming u,nder the local effect
loadings. It will be necessary for the designer to choose
rati?s of the transverse bottom to transverse top to, longi-
tudmal bottom to longitudinal top moments of resistance,
or to predetermine some of the values. This procedure is
illustrated by Example 2.3 at the end of this chapter.
Upper bound methods for beam and"§lab bridges
An upper bound method has been proposed for beam and
slab bridges by Nagaraja and Lash [88]. The type of
mechanism considered is shown in Fig. 2.24, and consists
of yield lines in the slabs, plus flexural hinges in the longi-
tudinal beams and torsional hinges in "the support dia-
phragms. The method of solution is similar to that described
above for a slab bridges, and consists of equating the work
,26
done by the loads to the energy dissipated in the yield lines
and the flexural and torsional hinges. The rotations in the
yield lines can be calculated from the geometry of the
mechanism, as can the rotations in the flexural and tor-
sional hinges. The ultimate moment of resistance of the
beams, and the ultimate torque that can be resisted by the
diaphragms, can be calculated by the methods described in
Chapters 5 and 6. Nagaraja and Lash [88] give the equa-
tions for various mechanisms, but it is again suggested
that, in practice, the calculations are carried out from 'first
principles.
Nagaraja and Lash [88] compared ultimate loads calcu-
lated by such an approach with those recorded from tests
on one-tenth scale models, and obtained ratios of experi-
mental to calculated ultimate live loads of 0.90 to 1.12
with a mean value of 1.03 for twelve tests.
Upper bound methods for box girders
Methods are available for carrying out upper bound
analyse~ of box girders by considering overall collapse
mechamsms, some of which involve distortion of the
cross-section. Spence and Morley [82] and Morley and
S~enc~ [89] have considere~ the combined flexural plus
dlstOl1lOnal collapse mechamsms of Fig. 2.25 for simply
supported single cell box girders with no internal dia-
phragms. The method of analysis is similar to that described
previously for slabs; i.e., the work done by the applied
loads is equated to the dissipation of energy in the mecha-
nism. For mechanisms (a), (b) and (c), energy is dissipated
in .yi~lding the longitudinal reinforcement at midspan, in
tWlstlng the box walls and end diaphragms, and in rotation
of the corner hinges. For mechanism (d), the webs do not
twist, but energy is dissipated in the shear discontinuities
along the corners near th~ loaded w~b. Spence and Morley
[82] tested thirteen model" girders without side cantilevers,
,and found that eight o( these failed in pure bending and
five exhibited failures involving distortion. The failure
loads of those failing in pure bending were 0.88 to 0.97 of
the theoretical pure bending failure load, and the failure
loads of those failing in the distortional mode were 0.85 to
t .00 of the theoretical failure' load calculated from consid-
eration of mechanism (a). The tests indicated that the dis-
placements at collapse were not sufficient for the webs,
flanges and diaphragms to yield in torsion, and it was thus
suggested that the appropriate dissipation terms be omitted
from the equations.
Morley and Spence [89] have indicated how continuous
single cell girders may be analysed. The loaded span fails
by the formation of a mechanism similar to one of those in
Fig. 2.25: the non-failing spans move outwards in order
that the longitudinal stresses in the failing span satisfy the
condition of zero axial force near to the, su·ppon:s. The dis- .
sipation of energy due to longitudinal extension and con-
traction in the support regions can then be calculated.
Cookson [90] has extended the work of Spenc~ and
Morley to multi-cell girders. Possible mechanisms are
shown in Fig. 2.26: the critical mechanisms are generally
those involving local failure of a single cell or a pair of
cells. These mechanisms involve shear discontinuities and
their analysis is quite complex. In addition, it appears that
Alllllys;s
(e) (b)
(c) (d)
• Flexural hinge
• S Flexural hinge plul
Ihear dllcontlnuity
(e) Displacement of 'loaded' web
FlA. 2.25(a)-(.) Sin~le cell box girder mechanisms (82. 89]
(a) Overall
- ----~~~---r------~---
(b) Local
"'i~. 2.26(a),(I)) Multi-cell box girder mechanisms (89. 90J
the amount of energy dissipated in the shear discontinuities
is less than that predicted theoretically. Further work thus
needs to be carried out before the method can be applied in
practice.
Model analysis and testing
The Code specifically permits the use of model analysis
and testing to determine directly the load effects in a struc-
lure. or 10 justify a particular theoretical analysis.
Testing may also be used to determine the ultimate resis-
tance of cross-sections which are not specifically covered
by the Code.
------...,,....-----
Examples
2.1 Yield line analysis of composite slab
bridge
The validity of applying yield line theory to composite slab
bridges has been demonstrated experimentally by Best and
Rowe [91].
The method will be used to calculate the number of
units of HB loading which would cause failure of a right
simply supported composite slab bridge of tOm span and
11.3 m breadth. The longitudinal and transverse sagging
moments of resistance per unit width have been calculuted
by the methods of Chapter 5 to be 856 and 70 kN Ill/m
respectively; and the hogging moments of resistance will
27](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-20-2048.jpg)
![~A
@ ~ -Sagging
~
E.... CD S ................. Hogging
..t
® ~ ~Unit
~ .
deflection
B -------~ B
E L ® ~ ®
j
CIC!
.... ~
--------~
(i) ~
E @ ~
... ~..t
® ~
I+--Y_--to!
f4-....... .... ?.!!_....._..... ._-+!+-____._...§.3m ..•.----+l.1
(a) Plan
11=1/4.1
4.1 m 1.8m 4.1 m ....oJ
1+--._...._._._....--...-.......-....•..- ...---+--------.---.,
(b) Section A-A
_____________tl_=_1_/Y__~
(c) Section B-B
2.27(a)·-(c) Example 2. I
he assumed to be zero. The centre line of th~ external
wheels of the HB vehicle cannot be less than 2 m from the
free edge of the slab. The self weight of the slab, can be
taken as 12 kN/m2 •. the superimposed dead load as
2.5 kN'/m2; and the parapets to apply line loadings of
3.5 kN/m along the free edges,
In view of the span. only one bogie of the HB. vehicle
will be consid~red. and the critical mechanism is that
shown in Fig. 2.27(a): it is sufficiently accurate to assume
that the HB bogie is positioned symmetrically about mid-
span. In practice. each of the mechanisms of Fig. 2.21
would be considered to determine which would be critical.
If the ~haded area of Fig. 2.27(a) deflects unity, then.
from Figs. 2.27(b') and (c),'the rotations in the longitudinal
and tnmsverse steel directions a~e 1/4.1, and lIy respec-
tively. where y is the unknown parameter which defines
2R
the geometry of the mechanism. The projected lengths of
the yield lines in the longitudinal and transverse steel direc-
tions are (5 + y) and 10 m respectively. The dissipation
of internal energy is thus given by (see equation (2.72»
Longitudinal steel = 2(856) (1/4.1) (5 + y)
= 2088 + 418y
Transverse steel ,= (70) (1/y) (10) = 700ly
:. Total dissipation == D = 2088 + 418y + (700M
The HB vehicle deflects unity and. thus. if the HB load is
P. the external work done is
E(HB) = P x I = P
The work done by' the uniformly distributed load is hest
calculated by dividing the slab into the nine regions shown
in Fig. 2.27(a); the work done in each region is then the
load intensity multiplied by the area of the region and by
the deflection of the centroid of the area. Hence, and not-
ing that region 9 does not deflect,
E(u.d.l) = (12 + 2.5) [2(5 x 4.1) (1/2) + (5, x 1.8) (1)
+ 4 (1/2 x 4.ly) (1/3) + (1.8.") (1/2)]
= 428 + 53y
, Similarly. the external work done by the parapet Iin,e load-
, ing is
E(k.e.l) = 3.5 [2(4.1) (1/2) + 0.8) (I)] = 21
:. Total external work done =E =P + 449 + 53y
Now p..,- D
:. p + 449 + 53,Y == 2088 + 418)' + (700ly)
or P =: 1639 + 365y + (7oo/y)
In order to find the value of y for a minimum P.
dP- :: 365 - (700/y2) = 0
dy
:. y = 1.3851n
This value is less thun 6.3 m. thus the proposed mechan-
ism would be able to form.
:. P = 2650 kN or 132 units of HB
*2.2 Yield line design of skew slab bridge
A solid slab highway bridge has a right span of 10m, 11
structural depth of 580 mm, a skew of 300
and .the cross-
section shown in Fig. 2.28(a). The specified highway load-
ing is HA and 45 units of HB. The nominal superimposed
dead load is equivalent to a uniformly distributed load of
2.5 kN/m2 • and the nominal parapet loading is 3.5 kN/m
IIlong each free edge. It is required to calculate, by yield
line theory. the moments of resistance to be provided by
hnttom reinforcement, placed parallel to the slab edges, for
load combination I (!lee Chapter 3). It will be aS!lumed that
no top steel is to be provided for strength.
In IIccordance with Chapter 3, the carriageway width is
9.3 m. and consists of three 3.1 m notional lanes.
It is explained in Chapter I that the design load effects
(the moments of resistance in this example) are given by
Yf. llluitiplied by the effects of YfL. Q..However. when
using yield line theory. it is necessary (see Chapter 4) to
calculate the design load effects from a 'design' load of
Yn Yo, Qk' The relevant nominal loads and partial safety
factors discussed in Chapter 3 are given in Table 2.2
Table 2.2 Example 2.2 'design' loads
Load Yf, Yfl. Nominal 'Design'
_._M M_..M....._____
kN/m2
Dead 1.15 1.2 13.92 kN/m2 9.2
Surfucing 1.15 1.75 2.5 kN/mz 5.03 kN/m2
Parapet 1.15 1.75 3.5 kN/m2 7.04 kN/m2
HA (alone) 1.1 I.S 9.68 kN/m2 16.0 kN/m2
plus plus
33.5 kN/m 55.3 kN/m
HA (with HB) 1.1 1.3 ditto 13.8 kN/m2
plus
47.9 kN/m
HB 1.1 1.3 450 kN/axle 644 kN/axle
Footway 1.1 1.5 4.0 kN/m2 6.6 kN/m2
-_.._-_.__.._-----_..__.__.-.
The longitudinal reinforcement is designed by consider-
ing mechanism (b) of Fig. 2.21 which is found to be more
critical than mechanism (a) under HA loading. The loads
and mechanism are shown in Fig. 2.28(b).
For unit d'eflection of the yield line, the total rotation in
the yield line = 2(1/5 sec 30) = 0.346
If the momentof resistance per unit length of the longitudi"
nal steel is m.. the dissipation is given by
D = m,(0.346) (13.3) == 4.6 m,
The external work components are
E(HB) -- 2(644/4) [(3.4715) -I- (3.97/5) + (4.47/5)
+ (4.97/5)] = 1087
E(k.e.l.) ,- (47.9·+ 47.9/3) (3.1 sec 30) (1) = 229
E(parapet) =: 4(7.04) (5 sec 30) (1/2) = 81
E(u.d.l.1) = 4(24.23) (5 x 0.5 sec 30) (1/2) = ]40
E(u.d.l.2) = 4(30.83) (5 x 1.5 sec 30) (1/2) = 534
E(u.d.l,3) = 2(24,23) (5 x 3.1 sec 30) (1/2) =434
E(u.d.I.4) = 2(38.03) (5 x 3.1. sec 30) (1/2) = 681
E(u.d.1.5) == 2(28.83) (5 x 3.1 sec 30) (/2) =516
:. Total work done == E = 3702
NowD =E
:. 4.6 ml = 3702
:. 1111 == 805 kN mlm
The transverse reinforcement is designed by considering
mechanism (c) of Fig. 2.21 which can be idealised tothut
shown in Fig. 2.28(c) [86, 87].
For unit deflection of the shaded area. the rotation in the
direction of the longitudnal steel is 1/4.874 =0.205
Projected length normal to longitudinal steel ==
5.1 + y sec 30 = 5. 1 + 1. 155y
Longitudinal steel dissipation ==
2(805) (0.205) (5.1 + 1.155y) = 1683 + 382y
Rotation in direction of transverse steel == Ily
Projected length normal to transverse steel = 10
If the moment of resistance per unit length of the trans-
verse steel is m2. the dissipation is given by (m2) (lly) (10)
=10m2/y
:.Total dissipation = D = 1683 + 382y + IOm2/Y
The external work components are (noting that it is
sufficiently accurate to assume that each wheel of the HB
vehicle displaces unity)
E(HB) = 2(644) (I) = 1288
E(parapet) = (7.04) [(1.8) (I) + 2(4.874) (1/2) 1=47
E(k.e.l.) = (47.9) (y) (1/2) = 24:v
E(u.d,l.l) = 2(24.23) (4.874 x 0.5) (1/2) = 59
E(u.d.l.2) = 2(30.83) (4.874 x 1.5) (1/2) = 225
E(u.d. .3) 2(24.23) (4.874 x 3.1) (112) = 366
E(u.d. .4) 4(38.03) (1/2 x 4.221)') (1/3) = 107)'
E(u.d.I.5) (38.03) (1.559)') (1/2) = 3Qv
E(u.d..6) (24.23) (1.8 x 3.1) (I) = 135
E(u.d. ,7) (30.83) (1.8 x 1.5) (I) = 83
E(u.d.I.R) (24.23) (.8 x 0.5) (I) = 22
:. Total work done = 2225 + 161y
Now D = E
:.1683 + 382)' + lOmh = 2225 + 161y
:. In! = 54.2y - 22. Il
29](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-21-2048.jpg)
![. J
0.5 '.5 '.0
Foot- Hard
way strip
I. ~14 _14 _14
3.65
Traffic
lane
3.65
Traffic
lane
L~L--'_~.f---_9'_3_c...!:~,,!.-:,_g.w,y==- 3-'· 1. --.-.------.,..----. Noti0 naI Ianes
5
5
I I
I I
/®/I I
I 1""I I
I I
(a) Cross-section
(b) Mechanism'
I
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I I
I
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I
I I
® I I
I I
I I
I I
I I
I I
y
t ::~L~(c) Mechanism 2
Fi~. 2.28(a)-(c) Example 2.2
In order to find the value of y for a maximum m
I
2,
1m2
dy = 54.2 - 44.2y = 0
.. y = 1.226 m
:his value is less than 3.58 m, thus the mechanism would
form as shown in Fig. 2.28(c).
:. m2 =33.2 kN mlm
I
I
I
I
-Sagging
'-""' '-""' Hogging
I
I
I
I
~ Unit deflection
The amount of reinforcement, that would develop this
moment, would be less than the Code minima discussed in
Ch~pte~ 10 and, thus, the latter value should be provided.
This Will generally be the case when designing in accor-
dance with yield line theory.
Finally, although no top steel is required to develop
adequate strength, some will be required in the obtuse cor-
ners to control cracking.
__ Sagging
..,.. ...... Hogging
'65kN wheel.....
O.2mO.3m O.2m
1----t-I----1
Fig. 2.29 Example 2.3
:=I0.16m
Hogging yield line
forms along upper
edge of beam flange
y
~2.3 Yield line design of top slab
A bridge deck consists of M-beams at 1.0 m centres with a
160 mm thick top slab. It is required to design the top slab
reinforcement, in accordance with yield line theory, to
withstand the HA wheel load.
It is necessary to pre-determine some of the reinforce-
ment areas, and it will thus be assumed that the Code
minimum area of 0.15% of high yield steel is provided in
both the top and bottom of the slab in a direction parallel
to the M-beams. Such reinforcement would provide sag-
ging and hogging moments of resistance per unit length of
5.35 kN m/m. Let the sagging and hogging moments of
resistance per unit length normal to the beams be m, and
m2 respectively.
The 'design' loads are (see Chapter 3 for details of
nominal loads and partial safety factors)
HA wheel load = (1.1) (1.5) (100) = 165 kN
Self weight = (1.15) (1.2) (3.84) = 5.30 kN/m2
Surfacing (say) = (1.15) (1.75) (2.5) = 5.03 kN/m2
Total u.d.l. = 10.33 kN/m2
The contact area of the HA wheel load is a square of
side 300 mm (see Chapter 3). Dispersal through the surfac-
ing will be conservatively ignored.
The collapse mechanism will be as shown in Fig. 2.29
in which the parameter y defines the geometry of the
mechanism. If the wheel deflects unity, the rotations
paraUel and perpendicular to the beams are (lIy) and
(1/0.2) respectively.
The internal dissipation of energy is
D = 2(m, + m2) (1/0.2) (0.3 + 2y) +
2(5.35 + 5.35)(1/y) (0.7)
= (m, + m2) (3 + 20y) + 14.981y
The external work done is
E = (165) (1) + 10.33 [2(0.3) (y) (1/2)
+ 2(0.3) (0.2) (1/2) + 8(1/2) (0.2) (y) (1/3)]
= 165.6 + 5.857y
NowD =E
:. (m, + m2) (3 + 2Oy) +14.98/y = 165.6 + 5.857y
. ( + ) _ 165.6 + 5.857y - 14.98/y
.. m, m2 - 3 + 20y
In order to find the value of y for a maximum (m, + m2)'
d(m, + m2) = 0
dy
From which, y = 0.239 m
:. (m, + m2) =13.4 kNm/m
Any values of m, and m2 may be chosen, provided that
they sum to at least 13.4 kNm/m and that they are not less
than the required global transverse sagging and hogging
moment of resistance respectively.
31'](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-22-2048.jpg)
![as live or superimposed dead load. Hence the greater
uncertainty associated with the latter loads is reflected.
in the values of the partial safety factors.
3. The value for a live load, such as HA load, is less
when the load is combined with other loads, such as
wind load in combination 2 or temperature loading in
combination 3, than when it acts alone, as in combina-
tion 1. This is because of the reduced probability that
a number of loads acting together will all attain their
nominal-values simultaneously. This fact is allowed
for by the partial safety factor Yf2' which was dis-
cussed in Chapter 1 and which is a component of YIL'
4. Avaluc of unity is specified for certain loads (e.g.
superimposed dead load) when this would result in a
more severe effect. This concept is discussed later.
5. The values for dead andsllperimposed dead load at
the ultimate limit state can b~ different to the tabulated
values, as is discussed later when these loads are con-
sidered in more detail.
Application 9f loads
General
The general philosophy governing the application of the
loads is that the worst effects of the loads should be
sought. In practice, this implies that the arrangement of ~he
loads on the bridge is dependent upon the load effect bemg
considered, and the critical section being considered. In
addition, the Code requires that, when the most severe
effect on a structural element can be diminished by the
presence of a load on a certain port~on .of the struc~re,
then the load is considered to act with Its least poSSible
magnitude. In the case of dead load this entails applying a
"ilL value of 1.0: it is emphasised that this value is applied
to all parts of the dead load and not solely to those p~rts
which diminish the load effect. In the case of supenm-
posed dead load and live load, these loads should n~t
be applied to those portions of the structure where !helr .
presence would diminish the load effect under consider-
ation.
influence lines are frequently used in bridge design and,
in view of the above, it can be seen that superimposed
dead load and live load should be applied to the ~dverse
parts of an influence line and not to relieving parts. It is
not intended that parts of parts of influence lines should be
loaded. For example, the loading shown in Fig. 3.1(a)
should not be considered.
Highway carriageways and lanes
Carriageway
The carriageway is defined as the traffic lanes plus hard
shoulders plus hard strips plus marker strips. If raised
kerbs are present, the carriageway width is the distance
between the raised kerbs. In the absence of raised kerbs,
34 .;
(allncorrect
~Influence
,ilin
•
(bl Correct
Fig. 3.1(a),(b) Influence line loading
the carriageway width is the distance between the safety
fences less the set-back for the fences: the set-back must
be in the range 0.6 to 1.0 m.
Traffic lane
The lanes marked on the running surface of the bridge are
referred to as traffic lanes. Hence traffic lanes in the Code
are equivalent to working lanes in BE 1177. However, the
traffic lanes in the Code have no significance in deciding
how live load is applied to the bridge.
Notional lanes
These are notional parts of the carriageway which are used
solely for applying the highway loading. They are equiv-
alent to the traffic lanes in BE 1177 and they are determined
in a similar manner, although the actual numerical values
are a little different for some carriageway widths.
Overturning of structure
.
The stability of a stru~iture against overturning is calculated
at the ultimate limit. state. The criterion is that the least
restoring moment due to unfactored nominal loads should
be greater than the greatest overturning moment due to
design loads.
Foundations
The soil mechanics aspects of foundations should be as-
sessed in accordance with CP 2004 [92], which is not
written in terms of limit state design. Hence these aspects
should be considered under nominal loads. However,
when carrying out the structural design of a foundation,
the reaction from the soil should be calculated for the
appropriate design loads.
Permanent loads
Dead load
The nominal dead loads will generally be calculated from
the usual assumed values for the specific weights of the -
materials (e.g. 24 kN/m3
for concrete). When such
assumed values are used it is necessary, at the ultimate
limit state, to adopt 'ilL values of 1.1 for steel and 1.2 for
concrete rather than the values of 1.05 and 1.15 respec-
tively given in Table 3.1. The latter values are only used
when the nominal dead loads have been accurately as-
sessed from the final structure. Such an assessment would
require the material densities to be confirmed and the
weight of. for example, reinforcement to be ascertained. It
is thus envisaged that in general the larger 'IlL values will
be adopted for design purposes. The emphasis placed on
checking dead loads in the Code is because the dead load
factor of safety is less than that previously implied by BE
1177.
As indicated earlier, when discussing the YrL values, it
is necessary to consider the fact that a more severe effect,
due to dead load at a particular point of a structure, could
result from applying a YIL value of 1.0 to the entire dead
loud rather than a value of, for concrete, 1.2.
~Superimposed dead load
The partial safety factor given in Table 3.1 for superim-
posed dead load appears to be rather large. The reason for
this is to allow for the fact that bridge decks are often
resurfaced, with the result that the actual superimposed
dead load can be much greater than that assumed at the
design stage [18]. However, by agreement with the
appropriate authority, the values may be reduced from
1.75 at the ultimate limit state, and 1.2 at the serviceability
limit state, to not less than 1.2 and 1.0 respectively. It is
then the responsibility of the appropriate authority to
ensure that the superimposed dead load assumed for design
is not exceeded in reality.
As for dead load, the possibility of a more severe effect,
due to applying a Yjl. value of 1.0 to the entire super-
imposed dead load, should be considered. In addition, the
removal of superimposed dead load from parts of the struc-
ture, where they would have a relieving effect, should be
considered. :
Filling material
The nominal loads due to fill should be calculated by con-
ventional principles of soil mechanics. The partial safety
factor of 1.5 at the ultimate limit state seems to be high,
particularly when compared with that of 1.2 for concrete.
However, the reason for the large value is that the pres-
sures on abutments, etc., due to fill. are considered to be
calculable only with a high degree of uncertainty, particu-
larly for the conditions after construction [18].
It seems reasonable to apply a factor of 1.5 when
considering the lateral pressures due to the fill; but,
when the vertical effects of the fill are considered, it seems
m~re log(c~l to treat the fill as a superimposed dead load
and to argue that a "ilL value of 1.2 should be adopted
because the volume of fill will be known reasonably
accurately.
/
Shrinkage and creep
Shrinkage and creep only have to be taken into account
when they are considered to be important. Obvious situ-
ations are where deflections are important and in the design
of the articulation for a bridge.
In terms of section design,proc~4ures exist in Part 4 of
the Code to allow for the effects of, shrinkage and creep on
the loss of prestress and in certain forms of composite con-
struction. These aspects are discussed in Chapters 7 and 8.
*"Differential settlement
In the Code, as in BE 1/77, the onus is placed upon the
designer in deciding whether differential settlement should
be considered in detail.
Transient loads
Wind
The clauses on wind loading are based upon model tests
carried out at the National Physical Laboratory and which
have been reported by Hay (93). The tests were carried out
in a constant airstream of 25m/s, and the model cross-
sections were very much more appropriate to steel than to
concrete bridges.
The clauses are very similar to those in BE 1177, and
the calculation procedure is as lengthy. However, it is em-
phasised that, according to the 'Code, it is not necessary to
consider wind loading in combination with temperature
loading. In addition, as is also the case in BE 1177, wind
loading does not have to be applied to the superstructure
ofa beam and slab or slab bridge having a span less than 20 m
and a width greater than 10m. However, a number of
overbridges have widths less than 10 m. and the exclusion
clause is not applicable to these.
In general the calculation procedure is as follows.
The mean hourly wind speed is first obtained for the
location under consideration from isotachs plotted on a
map of the British Isles.
The maximum wind gust speed and the minimum wind
gust speed are then calculated for the cases of live load
both on and off the bridge. The minimum gust speed is
appropriate to those areas of the bridge where the wind has
a relieving effect, and is used with the reduced Y,L values
of Table 3.1. The gust speeds are obtained by multiplying
the mean hourly wind speed by a number of factors, which
depend upon:
1. The return period: the isotachs are for a return period
of 120 years (the design life of the bridge), but the
Code permits a return period of 50 years to be adopted
for foot or cycle track bridges, and of 10 years for
erection purposes.
2. Funnelling: special consideration needs to be given to
bridges in valleys, etc.
35](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-24-2048.jpg)
![3. Gusting: a gust factor, which is dependent upon the
height above ground level and the horizontal loaded
length, is applied. The gust factor may be reduced for
foot or cycle track bridges according .to the height
above gi'oundlevel. ,
In addition the minimum wind gust speed is dependent
.upon an hourly speed factor which is a function of the
height above ground level.
The next stage of the calculation is to determine. the
transverse and longitudinal wind loads (which are dep~n
dent o.n the gust speed, an exposed area and a drag
coeffiCIent), and the vertical wind load (which is depen-
dent on the gust speed, the plan area and a lift coefficient).
This part of the calculation is probably the most com-
plex and requires a certain amount of engineering judge_
C
.ment to be made. This is because the Code gives cross-
sections for which drag coefficients may be obtained from
the Code, and also gives cross-sections for which drag
coefficients may not be derived and for which wind tunnel
tests should be carried out.
It should also be stated that an 'overall' depth' is
required to determine the exposed area and the drag
coefficient, but, in general, different values of the 'overall
depth' are used for the two calculations.
Finally the transverse, longitudinal and vertical win"
loads are considered in the following four combinations: .'
I.' Transverse alone
2. Transverse ± vertical
3. Longitudinal alone
4. 50% transverse + longitudinal ± 50% vertical
There are thus less combinations than exist in BE 1177.
Temperature
The clauses on temperature loading are based upon studies
carried out by the Transport and Road Research Labora-
tory and the background to the clauses has been described
by Emerson [94, 95]. The clauses are essentially identical
to those in BE 1177 except that temperature loading does
not have to be considered with wind loading.
There are, effectively, two aspects oftemperature load-
ing to be considered; namely, the restraint to the over-
all bridge movement due to temperature range, and the
effects of temperature differences through the depth of the
bridge.
"*Temperature range
The temperature range for a particular bridge is obtained
by first determining the maximum and minimum shade air
temperature, for the location of the bridge, from isotherms
plotted on maps of the British Isles. The isotherms were
derived from Meteorological Office data and. are for a
return period of 120 years (the design life of the bridge).
The shade air temperatures may be adjusted to those
appropriate to a return period of 50 years for foot or cycle
track bridges, for the design of joints and during erection.
An adjustment should also be carried out for the height
above mean sea level.
36
The minimum and maximum effective bridge tempera-
tures can then be obtained from tables which relate shade
air temperature to effective bridge temperature. The latter
is best· thought of as that temperature which controls the
overall longitudinal expansion or contraction of the bridge.
The tables referred to above were based upon data
obtained from actual bridges [94]. The effective bridge
temperatures are dependent upon the depth of surfacing,
~~.d,~orrection has to be made if this differs from the
'~', lOO"mi1f"lI~umecl!or,concrete bridges in the tables. Emer-
son [95] hassuggestelt~t such an adjustment should also
.take· account of the shape of the cross-section of the
bridge..
The effective bridge temperatures are used for two pur- .:,
poses.
First, when designing expansion joints, the movement to
be accommodated is calculated in terms of the expansion
from a datum effective bridge temperature, at the time the
joint is installed, up to the maximum effective bridge
temperature and down to the minimum effective bridge
temperature. The resulting movements are taken as the
nominal values, which have to be multiplied by the 'YfL
values of Table 3.1. The coefficient of thermal expansion
for concrete is given as 12 x lO-o/oC, except for limestone
agg~egate concrete when it is 7 x lO-o/oC. The author
would suggest also considering the latter value for light-
weight concrete [96].
Second, if the movement calculated as above is re-
strained, stress resultants are developed in the structure.
These stress resultants are taken to be nominal loads', but
this contradicts the definition of a load which, according to
the Code, includes 'imposed deformation such as those
caused by restraint of movement due to changes in temper-
ature'. The author would thus argue that the movement is
the load, and that anr stress re~ultants arising are load
effects. This differenc~ of approach is important when
designing a structure ,to resist the effects of temperature
and is elaborated in Chapter 13.
The Code indicates how to calculate the nominal loads
when the restraint to temperature movement is accom-
panied by flexure of piers or shearing of elastomeric bear-
ings.
Coefficients of friction are given for roller and sliding
bearings: these are used in conjunction with nominal dead
and superimposed dead load to calculate the nominal load
due to frictional bearing restraint. The values are the same
as those in BE 1177 and are based partly upon [97].
It should be noted that, as in BS 153, the effects of fric-
tional bearing restraint are considered in combination with
dead and superimposed dead load only. This is because the
resistance to movement of roller or sliding bearings is least
when the vertical load is a minimum. Hence movement
could take place under dead load conditions and. having
moved, the restraining force is relaxed [IS].
Temperature differences
Due to the diurnal variations in solar radiation and the rela-
tively small thermal conductivity of concrete, severe non-
linear vertical temperature differences occur through the
depth of a bridge. Two distributions of such differences
E r.t;LJ'l iM ~ i
o 0 I
~
+
~ E
0 ....
o
VI
h = overall depth of
concrete section
hs = depth of surfacing
2.5"C
(a) Positive
Fig. 3.2(a),(b) Temperature differences for J m structural depth
are given in the Code (see Fig. 3.2). The temperature dif-
ferences depend on the depth of concrete in the bridge:
those shown in Fig. 3.2 are for a depth,of 1 m. One of the
distributions is for positive temperature differences, and is
appropriate when there is a heat gain through the top sur-
face; and the other is for negative temperature differences,
and is appropriate when there is a heat loss from the top
surface. The temperature distributions are,composed of
four or five straight lines which approximate'the non-linear
distributions, which have been calculated theoretically and
measured on actual bridges by Emerson [95, 98]. The
approximation has been shown to be adequate for design
purposes by Blythe and Lunniss [99].
The Code distributions have been chosen to give the
greatest temperature differences that are likely to occur in
practice. It is not possible to think of them in terms of a
return period, but they are likely to occur more than once a
year 195].
The Code states that the effects of the temperature dif-
ferences in Fig. 3.2 should be regardf1d as nominal values
and that these effects, multiplied by 'YfL' should be
regarded as design effects. This again appears to be an
inconsistent use of terminology. The author would suggest
that the nominal loads are the imposed deformations due to
either internal or external restraint of the free movements
implied by! the temperature difference; and that the 'Yfl.
values should be applied to these to give design loads in
the form of design imposed deformations. Any stresses or
a.3°eL-_ _ _ _ _ _----'
(b) Negative
E
.c:1l)
1'11'1
00
VI
.c:E
Il)N
1'1 •
·0
o VI
.c: E
Il)N
N'.0
°Vl
E
-C:1l)
1'11'1
00
VI
stress resultants which are developed in response to the
imposed deformations would then be the load effects. Such
arguments regarding definitions may seem pedantic but
they are important when designing a structure to resist the
effects of temperature differences and are elaborated in
Chapter 13.
Regarding the 'YlL values, it should be noted from Table
3.1 that the serviceability limit state value is 0.8. This
means that the final effects are only SO% of those calcu-
lated in accordance with BE 1/77, which adopts the same
temperature difference diagrams. The reason for adopting
a 'YfL value of O.S at the serviceability limit state is not
clear but, in drafts of the Code prior to May 1977, values
of 1.0 and 1.2 for the serviceability and ultimate limit
states, respectively.. were specified. It thus appears that the
Part 2 Committee thought it reasonable to reduce each of .
these by 0.2 in view of the reduced probability of a severe
temperature difference occurring at the same time as a
bridge is heavily loaded with live load.
The temperature differences given in the Code were c,al-
culated for solid slabs, but it is considered that the inac-
curacy involved in applying them to other cross-sections is
outweighed by certain assumptions _made in the calcu-
lation. Measurements on box girders [100] and beam and
slab [101] construction have shown that the temperature
differenc~s are very similar to those predicted for a solid
slab of the same depth. .
In addition, the temperature differences given in the
37](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-25-2048.jpg)
![main body of the Code are for a surfacing depth of
100 mm. An appendix to Part 2 of the Code gives tempera-
ture differences for other depths of surfacing which are
based upon the work of Emerson [95].
Combination of temperature range and difference
A severe positive temperature difference can occur at any
time between May and August, and measurements have
shown that the lowest effective bridge temperature likely
to co-exist with the maximum positive temperature dif-
ference is 15°C [95].
A severe negAtiv~ tempernture difference can occur at
any time of the day, night or year. However, it is con-
sidered unlikely that a severe difference would occur
between about ten o'clock in the morning and midnight on,
or after, a hot sunny day. Thus, it is considered that a severe
negative difference is unlikely to occur at an effective
bridge temperature within 2°C of the maximum effective
bridge temperature [95].
The above co-existing effective bridge temperatures
have been adopted in the Code.
Exceptional loads
These include the loads due to otherwise unaccounted
effects such as earthquakes, stream flows, ice packs, etc.
The designer is expected to calculate nominal values of
stich loads in accordance with the probability approach
given in Part 1 of the Code and discussed in Chapter 1.
Snow loads should generally be ignored except for cer-
tain circumstances, such as when dead load stability could
be critical.
-tErection loads
At the serviceability limit state, it is required that nothing
should be done during erection which would cause damage
to the pennanent structure, or which would alter its
response in service from that considered in design.
At the ultimate limit state, the Code considers the loads
as either temporary or permanent and draws attention to
the possible relieving effects of the former.
The importance of the method of erection, and the pos-
sibility of impact or shock loadings, are emphasised.
As already mentioned, wind and temperature ,effects
during erection should generally be assessed for 10- and
50-year return periods respectively. For snow and ice
loading, a distributed load of 500 N/m2 will generally be
adequate; this loading does not have to be considered in
combination with wind loading.
Primary highway loading
General
The primary effects of highway loading are the vertical
loads due to the ma'ss of the traffic , and are considered as
static loads.
38
The standard highway loading consists of nonnal (HA)
loading and abnonnal (HB) loading. The original basis of
these loadings has been described by Henderson [102].
Both of these loadings are deemed to include an allowance
for impact.
It should be noted from Table 3.1 that, at the service-
ability limit state in load combination 1, the Y/L value for
HA loading is 1.2. This value was chosen [18] because it
was considered to reflect the difference between the uncer-
tainties of predicting HA loading ~nd dead load, which has
a value 6£1.0. Presumably, the HB value of 1.1 was
chosen to be between the HA and dead load values.
'*HA loading
HA loading is a formula loading which is intended to rep-
resent normal actual vehicle loading. The HA loading con-
sists of either a uniformly distributed load plus a knife
edge load or a single wheel load. The validity of represent-
ing actual vehicle loading by the formula loading has been
demonstrated by Henderson. [102], for elastic conditions,
and by Flint and Edwards [103]. for collapse conditions.
Uniformly distributed load The uniformly distributed
component of HA loading is 30 kN per linear metre of
notional lane for loaded lengths (L) up to 30 m and is
given by lSI (IlL)0, 475 kN per linear metre of notional
lane for longer lengths, but not less than 9 kN per linear
metre. The loading is compared with the BE 1/77 loading
in Fig. 3.3.
It can be seen that the two loadings are very similar;
however, the upper cut-off is now 30 kN/m at 30 m
instead of 31.5 kN/m at 23 m, and there is now a lower
cut-off of 9 kN/m at 379 m. The latter has been introduced
because of the lack of dependable traffic statistics for long
loaded lengths [18]. Figure 3.3 gives the loading per lineal'
metre of notional lane and the load intensity is always
obtained by dividing by the lane width irrespective of the
lane width. This is differ~nt to BE 1177 and means that,
for a loaded length less than 30 111, the load intensity can
be in the range 7.9 to 13.0 kN/m2
compared with the BE
1/77 range of 8.5 to 10.5 kN/m2
•
Loaded length The loaded length referred to above is the
length of the base of the positive or negative portion of the
influence line for a particular effect at the design point
under consideration. Thus for a single span member, the
loaded length for the span moment is the span. However,
for a two span continuous member, having equal spans of
20 m, as shown in Fig. 3.4, the loaded length for calculat-
ing the support moment would be 40 m and hence a load-
ing of 26.2 kN/m would be applied; and the loaded length
for calculating the span moment would be 20 m and a
loading of 30 kN/m would be applied.
For multispan members, each case will have to be con-
sidered separately. 'rhus the moment at SUpp0l1 B of the
four span member of Fig. 3.5 would be calculated by con-
sidering spans AB and BC loaded with loading appropriate
to a loaded length of 2L, or spans AB, BC and DE loaded
with loading appropriate to a loaded length of 3L: the
former is likely to be more severe for most situations. The
E 35
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.9 31.5
30~"""'Tl
25[
20
15
I
I
I
I
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23 30 50 100 150 200 250 300 350 379 400 450 500
5
o
Loaded length m
F'i~. 3.3 HA uniformly distributed load
A B C
~d~_ _• 20,,~__ 4-- 20m
=l
(a) Centre of span A-B.
(b) Support B
Fig. 3.4(a).(b) Influence lines for two spans
moment in span BC would be' calculated by considering
span BC loaded with loading appropriate to a loaded
length of L, or spans BC and DE loaded with loading
appropriate to a loaded length of 2L.
A B C 0
~ L
t L
t L
t
~--
(a) Support B
(b) Centre of span B-C
Fig. 3.5(a),(b) Influence lines for four spans
E
L
~
+
/.
Knife edge load It is emphasised that the knife edge part
of HA loading is not intended to represent a heavy ax.le,
but is merely a device to enable the same unifonnly dis-
tributed loading to be used to simulate the shearing and bend-
ing effects of actual vehicle loading [102]. The Code value
of the load is the same as that in BE 1/77, and is 120 kN
per notional lane.
The load per metre is always obtained by dividing by
the notional lane width and is thus in the range 31.6 to
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![~Il, I
52.2 kN/m, which should be compared with the BE 1177
range of 32.4 to 52.2 kN/m.
The knife edge load is generally positioned perpendicu-
lar to the notional lane except when considering supporting
members, in which case it is positioned in line with the
bearings, and when considering skew decks, in which case
it can be positioned parallel to the supporting members or
perpendicular to the free edges. This clause is thus more
precise than its equivalent in BE 1177, which requires, for
skew slabs, the knife edge load to be placed in a direction
which produces the' worst effect. It is understood that the
intention of the Code drafters was that the intensity of
l(nding ~hNlld be 120 kN divided by the skew width ,of a
notional lane when the knife edge loading is in a skew po- .
sition with respect to the notional lane. Hence, the total
load is always 120 kN per notional lane.
Wheel load The wheel lond is used mainly for local
effect calculations and the nominal load is a single load of
100 kN with a contact pressure of 1.1 N/mm2: the contact
area could thus be a circle of diameter 340 mm or a square
of side 300 mm. The wheel load is considered to disperse
through asphalt at a spread to depth ratio of 1 to 2 and
through concrete at 1 to 1 down to the neutral axis.
This loading is thus 'different to the BE 1177 load which
consists of two 112 kN wheel loads, with a contact pres-
sure of 1.4 N/mm2, and a 45° angle of dispersion through
both asphalt and concrete.
The change from two wheel loads to a single wheel load
may seem drastic; however, there is a requirement in the
Code that all bridges be checked under 25 units of HB
loading. It is envisaged that the worst effects of the single
100 kN wheel load, or at least 25 units of HB loading, will
be at least as onerous as those of the two 112 kN wheel
loads.
The reduction in contact pressure results in a greater
contact area, but the reduced dispersal through asphalt off-
sets this somewhat when the effective area at the neutral
axis is considered.
*HB loading
HB loading is intended to represent an abnormally heavy
vehicle. The nominal loading consists of a single vehicle
with 16 wheels arranged on four axles, as shown in
Fig, 3.6, which also shows the BE 1177 HB vehicle. It can
be seen that the transverse spacing of the wheels on an
axle has been rounded-off to 1.0 m, and that the overall
width of 'the vehicle is now given as 3.5 m. The'latter
point means that it is not necessary to specify a minimum
distance of the vehicle from a kerb, as is necessary in BE
1177. However, the most significant difference is that the
longitudinal spacing of the centre pair of axles is no longer
constant at 6. I m, but can be any of five values between 6
and 26 m inclusive. The reason for this is that the BE' 1177
vehicle originated in BS 153, which was intended for sim-
ply supported bridges (although, in practice, it was also
applied to continuous briclges) and the worst effects in a
simply supported bridge occur with the axles as close
together as possible. However, since the Code is intended
to be applied to any span configuration, a variable spacing
40
1 unit/axle =10kN/axie ......Wheel
t$'
+ + +
1.0
+ Effective
3.5m 1.0 m :
+ +
wheel pressure
+ + + =1.4N/mm2
1.0m
+ + + +
,l.aml. 6,11,16,21 or26m"J-8~
10,16,20,26 or 30 m
(a) 866400
1unit/axle = 10kN/axle
+
O.
+ +
0.9
+
/Wheel
.v +
+
+
+ Effective
wheel pressure
+ = 1.1 N/mm2
*
+
o.9m: + + +
~IJ~,a~m~I._____6:~.1~m~___.J.amPi
(b) BE 1177
Fig. 3.6(a),(b) HB londing
has been specified in order to calculate the worst effects at
all design points. As an example, the worst effects at an
interior support of a continuous bridge could occur with a
wide axle spacing. .
The comments on the contact pressure of the HA wheel
load and its dispersal are also pertinent to the wheels of the
HB vehicle.
The nominal HB loading is specified, as in BE 1177, in
terms of units of loading, with one unit being equivalent to
a total vehicle weight of 40 kN. The number of units for
all roads can vary from 25 to 45, and this is at variance to
BE 1177 in which no minimum is specified; but, unlike BE
1177, the number of units to be adopted for different types
of road are not specified:, Presumably, the Department of
Transport will issue a Q1emorandum containing guidance
on this point. '
"*A /. .pp Icatlon
HA loading The full uniformly distributed and knife edge
loads are applied to two notional lanes and one-third of
these loads to all other notional lanes. The wheel load is
applied anywhere on the carriageway. The applications are
thus identical to those of BE 1177.
HB loading Only one HB vehicle is, in general, required
to be considered on anyone superstructure, or any sub-
structure supporting two or more superstructures. The
vehicle can be either wholly in a notional lane, or can straddle
two notional lanes. If it is wholIyhl a notional lane. then
the knife edge component of HA loading for that lane is
completely removed. and the uniformly distributed com-
ponent is removed for 25 m in front to 25 m behind the
vehicle; the remainder of the lane is loaded with the
uniformly distributed loading component of HA having an
intensity appropriate to a loaded length which includes the
displaced length. The vehicle is thus considered to displace
part of the HA loading in one lane, but the adjacent lane is
still assumed to carry full HA loading. This is more severe
than the BE 1177 loading, where the adjacent lane carries
only one-third HA loading. . . ,
If the, vehicle straddles two lanes, then It IS conSIdered
to straddle either the two lanes loaded with full HA, or
one lane with full and one with one-third HA; in each case
the rules for omitting parts of the HA loading, which were
described in the last paragraph, are applied. These
arrangements of load are different to those of BE 1177.
The reason for the more severe arrangement of the
accompanying HA loading is that, in practice, queues of
heavy vehicles accumulate behind abnormal loads and,
when they overtake, they do so in a platoon [18].
lL shoull.! be noted fro111 Tuble 3.1 that, "hen HA lond-
ing is applied with HB loading, the '{fL values for HA are
the same as those for HB.
Verges. central reserves, etc. The accidental wheel load-
ing of BE 1177. for the loading of verges and central
reserves, has been replaced in the Code by 25 units of HB
loading: outer verges need only to be able to support any
four wheels of 25 units of HB loading.
Transverse cantilever slabs should be loaded with the
appropriate number of units of HB loading for the type of
road in one notional lane plus 25 units of HB in one other
notional lane. The latter is intended to be a substitute for
HA loading and has been introduced because the HA load-
ing no longer increases for spans less than 6.5 m, as it
does in BS 153. This is the only occasion when more than
one HB vehicle can act on a structure.
Secondary highway loading
General
The secondary effectsofhighway loading are loads parallel to
the carriageway due to changes in speed or direction of the
traffic.,One should note that each of the following secondary
loads is consideredseparately, and not incombinationwiththe
others. An associated primary load is applied with each ofthe
secondary loads.
Centrifugal load
This is a radial force applied at the surface of the road of a
curved bridge. The nominal load is the same as that in BE
1177 and is given by
F = 30000 kN
c r+ 150
(3.1)
where r is the radius of the lane in metres. Any number of
these loads at 50 m centres should be applied to any two
notional lanes. Each load Fe can be divided into two parts
of F)3 and 2 Fel3 at 5 m centres if these give a lesser
effect.
The, loading was based upon tests carried out at the
Transport and Road Research Laboratory [104].
The nominal primary load associated with each load Fe
is a vertical load of 300 kN distributed uniformly over the
notionall~n,e for a length of 5 m. If the centrifugal load is
divided, then the vertical load is divided in the same pro-
portion.
~ 700
___________________~~B~S~5-40-0----
"tl
.9 600
500
400
BE 1/77
o 3 10 12 20 30 40 50 60 70 ao 90 100
Loaded length rri
Fig. 3.7 HA braking load
Longitudinal braking
The longitudinal forces due to braking are applied at the
level of the road surface. The nominal HA braking load is
8 kN per metre of loaded length plus 200 kN, with a maxi-
mum value of 700 kN. This load is much greater than the
BE 1177 loading with which it is compared in Fig. 3.7.
The new loading is based ~pon the work of Burt [105] and
is much greater than the BE 1177 loading because of the
greater efficiency of modem brakes, which can achieve
decelerations which approach 1 g on dry roads [18]. The
large increase in braking load could be significant in terms
of substructure design. The HA braking load is applied, in
one notional lane, over the entire loaded length, and in
combination with full primary HA loading in that
lane.
It is assumed that abnormally heavy vehicles can only
develop a deceleration of 0.25 g'and, thus, the HB braking
force is taken to be 25% of the primary HB loading.and to
be equally distributed between the eight wheels of either
the front two or the back two axles. This load is thus very
similar to that in BE 1177 for 45 units of loading, but is
less severe for a smaller number of units.
Skidding
This is a new loading which has been introduced because a
coefficient of friction for lateral skidding of nearly 1.0 can
be developed under dry road conditions [18]. A single
nominal point load of 250 kN is considered to act in one
notional lane, in any direction, and in combination with
primary HA loading.
*Collision with parapets
The Code is not concerned with the design of the parapets.
which will presumably still be covered by BE 5 [106], but
only with the load transmitted to the member supporting
the parapets. The nominal load is thus similar to the BE
1177 loading, and is defined as the load to cause collapse
of the parapet or its connection to the supporting member.
whichever is the greater. The additional primary load-
ing assumed to be acting adjacent to the point of collision
consists of any four wheels of 25 units of the HB
vehicle.
41](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-27-2048.jpg)
![3.3 11.0
Hard
shoulder
3 No. traffice lanes Hard Central
Verge strip reserve
Parapet
{
~
0.1 surfacing
1.2 ~=====~:O.9'''U''"~ral_ 1·. concrete
Fig. 3.8 Loading example
*Cc/I/sion ~1/ith supports
In general, the Code recommends the provision of protec- c
tion of bridge supports from possible vehicle collision. The
nominal loads for highway bridge supports which should
oe considered are the loads transmitted by the guard rail of
150 and 50 kN, normal and parallel to. the carriageway
respectively, at 0.75 m above the carriageway level or at
the bracket attachment point. In addition, at the most
severe point between 1 and 3 m above the carriageway,
residual loads of 100 kN should be considered both normal
and parallel to the carriageway. The normal and parallel
loads should not be considered to act together. The above
loads are only two-thirds of those in BE 1177 and the nor-
mal and parallel components of the latter have to be
applied together. Thus, even allowing for the fact that, at
the ultimate limit state, the product Y/L Y/3 is about 1.44,
as opposed to the BE 1177 safety factor of 1.15, the Code
loading is less severe than the BE 1177 loading.
For a foot or cycle track bridge, the nominal collision
load is a single load of 50 kN applied in any direction up
to a height of 3 m above the carriageway. In view of the
safety factors of 1.44 and 1.15 mentioned above, this load-
ing is more severe than its BE 1177 equivalent.
Fatigue and dynamic loading
Fatigue loading is considered in Chapter 12.
ft has been found [107] that the stress increments due to
the dynamic effects of highway loading are within the
allowance made for impact in the nominal loading, and
thus it is not necessary to consider the effects of vibration.
Footway or cycle track loading
If the bridge supports only a footway or a cycle track, the
nominal load is 5 kN/m2
for loaded lengths up to 30 m,
above which the load of 5 kN/m2
is reduced in the ratio of
the HA uniformly distributed load, for the loaded length
under consideration. to that for 30 m. The loading for
loaded lengths greater than 30 m is thus less severe than
the BE 1177 loading.
The loading on elements supp0rting footways or cycle
tracks. in addition to a highway or railway, is 80% of that
mentioned above. However, if the footpath is wider than
2 m, the loading may be reduced further.
Greater reductions in loading are permitted if a main
structural member supports two or more highway traffic
lanes or railway tracks. in addition to foot or cycle track
42
1:40 fall
loading. It is not obvious whether a slab bridge was
intended to come under this category, but it would seem
more reasonable to apply the previous '80% rule' to slab
bridges.
It is necessary to consider the vibrations of foot and
cycle track bridges as explained in Chapter 12.
Railway loading
The railway loading was derived by a committee of the
International Union of Railways and its derivation is fully
explained in an appendix to Part 2 of the Code.
Example
In the following example, the notation is in accordance
with Part 2 of the Code, to which the various figure and
table numbers refer. Numbers in brackets are the Part 2
clause numbers. ,
It is required to cai~ulate the nomina! transient loads
which should be consi.d~red for a highway underbridge of
composite slab constrU~tion and having the cross-section
of Fig. 3.8, zero skew and a span of 15 m. The bridge is
situated in the Birmingham area at a site which is 150 m
above sea level and there are no special funnelling, gust or
frost conditions. The anticipated effective bridge tempera-
ture at the time of settfng the bearings is 16°C. Assume
open parapets.
Lanes
Carriageway width (3.2.9.1) = 3.3 + 11.0 + 0.6
= 14.9 m
Number of notional lanes (3.2.9.3.1) = 4
Width of each notional lane = 14~9/4 = 3.725 m
Wind
Since the bridge is less than 20 m span and greater than
10 m wide, it would be possible to ignore the effects of
wind on the superstructure; however. the wind loads will
be calculated in order to illustrate the calculation steps.
Mean hourly wind speed (Fig. 2) =v = 28 mlS
0.15m
O.25m
,-__~2.5°C
'".)~ ..
,,'f v-v-
'~ ~
Loadings
0.18m
0.20 In
O.20m
O.18m
6.15°C .£-_ _ _ _ _----1
j' ~
II "
. ~ 1/
(a) Positive (b) Negative
FIM. 3.9(a).(b)
;trJr d- C'
V J'r
Kt (5.3.2.1.2)= 1.0 V
- ~j'SI (5.3.2.1.3) - 1.0
S2 (Table 2) = 1.49 for 6 m above ground and loaded
length of 15 m.
Maximum gust speed (5.3.2.1) =vc =V KI SI S2
:. Vc =41.7 m/s
This value of Vc applies when there is no live load on the
bridge; 5.3.2.3 states thatvc:l>35 mls with live load.
Area A I is calculated for the unloaded and loaded condi-
tions, with L = 15 m for both cases.
Unloaded (5.3.3.1.2(a)(I}(i», d = 1.2 m (Table 4)
:. Al = 15 >< 1.2 = 18 m2
Loaded (5.3.3.1.2(b», d = 0.9 + 0.1 + 2.5 = 3.5 m
:.AI = 15 >< 3.5 = 52.5 m2
Drag coefficient (CD) is calculated for the unloaded and
loaded conditions. with b = 17.6 m.
Unloaded, d = 1.2 m (Table 5 (b»
hid = 17.6/1.2 = 14.7
Co = 1.0 (Fig. 5)
Superelevation = 1:40 = 1.43°
Increase Cp by (Note 4 to Fig. 5) 3 >< 1.43 = 4%
C/J = 1.04
Loaded. d = 2.5 m (Table 5 (b»
hid = 17.6/2.5 = 7.04
C" = 1.24 (Fig. 5)
Increase for superelevation, Co = 1.29
Dynamic pressure head (5.3.3) = q =0.613 v/ N/m2
Unloaded. q = 0.613 >< 41.72
/1000 = 1.07 kN/m2
.
Loaded. q = 0.613 >< 352
/1000 = 0.751 kN/m2
Nominal transverse wind load (5.3.3) = P, =q AI CD
Unloaded. ,P, = 1.07 >< 18 >< 1.04 = 20.0 kN
Loaded. PI =0.751 >< 52.5 >< 1.29 = 50.9 kN
Nominal longitudinal wind load (5.3.4) is the more severe
of:
PLs =0.25 q AI CD
with q = 1.07 kN/m2
, AI = 18 m2
, CD =1.3 (5.3.4.1)
:. PL., =0.25 x 1.07 x 18 x 1.3 =6.26 kN
or PLs + PLL
where P'~r =0.25 q AI CD
with q = 0.751 kN/m2
, AI = 18 m2
, CD = 1.3 (5.3.4.1)
and PLL =0.5 q Al CD
with q = 0.751 kN/m2, AI = 2.5 x 15 = 37.5 m2,
CD = 1.45(5.3.4.3)
Thus PLs + PLL = (0.25 x 0.751 x 18 x 1.3) +
(0.5 x 0.751 x 37.5 x 1.45)
=24.8 kN
Thus PL = 24.8 kN
Nominal vertical wind load (5.3.5) =PI' =q A3 CI.
A3 = 15 x 17.6 = 264 m2
C1J = ± 0.75
Unloaded, Pv = 1.07 x 264 x (± 0.75) = ± 212 kN
Loaded, Pv = 0.751 x 264 x (± 0.75) = ± 149 kN
Temperature range
Minimum shade air temperature (Fig. 7) =-20°C
Maximum shade air temperature (Fig. 8) =35°C
Height corrections (5.4.2.2) are (-0.5) (150/100) =-O.soC
and (1.0) (150/100) = ISC respectively.
Corrected minimum shade air temperature = -20.8°C
Corrected maximum shade air temperature = 36.5°C
Minimum effective bridge temperature (Table 10) =-12°C
Maximum effective bridge temperature (Table 11) =36°C
Take the coefficient of expansion (5.4.6) to be 12 x l(1f'rc.
Nominal expansion = (36 - 16) 12 x lcr x 15 = 3.6 mm
Nominal contraction =(16 + 12) 12 x 1(1f' x 15 =5.04 mm](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-28-2048.jpg)
![Temperature differences
The temperature differences obtained from Fig. 9 are
shown in Fig. 3.9. The positive differences can coexist
(5.4.5.2) with effective bridge temperatures in the range
15° to 36°C and the negative differences with effective
bridge temperatures in the range _12° to 34°C.
HA
Uniformly distributed load for a loaded length of 15 m is
30.0 kN/m of notional lane (6.2.1). Thus the intensity is
30.0/3.725 = 8.05 kN/m2
• Knife edge load (6.2.2) =
120 kN/notional lane. Thus the intensity is 120/3.725 =
32.2 kN/m.
The wheel load (6.2.S) would not be considered for this
bridge, but it is a 100 kN load with a circular contact area
of 340 mm diameter. It can be dispersed (6.2.6) through
the surfacing, at a spread-to-depth ratio of I: 2, and
through the structural concrete at 1 : 1 down to the neutral
axis. Thus, diameter at neutral axis is 0.34 + 0.1 + 0.9 =
1.34 m.
HB
Assume 45 units, then load per axle is 4S0 kN (6.3.1). For
a single span bridge. the shortest axle spacing of 6 m is
required. The contact area of a wheel is circular with an
effective pressure of 1. J N/mm2
(6.3.2) Contact circle
diameter = [(450/4)(1000)(4)/(1.1 IT)] Iz =361 mm
Disperse (6.3.3) as for HA wheel load.
Diameter at neutral axis = 0.361 + 0.1 + 0.9 = 1.36 m
It should be noted that both HA and HB wheel loads can
be considered to have square contact areas (6.2.S and
6.3.3). '
Load on central reserve and verge
Load is (6.4.3) 25 units of HB. i.e. 62.5 kN/wheel. .
44
Diameter of contact circle at neutral axis
=[(62.S)(1000)(4)/(1.1IT)]Iz /1000 + 0.1 + 0.9
=1.27 m
Longitudinal
For a loaded length of IS m, the HA braking load (6.6.1)
is 8 x lS~+,~2oo = 320 kN applied to one notional lane in
combination withcprimary HA loading.
The total HB braking load (6.6.2) is 0.25 (4 x 4S0) =
4S0 kN equally distributed between the eight wheels of
two axles 1.8 m apart. Applied with primary HB loading.
Skidding
Point load of 250 kN (6.7.1) in one notional lane, acting
in any direction. in combination with primary HA loading
(6.7.2).
Collision with parapet
Parapet collapse load (6.8.1) in combination with any four
wheels of 25 units of HB loading (6.8.2).
Footway
In order to demonstrate the calculation of footway loading,
it will be assumed that the I.S m wide central reserve of
Fig. 3.8 is replaced by a 3 m wide footway. The bridge
supports both a footwax and a highway and the nominal
load for a ISm 10adedJength is'(7.2.1) 0.8 x 5.0 =
4.0 kN/m2
• However, .b~cause the footway width is in
excess of 2 m, this loading may be reduced as follows.
Load intensity on first 2 In = 4.0 kN/m2
Load intensity on other 1 itt = 0.85 x 4.0
= 3.4 kN/m2
Average intensity = (2 x 4.0 + 1 x 3.4)/3
= 3.8 kN/m2
Chapter 4
Material properties and
design criteria
Material properties
Concrete
'Y:Characteristic strengths
As indicated in Chapter 1 material strengths are defined in
terms of characteristic strengths. In Part 4 of the Code the
characteristic cube compressive strength (f.·,,) of a concrete
is referred to as its grade, e.g. grade 40 concrete has a
characteristic strength of 40 N/mm2. Grades 20 to 50 may
be used for nOimal weight reinforced concrete and 30 to 60
for prestressed concrete.
~s .tress-stram curve
The general form of the short-term uniaxial stress-strain
curve for concrete in compression is shown by the solid
line of Fig. 4.1(a). For design purposes, it is assumed that
the descending branch of the curve terminates at a strain of
0.0035, and that the peak of the curve and the descending
branch can be 'replaced by the chain dotted horizontal line
at a stress of 0.67 feu. The resulting Code short-term
characteristic stress-strain curve is shown in Fig. 4.1 (b).
The elastic modulus shown on Fig. 4.1 (b) is an initial
tangellf value, and the Code also tabulates s"(!cantvaiues
which are used for elastic analysis (see Chapter 2) and
for serviceability limit state calculations as explained in
Chapter 7.
*Other properties
Poisson's ratio is given as 0.2. and the coefficient of ther-
mal expansion as 12 x 10-6
/
o
C for normal weight con-
crete. with a warlling that it can be as low as 7 x lO-6/oC
for lightweight and limestone aggregate concrete. These
values are reasonable wh~n compared with published data
[96, 108].
Certain other properties, such as tensile strengths, are
required for the design of prestressed and composite mem-
bers, but these properties are included as allowable stresses
rather than as explicit characteristic values.
The shrinkage and creep properties of concrete can be
evaluated fro'm datu contained in an appendix to Part 4 of
Stress
Stress
0.67fcu
0.0035
(a) Actual and idealised
I
I
I
I
I
I
I
I
I
I
5.517;;;; kN/mm2
I
"--_CL----'--==-_ _-J.._ Strain
2.4 x 10-4 17;;;; 0.0035
(b) Code characteristic
Fig. 4.t(a),(b) Concrete stress-strain curves
Strain
the Code. Approximate properties for use in the design
of prestressed and composite members are discussed in
Chapters 7 and 8.
Reinforcement
:!fCharacteristic strengths
The quoted characteristic strengths of reinforcement (f,,)
are 250 N/mm2
for mild steel; 410 N/mm2
for hot rolled
45](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-29-2048.jpg)
![---------.......Stress
fy " ,-_._------------
~~
O.Sfy
_ Mild or hot rolled high yield steel
--- Cold worked high yield steel
- - - Code characteristic
Strain
Fig. 4.2 Reinforcement stress-strain curves
high yield steel; 460 N/mm2
for cold worked high yield
steel. except for diameters in excess of 16 mm when it is
425 N/mm 2; and 485 N/mm2
for hard drawn steel wire.
*'Stress -strain curve
The general forms of the stress-strain curves for mild or
hot rolled high yield steel and for cold worked high yield
steel are shown by the solid lines of Fig. 4.2. The Code
characteristic stress-strain curve is the tri-Iinear lower
bound approximation to these curves, which is shown
chain dotted in Fig. 4.2.
Prestressing steel
*Characteristic strengths
Tables are given for the characteristic strengths of wire.
strand, compacted strand and bars of various nominal size.
Each tabulated value is given as a force which is the
product of the characteristic strength (fpu) and the area
(Ap.) of the tendon.
Stress-strain curve
The tri-linear characteristic stress-strain curves for normal
and low relaxation tendons and for 'as drawn' wire and 'as
spun' strand are shown in Fig. 4.3. They are based upon
typical curves for commercially available products.
Material partial safety factors
*Values
As was explained in Chapter 1, design strengths are
obtained by dividing characteristic strengths by appropriate
partial safety factors (Ym)' The Ym values appropriate to
the various limit states are summarised in Table 4.1. The
sub-divisions of the serviceability limit state are explained
later in this chapter when the design criteria are discussed.
The concrete values are greater than those' for steel
because of the greater uncertainty associated with concrete
properties.
The explanation of the choice of 1.0 for both steel and
concrete for analysis purposes, at both the serviceability
and ultimate limit state. is as follows.
When analysing a structure, its overall response is of
interest and, strictly, this is governed by material proper-
46
Stress
0.8fpu ---
_.....
/,/
/
/
/
/
/
/
/
/ .
, 200 kN/mm2 for wire, strand
/ 175kN/mm2 for bar, 19 wire strand
Strain
(a) Normal and low relaxation products
Stress
0.6fpu
,/
I,/
/,/,/
/
/ 200 kN/mm2 for wire
J/ 175kN/mm2for 19 wire strand
~
Strain
(b) 'As draINn' wire and '~s spun' strand
Fig. 4.3(a),(b) Prestressirt$ steel stress-strain curves
Table 4.1 Ym values
Limit state Concrete Steel
Serviceability
Analysis of structure 1.0 1.0
Reinforced concrete cracking 1.0 1.0
Prestressed concrete cracking 1.3 1.0
Stress limitations 1.3 1.0
Vibration 1.0 1.0
Ultimate
Analysis of structure 1.0 1.0
Section design 1.5 I. 15
Deflection 1.0 1.0
Fatigue 1.3 1.0
ties appropriate to the mean strengths of the materials. If
there is a linear relationship between loads and their
effects. the values of the latter are determined by the rela·
tive and not the absolute values of the stiffnesses. Conse-
quently the same effects are calculated whether the ma-
terial properties are appropriate to the mean or character-
istic strengths of materials. Since the latter, and not the
mean strengths, are used throughout the Code it is simpler to
use them for analysis; hence Y... values of 1.0 are specified,
Stress
0.45fcu ------- I
Stress
I
I
I
I
I
I
I
I
I
.. I
I
I
I
4.5w;.; kN/~m2 I
........... l~ ,..-'::--·--·--·----·-..-..··-·-·-O.ooJ~
2 x 10 v;;,u Strain
(a) Concrete
0.87f,,,, - ......-......
0.7f'1II ...-..- -- ,,
I
/
I
,
I
,
,
,,
I ,-1 See Fig. 4.3 (a)
j./"' I
--,..... /
L'i.0.005
ic) Normal and low relaxation
pre-stressing steels
.. _....-_....
Strain
Stress
0.87fy
Stress
0.B7fplI - ....
0.52fpu -.--
Material properties and design criteria
Tension
/
.-.f-.-.-.-~~~~I!!2'!...-
I
/
I
I
I
I
I
I I
~,k{2Q(JkN/mm'
0.002
(b) Reinforcing bars
,/
I
I
,
I
/
,I
fl
'.' See Fig. 4.3 (b)
./ V
,L _ __ ._
•Strain
. -_._--------+
0.005 Strain
(d) 'As drawn' wire anrl'as spun' strand
t
Fig. 4.4(a)-(d) Design stress-strain curves for ultimate limit state
For the same reason as above. Ym is taken to be 1,0
when analysing a section if the effect under consideration
is associated with deformations; examples of this are
cracking in reinforced concrete, deflection and vibration.
However, when the effect under consideration is asso-
ciated with a limiting stress. then a value of Ym should be
adopted which should reflect the uncertainty associated
with the particular material and the importance of the par-
ticular limit state.
The values of 1.5 for concrete and 1.15 for steel origi-
Il<lted in the Comite Europeen du Beton, now the Comite
Ruro-International du Beton (CEB), which chose these
values hecause, when used with the CEB partial safety fac-
tors for loads, they led to structures which were sensibly the
same as those designed using the European national codes
11091. It should be noted that. although Ym for concrete is
partially intended to renect the degree of control over the
production of concrete, a single value of 1.5 is adopted in
the Code irrcspective of the control, whereas the CEB
Recommendations [110] state that y... can vary from 1.4.
for accurate batching and control, to 1.6, for concrete
made without strict supervision.
When carrying out stress calculations at the service-
ability limit state. Ym values of 1.3 for concrete and 1.0 for
.. steel are used. These values also originated in the CEB
[ III]. Cracking in prestressed conl'rete is considered to be
a limiting stress effect because, as explained later in this
chapter. it involves a limiting tensile stress calcul.ation.
However, it is emphasised that the Y", value of 1.3 !or
concrete for limiting stress calculations never has to be
used Ily a designer because the stress limitations, which
are given in the Code as design criteria, are design values
which include the Y", value of 1.3.
A value of 1.3 for concrete is giv.en for fatigue calcu-
lations because it is the strength of a section which is of
interest. However, this value need never be used Ily 11
designer because there is not a requirement in the Code to
check the fatigue strength of concrete.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-30-2048.jpg)
![Design stress-strain curves
The Ym values referred to above are applied to the charac-
teristic strengths whenever they appear in a calculation.
Hence design stress-strain curves are obtained from the
characteristic curves (see Figs. 4.1 to 4.3) by dividing the
characteristic strengths (feu, fy, fpu) , whenever they occur,
by the' appropriate values of Ym' The design curves at the
ultimate limit state are of particular interest and are shown
in Fig. 4.4.
It should be noted from Fig. 4.1 that the concrete
reaches its peak compressive stress, and then starts to
("nish, at a strain of about 0.002. Once the concrete starts
to crush it is less effective in providing lateral restraint to
any compression reinforcement, and there is thus a possi-
bility of the latter buckling. Hence the design str~ss of
compression reinforcement is restricted to the stress equiv-
alent to a strain of 0.002 on the d'esign stress-strain
curve. This stress is
1.15 + 2t60
(4.1)
It lies in the range 0.718 fy to 0.784 fy for fy in the range
485 to 250 N/mm 2
•
Design criteria
In this chapter the design criteria are presented and dis-
cussed, but methods of satisfying the criteria are presented
in subsequent chapters.
The criteria are given in Part 4 of the Code under the
headings: ultimate limit state, serviceability limit state
and other considerations. The latter includes those criteria
which are not specified in the Code but which are,
nonetheless. important in design terms. The criteria are
listed in Table 4.2 from which it can be seen that there are
a great number of criteria to be satisfied and, if calcu-
lations had to be carried out for each, the design procedure
would be extremely lengthy. Fortunately, as explained
later in this chapter, some of the criteria can be checked by
'deemed to satisfy' clauses. Furthermore, as experience of
Table 4.2 Design criteria
Ultimate limit state
Rupture
Buckling
Overturning
Vinratidn
Serviceability limit .~/ate
Steel stress limitations
Concrete stress limitations
Cracking of prestressed concrete
Cracking of reinforced concrete
Vihration
Olher considerations
Deflections
Fatigue
Durahility
....._---_... __........ _._--
the Code is gained, it will be possible to identify those·
criteria which would not be critical for a particular situ-
ation.
Each criterion is now discussed.
Ultimate limit state
The criterion for rupture of one or more sections, buckling
or overturning is simply that these events should not occur.
A vibration criterion, which 'would be concerned with
vibrations to cause collapse of a bridge, is not given, but,
instead, compliance with the serviceability limit state
vibration criterion is deemed to satisfy the ultimate limit state
requirements.
Serviceability limit state
Steel stress limitations
*Reinforcement It is explained in Chapter 7 that it is gen-
erally only necessary to check cracks widths in highway
bridges under HA loading for load combination 1. This
means that there is an indirect check on reinforcement
stresses under primary HA loading but not under other
loads. In view of the fact that it is desirable to ensure that
the steel remains elastic under all serviceability conditions,
so that cracks which open under the application of oc-
casional loading will close when the loading is removed, it
was decided to introduce a separate steel stress criterion. It
can be seen from Fig. 4.2 that the steel stress-strain curve
becomes non-linear at a stress of 0.8 fy and this stress is,
thus, the Code criterio!,. It is unfoI'tunate that the introduc-
tion of this criterion complicates the design procedure for
reinforced concrete for two reasons:
I. As shown later in this Chapter, different Yp values
are given for the crack width and steel stress calcu-
. lations and this could cause confusion.
2. As explained in Chapter 7, deemed to satisfy rules for
crack control in slab bridges are given in the Code,
but the fact that steel stresses have to be calculated
counteracts to a large extent the advantages of the
deemed to satisfy rules.
f.Prestressing steel Since a stress limitation existed for
reinforcement, it was considered logical to specify similar
criteria for prestressing steel. Hence, with reference to
Fig. 4.3, stress limitations of 0.8 fpu, for normal and low .
relaxation products, and 0.6fp ,I' for 'as drawn' wire and ,
'as spun'strand, are given in the general section of Part 4
of the Code. These criteria imply that tendon stress incre-
ments should be calculated under live load. Since such a
calculation is not generally carried out in prestressed con-
crete design, a clause in the prestressed concrete section of
Part 4 of the Code effectively states that the criteria can be
ignored. Hence, to summarise, although prestressing steel
limiting stress criteria are stated in the Code, they can be
ignored in practice.
Concrete stress limitations
It should be noted that this heading does not cover tensile
stresses in prestressed concrete, which are co.ve~d. by
'cracking of prestressed concrete'. The stress hmltatlons
referred to as 'concrete stress limitations' include compres-
sive stresses in reinforced and prestressed con:rete, and
compressive, tensile and interface shear stresses tn c()mpo-
site construction.
*Compressive stresses in reinforced concrete In order to
prevent micro-cracking, spalling and unacceptable
amo;Jllts of creep occurring under serviceability con-
ditions, compressive stresses are limited to 0.5 feu.
1f:Compl'essive stresses in prestressed concrete The limit-
ing stresses for the serviceability limit state and at transfer
are given in Table 4.3. It is not necessary to apply the Ym
value of 1.3 to the stresses. The stresses are identical to
those in BE 2/73 except that a stress of 0.4 feu is now per-
mitted in support regions because such a region is subject
to a triaxial stress system, due to vertical restraint to the
compression zone from the support. In addition, the actual
flexural stress at a support of finite width is less than the
stress calculated assuming a concentrated support.
* . , t t' TheCompressive stresses In composIte cons ruc Ion
compression flange of a prestressed precast beam with an
in-situ concrete slab is restrained by the latter and placed
under a triaxial stress condition. It is thus permitted, under
such conditions, to increase the limiting stresses of Table
4.3(a) by 50%, but the increased stress should not exceed
0.5 feU' (The Code clause actually refers to the stresses of
Table 4.3(b) but it should be those of Table 4.3(a).) The
resulting stresses are of the same order as those of BE
2173. However, the Code implies that the increased stresses
may be used both when a precast flange is completely
encased in in-situ concrete (such as a composite slab) and
for sections formed by adding an in-situ topping to precast
beams. This is contrary to BE 2173, which only permits
increased stresses to be used for the first of these situ-
ations. The author would suggest that the approach sug-
gested in the CP 110 handbook [112] could be adopted
for flanges which have a topping but are not encased in
in-situ concrete: the suggestion is that, in such circum-
stances, an increase of only 25% of the Table 4.3(a) values
should be permitted.
Table 4.3 Limiting concrete compressive stresses in prestressed
concrete
(a) Serviceability limit state
Loading
Bending
Direct compression
(b) Transfer
Stress distribution
Triangular
Uniform
Allowable stress
0.33 feu (0.4 feu at supports)
0.25 feu
Allowable stress
0.5 fei
O.4fei
Material properties and design criteria
Stress
/
Hypothetical stress at
first observed cracking
I
------1
I
I
I
I
I
/
/
I
Fig. 4.5 Tensile stress-strain curve of restrained concrete
16
• Tests [113]
- Code •
e 14 -- Codex2.5 ••• • -- ..&-
~ 12 · ......z • ,.......
~
10 .......,... .
1ii 8
,;4'
J!
.~
6
~
~4
2
0L---~10~--~2~0----~30~--~4~0----~50~--~60
feuN/mm2
Fig. 4.6 Tensile stresses at first observed cracking
'*Tensile stress;es in composite construction When flexural
tensile stresses are induced in the in-situ concrete of a
composite member consisting of precast prestressed units
and in-situ concrete, the precast concrete adjacent to the
in-situ concrete restrains the latter and controls any cracks
which may form. Hence, the descending branch of the ten-
sile stress-strain curve can be made use of, and tensile
strains in excess of the cracking strain of concrete toler-
ated. In the Code this is achieved by specifying allowable
stresses which are in excess of the ultimate tensile strength
of concrete and are, thus, hypothetical stresses as illus-
trated in Fig. 4.5.
The Code stresses are identical to those in BE 2/73 and
are given in Table 4.4
Table 4.4 Limiting concrete flexural tensile stresses in in-situ
concrete
In-situ concrete grade
Tensile stress (N/mm2
)
25
3.2
30
3.6
40
4.4
50
5.0
The Table 4.4 values are extremely conservative, as can
be seen from Fig. 4.6 where they are compared with in-
ferred stresses at which cracking was first observed in tests
on composite planks reported by Kajfasz, Somerville and
Rowe [113]. The Code values could be obtained from the
lower bound to the experimental values by applying a par-
tial safety factor of 2.5. It is thus extremely unlikely that
cracking of the in-situ concrete would be observed, even at
the ultimate limit state, in a composite member designed in
accordance with the Code.
49](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-31-2048.jpg)
![N
5 • Experimental first slip [117!E
E x Experimental ultimate [118!
"'z(I) x
(I)
4~....en
• •
3 - •"
•2 -
Code
o sb--.,..J6b·
Fig. 4.7 Surface type I interface shear stresses
It is permissible to increase the stresses in Table 4.4 by
50%, provided that the permissible tensile stress in the
prestressed unit is reduced by the same numerical amount.
This is because more prestress is then required, and it is
known [114] that the enhancement of the tensile strain
capacity of the adjacent in-situ concrete increases as the
level of prestress at the contact surface increases.
'*'Inteiface shear in composite construction Three types of
surface are defined as follows:
I. Rough and no steel across the interface.
2. Smooth and at least 0.15% steel across the interface.
3. Rough and at least 0.15% steel across the interface.
These surface types originated in CP 116 and will lead
to considerable problems for bridge engineers because the
rough surface of types 1 and 3 requires the laitence to be
removed from the surface by either wet brushing or tool-
ing, and this is not the usual practice for precast bridge
beams: the top surfaces of the latter are usually left 'rough
as cast'.
The minimum link area of 0.15% and a Code detail-
ing rule, which states that the link spacing in composite
T-beams should not exceed four times the in-situ concrete
thickness, nor 600 mm, were based upon American Con-
crete Institute recommendations [115].
The allowable interface shear stresses for beam and slab
construction are given in Table 4.5; however, it should be
emphasised that surface type 1 is not permitted for beam
and slab bridge decks because it is always considered
Table 4.5 Limiting interface shear stresses
In-situ concrete Limiting interface shear stress (N/mm2) for
grade
Type 1 Type 2 Type 3*
25 0.38 0.36 1.22
30 0.45 0.38 1.25
40 0.54 0.42 1.32
50 0.59 0.46 1.38
60 0.64 0.50 1.45
* Increuse by 0.5 N/mm 2
per I% of links in excess of 0.15%
4
N
E 3
E.....
Z
(I)
2(I)
~....en
0
6N
E
E.....
z 5
~
tiS 4
3
2
o
X X x
• x
x e
x • ••
e Experimental surface type 2 [118!
x Experimental surface type 3 [118!
- - Code surface type 3
---Code surface type 2
x
•x x
e. •e
------------------------
x
x
O. 0.4 0.6 0.8
-1
1.0 1.2
Percent steel across interface
(a) Experimental serviceability stresses
)(
X)(
•
x
•
e x ~
ex
x
e e
•
" •e • x
------------------------0.2 0.4 0.6 0.8 1.0 1.2
Percent steel across interface
(b) Experimental ultimate stresses
Fig. 4.8(a),(b) Surface types 2 and 3 interface shear stresses
(feu = 25 N/mm2
)
necessary to provide li~ks. This is because the calculated
shear capacity of an unreinforced interface cannot be relied
upon under conditions of repeated loading as occur on
bridges. It can be seen that the Code approach to interface
shear design is very different to that of BE 2173 which is
based upon an adaptation of the CP 117 approach [11 6]. In
contrast, the Table 4.5 values were essentially chosen to
be a little less conservative than the CP 116 values. How-
ever, they are still extremely conservative for surface types
1 and 2.
In Fig. 4.7 the Code surface type 1 stresses are com-
pared with some test results of Hanson [117] and Saemann
and Washa [l18]. It can be seen that the Code stresses are
very conservative.
The Code surface types 2 and 3 stresses can be consid-
ered by examining the experimental results of Saemann
and Washa [118], who tested composite I-beams in which
the steel area across the interface and the shear span to
effective depth ratio were varied. In addition, three sur-
faces were tested, and two of these were equivalent to the
Code types 2 and 3. In Fig. 4.8 the test data for a cube
strength of about 25 N/mm 2
and the Code stresses are
compared at a serviceability criterion of a slip of
0.127 mm (as suggested by Hanson [117]) and at failure. It
can be seen that the Code type 3 stresses are reasonable;
but the Code type 2 stresses are very conservative and
should be more dependent on steel area than the type 3
stresses, whereas the Code allows no increase of the type 2
stresses for a steel area in excess of 0.15%. In view of this
~he author would suggest that surface type 3 be considered
to be applicable to the 'rough as cast' surface used in
bridge practice.
Finally, it is emphasised that it was not intended that
interface shear should be checked in composite, slabs
formed from precast inverted T-beams with solid infill. It
is understood that the limiting stresses given in the Cbde
for cOinposite slabs were intended for shallow slabs. The
type of situation where they would be ~pplicable in bridg.es
is where the LOp slab u1 a (leeK cunsisls oj precast umts
spanning between longitudinal beams, and the units act as
permanent formwork for in-situ concrete to form a com-
posite slab.
'*Cracking of prestressed concrete
The criteria for the control of cracking in prestressed con·
crete are presented in terms of limiting flexural tensile
stresses for three classes of prestressed concrete.
Class I No tensile stresses are permitted except for
I N/mm 2
under prestress pJus dead load, and at transfer.
These criteria are thus identical to those of BE 2/73.
Class 2 Tensile stresses are permitted but visible crack-
ing should not occur. Beeby [119] has suggested that the
flexural tensile strength of concrete is equal to 0.556 /l.u.
The appropriate partial safety factor to be applied is 1.3
and thus design values of the flexural tensile strength
should be given by 0.428 !feu which compares favourably
with the Code value of 0.45 ./lcu for pre-tensioned mem-
bers. However, only 80% of this value (i.e. 0.36 ./lcu)
should be taken for post-tensioned members, because tests
reported by Bate [120] indicated that cracks in a grouted
post-tensioned member widen at a greater rate than those
in a pre-tensioned member, and thus the design stress for
the former should be less than that for the latter. No refer-
ence is made in the Code to unbonded tendons and, thus,
the author would suggest the adoption of t~e Concrete
Society recommendations of 0.15 Ifeu and zero in sagging
and hogging moment regions respectively [121].
It is necessary to check that, under dead and superim-
posed dead load, a Class 2 member satisfies the Class 1
criterion in order to ensure that large span bridges, for
which dead load dominates, have an adequate factor of
safety against cracking occurring under the permanent
loading which actuaJIy occurs in practice.
At transfer of a Class 2 member, flexural tensile stresses
of 0.45 Ifei and 0.36 ./lei are pennitted for pre- and post-
tensioned members respectively, where lei is the concrete
cube strength at transfer.
Class 3 Cracking is permitted provided that the crack
widths do not exceed the design values for reinforced con-
crete. given later in Table 4.7. However, it is not neces-
sary to carry out a true crack width calculation. because
the Code giv~s limiting hypothetical flexural tensile stress-
es which are deemed to be equivalent to the limiting crack
Material propertIes and d.esign criteria
Table 4.6 Hypothetical flexural tensile stresses for Class 3
members
(a) Basic stresses
Tendon
type
Limiting
crack
Stress (N/mm2) for concrete
grade
width (mm)
Pre-tensioned 0.1
Grouted
post-tensioned
0.2
0.25
0.1
0.2
0.25
Pre-tensioned, 0.1
close to tension 0.2
face 0.25
(b) Depth factors
Depth (mm) EO 200
Factor 1.1
400
1.0
30
3.2
3.8
4.1
600
0.9
40
4.1
5.0
5.5
4.1
5.0
5.5
5.3
6.3
6.8
800
0.8
;;, 50
4.8
5.8
6.3
4.8
5.8
6.3
6.3
7.3
7.8
;;,1000
0.7
widths. The basic stresses are given in Table 4.6(a), and it
can be seen that they are referred to as hypothetical stress-
es because they exceed the, tensile strength of concrete
and so cannot actually occur. The basic stresses were
derived from tests on beams by Bate [120] and Abeles
[122], who calculated the hypothetical tensile stresses
present at different maximum crack widths observed in
the tests.
Beeby and Taylor [123] have shown that the hypotheti-
cal tensile stresses should decrease with an increase in
depth and, thus, the basic stresse&, have to be multiplied by
a depth factor from Table 4.6(b).
The presence of additional reinforcement in a prestress-
ed member increases the crack control properties and a
higher hypothetical tensile stress may be adopted. The
Code increases of 4 N/mm 2 per 1% of additional steel, for
pre-tensioned and grouted post-tensioned tendons, and of
3 N/mm2
per 1% of additional steel, for pre-tensioned ten-
dons close to the tension face, are based upon the tests of
Abeles [122]. It should be noted that the steel percentages
are based upon the area of tensile concrete and not the
gross section area.
A Class 3 member has to be checked as a Class I
member under dead and superimposed dead load for the
same reason as that given previously for Class 2.
At transfer, the flexural tensile stresses in a Class 3
member should not exceed the limiting stress appropriate
to a Class 2 member. This is in order to avoid cracking at
the ends of members.
Finally, the Code gives no guidance on unbonded ten-
dons, and the author would suggest that [123] be consulted
for such members.
tCracking of reinforced concrete
The design surface crack widths were assigned from con-
siderations of appearance and durability, and were based
partly upon the 1964 CEB recommendations [111]. They
51](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-32-2048.jpg)
![Table 4.7 Design crack widths
Conditions of exposure
Moderate
Surface sheltered from
severe rain and against
freezing while saturated
with water, e.g.
(I) Surfaces protected by
a waterproof membrane
(2) Internal surfaces,
whether subject to
conden~ation or not
(3) Buried concrete and
concrete continuously
under water
Severe
(I) Soffits
(2) Surfaces exposed to
driving rain, alternate
wetting and drying, e.g.
in contact with backfill
and to freezing while
wet
Very severe
(1) Surfaces subject to the'
effects of de-icing
salts or salt spray. e.g.
roadside structures and
marine structures
excluding soffits
(2) Surfaces exposed to the
!lclion of seawater with
abrasion or moorland water
having a pH of 4.5 or less
Design crack
width (mm)
0.25
0.20
0.10
are summarised in Table 4.7. and it should be noted that
different design crack widths are assigned for different
conditions of exposure; unlike BE 1173, which differen- .
tiates only between different types of loading. In addition,
for bridge decks, different design crack widths are
assigned for soffits, and for top slabs if the latter are pro-
tected by waterproof membranes. The fact that both of
these design crack widths are as, or more, onerous than the
BE 1173 values of 0.25 and 0.31 mm is counteracted by
the fact that different crack width formulae are adopted, as
explained in Chapter 7. However, as discussed in Chapter
9. the very severe exposure condition for roadside struc-
tures could prove to be exceptionally onerous. and to lead
to impracticably large areas of reinforcement if applied to
piers and abutments.
The philosophy of relating the design crack width to the
condition of exposure and, thus. indirectly to the amount
of corrosion. must now be viewed with some scepticism in
the light of research at Munich, from which Schiessl [124]
concluded that there was no significant relationship be-
tween corrosion and crack width or cover. This work has
bcen discussed by Beeby [125J.
Vibration
It is only necessary to consider foot and cycle track
bridges, and the criterion is discomfort to a user of the
52
bridge. The derivation of the Code criterion is described
fully by Blanchard, Davies and Smith [107], and a sum-
mary follows.
The effects of vibrations on humans are closely related
to acceleration and, thus, maximum tolerable accelerations
were plotted against frequency in order to derive a crite-
rion in terms of natural frequency. The maximum tolerable
accelerations were assessed from two criteria: discomfort
while standing on a vibrating bridge, and the impairance of
normal walking due to large amplitude vibrations. The
Code criterion lies approximately' midway between these
two criteria and is given in an appendix to Part 2 as an
acceleration of 0.5 110 Ill/S
2
where fo is the fundamental
natural frequency of the unloaded bridge.
Other considerations
Deflection
A specific criterion is not given for deflection in terms of
an absolute limiting deflection, or of span to depth ratios.
However, it is obviously necessary to calculate deflections
in order to ensure that clearance specifications are not vio-
lated and adequate drainage will obtain. Deflection calcula-
tions are also important where the method of construction
requires careful control of levels. and for bearing design.
Fatigue
The relevant criterion is essentially that there should be a
fatigue life of 120 years. When considering unwelded bars
an equivalent criterion in terms of a stress range is given in
the Code The criterion is that the stress range should not
exceed 325 N/mm2
for high yield bars nor 265 N/mm 2
for
mild steel bars. These ranges are'identical to those of
BE 1/73 except that thera'nge for mild steel bars is inde-
pendent of bar diameter. "
It is not clear why stress ranges which are dependent
upon bar type have been adopted, and it would appear
,more logical for the stress ranges to be dependent upon the
type of loading and the loaded length: indeed, such a
dependence was considered during the drafting stages of
the Code.
Durability
A durability criterion is not defined but, provided that the
requirements of the Code with regard to Ilmiting crack
widths, minimum covers and minimum cement contents
are complied with, durability should not be a problem.
Yfa values
In Chapter 3 the nominal loads and the values of the partial
safety factor Y[I_' by which these' loads are mUltiplied to
give design loads. are presented. Furthermore, it is
explained in Chapter I that the effects of the latter have to
be multiplied by a partial safety factor Yf3 in order to
obtain design load effects. The values of Yf3 are dependent
upon the material of the bridge and hence, for concrete
bridges, are given in Part 4 of the Code. It is thus con-
venient to introduce the Yf3 values and the design criteria
together in this chapter. It should be noted that, some of the
values given in the following are not necessarily stated
in the Code, but are either implied or intended by the
drafters.
Jltimate limit state
A value of 1.15 for dead and superimposed dead load is
stated for all methods of analysis since, for loads which
are essentially uniformly distributed over the entire struc-
ture. any analysis should predict the effects with reason-
able accuracy. It is suggested in Chapter 1 that. Yf3 could
be considered to be an adjustment factor which ensures
that designs to the Code would be similar to designs to
existing documents. It can be shown that, on this basis, a
value of 1.15 for dead and superimposed dead load is a
reasonable average value.
The Code states that, for imposed loads, Yf3 should be
related to the method of analysis and quotes a value of 1.1
fol' all m~thods of analysis (including yield line theory)
except for methods involving redistribution, in which case
a value of [1.1 + (~- 10)/200] is quoted, where ~ is the
percentage redistribution. These values do not seem
entirely logical since the Yp value for an upper bound
method should be greater than that for a lower bound
method, because the former is theoretically unsafe and the
latter theoretically safe. The drafters' reason for including
yield line theory with lower bound methods was that,
although it is theoretically an upper bound method, tests
[86. 87] show that it predicts safe estimates of the
strengths of actual slabs; but this is also true of other
methods of analysis and, indeed, lower bound methods
predict even safer estimates of the strengths of actual slabs
[1261. Furthermore. it seems illogical to have a Yp value of
1.1 for the extreme cases of no redistribution (elastic
analysis) and what might be considered as full redistri-
bution (yield line: theory), yet values greatet;.than 1.1 are
permitted for redistrib,ution in the range 10 tq 30%. i
Since the max~mum permitted value of ~ is 30%, Yp
cannot be greater than 1.2 and it thus always lies in the
range 1.1 to 1.2 for imposed load and is always 1.~J5 for
dead load. In order to simplify the calculations. the Code
allows. as an altetnative. the adoption of 1.15 for all loads
and all types of ~nalysis, provided that ~ do~s not exceed
20%. The reason I for the proviso is that the' value of Yp
calculated from the formula is greater than 1.15 for B
greater than 20%.
It should be noted that the formula for Yp. gives values
less than 1.1 when ~ is less than 10%, but it was intended
that a value of 1.1 should be used for these cases.
When using yield line theory. Yp should be applied to
the load effects <the required moments of resistance) in
(l(:cordance'with equation (1.2). However, different values
of YI'1 have tt) be applied to dead loads and imposed loads.
This causes problems because, when using yield line
theory, the effects of different types of load cannot be cal~
culated separately and then added together. Thus, although
strictly incorrect, it is necessary to apply Yp as indicated
in equation (1.3) when using yield line theory.
*'Serviceability limit state
The Yp values at the serviceability limit state are all unity
exeept for the following: a value of 0.83 is applied to the
effects of HA loading and of·0.91 to the effects of HB
loading (or of HA combined with HB) when checking the
cracking limit state under load combination 1. These val-
ues are not stated in the Code, but are implied because the
Code ~tates a value of 1.0 for the product Yfl· Yf2. Yp;
and, SlDce y,d= Yfl. Yf2) is 1.2 for HA loading and 1.1
for HB loading at the serviceability limit state for load
combination 1 (see Table 3.1), the implied values of YP
are 0.83 and 0.91 respectively.
In the previous paragraph. it is implied by the author
that Yp should be taken to be unity. when checking the
stress limitation limit state. [n fact. an appropriate Y:.
value is not explicitly stated in the Code but it was t~e
intention of the drafters that it should be unity.
The fact that. for load c,ombination 1, different Yp val-
ues are specified for the cracking limit state, to those for
the stress limitation limit state, causes problems in the cal-
culations for reinforced and prestressed concrete members
for the following reasons:
I . In the case of reinforced concrete. stress limitation
calculations are carried out for a load of. essentially.
1.2 HA or 1.1 HB, whilst crack width calculations are
carried out for a load of 1.0 HA or 1.0 HB. This
could obviously cause con(usion and it also compli-
cates the calculations.
2. In the case of prestressed concrete, tensile stress cal-
culations are considered at the cracking limit state
and, hence. under a load of 1.0 HA or 1.0 HB: whilst
compressive stress calculations are considered at the
stress limitation limit state under a load of 1.2 HA or
1.1 HB. Hence stresses of different signs on the same
member are checked under different loadings. In view
of the anomaly so created. the prestressed concrete
section of Part 4 of the Code states that compressive
stresses should be checked under the same load as
that specified for tensile stresses: in other words.
loads of 1.0 HA and 1.0 HB are used for calculatiflR
both compressive and tensile stresses in prestressed
concrete.
It should be stated that the calculations are not necessar-
ily as complicated as implied above because. strictly.
stresses and crack widths should be calculated under the
design loads. which are 1.2 HA and 1.1 HB. and the
stresses and crack widths so calculated should then be
multiplied by the appropriateY(l value (1.0 or 0.83 or 0.91)
to give the design load effects as explained in Chapter I.
However, the adoption of different Yf'J values seems to be
an unnecessary complication. when the same final result.
in design temlS. could have heen obtained by modifying
the design criteria. '](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-33-2048.jpg)
![Summary
An attempt is now made to summarise the implications of
the Code Y/3 values and design criteria by comparing the
number of calculations required for designs in accordance
with current documents and with the Code.
*Reinforced concrete
In accordance with BE 1/73, a modular ratio design is car-
ried out at the working load, and crack widths are checked
at the same load. However, in accordance with the Code,
stresses, crack widths and strength have to be checked at
different load levels; and thus three calculations, each at a
different load level, have to be carried out, as opposed to
two calculations, at the same load level, when designing in
accordance with BE 1/73.
54
~
Prestressed concrete
The number of calculations for designs in accordance with
the Code and in accordance with BE 2/73 are identical
because, in both cases, stresses have to be checked at one
load level and strength at another.
'*Composite construction."...}-
In addition to the comments made above regarding rein-
forced and prestressed concrete construction, the design of
composite members is complicated by the interface shear
calculation. The latter calculation is considered at the
stress limitation limit state (Y/3 = 1.0) and the load level is
thus different to that adopted for checking the stresses in,
and the strength of, the, generally, prestressed precast
members. Thus, three load levels have to be considered for
a composite member designed in accordance with the
Code, as opposed to two load levels for a design in accor-
dance with BE 2/73.
Chapter 5
Ultimate limit state - flexure
and in-plane forces
Reinforced concrete beams
*Assumptions
The following assumptions are made when analysing a
cross-section. to determine its ultimate moment of resis-
tance:
1. Plane sections remain plane.
2. The design stress-strain curves are as shown in
Fig. 4.4.
3. If a beam is reinforced only in tension, the neutral
axis depth is· limited to half.the effective depth in
order to. ensure that an over-reinforced failure involv-
ing crushing of the concrete, before yield of the ten-
sion steel, does not occur. This is because such a fail-
ure can be brittle, and th~re is little warning that it is
about to take place. A balanced design, in which the
concrete crushes and the tension steel yields simul-
taneously, is given by considering the strain diagram
of Fig. 5.1 in which Ey is the strain at which the steel
commences to yield in tension. This strain Js given,
by reference to Fig. 4.4(b), by
E = 0 002 0.87/y
y • + 200000
(5.1)
Ey is thus in the:: approximate range of 0.003 to 0.004
and, for a balanced design, the neiJtral axis depth (t)
is approximately half of the effective depth since,
from Fig. 5.1, the neutral axis depth is given by
0.0035d
x =0.0035 +Ey
(5.2)
4. The. tensile strength of concrete is ignored.
5.. Small axial thrusts, of up to O.1feu times the cross-
sectional area, are ignored, because they increase the
calculated moment of resistance. [112]. .
*'Simpli,fied concrete stress block
The parabolic-rectangular distribution of concrete com-
pressive stress, implied by the stress-strain curve of
Eu -
0.0035
d
ty
Fig. 5.1 Strains for a balanced design
Fig. 4.4(a), is tedious to use in practice for hand calcu-
lations. The Code thus permits tlJe approximate rectangular
stress block, with a constant stress of O.4leu, as shown
in Fig. 5.2, to be adopted. Beeby [127] has demon-
strated that,. for beams, the adoption of· the rectangu-
lar stress block results in steel areas which are essen-
tially identical to those using the parabolic-rectangular
curve.
Strain compatibility
The ultimate moment of a section can be determined by
using the strain compatibility approach which involves the
following steps.
1. Guess a neutral axis depth and, hence,determine the
strains in the tension and compression reinforcement
by assuming a linear strain distribution and an extreme
fibre strain of 0.0035 in the compressive concrete.
2. Determine from the design stress-strain curves the
steel stresses appropriate to the calculated steel
strains.
3. Calculate the net tensile and compressive forces at the
section. If these are not equal. to a reasonable accu-
racy, adjust the neutral axis depth and return to step I.
4. If the net tensile and compressive forces are equal.
take moments of the forces about a common point in
the section to obtain the ultimate moment of resis-
tance.
55](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-34-2048.jpg)
![Stress
Parabolic-rectangular
0.4Sfcu I- - -- - -;•.,.,.. - • _. _. _. - ._.- "
0.4feu I
;' Simplified rectangu ar
.I.
b _-I
As
000
------r +__
d
z
I ....~- . ------+Fs = 0.87f~s
I .~ ..
I.
I.
Fig. 5.1 Rectangular stress block
Design charts
0.0035
Strain
The strain compatibility method described above is· tedi-
OWl for analysis and is not amenable to direct design. Thus
design charts are frequently used and the CP 110 design
charts [128] are appropriate.
Design formulae
As an alternative to using strain compatibility or design
charts, the Code gives simplified formulae for hand calcu-
lations. The formulae are based upon the simplified
rectangular stress block discussed previously and their
derivations are now presented.
"*Singly reinforced rectangular beam
The stresses and stress resultants at failure are as shown in
Fig. 5.3.
For equilibrium
F" = Fs
O.4t.."bx =0.87f.A.•
:. x == 2.2f0_·
t.·ub
Since a rectangular stress block is assumed, the lever arm
(z) is given by
-d /2-d ~z - -x - - t. b
('U
or z =(I -l.lf0·)d
fc"bd
(5.3)
However. the Code restricts z to a maximum value of
0.95<1. It is not clear why the Code has this restriction, but
Beeby 1127] has suggested that it could be either that there
is evidence that the concrete at the top of a member tends
to be less well compacted than that in the rest of the
member, or that it is felt desirable to limit the steel strain
at failure (a maximum lever arm of 0.95d implies a maxi-
mum steel strain of 0.0315).
The ultimate moment of resistance (M,,) can be obtained
by taking moments about the line of action of the resultant
concrete force; hence
Fig. 5.3 Singly reinforced rectangular beam at failure
(5.4)
However, the Code restricts the neutral axis depth to a
maximum value of 0.5d, in which case the ultimate
moment of resistance can be obtained by taking moments
about the reinforcement; hence
Mu =O.4fcub(O.5d) (O.7Sd) =O.15fcubcf2 (5.5)
The ultimate moment of resistance should tie taken as the
lesser of the values given by equations (5.4) and (5.5).
Equations (5.3) to (5.5) are given in the Code as design
equations, but they are obviously more suited to analysing
a given section, rather than to designing a section to resist
a given moment. In view of this it is best, for design pur-
poses, to rearrange the equati~ns as follows. From (5.4)
A=~., O.87fvz (5.6)
Substitute in (5.3) and solve the resulting equation for z
Z = 0.5d(1 +Ji-L~J2 ) (5.7)
Thus the lever arm can be calculated from equation (5.7)
and, then, the steel area from (5.6).
The Code limits the application of the design equations
to situations in which lessthan 10% redistribution has been
assumed. This is becau~the neutral axis depth is limited
to 0.5d and, for this value, the relationship between
neutral axis depth and amount of redistribution, which is
discussed in Chapter 2, implies a maximum redistribution
of 10%. However, it is possible to derive the following
more general version of equation (5.5), which is appropri-
ate for any amount of redistribution (~)
Mu =0.4fcubd(0.6-~) (0.7 +0.5~) (5.8)
Doubly reinforced rectangular beam
The procedure for deriving the Code equations for doubly
reinforced beams is to assume that the neutral axis is at the
same depth (0.5d) as that for balanced design with no
compression reinforcement. Compression reinforcement is
then provided to resist the applied moment which is in
excess of the balanced moment given by equation (5.5).
and tension reinforcement provided such that equilibrium
is maintained. The strains, stresses and stress resultants are
as shown in Fig. 5.4.
lt is mentioned in Chapter 4 that the design stress of
compression reinforcement is in the range 0.718fv to
0.784fv: it is thus conservative always to take a value of
O.72!v: as in the Code and Fig. 5.4. It should be noted,
from -the strain diagram of Fig. 5.4, that, in order that the
~
b
~
I-0.0035 ~
d' ;;;:0.002
0 A~ 0 ~
dod' d
As
o o o
Section Strains
Fig. 504 Doubly reinforced rectangular beam at failure
compression reinforcement may develop its yield strain of
0.002, and hence its des~gn strength of O.72fy , it is necess-
ary that
0.0035 (0.5d - d')/O.Sd ." 0.q,02
or d'ld :!EO 0.214
Hence the Code states that
d'/d:!EOO.2 (5.9)
The ultimate moment of resistance can be obtained by
taking moments about the tension reinforcement; hence
Mil =0.2f""bd(0.75d) + O.72fyA': (d-d')
or Mil = 0.15fcllbcf2 + O;72f.A': (d-d') (5.10)
From which A,,' can be calculated directly. For equilibrium
O.87f,As =0.2fc"bd + O.72fyA': (5.11)
From which A can be calculated directly.s
Equations (5.10) and (5.11) are given in the Code, but it
should be noted that the final term of equation (5.10) is
incorrectly printed in the Code. The equations are again
restricted to less than 10% redistribution, but can be written
more generally as
eI' 3
d = --::; (0.6-13) (5.12)
M" = 0.4t.."bd2
(0.6-(3) (0.7 + 0.5(3) +
O.72fyAs' (d-d') (5.13)
0.87[,. A., = 0.4f(."bd(O.6-~) + O.72fA: (5.14)
*Flanged beams'
It is assumed in the Code that any compressive stresses in
the web concrete can be conservatively ignored: this is
valid provided that the flange thickness does not exceed
half the effective depth. The stresses and stress resultants
at failure are then as shown in Fig. 5.5.
The ultimate moment of resistance is taken to be the
lesser of the value calculated assuming the reinforcement
to be critical:
M" =0.87f)As (eI-hj2) (5.15)
and that assuming the concrete to be critical:
I~
0-"1
Fs = 0.87f~$
b
0.4feu
Stresses
0.7Sd
F; =O.72f~~
Fe =0.2feubd
-'~
_--h,- 0.4fcubh,
As
00
d
Fig. 5.5 Flanged beam at failure
Mu =O.4t.,,,bh, (d-hj2)
---0.87f~s
(5.16)
These equations are given in the Code and can be used for
design purposes by calculating the steel area from (5. 15),
and checking the adequacy of the flange using (5.16).
The breadth b shown in Fig. 5.5 is the effective flange
width. which is given by the lesser of (a) the actual width
and (b) the web width plus one-fifth of the distance be-
tween points of zero moment, for T-beams, or the web
width plus one-tenth of the distance between points of zero
moment, for L-beams. The distance between points of zero
moment may be taken as 0.7 times the span for continuous
spans, and it would seem reasonable to take a value of
0.85 times the span for an end span of a continuous
member.
The resulting effective widths are of the same order as
those calculated in accordance with CP 114. In addition. a
comparison with values obtained from Table 2 of Part 5 of
the Code indicates that, at mid-way between points of zero
moment, the Code is generally conservative.
Prestressed concrete beams
*Assumptions
The assumptions made for reinforced concrete are also
made for prestressed concrete; in addition it is assumed
that:
t. The stresses at failure in bonded tendons can be
obtained fr~'m either the appropriate design stress-
strain curve of Fig. 4.4 or from a table in the Code
57](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-35-2048.jpg)
![which gives the tendon stress and neutral axis depth at
failure as functions of the amount of prestress. The
table is based upon the test data of Bate [120], and is
very similar to the equivalent table of CP lIS. How-
ever, the Code neutral axis depths are 87% of the CP
110 values because the Code adopts a design tendon
strength of O.87[p", whilst CP lIS uses an ultimate
strength of [pu' Hence, to maintain equilibriu~?,a,.
smaller neutral axis depth has to be adopted b~cause
the same concrete compressive stress is used [n both
CP lIS and the Code.
2. The stresses at failure in unbonded tendons are
ubla!m:o lrom a table in the Code which gives the ten- .
don stress and neutral axis depth as functions of the I.
amount of prestress and the span to depth ratio. The
table is based upon the resll'lts of tests carried out by
Pannell [129], who concluded that unbonded beams
remain elastic up to failure except for a plastic zone,
the extent of which depends upon the length of the
tendon. Hence, the failure stress and neutral axis
depth depend upon the span to depth ratio.
3. In order to give warning of failure it is desirable that
cracking of the concrete should occur prior to either
the steel yielding aQd fracturing, or the concrete crush-
ing in the compression zone. This can be checked. by
ensuring that the strain at the tension face exceeds the
tensile strain capacity of the concrete. The latter c~n
be obtained by multiplying the design limiting tensile
stress of 0.45 ff,." for a Class 2 member (see Chapter
4) by 1.3 (the partial safety factor incorporated in the
formula), and dividing by the appropriate elastic modu-
lus given in the Code.
Strain compatibility
The strain compatibility method described for reinforced
concrete can be applied to prestressed concrete, but the
prestrain in the tendons should be added to the strain, cal-
culated from the strain diagram at failure, to give the total
strain. The latter strain is used to obtain the tendon stress
from the stress-strain curve.
Design charts
Design charts are given in CP 1l0 [130] for rectangular
prestressed beams, but these are of limited use to bridge
engineers, who are normally concerned with non-
rectangular sections. To the author's knowledge design
charts for non-rectangular sections are not generally aVllil-
able, but Taylor and Clarke [131] have produced some
typical charts for T-sections. These should be useful to
bridge engineers, because they can be applied to a number
of standard bridge beams.
Design formula
The Code gives a formula for calculating the ultimate
moment of resistance of a rectangular beam, or of a
58
b I_ O.4fcu ./
-~--r---
d
Fig. 5.6 Rectangular prestr~ssed beam at failure
x
d
Fps =Ap.fpb
Fig. 5.7 General prestressed be.am at failure
z=d-O.5x
flanged beam with the neutral axis within the flange. The
formula is obtained by taking moments of the tendon
forces at failure about the line of action of the resultant
concrete compressive force. Hence, with reference to Fig.
S.6,
(S .17)
The tendon stress (Jpb) and neutral axis depth (t) at fail-
ure are obtained from the tables mentioned previously.
Although these tables, and hence equation (S.17), are
intended for rectangular sections, it is possible to adapt
them to non-rectangular,sections (1)2]. This is achieved
by writing down an equil,ibrium equation in terms of the
unknown tendon stress arl~ unknown neutral axis depth.
Hence, with reference tofig. S.7,
Ap.!pb = 0.4/mAc
where Ac is, generally, a linear function of x: thus /pb is
also, generally, a linear function of x. A graph can be plot-
ted of/Pb against x for the section under consideration and,
on the same graph. the Code tabulated values of/pb and x
can be plotted; the required values of/pb and x can be read
off where the two lines cross.
Reinforced concrete plates
General
The design of reinforced concrete plates for bridge~ is
more complicated than for buildings because, in bridges.
the principal moment directions are very often inclined to
the reinforcement directions (e.g. skew slabs), and plates
are often subjected to both bending and in-plane effects
(e.g. the walls of box girders). .
The Code states that allowance should be made for the
fact that principal stress resultant directions and reinforce-
ment directions do not generally coincide by calculating
required ~resistiye stress resultants' such that adequate
.strength is provided in all directions at a point in a plate.
No guidance is given in the ~ode on the calculation pro-
cedure, but the Code statemelllimplies that it is necessary to
.' satisfy the relevant yield criterion. In the following, it is
shown how this can be done for plates designed to resist
bending. in-plane or combined bending and in-plane
. effects. It should be. noted that all stress resultants are in
terms of values per unit length.
Bending
.Orthogonal reinforcement
In general, it is required to reinforce in the x and y direc-
tions a plate element which is subjected to the bending
moments Mx, My and the twisting moment Mxy shown in
Fig. 2.9.
The yield criterion for a plate element subjected to bend-
ing only is simply a relationship between the amounts of
reinforcement in the element and the applied moments
(Mit. My, M..y) which would cause yield of the element. It
can be shown [132J that the yield criterion is
(5.18)
M*,r and M*y are the moments of resistance per unit
length. of the reinforcement in the x and y directions
respectively, calculated in the reinforcement directions.
These moments of resistance can be calculated by means
of the methods previously described for reinforced con-
crete beams. A combination of M.., My, Mxy satisfying
equation (5.18) would cause the slab element to yield.
A designer is generally interested in determining values
of M*,. and M*y to satisfy equation (5.1S) for known val-
ues of M.., Ml" M n ,. This could be done by choosing either
M*.•, or M*,.. 'and then calculating the other from the yield
criterion. However. it is more convenient to make use of
equations which give M*x and M. directly. Such equa-
tions can be derived by noting that the follo,wing expres-
sions for M*x and M*y satisfy the yield criterion
M*,r = Mx- MxyK
M*I' =My ,- MxyK-1
(S.19)
(S.20)
Any value of K can be chosen by the designer and thus
there is an infinite number of possible combinations of M"'x
and M* ' capable of resisting a particular set of applied
moments. However, a solution which minimises the total
amount of steel at a point is generally sought and, to a first
order approximation, the total amount of steel is propor-
·tional to (M"'.. + M",). Hence the value of K required to
give a minimum steel consumption can be obtained by dif-
ferentiating (M"'x + M*y) with respect to K and equating to
zero, thus
M*x + M*y = Mx + My- Mxy (K + K-')
o(M + M*y) = -M.,I' (l _ K'2) =0
oK
K=±!
The second derivative is
o2(M*x + M'''y) =,;", 2M K-3
oK2' . xy
When considering bottom reinforcement, the sign con-
vention is such that M*x and M*y must be P9sitive,'hence
minimum steel consumption coincides with a mathematical
minimum and the second derivative should be positive.
Hence, Mxy and K must be ofopposite sign and . c
MxyK =MxyK-1
=-IMxyl
Thus, for bottom reinforcement
M*x =Mx + IMxyl
M*y = My + IMxyl
) (S.21)
When considering top reinforcement, the sign conven-
tion is such that M*It and M*y must be negative, hence
minimum steel consumption coincides with a mathematical
maximum and the second derivative should be negative.
Hence, M.ty and K must be of the same sign and
M.tyK =MxyK",j =IMxyl
Thus, for top reinforcement
M*x =Mx -IMxyl
M*y, = My -IMxyl
) (5.22)
Equations (S.21) and (5.22) are for the optimum
amounts of reinforcement with reinforcement in both the x
and y directions. but it is possible forr II value of M*It and
M*y so calculated to have the wrong sign. This implies
that no reinforcement is required in the appropriate din~c
tion, and another set of equations should be !.I!sed whRch C£lfl
be derived as follows.
If Mx < - IMxyl , so that a negative value of M*,. is
calculated from the first of equations (5.2]). then no rein-
forcement is required in the bottom in the x direction.
Hence. M = 0 can be substituted into equation (S.19) to
obtain a value of K
o = MX-,MxyK
K = MxlMxy
This value of K is thensubstituted into equation (S.20) to
give
M*y = My - M2x/Mx
Mx must be negative and thus this equation is generally
written
M*y = My + IM2x/M..1 (S.23)
Similar equations can be derived for .the other possi-
bilities and the complete set of equations, including
(S.21) and (S.22), are known to bridge engineers as
Wood's equations [133]; although they were originally
proposed by Hillerborg [134]. The complete set of equa-
tions is given in Appendix A to this book as equatiol)s A I
to AS.
Skew reinforcement
A similar set of equations - (A9) to (A 16) of Appendix
A - can be derived for skew reinforcement in the x direction
S9](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-36-2048.jpg)
![and in a direction at an angle Q: measured clockwise from
the x direction. These equations are known to bridge
engineers as Armer's equations [135].
Practical considerations
Experimental verification The validity of the theory,
upon which equations Al to AI6 is based, has been
confinned by tests on slab elements; in additio!" model" '.-
skew slab bridges have been designed by the equations and
successfully tested by Clark [126], Uppenberg [136] and
Hallbjorn [137].
Failure direction The direction in which failure (i.e.
yield) of a slab element occurs can be determined from
theoretical considerations. Strictly, once M'"x, etc.. have
been calculated from equations (AI) to (A16), they should
be resolved into the failure direction and the section design
carried out in this direction. However, in practice, it is
usual to carry out the section designs independently in
each reinforcement direction. Theoretically, this can mean
that the concrete is overstressed because the principal con-
crete stress occurs in the failure direction and not neces-
sarily in a reinforcement direction. The author would suggest
that, in practice. this error can be ignored because under-
reinforced sections. in which the concrete' is not critical,
are generally adopted. and the greater ductility of a slab
compared with a beam is neglected in design. Slabs are
more ductile than beams because the ultimate strain cap-
acity ofthe concrete in the compression zone increases as the
section breadth increases [138]. Thus, although Morley
1139] and Clark [126] discuss the correct section design
procedure. the author would suggest that the existing prac-
tice of designing the sections in the individual steel direc-
tions be continued.
Minimum reinforcement When reinforcement is propor-
tioned in accordance with the required moments of resis-
lance. calculated from equations (A I) to (A 16), it will
sometimes be found that the reinforcement areas are less
than the Code minima discussed in Chapter 10. In such
situations it is necessary to increase the reinforcement area
to the minimum specified in the Code. When this is done it
is often theoretically possible to decrease the reinforcement
area in another direction. As an example. if the value of
M from equation (A I) implies a reinforcement area in
the x direction which is less than the minimum. then the
minimum area should obviously be adopted in this direc-
tion, and the reinforcement area in the y direction can then
he less than that implied by equation (A2). The equations
for carrying out such calculations have been presented by
Morley 11391. However, it is obviously conservative, in
the above example. to provide the area of reinforcement in
the.v direction implied by equation (A2).
Multi,,/(, load combinations Bridges have to be designed
for a number of different load positions and combinations
and. hence. at each design point, for a number of different
moment triads (M,. Ml" M,)). Many computer programs
are available which calculate M*.•, M*v or M'"'" for each
triad lind then output envelopes of maximum values of
60
M'"x' etc. Such an approach ignores the interaction of the'
multiple triads; but it is simple and conservative. How-
ever, it is possible to reduce the total amount of reinforce-
ment required at a particular point by considering the
interaction of the multiple triads. The interested reader is
referred to the work of Morley [139] and Kemp [140] for
further infonnation on such procedures.
In-plane for~
.,
Equations, very similar to those discussed previously for
bending and twisting moments, have been derived for cal-
culating the forces required to resist an in-plane force triad
consisting of two in-phlile forces per unit length (Nx,Ny )
and an in-plane shear force per unit length (Nxy)' The sign
convention adopted for the forces is shown in Fig. AI.
The requiredl'esistive forces are designated N*x etc.,
and are equ!valent to the appropriate reinforcement area
per unit length multiplied by the design stress of the rein-
forcement. The equations..;, (A17) to (A30) - for deriving
the values of the required resistive forces are given in
Appendix A, together with equations for the principal con-
crete forces per unit length. The principal concrete stresses
can be obtained from the latter by dividing by the plate
thickness. .
The equations for orthogonal reinforcement are gener-
ally referred to as Nielson's equations [141] and the equa-
tions for skew reinforcement have been presented by Clark
[142]. It should be noted that it is required that the values
of N, etc., in equations (A17) to (A30) should always
be zero or positive, which implies that the r~inforcement is
always in tension. An extended set of equations which
includes the possibility that compression reinforcement
may be required has been. presented by Clark [142].
The validity of the equations have been continned
experimentally only for situations in which all of the rein-
forcement yields in tension [141. 143].
Morley and Gulvanessian [144] have considered the
problem of providing a minimum area of reinforcement in
a specified direction but have not presented explicit equa-
tions, although they do describe a suitable computer pro-
gram.
Multiple load combinations could be considered by the
approaches of [139] and [ 140].
Combined bending and in-plane forces
The provision of reinforcement to resist combined bending
and in~plane forces is extremely complex. A design sol-
ution is usually obtained by adopting a sandwich approach
in which the six stress resultants are resolved into two sets
of in-plane stress resultants acting in the two outer shells
of the sandwich. Such an approach has been suggested by
Morley [145]. If the centroids of'the outer shells are
chosen to coi'ncide with the centroid~ of the reinforcement
layers. as shown in Fig. 5.8. then equations (AI7) to
(A30) for in-plane forces can be applied to the above two
sets of in-plane stress resultants. Such an approach is valid
because both equilibrium and the in-plane force yield
c{ .---...
0 0 0 0 o.
-
0 n '0 0 0
csl.
Actual section
Fig. 5.8 Equivalent sandwich plate
0
-'l
N'!..a.,
r
l!.
2
h
2
criteria are satisfied:. thus a safe lower bound design
rem~. .
With reference to Fig. 5.8, it can be seen that the m·
plane force Nx and the bending moment Mx are stat!cally
equivalent to forces NxB and NxTapplied at the centrOIds of
the bottom and top outer shells respectively. The values of
the latter forces are
Mx + Nx (h/2 - cr)
= h,-CB-Cr
- Mx + Nx (h/2 - CB)
NxT = h - CB - cr
Similarly the other stress resultants are
=_M...l.y-;+,.....N-'y~(~hl_2_-_c.:..<T)
NYB h. - CB- CT
_ - My + Ny (h/2 - CB)
NyT -
h - CB- CT
N
_ Mxy + Nxy (h/2 - cr)
xyB -
h-CB-CT
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
N _ - Mxy + Nxy (h/2 - CB) (5.29)
xyT - h-CB-CT
Equations (A17) to (A30) can be used to design re-
inforcement, in the bottom, to resist NxB, NyB, NxyB and, in
the top, to. resist NxT• Nyr, Nxyr'
It should be noted that the core of the sandwich is
assumed to make no contribution to the strength of the sec-
tion. However, Morley and Gulvanessian [l~] have
extended the method' to include the possibility of the core
contributing to the strength.
Prestressed concrete slabs
The Code states that prestressed concrete slabs should be
designed in accordance with the clauses for prestressed
concrete beams. In addition, 'due allowance should be
made in the distribution of prestress in the case of skew
slabs'. The latter point is intended to emphasise the fact
that, when a skew slab is prestressed longitudinally, some
of the longitudinal bending component of the prestress is
distributed in the form of transverse bending and twisting
moments. The result is that the prestress is less than that
calculated on the basis of a simple beam strip in the direc-
Ultimate limit state -flexure and in-plane forces
Equivalent sandwich
300
1200 1080
N"r c;f
cJ
N"s Cd
Cd
In-situ slab
Formwork
Precast M-beams
at 1.5 m centres.
Each beam has
31 No. 15.2 mm
low relaxation
strands
200
t15
125
60
+++ --.:
+ + + + 6Oy- Jl10~______________~______- J
Fig. 5.9 Ellample 5.1
tion of the prestress. Clark and West [146] have given guid-
ance on the resultant prestress to be expected in skew slab
bridges.
Regarding section design for prestressed slabs, it is
difficult to imagine how the beam clauses can be applied to
a general case. The author would suggest that the prestress
should be considered as an applied load at the ultimate
limit state, and a set of bending moments and in-plane
forces, due to the prestress, calculated and added algebraic-
ally to those due to the applied loads. Conventional rein-
forcement could then be designed to resist the resulting I
'out-of-balance' stress resultants by using the equations
given in the Appendix. Clark and West have designed, and
successfully tested, model skew solid [146] and voided
[147] slab bridges by such an approach.
Examples
'*5.1 Prestressed beam section strength
It is required to calculate the ultimate moment of resistance
of the pre-tensioned composite section shown in Fig. 5.9.
The initial prestress is 70% of the characteristic strength
and the' losses amount to 30%. The precast and in-situ
concretes are of grades 50 and 40 respectively. From Table
21 of the Code
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![60 mm from the face, is subjected to the following design
stress resultants at the ultimate limit state
N.. = - 240 kN/m
N,. = 600 kN/m
N:.y = 340 kN/m
Mx = -166.0 kNmlm
My = 24.0 kNmim
Mxy = 1.6 kNm/m
Design reinforcem~nt in the x and y directions if fy =
250 N/mm2andleu = 50 N/mm2
The design strengths are
Reinforcement = 250/1.15 = 217 N/mm2
Concrete = 0.4 X 50 = 20 N/mm
2
In equations (5.24) to (5.29)
CT = CB = 60mm
hl2 - CT =hl2 - Cs =65 mm
h-CB-CT= 130 mm
The statically e.quivalent stress resultants in the outer shells
of the sandwich plate of Fig. 5.8 are calculated, from equ-
ations (5.24) to (5.29), as, in N/mm units
..
N.tn = -1397
Nyn = 485
N.t .l •n = 182
Bottom reinforcement
From equation (AI7)
64
NxT = 1157
NyT = 115
NxyT = 158
N*xB =-1397 + 11821 =-1215
N*xS < 0, :. N*xS =0 and calculate N*yB
from equation (A20)
N*yB = 485 + 1(182)2/(-1397)1 = 509 N/mm
AYB = (509/217)103
= 2340 mm2/m
From equation (A21)
. -4B= -1397 + (182)2/(-1397) =-1421 N/mm
Bottom.concrete compressive stress = 14211120
=11.8 N/mm2
This is less than the design stress of 20 N/mm2
Top reinforcement
From equation (A17)
N*xT = 1157 + 158 = 1315 N/mm
AxT =(1315/217)103
= 6060 mm2
/m
From equation (A18)
N*yT = 115 + 1158 = 273 N/mm
Ay7' = (273/217)103
= 1260 mm2
/m
From equation (A19)
FcT = -21581 = -316 N/mm
Top concrete compressive stress = 316/120 = 2.6 N/mm2
This is less than the design stress of 20 N/mm2
•
..Chapter 6
Ultimate limit state - shear
and torsion
:*'Introduction
In this chapter, the Code methods for designing against
shear and torsion for reinforced and prestressed concrete
construction are discussed. The particular problems which
arise in composite construction are not dealt with in this
chapter but are presented in Chapter 8.
It should be noted that, in accordance with the Code, all
shear and torsion calculations, with the exception of inter-
face shearin composite construction, are carried out at the
ultimate limit state.
With regard to shear, calculations have to be carried
out, as at present, for both flexural shear and, where.
appropriate, punching shear. However, it should be noted
that BE 1173 requires shear calculations for reinforced
concrete to be carried out under working load conditions as
opposed to the ultimate limit state as required by the Code.
In addition, as explained later, the procedures for design-
ing shear reinforcement in beams differ between BE 1173
and the Code.
The design of prestressed concrete to resist shear is car-
ried out at the ultimate limit state in accordance with both
BE 2173 and the Code, and the calculation procedures for
each are very similar.
Design against torsion is notcovered in th~:Department
of Transport's current design documents and, in practice,
either CP 110 or the Australian Code of Practice is often
referred to for design guidance. The latter document is
written in terms of permissible stresses and working loads,
and thus differs from the Code approach of designing at
the ultimate limit state.
Shear in reinforced concrete
Flexural shear
Background
The design' rules for flexural shear in beams are based
upon the work of the Shear Study Group of the Institution
of Structural Engineers [148]. The background to the
rules, which are identical to those of CP 110, has been.
described by Baker, Yu and Regan [149] and by Regan
[150].
The general approach adopted by the Shear Study Group
was, first, to study test data from beams without shear
reinforceme.nt and, then, to study test data from beams
with shear reinforcement. .
*Beams without shear reinforcement
The data from beams without shear reinforcement indicate
that, for a constant concrete strength and longitudinal steel
percentage, the relationship between the ratio of the
observed bending moment at collapse (Me) to the calcu-
lated ultimate flexural moment (Mil) and the ratio of shear
span (a,.) to effective depth (d) is of the form shown in
Fig. 6.1(a). '
This diagram has four distinct regions within each of
which a different mode of failure occurs: region t, corbel
action or crushing of a compression strut which runs from
the load to the support: region 2. diagonal tension causing
splitting along a line joining the load to the support; region
3, a flexural crack develops into a shear failure crack;
region 4. flexure. These modes are illustrated in Fig. 6. t
(b-e).
From the design point of view, it is obviously safe to
propose a design method which results in the observed
bending moment at collapse (Me) always exceeding the
lesser of the calculated ultimate flexural moment and the
moment when the section attains its calculated ultimate
shear capacity. Such an approach can be developed as fol-
lows: if the shear force at failure of a point loaded beam is
V,., then
Now, for a particular concrete strength and steel percen-
tage, the ultimate flexural moment is given by
Mil = Kbd2
where K is a constant. Thus
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![1
/crushing
3 4
avId
(a) General relationship
(b) Region 1
(e) Region 2
(d) Region3
(e) Region4
Flexural
failure
Fig. 6.1(a)-(e) Shear failure modes in reinforced concrete
beams
Hence,
~ =(~d)(~") (6.1)
It is thus convenient to replot the test data in the form of a
graph of (Mjbcf2) against (ajd) as shown by the solid l~e
of Fig. 6.2. The dashed line is that calculated assummg
that flexural failures always occur (i.e. Mjbd2
), and the
chain dotted line is a line that cuts off the unsafe side of
the graph (those beams which fail in shear) ~d. can thus
be considered to be an 'allowable shear lme. The
significance of equation (6.1) can now be seen because the
term Vjbd is the slope of any line which passes through
the origin. Hence if Vjbd is chosen to be the slope of the
chain dotted line, tlien Vjbd can be considered to de~ne
an 'allowable shear' line which separates an unsafe region
to its left from a safe region to its right. Furthermore Vjbd
can be considered to be a nominal allowable shear stress
(ve) which acts over a nominal shear area (bd). It is
emphasised that, in reality, a constant shear s~ress does not
act over such an area, but it is merely conveOlent to choose
a shear area of (bd) and to then select values of vc such
that the allowable values of the moment to cause collapse
fall below the test values.
66
Flexure
... 2
Fig. 6.2 Shear design diagram
Table 6.1 Design shear stresses (vc N/mm2
)
100 A.
Concrete grade
bd 20 25 30 :;;.40
:e; 0.25 0.35 0.35 0.35 0.35
0.50 0.45 0.50 0.55 0.55
1.00 0.60 0.65' 0.70 0.75
2.00 0,80 0.85 0.90 0.95
j;!!: 3.00 0.85 0.90 0.95 1.00
The code contains a table w.hich gives values of Vc for
various concrete grades and longitudinal steel percentages.
Reference to the table (see Table 6.1) shows that vc is only
slightly affected by the concrete strength but is greatly
dependent upon the area of the longitudinal steel.. This is
because the latter contributes to the shear capacity of a
section in the following two ways:
1. Directly, by dowel action [151] which can contribute
15 to 25% of the total shear capacity [152].
2. Indirectly, by controlling crack widths which, in tum,
influence the amount of shear force which can be
transferred by the 'interlock of aggregate particles
across cracks. Aggregate interlock can contribute 33
to 50% of the total shear' capacity [152].
It should be noted that when using Table 6.1, the lon-
gitudinal steel area to be used is that which extends at least
an effective depth beyond the section under consideration.
The reason for this' is given later in this chapter.
It is not necessary to apply a material partial safety fac-
tor to the tabulated vc values because they incorporate a
partial safety factor of 1.25 and are thus design values. In
fact the Vc values can be considered to be, by adopting the
terminology of Chapter 1, design resistances obtained from
equation (1.8). An appropriate Ym value would lie between.
the steel value of 1.15 and the concrete value of 1.5
because the shear resistance of a section is dependent upon
both materials. It was decided that a value of 1.25 was
reasonable for shear resistance when compared with the
usual value of 1.15 for flexural resistance.
'#:.Seams with shear reinforcement
When the nominal shear stress exceeds the appropriate
tabulated value of Vc it is necessary to provide shear rei~
forcement to res~st the shear force in excess of (vcbd). ThiS
approach differs to that of BE 1/73 in which shear rein-
Stirrup
.stress
I
1
1
1
1
1
. 1
1
1
c-/~I
;SI
~I
2./
~1.
'111
rfl
~I
(jl
1
1
1
1 Carried :ly concrete
/+----- -..-...... --_. ~'--"'-"'C""
Fig. (;.3 Influence of shear reinforcement
forcement has to be designed to resist the entire shear
force when the BE 1173 allowable concrete shear stress is
exceeded, and two-thirds of the she.ar force when the allow-
able concrete shear stress is not exceeded. The justification
for designing reinforcement to resist only th~ excess sh~ar
force is that tests indicate that the stresses 10 shear rem-
forcement are extremely small until shear cracks occur,
after which, the stresses gradually increase as shown in
Fig. 6.3, Hence, the shear resistance of the shear rein-
forcement is additive to that of the section without shear
reinforcement (i.e. v"bd). This was confirmed by the Shear
Study Group who, for vertical stirrups, obtained a good
lower bound fit to test data in the form:
(6.2)
where i' = V/bd and V is the shear force at failures; h'v is
the characteristic strength of the shear reinforcement; and
Asi' and s" are the area and spacing of the shear reinforce-
ment.
A theoretical expression for A,,,. can be derived by con-
sidering the truss analogy shown in Fig. 6.4(a). Theoreti-
cally, the inclination (8) of the compression struts, can be
assumed to take any value, provided that s~ear remforce~
ment is designed in accordance with the chosen value of
e. However, the greater the difference between eand the
inclination of the elastic principal stress (4~0) when a shear
crack first forms, the greater is the implied amount of
stress redistribution between initial cracking and collapse.
In order to minimise the stress redistribution,e is chosen in
the Code to coincide with the inclination of the elastic
principal stress (45°).
For vertical equilibrium along section A-A and assum-
ing that only the excess shear force is resisted by the shear
reinforcement:
(v - v,.)bd =A,v,.fy>, (sin ex + cos ex) (d - d')lsj; (6.3)
where ex is the inclination of the shear reinforcement (stir-
rups or bent-up bars). The Code assumes t~at d - d'=' d,
hence equation (6.3) can be rearranged to give
II = Vc + fyl' (sin ex + cos ex) (A,.vlbs,,) (6.4)
Top steel
A
Stirrups ...
Compression" -
strut ~~;z:=~~===:+:::::::;~~.
B
(a) Stirrups
i
Bent-up bar . '. f
VVn1: 45"
(b) Bent-up bars
Fig. 6.4(a),(b) Truss analogy for shear
d' ,
L ...[
1. d.'
•• "7"
For the case of vertical stirrups, ex = 90° and equation
(6.4) becomes
v = Ve +fyv (A",.Ibsv) (6.5)
It can be seen that equations (6.5) and (6.2) which have
been obtained theoretically and from test data respectively
are in good agreement. For' design purposes it is necessary
to apply a material partial safety factor of 1.15 (the value
for reinforcement at the ultimate limit state) to !VI'; equa-
tions (6.4) and (6.5) can then be rearranged as:
Asv b(v - lie)
s;:- =O.87fYl" (sin ex +cos ex)
(6.6)
Asv _ b(v - v~)
s:: -- 0.87(v"
(6.7)
The latter equation appears in the Code, with 1,'1' restricted
to a maximum value of 425 N/mm 2
• This is because the
data considered by the Shear Study Group indicated that
the yield stress of shear reinforcement should not exceed
about 480 N/mm2 in order that it could be guaranteed that
the shear reinforcement would yield at collapse prior to
. crushing of the concrete. The Code value of 425 N/mm2
is
thus conservative. It is implied, in the above derivation,
that equation (6.6) can be applied to either inclined stirrups
or bent-up bars. Although this is theoretically correct, the
Code states that when using bent-up bars the truss analogy
of Fig. 6.4(b) should be used in which the compression
struts join the centres of the bends of the lower and upper
bars. This approach is identical to that of CP 114 and it is
not clear why, originally, the CP 110 committee and, sub-
sequently, the Code committee retained it. It is worth men-
tioning that Pederson [153] has demonstrated the validity
of considering the compression struts to be at an angle
other than that of Fig; 6.4(b). .
In view of the limited amount of test data obtained from
beams with bent-up bars used as. shear reinforcement and
because of the risk of the concrete being crushed at the
bends, the Code permits only 50% of the shear reinforce-
ment to be in the form of bent-up bars.
Finally, an examination of Fig. 6.4(a) shows that if the
shear strength is checked at section A- B, then the
assumed shear failure plane intersects the longitudinal
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![tension reinforcement at a distance equal to the effective
depth from the section A-B. Hence the requirement men-
.tioned previously that the value of A.• in Table 6.1 should
be the area of the longitudinal steel which extends at least
an effective depth beyond the section under consideration..
This argument is not strictly correct. but instead. as shown
in Chapter 10 when discussing bar curtailment. the area of
steel should be considered at a distance of half of the lever
arm beyond the section under consideration. Thus the
Code requirement is conservative.
Maximum shear stress (vu)
It is shown in the last section that the shear capacity of a
reinforced concrete beam can be increased by increasing
the amount of shear reinforcement. However. eventually a
point is reached when the shear capacity is no longer
increased by adding more shear reinforcement because the
beam is then over-reinforced in shear. Such a beam fails in
shear by crushing of the concrete compression struts of the
truss before the shear reinforcement.yields in tension. The
Code thus gives. in a table. a maximum nominal flexural
shear stress of 0.75 /h.,,(but not greater than 4.75 N/mm2
)
which is a design value and incorporates a partial safety
factor of 1.5 applit?d to /,.,,: hence the effective partial
safety factor applied to the nominal stress is M. Clarke
and Taylor r154] have considered data from beams which
failed in shear by cr.ushing of the web concrete. They
found that the ratios of the experimental nominal shear
stress to that given by 0.75 If,." were in the range 1.02 to
3.32 with a mean value of 1.90.
The upper limit of 4.75 N/mm 2
imposed by the Code is
to allow for the fact that shear cracks in beams of very
high strength concrete can occur through. rather than
around. the aggregate particles. Hence a smooth crack sur-
face can result across which less shear can be transferred
in the fornl of aggregate interlock [152J.
Short shear spans
An examination of Fig. 6.2 reveals that for short shear
spans (a..ld less than approximately. 2) the shear strength
increases with a decrease in the shear span. Hence, the
allowahle nominal shear stresses (vJ are very conservative
for short shear spans. In view of this. an enhanced value of
I', which is given by 1',(2dla,.) is adopted for a..ld less than
2. However. the enhanced stress should not exceed the
maximum allowahle nominal shear stress of O.7S!lc.". The
enhanced stress has been shown to be conservative when
compared with data from tests on beams loaded close to
supports and on corbels [1121.
"fMinimum shear reinforcement
If the nominal shear stress is less than 0.5 lI,.• the factor of
safety against shear cracking occurring is greater than
twice that against flexural failure occurring. This level of
safety is· considered to be adequate and the provision of
shear reinforcement is not necessary in such situations.
If the nominal shear stress exceeds O.S VC bu( is less than
v,. it is necessary h.J pr'bvide a minimum.amount of shear
I'c·inforcement. In order that the presence of shear rein-
68
forcement may enhance the strength of a member. it is.
necessary that it should raise the shear capacity above the
shear cracking load. The Shear Study Group originally
suggested. from considerations of the available test data. a
minimum value of 0.87 Iv,. A....Ibs•. equal to 601b/in2
(0.414 N/mm2
) in order to ensure that the shear reinforce-
ment would increase the shear capacity. Hence for IV!' =
250 N/mm2
(mild steel) and 425 N/mm2 (the greatest'per-
mitted in the Code for high yield steel for shear reinforce-
ment). A....Is,. = 0.0019b and 0.00112b respectively: these
values have been rounded up to 0.002b and O.OOI2b in the
Code so that each is equivalent to 0.87/y,' A...Ibs,. =
0.44 N/mm 2
• The value of 0.0012b for high yield steel is
also the minimum value given in CP 114.
It is also necessary to specify a maximum spacing of
stirrups in order to ensure that the shear failure plane can.
not form between two adjacent stirrups, in which case the
stirrups would not contribute to the shear strength. Figure
6.4(a) shows that the spacing should not exceed
[(d-d')(l + cot ex)]. This expression has a minimum value of
(d - d') when ex =90~ andt to simplify the Code clause. it
is further assumed that (d - d')'::::!0.75d. This spacing
was also shown. experimentally. to be conservative by
plotting shear strength against the ratio of stirrup spacing
to .effective depth for vario.us test data. It was observed
that the test data exhibited a reduction in shear strength for
a ratio greater than about LO lI49].
Finally. it is necessary for the stirrups to enclose all the
tension reinforcement because the latter contributes. in the
form of dowel action. about 15 to 25% of the total shear
strength [151. 152]. If the tension reinforcement is not
supported by being enclosed by stirrups. then the dowel
action tears away the concrete cover to the reinforcement
and the contribution of dowel action to the shear strength .
is lost because the rei~forcement can then no longer act as
a dowel. .
'*Shear at points of c;ntraflexure
A problem arises near to points of contraflexure of beams
because the value of Vc to be adopted is dependent upon
the area of the longitudinal reinforcement. It is thus neces-
sary to consider whether the design shear force is accom-
panie~ by a sagging or hogging moment in order to deter-
mine the appropriate area of longitudinal reinforcement.
A situation can arise in which. for example, the area of
top steel is less than the area of bottom steel and the maxi-
mum shear force (V,) associated with a sagging moment
exceeds the maximum shear force (V,,) associated with a
hogging moment. However. because the area of top steel
is less than the area of bottom steel, the value of v,. to be
considered with V" could be less than that to be considered
with Vs' Thus although V. is greater than V". it could be
the latter which results in the greater amount of shear rein-
forcement. It can thus be seen that it is always necessary to
consider the maximum shear force associated with II sag-
ging moment and the maximum shear force associated
with a hogging moment. A conservative alternative pro-
cedure, which would reduce the number of calculations.
would be to tonsider only the absol~te maxim~111 shear
forte and to usc II value ofv(' approp~iate to the lesser of
the top or bottom steel areas.
If either ofthese procedures is adopted then, generally,
more shear. reinforcement is required than when the calcu-
'lations are carried out .inaccordanc:e with BE 1173, in
which the allowable shear stress is not dependent upon the"
. area of the longitudinal reinforcement. The increase in:
·shear reinforcement in regions of contraflexure was the
subject of criticism during the drafting stages of the Code
and in order to mitigate the situation an empirical design
rule, which takes account of the minimum area of shcar
reinforcement which has to be provided. has been included
in the Code.
The design rule implies that, for the situation described
above. shear reinforcement should be designed to resist (a)
V, with a v" value appropriate to the bottom reinforcement
and (b) the lesser of Vhand 0.8 v,. with <I v,. value appro-
priateto the top reinforcement. The greater area of shear
reinforcement calculated from (a) and (b) should then be
provided. The rule should be interpreted in a similar man-
ner for other relative values of Vs' V" and of bottom and
top reinforcement.
:rile logic behind the above rule is not clear. Further-
more. it was based upon a limited number of trial calcu-
lations which indicated that it was conservative. However, it
can be shown to be unconservative in some circuOlstunces.
In view of this. the author would suggest that it would be
safer not to adopt the rule in practice.
Slabs
General The design procedure for slabs is essentially
identical to that for beams and was originally proposed for
building slabs designed in accordance with CP lIO. The
implications of this are now discussed.
v,. values The values of v,. in Table 6.1 were derived
from the results of tests on. mainly, beams. although some
one-way spanning slabs with no shear reinforcement were
also considered. The slabs had breadth to depth ratios of
about 2.5 to 4 and thus were. essentially. wide beams,
It is probably reasonable to apply the Vc values tobuild-
ing slabs because the design loading is. essentially,
uniform and the design procedure generally involves con-
sidering one-way bending in orthogonal directions parallel
to. the flexural reinforcement. Hence it is reasonable to
consider slabs as wide beams. However, it is not clear
whether the same' Vc values can be applied to slabs, in
more general circumstances. when the support conditions
and/or the loading are non-uniform. An additional problem
occurs when the flexural reinforcement is not perpendicu-
lar to the planes of the principal shear forces because it is
not then obvious what area of reinforcement should be
used in Table 6.1. Although. strictly. this situation also
arises in building slabs. it is ignored for design purposes.
It is not certain whether it can also be ignored in bridge
slabs, where large principal shear forces can act at large
angles to the flexural reinforcement directions. It should
also be noted that shear forces in bridge slabs can vary
rapidly across their widths. A decision then has to be made
as to whether to· design against the peak shear force or a
value averaged over a certain width.
None of the above problems is considered in the Code.
Ultlmate limit state - shear and torsion
A possible 'engineering solution' would .be todesigI1
against shear forces averaged over a width of shib. equal to
twice the effective depth, and to carry out the sheardesign
calculation for the shear forces acting on planes normal to
each flexural reinforcement direction. The latter suggestion
of considering beam strips in each of the flexural r~in
forcement directions can be shown to be. in general.
unsafe. This is because it is the stiffness. rather thari the
strength, of the flexural reinforcement which is of impor-
tance in terms of shear resistaIlce. The flexural reinforce-
,.Inent should be resolved into a direction perpendicular to
the plane of the critical shear crack, and it is explained in
Chapter 7 that. when considel'ing stiffness•.reinforcement
areas resolve in accordance withcos4
cx, where ex is the
orientation of the reinforcement to the perpendicular to the
critical crack. Thus the resolved arca and, hence, the
appropriate Vc' value could be much less than the values
appropriate to the steel directions. However, in those re-
gions of slabs where a flexural shear failure could possibly
occur, such as near to free edges, the suggested approach
should be reasonable. Urifortunately, there is no experi-
mental evidence to justify the approaches suggested above.
1(::.Enhallced v" values The basic vc values of Table 6_J m~y_
be enhanced by multipl~ a tabulated factor. (s. > 1)'
which increases as tbe ov~ralJ dep-th~cre~~~.LpLQY..W..e.~L··
that the overaltde,pth isj~.§.~!ban 300 mm, The reason for
this is that tests have shown that the shear strength of a
member increases as its depth decreases. Relevant test data
have been collated byTaylor [155] and are summarised in
Fig. 6.S in terms of the shear strength (Vu) divided by the
shear strength of an equivalent specimen of 250 mm depth
(V250); due allowance has been made for dead load. shear
forces. It should be noted that illl of the test specimens
were beams. whereas the Code applies the enhancement
factor to slabs.
It can be seen that the Code values give a reasonable
lower bound to the test data for overall depths less than
500 mm. For greater depths. Taylor observed that there is
a reduction in shear strength for large beams but that the
minimum stirrups required for beams should take care of
this. However. the Code does not require minimum stir-
rups to be provided for slabs unless more than 1% of com-
pression steel is present: this did nO.t cause a problem in
the drafting of CP 110 because building slabs are generally
thin; however, bridge slabs can be thick. Furthermore.
Taylor suggested that code allowable shear stresses should
be reduced by 40% if the depth to breadth ratio of a beam
exce,eds 4. Such a ratio could be exceeded in bridge beams
and the webs of box girders. These points are raised here
to emphasise that the values of vc and s.. were derived with
buildings in mind. It is not possible at present to state
whether the values are appropriate to bridges beduse of
the lack of data from tests on slabs subjected to the stress
conditions which occur in bridges.
*Shear reinforcement When the noininal shear stress
exceeds ;, Vc> shear reinforcement should be provided and
designed, as for beams, to resist the shear force in excess
of that which can be resisted by the concrete (;s vcd per
unit length). The required amount of shear reinforcement
69](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-41-2048.jpg)
![1.5
••
•~ •V250
1.0 •
• •
• •
•0.5
Test
Code
o 250 500 750
Fig. 6.5 Slab enhancement factor
should be calculated from equations (6.6) o.r (6.7), as
appropriate, with vc' replaced by ~ vc This ~esign
approach is probably reasonable for building slabs for the
reasons discussed previously: however, it is not clear
whether it is reasonable for slabs subjected to loadil).gs
which cause principal shears which are not aligned with
the flexural reinforcement. A possible design approach for
such situations is suggested earlier in this chapter.
*'Maximum shear stress The maximum nominal shear
stress in slabs is limited to 0.375 !!cu, which is half of that
for beams. It is understood that this was originally sug-
gested by the CP 110 committee because it was felt that,
in building slabs, the anchorage of stirrups could not be
relied upon. The author would suggest that, if this is cor-
rect, it would also be the case for top !?labs of bridge decks
but not necessarily for deeper slab bridges.
Furthermore, it is considered that shear reinforcement
cannot be detailed and placed correctly in slabs less than
200 mm thick, and shear reinforcement is consequently
considered to be ineffective in such slabs. Hence, the maxi-
mum nominal shear stress in a slab less than 200 mrn
thick is limited to 1;" Vc and not to 0.375 !feu.
Minimum shear reinforcement . Unlike beams, it is not
necessary to provide minimum shear reinforcement if the
nominal shear stress is less than l;, vc' It is understood that
this decision was rriade by the CP 110 committee because .
it was considered t~ be in accordance with nonnal practice
for building slabs. It would also appear to be reasonable
for bridge slabs.
The maximum stirrup spacing for slabs is the effective
depth. This is greater than the maximum spacing of 0.75d
for beams because the latter value was considered to be too
restrictive for building slabs. The test data referred to
when discussing the 0.75d value for beams suggest that a
spacing of d should be adequate for both beams and
slabs.
70
•
1000
•
1250 1500
Overall depth mm
-¥vOided or cellular slabs No specific rules are given in
the Code for designing voided or cellular slabs to resist
shear. However, when considering longitudinal shear, it is
reasonable to apply the solid slab clauses and to consider
the shear force to be resisted by the minimum web thick-
ness. With regard to transverse shear, designers, at pres-
ent, either arrange the voids so that they are at points of
low transverse shear force or use their own design rules.
Possible approaches to the design of cellular and voided
slabs to resist transverse shear forces are given in Appen-
dix B of this book.
Punching shear
Introduction
Prior to discussing the Code clauses for punching shear, it
should be stated that most codes of practice approach the
problem of designing against punching shear failure by
considering a specified allowable shear stress acting over a
specified surface at a specified distance from the load. It is
emphasised that the specified surfaces do not coincide with
the failure surfaces which occur in tests, This fact can
cause problems when code clauses are applied in circum-
stances different to those envisaged by those originally
responsible for writing the clauses.
The Code clauses are based very much on those in
CP 110, which were written with building slabs in mind,
and these clauses are noJ. summarised.
CP 110 clauses
Punching shear in CP 110 is considered under two separate
headings: namely, 'Shear stresses in solid slabs under con-
centrated loadings' and 'Shear in flat slabs'. The former
clauses are concerned with the punching of applied loads
through a slab, whereas the latter are concerned with
punching at columns acting monolithically with a slab.
Critical
Support
(a) Elevation
(b) Plan
Fig. 6.6(a),(b) Punching shear perimeter
The critical shear perimeter for both situations is given
as 1.5 times the slab overall depth from the face of the
load or column as shown in Fig. 6.6, and the area of con-
crete, deemed to be providing shear resistance, is the
length of the perimeter multiplied by the slab effective
depth. A constant allowable design shear stress is assumed
to act over this area. The perimeter was chosen so that the
allowable design shear stress could be taken to be the val-
ues of Vc given in Table 6.1. Hence, the perimeter was
chosen so that the same Vc values could be used for both
flexural and punching shear. However, in the original ver-
sion of CP 110 it was stated that, for flat slabs having
lateral stability and with adjacent spans differing by less
than 25%, the tabulated values of Ve should be reduced by
20%. This was because the design approach was to take
the design shear force to be that acting when all panels
adjacent to the column were loaded and, thus, it was
necessary to make an allowance for the non-symmetrical
shear distribution which would occur if patterned loading
were considered. The reduction of Ve by 20% was thus
intended to allow for patterned loading [112]. Subse-
quently, in 1976, the flat slab clause was amended so that,
at present, vc is not reduced but the design shear force is
increased by 25% to allow for the possible non-
symmetrical shear distribution. If the slab does not have
lateral stability or if the adjacent spans are appreciably dif-
ferent, it has always been necessary to calculate the
moment (M) transmitted by the slab and to increase the
design shear force (V) by the factor (1 + 12.5 MIVl),
where I is the longer of the two spans in the direction in
which bending is being considered.
Regan [156] has shown, by comparing with test data,
that the original CP 110 clauses were reasonable. It should
be noted that most of the tests were carried out on simply
supported square slabs under a concentrated load and there
are very few data for slabs loaded with a concentrated load
near to a concentrated reaction, as occurs near to a bridge
pier. Regan quotes only three tests of such a nature and
reports satisfactory prediction, by the CP 110 clauses, of
the ultimate strength.
It is generally the case that the flexural reinforcement in
the vicinity of a concentrated load or a column head is
different in the two directions of the reinforcement. The
Load Actual failure
~ ~. ,mfa"
zZfl3", " }tI I
== O.5h -.I l--- .1 .
I I Reinforcement
1== 1.5h I
I~
Fig. 6.7 Failure surface
Alternative failure
ZllllmlllrS:~ f
I~Shear reinforcement!
Fig. 6.8 Influence of shear reinforcement
question then arises as to what area of reinforcement to
adopt for determining Vc from Table 6.1. CP 110 allows
one to adopt the average of the reinforcement areas in the
two directions and tests carried out by Nylander and Sund~
quist [157] in which the ratio of the -steel areas varied
from 1.0 to 4.1 justify this approach. These tests essen-
tially modelled a pair of columns with line loads on each
side as could occur for a bridge.
When averaging the reinforcement areas in the two
directions, the area in each di:rection should include all of
the reinforcement within the loaded area and within an
area extending to within three times the overall slab depth
on each side of the loaded area. The reason for considering
the reinforcement within such a wide band is that the
actual failure surface extends a large distance from the
load as shown in Fig. 6.7. The validity of considering the
reinforcement in a large band has been confirmed by the
results of tests carried out by M'oe [158] in which the
same area of reinforcement was distributed differently. It
was found that the punching strength was essentially inde-
pendent of the reinforcement arrangement.
If the actual nominal shear stress (v) on the perimeter
exceeds the allowable value of ~ ve, it is necessary to
design shear reinforcement in accordance with
(6.8)
where CIA.v} is the total area of shear reinforcement and
Ucrll is the length of the perimeter. This equation can be
derived in a similar manner to equation (6.7). It can be
seen from Fig. 6.7 that, in order to ensure that the shear
reinforcement crosses the failure surface, it is necessary
for the reinforcement to be placed at a distance of about
0.5h to 1.5h from the face of the load. In fact, CP 110
requires the shear reinforcement calculated from equation
(6.8) to be placed at a distance of O.75h. However,
CP 110 also requires the same amount of reinforcement to
be provided at the critical perimeter distance of 1.5h;
hence twice as much shear reinforcement as is theoretically
required has to be provided. It appears that such a conser-
vative approach was proposed because of the limited range
of shear reinforcement details covered by the available test
data [158].
The presence ofshear reinforcement obviously strengthens .
71](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-42-2048.jpg)
![a slab in the vicinity of the shear reinforcement and can
thus cause failure to occur by the formation of shear cracks
outside the zone of the shear reinforcement as shown in
Fig. 6,8. CP 110 thus requires the shear strength to be
checked also at distances, in steps of O.75h, beyond 1.5h
and, if necessary, shear reinforcement should be provided
at these distances.
The minimum amount of shear reinforcement implied by
equation (6.8) has to be provided only if v> ;"vc and is
very similar to the amount originally proposed for beams
by the Shear Study Group. .
As is also the case for flexural shear in slabs, the maxi-
mum nominal punching shear stress should not exceed
0.375 flu.
*BS 5400 clause
The clause in the Code which covers punching ,shear is
identical to that in CP 110 which covers 'Shear stresses in
solid slabs under concentrated loadings'. Hence, the modi-
fications in CP 110 which allow for non-uniform shear
distributions in flat slabs are not included in the Code.
Instead. whether punching of a wheel through a deck or of
a pier (integral or otherwise) is being considered the,design
procedure is to adopt' the CP 110 perimeters and the.Table
6.1 values of Vc (modified by ;., if the depth permits), and
to design shear reinforcement using equation (6.8).
Such an approach is probably reasonable when consi~er
ing wheel loads or piers which are not integral with the
deck. However, when dealing with piers which are integral
with the deck, and thus non-symmetry of the shear dis-
tribution and moment transfer should be considered, it
could be that a modification, similar to that for flat slabs in
CP 110, should be made to either the Vc values or the
design shear force. However, this has not been included in
the Code ,and the implication is that the effects of non-
symmetry and moment transfer can be ignored.
Further problems, which are probably of more impor-
tance to the bridge engineer than to the building engineer,
are those caused by voids running parallel to the plane of a
slab and by changes of section due to the accommodation
of services. These problems are not considered by the
Code.
Some tests have been carried out by Hanson [159] on
the influence on shear strength of service ducts having
widths equal to the slab thickness and depths equal to 0.35
of the slab thickness. He concluded that provided the
ducts were not within two slab thicknesses of the load
there was no reduction in shear strength. However, it is
not clear whether such a rule would apply to slabs with
voids as deep as those which occur in bridge decks.
~ Shear in prestressed concrete
Flexural shear
Beam failure modes
Two different types of shear failure can occur in pre-
stressed concrete beams as shown in Fig. 6.9.
72
Web
eraekiny
dl2 '
~
(a) Shear failures
(b) Shearforee (V)
(e) Bending moment (M)
(~)1 f4dI2~
d~,-(~),-f
(d) MIVdiagram
Fig. 6.9(a)-(d) Shear failure' modes in prestressed concrete
beams
In regions uncracked in flexure, a shear failure is caused
by web cracks forming when the principal tensile stress
exceeds the tensile strength of the concrete. In regions
cracked in flexure, a shear failure is caused either by web
cracks or by a flexural crack developing into a shear fail-
ure. Hence, it is necessary to check both types of failure
and to take the lesser of the ultimate loads associated with
the two types of failut~ as the critical load.
*Sections uncr8cked}n flexure
The ultimate shear strength in this condition is designated
Ven, and the criterion of failure for a section with no shear
reinforcement is that the principal tensile stress anywhere
in the section exceeds the tensile strength of the concrete.
If the principal tensile stress is taken as positive and equal
to the tensile strength of the concrete, then for equilibrium
Is =Veo Ay/lb
and
-I, = ifcp +Ib)/2 - Jifep +Ib)2/4 +1/
combining the equations gives
Vco =~y III + ifcp +Ib)/,
In the above equation,
Veo = shear force to cause web cracking
I = second moment of area
b = width
Ay = first moment of area
Icp = compressive stress due to prestress
Ib = flexural compressive stress
fs shear stress
I, tensile strength of concrete.
In order to simplify calculations, it is assumed that the
principal. tensile stress is a maximum at the bea~ centroid,,:
in which case fb = 0, and, for a rectangular sectIOn, Ib/Ay
:::: 0.67 bh. Hence
V"n =0.67 bh /tr + [,.,,[, (6.8)
The above simplification was originally introduced into
the Australian Code [160]. It should be noted that it is
unsafe for I-beams to consider only the centroid but this is
mitigated by the fact that, for such beams, Ib/Ay :::< 0.8~h
as opposed to 0.67 bh. However, for flanged oeams m
which the neutral axis is within the flange, it is considered
to be adequate to check the principal tensile stress at the
junction of the web and flange. This simplification again
originated in the Australian Code [160]. .
Tests on beams of concretes made with rounded river
gravels as aggregate have indicated [160] that I, = 54Yl
(in Imperial units). However, the Australian Code adopted
4/fry] in. order to allo~ f~r ~trength ~eductions c.au~ed ~y
shrinkage cracking, mild fatigue loadmg and vanatlons In
concrete quality. If the latter value is converted to S.1.
units and it is assumed that leVI = O.Slcu, then I, =
0.297 4". A partial safety factor of 1.5 was then applied
to ["/1 to give the design value of 0.24 !fe" which appears
in the Cbde.
Since a partial safety factor of jT3 is applied to fr,
partial safety factors of (/1.5)2 and jf.5 are implied in
the first and second terms respectively under the square
root sign of equation (6.S). In order that a partial safety
factor of (/13)2 is implied for both terms, it is necessary
to apply a multiplying factor of 1/jT3==0.8 to IC/I' This
results in the following equation, which appears in the
Code
v"" = 0.67 bh I/? + 0.8 f,·,.!, (6.9)
Reynolds, Clarke and Taylor [161] have compared equa·
tion (6.9). without the partial safety factors, with test
results and found that the ratios Of the observed shear
forces causing web cracking to Veo were, with the excep-
tion of one beam which had a ratio of 0.68, in the range
0.92 to 1.59 with a mean of 1.13.
.w.rumjn..G.!it1.ci!~lliIOnLID:.~_lh~~-,"Jbl<,'yl<Jj.i.~l!L£"Q.l!!pQ!l~.!lt.>
QfJh~. pre.§tI~~s._~hSmlgJ:!e.J!J!<!~~LtQ Yc.'L!L~Qt.!!in the::-:W}!!L
shear resista,nce. However._Jb.e.Sod~jLllly__~rmits ~g%,gf
the vertical comR2n~lJQ.J?.e._l!9.de<ljlLQrQer J.Q_h~L<;Q.Il§i§
tenLwith .~!!J!t.iQ.!L(§~2L
*Sections cracked in flexure
A shea~ failure can occur in a prestressed beam by a flex-
ural crack developing into an inclined crack which even-
tually causes a shear failure. The position of the critical
Ilexural crack, relative to the load. varies, but it has been
shown 11591 that it can be assumed to be at half the effec-
tive depth from the load. .
Sozen and Hawkins lJ621 considered the loads' at'
which a flexure-shear crack formed in 190 tests and
showed that a good lower bound to the shear force (Vcr)
could be given by the following empirical equation in
Imperial units
V(,r = 0.6 lnl ';[;.,. + M,/(MIV - d/2) (6.10)
where M, is the cracking moment and M and V are the.
moment and shear force at the section under consideration.
If this equation is transformed to S.I. units, it is assumed
that fey] = 0.8 leu and a partial safety factor of 1.5 applied
tofeu, then
Ver = 0.037 bd liu + M,/(MIV - d/2)
It is obviously conservative to ignore the d/2 term, in
which case the following equation, which appears i11 the
Code, is o~!ained.
Vcr = 0.037 bd!feu + V(M/M) (6.11)
It should be noted that, in equation (6.11), d is the dis-
tance from the extreme compression fibre to the centroid of
the tendons.
If the modulus of rupture of the concrete is fro then the
cracking moment is given by
M, =if, +Ip,)l/y
where /,)/ is the tensile stress due to prestress at an extr~me
fibre, distance y from the centroid. ACI·ASCE Committee
323 [163] originally suggested that, in Imperial units,fr =
7.5 ./ley! and the Australian Code [160] subsequently
reduced this to 6 !f..I' I to allow for shrinkage cracking.
repeated loading and varia.tions in concrete quality. If the
latter expression is converted to S.l. units, it is assumed
that fcyl = 0.8 /1'11 and a partial safety factor of 1.5 is
applied to fe". then the Code design "alue of 0.37 /f,." is
obtained. Again, only SO% of the prestress should be taken
to give a consistent partial safety factor. Hence, the fol-
lowing design equation, which appears in the Code, is
obtained.
M, =(0.37 /1..'/1 + 0.8/p,) l/y (6.12)
A minimum value of Vcr of 0.1 bd /l.u is stipulated in the
Code. This value originated in the American Code as, in
Imperial units, 1.7 bd .fl.y ]' The reason for this value is
not apparent but if it is converted to S.l. units. it is
assumed that In'1 :::: O.8fCII and a partial safety factor of 1.5
is applied to I;..;" then the Code value is obtained.
The majority of the beams for which equation (6.10) was
found to give a good lower bound fit had relatively high
levels of prestress, with the ratio of effective prestress to
tendon characteristic strength (f,,,/fpII) in excess of 0.5.
These beams were thus representative of Class 1 or Class 2
beams. but not necessarily of Class 3 beams which can
have much lower levels of prestress. A modified expres-
sion for Vcr was thus derived for Class 3 beams which
gives a linear transition from the reinforced concrete shear
clauses (f,,,,//,1lI =0) to the Class 1 and Class 2 formula
(equation (6.11)) when .f;,/(,,,, = 0.6. In view of the two
terms of equation (6.11) it was proposed [1611 that for
Class 3 members.
Vcr = A + B (6.13)
where A depends Oil material strength and is analogous to
the shear force calculated from the I'e values of Table 6. I.
and B is the shear force to f1exurally crack the beam. Both
A and B are to be determined. The term, A, was written as
a function of V,. and the effective prestress (f,,,)
A = (I - nf"Jf"lI) v,.hd
73 .](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-43-2048.jpg)
![where n is to be determined. This function was chosen for
A because it reduces to the reinforced concrete equation
whenfpe= O.
. Equation (6.11) can be expanded to the following by
using equation (6.12)
_ !. V MoV (6 14)
Vrr - 0.037 bd !/"u + 0.371!cu y M + M .
where Mo =0.8fpl Ily is the moment to produce zero stress
at the level of the steel centroid. It is thus convenient to
write the term, B, of equation (6.13) as Mo VIM and Vcr
becomes for reinforced and all classes of·.prestressed
concrete
(6.15)
Forreinforcedconcrete.fp, =Mo =oand hence Vcr =vcbd,
which agrees with the reinforced concrete clauses. In order
that equations (6.14) and (6.15) for Classes 1 and 2 and
Class 3 respectively agree for fp,lfpu =0.6. it is necessary
that
0.037 bd I!c.. + 0.371!cu f~, = (1 - O.6n) vcbd
A shear failure is unlikely to occur if MIV > 4h and thus
it is conservative [161] to put MIV = 4h. It is further
assumed that d == h, fly = bh2
/6 (the value for a rec-
tangular section), feu = 50 N/mm2, Ve = 0.55 N/mm
2
(i.e. 0.5% steel) and thus n = 0.55. Hence. the fol-
lowing equation, which appears in the Code. is obtained
Vcr =(1...,.. 0.55fpelfplI) vcbd + MoVIM (6.16)
In view of the large number of simplifications made in
deriving this equation. Reynolds. Clarke and Taylor
[161] compared it. with the partial safety factors
removed. with observed Vcr values from 38 partially pre-
stressed beams. The ratios of the experimental ultimate
shear forces to V"r were. with the exception of one beam
which had a ratio ·of 0.77. in the rang~ 0.97 to 1.40 with a
mean of 1. 18. The exceptional beam had a cube strength
of only 20 N/mm2 and a high amount of web reinforce-
ment.
The total area of both tensioned and untensioned steel in
the tension zone should be used when assessing Ve from
Table 6.1; and, in equation (6.16). d should be the dis-
tance from the extreme compression fibre to the centroid of
the steel in the tension zone. The total area of tension steel
is used because the longitudinal steel contributes to the
shear strength by acting as dowel reinforcement and by
controlling crack widths. and thus indirectly influencing
the amount of aggregate interlock. Thus any bonded steel
can be considered. The Code also implies that unbonded
tendons should be considered. but it could be argued that
they should be excluded because they cannot develop
dowel strength and are less effective in controlling crack
widths.
When both tensioned steel of area A.,(I) and characteristic
strengthf,>u(/) and untensioned steel of areaA.,(u)and charac-
teristic strength[Vl.(u)are present.fp,.lfpu should be taken as.
by analogy. the ratio of the effective prestressingforce (PI)
divided by the total ultimate force developed by both the
tensioned and untensioned steels. i.e.
74
PAAs(/)fpu(t) + As(u)/yL(u»'
It will be recalled that the d/2 term which appears in
equation (6.10) was ignored in deriving equations (6.11)
and (6.16). An examination of the bending moment and
shear force diagrams of Fig. 6.9 reveals that the value of
MIV at a particular section is equal to the value of
(MIV-d/2) at a section distance dl2 from the particular sec-
tion. It is thus reasonable to consider a value of Vcrcalcu-
,.,,,'tated from equations(6.11) and (6.16) to be applicable
for a distance 0{'d72·in"'i£i?Uireation of increasing mo-
ment from the particular section under consideration.
Finally. contrary to the principles of statics, the Code
does not permit the vertical component of the forces in
inclined tendons to be added to Ver to give the total shear
resistance. this requirement was based upon the results of
tests on prestressed beams, with tendon drape angles of
zero to 9.95°. reported by MacGregor. Sozen and Siess
[164]. They concluded that the drape decreased the shear
strength. However. since. except at the lowest point of a
tendon. the effective depth of a draped tendon is less than
that of a straight tendon. equations (6.11) and (6.16) do
predict a reduction in shear strength for a draped. as com-
pared with a straight. tendon. It is not clear whether the
reduction in strength observed in the tests was due to the
tendon inclination or the reduction in effective depth. If it
is because of the latter. then the Code effectively allows
for the reduction twice by adopting equations (6.11) and
(6.16) and excluding the vertical component of the pre-
stress. Hence. the Code, although conservative, does seem
illogical in its treatment of inclined tendons.
*shear reinforcement
The shear force (Ve) which can be carried by the concrete
alone is the lesser of V;~ and Vcr. If Vc exceeds the applied
shear force (V) then. thioretically. no shear reinforcement
is required. However, the Code requires nominal shear
reinforcement to be provided, such that 0.87 fy"AsJbsv
;;:. 0.4 N/mm2, if V;;:. 0.5 Ve' These requirements were
taken directly from the American Code. Thus shear rein-
forcement need not be provided if V < 0.5 Ve' In addition
the Code does not require shear reinforcement in members
of minor importance nor where tests have shown that shear
reinforcement is unnecessary. The CP 110 handbook
[112] de'fines members of minor importance as slabs.
footings. pile caps and walls. However. it is not clear
whether such members should be considered to be minor
in bridge situations and the interpretation of the Code
. obviously involves 'engineering judgement'.
If V exceeds Vc then shear reinforcement should be pro-
vided in accordance with
Asv V - Ve
s: =0.87 [VI' d,
(6.17)
where dr is the distance from the extreme compression
fibre to the centroid of the tendons or to any longitudinal
bars placed in the comers of the links. whichever is the
greater. The amount of shear reinforcement provided
should exceed the minimum referred to in the last para-
graph. Equation (6.17) can be derived in the same way as
equation (6.7).
The basic maximum link spacing is 0.75 d
"
which is the
same as that for reinforced concrete beams. However. if V
< > 1.8 Ve, the maximum spacing should be reduced to
0.5 d,: the reason forthis is not clear. In addition, for any
value of V. the link spacing in flanged members should not
exceed four times the web width: this requirement presum-
ably follows from the ·CP 115 implication that special con-
siderations should be given to beams in which the web
depth to breadth ratio exceeds four.
Maximum shear force
,.)/".
In order to avoid premature crushing of the web concrete.
it is necessary to impose aJ;l upper limit to the maximum
shear force. The Code tabulates maximum design shear
stresses which are derived from the same formula
(0.75 ./Tcu) as those for reinforced concrete. The shear stress
is considered to act over a nominal area of the web breadth.
minus an allowance for ducts. times the distance from the
extreme compression fibre to the centroid of aU (tensioned
or untensioned) steel in the tension zone.
Clarke and Taylor [154] have considered prestressed
concrete beams which failed by web crushing and found
that the ratios of observed web crushing stress to the Code
value of 0.75 I1cIl were in the range 1.04 to 4.50 with.a
·meanof2.l3.
It has been suggested by Bennett and Balasooriya
[165] that beams with a web depth to breadth ratio in
excess of ten could exhibit a tendency to buckle prior to
crushing: such a ratio could be exceeded in a bridge. How-
ever. the test data considered by Clarke and Taylor [154]
induded specimens with ratios of up to 17; and Edwards
[!66] has tested a prestressed box girder having webs with
slenderness ratios of 33 and did not observe any instability
problems. It thus appears that web instability should not be
a problem in the vast majority of bridges.
It is mentioned previously that a reduced web breadth,
to allow for ducts, should be used when calculating shear
stresses. The Code stipulates that the reduced breadth
should be the actual breadth less either the duct diameter
for ungrouted ducts or two-thirds the duct diameter for
grouted ducts. These values were originally suggested by
Leonhardt [167] and have subsequently been shown to be
reasonable by tests carried out by Clarke- and Taylor
[154] on prisms with ducts passing through them.
It should be noted that when checking the maximum.
she..ar.Jo.rccJ.D.Y-.Y-e.rticJtlj:..QIDPOOe1lLQLpr.e.s.tm£SJbould be___
considered o!'!l:x_fqL~~Jj!lll~Jm~.t@.£:k~(U.~Y!'~ This
again defies statics but is consistent with the approach to
calculating Veo and Vcr.
Slabs
The Code states that the flexural shear resistance of pre-
stressed slabs should be calculated in exactly the same
manner as that of prestressed beams. except that shear
reinforcement is .not required in slabs when the applied
shear force is less than Yr' This recommendation does not
appear to be based upon test data and the author has the
same reservations about the recommendation as those dis-
cussed previously in connection with reinforced slabs.
For design purposes, it would seem reasonable to aver-
Ultimate limit state - shear and torsion
age shear forces over a width of slab equal' to twice the
effective depth, and to carry out.the shear design calcu-
lation for the shear forces acting on planes normal to the
tendons and untensioned reinforcement. A similar
approach, for reinforced slabs. is discussed elsewhere in
this chapter.
*Punching shear·
The Code specifiesa.Aifferent critical shear perimeter for
prestressed concrete than for reinforced concrete. The
perimeter for prestressed concrete is taken to be at a dis-
tance of half of the overall slab depth from the load.
The section should then be considered to be uncracked
and Veo calculated as for flexural shear. In other words, the
principal tensile stress at the centroidal axis around the
critical perimeter should be limited to 0.24 l1cu. It should
be noted.that Clause 7.4 of the Code refers to values of Veo
in Table 32 of the Code whereas it should read Table 31.
If shear reinforcement is required. it should be designed
in the same way as that for flexural shear.
The above design approach is. essentially. identical to
that of the American Code [168] and was originally
proposed by Hawkins. Crisswell and Roll [169]. They
considered data from tests on slab-column specimens and
slab systems. and found that the ratios of observed to cal- .
culated shear strength (with material partial safety factors
removed) were in the range 0.82 to 1.28 with a mean of
1.06. The data were mainly from reinforced concrete slabs
but 32 of the slabs were prestressed. In addition. the
specimens had concrete strengths and depths less than
those which would occur in bridges. However. the author
feels that the Code approach should be applicable to bridge
structures.
Finally, the reservations. expressed earlier when dis-
cussing reinforced concrete slabs. regarding non-uniform
shear distributions and the presence of voids are also
applicable to prestressed slabs.
Torsion - general
Equilibrium and compatibility torsion
In the introduction to this chapter it is stated that. accord-
ing to the Code. torsion calculations have to be carried out
only at the ultimate limit state.
An implication of this fact is that it is necessary to think
in terms of two types of torsion.
Equilibrium torsion
In a statically determinate structure. subjected to torsional
loading. torsional stress resultants must be present in order
to maintain equilibrium. Hence. such torsion is referred to
as equilibrium torsion and torsional strength must be pro-
vided to prevent collapse occurring. An example of
equilibrium torsion is that which arises in a cantilever
beam due to torsional loading.
It is assumed in the Code that the torsion reinforcement
provided to resist equilibrium torsion at the ultimate limit](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-44-2048.jpg)
![state is adequate to control torsional cracking at the ser-
viceability limit state.
Compatibility torsion
In a structure which is statically indeterminate, it .is
theoretically possible to provide no torsional strength, and
to prevent collapse occurring by designing more flexural
and shear strength than would be necessary if torsional
strength were provided. The explanation of this is that a
stress resultant distribution, within the structure, with zero
torsional stress resultants and which satisfies equilibrium
can always be found. Since such a distribution satisfies
equilibrium it leads to a safe lower bound design [27].
Although a safe design results from a stress resultant
distribution with no twisting moments or torques, it is
obviously necessary, from considerationsof compatibility,
for various parts of the structure to displace by twisting.
Hence, such torsion is refened to as compatibility torsion..
The torsion which occurs in bridge decks is, generally,
compatibility torsion and it would be acceptable, in an
elastic analysis, to assign zero torsional stiffness to a deck.
This would result in zero twisting moments or torques
throughout the deck and bending moments greater than
those which would occur jf the full torsional stiffness were
used.
In the above discussion it is implied that either zero or
the full torsional stiffness should be adopted. However, it
is emphasised that any value of torsional stiffness could be
adopted. As an example, Clark and West [170] have
shown that it is reasonable, when considering the end
diaphragms of beam and slab bridges, to adopt only SO%
of the torsional stiffness obtained by multiplying the elastic
shear modulus of concrete by Ji (see equation (2.43».
Finally, although it is permissible to assume zero tor-
sional stiffness at the ultimate limit state, which implies
the provision of no torsion reinforcement, it is necessary to
provide some torsion reinforcement to control any tor-
sional cracks which could occur at the serviceability limit
state. The Code assumes that the nominal flexural shear
reinforcement discussed earlier in this chapter is sufficient
for controlling any torsional cracks.
Combined stress resultants
The Code acknowledges the facl Ihat, at a particular point,
the maximum bending moment, shear and torque do not
generally occur under the same loading. Thus, when
Fig. 6.11 Torsional cracks
designing reinforcement to resist the maximum torque,
reinforcement, which is present and is in excess of that
required to resist the other stress resultants associated with
the maximum torque, may be used for torsion reinforce-
ment.
Torsion of reinforced concrete
Rect~ngular section
Torsional shear stress
Methods of calculating elastic and plastic distributions of
torsional shear stress are available for homogeneous sec-
tions having a variety of cross-sectional shapes, including
rectangular. The calculation of an elastic distribution is
generally complex, and that of a plastic distribution is gen-
erally much simpler. However, neither distribution is cor-
rect for non-homogeneous sections such as cracked struc-
tural concrete.
In order to simplify c~lculation' procedures, the Code
adopts a plastic distributio'n of torsional shear stress over
the entire cross-section. It is emphasised that such a dis-
tribution is assumed not because it is correct but merely for
convenience. The Code also gives allowable nominal tor-
sional shear stresses with which to compare the calculated
plastic shear stresses. The allowable values were obtained
from test data.
It can be seen that the above approach is similar to that
adopted for flexural shear, in which allowable nominal
flexural shear stresses, acting over a nominal area of
breadth times effective depth, were chosen to give agree-
ment with test data.
The plastic torsional shear stress distribution is best cal-
culated by making use of the sand-heap analogy [171] in
which the constant plastic torsional shear stress (VI) is
proportional to the constant slope ('lJ) of a heap of sand on
the cross-section under consideration. In fact
V, = T'lJIK (6.18)
where T is the torque and K is twice the volume of the
heap of sand.
The plastic shear stress for a rectangular section can thus
be evaluated from Fig. 6.10 as follows
Volume of sand-heap= (hmi,,/2)(hmax){'lJhmi,,/2)
- (2)(1/6)(h",i")(I1,,,i,,/2)('lJll",i,,/2)
'(1/4)'1' h2
mi" (hmax - hmi,,/3)
y,
A.v--+-+-
x,
(a) Cross-section
Fig. 6.12(a),(b) Space truss analogy
Thus, from equation (6.18),
2T
VI =h2
ml" (hmax - hm;" 13)
(6.19)
This equation is given in the Code.
-'*.,orsion reinforcement
Design If the applied torsional shear stress calculated
from equation (6.19) exceeds a specified value (V,ml,,) it is
necessary to provide torsion reinforcement. One might
expect VI",;" to be taken as the stress to cause torsional
cracking or that corresponding to the pure torsional
strength of a member without web reinforcement. In fact,
it is taken in the Code to be 2S% of the latter value. Such a
vaiue was originally chosen by American Concrete Com-
mittee 438 [172] because tests have shown [173, 174] that
the presence of such a torque does not cause a significant
reduction in the shear or flexural strength of a member.
The tabulated Code values of Vlmi" are given by
0.067 /f.," (but not greater than 0.42 N/mm2
) and are
design values which include a partial safety factor of 1.S
applied to feU' The formula is based upon that originally
proposed by the American Concrete Institute but has been
modified to (a) convert from cylinder to cu~ strength,
(h) allow for partial safety factor and (c) allow for the fact
Ihat the American Concrete Institute calculates VI from an
equation based upon the skew bending theory of Hsu [17S]
instead of the plastic theory.
If VI exceeds Vlmi,,' torsion reinforcement has to be pro-
vided in the form of longitudinal reinforcement plus closed
links. The reason for requiring both types of reinforcement
is that. under pure torsional loading, principal tensile stress-
es are produced at 45° to the longitudinal axis of a beam.
Hence. torsional cracks also occur at 4So and these tend to
form continuous spiral cracks as shown in Fig. 6.11. It is
necessary to have reinforcement, on each face, parallel and
normal to the longitudinal axis in order that the torsional
cracks can be controlled and adequate torsional strength
developed.
The amounts of reinforcement required are calculated by
considering, at failure, a space truss. This is analogous to
the plane truss considered for flexural shear earlier in this
Chapter. In the space truss analogy. a spiral failure surface
Compression strut
. y,
(b) Elevation on a larger face
is considered which, theoretically, could form at any
angle. However, for design purposes, it is considered to
fonn at the same angle (4S0) as the initial cracks in order
that the amount of stress redistribution· required prior to
collapse may be minimised. The failure surface assumed
by the Code is shown in Fig. 6.12 together with relevant
dimensions. .
If the two legs of a link have a total area of A.,v and' the
links are spaced at sv. then y1ls" links cross a line parallel
to a compression strut on a larger face of the member. If
the characteristic strength of the link reinforcement is fy,"
then the steel force at failure in each larger face is
Fy =fyv(A.,.,I2){y lls,,)
Similarly, the steel force at failure in each smaller face is
Fx =fyv(A.)2} (xlls,,)
The total resisting torque is
T =FyXl + FxYI
=fy,A,,,,xlylls,, (6.20)
At the ultimate limit state, fy" has to be divided by a partial
safety factor of I.IS to give a design stress of 0.87 fyl"
Hence, in the code, fyv in equation (6.20) is replaced by
0.87 fy," Furthermore, the Code introduces an efficiency
factor, which is discussed later, of 0.8 in order to obtain
good agreement between the space truss analogy and test
results. Hence, in the Code, equation (6,20) is presented
as
A.,. ;, T
Sv 0.8x)YI (0.87!v,')
(6.21)
The force Fy is considered to be the vertical component
of a principal force (F) which acts perpendicular to the
failure surface and thus F = F). ./2. The principal force
also has a horizontal (longitudinal) component FI./2
which, from above, is equal to F,,, A force of this mag-
nitude acts in each larger face and, similarly, a horizontal
(longitudinal) force of magnitude F., acts in each smaller
face. Thus the total horizontal (longitudinal) force. which
tends to elongate the section, is 2(Fx + Fv). This elongat-
ing force has to be resisted by the longitudinal reinforce-
ment: if the total area of longitudinal reinforcement is Ad.
77](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-45-2048.jpg)
![Diagonal thrust
(b) Section
..... Components of
diagonal thrustsc
'
Corners spall
Fig. 6.13(a),(b) Diagonal torsional compressive stresses
and it has a characteristic strength of/yL
AsUyL =2(F.. + Fy) =Asv/YV(XI + YI)/s"
Thus, in the Code,
A.•L ;;;=~:v(f) (Xl + Yl) (6.22)
Detailing The detailing of torsional reinforcement has
been considered by Mitchell and Collins [176] and the fol-
lowing Code rules are based very much on their work.
Diagonal compressive stresses occur in the concrete
between torsional .cracks, and such stresses near the edges
of a section cause the comers to spall off as shown in
Fig. 6.13. In order to prevent premature spalling it is
necessary to restrict the link spacing and the flexibility of
the portion of longitudinal bar between the links. Tests
reported by Mitchell and Collins [176] indicate that the
link spacing should not exceed (Xl + YI)/4 nor 16 times
the longitudinal comer bar diameter. The Code specifies
these spacings and also states that the link spacing should
not exceed 300 mm. The latter limitation is intended to
control cracking at .the serviceability limit state in large
members where the two other limitations can result in
large spacings. In addition, the longitudinal comer bar
diameter should not be less than the link diameter.
The characteristic strength of all torsional reinforcement
is limited to 425 N/mm2
primarily because such a restric-
tion exists for shear reinforcement. However, it is justified
by the fact that some beams, which were tested by Swann
[177] and reinforced with steel having yield stresses in
excess of 430 N/mm2
, failed at ultimate torques slightly
less than those predicted by the Code method of calcula-
tion [178]. The reason for this was that the large concrete
strains necessary to mobilise the yield stresS' of such high'
strength steel resulted in a reduction in the efficiency factor .
mentioned earlier and discussed in the next section of this
chapter.
Maximum torsional shear stress
The space truss of·Fig. 6.12 consists of the torsion re-I
inforcement acting as tensile ties pIllS concrete compressive
struts. As is also the case for flexural shear, it is necessary,
to limit the compressive stresses in the struts to prevent the
struts crushing prior to the torsion reinforcement yielding"
iu tension. This is achieved by limiting the nominal tor-
sional shear stress. However, the derivation of the limiting
values in the Code is connected with the choice of the
efficiency factor applied to the reinforcement in equation
(6.20).
It is mentioned elsewhere in this chapter that an effi-
ciency factor has to be introduced into equation (6.20) in
order to obtain agreement between test results and the pre-
dictions of the space truss analogy. Swann [178] found
that the efficiency factor decreases with an increase in the
nominal plastic torsional shear stress at collapse and also
decreases with a decrease in specimen size.
The dependence of the efficiency factor on stress is
explained by the fact that, if the nominal shear stress is
high, the stresses in the inclined compression struts be-
tween the torsional cracks of Fig. 6.11 are also high. Thus,
at high nominal stresses, a greater reliance is placed upon
'the ability of the concrete in compression to develop high
stresses at high strains than is the case for 'low nominal
stress. In view of the strain-softening exhibited by concrete
in compression (see Fig. 4.1 (a» it is to be expected that'
the efficiency factor spould decr~ase with an increase in
nominal stress.
The size effect is explained by the fact that spalling
occurs at the comers <>fa member in torsion as shown in
Fig. 6.13. This alters the path of the torsional shear flow
and can be considered to reduce the area of concrete' resist-:
ing the torque. As shown in Fig. 6.14, this effect is more
significant for a small than for a large section.
In order to formulate a simple design method which
takes the above points into account, Swann [178] proposed
that the efficiency factor should be chosen to be a value
which could be considered to result in an acceptably high '
nominal stress level in large sections. The reason for con-
sidering large sections was that these are least affected by
comer spalling.
Consideration of tests carried out on beams with a maxi-
mum cross-sectional dimension of about 500 mm indi-
cated that 0.8 was a reasonable value to take for the effi-
ciency factor. The nominal stress which could be attained
with this factor was 0.92 #Cu. If a partial safety factor of
1.5 is applied to te." a design maximum nominal torsional
shear stress (v,,,) of 0.75 If.,u is obtained. This value is
tabulated in the Code with an upper limit of 4.75 N/mm2 • It
should be noted that the stress is the same as the design
maximum nominal flexural shear stress. Thus, Swann
[178] essentially chose an efficiency factor to give the
same design maximum shear stress for torsional shear as
for flexural shear. Having established a nominal maximum
(a) Sections before spalling
(b) Sections after spalling
Fig. 6.14(a),(b) Torsional size effect [178]
stress and an efficiency factor for large sections, it was
decided to adopt the same efficiency factor ,<0.8) for .small
sections and to determine reduced nommal maximum
stresses for the latter by considering test data. It was f~und
that the stress of 0.92 ./!c.. had to be modified by multiply-
ing by (y /550) for sections where Yl < 550 mm. Hence the
design st:ess of 0.75 !fell also has to be multiplied by this
ratio.
It ,is emphasised that the Code requi~s. that both of the
following be satisfied:
I. The total (flexural plus torsional) shear stress (v + v,)
should not exceed 0.75 /fCII or 4.75 N/mm2
2. In the case of small sections (Yl < 550 mm} the tor-
sional shear stress should not exceed 0.75 v'/cu(y1/550)
or 4.75 (yI/550) N/mm2
•
T-, L- and I-sections
Torsional shear stress
The plastic shear stress for a flanged section could be
obtained from considerations of the appropriate sand-heap,
such as that shown in Fig. 6.15(a) for a T-section. How-
ever, the junction effects make the calculations ~a~her t~di.
ous and the Code thus permits a section to be dIVided mto
its component rectangles which are then considered indi-
vidually.
The manner in which the section is divided into rec-
tangles should be such that the function r.(hm/l~h'il/) is
maximised. This implies that, in general, the section should
be divided so that the widest of the possible compo-
nent rectangles is made as long as possible as shown in
Fig. 6.15(b). and (c). The total torque (T) applied to the
section is then apportioned among the component rec-
tangles such that each rectangle is subjected to' a torque:
,
b
h
A
(a) Actual
h
~ ~
(b) ,Idealised h, > bw (c) Idealised h, < bw
(d) R~inforcement
Fig. 6.JS(a)-(d) Torsion of T-sectiQn
T(hmqxh
3
min)
r.(hmax h3
min)
The above considerations imply that an anomalous situ-
ation arises in the treatment of flanged sections for the
following reason.
An examination of equation (2.43) shows that the divi-
sion into rectangles and the apportioning of the torque is
carried out on the basis of the approximate elastic stiffnesses
of the rectangles. However, the nominal torsional shear
stress in each rectangle is calculated using equation (6.19)
based upon plastic theory.
¥Torsion reinforcement
If the nominal torsional shear stress in any rectangle is less
than the appropriate V'mi" value discussed earlier in
connection with rectangular sections, then no torsion
reinforcement is required in that rectangle. Otherwise,
reinforcement for each rectangle should be designed in
accordance with equations (6.21) and (6.22), and should
be detailed so that the individual rectangles are tied
together as, for example, in Fig. 6.15(d).
Maximum torsional shear stress
The maximum nominal torsional shear stresses discussed
79
•](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-46-2048.jpg)
![Median line
x x
l J Shaded area
=Ao
Curvature
l-11...L.._ _-'-'(L....l":I..,.~
(c) Section!bro~ mombrane at X-X
-ll+-hwo/2
(a) Cross-section (b) Median line and enclosed area Ao
Fig. 6.16(a)-(c) Torsion of box section
L:___
Fig. 6.17 Box girder notation
in connection with rectangular sections should not be
exceeded in any individual rectangle;
Box sections
Torsional shear stress
The elastic torsional shear stress distribution is easily cal-
culated for a box section. Hence, the Code requires the
nominal stress to be calculated from the standard formula
for a thin-walled closed section [76]:
VI = Tl2h,.,c0o
where hw() is the wall thickness at the point where the shear
stress V, is determined andAo is the area within the median
line of the section as shown in Fig. 6.16.
The above equation can be derived by applying the
membrane analogy [76] in which the variable elastic tor-
sional shear stress is proportional to the variable slope ('jJ)
of a membrane inflated over the cross-section. The
mathematical expression used is identical to that for the
sand-heap analogy for plastic torsion (equation (6.18» butK
is now twice the volume under the membrane. With refer-
ence to Fig. 6.16 tlnd by ignoring the slight curvature of
the membrane (so that a section through the membrane has
straight edges), the slope at a particular point is
'i' = hlhwo '
where h = height of membrane, and the volume underthe
membrane is ""hAo. Thus, from equation (6.18), '
VI = T'i'/2hAo = Tl2hwl~o (6.23)
Equation (6.23) is, strictly, for thin-walled boxes'
whereas it could be argued that many concrete box sec-
tions are not thin. Maisel and Roll [41] have shown that
the error (6 v,)in calculating VI for a thick-walled box from
80
equation (6.23) is, with the notation of Fig. 6.17,
6VI /t
2
I'o(b I b -1 2d)
V, = 2Ao III,·· "hi, . "hOI' (6.24)
1frorsion reinforcement
It can be seen from equation (6.23) that the nominal tor-
sional shear stress at a point is dependent upon the wall
thickness at that point and thus varies around a box with
non-uniform wall thickness. If the nominal torsional shear
stress at any point exceeds the' V"ntn values discussed
eat'lier in connection with rectangular sections, then torsion
reinforcement must be provided.
The design of torsion reinforcement is complicated in
the Code by the fact that two sets of equations may be used
and the lesser of the amounts so calculated may be pro-
vided. The two sets of equations are (a) equations (6.21)
and (6.22) which were derived, for solid rectangular sec-
tions, from the space truss analogy and (b) equations (6.25)
and (6.26) below which were also derived from ,the space
truss analogy, but with the reinforcement assumed to be
concentrated along the median line of the box walls and
with the efficiency factor taken to be 1.0.
Asv T
-~
Sv Ao (0.87fyv)
(6.25)
A
Asv fyv (perimeter of Ao)
sL ~--r
Sv JyL 2
(6.26)
The reason for having two sets of equations is that
(6.21) and (6.22) were originally intended for beams of
relatively small cross-section and they can be over-
conservative for large thin-walled box sections in which Xl
is greater that about 300 mm [179].
Swann and Williams [179] have tested model reinforced
concrete box beams under pure torsional loading and found
that the observed ultimate torques exceeded the ultimate
torques calculated from either set of equations. It was also
found that equations (6,.21) and (6.22) were more conser-
vative than (6.25) and (6.26): this was to be expected since
the, models were large in the sense that they were models
of large prototype sections.
The consideration of a box girder subjected only to tor-
sion is rather academic sinc;e, in practice, flexural loading
is also generally present. There is thus an essentially con-
stant compressive stress (fco,') due to flexure over the
cross-sectional area (Ac) of one flange. Thus a compre~sive
in-plane force of AicQ" acts in conjunction with an in-plane
shear force of Acv,. Hence, from equation (AI7), the
Stress
Tendon design curve
~.1----.;1~4-0.00185
(a) Author's criterion
Fig. 6.18 Tendons as torsion reinforcement
V
•Strain
(a) Stress - resultants on entire cross-section
Fig. 6.19(a),(b) Combined stress-resultants
amount (A) of longitudinal reinforcement required is given
by
(O. 87tvdA = -Arico' + IAcv11
or
A = -A/ca,' + IAcv11
0.87fyl_ 0.87fvl.
This implies that the amount of reinforceU;ent calculated
from equation (6.22) or (6.26) may be reduced by
Aic",'! O.87/yl.' The validity of this approach to design has
been confinned by the tests of Swann and Williams [179]
and is consequently permitted by the Code. It should be
noted that the Code adopts for At' the area of section sub-
jected to flexural compressive stresses instead of simply
the flange area, and that the Code also refers to a stress fve
~h~W~k· .
The reduction in 'longitudinal steel area due to the effect
of flexural compressive stresses could, theoretically, be
applied to sections other than boxes but the Code limits the
reduction to box sections only.
Maximum torsional shear stress
The maximum nominal torsional shear stresses discussed
in connection with rectangular sections have been shown
to be reasonable for box sections by the tests of Swann and
Williams [179].
Ultimate limit state - shear and torsi'on
Stress
Idealised tendon curve
Idealised reinforcement curve
Strain
(b) Lampert's criterion
(b) Stress - res41tants at a point
*r . f 'orslon 0 prestressed concrete
General
The code essentially assumes that prestressed concrete
members can be designed to resist torsion by ignoring the
prestress and designing them as if they were reinforced.
Thus values of VI' Vtm/n' Vtu, As,' and A.L should be calcu-
lated in accordance with the equations 'for reinforced con-
crete presented earlier in this chapter. The validity of
such a~ approach has been checked, with bridges speci-
fically tn mtnd, by Swann and Williams [179]. who carried
out tests on model prestressed concrete box beams.
It has been mentioned that the greatest pennissible
characteristic strength for conventional torsion reinforce-
ment is 425 N/mm 2
• This value was chosen because the
large yield strains associated with higher strengths reduce
the efficiency factor in equation (6.21) to less than 0.8.
Thus, to achieve an efficiency factor of 0.8 when using
tendons as torsion reinforcement, it is necessary to ensure
that the additional tendon strain. required to mobilise the
tendons' ultimate strength, does not exceed the yield strain
of other reinforcement in the section having a character-
istic strength of 425 N/mm2
• If the latter is conservatively
assumed to be hot-rolled, the design yield strain is 0.87 x
425/200000 = 0.00185. Hence, the design ultimate stress](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-47-2048.jpg)
![,Predetermined position (yP zp)
F / of centroid of tendons
1 /
•
..i..____._..... __..___.__y
~ . .~
•
z
Fig. 6.20 Segmental box beam
(fpd) for the tendons should be the lesser of 0.87!pu.and (see
Fig. 6.18(a» !p~ plus the stress increment equivalent to a
strain increment of 0.00185, where f~e is the effective pre-
stress in the tendons.
Although the above argument seems logical, the result-
ing limiting stresses are not included in the Code. Instead,
the design stress is limited to the· lesser of 460 N/mm2
and
(O.87fpII - fp~). The specification of such stresses seems to
indicate that a criterion suggested by Lampert [180] and by
Maisel and Swann [181] was misinterpreted by the draft-
ers. The suggested criterion was that tendons and conven-
tional reinforcement should reach yield at about the same
stage of failure of the section, in order that excessively
large strains would not develop in one type of reinforce-
ment prior to yield of the other. This implies that (see
Fig.6.18(b»
f,lIl - [". == fl'
where [,,,1 is the tendon design stress at ultime.te. The draft-
ers took tv to be 460 N/mm2
(the greatest value likely to
be used)· although it exceeds the greatest value (425
N/mm2
) permitted for torsion reinforcement. Hence,
f~/(I ==f"e + 460 N/mm2
• However, fpd cannot exceed
0.87 fplI' thus, fpti should be taken as the lesser of
0.87 fpu and (fp~ +460 N/mm2
). It thus appears that the
limiting stresses given in the Code are not, as stated in the
Code, design stresses but the stress increment necessary to
raise the stress from the effective prestress to the design
stress. It should also be mentioned that it is inconsistent to
apply a partial safety factor to fplI but not to fy , and thus the
second limit should be (fpe+ 400 N/mm2
). Hence the stress
increment in the Code should be 400 N/mm2
•
In conclusion, the Code stress limits are stress illcre-
ments and not design stresses; moreover, the author would
suggest that the alternative criterion illustrated in
Fig. 6.18(a) is more logical.
Alternative design methods
The Code method of design of a section subjected to a
general loading is to consider the stress resultants acting on
the entire cross-section, as shown in Fig. 6.19(a), and then
to superpose the effects of any local actions. This method
is probably very suitable to small sections but for large
sections (and particularly for box beams) it could be consi-
dered desirable to vary the reinforcement over, for exam-
ple. the depth of a web. It is then necessary to consider the
stress-resultants acting at various points of the cross-
section al> shown in Fig. 6.19(b). Each point would gener-
ally be subjected to both in-plane and bending stress resul-
tants and thus the sandwich method discussed in Chapter 5
82
E
E
~
II
~
~--
•• •Fig. 6.al Example 6.1
6/26•• •
would be appropriate for design. A simplified version of
such an approach has been described by Swann and Wil-
liams [179] and Maisel and Swann [181].
The Code refers to 'other design methods' without
specifically mentioning any; however, the above approach
of considering the stress resultants at points of the cross-
section would be acceptable.
Segmental construction
In segmental construction it is not generally convenient to
provide continuous longitudinal conventional reinforce-
ment. Thus longitudinal tendons in excess of those needed
for flexure have to be provideg.
The Code states that the line of action of the longitudi-
nal elongating force should coincide with the centroid of
the steel actually provided. If the longitudinal steel
capacities required in each flange and web of a segmental
box beam are as shown in Fig. 6.20, then they can be
replaced by a force F, situated at (y, i), where:
F, = IFI
Y = IFly/IFi
i = IFiz/IFi
and each summation is for i = 1 to 4.
Thus the total ultimate capacity of the tendons needs to be
F, and their centroid needs to be at {J, n.
In practice, it is often simpler to calculate the necessary
total ultimate capacity of tendons situated at predetermined
positions such that their centroid is at, say, (yp. zp). In this
case F, must be such that the capacity of each flange or
web exceeds the appropriate value of F;. Hence, moments
should be taken about each web and flange in tum to give
four inequalities. For example. by taking moments about
web 2 .
(Y2 - y,,)F, ;!: F1(Y2 - YI) + F~(Y2 - Y~) + f4(Y2 - Y4)
Slab
,..,..------....
'" "1/ ,
A '7
t .. '
,/ 14600~ vY
I
I NO...·~ B I 7'
A
o I 4 ..... t
i Pier 'Critical ~
, . Iperimeter.
I I
I
I
, I
, '"" ,/
...... _-_.. _-_.....
(a) Plan
/1%(d= 1060)
I
i
-~I....
(b) Section A-A
Flg.6.22(a),(b) Example 6.2
The largest value of F, obtained from the four inequalities
is the total ultimate tendon capacity which must be pro-
vided. Swann and Williams [179] give a numerical exam-
ple of such a calculation, and they also present the results
of tests on two model segmental box beams against which
the above method of calculation was checked.
Examples
"*6.1 Flexural shear in reinforced concrete
Design stirrups of 250 N/mm2
characteristic strength to
resist an ultimate shear force of 540 kN applied to the sec-
tion shown in Fig. 6.21. The concrete is of grade 40.
Nominal applied shear stress = v =
540 x 103
/(300 x 600) = 3 N/mml
From Table 6 of Code or 0.75 ./fc.u, maximum allowable
shear stress =
v" = 4.75 N/mm2
v" > v. o.k.
Area of tension reinforcement A., = 2950 mml
lOOA,/bd = (l00 x 2950)/(300 x 600) = 1.64
From Table 5 of Code. allowable shear stress without
shear reinforcement
v,. = 0.88 N/mm2
From equation (6.7),
An· 300(3.00 - 0.88) ~ 2.92 mml/mm
-s-" = 0.87 x 250 -
The stirrup spacing should not exceed 0.75 d = 450 mm.
12 mmstirrups(2Iegs)at75 mmcentresgive3.02 mm2/mm.
~6.2 Punching shear in reinforced concrete
Design the slab shown in Fig. 6.22 to resist punching
shear if the characteristic strengths of the steel and con-
crete are 425 and 40 N/mm2
respectively and the col~mn
reaction at the ultimate limit state is 15 MN.
The critical .,e"nme.ter is at 1.5h = (1.5)(1100) = 1650 mm
The critical section is shown on Fig. 6.22.
Perimeter =(2)(600) + (2)(1200) + (217)(1650)
=14000 mm
'Average' effective depth, d =(3 x 980 + 1 x 1060)/4
= 1000 mm
Nominal applied shear stress,
Y =15 x 106
1(14 000 x 1000) =1.07 N/mm2
From Table 6 of Code or 0.75 lieu, maximum allowable
shear stress =Yu = 4.75 N/mm"';'
Only take half for slab, :. Yu = 2.38 NfmtnJ
Vu > v, o.k.
Average lOOAslbd = (3 + 1)/2 = 2
From Table 5 of the Code, allowable shear stress without
shear reinforcement Vc = '0.95 N/mm 2
From Table 8 of Code, t. = 1.00
v - ;"v.. = 1.07 - 0.95 = 0.12 N/mm 2 .
Thus shear reinforcement required, but must provide for at
least 0.4 N/mm2
From equation (6.8)
IAsv ;!:(0.4)(14 000)(1000)/(0.87)(425) = 15 100 mm2
This amount of reinforcement must be provided along a
perimeter 1.5h from the loaded area and also along a
perimeter 0.75h from the loaded area.
Length of perimeter at 0.75h = (2)(600) + (2)(1200) +
(2")(825) = 8780 mm
Thus on 1.5h perimeter provide 15100/14 = 1080 mm2/m.
and, on 0.75h perimeter, 15100/8.78 = 1720 mm2/m.
Now check on perimeter at (1.5 + 0.75)/1 = 2.25h.
Length of perimeter = (2)(600) + (2)(1200) + (211")(2475)
= 19200 mm
Nominal applied shear stress = v
= 15 x 106
/(19200 x 1000) = 0.78 N/mm2
v < !;"vc• thus no need to provide more shear reinforce-
ment.
*6.3 Flexural shear in prestressed concrete
The pretensioned box beam shown in Fig. 6.23 is sub-
jected to a shear force of 0.9 MN with a co-existing
moment of 3.15 MNm at the ultimate limit state. Design
shear reinforcement with a characteristicstrimgth of
250 N/mm 2. The initial prestress is 70% of the characteris-
tic strength and the losses amount to 30%. The concrete is
of grade 50 and the section has been designed to be class I.
From Table 32 of Code or 0.,75 /I.:". maximum. aJl()wabl~
shear stress
VII = 5.3 N/mm 2
83](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-48-2048.jpg)
![Chapter 7
Serviceability limit s~ate
-f .
Introduction
As explained in Chapter 4, the criteria which have to be
satisfied at the serviceability limit state are those of per-
mis~ible steel and con~rete stress, permissible crack width,
interface shear in composite construction and vibratio~. In
this chapter, methods of satisfying the criteria of permis-
sible steel and concrete stress, and of cra£k width in both
reinforced and prestressed concrete construction are pte-
sented. The additional criteria, associated with interface
shear and tensile stress in the in-situ concrete of composite
construction, are presented in Chapter 8.
Compliance with the vibration criterion is discussed,
together with other aspects of the dynamic loading of
bridges, in Chapter 12.
Reinforced concrete stress
limitations
""*General
As discussed in Chapter 4, the concrete compressive stres-
ses in reinforced concrete should not exceed 0.5feu and the
reinforcement, tensile or compressive, stresses should not
exceed O.8fy.
Although it is not stated in the Code, the above limit-
ations should be applied only to axial or flexural stresses.
Thus it is not necessary to consider flexural shear or tor-
sional shear stresses at the serviceability limit state.
In practice, axial and flexural stress calculations will
involve the application of conventional elastic modular
ratio theory. However, a modular ratio is not explicitly
stated in the Code, and it is necessary to calculate a value
from the stated modulus of elasticity of the reinforcement
(200 kN/mm2
) and the modulus of elasticity of the con-
crete. Short term values of the latter modulus are given in
the Code and these vary from 25 kN/mml
for a charac-
teristic strength of 20 N/mm2
to 36 kN/mml
for a charac-
86
teristic strength of 60 N/mml
• The Code is not explicit as
to whether these short term moduli shOuld be used when
calculating stresses or whether they should be adjusted to
give long-term moduli. However, the Code does state that
a long-term modulus, equal to half of the short term value,
should be used when carrying out crack width calculations.
In view of this requirement, and the fact that all structural
Codes of Practice have hitherto adopted a long-term mod-
ulus, the author would suggest that, for stress calculations
in accordance with the Code, a long-term modulus should
be adopted. Since a relatiyely weak concrete having a
characteristic strength of less than about 40 N/mm2
is gen-
erally adopted for reinforced concrete, the long-term mod-
ular ratio calculated from the Code would be between 13
and 16. These values are of the same order as the value
of 15 which is general~y adopted f~r design purposes at
present.
Tension stiffening
In the above discussion, conventional modular ratio theory
is mentioned. This theory generally considers a cracked
section and ignores the stiffening effect of the concrete in
tension between cracks (tension stiffening); hence, it over-
estimates stresses and strains as shown in Fig. 7.1. The
difference between lines AB and OC of Fig. 7.1 is a
measure of the tension stiffening effect and can be seen to
decrease with an increase in load above the initial cracking
load. This decrease results fr@m the development of further
cracks and the gradual breakdown of bond between the
reinforcement and the concrete. Hence the strain (E.) cal-
culated ignoring tension stiffening should be reduced by an
amount (E/S) to give the actual strain (Em).
The Code permits tension stiffening to be taken into
account in certain crack width calculations (as referred to
later in this chapter), but it is not clear whether one is
permitted to allow for it in permissible stress calculations.
However, test results [183] indicate that, at stresses of the
order of the steel stress limitation of 0.8fy, the tension stif-
fening effect is negligible. Therefore, it seems to be
reasonable to ignore tension stiffening when 'carrying out
stress calculations.
Load
N
Cracking
load
o
--Actual
----Calculated on
cracked stiffness
-;1
. "Ncl ; I
1"/ I
/1 I
"/ I I
/ t f
ts
~; N" (,.
;" 51 I
/ I I
/" I I
;
Fig. 7.1 Tension stiffening
"*Slabs
""""
B
/'c
"""
Stress
Strain
If the principal stresses in a slab do not coincide with the
reinforcement directions, it is extremely difficult to calcu-
late accurately the concrete and reinforcement stresses. As
the Code gives no guidance, the author would suggest the
following procedure:
1. Assume the section to be uncracked and calculate the
four principal extreme fibre stresses caused by the
stress resultants due to the applied loads.
2. Where a principal tensile stress exceeds the permis-
sible design value, assume cracks to form perpendicu-
lar to the direction of that principal stress. The per-
missible design value could be taken to be the Class 2
prestressed concrete limiting stress of 0.45 !feu (see
Chapter 4).
3. Consider each set of cracks in tum and calculate an
equivalent area of reinforcement perpendicular to
these cracks.
4. Using the equivalent area of reinforcement, calculate
the stresses in the direction perpendicular. to the cracks
by using modular ratio theory.
5. Compare the extreme fibre concrete compressive
stress with the allowable value of 0.5 feu.
6. If the calculated stress in the equivalent area of rein-
forcement is In, then calculate the stress in an i-th
layer of reinforcement, inclined at an angle Q' i to the
direction perpendicular to the cracks, from {; =bIn
where b is discussed later. Compare /; with the per-
missible value of O.8fY'
The calculation of the equivalent area of reinforcement
(step 3 above) is explained by considering a point in a
cracked slab where the average direct and shear strains,
referred to axes perpendicular and parallel to a crack (see
Fig. 7.2), are En, E" Ynt.
The strain in the direction of an i-th layer of reinforce-
ment at an angle IXi to the n direction is, by Mohr's circle
2 • 2 •
Ei = En COS 01.1 + Et sm 01.; - Ynt S1l1 IXi COS IX;
The steel stress is thus
li=E.• Ei
Fig. 7.2 Cracked slab
where Es is the elastic modulus of the steel. If the steel
area per unit width is Ai' the steel force per unit width is
FI=Atfl
If N such layers are considered, the total resolved steel
force in the II direction is
N
FII .. I FI cos2
IXI
t .. t
N
== £, I At (Ell cos4
IXI + E, sin2
IXi cos2 IXI-
1.. 1
Ynt sin IXi cos3
IXi)
The force FII can be considered in terms of an equivalent
area (A'l) of reinforcement per unit width in the n direc-
tion. Thus F,. =An Es En' Hence, by comparison with the
previous equation,
N E
An = I Ai (COS
4
IXi + E: sin
2
IXi COS2IXi -
i .. l
Ynt • . ·3
£" sm IXi COS IXi)
n
It is reasonable, at the serviceability limit state, to·
assume that the nand t directions will very nearly coincide
with the principal strain directions. Thus Ynl ==0 and the
third term in the brackets of the above equation can be
ignored.
There are now three cases to consider for a slab not
subjected to significant tensile in-plane stress resultants:
1. If the slab is cracked on one face only and in one
direction only, En» Et and the expression for An
reduces- to
N
An =. ~ AI cos
4
IXI
1= 1
(7.1)
2. If the slab is cracked in two directions on .the same
face, then Et will be of the same sign as En' If E, is
again taken to be zero, the calculated value ofA"will
be less than the true value. It is thus conservative to
use equation (7.1).
3. If the slab is cracked in two directions on opposite
faces, E, will be of opposite sign to En and could take](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-50-2048.jpg)
![any value. The precise value of E/En to adopt is very
difficult to determine, although some guidance is
given by Jofriet and McNeice [184]. As an approxi-
mation, it is reasonable to assume that, at the ser·
viceability limit state, it is unlikely that En < /E,/; it is
thus conservative to assume EnlE, = -1. In which
case
N
~ A/ cos2
(X/ (cos2
(X/ - sin2
(X/) ._
""-
(7.2)
/= I
It is implied in equation (7.2) that reinforcement in-
clined at more than 45" to the n direction should be
ignored.
By implication the stress transformation factor, 6, refel'-
red to earlier in this chapter should be taken as
cos2
0:/ if equation (7.1) is used to evaluate An and as
(C05
2
0:/- sin2
0:/) if equation (7.2) is used.
Little error is involved in adopting the above approxima-
tions for An when the reirtforcement is inclined at less than
about 25° to the perpendicular to the cracks. If, when
using the above approximations with reinforcement
inclined at more than 25", it is found that the reinforce-
ment stresses are excessive, it would be advisable toesti-
mate a more accurate value of EI'.IE, (as opposed to, the
above approximate values of zero and -1) as described in
r184].
Crack control in reinforced concrete
General
Statistical approach
The design crack widths given in Table 4.7, and discussed
in Chapter 4, are design surface crack widths and are
derived from considerations of appearance and durability.
In order to calculate crack widths it is necessary to
decide on an interpretation of the design crack width~ is it
a maximum, a mean or some other value? It is not possible
to think in terms of a maximum crack width but it is feas-
ible to predict a crack width with a certain probability of
exceedence. This can be illustrated by considering two
nominally identical beams having zones of constant bend-
ing moment. If each beam is subjected to the same loading
and the widths of all of the cracks within the respective
constant moment zones are measured, then distributions of
crack widths can be plotted as shown in Fig. 7.3. It is
found that, although the maximum width of crack
measured on each of the two nominally identical beams
may be very different the crack widths exceeded by a cer-
tain percentage of the results are quite similar. For this
reason, the design crack widths are defined as those having
a certain proQability of exceedence. '
The probability of exceedence that has been chosen in
the Code is 20% (i.e., 1 in 5 crack widths greater than
the design value), which is the same percentage as that
adopted in Lhe building code, CP 110. At first sight it may
appear very liberal to permit t in 5 crack widths to exceed
88
Noof
cracks -,
-Beam 1
"'~,Beam2
Width exceeded
by 11% of cracks
I
I
I
I
I
Crack width
."Ig. 7.3 Crack width distributions for nominally identical beams
the design value; however, the formula used to calculate
the '1 in 5' crack width is applicable only to the constant
moment zQnes of specimens tested under laboratory con-
ditions. The exceedence level in practice is much less than
20% and, for buildings, it has been estimated [182] that
the chance of a specified width being exceeded by any
single crack width will lie in the range'lQ-3 to 10-4• The
main reasons for this reduction in probability are:
1. The specified design loading under which cracking
should be checked is, ~ssentially, the full nominal
loading. This is greater than the 'average' loading
which occurs for a significant length of time.
2. In design, lower bound estimates of the material prop-
erties are used; thus the probability of the stiffness
being as low as is assumed in calculating the strains is
low.
3. ,Structural members are not generally subjected to
'uniform bending over any great length, and thus the
,:anly cracks in a member which have any serious
chance of being critical are those close to the critical
sections of the memb~r. These will be few in number
compared with the tptal population of ·cracks.
The above reduction in probability will be less dramatic
in bridges because the latter are subjected to repeated load-
ings which cause crack widths to gradually increase during
the life of the bridge. In addition, the design loading is
more likely to be achieved on a bridge than on a building.
However, considerably tess than 20% of the crack widths
in a bridge should exceed the design value if the design is
carried out in accordance with the Code formulae.
Crack control in the Code
General approach Although cracking due to such effects
as the restraint of shrinkage and early thermal movement is
a significant practical problem, it is cracking induced by
applied loading that is used as a basis for design in the
Code. However, the Code does require at least 0.3% of the
gross concrete area of mild steel or 0.25$ of high yield
steel to be provided to control restrained shrinkage and
early thermal movement cracks. These percentages are
rather less than those suggested by Hughes [185] and
should be used with caution.
The only crack widths that need to be calculated accord-
ing to the Code are those due to axi~1 and flexural stress.
resultants. Hence, flexural shear and torsional shear cracks
do not have to be considered explicitly since, as discussed
. in Chapter 6, it is assumed that the presence of nominal
links control the widths of such cracks.
Cracks due to applied loadings which cause axial and
flexural stress resultants are controlled by limiting the
spacing of the bars. This approach is identical to that
adopted in BE 1173. The bar spacing should not generally
exceed 300 mm, and should be such that the crack widths
in Table 4.7 are not exceeded midway between the bars
under the specified design loading. It should be noted that
the Code gives a different method of ensurin,s",that the
crack widths are not exceeded for each type of structural
del1l~Jll.
Loading Since the widths of cracks are controlled primar-
ily for durability purposes, it is logical to define the design
loading as that which can be considered to be virtually
permanent rather than that of occasional but more severe
loads. This is because, after the passage of occasional
severe loads, the crack widths return to their values under
permanent loading provided that the reinforcement
reinained elastic during the application of the occasional
load. Since the limiting reinforcement stress is 0.81", the
reinforcement will remain elastic.
Ai one time the drafters considered that 50% of HA
loading should be taken to be permanent, together with
pedestrian loading; dead load and superimposed dead load.
When the appropriate partial safety factors were applied,
the resulting design load was:
dead load + 1.2 (superimposed dead load) + 0.6 (HA) +
1.0 (pedestrian loading)
Subsequently, it, was decided to check crack widths
under full HA loading but the partial safety factors were
altered so that the design load became:
dead load + 1.2 (superimposed dead load) + 1.0 (HA) +
1.0 (pedestrian loading)
This design loading is given, in the general design section
of Part 4 of the Code, with the requirement that the wheel
load should be excluded except when considering top
flanges and cantilever slabs. In addition, for spans less
than 6.5 m, 25 units of HB loading with associated HA
loading should be considered. This additional loading was
introduced to comply with the requirement of Part 2 of the
Code that all bridges should be checked for 25 units of HB
loading. It is not clear whether top flanges and cantilever
slabs should be loaded with 25 units of HB when they span
less than 6.5 m. However, it would seem reasonable to
design such slabs for the more severe of the local effects of
the HA wheel load or 25 units of HB loading.
The above loadings are very similar to those in BE 1173,
with the exception that the latter document refers to 30
units of HB loading for loaded length less than 6.5 m. The
implication in BE 1173 appears to be that the 30 units of
HB loading should be applied to top flanges and cantilever
slabs.
Finally, the Code clause concerned with reinforced con-
crete walls requires that any relevant earth pressure load-
ings should be considered in addition to dead, superim-
posed dead and highway loadings.
Stiffnesses Although the design crack widths· and ioad-
ings, referred to previously, are given in the general design
section of Part 4 of the Code, the clauses concerned with
crack width calculations appear much later in the Code
under the heading: 'Spacing of reinforcement'. Guidance
on the stiffnesses to be adopted in the calculations also
appears under this heading.
In all crack width calculations the Code requires _the
elastic_~odulus of the· concrete to ,be a long-term v~lue
.equal to-halfof the tabulated. short-term value. Hence a
modular ratio Of about 13 to 16 would be used to calculate
the strains ignoring tension stiffening. In certain situations
which are discussed later in this chapter, the Code permit~
the strains ignoring tension stiffening to be reduced. It is
thus necessary to determine the strain EIS in Fig. 7.1. This
~s best achieved by initially considering an axially re-
lOforced cracked section (having a concrete area ofAc and a
steel area ofAs) which is axially loaded. At a crack all of
the applied force <tV) is carried by the steel:
N =E" El As
However, the average steel and concrete forces are given
by:
Ns =Es em As
Nc =Aclcm
where femis the average tensile stress in the concrete be-
tween the cracks.
But
N= Ns + Ne
:. E. El As =E" Em As + Aclem
.'. Em =El - AelemlEs As
Or
Ets =AclemlEs As
At the cracking load, fern is obviously equal to the tensile
strength (f,) of the concrete. At higher loads, tests [183,
186, 187] indicate that lemreduces in accordance with
lem =1,Iser 111
where fser is the steel stress at a crack at the cracking load
and 11 is the steel stress at the load corresponding to the
strain El'
For an axially reinforced and loaded section it is obvious
that Ae "'" bh where band h are the breadth and overall
depth respectively. However, for a flexural member, it is
necessary to define an effective area (Kbh) of concrete in
tension over which the average stress lem acts. Hence, for
flexure, and considering only surface strains
E,•. = Kbh Iser f, 1E. As11
In order that strains at any depth (at) from the compres-
sion face can be considered the above expression is modi-
fied to
Ets =Kbhf,erf, (at - x)IE"AsIl(h - x)
where x is the neutral axis depth.
In the Code, cracking is generally checked under HA
loading only and the values of (YfL Yf3) at the ultimate
limit state are 1.32 for dead load and 1.73 for HA loading,
giving an average of 1.53; hence 11 "'" 0.871,,/1.53 =0.571".
89](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-51-2048.jpg)
![However, implicit in the tension stiffening fonnula in the
Code is the assumption that I. =0.S8/yo This is because
the fonnula was originally derived for CP ItO, for which
I. =0.S8/y is correct. However, the difference between
this value of 11 and the 'correct' value for the Code is
negligible. Hence
tts = Kbh Iserlt(a' - x)/Es A.(0.S8 /y) (h - x)
Tests carried out by Stevens [188] indicated that, on
average,
KlserII0.S8E., = 1.2 X 10-3
N/mm2
•
It is emphasised that this constant has the dimensions of
N/mm2
• Thus
Ets =1.2bh(a' - x) 1O-31As (h - x) fy
Hence, the tension stiffening fonnula in the Code is
obtained as
Em = E1 - 1.2bh(a' - X) 10-3IA;(h - x)fy'
',-J
(7.3)
Beeby [189] has shown that equation (7.3) provides a
reasonable lower bound fit to the instantaneous results of
Stevens and a reasonable average fit to the latter's results
obtained after long tenn loading of two years' duration.
However, test results, under short tenn loading, reported
by Rao and Subrahmanyan [186], Clark .and Speirs [183]
and Clark and Cranston [187] show tension stiffening val-
ues about one-third of those of Stevens and of those pre-
dicted by equation (7.3). In addition, the latest CEB tem-
sion stiffening equation [110] is in reasonable accord [187]
with the data presented in [183], [186] and [187]. There is
thus evidence to suggest that equation (1.3) overestimates
tension stiffening. .
Finally, equation (7.3) was originally derived with
buildings in mind and thus the effects of repeated loading
were not considered. Bridges are subjected to repeated
loading and it is reasonable to assume that such loading
reduces the tension stiffening. There is a lack of experi-
mental data in this respect and, as an interim measure, it
might be sensible to adopt the CEB recommendation [1 to]
that tension stiffening under repeated loading should be
taken as 50% of that under instantaneous loading. Tests on
model solid [87, 126] lnd voided [71] slab bridges indicate
that, for HB loading, the tension stiffening, as a proportion
of the instantaneous value, is of the order of 60%, 50%
and 40% after tOOO, 2000 and 4000 load applications
respectively. These values are reasonably consistent with
the CEB value.
*Crack control calculations
Beams
Base, Read, Beeby and Taylor [190] have carried out tests
on 133 reinforced concrete beams. They found that there
was an average difference of only 13% between the crack
control perfonnances of plain and defonned bars, and thus
it is not necessary to have separate crack width fonnulae
for the two types of bar. This point is particularly valid in
90
Load ~
/ -Actual
'/ ----Calculated on
/ cracked stiffness
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
'I
Strain
Fig. 7.4 Tension stiffening: high steel percentage
view of the fact that crack widths are calculated only at
positions mid-way between bars where the bar type has the
least influence.
Base et al. found that the distributions of crack width
were Gaussian and that, for defonned bars, the mean crack
width (wm) could be predicte.d by
Wm =1.67 acr Em
where aer is the perpendicular distance from the point
where the crack width is to be predicted to the surface of
the nearest reinforcing bar. The standard deviation of the
crack width population at any strain was found to be
0.416 Wm.
It is mentioned previously in this chapter that the proba-
bility of exceedence adopted in the Code is 20%. In a
Gaussian distribution, a. value of mean"plus 0.842 standard
deviations is exceeded bf 20% of the population; thus the
design crack width is g~ven by
W = (1 + 0.842 x 0.416) 1.67acr Em = 2.3aer Em
However, in the Code, tension stiffening is not taken into
account for beams and it is thus assumed that Em = E1'
Hence the following Code equation is obtained
W =2.3aer El (7.4)
The reason for ignoring tension stiffening in beams is
that it is envisaged that reinforced concrete bridge beams
would be heavily reinforced; in which case the load-strain
relationship is as shown in Fig. 7.4. It can be seen that the
tension stiffening effect is a small proportion of the actual
strain and can be ignored.
Equation (7.4) is very similar to the equation in BE 1/73;
however, in the latter document the constant is not 2.3 but
is 3.3 for defonned bars and 3.8 for plain bars. The value
of 3.3 is appropriate to the 1% exceedence level [190],
which is a much more severe criterion than the 20% level
adopted in the Code. However, it should be remembered
that the design crack widths specified in BE 1173 and the
Code are also different. The BE 1173 value of 3.8 was
obtained by increasing the value of 3.3 for defonned bars
by 13% and rounding up [190].
Solid slab bridges
.At one stage in the drafting of the Code, it was hoped to
p~epare simple bar spacing rules which would obviate the
need to carry out crack width calculations. However, it
was found that, due to the large number of variables to be
considered, it was possible to produce such rules only for
. single sp!ln solid slab bridges. Consequently an exercise
was carried out [191] in which a range of solid slab bridges
was designed atthe ultimate limit state either by yield line"
theory or in accordance with elastic momentfiellds. The
bar spaCings required to control crack widths to the design
values were then determined. Bar spacing tables based
upon this exercise did appear in some drafts of the Code,
but it Was eventually decided to simplify the bar spacing
rules considerably. Consequently, the Code simply states
that the longitudinal bar spacing should not exceed
ISO mm and the transverse bar spacing should not exceed
300 mm. These values apply to continuous slabs in addi-
tion to single span slabs and, in view of this, the author
would suggest that 'longitudinal' be interpreted as primary
and 'transverse' as secondary in order to avoid excessive
cracking in certain situations. For example, the Code
implies that the spacing of the transverse bars in the region
of an interior support of a slab bridge continuous over dis-
crete columns could be 300 mm; however, in such situ-
ations the' transverse bending moments could be large
enough to cause excessive cracking if the bar spacing were
not considerably less than 300 mm. It is thus suggested
that the bar spacing rules should be interpreted with
'engineering judgement'.
From present design experience, it is known that, in
some situations, a longitudinal bar spacing of 150 mm
does not control the crack widths to the design values
unless the reinforcement is used at a stress less than the
maximum permitted in BE 1173 (= 0.56Iy)' The drafters
anticipated that when designing in these situations in
accordance with the Code, bar spacings less than 150 mm
would be forced upon the designer because of the large
amount of reinforcement that· would be. required to satisfy
the ultimate limit state criterion.
An examination of [191] shows that if bar spa~ings were
to be calculated for single span slab bridges; they would
generally be greater than the Code values of 150 and
3(}() mm, except for elastically designed slabs having skew
angles greater than about 30° and yield line designed slabs
having skew angles greater than about 15°. The Code val-
ues ·should thus' be used with caution if the skew angle
exceeds these values.
Finally, it is worth mentioning that the reason for having
bar spacing rules is to avoid can-ying out specific crack
width calculations. However. since the Code requires
stress calculations to be carried out at the serviceability
limit state, albeit at a different design load to that for the
crack width calculation, a large proportion of the data
required for a crack width calculation would already be
calculated. Thus, to a certain extent, the advantage of
quoting bar spacing rulel'; is lost and litlle extra effort is
involved in carrying out a complete crack width calcu-
lation by, for example, applying the general procedure sug-
gested by Clark [192, 193].
an =strass normal to
line of symmetry
.- - - . - _.1--.-
I 0' = sttess tangential
I t ofree surface
. of void or soff.it
(a) Elastic stresses In a quadrant
(b) Crack pattern (194]
Flg.7.S(a),(b) Cracks in voided slabs
Sidefece
Soffit
Slab bridges with longitudinal circular voids
Cracks due to longitudinal bending The spacing and
-widths of cracks due to longitudinal bending are very simi-
lar to those occurring, at the sam~ steel strain, in solid
slabs. Thus, the Code limits the spacing of longitudinal
reinforcement to 150 mm which is the same as the value
for solid slabs. This spacing was not checked for voided
slabs by either calculation or experiment; however, the
author would suggest that, with the same precautions as
discussed for solid slabs,it is a reasonable value to adopt
in practice.
Cracks due to transverse bending The stress raising
• effects of the voids result in the response of a voided slab
to transverse bending being very different to that of a solid
slab. Cracking due to transverse bending in voided slabs
has been described in some detail by Clark aod Elliott
[194] and their findings can be summarised as follows:
1. Linear elastic analyses indicated that:
(a) The peak stress occurs at the crown of the. void as
shown in Fig. 7.5(a). It is here, on the inside ·of the
void, that the first crack should initiate.
(b) The peak stress on the outer face does not occur at
the void centreline, but at approximately the quarter
points of the void spacing as shown in Fig. 7.5(a).
Thus cracks propagating from the outer face should
initiate at the quarter points.
2. The theoretical predictions were confinned by tests on
transverse strips of voided slabs. An actual crack pat-
tern is shown in Fig. 7.5(b).
91](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-52-2048.jpg)
![Reinforcement
/ / ...
...
'f
(a' Pattern 1
----:::-:.:---::
----Crack pattern 1
FIR. 7.6(R),(b) Phm views of crock pntlcrnR in ~(llid slah
.The tests indicated that in order to obtain a controlled
nack pallern. such as th~lt shown in Fig. 7.s(b), with no
cracks passing completely through the flange. it was
necessary to hnve at least I% of transverse reinforcement
in the fhmge. This reinforcement should he cnlculatcd as a
percentage of the mi/limum !lange area. This miilimum
reinforcement percentage requirement is given in the 'Code
f~'r .predominantly tension flanges, together with an upper
IIII1It of 1500 mm2
/m. The laller value was introduced to
avoid excessive amounts of reinforcement in slabs ~ith
very t!lick flange... Clark and Elliott I194J have suggested
that, In such cases, it may be preferable to consider the
minimum flange thickness as having two critical layers:
one Inyer would be adjacent to the outer face and the other
adjacent to the crown of the void. The thickness of each
layer .would he equal to twice the relevant cover plus the
ba.r .dlameter. and. each layer would he provided with a
nlJllImum of 1% of reinforcement. This suggestion is simi-
lar to that recommended by Holmberg lJ95J.
In the case of predominantly compression flanges,· the
Code requires that the area of transverse reinforcement
should be the lesser of 1000 mm2
/m or 0.7% of the
minimum !lange area. These values were chosen because
the test~ reported by Clark and Elliott 1194) indicated that
the moment at which su(:h steel would be stressed to
2.10 N/mm 2
would be greater than the cracking moment of
the section. Thus. if cra(:king did occur due to an unex-
pected severe loading situation. the reinforcement would
not yield suddenly and a controlled crack pattern would
occur.
In addition tn the ahove limitations on the area of trans-
verse .reint~)rcement. it is necessnry to limit the spacing of
the rCll1fOrCelllenf. The Code states that the spacing should
not exceed the solid slah value of 300· mm nor twice the
minimum !lange thickness. The latter crit.erinn was intro-
duced to discourage the use of large diameter hars if thin
flanges were adopted. since Stich flanges are sutijected to
p:n1icularly large stress concentrations. . '
Flanges
In order to discuss crack control in the !langes of beam and
~Iah. l'cllular slah anll hox hcum construction it is first
(b, Crack pattern 2superposed
on crack pattern 1.
Crack pattern 1
Crack pattern 2
/
necessary to consider. in general terms. cracking in slabs..
Beeby [196] has investig~ted cracking In slabs spanning
one ~~y and fu.und that there are two basic crackpatternll
(see I'Ig. 7.6):
I. A pattern controlled by the deformation imposed on
- the section. .
2. A pattern controlled by the proximity of the re-
inforcement. .
Beeby [196] proposed a theory which adequately pre-
dicts the properties of the two patterns and their interac-
tion. In addition. formulae for predicting the widths of
cracks at any point on a slab were derived. These formulae
are too complicnted for design purposes nnd thull Beeby
[182] reduced them to the following single design formula
w=
1 + K (0", - Cmi';)
2 (h - x) .
,
where em/" is the minimum cover to the tension steel and
KI and K2 are constants which depend upon the probahility .
of exceedence of the design crack width. The appropriate
values of KI and K2 are 3 and 2. respectively, for the 20%
prohability of exceedence adopted in the Code. Hence, the
following crack width equation, which isl given in the
Code. is obtained
(7.5)
It should be noted that the strain. allowing for tension
stiffening, is used because it is considered that slabs are
relatively lightly reinforced and have load- strain relation-
ships similar to that shown in Fig. 7.1 ..In such l~ases, the
tension stiffening effect is significant and should be calcu-
lated from equation (7.3): although. as mentioned earlier
in this chapter, the tension stiffening could be reduced hy
repeated loading.
Equation 17_5) wa~ved from tests in which the re-
inforcement was perpendicular to the cracks. The general
problem of CflIt-k control wht'n the reinforcement is not
perpendicular to the cracks has been wlIsidercd hy Clark
;% //////.
A, F t=
P ~I-'1-.
?:
(a' Right bridge (b) Skew bridge,
orthogonal
transverse
reinforcement
(c' Skew bridge,
skew transverse
reinforcement
.I ILongitudinal beams/webs
A, • Transverse reinforcement
Crack due to local
transverse bending
FiJ.l. 7.7(1I)-·(e) Cracking in fllInges
11921. This study showed that equation (7.5) is applicable
provided that the crack width calculation is carried out in a
direction perpendicular to the crack and that all of the re-
inforcement should be resolved to an equivalent area of steel
perpendicular to the crack. However, equation (7.3),
which predicts the tension stiffening effect. has not been
checked for reinforcement which is not perpendicular to
.the cracks. In view of the complex stress state in the con-
crete between the cracks. it would be advisable to ignore
tension stiffening when considering such arrangements of
reinforcement.
r.lIl1gitudinal reinforcement in .!7anges Although it is not
clear in the Code, it is unnecessary to calculate the spacing
of the longitudinal reinforcement in a flange because a
simple bar spacing rule is given.
A bar spacing rule for tension flanges was derived by
noting that, for HA loading, the design load at the ser-
viceability limit state is about 70% of that at the ultimate
limit state: thus
EI = 0.7 x O.R7f..1200 x IO~
If it is assumed that f", """ 0.8 fl' then
I'm"'" n.R x 0.7 x O.R7f,J200 X 10"
Hence, a reasonahleupper limi.t to Em is 0.001. Since the
strain is very nearly constant over the depth of a flange.,
the neutral axis depth 't) for the flange tends to infinity.
Hence, equation (7.5) reduces to w =3acr E",and, forE".=
o.onI and a design value of w of 0.2 mm (see Table 4.7).
an = 67 mm, which results in a bar spacing of about
150 mm. This maximum har spacing is given in the Code
forplJ!dominantly t~nsion flanges together with a value of
300 nllll for predominantly compression flanges. The latter
value simply complies with the general maximum bar
spacing given in the Code.
Trall,~verse reiltforcement ill flanges When drafting the
clauses for crack control in flanges. it was assumed that
the effects of local bending in the flanges would generally
dominate the transverse bending effects. Thus the major
principal moment in a flange would act very nearly per-
pendicular to the longitudinal beams or webs. Tests [R7,
126. 1921 on slab bridges and slab elements indicate that,
lit the serviceability limit state. it is rellsonable to assume
that the cracks are perpendicular.to the principal mome~t
direction. There is no reason to assume that the cracks in a
slab acting as a flange would not form in the same direc-
tion and, thus, they would be very nearly parallel to the
longitudinal beams or webs.
It can be seen from Fig. 7.7 that, in the case of aright
bridge or of a skew bridge in which the transverse rein-
forcement is perpendicular to the beams or webs, the rein-
forcement would be perpendicular to the service load
cracks. Hence, equations (7.3) and (7.5) can be applied, to
a unit width of slab perpendicular to the cracks, with A,
equal to the llreaof transverse reinforcement per unit
width. However. for a skew bridge in which the transverse
reinforcement is parallel to the supports and such that it
makes an angle (IX) to the perPendicular to the crack; as
shown in Fig. 7.7(c), equations (7.3) and (7.5) cannot be
applied directly. It is first necessary to calculate an effec-
tive area of reinforcement perpendicular to the crack hy
using either equation (7.1) or (7.2). IIHuet, equation (7.1)
is adopted in the Code because the drafters had top flanges
primarily in mind and it is most likely that these will be
cracked on one face only and in one direction. Further-
more, since the longitudinal reinforcement would generally](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-53-2048.jpg)
![be parallel to the longitudinal beam or webs, equation
(7.1) reduces to
An = At COS4~
whereAt is area of transverse reinforcement per unit width.
HenceAs , in equation (7.3), should be replaced by An and
the latter value used to calculate the neutral axis depth and
the strain in a direction perpendicular to the cracks.
Fi~aIly, the tension stiffening equation is not dependent
upon bar spacing because it was derived from tests on
beams; whereas, in the Code, it is used only for slabs. One
would expect tension stiffening in slabs to depend upon the
bar spacing and, indeed, test data indicate such a depen-
dence for large bar spacings. However, tests reported by·
Clark and Cranston [187] show that the influence of bar
spacing is insignificant provided that the bar spacing does
not exceed about 1.5 times the slab depth. Since such large
spacings are unlikely to occur in a bridge, the tension stif-
fening equation can be applied to flanges. However, the
reservations expressed earlier, regarding skew reinforce-
ment and repeated loading, should be considered.
Columns
If tensile. str~sse~I.!I~_ i~~o'luitttf;·llietT·~the column
should be consIdered as a beam for crack control purposes
and equation (7.4) used.
Walls
If tensile stresses occur in a reinforced concrete wall, then
it is obviously reasonable that the wall should be consid-
ered as a slab for crack control purposes. The Code takes
suc~ an approach and also distinguishes between the two
exp<:>~ure conditions (see Table 4.7) which are .applicable
to a"Vall.
Severe exposure This condition includes surfaces in con-
tact with backfill and is thus appropriate to. the. back faces
of retaining walls and wing walls. The design crack width
is 0.2 mm which is also the value for soffits. Hence the
Code permits the bar spacing rules for slab bridges, of 150
and 300 mm for longitudinal and transverse reinforcement,
respectively, to be adopted for walls subjected to a severe
exposure condition. However, it should be noted that
calculations for walls were not carried out to check
specifically that the spacings of 150 and 300 mm would
be reasonable.
Very severe exposure A leaf pier is an example of a wall
subject to the effects of salt spray; hence, its exposure
condition is classed by the Code as very severe. The bar
spacing rules for slab bridges are appropriate only to the
severe exposure condition and it is thus necessary to carry
out crack width calculations for walls if the exposure con-
dition is very severe. Thus equation (7.5) should be used,
in conjunction with equation (7.3).
Bases
Except for the statement that 'reinforcement need not be
provided in the side faces of bases to control cracking' , the
94
Code is rather vague regarding crack control in bases. The
relevant clause states that the method of checking crack
widths depends on the type of base and the design assump-
tions. Before discussing this statement, it should be men-
tioned that, although reinforcement is generally provided
in the side faces of deep members to control cracks for
aesthetic purposes, it is not necessary to do this in bases
because they are generally buried.
In drafting the Code clause on crack control in bases
it was intended that the various components of a base
should be checked for cracking in accordance with the
most appropriate of the procedures given for other struc-
tural elements. It was intended that 'beam components'
should be checked by applying equation (7.4), and that
'slab components' (e.g., spread footings) should be con-
sidered as follows:
1. If a moderate or severe exposure condition (see Table
4.7) is appropriate, then apply the bar spacing rules,
of 150 and 300 mm, for slab bridges.
2. If a very severe exposure condition is appropriate,
then apply equations (7.3) and (7.5).
The reasons for these recommendations are the same as
those discussed previously in connection with walls.
It is obvious that 'engineering judgement' is required
when checking crack widths in bases.
Prestressed concrete stress
limitations
*Seams
In Chapter 4, limiting ~alues of prestressing steel stresses
and concrete compressive and tensile stresses are given.
It is emphasised in Chapter 4 that, although prestressing
steel stress criteria are given in the Code, another clause
essentially states that these can be ignored because it is not
necessary to calculate tendon stress changes due to the
effects of applied loadings.
Concrete stresses can be calculated by applying conven-
tional elastic theory; but the calculation of tensile stresses
in Class 3 members requires some comment. Although a
Class 3 member is, by definition, cracked at the service-
ability limit state, it is considered to be uncracked for the
purposes of calculating stresses. It is permissible to do this
because the hypothetical tensile stresses in Table 4.6(a)
were calculated from test data by assuming elastic
uncracked behaviour. Hence, for stress calculation pur-
poses, a cracked Class 3 member is considered in exactly
the same way as an uncracked Class 1 or 2 member.
It should be noted that a disadvantage of the above
approach to design is that, because the section is actually
cracked in practice, the actual compressive stress exceeds
the value calculated for an uncracked section. Thus the
actual compressive stress could exceed the allowable com-
pressive stress from Table 4.3(a) although the calculated
compressive stress might not. Thus the author would sug-
gest that, when designing Class 3 members, it would be
good practice not to stress the concrete to its allowable
limit in compression.
Slabs
In an uncracked slab (Class 1 or 2), conventional elastic
theory can be applied in the usual way to calculate the
principal concrete stresses. However, the approach
adopted for cracked Class 3 beams, in which hypothetical
tensile stresses appropriate to an uncracked beam are cal-
culated, would not, in general, be correct for cracked
Class 3 slabs. This is because the hypothetical tensile
stresses in Table 4.6 were calculated from test data for
beams and, although probably applicable to slabs in which
the prestressed and non-prestressed steel are parallel to the
principal stress direction, they would not be applicable to
slabs in which these directions do not coincide. Test data
are not available for such situations and the author would
suggest the following interim measures which are based
upon consideration of equation (7.1).
1. Beeby and Taylor [123] have studied, theoretically
and experimentally, cracking in Class 3 members.
Their theoretical expression for the hypothetical ten-
sile stress is a function of the area of prestressing steel
perpendicular to the crack. Equation (7.1) suggests
that if the area of prestressing steel per unit width is
A ps and it is at an angle ~ to the major principal stress
direction. then the equivalent area of prestressing steel
perpendicular to the crack is Aps COS4~. A study of
Beeby and Taylor's work indicates that it is reason-
able to assume that a reduction of steel area from Aps
to A COS4~ results in a reduction of the hypotheticalps
tensile stress from, say, !ht to !ht COS2~. Thus the
author would suggest that, when the prestressing ten-
dons are at an angle ~ to the major principal stress
directions, the limiting hypothetical tensile stresses of
Table 4.6(a) should be multiplied by COS2~.
2. The depth factors of Table 4.6(b) should not be modi-
fied.
3. Any additional conventional reinforcement should be
considered in terms of an equivalent area of re-
inforcement perpendicular to the crack by using equa-
tion (7.1). This equivalent area should be used to cal-
culate the increase of hypothetical tensile stress which
is permitted when additional reinforcement is present.
4. If the final limiting hypothetical tensile stress is less
than the appropriate limiting tensile stress for a Class
2 member, the section will be uncracked and should
he treated as a Class 2 member.
Losses in prestressed concrete
"*Initial prestress
In order that excessive relaxation of the stress in the ten-
dons will not occur, the normal jacking force should not
exceed 70% of the characteristic tendon strength. How-
ever. in order to overcome the effects offricfdon, the jack-
ing force may be increased to 80% of the characteristic
tendon strength, provided that the stress- strain curve of
the tendon does not become significantly non-linear above
a stress of 70% of the characteristic tendon strength.
The above requirements are essentially identical to those
of BE 2173.
~oss due to steel relaxation
If experimental data are available, then the loss of pre-
stress in the tendon should be taken as the relaxation. after
1000 hours duration, for an initial load equal to the jacking
force at transfer. This value is taken because it is approxi-
mately equal to the relaxation, after four years, for an ini-
tial force of 60% of the tendon strength: this force is
roughly the average tendon force over four years [197].
In the absence of experimental data, the relaxation loss
should be taken as 8% for an initial prestress of 70% of the
characteristic tendon strength, decreasing linearly to 0%
for an initial prestress of 50% of the characteristic tendon
strength. These values were based upon tests on plain cold
drawn wire [112]. -
The Code also refers to losses given in Part 8 of the
Code, but this appears to be a mistake because no losses
are given in Part 8.
The Code losses are essentially the same as those of
BE 2173 except for the reduced loss which may be adopted .
if the initial jacking force is less than 70% of the charac-
teristic tendon strength.
Loss due to elastic deformation of the concrete
The elastic loss may be calculated by the usual modular
ratio procedure.
For post-tensioned construction, the elastic loss may be
calculated either, exactly, by considering the tensioning
sequence, or, approximately. by multiplying the final
stress in the concrete adjacent to the tendons by half of the
modular ratio. The latter procedure is an approximate
method of allowing for the progressive loss of prestress
which occurs as the tendon forces are gradually transferred
to the concrete.
Elastic losses calculated in accordance with the Code
will thus be identical to those calculated in accordance
with BE 2173.
Loss due to shrinkage of concrete
For a normal exposure condition of 70% relative humidity
the Code gives shrinkage strains of 200 and 300 micro-
strains, for post-tensioning and pre-tensioning, respectively.
These values are identical to the CP 115 values. However,
for a humid exposure condition of 90% relative humidity,
the Code also gives shrinkage strains of 70 and 100
microstrains, for post-tensioning and pre-tensioning.
respectively.
Loss due to creep of concrete
According to Neville r198]. the relationship hetween c~ep
of concrete and the stress to strength ratio is of the form
shown in Fig. 7.8. It can be seen that, for a particular
concrete. creep is directly proportional to stress for stress](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-54-2048.jpg)
![Stress
Strength
~0.7 -------------
... 0.3 ---
Creep
Fig. 7.8 Creep-stress/strength relationship
to strength ratios less than approximately 0.3. The Code
limit of one-third, below which creep can be considered to
be proportional to stress, is thus reasonable.
The Code gives values for the specific creep strain
(creep strain per unit stress) of:
1. For pre-tensioning; the lesser of 48 x 10-8
and
48 x 10-0
x 4O!fc,per N/mm2
•
2. For post-tensioning; the lesser of 36 x 10-8
and
36 x 10-8
x 4O!fc; per N/mm2
..
In the above, tel is the cube strength at the time of transfer.
The values of specific creep strain are identical to those in
BE 2/73 and CP 115.
If the compressive stress anywhere in the section
exceeds one-third of the cube strength at transfer, the
specific creep strain should be increased as indicated by
Fig. 7.8. The Code gives a factor, by which the above
specific creep strains should be multiplied, which varies
linearly from unity at a stress-to-strength ratio of one-thi~d
to 1.25 at a ratio ofone-half (the greatest allowable ratIo
under any conditions). This factor is less than that indi-
cated by tests [198] on uniaxial compression specimens
because, generally, it is parts, rather than the whole, of a
cross-section which are subjected to stresses in excess of
one-third of the cube strength.
The above requirements are identical to those of BE 2173
but it should be noted that the Code also states that half
of the total creep may be assumed to take place in the first
month after transfer and three-quarters in the first six
months.
Loss due to friction in duct
In post-tensioning systems, losses occur due to friction in
the duct caused by unintentional variations in the duct
profile ('wobble') and by curvature of t~e duct. The Code
adopts the conventional friction equation [199]
Px = Po exp(- Kx + IJXlrps)
where
P" prestressing force at a distance x from the jack
Po = prestressing force in the tendon at the jack
radius of curvature of ductr"..
K = wobble factor
/.L = coefficient of friction of tendon.
96
In the absence of specific test data, the Code gives a
general value of K of 33 x 10-4
per metre, but a value of
17x 10-4
may be used when the duct former is rigid or
rigidly supported.
The Code gives the following values for /.L:
1. 0.55 for steel moving on concrete.
2. 0.30 for steel moving on steel.
3. 0'.25 for steel moving on lead.
The above values of K and fA. ~ identical to those in
CP 115 and were originally based upon the test data of
Cooley [199]. However, the Construction Industry Re-
search and Information Association has now assessed all
of the available experimental data on K and /.L values and
has recommended [200]:
1. i.t values of 0.25 and 0.20 for steel moving on steel
and lead, respectively.
2. For other than long continuous construction, the K
values given in the Code.
3. A K value of 40 X 10-4
per metre for long continuous
construction because it has been suggested [201] that
the Code value of 33 x 10-4
underestimates friction
losses in such situations. The author presumes that a
value· of 20 x 10-4 would be adopted for rigidly sup-
ported ducts. .
Finally, where circumferential tendons are used, the fol-
lowing Il values are recommended in the Code:
1. 0.45 for steel moving in smooth concrete.
2. 0.25 for steel moving on steel bearers fixed to the
concrete.
3. 0.10 for steel moving on steel rollers.
These values are the same as those in CP 115 and were
originally suggested by Creasy [202].'
Other losses
The Code does not give specific data for calculating the
losses due to steam curing nor, in post-tensioning systems,
the losses due to friction in the jack and anchorages and
to tendon movement at the anchorages during transfer.
Instead reference is made, as in CP 115, to specialist
advice.
Deflections
General
It is explained in Chapter 1 that there is not a limit state of
excessive deflection in the Code since a criterion, in the
form of an allowable deflection or span-to-depth ratio, is
not given. However, in practice, it is necessary to calcu-
late deflections in order, for example,to calculate rotations
in the design of bearings. The Code thus gives methods of
calculating both short and long term curvatures in Appen-
dix A of Part 4.
The procedure is to calculate the curvature assuming,
Ecfc
Compressive stresses
in concrete
Tensile stresses-
in cracked
concrete Level of centroid of tension
!.~ reinforcement
1N/mm2 (short-term)
0.55 N/mm2 (long-term)
Fig. 7.9 Tension stiffening for curvature calculations
first, an uncracked section and / second, a cracked section.
The larger of the two curvatures is then~dopted. Lo~g
term effects of creep are allowed for byustng an effective
elastic modulus, for the concrete which is less than the
short-term modulus. Shrinkage is allowed for by separately
calculating the curvature due to shrinkage.
Short-term curvature
The short-term elastic moduli for concrete, which are tabu-
lated in the Code, should be used to calculate the short-
term curvature under imposed loading.
The calculation for the uncracked section is straightfor-
ward. However, the calculation for the cracked section is
more complicated because of the need to allow for tension
stiffening.
It is mentioned elsewhere in this chapter, that when cal-
culating crack widths in flanges, tension stiffening is
allowed for by subtracting a 'tension stiffening strain' from
the reinforcement strain calculated by ignoring tension stif·
fening. However, when calculating deflections, a different
approach is adopted: a triangular distribution of tens~le
stress is assumed in the concrete below the neutral aXIS,
with a stress of 1 N/mm2
at the centroid of the tension
reinforcement, as shown in Fig. 7.9.
The stress of 1 N/mm2
was derived from the test results
of Stevens [188] which are referred to earlier in this
chapter. .
The CP 110 handbook [112] implies that the peutral aXIs
should be calculated from the stress diagram of Fig. 7.9 by
employing a trial-and-error approach; but, as suggested by
Allen [203], it is simpler to adopt the following procedure,
which involves little error:
1. Calculate the neutral axis depth (,x) ignoring tension
stiffening. .
2. Calculate the extreme fibre concrete compressive
stress due to the applied bending moment.
3. Calculate the extreme fibre strain (Fe) by dividing the
stress by the elastic modulus of the concrete (EJ.
4. Calculate the curvature (E,JX).
Allen gives equations which aid the above calculations
for csctangular and T-sections. .
Long-term curvature
The curvature calculated above would increase under
long-term loading due to creep of the concrete. Creep in
compression is allowed for by dividing the short-term elas-
tic modulus of the concrete by (1 + <1» where <I> is a creep
coefficient. It is emphasised that <I> does not, in this chap-
ter, refer to the same coefficient as it does in Chapter 8. In
fact, the creep coefficient ~ referred to in Chapter 8 is iden-
tical to the creep coefficient </> referred to above. An
appropriate value of <I> can be determined from the data
given in Appendix C of the Code, provided that sufficient
prior knowledge of the concrete mix and curing conditions
are known. Since <I> depends .upon many variables, the
CP 110 handbook [112] gives a table of <I> values which
may be used in the absence of more detailed information.
As an aitc1'l1alive, a simplified method of obtaining <I> is
given by Parrott [204].
Creep of the concrete in tension is allowed for by re-
ducing the tensile stress in the concrete, at the level of the
centroid of the tension reinforcement, from its short-term
value of 1 N/mm2
to a long-term value of 0.55 N/mm 2•
The latter value was again derived from the test results of
Stevens [188].
The long-term curvature can be calculated by following
the same procedure as that given earlier for the short-tenn
curvature.
Shrinkage curvature
The Code gives the following expression for calculating
the curvature ('IjIs) due to shrinkage.
'IjIs =Po Er./d (7.6)
where d is the effective depth, Ec.. is the free shrinkage
strain and Po is a coefficient which depends upon the per-
centages of tension and compres~ion steel.
The free shrinkage strain can be determined from the
data given in Appendix C of the Code. However, as is also
the case for <1>, the free shrinkage strain depends upon
many variables. Thus the CP 110 handbook [112] suggests
a value of 300 x 10-/ for a section less than 250 mm
thick and a value of 250 x 10-/ for thicker sections. The
handbook also indicates how the shrinkage develops with
time. As an alternative, Parrott [204] gives a graph for
estimating shrinkage.
Values of the coefficient Po are given in a table in
Appendix A of the Code. The tabulated values are based
upon empirical equations derived by Branson [205] from
two sets of tt~st data. Hobbs [206] has compared the Code
values of Po with these data and also with an additional set
of data. He found that the Code values are conservative
and are particularly conservative, by a factor of about 2,
for lightly reinforced and doubly reinforced sections. In
view of this, Hobbs suggests an alternative procedure for
calculating shrinkage curvatures, which is based upon
theory rather than empirical equations, and which gives
good agreement with test data [206].
General calculation procedure
.
The following general procedure is suggested in the Code
for calculating the iong-telm curvature:](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-55-2048.jpg)
![Moment Short term
Short term Creep due to . • transient
permanent permanent Shrmkage,_
~4 .14 .,4 '"
Total
Permanent
Curvature
Fig. 7.10 Calculation of long term curvature
I. Calculate the instantaneous (i.e., short·term) curva-
tures under the total design load and under the peona-
nent design load. "
2. Calculate the long4 term curvature under the perlllilnent
design load.
3. Calculate the difference between the instantaneous
curvatures under the total and permanent design loads.
This difference is the curvature due to the transient
design loads. Add this value to the long-term curva-
ture under the permanent load.
4. Add the shrinkage curvature.
The above procedure is logical and its net effect is ilIus-
trated in Fig. 7.10. .
Examples
~7.1 Reinforced concrete
The top slab ofa beam and slab bridge has been designed
at the ultimate limit state and is shown in Fig. 7.1 J. The
Table 7.J Example 7.1. Nominal load effects
la) Local moments (kNmlm)
, ~
Load
Dead
Superimposed dead
HA wheel
45 HB units
25 HB units
(h) Glohal effects
Load
Transverse
0.45
0.22
10.8
11.7
7.56
Transverse
moment
(kNm/m)
Longitudlna~
0.0
0.0
7.20
7.65
5.26
Longitudinal
compre~,ive stress
in slab (N/mm2)
Top Bottom
Superimposed dead 0.0 0.33 0.14
1.60
0.25
2.55
1.48
HA 0.0 3.90
HA wheel 6.0 0.61
45 HB units 43.0 6.24
25 HB units 23.3 3.63
98
Transverse steel Y16-100
Longitudinal steel Y16-200
Fig. 7.11 Example 7.1
eharacteristic strengths of the reinforcement and concrete
are 425 and 30 N/mm2
, respectively.
. Mid~way between the beams, the local bending
moments given in Table 7.1 (a) act together with the global
~ffects given in Table 7.1(b). The effects due to HB
include those due to associated HA.
It is required to check that the slab satisfies the service-
ability limit state criteria under load combination 1.
*Genera/
From Table 2 of the Code, short-term elastic modulus of
concrete ::i::: 28 kN/mm2. Use a long-term value of 28/2 =
14 kN7mmB for both cracking and stress calculation
(Clause 5.3.3.2). Modular ratio = 200/14 = 14.3.
Each design load effect will be calculated as nominal
load effect x Yfl. x Yf3. Details of these partial safety fac-
tors are given in Chapters 3 and 4.
'*Cracking
Due to transverse bending
HA design moment
== (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0)
. + (l0.8)(l.2)(0:~3) + (6.0)(1.2)(0.83)
= 17.5 kNmlm
2S HB design moment
~ (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0)
+ (7.56)(1.1)(0.91) + (23.3)(1.1)(0.91)
=31.6 kNm/m
~rit,((atwoment == 31.6 kNmlm
Area of bottom ~teel = area of top steel = 2010 mm2/m
. Ifelastic neutral axis is at depth x, then by taking first
moments of area about the neutral axis:
(1/2)(1000) Xl + (13.3)(2010) (x - 38)
= (14.3)(2010)(152 - x)
:. x:::: 62.2 mm
.Second moment of area, I, about the neutral axis is
given by:
(113)(1000)(62.2)3 + (13.3)(2010)(24.2)2 +
(14.3)(2010)(89.8)2
= 0.328 x lOP mm 4
/m
Bottom steel stress =
(14.3)(31.6 x 106
)(89.8)/0.328 x 109
=.124 N/mm2
Soffit strain ignoring tension stiffening is. .
£1 = (124/200 x 10:1)(137.8/89.8) = 9.51 x 10-4
From equation (7.3), soffit strain allowing for tension stif-
fening is
Em = 9.51 X 10-4
- (1.2)(1000)(200)(10-3)/(2010 x 425)
= 6.70 x 10-4
Maximum crack' width occurs mid-way between bars
. where the distance (acr) to the nearest bar is given by
"acr = /502 + 482
- 8 = 61.3 mm.
From equation (7.5), crack width is
(3)(61.3)(6.70 x 10-
4
) = 0.094 mm
w = 1 + (2)(61.3-40)/(200-62.2)
From Table of the Code, allowable crack width is
0.20 mm.
!.JttDue to longitudinal bending The longitudinal bars are in
a region of predominantly compressive flexural stress and
thus the Code maximum spacing is 300 mm. This exceeds
the actual spacing of 200 mOl.
*Stresses
Limiting values The limiting stresses are 0.5 x 30 =
15 N/mm2
for concrete in compression, and 0.8 x 425 =
340 N/mm2 for steel in tension or compression.
Due to transverse bending
HA design moment
= (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0)
+ (10.8)(1.2)(1.0) + (6.0)(1.2)(1.0)
= 20.9 kNm/m
45 HB design moment
= (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0)
+ (11.7)(1.1)(1.0) + (43.0)(1.1)(1.0)
= 60.9 kNrnlm'
Critical moment = 60.9 kNmlm
From crack width calculations, x = 62.2 mm,
I = 0.328 X 109
mm4
/m
Maximum concrete stress
== (60.9 x 108
)(62.2)/0.328 x 109
= 11.5 N/mm2
<15 N/mm2
Steel stresses are
(14.3)(60.9 X 106
)(89.8)/0.328 x 109
= 238 N/mm2
tension
and
(14.3)(60.9 x 106
)(24.2)/0.328 x 109
= 64 N/mm2
compression
Both < 340 N/mmI
Due to longitudinal effects Maximum coml?ressive
stresses occur under 45 units of HB loading. Assume section
to be uncracked and ignore reinforcement.
Extreme fibre stresses due to nominal local moment
= ±(7.65 x 103
)(6)/(200)2 = ± 1.14 N/mm2
Net top fibre design stress is
(0.33)(1.2)(1.0) + (6.24)(1.1)(1.0) + (1.14)(1.1)(1.0)
= 8.51 N/mm2<15 N/mm2
1500
175
900
600
~I
Fig. 7.12 Example 7.2
Maximum tensile stress in reinforcement occurs under
the HA wheel load. The longitudinal global stresses under
the HA wheel load and superimposed dead load are:
Top = 0.61 + 0.33 = 0.94 N/mml
Bottom = 0.25 + 0.14 = 0.39 N/mm2
These are equivalent to an in-plane force of
200 (0.94 + 0.39)/2 = 133 kN/m, and a moment of
[(0.94 - 0.39)/2] (2002
) 10-3
/6 = 1.83 kNm/m. Total
nominal moment = 7.2 + 1.83 = 9.03 kNrnlm. If the
in-plane force is conserva~ively ignored, the design load
effect is a moment of (9.03)(1.2)(1.0) = 10.8 kNm/m.
Area of bottom steel = area of top steel = 1010 mm2
/m.
If elastic neutral axis is at depth x and is above the top
steel level
(112)(1000) x 2
= (14.3)(1010)[(136 - x) + (54 - x)]
:. x = 50.7 mm and is above the top steel.
Second moment of area =
(1/3)(1000)(50.7)3 + (14.3)(101Q)(85.32
+ 3.32)
= 0.149 X 109
mm4
/m
Bottom steel stress
= (14.3)(10.8 x 108
)(85.3)/0.149 x 109
=88 N/mm2
< 340 N/mm2
'*7.2. Prestressed concrete
A bridge deck is constructed from the pre-tensioned
I-beams shown in Fig. 7.12. Each beam is required to
resist the moments, due to nominal loads, given in Table
7.2. Determine the prestressing force and eccentricity
required to satisfy the serviceability limit state criteria,
under load combination 1, for each of the three classes of
prestressed concrete. Assume that the losses amount to
Table 7.2 Example 7.2. Design data
Moment (kNm) Stress (N/mm2
)
Load
Nominal Design Top Bottom
Dead 477 477 +3.67 - 5.24
Superimposed dead 135 162 + 1.25 - 1.78
HA .949 949 +7.30 - 10.42
HB + Associated
HA 1060 1060 +8.15 -11.64](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-56-2048.jpg)
![Chapter 8
Precast concrete and
composite construction
Precast concrete
The design of precast members in general is based upon
the design methods for reinforced or prestressed concrete
which are discussed in other chapters. Bearings and joints
for precast members are considered as part of this chapter.
Bearings
The Code gives design rules for two types of bearing: cor-
bels and nibs.
*Corbels
A corbel is defined as a short cantilever bracket with a
shear span to depth ratio less than 0.6 (see Fig. B.l(b».
The design method proposed in the Code is based upon
test data reviewed by Somerville [207]. The method
assumes, in the spirit of a lower bound design method, the
equilibrium strut and tie system shown in Fig. B.1(a). The
calculations are carried out at the ultimate limit state.
In order to assume such a strut and tie system it is first
necessary to preclude a shear failure. This is done by
proportioning the depth of the corbel, at the face of the
supporting member, in accordance with the clauses cover-
ing the shear strength of short reinforced concrete beains.
The force (F,) to be resisted by the main tension re-
inforcement can be determined by considering the equilib-
rium of the strut and tie system as foHows (the notation is.
in accordance with Fig. 8.1).
V =Fe sin ~
P, = H + Fe cos ~ = H + V cot ~ (B.1)
Somerville [207] suggests that ~ can be determined by
assuming a depth of concrete x having a constant compres-
sive stress of 0.4 feu and considering equilibrium at the
face of the supporting member, as shown in Fig. 8.l(b)
and (c). An iterative procedure is suggested in which x/d is
first assumed to be 0.4; in which case cot ~ = a.,lO.&l and
the compressive force in the concrete is given by
Fe = 0.4 f,'u bx cos P
102
'''~'' ,
v
H
Steel
F, .......-.ti~e~-".
(a) Strut and tie
v
A
F, ...-+---_~
d
/
x /
I
/
x~/
(b) Deterrnination of ~
F,
d-x/2
Fe cos ~
=O.4feu bx
(c) Horizontal forces at A-A
Fig. 8.I(a)-(c) Corbel strut and tie system
L..----......-r'---,Corner
shears
d
.£.d
3
Main
reinforcement
.', ··Horizontal
links
Bars to
anchor links
(a) Local bearing failure (b) Horizontal links
Fig. 8.2(a),(b) Corbel detailing
The horizontal component of this force is
0.4 feu bx COS2~
For equilibrium, the tensile force (F1) in the reinforcement
must equal this horizontal component of the concrete com.
pressive force; thus
F, = 0.4 feu bx cos2
~.
A new value of ~can then be calculated from
cot ~ =aj(d - x/2)
This procedure can be continued iteratively.
(8.2)
(8.3)
It should be noted that both Somerville [207] and the CP
110 handbook [112] give the last term of equation (8.2),
incorrectly, as cos ~ instead of cos2
~.
The area of reinforcement provided should be, not less
than '0.4% of the section at the face of the supporting
member. This requirement was determined:empirically
from the test data. It is important that the reinforcement is
adequately anchored: at the front face of the corbel this can
be achieved by welding to a transverse b~r or by bending
the main bars to form a loop. In the latter cas€( the bearing
area of the load should not project beyond the straight por-
tion of the bars, otherwise shearing of the comer of the
corbel could occur as shown in Fig. B.2(a).
Theoretically no reinforcement, other than that referred
to above, is required. However, the Code also requires
horizontal links, having a total area equal to 50% of that of .
the main reinforcement, to be provided as shown in
Fig. B.2(b). Horizontal rather than vertical links are
required because the tests, upon which the design method
is based, showed that horizontal links were more efficient
for values of ajd < 0.6.
The Code states that the above design method is appli-
cable for ajd < 0.6.The implication is thus that, if ajd
~ 0:6, the corbel should be designed as a flexural can~
tilever. However, the test data showed that the design
l.ink In
supporting~
member
Fig. 8.3 Nib
~ .
V(Line of action is
at outer edge of
loaded area)
'---4-...
Imaginary compressive
strut
method based upon a strut and tie ~ystem was applicable to
avid values of up to 1.5. In view of this the CP 110 hand-
book [112] suggests that, as a compromise, the method can
be applied to corbels havingavldvalues of up to 1.0.
Finally, in order to prevent a local failure under the
load, the test data indicated that the depth of the corbel at
the outer edge of the bearing should not be less than 50%
of the depth at the face of the supporting member.
*Nibs
The Code requires nibs less than 300 mm deep to be
designed as cantilever! slabs at the ultimate limit state to
resist ,a bending moment of Vav (see Fig. B.3). Clarke
[20B] has shown by tests that this method is safe, but that
an equilibrium strut and tie system of design is more
appropriate for nibs which project less than 1.5 times their
depth. The distance av is taken to be from the outer edge
of the loaded area (i.e. the most conservative position of
the line of action of V) to the position of the nearest vert-
ical leg of the links in the supporting member. The latter
position was chosen from considerations .of a strut and tie
system in which the inclined compressive strut is as shown
in Fig. B.3.
Detailing of the reinforcement is particularly important
in small nibs and the Code gives specific rules which are·
confirmed by the test results of Clarke [20B].](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-58-2048.jpg)
![I,,
---Code
-----Williams (205)
1.0
0.5
,,,
o 0.2
Fig. 8.4 Bearing stresses
Bearing stress
~~
0.4
Nonnally the compresSive stress at the ultimate limit state
between two contact surfaces should not exceed 40% of
the characteristic strength of the concrete.
If the bearing area is well-defined and binding re-
inforcement is provided near to the contact area, then a tri-
axial stress state is set up in the concrete under the bearing
area and stresses much higher than 0.4 feu can be resisted.
The Code gives the following equatio!l for the limiting
bearing stress (fb) at the ultimate limit state
= 1.5 feu
1 + 2yp,)Yo (8.4)
where ypo and Yo are half the length of the side of the
loaded area and of the resisting concrete block respec-
tively. This equation was obtained from a recommendation
of the Comite Europeen du Beton, but the original test data
is not readily available. However, Williams [209] has
reviewed all of the available data on bearing stresses and
found that a good fit to the data is given by
fb =0.78 feu (yp,)y?)-O.441 ·(8.5)
In Fig. 8.4, the latter expression, with a partial safety
factor of 1.5 applied to feu, is compared with the Code
equation. It can be; seen that the latter is conservative for
smal(Ioaded areas.'
Th~ Code recogrises that extremely high bearing stress-
es can be developed in certain situations: an example is in
concrete hinges [210], the design of which will probably
be covered in Part 9 of the Code. At the time of writing,
Part 9 has not been published and thus the Department of
104
0.6 O.S 1.0
Transport's Technical Memorandum on Freyssinet hinges
[211] will, presumably, be used in the interim period. This
document pennits average compressive stresses in the
throat of a hinge of up to 2 feu.
-¥Halving joints
Halving joints are quite common in bridge <,:onstruction and
the Code gives two alternative design methods at the ulti-
mate limit state: one involving inclined links (Fig. 8.5(a»
and the other involving vertical links (Fig. 8.5(b». Each
method is presented in the Code in tenns of reinforced
concrete, but can also be applied to prestressed concrete.
Inclined links
When inclined links are used, the Code design method
assumes the eqUilibrium strut and tie system, shown in
Fig. 8.5(c), which is based upon tests carried out by
ReynOlds [212]. The Code emphasises that, in order that
the inclined links may contribute to the strength of the
joint, they must cross the line of action of the reaction Fv'
For eqUilibrium, the vertical component of the force in
the links must equal the reaction (Fv). Hence
Fv =Asv fyv cos 9
where Am fyv and 9 are the total area, characteristic
strength and inclination of the links respectively. A partial
safety factor of 1.15 has to be applied to fyv and hence the
Code equation is obtained.
,Inclined links
Horizontal
reinforcement
----~~-r-------'''"--t,.. to resist moment
at root of half end
cantilever plus
any horizontal
forces
Main tension reinforcement
(a) Inclined links
- 1--'
TFv
Minimum end cover to main
tension reinforcement"
(b) Vertical links
Strut (concrete)
Tie
-
..
(main reinforcement)
L--_ _ _ _I
(c) Strut and tie system
Fig. 8.5(a)-(c) Halving joint
F,. =As,. (0.87 fVl') cos e (8.6)
It should be noted that, theoretically, any value of emay
be chosen.: However, the crack which initiates failure
forms at about 45° as shown in Fig. 8.5(a), and thus it is
desirable to incline the links at 45°. The adoption of such a
value of f) is implied in the Code.
Reynolds suggested [212] that, althougl} his tests
showed that it is possible to reinforce a joint ',so that the
maximum allowable shear force for the full beam section
could be carried, it would be prudent in 'practice to limit
the reaction at the joint to the maximum allowable shear
force for the reduced section (with an effective depth of
do). This limit was suggested in order to prevent over-
reinforcement of the joint and. hence, to ensure a ductile
joint.
One might expect that the above limit would imply Fv =
vllbd,,: instead, the Code gives a value of Fv =4vrbdo. The
reason for this is that, in a draft of CP 110, 4vc was the
maximum allowable shear stress in a beam and was thus
equivalent to VII' which occurs in the final version of CP
10 and the Code. However, the clause on halving joints
was written when 4"" was in CP 110 and was not subse-
quently altered when V" (= 0.75 It,,) was introduced (see
Chapter 6).
Finally, .the horizontal component of the tensile force in
the inclined links has to be transferred to the bottom main
tension reinforcement. Nonnally one would assume that
such transfer occurs by bond, but the Code assumes that
the transfer occurs in two ways: half by bond with the con-
crete, and half by friction between the links and bars. For
the latter to occur the links must be wired tightly to the
main bars. The division of the force transfer into bond and
friction was not based upon theory but was an interpreta-
tion of the test results. Since only half of the force has to
be t-rtlnsferred by bond, the anchorage length of the main
tension reinforcement (lsb) given in the Code is only half
of that which Would be calculated by applying the anchor-
age bond stresses discussed in Chapter 10.,
Vertical links
As an alternative to the inclined link system of Fig. 8.5(a),
a halving joint may be reinforced with vertical links as
shown in Fig. 8.6(b). The vertical links should be
designed by the method described in Chapter 6 and
anchored around longitudinal reinforcement which extends
to the end of the beam as shown in Fig. 8.5(b) [212].
Horizontal reinforcement in half end
The Code does not require flexural reinforcement to be
designed in half ends. Howt(ver, it would seem prudent to
design horizontal reinforcement to resist the moment at the
root of the half end cantilever. Such reinforcement should
also be designed to resist any horizontal forces.
Composite construction
General
In the context of this book, composite construction refers
to precast concrete acting compositely with in-situ con-
crete. Very often in bridge construction, the precast con-
crete is prestressed and the in-situ concrete is reinforced.
The design of composite construction is complicated not
only by the fact that, for example, shear and flexure cal-
culations have to be carried out for both the precast unit
and the composite member, but because additional calcu-
lations, such as those for interface shear stresses, have'also
to be carried out.
The various calcula'tions are now discussed individually.
Ultimate limit state
Flexure
The, flexural design of precast elements can be carried out
in accordance with the methods of Chapter 5. In the case
of a composite member, the methods of Chapter 5 may be
applied to the entire composite section provided that hori-
zontal shear can be transmitted, without 'excessive slip,
across the interface of the precast and in-situ concretes.
The criteria for interface shear stresses, discussed in Chap-
ter 4, ensure that excessive slip does not occur.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-59-2048.jpg)
![r In-situ
1
Composite
:-- .~.. - '.1-- _.-. - . - 'ceritroic:r-' ....- .._- ' •. . -_ .•.- .._- . -- ....- ... --1_-+#. - . - . - . - . _. -14----.1_
Precast
(a) Section (b) Shear stross
dueto Va1
Fig. 8.6(a)-(d) Shear in composite beam and slab section
*Shear
It is not necessary to consider interface' shear at the ulti-
mate limit state because the interface shear criteria dis-
cussed in Chapter 4, for the serviceability limit state, are
intended to ensure adequate strength at the ultimate limit
state in addition to full composite action at the service-
ability limit state.
Thus it is necessary to consider only vertical shear at the
ultimate limit state. The fact that interfac'e shear is chec!
at the serviceability limit state and vertical sh~ar at the
ultimate limit state causes a minor problem in the organis-
ation of the calculations and introduces the possibility of
errors being made. It is understood that in the proposed
amendments to CP 110, which are being drafted at the
time of writing, the interface shear criteria are different to
those in the Code, and the calculations will be carried out
at the ultimate limit state.
The design of a precast element to resist vertical shear
can be carried out in accordance with the methods
described in Chapter 6. However, the design of a com-
posite section to resist vertical shear is more complicated
and there does not seem to be an established method. The
latter fact reflects the lack of appropriate test data.
The appropriate Code clause merely states that the
design rules for prestressed and reinforced concrete should
be applied and, when in-situ concrete is placed between
precast prestressed units, the principal tensile stress in the
prestressed units should not anywhere exceed 0.24 !feu
(see Chapter 6). It is thus best to consider the problem
from first principles. In the folIowing, it will be assumed
that the precast units are prestressed and thus the problem
is one of determining the shear capacity of a prestressed-.
reinforced composite section when it is flexurally
uncracked (Vr~) and also when it is 'flexurally cracked
(VeT)' The terminology and notation are the same as those
of Chapter 6. It is emphasised that the suggested pro-
cedures are tentative and that test data are required.
There are two general cases to consider:
'I. Precast prestressed beams with an in-situ reinforced
concrete top slab to form a composite beam and slab
bridge. •
2. Precast prestressed beams with in-situ concrete placed
between and overthe beams to form a composite slab.
. .106
(c) Shear stress
dueto Vc2
(d) Total shear
stress
*Composite beam and slab It is suggested by Reynolds,
Clarke and Taylor [161] that Veo for a composite section
should be detennined on the basis of a limiting principal
tensile stress of 0.24 I!cu 9t the centroid of the composite
.section. This approach is thus similar to that for pre-
stressed concrete.
The shear force (Vel) due to self weight and construc-
tion l~ads produces a shear stress distribution in the precast
member as shown in Fig. 8.6(b). The shear stress in the
. precast member at the level of the composite centroid is!,.
The additional shear force (VeZ) which acts on the com-
posite section produces a shear stress distribution in the
composite section as shown in Fig. 8.6(c). The shear
stress at the level of the composite centroid, due to Vc2
, is
fs.
The total shear stress distribution is shown in
Fig. 8.6(d) and the shear stress at the level of the com-
posite centroid is (I. +f'.;~. '
Let the compressive stress at the level of the composite
centroid due to self weight and construction loads plus 0.8
of the prestress befcp' (The factor of 0.8 is explained in
Chapter 6.) Then the major principal stress at the c~m
posite centroid is given by
II =-f,./2 + 1(f'cpI2)2 + (I. + I's)2
This stress should not exceed the limiting value of f, =
0.24/!cu' Hence, on substitutingfl =f" and rearranging
f .• = II? + f,.,J,- I.
But 1'..=V,.zAy!Ib, where I, band Aji refer to the com-
posite section.
Hence
Ib
Vc2 = A)i [II/ + f,.,,!, - I.] (8.7)
Finally, the total shear capacity (Veo).is given by
Veo = Vet + Ve2 (8.8)
The above calculation would be carried out at the junc~
tion of the flange and web of the composite section if the
centroid were to occur in the flange (see Chapter 6).
The above approach, suggested by Reynolds, Clarke
and Taylor, seems reasonable .except, possibly, when the
section is subjected to a hogging bending moment. In such
.....
'-....~..
a situation the reinforced concrete in-situ flan~e might be
racked and it could be argued that the in-Situ concrete
~hould ;hen be ignored. However, as explained in Chapter
6; a significant amount of shear c~n be transmitted by
dowel action of reinforcement, which would be present
.in the flange, and by aggregate interlock a~ro~s the cracks.
It thus seems reasonable to include the m-sltu flange. as
part of a homogeneous section, whether or not the section
is subjected to hogging bending. ,
When calculating Vcr> which is the shear capacity of the
member cracked in flexure, it is reasonable, if the member
is subjected to sagging bending, to apply the prestress.ed
concrete clauses to the composite section because the tn-
situ flange would be in compression, However, when the
.section is subjected to hogging bending, it would be con-
servative to ignore the cracked in-situ flange and to carry
out the calculations by applying the prestressed concrete
clauses to the precast section alone. .
When the prestressed concrete clauses are applied to the
composite section, the author would suggest that the fol-
lowing amendments be made.
1. Classes 1 and 2:
(a) Replace equation (6.12) with
M, =Mb(l - y"I,Jyc!") + (8.8)
(0.37 /f.'"b + 0.8 [p,) I,.Iyc
where the subscripts b andc refer to the precast
beam and the composite sections respectively and
Mb is the moment acting on the precast beam
alone. Equation (8.8) is derived as follows. The
design value of the compressive stress at the bot-
tom fibre due to prestress and the moment acting
on the precast section alone is
0.8 Ip ,- MbYb11b
The additional stress to cause cracking is (see
Chapter 6)
0.37 /t:.uh + O.81p , - MhYb11h
Thus the additional moment, applied to the com-
posite section to cause cracking, is:
Ma = (0.37 /I."b + a.8lp, - MhYb1h)1e1Ye
The total moment is the cracking moment:
M, = Ma + Mb
which on simplification gives equation (8.8)
(b) Replace the term (£I !1c,,) in equation (6.11) with
(d 17 b +d· If .) where the subscripts band ih vJC14 I ..;JCUI ,
refer to the precast beam and in-situ concrete
respectively, and db and d; are defined in
Fig. 8.7(a). In this context db is measured to the
centroid of all of the tendons.
2. Class 3:
(a) Calculate Mo from
M" = Mb (1 - Ybl/v"/h) + O.8Ip ,I,./y,. (8.9)
which can be d~rived in a similar manner to equa-
tion (8.8).
(b) Replace the term (I,.d) in equation (6.16) with
dif
db
x
y·'liQ·····~(, ..', ".
X
Centroid of tendons, or
all steel in tension zone;
as appropriate,
(a) Beam and slab
Fig. 8.7(a),(b) Composite sections.
(vcbdb +vc/d;), where Vcb and Vc' are the nominal
shear stresses appropriate to Icub and lell' respec-
tively. db is measured to the centroid of all of the
steel in the tension zone.
Finally, it is suggested that the maximum allowable
shear force should be calculated from
V'4 =0.7Sb(dh/tllh + d, !f:.u,) (8.10)
db is measured to the centroid of all of the steel in the
tension zone.
The above suggested approach is slightly different to
that of BE 2173.
:If::Composite slab In order to 'comply.strictly. with the ~od.e
when calculating Veo for the composite sectIOn, the prmcl-
pal tensile stress at each point in the precast beams should
be checked. However, the author would suggest that it is
adequate to check only the principal tensile s~ress at the
centroid of the composite section. When carrytng out the
V calculation no consideration is given to whether the0 0 ' •
in-situ concrete between the beams is cracked. This IS
because the adjacent prestressed concrete restrains the in-
situ concrete and controls the cracking [113]. This effect is
discussed in Chapter 4 in connection with the allowable
flexural tensile stresses in the in-situ concrete. Provided
that the latter stresses are not exceeded, the in-situ and
precast concretes should act compositely.
It could be argued that, when subjected to hogging
bending, the in-situ concrete above the beams shoul,d be
ignored. However, the author would suggest that It be
included for the same reasons put forward for including it
in beam and slab composite construction.
The general Code approach differs from the approach.of
BE 2173, in which ?,~eas of plai~ in-situ concrete. w.h.lch
develop principal tensile stresses m excess of the hmltmg
value are ignored. .
When calculating Vcr, the in-situ concrete could be flex-
urally cracked before the precast concrete cracks. It is not
clear how to calculate Vcr in such a situation although tbe
author'feels that the restraint to the in-situ concrete pro-
vided by the precast heams should enab~e one to a~ply the
prestressed concrete clauses to the entire composite sec-
tion. However, in view of the lack of test data, the aut~or
would suggest either of the following two conservative
approaches.
1. Ignore all of the in-situ concrete and apply the pre-
stressed concrete clauses to the precast beams alone.
as proposed by Reynolds, Clarke and Taylor [161].
107](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-60-2048.jpg)
![2. Apply the reillforced concrete clauses to the entire
composite section.
Finally. it is suggested that the maximum allowable
shear force for a section (h" + hi) wide should be calcu-
. lated from
v" = O.75d(b,. /1,'"1> + hi /i.'"I) (8.11 )
where h" and b; are defined in Fig. 8.7(b).
The above suggested approach is different to that of
BE 2173 in which a modified form of equation (6.11) is
adopted for Vcr'
Serviceability limit state
*General
It is mentioned in Chapter 4 that the stresses which have to
he checked in a composite member are the compressive
and tensile stresses in the precast concrete, the com-
pressive and tensile stresses in the in-situ concrete
and the shear stress at the interface between the two con-
cretes.
For the usual case of a prestressed precast unit acting
compositely with in-situ reinforced concrete. the stress
calculations are complicated by the fact that different 10ld
levels have to be adopted for checking the various stresses.
This is because. as discussed in Chapter 4, different yp
values have to be applied when carrying out the various
stress calculations. It is explained in Chapter 4 thaI the
value of Yp implied by the Code is unity for all stress
calculation under any load except for HA and HB loading.
For the latter loadings, YP is unity for all stresscalcu-
lations except for the compressive and tensile stresses in the
prestressed concrete. when it takes the values of 0.83 and
0.91 for HA and HB loading respectively. It is emphasised
that the actual values to he adopted for the 'various stress
calculations are not 'stated in the Code. The values quoted
above can be deduced from the assumption that it was the
drafters' intention that. in general. Yp should be unity at
the serviceability limit state. However, there is an impor-
tant implication of this intention, which may have been
overlooked by the drafters. The tensile stresses in the in-
situ concrete have to be checked under a design load of
1.2 HA or 1.1 HB (because Yfl. is 1.2 and 1.1 respectively
and YP = 1.0) as compared with 1.0 HA or 1.0 HB when
designing in accordance with BE 2173. despite the allow-
able tensile stresses in the Code and BE 2173 being identi-
cal. This suggests that. perhaps. the drafters intended th~
Code design load to be 1.0 HA or 1.0 HB and, hence, the
Yrl values to be 0.83 and 0.91 respectively. This argument
also throws some doubt on the actual intended values of
YI~ to he adopted when checking interface shear stresses.
In conclusion. although it is not entirely clear what
value of Yf3 should he adopted for HA and HB loading,
and it could be argued that (Yf', Yp) should always be
taken to be unity. the following values of Yp will be
assumed.
I . HA - 1.0 except for stresses in prestressed concrete
when it is 0.83.
lOR
o
Average interface shear stress = v" = O.4fcubih;lbel
1--,--
Minimum
moment
section
O.4fcub;h;
Precast
,...,._-, .....L .......---...-----+1
Maximum
moment
. section
Fig. 8.8 Interface shear at ultimate limit state
2. HB - 1.0 except for stresses in prestressed concrete
when it is 0.91.
t Compressive and tensile stresses
Compressive and tensile stresses in either the precast or
In-situ concrete should not exceed the values discussed in
Chapter 4. Such stresses can be calculated by applying
elastic theory to the precast section or to the cOIl)posite
section as appropriate. The difference between the elastic
moduli of the two concretes should be allowed for if their
strengths differ by more than one grade.
It is emphasised that the allowable flexural tensile stress-
es of Table 4.4 for in-situ concrete are applicable only
when the in-situ concrete is in direct contact with precast
prestressed concrete. If the adjacent precast unit is not pre-
stressed then' the flexural cracks in the in-situ concrete
should be controlled by applying equation (7.4).
The allowable flexural tensile stresses in the in-situ con-
crete are interpreted differently in the Code and BE 2/73.
In the Code it is explicit,ly stated that they are stresses at
the contact surface; when~!ls in BE '2173 they are applicable
to all of the in-situ concrete. but those parts of the latter in.
which the allowable stress is exceeded are not included in
the composite section.
In neither the Code nor BE 2173 is it necessary to calcu-
late the flexural tensile stresses in any in-situ concrete
which is not considered, for the purposes of stress calcu-
lations, to be part of the composite section.
:¥Interface shear stresses
In terms of limit state design, it is necessary to check inter-
face shear stresses for two reasons.
I. It is necessary to ensure that, at the serviceability limit
state, the two concrete components act compositely.
Since shear stress can only be transmitted across the
interface after the in-situ concrete has hardened, the
loads considered when calculating the interface shear
stress at the serviceability limit state shoul~ consist
only of those loads applied after the concrete has hard-
ened. Thus the self weight of the precast unit and the
in-situ concrete should be considered in propped but
not in unpropped construction. In addition. at the ser-
viceability limit state it is reas~ahle to calculate the
interface shear stress by using elastic theory; hence
Vir = VS,llh,. (8.12)
CompreSSion Tension
-------&- - - -...
E;
~---------- ·1
-------------lI
1---e::~~~~~~~:..=.:t=-:.:::;-=============l -"""""Original
I positions
~!/
1 . "." .. ,..,." '"~'''
"I
I
I
I
I.I
I1--___-.£_---' __________________ ...1
I-----_____.!:.I?!!...-..____--.~
E; =free shrinkage of in-situ concrete
Eb/ = creep plus shrinkage of top of precast concrete
Ebb =creep plus shrinkage of bottom of precast concrete
Strains
Fig. 8.9 Differential shrinkage pIllS creep
I
= horizontal interface shear stress
= shear force at point considered
= first moment of area, about the neutral axis
of the transformed composite section, of the
concrete to one side of the interface
= second moment of area of the transformed
composite section
be = width of interface.
2. It is necessary to ensure adequate horizontal shear
strength at the ultimate limit state. The shear force per
unit length which has to be transmitted across the
interface is a function of the normal forces acting in
the in-situ concrete at the ultimate limit state. The lat·
ter forces result from the total desigll load at the ulti-
mate limit state. If a constant flexural stress of 0.4 feu
(see Chapter 5) is assumed at the point of maximum
moment, then the maximum nonnal force is
0.4Jeu bi hi where bi is the effective breadth of in-situ
concrete above the interface and hi is the depth of in-
situ concrete or the depth to the neutral:8xis if the
latter lies within the in-situ concrete. It is conservative
to assume that the normal force is zero at the point of
minimum moment, which will be considered to be dis-
tance I from the point of maximum moment. Hence an
interface shear force of 0.4feu bi hi must be transmit-
ted over a distance I (see Fig. 8.8); thus the average
interface shear stress is
(8.13)
A Technical Report of the Federation Internationale de
la Precontrainte [213] suggests that the average shear
stress should be distributed over the length I in pro-
portion to the vertical shear force diagram.
It can be seen from the above that, in order to be thorough,
two calculations should be carr.ied out:
I. An elastic calculation at the serviceability limit state
.which considers only those loads which are applied
after the in-situ concrete hardens.
Stresses
2. A plastic calculation at the ultimate limit state which
considers the total design load at the ultimate limit
state.
However, the Code does not require these two calcu-
lations to be carried out: instead a single elastic calculation
is carried out at the serviceability limit state which con-
siders the total design load at the serviceability limit state.
This is achieved by taking V in equation (8.12) to be the
shear force due to the total design load at the serviceability
limit state. This procedure, though illogical, is intended to
ensure that both the correct serviceability and ultimate
limit state criteria can be satisfied by means of' a single
calculation. '
Finally, as mentioned previously in this chapter, the
interface shear clauses in CP 110, which are essentially
identical to those in the Code, are being redrafted at the
time of writing. It is understood that the new clauses will
require the calculation to be carried out only at the ultimate
limit state by considering the total design load at this limit
state. This procedure, if adopted, would be more logical
than the existing procedure.
~Differential shrinkage
"When in-situ concrete is cast on an older precast unit,
much of the movement of the latter due to creep and shrink-
age will already have taken place, whereas none of the
shrinkage of the in-situ concrete will have occurred.
Hence, at any time after casting the in-situ concrete, there
will be a tendency for the in-situ concrete to shorten rela-
tive.to the precast unit. Since the in-situ concrete acts
compositely with the precast unit, the latter restrains the
movement of the former but is itself strained as shown in
Fig. If.9. Hence, stresses are developed in both the in-situ
and precast concretes as shown in Fig. 8.9. It is possible
to calculate the stresses from considerations of equili-
brium, and the necessary equations are given by Kajfasz, .
Somerville and Rowe [113].](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-61-2048.jpg)
![- 'l
Support
I itu concrete
~=ii$§~~~1Es22S£t:-Continuity reinforcement
(a) Beams supported on pier
'l
Support
I,
'95&S.-/ ity
fill reinforcement
ransversely pre-stressed
in-situ concrete crosshead
(b) Beams embedded in crosshead
Fig. 8.10(a),(b) Continuity in composite construction
it is emphasised that the above calculations need to ~e
carried out only at the serviceability limit state since the
stresses arise from restrained deformations and can thus be
ignored at tl)e ultimate limit state. The explanation of this
is given in Chapter 13 in connection with a discussion of
thermal stresses.
The Code does not give values of Yf'~ and Yt3 to be used
when assessing the effects of differential shrinkage at the
serviceability limit state; but it would seem to be reason-
able to use 1.0 for each.
The most difficult part of a differential shrinkage calcu-
lation is the assessment of the shrinkage strains of the two
concretes, and the creep strains of the precast unit. These
strains depend upon many variables and, if data from tests
on the concretes and precast units are not available, esti-
mated values have to be used. For beam and slab bridges
in a normal environment the Code gives a value of
100 x W'(' for the differential shrinkage strain, which is
defined as the difference between the shrinkage strain of
the in-situ concrete and the average shrinkage plus creep
strain of the precast unit. This value was based upon the
results of tests on composite T-beams reported by Kajfasz,
Somerv'ilIe and Rowe [113]. The test results indicated dif-
ferential shrinkage strains which varied greatly, but the
value quoted in .the Code is a reasonable value to adopt for
design purposes.
Although it is not stated in the Code, it was intended
that the current practice [6. 113J of ignoring differential
shrinkage effects in composite slabs, consisting of preten-
sioned beams with solid in-situ concrete infill, be con-
tinued.
Finally. the stresses induced by the restraint to differen-
tial shrinkage are relieved by creep and the Code gives a
reduction factor of 0.43. The derivation of this factor is
discussed later in this chapter.
110
Continuity
Introduction
A multi-span bridge formed of precast beams can be made
continuous by providing an in-situ concrete diaphragm at
each support as shown in Fig. 8.10(a). An alternative form
of connection in which the ends of the precast beams are
not "supported directly on piers but instead are embedded
in a transversely prestressed in-situ concrete crosshead,
with some tendons passing through the ends of the beams,
has been described by Pritchard [214] (see Fig. 8.10
(b».
A bridge formed by either of the above methods is stati-
cally determinate for dead load but statically indeterminate
for live load; and thus the in-situ concrete di!lphragm or
crosshead has to be designed to resist the hoggil'ng
moments which will occur at the. supports. The design
rules for reinforced concrete can be applied to the dia-
phragm, but consideration should be given to the following
points.
Moment redistribution
Tests have been carried out on half-scale models of con-
tinuous girders composed. of precast I-sections with an
in-situ concrete flange and support diaphragm, as in
Fig. 8.IO(a), at the Portland Cement Association in
America [215, 216]. It was found that, at collapse,
moment redistributions causing a reduction of support
moment of about 30% could be achieved. It thus seems to
be reasonable to redistribute moments in composite bridges
provided that the Code upper limit of 30% for reinforced
concrete is not exceeded.
'*Flexural strength
The Code permits the effect of any compressive stresses
due to prestress in the ends of the precast units to be
ignored when calculating the ultimate flexural -strength of
connections such as those in Fig. 8.10.
This recommendation is based upon the results of tests
carried out by Kaar, Kriz and Hognestad [215]. They car-
ried out tests on continuous girders with three levels of
prestress (zero, 0.42 fcy/ and 0.64 fey') and three percen-
tages of continuity reinforcemeflt (0.83, 1.66 and 2.49). It
was found to be safe to ignore the precompression in the
precast concre(e except for the specimens with 2.49% re-
inforcement. In addition, it was found that the difference
between the flexural strengths calculated by, first, ignoring
and. second, including the precompression was negligible
except for the highest level of prestress. Kaar, Kriz and
Hognestad thus proposed that the precompression be
ignored provided that the reinforcement does not exceed
1.5%. and the stress due to prestress does not exceed
0.4 fc)'1 (i.e. about 0.32.r'II)' Although the Code does not
quote these criteria, they will generally be met in prac-
tice.
In addition to the above tests. good agreement between
calculated and observed flexural strengths of continuous
connections involving inverted T-beams with ad.ded in-situ
concrete has been reported by Beckett [217).
Shear strength
The shear strength of continuity connections of the type
shown in Fig. 8.1O(a) has been investigated by Mattock
and Kaar [218]. who tested fifteen half-scale models. The
reinforced concrete connections contained no shear rein-
forcement, but none of the specimens failed in shear in the
connection. The actual failures were as follows: thirteen by
shear in the precast beams, one by flexure of a precast
beam and one by interface shear.
When designing, to resist shear, the end of a precast
beam which is to form part of a continuous composite
bridge, it should be rcmcmbercu that the end of the beam
will be subjected to hogging bending. Thus flexural cracks
could form at the top of the beam; consequently this region
should be given consideration in (he shear design.
Sturrock [219] tested models of continuity joints which
simulated the type shown in Fig.8.10(b). The tests
showed that no difficulty should be experienced in re-
inforcing the crosshead to ensure that a flexural failure,
rather than a shear failure in either the joint or a precast
beam away from the joint, would occur.
Serviceability limit state
Crack widths and stresses in the reinforced concrete dia-
phragm can be checked by the method discussed for re-
inforced concrete in Chapter 7.
Since the section in the vicinity of the ends of the pre-
cast beams is to be designed as reinforced concrete, it is
almost certain that tensile stresses will be developed at the
tops of the precast beams. If the beams are prestressed, the
Code implies that these stresses should not exceed the
allowable stresses, for the appropriate class of prestressed
member, given in Chapter 4, This means that no tensile
stresses are permitted in a Class 1 member: this seems
rather severe in view of the fact that any cracks in the
in-situ concrete will be remote from the tendons. Conse-
quently, the CP 110 handbook [112] suggests that the ends
of prestressed units, when used as shown in Fig. 8.10,
should always be considered as Class 3 and, hence, crack-
ing permitted at the serviceability limit state. :
Shrinkage and creep ,
The deflection of a simply supported composite beam
changes with time because of the effects of' differential
shrinkage and of creep due to self weight and prestress.
Hence, the ends of a simply supp0l1ed beam rotate as a
function of time, as shown in Fig. 8.11 (a), and, since
there is no restraint to the rotation, no bending moments
are developed in the beam. But, in the case of a beam
made continuous by providing an in-situ concrete dia-
phragm, as shown in Fig. 8.10, the diaphragm restrains the
rotation at the end of the beam and bending moments are
developed as shown in Fig. 8. II(b).
A positive rotation occurs at the end of the beam
because of creep due to prestress and thus a sagging
moment is developed at the connection. The sagging
moment is relieved by the fact that negative rotations occur
as a result of both differential shrinkage and creep due to
self weight, which cause hogging moments to be
(a) Simply supported
(b) Continuous
Fig. 8.11 Long-term effects
developed at the connection. Nevertheless, a net sagging
moment at the connection will generally be developed.
Since this sagging moment exists when no imposed load-
ing is on the bridge, bottom reinforcement, as shown in
Fig. 8.IO(a), is often necessary, together with the usual
top reinforcement needed to resist the hogging moment
under imposed loading. The provision of bottom rein-
forcement has been considered by Mattock [220].
In order to examine the influence of creep and shrinkage
on connection behaviour, Mattock [220] tested two con-
tinuous composite beams for a period of two years. One
beam was provided with bottom reinforcement in the dia-
phragm and could thus transmit a significant sagging
moment, whereas the other beam had no bottom re-
inforcement. The latter beam cracked at the bottom of the
diaphragm after about one year and the behaviour of the
beam under its design load was adversely affected.
Mattock found that the observed variation due to creep
and shrinkage of the centre support reactions could be pre-
dicted by the 'rate of creep' approach [198]. He thus sug-
gested that this approach should be used to predict the
support moment, which is of particular interest in design.
The 'rate of creep' approach, which is adopted in the
Code, assumes that under variable stress the rate of creep
at any time is independent of the stress history. Hence, if
the ratio of creep strain to elastic strain at time t is ~ and
the stress at time 1 is J, then the increment of creep strain
( bEe) in time &1 is given by
(8.14)
whereE is the elastic modulus. <In the continuity problem
under consideration, it is more convenient to work in terms
of moment (M) and curvature (t!); and thus, by analogy
with stress and strain
bt! = (MI El)b~ (8.15)
The effects of creep and shrinkage are discussed in
detail in reference [220]. In the following analyses, they
are treated less rigorously but with sufficient detail to illus-
trate the derivation of the relevant formulae in the Code.
Creep due to prestress The general method is illustrated
here by considering the two span continuous beam shown
in Fig. 8. 12. The beam has constant flexural stiffness and
III](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-62-2048.jpg)
![Maximum curvature = ~
/ M ~
if--~--~--""tJir----~i
L L..............._----....._._----_...-.j
Fi~. 8.12 Effect of creep on continuous beam
equal spans. The prestressing force is P and the maximum'
eccentricity in each span is e. Hence the maximum pre-
stressing moment in each span is Pe. At time t, let the
curvatures associnted with tl;.;;:;e moments be jJ and the
restraint moment at the internal support be M.
In time Of, the curvature'I' will increase due to creep by
6'1', where
ll~, =,,6~ =(Pe/El)o~
If the spans were freely supported, the change in end ro-
tation of span 1 at the internal support would be
k6,~ =k (Pe/El)o~
and of span 2 would be
-ko~, = -k (Pe/EI)6B
(8.16)
(8.17)
where k depends upon the tendon profile. For a straight
profile k = L, and for a parabolic profile k = 2L/3.
The rotation due to M if the spans were simply supported
would be ± 2MLl3EI with the negative and positive signs
being taken for spans 1 and 2 respectively. In time Of,
these rotations would change due to creep by
(-2MLl3E/) op
and
(+ 2MLl3E/)6P
(8.18)
(8.19)
Also in time Of, the restraint moment would change by
~M. The rotation, due to this change, at the ends of spans
I and 2 respectively would be
(- 20MLl3EI)
and
(+ 20MLl3El)
(8.20)
(8.21)
Since the two spans are joined at the support, their net
changes of rotation must be equal. The net change of ro-
tation for span 1 is obtained by summing expression (S.16),
(8.18)' and (S.20); and, for span 2, by sU1'Jlming expres-
sions (S. 17), (S.19) and (S.21). If the two net changes of
rotation are equated and the resulting equation re-
arranged, the following differential equation is obtained
elM + M = 3kPe
dl3 2
The right hand side of this equation is the sagging restraint
moment which would result, in the absence of creep, if the
beam were cast and prestressed as a monolithic continuous
beam. This moment will be designated M". Hence
dM
dP + M =Mr
The solution of this equation with the boundary condition
112
that, when t = 0, P= 0, M = 0 is
M = Mp [l - exp (-B)]
In the Code, the expression [1 - exp (- B)] is designated
<1>1; thus
M = M"CPI (S.22)
Although equation (8.22) has been derived for a two-span
beam, it is completely general and is applicable to any
span arrangement, provided th~t the appropriate value of
Mp is used. Hence, fol' any continuous beam, the restraint
moments at any time can be obtained by calculating the
restraint moments which would occur if the beam were
cast and prestressed as a monolithic continuous beam and
by then multiplying these moments by the creep factor <PI.
The ahove analysis implies that the prestress moment
(Pe) should be calculated by considering the prestressing
forces applied to the entire composite section. Hence the
eccentricity to be used should be that of the prestressing
force relative to the centroid ofthe composite section. This
is explained as follows. Consider the composite beam
shown in Fig. S.13. The eccentricities of the prestressing
force P are e'l and ec with respect to the precast beam and
composite centroids respectively. In time t after casting the
in-situ concrete, creep of the precast concrete will cause
the axial strain of the precast beam to change by (P/A,,)~
and its curvature to change by (Pe,,!EbI,,) B, where the sub-
script b refers to the precast beam. The in-situ concrete is
initially unstressed and thus does not creep. In order to
maintain compatibility between the two concretes it is
necessary to apply an axial force of P~ and a moment of
p..,,~ to the precast beam. However, since the composite
section must be in a state of internal equilibrium. it is now
necessary to apply a capcelling force of P~ and a cancel-
ling moment of Peb~ t< the composite section. The net
moment applied to the composite section and which pro-
duces a curvature of the composite section is thus
(PeI>B) + (PP) (ec - e,,) = PeeB
Hence. the eccentricity used should be that relative to the
centroid of the composite section. A more rigorous proof
is given in reference [220].
The creep factor Bis dependent upon a great number of
variables. Appendix C of Part 4 of the Code gives data for
the assessment of the following effects on ~ : relative
humidity, age at loading, cement content, water-cement
ratio, thickness of member and time under load. Unfortu-
nately, the basic data required to assess these effects are
not generally known at the design stage. For design pur-
poses, one is interested in 13ce, which is the value towards
which ~ would eventually tend. In the absence of more
precise data, Mattock [220] suggested that ~e(' should be
taken as 2. This value implies that the creep factor CPI to
be applied to the restraint moment due to creep is 0.S7. In
practice the value of ~C'e is likely to be between 1.5 and
2.5. and the adoption of the average value of 2 for B,.,.
implies a maximum error of 10% in the value of CPl.
If ~ is calculated from the data in Appendix C of the
Code, it should be remembered that its value should be
based on the increase in creep strain from the time that the
heam is made continuous by casting the in-situ concrete,
and not from the time of prestressing.
Precast concrete and composite construction
Precast /
/
Potential p,
~p ~~ ~Composite
centroid
Precast
centroid
Tendon
-- -- - - -- -- - --1-- ---
/
-----------r·----
/
/
---------~-------
~trai~--E----Pp~--------Pjf-1------ .I'P ._
~J-~~ - ~ -------. ------------
_.:1___ Pec~
centroid L-.._ _ _ _ _-L.._ _ _ __
Pebl3
Potential creep strains Stress resultants
applied to precast
section
Stress resultants
applied to
composite section
Net stress
resultants
applied at
composite
centroid
. Fig. 8.13 Effect of creep on composite section
Creep due to dead load By an analysis similar to that
given abpve, it can be shown that the restraint moment due
to creep under dead load is given by
M =M.A>I (8.23)
Md is the hogging restraint moment which would occur if
the beam were cast as a monolithic continuous beam; i.e.
the restraint moment due to the combined dead loads of the
precast beam and in-situ concrete applied to the composite
section. For the two-span beam considered in the last sec-
tion, Md is a hogging moment of magnitude (WL2/8) where
w is the total (precast plus in-situ) dead load per unit
length. CP1 can again be taken as 0.87.
Shrinkage Before considering the effects of shrinkage on
the restraint moments of a continuous beam, the relief of
shrinkage stresses, in general, due to creep is examined. In
the analysis which follows, it is assumed that the relation-
ships between creep strain per unit stress (specific creep)
and time, and between shrinkage strain (E.) and time, have
the similar forms shown in Fig_ 8.14, so that,
E., = K(E)f)
where K is a constant. Hence, using equation (8.14)
E, = KP/E and OE. = KoBlE (8.24)
Consider a piece of concrete which is restrained against
shrinkage so that a tensile shrinkage stress is developed.
At time t, let this stress bef. In time Ot, let.'the increase of
shrinkage be OE.. and the change of the shrinkage stress be
6[. Also in time Ot, there would be a potential creep strain
of (f/E)O~. Thus the net potential change of strain, which
results in a change of shrinkage stress, is
6E., - (f/E)b~
Hence, the change of stress is
or
df f E dEs
dff + = dB
Hence, using equation (8.24)
df
d~ + f= K
Shrinkage (~.)
and specific
creep (~/E)
strain
'Off< --------i~~~~----
~cc/E
Time(t)
Fig. 8.14 Specific creep-time and shrinkage-time curves
The solution of this equation with the boundary condition
that, when t = 0, ~ = 0, f = 0 is
f =K [1 - exp (- B)1 =K<I>I
Hence, using equation (S.24)
f =EE 11• P
In the Code, the expression CPl/~ is designated cP ; thus
f= EE,cp (8.25)
Hence, the shrinkage stress, at any time t, is the shrinkage
stress (EE.,) which would occur in the absence of creep
multiplied by the creep factor cp. If the limiting value (~,.c)
of ~ is again taken as 2, then cP = 0.43. This factor is
referred to earlier in this chapter in connection with the
relief of differential shrinkage stresses.
Differential shrinkage The general method is again illus-
trated by considering a two-span continuous beam which is
symmetrical about the internal support. Due to the differ-
ential shrinkage between the precast beam and the in-situ
flange, at any time t, there will be a constant curvature ('jJ)
imposed throughout the length of the beam, and the
change (o,,) of curvature in time Ot can be calculated by
considering Fig. 8.15. In time Ot, the differential shrink-
age strain will change by OE,. If it is required that the
precast beam and the flange stay the same length, it is
necessary to apply to the centroid of the flange a tensile
force of
of = OEs Ee! Ac!](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-63-2048.jpg)
![In-situ
flange
centroid
Precast
:nn'nn!:'t••4-------------------,---
centroid
8Facent
j---~
Potential shrinkage strain Force
applied
Force
applied
Net stress
resultants
applied at
composite
centroid
•to in-situ
flange
to
composite
section
lrig. 8.15 Differential shrinkage
where Act and Eel are the area and elastic modulus respec-
tively of the flange concrete. Since the composite section
must be in a state of internal equilibrium, it is now neCes-
sary to apply a cancelling compressive force of bF to the
composite section. This force has an eccentricity of ace"t
with respect to the centroid of the composite section.
Hence a moment bFace~t is effectively applied to the com-
posite section and this moment induces the curvature1jl.
Thus
6~, = bFacentlEl = bEsEctActaee,,/EI
where the EJ value is appropriate to the composite section.
If at time t, the restraint moment at the internal support is
M and it changes by bM in time bt, then an analysis,
identical to that presented earlier in this chapter for creep
due to prestress, results in the following differential equa-
tion
dM 3 dE
dj3 + M =2"E'i A~f aUnt d;
Using equation (8.24)
dM 3 K
dB +M = '2 EctActacmt E
The solution of this equation with the boundary condition
that, when t = 0, ~ = 0, E. = 0, M = 0 is
M= tEctActacent ~ [1 -exp(-~)]
Using equation (8.24)
M - 1. [1 - exp(-~)]
- 2 (EsEctActaw,,) ~
or
(8.26)
(8.27)
where Me., is the hogging restraint moment which would
occur in the absence of creep. Although equation (8.26)
has been derived for a two-span beam, equation (8.27) is
completely general and is applicable to any span arrange-
ment. The Code again assumes a value of 0.43 for <1>. For
design' purposes, one is interested in Ee/iff' which is the
value towards which E.• would eventually tend. Appendix
C of the Code gives data which enable the effects on
114
shrinkage of the following to be assessed: relative humid-
ity, cement content, water-cement ratio, thickness of
member and time. Hence, the differential shrinkage strain
can be assessed: a typical value would be about 200 X
10-6 • This value is much greater than the recommended
value of 100 X 10-6
quoted earlier in this chapter, when
differential shrinkage calculations for beam and slab
bridges were discussed. This is because the latter value
allows for the creep strains of the precast beam whereas,
when carrying out the restraint moment calculations, the
creep and shrinkage effects' are treated independently
(compare Figs. 8.9 and 8.15).
Finally, it should be noted that the Code states that Mcs
can be taken as
Me. = Ediff EctAct acent <I> (8.28)
for anilnternal support. This is appr.oximately correct for a
large number of spans (5 qr more) but it will underestimate
the restraint moment for, beams with fewer spans. In gen-
eral the value of Me. calculated from equation (8.28) needs
to be multiplied by a constant ~hich depends upon the
,span arrangement. Equation (8.26) shows that the constant
. ':.'js 3/2 for a two-span beam with equal span lengths.
:< Appropriate values for other numbers of equal length spans
are:
1. Three spans: 1.2 for both internal supports.
2. Four spans: 1.29 for first internal supports, 0.86 for
centre support.
3. Five or more spans: 1.27 for first internal supports,
1.0 for all other supports.
Values for unequal spans would have to be calculated
from first principles.
Combined effects of creep and differential shrinkage The
net sagging restraint moment due to creep under prestress
and dead load and due to differential shrinkage can be
obtained by summing equations (8.22), (8.23) and (8.27)
tl,vith due account being taken of sign. Hence '
.~;~
-; M =(M" - Me/)<I>1 - Mc,,<I> (8.29)
Examples of these calculations are given in [113], [216]
and [219]. The calculations need to be carried out only at
the serviceability limit state since the restraint moments are
_____ .:1 I
g
N
...
1000
+
+
+
+
+ + l'-_______+_______-/ J51 ~I ~
+ + + + + +
+ + ... + + +
Quarter span
Fig. 8.16 Composite beam cross-sections
due to imposed deformations and can be ignored at the
ultimate limit state.
Code flotation The notation adopted in the Code for the
creep factors is confusing. Appendix C of Part 4 of the
Code adopts the symbol <I> for the creep factor which, in
this chapter, has been given the symbol ~ (or ~cc after a long
period of time). However, the main body of Part 4 of the
Code uses ~c'(" <I> and <1>1 in the same sense as they are used
in this chapter. Thus care should be exercised when assess-
ing a creep factor from Appendix C of the Code for use in
composite construction calculations.
The notation adopted for the creep factors is also refer-
red to in Chapter 7 in connection with the calculation of
long-term curvatures and deflections.
'*Example - Shear in composite
construction .
A hridge deck consists of pretensioned precast standard
M8 beams at 1 m centres acting compositely with a
160 mm thick in-situ concrete top slab. The characteristic
strengths of the shear reinforcement to he designed and of
the precast and in-situ concretes are 250 N/mm2, 50
N/mm 2
and 30 N/mm2
respectively. Foq.r tendons are
deflected at the quarter points and the tendon patterns at
mid-span and at a support are shown in Fig. 8.16. The
total tendon force after all losses have occurred is
3450 kN. The span is 25 m, the overall beam length
25.5 m and the nominal values per beam of the critical
shear forces and moments at the support and at quarter-
span for load combination I are given in Table 8.1.
Design reinforcement for both vertical and interface
shear at the two sections.
Support
Table 8.1 Nominal values of stress resultants
Support Quarter span
Load
Shear force Moment Shear force Moment
(kN) (kNm) (kN) (kNm)
Self weight 163 0 81 763
Parapet 27 0
,
14 132
Surfacing 29 0 15 135
HB + associated
HA 332 0 196 1333
Section properties
The modular ratio for the in-situ concrete is /30/50 =
0.775. Reference [36] gives section properties for the pre-
cast and composite sections; these are summarised in the
upper part of Table 8.2. The composite values are based
upon a modular ratio of 0.8 which is only slightly differ-
ent to the correct value of 0.775.
Table 8.2 Section properties
Property
Area (mm2)
Height of centroid above bottom
fibre (mm)
Second moment of area (mm4)
First moment of a;ka above
composite centroid (mm")
First moment of area above
interface (mm")
Precast Composite·
393450
454 642
65.19 x 09 124.55>.< JOP
44.4 x OP 116.0 X lOR
,-----,----------](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-64-2048.jpg)
![Chapter 9
Substructures and
foundations
Introduction
The Code does not give design rules which are specifically
concerned with bridge substructures. Instead, design rules,
which are based upon those of CP 110, are given for col-
umns, walls (both reinforced and plain) and bases. In ad-
dition, design rules for pile caps, which did not originate in
CP 110, are given. The CP 110 clauses were derived for
buildings and, thus, the column and wall clauses in the
Code are also more relevant to buildings than to bridg~s.
In view of this, the approach that is adopted in this chapter
is, first, to give the background to the Code clauses and,
then, to discuss them in connection with bridge piers, col-
umns, abutments and wing walls. However, these struc-
tural elements are treated in general terms only, and, for a
full description of the various types of substructure and of
their applications, the reader is referred to [221].
The author anticipates that the greatest differences in
section sizes and reinforcement areas, between designs
carried out in accordance with the current documents and
with the Code, will be noticed in the design of substruc-
tures and foundations. The design of the latter will also
take longer because, as explained later in this chapter,
more analyses are required for a design in accordance with
the Code because of the requirement to check stresses and
crack widths at the serviceability limit state in addition to
. strength at the ultimate limit state. In CP 110, from which
the s.ubstructure clauses of the Code were derived, it is
only necessary to check the strength at the ultimate limit
state, since compliance with the serviceability limit state
criteria is assured by applying deemed to satisfy clauses.
The column and wall clauses of CP 110 were derived with
this approach in mind. The fact that the CP 110 clauses
have been adopted in the Code without, apparently, all~w
ing for the Code requirement that stresses and crack widths
at the serviceability limit state should be checked, has led
to a complicated design procedure. This complication is
mainly due to the fact that it has not yet been established
which limit state will govern the design under a particular
set of load effects. Presumably, as experience in using the
Code is gained, it will be possible to indicate the most
likely critical limit state for a particular situation.
118
Columns
r
*General
tJ{:..Definition
A column is not defined in the Code; but a wall is defined
as having an aspect ratio, on plan, greater than 4. Thus a
column can be considered as a member with an aspect
ratio not greater than 4.
..:J(;.Effective height
The Code gives a table of effective heights (Ie), in terms of
the clear height (10)' which is intended only to be a guide
for various end conditions..The table, which is here sum-
marised in Table 9.1, is based upon a similar table in
CP 110 which, in turn, WilS based upon a table in CP 114.
The effective heights have been derived mainly with
framed buildings in miri'd and do not cover, specifically,
the types of column which oCCur in bridge construction.
Indeed, in view of the variety of different types of articula-
tion which can occur in bridges, it would be difficult to
produce a table covering all situation.s. It would thus
appear necessary to consider each particular case individ-
ually by examining the likely buckling mode. In doing this,
consideration should be given to the way that movement
can be accommodated by the bearings, the flexibility of
the column base (and soil in which it is founded), and
whether the articulation of the bridge is such that the col-
umns are effectively braced or can sway. Some of these
aspects are discussed by Lee [222].
Table 9.1 Effective heights of columns
Column type
Braced, restrained in direction at both ends
Braced, partially restrained in direction at
one or both ends '
Unbraced or partially braced, restrained
in direction at one end, partially restrained
at other
Cantilever
/,J/"
0.75
0.75-1.0
1.0-2.0
2.0-2.5
N
Ultimate -- --------------------
Serviceability
I
1--_ _ _--'-1.__________..,...
1
Fig. 9.1 Lateral deflection
Reference is made in Table 9.1 to braced and unbraced
columns: the Code states that a column is braced, in a
particular plane, if lateral stability to the structure as a
whole is provided in that plane. This can be achieved by
designing bracing or bearings to resist all lateral forces.
~Slenderness limits
A column is considered to be short, and thus the effects of
its lateral deflection can be ignored, if the slenderness
ratios appropriate to each principal axis are both less than
12. The slenderness ratio appropriate to a particular axis is
defined as the effective height in respect to that axis
divided by the overall depth in respect to that axis. The
overall depth should be used irrespective of the cross-
sectional shape of the column.
If the slenderness ratios are not less than 12, the column
is defined as slender and lateral deflection has to be con-
sidered by using the additipnal moment concept which is
explained later. The limiting slenderness ratio is taken as
12 because work carried out by Cranston [223] indicated
that buckling is rarely a significant design consideration for
slenderness ratios less than 12. This work formed the basis
of theTP 110 clauses for slender columns.
It is possiSle for 'slender columns to buckle bey combined
lateral bending and twisting: Marshall [224] reviewed all
of the available relevant test data and concluded that lateral
torsional buckling will not influence collapse provided
that, for simply supporteg ends,
I" :s;;; 500 b2
/h
where h is the depth in the plane under consideration and b
is the width. Cranston [223] suggested that this limit
should be reduced, for design purposes, to
I,,:S;;; 250 b% (9.1)
Cranston also suggested the following limit for columns
for which one end is not restrained against twisting, and
this limit has been adopted in the Code for cantilever col-
umns.
(9.2)
In addition, the Code requires that I" should not exceed
60 times the minimum column thickness. This limit is
stipulated because Cranston~s study did not include col-
umns having a greater slenderness ratio. This. limit on 10 is,
generally, more onerous than that implied by equation
(9.1) and, thus, the latter equation does not appear in the
Code.
It should be noted that, for unbraced columns (which
frequently occur in bridges), excessive lateral deflections
can occur at the serviceability limit state for large slender-
ness ratios. This is illustrated in Fig. 9.1. An analysis,
which allows for lateral deflections, is not required at the
serviceability limit state and, c9nsequently, CP 110sug-
gests a slenderness ratio limit of 30 for unbraced columns.
This limit does not appear in the Code, but it would seem
prudent to apply a similar limit to bridge columns, unless
it is intended to consider, by a non-linear analysis, lateral
deflections at the se.rviceability limit state.
In the above discussion of slenderness limits, it is
implicitly assumed that the column has a constant cross-
section throughout its length. However, many columns
used for bridges are tapered: for such columns, data given
by Timoshenko [225] indicate that, generally, it is con·
servative to calculate the slenderness ratio using the aver·
age depth of column.
Ultimate limit state
Short column
'*Axial load Since lateral deflections can be neglected in. a
short column, collapse of an axially loaded column occurs
when all of the material attains the ultimate concrete com-
pressive strain of0.0035 (see Chapter 4). The design stress-
strain curve for the ultimate limit state (see Fig. 4.4)
indicates that, at a strain of 0.0035, the compressive stress
in the concrete is 0.45 ie,.. Similarly Fig. 4.4 indicates
that, at this strain, the compressive stress in the reinforce-
ment is the design stress which can ;'ary from 0.718 fy to
0.784Jy (see Chapter 4). The Code adopts an average
value of 0.75 fy for the steel stress. It is not clear why the
Code adopts the average value for columns, but a
minimum value of O.72fy for beams (see Chapter 5),
If the areas of concrete and steel are Ac and A.•r respec-
tively, then the axial strength of the column is .
N =0.45 fruAc + 0.75 ivA,,· (9.3)
The Code recognises me fact that some eccentricity of load
will occur in practice and, thus, the Code requires a
minimum eccentricity of 5% of the section depth to be
adopted. It is not necessary to calculate the moment due to
this eccentricity, since allowance for it is made by r,e-
ducing the ultimate strength, obtained from equation (9.3),
by about 10%. This reduction leads to the Code formula
for an axially loaded column:
N =0.4 fr,,Ac + 0.67 i,A.,r (9.4)
Axial load plus uniaxial bending When a bending
moment is present. three possible methods of design are
given in the Code:
1. For symmetrically reinforced rectangular or circular
columns, the design charts of Parts 2 and 3 of CP 10](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-66-2048.jpg)
![It N
I
-e= MIN
0.4fCI
Elevation
i
I
-IP'oo
I
I,
I
i
~
h ,.~
Fig. 9.2 Plain column section
[128. 130] may be used. These charts were prepared
using the rectangular - parabolic stress-strain curve
for concrete and the tri-linear stress-strain curve for
reinforcement discussed in Chapter 4. Allen [203]
gives useful advice on the use of the design charts.
2. A strain compatibility approach (see Chapter 5) can be
adopted for any cross-section. An area of reinforce-
ment is first proposed and then the neutral axis depth
is guessed. Since the extreme fibre compressive strain
is 0.0035, the strains at all levels are then defined.
Hence, the stresses in the various steel layers can be
determined from the stress-strain curve. The axial
load and bending moment that can be resisted by the
column can then be determined. These values can be
compared with the design values and, if deficient, the
area of reinforcement and/or neutral axis depth modi-
fied. The procedure is obviously tedious and is best
performed by computer.
3. The Code gives formulae for the design of rectangular
columns only. The formulae, which are described in
the next section. require a 'trial and error' design
method which can be tedious. An example of their use
is given by Allen [203]. Although the Code formulae
are for rectangular sections only, similar formulae
could be derived for other cross-sections.
In conclusion, it can be seen that a computer, or a set of
design charts, is required for the efficient design of col-
umns subjected to an axial load and a bending moment.
-fCode formulae Consider an unrein/orced section at col-
lapse under the action of an axial load (N) and a bending
moment (M). If the depth of concrete in compression is d,.,
as shown in Fig. 9.2, then by using the simplified reclan-
120
II
?M
1 III
I
• •
1A.2 A;,
• •
~d2 __._ .--1£.......-...._._..~d'
'II···_··_···········-F---·---..----.
I,
I
0.0035
I
~t---..dc •..._ ..-.j
Section
elevation
Section
plan
Strains
0.4fcu Stresses
O.72fy
,
1 Fe ~;
I
t.
Fig. 9.3(a),(b) Reinforced "column section
Stress
resultants
gular stress block for concrete with a constant stress of
0.4 feu:
Eccentricity (e) = MIN = (hI2) - (d,./2)
:. de =h-2e (9.5)
For equilibrium
N =0.4 feubdc =0.4 !c."b (h - 2e) (9.6)
Hence, only nominal reinforcement is required if the axial
load does not exceed the value of N given by equation
(9.6). However, it should be noted from this equation that,
when e > h12, N is negative. Hence, equation (9.6) should
not be used for e > h12; however. the Code specifies the
more conservative limit of e = h/2 - d I, where d I is the
depth from the surface to the reinforcement in the more
highly compressed face.
When N exceeds the value given by equation (9.6), it is
necessary to design reinforcement. At failure of a re-
inforced concrete column, the strains, stresses and stress
resultants are as shown in Fig. 9.3. It should be noted that
the Code now takes the design stress of yielding compres-
sion reinforcement to be its conservative value of 0.72 f}"
•M.
N
i
._._..__.....
My'
(a) Biaxial interactioli diagram
Fig. 9.4 Biaxial bending interaction''lilagraln
For equilibrium
N = 0.4 !c'l/hd,. + 0.72 t,A:,I +/,2A"'2 (9.7)
and, by taking moments about the column centre line.
M =0.2 !c,"bde (/1 - de) + O.72 fvA.~, (h/2 - d')
-J.~2A.'2 (h/2 - d2) (9.8)
These equations are difficult to apply because the depth
(tic) of concrete in compression and the stress (/,2) in the
reinforcement in the tension (or less highly compressed)
face are unknown. The design procedure is thus to assume
values of d,. and f,2' then calculate A.:, and A.,2 from equa-
tion (9.7). and check that the value of M calculated from
equation (9.8) is not less than the actual design moment. If
M is less than the actual design moment. the assumed val-
ues of d,. and /,2 should be modified and the procedure
repeated. Guidance on applying this procedure is given by
Allen [2031. However. the Code does not allow de to be
taken to be less than 2d I. From Fig. 9.3 it can be shown
that. at this limit, the strain in the more highly compressed
reinforcement is 0.00175, which is less than the yield
strain of 0.002 (see Chapter 4). However. serious errors in
the required quantities of reinforcement should~not arise by
assuming that the stress is always 0.72 fy. as in Fig. 9.3.
When the eccentricity is large (e > h/2 - d2) and thus the
reinforcement in one face is in tension, the Code permits a
simplified design method to be used in which the axial
load is ignored at first and the section designed as a beam.
.The required design moment is obtained by taking
.moments about the tension reinforcement. Thus. from
·Fig.9.3.
,M + N(d - h/2) = Fe (d - d,J2) + F,: (d - d') (9.9)
:The right-hand side of this equation is the ultimate moment
lof resistance (Mu) of the section when considered as a
'beam. Hence, the section can be designed. as a beam. to
resist the increased moment (Ma) given by the left-hand
side of equation (9.9). i.e.
Ma = M + N(d - h/2)
Now, d - h/2 = hl2 - d2•
(9. to) in the lorm
Mil = M + N(h12 - d2)
«l~~~5r:c~;,
and the Code gives equ~ti~n~ I
C' ",(9~j:'
''''''~
Actual
-'-"-Idealised
(.~'!.)lln + (My )an= 1
Mu. Muy
(b) Section ABC
If force equilibrium is considered.
N =Fc + F: - F"
or
F, =(Fc + F'.,) - N (9.12)
In the absence of the axial force N,
Fa =Fc + F,: =Fi,
where Fb is the tension steel force required for the section
considered as a beam. Hence. from equation (9.12). when
the axial force is present
F.r =Fb - N
The area of tension reinforcement is obtained by dividing
F" by the tensile design stress of 0.87 fy; hence
A., =Ab - NIO.87 fy (9.13)
wh~re A., and Ah are the required areas of tension re-
inforcement for the column section-and a beam respectively.
Hence. the section can be designed by, first. designing it
as a beam to resist the moment Mn from equation (9.]])
and then reducing the area .of tension reinforcement by
(!VIO.87 fy).
Axial load plus biaxial bending If a .column of known
dimensions and steel area is analysed rigorously. it is pos-
sible to construct an interaction diagram which relates fail-
ure values of axial load (N) and moments (Mx• My). about
the major and minor axes respectively. Such a diagram is
shown in Fig. 9.4(a),. where N"zrepresents the full axial
load carrying capacity'given by equation (9.3). A section,
parallel to the MxMy plane. through the diagram for a par-
ticular value of NIN"z would have the shape shown by
Fig. 9.4(b), where. assuming an axial load N. Mil.• and Mil}'
are the maximum moment capacities for bending about the
major and minor axes respectively.
The shape of the diagram in Fig. 9.4(b) varies according
to the value ofNINuz but can be represented approximately
by
(MxIMI/..>'''n + (MvlMlly)"'n = I (9.14)
(¥n is a function ofNINllz. Appropriate values are tabulated
in the Code.
·· ..When,designing a column subjected to biaxial bending.'
it is first necessary to assume a reinforcement area. Values
4~~~.:'~~., '. :.~.~~-,~t~~'~l.
~, .,', .". ",~;j~~it
,fr~ 121](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-67-2048.jpg)
![N
I
I
I
I
I
I
e I
-.!~
I
I
I
I
I
I
N
Initial
eccentricity
M; =Ne;
Initial
moment
Fig. 9.5 Additional moment
N
Additional
eccentricity
of N"" Mil.• and Mlly can then be calculated from first prin-
ciples or obtained from the design charts for uniaxial bend-
ing. It is then necessary to check that the left-hand side of
equation (9.14) does not exceed unity.
"*Slender columns
General approach When an eccentric load is applied to
any column, lateral deflections occur. These deflections
are small for a short column and can be ignored, but they
can be significant in the design of slender columns. The
deflections and their effects are illustrated in Fig. 9.5,
where it can be seen that the lateral deflections increase
the eccentricity of the load and thus produce a moment
(M"dd) which is additional to the primary (or initial)
moment (M;). Hence, the total design moment (M,) is
given by
M,= M;+ Madd
where
M;= Ne;
M"dtl = Neatld
(9.15)
(9.16)
(9.17)
e,. and e"dd are the ipitial and additional eccentricities
respectively.
Since the section design is carried out at the ultimate
limit state, it is necessary to assess the additional eccen-
tricity at collapse. The additional eccentricity is the .lateral
deflection. and the latter can be determined if the distri-
hution of curvature along the length of the column can be
calculated. The distribution of curvature for a column
subjected to an axial load and end moments is shown in
Fig. 9.6. where 'I'" is the maximum curvature at the centre
of the column at collapse. The actual distribution ofcurva-
ture depends upon the column cross-section, the extent of
cracking and of plasticity in the concrete and reinforce-
ment. However. it can-be seen. from Fig. 9.6. that it is
unconservative to assume a triangular distribution, and
conservative to assume a rectangular distribution. For
these two distributions the central deflections are given by,
respectively:
('"dd =1;'1',/12
('",Id = 1;11',,/8
122
Additional
moment
M N
'-V
Total
moment
Fig. 9.6 Curvature distributions
M,= M;+ Madd
--Actual
--- Conservative
--- Unconservative
, ,
Thus, Cranston [223] ~uggested that, for design pur-
poses. a reasonable value to adopt would be:
eadd =1;,p1l110 (9.18)
'jill can be determined if the strain distribution at col-
lapse can be assessed. It is thus necessary to consider the
mode of collapse. Unless the slenderness ratio is large, it
is unlikely that a reinforced concrete column will fail due
to instability, prior to material failure taking place [223].
Hence, instability is ignored initially and, for a balanced
section in which the concrete crushes and the tension steel
yields simultaneously, the strain distribution is as shown in
Fig. 9.7.
The additional moment concept used in the Code is based
upon that of the C.E.B. [226] in which the short-term con-
crete crushing strain (Eu) is taken as 0.0030. In order to
allow for long-term effects under service conditions, the
latter strain has to be multiplied by a creep factor which
Cranston [223J suggests should be conservatively taken as
l.25. Hence. Ell = 0.00375. The strain (E..) in the rein-
forcement is that appropriate to the design stress at the
ultimate limit state. Since the characteristic strength of the
reinforcement is unlikely to exceed 460 N/mm2 • E.. can be
conservatively taken as 0.87 x 460/200 x 103 = 0.002.
Hence, the curvature is given by
'jill = (0.00375 + 0.002)/d = O.00575id (9.19)
M
N
"'.
d
Fig. 9.7 Collapse strains for billanced section
It is now necessary to consider the possibility of an insta-
bility failure as opposed to a material failure. In such a
situation, the strains are less than their ultimate values and,
hence, the curvature is less than that given by equation
(9.19). The C.B.B. Code [226] allows for this by reducing
the curvature obtained from equation (9.19) by the follow-
ing empirical amount
le/50 000h2
It also assumed that d =11, so that the curvature is obtained
finally as
'I'" = (0.00575 -1)50 OOOh)/h (9.20)
If this curvature is substituted into equation (9.18), then
the following expression for the lateral deflection (or ad-
ditional eccentricity) is obtained
e,l/Itt = (hI1750)(/~/h)2 (1- 0'()035 [)h) (9.21)
It should be noted that for a slenderness ratio of 12 (i.e.
just slender), e"dd = O.OSh, but for the maximum permitted
slenderness ratio of 60, eadtl = 1.63h. Hence, for very
slender columns, the additional eccentricity and, hence,
the additional moment can be very significant in design
terms.
Minor axis bending If h is taken as the depth with respect
to minor axis bending. then the additional eccentricity is
given by equation (9.21). Hence. the total design moment,
which is obtained from equations (9.15), (9.17) and
(9.21). is given in the Code as
M, = M; + (Nh/1750) (/,.Ih)2 (1- O.00351)h) (9.22)
~M,
Loading Lateral Moments
deflection
Fig. 9.8 Effect of unequal end moments
1, is taken as the greater of the effective heights with
respect to the major and minor axes. It should be noted
that M/ should not be taken to be less than 0.05 Nh, in
order to allow for the nominal minimum initial eccentricity
of 0.05h.
The column should then be designed, by anyone of the
methods discussed earlier felr short columns, to resist the
axial load N and the total moment M"
In Fig. 9.5, the maximum additional moment occurs at
the same location as the maximum initial moment. If these
maxima do not coincide, equation (9.15) is obviously con-
servative. Such a situation occurs when the moments at the
ends of the column are different, as shown in Fig. 9.8.
To be precise, one should determine the position where
the maximum total moment occurs and then calculate the
latter moment. However, in order to simplify the calcu-
lation for a braced column, Cranston [223] has suggested
that the initial moment, where the total moment is a maxi-
mum, may be taken as
M; =0.4 M + 0.6 M2
but
M/ <t 0.4 M2
(9.23)
where M1 and M2 are the smaller and larger of the initial
end moments respectively. For a column bent in double
curvature. M is taken to be negative.
It is possible for the resulting total moment (M,) to be
less than Mi. In such '3 situation, it is obviously necessary
to design to resist M2 and thus M, should never be taken to
be less than M2•
For an unbraced column, the Code requires the total
moment to be taken as the sum of the additional moment
and the maximum initial moment. This can be very con-
servative for certain bridge columns which are effectively
fixed at both the base and the top, but which can sway
under lateral load or imposed deformations (e.g. tempera-
ture movement).
Major axis bending A column which is loaded eccentric-
ally with respect to its major axis can fail due to large
additional moments developing about the minor axis. This·
is because the slenderness ratio with respect to the minor
axis is greater than that with resp~ct to' the major axis.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-68-2048.jpg)
![N ~+----------4---+Y b
M;
x
h
Fig. 9.9 Major axis bending
Hence, with reference to Fig. 9.9 the column should be
designed for biaxial bending to resist the following
moments
M,x =M/ + (Nh/1750) (lexfh)2 (1- 0.0035Iexfh)
M,.I' =(Nb/1750) (ley/b)2 (1- O.0035Ieyfb)
(9.24)
(9.2~)
where M,.~ and M,y are the total moments about the major
(x) axis and minor (y) axis respectively, and lex and ley are
the effective heights with respect to these axes.
Cranston [223] has shown that, for a braced column with
II :t> 3b, it is conservative to design the column solely for
bending about the major axis, but the slenderness ratio
should then be calculated with respect to, the minor'axis.
Hence, in such situations, the Code permits the column to
be designed to resist the axial load N and the following
total moment about the major axis.
M, = Mi + (Nh/1750)(I)b)2 (1- 0.00351)b) (9.26)
where Ie is the greater of lex and ley.
Biaxial bending When subjected to biaxial bending, a
column should be designed to resist the axial load Nand
moments (Mx = MIX! My = M,y) such that equation (9.14)
is satisfied. The total moments about the major and minor
axes respectively are
M,x = Mix + (Nh/1750) (lexfh? (1- 0.0035 lex/h)
M,y = Miy + (Nb/1750) (ley/b)2 (1 - 0.0035 ley/b)
(9.27)
(9.28)
where Mix and Miy are the initial moments with respect to
the major and minor axes respectively.
Serviceability limit state
General
The design of columns in accordance with the Code is
complicated by the fact that it is necessary to check stress-
es and crack widths at the serviceability limit state in
addition to carrying out strength calculations·at the ulti-
mate limit state.
fStresses
At the serviceability limit state, the compressive stress in
the concrete has to be limited to 0.5 fm and the reinforce-
ment stresses to 0.8 f).. Thus, for an axially loaded col-
124
J'
.....
O.~
~O.5fcu
O.87fJ '
~O.4fcu
(a) Section
,(b) Limiting stresses
at serviceability
limit state
(c) Limiting stresses
at ultimate limit
state
Fig. 9.10(a)-(c) Stress comparison
umn, the design resistance at the serviceability limit state
is, usually, '
Ns =0.5 fcuAc + (0.5 fcuEs/Ec)Asc
This value generally exceeds the design resistance at the
. ul'timate limit state, as given by equation (9.4). Since the
design load at the ultimate limit state exceeds that at the
serviceability limit state, it can be seen that ultimate will
generally be the critical limit state when the loading is pre-
dominantly axial.
When the loading is eccentric to the extent that one face
is in tension, the stress conditions at the ultimate and ser-
viceability limit states will be as shown in Fig. 9.10. Since
the -average concrete stress at the serviceability limit state
(0.25 feu) is much less than that at the ultimate limit state
(0.4 feu), it is likely th,at, with regard to concrete stress, the
serviceability limit state will be critical.
*Crack widths
The Code considers that if a column is designed for an
ultimate axial load in excess of 0.2 fcuAn it is unlikely that
flexural cracks will occur. For smaller axial loads, it is
necessary to check crack widths by considering the column
to be a beam and by applying equation (7.4). From Table
4.7, it can be seen that, since a column could be subjected
to salt spray, the allowable design crack width could be as
small as 0.1 mOl. Hence, equation (7.4) implies a maxi-
mum steel strain of about 1000 X 10-6 , or a stress of
about 200 N/mm2
• For high yield steel, this stress is equi-
valent to about 0.48/1' Hence, when crack control is con-
sidered, the reinforcement stress in Fig. 9.IO(b) is limited
to much less than 0.81", and thus crack control could be
the critical design criterion for columns with a large eccen-
tricity of load.
Summary
Ultimate is likely to be the critical limit state for a column
which is either axially loaded or has a small eccentricity of
load. However, due to the fact that either the limiting
compressive stress or crack width could be the critical
design criterion, serviceability is likely to be the critical
limit state for a column with a large eccentricity of load. It
appears that, in order to simplify design, studies should be
carried out with a view to establishing guidelines for iden-
tifying the critical limit state in a particular situation.
Reinforced concrete walls
General
Definitions
Rctuining wulls, wing w!llIs and similar structures which,
primm'i!y, are subjected to bending should be considered
us slabs and designed in accordance with the methods of
Chapters 5 and 7. The following discussion is concerned
with walls subjected to significant axial loads.
In terms of the Code, a reinforced concrete wnll is a
vertical load-bearing member with an aspect ratio, on plan,
greater than 4; the reinforcement is assumed to contribute
to the strength, and has an area of at least 0.4% of the
cross-sectional area of the wall. This definition thus covers
reinforced concrete abutments. The limiting value of 0.4%
is greater than that specified in CP 114 because tests have
shown that the presence of reinforcement in walls reduces
the in-situ strength of the concrete [227]. Hence,under
axial loading, a plain concrete wall can be stronger than a
wall with a small percentage of reinforcement.
Slenderness
The slenderness ratio is the ratio of the effective height to
the thickness of the wall.
A short wall has a slenderness ratio less than 12. Walls
with greater slenderness ratios are considered to be slen-
der.
In general, the slenderness ratio of a braced wall should
not exceed 40, but, if the area of reinforcement exceeds
1%, the slenderness ratio limit may be increased to 45.
These values are more severe than those' for columns
because walls are thinner than columns, and thus deflec-
tions are more likely to lead to problem~. If lateral stability
is not provided to the structure as a whole, then a wall is
considered to be unbraced and its slenderne~s ratio should
not exceed 30. This rule ensures that deflections will not
he excessive.
The above slenderness limits were obtained from
CP 110 and were thus derived with shear walls and in-fill
panels in framed structures in mind. They are thus not
necessarily applicable to the types of wall which are used
in bridge construction. However, the slenderness ratios
should not result in any further design restrictions com-
pared with existing practice.
Analysis
The Code requires that forces and moments in reinforced
concrete walls should be determined by elastic analysis.
When considering bending perpendicular to an axis in
the plane of a wall, a nominal minimum eccentricity of
0.05h should be assumed. Thus a wall should be designed
for a moment per unit length of at least 0.05 nwh where nw
is the maximum load per unit length.
Ultimate limit state
Short walls
'*Axialload An axially loaded wall should be designed in
accordance with equution (9.4).
'*Eccentric loads If the load is eccentric such that it pro-
duces bending about an axis in the plane of a wall, the
wall should be designed on a unit length basis to resist the
combined effects of the axial loud per unit length and the
bending moment per unit length. The design could be car.
ried out either by considering the section of wall as an
eccentrically loaded column of unit width or by using the
'sandwich' approach, described in Chapter 5, for design-
ing against combined bending and in-plane forces.
If the load is also eccentric in the plane of the wall, an
elastic analysis should be carried out, in the plane of the
wall, to determine the distribution of the in-plane forces
per unit length of the wall. The Code states that this
analysis may be carried out assuming no tension in the
concrete. In fact, any distribution of tension and com.
pression, which is in equilibrium with the applied loads
could be adopted at the ultimate limit state since, as ex-
plained in Chapter 2, a safe lower bound design would
result.
Each section along the length of.. the wall should then be
designed to resist the combined effects of the moment per
unit length at right angles to the wall and the compression,
or tension, per unit length of the wall. The design could be
carried out by considering each section of the wall as an
eccentrically loaded column or tension member of unit
width, or by using the 'sandwich' approach.
Slender walls
The forces and moments acting on a slender wall should be
determined by the same methods previously described for
short walls. The pol1ion of wall, subject to the highest
intensity of axial load, should then be designed as a slen-
der column of unit width.
Serviceability limit state
Stresses
The comments made previously regarding stress calcu-
lations for columns are also appropriate to walls.
Crack widths
Walls should be considered as slabs for the purposes of
crack control calculations, and the details of the Code
requirements are discussed in Chapter 7.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-69-2048.jpg)
![Plain concrete walls
"*General
A plain concrete wall or abutment is defined as a vertical
load-bearing member with an aspect ratio, on plan, greater
than 4; any reinforcement is not assumed to contribute to
the strength.
If the aspect ratio is less than 4, the member should be
considered as a plain concrete column.. The following.
design rules for walls can also be applied to columns, but,
as indicated later, certain design stresses need modifi-
cation.
The definitions of 'short', 'slender', 'braced' and
'unbraced', which are given earlier in this chapter for re-
inforced concrete walls, are also applicable to plain con-
crete walls.
The clauses, concerned with slenderness and lateral
support of plain walls, were taken directly from CP 110
which in tum were based upon those in CP 111 [228]. In
order to preclude failure by buckling the slenderness ratio .
of a plain wall should not exceed 30 [229]. The effective
heights given in the Code are summarised in Table 9.2.
Table 9.2 Effective heights of plain walls
Wall type
Unbraced. lateralty spanning structure at top
Unbraced, no laterally spanning structure at top
Braced against lateral movement and rotation
Braced against lateral movement only
* l., = distance between centres of support
I 1" = distance between a support and a free edge
1.5
2.0
0.75* or 2.0t
1.0* or 2.5t
In order to be effective, a lateral support to a braced
wall must be capable of transmitting to the structural ele-
ments, which provide lateral stability to the structure as a
whole, the following forces:
I. The static reactions to the applied horizontal forces.
2. 2.5% of the total ultimate vertical load that the wall
has to carry.
A lateral support COUld be a horizontal member (e.g., a
deck) or a vertical member (e.g., other walls), and may be
considered to provide rotational restraint if one of the fol-
lowing is satisfied:
I . The lateral support and the wall are detailed to provide
bending restraint.
2. A deck has a bearing width of at least two-thirds of
the wall thickness, or a deck is connected to the wall
by means of a bearing which does not allow rotation
to occur.
~. The wall supports, at the same level, a deck on each
side of the wall.
Forces
Members, which transmit load to a plain wall, may be
considered simply supported in order to calculate the re-
action which they transmit to the wall.
126
I.. h
~I
h-2e"
Fig. 9•.11 Eccentrically loaded short wall'at collapse
If the load is eccentric in the plane of the wall, the
eccentricity and distribution of load along the wall should
be calculated from statics. When calculating the distribu-
tion of load (i.e., the axial load per unit length of wall),
the concrete should be assumed to resist no tension.
If a number of walls resist a horizontal force in their
plane, the distributions of load between the walls should
be in proportion to their relative stiffnesses. The Code
clause concerning horizontal loading refers to shear con-
nection between walls ahd was originally written for
CP 110 with shear walls..in buildings in mind. However,
the clause could be applied, for example, to connected
semi-mass abutments.
When considering eccentricity at right angles to the
plane of a wall, the Code states that the vertical load
transmitted from a deck may be assumed to act at one-third
the depth of the bearing area back from the loaded face. It
appears from the CP 110 handbook [112] that this
requirement was originally intended for floors or roofs of
buildings bearing directly on a wall. However, the inten-
tion in the Code is, presumably, also to apply the require-
ment to decks which transmit load to a wall through a
mechanical or rubber bearing.
Ultimate limit state
Axial load plus bending normal to wall
Short braced wall The effects of lateral deflections can
be ignored in a short wall and thus failure is due solely to
concrete crushing. The concrete is assumed to develop a
constant compressive stress of "Awfcu at collapse, where "A...
is a coefficient to be discussed. The concrete stress dis-
tribution at collapse of an eccentrically loaded wall is as
shown in Fig. 9. t t. The centroid of the stress block must
design criterion, serviceability is likely to be the critical
limit state for a column with a large eccentricity of load. It
appears that, in order to simplify design, studies should be
carried out with a view to establishing guidelines for iden-
tifying the critical limit state in a particular situation.
Reinforced concrete walls
General
Definitions
Retaining wulls, wing walls and similar structures which,
primarily, are subjected to bending should be considered
as slabs and designed in accordance with the methods of
Chapters 5 and 7. The following discussion is concerned
with walls subjected to significant axial loads.
In terms of the Code, a reinforced concrete wall is a
vertical load-bearing member with an aspect ratio, on plan,
greater than 4; the reinforcement is assumed to contribute
to the strength, and has an area of at least 0.4% of the
cross-sectional area of the wall. This definition thus covers
reinforced concrete abutments. The limiting value of 0.4%
is greater than that specified in CP 114 because tests have
shown that the presence of reinforcement in walls reduces
the in-situ strength of the concrete [227]. Hence,under
axial loading, a plain concrete wall can be stronger than a
wall with a small percentage of reinforcement.
Slenderness
The slenderness ratio is the ratio of the effective height to
the thickness of the wall.
A short wall has a slenderness ratio less than 12. Walls
with greater slenderness ratios are considered to be slen-
der.
In general, the slenderness ratio of a braced wall should
not exceed 40, but, if the area of reinforcement exceeds
1%, the slenderness ratio limit may be incre,ased to 45.
These values are more severe than those for columns
because walls are thinner than columns, and thus deflec-
tions are more likely to lead to problems. If lateral stability
is not provided to the structure as a whole, t~en a wall is
considered to be unbraced and its slenderness ratio should
not exceed 30. This rule ensures that deflections will not
be excessive.
The above slenderness limits were obtained from
CP 110 and were thus derived with shear walls and in-fill
panels in framed structures in mind. They are thus not
necessarily applicable to the types of wall which are used
in bridge construction. However, the slenderness ratios
should not result in any further design restrictions com-
pared with existing practice.
Analysis
The Code requires that forces and llIoments in reinforced
concrete walls should be determined by elastic analysis.
When considering bending perpendicular to an axis in
the plane of a wall, a nominal minimum eccentricity of
0.05h should be assumed. Thus a wall should be designed
for a moment per unit length of at least 0.05 nwh where nw
is the maximum load per unit length.
Ultimate limit state
Short walls
Axialload An axially loaded wall should be designed in
accordance with equation (9.4).
Eccentric loads If the load is eccentric such that it pro-
duces bending about an axis in the plane of a wall, the
wall should be designed on a unit length basis to resist the
combined effects of the axial load per unit length and the
bending moment per unit length. The design could be car-
ried out either by considering the sec.tion of wall as an
eccentrically loaded column of unit width or by using the
'sandwich' approach, described in Chapter 5, for design-
ing against combined bending and in-plane forces.
If the load is also eccentric in the plane of the wall, an
elastic analysis should be carried out, in the plane of the
wall, to determine the distribution of the in-plane forces
per unit length of the wall. The Code states that this
analysis may be carried out assuming no tension in the
concrete. In fact, any distribution of tension and com-
pression, which is in equilibrium with the applied loads
could be adopted at the ultimate limit state since, as ex-
plained in Chapter 2, a safe lower bound design would
result.
Each section along the length of.. the wall should then be
designed to resist the combined effects of the moment per
unit length at right angles to the wall and the compression,
or tension, per unit length of the wall. The design could be
carried out by considering each section of the wall as an
eccentrically loaded column or tension member of unit
width, or by using the 'sandwich' approach.
Slender walls
The forces and moments acting on a slender wall should be
determined by the same methods previously described for
short walls. The po~ion of wall, subject to the highest
intensity of axial load, should then be designed as a slen-
der column of unit width.
Serviceability limit state
Stresses
The comments made previously regarding stress calcu-
lations for columns are also appropriate to walls.
Crack widths
Walls should be considered as slabs for the purposes of
crack control calculations, and the details of the Code
requirements are discussed in Chapter 7.](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-70-2048.jpg)
![Plain concrete walls
General
A plain concrete wall or abutment is defined as a vertical
load-bearing member with an aspect ratio, on plan, greater
than 4; any reinforcement is not assumed to contribute to_..···
the strength. ",,' #..-'-" '"'~.
If the aspect ratio is less than 4, the member should be
considered as a plain concrete column. The following
design rules for walls can also be applied to columns, but,
as indicated later, certain design stresses need modifi·
cation.
The definitions of 'short'. •slender' , 'braced' Ilnd
'unbraced', which are given earlier in this chapter for reo
inforced concrete walls, are also applicable to plain con·
crete walls.
The clauses, concerned with slenderness and lateral
support of plain walls, were taken directly from CP 110
which in tum were based upon those in CP 111 [228]. In
order to preclude failure by buckling the slenderness ratio
of a plain wall should not exceed 30 [229]. The effective
heights given in the Code are summarised in Table 9.2. .
Table 9.2 Effective heights of plain walls
Wall type
Unbraced, laterally spanning structure at top
Unbraced, no laterally spanning structure at top
Braced against lateral movement and rotation
Braced against lateral movement only
* 10 = distance between centres of support .
I I" = distance between a support and a free edge
1.5
2.0
0.75~ or Z.ot
1.0* or 2.5t
In order to be effective, a lateral support to a braced
wall must be capable of transmitting to the structural ele-
ments, which provide lateral stability to the structure as a
whole, the following forces:
I. The static reactions to the applied horizontal forces.
2. 2.5% of the total ultimate vertical load that the wall
has to carry.
A lateral support COUld be a horizontal member (e.g., a
deck) or a vertical member (e.g., other walls), and may be
considered to provide rotational restraint if one of the fol-
lowing is satisfied:
1. The lateral support and the wall are detailed to provide
bending restraint.
2. A deck has a bearing width of at least two-thirds of
the wall thickness, or a deck is connected to the wall
by means of a bearing which does not allow rotation
to occur.
3. The wall supports, at the same level, a deck on each
side of the wall.
Forces
Members, which transmit load to a plain wall, may be
considered simply supported in order to calculate the re-
aciion which they transmit to the wall.
126
I .'~,
I
i nw
,
I,
I
...'
I....
I
I,
I
f4---.---'-h'------""'i~1
","~__h_-2_e"-,,'_.~
Fig. 9.11 Eccentrically loaded short wall at collapse ..
If the load is eccentric in the plane of the wall, the
eccentricity and distribution of load along the wall should
be calculated from statics. When calculating the distribu-
tion of load (Le., the axial load per unit length of wall),
the concrete should be assumed to resist no tension.
If a number of walls resist a horizontal force in their
plane, the distributions of load between the walls should
be in proportion to their relative stiffnesses. The Code
clause concerning horizontal loading refers to shear con-
nection between walls aqd was originally written for
CP 110 with shear walls ,in buildings in mind. However,
the clause could be applied, for example, to connected
semi-mass abutments.
When considering eccentricity at right angles to the
plane of a wall, the Code states that the vertical load
transmitted from a deck may be assumed to act at one-third
the depth of the bearing area back from the loaded face. It
appears from the CP 110 handbook [112] that this
requirement was originally intended for floors or roofs of
buildings bearing directly on a wall. However, the inten-
tion in the Code is, presumably, also to apply the require-
ment to decks which transmit load to a wall through. a
mechanical or rubber bearing.
Ultimate limit state
Axial load plus bending normal to wall
Short braced wall The effects of lateral deflections can
be ignored in a short wall and thus failure is due solely to
concrete crushing. The concrete is assumed to develop a
constant compressive stress of "•.feu at collapse, where "w
is a coefficient to be discussed. The concrete stress dis-
tribution at collapse of an eccentrically loaded wall is as
shown in Fig. 9.11. The centroid of the stress block must
i' (a) Braced-code (b) Braced
Fig. 9.12(a)-(c) Lateral deflection of a slender wall
coincide with the line of action, of the axial load per unit
length of wall (nw), which is at an eccentricity of ex. Hence
the depth of concrete in compression is
2(h12 - ex) =h - 2ex
Thus the maximum possible value of nw is given by
(9.29)
The coefficient "w varies from 0.28 to 0.5. It is tabulated
in the Code and depends upon the following:
I. Concrete strength. For concrete grades less than 25,
lower values of "ware adopted than f<?r concrete
grades 25 and above. This is because of the difficulty
of controlling the quality of low grade concrete in a
wall. Hence, essentially, a higher 'value of Ym is
adopted for low grades than for high grades.
2. Ratio of clear height between supports to wall length.
Tests reported by Seddon [229] have shown that the
stress in a wall at failure increases as its height to
length ratio decreases. This is because the base of the
wall and the structural member(s) bearing on the wall
restrain the wall against lengthwise expansion. Hence,
a state of biaxial compression is induced in the wall
which increases its apparent strength above its uni-
axial value. The biaxial effect decreases with distance
from the base or bearing member and, thus, the aver-
age stress, which can be developed in a wall,
increases as the height to length ratio decreases.
CP III permits an increase in allowable stress which
varies linearly from 0%, at a height to length ratio of
1.5, to 20%, at a ratio of 0.5 or less. Similar increases
have been adopted in the Code.
ea ex1
nw
Lateral ~
load I
I
/
I,
I,
I,
I,
(c) Unbraced
3. Ratio of wall length to thickness. It is not clear why
the Code requires "'IV to be reduced, when the ratio of
wall length to thickness is less than4 (Le., when the
wall becomes a column). The reduction coefficient
varies linearly from 1.0 to 0.8 as the length to thick-
ness ratio reduces from 4 to I: The reason could be to
ensure that the value of "w does not exceed 0.4 when
the aspect ratio is 1, because 0.4 is the value adopted
for reinforced concrete columns and beams.
Slender braced wall At the base of a wall, the eccen-
tricity of loading is assumed to be zero. Thus the eccentricity
varies linearly from zero at the base to ex at the top. A
slender wall deflects laterally under load in the same man-
ner as a slender column. The lateral det1ection increases
the eccentricity of the load and the Code takes the net
maximum eccentricity, to be (0.6ex +eo), as shown in
Fig. 9.12(a). The additional eccentricity (eo) is taken,
empirically. to be 1;/2500h, where I, and h are the effec-
tive height and thickness of the wall respectively. It should
be noted that the Code mistakenly gives the additional
eccentricity as 1./2500h. If ex in equation (9.29) is replaced
by (O.6ex + eo). the following equation is obtained for the
ultimate strength of a slender braced wall:
(9.30)
The above assumption of zero eccentricity at the base of
a braced wall is based upon considerations of walls in
buildings [II2}. In the case, for example, of an abutment
an eccentricity could exist at the bottom of the wall as .
shown in Fig. 9. 12(b). If the eccentricities at the top and
bottom are ed and e,.2 respectively. the author would sug-
127](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-71-2048.jpg)
![I+-h/2 N
V I
N I V
)ii21
I.
Fig. 9.13 Shear normal to wall
gest that, by analogy with equation (9.23), the maximum
net eccentricity should be taken as the greater of >
(0.4 ext + 0.6 ex2 + eu)
and
(0.6 e.tl + 0.4 ex}. + ea)
The. appropriate net eccentricity should then be substituted
for e.r in equation 9.29.
Slender unbraced wall The lateral deflection of a slender
unbrl,lced wall is shown in Fig. 9.12(c). The net eccen-
tricities, fro,!! the wall centre line. at the top and bottom of
the wall are e.,! and (e.r 2 + ea) respectively. .The Code
requires every section of the wall to be capable o.f resisting
the load at each of these eccentricities. Hence, by repla-
cing ex in equation (9.29) by each of these eccentricities,
the ultimate strength of a slender unbraced wall is the
lesser of:
11",.= (h - 2ex l) f...wt.." (9.31)
and .
II ... =(h - 2ex2 - 2ea) f...w/Cl( (9.32)
-t.Shear
In general the total shear force in a horizontal plane should
not exceed one-quarter of the associated vertical load. The
reuson for this requirement is not clear but, since the
requirement was taken from CP 110. it was intended pre-
sumably for shear walls bearing on a footing or a floor.
Thus it appears that the design criterion was taken to be
shear friction with a coefficient of friction of 0.25.
A shear force at right angles to a wall arises from a
change in bending moment .down the wall. The maximum
moments at the ends of a wall occur when the load is at its
greatest eccentricity of h12. The maximum change of
i2R
moment over the length of the wall occurs when the eccen-
tricities at each end are of opposite sign, as shown in
Fig. 9.13, and is given by N(h12 + hI2)= Nh. Hence, the
constant shear force throughout the length of the wall is
V =Nhll,.
In order that V does not exceed 0.25N, it is necessary that
l)h should exceed 4. In fact, the Code states that it is not
necessary to consider shear forces.normal to the wall ifl,/h
exceeds 6. The Code is thus conservative in this respect.
When considering shear forces in the plane of the wall,
it is necessary to check that the total shear force does not
exceed 0.25 of the associated total vertical load, and that
the average shear stress does not exceed 0.45 N/mm 2 for
concrete of grade 25 01' above, or 0.3 N/mm2 for lower
grades of concrete. The reason for assigning these allow-
able stresses is not apparent.
e(Searing
The bearing stress under a localised load should not exceed
the limiting value given by equation (8.4).
Serviceability limit state
"*Deflection
The Code states that excessive deflections will not occur in
a cantilever wall if its height-to-Iength ratio does not
exceed 10. The basis of this criterion is not apparent, but
the CP 110 handbook [112] adds that the ratio can be
increased to 15 if tension does not develop in the wall
under lateral loading.
-tCrack control
It is necessary to control cracking due to both applied load-
ing (flexural cracks) and the effects of shrinkage and
temperature.
Flexural cracking Reinforcement, specifically to control
flexural cracking, only has to be provided when tension
occurs over at least 10% of the length of a wall, when
subjected to bending in the plane of the wall. In such situ- .
ations, at least 0.25% of high yield steel or 0.3% of mild
steel should be provided in the area of wall in tension: the
spacing should not exceed 300 mm. These percentages are
identical to those discussed in the next section when con-
sidering the control of cracking due to shrinkage and
temperature effects. The spacing of 300 mm is in accor-
dance with the maximum spacing discussed in Chapter 7.
Shrinkage and temperature effects In order to control
cracking .due to the restraint of shrinkage and temperature
movements, at least 0.25% of high yield steel or 0.3% of
mild steel should be provided both horizontally and verti-
cally. These percentages are identical to those for water-
retaining structures in CP 2007 [230], but it should be
noted that they are much Jess than those given in the new
standard for water retaining structures (BS 5337) [231],
and are also much less than those. suggested by Hughes
[1851. The author would thus suggest that the values of
0.25% and 0.3% should be used with caution.
Bridge piers and columns
Hurlier in this chapter, the Code clauses concerned with
c()lumns and reinforced walls are presented and brief men-
tion lTude of their application. In the following discussion
the design of bridge piers and columns in accordance with
the Code i$ considercd briefly and compared with present
practice.
*Effective heights
The Code clauses concerning effective heights are
intended. primarily, for buildings and are not necessarily
applicahle to bridge pier!; and column!;. However, this criti-
cism is equally applicable to the effective heights given
in the existing design document (CP 114). Thus, there is
flO difference in the a!;sessment of effective heights in
accordance with the Code and with CP 114.
Slender columns and piers
('r 114 defines a slender column as one with a slenderness
ratio in excess of 15. whereas the Code critical !;lenderness
ratio is 12. This means that some columns. which could be
considered to he short at present, would have to be con-
sidered as slender when designed i'n accordance with the
C()de.
ep 114 allows for slenderness by applying a reduction
factor to the calculated permissible load for a short column.
The redlll,tion factor is a function of the slenderness ratio.
This upproach is simple. but does not reflect the true
Ix~haviour of a slender column at collapse. Thus the reduc-
tion fac!!',r approach has not been adopted in: the Code:
inst('ad. the additional moment concept. which is described
earlier in this chapter. is used. Use of the latter concept
requires more lengthy calculations. and thus the design of
slender {'olumns. in accordance with the Cod~, will take
longer than their design in accordance with CP 114.
Design procedure
In accordance with CP I 14. ollly olle calculation has to be
IIndertaken-- the permissible load has to he checked under
working load ('onditions. However. in accordance with the
Code;' three calculations. each under a different load con-
dition. have to be carried out. These calculations are con-
cerned with strength at the ultimate limit state. stresses at
the serviceahility limit stilte and. if appropriate, crack
width at the serviceability limit state. Hence, the design
procedure will he much longer for II column designed in
accordance with the Code.
A further problem arises when applying the Code: it is
not clear in advance which of the three design calculations
will be critical. However, it is likely that ultimate will be
the critical limit state for a column, which is either axially
loaded or is· subjected to a relatively small moment. For
columns subjected to a large moment, either the limiting
concrete compressive stress at the serviceability limit state,
or the limiting crack width at the serviceability limit state
could be critical. If the latter criterion is critical then it
may be necessary to specify columns with greater cross-
sectional areas than are adopted at present. This is because
a very large amount of reinforcement would be required to
control the cracks. For a column size currently adopted,
the required amount of reinforcement may exceed the maxi-
mum amount permitted by the Code. This possibility is
increased by the fact (see Chapter 10) that the maximum
amount of reinforcement permitted in a vertically cast col-
umn is 6% in the Code as compared with 8% in CP 114.
Bridge abutments and wing walls
The design of abutments and wing walls in accordance
with the Code is very different to their design to current
practice. A' major difference is the number of analyse!!
which need to be carried out. At present a single analysis
covers all aspects of design but, in accordance with the
Code, five analyses. each under a different design load.
have to be carried out for the following five design aspects:
I. Strength at the ultimate limit state.
2. Stresses at the serviceability limit state.
3. Crack widths at the serviceability limit state; but
deemed to satisfy rules for bar spacing are appropriate
in some situations (see Chapter 7).
4. Overturning. The Code requires the least restoring
moment due to unfactored nominal loads to exceed the
greatest overturning moment due to the design loads
(given by the effects of the nominal loads multiplied
by their appropriate YfL values at· the ultimate limit
state).
5. Factor of safety against sliding and soil pressures due
to unfactored nominal loads in accordance with
CP 2Q04 [92J.
A further important difference in design procedures
occurs when considering the effects of applied defor-
mations described in the Code and in the present documents.
In the latter. all design aspects are considered under work-
ing load conditions. and thus the effects of applied defor-
mations (creep. shrinkage and temperature) need to he
considered for all aspects of design. However. as
explained in Chapter 3, the- effects of applied defor-
mations can be ignored under collapse conditions. Thus Part
4 of the Code pemlits creep, shrinkage and temperature
effects to he ignored at the ultimate limit state. The impli-
cation of this is that less main reinforcement would be
required in an abutment designed to the Code than one
designed to the existing documents.
Although the effects of applied deformations can he
129](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-72-2048.jpg)
![ignored at the ultimate limit state, they have to be con-
sidered at the serviceability limit state. The effects of
applied deformations thus contribute to the stresses at the
serviceability limit state. Since less reinforcement would
be present in an abutment designed to the Code than one
designed to the existing documents, the stresses at the ser·
viceability limit state would be greater in the former abut·
ment. However, it is unlikely that they would exceed the
Code limiting stresses of 0.8t" and O.S /,." for reinforce~
men! and concrete respectively.
The main bar spacings will generally be greater for
abutments designed in accordance with the Code than for
those designed in accordance with the present documents.
This is because the Code maximum spacing of ISO mm
(see Chapter 7) will generally be appropriate for abut·
ments, whereas spacings of about 100 mm are often neces·
sary at present. The Code should thus lead to less,conges-
tion of main reinforcement.
Finally, the Code does permit the use of plastic methods
of analysis, and the design of abutments and wing walls is
an area where plastic methods could usefully be applied.
In particular, Lindsell [2321 has tested a model abutment
with cantilever wing walls. and has shown that yield line
theory gives reasonable estimates of the loads at collapse
of the abutment and of the wing walls, An alternative plas-
tic method of design is the Hillerborg strip method, which
is applied to an abutment in Example 9.2 at the end of this
chapter.
Foundations
General
A foundation should be checked for sliding and soil bear-
ingpressure in accordance with the principles of CP 2004
1921. The latter document Is written in terms of working
stress design and thus unfactored nnminalloads should be
used when checking sliding and soil bearing pressure.
Unwever. when carrying out the structural design of a
foundation. the design loads appropriate to the various
limit stutes should be adopted. Uence. more calculations
hav!.' to be cltrried out when designing foundations in
Ilcclll'dance with the Code ihan for those designed in
accordance with the current documents,
III .thc absence of a more accurate method. the Code
permits the u!lual assumption 01' a linear distribution of
hearing stress under a foundation.
Footings
Ultimate limit state
,.'lou"(1 The critical secticlll for hending ill taken at the
fal'e of the column or wall as shown in Fill. 9,14(a), Re-
inf()rcement should he designed for the total moment at this
l'riticul se<.'tion and. except for·the reinforcement parallel to
the shorter side of a rectanllular footing. it should be
l.lO
I,
I,
I
---d-I
I.) Flexure
,
I,
I,
~-..-.-.-..-.t-.,
I,
~....,D I,
I,
I,
Iotd~
Ibl Flexural shear
Critical
aactlona
Crltlca'
section.
( 1.Sh 'b-._-_..__l-- ~ectlon'-0--' .Critical
I i ' :1
............ /'.
leI Punching shear
Fig. 9.14(.)-(t) Crilical sections for footing
spread uniformly across the base. Reinforcement parallel
to the shorter' side should be distributed as !lhown in
Fig, 9.1!1i. The latter requirement ill empirical and was
based upon a lIimilar requirement in the ACI Code 116RI,
The Code is thus more precise than CP 114 with regard to
the distribution of reinforcement.
Flexural shtar The total shear on a section. at a distance
equal to the effective depth from the face of the column or
wall (see Fig. 9.14(b». should be checked in accordance
with the method given in Chapter 6 for flexural shear in
beams. These requirements, when allowance is made for
the different dellign loads. are very similar to those of
BE 1/73.
It is worth mentioning that the Code clause is identical
to that in CP 110. except that the latter document require!!
~________._.!.L.____ ,.~
:I
I
I
I
I
I
I
I
I
" I" 'I,' ," ~ •
~.............._-toJ-_..............................
(~;~~.~) ..~~ (r~1!l) A.
h '" overall slab depth
I
I
I
I
I
I
I
I
I
I
... .'. . ,. Plreaifot"" ,,.
.14
~ reinforcement
(~1 -1) A.
~-;Ti 2"
A." total area of reinforcement parallel to shorter side
1~1" /1 .._
Central band width
Fig. 9.15 Distribution of reinforcement in rectangular footing
the critical section to be at Ilh times the effective depth
from the face. This critical section was adopted in CP 110
because a critical section, at a distance equal to the effec-
tive depth from the loaded face, would have resulted in
much deeper foundations than those previously required in
accordance with CP 114.
Punching shear The critical perimeter and design method
discussed for slabs in Chapter 6 should be used for foot-
ings. The perimeter is shown in Fig, 9.14(c),
'*serviceability limit state
Stresses It is necessary to restrict the stresses to the limit-
ing values of 0.81y and 0.5 fe" in the reinforcement and
concrete respectively.
Crack widths As discussed in Chapter 7, footings should
be treated as slabs when considering crack control.
Piles
The Code does not give specific design rules, for piles.
However, once the forces acting on a pile have been assess-
ed, the pile can be designed as a column in accordance
with CP 2004 and the Code.
Pile caps
Ultim~te limit stat~
The reinforcement in a pile cap may be designed either by
bending theory or truss analogy. The shear strength then
has to be checked. . .
Bending theory When applying the bending theory
[233], the pile cap is considered to act as a wide beam in
each direction. The total bending moment at any section
Concrete
strut
1..oo"'=!--,I,..4~-Reinforcement
~~-i-"""_Q"" tie
Fig. 9.16' Ttussanalogy for piJecap
can be obtained at that section, and the total amount of
reinforcement at the section determined from simple bend-
ing theory as described in Chapter S. Such a design
method is not correct because a pile cap acts as a deep.
rather than a shallow, beam; however, the method has
been shown by tests to result in adequate designs [234].
This is probably because most pile caps' fail in shear and
the method of design of the main reinforcement is, largely,
irrelevant [234]. The total I1mount of reinforcement cal-
culated at a' section should be unifonnly distributed across
the section.
Truss analogy The truss analogy assumes a strut and tie
system within the cap, and is in the spirit of a lower bound
method of design. The strut· and tie system for a four-pile
cap is shown in Fig. 9.16. Formulae for detennining the
forces in the ties for various arrangements of piles are
given by Allen [203] and Yan [235]. It can be seen from
Fig. 9,16 that, because of the assumed structural action,
the reinforcement, calculated from the tie forces, should be
concentrated in strips over the piles. However, since it is
considered good practice to have some reinforcement
throughout the cap, the Code requires 80% of the rein-
forcement to be concentrated in strips joining the piles and
the remainder to be unifonnly distributed throughout the
cap.
Tests carried out by Clarke [234] have demonstrated the
adequacy of the truss analogy,
Flexural shear The Code requires flexural shear to be
checked across the full width of a cap at a section at the
face of the coluf!1n, as shown in Fig, 9.17(a). It should be
noted that the critical section is not intended to coincide
with the actual failure plane, but is chosen merely because
it is convenient for design purposes.
The question now arises as to what allowable design
shear stress should be used in association with the above
critical section. Tests carried out by Clarke [234] have
indicated that the basic design stresses given in Table 6.1
should be used, except for those partsof the critical sec-
tion which are crossed by flexural reinforcement which is
fully anchored by passing over a pile. For the latter parts
of the critical section, the basic design shear stresses
should be enhanced to allow for the increased shear resis-
tance due to the short shear span (see Chapter 6). The
enhancement factor (2dfav) where d is the effective depth
and av , is the shear span which, in the present context. is](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-73-2048.jpg)
![C~~ical section
r./___~_____?' / '
q 1
) .
--- ~-::-- - - -
Enhance Vc
__ -T-::;-- -- over these
I I lengths
, /'
p.---~----
/ -,
I " .... ,J
2600kN
280kN
Bearing which
permits only
rotation
d~ X _14 P _I . __,_.._ "_"" --<.,.."...._,
d ;.. effective
depth of
cap
(a) Flexural shear
#
,,/..U' / Critical
~,> '/_...----+- section
. / I '
'< I I
, ,/
~ ~---
(b) Punching shear-Code
(c) Punching shear-actual
Fig; 9.17(a)-(c) Shear in pile caps
taken as the distance between the face of the column and
the nearer edge of the piles, viewed in elevation, plus 20%
of the pile diameter. The Code states that the reason for
adding 20% of the pile diameter is to allow for driving
tolerances. However, Clarke [234] has suggested the same
additional distance in order to allow for the fact that the
piles are circular, rather than rectangular, and thus the
'average' shear span is somewhat greater than the clear
distance between pile and critical section. The value of
20% of the pile diameter, chosen by Clarke [234], is simi-
lar to the absolute value of 150 mm suggested by Whittle
and Beattie [233] to allow for dimensional errors. How-
ever, the result is that the allowable design stress varies
along the critical section, as shown in Fig. 9.17(a), and
the total shear capacity at the section should be obtained
by summing the shear capacities of the component' parts of
the section. '
It is understood that, in a proposed amendment to
CP 110, the critical section for flexural shear is located at
20% of the diameter of the pile inside the face of the pile.
Thus the critical section is at the distance av , defined in
Fig 9.17(a), from the face of the column. This criticalsec-
tion is more logical than that defined in the Code, but the
132
8m
O,6m
H
I,m D
Fig. 9.18 Bridge column
resulting shear force at the critical section will be only
marginally different.
Punching shear Clarke [:~34] suggests that punching of
the column through the cap:' need only be considered if the
pile spacing exceeds four times the pile diameter. which is
unlikely; thus the Code only requires punching of a pile
through the cap to be considered.
The critical section given in the Code for punching of a
comer pile is extremely difficult to interpret and originates
from the 1970 CEB recommendations [226]. The relevant
diagram in the latter document shows that the correct
interpretation is as shown in Fig. 9. 17(b). The Code does
not state what value of allowable design shear stress
should be used with the critical section. In view of this,
the author would suggest using the value from Table 6.1
which is appropriate to the average of the two areas of
reinforcement which pass over the pile, This suggestion is
not based upon considerations of the Code section of
Fig. 9.17(b) but of the section which would actually occur
as shown in Fig. 9.17(c). The basic shear stress, obtained
from Table 6.1 should then be enhanced by (2dlav), where
. av should be taken as the distance from the pile to the
critical section (I.e. dI2). Thus, in all cases in which failure
could occur along the Code critical section. the enhance-
ment factor would be equal to 4.
It is understood that, in a proposed amendment to
CP 110, punching shear is checked by limiting the shear
stress calculated on the perimeter of the column to
0.8/hu' This limiting shear stress is very similar to the
maximum nominal shear stress of 0.75 JTcu which is
specified in the Code (see Chapter 6).
Serviceability limit state
Realistic values of stresses and crack widths in a pile cap,
at the serviceability limit state, could only be assessed by
carrying out a proper analysis; such an analysis would
probably need to be non-linear to allow for cracking. Since
it is difficult to imagine serviceability problems arising in a
cap which has been properly designed and detailed at the
ultimate limit state, a sophisticated analysis at the ser-
viceability limit state cannot be justified. Thus, the author
would suggest ignoring the serviceability limit state criteria
for pile caps.
Examples
'"*9.1 Slender column
A reinforced concrete column is shown in Fig. 9.18. The
loads indicated are design loads at the ultimate limit state.
Design reinforcement for the column, at the ultimate
limit state, if the characteristic strengths of the reinforce-
ment and concrete are 42S N/mm2 and 40 N/mmD
respec-
tively. Assume that the articulation of the deck is (a) such
that sidesway is prevented and (b) such that sidesway can
occur.
·'f:.No sidesway
With sidesway prevented, the column can be considered to
be braced. Consider the column to be partially restrained
in direction at both ends, and, from Table 9.1, take the
effective height to be the same as the actual height, I.e.. Ie
= 8 m.
Slenderness ratio = 8/0.6 = 13.3. This exceeds 12, thus
the column is slender.
Assume a minimum eccentricity of O.OSh = 0.03 m for
the vertical load.
Initial moment at top of column = Ml = 0'
Initial moment at bottom of column = M2
= 280 x 8 + 2600 x 0.03 = 2318 kNm
Since the column is braced, the initial moment to be added
to the additional moment is, from equation (9.23),
Mi = (0.4)(0) + (0.6)(2318) = 1391 kNm
From equation (9.22), the total moment is
M, =
1391 + (2600 x 0.6/1750)(13.3)2(1 - 0.0035 x 13.3)
= 1391 + 150 = 1541 kNm
However,this moment is less than M2 , thus design to
resist
M, = M2 = 2318 kNm.
M/bh2 = 2318 x 108
/(1200 x 6002) = 5.37 N/mm2
Nlhh = 2600 x 03/(1200 x 600) = 3.61 N/mm2
Assume 40 mm bllrs with 40 mm cover in each face, so
that d/h = 540/600 = 0.9.
Substructures and foundations
Hence use Design Chart 84 of CP 110: Part 2 [128]: from
which
100 As,!bh = 2.8
:. Asc = 2.8 x 600 x 1200/100 = 20160 mml
Use 16 No. 40 mm bars (20160 mm2) with 8 bars in each
face.
:Jf.Sidesway
Since sidesway can occur, consider the column to be a
cantilever with effective height equal to twice the actual
height, i.e. Ie = 16 m.
SJr.ndenless ratio == 16/0.6 = 26.7
The initial and additional moments are both maximum at
the base. Hence,
M; = 23,18 kNm and, from equation (9.22), the total
moment IS
M, =
2318 + (2600 x 0.6/1750)(26.7)2 (1 - 0.0035 x 26,7)
= 2318 + 576 =2894 kNm !
Mlbh2
=2894 x 106
/(1200 x 6002
) = 6.70 N/mm2
Nlbh = 3.61 N/mm2
(as before)
From Design Chart 84 of CP 110: Part 2,
100 As,!bh = 3.6
:. Asc = 3.6 x 600 x 1200/100 = 25 920 mml
Use 22 No. 40 mm bars (27 '720 mml) with II bars in
each face.
. It should be noted that the above designs have been car-
ried out only at the ultimate limit state. In an actual
design, it' would be necessary to check the stresses and
.crac~ widths at the serviceability limit state by carrying out
elastic analyses of the sections.
'¥ 9.2 Hillerborg strip method applied to an
abutment ..
A reinforced concrete abutment is 7 m high and 12 m
wide. At each end of the abutment there is a wing wall
which is structurally attached to the abutment.
The lateral loads acting on the abutment are the earth
pressure, which varies from zero at the top to 5H kN/m2
at a depth H; HA surcharge, the nominal value of which is
10 kN/m2
(see Chapter 3); and the HA braking load
which acts at the top of the abutment and may be taken t~
have a nominal value of 30 kN/m width of abutment.
The Hillerborg strip wethod will be used to obtain a
lower bound moment field for the abutment. .
At the ultimate limit state, the design loads (nominal
load x YfL x YfJ) are:
Earth pressure 5H X 1.5 x 1.15 = 8.625H kN/m2
HA surcharge = 10 x 1.5 x 1.10 = 16,5 kN/m2
HA braking = 30 x 1.25 x 1.10 = 41.25 kN/m
The wing walls and abutment base are considered to
provide fixity to the abutment, which will thus be designed
as if it were fixed on three sides and free on the fourth.
The load distribution is chosen to be as shown in Fig. 9.19
(see also Chapter 2). Thus at the top of the abutment all of
the load is considered to be carried in the y direction; at the
centre of the base, all of the load is considered to be car-
ried in the x direction; in the bottom comers, the load is](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-74-2048.jpg)
![,....1..;,.6;.,;;.5,;.;,kN;..;.;/-{'m;.;...2__.... 41.25kN/m
2S.1kN/m2
4m
7m
---~-r--I
." 1~=11..1
.' '_I~ f~1
2.5m tSm
51.0kN/m2
---,--r-B~----~~~--------~~--~--4-~~B 68.3kN/m2
1m 1.5m
76.9kN/m2
t~-~: ~
Pressure distribution
- - Free edge
~Flxededge
-- - Load dispersion line
---- Zero moment line
--- Typical strip'
'(ii-'/'.(ii!: Strong edge band
Fig. 9.19 Abutment
"':467kNm~' . .~-467kNm
~+374kNm
Fig. 9.20 Strip AA
considered to be shared equally between the x and ydirec-
tions. It is emphasised that any distribution of load could
be choseiland that shown in Fig. 9.19 iSinerely one pos-
sibility.
. In order .that the resulting moments do not depart too
much from the. elastic. moments. and thus serviceability
problems do not arise, the zero moment lines shown in
Fig. 9.19 are chosen.
Typical strips AA, BB, CC, DD and EE of unit width
are now considered.
StripAA
Tile loading and bending moments are shown in Fig. 9.20.
Strip 88
The loading and bending moments ate shown in Fig. 9.21.
Strip CC
. ." . .
Strip EE. which carries the braking load. earth pressure
and surcharge at the top of the abutment,must also pro-
134
34.1SkN/m 34.15kN/m
~~~--~
6m
-137kNm~ , ~-137kNm
~------:::;;;>
+17kNm
Fig. 9.21 Strip BB
0.5 m+______
O.5m
-¥-------
3m
51.0kN/m-¥-._---
1.5m
1.5m
76.9kN/m
Fig. 9.22 Strip CC
.Ji=12:4kN
-187kNm
+43kNm
vide a reaction to strip CC. Hence. the loading and bend-
ings moments for strip CCare shown in Fig. 9~22; The
reaction (R) can first be obtained by taking moments about
the point of zero moment, and then the bending moment
diagram can be calculated.
"~..,.,,
- -_._---_.__.
O.5m R=6.2kN
O.5m
3m
25.5kN/m
1.5 III
1.5m
....f. "........-....-.----..- ~m:'r77
38.5kN/m
Fi;. 9.23 Strip DD
6.2kN/m 12.4kN/m 6.2kN/m
e ooeocoo
25.1 +41.25 = 66.35kN/m
~ . .~
2m1m 6m. 1m2m
i4=-~----- "I' 'I. iI
+22kNm
Reactions from
K direction strips
Earth pressure,
surcharge
and braking
.,~7~~~,~~~ /1-763kNm
". ~
'.' ..: +627*Nm .
l<'lg. 9.24 Strip EE
Strip DO
Strip DD is similar to strip CC and its loading and bending
moments are shown hi Fig. 9.23.
Strip EE
Strip EE acts as a strong edge band (1 m wide) which not
only supports the surcharge, earth pressure and braking
loads but also supports the ends of typical strips CC and
DO. Thus the loading and bending moments are as shown
in Fig. 9.24. The loading is taken, conservatively, as that'
at a depth of 1 m.
*9.3 Pile cap
Design the four-pile cap shown in Fig. 9.25 if the charac·
teristic strengths of the reinforcement and concrete are
425 N/mm2
and 30 N/mm2
respectively. The design load
at the ultimate limit state is 5200 kN.
Load ~r pile = 5200/4 = 1300 kN
The ca will be designed by both bending theory and truss
analog methods.
"fBendingtheory
Bending
Total bending moment at column centre line
::: 2 x 1300 x 0.75 = 1950 kNm
Assume effective depth = d = 980 mm
From equation (5.7), the lever arm is
O.4m
O.76m
O.76m
O.4m
Substructures and foundations
f
~
0.3m I
----~.3~ --l-~--r------~·--
"....,..........,~
/ .,I
I -I- I
I
, I
',_...-;'
I
~ m___-tI .....--.....
0.3 m 0.3 mO.2 ml "
I I
I I+-'
i /I ' .... ...-;1"
I
~~~O.74m~+~-0~.7~6~m~·4+~-~O.~76~m--~+~O.4m~
16200kN
1.1 m
Fla. 9.25 Pile cap
(
/ 5 x 1950 X lOll )
Z =0.5 x 980 1 + vI - 30 x 2300 x 9802
=943 mm
But maximum allowable z = 0.95d = 0.95 x 980
= 931 mm
Thus z = 931 mm
From equation (5.6), required reinforcement area is
As = 1950 x 108
/(0.87 x 425 x 931) = 5665 mm'
Use 19 No. 20 mm bars' (5970 mm2
)
Flexural shear
100 A/bd = (100 x 5970)/(2300 x 980) =0.26
From Table 5 of Code, allowable shear stress without
shear reinforcement = Vi" = 0.36 N/mm2
• This stress may
be enhanced by (2dla,,) for those parts of the critical sec-
tion il1dicated in Fig. 9.17(a).
a" = 200 + 0.2 x 500 = 300 mm
Enhancement factor = 2 x 980/300 = 6.53
Enhanced Vc = 6.53 x 0.36 = 2.35 N/mm2
Shear capacity of critical section
= «2)(2.35)(500) + (0.36)(2300 - 2 x 500)] 980 x lO-a
= 2760 kN
Actual shear force = 2 x 1300 = 2600 kN < 2760 kN
:. O.K.
135](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-75-2048.jpg)
![Punching shear
Tile critical section shown in Fig. 9.17(b) would occur
under the column in this example. Thustake critical section
11t' cotner of column, as shown in Fig. 9:25, at
[(0.75 - 0.3)/2 - 0.25] = 0.386 m from pile. The latter
value will be assumed for avo
. .' From Fig. 9.17(b), length of perimeter is 980 + 500
= 1480 mm
As for flexutal shear~. v,: = Q,36 N/mm 2 .
Enhancement factor ~ (2 x 980/386) = 5.08
Enhanced Vc = 5.08 x 0.36 = 1.83 N/mm2
Shear capacity of critical sectiQn
= 1;83 x 1480 x 980 x 1O~3 = 2650 leN
Actual shear force = 1300 kN< 2650 kN :. O.K.
. :¥. Truss analogy
Truss
FQr equilibrium, the force in each of the reinforcement ties
of Fig. 9.16 is NIISd.
= (5200 x 1500)/(8 x 980) = 995 kN
Required reinforcement area is
A" = 995 x lOa/(0.87 x 425) = 2690 mm2
Since there at two ties in each direction, the total re~
inforcement area in each direction is 2 x 2690 =
5380 mm
2
• It can be seen that the truss theory requires less
reinforcement than the bending theory, and this is gener~
ally the case.
80% of the tie reinforcement should be provided over the
piles, i.e.
0.8 x 2690 = 2150 mm2
Use 7 No. 20mm bars (2200 mm2
) over the piles.
The remaining 20% (540 mm2) should be placed between
136
the cap centre line and the piles, i.e. 2 x 540 ::::'
1080 mm2
should be placed between the piles. Use 4 No.
20 mm bars (1260 mm2
) between the piles.
'*Flexural shear
Over a pile, 100 Aslbd = (100 x 2200)/(500 x 980) .
= 0.45.
From Table 5 of Code, Vc = 0.51 N/mm2
Enhancement factor = 6.53 (as for bending theory)
Enhanced Vc = 6.53 x 0.51 = 3.33 N/mm2
Between piles, 100 AJbd = (100 x 1260)/(1000 x 980)
=0.13.
From Table 5 of Code, Vc = 0.35 N/mm2
Assume no reinforcement outside piles, thus
Vc = 0.35 N/mm2
Shear capacity of critical section
=[(2)(3.33)(500) + (0.35)(2300 - 2 x 500)] 980 X 10"a
= 3710 kN' .
Actual shear force = 2 x 1300 = 2600 kN < 3710 kN
:. O.K.
*Punching shear
Length of critical section = 1480 mm (as for bending
theory)
As for flexural shear, Vc'= 0.51 N/mm2
Enhancement factor = 5.08 (as for bending theory)
Enhanced Vc = 5.08 x 0.51 = 2.59 N/mm2 .
Shear capacity of critical section
= 2.59 x 1480 x 980 x lO-a = 3760 kN
Actual shear force = 1300 kN < 3760 kN :. O.K.
It c~m be seen from the above calculations that the truss
theory design results in a greater shear capacity than does
the bending theory design.
Chapter 10
Detailing
Introduction
In this chapter, the Code clauses, concerned with con-
siderations affecting design details for both reinforced and
prestressed concrete, are discussed and compared with
those in the existing design documents.
Table 10.1 Nominal covers
Conditions of exposure
Nominal cover (mm) for
concrete grade
25 30 40 ~50
Moderate
Surfaces sheltered from severe 40
rain and against freezing
whilst saturated with water,
e.g.
( I) surfaces protected by a
waterproof membrane;
(2) internal surfaces whether
subject to condensation
or not;
(3) buried concrete and
concrete continuously
under water
Severe
(I) Soffits
(2) Surfaces exposed to
driving rain, alternate
wetting and drying; e.g.
in contact Vith back-fill
and to freezing whilst wet
Very severe
30 25 20
30 25
(I) Surfaces subject to the NtA 50* 40* 25
effects of de-icing salts
or salt spray, e.g. roadside
structures and marine
structures
(2) Surfaces exposed to the NtA NtA 60 50
action of sea water with
abrasion or moorland water
having a pH of 4.5 or less.
* Only applicable if the concrete has entrained air (see text)
Reinforced concrete
*'Cover
The cover to a particular bar should be at least equal to the
bar diameter, and is also depen.dent upon the exposure con-
dition and the concrete grade as shown by Table 10.1.
These values are very similar to those given in Amend-
ment 1 to BE 1173: however, there are two important dif-
ferences.
First, the Code considers all soffits to be subjected to
severe exposure conditions, whereas BE 1173 distinguishes
between sheltered soffits and exposed ·soffits. Thus. for the
soffit of a slab between precast beams, the Code would
require, for grade 30 concrete, a minimum cover of
40 mm, whereas BE 1173 would require only 30 mm.
Hence, top slabs in beam and slab construction may need
to be thicker than they are at present.
Second, for roadside structures subjected to salt spray
and constructed with grade 30 or 40 concrete, the Code
requires the concrete to have entrained air. BE 1173 does
not have this requirement. The Code thus requires a
dramatic change in current practice. The footnote to Table
10.1 appears in the Code with a reference to Part 7 of the
Code. However, Part 7 refers only to the permitted vari-
ation in specified air content without giving the latter, but
Clause 3.5.6 of Part 8 of the Code does specify air contents
for various maximum aggregate sizes.
Bar spacing
if·Minimum distance between bars
For ease of placing and compacting concrete, the Code
relates the minimum distance between bars to the maxi-
mum aggregate size. The Code clauses were taken from
CP 116 and are thus more detailed than those in CP 114,
although they are very similar in implication.
I.n addition to rules for single bars and pairs of bars, the
Code -gives rules for bundled bars since the latter are
allowable.
137](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-76-2048.jpg)
![Maximum spacing of bars in tension
In order to 'cbntrol crack widths. to the 'val~es given in
Table 4,7, thernaximul11 spacing of bars has to be Ihnited;
. The procedures for cal~ulating maximum har spacings are
discussed in Chapter 7. The Code also stipulates that, in
no circumstances, should the spacing exceed 300,mm.
This was considered a .reas()nablema~iinumspacingto
ensure that, in all reinforcedconcrete bridge members, the .
bars wouldbesufficientlyclbse together for them to be
assumed ·to·fonn a'smeared' layer of reinforcement,.. rather
than act as individual bars.
Minimum reinforcement areas
*Shrinkage andtemperatur~'reinfbrce~ent'
In those parts of a structure where cr@cking could occur
due ,to restraint to. shrinkage or thermal movements,at
least 0.3% of mild. steel or 0.25% of high yield steel
should be provided. These values are less than those sug-
gested .by Hughes [185] and the author would suggest that
they.be used with caution. The Code values originated in
CP 2007.
"*Beams and slabs
.A minimum area of tension reinforcement is required in, a
beam or slab in order to ensure that the cracked strength of
the. section exceeds its uncracked strength; otherwise, any
reinforcement would yield as soon as cracking occurred,
and extremely wide cracks would result.
The cracking moment of a rectangular concrete beam is
given by
M, =f,bh2
/6
where !tis the tensile strength of the concrete, and band h
are the breadth and overall depth respectively. If the beam
is reinforced with an area of reinforcement (As)at an effec-
,tive depth (d) and having a characteristic strength (fy), the
ultimate moment ofresistance is given by
M,. =:=fyA,.•z
where z is the lever arm~
Since it is required that Mu~ M"then
f,.A.z ~ f,hh2/6
.01' .'
(
h2
). '... f
,- ==16.7:IJ..
d. .. fy
Beeby ri191has shownthatji"" 0.556)];,;: thus, for the
maximum allowable value of Icuof' 50 N/mm2, ft =
3..9N/mm2. Hence, forfy =' .250 N/mm2 and 410 Nlmm2
re'spectively, the required minimum reinforcement percen-
tages are 0.26 and 0.16. These values agree very well with
. the Code values of 0.25 and 0.15 respectively. The latter
.values cannot be compar~9directly with thos~ in CP 114
because the CP 114 values are expressed as a percentage
of the gross section, rather than the effective section.
However; the· Code will generally require greater
minimum areas of reinforcement than does CP 114..
138
load
Column service ,.",~~.,..,.,~,..,..,..,..;".,..,....,_......,;....
load
load Carried
~~~~~~~~~~~b';'com:ret& '
.......... '-... ·Time
Fig. 10.1 Load transfer in column 'under service' load conditions
The above minimum reinforcement areas are. given in " .
the Code under the heading 'Minimum area of main i.
reinforcement' '. but, since a minimum .area of secondary i
re.inforcement is not speCified for solid slabs, it would
seem prudent also to apply the Code values to secondary
reinforcement.
Voided slabs
Although a minimum area of secondary reinforcement in 'i
solid slabs is not specified, values are given for voided
slabs. These values are discussed in detail in Chapter 7.
Columns
Under long-term service load conditions, load is transfer-
red from the concrete to the reinforcement as shown in
Fig. 10.1. The load transfer occurs because the concrete
creeps and shrinks. If the area of reinforcement is very
small. there is a danger of the reinforcement yielding
under service load conditions. In order to prevent yield,
ACI Committee 105 [236] proposed a minimum rein~
forcement area of 1%. This value is adopted in the Code
and is a little greater than that (0:8%) in CP 114. How-
ever, if a column is lightly loaded, the area of reinforce-
mentis allowed' to be. less than 1% but not less than
(0.15 Nlfy), where Nis the ultimate axial load and fy is the
characteristic strength of the reinforcement. This require-
ment is intended to cover a case where a column is made
much larger than is necessary to carry the load.
In order to ensure the stability of ,a reinforcement cage
pdor to casting, the Code requires (as does CP114) the
main bar diameter to be at least 12 mm. In addition, the
Cbde require~ at least six main bars· for circular columns
and four bars for rectangular columns..
t Walls
It is explained in Chapter 9 that a reinforced concrete wall,
, which carries a significant axial Imid, should have at least. .
0.4% vertical reinforcement. This requirement is necessary
because !imaller amounts of reinforcement .canresult in ,a
reinforced wall which is weaker than a plain concrete wall
[227].
Links
Links are generally present in a member for two reasons:
to act as shear or torsion reinforcement, and to restrain
main compression bars.
The minimum requirements for links to act as shear
reinforcement are discussed in detail in Chapter 6: In the
following, the requirements for a link to restrain a com-
pression bar are discussed.
Beams and columns The link diameter should be at least
one-quarter of the diameter of the largest compression bar,
and the links should be spaced at a distance which is not
greater than twelve times the diameter of the smallest
compression bar. These requirements are the same as those
in CP 114. except that the latter also requires the link spac-
ing in columns not to exceed the least lateral dimension of
the column nor 300 mm, and the link diameter not to be
less than 5 mm. The latter requirement is automatically
sulil'oli~J by the fad lltul the slllullest available bar has a
diameter of 6 mm (although it is now difficult to obtain
reinforcement of less than 8 mm diameter).
Walls and slabs When the designed amount of compres-
sion reinforcement exceeds 1%, links have to be provided.
The link diameter should not be less than 6 mm nor one-
quarter of the diameter of the largest compression bar. In
the direction of the compressive force, the link spacing
should not exceed 16 times the diameter of the compres-
sion bar. In the cross-section of the member, the link spac-
ing should not exceed twice the member thickness. These
requirements were taken from the ACI Code [168] and are
different to those ofCP 114.
Maximum steel areas
In order to ease the placing and compacting of concrete,
the amount of reinforcement in a member must be
restricted to a maximum value. The Code values are as
follows.
Beams and slabs
Neither the area of tension reinforcement nor that of com-
pression reinforcement should exceed 4%. CP 114 requires
only that the area of compression reinforcement should not
exceed 4%.
Columns
The amount of longitudinal reinforcement should not
exceed 6% if vertically cast, 8% if horizontally cast nor
10% at laps. The CP 114 amount is always 8%. Hence,
the Code is 1110re restrictive with regard to vertically cast
columns, and this fact, coupled with the small allowable
design crack width, could result in larger columns - as
discussed in Chapter 9.
Walls
The area of vertical reinforcement should not exceed 4%.
No limit is given in CP 114.
Bond
General
All bond calculations in accordance with the Code are car-
ried out at the ultimate limit state.
t
T+'OT
~______~'OX~______~~I
Fig. 10.2 Local bond
The Code, like Amendment 1 to BE 1/73, recognises
two types of deformed bars:
1. Type 1, which are, generally. square twisted.
2. Type 2, which have, geneFaHy, transverse ribs.
Type 2 bars have superior bond characteristics to type 1
bars. However, unless it is definitely known at the design
stage which type of bar is to be used on site, it is necessary
to assume type 1 for deSign purposes.
Local bond
Consider a beam of variable depth subjected to moments
which increase in the same direction as the depth
increases, as shown in Fig. 10.2.
The tension steel force at any point (x) is T, where
T= Mlz
and M and z are the moment and lever arm respectively at
x. The rate of change of T is
dT _ z(dM/dx) - M(dz/dx)
dx - Z2
But dMIdx is the shear force (V) at x and the Code assumes
dzldx "'" tan e•.
Hence
dT V - M tan eslz
dx z
But dTldx is also equal to the bond force per unit length;
which is fbs (l:us), where fb. is the local bond stress and
(l:us) is the sum of the perimeters of the tension rein-
forcement. Hence
V - M tan fj,/z
A. (l:us ) = z S
or
V - M tan fJ.lz
A, = (l:us)z (10.1 )
The Code assumes that z"'" d (and adjusts the allowable
values of fbs accordingly). If M increases in the opposite
direction to which d increases, the negative sign in equa-
tion (10.1) becomes positive. Hence, the following Code
equation is obtained
I _ V + M tan 0.,1d
Jb.• - (l:us)d
(10.2)
In addition to the modification to allow for variable
depth. this equation differs to that in CP 114 because the
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![CP 114 equation is written in terms of the lever arm rather
than the effective depth.
The allowable local bond stresses at the ultimate limit
state depend on bar type and concrete strength: they are
given in Table 10.2. The bond stresses for plain, type 1
deformed and type 2 deformed are in the approximate ratio
1 : 1.25 : 1.5. It is understood thl1t the tabulated values
were obtained by considering the test data of Snowdon
[237] and by scaling up the CP 114 values, for plain bars
at working load conditions, to ultimate load conditions.
Snowdon's tests on 150 mm lengths of various types of
bar indicated that the bond stresses developed by plain,
square twisted (type 1) and ribbed (type 2) bars were in the
approximate ratio 1 : 1.3: 3.5. Hence the Code ratio is c
reasonable for type 1 deformed bars but can be seen to be
conservative for type 2 deformed bars. However, Snowdon
found that the advantage of the latter bars over plain bars
decreased with an increase in diameter, particu)a1'1y with
low strength concrete.
Table 10.2 Ultimate local bond stresses
Local bond stress (N/mm2) for concrete
grade
Bar type
30 :s; 4020 25
Plain 1.7 2.0 2.2 2.7
Deformed Type 1 2.1 2.5 2.8 3.4
Deformed Type 2 2.6 2.9 3.3 4.0
The Code local bond stresses are about 1.5 to 1.6 times
those in BE 1/73 if overstress is ignored. However, it
should be remembered that the Code bond stresses are
intended for use at the ultimate limit state with a steel
stress of 0.87 f)., whilst the BE 1/73 bond stresses are
intended to be used at working load with a steel stress of
about 0.56 I,.. The ratio of these steel stresses is 1.55, and
thus the result of carrying out local bond calculations in
accordance with the Code and with BE 1173 should be
about the same.
The author understands that, in a proposed amendment
to CP 110, local bond calculations are not required at all.
If this proposal is adopted, then local bond calculations
will, presumably, be omitted from the Code also.
Anchorage bond
The conventional expression for the anchorage length (L)
of a bar, which is required to develop a certain stress (j,)
can be obtained from any standard text on reinforced con-
crete, and is
(10.3)
where rna is the average anchorage bond stress and cp is the
bar diameter.
Since bond calculations are carried out at the ultimate
limit state, the reinforcement stress (j,) is the design stress
at the ultimate limit state and is 0.87 f" for tension bars
and O.72/y for compression bars. However, if more than
the required amount of reinforcement is provided. then the
140
stress in the bar is less than the design stress and lower
values ofis, than those given above, may be used.
The allowable average anchorage bond stresses (fba)
depend upon bar type, concrete strength and whether the
bar is in tension or compression. Higher values are permit-
ted for bars in compression because some force can be
transmitted from the bars to the concrete by end bearing of
the bar. The allowable anchorage bond stresses are given
in Table 10.3. The stresses for plain, type 1 deformed and
type 2 deformed bars are .in the approximate ratio
1 : 1.4 : 1.8, and those for bars in compression are about
25% greater than those for bars in tension. It is understood
that the values for bars in tension were obtained by consid-
ering the test data of Snowdon [237] and by scaling up the
CP 114 values, for plain bars at working load conditions,
to ultimate load conditions. Snowdon's tests indicated that
the anchorage lengths for plain, square twisted (type 1)
and ribbed (type 2) bars were in the approximate ratio
1 : 1.4 : 2. The Code ratio agrees very well with Snow- .
don's results. The Increase of 25%, when bars are in com·
pression, was taken from that implied in CP 114.
Table 10.3 Ultimate anchorage bond stresses
Bar type
Plain, in tension
Plain, in compression
Deformed, type I, in tension
Deformed, type I, in
compression
Deformed, type 2, in tension
Defon:ned, type 2. in .
compression
Anchorage bond stress
(N/mm2) for
concrete grade
20 25 30
1.2 1.4 1.5
1.5 1.7 1.9
1.7 1.9 2.2
2.1 2.4 2.7
2.2 2.5 2.8
2.7 3.1 3.5
;:!!: 40
1.9
2.3
2.6
3.2
3.3
4.1
The Code average bond stresses for plain bars are about
1.5 those in BE 1/73, if overstress is ignored. However,
since the ratio of steel stresses is 1.55 (see previous dis-
cussion of local bond stresses), anchorage lengths for plain
bars will be about the same whether calculated in accor-
dance with the Code or BE 1173. But anchorage lengths
for deformed bars will be shorter by about 13% for type I
bars and 29% for type 2 bars. This is because BE 1173
allows increases in bond stress. above the plain bar values,
of only 25% for type I bars and 40% for type 2 bars.
These percentages compare with 40% and 80%. respec-
tively, in the Code.
Bundled bars
The Code permits bars to be bundled into groups of two,
three or four bars. The effective perimeter of a group of
hars is obtained by calculating the sum of the perimetcrs
of the individual bars and, then, by multiplying by a reduc-
tion factor of 0.8. 0.6, or 0.4 for groups of two, three or
four bars respectively. The resulting perimeter, so calcu-
lated, is less than the actual exposed perimeter of the
group of bars to allow for difficulties in compacting con-
crete around groups of bars in contact.
Steel
force
T
..
~--~--+4--~-.--.-.I 1
---~
1
Mlz '
I
-+-_--====-1
1
,
1
t. beam
Fig. 10.3 Steel force diagram
"JY:; Lap lengths
In general, as in CP 114, a lap length should be not less
than the anchorage length calculated from equation (10.3).
However. for deformed bars in tension, the lap length
should be 25% greater than the anchorage length. This
requirement is to allow for the stress concentrations which
occur at each end of a lap, and which result in splitting of
the concrete along the bars at a lower load than would
occur for a single bar in a pull-out test [238]. Such split-
ting does not occur with plain bars, which fail in bond by
pulling out of the concrete.
In addition to the above requirements, the Code requires
the followTng minimum lap lengths to be provided for a
bar of diameter <p:
I. Tension lap length 4: 25 <P + 150 mm
2. Compression lap length 4: 20 cp + J50 mm
These minimum lengths are much ITIore conservative than
those in CP 114 for small diameter bars, and slightly less
conservative for large diameter bars.
Bar curtailment and anchorage
General curtailment
As in CP 114, a bar should extend at least twelve dia-
meters beyond the PQint at which it is no longer needed to .
carry load.
A bar should also be extended a minimum distance to
allow for the fact that. in the presence of ~hear, a bar at a
particll!ar section has to carry a force greater than that cal-
culated by dividing the bending moment (M) at the section
by the lever arm (z). A rigorous analysis [239] of the truss
of Fig. 6.4(a), rather than the simplified analysis of Chap-
tcr 6, shows that the total force, which has to be carried by
the main tension reinforcement at a section where the
moment and shear force are M and V, respectively, is
T = Mlz + (VI2) (cot () - cot ex)
The Code assumes 0 = 45°. thus
T = Mlz + (VI2) (1 - cot ex) (10.4)
In Fig. 10.3, the distributions of tension force due to
, bending (Mlz) and total tension force (T) are plotted for a
general case. It can be seen from Fig. 10.3 that the
increase in steel force due to shear can be allowed for by
de~igning the reinforcement at a section to resist only the
moment at that section, and by extending, the reinforce-
ment ~eyond that section by the distance6x in Fig. 10.3.
The dIstance 6x can be found by equating the total steel
force at a section at x to the steel force due to moment only
at a section at (x +6 x).
The maximum increase in steel force due to the shear
force, and, hence, the maximum value of6x occurs when
cot IX is zero (i.e. vertical stirrups) and equation (10.4)
becomes
T = Mlz + VI2 (l0.5)
For a central point load (2W) 'on a beam of span I, the
moment and shear force at x are:
M.t = Wx
Vx = W
From equation (10.5)
T.r =Wxlz + WI2
The moment at (x + 6X) is
M~·+D.X =W(x + 6X)
t::.x can be found from
Tx = Mx~D.xlz
Th'us
WXlz + ~/2=W(x + 6Jc)/z
From which
I1x == il2
If this analysis is repeated for a uniformly distributed load.
it can be shown that6x is, again, about z/2. Hence, if the
longitudinal reinforcement is designed solely to resist the
moment at a section, the reinforcement should be extended
a distance zl2 beyond that section. However, the Code.
conservatively, takes the extension length to be the effec-
. tive depth.
.Curtailment in·tensionzones
In addition to the above general requirements, the Code
requires anyone of the following conditions to be met
before a bar is curtailed in a tension zone. .
1. In order to control the crack width at the curtailment
point, a bar should extend at least an anchorage
length, calculated from equation (10.3) with f, =
0.87 fy, beyond the point at which it is no longer
required to resist bending.
2. Tests, such as those carried out by Ferguson and Mat-
loob [2401, have shown that the shear capacity of a
section with curtailed bars can be up to 33% less than
that of a similar section in which the bars are not cur-
tailed. In order to be conservative, the Code requires
the shear capacity at a section. where a bar is curtailed,
to be greater than twice the actual shear force.
3. In order to control the crack width at the curtailment
point, at least double the amount of reinforcement
required to resist the moment at that section should be
provided. This requirement was taken from the ACI
Code [168].
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![o
(a) Bearing force
Bb = $
14-14-"-~
o o
f.---- $_.. ___+1
(b) Definitions of s"
Fig. IO.4(a),(b) Bearing force at bend
o
The above requirements are far more complicated than
those of CP 114, unless one chooses to apply option 1 and
continue a bar for a full anchorage length.
*Anchorage at a simply supported end
The Code requires one of the following conditions to be
satisfied.
1. As in CP 114,. a bar should extend for an anchorage
length equivalent to twelve times the bar diameter;
and no bend or hook should begin before the centre of
the support,
2. If the support is wide and a bend or hook does not
begin before dl2 from the face of the support (where d
is the effective depth of the member), a bar should
. extend from the face of a support for an anchorage
length equivalent to (dI2' + 12 <1», where II> is the bar
diameter.
3. Provided that the local bond stress at the face of a
support is less than half the value in Table 10.2, a
straight length ofbar should extend, beyond the centre
line of the support. the greater of 30 mmor one-third
of the support width. This clause was originally writ-
ten. for CP 110, to covet small precast units [112].
and it is not clear whether the clause is applicable to
bridges.
Bearing stresses inside bends
The bearing stress on the concrete inside a bend of a bat of
diameter 11>, which is bent through an angle II> with a radius'
r. should be calculated by assuming the resultant force (R)
142
on the concrete is uniformally spread over the length of the
bend. Hence, with reference to Fig. 1O.4(a) the resultant
force is
R =2Fbt sin (8/2)
where FbI is the tensile force in the bar at the ultimate limit
state. The'bearing area is
II> [2r sih (8/2)]
Thus the bearing stress fb is given by
fb ,= 2Fbl sin (8/2)/<1> [2r sin (8/2)]
... fb =.Pb/rep
which is the equation given in the Code.
(10.6)
The bearing stress calculated from equation (10.6)
should' not exceed the allowable value given by equation
(8.4). In the present context, the bar diameter is the length
of the loaded area and thus, in equation (8.4), Ypo =11>/2.
Similarly the bar spacing (ab) is the length of the resisting
concrete block and thus, in equation (8.4), Yo =abl2. On
substituting into equation (8.4), the following Code ex-
pression is obtained.
1.5 feu
1 +2(j)/ab
However, for a bar adjacent to a face of a member, as
shown in Fig. 1O.4(b), the length of the resisting concrete
block is (c + II> + abI2), where c is the side cover. Thus Yo
should be taken as(c + II> + abI2)/2; but the Code, by
redefining ab as (c + 11», implies that
Yo = (c + 11»/2
The Code thus seems. to be conservative in this situation. It
appears that the Code requirements were based upon those
of the CEB [226]. which,in fact, defines ab as (c + 11>/2)
for a bar adjacent to a faGe of a member.
The Code definitions of ab are summarised in
Fig. lO.4(b).
It is not necessary to carry outthese bearing stress caicu-
lations if a bar is not assumed to be stressed beyond the
bend. Hence, bearing stress calculations are not required
for standard end hooks or bends. .
Prestressed concrete
The following points concerning detailing in prestressed
concrete are intended to be additional to those discussed
previously for reinforced concrete.
Cover to tendons
Bonded tendons
As in BE 2/73, th~ Code requires the covers to bonded
tendons to be the same as those to bars in reinforced con-
crete. Hence. the comments made earlier in this chapter
regarding the reinforced concrete covers are relevant.
Tendons in ducts.
Again. as in BE 2/73, the Code specifies a minimum cover
of 50 mm to a duct. In addition a table, which is identical
to that in BE 2/73. is provided (in Appendix D of Part 4 of
the Code) for covers to curved tendons in ducts.
11:xternal tendons
Part 4 of the Code refers to Clause 4.8.3 of Part 7 of:>the
Code for the definition of an external tendon. However,
the latter clause does not exist in Part 7, but exists in Part
8 of the Code as Clause 5.8.3. It defines an external ten-
don as one which 'after stressing and incorporation in the
work, but before protection, is outside the structure'. This
definition is essentially the same as that in BE 2/73.
As in BE 2/73, the Code requires that, when external
tendons are to be protected by dense concrete, the cover to
the tendons should be the same as if the tendons were
internal. In addition, the protective concrete should be
anchored, by reinforcement, to the prestressed member,
and should be checked for cracking. The Code is not
specific regarding how the latter check should be carried
alit: BE 2/73 refers to the reinforced concrete crack width
formula of BE 1/73. When using the Code, the author
would suggest that equation (7.4) for beams should be
used.
Spacing of tendons
Bonded tendons
The minimum tendon spacing should comply with the
.minimum spacings specified for reinforcing bars. The
latter spacings are similar to those of CP 116 and, as
explained earlier in this chapter, are similar to those
specified in BE 2/73. In addition, BE 2/73 requires com-
pliance with the maximum spacings specified in CP 116,
whereas the Code does not refer to maximum spacings.
Tendons in ducts
The Code gives a number of requirements -for the clear
distance between ducts; these requirem~nts are identical to
those in BE 2/73.
'*Transmission length in pre-tensioned
members
In both the Code and BE 2/73, the transmission length is
defined as the length required to transmit the initial pre-
stressing force in the tendon to the concrete.
The transmission length depends upon a great number of
variables (e.g. concrete compaction and strength, tendon
type and size) and. ifpossible, should be determined from
tests carried out under site or factory conditions, as
appropriate. If such test data are not available, the Code
gives recommended transmission lengths for wire and for
strand. The Code implies values which are identical to
those in BE 2/73.
The suggested transmission lengths for wires are based
i::::t
---------- --- ----!~~;:,:~~::re
this length
----------
- h"._ Stress
distribution
Fig. 10.5 Splitting at end of prestressed member
on data from tests carried out in the laboratory and on site
by Base [242]; and those for strands are based on data
from tests carried out only in the laboratory by Base [243].
The CP 110 handbook [112] warns that the transmission
lengths for strands,· which were based upon laboratory
data, could be exceeded on site; it also warns that they
should not be used for compressed strands, for which
transmission lengths can be nearly tw~ce those given in the
Code.
In members which have the tendons arranged vertically
in widely spaced groups, the end section of the member
acts like a deep beam when turned through 90° (see
Fig. 10.5). This is due to tile fact that, towards the end of
the member, concentrated loads are applied by the ten-
dons, and, at some distance from the end, the prestress is
fully distributed over the section. Hence, the end face of
the member is in tension and a crack can form, as shown
in Fig: 10.5. Vertical links should be provided to control
the crack, and Green [241] suggests that, by analogy with
a deep beam, the required area of the vertical reinforce-
ment (As) should be calculated from:
As =0.2hfebJfs < O.04Pklh
where
(10.7)
h = vertical clear distance between tendon groups
bw = width of web, or end block, at a distance h from
the end of the member
ie = average compressive stress between the tendons at
a distance h from the end of the member .
is = permissible reinforcement stress (O.87iy )
Pk = total initial prestressing force
"
In Fig. 10.5 and the above discussion, the prestressing
forces have been considered to be applied at the end of the
member, whereas, of course, they are transferred to the
concrete over the transmission length, which is typically of
the order of 400 mm. Thus the deep beam analogy tends to
overestimate the tendency to crack, and equation (10.7)
should be conservative.
End blocks in post-tensioned members
General
In a post-tensioned member, the prestressing forces are
applied directly to the ends of the member by means of
relatively small anchorages. The forces then spread out
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![1
End block
Compressed
wedge of
concrete
t t
Splitting action
t t t
(a) Splitting action
Fig. 10~6(a).(b) End block with symmetrical anchorage
(1m 0.8
r::v:
0.6
0.4
0.2
Tensile
stress
(1m
Co",P"~~::.Tstress ..-~:
Distance
from loaded
face
(b) Transverse stress distribution along
block centre line
oL-----~0~.1.---~0~.2r-----0~.~3----~0~.A4-----n0~.5-----n~----~,----,~----~0~~9----~1.6
ypo/Yo
Fig. 10.7 Ma."(imum transverse tensile stress in end block [244]
over the cross-section of the member and. in this region of
spread (the end block), high local stresses occur. In par-
ticular, large transverse bursting stresses occur: it is easiest
to examine these st.resses by considering an end block
subjected to a single symmetrically placed prestressing
force.
Single anchorage
Single symmetricallY placed anchorages have been studied
both theoretically and experimentally [244]. Qualitatively,
the structural behaviour consists of a cone of compressed
concrete being driven into the end block and, thus, causing
splitting of the end block as shown in fig. 10.6(a). The
splitting action cailses transverse bursting stresses which
are greatest across a horizontal or vertical section through
the axis o'fthe end block. The distribution of transverse
stresses along such a section is of the form shown in
Fig. 1O~6(b),where it can be seen that compressive stress-
es exist near to the loaded face. but at a distance of about
0.2 y" from the loaded face (where 2 Yo is the length of the
side of the end block) the stress becomes tensile. The ten-
sile stress reaches a maximum of ltin at about 0.5 Yo from
the loaded face, and then decreases to nearly zero at about
2yo from the loaded face (Le.at a distance equal to the
144
length of the side of the:, end block) [244]. It can be seen
that it is reasonable to approximate the actual stress dis-
tribution to the triangular stress distribution shown in
Fig. 1O.6(b).
The maximum stress is mainly dependent upon the ratio
of the length of the side of the loaded area (2 Ypo) to that of
the end block (2 Yo). The ratio of maximum transverse ten-
sile stress (f,m) to the average compressive stress, over the
total cross-sectional area of the end blo.ck (fal'e), is plotted
againstYpd.vo in Fig. 10.7; the relationship was determined
experimentally [244].
The bursting tensile stresses have to be resisted by re-
inforcement, and thus the total bursting tensile force to be
resisted is of primary interest. The bursting tensile force
can be obtained by integrating a number of stress dia-
grams, similar to that of Fig. 10.6(b). As one might expect,
the bursting tensile force (Fbsl) is mainly dependent upon'
YP(jyo' The ratio of Fbsl to the maximum prestressing ten-
don force (Pk) is plotted against YpdYo in Fig. 10.8. In this
figure, the range of values obtained from various theories
[244], values determined from tests [244] and the Code
values are given. It can be seen that the experimental val-
ues exceed the theoretical values, and that the Code values
have been chosen to lie between the theoretical and
experimental values.
Fbst 0.4
Pk
-
0.3
0.2
0.1
I
0.3
I
0.2
I
0.40'---*
Fig. 10.8 Bursting tensile force
The tendon force for use in determining the bursting
force should be the greatest load that the tendon will carry
during its life. This will be the jacking load for a bonded
tendon, and the greater of the jacking load .and the tendon
load at the ultimate limit state for an unbonded tendon.
The latter load may be assessed, as explained in Chapter 5,
from Table 30 of the Code. However, the Code clause on
end blocks refers to Tables 20 to 23 of the Code: these
tables give characteristic strengths of tendons, and it is not
clear whether the reference is an error, or whether it is
intended that the load at the ultimate limit state should be
taken as that equivalent to the characteristic strength of the
tendon. It would seem more appropriate to use a load
assessed from Table 30.
The bursting tensile force, calculated from the Fh.•1Pk
ratios given in the Code and plotted in Fig. 10.8, should be
resisted by reinforcement. From Fig. 10.6(b) it can be
seen that this reinforcement should be distributed in a
region extending from 0.2)'" to 2),,, from the loaded face of
the end block. The reinforcement should be designed at the
ultimate limit state and, thus, its design stress is 0.87/Y'
However, in order to control cracking, the reinforcement
stress should be limited to a value correspond.ing to a strain
of 0.001 (Le. 200 N/mm 2
) if the cover to the reinforce-
ment is less than 50 mm.
If the end block is rectangular. the'value of ypjyo is
different in the two principal directions. Hence, Fh..1 should
be determined in each of the principal directions and rein-
forcement proportional accordingly. But, for detailing pur-
poses, it is generally more convenient to use the greater
area of reinforcement in both directions.
The above design method, in which all of the bursting
tensile force is resisted by reinforcement, is the method
given in the Code. However, the Code does permit the
adoption of alternative design methods, in which some of
the bursting tensile force is resisted by the concrete, to be
adopted. One such method is that suggested by Zielinski
and Rowe [244]; when using this method, the values of
F".,/Pk given in the Code should not be used but, instead,
the test values given in Fig. 10.8 should be used. This
design procedure is. first for the appropriate value of
YllI,!Y", to obtain the maximum tensile stress frqm Fig. 10.7
I
0.5
I
0.6
Tensile·
stress
(1m
- .....--Tests
-·-·-Code
')7"..-7.17)7 Range of
~U~.:::; theories
I
0.8
I
0.7
~~~~~~LU~~<~~~~_~~_--+
Distance
~ Provide reinforcement --.'
in this region
Fig. 10.9 Stresses resisted by concret~ end reinforcement
from loaded
face
and the bursting tensile force from Fig. 10.8. The ideal-
ised triangular stress distribution diagram of Fig. ]0.6(b)
is then constructed and the pennissible tensile stress in the
concrete (flp) superimposed on the diagram as shown in
Fig. 10.9. Only those areas where the stress exceeds the
permissible tensile stress of the concrete need to be re-
inforced, as shown in Fig. 10.9. The bursting tensile force
to be resisted by reinforcement (f.) as a fraction of the
total bursting tensile force (Fhsl) is equal to the ratio of the
area of the shaded part of the stress diagram to the total
area; hence.
(10.8)
The permissible steel stress used to calculate the
required area' of reinforcement is usually chosen to be
140 N/mm2
• The strain associated with this stress is gener-
.ally too small to cause observable cracking of the concrete.
Regarding the value to be taken for the permissible tensile
stress in the concrete, BE 2173 states that it should be the
cylinder splitting strength of the concrete divided by 1.25,
and this document also gives values of cylinder splitting
strength for various grades of concrete.
The above two design methods lead to similar amounts
of reinforcement because. although in the second method
some of the bursting tensile force is resisted by the con-
crete. the total bursting tensile force to be resisted is
greater (see Fig. ]0.8).
BE 2173 also permits the use of either of the two design
methods.
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![Y01
Y01
Y02
Y02
Y03
Y03
.'ig. 10.10 Multiple anchorages
Multiple anchorages
Prism for
·-anchorage 1
.-Anchorage 1
.'_ Anchorage 2
Prism for
-.- anchorage 2
." Prism for
anchorage 3
Anchorage 3
Very often the total prestressing force is applied to the end
of a member by a numb~r of anchorages. Tests have been
carried out on end blocks with mUltiple anchorages by
Zielinski and Rowe [246], and the results indicate that
each anchorage may be associated with a prism of con-
crete, which acts like an end block for the particular
anchorage. as shown in Fig. 10.10. Each prism is symmet-
rically loaded by its anchorage, and its vertical dimension
is the lesser of twice the distance from the centre line of its
anchorage to the centre line of the nearer adjacent anchor-
age, and twice the distance from the centre line of its
anchorage to the edge of the concrete.
The bursting tensile force and the required amount of
reinforcement in each prism can be assessed by either of
" the methods for single anchorages described earlier.
~n addition, the individu~1 pristn~ sh?uld ~ tied together
.reinforcement. No gUIdance IS gIven In the Code on
~<tzsign such reinforcement, but a method has been
'~rke [246]. which is based on the French
.~
'<:ses
~
~
.~~
~
~loaded face of an end block
~ §i ~.
t.h.& .If-.*"'- ~ ~
con~ (I~ ~ h'
splitt~~b~ .....~SI:::¢C
SPlitti~J>'()~$ ~~($'
~~es arise for similar
"'~in this chapter, in
~n pre-tensioned
~qy be used toaregreal .....t- ~ ;$ !$
• ~ <&-~1lJ o·~
the aXIs e ~ b~ ~v b
stresses alt..1 0<;: ~~.,# .$
F· 10 6(b)·~ (IV '(). 1lJ'<i .....1lJ'<i
Ig. ~ • C5 b ~ ~
es exist near tf'.# ~ ~o
0.2 y" from the'$' .s ~""... ,,'I,;
side of the end b'r a-
sile stress reaches .)iJ'
.~'
the loaded face, ana."
2y-o from the loaded
144
~dbY
dded
/
_,orage,
.flmsverse
Crack
A--- 1-='--_. -- -_.- .-. - ·--A
L,
L
Reinforcement
~~_-"..~"",",,___ Transverse stress
across AA
Compression
Fig. 10.11 Spulling tcnsile stresses due to eccentric prestress
stress distribution along a line parallel to the axis of an
eccentric prestressing force is as shown in Fig. 10.11
[2481. Figure 10.11 also shows the crack which can occur
at the loaded face. Green [241] suggests that vertical re-
inforcement sufficient to resist a force of O.04Pk should be
placed as near as possible to the loaded face of the member
to control the crack. It would seem prudent to provide a
minimum amount of such reinforcement in all end blocks
whether or not the prestressing force is highly eccentric.
Further guidance on resisting spalling tensile stresses,
based upon the French C~de [247], is given by Clarke
[246].
End block and beam of different shapes
Prestressed concrete beams are generally of a non-
rectangular shape with flanges; however, the end blocks
are often made rectangular. As one might expect, stress
concentrations occur where the section changes from
rectangular to non-rectangular, and, at the junction, a
second region of transverse burst~ng tensile stress occurs.
Tests [246] indicate that; at the junction of end block and
beam, reinforcement sho~ld be provided to resist a burst-
ing tensile force equal to 70% of that calculated within the
end block.
Summary
The steps in the design of a general end block can be
summarised as follows.
1. Design reinforcement to prevent bursting of the indi-
vidual prisms of concrete associated with each
anchorage.
2. Design reinforcement to tie together the individual
prisms.
3. Design reinforcement to resist the spalling tensile
stresses between, and at a distance, from anchorages.
4. Design reinforcement to prevent bursting at the junc-
tion of a rectangular end block and a non-rectangular
beam.
Clarke [246] gives a detailed worked example which
illustrates the above design procedure.
Chapter 11
Lightweight aggregate
concrete
~Introduction
The design recommendations which are discussed in pre-
vious chapters are intended only for concretes made from
normal weight aggregates. AU naturaUy occurring aggre-
gates, with the exception of pumice, are of normal weight
but, as such aggregates become scarce, it is likely that
manufactured aggregates will become more popular than
they are at present. The majority of manufactured aggre-
gates (e.g., expanded shale and clay, foamed blast furnace
slag and sintered pulverised fuel ash) are lightweight.
In addition to the above consideration of the future
availability of natural aggregates, the use of lightweight
aggregate concrete has obvious advantages where ground
conditions are poor, and there is a need to reduce, as much
as possible, dead loads and, thus, foundation loads.
Lightweight aggregate concrete has been used through-
out the world for both reinforced and prestressed concrete
construction, but it has been used far less in Great Britain
than in some other countries. This is particularly true of
bridge construction [280].
The first lightweight aggregate concrete road bridge
built in Great Britain was the Redesdale Bridge, which
was constructed by the Forestry Commission in North-
umberland. This bridge has a single span of 16.8 m. It is
constructed of prestressed precast invet;ted T-beams with
in-situ concrete in-fill to form a composite slab. The beams
were cast from a concrete composed of ,sintered pul-
verised fuel ash (Lytag) coarse aggregate and natural sand
fine aggregate, with a wet density of 1890 kg/m3, to give a
minimum strength at transfer of 35 N/mm2
and a minimum
28-day strength of 48 N/mm 2. As explained later, greater
losses of prestress occur with lightweight, as compared
with normal weight, aggregate concrete; for this bridge,
the losses were assumed to be 40% greater at transfer, and
30% greater finally.
The .Transport and Road Research Laboratory carried
out tests on beams identical to those used in the Redesdale
Bridge and found that they behaved satisfactorily under
both static and repeated loading [249].
Although the Redesdale Bridge was the first lightweight
aggregate concrete road bridge in Great Britain, it does not
form part of a public highway. The first lightweight aggre-
gate concrete brid,ge to be built over a public highway. the
Glasshouse Wood Footbridge at Kenilworth having a span
of 31.S m. was designed by the Warwickshire Sub-Unit of
the Midland Road Construction Unit. This bridge was
opened in 1974. Lytag was used for both coarse and fine
aggregates: the 28-day strength was 45.7 N/mm 2 and the
design air-dry density was 1700 kg/m3.
. Examples are to be found in Staffordshire of composite
slab motorway bridges with spans of about II m, which
were constructed with normal weight aggregate precast
concrete inverted T-beams with lightweight aggregate
(Lytag) in-situ in-fiU concrete. Lightweight aggregate con-
crete as in-fiU for composite slabs has also been used else-
where in Great Britain.
Recently Kerensky, Robinson and Smith [250] have
reported the successful completion', in 1979, of the Friar-
ton Bridge over the River Tay at Perth. This is a steel-
concrete composite bridge consisting of a steel box girder
with a composite lightweight aggregate concrete deck slab.
Lytag was used for both coarse and fine aggregates, and
the design strength and air-dry density were 30 N/mm2
and
1680 kg/m3 respectively.
In the following, the structural properties of lightweight
aggregate concrete, and how these are dealt with by the
Code, are discussed. At present, the requirements for the
structural use of lightweight aggregate concrete in highway
structures are covered by BE 11 [251]. However, this
document limits the, j.lse of lightweight aggregate concrete
to in-filling (such as between inverted T-beams), and only
gives data on density, modulus of elasticity and allowable
tensile stresses for in-fill concrete.
*Durability
The durability of lightweight aggregate concrete can be
very good, as was demonstrated by one of the early con-
crete ships - the Selma. This ship was constructed of a
concrete with expanded shale aggregate and the reinforce-
ment cover was only 10 mm. The reinforcement was
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![still in excellent condition after forty years in service
[96].
In lIon-marine environments, it is thought that, because
of the greater porosity of lightweight aggregates, which
permits the relatively easy diffusion of carbon dioxide
through the concrete, carbonation of the concrete may
. occur to a greater depth when lightweight aggregates are
used. The Code thus requires the cover to the reinforce-
ment to be 10 mm greater than the appropriate value
obtained from Table 10.1 for normal weight concrete.
However, this requirement may be conservative because
tests, carried out by Grimer [252], in which specimens of
five different lightweight aggregates and one normal
weight aggregate were exposed to a polluted atmosphere"
for six years, showed that the effect of the type of aggre-
gate on the rate of penetration of the carbonation front was
small in comparison with the effect of mix proportions.
Strength
1:Compressive strength
The minimum characteristic strengths permitted by. the
Code when using lightweight aggregate concrete '~e
15 N/mml, 30 Nlmml and 40 N/mml for reinforced,
post-tensioned and pre-tensioned construction respectively.
It should be noted that BE 11 requires a minimum strength
of 22.5 N/mm2
for in-fill concrete. These strengths can be
attained readily with lightweight aggregates [96], and
details of mixes suitable for prestressed concrete have been
given by Swamy et al. [253].
One important difference between concretes made with
lightweight and normal weight aggregates is that the gain
of strength with age may be different. .In particular the
gain in strength with certain lightweight aggregates may be
very small for rich mixes [254].
Tensile strength
The.tensile strength of any concrete is greatly influenced
. by the moisture content of the concrete, because drying
reduces the tensile strength. The flexural tensile strength
tends to be reduced by drying more than the direct tensile
strength.
Curing conditions affect the tensile strength of light-
weight aggregate concrete more than normal weight aggre-
gate concrete. Although the tensile strengths are. similar for
moist cured specimens, the tensile strength of lightweight
aggregate concrete when cured in dry conditions can be up
to 30% less than that of comparable normal weight con-
crete [96].
The relatively reduced tensile strength does not influ-
ence the design of reinforced concrete, but has to be
allowed for in the design of prestressed concrete. No
specific guidance is. given in the Code, but the CP 110
handbook [112] suggests that all allowable tensile stress-
es, referred to in the prestressed concrete clauses for nor-
148
mal weight concrete, should be multiplied by 0.8. This
value seems reasonable in view of the reduction in tensile
strength referred to in the last paragraph. The implications
of the factor of 0.8 are:
1. The allowable tensile stresses for Class 2 pre-
tensioned and post-tensioned members are 0.36 !feu
and 0.29 I!cu respectively, instead of 0.45 /!CU and
0.36 /Tcu, respectively, for normal weight aggregate
concrete (see Chapter 4).
2. The basic allowable hypothetical tensile stresses for
Class 3 prestressed members, which are given in
Table 4.6(a), should be multiplied by 0.8.
3. The allowable concrete flexural tensile stresses for
in-situ concrete, when used in composite construction,
which are given in Table 4.4, should be multiplied by
0.8. The allowable stresses, so obtained, are very
close to those given in BE 11.
4. When calculating the shear strength of a prestressed
member uncracked in flexure, the design tensile
strength of the concrete (fi) should be taken as
0.19/!cu, instead of 0.24 0.. for normal weight
aggregate concrete (see Chapter 6).
5. When calculating the shear strength of a Class 1 or 2
'prestressed member cr,cked in flexure, the design
flexural tensile strength of the concrete (fr) should be
taken as 0.30 !feu, instead of 0.37 .ftcu for normal
weight aggregate concrete (see Chapter 6). Hence,
equation (6.12), for the cracking moment, becomes
Mt = (0.3 /leu + 0.8 fpt)lly (11.1)
"'t Shear strength
The shear cracks, which develop in members of light-
weight aggregate concrete, frequently pass through the
aggregate rather than around the aggregate, as occurs in
members of normal weight aggregate concrete. Hence, the
surfaces of a shear crack tend to be smoother for light-
weight aggregate concrete, and less shear force can be
transmitted by aggregate interlock across the crack (see
Chapter 6). Since aggregate interlock can contribute 33%
to 50% of the total shear capacity of a member [152], the
shear strength of a lightweight aggregate concrete member
can be appreciably less than that of a comparable normal
weight concrete member.
Tests carried out by Hanson [255] and by Ivey and
Buth [256] on beams without shear reinforcement have
shown that, for a variety of lightweight aggregates, it is
reasonable to calculate the shear strength of a lightweight
aggregate concrete member by multiplying the shear
strength of the comparable normal weight aggregate con-
crete member by the following factors.
1. 0.75 if both coarse and fine aggregates are light-
w e i g h t . /
2. 0.85 if the coarse aggregate is lightweight and the fine
aggregate is natural sand.
In the Code an average value of 0.8 has been adopted
for any lightweight aggregate concrete, and, by analogy,
the same value has been adopted for torsional strength.
Hence:
1. The nominal allowable shear stresses for reinforced
concrete (vc), which are given in Table 6.1, should be
multiplied by 0.8.
2. The maximum nominal flexural,. or torsional, shear
stress should be 0.6 ./f.,u (but not greater than
3.8 N/mm2 and 4.6 N/mm2
for reinforced and pre-
stressed concrete respectively), instead of 0.75 lieu
(but not greater than 4.75 N/mml and 5.8 N/mml
for
reinforced and prestressed concrete respectively) for
normal weight aggregate concrete (see Chapter 6).
Only half of these values should be used for slabs (see
Chapter 6).
3. When calculating the shear strength of a Class 1 or 2
prestressed member cracked in flexure, although it is
not stated explicitly in the Code, the first term of the
right-hand side of equation (6.11) should be taken as
0.03 bd v'.fcu,instead of 0.037 bd./f.,u as used for nor-
mal weight aggregate concrete.
4. The limiting torsional stress (Vtmln) above which tor-
sion reinforcement has to be provided should be
0.054 ./!cu (but not greater than 0.34 N/mml), instead
of 0.067 ./Tcu (but not greater than 0.42 N/mmD
) as
used for normal weight aggregate concrete (see Chap-
ter 6).
5. Although not stated in the Code, it would seem pru-
dent to multiply the basic limiting interface shear
stresses for composite construction, which· are given
in Table 4.5, by 0.8.
Bond strength
Shideler [257] has carried out comparative pull-out tests
on deformed bars embedded in eight different types of
lightweight aggregate concrete and one normal weight
aggregate concrete. The average ultimate bond stresses
developed with the lightweight aggregate concretes were,
with the exception of foamed slag, at least 76% of those
developed with the normal weight aggregate concrete. The
Code reduction factor of 0.8 to be applied, for deformed
bars, to the allowable bond stresses of Tables 10.2 and
10.3 thus seems reasonable.
Short and Kinniburgh [258] have reported the results
of pull-out and 'bond beam' tests in which plain bars were
embedded in three different types of lightw~ight aggregate
concrete and in normal weight aggregate concrete. The
average ultimate bond stresses developed with the light-
weight aggregate concretes were 50% to 70% of those
developed with the normal weight aggregate concrete. The
Code reduction factor of 0.5 to be applied, for plain bars,
to the allowable bond stresses of Tables 10.2 and 10.3 thus
seems reasonable.
Shideler's tests with foamed slag aggregate indicated
that the average bond stress could be as low as 66% for
horizontal bars due to water gain forming voids in the con-
crete under the bars. The Code thus advises that allowable
bond stresses should be reduced still further (than 20% and
50%) for horizontal reinforcement used with formed slag
aggregate: appropriate reduction factors would seem to
be 0.65 and 0.33 for deformed and plain bars respec-
tively.
The lower bond stresses developed with lightweight
aggregate concrete imply that transmission lengths of pre-
tensioned tendons are greater than the values discussed in
Chapter 10 for use with normal weight aggregate concrete.
The Code gives no specific advice on transmission lengths
for lightweight aggregate concrete, but the CP 110 hand-
book [112] suggests that they should be taken as 50%
greater than those· for normal weight aggregate concrete.
This increase seems reasonable, 'as an upper limit, when
compared with test data collated by Swamy [259].
Bearing strength
It can be seen from Fig. 10.6(a), which illustrates an end
block of a post-tensioned member, that a bearing failure is,
essentially, a tensile splitting failure. Hence the allowable
bearing stresses for lightweight aggregate concretes should
reflect their reduced tensile strength discussed earlier in
this chapter. Since the tensile strength of lightweight
aggregate concrete can be up to 30% less than that of
comparable normal weight aggregate concrete [96], the
Code requires the limiting t?earing stress for lightweight
aggregate concrete to be two-thirds of that calculated from
equation (8.4).
The Code implies that the above reduction should be
applied only when considering bearing stresses inside
bends of reinforcing bars, but it would seem prudent to
apply the reduction to all bearing stress calculations
involving lightweight aggregate concrete.
Movements
Thermal properties
Lightweight aggregate concrete has a cellular structure
and, thus, its thermal conductivity can be as low as one-
fifth of the typical value of 1.4 W/moC for normal weight
aggregate concrete [96]. The reduced thermal conductivity
is of great benefit in buildings, because it provides good
thermal insulation. However, for bridges, it implies that
the differential temperature distributions are more severe
than those discussed in Chapters 3 and 13 for normal
weight aggregate concrete.
Although differential temperature distributions are more
severe with lightweight aggregate concrete, their effects
are mitigated by the fact that the coefficient of thermal
expansion can be as low as 7 x lO-s/oC [96], as compared
with approximately 12 x 1O-o/oC for normal weight aggre-
gate concrete. The lower coefficient of thermal expansion
also means that overall thennal movements of a bridge are
less when lightweight aggregate concrete is used. This
fact, coupled with the lower elastic modulus of lightweight
aggregate concrete (see next section), means that thennal
stresses, which result from restrained thermal movements,
are less than for normal weight aggregate concrete.
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![Elastic modulus
The elastic modulus of lightweight aggregate concrete can
range from 50% t075% of that of normal weight aggregate
concrete of the same strength [254]. The higher values are
associated with fGamed blast furnace slag aggregate and
the lower values with expanded clay aggregate [254].
It is mentioned in Chapters 2 and 4 that the Code gives a
table of short-term elastic moduli for normal weight aggre-
gate concretes. These values are in good agreement with
the following relationship, suggested by Teychenne, Parrot
and Pomeroy [20] from considerations of test data.
E - 9 1f, 0.33c - . cu
where Ec is the elastic modulus in kN/mml
and fCIl is the
characteristic strength in N/mm2
• The latter authors further
suggested, from considerations of test data, that the elastic
modulus of a lightweight aggregate concrete with a density
of Dr (kg/m3) could be predicted from
(11.2)
Equation (11.2) was based upon data from sixty mixes
covering four different lightweight aggregates with con-
crete densities in the range 1400 kg/ms to 2300 kg/m3.
The Code states that the elastic modulus of a lightweight
aggregate concrete, with a density in the above range, pan
be obtained by multiplying the elastic modulus of a normal
weight aggregate concrete by (D,/2300)2
• The resulting elas-
tic modulus will thus agree closely with that predicted by
equation 01.2). They will also be within 20% of those
specified in BE 11.
The reduced elastic modulus of lightweight aggregate
concrete has the following design implications.
1. Stresses arising from restrained shrinkage or thermal
movements are less than for normal weight aggregate
concrete.
2. Elastic losses in a prestressed member can be up to
double those in normal weight aggregate concrete
members.
3. Lateral deflections of columns are greater than for
normal weight aggregate concrete. Hence stability
problems are more likely to occur, and additional
moments (see Chapter 9) are greater. These factors
are allowed for in the Code by:
150
(a) Defining a short column as one with a slenderness
ratio ofnot greater than 10, as compared with the
critical ratio of 12 for normal weight aggregate
concrete columns (see Chapter 9);
(b) Substituting the divisor of 1750, in the additional
moment parts of equations (9.21) to (9.28), by
a divisor of 1200. Hence, the additional moment is
increased by nearly 50%. The requirement to
reduce the divisor from.t 750 to 1200 implies that
Creep
the assumed extreme fibre concrete strain at fail-
ure for lightweight aggregate concrete is about
0.00633 as compared with 0.00375 for normal
weight aggregate concrete (see Chapter 9). This
increase is reasonable in view of the reduced elas-
tic modulus and the greater creep of lightweight
aggregate concrete (see next section).
The data on creep of lightweight aggregate concrete, as
compared with that of normal weight aggregate concrete,
are conflicting. Although creep of lightweight aggregate
concrete can be up to twice that of normal weight aggre-
gate concrete [96], it has also been observed [260] that less
creep may occur with structural lightweight aggregate con-
crete as compared with normal weight aggregate concrete.
As is true of all other concretes, creep of lightweight
aggregate concrete depends upon a great number of fac-
tors, and it is desirable to obtain test data appropriate to
the actual conditions under consideration. In lieu of such
data, S.pratt [96] suggests that creep of lightweight aggre-
gate concrete should be assumed to be between 1.3 and
1.6 times that of normal weight concrete under the same
conditions. In similar circumstances, the CP ItO handbook
[112] suggests that the loss of prestress due to concrete
creep should be assumed to be 1.6 times that calculated for
normal weight aggregate concrete.
Shrinkage
Great variations occur in the shrinkage values for light-
weight aggregate concrete; values up to twice those for
normal weight aggregate'foncrete ha~e been reported [96].
In the absence of data pe.rtaining to the actual conditions
under consideration. Spratt [96] suggests the adoption of
an unrestrained shrinkage strain of between 1.4 and 2.0
times that of normal weight aggregate concrete under the
same conditions. In similar circumstances, the CP ItO
handbook [112] suggests that the loss of prestress due to
shrinkage should be assumed to be 1.6 times that calcu-
lated for normal weight aggregate concrete.
Losses in prestressed concrete
From the previous considerations, it is apparent that total
losses in prestressed lightweight aggregate concrete may
be up to 50% greater than those in prestressed normal
weight aggregate concrete. This is because of the smaller
elastic modulus and the greater creep and shrinkage of
lightweight aggregate concrete.
Chapter 12
Vibratio'n and fatigue
Introduction
In this chapter, the dynanlic aspects of design are con-
sidered in .terms of vibration and fatigue. Hence, reference
is made to Parts 2, 4 and 10 of the Code.
Vibration
Design criterio~
It is explained in Chapter 3 that it is not necessary to con-
sider vibrations of highway bridges, because the stress
increments due to the dynamic effects are within the
allowance made for impact in the nominal highway load-
ings [t07]. In addition, vibrations of railway bridges are
allowed for by multiplying the nominal static standard
railway loadings by a dynamic factor. Hence, specific
vibration calculations only have to be carried out for
footbridges and cycle track bridges.
It is explained in Chapter 4 that the appropriate design
criterion for footbridges and cycle track bridges is that of
discomfort to a user; this' is quantified i~ the Code as a
maximum vertical acceleration of 0.5 110 mls·, where fo is
the fundamental natural frequency in Hertz of,tpe unloaded
bridge [107]. .
Compliance
Introduction
The above design criterion is given in Appendix C of Part
20f the Code, which also gives methods of ensuring
compliance with the criterion. It should be noted that the
criterion and the methods of compliance are the same as
those in BE 1177. The background to the compliance rules
has been given by Blanchard, Davies and Smith [107].
They considered a pedestrian, with a static weight of
0.7 kN and a stride length of 0.9 m. walking in resonance
,
,
0.6
"
',9>.
'0$
',
0.2
, .
, I
,,
"I
I
I
I.
I
o~----~~----~~__~~~____~
1.0 1.1 1.2 ,,:: 1.3 1.4
Fig. 12.1 Attenuation factor
t
Natural freq~ency
{pacing frequency
with the natural frequency of a footbridge. It was found
that it was possible to excite a bridge in this way if the
natural frequency did not exceed 4 Hz, since the latter
value is a reasonable,upperlirilit, of applied pacing fre-
quency (frequencies above,3 Hz representing running).
,Thus, if the natural frequency exceeds 4 Hz, resonant
vibrations do not occur; ,however,it is still necessary to
calculate the amplitude of. the non-resonant vibrations
which do occur when a pedestrian strides at the maximum
possible frequency of 4 Hz. Analyses were carried out to
determine an attenuation factor defmed as the ratio of the
maximum acceleration when walking below the resonant
frequency to that when walking at the resonant frequency.
The results of such an analysis have the form shown in
Fig. 12.1. It can be seen that the attenuation factor drops
rapidly and, at a frequency ratio of 1.25 (for which the
natural frequency is 5 Hz if the pacing frequency is 4 Hz),
the attenuation factor is very small. It is thus considered
151](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-83-2048.jpg)
![f t K= 1.0
I- -I
f I
f t
14
-I- -I
K=:0.7
f 11 f f 11
I- -I- _14
t MI K
1+--7-'----IoI+----=..,;.--:-+l4--~-+i-1 1.0 .. 0.6
0.8 0.8
';;0.,6 0.9
F'g. rd Configuration factor (K).1" ' . , ' , ' ", ..
, time
Fig. 12.3 Decay due to damping
very ,difficult to excite bridges with natural frequencies
greater than 5 Hz, and their vibration may be ignored.
Hence the C~de states tha,t, if the natural frequency of the
unloaded bndge exceeds 5 Hz, the vibration criterion is
deemed to, be satisfied.
If the natural frequency lies between 4 Hz and 5 HZ,"
Blanchard, Davies and Smith [107] suggest that the maxi-
mum bridge acceleration should first be calculated using
the natural frequency. This maximum acceleration should
then be multiplied by an attenuation factor which varies
linearly from 1.0 at 4 Hz to 0.3 at 5 Hz, as shown in
~ig. 12.1, to give the maximum acceleration due to a pac-
Ing frequency of 4 Hz. This approach has been adopted in
the Code.
, In th~ above discussion it is implied that it is necessary
to conSider only a single pedestrian crossing a bridge. This
requireme~t ,was proposed by Blanchard, Davies and
Smith [107] by considering some existi~g bridges: for each
bridge the number of pedestrians required to produce a
maximum acceleration just equal to the allowable value of
0:5 If" was calculated. It was concluded that, in order that
the more sensitive, of the existing bridges could be con-
sidered to be just acceptable to the Code' vibration
criterion, theappliep loading should be limited to a single
pedestrian. ." .
Calculat;on of natural frequency
The Code requires that the natural frequency of a concrete
bridge should be determined by ,{,:onsidering ihe uncracked
secti?!l (neglectingthereinforceInent), ignoring shear lag,
but Including the stiffness of parapets. The, Code also
requires· the,shoft-term elastic modulus of concrete to be
used. It would seem appropriate to use the dynamic
modulus, and Appendix B of Part 4 of the Code tabulates
such moduli for various concrete strengths. The tabulated
values are in good agreement with data presented by
Neville [108]. "
Wills [2~1] has used the material and section properties,
referred to In the last paragraph, to obtain good agreement
152
6
4
2
0;~--~1~O~--~20~--~3~0-----L----~
4050
Main span (I) metres
Fig. 12.4 Dynamic response factor ('IjI) for 6 = 0.05
between' predicted and observed natural 'frequencies of
existing bridges. '
The natural frequency of a bridge should be ,calculated,
by including superimposed dead load but excluding pedes~
trian live loading.
For bridges of constant cross-section and up to four
spans, the fundamental natural frequency can be obtained
easily by using, for example, tables presented by Gorman
[262]. The fundamental natural frequency ifo) is given by
[262]
f =o
where
!Ef
/p:4
EI = flexural rigidity •
A = cross-sectional area
P = density
.
,. i,
L = length of bridge ,
(12.1)
~ parameter dependent· on' span arrangement' and
lengths, support conditions and the vibration
mode.
For bridges of varying cross-section, it is necessary to
us
7either a compu~er ~rogram, such as that adopted by
WIlls [261], or a sImphfied analysis based on a uniform
cross-section. In the latter approach, a bridge of varying'
cross-section is replaced by a bridge having a constant
~ross-section with a mass per unit length and flexural rigid-
Ity. equal to the weighted means of the actual masses per
umt length and flexural rigidities of the bridge. Equation
(12.1) can then be used. Wills [263] explains this pro-
cedure and shows that it leads to satisfactory estimates ofthe .
natural frequencies of bridges having cross-sections which
vary significantly. Nine bridges were considered and the
calculated frequencies were within 6% of the measured
frequencies except for one bridge, which had an error of
15%.
. " .,'
Forcei1' ' ,
, ~----"'T-----i-"+~
".
.
25% of
static
load
,"
,...-
A" "
.
, .
Static ~__'~U~.--_J_---1---4-'--':¥-~'='"---
, load
....
Fig. 12.5 Moving pulsating force
Calculation of acceleration
Simplified method For a bridge of constant cross-section
and up to three symmetric spans (as shown in Fig. 12.2),
the Code gives the following formula for calculating the
maximum vertical acceleration (a):
a =4rrF0 Y.• k,,
where
fo = fundamental natural frequency (Hz)
Y. = static deflection (m)
K = configuration factor
'' dynamic response factor
(12.2)
Iffo lies inthe range 4 Hz to 5 Hz, the acceleration calcu-
lated from equation (12.2) should be reduced by applying
the attenuation factor discussed earlier in this chapter.
Equation (12.2) was derived by Blanchard, pavies and
Smith [107] and represents, in a simple form, tbe results of
a study of a number of bridges with different span
arrangements. For each bridge, a numerical solution to its
governing equation of motion was obtained.
The static deflection (y.) should be calculated, at the
midpoint of the main span, for a vertical load of O.7kN,
which represents a single pedestrian.
The configuration factor (K) depends upon the number
and lengths of the spans; values are given in Fig. 12.2.
Linear interpolation may be used for intermediate values of
1111 for three span bridges.
The dynamic response factor ('I) depends upon the main
span length and the damping characteristics of the bridge.
If a bridge is excited, the amplitude of the vibration gradu-
ally decreases due to damping, as shown in Fig. 12.3. The
damping is expressed in terms of the logarithmic dec-
rement «(), which is the natural logarithm of the ratio of
the amplitude in one cycle to the amplitude in the following
cycle. The Code suggests that, in the absence of more pre-
cise data, the logarithmic decrement, of both reinforced
-One foot
---Actual combined
-·-Idealised combined
Right
Time,
and prestressed concrete footbr~dges, should be assumed to
be 0.05. In the past, a larger value has often been taken for
reinforced concrete than for prestressed concrete, because
it was felt that cracks in reinforced concrete dissipate
energy and thus improve the damping. However, accord-
ing to Tilly [264], this view is not supported by experi-
mental data. Furthermore, when considering a bridge, it is
the overall damping, due to energy dissipation at joints,
etc, in addition to inherent material damping. which is of
interest. Tests on existing concrete footbridges have indi-
cated logarithmic decrements in the range 0.02 to 0.1, and
thus the Code value of 0.05 seems reasonable [264].
The relationship between the dynamic response factor
(,,) and the main span length (I) is given graphically in the
Code. The relationship is shown in Fig. 12.4 for the
logarithmic decrement of 0.05 suggested in the Code.
General method For bridges with non-uniform cross-
sections, and/or unequal side spans, and/or more than three
spans, the above simplified method of determining the
maximum vertical acceleration is inappropriate. In such
situations, it is necessary to analyse the bridge under the
action of an applied,.moving and pulsating point load,
which represents a pedestrian crossing the bridge;. The
amplitude of the point load was cho~n so that, when
applied to a simply supported singli~ span bridge, it pro-
duces the same response as that produced by a pedestrian
walking across the bridge [107]. In Fig. 12.5, the force-
time relationship is shown for a single foot. The relation-
ship obtained by combining consecutive single foot rela-
tionships is also shown. It can be seen that the combined
effect can be represented by a sine wave with an amplitude
of about 25% of the static single pedestrian load of
0.7 kN. In fact the Code takes the amplitude to be 180 N
and, as discussed earlier in this chapter, the pacing fre-
quency is taken -to be equal to the natural frequency of the
bridge (fo). Hence the pulsating point load (Fin Newtons).
given in the Code, is:
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![F == 180 sin (211 it,1) (12,3)
where T is the time in. seconds.
It is mentioned earlier in this Chapter that the assumed
stride length is 0.9 m. Thl,ls, ifthe.pacing frequency is
bars, the implications of not considering concrete and pre-
stressing tendons are discussed briefly in a simplistiC man- '.
nero
,t;,Hz, the required velocity (v, in mls) of the pulsating point Concrete
load is given by: .
Concrete in compression can withstand,. for .2 million.'
. V, =0.9 fo (12.4) cycles of repeated loading, a maximum stress' of about
.Wills [261] discusses two metbadsofailalysing a bridge 60% of the static strength if the minimum stress in acycle
.'under the above moving pulsating point.load. On~_Jllethod·-···· is zei'0-·t16~~ The maximum. stress which can be. tolerated
requit(:sa large amount of computer stor~ge.<space and the increases as tfie minimum stress increases. Since Part 4 of
other more approximate method requires much less storage the Code:: specifies a limiting compressive stress of 0.5 feu ..
space'. Wills [261J shows that the approximate method is for concrete at the serviceability limit state (see Chapter
adequate for many footbridges. 4), it is very unlikely that fatigue failure of concrete in
compression would occur. It is thus reasonable for the
Forced vibrations
Up to now in this chapter, only those vibrations which
. result from normal pedestrian use of a footbridge have
. been considered. However, it is also necessary to consider
the possibility of damage arising due to vandals deliber-
ately causing resonant oscillations. It was not possible
[107] to quantify a loading or a criterion for this action;
thus the Code merely gi~es a warning that reversals of load
effects can occur. However, the Code does suggest that,
for prestressed concrete, the section should be provided
with . unstressed reinforcement capable of resisting a
reverse moment of 10% of the static live load moment.
Fatigue
'General
Code approach . .
Although Part 1 of the Code refers to fatigt,le .under the
heading 'ultimate limit state', it is the repeated apph-
cation of working loads which cause deterioration to a stage
where failure occurs; Alternatively, the working loads may
cause minor fatigue damage; which could result in the
bridge being considered unserviceable. Hence, fatigue cal-
culations !lre.catriedout separately from the calculations to
check compliance with the ultimate.and serviceability limit
state criteria:' in Part 4 ()f the Code, fatigue is dealt with
under 'Other considerations'. The design fatigue loading
is specified in Part 10 of the Code and, since it is a design
loading, partial safetyfactors (YjLalid Yf3) do not have to
be applied. '. . .
When.deterrniningth~response ora.bridge .tofatigue
10ading,Part lof the Code requires the use of a linear
elastiC method with>the elastic modulus·of concrete equal
to its short-term value.
With regard to .concrete bridges, Part 4 .of the Code
requires only the fl1tigue strength (or life) of reinforcing
hars to be assessed. Thus concrete and prestressing ten-
dons do not have to be considered in fatigue calculations.
Befote presenting the Code requirements for reinforcing
154
Code not to require calculations for assessing the fatigue
life of concrete in compression.
It should be noted that 2 million cycles of loading
during the specified design life of 120 years are equiv- .
alent to about SO applications of the ·full design load per
day. . .
For concrete in tension, cracking occurs at a lower stress
under repeated loading than under static loading. Cracking
does not occur, in less than 2 million cycles of repeated
loading, if the maximum t~nsile stress does not exceed
about 60% of the static tensile strength, if the niinimum
"stress is zero or compressive [265, 266]. The maximum
tensile stress which can be tolerated without cracking .
increases if the minimum stress is tensile. The reduced
resistance to cracking of concrete subjected to repeated
loading has two implications. .
1. Shear cracks may form at a lower load, with a pos-
sible decrease in shear strength.
2. Flexural cracks may form at a .lower load, resulting in
either cracks in Class 1 or 2 prestressed members, or
cracks wider than thee. allowable values in reinforced
concrete or Class 3 pre'stressed concrete members.
With the partial safety factors for loads and for material'
properties.that ha.ve been adopted in the Code at the ulti~
mate limit state, it is unlikely that principal tensile stresses
under working load conditions.would be great enough to
cause fatigue shear cracks. Thus a bridge, designed ,to
resist static shear in accordance with the Code, should
exhibit adequate shear resistance when subjected .to reo
peated loading.
The interface shear str~ih of' composite members
should also be adequate under repeated loading. Badoux
and Hulsbos [279] have tested composite beams under 2
million cycles of loading. The test specimens were essen-
tially identical to those, tested under static loading. by
Saemann and Washa [118], which.are discussed. in Chapter
4; It was found that, under repeated loading,. the interface
shear strength was reduced. However, the allowable inter-
face shear stresses, which were proposed in [279] for re-
peated loading, exceed the allowable stresses given in the
Code for static loading at the serviceability limit state (see
Chapter 4). Since the latter stresses have to be checked
under the full design load at the serviceability limit state
(i.e. under dead plus imposed loads) it is very unlikely that
interface shear fatigue failure would occur. The require-
. ment to check interface shear stresses under the full design
load is discussed fully in Chapter 8.
Flexural cracking under repeated loading is now con-
sidered.
Reinforced concrete Repeated loading causes cracks to
form at a lower load than under static loading and, subse-
quently, the cracks are wider. However, the author would
suggest that the breakdown of tension stiffening under
repeated .loading has a greater influence. on crack widths
than does the reduction in the load at which cracking
occu~s. Tension stiffening under repeated loading is dis-
c:.:::;sed in Chapter 7, where the author suggests that, as an
interim measure, tension stiffening under repeated loading
should be taken as 50% of that under static loading.
Class I prestressed concrete Flexural tensile stresses are
not permitted under service load conditions and thus re-
peated loading cannot cause cracking.
Class 2 prestressed concrete No flexural tensile stresses
are permitted under dead plus superimposed dead loads
(see Chapter 4), and thus the minimum flexural stress is
always compressive. For repeated loading, the maximum
tolerable tensile stress, appropriate to a compressive
minimum stress, is about 60% of the static tensile strength.
Since the flexural tensile stress permitted by the Code may
be up to 80% of the static flexural tensile strength (see
Chapter 4), it is possible that flexural fatigue cracking
could occur in a Class 2 member designed in accordance
with the Code. It is significant that, iIi CP 115, the allow-
able tensile stresses for repeated loading are about 65% of
those for non-repeated loading; the latter stresses are very
similar to the Code values. It would thus seem prudent to
take about 65% of the Code values when considering re-
peated ioading.
Class 3 prestressed concrete A Class 3 member is
designed to be cracked under the serviceability limit state
design load. Repeated loading may cause the cracks to be
wider than under static loading. However, the permissible
hypothetical tensile stresses in the Code (see Chapter 4)
are conservative in comparisionwith test data [120, 122,
123], and thus excessive cracking under repeated loading
should not occur.
Prestressing tendons
The allowable .concrete flexural, compressive and tensile
stresses specified in the Code imply that the stress range,
under service load conditions, of a prestressing tendon in a
Class 1 or 2 member cannot exceed about 10% of the ulti-
mate static strength.
If it is assumed that the effective prestress in a tendon is
about 45% to 60% of its ultimate strength, then test results
indicate that it may be conservatively assumed that 2 mil-
lion cycles of stress Clm be withstood by a tendon without
failure, providing that the stress range does not exceed
about 10% of the ultimate static strength [265]. It is thus
unlikely that fatigue failure of a prestressing tendon, in a
Class 1 or 2 member. would occur under service load con-
ditions.
In a Class 3 member, designed for the m/lXimum.allow-
able hypothetical tensile stress of 0.25 feu (see Chapter 4),
stress ranges in tendons could be. up to 15% of the ultimate
static strength of a tendon. Fatigue failure of tendons are
thus possible in some Class 3 members.
However, it should be emphasised that, in the vast
majority of Class 3 members, tendon stress ranges ",ill be .
much less than the values quoted above and fatigue fail-
ures would then be unlikely. Nevertheless, it would seem
prudentto ensure that,. for all classes, tendon s~ss.ranges
do not exceed 10% of the ultimate static strength of the
tendon.
It should be noted that the conservative tolerable stress
range of 10% of the ultimate static strength, which is
quoted in the last paragraph, is based upon tests on pre-
stressing tendons in air. However, the work of Edwards
[267, 268] has shown that tendon fatigue strength can be
less when embedded in concrete than in air: for 7-wire
strand, the stress range in concrete was about 8% of the
ultimate static strength as compared with about 13% in air.
Fatigue failure of anchorages and couplers, rather than
of a tendon, should also be considered, since they can
withstand smaller stress ranges [265]. Particular attention
should be given to the possibility of an anchorage fatigue
failure when unbonded tendons are used, because stress
changes in the tendon are transmitted directly to the
anchorages.
Code fatigue highway loading
A table in Part 10 of the Code gives the total number of
commercial vehicles (above 15 kN unladen weight) per
year which should be assumed to travel in each lane of
various types of road. The number of vehicles varies from
0.5 >< 108 for a single two-lane all purpose road to
2 >< 108 for the slow lane of a dual three-lane motorway.
Part 10 of the Code also gives a load spectrum for
commercial vehicles showing the proportions of vehicles
having various gross weights (from 30 kN to 3680 kN)
and various axle arrangements. The load spectrum depicts
actual traffic data in terms of twenty-five typical com-
mercial vehicle groups.
It is obviously tedious in design to have to consider a
number of different axle arrangements, and thus it was
decided to specify a standard fatigue vehicle. The intention
was that each type <?f commercial vehicle in the load spec-
trum would be represented by a vehicle having the same
gross weight as the actual vehicle, but having the axle
arrangement of the standard fatigue vehicle.
The standard fatigue vehicle was chosen to give the
same cumulative fatigue damage, for welded connections
in steel bridges, as do the actual vehicles. However~ John-
son and Buckby [24] have emphasised that equivalence of
cumulative fatigue damage does not occur for shear con-
nectors in composite (steel-concrete) construction. This is
because fatigue damage in welded steel connections is
proportional to the third or fourth po'wer of the stress range
whereas, for a shear connector. it is proportional to the
eighth power. Fatigue damage of unwelded reinforcing
bars is proportional to stress range to the power 9.5 (see
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![Each wheel load 20kN on 200 square or
225 diameter contact area
~/:: :t:3'~0"~0".. '.j:; + +f
1.1800 ~I_' . 600g' '.. ~I. 1800.1
conditions, .and hence the stress range in a. link is always
very small.
Welded bars
General Part 4 of the Code permits the use of welded
bars, Pl'Qyi~ed that the· following Jour requirements' ate.:·
FI~.iiI6;~jit!lclar~ fatig~evehide . m e t : ·
next.section), .and thus the, standard f~tigue vehicle would 1. Welding must be carried out in accordance with Parts,
no~give'e.~Il'yalenceof·.fatigue•.damaioforQJlwelded..b~~....'.....~~-,~·ind:·~i·~f.·tbe· C~.T~k)'feldin3· iJ·.··not.~nnitted .
Howevel",.:it is·appropriate for welded bats. ..",..."""-' 'sinceite~h-reduce fatigue sttengthconsiderably; for
The standard fatigue vehicle Spe9ified .in Part 10 of the example, tack· welding of stirrups to main bars can
Codeha$ agrossweig~t of 320kN, and COrlsi,$ts of four reduce .the fatigue strength of the latter by about 35% ..'
wheel~ on each Of four axles. The vehicle is similar to the (or fatigue life by 75%) [270]..
shortest HB vehicle (see Fig. 3.6) but the transverse .spac- c 2. 'The welded bar is not part of a top slab. This is, pre-
.. . f th .. hi' d'f~ ". h . F' 1:Z 6 '.' sumably, because such bars are subjected to the loeal
109 ~•..... ew ee 8. IS .' I erent, aSs ownm Ig. ". ...•. efflec·t·s' 'o'f' concentrated' wheel loads I'n ad'dl'tl'on to"th"e'
;'Part 10 ·of ihe Code also gives 1:1 simplified load, spec-
trum for use with the standard fatigue veHicle, The sim- global effects of the standard fatigue vehicle.
p1lfied spectrum gives the proportions of total 2.ommercial 3. The stress range is within that permitted by Part 10 of
vehicles for various multiples of the standard fatigue the Code. This pOint is discu~sed in the following sec-
vehicle gross weight of 320 kN. . tions of this chapter.
Compliance clauses for reinforcing bars
Unwelded bars
Part 10 of the Code does nllt give a compliance clause for
unwelded bars: instead maximum allowable stress ranges
under the normal 'static' design loads at the serviceability
limit state are specified in Part 4 of the Code. These ranges
are 325 N/mm2
for high yield bars and 265 N/mm Z for
mild steel bars, and are taken from BE 1173.
T~eauthor is. unaware of the origin of these stress
ranges,but their implications are now considered in a sim-
plistic manner.
Moss [269] has reported that the stress range (or) -
cycles to failure (IV) relationship for a variety ofbigh yield
bars, when the bars are always in tension, is
No/ 5
=1.8x 1029 (12.5)
Thus,,for a stress ra:hge of 325 N/mrh 2, the fatigue life (IV)
is 0.2~X 108
cyclefis. However, a stress range as large as
325 N/mm2 is only likely to occur when there is a reversal
of stress during the' stress '.cycle (I.e. the stress changes
from compression to tension). In sjJch cirCllmstances the
stress range fora particular number of cycles'may be up to
one-third greater th~nthat predicted by equation (12;5).
Hence, the upper limit to equation 02.5) can be expressed
as
Nor
9
:!'> =.~.8 ..x 1 ( l 3 o ( 1 2 . 6 ) .
Thus,lor .....~. ,',stress!an~~' of ·325 N/tltm2, .. ihe{atigue ,life
from. equatlOll (12.'5) IS 3.8 X 108
cycles. Hence, the
fatigue life ofa bar, designed to have the maximum stress
range ofPart 4 of the Code, is likely to be of the order of
0.25 x 108
to 3.8 X 108
cycles. Thus, the specified stress
ranges seem. reasonable.' However, it shQuld be remem-
bered that bending a bar can reduce its fatigue strength by
up to 50% [265) ~ This fact should not affect sheilr links,
since shear cracks are unlikely to occur under service load
156
4. Lap welding should not be used, because adequate
control cannot be exercised over the profile of the root
beads.
Or - N relationships Part 10 of the Code gives Or -N
relationships for the following cases: ' .
1. Welded intersections in fabric, or between hot rolled
bars.
2. Butt weld between the ends of hot rolled bars.
3. Butt weld between the end of a hot rolled bar and the
surface of a plate..
4. Fillet weld between the end of a hot rolled bar .and the
surface ofa plate.
, .
The appropriate design Or .,. N relationship, which corre-
sponds to a 2.3% probabilit~ of failureis:
02.7) .
where K is 1.52 for cases 1 and 2, 0.63 for case 3 and
0.43 for case 4. These relationships were derived, princi-
pally, for structural steelwork connections and their use is
thus restricted in the Code to hot rolled bars. ijowever,
Moss [269] has tested butt welded connections in both hot
rolled and cold worked bars, and he found that the Code
relationship is a satisfactory lower bound fit to his test
data.
The Code gives three methods of using equation 02.7)
to assess fatigue life. The methods are now described
briefly in ascending order of accuracy and complexity.
Full descriptions are 'given in Part 10 of the Code..
1" . , " , i, 1 '" " . :, >', .' ',"
Assessment without damage calculation Equation (12.7) .
has been used in conjunction with the simplified load spec-
trum, referred to earlier in this chapter, to produce graphs
of limiting stress range (OH) against loaded length (L) for
various. categories of road. The loaded length is defined as
the base length of the loop of the point load influence line
which contains the greatest ordinate, as shown in
Fig. 12.7. A full description of the derivation of the
•
Greatest + ve or - ve ordinate
Loaded length (L)
Fig. 12:7 Loaded length for fatigue c'alculation
graphs is given by Johnson and Buckby [24], and the
graphs are given in Part 10 of the Code.
The graph of 0H against L for a butt weld between the
ends of two bars is shown in Fig. 12.8 for a dual three-
lane motorway.
The stresses due to .the standard (320 kN) fatigue
vehicle as it crosses the bridge in each slow lane and each
adjacent lane in tum should be calculated. At a particular
design point on the bridge, the. stress range is the greatest
algebraic difference between these calculated stresses,
irrespective ,of whether the maximum and minimum stres-
ses result from the vehicle being in the same lane. The
stres~ range, so calculated, should not exceed the appro-
priate limiting stress range (OH)'
In some situations (e.g. at an expansion joint) the
stresses calculated due to the passage of the standard
fatigue vehicle should be increased by an impact factor,
which can be up to 25%.
It is emphasised that a load spectrum is not used with
this simplified method of compliance.
Damage calculation. single vehicle method As with the
previous method, a load spectrum is no't used: instead, the
standard fatigue vehicle is applied to each slow and each
adjacent lane in tum. However, instead of just considering
the greatest stress range (as is done in the previous
method), the fatigue damage due to each stress cycle is
assessed by means of a 'damage chart' given in Part 10
of the Code. The total damage is assumed to be the sum of
the damages contributed by each individual stress cycle.
The derivation of the damage charts and an example of
20r-~--__~~~~~~~~~~
1 2 5 10 20 50 100 200
Loaded length L (m)
Fig. 12.8 Limiting stress range for butt welded bars for dual
three-lane motorway
their use are given in Appendices C and D, respectively,
of Part 10 of the Code.
Damage calculation. vehicle spectrum method. This
method requires each vehicle in the load spectrum to be
traversed along each lane, and consideration should be
given to vehicles occurring simultaneously in one or more
lanes. From the resulting stress histories, a stress spec-
trum, which gives the number of occurrences of various
stress ranges, can be produced by the methods described in
Part 10 of the Code. The fatigue damage due to each stress
range can then be calculated from'the Or - N curves given
in the Code. The cumulative damage is assessed by using
Miner's rule, which states that the cumulative damage is
equal to the sum of the individual damages [271]. This
summation should not exceed unity if the fatigue life is to
be considered acceptable.
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![Chapter 13
Terylperature loading:
"ntroduction .', . '~ ;, " .,: ~ .'
A~ explained inChapter 3, it is necessary to considertwo
aspects of temperature loading: overall temperature move-
ments, and differential temperature effects.' The overall
movements ate discussed first.
.Having locllted the point on the structure which does not
move (the stagnant paint), it is simple to calculate the
overall movement at any other point of the structure. If the
articulation of the bridge is not complicated, it is possible
to locate the stagnant point by inspection (e.g. at a fixed
. bearing). However, in .general, it is necessary to consider
the relative stiffnesses of the deck, piers'and foundations
in order to calculate the stagnant point. This calculation is
discussed in detail by Zederbaum [272]. If any of the
overall movements are restrained then stress resultants are
induced in the structure. These stress resultants can be
simply calculated from a knowledge of the restrained
movements.
Since these overall movement calculations are well
known and are not contentious, they are not discussed
further in this chapter.
In contrast, many bridge engineers are uncertain as to /'
how to include differential temperature effects into the
design procedure. In particular, it is not clear how to deal
with cracked Sections, and there are differing views ori the
method of calculation of differential temperature effects at
the ultimate limit state. The Code gives no guidance what-
soever on these matters. In view of this, the remainder of
this, chapter is concerned with differential .temperature
effects, and design procedures are suggested. These pro-
cedur~s are; to a certain extent, based upon current un-
completed research at the University of Birmingham and,
thus, they should be considered as interim measures until'
the research is completed.
. , . " . ' .
Servic~abiHty limit state
Compohent effects
Introduction
In the following a general non-linear differential temper-
ature distribution is considered to be applied to a general
section, as shown inFig; 13. L
158
It is.important to realise that, when a temperature dis-
tribution isappJied to a stnicture, temperatute~iriduced
strains occur, but temp'rature-induced stresses result
only if such strains are restrained, either externally or
internally. .' . .
There are two basic approaches to the determination of
the effects of a non-linear temperature distribution, such as
that shown in Fig. 13.1: the strain method or the stress
method. The two methods are based upon the same
assumptions of structural behaviour, and their'end results
are identical. In view of the fact that temperature loading
is, in structural terms, an applied deformation, the author
prefers the strain method; however, many bridge engineers
prefer the stress method because they are used to working
in terms of stresses, and because it is computationally
more convenient for statically indeterminate structures.
Each method is now presented.
In·the following all stresses, strains and stress resultants
are positive when tensile.
Strain method
Consider the general section and temperature· distribution
of Fig. 13.1.'At a distance z above the bottom of the sec-
tion, the temperature istz " the section breadth is bz, the
elastic modulus is Ez and the coefficient of expansion is ~z;
each is a function of z. The coefficient of expansion may
vary with depth due to different materials existing in the
section. The elastic modulus may vary for two reasons:
1. Different materials existing in the section.
2. Sttesses, due to co-existing force loading, varying
through the depth ofthe section and, thus, being on
different points of the material stress-strain curve.
The potential thermal strain (t;) at Z, in the absence of
any restraint, is
(13.1)
If it is assumed. that plane sections remain plane, then the
section must take up the strains indicated by the dashed
line of Fig. 13.2(a). The latter strains can be defined in
terms of a strain (Eo) at Z = 0, and a curvature ('11). At any
level the difference betwe.en the potential thermal strain and
the strain that actually occurs is a strain which induces a
thermal stress. Since. no external: force is applied to the
. sections, these thermal stresses m~t be self-equilibrating.
'.
z
General
section
Fig. 13.1 Temperature distribution
h
z
(a) . Potential and
final strains
h
Potentla.1
Fig. 13.2(a)-(d) Thermal strain method
Temperature
distribution
(b) Strain
Hence, in the absence of any external restraint, the section
experiences an axial strain, a curvature amI a set of self-
equilibrating stresses, as shown in Fig. 13.2.
At distance z above the bottom of the section, the stress-
inducing strain is
,
Eo + 'IIz - ~ztz
Thus, the stress at z is
f= Ez (Eo + 'IjI z - ~ztz) (13.2)
For force and moment eqilibrium respectively:
(13.3)
S: fbzzdz =0 (13.4)
If equation (13.2) is substituted into equations (13.3) and
(13.4), Eo and '11 can be obtained as
Eo =(F3F4 - F2FS)/FIF3 - Fl)
'11 =(FIFs - F2F4)/(FIF3 - F22)
(13.5)
(13.6)
. '~,:.
(e) Curvature
where
F4 = Soh '" E b t d"z z z z Z
Potential
thermal strains
(d) Self-equilibrating
stresses
(13.7)
Having obtained Eo and '11, the self-equilibrating stresses
can be obtained from equation (13.2).
Uncracked section For the case of a section which is
uncracked, and in which the concrete stresses are
sufficiently low for the elastic modulus to be considered
constant throughout the section, equations (13.5) and
(13.6) reduce to
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![130
Elevation
Fig. J,3;3~ta~k~ds~ction
200
2or4Y16
Section
£O:::;~I [If: bztzdz - A.iS: bztz(z - i) dZ] (13.8)
'v= ~cr: bztz (z - z) dz (13.9)
where Ais the cross-sectional area, Z is the distance of the
section "centroid from' the bottom of the section I is the
secotldmoment of area about the, centroid and 'c¥c is the
~oefficient of expansion of concrete. The strain at the cen-
troid of the section (e) is given by
£';" £" + 'IjI i
¥' :. ~;= C¥c,Sh b'};tzdz (13.10)
A 0, ,,"
.,' ,
Cracked section Tests carried out by Church [273] at the
Transport and Road Research Laboratory, on both cracked
and uncracked sections under the thermal loading shown in
Fig. 13.3, have shown that the free thermal curvature of a
cracked section is, typically, 20% greater than that of an
uncracked section. Furthermore, the thermal curvature can
be, calculatedfrom equation (13.6) by assuming the elastic
modulus of the concrete, within the hpight of the crack, to
be, zero. If it is assumed that the crack extends to the
neutral ~xis of the cracked transformed section, and the
thermal stresses dQ not affect' the position of the neutral
axis, then equations (13.9) and (13.10) become
'IjI ==E:I[~..E.rAl,'(h :"'d- z) +
, c¥cEc S~ bztz (z "" i)dZJ (13.11)
"E = E~[C¥".E.rA.'t.,+ c¥cEc S:bztzdZ]
whe,re
. "'J:::: ,"
~,; ,""" ,coefficieIltof eXpansion of reinforcement'
E.. = elastic modulus of reinforcement
A" = area ofreinf()ccement
t.. , = temperature at level of reinforcement
d:::;effective depth ofreinforcement
160
"I
25°C
A :::; area of cracked transformedsectio~'
I = second moment of area of crllckedtransformed
section about its centroid '
Z = ~istance of centroid of crackedtra~sfo~ed ~."
hon from the bottom of the section. " " .' " ,
. T~sts have not yet been ca~ried out under temperature
distrIbutions which cause the cracked partofthesectiooto'
be hotter than the uncracked part. However, _nsuch cit"
cumstances, the first of equations (13.11) predicts for
temperature differences of the shape shown in Fig. i3.3,
that the free thermal curvature is, typically, 20% less than
that of an uncracked section. . ,
. However, when the Code temperature distribution, with
It~ non-ze~o tem~rature at the bottom of the secti()n.'(see
Fig. 3.2) IS considered, it is found' that the differences
between the calculated free, thermal curvatures 'for the
uncracked and cracked section are much less than 20%
(see E~ample 13.1). Thus the :'author w6uld' suggest that,
for deSign purposes, the free thermal curvature could 00 .
calc~lated from equation (13.9) (which is foran uncracked
sectIOn), irrespective of whether the section is cracked or
uncracked.
Stress method
In the stress method,the section is, at first, assumed to be
fully re~trai~ed, so that" no, displacements. take place, as
shown In Fig. 13.4(a). The streSs fa, at t, due to the
restraint is LJ f2. '" L· I::,.t . iX.
fa =- Ez c¥ztz fo -;: ~.z, E '" t, t- cI - E
Hence, the restraining forceis
F =fh "b dz. }o Z
a
If no longitudinal restraint is,present,iherestrainitlg'
force must be released by application of a releasing force
Fr = -F. The stress (f,) at z due to the releasing force is ,
f, = -E.z J:fobzdz/f: Ezbzdz ,(13.14)
The net stress at z is nOW ifo + jJ).
Potential
+
z
+
h
(a) Potential and
restrained strains
(ti) Restrained
strlilises
(c) Stresses
dueto
relaxing
force
(d) Stresses
dueto
relaxing
moment
(e) Self-equilibrating
stresses
Fig. 13.4(8)-(e) Thermal stress method
The restraining moment (M) can be obtained by taking
moment~ about z :::; 0; if a positive moment is sagging:
M :::; - J: (fa + ft)bzzdz
(13.15)
If no moment restraint is. present, the restraining
moment tnust be released by application of a releasing
moment M, = -M. The stress (f2) at z due to the releasing
moment is
Ii =-Eiz - i) (-M) / f: Ezbz(z - z)2dz
(13.16)
If the section is uncracked, equations (13.12) to (13.16)
become
(13.17)
Ii == -(z:'" i)(-M)II
, .
If the section is cracked, the elastic modulus of the con-
crete, within the height of the crack, should be assumed to
be zero. The equations for F and M then become:
rh'
F:::; -<XsE,.Asts -C¥cEc bztzdz• f
M =c¥sE,.Asts (h - d - z) +
c¥cEc .I ~ bztz(z - z)dz 1
(13.18)
This approach is very similar to that adopted by Hambly
[274] for a cracked section. ....
It can be seen from Fig. 13.4 that the self-equilibrating
stresses can be calculated from
(13.19)
The stresses calculated from equation (13.19) are identi-
cal to those calculated, by the strain method, from equa-
tion (13.2).
Externally unrestrained structure
In a structure which is externally unrestrained, such as a
simply supported beam, the strain £0 and the curvature
'IjI can occur freely. Thus the only stresses in the structure
ar~ the self-equilibrating stresses calculated from equation
(13.2) or (13.9).
However, these stresses do not occur at the free ends of·,
the structure, where plane sections distort and do not reinain
plane. Hence the stresseS build-up from zero to the values
given by equations (13.2) and (13.19). Such a build-up of
stress implies that, in order to maintain equilibrium,
longitudinal shear stresses occur near to the ends of a
member. The calculation of these shear stresses is demon-
strated in Example 13.2.
Externally restrained structure
If the strain £"or the curvature 'IjI is prevented from occur-
ring by the presence of external restraints, then secondary
stresses occur in addition to the self-equilibrating stresses.
The secondary stresses can be calculated by using either a
compatibility method or an eqUilibrium method. The
author feels that the compatibility method 'explains the
actual behaviour better, but the equilibrium method is gen-
erally preferred because it is computationally more con-
venient. The t~o methods are compared in the following
section by considering a two-span beam.
It should be emphasised that bridge decks are two-
dimensional in plan, and thus the transverse effects of tem-
perature loading should also be considered. However, the
same principles, which are illusttated for a one-dimen-
sional structure, can also be applied to two-dimen-
sional structures (see reference [99]).
Compatibility method
The supports of the two-span beam shown in Fig. 13.5 are
assumed to permit longitudinal movement, so that the
strain £0 can occur. However, the free thermal curvature
'IjI is prevented from occurring by the centre support. If the
centre support were absent, the beam would take up the
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![(a) Fr~eDefl.ected shape
t
·lR=3£ItI;//
J21
(b) Force applied to give zeJO displacement atcehtre support
<~.. ,.sel' . ~....
..
(e) .Thermal moments
. Fig.. 13.S(a)-(c) Compatibility method
~r~e thermal curvature, and the displacement at the pos-
~~on .of the centre support would be [2~/2, as shown in
Ig. 13.5(a). It is now necessary to apply a vertical force
R.at the centre support; as shown in Fig. I3.5(b), to
~estore the beam at this point to the level of the centre
support. Hence, for the two-span beam,
R== 48 ErW~/2) / (2/)3 =3EI~/l
T~is force induces the thermal moments shown in
Fig. 13.5«:); the maximum moment, at the support, is
M.i=.R(2/)/4 == 1.5 EN (13.20)
un~f the sect~on is uncracked, the EI value of the
th crack~d ~ectlon should be used in equaribn (13.20). If
,e sechon IS cracked, the author would suggest that the EI
v<llue.of the cracked transformed section should be used in
equatlon~I3.20). In addition, this value of Eland the
neut.ral aXIs depth appropriate to the cracked transformed
section should be used to calculate the secondary stresses
due to the thermal moment. However, although~,could b~
calculated .from equations (13,6) and (13 II) 't' ·b
bl ffi···· . ,I IS pro -
a . y Sl.! ,Clently. accurate to calculate ~ from equation
~13.?) usmg .the second moment of area of the uncracked
st:ctlOn,~s discussed earlier in this chapter.
. ,
Equilibrium method
At e~chsup~rt,the beam is first assumed to be fully
r~stramed ,agamst rot~ti?n but not against longitudinal
~ovement. Thus restral?mg moments (M), given by equa~
.tlon (~3: 15)and shown In Fig. 13.6(a); are set up. The. net
restra~n!ng moments are shown in Fig. I3.6(b), and the
r~st:ammgmoment diagram is shown in Fig. I3.6(c).
Sm~~n~extemaJmoments are applied,itis necessary, for
~qulll~num. to. cancel the end restraining moments by
::Plymg rel~a~mg moments which are equal and opposite
i tlte re~~~~mmgmOmeflts~ as .shown in Fig. 13.6(d). If
be beam IS analysed under the effects of the teleaslng
1: .
I,:
162
M
~)
(a) Restraining moments
)
M
(b) Net restrilining moments
(c) ReStraining moment diagram
(d) Releasing moments
(e) Releasing moment diagram
~
(f) Net thermal moment = (c) + (e)
Fig. 13.6(a)-(f) Equilibrium method
~oment~, the moment distribut.ion s.hown i.n F..ig.. 1~.6(e)..
IS obtamed. The final thermal moments·h .
F' 13 ' S ow In·
Ig. .6(f), are obtained by summing the restraint ._
ments and the distributed releasing moments. The maxi-
mum thermal moment is 1.5M;· by using equations
(l3.12?, (1~.I4) and (13.15), this moment can be shown
to be ~dentlcal to that given,by the compatibility method
(equation (13.20».'
If the secti?n is uncr~cked, the properties •of the
uncracked section should be used to calculate the s·e·· _
d
. . . . . con
ary stresseS due to the thermal moments.
It is m~ntioned earlier in this chapter that, if a section is
cracked, ItS response to thermal loading Catl be calculated
by assuming. the elastic modulus of the concrete, within
the crack heIght, to be zero. Thus, the restraint moment
sho.Uld be caIc~lated f~om the second of equations (13.18)
which was denved usmg thisassumptton. Thepropei1ies .
Of. the cracked transformed section shOUld be used to I u-
1 t . h· ca c
a e t e secondary stresses due to the thernial mame t. . . . n s.
Ultimate limit state
. .
.. ' .
~~en c~~sidering thermal effects under ultimate load con- .
?1t1~ns It IS essen~ial to bearin mind that the tIiermalload-
mg I~ a deformatIOn rather thari a force. T~e significarice
of this can be seen froin Fig. 13.7, whichcompaces the
respon~e of a material ,to an applied St~ssanp an applied
..Attain at lioth tM servIceability and ultimate limit states.
I'
D=D~ad load
L=Live load
T=Temperature load
(a) Serviceability
(b) Ultimate
Fig. 13.7(a),(b) Effects of applied strains
E
Although Fig. 13.7.is presented in terms of stress-strain
curves,the following discussion is equally applicable to
load-deflection or moment-rotation relationships.
It can be seen from Fig. 13.7 that, at the serviceability
limit state, the applied thermal strain results in a relatively
large thermal stress, but, at the ultimate limit state, only a
small thermal stress arises.
In view of the comments made in the last paragraph, it
is essential to consider carefully what the IQads are, and
what load effects result (see also Chapter 3). The author
would suggest that the· applied thermal strains or displace-
ments should be iriterpreted as being nominal loads. These
strains or displacements should be multiplied by the
appropriate '{tL values to give the design loads. The design
load effects are then the final strains.or displacements, and
the stresses or stress resultants which arise from any
restrain'ts. At the ultimate limit state, the stress or stress
resultant design load effects arte very small and, if full
plasticity is assumed, are zero. However, due consider-
ation should be taken at the ultimate limit state of the
magnitudes of the thermal strains or displacements.
Thus, whereas at the serviceability limit state, it is
necessary to limit the .totaI (dead + live + thermal) stress
so that it is less than the specified permissible stress; at the
ultimate limit state, one is more concerned about strain
capacity (i.e. ductility) and it is only necessary to limit the
total (dead + live + thermal) strain so that it is less than
the strain capacity of the material.
To summarise, the author would suggest that thermal
stresses or stress resultants can be ignored at the ultimate
limit state provided that it can be demonstrated that the
structure is sufficiently ductile to absorb the thermal
strains.
The strains associated with the self-equilibrating stresseb
are, typically, of the order.of 0.0001. Such strains are·.very
small compared with .the strain capacity of concrete in
compression, which the Code assumes to be 0.0035 (see
Chapter 4). It thus seems reasonable to ignore, at the ulti-
mate limit state, the strains associated with the self-
equilibrating stresses.
One would expect structural concrete sections to possess
adequate ductility. in terms of rotation capacity, so that
thermal moments could be ignored at the ultimate limit
state. However, it is not clear whether they are also
sufficiently ductile in terms of shear behaviour. Tests,·
designed to examine these problems, are in progress at the
University of Birmingham. To date, tests have been carried
out on simply . supported beams under various combi-
nations of force and thermal 'loading. It has been found that
temperature differences as large as 30c
C. through the depth
of a beam, with peak temperatures of up to 50"C do not
affect the moment of resistance or rotation capacity [273].
Tests on statically indeterminate beams are about to com-
mence to ascertain whether adequate ductility, in
terms of bending and shear, is available to redistribute
completely the thermal moments and shear forces, which
arise from the continuity.
Design procedure
General
The logical way to allow for temperature effects in the
design procedure is to check ductility at the ultimate limit
state, and provide nominal reinforcement to control crack-
ing, which may occur due to the temperature effects, at the
serviceability limit state [275, 276]. However, the Code
does not permit such an approach; thus, the following
design procedure is suggested by the author.
Ultimate limit state
1. Calculate the free thermal curvature and self-
equilibrating stresses using the uncracked section.
2. Because of material plasticity, ignore the self-
equilibrating stresses.
3. Calculate, from the free thermal curvature, the ro-
tation required, assuming full plasticity of the section,
such that no thermal continuity moments occur.
4. If the required rotation is less than the rotation cap-
acity, ignore the thermal continuity moments.
5. If the required rotation exceeds the rotation capacity,
add the thermal continuity moments to the moments
due to the other loads and design accordingly.
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![1000
1000
.100 surfacing
ioooomm2
000 0
(a) Section (b) Nominal
temperatures (OC)
(c) Design
temperatures (OC)
(d) Free thermal
strains (x 108)
Fig. 13.8(~)':'(d) Example 13.1
It should be noted that step 4 assumes adequate ductility
in shear in addition to adequate rotation capacity. How"
ever, experimental evidence of adequate ductility in shear
is not available at present.
Serviceability limit'state
I . Calculate the free thermal curvature (or the restraining"
moment) and the self"equilibrating stresses, using the
uncracked or cracked section as appropriate.
2. Calculate the secondary stresses due to any external
restraints.
3. Compare the total stresses with the allowable values.
In the case of a prestressed member designed as Class
1 for imposed force loadings, it would seem reason"
. able to adopt the Class 2 allowable stresses when con"
sidering thermal loading in addition to the other load"
ings. Similarly, it would seem reasonable to adopt the
Class 3 allowable stresses, under thermal loading, for
a member designed to Class 2 for imposed force load"
ings.
It is worth mentioning that, for a reinforced concrete
section, the effects of thermal loading at the serviceability
limit state'are less onerous when deSigning to the Code
than to the present documents. This is because the Code
does not require crack widths to be checked under thermal
loading. and the Code allowable stresses (0.5 feu for con"
crete and 0.8/y for reinforcement) are greater than those
specified in BE 1/73 (see Chapter 4).
Examples
13.1 Uncracked and cracked rectangular
section
It is required to determine the response at the serviceability
limit state of the section shown in Fig. 13.8(a) to the
application of a differential temperature distribution. The
164
section is identical to that considered by Hambly [274].
The nominal positive temperature difference distribution,
obtained from Figure 9 of Part 2 of the Code, is shown in
Fig. 13.8(b). The concrete is assumed to be grade 30;
thus, from Table 2 of Part 4 of the Code, the short"term
elastic modulus of the concrete is 28 kN/mml
• (It is not
yet clear what value of elastic modulus to adopt, but the
short"term value seems more appropriate than the long"
term value.) The coefficients of thermal expansion of steel
and concrete are each assumed to be 12 x 1O"8fC.
The nominal temperature differences in Fig. 13.8(b)
first have to be multiplied by a partial safety factor (Y,L) of
0.8 (see Chapter 3) to give the design temperature differ"
ences of Fig. 13.8(c). The free thermal strains appropriate
to the design temperature differences are shown in
Fig. 13.8(d); these strains are the design loads.
In the following analysis; the section is considered to be
both uncracked and cracked}, and both the strain and stress
methods are demonstrated.,'
Uncracked
Cross-sectional area = A = 1000 x 1000
= 1 X 108
mml
Height of neutral axis = z.= 500 mm
Second moment of area = I = 1000 x 10003
/12
=83.33 x 10' mm4
f: bztzdz = 1000 [(6.6) (150) +
(1.2) (250) + (1) (200)] = 1.49 x 108
= 1000[(6.6) (150) (440.9) + (1.2) (250) (266.7) -
(1) (200) (433.3)] = 0.43 x 10'
Strain method
From equation (13.10), the axial strain is
£ = (12 X 10"8) (1.49 x 108)11 x 108
= 17.9 X 10"8
150
250
·400
200
-1.0
-2.7 50
(e) Uncrecked (b) Cracked
Fli. 13.9(a),(b) ~lf"equilibrating stresses (N/mm')
Prom equation'(13.9), the curvature is
11' .. (12 x 10·') (0043 x 10')183.33 x 10"
III 61.9 x 10·' mm"l
Bottom fibre strain ::: Eo" 8 ... 'IjIt
= 17.9 x 10·'-
(61.9 x 10"") (500)
::: -13.1 x 10"'
696
'the self-equilibrating stresses, shown in Fig. 13.9(a), can
now be obtained from equation (13.2).
Slress method
From equation (13.17), the restraining force and moment
are:
F =-(12 x 10"') (28 x 10') (1.49 x 10')
= -0.501 x lOIN
M = (12 x 10·')(28 x 10') (0.43 X 10")
= 0.144 x 10' Nmm
The self"equilibrating stresses can now be obtained, from
equation (13.19): they are identical to those, calculated by
the strain method, in Fig. 13.9(a).
Cracked
With an ela"tic modulus of the reinforcement of
200 ~/mm', 'the neutral axis depth, is found to be
.304 mm: Thus! =696 mm.
Area of transformed section .. A
.. (1000) (304) + (200128) (10 000)
'. 0:3754 'x 10' mm'
Second moment uf area of transformed section .. 1
.. (1000) (304)·/3 + (200128) (10 000) (646)'
.. 39.17 x 10' mm4
f: E,b,t,d:
-1£,.4;1. +Ii E,.bt,d:
.= (200 X 103
) (10 000) 0.5)
+ (28 x 103)(1000)[(6.6) (150) + (1.661) (154)]
= 37.88 x 10'
~ E,As/6 (h - d - 2) + S~ Ej:Jtzzdz
=(200 x 103) (10 000) (1.5) (-646) +
(28 x 103) (1000) [(6.6) (150) (245) +
(1.661) (154) (88.4)]
= 5.487 X 1011
Strain method
From equations (13.11), the axial strain and curvature are
(12 x 10"8) (37.88 x 101)
! =(28 x 103) (0.3754 X 108)
= 43.2 X 10"8
(12 x 10-6)(5.487 >(1011
)
'IjI = (28 x 103) (39.17 X 10")
=60.0 x 10"9 mm" I
Thus the cracked free thermal curvature is only 3% less
than the uncracked free thermal curvature.
Bottom fibre strain = Eo =i! - 'IjI i
::: 43.2 X 10"8 - (60 X 10"1) (696)
= 1.4 x 10"8
The self"equilibratlng stresses, shown in Fig. 13.9(b). can
now be obtained from equation (13.2). It can be seen that,
after cracking, the extreme fibre compressive stress is
reduced by 17% but the peak tensile stress is increased by
29%; however, the tensile stress is small.
Stress method
From equations (13. 18), the restraining forge and moment
are:
F =-(12 X 10"8) (37.88 x 10')
= -0.454 x lOG N
M =(12 x 10"6)(5.487 x 1011)
=65.8 X 108
Nmm
The self"equilibrating stresses can noW be obt~ll1ed from
equation (13.19): they are identical to those calculated by
the strain method. in Fig. 13.9(b).
Design load effects
If no extemal restraints are applied, the load effects are the
axial strain, the curvature and the self"equilibrating stres·
ses. These should be multiplied by the appropriate value of
.Y/3to give the 'design load effects: In fact, ypls unity at the
. serviceability limit state (see Chapter 4). Thus the stresses
shown in Pig. 13.9 are the stresses which should be added
to the dead and live load stresIes, aM the net' stresses
coml,ared with the allowable values.
16~](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-90-2048.jpg)
![2.2
0.75
Fig. 13.10(a):-(c) Example 13.2
',' '", .
13.2B6)(:girder )i
A prestresSttdconcrete co~tinuous .viaJ~ct~~ith spans of
45 m, ha~ the cross"section shown in Pig. 13.10(a). The
concrete IS of grade SO. It is required tOdetermfue the
longitudinal effe9ts of a positive temperature,distribution at
both the serviceability and ultimate limit stat~s.', :i
General
;1
;.
•
Since the sectionis prestressed, it isassumedtobe ~ncracked.
Cross"sectional area = A = (12) (0.25) + (5)(0;4) +
(2) (0.75) (1.5S)
= 7.325 ml
'1
First moments of area about ~offit to determine i:
Z = [(12) (0.25) (2.075) + (5) (0.4) (012) +,
(2) (0.75) (1.55) (1.175)]/7.3'25 •
= 1.277 m
Second mo.m~nt of area =; I
= (12) (0:25)~/I2 + (12) (0.25) (0.798)2 +
(5)(0.4)3/12 + (5) (0.4) (1.077)2 +
(2)(0.75) (1.55)3/12 +
= 4.762 m4
,,(2) (0.75) (1.55) (0; 102)2
Prom Table 2 of ,Part 4 of the Code, elastic modulus of
concre~e =: E == 34 X 103 N/mm2
• The, coefficient of
expansion JS 12 x W-8/°C.
(: ~.,
Serviceability limit state ", '
T~e norri~nal" tempera~~re ',difference ~istribution, from
Figure 9 ?f Part 2 of ttt;e Cdde, is shown in Pig. d.l0(b).
These te~peratures h~vetObe multiplied by 0.8 (see
Exa~ple 13.1) to obtain the design temperature differences
of Fig. I3.W(c).
166
f ~
(bl Nominal (el Design
'temperatures temperatures
, ~) ~"
I~' .. ~t. .
II: (12) (<t'~1?5) (6~6) +:(12) (0.1) (L92) +
(2) (0.75~ (0:15) (0.72) + (5) (0.2) (1.0)
= 15.35 'y:
= (12) (0.15) (6.6) (~.8639) +
(12) (0.1) (1.92) (0.7276) +
(2) (0.75) (0.15) (0.72) (0.623) +
(5) (0.2) (1.0) (- 1.21)
= 10.83
Using equations (13.9) and (13.10) (for the strain method)
the curvature and axial strain are: • '
", = (12 X 10"8) (10.83)/4.762'
= 27.3 X 10"8 mOl
f: = (12 X 10"8) (15.35)17.325
J.
, '= 25.1 )f10"8
The soffit strain is
Eo = £ -'IIi =25.1 X 10-6 _'
, (27.3 X 10"8) (1.277) =-9.7 x 10"'
The self"equilibrating stresses, shown in Pig. 13.11(a),
can now beobtained from equation (13.2). If it is assumed
that the articulation is such that the axial strain can occur .
freely, then the only secondary stresses to occur are those
due to the restraint to the curvature. This restraint produces
the thermal be~ding moment diagram,' shown. in Pig.
13.12. The maximum thermal moment occurs at the first
interior support and is ' , ' ,
Ms = 1.27EI '" == (1.27)(34 x lQ6) (4.762) x
(27.3 x 10-6)
=5614 kNm
The secondary stresses due to this moment are shown in
Pig. I3.1l(b). The net stresses, which are obtained by
adding the secondary. stresses to the self"equilibrating
stresses, are shown in Pig. I3.11(c): '
-2.7
, 1.1
(a) Self-equilibrating
0.15
0.25
1.6
0.2
FIR. 13.11(a),(c) Thermal stresses (N/mm')
•
(bl Flexural
secondary
0.2~r
1.3
-3.8
-0.3 ;
(c) Net
0.4
1.0
1.2
Free thermal curvature =til
1 1 r r r r I
EItII
1.27EltII 1.27EIw
Fig. 13.12 Thermal moments in mUlti-span viaduct
Free thermal curvature
. PI..t;,~K<>'~hinges ~~~
r
Failing span
"
1/2 _I_ 1/2 ~I
F.g. 13.13 Thermal rotations in multi-span vi~duct
Longitudinal shear stresses At the free end of the via-
duct, the self-equilibrating stresses do not occur, but
longitudinal shear stresses occur in the zone within which
the self-equilibrating stresses build-up. The shear stress is
greatest at the web-cantilever junction. The average com-
pressive stress in the cantilever, away from the free end,
is, from Pig. I3.11(a), (2.7 - 0.6)/2 = 1.1 N/mm2
• In
accordance with St. Venant's principle this stress will be
assumed to build-up over a length, along the span, equal to
the breadth of the cantilever (i.e. 3.5 m). Then the average
longitudinal shear stress at the web-cantilever junction is
equal to the average compressive stress in the cantilever
(i.e. 1.1 N/mm2
).
Transverse effects In practice, the transverse effects of
the temPerature distribution should also be investigated.
Ultimate limit state
The partial safety factor YfL at the ultimate limit state is
1.0 (see Chapter 3). ,Hence the free thermal curvature at
the ultimate limit state is (27.3 x 10-8) (1.0(0.8) =
34.1 x 10-6 mOl. The worst effect that this pOsitive curva-
ture can have on the structure at tIle ultimatelimit~ state is
to cause rotation in a plastic hinge at the centre of Ii failing
span, as shown in Pig. 13.13. This is because such a
thermal rotation is of the same sign as the rotation due to
force loading. (If a negative temperature difference, dis-
167 .](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-91-2048.jpg)
![tribution were being considered, the thermal rotation at a
support hinge would be calculated.)
The total thermal rotation (a,) is given by
a, ='11112
where I is the distance between hinges. Thus a, =
(34.1 x 10-8
)(22.5) = 0.77 X 10-8
radians. To obtain the
design load effect this rotation must be multiplied
by' . Yp, which is L 15 at the ultimate limit state. Thus
the design rotation is 1.15 x 0.77 x 10-3
= 0.88 X 10-3
radians.
168
The Code does not give permissible rotations, but the
CEB Model Code [110] gives a relationship between per-
missible rotation and the ratio of neutral axis depth to
effective depth. It is unlikely that the permissible rotation
would be less than 5 x 10-3
radians. This value is much
greater than the design rotation. Thus,' unless the moment
due to the applied forces is considerably redistributed away
from mid-span, the section has sufficient ductility to
enable the thermal moments at the ultimate limit state to be
ignored.
Appendix A
Equations for plate design
Sign conventions
The positive directions of the applied st,ress resultants ~r
unit length and the reinforcement directtonex are shown 10
Fig. AI.
Stress resultants with the superscript* are the required
resistive stress resultants per unit length in the reinforce-
ment directions x. y for orthogonal reinforcement and x,
ex for skew reinforcement.
The principal concrete force per unit length (Fe) which
appears in equations (AI9), (A21), (A23), (A26), (A28)
and (A30) is tensile when positive.
Bending
brthogonal x I y reinforcement
Bottom
Generally
M; = Mx + IMxyl
M: = My + IMxyl
If M; < ()
M;=O
M* = M + M~y y y Mx
If M; < 0
M; =0
M; =Mx+I~~1
Top
Generally
M; = Mx -IMxyl
(AI)
(A2)
(A3)
(A4)
(AS)
Fig. A.I Slab element
M; = My -IMxyl
If M:> 0
M;= 0
M; = My
If M;> 0
M;=O
-1~M~x'
M;. = Mx -I~
Skew x I ex reinforcement
Bottom
Generally
M: = Mx + 2Mxy cot ex + My cot
2
ex +
Mxy.+ My cot ex Ism ex .
M Mxy + My cot ex
M*ex =sin! ex + sin ex
If M:< 0
M;=O
(A6)
(A7)
(AS)
(A9)
(AlO)
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![* :::_1_.(M +1 Mx + M cot (X)Z ) (All)
Me'( sinz (X .y (Mx + 2Mxy cot (X + My cot (X
If M;< 0
M~ =0
M: = Mx + 2Mxy cot (X + My cot 2(X +
Top
Generally
I(MXY +M~y cot(Xfl
M; = Mx + 2Mxy cot (X + My cot (X -
IMxy +.My cot (X Ism (X
M~ = My. -IMxy +,My cot (X Isin2 (X sm (X
If M;> 0
M;=O
(A12)
¥(A13)
(A14)
M~ = 1 (M _/ (Mxy + My cot (X)Z I)sin2(X y (Mx + 2Mxy
cot (X + My cot2(X) (A15)
If M*" > 0
M~=O
M: = Mx + 2Mxy cot (X + My cot2(X -
I
(Mxy + My cot (X)Z/
My (A16)
In-plane forces
Orthogonal X, Y reinforcement
Generally
N: = Nx + INxyl
N; = Nv + IN.•y I
/.;, = -2 INxyl
If N. < 0
Ni =0
170
(AI7)
(AIS)
(A19)
Skew X, ~ reinforcement
Generally
N: = Nx + 2Nxy cot (X + Ny cotZ
(X +
INxy + Ny cot (X Ism (X
N' = .Ny . + INxy +,Nycot (X I0< 5m2(X sm (X
Fc =-2 (Nxy + Ny cot (X) (cot(X ± cosec (X)
(A20)
(A2l)
(A22)
(A23)
(A24)
(A25)
(A26)
In equation (A26), the sign in the last bracket is the same
as the sign of (Nxy + Nycot (X).
If N; < 0
N;=O
N; =_1_ (N +1 (Nxy +Ny cot (X)Z
sinz (X y Nx + 2Nxy cot (X t Ny cot
Fc =(Nx + Nxy cot (X)Z + (Nxy + N~ cot (X)Z
Nx + 2Nxy cot (X + Ny cot (X
If N~ < 0
N:=O
N; = Nx + 2Nxy cot (X + Ny coe ~ +
I
(NXY + Ny cot (X)21
Ny
(X ) (A27)
(A28)
(A29)
(A30)
....-~---.- ..-,.~ ...~.- -_.._--_._--_.•.. _.....-...--..-....... .
Appendix B
Transverse shear in cellular
and voided slabs
Introduction
It is mentioned in Chapter 6 that no rules are given in the
Code for the design of cellular or voided slabs to resist
transverse shear. In this Appendix, the author suggests
design approaches at the ultimate limit state.
All 'stress resultants in the following are per unit length.
Cellular slabs
General
The effect of a transverse shear force (Qy) is to deform
the webs and flanges of a cellular slab, as shown in
Fig. B.l(b). Such deformation is generally referred to as
Vierendeel truss action. The suggested design procedure is
initially to consider the Vierendeel effects separately from
those of global transverse bending, and then to combine
the global and Vierendeel effects.
Analysis of Vierendeel tru~s
Points of contraflexure may be assumed ,at the mid-points
of the flanges and webs. Assuming the point of contraflex-
ure in the web to be always at its mid-point implies that
the. stiffnesses of the two flanges are always equal, irres-
pective of their thicknesses and amounts of reinforcement.
However, a more precise idealisation is probably not
justified. The shear forces are assumed to be divided
equally between the two flanges to give the loading, bend-
ing moment and shear force diagrams of Fig. B.l.
Design
, Webs
A(;eb can be de~igned as a slab, in accordance with the
methods described in Chapters 5 and 6, to resist the bend-
ing moments and shear forces of Figs. B.l(c) and (d),
respectively.
Flanges
The flanges can be designed as slabs, in accordance with
the method descri~ed in Chapter 6, to resist the shear'
forces of Fig. B.l(d). .
In addition to the Vierendeel bending moments, the
global transverse bending moment (My) induces a force of
Mylhe, where he is the lever arm shown in Fig. B.1(a), in
both the compression and tension flanges. Thus, each
flange should be designed as a slab eccentrically loaded by
a moment Q"s/4 (from Fig. B.l(c» and either a compres-
sive or tensile force, as appropriate, of Mylz.
Voided slabs
General
The effect of a transverse shear force is to deform the webs
and flanges of a voided slab in a similar manner to those of
a cellular slab. However, since the web and flange thick-
nesses of a voided slab vary throughout their lengths,
analysis of the Vj~rendeel effects is not readily carried out.
In view of this, a method of design is suggested which is
based on considerations of elastic analyses of voided slabs
and the actual behaviour of transverse strips of reinforced
concrete voided slabs subjected to shear [277]. The sug-
gested ultimate limit state method is virtually identical to
an unpublished working stress method proposed by Elliott
[278] which, in tum, is based upon the test data and design
recommendations of Aster [277J. Although Aster's tests to '
failure were conducted on transverse strips of voided slabs,
a similar failure mode has been observed in a test on a '
model voided slab bridge deck by Elliott, Clark and ~
Symmons [71].
The design procedure considers, independently, possible<!
cracks initiating on the outside and inside of a void due to L
the Vierendeel effects of the transverse shear. The latter
)
"
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![Idealised Vierendeel truss
0.)2
~
~
I~
0
r;-
.., I~
Oyl2
/
~
s
(a) Section
OyS/2 - ___
0.)2
i.....
:~
-/ ,
Oyl2
""--
---_ OyS/2
(e) Bending moments
Fig. B.I(a)-(d) Cellular slab
Oyl2
d
Possible
Oy/2j crack ~ Oyl2
~ ~~~~--------~
o
Compression
;.:..:..:5
he (
",,~ ..,~.,
AI
0.)2
Oyl2
00
(b) Loading
----.--~
------~~~--~
(d) Shear forces
CDI '.
I I :,
I I,
I.Htdl4
I I
I I
I I
I I
I I
0:
Elastic bottom~____-I-_~
Bottom
reinforcement
strain
I
I
fibre stresses
~~
(a) Elastic stresses
Fig. B.2(a),(b) Vierendeel stresses in voided slab [277]
effects and the global transverse bending effects are then
combined.
.In the following, the global transverse moment (My),
co-existing with the transverse shear force (Qy) , is
assumed to be sagging.
172
(b) Measured reinforcement strains
Bottom flange design
Elastic analysis of the uncracked section shows that the
distribution of extreme fibre stress, due to Vierendeel
action, is as shown in Fig. B.2(a) [277]: the peak stress
Oyl2
000.)2
Fig. B.3 Bottom flange of voided slab
00
Fig. B.4 Top flange of voided slab
occurs at about the quarter-point of the void (i.e., at d/4
from the void centre line, where d is the void diameter).
Thus a crack may initiate, from the bottom face of the
slab, at this critical section.
It has also been observed that peak bottom flange rein-
forcement strains, in cracked concrete slab strips, occur at
about d/4 from the void centre line. This is illustrated by
Fig. B.2(b), which shows some of Aster's. measured bot-
tom reinforcement strains in a reinforced 'concrete trans-
verse strip.
Fig. B.2(a) shows that the Vierendeel bending stress at
the centre line of the void is zero; hence, only a shear
force acts at this section, as shown in Fig.' B.3. It is con~
servative, with regard to the design of the reinforcement in
the bottom flange, to assume that the shear force (Qy) is
shared equally between the two flanges. In fact, less than
Q,J2 is carried by the bottom flange because it is cracked.
Thus the Vierendeel bending moment at the critical sec-
tion, d/4 from the void centre line, is:
Mv =(Q,J2) (d/4) =Qyd/8 (B.l)
The bottom flange reinforcement is also subjected to a
tensile force of (Mylz), where My in this case is the maxi-
mum 'global transverse moment and z is the lever arm for
global bending shown in Fig. B.3. The resultant compres-
sive concrete force (C) in the top flange is considered to
act at mid-depth of the minimum flange thickness (t),
because the design is being carried out at the ultimate limit
i Reinforcement
Critical section
Critical
secfion
O.5t
z
state and the concrete can be considered to be in a plastic
condition. •
The bottom flange reinforcement should be designed for
the combined effects of the force M,Jz and the Vierendeel
momentMv• The section depth should be that at the critical
section.
Top flange design
The extreme top fibre stress distribution, due to Vierendeel
action, is similar in form to that, shown in Fig. B.2(a), for
the bottom fibre. Thus, due to Vierendeel action, a crack
may initiate, from the top face of the slab, at the critical
section (distance d/4 from the void centre line). The
Vierendeel bending stress is again zero at the centre line of
the void, but it is now conservative, with regard to the design
of the reinforcement in the top flange, to assume that all of
the shear force is carried by the top flange. This assump-
tion implies that the bottom flange is severely cracked due
to global transverse flexure and cannot transmit any shear
by aggregate interlock or dowel action.
The Vierendeel bending moment at the critical section is
(see Fig. B.4):
(B.2)
The top flange is also subjected to a compressive force
of (M,Jz) which counteracts the tension induced in the top
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![Maximum tensile stress'" KQylh
d/h = 0.800
d/h =0.775
d/h =0;750
d/h = 0.725
d/h =0.650
200)-----~----~~----~----_+·----~1
2 3 4 5
( ) M
. My/Qyh
a aXlmum tensile stress at face of void
h
(c) Section
Fig. B.S(a)-(e) Maximum tensile stress at face of void
flange reinforcement by the Vierendeel moment My.
Hence, the greatest tension in the reinforcement is
obtained when My is a minimum.
The top flange should be designed as an eccentrically
loaded column (see Chapter 9) to resist the compressive
force (Mylz), which acts at t/2 from the ~QP face, and the
moment Mv. The depth of the column should be taken as
the flange thickness at the critical section.
Detailing of flange reinforcement
The areas of flange reinforcement provided should exceed
the Code minimum values discussed in Chapter to, and
the bar spacings should be less than the Code maximum
values discussed in Chapters 7 and to.
Web design
It is desirable to design the section so that the occurrence
of cracks initiating from the inside of a void is prevented,
because it is difficult to detail reinforcement to control
such cracks.
Elliott [278] has produced graphs which give the maxi-
mum tensile stress on the inside of a void due to com-
bined transverse bending and shear: it is conservatively
ass~m~d that all of the· shear is carried by the top flange.
Elltott s graphs are. reproduced in Fig. B.S.
The maximum tensile stress obtained from Fig. B.5(a)
should be compared with an allowable tensile stress. The
author would suggest that the latter stress should be taken
as 0.45 /Tcu: the derivation of this value, which is the
174
20~____~____~.-____*-____-+____~
c
0 2. 3 4 5
(b) Location of maximum tensile stress MylQyh
Effective
depth
14
s ~I
Fig. B.6 Vertical web reinforcement in voided slab
Code allowable flexural tensile stress for a Class 2 pre-
tensioned member, is given in Chapter 4.
Tensile stresses less than and greater than the allowable
stress now have to be considered.
Tensile stress less than allowable
Cracking at the inside of a void would not occur in this
situation, and vertical reinforcement in the webs should be
provided.
The work of Aster [277] indicates that the design can
be carried out by considering the Vierendeel truss of
Fig. B.l(b), for which the horizontal shear force at the
point of contraflexure in the web is Q"slhe. The critical
section for Vierendeel bending of a web is considered to
be at d/4 above the centre line of the void, .as shown in
Fig. B.6. The Vierendeel bending moment at this critical
section is:
-------_.- • _ _ _0 __- " _ _ • • • • _ _- - -
h"
C=Concrete strut
T =Reinforcement tie
.lg. B.7 lncli~ed web reinforcement in voided slab
(B.3)
Reinforcement at the critical section, with the effective
depth shown in Fig. B.6, can be designed to resist the
momentMv·
The vertical reinforcement in the web is most con-
veniently provided in the form of vertical links, as shown
in Fig. B.6-; however, only one leg of such a link may be
considered to contribute to the required area of reinforce-
ment. This area should be added to that required to resist
the longitudinal shear to give the total required area of link
reinforcement.
Tensile stress greater than allowable
If the tensile stress obtained from Fig. B.5(a) is greater
than the allowable stress, cracking will occur on the inside
of the void. In this situation, it is preferable to reduce the
size of the voids, so as to reduce the tensile stress, or to
alter the positions of the voids in the deck, so that they are
not in areas of high transverse shear. If cracking is not
precluded by either of these means,· it is necessary to
design the voided slab so that reinforcement crosses the
crack, which initiates on the inside of the void. This can
be done either by providing inclined reinforcement in the
webs, or by providing a second layer of horizontal re-
inforcement in the flange, close to the void.
Inclined reinforcement The forces acting in -a web are
shown in Fig. B.7. The horizontal shear force at the point
of contraflexure of the web is Q,slhe (see discussion of
vertical web reinforcement). For horizontal equilibrium
,
(T + C) cos IX =Q,sllze
But T = C, from vertical equilibrium; thus
T =Q,sI2he cos IX (B.4)
Critical section
for bottom laver
of top flange
reinforcement
Fig. B.8 Additional horizontal reinforcement in voided slab
Inclined reinforcement should be designed to resist this
force. The reinforcement could take the form of. for
example, inclined links or bars: the latter should be
anchored by lapping with the top arid bottom flange re-
inforcement.
Additiona/horizontal reinforcement As an alternative to
inclined reinforcement an additional layer of horizontal
reinforcement may be provided as shoWn in Fig. B.S. The
critical section for designing this reinforcement should be
taken as the position of maximum tensile stress, obtained
from Fig. B.5(b). The latter figure gives the position in
terms of the angular displac~ment (cp): its horizontal dis-
tance from the void centre line is thus d sin cj) 12. It is
conservative to assmne that all of the transverse shear
force is carried by the top flange and thus, from Fig. B.S,
the Vierendeel bending .moment at the critical section is:
My =Q,.d sin cj)/2 (B.S)
The top flange is also subjected to a compressive force
of (Mlz), which counteracts the tension induced in the
reinforcement by the Vierendeel moment My. Hence, the
greatest tension in the reinforcement is obtained when My
is a minimum.
The critical section should be designed as an eccen-
trically loaded colm (see Chapter 9) to resist the com-
pressive force (Mylz),which acts at tl2 from the top face;
and the moment My. The depth of the column should be
taken as the flange thickness at the critical section.
Effect of global twisting moment
A global twisting moment induces forces in the flanges;
these forces can be taken into account in the suggested
design methods by replacing My throughout by M;
(obtained {lom the appropriate equation of Appendix A).
175](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-95-2048.jpg)
![bending, 59-60, 169-70
bridge, 11
cellular, 12-13, 17, 19,70, 171
composite, 11, 12-13,27-9,51, 107-8, 110
cracking, 48, 91, 92-3
Hillerborg strip method, 20
in~planeforces, 6O~1, 170
membrane action, 26
prestressed, 61, 95
shear, 69-72, 75
skew, 23, 24, 29-30, 61, 63
stiffness, 16, 18-19
stresses,' 87~8
voided, 11-12, 16-17, 19,70,72,91-2, 171-5
yield line theory; 22-6, 27-31, 130
snow load, 38
stiffness, 9, 10
axial,9
. flexural, 9, 14, 16-19
for grillage analysis, 18-19
for plate analysis, 16-18
of beam and slab, 17, 19
of cellular slab, 17, 19
of discrete boxes, 17, 19
of slab, 16, 18-19
of voided slab, 16-17, 19
shear, 9, 12, 14, 16-19
torsional, 9, 10, 13, 14, 16-19
stress limitation, 48-51
concrete in compression, 49, 86:"'8, 94-S, 98-101,
108
concrete in tension, 49-50, 51, 94-5, 98-101, 108
interface shear, 50-1, 108-9, 117
prestressing tendon, 48
reinforCement, 48, 86-8, 98-9
stress-strain
concrete, 45, 49
design, 48
prestressing tendon, 46
reinforcement, 46
superimposed dead load,.32, 35
temperature, 32, 36-8, 158-68
combination of range and difference, 38
difference, 36-8, 158-68
frictional bearing restraint, 36
range, 36
seJ:Viceability limit state, 158.,..62, 164-7
ultimate limit state, 162-3, 163-4, 167-8
. tension stiffening, 86, 89-90, 97 .
beam, 90
.flange, 92-3,94
top hat beam, 13
torsion, 75-.83
186
box section, 80-1, 85
compatibility, 76
cracking, 76, 77, 78
equilibrium, 75-6
flanged beam; 79-80
prestf!;lssed concrete, 81-3, 85 ./
rectangular section, 76-9, 84-5
reinforced concrete, 76-81, 84...85
reinforcement, 77~8, 79,80-1
segmental construction, 82-3
traffic lane, 34
U-beam,13
ultimate limit state, 4
analysis, 10
beam.55-8,65~9,7S-83
column, 119-24
.... ' composite construction, 105'-8
design criteria, 48
fleXUre, 5'-60, 61"':3
footing, 130-1
in-plane forces, 60-1, 63-4
pile cap, 131..,.3
plate, 58-61
shear, 65-15, 171-5
slab, 58-61, 69-75 .
temperature effects, 162-3, 163-4, 167-8
torsion, 75-83 .
upper bound method, 19,20, 22-7
verge loading, 41
verification, 7
vibration, 4, 42, 48, 52, IS1-4
voided slab, 11-12
cracking, 91-2
shear, 70, 72, 171-S
stiffness, 16-17, 19
wall, 125-9
cracking, 94, 125, 128-9
effective height, 126
plain, 126-9
reinforced, 125
retaining, 94
shear, 128 t.
short, 125, 126-7
slender, 125, 127-8
slenderness, 125, 126
stresses, 125
wing, 94, 129-30
welding, 156
wheel load
dispersal, 40
effective pressure, 40
HA, 38,40
HB,40
wind load, 32, 35-6
wing wall, 94, 129-30
working lane, 34
working stress design, 2-3
yield line theory, 10,22-6, 53
abutment, 130
composite slab, 27-9
slab bridge, 23-6, 27-30
top slab, 26, 31
wing wall, 130
•
~ ...•..-.----- ..-...•~-.-~--..._.'. . .._---_......_._...._..__._------_....•........_..._......._...._---------
Chapter 1
Introduction
Code format (p.1)
All ten parts of the Code have now been published.
However. Part 9 on bearings has been published in two
sections: 9.1 Code of Practice for design of bridge
bearings; and 9.2 Specification for materials, manufac-
ture and installation of bridge bearings. Since Purt 9 has
now been published. Appendix F of Part 2. which
covered bearings. has been deleted.
Highway bridges (p.2)
The Department of Transport (DTp) has published an
implementation document for each Part of the Code
except Parts 7 and 8, for which the DTp requires its
own specification document PO) to be used. However.
at the time of writing, the latter document is being
revised.
The relevant DTp implementation documents are:
BD 15/82. General principles for the design and con-
struction of bridges. Use of BS 5400 : Part
1 : 1978 [281].
BD 14/82. Loads for highway bridges. Use of
BS 5400 : Part 2 : 1978. (Including amend-
ment No.1) [282]. '
BD 24/84. Design of concrete bridges. Use of
BS 5400 : Part 4 : 1984 [2H3J.
BD 20/83. Bridge. bearings. Use of BS 540() : Part
9 : 1983 (284).
BD 9/81. Implementation of BS )400 : Part
to : 1980. Code of Practice for fatigue
[285].
BA 9/81. Usc of BS 5400 : Part 10: 19XO: Code of
Practice for fatigue. (Induding Amendment
No.1) (286).
Characteristic strengths (p.4)
In addition to characteristic strengths, characteristic
stresses have now been introduced. These are defined
as the stresses at which material stress-strain curves
become non-linear.
Design load effects (p.5)
Values of Yj3 in Part 4 are no longer dependent on the
type of loading.
1](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-101-2048.jpg)
![Chapter 2
Analysis
General requirements (p.S)
Part 4 now states that the ef(ects of shear lag on
structural ari~lysis may be neglected. except for cable-
stayed superstructures. at both the serviceability and
ultimate limit states.
Serviceability limit state (p.S)
Axial. torsional and shearing stiffnesses may now be
based on the concrete section ignoring the presence of
reinforcement (as before). or the gross section includ-
ing the reinforcement on a m~dular ratio basis.
The tabulated short'-term elastic modulus of concrete
shOLJld be used for analysing a structure under applied
forces. However: when analysing a structure under
applied deformations. a modulus intermediate between
the tabulated value and half that value should now be
used. The; value adopted should reflect the proportions
of permanent and short-term deformations.
Ultimate limit state (p.10)
In BO JS/S2. the DTp seems to imply that plastic
methods of analysis arc those based only on considera-
tions of collapse mechanisms. In addition, Part 4
permits the usc of plastic methods only with the
agreement of the relevant bridge authority.
The stiffnesses used in an elastic analysis at the
ultimate limit state should be based on the same section
properties as those used at the serviceability li'mit state.
The load effects due to restraint of tors,ional and
distortional warping may be neglected for longitudinal
memhers.
Local effects (p.1 0)
The statement that the worst loading case occurs in the
regions of sagging moment under load combination 1
2
has been removed..It is now necessary to determine the
most adverse combination of global and local effects.
Hillerborg strip method (p.20)
This method is no longer referred to in the Code.
Moment redistribution (p.20)
The clauses on moment redistribution in Part 4 : 1984
are very much different to those in Part 4 : 1978. The
following four conditions now have to be met.
1. Adequate rotation capacity has to be verified at
sections where moments are reduced either by
reference to test data or by calculating the rotation
capacity as the lesser of: .
0.008 + 0.035 (0.5 - dcld,,) (2.68a)
or, for reinforced concrete
0.6 t1> /(d - de) ' ( 2 . 6 8 b )
or. for prestressed concrete
to/(d - de) (2.68c)
but {O, and *0.015
where de' = calculated depth of concrete in com-
pression at the ultimate limit state
d,. =effective depth of a rectangular section. or the
overall depth of a compression flange
d ~ effective depth to tension reinforcement
<1> = diameter of smallest tensile reinforcing bar.
It should be noted that rota,tion capacity can be
exhausted either due to crushing of concrete or due
to fracture of reinforcemeht or prestressihg steel.
Expression (2.68a) is' a linear approximat,ion to
the relationship given in the CEB Model Code
(1 lO). which was based. on a review of t~I',t J~at(,l
carried out by Siviero [287]. The expression results
in a conservative estimate of the rotation which can
occur prior to crushing of the concrete in the
compression zone.
Expressions (2.68b) and (2.68c) generally give,
conservative estimates of the rotation which can~
develop prior to fracture of reinforcings~c:cJor
"
, i
...........~.-....~.......;.........-..........~'-...'..
;1/0111/1',.11'1'. ':"J"'.'i
prestressing steel respectively. It is understood that ml == 822 kN m/m '
the expre$sions were derived from considerations y .. 1.251 m
of the ultimate elongations and gauge lengths spe- "'2 =35.7 kN mlm
cified in the British Standards for the various types It should also be noted that in accordance with
of reinforcing and prestressing steels, and from BOll4/82 the OTp would require full HA to be applied
data provided by manufacturers of prestressing to the third lane instead of the HAl3 loading required
s t e e l s . , . .. ' " .~, , ", by Part 2 of the Code (see Fig. 2.28). In this case it is
2. The effects of th"~ r"distributlonof't()flgitudinal '!oundJhat:
moments on the transverse Inress resultants should"" ......"",.
, be assessed by means of It non-linear analysis. The ml .. 884 kN'~/m
author would suggest that a linear analysis could y II1II 0.861 m '
also be used all indicated on page 22. h12:= 18.9 kN m/m
3. The greater of the shearllund reactions before and The required"transvers(;!- momcnt of resistance has
after redistribution should be used for design. as therefore decreas'ed substantially because the change in
suggested on page 22. loading affects me~anilim (b) but not (c). and the
4. The overall depth of the member should not ex- longitudinal moment of resistance has increased. This
ceed 1.2 m. A depth limit has been introduced illustrates the extreme sensitivity of yield line d~sign to
because the available t"ltt data (287) on rotation changes in loading. , '"
capacity do not cover dtH'!p memberA, and thore is The unltll for the parapet loading in Table 2.2 should
evidence to sugcst Chill rotation tmpaeity reduces be correctad to kN/m.
with an Inerea!!e In depih.
2.2 Vleld line design of skew slab bridge
(p.29)
Part '* h6W requlr@s Vtt bo Hike" ali 1.U for all imlds
when ulilng Ii plil§tie method of linalyslli at th@ ultimate!
limit litat@, Heneo; eDeh f:fltfy 1ft the !UHiOi'lti ~ofuffin of
Tllhl~ 2,2 lihould ftbW be t, IS, und tht! Idiliiln' lOads in
tht! fiHUll:blumH lil'iould.b6 multiplied by (1, 1~/t.l) fot
HAl HI lind footway loadings, Ali Ii r"!lull of this
Inef@ii!l&! in Ibattlngl It ill found thai:
• ".
2.3 Vield line dellgn of top Ilab (p,31)
As for E:xample 2.2,'tht% value of Yp.. upplled to the HA
wh@@lload shoUld floW b@ L15 ifiliiOlld gf LL Htifice,
tlu~ d"lIign load Is HOW i7~,5 kHI As a ft!sult of this
hu~r@a§e in loading, it is found that:
y == 0.231 m
(m, "f hi;!) == 14.4 kN m/Ift ,
•](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-102-2048.jpg)
![Chapter 3 .
Loadings
General (p.32)
~mcndment No.1 to Part 2 of the Code was published
in March 1983. The amendment corrected and revised.
the text. but the technical content· was un~ltered:
Hence. Ch~pter 3 of the m,ain text still' refle'cts .the
current Purt 2. However. the DTp has issued its own
umendments to Part 2'in the form of BD 14/82 [282]. In
the following sections, these amendments are discus-
sed. It is emphasised that the amendments are those of
. the DTp and not BSt Nevertheless, Part 2 is undqr
revision at the time of writing, and the DTp's amend-
ments will probably be incorporated in the revision.
Partial safety factors (p.33)
The DTp has amended some values of Y/I. in Table 3.1.
The changes are discussed later.
Superimposed dead load (p.35)
The partial safety factors of ].75 and 1.2 given in Table
3.1 for superimposed dead load are applied only to the
deck surfacing load. Smaller partial safety factors of 1.2
and ].0 at ultimate and serviceability limit states re-
spectively are applied to other superimposed dead
loads.
Differential settlement (p.35)
The nominal. value of differential settlement assumed
should have a 95% probability of not being exceeded
(juring the design life of the structure. The partial safety
factors arc now specified as1.2 and].O at ultimate and
serviceability limit statesr~s~ectively..
4
Temperature'range (p.36)
Coefficients of thermal expansion are given for con-
cretes made with various natural aggregates. The
values. which allow for the presence of reinforcement" .
vary from 9'>< lO-·lIrC for limestone to 13.5 x lO-I,/oC
for chert.
Erection loads (p.38)
A partial safety factor of 1.0 is specified at the ser-
viceabilitylimit state.
HA loading (p.38)
Type HA loading is uncJer review at the time of writing,
and its value has to be agreed with the DTp for loaded
lengths in excess of 40 rri.
HB loading (p.40)
The number of units of HB loading is specified as: 45
for motorways and trunk roads; 37.5 for principal
roads; 30 for other public roads; and 25 for accom-
modation roads and byways.
Applica{;on (p.40)
HA loading The full uniformly distributed and knife-
edge loads arc applied to two notional lanes and O.h
times these loads to all other notional lanes. This
loading is more severe than the one-third HA in the'
other lanes specified in the Code.
/IB loading The HB vehicle can occupy any notional
lane or can straddle any two notional lanes. Hence. for
a bridge with four notional lanes. it is necessary to
consider HB loading in one lane. full HA loading in
two other lanes. and 0.6 HA loading in the remaining
lane. Hence, this loading can be much more severe
than that specified in Part 2 of the Code.
:!
'"' , ~
Collision with parapets (p,41)
The moment and shear transmilled to the member
supporting the parapet should be taken as the moment
and shear resistance of the parapet at the ultimate limit
state calculated in accordance with the Code.
The partial safety factors have been increased from
1.25 and 1.0 to 1.5 and 1.2 at the ultimate and
serviceability limit states respectively.
Collision with supports (p.42)
The normal and parallel components of the loads now
have to be applied concurrently. as was the case in
BE 1/77. In addition the partial safety factor at the
ultimate limit state h:ts been increased from. 1.25 to 1.5.
It is no longer necessary to consider the. serviceability
limit state. These change~ imply that the loading is now
very similar to the BE )n7 loading.
For superstructures with a headroom clearance of
less than 5.7 m above a carriageway. a single nominal
collision load of 50 kN, in any direction between
horizontal and vertical. has to be applied to the super-
structure soffit.
5](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-103-2048.jpg)
![Chapter 4
Material properties and
design criteria
Concrete (p.45)
Characteristic strengths (p.45)
The lowest grade that can be used for reinforced
concrete has been increased from 20 to 25.
Stress-strain curve (p.45)
The strain at which the parabola joins the horizontal
line in Fig. 4.1(b) is how defined as 2.44 x 10-4
VTcu.
Other properties (p.45)
The characteristic stresses in compression and tension
are 0.5 feu and 0.56 V'1:u respectively.
Reinforcement (p.45)
Characteristic strengths (p.45)
In accordance with the revisions to the British Stan-
dards for reinforcement, the quoted characteristic
strength of both hot rolled and cold worked reinforcing
bars is 460 N/mm2.
Stress-strain curve (p.46)
The characteristic stress in tension or compression is
0.75 fyo The characteristic stress is intended to be the
stress at which the stress-strain curve becomes non-
linear. Figure 4.2 indicates that this stress should be
0.8/y. This anomaly has probably arisen because, it is
understood, the value of 0.75 /y was obtained by
applying the BE In3 overstress factors to the BE In3
allowable tensile stresses.
The first kink in the stress-strain curve of Fig. 4.4(b)
should be labelled 0.7 fy.
Characteristic strengths (p.46)
A table is given for the characteristic strength of
compacted strand for which a British Standard is not
available. For other types of prestressing steel, the
relevant British Standards are quoted.
6
Values (p.46)
The values of the partial safety factors at the ultimate
limit state are still 1.5 and 1.15 for concrete and steel
respectively.
However, new values which are applied to the
characteristic stresses have been introduced at the
serviceability limit state. The value for reinforcement is
1.0, and the values for concrete are given in Table
4.1A. The higher values arise for prestressed concrete
and for uniform stress distributions because more crf;ep
occurs under these conditions. The reason for the
larger value in tension for post-tensioned, as opposed
to pre-tensioned, construction is explained on page·51.
Table 4.1A Ym values for concrete at serviceability
Type ofstress
Triangular compression
Uniform compression
Tension
• Pre-tensioned
+ Post-tensioned
Type ofconstruction
Reinfon:ed Prestressed
1.00
1.33
1.25
1.67
1.25*
1.55+
Reinforcement (p.48) The limiting stress is equal to
the characteristic stress of 0.75 fy because the partial.
safety factor is 1.0. Due to other changes, the complica-
tions referred to in the main text no longer exist.
Prestressing steel (p.48) No reference is now made to
a stress limitation.
Compressive stresses in reinforced concrete
(p.49) When the partial safety factors in Table 4.1A
are applied to the characteristic stress of 0.5 fe", limit-
ing stresses of 0.5 feu and 0.38 feu are obtained for
triangular and uniform stress distributions respectively.
Compressive stresses in prestressed concrete
(p.49) When the partial safety factors in Table 4.1A
are applied to the characteristic stress of 0.5 feu, Iimit-
ing stresses of 0.4 j~u and 0.3 fnl arc obtained for
triangular and uniformstress distributions respectively.
- These stresses are greater than those in Table 43(a).
However, as explained later. the loads under which the
stresses are checked are also greater. In addition it
should be. noted that a higher stress at a support is no
longer permitted; The transfer stresses are the same as
those in Table 4.3(b), except that upper limits of 0.4 fe/l
and 0.3 fm are imposed for triangular and utfiT'orm
stress distributions respectively. Thi,si it to prevent
transfer stresses in excess of the limiti'll'gservice stresses
occurring, when a member is post-tensioned at such an
age that its concrete strength exceeds about three-
quarters ~f the characteristic strength.
Compressive stresses in composite construction
(p.49) The permitted increase in limiting compressive
stress is reduced from 50% to 25%, as was the case in
BE 2173. ..
Tensile stresses in composite construction (p.49) The
values given in Table 4.4 are only applicable when
tension is induced by sagging moments due to imposed
service loading.
Interface shear in composite construction (p.50) The
relevant clauses have changed almost entirely. In par-
ticular, shear calculations are now carrid out at the
ultimate limit state, which is far more logical than the
1978 procedure.
Two types of surface are now defined:
1. Roughened by wet brushing or subsequent tooling.
2. Laitance removed by jetting with air and/or water.
'Rough as cast' surfaces are classified as type 2.
The design approach is now similar to the CP 117
approach, which was adapted for BE 2/73, and isoow
compatible with that in BS 5400 : Part 5 : 1979 Jor
steel/concrete composite construction. The new Part 4
method is based on the shear friction approach of
Mattock and Hawkins [288]. Ftom test data they found
that the maximum shear stress (vuL that could be
transferred across a pre-existing crack was given by:
Vu =1.38 N/mm2 + 0.8 pfy *0.3 f; (4.18)
where p is the area of reinforcement per unit area with
yield stress h. crossing the shear plane~ If the cylinder
strength f: is assumed to be 0.8 feu, then the upper limit
of Vu becomes 0.24 feu. If the material partial safety
factors of 1.15, 1.5 and 1.5 are applied toIy, feu and
1.38 N/mm2(since the latter stress is that which can be
transmitted by concrete alone)' respectively, then the.
design shear strength (Vdu) is obtained as:
Vdu =0.92 N/mm2 + 0.7 pfy *0.16 feu (4:1b)
From considerations of tests on composite beams,
Johnson [289] proposed that the upper limit of Vdu
should be reduced to 0.15 feu, and this value is adopted
in the Code. The stress of 0.92 N/mm2 represents the
shear stress which can be transmitted by the concrete
alo.ne. This stress should be a function of concrete
strength. and the F1P 1213] has proposed that it should
he taken as 0.025/..". The values so far discussed arc
appropriate to a type 1 surface (or monolithic concrete)
and have. essentially, been adopted in the Code.
For a type 2 surface, the FIP has proposed that the
shear stress transmitted by the concrete alone should be
taken as 0.015 fC/I' This stress is 60% of that for a type 1
surface. and the same'percentagehas been adopted for
the upper limit of shear stress to give 0.09/"/1' .
. The Code actually refers to the shear force per unit
length (V,). and gives the following design equation:
V, =v, L.• +0.7 Aefy ;}kdeu Ls (4.1c)
where La is the length of the shear plane, and A(. is the
arell of steel per unit length crossing the shear plane
(excluding co-existent bending steel). Values of V, are
given in Tabl~ 4.5A. k t is 0.15 for monolithic concrete
or a type 1 surface, and 0.09 for a type 2 surface.
Table 4.SA V,values (N/mm2)
Surface
1
2
Conc:rete grade
0.50
0.30
2S
0.63
0.38
30
0.75
0.45
iii!: 40
0.80
0.50
It can be seen that now the shear capacity can always
be increased by providing additional reinforcement.
Hence, the problems discussed on page 51 no longer
arise. The modified stresses also give a better lower
bound to the test data of Saemann and Washa [118]
shown in Fig. 4.8(b).
It is now stated in the Code that interface shear
calculations do not have to be carried out for a compo-
site slab formed from precast inverted T-beams with
solid infill.
Cracking ofprestressed concrete (p.51) .
The flexural tensile stress limitations for a Class 2
member are now obtained by dividing the characteristic
tensile stress of 0.56 VTcu by the appropriate partial
safety factor given in Table 4.1A (i.e. 1.25 for pre-
tensioned construction, and 1.55 for post-tensioned
construction). The resulting limiting stresses are
0.45 VTcu and 0.36 VTcu, which are identical to those
in Part 4 : 1978.
It is now stated that members have to be designed for
Class 1 under load combination 1, and Class 2 or 3
under load combinations 2 to·S. However, only 25 units
of HB loading need to .be considered in combination 1.
In addition, all live loading may be ignored for Iightiy
trafficked highway bridges and railway bridges where
the live loading is controlled. The DTp [283] has
defined lightly trafficked struCtures to be accommoda-
tion bridges, bridleway bridges, and foot/cycle track
bridges. The DTp also requires that Class 3 prestressed
concrete should not be used.
At transfer, the flexural tensile stress is limited to
1 N/mm2for all classes.
Limiting stresses based on the FIP recommendations](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-104-2048.jpg)
![[290] ate now given for joints in post-tensioned seg-
ment,11 construction. For cement mortar joints. the net
strt!ssl~s should be compressive and not less than 1.5 Nt
mm2
at the serviceability limit stale. For resin mortar
joints, the net stresses should be compressive at the
serviceability limit state. I!1 addition, for resin, joints,
during the jointing operation, the average stress should
be between 0.2 N/mOl2 and 0.3 Nln~!m2, but nowhere·
less than 0.15 Ntmm2,' and' ll~~ di,fEe{encc I(lctweeOl
stresses across the section sl!toouldi oot cxcc<:(,I; @,'.5 Nt
mOl2• '
Cracking of reinforced concrete (p,51)
The design cfat'k widths in Table 4.7 have been altered
slightly. The value for 'Severe,' exposure has been
increased to 0.25 mm. The exposurc condition referred
to as 'Very severe (2)' is now referred to as 'Extreme'
and still has a design crack width of 0.1 mm. However,
the uesign crack width for the 'Very severe (1)' expo-
sure has been increased to 0.15 mm.
The Hbove increases in design surtace crack widths
should not be viewed as relaxations of standards.
because the covers have also been increased (see
Chapter lO). ,
It should also be noted that buried concrete is now
considered to he in a severe. rather than a moderate,
environment.
Ultimate limit state (p.S3)
The vHlucs of Yf3 have now been simplified consider-
ably, and are no longer dependent on load type. A
value of 1.1 should be used for all loads unless a plastic
method of analysis is adopted, when 1.15 should be
used.
Serviceability limit state (p.53)
The values hHVC again been simplified considerably and
are now always 1.0. This implies that the design
highway or railway loading under load combination 1 is
8
greater th:H1 the nominal load for al des!g,11 triter,ia;
e.g. the design HA load is now 1.2 HA. l.htsexplHlIls
why the limiting concrete compressive stresses in l,re..
stressed concrete of Table 4.3 have. been increased by
20%. In Part 4 : 1978. stresses in prestrt:.ssed c~mcrete
were checked under 1.0 HA, whereas now they are
checked under 1.2 HA. By increasing the limiting
stress. by 20'),;, (the same increase as that applicu to the
HA loading), the drafters appear to have assumcd that
4lead load stresses are always balanced by stresscs·due
to prestress. It should be noted that under load con~
binations 2 and 3, the designHA load is 1.0 HA rather
thHn 1.2 HA. Hence. larger compressive stresses are
now permitted under load comrinations 2 anu 3 than .
was the case in Part 4 : 1978.
Although the limiting compressive st'resses for pre-
stressed concrete have been increased to allow for the
increased loading under 10Hd combination I, it seems
anomHlous that the limiting tensile stresses have not
also been increased.
Reinforced concrete (p.54)
Only two load levels now need consideration, since
stresses and cmck widths arc now checked at the same
design load, However, in lo~d combination I, only a
maximum of 25 units of HB loading have to be
considered for crack widths, whereas up to 45 units
could be required for stresses.
Prestressed concrete (p.S4)
In load combination 1, only a maximum of 25 units of
HB loading have to be ~onsidered for tensile stresses,
wherea~ up to 45 units co~ld be required for compress-
ive stresses.
Composite construction (p.S4)
Interface shear is now considered at the ultimate limit
state. This simplifies the calculations considerably.
Chspter5
Ultimate limit state - flexure
and in-plane forces
Assumptions (p.55)
In order to ensure a ductile failure, the strain at the
centroid of the tension reinforcement must now exceed
the value given by equation (5.1), However. if the
ultimate moment of resistance of the section is at least
1.15 times the required value it is not necessary to
satisfy the above requirement. The reason for this is
given latcr in the discussion of the prestressed concrete
assumptions of page 57.
Simplified concrete stress block (p.55)
The simplified rectangular stress block can only be used
for a rectangular section or a flanged section with the
neutral axis within the flange.
Singlyreinforced rectangular beam (p.56)
In equation (5.8), correct d to d2
•
Flanged beams (p.57)
At the ultimate limit state, it is now permitted to take
the full flange width as the effective wid~h. This allows
for plastic redistribution of stresses at coilapse.
Assumptions (p.57)
2. The Code no longer covers unbonded tendons.
Hence, the table of failure stresses has been re-
moved.
3. It is now necessary to ensure a ductile failur~ by
checking that the strain in the outermost tendon
exceeds the following value:
fpl4
£ = 0.005 +--
E.,Ym
(5.14a)
If the outermost layer of tendons provides Jess than
25% of the total tendon area, th.e above require-
ment should also be met at the centroid of the
outermost 25% of tendon area. Equation (5.14a) .
.", ',-........ ,!'"' ..... "
can be derived from considerations of the strcss-
strain curves given in Figs 4.4(c) and (d) in the
same manner as equation (5.1) was derived. This
ductility requirement can be very restrictive be-
cause it is satisfied by few' currently designed
standard bridge beams. As a result it is permissible
to ignore the requirement provided that the ulti-
mate moment of resistance of the section is at least
1.15 times the required value, The 1.'15 factor
ensures that the vast majority of current designs
would be acceptable to the new Code,
Orthogonal reinforcement (p.59)
Insert = before M;yin equation (5.18),
5.1 Prestressed beam section strength
(p.61 )
In this example, the characteristic tendon strength
would now be obtained from the relevant British
Standard rather than a Code table.
The outermost tendon strain is 0.014 which exceeds
tbe value of 0.0121 obtained from equation (5.14a),
However, it is not now permissible to use the simplified
rectangular stress block because the neutral axis is not
in the flange. Hence, a computer program would be
needed to analyse the section in practice,
In the strain compatibility approach the neutral axiS'
depth is incorrectly stated as 330 mm instead of the
correct value of 339 mm which was used in the subse-
quent calculations.
Table 29 in Part 4 : 1978 has been renumbered to
Table 27 in Part 4 : 1984,
5.2 Slab (p.63)
Three corrections in right-hand column:
(i) Line 4: change M; > 0 to M; > 0
(ii) Line 6: change (0.9) to (-0.9)
(iii) Last two lines: transpose the words 'top' and
'bottom'.
9](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-105-2048.jpg)
![Chapter 6
Ultimate limit state - shear
and torsion
Introduction (p.65)
Interface shear calculations in composite construction
are now carried out at the ultimate limit state.
Beams without shear reinforcement (p.65)
The design shear stresses given in Table 6.1 have been
altered slightly. The new values are derived from the
following equation: '
_ 0.27 (lOOA)Ih '( .IIIVc - - - ----.J vcu)
Ym bd
(6.18)
The partial safety factor Ym is 1.25, and IC'u should not
be taken as greater than 40 N/mm2
•
The depth factor for slabs, discussed on page 69, is
now also applicable to beams. Thus, for both slabs and
beams the shear stress which can now be resisted in the
absence of shear reinforcement is ~ v,"
Beams with shear reinforcement (p.66)
When the nominal applied shear stress exceeds ~ vC" it
is now necessary to provide shear reinforcement to
resist the shear force in excess of (~ v(' - 0.4) bd rather
than Ve b d. Hence, equations (6.6) and (6.7) become:
b(v + 0.4 - ;'' v,.)
A.I
·,. =--'-----=:::......!.<~
0.87 fy ,' (sin a + cos a)
A.". = b(v + 0.4 - !;.. ve)
0.87!,.v
(6.6A)
(6.7A)
It is understood [291J that the reason for introducing
the stress of 0.4 N/mm is to allow for the reduction of
aggregate interlock under repeated loading. It appears
that the stress of 0.4 N/mm2 was not based on test data,
but was chosen because it is approximately the section-
al shear stress which can be resisted by the minimum
shear reinforcement required by the Code (see page
68). This requirement to overdesign shear reinforce-
ment was originally introduced by British Rail (292)'
In equation (6.7A),lvv is now limited to a maximum
10
"" r
value of 460 N/mm2 instead of 425 N/mm2
• This value
is still less than 480 N/mm2
and is thus conservative (sec
page 67).
There is now a requirement that the area of longitu-
dinal reinforcement in the tensile zone should be at
least:
v
A.f =2 (0.87 /y)
(6.78)
The derivation of this equation is given in full e1se- .
where [293]. However, it can be seen from equa~ion
(10.5) that the longitudinal reinforcement should really
resist a force of VI2 in excess of the force due to
bending alone.
When designed shear reinforcement is necessary in a
flanged beam, the longitudinal shear resistance at the
flange/web junction has to be checked in accordance
with the method explained in the amendments to
Chapter 8.
Minimum shear reinforcerrent (p.68).,
Minimum shear reinforcement now has to be provided
in all beams as follows:
A.•v 0.4 b
-;.:- =0.87fyv
(6.7b)
The values given by this equation are very similar to
those in Part 4 : 1978..
Shear at point of contraflexure (p.68)
The empirical design rule of Part 4 : 1978, which is
discussed on page 69 has been omitted from Part
4 : 1984.
Enhanced Ve values (p.69) The enhancement factor
(;,,) is now given as a function of effective depth. rather
than overall depth:
1/4 J- 7;,. =(500/d) .,.0. (6.7~r .
. <.
A fourth root relationship was proposed by Reg~I1"
(150). Equation (6.7c) fits the test data of Fig. 6.5
better than the Part 4 : 1978 relationship.
~ 1-5dx~ ~ 1-5dx~
"5dI~ -------~
r I I
: ~ I
I ~ I
I: :1'Sdy :.. _______ ~
'tSdyI~ -------~~~i:ii~~ter
I ® I
I
:''''-''-+-support
"5d I I or load
Y I I
'-_.___ •.~ ..... _.J
(a) (entre
~ 1'5dK~ ~ "5dx~ ex
Shortest ,-
straight line 1'1'
touching ,-
lSdI; --------
Y I
I~
1-
sa
ri --------
r I
:~
Unsupported
edge
loaded ",'-
area~/
/~
'~I~________ /
(b) Edge (c) (ornerO) (d) (orner Oil
6.6A New punching shear perimeters
An important point to note is that the enhancement
factor from equation (6.7c) is less than unity for slabs
with an effective depth greater than 500 mm. This fact,
coupled with the requirement to resist an additional
nominal stress of 0.4 N/mm2, will result in an increase
of shear reinforcement in slab bridges.
Shear reinforcement (p.69) Equation (6.7A) is now
applicable to slabs.
Maximum shear stress (p.70) The m,!ximum shear
stress in slabs at least 200 mm thick is now the same as
that for beams.
Voided or cellular slabs (p.70) It is now,stated that the
longitudinal ribs of voided slabs should be designed as
beams. Transverse shear effects should be resisted by
transverse flexural reinforcement: the method sug-
gested in Appendix B of the main text could be used.
However, it is now also stated that the top and bottom
flanges should be designed as solid slabs, each to carry
a part of the global transverse shear force and any shear
forces due to torsional effects, proportional to the
flange thicknesses. The author does not see the need
for designing the flanges of a circular voided slab as
individual solid slabs if the design method suggested in
Appendix B is adopted. In addition, it is not clear how
to include the shear forces due to torsion in the shear
design of the flange since the torsional shear flow in the
flange is perpendicular to the flexural shear flow.
BS 5400 clause (p.72)
The critical perimeter is now situated at 1.5 times the
effective depth from the face of the load or column.
Since the effective depth is different in the two steel
directions, the critical perimeter is also at different
distances in these two directions. It is also stated that
the perimeter is rectangular whatever the shape of the
loaded area (see Fig. 6.6A(a». There is little differ-
ence in the lengths of the 1978 and 1984 perimeters for
rectangular loaded areas; but, for a circular loaded
area, the new perimeter is longer and this results in a
greater punching shear strength. These changes origin-
ated in the revisions to the building code [294].
The total shear capacity of the concrete is taken as
the sum of the shear capacities of each straight portion
of the critical perimeter. The flexural reinforcement
perpendicular to each portion should be used to evalu-
ate the appropriate value of Vc for each portion. The
effective area of flexural reinforcement should now
include all of the reinforcement within the width of the
loaded area and a width extending to within three times
the effective depth on each side of the loaded area.
For the design of shear reinforcement. equation (6.8)
has now been arranged in the form:
0.4 ~ b d :::; (~ A".) (0.87f,.,.) ~ V - V, (6.8A)
where ~ b d is the area of the critical section. V is the
applied shear force at the ultimate limit state. and V<, is
the total shear capacity in the absence of shear rein-
forcement.
11](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-106-2048.jpg)
![All important addition in Part 4 : 19t-i4 is the pro-
vision of design rules for thc punching of loads ncar to
frec edges. The code perimeters and the layers of
flexural reinforcement which arc considered to be in
tension arc shown in Fig.6.6A(b - d). The author has
discussed the question of which layers of steel arc in
tension elsewhere [2')5], and is of the opinion that the
stccl in the unloaded face is always in tension in case
(b), and also in caSl' (c) eXCl~rt when both e. and c.I' in
Fig. h.6A(c) arc less than the slab depth. In the latter
case, all steel layers are in tension, and therefore the
lesser of the top and bottom reinforcement areas in a
particular direction should be used to determine tht:~
allowable shear stress approprime to that direction.
For cases (b) and (c), the total shear capacity of the
concrete is taken as 80%) of the sum of the shear
capacities of the three or two portions, respectively, of
the critical perimeter. It is understood that the reduc-
tion of 2()(X) is to allow for the fact that it is not certain
whether the design method is strictly applicable to
supports ncar free edges. However, it is of interest to
note that in thl' building codes [15, 294], a similar
reduction of 20% allows for moment transfer at edge
and corner columns [296]. Since slab bridges arc gener-
ally seated on bearings which do not permit moment
transfer, it is doubtful whether the 20% reduction is
appropriate.
Case (d), essentially, depicts a flexural shear failure,
and the layers of tension reinforcement are shown
correctly. The shear capacity of the concrete is calcu-
lated for the critical section (of length b), with the
effective depth and area of flexural steel, used to
determine v,,, taken as the average of the values in the
two reinforcement directions. If shear reinforcement is
required for this case, then it is illogical to place the
shear reinforcen:tent as required by the Code. i.e. on
the critical perimeter and at a distance of 0.75 d inside
it, since the latter reinforcement would make no con-
tribution to the shear strength. The author would
suggest that shear reinforcement should be designed in
accordance with equation (6.8A), and placed within a
distance from the load equal to an effective depth. as
shown in Fig. 6.6a.
It should be noted that the Code now states that a
group of concentrated loads should be considered both
singly and in combination.
Finally, it is not necessary to consider punching
through the flange of a circular voided slab, because the
relative dimensions of wheel loads and flange thick-
nesses make it highly unlikely that such a failure would
occur. However, it is necessary to consider punching
through the slab as a whole. No guidance is given in the
Code as to how to carry out such a calculation.
However, research is being carried out at the Univer-
sity of Birmingham with a view to proposing a method
of calculation.
Shear in prestressed concrete (p.72)
Additional clauses are now given on shear in trans-
12
,I
,I
/ /
,I
,I /
Links !n this I' I' ,1,1
region "'--7-/~,I
/ ,I
,I
/ /
/ J
,I /
( (
6.6a Shear reinforccmentfor perimeter (d)
Critical
perimeter
mission lengths. !ind on segmental construction
Within tht.' transmission length of a pre-tensioned
member, the shear capacity should be taken as the
greater of the values obtained by assuming that: (a) thc
section is reinforced and all tendons ignored: and (b)
the section tS prestressed, with the appropriate value of .
prestress at the considered section determined by lIsing , '
a linear variation of prestress over the transmission
length.
For a post-tensioned member of segmental construc-
tion, the shear force at the ultimate limit state should
not exceed:
v =0.7 "IlL Ph tan 02 (6.8a) .'
where Ph is the horizontal component of the prestres-
sing force after all losses and,. YII. is the partial safety
factor of 0.87 which is appliedto this force. Equation
(6.8a) is based on the shear-friction theory [297], and
U2 is the angle of friction of the'joint between segments.
Tan 02 can vary from 0.7 for a smooth unprepared joint
to 1.4 for a castellated joint. These values were prop-
osed by Mast [297] from considerations of test data.
Th~ 0.7 factor in equation (6.8a) is merely an additional
factor of safety.
Sections uncracked in flexure (p.72)
There is now a general requirement that when calcu-
lating the shear strength of a prestressed member, the
prestressing force after losses should be multiplied by a
partial safety factor of 1.15 where it adversely affects
the shear strength and 0.87 in other cases. It is not
stated whether a single value o(the partial safety factor
should be used throughout a calculation, or whether
the factor can take different values. A single value
would seem more logical, and the author understands
that this was the intention of the drafters.
Because of the va!iable partial safety factor, equa-
tion (6.8) on page 73 rather than equation (6.9) appe'ars
in the Code, but fcp is calculated from the prestressing
force multiplied by the appropriate partial safety factor
(0.87).
- ------ --- -- - -----_.-.<---"..---------- ------
.1 ....".,.
Sections c~acked In flexure (1'.73),
'" .,Sttlte it is now neCeSS<lfY to api;ly a partial safety 'f~ctor
to the prctsttessingfQrce•.the following changes /have
hl!len made III the Code: '• ...._' ,", .. I
l. In e4u3tion(6.12r MI has changed to Me; and
appears as
Mrr ='(0.37 VJ:., +J~,) II)' (6.12A)
J~" is calcula,ted fronlthe factmc:d value of the.
prestressing forc~, (n~rtiul safety.fm:tor of 0.81).
2. In equatiOn (6'tt~Jl KIn =!"II'y. an~ ~Otlt/", and!,,(.
are calculated '''()m the factored value of the pre-
stressing forcl!. tit'should be noted that the minimum
vltiue of V(., ~ould he obtained with either value
(O.~7 or 1.15) ~)fthe partial safety factor.
3. Although not sUIted, one should. pr~sumahly~ also
take P, to he the factored prestressing force :when ,
considering comhined tensioned and unt¢nsl()ned
steel.
Shear reinforcement (p. 74)
It is now necessllry to 'provide at .·Ieastthe minimum
amount of s~car reinforcement in ~II members.
As is also the case for reinforced concrete. designed
shear reinforcement 'has to resist an excess' nominal
shear stress of 0.4 N/mm2
• 'Thus equation (6.17)
appears as:
}sv V + 0.4 b d, - Vc
s:- =' d.87/yv a,
(().17A)
The area of longitudinal steel (reinforcement plus
tendons) should exceed the value given by equation
(6.78). However, the author suggests that this equation
should be modified to allow for the effects of the
prestress, which counteract the longitudinal tension
due to shear. Thus, the area of longitudinal reinforce-
ment required is less than that given by equation (6~7a).
Consequently, a more realistic equation for prestressed
concrete is:
Iv pe(.) 1 (6.16a)
A.• =2 -d; 0.H7J;.
where P is the axial. component of t~e resultant effec-
tive prestressing force (with the partial safely factor of
0.87 applied), and e(. is the distance of theJine of action
of this force from the extreme compression fibre at the
ultimate limit state. The force Pe,!d, in equation (6.16a)
acts at depth d" and is one of a pair of forces (the other
acts at the extreme compression fibre) which are stati-
cally equivalent to the eccentric prestressing force P.
When designed shear reinforcement is necessary in a
flanged beam. the longitudinal shear resistance at the
flange/Veb junction has to be checked in accordance
with the method explained in the amendments to
Chapter 8.
Punching shear (p.75)
The Code refers to values of Veo given in Table 28 of
, II
the Code. ht fact. iable 28 gives values of maximum'
shettt sHess. Ti,e Code should refet· tu "hhlEt31 of Part '
4 : 1978. but this table has not been included in't'art '
4 : 1984.
torsion reinforcement (p.77)
Th~ notation has now changed:
ASI ::: area of one leg of a closed link
AsL :::: atell of one bar of longitudinal reiuforcel1lent
sL =spacing .01' longitudinal rein(ot'ccll1ent
,i, Hence, equations (6.21) and (6.22) appear'in th~ Code
as: ':
As, ~ T (6.21 A) ,
Tv- !1,.6xlJ'1 (0..8.71;-:>
ASL~,_A,SI (:flY) (6.22A)
SL 'Sv fyL.
Huwever, it is not clear what value should he taken for
sl.in the situation shown in Fig. 6.12(a). Presumably.
one "takes the average of the horrzontal and vertical
spacirlgs. If thi!iis done. equation (6.22A) is. essen- :
tially"".iQ~ntical to equation (6.22).
Delai/ill8·.(p.78) Thl~ charactellistk strength of torsion I
reinforcement should, noll be· assumed greater than
46() N/mm2
• This value stighdy exceeds the critical '
value of 430 N/mm2 found by Swann P·77.1 (s.ee page
78).
.The area of links or tongitudtnal torsion reinforce-
ment may be reduced by up to 2()'%. provided that the
following product remain~ uncha;'1ged:
A.vv Asl•
--Sy 5L
The implication of this detailing rule is that, the spiral
failure surface (see Fig. 6. 12{b» would form at an angle
other than 45°, which is the valuc assumed in the
dcrivatio~ of equations (6.21A) and (6.22A).
Torsionreinforcement (p.79)
Equations (6.21A) and (6.22A) should be used for each
individual rectangular section requiring torsion rein-
forcement.
Torsion reinforcement (p.80)
The equations for rectangull;r sections are no longer
included in the Code. Hence, only equations. (6.25)
and (6.26). suitably modified to allow for the previously
mentioned change of notation. have to he c<lOsidered.
The required area of links is given by:
}s" ~ 1"
S; .... 2 Au (0.87 fYI')
(6.25A)
and the required area·of longitudinal steel is given by
equation (6.22A).
It is now permitted to reduce the amount of longitu-
dinal reinforcement in flexural compressive zones, as
discussed on page 81, for all sections (not just box
13](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-107-2048.jpg)
![Chapter 7
Serviceability limit state
Introduction (p.86)
Interface shear calculations in composite construction
are now carried out at the ultimate limit state,
General (p.86)
It is emphasised that' stresses in reinforced concrete
need to be checked only in the following situations: ..
1. Where the effects of applied deformations arc not
considered at the ultimate limit state. The author
assumes that the drafters had in mind the situation
in which the full elastic effects of applied deforma-
tions, with no relaxation due to the plastic be-
haviour of materials, are considered at the ultimate
limit state. Hence, if applied deformations are
considered elastically at the ultimate limit state,
less calculations have to be carried out. However,
the true nature of the effects of applied defor-
mations are confused by this design approach,
since they are actually more significant at the
serviceability rather than the ultimate limit state.
2. Where a plastic method of analysis has been used
at the ultimate limit state, because such methods
imply redistribution of elastic stresses.
3. Where combined global and local effects are con-
sidered separately at the ultimate limit state. Such
a situation would arise if a top slab were designed
to resist local wheel load· effects using yield line
theory.
The concrete compressive stresses should not exceed
0.5fcu and 0.38fcu for triangular and uniform stress
distributions respectively. Reinforcement. tensile or
compressive stresses should not exceed 0.75fy.
The value assumed for the elastic modulus of the
concrete should be an appropriate value between the
long-term and short-term values depending on the ratio
of permanent to short-term effects. This implies a
different modular ratio for each load case and load
combination.
16
Slabs (p.87)
At the top of page g8. I!."h-" should be corrected to l',11'1/
in two places: (a) immediately ahove equation (7.2):
and (b) towards the end of the last paragraph.
General approach (p.SS) It is now necessary to pro
vide at least the following amount of reinforcement to
control cracks due to the restraint of shrinkage and
early thermal movements wherever restraint occurs:
A.v =k, (A,. _. 0.5 A C{1,f (7.2a)
where k, = 0.005 or 0.006 for high-strength or mild
steel respectively, AI" is the gross cross-sectional area.
and Am, is the area of the concrete core section more
than 250 mm away from all surfaces. Equation (7.2a)
results in up to twice as much reinforcement as that
required by Part 4 : 1978. The reason for not taking As
as a proportion of the gross cross-sectional area is that
the cracks develop progressively from the surface.
Hence, it is only necessary to reinforce the 'surface
zones', which are each:assumed to be 250 mm thick
[231]. Thus, equation (7.2a) is conservative since only
half of the core area is subtracted from the total area.
The spacing of the reinforcement should not exceed
150 mm.
Loading (p.89) The design highway loading is now
increased to 1.2 HA and 1.1 (25 units of HB) irrespec-
tive of the span.
Stiffnesses (p.89) The value assumed for the clastic
modulus of the concrete should be an appropriate value
between the long-term and short-term values. depend·
ing on the ratio of permanent to short-term effects.
This implies a different modular ratio for each load casc
and load combination.
The tension stiffening formula has been amended to:
E",=EI- [3.8bl!(a
l
-d(.)] [1-.;t IO'~ *£1 (7.3A)
E.,AAh-d..) I( J
Equation (7.3A) differs from equation (7.3) in the
following ways:
I:.
I. Notation: hi replaces h, and d,. replaces x.
2. lnstead of the service load stcel stress being
assumed equal to O.5g/I
., the actual calculated value
(E.•E.,) is used. Hence, E,~ rather than fl" appears in
the equation. .
3. The second term in square brackets, in which Mq
and MI( arc the live and permanent load moments
respectively. has been introduced. to allow for the
breakdown of tension stiffening due to .repeated
live loads.
For zero live loads. the two equations give the same
result for a service load stress of 0.63/).. ..
Crack control calculations (p.90)
Crack width calculations now have to be carried out for
all structural clements. including solid slab bridges. A
single crack width equation is now given. This is
equation (7.5) with the notation changed such that C"om
replaces e",ill and de· replaces x. C"(I'" is the required
nominal cover. It is emphasised that crack width cal-
culations arc now carried out at an imaginary surface at
a distance from the outermost bars equal to the re-
quired nominal cover. Hence, with respect to calcu-
lated crack widths a designer is no longer penalised for
providing cover greater than that specified in the Code.
It is not clear ·from the Code what value should be
taken for C"om when calculating the width of a crack
perpendicular to an inner layer of bars. Presumably it is
the nominal cover plus the diameter of the outer layer
of bars.
For a section entirely in tension (e.g. a flange of a
box girder). equation (7.5) reduces to (see page 93):
w =3acr E", (7.Sa)
This equation is now given in the Code.
Specific guidance is now given on the combination of
global and local effects for crack width calculations.
The strain Em may be taken as the sum of the strains
calculated separately for the global and local effects.
Equation (7.Sa) may then be used to calculate the crack
width. However, if the global effect induc~s compress-
ion in a slah or flange, the crack width can be calculated
from equation (7.5) with the depth of concrete in
compression calculated by considering only the local
bending moment. These approaches to calculating
crack widths ar,. more conservative than that based on
the strain due to combined glohal and local effects.
Beams (p.94)
Limiting values of prestressing steel stresses are no
longer stated. Under HB loading. compressive and
tensile stresses may have to be checked under different
load levels in load combination 1 (see the discussion to
the amendments of Chapter 4).
Initial prestress (p.95)
Immediately after anchoring, the tendon force should
not exceed 70% or 75% of the characteristic tendon
strength for pOSt-tensioned and pre-tensioned tendons
respectively.
Losses due to steel relaxation (p.95)
The reference to Part 8 of the Code and the quoted loss
in the range zero to 8% has been removed. Instead
reference is made to the appropriate British Standard.
7.1 Reinforced concrete (p.98)
The characteristic strengths are changed to 460 N/mm2
and 40 N/mm2
• If grade 30 concrete were used then it .
would have to be air-entrained (see Chapter 10) unless
the slab were supported on permanent formwork.
General (p.98)
From Table 3 of the Code, short-term elastic modulus
of concrete =31 kN/mm2
. Hence, long term value :=
3112"" IS.5 kN/mm2
. .
The value for design depends on the ratio of perma-
nent to short-term effects. It is not clear how this ratio
can be calculated when both in-plane and bending
effects have to be considered. However, for a top slab,
where the dominant effects are due to wheel loads. it
would seem reasonable to use the short-term elastic
modulus. Hence, modular ratio = 200/31 =6.45,
The values of yp for cracking are now 1.0 for HA and
HB loading. Hence, the design loads are increased.
Cracking (p.98)
Due to transverse bending (p.98)
HA design moment
=(0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) +
(10.8)(1.2)(1.0) + (6.0)(1.2)(1.0)
=20.9 kNmlm
2SHB design moment
::: (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) +
(7.56)(1.1)(1.0) + (23.3)(1.1)(1.0)
= 34.7 kNmlm
Critical moment = 34.7 kN mlm
With modular ratio of 6.4S:
de ::; 49.2 mm
1= 0.178 X 109
mm4
/m
Required nominal cover = 35 mm (see Chapter 10
amendments). Thus crack width calculation carried out
at a surface 35 mm from transverse steel, rather than at
the soffit.
Ratio of live to permanent moments> 1, thus equation
(7.3A) reduces to Em = EI
Strain at 35 mm below transverse steel is:
£1 =(34.7 x 106
)(145.8)/(31 x 103
)(0.178 x 109
)
= 9.17 X 10-4
... E", =9.17 X 10-4
acr = YS02 + 432
- 8 = 57.9 mm
From eq~tion (~:~), crack width is
"
C,'"
" 1., , t ' . 17V.l I ,](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-109-2048.jpg)
![= t007 X 10" (642 - 454)/(65.19 x }(}'1)
= 2.90 N/mm2
(tension)
With the partial safety factor of 0.87 applied" to the
stress due to prestress, the total stress at the composite
centroid to be used in equation (8.7) is:
f cp =(0.87)(5.14) - 2.90 = 1.57 N/m.m2
Design shear force at the ultimate limit state acting on
the precast section alone is
Vel = 81 x 1.2 x 1.1 =107 kN
Shear stress at composite centroid is
Is = (107 x 103
)(44.4 x 106
)/(65.19 x 1()9)(160)
= 0.46N/mm2
Additional shear force (Vc-2) which can be carried by
the composite section before the principal tensile stress
at the composite centroid reaches 1.70 N/mm2is:
V =(124.55 x 109)(160) x
c-2 116 X 106
(Y(1.7)2 + (1.57)(1.7) - 0.46) 10--'
=326 kN
Vrll = Vd + Vc-2 =107 + 326 =433 kN
The section is next co~sidered cracked in flexure. The
cracking moment is now calculated from equation (8.8)
on page 107 with the partial safety factor of 0.87 (rather
than 0.8) applied to the extreme tension fibre stress due
to the prestress (Jpt) of 17.54 N/mm2• The notation for
cracking moment is now Mc-,;.hence
Mc-, = (1007 x 106
) x (1 - 454 x 124.55 x 109/642 x
65.19 x 109
) + (0.37 V3() + 0.87 x 17.54)
124.55 x 109
/642
= 3114 x t<1' N mm =3114 kN m
Design nioment at the ultimate limit state is
M = 1007 + (132 x 1.2 XLI) + (135 x 1.75 x 1.1) +
(1333 x 1.3 x 1.1)
=3347 kN m
Design shear force at the ultimate limit state is
V= 107 + (14 x 1.2 x 1.1) + (15 x 1.75 xLI) + (196
x 1.3 XLI)
=435 kN
From the modified form of equation (6.11), suggested
on page 107
Vc-, = (0.037)(160)(1111 V50 + 130 V30) 10--' +.
435 (3114/3347) =455 kN
Vc =Vto =433 kN
V > Ve, but minimum link requirement governs.
Hence, provide minimum links, as on page 117, of 10
mm (2 legs) at 500 mm centres (314 mm2/m).
22
Interface shear (p.117)
At support (p. 117)
Design shear force at the ultimate limit state which acts
after the top slab hardens is
(27 x 1.2 x 1.1) + (29 x 1.75 x 1.1)+
(332 x 1.3 x 1.1) .
=566 kN
From equation (8.12), interface shear stress is
=(566 :x: 10")(71.6. x Ht')= 1.08 N/mm2
v" (124.55 x 109)(300)
Longitudinal shear force per unit length is
V, = 1.08 x 300 :::; 325 N/mm
For a rough as cast surface (i:e: Type 2),k, from Table
31 of Code is 0.09 and. for grade 30 concrete, V, =: 0.45
N/mm2 (see also Table 4.5A). From equation (4.lc),
longitudinal shear force per unit length should not
exceed
0.09 x 30 x 300 =810 N/mm > V, .'. O.K.
From equation (4.lc) rearranged, required area of
reinf<?rcement per unit length is
AI' =(325 - 0.45 x jOO)/(0.7 x 250)
= 1.09 mm2
/mm = 1090 mm2/m
Required minimum =0.15% of c9ntact area
=(0.151100)(300) =0.45 mm2/mm
< 1.09 mm2
/mm
10 mm links (2 legs) at 150 mm centres (1050 mm2/m)
provided .lor vertical shear is not sufficient for interface
shear which requires 1090 mm2
/m. Hence. use 10 mm
links (2 legs) at 125 mm ~ntres (1260 mm2/m).
At quarter span (p. 117)
De~ign shear force at the ultimate I.imit state which acts
after the top slab hardens is
(14 x 1.2 x 1.1) + (15 x 1.75 x 1.1) +
(196 x 1.3 xLI)
=328 kN
For which V, = 189 N/mm, and
A,. =(189 - 0.45 x 3(0)/(0.7 x 250)
=0.309 mm2
/mm
Hence, minimum of 0.15% (0.45 mm2/mm) governs for
interface shear. This amount exceeds that provided for
vertical shear (314 mm2
/m). Hence. provide 10 mm
links (2 legs) at 350 mm centres which gives 449
mm2
/m.
Chapter 9
Substructures and
foundations
General (p.118)
A clause on the shear resistance of columns has been
introduced into Part 4 : 1984. A column should be
designed to resist shear as if it were a beam except that
the ultimate shear stress (!;..v,.) resisted by the concrete
can be multiplied by (1 + 0.05 N1Ac). N is the ultimate
axial load in newtons and AI' is the cross-sectional area
in mm2
• The multiplier is a more conservative version
of that used in North American practice (168].
A column subjected to biaxial shear should be de-
signed such that:
(9.0a)
where Vx and Vy are the applied shear forces, and V/I.t
and VIIV are the corresponding shear capacities. There
appearS to be no experimental or theoretical evidence
to support this interaction formula, but it would seem
to be conservative.
No guidance is given in the Code regarding the
effective shear area of a circular column nor the area of
reinforcement in a circular column which should be
used to determine the allowable shear stress from
equation (6.1a). The American code [168] considers
only that tension reinforcement which is in the half of
the member opposite the extreme compression fibre,
and takes the effective depth as the distance from the
extreme compression fibre to the centroid of this
reinforcement. The web width is taken as the diameter
of the column. These recommendations are based on
tests by Farodji and Diaz de Cassio [298J who found
that the ACI equations for rectangular sections could
be applied to circular sections if the external diameter
were used for the effective depth, and the gross section
area used for the product bd [299]
Definition (p.118)
A column is now defined as a member with an aspect
ratio not greater than 4.
Effective height (p.118)
A more precise table of effective heights is now given,
which replaces Table 9.1. The new table is summarised
in Fig. 9.0a. It should be noted that the rotational
restraint at a rotationally restrained end is assumed to
be at least four times the column stiffness for cases 1,2
and 4 to 6, and eight times for case 7. The former value,
which is generally conservative, was chosen to give the
same effective heights for cases 1 and 2 as those used
hitherto. However, -the effective height of a cantilever
column is extremely sensitive to the rotational res-
traint; hence, the restraint value was doubled for case 7
to give a more realistic value [300]. The effective
heights can be derived analytically [225]. For case 4, it
has been assumed that (a) under rotation the axial force
remains perpendicular to the mid-plane of the elas-
tomeric bearing, and (b) under lateral displacement the
axial force acts through the centre line of the bearing
[300].
Slenderness limits (p.119)
The upper limit of slenderness ratio (l.Jh) is generally
now 40, since this is considered a practical upper limit.
However, when sway can occur, the upper limit.is 30,
as suggested on page 119, and equation (9.2) is no
longer used.
Axial load (p.119) Equation (9.4) is now omitted
from the Code.
Code formulae (p.120) Equations (9.5) to (9.13) are
still appropriate, if the following modifications are
applied:
1. In equation (9.6), the term 0.4f('l/ should be re-
placed twice by 0.45,t;·u' The latter value is correct
for zero eccentricity and zero reinforcement, and is
a conservative value for larger eccentricities when
the minimum reinforcement, discussed on page
138, is provided.
2. In equations (9.7) and (9.8), the term O.72fy should
be replaced with fy,' which is equal to expression
23](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-112-2048.jpg)
![CaseJ
1,,:::0'7/0
Case 2
.Ie =.0.8510
Case 6
11'=1'5/0
9.00 Encctivl~ hcight~
(4.1) on page 4X. Hence, the conservative value of
0.72/1' is no longer lIsed for all steels: instead, for
current steel strengths; the appropriate values offl"
arc O.7X4J; for mild steel and (I.725fy for high- .'
strength steel.
Axial load pillS biaxial bending (p.121) Nil: is now
c"lculat~d from equation (9.3) with the term 0.75fl'
replaced with fv(' (see the last paragraph f~r the ex-
planation).
Slender columns (p. 122)
The nomin~1 minimum initial eccentricity in .Part
4 : 1978 of O.05h to allow for construction tolerances
has been found unnecessarily large for the size of
columns usually used in bridges. Hence, it now has an
upper limit of 20 mm. In addition for biaxial bending,
the nominal eccentricity is now 0.3 times the overall
depth of the cross-section in the appropriate. plane of
bending (but ::}20 mm).
The notation used fo), the dimensions of a column
caused confusion in Part 4: 1978. Hence. in Part
4 : 1984 the notation .shown in Fig. 9.9aisuscdconsis-
tcntly. As a result:
I. In l'quation (9.22). It should be replaced with hr'
2. If 11I'::}3 h.. equation (lJ.26) may be used with band
" replaced with ht' and hI' respectively
3~ In equations (9~27).and·(9.28),band h should oc
replaced with h, and hI' respectively.
Stresses (p. 124)
The concrete stress is now limited tqO.38J;/i instead of
0.5/;/1 under axial loading. Thus, the design resistance
at serviceability n() longer exceeds that at ultimate..
.However. the ultimate limit state is still likely to be
critical when the loading is predominantly axial.
24
I
Case 7
Ie =2-3 /0
I
I
~
No restraint to
sideswayin
cases 6 and 7,
hy x- - +- - - -x
I
1
9.9. Column section notation
Crack widths (p. 124)
The design crack width for very severe exposures has
been increased to 0,15 mm and a different crack width
formula introduced. with the result that crack control is
now less likely to be the critical design criterion for a
column with a large eccentricity of load.
Axialloati (p.125) No reference is. now made to walls
subjected only to axial loading.
Eccl'Il(r;c load (p.125) The reference to 'no tension'
when. determining the distribution of in-plane forces
per unit length has now beenomitted,
General (p.126)
In order to reflect current practice, the Code clauses
arc only appropriate to walls having a height to average
thickness ratio not exceeding five. In view of this
restriction. all references to effective height and to
shori and slender walls arc now omitted. Unbraced
walls are now referred to as walls unrestrained in
,'losition. ';':'.,
Axial load plus bending normlll to wall (p.126)
The coefficient All' has been considerably simplified and
Call now take one of two values: 0.35 for grade 15 or 20
concrete, and 0.4 for grades 25 and above.
Slender.walls are no longer covered by the Code.
Shear (p.128)
No reference is now made to shear forces at right angles
to a wall. However. it is understood that the clauses for
shear forces in the plane of a wall. which arc discussed
on page 128. should also be applied to shear forces'at
right-angles to a wall.
Bearing (p.128)
Equation (8.4A) should be used
Deflection (p.128)
Since the height to thickness ratio is now limited to five,
deflections need not be considered,
Crack control (p. 128)
Reinforcement has to be provided only to control
shrinkage and temperature cracks. The required
amount of reinforcement is given by equation (7.2a).
• and its spacing should not exceed 150 mm.
Effective heights (p.129)
The new effective heights have been derived specifi-
cally for bridges.
Bridge abutments and wing walls
(p.129)
There are no longer deemed to satisfy ·rules for the bar
spacings.
Flexure (p. UO) Reinforcement should be distributed
evenly across the width of the footing unless the width
exceeds 1.5 (h('ol + 3d). where bcol is the width of the
column. In such a case. two-thirds of the reinforcement
should be concentrated in the width (bcol + 3d) centred
on the column. Hence. Fig. 9.15 is no longer appro-
priate. and Part 4 : 19H4 requires a greater concen-
tration of reinforcement around the column than did
Part 4 : 1Y78.
Pllnching shear (p.l31) The appropriate perimeters
arc now those shown in Fig. 6.6A.
Serviceability limit state (p. 131)
The limiting steel stress is now O.75J~.
NeXlifal shear (p.DI) Any section across the full
width riow has to be checked, as opposed to just a
section at the face of the column,
Punching shear (p.132) The perimeter shown in Fig.
9.17(b) is no longer used: instead. the perimeters
shown in Fig. 6.6A sholiid he ustld for the punching bf
both the columns and the piles.
9.1 Slender column (p.133)
A characteristic strength of 425 N/mm2
is retained for
the rC,inforcemcnt, because, at the time of vriting,
column design charts are n()t generally available for
460 N/mm2 reinforcement. Calculations are presented
only for the ultimate limit state.
No sidesway (p.133)
Case 2 of Fig. 9.0a is appropriate. Hem:e
Ie .-:.::. 0.85 tv ;:: 0.85 X 8 =: 6.8 m
Slenderness ratio = 6.8/0.6 :: 11.3. This do.es not
exceed 12. thus the column is short.
.The minimum eccentricity is the lesser of 0.05 x 0.6 =
0.03 m and 20 mm (i.e. 20. mm).
Moment at base == 280 x 8 + 2600 x 0.02= 2292 kN m
Mlbh'1. := Mlhyhx 2 = 2292 x 106
/(1200 x 60(2)
=5.31 N/mm2
Nlbh =- N1hvhx := 2600 x m1
/(l200 x 6(0)
. =3.61 N/mm2
.
Assume 40 mrn main bars, 10 mm links and 50 mm
cover for very severe exposure (see amendments to
Chapter 10), so that dlhx = 520/600 = 0.867. Hence.
use Design Chart 85 of CPll0 : Part 2 112g]: from
which
100 A.,lhyh.t == 3.1
... AS( = 3.1 x 1200 x flOo/I00 .::: 22.320 mm2
.Use 18 No. 40 mm bars (22680 mm2
) with 9 bars in
each face.
Sidesway (p.133)
Case 7 of Fig. 9.0a is appropriate. Hence
Ie == 2.3/" :=:: 2.3 X 8 =18.4 m
Slenderness ratio = 18.4/0.6 == 30.7. It is emphasised
that this ratio exceeds the Part 4 : 1984 limitation of 30;
however. in order to provide a design comparative with
that of Part 4: 1978. the section size will not oe
increased. The column is slender, and the initial and
additional moments are both maxima at the base.
l.i~ence .
¥l == 280 x 8::= 2240 kN m and, from equation (9.22).
t'he total moment is
Ntl == 2240 + (2600 x 0,(11750)(30.7)2 (I -·0.0035 x
30.7)
== 2240 + 750 := 2990 kN m
25](https://image.slidesharecdn.com/concrete-bridge-design-to-bs5400-190313042802/75/Concrete-bridge-design-to-bs5400-113-2048.jpg)
![M/hh2 =M/hyh.~ ";'~99(} XlOh 1(1200X 6()02)
= 6.92 N/mm2
NIbil =Nllzyhx= 3.61 N/mm2
(as before)
From DesigltChart 85 of CPllO : Part 2,
100 A.,,lhvhx =4.4
.. .. An =' 4.4 X 1200 X 6001100 ~ 31680 mm2
.. llse. 26 No. 40.mmbars{32700mm2Ywith 13 bars in .
..' eac~ fa~e. .... .... .........•.... ,.' ...... ... ' ::........ .... ...•...• . . .
. Sheaf
~hear reinforcement will be designed only fol' the no
. ,sl~esway case, w~ich has the.le.ast amOUnt oftension
.remforccrnentofthe two cas¢~ c<)nsidered,.. .
The t.ensionsteel consists' of 9 No. 40 mm bars
(11 300 mm2
).
100 As,lbd = 100 x 11300/(1200 x 520) = 1.81
From equation(6.1al-- '"
Vc: =(O.27/L2S)(1.81)1~ (40)IIJ ..
. =0.9 N/mm2 '. . .
From equation (6.7c)
~' =(SO(}/S20)1i4 =0.99
Axial load enhancement factor
= 1 + 0.05 x 2600 x 103/(600 x 12(0) =1.18
Thus, enhanced value of t-v(.
= 1.18 x 0.99 x 0.9 = 1.05 N/mm2
Applied s~ear stress"'; v'= 280 x 103/(1200 x 520)
=0.449 N/mm2
< 1.0S N/mm2, thus only nominal links required.
9.2 Hillerborg strip method applied to an
abutment (p.133)
Since the Hillerborg strip method is a plastic method,
the value ofYf3 should be 1.15 for all loads. Hence, the
HA surcharge and braking design loads shoul.d be
. d 2
mcrease t~ 17.25 kN/m and 43.1 kN/m respectively.
The me.thod of calculation thereafter is unchanged.
In FIg. 9.19, the x and y axes are vertical and
horizontal respectively.
9.3 Pile cap (p.135)
The reinforcement characteristic strength is increased
!o 460 N/mm
2
. The concrete. characteristic strength .is
Increased to 40 N/mm2 to avoid the use of air-entrained
concrete (see amendments to Chapter 10).
Bending theory (p.135)
Bending (p.135) Because of the increased characteris-
tic.strengthof the steel, only 17 No; 20 mlTi bars
(5340 ,mm2) are required.
Flexural shear (p.135)
100 A,Ibd =(100 x 5340)/(2300 x 980) =0.237
From equation (6.18)
v(' =(0.27/1.25)(O.237)1fJ (40)11.l =0.457 N/mm2
From equation (6.7c) ' .
26
. t .
cap
9.258 Punching perimete.rJor Example 9,~
~! =(500/980)114 =0.845
Cr~tical sect~on is where a" is aslargc as possible; thus,
as In the maIO text, a" =300 mm, and the enhancement
factor is 6.53. Thus, the shear capacity of the critical
section
= 0.845·x 0.457 [(6.53)(2 x 500) + (2300 - 2 x500)1
. x 980 x 10-'
=2960 kN
Actual shear force =2 x 1300 =2600 kN < 2960 kN
.·.O.K.
Punching shear (p.136) Only perimeter (d) of Fig.
6.6A can be accommodated. The dimensions of the
perimeter are shown in Fig. 9:25a. In one direction, d
= 980 mm and 100 A,Ibd = 0.237. If the same rein-
forcement is provided in the orthogonal direction with
an effective depth of 1000 nlp1, then 100 A,Ibd =0.232.
Average 100 A,Ibd = 0.235
From equation (6.18)
Vc =(0.27/1.25)(0.235)1/1(40)IA = 0.456 N/mm2
Average d = 990 mm
From equation (6.7c)
!;.. =(500/990)1/4 =0.843
The perimeter is 386 mm from the corner of the column
(see pag,e 136). Hence, the shortshear span enhance-
ment factor = 2 x 990/386 =5.13.
Length of perimeter 'crossed by reinforcement
anchored over pile
= 1.63 - 2(0.15) V2== 1.21 m
Shear capacity
=0.843 x 0.456 [5.13 x 1210 + (1630 - 1210)] 990 x
10-3
== 2520kN
Actual shear force = 1300 kN < 2520 kN
.·.O.K.
Truss analogy (p.136)
Truss (p.136) In spite of the increased characteristic
strength of the reinforcement, the main steel provided
is the same as in the main text. .
e
"'1
Flexural shear (p.136)
Over a pile, v,. from equation (6.1a) is
Vi' =.: (0.27/1.25)(0.45)1/1
(40)'11 =0.566 N/mm
2
As before, the short shear span enhancement factor is
6.53.
Between and outside piles. v.. = 0.39 N/mm
2
(Tahle 8
of Code).
Depth factor. as in the bending theory calculations.
= 0.845
Shear capacity
= 0.845 [(6.53)(0.566)(2 x 5(0) + (0.39)(2300 - 2 x
500)]980 xlO~:.3
=3480 kN
Act~al shear force = 2 x 1300 =2600 kN < 3480 kN
.·.O.K.
j
Punchillg shear (p.136) The critical perimeter is
shown in Fig. 9.25a.
Average d =990 mm. There will be 9 bars (2830 mm2
)
crossing the perimeter in each of the orthogonal direc-
tions, thus 100 A/bd = (100 x 2830)/(1150 x 990)
=0.249
From equation (6.1a)
Vc =(0.27/1.25)(0.249)'11
(40)1;., == 0.465 N/mm2
For the average d. the depth factor = 0.843. and the
short shear span enhancement factor == 5.13.
Shear capacity
-'''·''.'''J..,JL~43 x 0.465 [5.13 x 1210 + (1630 - 1210)] 990
",..... x 10-3
:;; 2570 kN
Actual shear force == 1300 kN < 2570 kN
,'. O.K.
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![Chapter 10
Detailing
Cover (p.137)
It is now emphasised in the Code that, although ttH'
nominal cover is used for dcsign and is indkuted on the
dnlwings, thc actual cover can be up to 5 mm less,
The nominal covers in Purt 4: 197H have hecn
increased by at least 5 mm. and air-entrained concrete
is now required in morc'situations, Thc Part 4 : 19~4
values arc given in Table 1O.lA which sup(:rsedes Tublc
10.1, However, the nominal cover should also I'll.' not
less than the bar size or maximum aggregute size plus 5
mill. The mure l)iI~lOull ~overs, etc, rcnc~t the durabil·
ity problems which have bcen exp~~ricnced in the UK in
recent years.
Table IO.IA Nominal covers
Nominal cover (mm)
Envlronm~nt
ror concrete grade
15 30 40 ~SO
Extreme
Exposed to abrasive action by l-ea
water, or to water with a pH ~4.5 6.'i • 55
Very severe
Directly afft'ctcd by dc-icing
',I,
salts. or sea water spray ;. 50" 40
Severc
Exposed to driving ran, or
altcrnatc wetting and drying 45' 35 30
Moderate
Shcltered. orpermancntly
saturated with water with a pH >4.5 45 35 30 25
• Air-entrained concrete needed if liahle to freezing whilst
wet.
t Grade 30 permitted only for parapetsif air entrained and
60 mmcover.
Minimum distance between bars (p. 137)
Thc minimum distance betwecn rows of bars in in-situ
.members has been increased to the maximum aggre-
gate size,
28
Shrinkage and temperature reinforcement (p, 138)
The arcas ()f reinforcement rcqulrell to t:ontrol shink·
agc and temperature cracks havl~ heen, essentially.
doubled to the values givl~n by equation (7.2a),
• Beams and slabs (p, 138)
Correcl the term (/t 2
/d)oin the last equation on page 13H
to (h/df,
Minimum areas of secondary reinforcement in pre-
(10mmantly tensile areas of solid slahs are now given LIS
O.12cyo of high strength stcel or 0.15'Yc, of mild steel.
These percentages are numerically the same as those in
the building codes (7.15). but are expressed in terms of
the effective area rather than the gross area, Thus
smaller minimum areas of secondary reinforcement are
required than in accordan~ with CP114.
The percentages of secO,ndary reinforcement men-
tioned in the last paragr~ph should also be PfQ,vided
where the main reinforcement resists compressfon. The
diameter of the secondary reinforcement should not he
less than one-quarter of that of the main bars, and the
spacing should not exceed 300 mm, However, as ex-
plained on page 139, links should be provided if the
area of compression reinforcement exceeds 1%. The
clauses concerned with the restraint of compression
hars in s.labs (and as mentioned later, walls) are consis-
tent with those in the new building. code (BSHI!O)
1294].
It is now necessary to provide longitudinal reinforce-
ment to control cracking at the side of heams where thc
depth of the side face exceeds 600 mm. Steel having an
area of 0.05% of the effective section area should he
provided in each face with a spacing not greater than
300 mm.
Walls (p.138)
The above requirements for secondary reinforccment
in slabs. where the main reinforcement resists com-
pression, are also applicable to walls.
•
Lap lengths (p.141)
It is no longer necessary to increase the lap length by
25% for deformed bars in tension.
However. it is known that the bond strength of top
cast bars is less than that of bottom cast bars [301], and
that the mode of failure of a lap changes from slip to
splitting of the concr~te cover when the cover is less
than about 2.5 times the bar diamcters [302, 303},
Hence, longer lap lengths are required in such situ-
ations. Thus. the Code requires the lap length to be
increased by 40% if any of the following conditions
apply:
1. The bars are in the top of the section as cast. and
the nominal cover is less than twice the bar diam-
eter.
2, The clear distance between the lap and another
pair of lapped bars is less than 150 mm.
3. The lap is in a corner, and the nominal cover to
either face is less than twice the bar diameter.
The lap lengthshould be increascd by 1000t'c, if either
conditions 1 and 2 or 1 and 3 occur together,
Anchorage at a simply supported end (p.142)
Condition 3 on page 142 has heen removed from the
Code.
External tendons (p. 143)
The Code no longer deals with external tendons.
Transmission length in pre-tensioned
members (p. 143)
The following formula is now given for the transmission
length (I,) wh(~re the initial stress does not exceed 75'X,
of the characteristic strength of the tendon. and the
concrete strength at transfer is at least 30 N/mm2:
I, ::: k, <l>tv'r., (10.6a)
where <t> is th,~ nominal diameter of the tendon in mm.
and k/ is a coeffident (in the range 240 to 6(0) which
depends on the type of tendon. This formula results in
transmission length!. very similar to thos(~ in Part
4 : 1978 for wires, but thc transmission lengths for
strand arc ahout 40(Yc, greater than those in Part
4 : 1978. This increase reflects currt:!nt test data.
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Concrete bridge-design-to-bs5400
This document provides a summary of a book on concrete bridge design according to BS 5400. The book aims to provide guidance on applying the limit state design code for concrete bridges by explaining its clauses and comparing them to previous design standards. It discusses analysis methods, loadings, material properties, design criteria, and worked examples to illustrate the code's application to bridge elements like beams, slabs, foundations and composite construction.
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- 1. Concrete bridge .design toBS 5400 L. A. Clark
- 2. Constructloo Press an imprintof Longman Group Limited ·;f. • Longman House, Burnt Mill,Harlow, Essex CM20 2JE, England ' Associated companies throughout the World Published in the United States ofAmerica by Longman Inc., New York © L. A. Clark. 1983 , I 0,•• ,'" ' ", . All rights reserved. No part of this pUblication may be' reproduced, stored, iit a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying. recording, or otherwise. without the prior permission of the Copyright owner. First published 1983, AMD 1986 BrItish Library Cataloguing In Publication Datal Clark. L.A. ' Concrete bridge design to BS 5400. I. Bridges, Concrete - Design and construction I. Title 624' .25 TG33S ISBN 0-86095-893-0 Set in 10/12 Times Roman (VIP) Printed in Great Britain by The Pitman Press Ltd., Bath ." .:, r ' .~ '" . ' ',': , ~, . Preface BS 5400: Part 4, which deals with the .design of concrete bridges, was published in 1978. However, the Department of Transport, as a client, did not permit its use unless amended in accordance with the Depart- ment's own implementation document published in 1983. The majority of the amendments contained in the latter document, together with other revisions, were incorporated in the revised BS 5400 : Part 4, which was published in 1984. This supplement to Concrete Bridge Design to BS 5400 updates the main text and brings it into line with the 1984 revision. The opportunity has also been taken to correct some minor typographical errors in the main text. The supplement contains only those headings of the main text which require amendments, together with their relevant main text page numbers. The main text pages relevant to a particular supplement page are indicated at the top of each supplement page. Amended equations, tables and figures are indicated by placing the letter A after the original number; e.g. equation (6.21A) is the amended version of equation (6.21). New equations, tables and figures are numpered in an a,b,c sequence following on from the last number referred to in the main text, and are printed in bold typeface; e.g. equation (6.7b) is the second new equa- tion after equation (6.7) in the main text. Examples have either been completely reworked or had minor changes indicated. L. A. Clark May 1985 '
- 3. Acknowledge'ments I thank manyof my former colleagues at the Cement and Concrete Association for the contributions which they have indirectly made to this book through the discuNsions which I had with them. I am particularly indebted to George Somerville and Gordon Elliott who, each in his own particular way, encouraged my interest in concrete bridges. In addition, it would not have been possible for me to write this book without the benefit of the numerous dis(~ussi()ns which I have had with bridge engineers throughout the llnited Kingdom - I am grateful to each of them. My thanks are due to Peter Thorogood and Jim Church Publisher's acknowledgements Figures 4.5, 4.6, 4.7, 4.8,8.4 and 10.8 were originally prepared by the author for the Bridge Engineering Stan- danls Division of the Department of Transport under con- tract. These figures, together with references, to the requirements of the Department of Transport's Design vi who read parts of the manuscript and made many construc- tive criticisms; also to Julie Hill who, with a small con- tribution from Christine Cope, carefully and efficiently typed the manuscript. Finally, prior to writing this book, I had wondered why it is usual for an author to thank his wife and family - now I know! Thus, , wish to thank my wife and daughters for their patience and understanding during the past three years. L. A. Clark June, 1981 Standards, are reproduced with the pennission of the Con- troller of Her Majesty's Stationery Office. Extracts from British Standards are included by permission of the British Standards Institute, 2 Park Street, London WIA 2BS, from whom complete copies can be obtained. Contents Preface Notation Chapter 1. Introduction The New Code Development of design standards for concrete structures Philosophy of limit state design Summary Chapter 2. Analysis General requirements Types of bridge deck Elastic methods of analysis Elastic stiffnesses Plastic methods of analysis Model analysis and testing Examples Chapter 3. Loadings General Loads to be considered Load combinations Partial safety factors Application ofloads Pemianent loads Transient loads Example Chapter 4. Material properties and design criteria Material properties Material partial safety factors Design criteria Yp values Summary ix xi ·1 ' 1 2 4 7 9 9 11 13 16 19 27 27 32 32 32 32" 33 34 34 35 42 45 45 46 48 52 54 Chapter 5. Ultimate limit state - flexure and in-plane forces 55 Reinforced concrete beams 55 Prestressed concrete beams 57 Reinforced concrete plates 58 Prestressed concrete slabs 61 Examples 61 Chapter 6. Ultimate limit state - shear-and torsion 65 Introduction 65 Shear in reinforced concrete 65 Shear in prestressed concrete 72 Torsion - general 75 Torsion of reinforced concrete 76 Torsion of prestressed concrete 81 Examples 83 Chapter 7. Serviceability limit state 86 Introduction 86 Reinforced concrete stress limitations 86 Crack control in reinforced concrete 88 Prestressed concrete stress limitations 94 Deflections 96 Examples 98 Chapter 8. Precast concrete and composite construction 102 Precast concrete 102 Composite construction 105 Example - Shear in composite construction 115 Chapter 9. Substructures and foundations 118 Introduction 118 Columns 118 Reinforced concrete walls 125 Plain concrete walls 126 Bridge piers and columns 129 vii
- 4. --,. Bridge abutments andwing walls 129 Foundations 130 Examples 133 Chapter 10. Detailing 137 Introduction 137 Reinforced concrete 137 Prestressed concrete 142 Chapter 11. lightweight aggregate concrete 147 Introduction 147 Durability 147 Strength 148 Movements 149 Chapter 12. Vibration and fatigue 151 Introduction 151 Vibration 151 Fatigue 154 viii Chapter 13. Temperature loading Introduction Serviceability limit state Ultimate limit state Design procedure Examples Appendix A. Equations for plate design Sign conventions Bending In-plane forces Appendix B. Transverse shear in cellular and voided slabs Introduction Cellular slabs Voided slabs References Index 158 158 158 162 163 164 169 169 169 170 171 171 171 171 176 183 Preface During the last decade, limit state design has been intro- duced, both nationally and internationally, into codes of practice for the design of concrete structures. Limit state design in British codes of practice first appeared in 1972 in the building code (CP 110). Since then it has been used in the water retaining structures code (BS 5337) in 1976, the masonry code (BS 5628) in 1978 and, finally, the bridge code (BS 5400) in 1978. The introduction of limit state design to the design of concrete bridges constitutes a radi- cal change in design philosophy because the existing design documents are written, principally, in terms of a working load and permissible stress design philosophy. Thus, the use of BS 5400 may change design procedures, although it is unlikely to change significantly the final sec- tion sizes adopted for concrete bridges. This is due to the fact that the loadings and design criteria were chosen so that, in general. bridges designed to BS 5400 would be similar to bridges designed to the then existing design documents. In view of the different design methods used in BS 5400, a number of bridge engineers have expressed the need for a document which gives guidance in the use of this code of practice. The present book is an attempt to meet this need; its aim is to give the background to the various clauses of BS 5400, which are concerned with concrete bridges, and to compare them with the corres- ponding clauses in the existing design documents. After tracing the history of limit state design and explaining its terminology, the analysis:, loading and design aspects of BS 5400 are discussed. BS 5400 permits the use of plastic methods of analysis. However, bridge engineers have complained that there is a lack of guidance in BS 5400 on the use of plastic methods. Therefore, applications of plastic methods are discussed in Chapter 2. In contrast, the reader is assumed to be familiar with current methods of elastic analysis and so these methods are discussed only briefly. However, the evalu- ation of elastic stiffnesses for various types of bridge deck is discussed in some detail. The loadings in BS 5400 differ from those in the exist- ing design documents. The two sets of loadings are com- pared in Chapter 3, where it can be seen that some load- ings differ only slightly whereas others differ significantly. Compared with those of existing documents, the design criteria of BS 5400, and the methods of satisfying them, are very different for reinforced concrete, but very similar for prestressed concrete. These differences are discussed in Chapters 4 to 12. Worked examples are given at the ends of most chap- ters. These examples illustrate the applications of various clauses of BS 5400. Many bridge engineers have expressed the view "that BS 5400 does not deal adequately with certain aspects of concrete bridge design. Thus, in addition to giving the background to the BS 5400 clauses and suggesting interpretations of them in ambiguous situations, this book suggests procedures for those aspects of design which are not covered adequately; e.g. shear in composite construc- tion, transverse shear in voided slabs, and the incorpor- ation of temperature loading· into the design procedure. It is hoped that this book will assist practising concrete bridge engineers in interpreting and applying BS 5400. Also it is hoped that it will be of use to undergraduate and postgraduate students taking courses in bridge engineering. L. A. Clark June 1981 ix
- 5. Notation The principal symbolsused in this book are as follows. Other symbols are defined in the text. Ar area of concrete Ae, area of flange of composite be'am A,.. area of tendon A. area of tension reinforcement A; area of compression reinforcement in beam A... area of reinforcement in column A.I• area of longitudinal torsion reinforcement A•• area of shear reinforcement A, area of transverse reinforcement in flange A.. area within median line of box Q span: acceleration Q' distance measured from compression face of beam a" bar spacing art'''' distance between centroids of compressive flange and of composite section a" perpendicular distance from crack a,. shear span b breadth b. width of interface in composite section C torsional inertia: compressive force: coefficient CD drag coefficient C,. lift coefficient e cover em/It minimum cover 0 internal dissipation of energy Dr density of concrete 0 .. 0" D.,., D. plate bending stiffnesses per unit length d effective depth: void diameter d' depth to compression reinforcement in beam dr depth of concrete in compression d, effective depth in shear d.. effective depth of half end E elastic modulus; work done by external loads Er elastic modulus of concrete E•., elastic modulus of flange of composite beam E, elastic modulus of steel e eccentricity e, initial .:olumn eccentricity t.dd additional column eccentricity F force Fin, F", F. F. P, F, f f.w Ib fbG fb. fa I.... 1«1 leu fey' I", I. 'ph fpd fp, fp, fpu f. I. lit, ",,'"flp Iy 'yl. f,. I.. I. G H h h" h, h, 11""., h""", bursting force tensile force in bar at ultimate limit state concrete force; centrifugal force steel force force in compression reinforcement tie force stress average compressive stress in end bloek bearing stress average anchorage bond stress loeal bond stress concrete strength at transfer average concrete tensile stress between cracks compressive stress due to prestress characteristic strength of concrete cylinder compressive strength of concrete hypothetical tensile stress characteristic strength tendon stress at failure design stress of tendon when used as torsion rein- forcement effective prestress tensile stress due to prestress at an extreme concrete fibre characteristic strength of tendon flexural strength (modulus of rupture) of concrete shear stress steel stress at a crack at cracking load design tensile strength of concrete maximum tensile stress in end block permissible concrete tensile stress in end block characteristic strength of reinforcement characteristic strength of longitudinal torsion rein- forcement characteristic strength of link reinforcement fundamental natural frequency of unloaded bridge steel stress ignoring tension stiffening shear modulus ; depth of back-~II overall depth or thickness bottom flange thickness lever arm of cellular slab T-beam flange thickness minimum and maximum dimensions of rectangle . xi
- 6. ~ h, top flangethickness n· h", web thickness r h",o box wall thickness rp. second moment of area S· I, second moment of area of flange S. /.. second moment of area of web I., second moment of area of longitudinal section S.. Sy I" second moment of area of transverse section S. J. longitudinal torsional inertia S2 i" transverse torsional inertia s K coefficient; factor T K, flange stiffness Kw web :;tiffnc:;5 I, k torsional constant; factor Ucrl' L span; length u. I., transmission length ill box girder V span; length V. /" I... I... effective height of column Vr.r /',1> anchorage length at half joint /" clear height of column V.o /1 length of side span V" M moment v MaJd additional column moment vr ME initial column moment v" M, total column moment; cracking moment v, Mx , My, M.y plate ,bending and twisting moments per unit length V'mi" M;, My~ Mo<' plate moments of resistance per unit length Vru Mil moment of resistance of beam or column W Mv Vierendeel bending moment W M" moment to produce zero stress at level of steel of Wr'IV", prestressed beam M"M2 larger and smaller column end moments W,. m plate yield moment per unit length x m" corner moment per unit length X,Y, z "," nonnal moment of resistance per unit length of yield XI.YIline N axial load; number of cycles of stress to cause fatigue Y failure Y"o N,. concrete force Y. N.. steel force; axial load capacity of column at service- Yn ability limit state z Nil: axial load capacity of column at ultimate limit state i N..; Ny. N.), plate in-plane forces per unit length IX N:. N;, N! plate resistive forces per unit length flU' maximum wall load per unit length lX,. I' point load; prestressing force CII" l'k total initial prestressing force; maximum tendon force CII.• Pr effective prestressing force IX, P" longitudinal wind load p I',.,. longitudinal wind load on live load p,." 1'1,.• longitudinal wind load on superstructure Yrl. Yf2. Yfl. 1', transverse wind load Yp 1',. vertical wind load YR r, prestressing force at distance x from jack Ytn~. Ym2, Ym P" prestressing force at jack Y"I' Yn2. y" f.' uniformly distributed load Y.. Y.'· () load /) ()* design load ()k characteristic load (),. Q,. plate shear forces per unit length q. uniformly distributed load f.t· U reaction EC.f xii '" design resistance radius radius of curvature of duct design load effect first moment of area of flange in composite con- struction plate shear stiffnesses per unit length funnelling factor gust factor spacing tension force; torque; time thickness; time temperature at distance z above soffit lengthof punching shear critical perimeter circumference of bar shear force shear force carried by concrete shear force carried by concrete in f1exurally cracked prestressed beam shear force to cause web cracking maximum IllIowuble shear force shear stress; mean hourly wind speed allowable shear stress; maximum wind gust speed interface shear stress torsionnl shear stress value of torsional shear stress above which torsion reinforcement is required . maximum aU(iwable torsional shear stress load displacemen't; crack width flange and web warping force~ per unit length in box girder mean crack width neutral axis depth rectangular co-ordinates link dimensions parameter defining yi~ld line patlern half length of side of loaded area static deflection half len'gth of side of resisting concrete block lever arm distance of section centroid from soffit angle; Hillerborg load proportion: parameter defin- ing yield line pattern coefficient of expansion of concrete index used in biaxial bending of column coefficient of expansion of steel coefficient of expansion at distance z from soffit percentage redistribution; creep fartor; angle creep factor partial safety factors applied to loads partial safety factor applied to load effects gap factor partial safety factors applied to material strengths consequence factors plate shear strains deflection; stress transfonnation factor: logarithmic decrement strain strain at section centroid creep strain free shrinkage strain Ediff Em En_ E,I Yn' E. E" E. £y En fl " 1;. a 8" 0, differential shrinkage strain k moment reduction factor strain allowing for tension stiffening ~w stress block factor for plain concrete walls direct and shear strains ,.,. coefficient of friction steel design strain; shrinkage strain v Poisson's ratio tension stiffening strain p density ultimate concrete strain PD shrinkage coefficient yield strain of reinforcement a, stress range soffit strain aH limiting stress range strain ignoring tension stiffening cjI bar diameter; creep coefficient; angle depth of slab factor rotation; angle cjIl creep coefficient 11' slope; curvature; dynamic response factor nonnal rotation in yield line 11', shrinkage curvature thennal rotation 11'. maximum column curvatul'e at collapse xiii
- 7. Chapter 1 Introduction The NewCode Background Rilles for the design of bridges have been the subject of continuous amendment and development over the years, amI a significant development took place in 1967. At that time, a mee.ting was held to discuss the revision of British Standard BS 153 I. t], on which many bridge design docu- ments were based [2·1. It was suggested that a unified code of practice should be written in terms of limit state design which would cover steel, concrete and composite steel- concrete bridges of any span. A number of sub-committees were then formed to draft various sections of such a code; the work of these sub-committees has culminated in Brit- ish Standard 5400 which will, henceforth. be referred to in this book as the Code. The author understands that the Code Committee did not intend to produce documents which would result in significant changes in design practice but. rather, intended that bridges designed to the Code would be broadly similar to those designed to the then current documents. In addi- tion the suh-committees concerned with the various ma- terials and types of bridges had to produce documents which would be compatible with each other. In subsequent chapters, the background to the Code is given in detail and suggestions made as to its interpretation in praL'ticc. The remainder of this first chapter is concerned with general aspects of the Code. "*Code format The Code consists of the ten parts listed in Table 1. I and. at the time of writing, all except Parts 3 and 9 have been published: drafts of these parts are available. Hence. sufficient documents have been published to design con- crete bridges. It sh(luld be noted that BS 5400 is both a Code pf Practi<:e and a Specification. However. not all aspects I of the design and construction of bridges are covered; exceptions worthy of mention are the design of " Table I.IBS 5400 - the ten parts Part Contents I 2 :~ 4 5 6 7 General statement Specification for loads Code of practice for design of steel hridges Code of practice for design of concrete bridges Code of practice f(lr design of composite bridges Specification for materials and workmanship. steel Specification for materials and workmanship. concrete. reinforcement and prestressing tendons Recommendations for m"terials and workmllnship. concrete, reinforcement and prestressing tendons 9 10 Code of practice for bearings Code of practice for fatigue parapets and such constructional aspects as expansion joints and waterproofing. The contents of the individual parts are now sum- marised. Part 1 The philosophy of limit state design is presented and the methods of analysis which may be adopted are stated in general terms. Part 2 Details are given of the loads to be considered for all types of bridges. the partial safety factors to be applied to each load and the load combinations to be adopted. Part 3 Design rules for steel bridges are given but reference is not made to Part 3 in this book. At the time of writing it is in draft form. Part 4 Design rules for reinforced. prestressed and '~()rnposile (precast plus in-situ) concrete bridges are given in terms of material properties, design criteria and methods of com- pliance.
- 8. Part 5 ~esign rulesfor steel-concrete composite bridges are given and some of these are referred to in this book. material from the Code both during the Code's drafting stages and since publication. Railway bridges There should be fewer problems in implementing the Code The. speci.fication of materials and workmanship in con. for railway 'bridges than for highway bridges, because Brit· ?~ctlon with structural steelwork are given, but reference,,1sh Rail have been using limit state design since 1974 [14]. IS not made to Part 6 in this book. ....;" ....~.~.w. ".,," Part 6 Part 7 The, speci.fication of materials and workmanship in con- nectl~n "':Ith concrete, reinforcement and prestressing ten- dons IS gIven. Part 8 Recommendations are given for the application of Part 7. Part 9 The design, lesting and specification of bridge bearings are l'overed. At the time of writing Part 9 is in draft form but some material on bearings is included in Part 2 as an appendix which will eventually be superseded by Part 9. Part 10 l.:o.ading~ for fatig.ue calculations and methods of assessing f,lllgue life are gIven. Part 10 is concerned mainly with s,tcel and steel-concrete composite bridges but some sec- tlons are referred 10 in this book. I~plementation of Code for concrete bndges Highway bridges From the previous discussion, it can be seen that, if the Code were to be adopted for concrete highway bridges: , I. Part 2. would replace the Department of Transport'~ Techmcal Memorandum BE 1/77 [3] and British Standard BS 153 Part 3A r4 ] . ' 2. Part 4 would replace the Department of Transport's Technical Memoranda BE 1/73 [5J and BE 2/73 [6] and Codes of Practice CP 114 [7], CP 115 [8] and CP 11619]. ' .1, Parts 7 and 8 would replace the Department of Trans- port's Specification for road and bridge works [10]. 4. Part 9 would replace the Department of Transport's Memoranda BE 1/76 [II] and 1M II [12]. . At the time of writing, the Department of Transport's views on the implementation of the Code are summarised in their Departmental Standard BD 1/78 [13]. This es- sentially, states that the Code will, in due course', be ,~upp.lemented by Departmental design and specification rCl.Jlllfcmcnts.ln addition, implementation is to be phased o;('r an ~nst~ted. period of time, with an initial stage of trial applicatIons of thc Code to selected schemes. How- ~:cr, ~t should be noted that several of the Department's rcl'hnlcal Memoranda hnve been updated to incorporate 2 Development of design standards for concrete structures Before explaining the philosophy of limit state design it is instructive to consider current design procedures for con- crete bridges and to examine the trends that have taken place in the development of codes of practice for concrete , structures in general. Current design procedures for concrete bridges are based primarily on the requirements of a series of Technical Memoranda issued by the Department of Transport (e.g. BE 1173, BE 2/73 and BE 1/77); these in tllrn are based on current Codes of Practice for. buildings (e.g. CP 114, CP 11"5 and CP 116), with some important modifications which reflect problems peculiar to bridges. ' Essentially, trial structures are analysed elastically to determine maximum values of effects due to specified working loads. Critical sections are then designed on a ,~modular ratio basis to ensure that certain specified stress ,;,Iimitations for both steel and concrete are not exceeded. Thus, the approach is basically one of working loads and permissible stresses, although there are also requirements to check crack widths in reinforced concrete structures and pto check the ultimate strength of prestressed concrete struc- tures. This design pro~ess has three distinguishing features - it is based on ~ Permissible working stress phil- osophy, it assumes elastic material properties and it is deterministic. Each of these, features will now be discussed with reference to Table 1.2 which summarises the basic requirements of the various structural concrete building codes since 1934. Permissible stresses The permissible working stress design equation is: stress due to working load ~ permissible working stress where, permissible working stress = material 'failure' stress safety factor Thus stresses are limited at the workinR load essentially to provide an adequate margin of safety against/ai/lire. Such a design approach was perfectly adequate whilst matedal strengths were low and the safety factor high because the pennissible working stresses were sufficiently low for ser- viceability considerations (deflections and cracking) not to be critical. In Ihis respect the most important consideration is that of the permissible steel stress and Table 1.2 shows Table 1.2 Summary of basic requirements from various Codes of Practice for structural concrete Code Basis of analysis Steel stress Umltatlon Required load factor ' Additional design and design (N/mml) requirements OSIR Elastic analysis, with 140 for beams None for beams None (1934) variable modular ratio 100 for columns 3.0 for columns and permissible stresses f>0.45 fy CP 114 As above, but m = 15 190 in tension For columns: Warning against (1948) 140 in cQmpression 2.0 for steel excessive deflecti()Ds f>0.50 fy 2.6 for concrete ····c CP 114 Either elastic analysis 210 in tension 2.0 for steel Span/depth ratios (1957) or load factor method 160 in compression 2.6 for concrete given for beams and slabs. f>0.50 fy Warning against cracking CP 115 Both elastic and ultimate Cracking avoided by 1.50 + 2.5L (1959) load methods required limiting concrete tension Warning against or 2 (0 + L) excessive deflections CP 114 Either elastic analysis 230 In tension 1.8 for steel More detailed span/depth (as amended or load factor me.thod 170 in compression 2.3 for concrete ratios for deflection. 1965) f>0.55 £y Warning against cracking CP 110 Limit state design No direct limit 1.6-1.8 for steel Detailed span/depths or (1972) methods set, except by cracking 2.1-2.4 for concrete calculations for deflection. and deflection requirements that the ratio of permissible steel stress to steel yield strength has gradually increased over the years (i.e, the safety factor has decreased). This, combined with the introduction of high-strength reinforcement, has meant that permissible steel stresses have risen to a level at which the serviceability aspects of design have now to be considered specifically. Table 1.2 shows that, as permissible steel stresses have increased, more attention has been given in building codes to deflection and cracking. In bridge design, the consideration of the serviceability aspects of design was reflected in the introduction of specific crack control requirements in the Department of Transport documents. It can thus be seen that the original simplicity of the permissible working stress design philosophy has been lost by the necessity to carry out further calculations at the working load. Moreover, and of more concern, the working stress designer is now in a position in which he is using a design process in which the purposes of the various criteria are far from self evident. Elastic material behaviour, It has long been recognised that steel and concrete exhibit behaviour of a plastic nature at high stresses. Such behaviour exposes undesirable features of working stress design: beams designed on a working stress basis with identical factors of safety applied to the stresses, but with different steel percentages, have different factors of safety against failure, and the capacity of an indeterminate struc· ture to redistribute moments cannot be utilised if its plastic properties are ignored. However, it was not until 1957, with the introduction of the load factor method of design in ,CP 114, that the plastic properties of materials were recognised, for all structural members, albeit disguised in a workin~ stress format. The concept of considering the elas- tic response of a structure at its working load and its plas-! tic response at the ultimate load was first codified in the Specific calculations for crack width required prestressed concrete code (CP 115) of 1959; and this code may be regarded as the first British limit state design code. Deterministic design The existing design procedure is deterministic in that it is implicitly assumed that it is possible to categorically state that, under a specified loading condition, the stresses in the materials, at certain points of the structure, will be of uniquely calculable values. It is obvious that, due to the inherent variabilities of both loads and material properties, it is not possible to be deterministic and that a probabilistic approach to design is necessary. Statistical methods were introduced into CP 115 in .1959 to deal with the control of concrete quality, but were not directly involved in the design process. Limit state design The implication of the above developments is that it has been necessary: 1. To consider more than one aspect of design (e.g. strength, deflections and cracking). 2. To treat each of these aspects separately. 3. To consider the variable nature of loads and material properties. The latest building code, CP 110 [15], which introduced limit state design in combination with characteristic values and partial safety factors. was the culmination of these trends and developments. CP 110 made it possible to treat each aspect of design separately and logically, and to recognise the inherent variability of both loads and material properties in a more formal way. Although, with its intro- duction into British design practice in CP 110 in 1972, limit state design was considered as a revolutionary design , 3
- 9. approach, it couldalso be regarded as the fonnal recog- nition of trends which have been developing since the first national code was written. Generally, design standards for concrete bridges have tended to follow, either explicitly or with a slightly con- servative approach, the trends in the building codes. This is also the case with BS 5400 Part 4 which, while written in terms of limit state design and based substantially on CP Ito, exhibits some modifications introduced to meet. the particular requirements of bridge structures. Philosophy of limit state design What is limit state design? Limit state design is a design process which aims to ensure that the structure being designed will not become unfit for the use for which it is required during its design life. The structure may reach a condition at which it becomes unfit for use for one of many reasons (e.g. collapse or excessive cracking) and each of these conditions is referred to as a limit state. In Jimit state design each limit state is examined separately in order to check that it is.. not attained. Assessment of whether a limit state is attained could be made on a deterministic or a probabilistic basis. In CP Ito and the Code, a probabilistic basis is adopted and. thus, each limit state is examined in order to check whether there is an acceptable probability of it not being achieved. Different 'acceptable probabilities' are associated with the different limit states, but no attempt is made to quantify these in the Code; in fact, the partial safety factors and design criteria, which are discussed later, are chosen to give similar levels of safety and serviceability to those obtained at present. However, typical levels of risk in the design life of a structure are taken to be 10-6 against collapse and 10-2 against unserviceability occurring. Thus the chance of collapse occurring is made remote and much less than the chance of the serviceability limit state being reached. Limit state design principles have been agreed interna- tionally and set out in International Standard ISO 2394 116); this document forms the basis. of the limit state design philosophy of BS 5400 which is presented in Part 1 of the Code and is now explained. Limit states As impJiedpreviously, a limit state is a condition beyond which a structure. or a part of a structure, would become less than completely fit for its intended use. Two limit states are considered in the Code. Ultimate limit state This corresponds to the maximum load-carrying capacity of the structure or a section of the structure, and could be attained by: 4 1. Loss of equilibrium when a part or the whole of the structure is considered as a rigid body. 2. A section of the structure or the whole of the structure reaching its ultimate strength in terms of post-elastic or post-buckling behaviour. 3. Fatigue failure. However, in Chapter 12, it can be seen that fatigue is considered not under ultimate loads but under a loading similar to that at the ser- viceability limit state. """""'" ,- Serviceability limit state This denotes a condition beyond which a loss of utility or cause for public concern may be expected, and remedial action required. For concrete bridges the serviceability limit state is, essentially, concerned with crack control and stress limitations. In addition, the serviceability limit state is concerned with the vibrations of footbridges; this aspect is discussed in Chapter 12. .. Design life This is defined in Part 1 of the Code as 120 years. How- ever, the Code emphasises that this does not necessarily mean that a bridge designed in accordance with it will no longer be fit for its purpose after 120 years, nor that it will continue to be serviceable for that length of time, without adequate and regular inspection and maintenance. Characteristic and nominal loads It is usual in limit state design to d~fine loads in terms of their characteristic valu~s, which are"defined as those loads with a 5% chance of heing exceeded, as illustrated in Fig. 1.1(a). However,Jor bridges, the statistical data required to derive the ch,aracteristic values are not avail- able for all loads; thus, the loads are defined in terms of nominal values. These have been selected on the basis of the existing data and are, in fact, very similar, to the loads in use at the time of writing the Code. For certain bridge loads, such as wind loads, statistical distributions are available; for these a return period of 120 years has been adopted in deriving the nominal loads, since 120 years is the design life specified in the Code. It is emphasised that the term 'nominal load' is used in the Code for all loads whether they are derived from statis- tical distributions or based on experience; Values of the nominal loads are assigned the general symbol Qk. They are given in Part 2 of the Code because they are appro- priate to all types of bridges. '*Characteristic strengths The characteristic strength of a material is defined as that strength with a 95% chance of being exceeded (see Fig. 1.I(b». Since statistical data concerning material properties are generally available, characteristic strengths / Characteristic load Ok I I I I I I I I I _..-----..Load (a) L.uoe! Fig. 1.1(a),(b) Characteristic values ... can be obtained and this term is thus adopted in the Code. Characteristic strengths are assigned the general symbol fk and are given, for concrete bridges, in Part 4 of the Code.. Design loads At each limit state, a design load is obtained fl:01l1 each nominal load by multiplying the latter by a p~rtlal safety factor (Yfd. The design load (Q*) is thus obtamed from . (1.1) Q* = YfI" Qk The partial safety factor, YrL ' is a function of two other partial safety factors: . ., . Y which takes account of the posslblltty of unfavourable n, . . I I deviation of the loads from their nomma va ues; Yj2' which takes account of the reduce~ probabil~ty th~t various loadings acting together wtll all attam their nominal values simultaneously. . It is emphasised that values ofYfl and Yf2are not given m the Code, but values oCyf/_are given in Part 2 of the Code. They appear in Part 2 because they are applicable to all bridges; they are discussed in Chapter 3. It should be stated here that the value of Yfl- is dependent upon a number of factors: I. Type of loading: it is obviously greater for} a hfighlY variable loading such as vehicle loading t lan or a reasonably well controlled loading such as. dead load. This is because in the former case there IS a gre~ter chance of an unfavourable deviation from the nommal value. 2. Number of loadings acting together: thefvallue rorda particular load decreases as the nu~ber .0 . o~ ler . o~ s acting with the load under consideratIOn...ncreases. This is because of the reduced probablltty o.f all of the loads attaining their nominal values sunul- taneously. 3. Importance of the limit state: the value ~01' a tru:~ ticular load is greaterwhen considering t~e ultl~ate ~m~ state than when considering the servlceabthty hmlt state because it is necessary to have a smaller proba- bility of the former being reached. Characteristic strength fk I I I I , I I I I (b) Strength *Oesign load effects Introduction Strength The design load effects are the moments, shears, etc., hich must be resisted at a particular limit state. The~ are :btained from the effects of the design loads by multiply- ing by a partial safety factor YjJ. The design load effects (8*) a1'e thus obtained from . S* :: YjJ (effects of Q*) (1.2) = Yp (effects ofYfL Qk) If linear relationships can be assumed between load .and load effects, the design load effects can be determmed from (1.3) S* = (effects of YP YfL Qk) It can be seen from Fig. 1.2 that equations. (1.2,> and (1 3) give the same value of S* wl!en the relatlo~s~IP be- t~een load and load effect is linear, but not ~hen It IS non- linear. In the latter case, the point in the .deslgn process~ at which Yt:. is introduced, influences the final value of S . As is discussed in Chapter 3, elastic analysis wi~1 gener- ally continue to be used for concrete bridge design. and . thus equation (1.3) will very often be the one used. . The partial safety factor, Yp, takes account of any 10- accurate assessment of the effects of loadi~g. u~for~seen stress distribution in the structure and. variations 10 dimen- sional accuracy achieved in constructIOn. . f h v,I of yare dependent upon the mat~nal 0 t e a ues p . . P rt 4 of the bridge and. for concrete bridges. are glven!n a Cocie The numerical values are discussed 10 Chapter 4. In . 'd I of yareaddition to the material of the bn ge. va ues p dependent upon: I. Type of loading: a lower value is fusled df~r a(nS;;~e:~ tially uniformly distributed type o· oa 109 dead load) than for a concentrated loading because the effects of the latter can be analysed less accurately. Method of analysis: it is logical to adopt ~ larger,value 2. for an analysis which is known .to ~e maccurate or unsafe. than for an analysis which IS known to be highly accurate or conservative. . 3 Importance of the limit state: the consequences of the . . effects for which Yr!o is intended to allow are more 5
- 10. Load effect Effect of 'If3'IfL Ok & 'If3 (effect of 'IfL Ok) Effect of 'IfL Ok1----.( 'IfL Ok 'If3 '1ft. Ok Load (a) Linear Fig. 1.2(a),(b) Loads and load effects important at the ultimate than the serviceability limit state and thus a larger value .should be adopted for the former. It should be stated that the concept of using Y/3 can create problems in design. The use of Y(3' applied as a general multiplier to load effects to allow for analysis accuracy, has been criticised by Beeby and Taylor [17]. They argue, from considerations of framed structures, that this concept is not defensible on logical grounds, since: I. For determinate structures there is no inaccuracy. 2. For many indeterminate structures, errors in analysis are adequately covered by the capability of the struc- ture to redistribute moment by virtue of its ductility and hence Yp should be unity. 3. Parts of certain indeterminate structures (e.g. columns in frames) have limited ductility and thus limited scope for redistribution. This means large errors in analysis can arise, and Yp should be much larger than the suggested value of about 1.15 discussed in Chap- ter 4. 4. There are structures where errors in analysis will lead to moment requirements in an opposite sense to that indicated in analysis. For example, consider the beam of Fig. 1.3: at the support section it is logical to apply Yp to the calculated bending moment, but at section x-x the calculated moment is zero and Y/3 will have zero effect. In addition, at section Y- Y, where a pro- vision for a hogging moment is required, the appli- cation of Yp. will merely increase the calculated sag- ging moment. . The above points were derived from considerations of framed building structures, but are equally applicable to bridge structures. In bridge design, the problems are further complicated by the fact that, whereas in building design complete spans are loaded, in bridge design posi- tive or negative parts of influence lines are loaded: thus if the influence line is not the 'true' line then the problem discussed in paragraph 4 above is exacerbated. because the designer is not even sure that he has the correct amount of load on the bridge. In view of these problems it seems sensible, in practice, to look upon Yp. merely as a means of raising the global load factor from Yo. Ym to an acceptably higher value of 6 Load effect 'If3 (effect Of'lfL Ok) I---~ Effect of 'If3 'IfL Okl----If----7I~ Effect of 'IfL O"I------J' 'YfL Ok 'Yf3'/fL Ok Load (b) Non-linear - 'True' bending moments --- Calculated bending moments - - - 'If3 x calculated bending moments X y X Fig. 1.3 Influence of Y/'J on continuous beams Yp YfL Ym (Ym is defined in the next section). Indeed, in early drafts of the Code, Yp was called Y1I (the gap factor) and Henderson, Burt and Goodearl [18] have stated that the latter was 'not statistical but intended to give a margin of safety for the extreme circumstances where the lowest strength may coincide with the most unlikely severity of loading'. However. it could be argued that Yg was also required for another reason. The YfL values are the same for all bridges, and the. 1m values for a particular material, which are discussed in, the next section, are the same, irrespective of whether that material is used in a bridge of steel, concrete or composite construction. An additional requirement is that, for each type of construction, designs in accordance with the Code and in accordance with the existing documents should be similar. Hence,'-it is neces- sary to introduce an additional partial safety factor (Y/I or Yp.) which is a function of the type of construction (steel, concrete or composite). Thus Yf3 has had a rather confusing and debatable his- tory! Design strength of a material At each limit state, design strengths are obtained from the characteristic strengths by dividing by a partial safety fac- tor (y",): design strength = 'klym (1.4) The partial safety factor. Ym' is a function of two other partial safety factors: Yml' which covers the possible reductions in the strength of the materials in the'structure as a Whole as corn- pared with the characteristic value deduced from the control test specimens; which covers possible weaknesses of the structure Ym2' d . . th arising from any cause other than the ~e uctlO~ .In e strength of the materials allowed for to YmI' Includ- ing manufacturing tolerances; It is emphasised that individual values of Yml and Ym2 are not given in the Code but that values of Ym' for con- crete bridges, are given in Part 4 of the Code; they are discussed in Chapter 4. The values of Ym are dependent , ...'.'" upon: . Material: concrete is a more variable material than steel and thus has a greater Ym value. 2. Importance of limit state: greater values are used at the ultimate than at the serviceability limit state. because it is necessary to have a smaller probability of the former being reached. Design resistance of a structure or a structural element The design resistance of a structure at a particular limit state is the maximum load that the structure can resist without exceeding the design criteria appropriate to that limit state. For example, the design resistance of a struc- ture could be the load to cause collapse of the structure. or to cause a crack width in excess of the allowable value at a point on the structure. Similarly the design resistance ofa structural element is the maxim~m effect that the element can resist without exceeding the design criteria. In the case of a beam. for example, it could be the ultimate mome~t of resistance. or the moment which causes a stress In excess of that allowed. The design resistance (R*) is obviously a function of the characteristic strengths (tk) of the materials and of the par- tial safety factors (Ym): R* = function (/kly",) (1.5) As an example. when considering the ultimate moment of ~esistance (Mil) of a beam R* = Mil (1.6) and (see Chapter 5) . ( at(f(Yrm)A.,)' functlon ([k/y",) =(fvfy".,)A., d - (fc.,lY",c)b (1.7) where 'v,ICII =tk of steel and concrete. respecti~ely y"••, Y"". =Ym of steel and concrete, respectively A., =steel area b =beam breadth d =beam effective depth at =concrete stress block parameter . strengths. In such situations either values of R* or values of the function of/k are given in the Code. An example is the treatment of shear, which is discussed fully in Chapter 6: R* values for various values oflk are tabulated as allow- able shear stresses. Consequence factor In addition to the partial safety factors YfL, Yp. and y~", which are applied to the loads" load effects and material properties. there is another partial safety factor (y,,) which is mentioned in Part 1 of the Code. Y" is a function of two other partial safety factors: V"I. which allows for the nature of the structure and its behaviour; Y"z. which allows for the social and economic con- sequences of failure. logically. Y"l should be greater when failure occurs suddenly. such as by shear or by buckling, than when it occurs gradually. such as in a ductile flexural failure. However, it is not necessary for a designer to consider Y,,1 when using the Code because, when necessary, it has been Included in the derivation of the Ym values or of the func- tions of Ilc used to obtain the design resistances R*. Regarding Y"2, the consequence of failure of one large bridge would be greater than that of Jne small bridge and hence Y"2 should be larger for the former. However, th~ Code does not require a designer to apply Y"2 values: It argues that the total consequences of failure are the same whether the bridge is large or small, because a greater number of smaller bridges are constructed. Thus, it is assumed that. for the sum of the consequences, the risks are broadly the same. Hence, to summarise, neither Y"1 nor Y"2 need be con- sidered when using the Code. ' Verification of structural adequacy For a satisfactory design it is necessary to check that the design resistance exceeds the design load effects: R* ;r,S· or function (/k, Ym) ;r, function (Qk' YfL' yp.) (1.9) (1.10) This inequality simply means that adeq~ate l~a~ carrying capacity must be ensured ~t t~e ultimate h.mlt state and that the various design cntena at the servIce- ability limit state must be satisfied. Summary However, in some situations the design resistance is calculated from The main difference in the approach to concrete brid~e design in the Code and in the current design do~uments IS the concept of the partial safety fact?rs apphe~ ,'0 the loads load effects and material properties. In addItion. as is sh~wn in Chapter 4, some of the design criteria are dif- ferent. However. concrete bridges designed to the Code R* = [,function (jic))/y", (1.8) where Y"i is now a partial safety factor applied to the ~si~ tance (e.g. shear strength) appropriate to charactensttc 7
- 11. should be verysimilar in proportions to those designed in recent yearsj)ecause the design criteria and partial safetY factors have been chosen to ensure that 'on average' this will occur. There is thus no short-term ~dvantage to be gained from using the Code and, indeed, initially there will be the dis- advantage of an increase in design time due to unfamili- arity. Hopefully, the design time will decrease as designers " .' 8 become familiar with the Code and can recognise the critical'limit state for a particular design situation. The advantage of the limit state format, as presented in the Code, is that it does make it easier to incorporate new data on loads, materials, methods of analysis and structural behaviour as they become available. It is thus eminently suitable for future development based on the results of experience and research. Chapter 2 Analysis "* I .Genera reqUirements The general requirements concerning methods of analysis are set out in Part 1 of the Code, and more specific requirements for concrete bridges are given in Part 4. *"Serviceability limit state Part 1 permits the use of linear elastic methods or non- linear methods with appropriate allowances for loss of stiffness due to cracking, creep, etc. The latter methods of analYSis must be used where geometric changes signifi- cantly modify the load effects; but such behaviour is unlikely to occur at the serviceability limit state in a con- crete bridge. Although non-linear methods of analysis are available for concrete bridge structures [19], they are more suited to checking an existing structure, rather than to direct design; this is because prior knowledge of the reinforcement at each section is required in order to determine the stiff- nesses. Thus the most likely application of such analyses is that of <:hecking a structure at the serviceability limit state, when it has already been designed by another method at the ultimate limit state. Hence, it is anticipated that analysis at the serviceability limit state, in accordance with the Code, will be identical to current working load linear elastic analysis. Part 4 of the Code gives the following guidance on the stiffnesses to be used in the analysis at the serviceability limit state. The flexural stiffness may be based upon: 1. The concrete section ignoring the presence of re- inforcement. 2. The gross section including the reinforcement on a ,modular ratio basis. 3. The transformed section consisting of the concrete in compression combined with the reinforcement on a modular ratio basis./ However, whichever option is chosen, it should be used consistently throughout the structure. Axial, torsional and shearing stiffllesses may be based upon the concrete section ignoring the presence of the re- inforcement. The reinforcement can be ignored because it is difficult to allow for it in a simple manner, and it is con- sidered to be unlikely that severe cracking will OCClrr due to these effects at the serviceability limit state. Strictly, the moduli of elasticity and shear moduli to be used in determining any of the stiffnesses should be those appropriate to the mean strengths of the materials, because when analysing a structure it is the overall response which • is of interest. If there is a linear relationship between loads and their effects, the values of the latter are determined by the relative and not the absolute values of the stiffnesses. Consequently, the same effects are calculated whether the material properties are appropriate to the mean or charac- teristic strengths of materials. Since the latter are used throughout the Code, and not the mean strengths, the Code permits them to be used for analysis. Values for the short term elastic modulus of normal weight concrete are given in a table in Part 4 of the Code, and Appendix A of Part 4 of the Code states that half of these values should be adopted for analysis purposes at the serviceability limit state. The tabulated values have been shown [20] to give good agreement with experimental data. Poisson's ratio for concrete is given as 0.2. The elastic modulus for rein- forcement and prestressing steel is given as 200 kN/mmz, except for alloy bars to BS 4486 [21] and 19-wire strand to BS 4757 section 3 [22], in which case it is 175 kN/mm z . It is also stated in Part 4 that shear lag effects may be of importance in box sections and beam and slab decks having large flange width-to-Iength ratios. In such cases the designer is referred to the specialist literature, such as Roik and Sedlacek [23], or to Part 5 of the Code, which deals with steel-concrete composite bridges. Part 5 treats the shear lag problem in terms of effective breadths, and gives tables of an effective breadth parameter as a function of the breadth-to-Iength ratio of the flange, the longitudinal location of the section of interest, the type of loading (dis- tributed or concentrated) and the support conditions. The tables were based [24] on a parametric study of shear lag in steel box girder bridges [25]. However, they are con- sidered to be applicable to concrete flanges of composite bridges [26] and, within the limitations of the effective 9
- 12. I c' , ' I I j,: should be very similar ;. recent yearsl>ecause f' factors have been c' will occur. There is thus using the Coe' advantage C' arity. Hor pplicable to concrete ode permits the s of analysis. ed upon con- ~n-elastic dis- _.,(S. Although such _ -.If concrete bridge structure -J and the HiUerborg strip method for ... .,clvisaged that the vast majority of structures __,-.:;ofitlnue to be analysed elastically at the ultimate fimit state. However, a simple plastic method could be used for checking a structure at the ultimate limit state when it has already been designed at the serviceabiIity limit state. Such an approach would be most appropriate to prestressed con- crete structures. A design approach which is permitted in Part 4 of the Code, and which is new to bridge design, although it is well established in building codes, is redistribution of elas- tic moments. This method is discussed later in this chap- ter. The stiffnesses to be adopted at the ultimate limit state may be based upon nominal dimensions of the cross- sections, and on the elastic moduli; or the stiffnesses may be modified to allow for shear lag and cracking. As for the serviceability limit state, whichever alternative is selected, it should be used consistently throughout the structure. Part 4 of the Code also permits the designer to modify elastic methods of analysis where experiment and experi- ence have indicated that simplifications in the simulation of the structure are possible. An example of such a sim- plification would be an elastic analysis of a deck in which the torsional stiffnesses are put equal to zero, although they would be known to have definite values. Such a sim- plification would result in a safe lower bound design, as explained later in this chapter. and would avoid the com- mon problems of interpreting and designing against the torques and twisting moments output by the analysis. However, the author is not aware of any experimental data which, at present, justify such simplifications. Elastic analysis at the ultimate limit state The validity of basing a design against collapse upon an elastic analysis has been questioned by a number of de- signers, it being thought that this constitutes an anomaly. In particular, for concrete structures, it is claimed that such an approach cannot be correct because the elastic analysis would generally be based upon stiffnesses calculated from the uncracked section, whereas it is known that, at col- lapse, the structure would be cracked. Although it is an- ticipated that uncracked stiffnesses will usually be adopted for analysis, it is emphasised that the use of cracked trans- formed section stiffnesses are permitted. In spite of the doubts that have been expressed, one 10 "* should note that it is perfectly acceptable to use an elastic analysis at the ultimate limit state and an anomaly does not arise, even if uncracked stiffnesses are used. The basic reason for this is that an elastic solution to a problem satisfies' equilibrium everywhere and, if a structure is designed in accordance with a set of stresses (or stress resultants) which are in eqUilibrium and the yield stresses (or stress resultants) are not exceeded anywhere, then a safe lower bound design results. Clark has given a detailed, explanation of this elsewhere [27]. It is emphasised that the elastic solution is merely one of an infinity of possible equilibrium solutions. Reasons for adopting the elastic solution based upon uncracked stiff- nesses, rather than an inelastic solution, are: I. Elastic solutions are readily available for most struc- tures. 2. Prior knowledge of the reinforcement is not required. 3. Problems associated with the limited ductility of struc- tural concrete are mitigated by the fact that all c~itical sections tend to reach' yield simultaneously; thus stress redistribution, which is dependent upon ductility, is ,minimised. 4. Reasonable service load behaviour is assured. LocaI effects When designing a bridge deck of box beam or beam and slab construction, it is necessary to consider, in addition to overall global effects, the local effects induced in the top slab by wheel loads. Part 4 states that the local effects may be calculated elastically, with due account taken of any fixity existing between the slabs and webs. This conforms with the current pr~ctice of assuming full fixity at the slab and web junctions a:lld using either Pucher's influence sur- faces [28] or West~igaard's equations [29]. As an alternativ~ to an elastic method at the ultimate limit state, yield line theory, which is explained later, or another plastic analysis may be used. The reference to another plastic analysis was intended by the drafters to permit the use of the Hillerborg strip method, which is also explained later. However, this method is not readily ap- .plicable to modern practice, which tends to omit transverse diaphragms, except at !!Upports, with the result that top slabs are, effectively, infinitely wide and supported on two sides only. In order to reduce the number of load positions to be considered when combining global and local effects, it is pemlitted to assume that the worst loading case for this particular aspect of design occurs in the regions of sagging moments of the structure as a whole. When making this suggestion, the drafters had transverse sagging effects primarily in mind because these are the dominant structural effects in design terms. However, the worst loading case for transverse hogging would occur in regions of global and local hogging, such as over webs or beams in regions of global transverse hogging; whereas the worse loading case for longitudinal effects could be in regions of either global compression or tension in the flange, in combi- nation with the local longitudinal bending. [-~----- (a) Solid slab [0000000000 I (b) Voided slab [0000.0_00_0.0 1 (c) Cellular slab UUO'JO(d) Discrete boxes I I I I I I I I I I(e) Beam and slab Fig. 2.1(a)-(g) Bridge deck types Types of bridge deck General Prior to discussing the available methods of analysis for bridge decks, it is useful to consider the various types of deck used in current practice, and to examine how these can be divided for analysis purposes. In' Fig. 2.1, the variolls cross-sections are shown dia- grammatically. Those in Fig. 2.1 (a-e) are generally analysed as two-dimensional infinitesimally thin structures, and the effects of down-stand beams or webs are ignored; whereas those in Fig. 2.1 (f and g) are generally analysed as three-dimensional structures, and the behaviour of the actual individual plates which make up the cross-section considered. The choice of a type of deck for a particular situation obviously depends upon a great number of considerations, such as span, site conditions, site location .and availability of standard sections, materials and labour. These points are referred to by a number of authors [30-33] and only brief discussions of the various types of deck follow. (f) Box girders I(g) Widely spaced beam and slab Solid slabs I Solid slab bridges can be either cast in-situ, of reinforced or prestressed construction, or can be of composite con- struction, as shown in Fig. 2.2. In the latter case precast prestressed beams, with bottom flanges, are placed ad- jacent to each other and in-situ concrete placed between a~d over the webs of the precast beams to form a composite slab. The precast beams are often of a standardised form [34,35]. Solid slabs are frequently the most economic form of construction for spans up to about 12 m, for reinforced concrete in-situ construction, and up to about 15 m, for composite slabs using prestressed precast beams. The latter are available for span ranges of 7-16 m [34] and 4-14 m [35]. It is obviously valid to analyse either in-situ or com- posite slabs as thin plates. Voided slabs For spans in excess of about 15 m the self weight effects of solid slabs become prohibitive, and voids are introduced 11 '
- 13. .,"1 ~ , In-situ concrete Precastbeam Fig. 2.2 Composite solid slab Fig. 2.3 Continuous slab bridge to reduce these effects. It is often necessary in continuous slab bridges to make the centre span voided as shown in Fig. 2.3, i~ order. ~o prevent uplift at the 'end supports under certam condItions of loading. Voided slabs are gen- erally used fol' spans of up to about 18 m and 25 m for reinforced and post-tensioned construction respectively. It should be mentioned that the cost of forming the voids by means of polystyrene, heavy cardboard thin wood or thin metal generally exceeds the cost of the concrete replaced. Hence, economies arise only from the reduction in the self weight effects, and, in the case of prestressed construction, from the reduced area of concrete to be !>tressed. Voided slabs can be either cast in-situ, of reinforced or ~restressed construction, or can be of composite construc- tion as shown in Fig. 2.4. The latter are constructed in a similar manner to solid composite slabs, but void formers are placed between the webs of the precast beams prior to placing the in-situ concrete. The presence of voids in a slab reduces the shear stiff- ness of the slab in a direction perpendicular to the voids. The implication of this is that it is not necessarily valid to analyse such slabs by means of a conventional thin plate an~lysi~ which ignores shearing deformations. Analyses whIch ll1cJude the effects of shearing deformations are available and are discussed later in this Chapter. However, for the majority of practical voided slab cross-sections these effects can be ignored. Cellular slabs Th~ i~troduction of rectangular, as opposed to circular, vOIds 111 a slab obviously further reduces the self weight effects; but causes greater shear flexibility of the slab, and can result in construction problems for in-situ cellular slabs due to the difficulty of placing the concrete beneath the voids. The in-situ construction problems can be ove~come by using precast prestressed beams in combi~ation with in-situ concrete to form a composite cellular slab as shown in Figs. 2.5 to 2.7. ,,' The M-beam form of construction shown in Fig. 2.5 can be used for spans in the range J 5 to 29 m [36], but has not proved to be popular because of the two stages of in-situ In-situ concrete Precast beam Fig. 1.4 Composite'v(}ided slab Fig. 2.S Composite cellular slab using M-beams In-situ concrete Holes for transverse pre-stress Fig. 2.6 Composite cellular slab using box beams ' In-situ concrete Precast top hat beam (a) Bridge cross section (b) Beam detail Fig. 2.7(a),(b) Composite cellular slab using top hat beams [38] In-situ concrete Fig. 2.8 U-beam deck concreting required on site and the necessity to thread transverse reinforcement through holes at the bottom of the webs of the precast beams [37]. There are a variety ot slundanl box beam seCIiOIlS [34,35] (see Fig. 2.6) which can be used for spans in the range 12 to 36 m, but the need for transverse prest.rcssing tendons thrf)Ugh the dec.' creates site prohlE-lns, The precast top hat, beam (see Fig. 2.7) which was developed by G. Maunsell and P1II1ner [38] has the ndvan- tage that no transverse rcinl'orcc'meJlt'or prestressing ten- dons have to be threaded through the beam~. In all forms or celhtlat slnb construction a considerahle proportion of the cross,section i~ voided. It is generally necessary to adopt either a thin plate analysis which allows for the effects of shearing deformations, or ,to apply nn appropriate modification to an Ilnalysis which ignores them, as mentioned later in this chapter. Discrete box beams Discrete box beam decks can be constructed by casting an in-situ top slab on precast prestressed U-beams [35,39] as shown in Fig. 2.8. The advantage of such a form of con· struction is that it is not necessary to thread transverse reinforcement or prestressing tendons through the beams. In addition, some benefit is gained from the torsional stiff- ness of a closed box section, although this effect is not as beneficial as it would be if the beams were connected through the bottom, in addition to the top, flanges. The structural behaviour of such decks is extremely complex due to the fact that the cross-section consists of alternate flexible top slabs and stiff boxes.,Strictly, such decks should be analysed by methods which consider the behaviour of the individual plates ,!hich make up the cross-section but, in practice, they are often analysed by means of a grillage representation. Beam and slab Beam and slab type of construction, consisting of precast prestressed beams in combination with an in~situ top slab, is frequently used for all spans; and precast beams are available which can be used for span ranges of 12 to 36 m, in the case of I-beams [34], and 15 to 29 m, in the case of M-beams [36]. Beam and slab bridges are generally analysed as plane grillages. This is not strictly correct because the neutral surface is' a curved, rather than a plane, surface but, in practice, it lis reasonable to consider it as a plan~. Ox x o + iJO~ dx It <1x / aOOy+ ...,..._Y- dy dy Fig. 2.9 Stress resultants acting on It plate element Box girders There is a wide range of box girder cross-sections and ,methods of construction. The latter include precast or in- situ, reinforced or prestrellsed and the use of segmental construction. Box girders are generally adopted for spans in excess of about 30 m and a useful review of the various structural forms has been carried out by Swann [40]. The structural behaviour of box girders and methods for their analysis have been discussed by Maisel and Roll [41]. Elastic methods of analysis General It is assumed that the reader is familiar with the, elastic methods of analysis currently used in practice and only a brief review of the various methods follows. Orthotropic plate theory An orthotropic plate is one which has different stiffnesses in two orthogonal directions. Thus a voided slab is octho- tropic, and a beam and slab deck, when analysed by means of a plate analogy, is also orthotropic. It is emphasised that bridge decks are generally orthotropic due to geometric rather than material differences in two orthogonal direc- tions. If a plate element subjected to an intensity of loading of q is considered in rectangular co-ordinates x,y which co- incide with the directions of principal orthotropy, then the bending moments per unit length (Mxo My), twisting moment per unit length (Mxy) and shear forces per unit length (Qx, Qy) which act on the element are shown in Fig. 2.9. 13.
- 14. For equilibrium ofthe element it is required that [42] '0 2 Mx_ 2 a2MXY + a2 My = -q (2.1) 'Ox2 ax 'Oy al One should note that the equilibrium equation (2.1) applies to any plate and is independent of the plate stiff- nesses. The constitutive relationships in terms of the plate dis- placement (w) in the Z direction and the shear strains (Yx, Yy) in the x and y directions respectively are [4~]. Mx =- Dx .2.. ( a.w _ Yx) - DJ ~ (~w-YY) .. (2.2) ax, a,y 0)' oy M D 2-.(aw - Y) - DJ 2-.(aw - Yx) (2.3) y =- y ay ay y ax ax M.<y = -Dxy [;y(~; -Yx) + ;x (~; -yy)J (2.4) Qx =SxYx Q.v = SyYy (2.5) (2.6) where Dx, Dy are the flexural stiffnesses per unit length, DI is the cross-flexural stiffness per unit length, Dxy is the torsional stiffness per unit length and Sx. Sy are the shear stiffnesses per unit length. In conventional thin plate theory, it is assumed that Sx = S =00 or Yx = Yy = 0, i.e. that the plate is stiff in shear: it i; thelt possible to combine equations (2.2) to (2.4) to give the following fourth order governing differential equation for a shear stiff plate [42] a4w ' a4w a4w_ Dx ax4 + 2(DI + 2Dxy) ax2al + Dy ay4 - q (2.7) If the plate has finite values of the shear stiffnesses then Ubove and Batdorf [44] have shown how it is possible to obtain a sixth order. governing differential equation. How- ever, as discussed later, it is reasonable for many bridge decks to assume that one of the shear stiffnesses is infinite and the other finite: it is then possible to obtain fairly simple solutions to the goveming equations: Series solutions Many bridge decks are essentially prismatic rectangular plates which are simply supported along two edges, and, in such situations, it is possible to solve equation (2.7) by making use of Fourier sine series for the displacement (w) and the load (q) as follows (see Fig. 2.10) w = ~ Ym(Y) sin m1l'x m=1 L (2.8) q = ~ qm(y) sin !!!.!!!. m= I L (2.9) These expressions are chosen because they satisfy the simply supported boundary conditions at x = 0 and L. Any 14 Fig. 2.10 Rectangular bridge deck boundary conditions on the two other edges can be dealt ! Owith in the analysis. . The series solution for bridge decks was originally due to Guyon [45] and Massonnet [46], and was then developed by Morice and Little [47] who, together with Rowe, produced design charts which enable the calcula- tions to be carried out by hand [48]. Cusens and Pama [49] have published a more general treatment of the method which extends its range of application and have also pre- sented design charts for calculations by hand. In addition to the above charts, computer programs exist for performing series solutions such as the Department of Transport's program ORTHOP [50]. It is emphasised that simple series solutions cannot be obtained for non-prismatic decks in which the cross- section varies longitudinally; nor for skew decks, because it is not possible to satisfy the skew boundary conditions. Although the series solutions are· for single span simply supported decks, it is also possible to apply them to decks which are continuous over discrete supports by using a flexibility approach in which the di~crete supports are con- sidered to be redundanci,es and zero displacement imposed at each [49]. This apprdl).ch is used in the ORTHOP pro- gram referred to above ' Series solutions for shear deformable plates If the bridge deck shown in Fig. 2.10 is considered to be a voided or cellular slab, with the voids running in the span direction x, then it is reasonable to consider the deck to be shear stiff longitudinally (Sx = 00) but to be shear deform- able transversely. In such a case it is possible to combine equations (2.1) to (2.6) so that a series solution can be obtained. This has been done by Morley [51] by represent- ing the transverse shear, force by the following Fourier sine series, in addition to using equations (2.8) and (2.9) m= J Qym sin m1l'x L (2.10) This representation requires that, at the supports, Qy = O. and the method is only applicable if there are rigid end diap,hragms at the supports. Morley [51] presents design charts which enable solutions to be obtained by hand, and Elliott [52] has published a computer program which solves the same problem. An alternative approach has been presented by Cusens and Pama [49]. Folded plate method The cellular slab, the discrete boxes or the box girders shown in Fig. 2.1 can be considered to be composed of a number of individual plates which span from abutment to abutment and are joined along their edges to adjacent plates. Such an assemblage of plates can be solved by the folded plate method which was originally due to Goldberg and Leve [53], and was subsequently developed by De Fries-Skene and Scordelis [54] into the form in which it is incorporated into the Department of Transport's computer program MUPDI [55]. The folded plate method considers both in-plane and bending effects in each plate and can thus deal with local bending and distortional effects. It is a powerful method, but the bridge must be right and have simple supports at which there are diaphragms which can be considered to be rigid in their own planes but flexible out of plane; in addi- tion, the section must be prismatic because the solution is obtained in terms of Fourier series. Continuous bridges can be considered in the same way as that discussed previously for the series solution of plates. Finite elements In the finite element approach, a structure is considered to be divided into a number of elements which are connected at specified nodal points. The method is the most versatile of the available methods and, in principle, can solve almost any problem of elastic bridge deck analysi~. The reader is referred to one of the standard texts on fintte ele- ment analysis for a full description of the method. There are a great number of finite element programs available which can handle a variety of structural forms. In addition, there is a great variety of eleme~t shapes and types: the latter include one-dimensional beam elements, two-dimensional plane stress and plate bending elements, and three-dimensional shell elements. The following Department of Transport programs are readily available: t. STRAND 2 [56] is for the analysis of reinforced and prestressed concrete slabs and uses a triangular plate bending element in combination with a triangular plane stress element: in addition, beam elements, which are assumed to have the same neutral axis as that of the plate, can be used. 2. QUEST [57] is intended for the analysis of box gir- ders and uses quadrilateral thin shell elements which consider both bending and membrane stress resultants. 3. CASKET [58] is a general purpose finite element program with facilities for plane stress, plate bending, beam. plane truss, plane frame, space truss and space frame elements, which are all compatible with each other. Analysis Finite strips The finite strip method is a particuhir type of finite element analysis in which the elements consist of strips which run the length of the structure and are connected along the strip edges. The method is thus particularly suited to the analysis of box girders and cellular slabs since they can be naturally divided into strips. The in-plane and out-of-plane Ciisplacements within a strip are considered separately, and are represented by Fourier series longitudinally and polynomials transversely. Since Fourier series are used longitudinally, the method is only applicable to right prismatic structures with simply supported ends. However, intermediate supports can be considered in the same way as that discussed previously for the series solution of plates. The finite strip method was originally developed by Cheung [59] who adopted a third order polynomial for the out-of-plane displacement function, and specified two degrees of freedom (vertical displacement and rotation) along the edges of each strip. The in-plane displacement function is a first order polynomial, which implies a linear distribution of in-plane displacement across a strip, and there are two degrees' of freedom (longitudinal and transverse displacements) along the edges of each strip. The use of a third order polynomial for the out-of-plane displacement function results in discontinuities of trans- verse moments at the strip edges, because only compati- bility of deflection and slope is ensured. Hence, a large number of strips is required in order to obtain an accurate prediction of transverse moments. However, this can be overcome by introducing a fifth order polynomial, which ensures compatibility of curvature in addition to deflection and slope. However, two additional 'degrees of freedom' have to be introduced in order to determine the constants of the polynomial; these 'degrees of freedom' are the cur- vatures at the strip edges [60]. An alternative formulation. which also uses a fifth order polynomial, involves the introduction of an auxiliary nodal line in each strip and only has the two degrees of freedom of deflection and slope [49]. The auxiliary nodal line technique can also be adopted for the in-plane effects, and a second order polynomial is then used for the in-plane displacement function. Grillage analysis In a grillage analysis, the structure is idealised as a grillage of interconnected beams. The beams are assigned flexural and torsional stiffnesses appropriate to the part of the struc- ture which they represent. A generalised slope-deflection procedu(.e, or a matrix stiffness method. is then used to calculate the vertical displacements and the rotations about two horizontal axes at the joints. Hence the bending moments, torques and shear forces of the grillage beams at the joints can be determined. Since the grillage method represents the structure by means of beams, and cannot thus simulate the Poisson:s 15·
- 15. ratio effects ofcontinua, it should, strictly, be used only for grillage structures. Nevertheless, the grillage method is a very popular method of analysis among bridge engineers, and it has been applied to the complete range of concrete bridge structures [61]. When applied to voided or cellular slabs or to box girders, a shear deformable grillage is fre- quently used [61] in which shear stiffnesses, as well as flexural and torsional stiffnesses, are assigned to the grill- age members; and the slope-deflection equati~ns, or stiff- ness matrices, modified accordingly. I Guidance on the simulation of various tyJes of bridge deck by a grillage is given by Hambly [61~ and West [62]. Influence surfaces A number of sets of influence surt'aces have been produced in tabulated and graphical form for the analysis of isotropic plates, and these are extremely useful in the preliminary design stage of orthotropic, as well as isotropic, plates. The influence surfaces have, g(lnerally, been derived experimentally or by means of finite difference tech- niques. Influence surface va.1ues are available for rectangular isotropic slabs [28], skew simply supported isotropic slabs [63, 64, 65]. skew simply supported torsionless slabs [66J, and skew continuous isotropic slabs [67, 68, 69]. Elastic stiffnesses Plate analysis When idealising a bridge deck as an orthotropic plate and using a series solution, finite plate elements or finite strips, the following stiffnesses are suggested for the various types of deck. Whenever a composite section is being con- sidered, due allowance should be made for any difference between the elastic moduli of the two concretes by means of the usual modular ratio approach. It is assumed throughout that the longitudinal shear stiffness (Sx) is infinite. Solid slab D,. = Dr = 12(I-v2) Gh·1 D"y = 12 DI = vD)' S,. = 00 (2.11) (2.12) (2.13) (2.14) where h is the slab thickness, E and v are the elastic modulus and Poisson's ratio respectively of concrete, and G is the shear modulus which is given by E G = 2(1 + v) (2.15) Second moment of area = I" Fig. 2.11 Voided slab geometry Voided slab Elliott and Clark [70] have reported the results of finite element analyses which were carried out to detemline the flexural and torsional stiffnesses of voided slabs, and which were checked experimentally. It was foueud th~~, for a Poisson's ratio of 0.2, which is a reasonabl~ vahIliC to adopt for concrete, the following equations for the~iff nesses gave. values which agreed closely wi(h' those of the analyses and the experiments. Wi~h the notation of Fig. 2.11, Dx = Elx (2.16~ s(l- v2 ) Dy = ~~3 [1- 0.95 (~ r] (2.17) D = - 1-084 -Gh 3 [ . (d r]~ 12 . h (2.18) DI = vDy (2.19) The author is not aware of any reliable published dat;!l on the transverse shear stiffness of voided slabs. However, an idea of the significal)c'e of transverse shear flexibility can be obtained from theresults of a test on a model void- ed slab bridge, with a depth of void to slab depth ratio of 0.786, reported by Elliott, Clark andSymmons [71]. The experimental deflections· and strains, under simulated highway loading, were compared with those predicted by shear-stiff orthotropic plate analysis. It was found that the observed load distribution was slightly inferior' to the theoretical distribution in the uncracked state, bur wa~ similar after the bridge had been extensively cracked due to longitudinal bending. In design terms, it thus seems reasonable to ignore the effects of transverse shear flexi- bility and to take the transverse shear stiffness 2S infinity. Since the model slab had a void ratio which is very close to the practical upper limit in prototype slabs, it is sug- gested that the transverse shear stiffness caIn be taken as infinity for all practical voided slabs. However, if it is desired to specify a finite value of the transverse shear stiffness, then approximate calculations carried Ot!l~ by Elliott [72] suggest that, for practical void ratios, it is about 15% of the value of a solid slab of the same overall depth (h). The solid slab value is given by [73] 8,. =2Gh 6 (2.20) Second moment £&:£-g1+ s ~I ~-.~---I Fig. 2.12 Cellular slab geometry Hence, for a voided concrete slab 5 Gh Sy =< 0.15 x '6 Gh ="8 Cellular slab Second moment of area =Iy -9-j4 1 ~ (2.21) Elliott [52] has suggested the following values for rela~ tively thin-walled cellular slabs having an area of vOi.d n~t less than about .one-third of the gross area. The notation IS illustrated in Fig. 2.12. DI = v Dy (2.22) (2.23) (2.24) (2.25) The transverse shear stiffness can be obtained by con- sidering the distortion of a cell and ass~ming that points of contraflexure occur in the flanges midway between the webs as shown in Fig. 2.13 (as originally suggested by Holmberg [74]). The shear stiffness can then be shown to be s = 24 Kw 12 + KI + K2 .I' S 12 + 4(KJ + K2) + KIK2 where K Kw I = Kfl K KII' 2 = Kf2 Elw KII• = h(l _ v2) K - EI[1 fI - s(1- vi) Elf? Kj2 = $(1- v:Z) Discrete boxes With the notation of Fig. 2.14 D = Elx x $ (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) (~.33) (2.34) If1 Fig, 2.13 Cellular distortion Second Second moment moment ~[Jl 0 U ~~ '~I+--.!..Z_~' s s (e) Longitudinal (b) Transverse Fig. 2.14(a),(b) Discrete box geometry It is suggested that the torsi~nal stiffness be calculated by assuming that the 'longitudinal torsional stiffness' is equal to that of the shaded section shown in Fig. 2.14(a), and the 'transverse torsional stiffness' is zero. If the sections of top slab which do not form part of the box section are ignored, the 'longitudinal torsional stiffness' is given by the usual thin-walled box formula. GI. a G (¢,) (2.35) where Ao is the area within the median line of the box, t is the box thickness at distance I from an origin, and the integration path is the median line. Thus the 'longitudinal torsional stiffness per unit width' is given by GJx _ G (4A~ )s-s fdl t (2.36) Since the 'transverse torsional stiffness' is taken to be zero, the total torsional sHffness per unit width is also given by equation (2.36). Clark [75] has demonstrated that Dxy is one-quarter of the total torsional stiffness per unit width calculated as above; hence G (4A3)Dx)' = 4$ f~ (2.37) Rather than calculat~ a specific value for the transverse shear stiffness (Sv), it is suggested that the torsional stiff- ness (D."v) be modified by applying the relevant reduction factors given by Cusens and Pama [49]. Beam and slab With the notation of Fig. 2.15 .7
- 16. hI { (a) longitudinalsection o Second moment of area =Iy ......~"""'. rt( S Torsional j.-----!.y-----aJ~1S .. inertia =Jy ~~----~~y----_~~I~~-----S~y~--~. ~I (b) Transverse section Fig. 2.15(a),(b) Beam and slab geometry Dx = Elx s,. D = Ely y Sy = 00 (2.38) ,(2.'39) (2.40) (2.41) (2.42) T~e ~orsional inertias (Jx and Jy) of the individual longltudmal and, if present, transverse beams can be calcu- lated by dividing the actual beams into a number (n) of ~om~nent rectangles as shown in Fig. 2.16. The torsional I~ertla of the ith component rectangle of size b by h, . given by I , IS J, = k b~hi bl ~ h; ) J, = k b; M bl ~ h; (2.43) The coefficient (k) is dependent upon the aspect ratio of the rectangle, where the aspect ratio is always greater than or equal to unity and is defined as b;lh; or h;lb; as appropri- ate .. Valu~s o~ k are given in Table 2.1 [76]. The total torsional me~tla. of a beam is then obtained by summing those of the mdlvidual rectangles n .lr or Jv = l.: J; ;= I Table 2.1 Torsional inertia constant for rectangles Aspect ratio k 1.0 l.l 1.2 1.3 1.4 1.5 1.8 2,0 2.3 18 0.141 0.154 0.166 0.175 0.186 0.196 0.216 0.229 0.242 Aspect ratio 2.5 2.8 3.0 4.0 5.0 6.0 7.5 10.0 00 k 0.249 0.258 0.263 0.281 0.291 0.298 0.305 0.312 0.333 (2.44) (a) Actual (b) Component rectangles Fig. 2.16(a),(b) Torsional inertia of beam ,The I~tter calculation is an approximation and, in fact, ~nderestlmates the tru~ torsional inertia because the junc- tion effects, where adJacen't rectangles join, are ignored. ~ack~on [77] has suggested a modification to allow for the Junction eff~cts, .but it is not generally necessary to carry out the modificatIOn for practical sections [62]. Grillage analysis General Guidance on the evaluation of elastic stiffnesses for vllri- O~IS types of deck, for use with grillage analysis, has been glv~n by Hambly [.61J and West [62]. In general, Ham- bly ~ recom.mend.atlOns are modifications of those given prevI?usl.y In t~IS chapter for plate analysis, whereas ~est s differ qUite considerably when calculating torsional stlffnesses. It. should.be noted that an orthotropic plate has a single torsIOnal st.lffness (Dx~), whereas an orthogonal grillage can have different torsIOnal stiffnesses (GC GC)' th d' t' f . X> Y In e Irec Ions 0 Its two sets of beams: Furthermore sl'nc '11 . ' ' e a gn .age ~annot simulate ttlr Poisson effect of a plate, Pois- son s r.atto does not appear in the expressions for the flex- ural stlffnesses, and a grillage stiffness equivalent to D does nO.t occur: ~t is .again assumed that the longitudinall shear stiffness IS Infimte. . The following general recommendations are a combina- tion of those of Hambly [61] and West [62] and of the author's personal views. Reference should be made to Hambly [61] for more detailed information concerning edge beams and other special cases. The recommendations are given in terms of inertias rather than stiffnesses because this is the form in which the input to a grillag~ analysis program is generally required. . As for orthotropic plate stiffnesses, discussed earlier, differences between the elastic moduli of the concretes in a composite section should be taken into account by the modular ratio approach. Solid slab The inertias of an individual grillage beam should be obtained from the following inertias per unit width by multiplying them by the breadth of slab represented by the grillage beam. Longitudinal flexural h3 transverse flexural = 12 (2.45) h3 = 6(2.46) 8eam and slab Longitudinal torsional = transverse torsional Transverse shear area = (l(), (2.47) Voided slab The inertias of an individual grillage beam should be obtained from the following inertias per unit width by multiplying by the breadth of slab represented by the gril- lage beam: the notation is given in Fig. 2.11. Longitudinal flexural = Ixs " Transverse flexural == ~; [1- O.95(~r] Longitudinal torsional =transverse torsional == rll-O.84(~n h Transverse shear area == 00 or '8 Cellular slab (2.48) (2.49) (2.50) (2.51) The inertias of an' individual grill,age beam should be obtained from the following inertias per unit width by multiplying them by the breadth of slab represented by the grillage beam: the notation is given in Figs. 2.12 and 2.13. Longitudinal flexural I :::..!. s (2.52) Transverse flexural == 1, (2.53) Longitudinal torsional == transverse torsional =21" (2.54) Transverse shear area = 24K... _____12__+_K~I~+__K;2____ where K _ K... ,--Kn K~ ==~!':. " Kf2 1... K.. == h If1 Kfl ==--s S Discrete boxes 12 + 4 (K, + K2) + K,K2 (2.55) (2.56) (2.57) (2.58) (2.59) (2.60) Various writers l39. 61. 621 have proposed different methods of simulating a deck of discrete boxes by means of a grilIage. It is not clear which is the most appropriate simulation to adopt in a particular situation, but whichever simulation is chosen the author would recommend calcu- lating tile stiffnesses as suggested by the proponent of the chosen simulation. A longitudinal grillage beam can represent part of the top slab plus either a single physical beam, or a number of physical beams. If a longitudinal grillage beam represents It physical beams. where II is not necessarily an integer. then its inertias are given. with the notation of Fig. 2.15, by Flexural = nIx' Torsional = n J.. + -T (2.61) (2.62) ( S 11 3 ) A transverse grillage beam can represent either part of the top slab. or part of the top slab with a transverse physi- cal diaphragm. The shear area it taken to be infinity and the flexural and torsional inertias to be, with the notation of Fig. 2.15, and SR being the spacing of the tranverse grillage beams. Flexural = Sil 1"S,.. ( J h 3 ) Torsional = s/I i;.+6 Plastic method~ of analysis Introduction (2.63) (2.64) In this section, examples of plastic methods of analysis which could he used in bridge design are given. Howev('1'. as mentioned previously. it is unlikely that. with the poss- ible exception of yield line theory for slabs. such methods will be incorporated into design procedures in the nellr future. Before discussing the xarious methods. it is neces- sary to introduce some concepts used in the theory of phls- dcity and limit analysis. Limit analysis It is useful at this stage to distinguish between the terms limit analysis and limit state design. Limit analysis is il means of assessing the ultimate collapse load of a strUl:- ture. whereas limit state design is a design procedure which aims to achieve both acceptable service load behaviour and sufficient strength. Thus limit analysis can be used for calculations at the ultimate limit state in a limit state design procedure. A concept within limit analysis is that it is often not possible to calculate a unique value for the coIlapse load (If a structure: this is alien to one' s experience with the theory of elasticity. where a single value of the load. required Ii) produce a specific stress, at a particular point in a strur' ture, can be calculated. In limit analysis, all that it is gel1' eraIly possible to state is that the coIlapse load is between two values. known as upper and lower bounds to the col- lapse load. For certain structures coincidental upper and lower hounds can be obtained. and thus the unique value of the coIlapse load can he determined. However, this is not the general case and, for a vast number of commonly 14
- 17. occurring structures, coincidentalupper and lower bounds have not been determined. It is thus necessary to consider two distinct types of analysis within limit analysis; namely, upper and lower bound methods. An upper bound method is unsafe in that it provides a value of the collapse load which is either greater than or equal to the true collapse load. The procedure for calcu- lating an upper bound to the collapse load can be thought of in terms of proposing a valid collapse inechanismand equating the int~rnal plastic work to the work done by the external loads. A lower bound method is safe in that it provides a value ot' the collapse load which is either less than or equal to the true collapse load. A lower bound to the collapse load is the lond corresponding to any statically admissible stress (or stress resultant) field which nowhere violates the yield criterion. A statically admissible stress field is one which everywhere satisfies the equilibrium equations for the structure. The expression 'nowhere violates the yield criterion' essentially means that the section strength at each point of the structure should not be exceeded. It is important to note that neither deformations in any form nor stiffnesses are mentioned when considering lower bound methods. Lower bound methods As has been indicated earlier, any elastic method of analysis is a lower bound method, in terms of limit analysis. because it satisfies equilibrium. There are other lower bound methods available which employ inelastic stress distributions; but these have been developed gener- ally with buildings, rather than bridges, in mind. The inelastic lower bourid design of bridges is complicated by the more complex boundary conditions, and the fact that bridges are designed for different types of moving concen- trated loads. In the following, lower bound methods which could be adopted for bridges are described. Hillerborg strip method One inelastic lower bound method, which is mentioned specifical1y in the Code. is the Hillerborg strip method [78] for slabs, in which the two dimensional slab problem is reduced to one dimensional beam design in two directions. This is achieved by the designer choosing to make Mxy = 0 throughout the slab. Thus the slab equilibrium equation (2.1) reduces to ,iM a2 M ax{ + ~ = -q (2.65) It is further assumed that, at any point, the load intensity q can be split into components exq in the x direction and (1 - ex) q in the y direction, so that equation (2.65) can be split into two equilibrium equations a 2 M . )-~ = -exq (2.66) a2M~, = -(1 _ ex)q ay2 20 Equations (2.66) are the equilibrium equations for beams running in the x and y directions. The designer is free to choose any value of ex that he wishes, but zero or unity is frequently chosen. The above simplicity of the strip method breaks down when concentrated loads are considered, because it is necessary to introduce one of the following: complex moment fields including twisting moments [78]; strong ban~~~.~ement [79]; or one of the approaches sug- gested by Kemp [S0;81L. A:further problem in applying the strip method to bridge design is that difficulties arise with slabs supported only on two opposite edges, as OCCurS with slab bridges and top slabs in modem construction. where the tendency is to omit transverse diaphragms. In view of the above comments it is considered unlikely that the method will be used in bridge deck design, but it is possible that it could be used for abutment design as discussed in Chapter 9. Lower bound design of box girders A lower bound approach to the design of box girders has been suggested by Spence and Morley [82]. The basis of the method is that, instead of designing to resist elastic distortional and warping stresses. which have a peaky longitudinal distribution, it is assumed that, in the vicinity of an eccentric concentrated load, there is a transmission zone, having a length a little greater than that of the load, which is considered to act as a diaphragm and to transfer the load into pure torsion, bending and shear in the remainder of the beam. Thus, outside the transmission zone, the beam walls are subjected to only in-plane stress resultants. The design procedure is to statically replace an eccentric load by equivalent pairs of symmetric and anti-symmetric loads over the webs. The anti-symmetric loads, W in Fig. 2.17, result in a dJstortional couple. The length (L,) of the transmission zone 'i,s then chosen by the designer and equilibrium of the zone under the warping forces (wf and ww) and corner moments (mJ considered. It follows [82] that these can be calculated from Wb me = 8L, 2mc = wfb = wwc (2.67) (2,68) The beam, as a whole, is then analysed, for bending and torsion, as if a rigid diaphragm were positioned at the load: this results in a set of in-plane forces in the walls of the box. These in-plane forces are superimposed on the corner moments and warping moments, and reinforcement designed as described in Chapter 5. 1fM d"b .oment re Istn utlon A lower bound method of analysis. which has been permit- ted for building design for some time, is the redistribution of elastic moments in indeterminate structures. The Code permits this method to be used for prismatic beams, w w, m~4-r__~ w:ft. ~4t_W...;_'~_-_-_-_-----+-.1: me w, ~ b (a) Wall moments and forces (bl Transmission zone FIt.;. 1.17(3)· (~)Lower bound design of box girder [82] Although the Code dnes not state the fact, the restriction to pris!I1utic beams was intended to preclude redistribution of moments in all structures and structural elements except 'small' beams, such as are used for the longitudinal beams of beam and slab construction. This was because it was thought that there was a lack of knowledge of redistri- bution in deep members such as box girders. The concept of moment redistribution can be illustrated by considering an encastre beam of span L carrying an ultimate uniformly distributed load of W. The elastic bend- ing moment diagram, assuming zero self weight, is shown in Fig, 2. 18(b), If the beam is designed to resist a hogging moment at the supports of )'WLlt2, instead of WLl12, then the beam will yield at the supports at a load of), Wand, in order to carry the design load of W, the mid-span section must be designed to resist a sagging moment of [(WLl24) + (WLlI2) -(),WLlI2)], as shown in Fig. 2. t8(c). A similar argument would obtain if the mid-span section were designed for a moment less than WLl24. If the beam could be considered to exhibit true plastic behaviour. with unlimited ductility, then any value of ),could be chosen by the designer. However, concrete has limited ductility in terms of ultimate compressive strain, and this limits the ductility of a beam in terms of rotation capacity. As), decreases, the amount of rotation, after in- itial yield, is increased. Thus, ), should not be so small that the rotation capacity is exhausted. It should also be noted that the rotation capacity required at collapse is a function of the difference between the elastic moment and the reduced design moment. Thus. it is convenient to think in terms of this difference. and to definite the amount of redistribution as J) = 1 - ), . An upper limit to /3has to be imposed, when there is II reduction in moment, because of the limited ductility discussed above. The amount of redistribution permitted also has to be limited for another reason: although a beam designed for a certain amount of redistribution will develop adequate strength. it could exhibit unsatisfactory serviceability limit state behaviour. since, at this stage, the beam would hehave essentially elastically, As the difference between the ultimate elastic moment and the reduced design moment increases. the behaviour at the serviceability limit state. in terms of stiffness (and. thUS, cracking) deterior- ates. Hence. the amount of redistribution must be restricteD. The t~()de states four conditions which must be fulfilled when redistributing moments, and these are now discussed. w (cl Web (al Loading ~ ..../1-WLl12 ~ WLl24 (b) Elastic /Elastic -WLt12 ~RedlstrlbutCld .d-AWLi12 I WLi12-AWL:12 ..... ~.",; ..... _--," WL/24 + WL/12 -. AWLi12 (c) Redistribution ~'~BMz1 A.... ,..~~ fWL'8 Redistributed - - - - _........ .r (d) Overall equilibrium /Elastic ultimate ~Elastic service ~ T _. O.7WLi12 I':.> O.7WLt2~ .+_Redistribution ~ IWL,12 ...... ~,. RedistributeCi/........ - - - - - .... (e) Serviceability conditions ··300kNm ~200kNm 2BOkNm (f) Elastic ultimate moment envelope Fig. 2.18(8)-(f) MOlllent redistribution First. overaiI equilibrium must be maintained by keep- ing the range of the bending moment diagram equal to the 'free' bending momcnt (see Fig. 2.18(d». Second, if the beam is designed to resist the redistri- 21
- 18. buted moments shownin Fig. 2.18(d), then, in the regions A- B, sagging reinforcement would be provided. How- ever,. these.· rc;gions would be subjected to hogging moments af 'the -serviceability Iiinit state, where elastic I:onditions obtain. The Code thus requires every section to be designed to resist a moment of not less than 70%, for reinforced concrete, nor 80%, for prestressed concrete, of the moment, at that section, obtained from an elastic moment envelope covering a1l combinations of ultimate loads. For the single load case considered in Fig. 2.18(a), this implies that the resistance moment at any section should be not less than (for reinforced concrete) that appropriate to the chain dotted line of Fig. 2.18(e). The values of 70% and 80% originated in CP 110, where the ratio of (Y!I_ Yf3) at the serviceability limit state to that at the ultimate limit state is in the range 0.63 to 0.71. By provid- ing reinforcement, or prestress, to resist 70 or 80% respec- tively of the maximum elastic moments, it is ensured that clastic behaviour obtains up to about 70 or 80% of the ultimate load, i.e. at the service load. In the Code, the service load is in the range 0.58 to 0.76 of the ultimate load and, hence, the limits of 70 and 80% are reasonable. Third, the Code requires that the moment at a particular section may not be reduced by more than 30%, for re- inforced concrete, nor 20%, for prestressed concrete, of the maximum moment at any section. Thus, in Fig. 2.18(f), the moment at any section may not be reduced by more than 90 kNm for reinforced concrete. This seems illogi- cal, since it is the reduction in moment, expressed as a percentage of the moment at the section under conside,ra- lion, which is important when considering limited duc- tility. It is unclear why the CP 110 committee used the moment at any section. However, this condition is always over-ruled by the second condition which implies that the . moment at a particular section may not be reduced by more than 30 or 20% of the maximum moment at that section. There is no limit to the amount that the moment at a section can be increased, because this does not increase the rotation requirement at that section, and the third con- dition is intended to restrict the rotation which would occur at collapse. Beeby [83J has suggested that the Code limits were not derived from any particular test data, but were thought to be reasonable. However, they can be shown to be conservative by examining test data. The fourth, and final; condition imposed by the Code is that the neutral axis depth must not exceed 0.3 of the effective depth, if the full allowahle reduction in moment has been made. As the neutral axis depth is increased; the IImount of permissible redistribution reduces linearly to zero when the neutral axis depth is 0.6 of the effective depth, for reinforced concrete, and 0.5, for prestressed ~·()ncrete. The reason for these limits is that the rotation at failure, when crushing of the concrete occurs, is inversely proportional to the neutral axis depth. Thus, if the concrete l:rushing strain is independent of the strain gradient acros.s the section, the rotation capacity and, hence, the permis- sible redistribution increases as the neutral axis depth decreases. Beeby [83] has demonstrated that the Code limits on neutral axis dcpth are conservative. A general point regarding moment redistrihution is that, when this is carried out, the shear forces are also redistrib- 22 uted: the. author would suggest designing against the greater of· the non-redistributed and redistributed shear forces. Moment redistribution has been described in some detail because it is: a new concept in bridg~ design documents, but it must be stated that it is difficult to conceive how it can .be simply applied in practice to bridges. This is,. because, in order to maintain eqJJilibrium by satisfying. equation.(2.l), any l'~distribution of longitudinal moments should be accompanied. by redistribution of transverse and- . twisting momcmts. It would appear that redistribution in a deck can only be achieved by applying an imaginary 'load- ing' which causes redistribution. For example, longitudinal support moments could be reduced, and.span moments , increased~' by applying: an imagInary .'loading' consisting of a displac.ement of the suppC)rt; the moments due to this imaginary· 'loading" .wOIiid 'then be added to those of the conventional loadfiigs., ' .. ":" ", ~ : -......~ ""~ ... ,. , Upper boyn~ ·rnetho.ds' Upper bound m~thods are more suitable for analysis (i.e. calculating thel,ll~imate .strengtr of an existing structure) than fqr~e.sig~.;ho~e.ye,~"as will ~e}een, it is possible to use an upper.boun9 .method (yield line theory) to design slab bridges ~n,d .t<ip,slat>s.. . Introduction to. yield lin((]theory. The read~r is r~ferre~to 'one of the specialist texts. e.g. Jones and WoO(([~4),fdt'a·detai1ed ~xplanation of yield line theory, since onlf sUfficient background to apply the method is giver; in'the 'following. The first stage in the Yi.tM line analysis of a slab is to propose a valid collapse irtechanism consisting of lines, along whichyieid· of the.. r~inforcemef!t takes place. and rigid regions betwe~n the yield Hnes. A possible mecha- nism for a slab bridge subjected to a point, load is shown in Fig. 2.) 9; it can ·be seen tllat the geometry of the ykld line pattern can be specified i!1 terms of the parameter <x. In this pattern, only a single parameter is required to define the geometry. but, in a more general case. there could be a number ·of parameters; thus, in general, if there are 11 parametets, each will be designated <Xi where i takes the values of one to/f.:, . ,. A point on the slao is then given a unit deflection; the deflection at any other point, and the rotations in the yield lines, can be 'calculated in tenns of (Xi' In the slab bridge example of Fig.. 2.19, it would be most convenient to con- sider the point of application 'of the 10adP to have unit deflection. '- . Once the detlectioris at every point are known, the work done by the external loads, in moving through their deflec- tions, can be calculated from E= l:[JJ pb d~.dyl (2.69) . ,each rigid region where p is the load per unit area on an element of slab of side d.O(, dy and ·6,' which is a funCtion of OIi. is the deflec- tion of the element. inc 'su'mmation in equation (2.69) is / ISagging yield IIne5.---- / ~ Hogging ~/Yieldline ~ ~ ~ ~ S L 2" L ~f ~-..- ..--.•..... ......_--.-.-....!.-......-...-.-.....-.----.--.--..-~ Fig. 1.19 Slab bridge mechanism carried out for each rigid region between yield lines, and the integration for each rigid region is carried out over Its arclt. In practice, one is generally concerned with point loads. line loads and uniformly distributed loads. For a point load equation (2.69) reduces to the I.oad mult.ipli.ed hy its deflection, and for a line load, or a Uniformly dlstnb· uted load, it reduces to the total load multiplied by the deflection of its centroid. These calculations are illustrated in the examples at the end of this chapter. Similarly, once the rotations in the yield lines are known, the internal dissipation of energy in the yield lines can be calculated from D =l: 11m" 0" dl) (2.70) each line where 0", which is a function of «i, is the normal rotation in a particular yield line,m" is the normal moment of resis- tance per unit length of the yield line and I is the distance along the yield line. The summation in equation (2.70) is carried out for each yield nne, and the integration for each yield line is carried out over its length. ..: In general, a yield line will be crossed by a number of sets of reinforcement, each at an angle «Pi to the normal to the yield line. as shown in Fig. 2.20. If the moment of resistance per unit length of the jth .set of reinforcement is m·, in the direction of the reinforcement, then its contri- b~tion to the moment of resistance normal to the yield line ismj cos2 «PI' Hence. the total normal moment of resistance is given by mIl = l: mi cos 2 «P; (2.71) Since «Pi are each functions of <X;. so also is mil' In practice, it is often easier to calculate the dissipation of energy in a yield line by considering the rotations (0,) of thc yield line in the direction of each set of reinforce- ment, and by considering the projections (I,) of the yield line in tl}c directions normal to each set of reinforcement. The dissip,ation of energy is then given by 1> =l: [l:fmj OJ d/j] (2.72) each line Vield Line / Fill. 1.10 Normal moment in yield line The use of this equation is illustrated in the examples at the end of this chapter, The next stage in the analysis is to equate the external work dOlle to the internal dissipation- of energy. and to arrange the resulting equation as an expression for the applied load (P in the example of Fig. 2,19) as a function of «i' The minimum value of P for the proposed collapse mechanism can then be found by differentiating the expression for P with respect to each of «/. and equating to zero. The resulting set of n simultaneous equa- tions can be solved to give <X; and hence P. It is emphasised that, although the resulting value of P is the lowest value for the chosen mechanism, it is not necessarily the lowest value that could be obtained for the slab. This is because there could be an alternative mechan- ism which would give a lower minimum value of P. It is thus necessary to propose a number of different collapse . mechanisms and to carry out the above calculations and minimisation for each mechanism. A major drawback of yield line theory is that the engineer cannot be sure. even after he has examined a number of mechanisms. that there is not another mechan- ism which would give a lower value of the collapse load. The engineer is thus dependent on his experience. or that of others, when proposing collapse mechanisms: fortu- nately, the critical mechanisms have been documented for a number of practical situations. .Yield line analysis of slab bridges Yield line theory can be used for calculating the ultimate strength of a slab bridge which has been designed by another method. The various possible· critical collapse mechanisms, and their equations, have been documented by Jones (85) for the general case of a simply supported skew slab subjected to either a uniformly distributed load or a single point load. Granholm and Rowe (861 give guid- ance on choosing the critical mechanism for a simply supported skew slab bridge subjected to a uniformly distrib- uted load plus a group of point loads (i.e. a vehicle), and they also give the equations for such loadings. Clark l871 has extended these solut,ions to allow for different uniforn, load intensities on various areas of the bridge and for the application of a knife edge load. Although the above
- 19. Concrete bridge designto BS 5400 (a) (b) (d) Fig. 2.21(a)-(e) Skew slab bridge mechanisms authors give general equations for the various mechanisms and loadings, it is very often simpler, in practice, to work from first principles, as shown in the examples at the end of thischapter~ The possible critical mechanisms which should ,be examined when assessing the ultimate strength of a simply supported skew slab bridge are those shown in Fig. 2.21. In'the case of a continuous slab, similar mechanisms to those of Fig. 2.21 would form but, in each case, there would also be a hogging yield line at the interior support as shown in Fig. 2.22(a) and (b). Alternatively, a local mechanism could develop around the interior piers as shown in Fig. 2.22(c). Although the!>e mechanisms are not considered explicitly in the literature, similar mechanisms are analysed in [841 and [85] and thus guidance is avail- able in these texts. Yield line des/gn of slab bridges Although yield line theory is more suitable for analysis, it, can be used for the design of slab bridges. When used for desi'gn, as opposed to analysis, the calculation procedure is very similar. A colIapse mechanism is proposed; the ex- ternal work done calculated from equation (2.69), in terms of the known Joads p; the internal dissipation of energy cal- culated, frolll:. equation (2.72), in terms of the unknown' moments of resistance mj; and the external work done equated to the internal dissipation of energy. However,in- steadofminimising the load with respectto the parameters (X; which define the yield line pattern, as is done when imalys- ing a slab, the moments of resistance mj are maximised with respect to the parameters (Xi' Hence, in general, there are, an infinite number of possible design solutions which result in different. rcliltive values of the moments of resis- tance. In practice, the designer cho().~es ratios for the'" moments ofresistance, and it is usual to choose ratios which do not depart too much from the ratios of the equivalent elastic moments. Sometimes this is not necessary, because 24 (e). (e) the equations for one possible colIapse mechanism invoives only one set of reinforcement. It is thus possible to simpl.ify the design procedure because it is possible to calculate mj for that set of reinforcement directly. The values ofm; for the other sets of reinforcement can then be calculated by considering alternative mechanisms. As an example, if the transverse reinforcement is parallel to the abutments, mechanisms (a) and (b) of Fig. 2.21 involve only the longitudinal reinforcement. Hence, a suggested design procedure for simply supported skew slab bridges is to first provide suffici~nt longitudinal reinforcement to prevent mechanisms (a) and (b) fornling under their appropriate design load,S.The ~mount of transverse rein- forcement required to 'prevent mechanisms (c) and (e) forming can then be calculated by setting up equations (2.69) and (2:72), equating them, and maximising the unknown transverse moment of resistance with respect to the parameters (Xi defining the colIapse mechanism. This procedure is illustrated in Example 2.2 at the end of this Chapter. .' ',' Clark [87] has designed model skew siab bridges by this method and then tested them to failure: it was found that the ra,tos 9f exp~rimental'to calculated 'ultimate load were l.07 to 1,25 with 'a mean value of I:16 for six tests. Gnmholm and Rowe [86] aiso tested' model skew slab bridges and obtained ~aluesof the above ratio of 0.95 to 1.12 with a mean value of 1.08 for eleven tests. Thus, although yield line theory is theoretically an upper bound method. it can be seen, generally. to result in a safe esti- mate of the ultimate load and, thus, to a safe design if used as. a de~ig~l method. ' It should be emphasised that, 'because of the values of the, p~rti~1 ;afcty facto~s that have been adopted in the Code, i~ fill often b.e found that the calculated amount of transverse reinforcement is less than the Code minima of 0.15% and 0.25% for high' yield and mild steel respec- tively.. When this occurs, the latter values should ob- viously be provided. I f I~ / I~ (a) Fi~. 2.22(a)-(c) Continuous slab bridge mechanisms (a) Top slab remote from diaphragm (e) Cantih:iv!ilr slab remote from transverse beam I I I -r I I I I I I I I I I I I I Beem or web Fig. 2.2~(a)-(d) Top slab and cantilever slab mechanisms (e) I I I I I I I I I I I I__J f f~ (b) I I I I 'I I I I I I I I I I I I I I I I I I I I IL__ ------------~----- Diaphragm (b) Top slab near diaphragm I -I--Beam or web I I I I Freeedge~ I I I __ - II __ ..----- ..---1 I~ 1 I~ I~I I ----------~-------- Transverse beam (d) Cantilever slab near transverse beam
- 20. Torsional hinge / -, r--,r--, --, r--,t- Il I II II II II I I( II II II I " II II II I I~ Slab yield lines Longitudinal beam II I I~II I I) II I I( II I I) Flexural __~...-t hinge II II Ie II I IJ II I Ie II I P II II I( II II I' II II I~ II II I II II I~ II II_JL __ J L ____ JL __ Jl_ Support diaphragm Fig. 2.24 Beam and slab mechanism [88] Load in this region For a continuous slab, the design procedure is similar, except that the ratio of longitudinal hogging to sagging moment of resistance must be chosen by the designer. It is suggested that this ratio should be chosen to be similar to the ratio of the elasti~ support and span moments. The longitudinal reinforcement can then be determine from mechanism (a) of Fig. 2.22, and the transverse reinforce- ment from mecahnisms (b) and (c). Application to top slabs Yield line theory can be used for the analysis and design of top slabs in beam and slab, cellular slab and box girder construction, and also for cantilever slabs. One should ~ote that the application to top slabs of the simple yield hne theory outlined earlier is conservative because it con- siders only flexural action, and the beneficial effects of membrane action are ignored; as, indeed, they are also for elastic design. Possible collapse mechanisms for top slabs and can- tilever slabs subjected to a uniformly distributed load plus a point load are shown in Fig. 2.23. - A possible design procedure is to calculate the amounts of longitudinal and transverse reinforcement required to resist the global bending effects, and then to determine the additional amounts of reinforcement required to prevent the mechanisms of Fig. 2.23 forming u,nder the local effect loadings. It will be necessary for the designer to choose rati?s of the transverse bottom to transverse top to, longi- tudmal bottom to longitudinal top moments of resistance, or to predetermine some of the values. This procedure is illustrated by Example 2.3 at the end of this chapter. Upper bound methods for beam and"§lab bridges An upper bound method has been proposed for beam and slab bridges by Nagaraja and Lash [88]. The type of mechanism considered is shown in Fig. 2.24, and consists of yield lines in the slabs, plus flexural hinges in the longi- tudinal beams and torsional hinges in "the support dia- phragms. The method of solution is similar to that described above for a slab bridges, and consists of equating the work ,26 done by the loads to the energy dissipated in the yield lines and the flexural and torsional hinges. The rotations in the yield lines can be calculated from the geometry of the mechanism, as can the rotations in the flexural and tor- sional hinges. The ultimate moment of resistance of the beams, and the ultimate torque that can be resisted by the diaphragms, can be calculated by the methods described in Chapters 5 and 6. Nagaraja and Lash [88] give the equa- tions for various mechanisms, but it is again suggested that, in practice, the calculations are carried out from 'first principles. Nagaraja and Lash [88] compared ultimate loads calcu- lated by such an approach with those recorded from tests on one-tenth scale models, and obtained ratios of experi- mental to calculated ultimate live loads of 0.90 to 1.12 with a mean value of 1.03 for twelve tests. Upper bound methods for box girders Methods are available for carrying out upper bound analyse~ of box girders by considering overall collapse mechamsms, some of which involve distortion of the cross-section. Spence and Morley [82] and Morley and S~enc~ [89] have considere~ the combined flexural plus dlstOl1lOnal collapse mechamsms of Fig. 2.25 for simply supported single cell box girders with no internal dia- phragms. The method of analysis is similar to that described previously for slabs; i.e., the work done by the applied loads is equated to the dissipation of energy in the mecha- nism. For mechanisms (a), (b) and (c), energy is dissipated in .yi~lding the longitudinal reinforcement at midspan, in tWlstlng the box walls and end diaphragms, and in rotation of the corner hinges. For mechanism (d), the webs do not twist, but energy is dissipated in the shear discontinuities along the corners near th~ loaded w~b. Spence and Morley [82] tested thirteen model" girders without side cantilevers, ,and found that eight o( these failed in pure bending and five exhibited failures involving distortion. The failure loads of those failing in pure bending were 0.88 to 0.97 of the theoretical pure bending failure load, and the failure loads of those failing in the distortional mode were 0.85 to t .00 of the theoretical failure' load calculated from consid- eration of mechanism (a). The tests indicated that the dis- placements at collapse were not sufficient for the webs, flanges and diaphragms to yield in torsion, and it was thus suggested that the appropriate dissipation terms be omitted from the equations. Morley and Spence [89] have indicated how continuous single cell girders may be analysed. The loaded span fails by the formation of a mechanism similar to one of those in Fig. 2.25: the non-failing spans move outwards in order that the longitudinal stresses in the failing span satisfy the condition of zero axial force near to the, su·ppon:s. The dis- . sipation of energy due to longitudinal extension and con- traction in the support regions can then be calculated. Cookson [90] has extended the work of Spenc~ and Morley to multi-cell girders. Possible mechanisms are shown in Fig. 2.26: the critical mechanisms are generally those involving local failure of a single cell or a pair of cells. These mechanisms involve shear discontinuities and their analysis is quite complex. In addition, it appears that Alllllys;s (e) (b) (c) (d) • Flexural hinge • S Flexural hinge plul Ihear dllcontlnuity (e) Displacement of 'loaded' web FlA. 2.25(a)-(.) Sin~le cell box girder mechanisms (82. 89] (a) Overall - ----~~~---r------~--- (b) Local "'i~. 2.26(a),(I)) Multi-cell box girder mechanisms (89. 90J the amount of energy dissipated in the shear discontinuities is less than that predicted theoretically. Further work thus needs to be carried out before the method can be applied in practice. Model analysis and testing The Code specifically permits the use of model analysis and testing to determine directly the load effects in a struc- lure. or 10 justify a particular theoretical analysis. Testing may also be used to determine the ultimate resis- tance of cross-sections which are not specifically covered by the Code. ------...,,....----- Examples 2.1 Yield line analysis of composite slab bridge The validity of applying yield line theory to composite slab bridges has been demonstrated experimentally by Best and Rowe [91]. The method will be used to calculate the number of units of HB loading which would cause failure of a right simply supported composite slab bridge of tOm span and 11.3 m breadth. The longitudinal and transverse sagging moments of resistance per unit width have been calculuted by the methods of Chapter 5 to be 856 and 70 kN Ill/m respectively; and the hogging moments of resistance will 27
- 21. ~A @ ~ -Sagging ~ E....CD S ................. Hogging ..t ® ~ ~Unit ~ . deflection B -------~ B E L ® ~ ® j CIC! .... ~ --------~ (i) ~ E @ ~ ... ~..t ® ~ I+--Y_--to! f4-....... .... ?.!!_....._..... ._-+!+-____._...§.3m ..•.----+l.1 (a) Plan 11=1/4.1 4.1 m 1.8m 4.1 m ....oJ 1+--._...._._._....--...-.......-....•..- ...---+--------.---., (b) Section A-A _____________tl_=_1_/Y__~ (c) Section B-B 2.27(a)·-(c) Example 2. I he assumed to be zero. The centre line of th~ external wheels of the HB vehicle cannot be less than 2 m from the free edge of the slab. The self weight of the slab, can be taken as 12 kN/m2 •. the superimposed dead load as 2.5 kN'/m2; and the parapets to apply line loadings of 3.5 kN/m along the free edges, In view of the span. only one bogie of the HB. vehicle will be consid~red. and the critical mechanism is that shown in Fig. 2.27(a): it is sufficiently accurate to assume that the HB bogie is positioned symmetrically about mid- span. In practice. each of the mechanisms of Fig. 2.21 would be considered to determine which would be critical. If the ~haded area of Fig. 2.27(a) deflects unity, then. from Figs. 2.27(b') and (c),'the rotations in the longitudinal and tnmsverse steel directions a~e 1/4.1, and lIy respec- tively. where y is the unknown parameter which defines 2R the geometry of the mechanism. The projected lengths of the yield lines in the longitudinal and transverse steel direc- tions are (5 + y) and 10 m respectively. The dissipation of internal energy is thus given by (see equation (2.72» Longitudinal steel = 2(856) (1/4.1) (5 + y) = 2088 + 418y Transverse steel ,= (70) (1/y) (10) = 700ly :. Total dissipation == D = 2088 + 418y + (700M The HB vehicle deflects unity and. thus. if the HB load is P. the external work done is E(HB) = P x I = P The work done by' the uniformly distributed load is hest calculated by dividing the slab into the nine regions shown in Fig. 2.27(a); the work done in each region is then the load intensity multiplied by the area of the region and by the deflection of the centroid of the area. Hence, and not- ing that region 9 does not deflect, E(u.d.l) = (12 + 2.5) [2(5 x 4.1) (1/2) + (5, x 1.8) (1) + 4 (1/2 x 4.ly) (1/3) + (1.8.") (1/2)] = 428 + 53y , Similarly. the external work done by the parapet Iin,e load- , ing is E(k.e.l) = 3.5 [2(4.1) (1/2) + 0.8) (I)] = 21 :. Total external work done =E =P + 449 + 53y Now p..,- D :. p + 449 + 53,Y == 2088 + 418)' + (700ly) or P =: 1639 + 365y + (7oo/y) In order to find the value of y for a minimum P. dP- :: 365 - (700/y2) = 0 dy :. y = 1.3851n This value is less thun 6.3 m. thus the proposed mechan- ism would be able to form. :. P = 2650 kN or 132 units of HB *2.2 Yield line design of skew slab bridge A solid slab highway bridge has a right span of 10m, 11 structural depth of 580 mm, a skew of 300 and .the cross- section shown in Fig. 2.28(a). The specified highway load- ing is HA and 45 units of HB. The nominal superimposed dead load is equivalent to a uniformly distributed load of 2.5 kN/m2 • and the nominal parapet loading is 3.5 kN/m IIlong each free edge. It is required to calculate, by yield line theory. the moments of resistance to be provided by hnttom reinforcement, placed parallel to the slab edges, for load combination I (!lee Chapter 3). It will be aS!lumed that no top steel is to be provided for strength. In IIccordance with Chapter 3, the carriageway width is 9.3 m. and consists of three 3.1 m notional lanes. It is explained in Chapter I that the design load effects (the moments of resistance in this example) are given by Yf. llluitiplied by the effects of YfL. Q..However. when using yield line theory. it is necessary (see Chapter 4) to calculate the design load effects from a 'design' load of Yn Yo, Qk' The relevant nominal loads and partial safety factors discussed in Chapter 3 are given in Table 2.2 Table 2.2 Example 2.2 'design' loads Load Yf, Yfl. Nominal 'Design' _._M M_..M....._____ kN/m2 Dead 1.15 1.2 13.92 kN/m2 9.2 Surfucing 1.15 1.75 2.5 kN/mz 5.03 kN/m2 Parapet 1.15 1.75 3.5 kN/m2 7.04 kN/m2 HA (alone) 1.1 I.S 9.68 kN/m2 16.0 kN/m2 plus plus 33.5 kN/m 55.3 kN/m HA (with HB) 1.1 1.3 ditto 13.8 kN/m2 plus 47.9 kN/m HB 1.1 1.3 450 kN/axle 644 kN/axle Footway 1.1 1.5 4.0 kN/m2 6.6 kN/m2 -_.._-_.__.._-----_..__.__.-. The longitudinal reinforcement is designed by consider- ing mechanism (b) of Fig. 2.21 which is found to be more critical than mechanism (a) under HA loading. The loads and mechanism are shown in Fig. 2.28(b). For unit d'eflection of the yield line, the total rotation in the yield line = 2(1/5 sec 30) = 0.346 If the momentof resistance per unit length of the longitudi" nal steel is m.. the dissipation is given by D = m,(0.346) (13.3) == 4.6 m, The external work components are E(HB) -- 2(644/4) [(3.4715) -I- (3.97/5) + (4.47/5) + (4.97/5)] = 1087 E(k.e.l.) ,- (47.9·+ 47.9/3) (3.1 sec 30) (1) = 229 E(parapet) =: 4(7.04) (5 sec 30) (1/2) = 81 E(u.d.l.1) = 4(24.23) (5 x 0.5 sec 30) (1/2) = ]40 E(u.d.l.2) = 4(30.83) (5 x 1.5 sec 30) (1/2) = 534 E(u.d.l,3) = 2(24,23) (5 x 3.1 sec 30) (1/2) =434 E(u.d.I.4) = 2(38.03) (5 x 3.1. sec 30) (1/2) = 681 E(u.d.1.5) == 2(28.83) (5 x 3.1 sec 30) (/2) =516 :. Total work done == E = 3702 NowD =E :. 4.6 ml = 3702 :. 1111 == 805 kN mlm The transverse reinforcement is designed by considering mechanism (c) of Fig. 2.21 which can be idealised tothut shown in Fig. 2.28(c) [86, 87]. For unit deflection of the shaded area. the rotation in the direction of the longitudnal steel is 1/4.874 =0.205 Projected length normal to longitudinal steel == 5.1 + y sec 30 = 5. 1 + 1. 155y Longitudinal steel dissipation == 2(805) (0.205) (5.1 + 1.155y) = 1683 + 382y Rotation in direction of transverse steel == Ily Projected length normal to transverse steel = 10 If the moment of resistance per unit length of the trans- verse steel is m2. the dissipation is given by (m2) (lly) (10) =10m2/y :.Total dissipation = D = 1683 + 382y + IOm2/Y The external work components are (noting that it is sufficiently accurate to assume that each wheel of the HB vehicle displaces unity) E(HB) = 2(644) (I) = 1288 E(parapet) = (7.04) [(1.8) (I) + 2(4.874) (1/2) 1=47 E(k.e.l.) = (47.9) (y) (1/2) = 24:v E(u.d,l.l) = 2(24.23) (4.874 x 0.5) (1/2) = 59 E(u.d.l.2) = 2(30.83) (4.874 x 1.5) (1/2) = 225 E(u.d. .3) 2(24.23) (4.874 x 3.1) (112) = 366 E(u.d. .4) 4(38.03) (1/2 x 4.221)') (1/3) = 107)' E(u.d.I.5) (38.03) (1.559)') (1/2) = 3Qv E(u.d..6) (24.23) (1.8 x 3.1) (I) = 135 E(u.d. ,7) (30.83) (1.8 x 1.5) (I) = 83 E(u.d.I.R) (24.23) (.8 x 0.5) (I) = 22 :. Total work done = 2225 + 161y Now D = E :.1683 + 382)' + lOmh = 2225 + 161y :. In! = 54.2y - 22. Il 29
- 22. . J 0.5 '.5'.0 Foot- Hard way strip I. ~14 _14 _14 3.65 Traffic lane 3.65 Traffic lane L~L--'_~.f---_9'_3_c...!:~,,!.-:,_g.w,y==- 3-'· 1. --.-.------.,..----. Noti0 naI Ianes 5 5 I I I I /®/I I I 1""I I I I (a) Cross-section (b) Mechanism' I I I I I I I I I I I I I I ® I I I I I I I I I I I I y t ::~L~(c) Mechanism 2 Fi~. 2.28(a)-(c) Example 2.2 In order to find the value of y for a maximum m I 2, 1m2 dy = 54.2 - 44.2y = 0 .. y = 1.226 m :his value is less than 3.58 m, thus the mechanism would form as shown in Fig. 2.28(c). :. m2 =33.2 kN mlm I I I I -Sagging '-""' '-""' Hogging I I I I ~ Unit deflection The amount of reinforcement, that would develop this moment, would be less than the Code minima discussed in Ch~pte~ 10 and, thus, the latter value should be provided. This Will generally be the case when designing in accor- dance with yield line theory. Finally, although no top steel is required to develop adequate strength, some will be required in the obtuse cor- ners to control cracking. __ Sagging ..,.. ...... Hogging '65kN wheel..... O.2mO.3m O.2m 1----t-I----1 Fig. 2.29 Example 2.3 :=I0.16m Hogging yield line forms along upper edge of beam flange y ~2.3 Yield line design of top slab A bridge deck consists of M-beams at 1.0 m centres with a 160 mm thick top slab. It is required to design the top slab reinforcement, in accordance with yield line theory, to withstand the HA wheel load. It is necessary to pre-determine some of the reinforce- ment areas, and it will thus be assumed that the Code minimum area of 0.15% of high yield steel is provided in both the top and bottom of the slab in a direction parallel to the M-beams. Such reinforcement would provide sag- ging and hogging moments of resistance per unit length of 5.35 kN m/m. Let the sagging and hogging moments of resistance per unit length normal to the beams be m, and m2 respectively. The 'design' loads are (see Chapter 3 for details of nominal loads and partial safety factors) HA wheel load = (1.1) (1.5) (100) = 165 kN Self weight = (1.15) (1.2) (3.84) = 5.30 kN/m2 Surfacing (say) = (1.15) (1.75) (2.5) = 5.03 kN/m2 Total u.d.l. = 10.33 kN/m2 The contact area of the HA wheel load is a square of side 300 mm (see Chapter 3). Dispersal through the surfac- ing will be conservatively ignored. The collapse mechanism will be as shown in Fig. 2.29 in which the parameter y defines the geometry of the mechanism. If the wheel deflects unity, the rotations paraUel and perpendicular to the beams are (lIy) and (1/0.2) respectively. The internal dissipation of energy is D = 2(m, + m2) (1/0.2) (0.3 + 2y) + 2(5.35 + 5.35)(1/y) (0.7) = (m, + m2) (3 + 20y) + 14.981y The external work done is E = (165) (1) + 10.33 [2(0.3) (y) (1/2) + 2(0.3) (0.2) (1/2) + 8(1/2) (0.2) (y) (1/3)] = 165.6 + 5.857y NowD =E :. (m, + m2) (3 + 2Oy) +14.98/y = 165.6 + 5.857y . ( + ) _ 165.6 + 5.857y - 14.98/y .. m, m2 - 3 + 20y In order to find the value of y for a maximum (m, + m2)' d(m, + m2) = 0 dy From which, y = 0.239 m :. (m, + m2) =13.4 kNm/m Any values of m, and m2 may be chosen, provided that they sum to at least 13.4 kNm/m and that they are not less than the required global transverse sagging and hogging moment of resistance respectively. 31'
- 23. Chapter 3 Loadings '*General As explainedin Chapter 1, nominal loads are specified in Part 2 of the Code, together with values of the partial safety factor YfL, which are applied to thenominal loads to obtain the design loads.. It should be noted that, in the Code, the term 'load' includes both applied forces and imposed deformations, such as those caused by restraint of movements due to changes in temperature. . The nominal loads are very similar to those which appear as working loads in the present design documents BS 153 and BE 1177. Loads to be considered The Code divides the nominal loads into two groups: namely. permanent and transient. Permanent loads Permanent loads are defined as dead loads, superimposed dead loads, loads due to filling materials, differential set- tlement and loads derived from the nature of the structural material. In the case of concrete bridges, the latter refers to shrinkage and creep of the concrete. Transient loads All loads other than the permanent loads referred to above are transient loads: these consist of wind loads, tempera- ture loads, exceptional loads, erection loads. the primary and secondary highway loadings, footway and cycle track loadings, and the primary and secondary railway load- ings, Primary highway and railway loadings are vertical live loads, whereas the secondary loadings are the live loads due to changes in speed or direction. Hence the secondary 32 highway loadings include' centrifugal, braking, skidding and collision loads; and the secondary railway loadings include lurchmg, nosing, centrifugal, traction and braking loads. Load combinations There are three principal (combinations 1 to 3) and two secondary (combinations 4 and 5) combinations of load. Combination 1 The loads to be considered are the permanent loads plus the, appropriate primary Il~e loads for highway and foot- ,way or cycle track bridges; or the permanent loads plus the appropriate primary and secondary live loads for railway bridges. Combination 2 The loads to be considered are those of combination 1 plus wind loading plus erection loads when appropriate, Combination 3 The loads to be considered are those of combination 1 plus those arising from restraint of movements, due to tempera- ture range and differential temperature distributions, plus erection loads when appropriate. Combination 4 This combination only applies to highway and footway or cycle track bridges. The loads to be considered for highway bridges are the Table 3.1 Y/L Load Dead - steel - concrete Superimposed dead Reduced value for dead and superimposed dead if a more severe effect results WinJ ,. timing erectioll _ plus dead plus superimposed dead _ plus all other combinlltion 2 loads .- relieving effect Temperature - range - friction at bearings - difference Differential settlement Earth pressure':" fill or surcharge - relieving effect Erection Highway primary - HA alone - HA with HB or HB alone Highway secondary - centrifugal - longitudinal HA - longitudinal HB - skidding - parapet coliision - support collision Foot/cycle track Railway Limit state u s u s u s u u s u s u s u s u s u s u s ~ 1 u s u u u s u s u s u s u s u s u s u s u s u s permanent loads plus a secondary live load with i~s associ- ated primary live load: each of the seconda~y hve I~lad,s , I . . 'h pter are conSIdered mdl-discussed laler In t liS c a vidually, . I track The loads 10 he considered for footway or cyc e . hridges are the permanent loads plus the secondary bve load of a vehicle colliding with a support. Combination 5 , The loads to be considered are the permanent loads plus the loads due to friction at the bearings. Y/L for combination 1 1.05 1.00 1.15 1.00 1.75 1.20" 1.00 not specifled 1.50 1.00 1.00 1.50 1.20 1.30 1.I0 1.50 1.00 1.40 1.10 2. 1.05 1'.00 1.15 1.00 1.75 1.20 1.00 .10 1.00 1.40 .00 1.10 1.00 1.00 1.00 1.50 1.00 1.00 1.15 ,1.25 1.00 1.10 1.00 1.25 1.00 1.20 1.00 3 1.05 1.00 1.15 1.00 1.75 1.20 1.00 1.30 1.00 1.00 0,80 1.50 1.00 1.00 1.15 1.25 1.00 1.0 1.00 1.25 1.00 1.20 1.00 *Partial safety factors 4 1.05 1.00 1.15 1.00 1.75 1.20 1.00 1.50 1.00 1.00 1.50 1.00 1.25 1.00 1.0 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 5 1.05 1.00 1.15 1.00 1'.75 1.20 1.00 1.30 1.00 1.50 1.00 1.00 The values of the partial safety factor Yo. to be applic~ a~ the ultimate and serviceability limit states for t~e ~a~IOl~~ load combinations arc given in .Table 3,1. Thc Illdlvld~", d'.. "ed at this Juncture. but the follOWingvalues are not ISCUSS .' general points should be noted: 1. Larger values are specified for the ultimate than for the serviceability limit state. " The values arc less for reasonably w~1I defined .lo~ld~. 2. such as dead load, than for more vanable loads. such 33
- 24. as live orsuperimposed dead load. Hence the greater uncertainty associated with the latter loads is reflected. in the values of the partial safety factors. 3. The value for a live load, such as HA load, is less when the load is combined with other loads, such as wind load in combination 2 or temperature loading in combination 3, than when it acts alone, as in combina- tion 1. This is because of the reduced probability that a number of loads acting together will all attain their nominal-values simultaneously. This fact is allowed for by the partial safety factor Yf2' which was dis- cussed in Chapter 1 and which is a component of YIL' 4. Avaluc of unity is specified for certain loads (e.g. superimposed dead load) when this would result in a more severe effect. This concept is discussed later. 5. The values for dead andsllperimposed dead load at the ultimate limit state can b~ different to the tabulated values, as is discussed later when these loads are con- sidered in more detail. Application 9f loads General The general philosophy governing the application of the loads is that the worst effects of the loads should be sought. In practice, this implies that the arrangement of ~he loads on the bridge is dependent upon the load effect bemg considered, and the critical section being considered. In addition, the Code requires that, when the most severe effect on a structural element can be diminished by the presence of a load on a certain port~on .of the struc~re, then the load is considered to act with Its least poSSible magnitude. In the case of dead load this entails applying a "ilL value of 1.0: it is emphasised that this value is applied to all parts of the dead load and not solely to those p~rts which diminish the load effect. In the case of supenm- posed dead load and live load, these loads should n~t be applied to those portions of the structure where !helr . presence would diminish the load effect under consider- ation. influence lines are frequently used in bridge design and, in view of the above, it can be seen that superimposed dead load and live load should be applied to the ~dverse parts of an influence line and not to relieving parts. It is not intended that parts of parts of influence lines should be loaded. For example, the loading shown in Fig. 3.1(a) should not be considered. Highway carriageways and lanes Carriageway The carriageway is defined as the traffic lanes plus hard shoulders plus hard strips plus marker strips. If raised kerbs are present, the carriageway width is the distance between the raised kerbs. In the absence of raised kerbs, 34 .; (allncorrect ~Influence ,ilin • (bl Correct Fig. 3.1(a),(b) Influence line loading the carriageway width is the distance between the safety fences less the set-back for the fences: the set-back must be in the range 0.6 to 1.0 m. Traffic lane The lanes marked on the running surface of the bridge are referred to as traffic lanes. Hence traffic lanes in the Code are equivalent to working lanes in BE 1177. However, the traffic lanes in the Code have no significance in deciding how live load is applied to the bridge. Notional lanes These are notional parts of the carriageway which are used solely for applying the highway loading. They are equiv- alent to the traffic lanes in BE 1177 and they are determined in a similar manner, although the actual numerical values are a little different for some carriageway widths. Overturning of structure . The stability of a stru~iture against overturning is calculated at the ultimate limit. state. The criterion is that the least restoring moment due to unfactored nominal loads should be greater than the greatest overturning moment due to design loads. Foundations The soil mechanics aspects of foundations should be as- sessed in accordance with CP 2004 [92], which is not written in terms of limit state design. Hence these aspects should be considered under nominal loads. However, when carrying out the structural design of a foundation, the reaction from the soil should be calculated for the appropriate design loads. Permanent loads Dead load The nominal dead loads will generally be calculated from the usual assumed values for the specific weights of the - materials (e.g. 24 kN/m3 for concrete). When such assumed values are used it is necessary, at the ultimate limit state, to adopt 'ilL values of 1.1 for steel and 1.2 for concrete rather than the values of 1.05 and 1.15 respec- tively given in Table 3.1. The latter values are only used when the nominal dead loads have been accurately as- sessed from the final structure. Such an assessment would require the material densities to be confirmed and the weight of. for example, reinforcement to be ascertained. It is thus envisaged that in general the larger 'IlL values will be adopted for design purposes. The emphasis placed on checking dead loads in the Code is because the dead load factor of safety is less than that previously implied by BE 1177. As indicated earlier, when discussing the YrL values, it is necessary to consider the fact that a more severe effect, due to dead load at a particular point of a structure, could result from applying a YIL value of 1.0 to the entire dead loud rather than a value of, for concrete, 1.2. ~Superimposed dead load The partial safety factor given in Table 3.1 for superim- posed dead load appears to be rather large. The reason for this is to allow for the fact that bridge decks are often resurfaced, with the result that the actual superimposed dead load can be much greater than that assumed at the design stage [18]. However, by agreement with the appropriate authority, the values may be reduced from 1.75 at the ultimate limit state, and 1.2 at the serviceability limit state, to not less than 1.2 and 1.0 respectively. It is then the responsibility of the appropriate authority to ensure that the superimposed dead load assumed for design is not exceeded in reality. As for dead load, the possibility of a more severe effect, due to applying a Yjl. value of 1.0 to the entire super- imposed dead load, should be considered. In addition, the removal of superimposed dead load from parts of the struc- ture, where they would have a relieving effect, should be considered. : Filling material The nominal loads due to fill should be calculated by con- ventional principles of soil mechanics. The partial safety factor of 1.5 at the ultimate limit state seems to be high, particularly when compared with that of 1.2 for concrete. However, the reason for the large value is that the pres- sures on abutments, etc., due to fill. are considered to be calculable only with a high degree of uncertainty, particu- larly for the conditions after construction [18]. It seems reasonable to apply a factor of 1.5 when considering the lateral pressures due to the fill; but, when the vertical effects of the fill are considered, it seems m~re log(c~l to treat the fill as a superimposed dead load and to argue that a "ilL value of 1.2 should be adopted because the volume of fill will be known reasonably accurately. / Shrinkage and creep Shrinkage and creep only have to be taken into account when they are considered to be important. Obvious situ- ations are where deflections are important and in the design of the articulation for a bridge. In terms of section design,proc~4ures exist in Part 4 of the Code to allow for the effects of, shrinkage and creep on the loss of prestress and in certain forms of composite con- struction. These aspects are discussed in Chapters 7 and 8. *"Differential settlement In the Code, as in BE 1/77, the onus is placed upon the designer in deciding whether differential settlement should be considered in detail. Transient loads Wind The clauses on wind loading are based upon model tests carried out at the National Physical Laboratory and which have been reported by Hay (93). The tests were carried out in a constant airstream of 25m/s, and the model cross- sections were very much more appropriate to steel than to concrete bridges. The clauses are very similar to those in BE 1177, and the calculation procedure is as lengthy. However, it is em- phasised that, according to the 'Code, it is not necessary to consider wind loading in combination with temperature loading. In addition, as is also the case in BE 1177, wind loading does not have to be applied to the superstructure ofa beam and slab or slab bridge having a span less than 20 m and a width greater than 10m. However, a number of overbridges have widths less than 10 m. and the exclusion clause is not applicable to these. In general the calculation procedure is as follows. The mean hourly wind speed is first obtained for the location under consideration from isotachs plotted on a map of the British Isles. The maximum wind gust speed and the minimum wind gust speed are then calculated for the cases of live load both on and off the bridge. The minimum gust speed is appropriate to those areas of the bridge where the wind has a relieving effect, and is used with the reduced Y,L values of Table 3.1. The gust speeds are obtained by multiplying the mean hourly wind speed by a number of factors, which depend upon: 1. The return period: the isotachs are for a return period of 120 years (the design life of the bridge), but the Code permits a return period of 50 years to be adopted for foot or cycle track bridges, and of 10 years for erection purposes. 2. Funnelling: special consideration needs to be given to bridges in valleys, etc. 35
- 25. 3. Gusting: agust factor, which is dependent upon the height above ground level and the horizontal loaded length, is applied. The gust factor may be reduced for foot or cycle track bridges according .to the height above gi'oundlevel. , In addition the minimum wind gust speed is dependent .upon an hourly speed factor which is a function of the height above ground level. The next stage of the calculation is to determine. the transverse and longitudinal wind loads (which are dep~n dent o.n the gust speed, an exposed area and a drag coeffiCIent), and the vertical wind load (which is depen- dent on the gust speed, the plan area and a lift coefficient). This part of the calculation is probably the most com- plex and requires a certain amount of engineering judge_ C .ment to be made. This is because the Code gives cross- sections for which drag coefficients may be obtained from the Code, and also gives cross-sections for which drag coefficients may not be derived and for which wind tunnel tests should be carried out. It should also be stated that an 'overall' depth' is required to determine the exposed area and the drag coefficient, but, in general, different values of the 'overall depth' are used for the two calculations. Finally the transverse, longitudinal and vertical win" loads are considered in the following four combinations: .' I.' Transverse alone 2. Transverse ± vertical 3. Longitudinal alone 4. 50% transverse + longitudinal ± 50% vertical There are thus less combinations than exist in BE 1177. Temperature The clauses on temperature loading are based upon studies carried out by the Transport and Road Research Labora- tory and the background to the clauses has been described by Emerson [94, 95]. The clauses are essentially identical to those in BE 1177 except that temperature loading does not have to be considered with wind loading. There are, effectively, two aspects oftemperature load- ing to be considered; namely, the restraint to the over- all bridge movement due to temperature range, and the effects of temperature differences through the depth of the bridge. "*Temperature range The temperature range for a particular bridge is obtained by first determining the maximum and minimum shade air temperature, for the location of the bridge, from isotherms plotted on maps of the British Isles. The isotherms were derived from Meteorological Office data and. are for a return period of 120 years (the design life of the bridge). The shade air temperatures may be adjusted to those appropriate to a return period of 50 years for foot or cycle track bridges, for the design of joints and during erection. An adjustment should also be carried out for the height above mean sea level. 36 The minimum and maximum effective bridge tempera- tures can then be obtained from tables which relate shade air temperature to effective bridge temperature. The latter is best· thought of as that temperature which controls the overall longitudinal expansion or contraction of the bridge. The tables referred to above were based upon data obtained from actual bridges [94]. The effective bridge temperatures are dependent upon the depth of surfacing, ~~.d,~orrection has to be made if this differs from the '~', lOO"mi1f"lI~umecl!or,concrete bridges in the tables. Emer- son [95] hassuggestelt~t such an adjustment should also .take· account of the shape of the cross-section of the bridge.. The effective bridge temperatures are used for two pur- .:, poses. First, when designing expansion joints, the movement to be accommodated is calculated in terms of the expansion from a datum effective bridge temperature, at the time the joint is installed, up to the maximum effective bridge temperature and down to the minimum effective bridge temperature. The resulting movements are taken as the nominal values, which have to be multiplied by the 'YfL values of Table 3.1. The coefficient of thermal expansion for concrete is given as 12 x lO-o/oC, except for limestone agg~egate concrete when it is 7 x lO-o/oC. The author would suggest also considering the latter value for light- weight concrete [96]. Second, if the movement calculated as above is re- strained, stress resultants are developed in the structure. These stress resultants are taken to be nominal loads', but this contradicts the definition of a load which, according to the Code, includes 'imposed deformation such as those caused by restraint of movement due to changes in temper- ature'. The author would thus argue that the movement is the load, and that anr stress re~ultants arising are load effects. This differenc~ of approach is important when designing a structure ,to resist the effects of temperature and is elaborated in Chapter 13. The Code indicates how to calculate the nominal loads when the restraint to temperature movement is accom- panied by flexure of piers or shearing of elastomeric bear- ings. Coefficients of friction are given for roller and sliding bearings: these are used in conjunction with nominal dead and superimposed dead load to calculate the nominal load due to frictional bearing restraint. The values are the same as those in BE 1177 and are based partly upon [97]. It should be noted that, as in BS 153, the effects of fric- tional bearing restraint are considered in combination with dead and superimposed dead load only. This is because the resistance to movement of roller or sliding bearings is least when the vertical load is a minimum. Hence movement could take place under dead load conditions and. having moved, the restraining force is relaxed [IS]. Temperature differences Due to the diurnal variations in solar radiation and the rela- tively small thermal conductivity of concrete, severe non- linear vertical temperature differences occur through the depth of a bridge. Two distributions of such differences E r.t;LJ'l iM ~ i o 0 I ~ + ~ E 0 .... o VI h = overall depth of concrete section hs = depth of surfacing 2.5"C (a) Positive Fig. 3.2(a),(b) Temperature differences for J m structural depth are given in the Code (see Fig. 3.2). The temperature dif- ferences depend on the depth of concrete in the bridge: those shown in Fig. 3.2 are for a depth,of 1 m. One of the distributions is for positive temperature differences, and is appropriate when there is a heat gain through the top sur- face; and the other is for negative temperature differences, and is appropriate when there is a heat loss from the top surface. The temperature distributions are,composed of four or five straight lines which approximate'the non-linear distributions, which have been calculated theoretically and measured on actual bridges by Emerson [95, 98]. The approximation has been shown to be adequate for design purposes by Blythe and Lunniss [99]. The Code distributions have been chosen to give the greatest temperature differences that are likely to occur in practice. It is not possible to think of them in terms of a return period, but they are likely to occur more than once a year 195]. The Code states that the effects of the temperature dif- ferences in Fig. 3.2 should be regardf1d as nominal values and that these effects, multiplied by 'YfL' should be regarded as design effects. This again appears to be an inconsistent use of terminology. The author would suggest that the nominal loads are the imposed deformations due to either internal or external restraint of the free movements implied by! the temperature difference; and that the 'Yfl. values should be applied to these to give design loads in the form of design imposed deformations. Any stresses or a.3°eL-_ _ _ _ _ _----' (b) Negative E .c:1l) 1'11'1 00 VI .c:E Il)N 1'1 • ·0 o VI .c: E Il)N N'.0 °Vl E -C:1l) 1'11'1 00 VI stress resultants which are developed in response to the imposed deformations would then be the load effects. Such arguments regarding definitions may seem pedantic but they are important when designing a structure to resist the effects of temperature differences and are elaborated in Chapter 13. Regarding the 'YlL values, it should be noted from Table 3.1 that the serviceability limit state value is 0.8. This means that the final effects are only SO% of those calcu- lated in accordance with BE 1/77, which adopts the same temperature difference diagrams. The reason for adopting a 'YfL value of O.S at the serviceability limit state is not clear but, in drafts of the Code prior to May 1977, values of 1.0 and 1.2 for the serviceability and ultimate limit states, respectively.. were specified. It thus appears that the Part 2 Committee thought it reasonable to reduce each of . these by 0.2 in view of the reduced probability of a severe temperature difference occurring at the same time as a bridge is heavily loaded with live load. The temperature differences given in the Code were c,al- culated for solid slabs, but it is considered that the inac- curacy involved in applying them to other cross-sections is outweighed by certain assumptions _made in the calcu- lation. Measurements on box girders [100] and beam and slab [101] construction have shown that the temperature differenc~s are very similar to those predicted for a solid slab of the same depth. . In addition, the temperature differences given in the 37
- 26. main body ofthe Code are for a surfacing depth of 100 mm. An appendix to Part 2 of the Code gives tempera- ture differences for other depths of surfacing which are based upon the work of Emerson [95]. Combination of temperature range and difference A severe positive temperature difference can occur at any time between May and August, and measurements have shown that the lowest effective bridge temperature likely to co-exist with the maximum positive temperature dif- ference is 15°C [95]. A severe negAtiv~ tempernture difference can occur at any time of the day, night or year. However, it is con- sidered unlikely that a severe difference would occur between about ten o'clock in the morning and midnight on, or after, a hot sunny day. Thus, it is considered that a severe negative difference is unlikely to occur at an effective bridge temperature within 2°C of the maximum effective bridge temperature [95]. The above co-existing effective bridge temperatures have been adopted in the Code. Exceptional loads These include the loads due to otherwise unaccounted effects such as earthquakes, stream flows, ice packs, etc. The designer is expected to calculate nominal values of stich loads in accordance with the probability approach given in Part 1 of the Code and discussed in Chapter 1. Snow loads should generally be ignored except for cer- tain circumstances, such as when dead load stability could be critical. -tErection loads At the serviceability limit state, it is required that nothing should be done during erection which would cause damage to the pennanent structure, or which would alter its response in service from that considered in design. At the ultimate limit state, the Code considers the loads as either temporary or permanent and draws attention to the possible relieving effects of the former. The importance of the method of erection, and the pos- sibility of impact or shock loadings, are emphasised. As already mentioned, wind and temperature ,effects during erection should generally be assessed for 10- and 50-year return periods respectively. For snow and ice loading, a distributed load of 500 N/m2 will generally be adequate; this loading does not have to be considered in combination with wind loading. Primary highway loading General The primary effects of highway loading are the vertical loads due to the ma'ss of the traffic , and are considered as static loads. 38 The standard highway loading consists of nonnal (HA) loading and abnonnal (HB) loading. The original basis of these loadings has been described by Henderson [102]. Both of these loadings are deemed to include an allowance for impact. It should be noted from Table 3.1 that, at the service- ability limit state in load combination 1, the Y/L value for HA loading is 1.2. This value was chosen [18] because it was considered to reflect the difference between the uncer- tainties of predicting HA loading ~nd dead load, which has a value 6£1.0. Presumably, the HB value of 1.1 was chosen to be between the HA and dead load values. '*HA loading HA loading is a formula loading which is intended to rep- resent normal actual vehicle loading. The HA loading con- sists of either a uniformly distributed load plus a knife edge load or a single wheel load. The validity of represent- ing actual vehicle loading by the formula loading has been demonstrated by Henderson. [102], for elastic conditions, and by Flint and Edwards [103]. for collapse conditions. Uniformly distributed load The uniformly distributed component of HA loading is 30 kN per linear metre of notional lane for loaded lengths (L) up to 30 m and is given by lSI (IlL)0, 475 kN per linear metre of notional lane for longer lengths, but not less than 9 kN per linear metre. The loading is compared with the BE 1/77 loading in Fig. 3.3. It can be seen that the two loadings are very similar; however, the upper cut-off is now 30 kN/m at 30 m instead of 31.5 kN/m at 23 m, and there is now a lower cut-off of 9 kN/m at 379 m. The latter has been introduced because of the lack of dependable traffic statistics for long loaded lengths [18]. Figure 3.3 gives the loading per lineal' metre of notional lane and the load intensity is always obtained by dividing by the lane width irrespective of the lane width. This is differ~nt to BE 1177 and means that, for a loaded length less than 30 111, the load intensity can be in the range 7.9 to 13.0 kN/m2 compared with the BE 1/77 range of 8.5 to 10.5 kN/m2 • Loaded length The loaded length referred to above is the length of the base of the positive or negative portion of the influence line for a particular effect at the design point under consideration. Thus for a single span member, the loaded length for the span moment is the span. However, for a two span continuous member, having equal spans of 20 m, as shown in Fig. 3.4, the loaded length for calculat- ing the support moment would be 40 m and hence a load- ing of 26.2 kN/m would be applied; and the loaded length for calculating the span moment would be 20 m and a loading of 30 kN/m would be applied. For multispan members, each case will have to be con- sidered separately. 'rhus the moment at SUpp0l1 B of the four span member of Fig. 3.5 would be calculated by con- sidering spans AB and BC loaded with loading appropriate to a loaded length of 2L, or spans AB, BC and DE loaded with loading appropriate to a loaded length of 3L: the former is likely to be more severe for most situations. The E 35 Z¥. '0 CQ .9 31.5 30~"""'Tl 25[ 20 15 I I I I I10 I I - __ _ 9 -, ,,- .11_ -., - - - -'. - - ,-" -,. _._, - - --" - - - - - - --- - - -,-- -,,' -" -- -- - -, -"-" "'- -~,-::-.:::---::::-~.--:;.-~-II"I::':=~----- t t l ' ---------_ I I I I I I I I I I I I I : I I I I ..l..--L"_."",,,,,..,,._..,, L., ....____....l.--_-::-!I:-:-_ _-=~I ____----1-... I L_...J=--__...1-__.----.I 23 30 50 100 150 200 250 300 350 379 400 450 500 5 o Loaded length m F'i~. 3.3 HA uniformly distributed load A B C ~d~_ _• 20,,~__ 4-- 20m =l (a) Centre of span A-B. (b) Support B Fig. 3.4(a).(b) Influence lines for two spans moment in span BC would be' calculated by considering span BC loaded with loading appropriate to a loaded length of L, or spans BC and DE loaded with loading appropriate to a loaded length of 2L. A B C 0 ~ L t L t L t ~-- (a) Support B (b) Centre of span B-C Fig. 3.5(a),(b) Influence lines for four spans E L ~ + /. Knife edge load It is emphasised that the knife edge part of HA loading is not intended to represent a heavy ax.le, but is merely a device to enable the same unifonnly dis- tributed loading to be used to simulate the shearing and bend- ing effects of actual vehicle loading [102]. The Code value of the load is the same as that in BE 1/77, and is 120 kN per notional lane. The load per metre is always obtained by dividing by the notional lane width and is thus in the range 31.6 to 39
- 27. ~Il, I 52.2 kN/m,which should be compared with the BE 1177 range of 32.4 to 52.2 kN/m. The knife edge load is generally positioned perpendicu- lar to the notional lane except when considering supporting members, in which case it is positioned in line with the bearings, and when considering skew decks, in which case it can be positioned parallel to the supporting members or perpendicular to the free edges. This clause is thus more precise than its equivalent in BE 1177, which requires, for skew slabs, the knife edge load to be placed in a direction which produces the' worst effect. It is understood that the intention of the Code drafters was that the intensity of l(nding ~hNlld be 120 kN divided by the skew width ,of a notional lane when the knife edge loading is in a skew po- . sition with respect to the notional lane. Hence, the total load is always 120 kN per notional lane. Wheel load The wheel lond is used mainly for local effect calculations and the nominal load is a single load of 100 kN with a contact pressure of 1.1 N/mm2: the contact area could thus be a circle of diameter 340 mm or a square of side 300 mm. The wheel load is considered to disperse through asphalt at a spread to depth ratio of 1 to 2 and through concrete at 1 to 1 down to the neutral axis. This loading is thus 'different to the BE 1177 load which consists of two 112 kN wheel loads, with a contact pres- sure of 1.4 N/mm2, and a 45° angle of dispersion through both asphalt and concrete. The change from two wheel loads to a single wheel load may seem drastic; however, there is a requirement in the Code that all bridges be checked under 25 units of HB loading. It is envisaged that the worst effects of the single 100 kN wheel load, or at least 25 units of HB loading, will be at least as onerous as those of the two 112 kN wheel loads. The reduction in contact pressure results in a greater contact area, but the reduced dispersal through asphalt off- sets this somewhat when the effective area at the neutral axis is considered. *HB loading HB loading is intended to represent an abnormally heavy vehicle. The nominal loading consists of a single vehicle with 16 wheels arranged on four axles, as shown in Fig, 3.6, which also shows the BE 1177 HB vehicle. It can be seen that the transverse spacing of the wheels on an axle has been rounded-off to 1.0 m, and that the overall width of 'the vehicle is now given as 3.5 m. The'latter point means that it is not necessary to specify a minimum distance of the vehicle from a kerb, as is necessary in BE 1177. However, the most significant difference is that the longitudinal spacing of the centre pair of axles is no longer constant at 6. I m, but can be any of five values between 6 and 26 m inclusive. The reason for this is that the BE' 1177 vehicle originated in BS 153, which was intended for sim- ply supported bridges (although, in practice, it was also applied to continuous briclges) and the worst effects in a simply supported bridge occur with the axles as close together as possible. However, since the Code is intended to be applied to any span configuration, a variable spacing 40 1 unit/axle =10kN/axie ......Wheel t$' + + + 1.0 + Effective 3.5m 1.0 m : + + wheel pressure + + + =1.4N/mm2 1.0m + + + + ,l.aml. 6,11,16,21 or26m"J-8~ 10,16,20,26 or 30 m (a) 866400 1unit/axle = 10kN/axle + O. + + 0.9 + /Wheel .v + + + + Effective wheel pressure + = 1.1 N/mm2 * + o.9m: + + + ~IJ~,a~m~I._____6:~.1~m~___.J.amPi (b) BE 1177 Fig. 3.6(a),(b) HB londing has been specified in order to calculate the worst effects at all design points. As an example, the worst effects at an interior support of a continuous bridge could occur with a wide axle spacing. . The comments on the contact pressure of the HA wheel load and its dispersal are also pertinent to the wheels of the HB vehicle. The nominal HB loading is specified, as in BE 1177, in terms of units of loading, with one unit being equivalent to a total vehicle weight of 40 kN. The number of units for all roads can vary from 25 to 45, and this is at variance to BE 1177 in which no minimum is specified; but, unlike BE 1177, the number of units to be adopted for different types of road are not specified:, Presumably, the Department of Transport will issue a Q1emorandum containing guidance on this point. ' "*A /. .pp Icatlon HA loading The full uniformly distributed and knife edge loads are applied to two notional lanes and one-third of these loads to all other notional lanes. The wheel load is applied anywhere on the carriageway. The applications are thus identical to those of BE 1177. HB loading Only one HB vehicle is, in general, required to be considered on anyone superstructure, or any sub- structure supporting two or more superstructures. The vehicle can be either wholly in a notional lane, or can straddle two notional lanes. If it is wholIyhl a notional lane. then the knife edge component of HA loading for that lane is completely removed. and the uniformly distributed com- ponent is removed for 25 m in front to 25 m behind the vehicle; the remainder of the lane is loaded with the uniformly distributed loading component of HA having an intensity appropriate to a loaded length which includes the displaced length. The vehicle is thus considered to displace part of the HA loading in one lane, but the adjacent lane is still assumed to carry full HA loading. This is more severe than the BE 1177 loading, where the adjacent lane carries only one-third HA loading. . . , If the, vehicle straddles two lanes, then It IS conSIdered to straddle either the two lanes loaded with full HA, or one lane with full and one with one-third HA; in each case the rules for omitting parts of the HA loading, which were described in the last paragraph, are applied. These arrangements of load are different to those of BE 1177. The reason for the more severe arrangement of the accompanying HA loading is that, in practice, queues of heavy vehicles accumulate behind abnormal loads and, when they overtake, they do so in a platoon [18]. lL shoull.! be noted fro111 Tuble 3.1 that, "hen HA lond- ing is applied with HB loading, the '{fL values for HA are the same as those for HB. Verges. central reserves, etc. The accidental wheel load- ing of BE 1177. for the loading of verges and central reserves, has been replaced in the Code by 25 units of HB loading: outer verges need only to be able to support any four wheels of 25 units of HB loading. Transverse cantilever slabs should be loaded with the appropriate number of units of HB loading for the type of road in one notional lane plus 25 units of HB in one other notional lane. The latter is intended to be a substitute for HA loading and has been introduced because the HA load- ing no longer increases for spans less than 6.5 m, as it does in BS 153. This is the only occasion when more than one HB vehicle can act on a structure. Secondary highway loading General The secondary effectsofhighway loading are loads parallel to the carriageway due to changes in speed or direction of the traffic.,One should note that each of the following secondary loads is consideredseparately, and not incombinationwiththe others. An associated primary load is applied with each ofthe secondary loads. Centrifugal load This is a radial force applied at the surface of the road of a curved bridge. The nominal load is the same as that in BE 1177 and is given by F = 30000 kN c r+ 150 (3.1) where r is the radius of the lane in metres. Any number of these loads at 50 m centres should be applied to any two notional lanes. Each load Fe can be divided into two parts of F)3 and 2 Fel3 at 5 m centres if these give a lesser effect. The, loading was based upon tests carried out at the Transport and Road Research Laboratory [104]. The nominal primary load associated with each load Fe is a vertical load of 300 kN distributed uniformly over the notionall~n,e for a length of 5 m. If the centrifugal load is divided, then the vertical load is divided in the same pro- portion. ~ 700 ___________________~~B~S~5-40-0---- "tl .9 600 500 400 BE 1/77 o 3 10 12 20 30 40 50 60 70 ao 90 100 Loaded length rri Fig. 3.7 HA braking load Longitudinal braking The longitudinal forces due to braking are applied at the level of the road surface. The nominal HA braking load is 8 kN per metre of loaded length plus 200 kN, with a maxi- mum value of 700 kN. This load is much greater than the BE 1177 loading with which it is compared in Fig. 3.7. The new loading is based ~pon the work of Burt [105] and is much greater than the BE 1177 loading because of the greater efficiency of modem brakes, which can achieve decelerations which approach 1 g on dry roads [18]. The large increase in braking load could be significant in terms of substructure design. The HA braking load is applied, in one notional lane, over the entire loaded length, and in combination with full primary HA loading in that lane. It is assumed that abnormally heavy vehicles can only develop a deceleration of 0.25 g'and, thus, the HB braking force is taken to be 25% of the primary HB loading.and to be equally distributed between the eight wheels of either the front two or the back two axles. This load is thus very similar to that in BE 1177 for 45 units of loading, but is less severe for a smaller number of units. Skidding This is a new loading which has been introduced because a coefficient of friction for lateral skidding of nearly 1.0 can be developed under dry road conditions [18]. A single nominal point load of 250 kN is considered to act in one notional lane, in any direction, and in combination with primary HA loading. *Collision with parapets The Code is not concerned with the design of the parapets. which will presumably still be covered by BE 5 [106], but only with the load transmitted to the member supporting the parapets. The nominal load is thus similar to the BE 1177 loading, and is defined as the load to cause collapse of the parapet or its connection to the supporting member. whichever is the greater. The additional primary load- ing assumed to be acting adjacent to the point of collision consists of any four wheels of 25 units of the HB vehicle. 41
- 28. 3.3 11.0 Hard shoulder 3 No.traffice lanes Hard Central Verge strip reserve Parapet { ~ 0.1 surfacing 1.2 ~=====~:O.9'''U''"~ral_ 1·. concrete Fig. 3.8 Loading example *Cc/I/sion ~1/ith supports In general, the Code recommends the provision of protec- c tion of bridge supports from possible vehicle collision. The nominal loads for highway bridge supports which should oe considered are the loads transmitted by the guard rail of 150 and 50 kN, normal and parallel to. the carriageway respectively, at 0.75 m above the carriageway level or at the bracket attachment point. In addition, at the most severe point between 1 and 3 m above the carriageway, residual loads of 100 kN should be considered both normal and parallel to the carriageway. The normal and parallel loads should not be considered to act together. The above loads are only two-thirds of those in BE 1177 and the nor- mal and parallel components of the latter have to be applied together. Thus, even allowing for the fact that, at the ultimate limit state, the product Y/L Y/3 is about 1.44, as opposed to the BE 1177 safety factor of 1.15, the Code loading is less severe than the BE 1177 loading. For a foot or cycle track bridge, the nominal collision load is a single load of 50 kN applied in any direction up to a height of 3 m above the carriageway. In view of the safety factors of 1.44 and 1.15 mentioned above, this load- ing is more severe than its BE 1177 equivalent. Fatigue and dynamic loading Fatigue loading is considered in Chapter 12. ft has been found [107] that the stress increments due to the dynamic effects of highway loading are within the allowance made for impact in the nominal loading, and thus it is not necessary to consider the effects of vibration. Footway or cycle track loading If the bridge supports only a footway or a cycle track, the nominal load is 5 kN/m2 for loaded lengths up to 30 m, above which the load of 5 kN/m2 is reduced in the ratio of the HA uniformly distributed load, for the loaded length under consideration. to that for 30 m. The loading for loaded lengths greater than 30 m is thus less severe than the BE 1177 loading. The loading on elements supp0rting footways or cycle tracks. in addition to a highway or railway, is 80% of that mentioned above. However, if the footpath is wider than 2 m, the loading may be reduced further. Greater reductions in loading are permitted if a main structural member supports two or more highway traffic lanes or railway tracks. in addition to foot or cycle track 42 1:40 fall loading. It is not obvious whether a slab bridge was intended to come under this category, but it would seem more reasonable to apply the previous '80% rule' to slab bridges. It is necessary to consider the vibrations of foot and cycle track bridges as explained in Chapter 12. Railway loading The railway loading was derived by a committee of the International Union of Railways and its derivation is fully explained in an appendix to Part 2 of the Code. Example In the following example, the notation is in accordance with Part 2 of the Code, to which the various figure and table numbers refer. Numbers in brackets are the Part 2 clause numbers. , It is required to cai~ulate the nomina! transient loads which should be consi.d~red for a highway underbridge of composite slab constrU~tion and having the cross-section of Fig. 3.8, zero skew and a span of 15 m. The bridge is situated in the Birmingham area at a site which is 150 m above sea level and there are no special funnelling, gust or frost conditions. The anticipated effective bridge tempera- ture at the time of settfng the bearings is 16°C. Assume open parapets. Lanes Carriageway width (3.2.9.1) = 3.3 + 11.0 + 0.6 = 14.9 m Number of notional lanes (3.2.9.3.1) = 4 Width of each notional lane = 14~9/4 = 3.725 m Wind Since the bridge is less than 20 m span and greater than 10 m wide, it would be possible to ignore the effects of wind on the superstructure; however. the wind loads will be calculated in order to illustrate the calculation steps. Mean hourly wind speed (Fig. 2) =v = 28 mlS 0.15m O.25m ,-__~2.5°C '".)~ .. ,,'f v-v- '~ ~ Loadings 0.18m 0.20 In O.20m O.18m 6.15°C .£-_ _ _ _ _----1 j' ~ II " . ~ 1/ (a) Positive (b) Negative FIM. 3.9(a).(b) ;trJr d- C' V J'r Kt (5.3.2.1.2)= 1.0 V - ~j'SI (5.3.2.1.3) - 1.0 S2 (Table 2) = 1.49 for 6 m above ground and loaded length of 15 m. Maximum gust speed (5.3.2.1) =vc =V KI SI S2 :. Vc =41.7 m/s This value of Vc applies when there is no live load on the bridge; 5.3.2.3 states thatvc:l>35 mls with live load. Area A I is calculated for the unloaded and loaded condi- tions, with L = 15 m for both cases. Unloaded (5.3.3.1.2(a)(I}(i», d = 1.2 m (Table 4) :. Al = 15 >< 1.2 = 18 m2 Loaded (5.3.3.1.2(b», d = 0.9 + 0.1 + 2.5 = 3.5 m :.AI = 15 >< 3.5 = 52.5 m2 Drag coefficient (CD) is calculated for the unloaded and loaded conditions. with b = 17.6 m. Unloaded, d = 1.2 m (Table 5 (b» hid = 17.6/1.2 = 14.7 Co = 1.0 (Fig. 5) Superelevation = 1:40 = 1.43° Increase Cp by (Note 4 to Fig. 5) 3 >< 1.43 = 4% C/J = 1.04 Loaded. d = 2.5 m (Table 5 (b» hid = 17.6/2.5 = 7.04 C" = 1.24 (Fig. 5) Increase for superelevation, Co = 1.29 Dynamic pressure head (5.3.3) = q =0.613 v/ N/m2 Unloaded. q = 0.613 >< 41.72 /1000 = 1.07 kN/m2 . Loaded. q = 0.613 >< 352 /1000 = 0.751 kN/m2 Nominal transverse wind load (5.3.3) = P, =q AI CD Unloaded. ,P, = 1.07 >< 18 >< 1.04 = 20.0 kN Loaded. PI =0.751 >< 52.5 >< 1.29 = 50.9 kN Nominal longitudinal wind load (5.3.4) is the more severe of: PLs =0.25 q AI CD with q = 1.07 kN/m2 , AI = 18 m2 , CD =1.3 (5.3.4.1) :. PL., =0.25 x 1.07 x 18 x 1.3 =6.26 kN or PLs + PLL where P'~r =0.25 q AI CD with q = 0.751 kN/m2 , AI = 18 m2 , CD = 1.3 (5.3.4.1) and PLL =0.5 q Al CD with q = 0.751 kN/m2, AI = 2.5 x 15 = 37.5 m2, CD = 1.45(5.3.4.3) Thus PLs + PLL = (0.25 x 0.751 x 18 x 1.3) + (0.5 x 0.751 x 37.5 x 1.45) =24.8 kN Thus PL = 24.8 kN Nominal vertical wind load (5.3.5) =PI' =q A3 CI. A3 = 15 x 17.6 = 264 m2 C1J = ± 0.75 Unloaded, Pv = 1.07 x 264 x (± 0.75) = ± 212 kN Loaded, Pv = 0.751 x 264 x (± 0.75) = ± 149 kN Temperature range Minimum shade air temperature (Fig. 7) =-20°C Maximum shade air temperature (Fig. 8) =35°C Height corrections (5.4.2.2) are (-0.5) (150/100) =-O.soC and (1.0) (150/100) = ISC respectively. Corrected minimum shade air temperature = -20.8°C Corrected maximum shade air temperature = 36.5°C Minimum effective bridge temperature (Table 10) =-12°C Maximum effective bridge temperature (Table 11) =36°C Take the coefficient of expansion (5.4.6) to be 12 x l(1f'rc. Nominal expansion = (36 - 16) 12 x lcr x 15 = 3.6 mm Nominal contraction =(16 + 12) 12 x 1(1f' x 15 =5.04 mm
- 29. Temperature differences The temperaturedifferences obtained from Fig. 9 are shown in Fig. 3.9. The positive differences can coexist (5.4.5.2) with effective bridge temperatures in the range 15° to 36°C and the negative differences with effective bridge temperatures in the range _12° to 34°C. HA Uniformly distributed load for a loaded length of 15 m is 30.0 kN/m of notional lane (6.2.1). Thus the intensity is 30.0/3.725 = 8.05 kN/m2 • Knife edge load (6.2.2) = 120 kN/notional lane. Thus the intensity is 120/3.725 = 32.2 kN/m. The wheel load (6.2.S) would not be considered for this bridge, but it is a 100 kN load with a circular contact area of 340 mm diameter. It can be dispersed (6.2.6) through the surfacing, at a spread-to-depth ratio of I: 2, and through the structural concrete at 1 : 1 down to the neutral axis. Thus, diameter at neutral axis is 0.34 + 0.1 + 0.9 = 1.34 m. HB Assume 45 units, then load per axle is 4S0 kN (6.3.1). For a single span bridge. the shortest axle spacing of 6 m is required. The contact area of a wheel is circular with an effective pressure of 1. J N/mm2 (6.3.2) Contact circle diameter = [(450/4)(1000)(4)/(1.1 IT)] Iz =361 mm Disperse (6.3.3) as for HA wheel load. Diameter at neutral axis = 0.361 + 0.1 + 0.9 = 1.36 m It should be noted that both HA and HB wheel loads can be considered to have square contact areas (6.2.S and 6.3.3). ' Load on central reserve and verge Load is (6.4.3) 25 units of HB. i.e. 62.5 kN/wheel. . 44 Diameter of contact circle at neutral axis =[(62.S)(1000)(4)/(1.1IT)]Iz /1000 + 0.1 + 0.9 =1.27 m Longitudinal For a loaded length of IS m, the HA braking load (6.6.1) is 8 x lS~+,~2oo = 320 kN applied to one notional lane in combination withcprimary HA loading. The total HB braking load (6.6.2) is 0.25 (4 x 4S0) = 4S0 kN equally distributed between the eight wheels of two axles 1.8 m apart. Applied with primary HB loading. Skidding Point load of 250 kN (6.7.1) in one notional lane, acting in any direction. in combination with primary HA loading (6.7.2). Collision with parapet Parapet collapse load (6.8.1) in combination with any four wheels of 25 units of HB loading (6.8.2). Footway In order to demonstrate the calculation of footway loading, it will be assumed that the I.S m wide central reserve of Fig. 3.8 is replaced by a 3 m wide footway. The bridge supports both a footwax and a highway and the nominal load for a ISm 10adedJength is'(7.2.1) 0.8 x 5.0 = 4.0 kN/m2 • However, .b~cause the footway width is in excess of 2 m, this loading may be reduced as follows. Load intensity on first 2 In = 4.0 kN/m2 Load intensity on other 1 itt = 0.85 x 4.0 = 3.4 kN/m2 Average intensity = (2 x 4.0 + 1 x 3.4)/3 = 3.8 kN/m2 Chapter 4 Material properties and design criteria Material properties Concrete 'Y:Characteristic strengths As indicated in Chapter 1 material strengths are defined in terms of characteristic strengths. In Part 4 of the Code the characteristic cube compressive strength (f.·,,) of a concrete is referred to as its grade, e.g. grade 40 concrete has a characteristic strength of 40 N/mm2. Grades 20 to 50 may be used for nOimal weight reinforced concrete and 30 to 60 for prestressed concrete. ~s .tress-stram curve The general form of the short-term uniaxial stress-strain curve for concrete in compression is shown by the solid line of Fig. 4.1(a). For design purposes, it is assumed that the descending branch of the curve terminates at a strain of 0.0035, and that the peak of the curve and the descending branch can be 'replaced by the chain dotted horizontal line at a stress of 0.67 feu. The resulting Code short-term characteristic stress-strain curve is shown in Fig. 4.1 (b). The elastic modulus shown on Fig. 4.1 (b) is an initial tangellf value, and the Code also tabulates s"(!cantvaiues which are used for elastic analysis (see Chapter 2) and for serviceability limit state calculations as explained in Chapter 7. *Other properties Poisson's ratio is given as 0.2. and the coefficient of ther- mal expansion as 12 x 10-6 / o C for normal weight con- crete. with a warlling that it can be as low as 7 x lO-6/oC for lightweight and limestone aggregate concrete. These values are reasonable wh~n compared with published data [96, 108]. Certain other properties, such as tensile strengths, are required for the design of prestressed and composite mem- bers, but these properties are included as allowable stresses rather than as explicit characteristic values. The shrinkage and creep properties of concrete can be evaluated fro'm datu contained in an appendix to Part 4 of Stress Stress 0.67fcu 0.0035 (a) Actual and idealised I I I I I I I I I I 5.517;;;; kN/mm2 I "--_CL----'--==-_ _-J.._ Strain 2.4 x 10-4 17;;;; 0.0035 (b) Code characteristic Fig. 4.t(a),(b) Concrete stress-strain curves Strain the Code. Approximate properties for use in the design of prestressed and composite members are discussed in Chapters 7 and 8. Reinforcement :!fCharacteristic strengths The quoted characteristic strengths of reinforcement (f,,) are 250 N/mm2 for mild steel; 410 N/mm2 for hot rolled 45
- 30. ---------.......Stress fy " ,-_._------------ ~~ O.Sfy _Mild or hot rolled high yield steel --- Cold worked high yield steel - - - Code characteristic Strain Fig. 4.2 Reinforcement stress-strain curves high yield steel; 460 N/mm2 for cold worked high yield steel. except for diameters in excess of 16 mm when it is 425 N/mm 2; and 485 N/mm2 for hard drawn steel wire. *'Stress -strain curve The general forms of the stress-strain curves for mild or hot rolled high yield steel and for cold worked high yield steel are shown by the solid lines of Fig. 4.2. The Code characteristic stress-strain curve is the tri-Iinear lower bound approximation to these curves, which is shown chain dotted in Fig. 4.2. Prestressing steel *Characteristic strengths Tables are given for the characteristic strengths of wire. strand, compacted strand and bars of various nominal size. Each tabulated value is given as a force which is the product of the characteristic strength (fpu) and the area (Ap.) of the tendon. Stress-strain curve The tri-linear characteristic stress-strain curves for normal and low relaxation tendons and for 'as drawn' wire and 'as spun' strand are shown in Fig. 4.3. They are based upon typical curves for commercially available products. Material partial safety factors *Values As was explained in Chapter 1, design strengths are obtained by dividing characteristic strengths by appropriate partial safety factors (Ym)' The Ym values appropriate to the various limit states are summarised in Table 4.1. The sub-divisions of the serviceability limit state are explained later in this chapter when the design criteria are discussed. The concrete values are greater than those' for steel because of the greater uncertainty associated with concrete properties. The explanation of the choice of 1.0 for both steel and concrete for analysis purposes, at both the serviceability and ultimate limit state. is as follows. When analysing a structure, its overall response is of interest and, strictly, this is governed by material proper- 46 Stress 0.8fpu --- _..... /,/ / / / / / / / / . , 200 kN/mm2 for wire, strand / 175kN/mm2 for bar, 19 wire strand Strain (a) Normal and low relaxation products Stress 0.6fpu ,/ I,/ /,/,/ / / 200 kN/mm2 for wire J/ 175kN/mm2for 19 wire strand ~ Strain (b) 'As draINn' wire and '~s spun' strand Fig. 4.3(a),(b) Prestressirt$ steel stress-strain curves Table 4.1 Ym values Limit state Concrete Steel Serviceability Analysis of structure 1.0 1.0 Reinforced concrete cracking 1.0 1.0 Prestressed concrete cracking 1.3 1.0 Stress limitations 1.3 1.0 Vibration 1.0 1.0 Ultimate Analysis of structure 1.0 1.0 Section design 1.5 I. 15 Deflection 1.0 1.0 Fatigue 1.3 1.0 ties appropriate to the mean strengths of the materials. If there is a linear relationship between loads and their effects. the values of the latter are determined by the rela· tive and not the absolute values of the stiffnesses. Conse- quently the same effects are calculated whether the ma- terial properties are appropriate to the mean or character- istic strengths of materials. Since the latter, and not the mean strengths, are used throughout the Code it is simpler to use them for analysis; hence Y... values of 1.0 are specified, Stress 0.45fcu ------- I Stress I I I I I I I I I .. I I I I 4.5w;.; kN/~m2 I ........... l~ ,..-'::--·--·--·----·-..-..··-·-·-O.ooJ~ 2 x 10 v;;,u Strain (a) Concrete 0.87f,,,, - ......-...... 0.7f'1II ...-..- -- ,, I / I , I , , ,, I ,-1 See Fig. 4.3 (a) j./"' I --,..... / L'i.0.005 ic) Normal and low relaxation pre-stressing steels .. _....-_.... Strain Stress 0.87fy Stress 0.B7fplI - .... 0.52fpu -.-- Material properties and design criteria Tension / .-.f-.-.-.-~~~~I!!2'!...- I / I I I I I I I ~,k{2Q(JkN/mm' 0.002 (b) Reinforcing bars ,/ I I , I / ,I fl '.' See Fig. 4.3 (b) ./ V ,L _ __ ._ •Strain . -_._--------+ 0.005 Strain (d) 'As drawn' wire anrl'as spun' strand t Fig. 4.4(a)-(d) Design stress-strain curves for ultimate limit state For the same reason as above. Ym is taken to be 1,0 when analysing a section if the effect under consideration is associated with deformations; examples of this are cracking in reinforced concrete, deflection and vibration. However, when the effect under consideration is asso- ciated with a limiting stress. then a value of Ym should be adopted which should reflect the uncertainty associated with the particular material and the importance of the par- ticular limit state. The values of 1.5 for concrete and 1.15 for steel origi- Il<lted in the Comite Europeen du Beton, now the Comite Ruro-International du Beton (CEB), which chose these values hecause, when used with the CEB partial safety fac- tors for loads, they led to structures which were sensibly the same as those designed using the European national codes 11091. It should be noted that. although Ym for concrete is partially intended to renect the degree of control over the production of concrete, a single value of 1.5 is adopted in the Code irrcspective of the control, whereas the CEB Recommendations [110] state that y... can vary from 1.4. for accurate batching and control, to 1.6, for concrete made without strict supervision. When carrying out stress calculations at the service- ability limit state. Ym values of 1.3 for concrete and 1.0 for .. steel are used. These values also originated in the CEB [ III]. Cracking in prestressed conl'rete is considered to be a limiting stress effect because, as explained later in this chapter. it involves a limiting tensile stress calcul.ation. However, it is emphasised that the Y", value of 1.3 !or concrete for limiting stress calculations never has to be used Ily a designer because the stress limitations, which are given in the Code as design criteria, are design values which include the Y", value of 1.3. A value of 1.3 for concrete is giv.en for fatigue calcu- lations because it is the strength of a section which is of interest. However, this value need never be used Ily 11 designer because there is not a requirement in the Code to check the fatigue strength of concrete.
- 31. Design stress-strain curves TheYm values referred to above are applied to the charac- teristic strengths whenever they appear in a calculation. Hence design stress-strain curves are obtained from the characteristic curves (see Figs. 4.1 to 4.3) by dividing the characteristic strengths (feu, fy, fpu) , whenever they occur, by the' appropriate values of Ym' The design curves at the ultimate limit state are of particular interest and are shown in Fig. 4.4. It should be noted from Fig. 4.1 that the concrete reaches its peak compressive stress, and then starts to ("nish, at a strain of about 0.002. Once the concrete starts to crush it is less effective in providing lateral restraint to any compression reinforcement, and there is thus a possi- bility of the latter buckling. Hence the design str~ss of compression reinforcement is restricted to the stress equiv- alent to a strain of 0.002 on the d'esign stress-strain curve. This stress is 1.15 + 2t60 (4.1) It lies in the range 0.718 fy to 0.784 fy for fy in the range 485 to 250 N/mm 2 • Design criteria In this chapter the design criteria are presented and dis- cussed, but methods of satisfying the criteria are presented in subsequent chapters. The criteria are given in Part 4 of the Code under the headings: ultimate limit state, serviceability limit state and other considerations. The latter includes those criteria which are not specified in the Code but which are, nonetheless. important in design terms. The criteria are listed in Table 4.2 from which it can be seen that there are a great number of criteria to be satisfied and, if calcu- lations had to be carried out for each, the design procedure would be extremely lengthy. Fortunately, as explained later in this chapter, some of the criteria can be checked by 'deemed to satisfy' clauses. Furthermore, as experience of Table 4.2 Design criteria Ultimate limit state Rupture Buckling Overturning Vinratidn Serviceability limit .~/ate Steel stress limitations Concrete stress limitations Cracking of prestressed concrete Cracking of reinforced concrete Vihration Olher considerations Deflections Fatigue Durahility ....._---_... __........ _._-- the Code is gained, it will be possible to identify those· criteria which would not be critical for a particular situ- ation. Each criterion is now discussed. Ultimate limit state The criterion for rupture of one or more sections, buckling or overturning is simply that these events should not occur. A vibration criterion, which 'would be concerned with vibrations to cause collapse of a bridge, is not given, but, instead, compliance with the serviceability limit state vibration criterion is deemed to satisfy the ultimate limit state requirements. Serviceability limit state Steel stress limitations *Reinforcement It is explained in Chapter 7 that it is gen- erally only necessary to check cracks widths in highway bridges under HA loading for load combination 1. This means that there is an indirect check on reinforcement stresses under primary HA loading but not under other loads. In view of the fact that it is desirable to ensure that the steel remains elastic under all serviceability conditions, so that cracks which open under the application of oc- casional loading will close when the loading is removed, it was decided to introduce a separate steel stress criterion. It can be seen from Fig. 4.2 that the steel stress-strain curve becomes non-linear at a stress of 0.8 fy and this stress is, thus, the Code criterio!,. It is unfoI'tunate that the introduc- tion of this criterion complicates the design procedure for reinforced concrete for two reasons: I. As shown later in this Chapter, different Yp values are given for the crack width and steel stress calcu- . lations and this could cause confusion. 2. As explained in Chapter 7, deemed to satisfy rules for crack control in slab bridges are given in the Code, but the fact that steel stresses have to be calculated counteracts to a large extent the advantages of the deemed to satisfy rules. f.Prestressing steel Since a stress limitation existed for reinforcement, it was considered logical to specify similar criteria for prestressing steel. Hence, with reference to Fig. 4.3, stress limitations of 0.8 fpu, for normal and low . relaxation products, and 0.6fp ,I' for 'as drawn' wire and , 'as spun'strand, are given in the general section of Part 4 of the Code. These criteria imply that tendon stress incre- ments should be calculated under live load. Since such a calculation is not generally carried out in prestressed con- crete design, a clause in the prestressed concrete section of Part 4 of the Code effectively states that the criteria can be ignored. Hence, to summarise, although prestressing steel limiting stress criteria are stated in the Code, they can be ignored in practice. Concrete stress limitations It should be noted that this heading does not cover tensile stresses in prestressed concrete, which are co.ve~d. by 'cracking of prestressed concrete'. The stress hmltatlons referred to as 'concrete stress limitations' include compres- sive stresses in reinforced and prestressed con:rete, and compressive, tensile and interface shear stresses tn c()mpo- site construction. *Compressive stresses in reinforced concrete In order to prevent micro-cracking, spalling and unacceptable amo;Jllts of creep occurring under serviceability con- ditions, compressive stresses are limited to 0.5 feu. 1f:Compl'essive stresses in prestressed concrete The limit- ing stresses for the serviceability limit state and at transfer are given in Table 4.3. It is not necessary to apply the Ym value of 1.3 to the stresses. The stresses are identical to those in BE 2/73 except that a stress of 0.4 feu is now per- mitted in support regions because such a region is subject to a triaxial stress system, due to vertical restraint to the compression zone from the support. In addition, the actual flexural stress at a support of finite width is less than the stress calculated assuming a concentrated support. * . , t t' TheCompressive stresses In composIte cons ruc Ion compression flange of a prestressed precast beam with an in-situ concrete slab is restrained by the latter and placed under a triaxial stress condition. It is thus permitted, under such conditions, to increase the limiting stresses of Table 4.3(a) by 50%, but the increased stress should not exceed 0.5 feU' (The Code clause actually refers to the stresses of Table 4.3(b) but it should be those of Table 4.3(a).) The resulting stresses are of the same order as those of BE 2173. However, the Code implies that the increased stresses may be used both when a precast flange is completely encased in in-situ concrete (such as a composite slab) and for sections formed by adding an in-situ topping to precast beams. This is contrary to BE 2173, which only permits increased stresses to be used for the first of these situ- ations. The author would suggest that the approach sug- gested in the CP 110 handbook [112] could be adopted for flanges which have a topping but are not encased in in-situ concrete: the suggestion is that, in such circum- stances, an increase of only 25% of the Table 4.3(a) values should be permitted. Table 4.3 Limiting concrete compressive stresses in prestressed concrete (a) Serviceability limit state Loading Bending Direct compression (b) Transfer Stress distribution Triangular Uniform Allowable stress 0.33 feu (0.4 feu at supports) 0.25 feu Allowable stress 0.5 fei O.4fei Material properties and design criteria Stress / Hypothetical stress at first observed cracking I ------1 I I I I I / / I Fig. 4.5 Tensile stress-strain curve of restrained concrete 16 • Tests [113] - Code • e 14 -- Codex2.5 ••• • -- ..&- ~ 12 · ......z • ,....... ~ 10 .......,... . 1ii 8 ,;4' J! .~ 6 ~ ~4 2 0L---~10~--~2~0----~30~--~4~0----~50~--~60 feuN/mm2 Fig. 4.6 Tensile stresses at first observed cracking '*Tensile stress;es in composite construction When flexural tensile stresses are induced in the in-situ concrete of a composite member consisting of precast prestressed units and in-situ concrete, the precast concrete adjacent to the in-situ concrete restrains the latter and controls any cracks which may form. Hence, the descending branch of the ten- sile stress-strain curve can be made use of, and tensile strains in excess of the cracking strain of concrete toler- ated. In the Code this is achieved by specifying allowable stresses which are in excess of the ultimate tensile strength of concrete and are, thus, hypothetical stresses as illus- trated in Fig. 4.5. The Code stresses are identical to those in BE 2/73 and are given in Table 4.4 Table 4.4 Limiting concrete flexural tensile stresses in in-situ concrete In-situ concrete grade Tensile stress (N/mm2 ) 25 3.2 30 3.6 40 4.4 50 5.0 The Table 4.4 values are extremely conservative, as can be seen from Fig. 4.6 where they are compared with in- ferred stresses at which cracking was first observed in tests on composite planks reported by Kajfasz, Somerville and Rowe [113]. The Code values could be obtained from the lower bound to the experimental values by applying a par- tial safety factor of 2.5. It is thus extremely unlikely that cracking of the in-situ concrete would be observed, even at the ultimate limit state, in a composite member designed in accordance with the Code. 49
- 32. N 5 • Experimentalfirst slip [117!E E x Experimental ultimate [118! "'z(I) x (I) 4~....en • • 3 - •" •2 - Code o sb--.,..J6b· Fig. 4.7 Surface type I interface shear stresses It is permissible to increase the stresses in Table 4.4 by 50%, provided that the permissible tensile stress in the prestressed unit is reduced by the same numerical amount. This is because more prestress is then required, and it is known [114] that the enhancement of the tensile strain capacity of the adjacent in-situ concrete increases as the level of prestress at the contact surface increases. '*'Inteiface shear in composite construction Three types of surface are defined as follows: I. Rough and no steel across the interface. 2. Smooth and at least 0.15% steel across the interface. 3. Rough and at least 0.15% steel across the interface. These surface types originated in CP 116 and will lead to considerable problems for bridge engineers because the rough surface of types 1 and 3 requires the laitence to be removed from the surface by either wet brushing or tool- ing, and this is not the usual practice for precast bridge beams: the top surfaces of the latter are usually left 'rough as cast'. The minimum link area of 0.15% and a Code detail- ing rule, which states that the link spacing in composite T-beams should not exceed four times the in-situ concrete thickness, nor 600 mm, were based upon American Con- crete Institute recommendations [115]. The allowable interface shear stresses for beam and slab construction are given in Table 4.5; however, it should be emphasised that surface type 1 is not permitted for beam and slab bridge decks because it is always considered Table 4.5 Limiting interface shear stresses In-situ concrete Limiting interface shear stress (N/mm2) for grade Type 1 Type 2 Type 3* 25 0.38 0.36 1.22 30 0.45 0.38 1.25 40 0.54 0.42 1.32 50 0.59 0.46 1.38 60 0.64 0.50 1.45 * Increuse by 0.5 N/mm 2 per I% of links in excess of 0.15% 4 N E 3 E..... Z (I) 2(I) ~....en 0 6N E E..... z 5 ~ tiS 4 3 2 o X X x • x x e x • •• e Experimental surface type 2 [118! x Experimental surface type 3 [118! - - Code surface type 3 ---Code surface type 2 x •x x e. •e ------------------------ x x O. 0.4 0.6 0.8 -1 1.0 1.2 Percent steel across interface (a) Experimental serviceability stresses )( X)( • x • e x ~ ex x e e • " •e • x ------------------------0.2 0.4 0.6 0.8 1.0 1.2 Percent steel across interface (b) Experimental ultimate stresses Fig. 4.8(a),(b) Surface types 2 and 3 interface shear stresses (feu = 25 N/mm2 ) necessary to provide li~ks. This is because the calculated shear capacity of an unreinforced interface cannot be relied upon under conditions of repeated loading as occur on bridges. It can be seen that the Code approach to interface shear design is very different to that of BE 2173 which is based upon an adaptation of the CP 117 approach [11 6]. In contrast, the Table 4.5 values were essentially chosen to be a little less conservative than the CP 116 values. How- ever, they are still extremely conservative for surface types 1 and 2. In Fig. 4.7 the Code surface type 1 stresses are com- pared with some test results of Hanson [117] and Saemann and Washa [l18]. It can be seen that the Code stresses are very conservative. The Code surface types 2 and 3 stresses can be consid- ered by examining the experimental results of Saemann and Washa [118], who tested composite I-beams in which the steel area across the interface and the shear span to effective depth ratio were varied. In addition, three sur- faces were tested, and two of these were equivalent to the Code types 2 and 3. In Fig. 4.8 the test data for a cube strength of about 25 N/mm 2 and the Code stresses are compared at a serviceability criterion of a slip of 0.127 mm (as suggested by Hanson [117]) and at failure. It can be seen that the Code type 3 stresses are reasonable; but the Code type 2 stresses are very conservative and should be more dependent on steel area than the type 3 stresses, whereas the Code allows no increase of the type 2 stresses for a steel area in excess of 0.15%. In view of this ~he author would suggest that surface type 3 be considered to be applicable to the 'rough as cast' surface used in bridge practice. Finally, it is emphasised that it was not intended that interface shear should be checked in composite, slabs formed from precast inverted T-beams with solid infill. It is understood that the limiting stresses given in the Cbde for cOinposite slabs were intended for shallow slabs. The type of situation where they would be ~pplicable in bridg.es is where the LOp slab u1 a (leeK cunsisls oj precast umts spanning between longitudinal beams, and the units act as permanent formwork for in-situ concrete to form a com- posite slab. '*Cracking of prestressed concrete The criteria for the control of cracking in prestressed con· crete are presented in terms of limiting flexural tensile stresses for three classes of prestressed concrete. Class I No tensile stresses are permitted except for I N/mm 2 under prestress pJus dead load, and at transfer. These criteria are thus identical to those of BE 2/73. Class 2 Tensile stresses are permitted but visible crack- ing should not occur. Beeby [119] has suggested that the flexural tensile strength of concrete is equal to 0.556 /l.u. The appropriate partial safety factor to be applied is 1.3 and thus design values of the flexural tensile strength should be given by 0.428 !feu which compares favourably with the Code value of 0.45 ./lcu for pre-tensioned mem- bers. However, only 80% of this value (i.e. 0.36 ./lcu) should be taken for post-tensioned members, because tests reported by Bate [120] indicated that cracks in a grouted post-tensioned member widen at a greater rate than those in a pre-tensioned member, and thus the design stress for the former should be less than that for the latter. No refer- ence is made in the Code to unbonded tendons and, thus, the author would suggest the adoption of t~e Concrete Society recommendations of 0.15 Ifeu and zero in sagging and hogging moment regions respectively [121]. It is necessary to check that, under dead and superim- posed dead load, a Class 2 member satisfies the Class 1 criterion in order to ensure that large span bridges, for which dead load dominates, have an adequate factor of safety against cracking occurring under the permanent loading which actuaJIy occurs in practice. At transfer of a Class 2 member, flexural tensile stresses of 0.45 Ifei and 0.36 ./lei are pennitted for pre- and post- tensioned members respectively, where lei is the concrete cube strength at transfer. Class 3 Cracking is permitted provided that the crack widths do not exceed the design values for reinforced con- crete. given later in Table 4.7. However, it is not neces- sary to carry out a true crack width calculation. because the Code giv~s limiting hypothetical flexural tensile stress- es which are deemed to be equivalent to the limiting crack Material propertIes and d.esign criteria Table 4.6 Hypothetical flexural tensile stresses for Class 3 members (a) Basic stresses Tendon type Limiting crack Stress (N/mm2) for concrete grade width (mm) Pre-tensioned 0.1 Grouted post-tensioned 0.2 0.25 0.1 0.2 0.25 Pre-tensioned, 0.1 close to tension 0.2 face 0.25 (b) Depth factors Depth (mm) EO 200 Factor 1.1 400 1.0 30 3.2 3.8 4.1 600 0.9 40 4.1 5.0 5.5 4.1 5.0 5.5 5.3 6.3 6.8 800 0.8 ;;, 50 4.8 5.8 6.3 4.8 5.8 6.3 6.3 7.3 7.8 ;;,1000 0.7 widths. The basic stresses are given in Table 4.6(a), and it can be seen that they are referred to as hypothetical stress- es because they exceed the, tensile strength of concrete and so cannot actually occur. The basic stresses were derived from tests on beams by Bate [120] and Abeles [122], who calculated the hypothetical tensile stresses present at different maximum crack widths observed in the tests. Beeby and Taylor [123] have shown that the hypotheti- cal tensile stresses should decrease with an increase in depth and, thus, the basic stresse&, have to be multiplied by a depth factor from Table 4.6(b). The presence of additional reinforcement in a prestress- ed member increases the crack control properties and a higher hypothetical tensile stress may be adopted. The Code increases of 4 N/mm 2 per 1% of additional steel, for pre-tensioned and grouted post-tensioned tendons, and of 3 N/mm2 per 1% of additional steel, for pre-tensioned ten- dons close to the tension face, are based upon the tests of Abeles [122]. It should be noted that the steel percentages are based upon the area of tensile concrete and not the gross section area. A Class 3 member has to be checked as a Class I member under dead and superimposed dead load for the same reason as that given previously for Class 2. At transfer, the flexural tensile stresses in a Class 3 member should not exceed the limiting stress appropriate to a Class 2 member. This is in order to avoid cracking at the ends of members. Finally, the Code gives no guidance on unbonded ten- dons, and the author would suggest that [123] be consulted for such members. tCracking of reinforced concrete The design surface crack widths were assigned from con- siderations of appearance and durability, and were based partly upon the 1964 CEB recommendations [111]. They 51
- 33. Table 4.7 Designcrack widths Conditions of exposure Moderate Surface sheltered from severe rain and against freezing while saturated with water, e.g. (I) Surfaces protected by a waterproof membrane (2) Internal surfaces, whether subject to conden~ation or not (3) Buried concrete and concrete continuously under water Severe (I) Soffits (2) Surfaces exposed to driving rain, alternate wetting and drying, e.g. in contact with backfill and to freezing while wet Very severe (1) Surfaces subject to the' effects of de-icing salts or salt spray. e.g. roadside structures and marine structures excluding soffits (2) Surfaces exposed to the !lclion of seawater with abrasion or moorland water having a pH of 4.5 or less Design crack width (mm) 0.25 0.20 0.10 are summarised in Table 4.7. and it should be noted that different design crack widths are assigned for different conditions of exposure; unlike BE 1173, which differen- . tiates only between different types of loading. In addition, for bridge decks, different design crack widths are assigned for soffits, and for top slabs if the latter are pro- tected by waterproof membranes. The fact that both of these design crack widths are as, or more, onerous than the BE 1173 values of 0.25 and 0.31 mm is counteracted by the fact that different crack width formulae are adopted, as explained in Chapter 7. However, as discussed in Chapter 9. the very severe exposure condition for roadside struc- tures could prove to be exceptionally onerous. and to lead to impracticably large areas of reinforcement if applied to piers and abutments. The philosophy of relating the design crack width to the condition of exposure and, thus. indirectly to the amount of corrosion. must now be viewed with some scepticism in the light of research at Munich, from which Schiessl [124] concluded that there was no significant relationship be- tween corrosion and crack width or cover. This work has bcen discussed by Beeby [125J. Vibration It is only necessary to consider foot and cycle track bridges, and the criterion is discomfort to a user of the 52 bridge. The derivation of the Code criterion is described fully by Blanchard, Davies and Smith [107], and a sum- mary follows. The effects of vibrations on humans are closely related to acceleration and, thus, maximum tolerable accelerations were plotted against frequency in order to derive a crite- rion in terms of natural frequency. The maximum tolerable accelerations were assessed from two criteria: discomfort while standing on a vibrating bridge, and the impairance of normal walking due to large amplitude vibrations. The Code criterion lies approximately' midway between these two criteria and is given in an appendix to Part 2 as an acceleration of 0.5 110 Ill/S 2 where fo is the fundamental natural frequency of the unloaded bridge. Other considerations Deflection A specific criterion is not given for deflection in terms of an absolute limiting deflection, or of span to depth ratios. However, it is obviously necessary to calculate deflections in order to ensure that clearance specifications are not vio- lated and adequate drainage will obtain. Deflection calcula- tions are also important where the method of construction requires careful control of levels. and for bearing design. Fatigue The relevant criterion is essentially that there should be a fatigue life of 120 years. When considering unwelded bars an equivalent criterion in terms of a stress range is given in the Code The criterion is that the stress range should not exceed 325 N/mm2 for high yield bars nor 265 N/mm 2 for mild steel bars. These ranges are'identical to those of BE 1/73 except that thera'nge for mild steel bars is inde- pendent of bar diameter. " It is not clear why stress ranges which are dependent upon bar type have been adopted, and it would appear ,more logical for the stress ranges to be dependent upon the type of loading and the loaded length: indeed, such a dependence was considered during the drafting stages of the Code. Durability A durability criterion is not defined but, provided that the requirements of the Code with regard to Ilmiting crack widths, minimum covers and minimum cement contents are complied with, durability should not be a problem. Yfa values In Chapter 3 the nominal loads and the values of the partial safety factor Y[I_' by which these' loads are mUltiplied to give design loads. are presented. Furthermore, it is explained in Chapter I that the effects of the latter have to be multiplied by a partial safety factor Yf3 in order to obtain design load effects. The values of Yf3 are dependent upon the material of the bridge and hence, for concrete bridges, are given in Part 4 of the Code. It is thus con- venient to introduce the Yf3 values and the design criteria together in this chapter. It should be noted that, some of the values given in the following are not necessarily stated in the Code, but are either implied or intended by the drafters. Jltimate limit state A value of 1.15 for dead and superimposed dead load is stated for all methods of analysis since, for loads which are essentially uniformly distributed over the entire struc- ture. any analysis should predict the effects with reason- able accuracy. It is suggested in Chapter 1 that. Yf3 could be considered to be an adjustment factor which ensures that designs to the Code would be similar to designs to existing documents. It can be shown that, on this basis, a value of 1.15 for dead and superimposed dead load is a reasonable average value. The Code states that, for imposed loads, Yf3 should be related to the method of analysis and quotes a value of 1.1 fol' all m~thods of analysis (including yield line theory) except for methods involving redistribution, in which case a value of [1.1 + (~- 10)/200] is quoted, where ~ is the percentage redistribution. These values do not seem entirely logical since the Yp value for an upper bound method should be greater than that for a lower bound method, because the former is theoretically unsafe and the latter theoretically safe. The drafters' reason for including yield line theory with lower bound methods was that, although it is theoretically an upper bound method, tests [86. 87] show that it predicts safe estimates of the strengths of actual slabs; but this is also true of other methods of analysis and, indeed, lower bound methods predict even safer estimates of the strengths of actual slabs [1261. Furthermore. it seems illogical to have a Yp value of 1.1 for the extreme cases of no redistribution (elastic analysis) and what might be considered as full redistri- bution (yield line: theory), yet values greatet;.than 1.1 are permitted for redistrib,ution in the range 10 tq 30%. i Since the max~mum permitted value of ~ is 30%, Yp cannot be greater than 1.2 and it thus always lies in the range 1.1 to 1.2 for imposed load and is always 1.~J5 for dead load. In order to simplify the calculations. the Code allows. as an altetnative. the adoption of 1.15 for all loads and all types of ~nalysis, provided that ~ do~s not exceed 20%. The reason I for the proviso is that the' value of Yp calculated from the formula is greater than 1.15 for B greater than 20%. It should be noted that the formula for Yp. gives values less than 1.1 when ~ is less than 10%, but it was intended that a value of 1.1 should be used for these cases. When using yield line theory. Yp should be applied to the load effects <the required moments of resistance) in (l(:cordance'with equation (1.2). However, different values of YI'1 have tt) be applied to dead loads and imposed loads. This causes problems because, when using yield line theory, the effects of different types of load cannot be cal~ culated separately and then added together. Thus, although strictly incorrect, it is necessary to apply Yp as indicated in equation (1.3) when using yield line theory. *'Serviceability limit state The Yp values at the serviceability limit state are all unity exeept for the following: a value of 0.83 is applied to the effects of HA loading and of·0.91 to the effects of HB loading (or of HA combined with HB) when checking the cracking limit state under load combination 1. These val- ues are not stated in the Code, but are implied because the Code ~tates a value of 1.0 for the product Yfl· Yf2. Yp; and, SlDce y,d= Yfl. Yf2) is 1.2 for HA loading and 1.1 for HB loading at the serviceability limit state for load combination 1 (see Table 3.1), the implied values of YP are 0.83 and 0.91 respectively. In the previous paragraph. it is implied by the author that Yp should be taken to be unity. when checking the stress limitation limit state. [n fact. an appropriate Y:. value is not explicitly stated in the Code but it was t~e intention of the drafters that it should be unity. The fact that. for load c,ombination 1, different Yp val- ues are specified for the cracking limit state, to those for the stress limitation limit state, causes problems in the cal- culations for reinforced and prestressed concrete members for the following reasons: I . In the case of reinforced concrete. stress limitation calculations are carried out for a load of. essentially. 1.2 HA or 1.1 HB, whilst crack width calculations are carried out for a load of 1.0 HA or 1.0 HB. This could obviously cause con(usion and it also compli- cates the calculations. 2. In the case of prestressed concrete, tensile stress cal- culations are considered at the cracking limit state and, hence. under a load of 1.0 HA or 1.0 HB: whilst compressive stress calculations are considered at the stress limitation limit state under a load of 1.2 HA or 1.1 HB. Hence stresses of different signs on the same member are checked under different loadings. In view of the anomaly so created. the prestressed concrete section of Part 4 of the Code states that compressive stresses should be checked under the same load as that specified for tensile stresses: in other words. loads of 1.0 HA and 1.0 HB are used for calculatiflR both compressive and tensile stresses in prestressed concrete. It should be stated that the calculations are not necessar- ily as complicated as implied above because. strictly. stresses and crack widths should be calculated under the design loads. which are 1.2 HA and 1.1 HB. and the stresses and crack widths so calculated should then be multiplied by the appropriateY(l value (1.0 or 0.83 or 0.91) to give the design load effects as explained in Chapter I. However, the adoption of different Yf'J values seems to be an unnecessary complication. when the same final result. in design temlS. could have heen obtained by modifying the design criteria. '
- 34. Summary An attempt isnow made to summarise the implications of the Code Y/3 values and design criteria by comparing the number of calculations required for designs in accordance with current documents and with the Code. *Reinforced concrete In accordance with BE 1/73, a modular ratio design is car- ried out at the working load, and crack widths are checked at the same load. However, in accordance with the Code, stresses, crack widths and strength have to be checked at different load levels; and thus three calculations, each at a different load level, have to be carried out, as opposed to two calculations, at the same load level, when designing in accordance with BE 1/73. 54 ~ Prestressed concrete The number of calculations for designs in accordance with the Code and in accordance with BE 2/73 are identical because, in both cases, stresses have to be checked at one load level and strength at another. '*Composite construction."...}- In addition to the comments made above regarding rein- forced and prestressed concrete construction, the design of composite members is complicated by the interface shear calculation. The latter calculation is considered at the stress limitation limit state (Y/3 = 1.0) and the load level is thus different to that adopted for checking the stresses in, and the strength of, the, generally, prestressed precast members. Thus, three load levels have to be considered for a composite member designed in accordance with the Code, as opposed to two load levels for a design in accor- dance with BE 2/73. Chapter 5 Ultimate limit state - flexure and in-plane forces Reinforced concrete beams *Assumptions The following assumptions are made when analysing a cross-section. to determine its ultimate moment of resis- tance: 1. Plane sections remain plane. 2. The design stress-strain curves are as shown in Fig. 4.4. 3. If a beam is reinforced only in tension, the neutral axis depth is· limited to half.the effective depth in order to. ensure that an over-reinforced failure involv- ing crushing of the concrete, before yield of the ten- sion steel, does not occur. This is because such a fail- ure can be brittle, and th~re is little warning that it is about to take place. A balanced design, in which the concrete crushes and the tension steel yields simul- taneously, is given by considering the strain diagram of Fig. 5.1 in which Ey is the strain at which the steel commences to yield in tension. This strain Js given, by reference to Fig. 4.4(b), by E = 0 002 0.87/y y • + 200000 (5.1) Ey is thus in the:: approximate range of 0.003 to 0.004 and, for a balanced design, the neiJtral axis depth (t) is approximately half of the effective depth since, from Fig. 5.1, the neutral axis depth is given by 0.0035d x =0.0035 +Ey (5.2) 4. The. tensile strength of concrete is ignored. 5.. Small axial thrusts, of up to O.1feu times the cross- sectional area, are ignored, because they increase the calculated moment of resistance. [112]. . *'Simpli,fied concrete stress block The parabolic-rectangular distribution of concrete com- pressive stress, implied by the stress-strain curve of Eu - 0.0035 d ty Fig. 5.1 Strains for a balanced design Fig. 4.4(a), is tedious to use in practice for hand calcu- lations. The Code thus permits tlJe approximate rectangular stress block, with a constant stress of O.4leu, as shown in Fig. 5.2, to be adopted. Beeby [127] has demon- strated that,. for beams, the adoption of· the rectangu- lar stress block results in steel areas which are essen- tially identical to those using the parabolic-rectangular curve. Strain compatibility The ultimate moment of a section can be determined by using the strain compatibility approach which involves the following steps. 1. Guess a neutral axis depth and, hence,determine the strains in the tension and compression reinforcement by assuming a linear strain distribution and an extreme fibre strain of 0.0035 in the compressive concrete. 2. Determine from the design stress-strain curves the steel stresses appropriate to the calculated steel strains. 3. Calculate the net tensile and compressive forces at the section. If these are not equal. to a reasonable accu- racy, adjust the neutral axis depth and return to step I. 4. If the net tensile and compressive forces are equal. take moments of the forces about a common point in the section to obtain the ultimate moment of resis- tance. 55
- 35. Stress Parabolic-rectangular 0.4Sfcu I- --- - -;•.,.,.. - • _. _. _. - ._.- " 0.4feu I ;' Simplified rectangu ar .I. b _-I As 000 ------r +__ d z I ....~- . ------+Fs = 0.87f~s I .~ .. I. I. Fig. 5.1 Rectangular stress block Design charts 0.0035 Strain The strain compatibility method described above is· tedi- OWl for analysis and is not amenable to direct design. Thus design charts are frequently used and the CP 110 design charts [128] are appropriate. Design formulae As an alternative to using strain compatibility or design charts, the Code gives simplified formulae for hand calcu- lations. The formulae are based upon the simplified rectangular stress block discussed previously and their derivations are now presented. "*Singly reinforced rectangular beam The stresses and stress resultants at failure are as shown in Fig. 5.3. For equilibrium F" = Fs O.4t.."bx =0.87f.A.• :. x == 2.2f0_· t.·ub Since a rectangular stress block is assumed, the lever arm (z) is given by -d /2-d ~z - -x - - t. b ('U or z =(I -l.lf0·)d fc"bd (5.3) However. the Code restricts z to a maximum value of 0.95<1. It is not clear why the Code has this restriction, but Beeby 1127] has suggested that it could be either that there is evidence that the concrete at the top of a member tends to be less well compacted than that in the rest of the member, or that it is felt desirable to limit the steel strain at failure (a maximum lever arm of 0.95d implies a maxi- mum steel strain of 0.0315). The ultimate moment of resistance (M,,) can be obtained by taking moments about the line of action of the resultant concrete force; hence Fig. 5.3 Singly reinforced rectangular beam at failure (5.4) However, the Code restricts the neutral axis depth to a maximum value of 0.5d, in which case the ultimate moment of resistance can be obtained by taking moments about the reinforcement; hence Mu =O.4fcub(O.5d) (O.7Sd) =O.15fcubcf2 (5.5) The ultimate moment of resistance should tie taken as the lesser of the values given by equations (5.4) and (5.5). Equations (5.3) to (5.5) are given in the Code as design equations, but they are obviously more suited to analysing a given section, rather than to designing a section to resist a given moment. In view of this it is best, for design pur- poses, to rearrange the equati~ns as follows. From (5.4) A=~., O.87fvz (5.6) Substitute in (5.3) and solve the resulting equation for z Z = 0.5d(1 +Ji-L~J2 ) (5.7) Thus the lever arm can be calculated from equation (5.7) and, then, the steel area from (5.6). The Code limits the application of the design equations to situations in which lessthan 10% redistribution has been assumed. This is becau~the neutral axis depth is limited to 0.5d and, for this value, the relationship between neutral axis depth and amount of redistribution, which is discussed in Chapter 2, implies a maximum redistribution of 10%. However, it is possible to derive the following more general version of equation (5.5), which is appropri- ate for any amount of redistribution (~) Mu =0.4fcubd(0.6-~) (0.7 +0.5~) (5.8) Doubly reinforced rectangular beam The procedure for deriving the Code equations for doubly reinforced beams is to assume that the neutral axis is at the same depth (0.5d) as that for balanced design with no compression reinforcement. Compression reinforcement is then provided to resist the applied moment which is in excess of the balanced moment given by equation (5.5). and tension reinforcement provided such that equilibrium is maintained. The strains, stresses and stress resultants are as shown in Fig. 5.4. lt is mentioned in Chapter 4 that the design stress of compression reinforcement is in the range 0.718fv to 0.784fv: it is thus conservative always to take a value of O.72!v: as in the Code and Fig. 5.4. It should be noted, from -the strain diagram of Fig. 5.4, that, in order that the ~ b ~ I-0.0035 ~ d' ;;;:0.002 0 A~ 0 ~ dod' d As o o o Section Strains Fig. 504 Doubly reinforced rectangular beam at failure compression reinforcement may develop its yield strain of 0.002, and hence its des~gn strength of O.72fy , it is necess- ary that 0.0035 (0.5d - d')/O.Sd ." 0.q,02 or d'ld :!EO 0.214 Hence the Code states that d'/d:!EOO.2 (5.9) The ultimate moment of resistance can be obtained by taking moments about the tension reinforcement; hence Mil =0.2f""bd(0.75d) + O.72fyA': (d-d') or Mil = 0.15fcllbcf2 + O;72f.A': (d-d') (5.10) From which A,,' can be calculated directly. For equilibrium O.87f,As =0.2fc"bd + O.72fyA': (5.11) From which A can be calculated directly.s Equations (5.10) and (5.11) are given in the Code, but it should be noted that the final term of equation (5.10) is incorrectly printed in the Code. The equations are again restricted to less than 10% redistribution, but can be written more generally as eI' 3 d = --::; (0.6-13) (5.12) M" = 0.4t.."bd2 (0.6-(3) (0.7 + 0.5(3) + O.72fyAs' (d-d') (5.13) 0.87[,. A., = 0.4f(."bd(O.6-~) + O.72fA: (5.14) *Flanged beams' It is assumed in the Code that any compressive stresses in the web concrete can be conservatively ignored: this is valid provided that the flange thickness does not exceed half the effective depth. The stresses and stress resultants at failure are then as shown in Fig. 5.5. The ultimate moment of resistance is taken to be the lesser of the value calculated assuming the reinforcement to be critical: M" =0.87f)As (eI-hj2) (5.15) and that assuming the concrete to be critical: I~ 0-"1 Fs = 0.87f~$ b 0.4feu Stresses 0.7Sd F; =O.72f~~ Fe =0.2feubd -'~ _--h,- 0.4fcubh, As 00 d Fig. 5.5 Flanged beam at failure Mu =O.4t.,,,bh, (d-hj2) ---0.87f~s (5.16) These equations are given in the Code and can be used for design purposes by calculating the steel area from (5. 15), and checking the adequacy of the flange using (5.16). The breadth b shown in Fig. 5.5 is the effective flange width. which is given by the lesser of (a) the actual width and (b) the web width plus one-fifth of the distance be- tween points of zero moment, for T-beams, or the web width plus one-tenth of the distance between points of zero moment, for L-beams. The distance between points of zero moment may be taken as 0.7 times the span for continuous spans, and it would seem reasonable to take a value of 0.85 times the span for an end span of a continuous member. The resulting effective widths are of the same order as those calculated in accordance with CP 114. In addition. a comparison with values obtained from Table 2 of Part 5 of the Code indicates that, at mid-way between points of zero moment, the Code is generally conservative. Prestressed concrete beams *Assumptions The assumptions made for reinforced concrete are also made for prestressed concrete; in addition it is assumed that: t. The stresses at failure in bonded tendons can be obtained fr~'m either the appropriate design stress- strain curve of Fig. 4.4 or from a table in the Code 57
- 36. which gives thetendon stress and neutral axis depth at failure as functions of the amount of prestress. The table is based upon the test data of Bate [120], and is very similar to the equivalent table of CP lIS. How- ever, the Code neutral axis depths are 87% of the CP 110 values because the Code adopts a design tendon strength of O.87[p", whilst CP lIS uses an ultimate strength of [pu' Hence, to maintain equilibriu~?,a,. smaller neutral axis depth has to be adopted b~cause the same concrete compressive stress is used [n both CP lIS and the Code. 2. The stresses at failure in unbonded tendons are ubla!m:o lrom a table in the Code which gives the ten- . don stress and neutral axis depth as functions of the I. amount of prestress and the span to depth ratio. The table is based upon the resll'lts of tests carried out by Pannell [129], who concluded that unbonded beams remain elastic up to failure except for a plastic zone, the extent of which depends upon the length of the tendon. Hence, the failure stress and neutral axis depth depend upon the span to depth ratio. 3. In order to give warning of failure it is desirable that cracking of the concrete should occur prior to either the steel yielding aQd fracturing, or the concrete crush- ing in the compression zone. This can be checked. by ensuring that the strain at the tension face exceeds the tensile strain capacity of the concrete. The latter c~n be obtained by multiplying the design limiting tensile stress of 0.45 ff,." for a Class 2 member (see Chapter 4) by 1.3 (the partial safety factor incorporated in the formula), and dividing by the appropriate elastic modu- lus given in the Code. Strain compatibility The strain compatibility method described for reinforced concrete can be applied to prestressed concrete, but the prestrain in the tendons should be added to the strain, cal- culated from the strain diagram at failure, to give the total strain. The latter strain is used to obtain the tendon stress from the stress-strain curve. Design charts Design charts are given in CP 1l0 [130] for rectangular prestressed beams, but these are of limited use to bridge engineers, who are normally concerned with non- rectangular sections. To the author's knowledge design charts for non-rectangular sections are not generally aVllil- able, but Taylor and Clarke [131] have produced some typical charts for T-sections. These should be useful to bridge engineers, because they can be applied to a number of standard bridge beams. Design formula The Code gives a formula for calculating the ultimate moment of resistance of a rectangular beam, or of a 58 b I_ O.4fcu ./ -~--r--- d Fig. 5.6 Rectangular prestr~ssed beam at failure x d Fps =Ap.fpb Fig. 5.7 General prestressed be.am at failure z=d-O.5x flanged beam with the neutral axis within the flange. The formula is obtained by taking moments of the tendon forces at failure about the line of action of the resultant concrete compressive force. Hence, with reference to Fig. S.6, (S .17) The tendon stress (Jpb) and neutral axis depth (t) at fail- ure are obtained from the tables mentioned previously. Although these tables, and hence equation (S.17), are intended for rectangular sections, it is possible to adapt them to non-rectangular,sections (1)2]. This is achieved by writing down an equil,ibrium equation in terms of the unknown tendon stress arl~ unknown neutral axis depth. Hence, with reference tofig. S.7, Ap.!pb = 0.4/mAc where Ac is, generally, a linear function of x: thus /pb is also, generally, a linear function of x. A graph can be plot- ted of/Pb against x for the section under consideration and, on the same graph. the Code tabulated values of/pb and x can be plotted; the required values of/pb and x can be read off where the two lines cross. Reinforced concrete plates General The design of reinforced concrete plates for bridge~ is more complicated than for buildings because, in bridges. the principal moment directions are very often inclined to the reinforcement directions (e.g. skew slabs), and plates are often subjected to both bending and in-plane effects (e.g. the walls of box girders). . The Code states that allowance should be made for the fact that principal stress resultant directions and reinforce- ment directions do not generally coincide by calculating required ~resistiye stress resultants' such that adequate .strength is provided in all directions at a point in a plate. No guidance is given in the ~ode on the calculation pro- cedure, but the Code statemelllimplies that it is necessary to .' satisfy the relevant yield criterion. In the following, it is shown how this can be done for plates designed to resist bending. in-plane or combined bending and in-plane . effects. It should be. noted that all stress resultants are in terms of values per unit length. Bending .Orthogonal reinforcement In general, it is required to reinforce in the x and y direc- tions a plate element which is subjected to the bending moments Mx, My and the twisting moment Mxy shown in Fig. 2.9. The yield criterion for a plate element subjected to bend- ing only is simply a relationship between the amounts of reinforcement in the element and the applied moments (Mit. My, M..y) which would cause yield of the element. It can be shown [132J that the yield criterion is (5.18) M*,r and M*y are the moments of resistance per unit length. of the reinforcement in the x and y directions respectively, calculated in the reinforcement directions. These moments of resistance can be calculated by means of the methods previously described for reinforced con- crete beams. A combination of M.., My, Mxy satisfying equation (5.18) would cause the slab element to yield. A designer is generally interested in determining values of M*,. and M*y to satisfy equation (5.1S) for known val- ues of M.., Ml" M n ,. This could be done by choosing either M*.•, or M*,.. 'and then calculating the other from the yield criterion. However. it is more convenient to make use of equations which give M*x and M. directly. Such equa- tions can be derived by noting that the follo,wing expres- sions for M*x and M*y satisfy the yield criterion M*,r = Mx- MxyK M*I' =My ,- MxyK-1 (S.19) (S.20) Any value of K can be chosen by the designer and thus there is an infinite number of possible combinations of M"'x and M* ' capable of resisting a particular set of applied moments. However, a solution which minimises the total amount of steel at a point is generally sought and, to a first order approximation, the total amount of steel is propor- ·tional to (M"'.. + M",). Hence the value of K required to give a minimum steel consumption can be obtained by dif- ferentiating (M"'x + M*y) with respect to K and equating to zero, thus M*x + M*y = Mx + My- Mxy (K + K-') o(M + M*y) = -M.,I' (l _ K'2) =0 oK K=±! The second derivative is o2(M*x + M'''y) =,;", 2M K-3 oK2' . xy When considering bottom reinforcement, the sign con- vention is such that M*x and M*y must be P9sitive,'hence minimum steel consumption coincides with a mathematical minimum and the second derivative should be positive. Hence, Mxy and K must be ofopposite sign and . c MxyK =MxyK-1 =-IMxyl Thus, for bottom reinforcement M*x =Mx + IMxyl M*y = My + IMxyl ) (S.21) When considering top reinforcement, the sign conven- tion is such that M*It and M*y must be negative, hence minimum steel consumption coincides with a mathematical maximum and the second derivative should be negative. Hence, M.ty and K must be of the same sign and M.tyK =MxyK",j =IMxyl Thus, for top reinforcement M*x =Mx -IMxyl M*y, = My -IMxyl ) (5.22) Equations (S.21) and (5.22) are for the optimum amounts of reinforcement with reinforcement in both the x and y directions. but it is possible forr II value of M*It and M*y so calculated to have the wrong sign. This implies that no reinforcement is required in the appropriate din~c tion, and another set of equations should be !.I!sed whRch C£lfl be derived as follows. If Mx < - IMxyl , so that a negative value of M*,. is calculated from the first of equations (5.2]). then no rein- forcement is required in the bottom in the x direction. Hence. M = 0 can be substituted into equation (S.19) to obtain a value of K o = MX-,MxyK K = MxlMxy This value of K is thensubstituted into equation (S.20) to give M*y = My - M2x/Mx Mx must be negative and thus this equation is generally written M*y = My + IM2x/M..1 (S.23) Similar equations can be derived for .the other possi- bilities and the complete set of equations, including (S.21) and (S.22), are known to bridge engineers as Wood's equations [133]; although they were originally proposed by Hillerborg [134]. The complete set of equa- tions is given in Appendix A to this book as equatiol)s A I to AS. Skew reinforcement A similar set of equations - (A9) to (A 16) of Appendix A - can be derived for skew reinforcement in the x direction S9
- 37. and in adirection at an angle Q: measured clockwise from the x direction. These equations are known to bridge engineers as Armer's equations [135]. Practical considerations Experimental verification The validity of the theory, upon which equations Al to AI6 is based, has been confinned by tests on slab elements; in additio!" model" '.- skew slab bridges have been designed by the equations and successfully tested by Clark [126], Uppenberg [136] and Hallbjorn [137]. Failure direction The direction in which failure (i.e. yield) of a slab element occurs can be determined from theoretical considerations. Strictly, once M'"x, etc.. have been calculated from equations (AI) to (A16), they should be resolved into the failure direction and the section design carried out in this direction. However, in practice, it is usual to carry out the section designs independently in each reinforcement direction. Theoretically, this can mean that the concrete is overstressed because the principal con- crete stress occurs in the failure direction and not neces- sarily in a reinforcement direction. The author would suggest that, in practice. this error can be ignored because under- reinforced sections. in which the concrete' is not critical, are generally adopted. and the greater ductility of a slab compared with a beam is neglected in design. Slabs are more ductile than beams because the ultimate strain cap- acity ofthe concrete in the compression zone increases as the section breadth increases [138]. Thus, although Morley 1139] and Clark [126] discuss the correct section design procedure. the author would suggest that the existing prac- tice of designing the sections in the individual steel direc- tions be continued. Minimum reinforcement When reinforcement is propor- tioned in accordance with the required moments of resis- lance. calculated from equations (A I) to (A 16), it will sometimes be found that the reinforcement areas are less than the Code minima discussed in Chapter 10. In such situations it is necessary to increase the reinforcement area to the minimum specified in the Code. When this is done it is often theoretically possible to decrease the reinforcement area in another direction. As an example. if the value of M from equation (A I) implies a reinforcement area in the x direction which is less than the minimum. then the minimum area should obviously be adopted in this direc- tion, and the reinforcement area in the y direction can then he less than that implied by equation (A2). The equations for carrying out such calculations have been presented by Morley 11391. However, it is obviously conservative, in the above example. to provide the area of reinforcement in the.v direction implied by equation (A2). Multi,,/(, load combinations Bridges have to be designed for a number of different load positions and combinations and. hence. at each design point, for a number of different moment triads (M,. Ml" M,)). Many computer programs are available which calculate M*.•, M*v or M'"'" for each triad lind then output envelopes of maximum values of 60 M'"x' etc. Such an approach ignores the interaction of the' multiple triads; but it is simple and conservative. How- ever, it is possible to reduce the total amount of reinforce- ment required at a particular point by considering the interaction of the multiple triads. The interested reader is referred to the work of Morley [139] and Kemp [140] for further infonnation on such procedures. In-plane for~ ., Equations, very similar to those discussed previously for bending and twisting moments, have been derived for cal- culating the forces required to resist an in-plane force triad consisting of two in-phlile forces per unit length (Nx,Ny ) and an in-plane shear force per unit length (Nxy)' The sign convention adopted for the forces is shown in Fig. AI. The requiredl'esistive forces are designated N*x etc., and are equ!valent to the appropriate reinforcement area per unit length multiplied by the design stress of the rein- forcement. The equations..;, (A17) to (A30) - for deriving the values of the required resistive forces are given in Appendix A, together with equations for the principal con- crete forces per unit length. The principal concrete stresses can be obtained from the latter by dividing by the plate thickness. . The equations for orthogonal reinforcement are gener- ally referred to as Nielson's equations [141] and the equa- tions for skew reinforcement have been presented by Clark [142]. It should be noted that it is required that the values of N, etc., in equations (A17) to (A30) should always be zero or positive, which implies that the r~inforcement is always in tension. An extended set of equations which includes the possibility that compression reinforcement may be required has been. presented by Clark [142]. The validity of the equations have been continned experimentally only for situations in which all of the rein- forcement yields in tension [141. 143]. Morley and Gulvanessian [144] have considered the problem of providing a minimum area of reinforcement in a specified direction but have not presented explicit equa- tions, although they do describe a suitable computer pro- gram. Multiple load combinations could be considered by the approaches of [139] and [ 140]. Combined bending and in-plane forces The provision of reinforcement to resist combined bending and in~plane forces is extremely complex. A design sol- ution is usually obtained by adopting a sandwich approach in which the six stress resultants are resolved into two sets of in-plane stress resultants acting in the two outer shells of the sandwich. Such an approach has been suggested by Morley [145]. If the centroids of'the outer shells are chosen to coi'ncide with the centroid~ of the reinforcement layers. as shown in Fig. 5.8. then equations (AI7) to (A30) for in-plane forces can be applied to the above two sets of in-plane stress resultants. Such an approach is valid because both equilibrium and the in-plane force yield c{ .---... 0 0 0 0 o. - 0 n '0 0 0 csl. Actual section Fig. 5.8 Equivalent sandwich plate 0 -'l N'!..a., r l!. 2 h 2 criteria are satisfied:. thus a safe lower bound design rem~. . With reference to Fig. 5.8, it can be seen that the m· plane force Nx and the bending moment Mx are stat!cally equivalent to forces NxB and NxTapplied at the centrOIds of the bottom and top outer shells respectively. The values of the latter forces are Mx + Nx (h/2 - cr) = h,-CB-Cr - Mx + Nx (h/2 - CB) NxT = h - CB - cr Similarly the other stress resultants are =_M...l.y-;+,.....N-'y~(~hl_2_-_c.:..<T) NYB h. - CB- CT _ - My + Ny (h/2 - CB) NyT - h - CB- CT N _ Mxy + Nxy (h/2 - cr) xyB - h-CB-CT (5.24) (5.25) (5.26) (5.27) (5.28) N _ - Mxy + Nxy (h/2 - CB) (5.29) xyT - h-CB-CT Equations (A17) to (A30) can be used to design re- inforcement, in the bottom, to resist NxB, NyB, NxyB and, in the top, to. resist NxT• Nyr, Nxyr' It should be noted that the core of the sandwich is assumed to make no contribution to the strength of the sec- tion. However, Morley and Gulvanessian [l~] have extended the method' to include the possibility of the core contributing to the strength. Prestressed concrete slabs The Code states that prestressed concrete slabs should be designed in accordance with the clauses for prestressed concrete beams. In addition, 'due allowance should be made in the distribution of prestress in the case of skew slabs'. The latter point is intended to emphasise the fact that, when a skew slab is prestressed longitudinally, some of the longitudinal bending component of the prestress is distributed in the form of transverse bending and twisting moments. The result is that the prestress is less than that calculated on the basis of a simple beam strip in the direc- Ultimate limit state -flexure and in-plane forces Equivalent sandwich 300 1200 1080 N"r c;f cJ N"s Cd Cd In-situ slab Formwork Precast M-beams at 1.5 m centres. Each beam has 31 No. 15.2 mm low relaxation strands 200 t15 125 60 +++ --.: + + + + 6Oy- Jl10~______________~______- J Fig. 5.9 Ellample 5.1 tion of the prestress. Clark and West [146] have given guid- ance on the resultant prestress to be expected in skew slab bridges. Regarding section design for prestressed slabs, it is difficult to imagine how the beam clauses can be applied to a general case. The author would suggest that the prestress should be considered as an applied load at the ultimate limit state, and a set of bending moments and in-plane forces, due to the prestress, calculated and added algebraic- ally to those due to the applied loads. Conventional rein- forcement could then be designed to resist the resulting I 'out-of-balance' stress resultants by using the equations given in the Appendix. Clark and West have designed, and successfully tested, model skew solid [146] and voided [147] slab bridges by such an approach. Examples '*5.1 Prestressed beam section strength It is required to calculate the ultimate moment of resistance of the pre-tensioned composite section shown in Fig. 5.9. The initial prestress is 70% of the characteristic strength and the' losses amount to 30%. The precast and in-situ concretes are of grades 50 and 40 respectively. From Table 21 of the Code 61
- 38. N 1600 ~ 1400141~}~!!!'!!':' ________~______ z ell Xl 1200...ci) 1000 800 600 400 11l~~!!,!!,2 I I I I I I I I I I I I 200kN/mm2 I 1°.00669 I 6 10 10.0121,g 20 Strain >< 103 Fig. S.10 Design stress-strain curve for IS.2 mm low relaxation strand A/J,y =138.7 mm2 A"./PII =227.0 kN .~. !,JlI =227.0 X 103 /138.7 =1637 N/mm2 Effective prestress =(0.7) (0.7) (1637) =802 N/mml :, prestrain in tendons = 802/200 X 103 = 4.01 x 10 -3 The design stress-strain curve for the tendons at the ulti- mate limit state is given in Fig, 5.10. Strain compatibility approach rU, By t1)a and error the neutral axis depth has been found to ~I/ > be ~mm. The strain distribution is thus as shown in Fig. 5.11, where the total strains at the tendon levels and the tendon stresses are £, = -(0.0035 x 49/339) + (4.01 x 10-3 ) = 0.0035 fl = (0.0035) (200 x 103 ) = 700 N/mml £2 = (0.0035 x 921/339) + (4.01 x 10-3 ) = 0.0135 f2 = 1424 N/mm2 It £1 = (0.0035 x 9711339) + (4.01 x IO-a) = 0.0140 ,(1 = 1424 N/mm2 The tensile forces in the tendons are TI = (2) (138.7) (700 x 10-3) = 194 T2 = (14) (138.7) (1424 x 10-3 ) = 2765 T3 = (15) (138.7) (1424 x 10-3 ) = 2963 IT = 5922 kN The compressive forces in the concrete are calculated for zones 1 to 4 of Fig. 5.11 (the effective breadth is the actual breadth since the distance between points of zero moment would be at least 20 m). CI = (1200 x 200) (0.4 x 40)10-3 = 3840 C2 = (300 x 170) (0.4 x 40)10-3 = 816 C3 = (300 x 45) (0.4 x 50)10-3 = 270 C4 = (400 x 124) (0.4 x 50)10-3 = 992 IC = 5918 kN 921 971 ® ® 14 1600 ,14 300 "I I J_.tt J 14 400 ~ Fig. S.U Strain distribution IT = IC, thus take moments about neutral axis Tl (194) (-49)10-3 = -10 Tz (2765) (921)10-3 = 2547 Ta (2963)(971)10-3 = 2877 Cl (3840) (239)10-3 = 918 •Cz (816) (254)10-3 = 207 C3 (270)(146.5)10-3 =", 40 C4 (992) (62)10-3 = 62 1: ='6641 kN m Code table approach 170 46 124 21 Centroid of tendons in tension zone is at d from top of slab, where d = (14 x 1260 + 15 x 1310)/29 = 1286 mm For equilibrium, and ignoring Tt T2 .... T3 =Cl + C2 + C3 + C4 or Aps/pb= (3840 + 816 + 270)103 + (0.4 x 50) (400) (x - 215) where x is the unknown neutral axis depth and Aps = 29 x 138.7 =4022.3 mm2 .. fpb =797 + 1.99x .. (0.87/PIl) (fpbI0.87/PII) =797 + 1.99(xld)d .. 1424 (j~bI0.87/p,.) =797 + 1.99(xld) 1286 .. /pJO.87/plI =0.560+1.80(xld) In Fig. 5.12, the latter expression is plotted together with the Code Table 29 values. It can be seen that the intersec- tion occurs at/pJO.87/pu = 1.0 andxld = 0.244. Hence/ph = 1424 N/mm2 and x =314 mm. , , 1.1fpb '. 'o-an. pu Section equation Code Table 29 . 0.80!-·~.,r.,..J~b~""'-""-';,O:l:.~4iC!"-,~O."E'5':""".,~O;l;.6~-;;0~.7~'~O;8 ,',; ',' xld Fig. 5.12 Graphical solution T2, T3, Cl, C2 and C3 are as calculated for the strain com- patibility approach and C. = (400 x 99) (0.4 X 50,)10-3 =792kN. . " . Moments.about the neutral axis T2 (2765) (946)10-3 = 2616 Ta (2963) (996)10-3 ,.;. 2951 CI (3840) (214)10-3 = 822 C2 (816) (229)10-3 = 187 C3 (270) (121.5)10-3 = 33 c. (792) (49.5)10-3 = 39 1: = 6648 kN m This value is within 0.1% of the value calculated using strain compatibility. *'5.2 Slab The design applied moment triad, at the ultimate Jimit state, in the obtuse comer of a reinforced concrete skew slab bridge is (with the axes shown in Fig..5.13) Mx = -2.484 MNmlm M),= 1.139 MNmlm Mxy = .c..0.900 MNmlm Obtain the requited moments Of resistance in the re- inforcement directions, if the latter are (a) parallel and per- pendicularto the abltments and (b) parallel to the slab edges. In the following the equation ntimbers are those of Appendix A. Orthogonal reinforcement Bottom.reinforcement From equationA 1 M"'x:::;: -2.484+ 1-0.91 = ...,1.584 M'"x < 0, :. M*Jt = oand calculate M*y from equation (A3) M*y = 1.139 +1(-0.9)2/(-:-2,484)1 =1.465 MNmlm Top reinforcement From equation (AS) Ultimate limit state -flexure and in-plane forces ~.: a Y, ... ' ',. Fig. 5.13 Ske~ slab axes M'I':c= ...,2.484-1-0.91 =-3,.384 From equation (A6) M"'y = 1.139-1-0.91 =0.239 M'"Jt > 0, :. M"'). = 0 and calculate M*x from equation (AS) M*x =-2.484-1(0.9)2/1.1391 =-3.195 MNlfi/m Skew reinforcement From Fig. 5.13 it can be seen that ~ =i35° Bottom reinforcement From equation (A9) M"':c =-2.484 + 2(-0.9) (-I) + 1.139(-1)2 + =3.336 MNmlm From equation (AIO) -0.9 + 1.139(-1) 1 l//'i Mol< =1..:n2. + -0.9 + 1.13?(-l) " (1//2)2 11/2 =5.159 MNmlm Top reinforcement From equation (AI3) M"':c = -2.484 + 2(-0.9) (-1) + 1.139 (_1)2_ =,-2.427 MNmlm From equation (AI"4) 1 -0.9 + Ll39(...,1) llji '. M'" - 1.139 -0.9 + 1.139hl) I" - (1//2)2 - 11/2 = -0.605 MNmlm It should be noted that reinforcement is required in each direction in both the top and bottom of the slab Vl/hen skew reinforcement is used. However, reinforcement is required only transversely; in the bottom, and longitudinally. ill. the top of the slab, when orthogonal reinforcement is used. 5.3 Box ,girder wall A wall of a box girder, 250 mm thick, and with the cen-.- troid of the reinforcement in each face at a distance of 63
- 39. 60 mm fromthe face, is subjected to the following design stress resultants at the ultimate limit state N.. = - 240 kN/m N,. = 600 kN/m N:.y = 340 kN/m Mx = -166.0 kNmlm My = 24.0 kNmim Mxy = 1.6 kNm/m Design reinforcem~nt in the x and y directions if fy = 250 N/mm2andleu = 50 N/mm2 The design strengths are Reinforcement = 250/1.15 = 217 N/mm2 Concrete = 0.4 X 50 = 20 N/mm 2 In equations (5.24) to (5.29) CT = CB = 60mm hl2 - CT =hl2 - Cs =65 mm h-CB-CT= 130 mm The statically e.quivalent stress resultants in the outer shells of the sandwich plate of Fig. 5.8 are calculated, from equ- ations (5.24) to (5.29), as, in N/mm units .. N.tn = -1397 Nyn = 485 N.t .l •n = 182 Bottom reinforcement From equation (AI7) 64 NxT = 1157 NyT = 115 NxyT = 158 N*xB =-1397 + 11821 =-1215 N*xS < 0, :. N*xS =0 and calculate N*yB from equation (A20) N*yB = 485 + 1(182)2/(-1397)1 = 509 N/mm AYB = (509/217)103 = 2340 mm2/m From equation (A21) . -4B= -1397 + (182)2/(-1397) =-1421 N/mm Bottom.concrete compressive stress = 14211120 =11.8 N/mm2 This is less than the design stress of 20 N/mm2 Top reinforcement From equation (A17) N*xT = 1157 + 158 = 1315 N/mm AxT =(1315/217)103 = 6060 mm2 /m From equation (A18) N*yT = 115 + 1158 = 273 N/mm Ay7' = (273/217)103 = 1260 mm2 /m From equation (A19) FcT = -21581 = -316 N/mm Top concrete compressive stress = 316/120 = 2.6 N/mm2 This is less than the design stress of 20 N/mm2 • ..Chapter 6 Ultimate limit state - shear and torsion :*'Introduction In this chapter, the Code methods for designing against shear and torsion for reinforced and prestressed concrete construction are discussed. The particular problems which arise in composite construction are not dealt with in this chapter but are presented in Chapter 8. It should be noted that, in accordance with the Code, all shear and torsion calculations, with the exception of inter- face shearin composite construction, are carried out at the ultimate limit state. With regard to shear, calculations have to be carried out, as at present, for both flexural shear and, where. appropriate, punching shear. However, it should be noted that BE 1173 requires shear calculations for reinforced concrete to be carried out under working load conditions as opposed to the ultimate limit state as required by the Code. In addition, as explained later, the procedures for design- ing shear reinforcement in beams differ between BE 1173 and the Code. The design of prestressed concrete to resist shear is car- ried out at the ultimate limit state in accordance with both BE 2173 and the Code, and the calculation procedures for each are very similar. Design against torsion is notcovered in th~:Department of Transport's current design documents and, in practice, either CP 110 or the Australian Code of Practice is often referred to for design guidance. The latter document is written in terms of permissible stresses and working loads, and thus differs from the Code approach of designing at the ultimate limit state. Shear in reinforced concrete Flexural shear Background The design' rules for flexural shear in beams are based upon the work of the Shear Study Group of the Institution of Structural Engineers [148]. The background to the rules, which are identical to those of CP 110, has been. described by Baker, Yu and Regan [149] and by Regan [150]. The general approach adopted by the Shear Study Group was, first, to study test data from beams without shear reinforceme.nt and, then, to study test data from beams with shear reinforcement. . *Beams without shear reinforcement The data from beams without shear reinforcement indicate that, for a constant concrete strength and longitudinal steel percentage, the relationship between the ratio of the observed bending moment at collapse (Me) to the calcu- lated ultimate flexural moment (Mil) and the ratio of shear span (a,.) to effective depth (d) is of the form shown in Fig. 6.1(a). ' This diagram has four distinct regions within each of which a different mode of failure occurs: region t, corbel action or crushing of a compression strut which runs from the load to the support: region 2. diagonal tension causing splitting along a line joining the load to the support; region 3, a flexural crack develops into a shear failure crack; region 4. flexure. These modes are illustrated in Fig. 6. t (b-e). From the design point of view, it is obviously safe to propose a design method which results in the observed bending moment at collapse (Me) always exceeding the lesser of the calculated ultimate flexural moment and the moment when the section attains its calculated ultimate shear capacity. Such an approach can be developed as fol- lows: if the shear force at failure of a point loaded beam is V,., then Now, for a particular concrete strength and steel percen- tage, the ultimate flexural moment is given by Mil = Kbd2 where K is a constant. Thus 65
- 40. 1 /crushing 3 4 avId (a) Generalrelationship (b) Region 1 (e) Region 2 (d) Region3 (e) Region4 Flexural failure Fig. 6.1(a)-(e) Shear failure modes in reinforced concrete beams Hence, ~ =(~d)(~") (6.1) It is thus convenient to replot the test data in the form of a graph of (Mjbcf2) against (ajd) as shown by the solid l~e of Fig. 6.2. The dashed line is that calculated assummg that flexural failures always occur (i.e. Mjbd2 ), and the chain dotted line is a line that cuts off the unsafe side of the graph (those beams which fail in shear) ~d. can thus be considered to be an 'allowable shear lme. The significance of equation (6.1) can now be seen because the term Vjbd is the slope of any line which passes through the origin. Hence if Vjbd is chosen to be the slope of the chain dotted line, tlien Vjbd can be considered to de~ne an 'allowable shear' line which separates an unsafe region to its left from a safe region to its right. Furthermore Vjbd can be considered to be a nominal allowable shear stress (ve) which acts over a nominal shear area (bd). It is emphasised that, in reality, a constant shear s~ress does not act over such an area, but it is merely conveOlent to choose a shear area of (bd) and to then select values of vc such that the allowable values of the moment to cause collapse fall below the test values. 66 Flexure ... 2 Fig. 6.2 Shear design diagram Table 6.1 Design shear stresses (vc N/mm2 ) 100 A. Concrete grade bd 20 25 30 :;;.40 :e; 0.25 0.35 0.35 0.35 0.35 0.50 0.45 0.50 0.55 0.55 1.00 0.60 0.65' 0.70 0.75 2.00 0,80 0.85 0.90 0.95 j;!!: 3.00 0.85 0.90 0.95 1.00 The code contains a table w.hich gives values of Vc for various concrete grades and longitudinal steel percentages. Reference to the table (see Table 6.1) shows that vc is only slightly affected by the concrete strength but is greatly dependent upon the area of the longitudinal steel.. This is because the latter contributes to the shear capacity of a section in the following two ways: 1. Directly, by dowel action [151] which can contribute 15 to 25% of the total shear capacity [152]. 2. Indirectly, by controlling crack widths which, in tum, influence the amount of shear force which can be transferred by the 'interlock of aggregate particles across cracks. Aggregate interlock can contribute 33 to 50% of the total shear' capacity [152]. It should be noted that when using Table 6.1, the lon- gitudinal steel area to be used is that which extends at least an effective depth beyond the section under consideration. The reason for this' is given later in this chapter. It is not necessary to apply a material partial safety fac- tor to the tabulated vc values because they incorporate a partial safety factor of 1.25 and are thus design values. In fact the Vc values can be considered to be, by adopting the terminology of Chapter 1, design resistances obtained from equation (1.8). An appropriate Ym value would lie between. the steel value of 1.15 and the concrete value of 1.5 because the shear resistance of a section is dependent upon both materials. It was decided that a value of 1.25 was reasonable for shear resistance when compared with the usual value of 1.15 for flexural resistance. '#:.Seams with shear reinforcement When the nominal shear stress exceeds the appropriate tabulated value of Vc it is necessary to provide shear rei~ forcement to res~st the shear force in excess of (vcbd). ThiS approach differs to that of BE 1/73 in which shear rein- Stirrup .stress I 1 1 1 1 1 . 1 1 1 c-/~I ;SI ~I 2./ ~1. '111 rfl ~I (jl 1 1 1 1 Carried :ly concrete /+----- -..-...... --_. ~'--"'-"'C"" Fig. (;.3 Influence of shear reinforcement forcement has to be designed to resist the entire shear force when the BE 1173 allowable concrete shear stress is exceeded, and two-thirds of the she.ar force when the allow- able concrete shear stress is not exceeded. The justification for designing reinforcement to resist only th~ excess sh~ar force is that tests indicate that the stresses 10 shear rem- forcement are extremely small until shear cracks occur, after which, the stresses gradually increase as shown in Fig. 6.3, Hence, the shear resistance of the shear rein- forcement is additive to that of the section without shear reinforcement (i.e. v"bd). This was confirmed by the Shear Study Group who, for vertical stirrups, obtained a good lower bound fit to test data in the form: (6.2) where i' = V/bd and V is the shear force at failures; h'v is the characteristic strength of the shear reinforcement; and Asi' and s" are the area and spacing of the shear reinforce- ment. A theoretical expression for A,,,. can be derived by con- sidering the truss analogy shown in Fig. 6.4(a). Theoreti- cally, the inclination (8) of the compression struts, can be assumed to take any value, provided that s~ear remforce~ ment is designed in accordance with the chosen value of e. However, the greater the difference between eand the inclination of the elastic principal stress (4~0) when a shear crack first forms, the greater is the implied amount of stress redistribution between initial cracking and collapse. In order to minimise the stress redistribution,e is chosen in the Code to coincide with the inclination of the elastic principal stress (45°). For vertical equilibrium along section A-A and assum- ing that only the excess shear force is resisted by the shear reinforcement: (v - v,.)bd =A,v,.fy>, (sin ex + cos ex) (d - d')lsj; (6.3) where ex is the inclination of the shear reinforcement (stir- rups or bent-up bars). The Code assumes t~at d - d'=' d, hence equation (6.3) can be rearranged to give II = Vc + fyl' (sin ex + cos ex) (A,.vlbs,,) (6.4) Top steel A Stirrups ... Compression" - strut ~~;z:=~~===:+:::::::;~~. B (a) Stirrups i Bent-up bar . '. f VVn1: 45" (b) Bent-up bars Fig. 6.4(a),(b) Truss analogy for shear d' , L ...[ 1. d.' •• "7" For the case of vertical stirrups, ex = 90° and equation (6.4) becomes v = Ve +fyv (A",.Ibsv) (6.5) It can be seen that equations (6.5) and (6.2) which have been obtained theoretically and from test data respectively are in good agreement. For' design purposes it is necessary to apply a material partial safety factor of 1.15 (the value for reinforcement at the ultimate limit state) to !VI'; equa- tions (6.4) and (6.5) can then be rearranged as: Asv b(v - lie) s;:- =O.87fYl" (sin ex +cos ex) (6.6) Asv _ b(v - v~) s:: -- 0.87(v" (6.7) The latter equation appears in the Code, with 1,'1' restricted to a maximum value of 425 N/mm 2 • This is because the data considered by the Shear Study Group indicated that the yield stress of shear reinforcement should not exceed about 480 N/mm2 in order that it could be guaranteed that the shear reinforcement would yield at collapse prior to . crushing of the concrete. The Code value of 425 N/mm2 is thus conservative. It is implied, in the above derivation, that equation (6.6) can be applied to either inclined stirrups or bent-up bars. Although this is theoretically correct, the Code states that when using bent-up bars the truss analogy of Fig. 6.4(b) should be used in which the compression struts join the centres of the bends of the lower and upper bars. This approach is identical to that of CP 114 and it is not clear why, originally, the CP 110 committee and, sub- sequently, the Code committee retained it. It is worth men- tioning that Pederson [153] has demonstrated the validity of considering the compression struts to be at an angle other than that of Fig; 6.4(b). . In view of the limited amount of test data obtained from beams with bent-up bars used as. shear reinforcement and because of the risk of the concrete being crushed at the bends, the Code permits only 50% of the shear reinforce- ment to be in the form of bent-up bars. Finally, an examination of Fig. 6.4(a) shows that if the shear strength is checked at section A- B, then the assumed shear failure plane intersects the longitudinal 67
- 41. tension reinforcement ata distance equal to the effective depth from the section A-B. Hence the requirement men- .tioned previously that the value of A.• in Table 6.1 should be the area of the longitudinal steel which extends at least an effective depth beyond the section under consideration.. This argument is not strictly correct. but instead. as shown in Chapter 10 when discussing bar curtailment. the area of steel should be considered at a distance of half of the lever arm beyond the section under consideration. Thus the Code requirement is conservative. Maximum shear stress (vu) It is shown in the last section that the shear capacity of a reinforced concrete beam can be increased by increasing the amount of shear reinforcement. However. eventually a point is reached when the shear capacity is no longer increased by adding more shear reinforcement because the beam is then over-reinforced in shear. Such a beam fails in shear by crushing of the concrete compression struts of the truss before the shear reinforcement.yields in tension. The Code thus gives. in a table. a maximum nominal flexural shear stress of 0.75 /h.,,(but not greater than 4.75 N/mm2 ) which is a design value and incorporates a partial safety factor of 1.5 applit?d to /,.,,: hence the effective partial safety factor applied to the nominal stress is M. Clarke and Taylor r154] have considered data from beams which failed in shear by cr.ushing of the web concrete. They found that the ratios of the experimental nominal shear stress to that given by 0.75 If,." were in the range 1.02 to 3.32 with a mean value of 1.90. The upper limit of 4.75 N/mm 2 imposed by the Code is to allow for the fact that shear cracks in beams of very high strength concrete can occur through. rather than around. the aggregate particles. Hence a smooth crack sur- face can result across which less shear can be transferred in the fornl of aggregate interlock [152J. Short shear spans An examination of Fig. 6.2 reveals that for short shear spans (a..ld less than approximately. 2) the shear strength increases with a decrease in the shear span. Hence, the allowahle nominal shear stresses (vJ are very conservative for short shear spans. In view of this. an enhanced value of I', which is given by 1',(2dla,.) is adopted for a..ld less than 2. However. the enhanced stress should not exceed the maximum allowahle nominal shear stress of O.7S!lc.". The enhanced stress has been shown to be conservative when compared with data from tests on beams loaded close to supports and on corbels [1121. "fMinimum shear reinforcement If the nominal shear stress is less than 0.5 lI,.• the factor of safety against shear cracking occurring is greater than twice that against flexural failure occurring. This level of safety is· considered to be adequate and the provision of shear reinforcement is not necessary in such situations. If the nominal shear stress exceeds O.S VC bu( is less than v,. it is necessary h.J pr'bvide a minimum.amount of shear I'c·inforcement. In order that the presence of shear rein- 68 forcement may enhance the strength of a member. it is. necessary that it should raise the shear capacity above the shear cracking load. The Shear Study Group originally suggested. from considerations of the available test data. a minimum value of 0.87 Iv,. A....Ibs•. equal to 601b/in2 (0.414 N/mm2 ) in order to ensure that the shear reinforce- ment would increase the shear capacity. Hence for IV!' = 250 N/mm2 (mild steel) and 425 N/mm2 (the greatest'per- mitted in the Code for high yield steel for shear reinforce- ment). A....Is,. = 0.0019b and 0.00112b respectively: these values have been rounded up to 0.002b and O.OOI2b in the Code so that each is equivalent to 0.87/y,' A...Ibs,. = 0.44 N/mm 2 • The value of 0.0012b for high yield steel is also the minimum value given in CP 114. It is also necessary to specify a maximum spacing of stirrups in order to ensure that the shear failure plane can. not form between two adjacent stirrups, in which case the stirrups would not contribute to the shear strength. Figure 6.4(a) shows that the spacing should not exceed [(d-d')(l + cot ex)]. This expression has a minimum value of (d - d') when ex =90~ andt to simplify the Code clause. it is further assumed that (d - d')'::::!0.75d. This spacing was also shown. experimentally. to be conservative by plotting shear strength against the ratio of stirrup spacing to .effective depth for vario.us test data. It was observed that the test data exhibited a reduction in shear strength for a ratio greater than about LO lI49]. Finally. it is necessary for the stirrups to enclose all the tension reinforcement because the latter contributes. in the form of dowel action. about 15 to 25% of the total shear strength [151. 152]. If the tension reinforcement is not supported by being enclosed by stirrups. then the dowel action tears away the concrete cover to the reinforcement and the contribution of dowel action to the shear strength . is lost because the rei~forcement can then no longer act as a dowel. . '*Shear at points of c;ntraflexure A problem arises near to points of contraflexure of beams because the value of Vc to be adopted is dependent upon the area of the longitudinal reinforcement. It is thus neces- sary to consider whether the design shear force is accom- panie~ by a sagging or hogging moment in order to deter- mine the appropriate area of longitudinal reinforcement. A situation can arise in which. for example, the area of top steel is less than the area of bottom steel and the maxi- mum shear force (V,) associated with a sagging moment exceeds the maximum shear force (V,,) associated with a hogging moment. However. because the area of top steel is less than the area of bottom steel, the value of v,. to be considered with V" could be less than that to be considered with Vs' Thus although V. is greater than V". it could be the latter which results in the greater amount of shear rein- forcement. It can thus be seen that it is always necessary to consider the maximum shear force associated with II sag- ging moment and the maximum shear force associated with a hogging moment. A conservative alternative pro- cedure, which would reduce the number of calculations. would be to tonsider only the absol~te maxim~111 shear forte and to usc II value ofv(' approp~iate to the lesser of the top or bottom steel areas. If either ofthese procedures is adopted then, generally, more shear. reinforcement is required than when the calcu- 'lations are carried out .inaccordanc:e with BE 1173, in which the allowable shear stress is not dependent upon the" . area of the longitudinal reinforcement. The increase in: ·shear reinforcement in regions of contraflexure was the subject of criticism during the drafting stages of the Code and in order to mitigate the situation an empirical design rule, which takes account of the minimum area of shcar reinforcement which has to be provided. has been included in the Code. The design rule implies that, for the situation described above. shear reinforcement should be designed to resist (a) V, with a v" value appropriate to the bottom reinforcement and (b) the lesser of Vhand 0.8 v,. with <I v,. value appro- priateto the top reinforcement. The greater area of shear reinforcement calculated from (a) and (b) should then be provided. The rule should be interpreted in a similar man- ner for other relative values of Vs' V" and of bottom and top reinforcement. :rile logic behind the above rule is not clear. Further- more. it was based upon a limited number of trial calcu- lations which indicated that it was conservative. However, it can be shown to be unconservative in some circuOlstunces. In view of this. the author would suggest that it would be safer not to adopt the rule in practice. Slabs General The design procedure for slabs is essentially identical to that for beams and was originally proposed for building slabs designed in accordance with CP lIO. The implications of this are now discussed. v,. values The values of v,. in Table 6.1 were derived from the results of tests on. mainly, beams. although some one-way spanning slabs with no shear reinforcement were also considered. The slabs had breadth to depth ratios of about 2.5 to 4 and thus were. essentially. wide beams, It is probably reasonable to apply the Vc values tobuild- ing slabs because the design loading is. essentially, uniform and the design procedure generally involves con- sidering one-way bending in orthogonal directions parallel to. the flexural reinforcement. Hence it is reasonable to consider slabs as wide beams. However, it is not clear whether the same' Vc values can be applied to slabs, in more general circumstances. when the support conditions and/or the loading are non-uniform. An additional problem occurs when the flexural reinforcement is not perpendicu- lar to the planes of the principal shear forces because it is not then obvious what area of reinforcement should be used in Table 6.1. Although. strictly. this situation also arises in building slabs. it is ignored for design purposes. It is not certain whether it can also be ignored in bridge slabs, where large principal shear forces can act at large angles to the flexural reinforcement directions. It should also be noted that shear forces in bridge slabs can vary rapidly across their widths. A decision then has to be made as to whether to· design against the peak shear force or a value averaged over a certain width. None of the above problems is considered in the Code. Ultlmate limit state - shear and torsion A possible 'engineering solution' would .be todesigI1 against shear forces averaged over a width of shib. equal to twice the effective depth, and to carry out the sheardesign calculation for the shear forces acting on planes normal to each flexural reinforcement direction. The latter suggestion of considering beam strips in each of the flexural r~in forcement directions can be shown to be. in general. unsafe. This is because it is the stiffness. rather thari the strength, of the flexural reinforcement which is of impor- tance in terms of shear resistaIlce. The flexural reinforce- ,.Inent should be resolved into a direction perpendicular to the plane of the critical shear crack, and it is explained in Chapter 7 that. when considel'ing stiffness•.reinforcement areas resolve in accordance withcos4 cx, where ex is the orientation of the reinforcement to the perpendicular to the critical crack. Thus the resolved arca and, hence, the appropriate Vc' value could be much less than the values appropriate to the steel directions. However, in those re- gions of slabs where a flexural shear failure could possibly occur, such as near to free edges, the suggested approach should be reasonable. Urifortunately, there is no experi- mental evidence to justify the approaches suggested above. 1(::.Enhallced v" values The basic vc values of Table 6_J m~y_ be enhanced by multipl~ a tabulated factor. (s. > 1)' which increases as tbe ov~ralJ dep-th~cre~~~.LpLQY..W..e.~L·· that the overaltde,pth isj~.§.~!ban 300 mm, The reason for this is that tests have shown that the shear strength of a member increases as its depth decreases. Relevant test data have been collated byTaylor [155] and are summarised in Fig. 6.S in terms of the shear strength (Vu) divided by the shear strength of an equivalent specimen of 250 mm depth (V250); due allowance has been made for dead load. shear forces. It should be noted that illl of the test specimens were beams. whereas the Code applies the enhancement factor to slabs. It can be seen that the Code values give a reasonable lower bound to the test data for overall depths less than 500 mm. For greater depths. Taylor observed that there is a reduction in shear strength for large beams but that the minimum stirrups required for beams should take care of this. However. the Code does not require minimum stir- rups to be provided for slabs unless more than 1% of com- pression steel is present: this did nO.t cause a problem in the drafting of CP 110 because building slabs are generally thin; however, bridge slabs can be thick. Furthermore. Taylor suggested that code allowable shear stresses should be reduced by 40% if the depth to breadth ratio of a beam exce,eds 4. Such a ratio could be exceeded in bridge beams and the webs of box girders. These points are raised here to emphasise that the values of vc and s.. were derived with buildings in mind. It is not possible at present to state whether the values are appropriate to bridges beduse of the lack of data from tests on slabs subjected to the stress conditions which occur in bridges. *Shear reinforcement When the noininal shear stress exceeds ;, Vc> shear reinforcement should be provided and designed, as for beams, to resist the shear force in excess of that which can be resisted by the concrete (;s vcd per unit length). The required amount of shear reinforcement 69
- 42. 1.5 •• •~ •V250 1.0 • •• • • •0.5 Test Code o 250 500 750 Fig. 6.5 Slab enhancement factor should be calculated from equations (6.6) o.r (6.7), as appropriate, with vc' replaced by ~ vc This ~esign approach is probably reasonable for building slabs for the reasons discussed previously: however, it is not clear whether it is reasonable for slabs subjected to loadil).gs which cause principal shears which are not aligned with the flexural reinforcement. A possible design approach for such situations is suggested earlier in this chapter. *'Maximum shear stress The maximum nominal shear stress in slabs is limited to 0.375 !!cu, which is half of that for beams. It is understood that this was originally sug- gested by the CP 110 committee because it was felt that, in building slabs, the anchorage of stirrups could not be relied upon. The author would suggest that, if this is cor- rect, it would also be the case for top !?labs of bridge decks but not necessarily for deeper slab bridges. Furthermore, it is considered that shear reinforcement cannot be detailed and placed correctly in slabs less than 200 mm thick, and shear reinforcement is consequently considered to be ineffective in such slabs. Hence, the maxi- mum nominal shear stress in a slab less than 200 mrn thick is limited to 1;" Vc and not to 0.375 !feu. Minimum shear reinforcement . Unlike beams, it is not necessary to provide minimum shear reinforcement if the nominal shear stress is less than l;, vc' It is understood that this decision was rriade by the CP 110 committee because . it was considered t~ be in accordance with nonnal practice for building slabs. It would also appear to be reasonable for bridge slabs. The maximum stirrup spacing for slabs is the effective depth. This is greater than the maximum spacing of 0.75d for beams because the latter value was considered to be too restrictive for building slabs. The test data referred to when discussing the 0.75d value for beams suggest that a spacing of d should be adequate for both beams and slabs. 70 • 1000 • 1250 1500 Overall depth mm -¥vOided or cellular slabs No specific rules are given in the Code for designing voided or cellular slabs to resist shear. However, when considering longitudinal shear, it is reasonable to apply the solid slab clauses and to consider the shear force to be resisted by the minimum web thick- ness. With regard to transverse shear, designers, at pres- ent, either arrange the voids so that they are at points of low transverse shear force or use their own design rules. Possible approaches to the design of cellular and voided slabs to resist transverse shear forces are given in Appen- dix B of this book. Punching shear Introduction Prior to discussing the Code clauses for punching shear, it should be stated that most codes of practice approach the problem of designing against punching shear failure by considering a specified allowable shear stress acting over a specified surface at a specified distance from the load. It is emphasised that the specified surfaces do not coincide with the failure surfaces which occur in tests, This fact can cause problems when code clauses are applied in circum- stances different to those envisaged by those originally responsible for writing the clauses. The Code clauses are based very much on those in CP 110, which were written with building slabs in mind, and these clauses are noJ. summarised. CP 110 clauses Punching shear in CP 110 is considered under two separate headings: namely, 'Shear stresses in solid slabs under con- centrated loadings' and 'Shear in flat slabs'. The former clauses are concerned with the punching of applied loads through a slab, whereas the latter are concerned with punching at columns acting monolithically with a slab. Critical Support (a) Elevation (b) Plan Fig. 6.6(a),(b) Punching shear perimeter The critical shear perimeter for both situations is given as 1.5 times the slab overall depth from the face of the load or column as shown in Fig. 6.6, and the area of con- crete, deemed to be providing shear resistance, is the length of the perimeter multiplied by the slab effective depth. A constant allowable design shear stress is assumed to act over this area. The perimeter was chosen so that the allowable design shear stress could be taken to be the val- ues of Vc given in Table 6.1. Hence, the perimeter was chosen so that the same Vc values could be used for both flexural and punching shear. However, in the original ver- sion of CP 110 it was stated that, for flat slabs having lateral stability and with adjacent spans differing by less than 25%, the tabulated values of Ve should be reduced by 20%. This was because the design approach was to take the design shear force to be that acting when all panels adjacent to the column were loaded and, thus, it was necessary to make an allowance for the non-symmetrical shear distribution which would occur if patterned loading were considered. The reduction of Ve by 20% was thus intended to allow for patterned loading [112]. Subse- quently, in 1976, the flat slab clause was amended so that, at present, vc is not reduced but the design shear force is increased by 25% to allow for the possible non- symmetrical shear distribution. If the slab does not have lateral stability or if the adjacent spans are appreciably dif- ferent, it has always been necessary to calculate the moment (M) transmitted by the slab and to increase the design shear force (V) by the factor (1 + 12.5 MIVl), where I is the longer of the two spans in the direction in which bending is being considered. Regan [156] has shown, by comparing with test data, that the original CP 110 clauses were reasonable. It should be noted that most of the tests were carried out on simply supported square slabs under a concentrated load and there are very few data for slabs loaded with a concentrated load near to a concentrated reaction, as occurs near to a bridge pier. Regan quotes only three tests of such a nature and reports satisfactory prediction, by the CP 110 clauses, of the ultimate strength. It is generally the case that the flexural reinforcement in the vicinity of a concentrated load or a column head is different in the two directions of the reinforcement. The Load Actual failure ~ ~. ,mfa" zZfl3", " }tI I == O.5h -.I l--- .1 . I I Reinforcement 1== 1.5h I I~ Fig. 6.7 Failure surface Alternative failure ZllllmlllrS:~ f I~Shear reinforcement! Fig. 6.8 Influence of shear reinforcement question then arises as to what area of reinforcement to adopt for determining Vc from Table 6.1. CP 110 allows one to adopt the average of the reinforcement areas in the two directions and tests carried out by Nylander and Sund~ quist [157] in which the ratio of the -steel areas varied from 1.0 to 4.1 justify this approach. These tests essen- tially modelled a pair of columns with line loads on each side as could occur for a bridge. When averaging the reinforcement areas in the two directions, the area in each di:rection should include all of the reinforcement within the loaded area and within an area extending to within three times the overall slab depth on each side of the loaded area. The reason for considering the reinforcement within such a wide band is that the actual failure surface extends a large distance from the load as shown in Fig. 6.7. The validity of considering the reinforcement in a large band has been confirmed by the results of tests carried out by M'oe [158] in which the same area of reinforcement was distributed differently. It was found that the punching strength was essentially inde- pendent of the reinforcement arrangement. If the actual nominal shear stress (v) on the perimeter exceeds the allowable value of ~ ve, it is necessary to design shear reinforcement in accordance with (6.8) where CIA.v} is the total area of shear reinforcement and Ucrll is the length of the perimeter. This equation can be derived in a similar manner to equation (6.7). It can be seen from Fig. 6.7 that, in order to ensure that the shear reinforcement crosses the failure surface, it is necessary for the reinforcement to be placed at a distance of about 0.5h to 1.5h from the face of the load. In fact, CP 110 requires the shear reinforcement calculated from equation (6.8) to be placed at a distance of O.75h. However, CP 110 also requires the same amount of reinforcement to be provided at the critical perimeter distance of 1.5h; hence twice as much shear reinforcement as is theoretically required has to be provided. It appears that such a conser- vative approach was proposed because of the limited range of shear reinforcement details covered by the available test data [158]. The presence ofshear reinforcement obviously strengthens . 71
- 43. a slab inthe vicinity of the shear reinforcement and can thus cause failure to occur by the formation of shear cracks outside the zone of the shear reinforcement as shown in Fig. 6,8. CP 110 thus requires the shear strength to be checked also at distances, in steps of O.75h, beyond 1.5h and, if necessary, shear reinforcement should be provided at these distances. The minimum amount of shear reinforcement implied by equation (6.8) has to be provided only if v> ;"vc and is very similar to the amount originally proposed for beams by the Shear Study Group. . As is also the case for flexural shear in slabs, the maxi- mum nominal punching shear stress should not exceed 0.375 flu. *BS 5400 clause The clause in the Code which covers punching ,shear is identical to that in CP 110 which covers 'Shear stresses in solid slabs under concentrated loadings'. Hence, the modi- fications in CP 110 which allow for non-uniform shear distributions in flat slabs are not included in the Code. Instead. whether punching of a wheel through a deck or of a pier (integral or otherwise) is being considered the,design procedure is to adopt' the CP 110 perimeters and the.Table 6.1 values of Vc (modified by ;., if the depth permits), and to design shear reinforcement using equation (6.8). Such an approach is probably reasonable when consi~er ing wheel loads or piers which are not integral with the deck. However, when dealing with piers which are integral with the deck, and thus non-symmetry of the shear dis- tribution and moment transfer should be considered, it could be that a modification, similar to that for flat slabs in CP 110, should be made to either the Vc values or the design shear force. However, this has not been included in the Code ,and the implication is that the effects of non- symmetry and moment transfer can be ignored. Further problems, which are probably of more impor- tance to the bridge engineer than to the building engineer, are those caused by voids running parallel to the plane of a slab and by changes of section due to the accommodation of services. These problems are not considered by the Code. Some tests have been carried out by Hanson [159] on the influence on shear strength of service ducts having widths equal to the slab thickness and depths equal to 0.35 of the slab thickness. He concluded that provided the ducts were not within two slab thicknesses of the load there was no reduction in shear strength. However, it is not clear whether such a rule would apply to slabs with voids as deep as those which occur in bridge decks. ~ Shear in prestressed concrete Flexural shear Beam failure modes Two different types of shear failure can occur in pre- stressed concrete beams as shown in Fig. 6.9. 72 Web eraekiny dl2 ' ~ (a) Shear failures (b) Shearforee (V) (e) Bending moment (M) (~)1 f4dI2~ d~,-(~),-f (d) MIVdiagram Fig. 6.9(a)-(d) Shear failure' modes in prestressed concrete beams In regions uncracked in flexure, a shear failure is caused by web cracks forming when the principal tensile stress exceeds the tensile strength of the concrete. In regions cracked in flexure, a shear failure is caused either by web cracks or by a flexural crack developing into a shear fail- ure. Hence, it is necessary to check both types of failure and to take the lesser of the ultimate loads associated with the two types of failut~ as the critical load. *Sections uncr8cked}n flexure The ultimate shear strength in this condition is designated Ven, and the criterion of failure for a section with no shear reinforcement is that the principal tensile stress anywhere in the section exceeds the tensile strength of the concrete. If the principal tensile stress is taken as positive and equal to the tensile strength of the concrete, then for equilibrium Is =Veo Ay/lb and -I, = ifcp +Ib)/2 - Jifep +Ib)2/4 +1/ combining the equations gives Vco =~y III + ifcp +Ib)/, In the above equation, Veo = shear force to cause web cracking I = second moment of area b = width Ay = first moment of area Icp = compressive stress due to prestress Ib = flexural compressive stress fs shear stress I, tensile strength of concrete. In order to simplify calculations, it is assumed that the principal. tensile stress is a maximum at the bea~ centroid,,: in which case fb = 0, and, for a rectangular sectIOn, Ib/Ay :::: 0.67 bh. Hence V"n =0.67 bh /tr + [,.,,[, (6.8) The above simplification was originally introduced into the Australian Code [160]. It should be noted that it is unsafe for I-beams to consider only the centroid but this is mitigated by the fact that, for such beams, Ib/Ay :::< 0.8~h as opposed to 0.67 bh. However, for flanged oeams m which the neutral axis is within the flange, it is considered to be adequate to check the principal tensile stress at the junction of the web and flange. This simplification again originated in the Australian Code [160]. . Tests on beams of concretes made with rounded river gravels as aggregate have indicated [160] that I, = 54Yl (in Imperial units). However, the Australian Code adopted 4/fry] in. order to allo~ f~r ~trength ~eductions c.au~ed ~y shrinkage cracking, mild fatigue loadmg and vanatlons In concrete quality. If the latter value is converted to S.1. units and it is assumed that leVI = O.Slcu, then I, = 0.297 4". A partial safety factor of 1.5 was then applied to ["/1 to give the design value of 0.24 !fe" which appears in the Cbde. Since a partial safety factor of jT3 is applied to fr, partial safety factors of (/1.5)2 and jf.5 are implied in the first and second terms respectively under the square root sign of equation (6.S). In order that a partial safety factor of (/13)2 is implied for both terms, it is necessary to apply a multiplying factor of 1/jT3==0.8 to IC/I' This results in the following equation, which appears in the Code v"" = 0.67 bh I/? + 0.8 f,·,.!, (6.9) Reynolds, Clarke and Taylor [161] have compared equa· tion (6.9). without the partial safety factors, with test results and found that the ratios Of the observed shear forces causing web cracking to Veo were, with the excep- tion of one beam which had a ratio of 0.68, in the range 0.92 to 1.59 with a mean of 1.13. .w.rumjn..G.!it1.ci!~lliIOnLID:.~_lh~~-,"Jbl<,'yl<Jj.i.~l!L£"Q.l!!pQ!l~.!lt.> QfJh~. pre.§tI~~s._~hSmlgJ:!e.J!J!<!~~LtQ Yc.'L!L~Qt.!!in the::-:W}!!L shear resista,nce. However._Jb.e.Sod~jLllly__~rmits ~g%,gf the vertical comR2n~lJQ.J?.e._l!9.de<ljlLQrQer J.Q_h~L<;Q.Il§i§ tenLwith .~!!J!t.iQ.!L(§~2L *Sections cracked in flexure A shea~ failure can occur in a prestressed beam by a flex- ural crack developing into an inclined crack which even- tually causes a shear failure. The position of the critical Ilexural crack, relative to the load. varies, but it has been shown 11591 that it can be assumed to be at half the effec- tive depth from the load. . Sozen and Hawkins lJ621 considered the loads' at' which a flexure-shear crack formed in 190 tests and showed that a good lower bound to the shear force (Vcr) could be given by the following empirical equation in Imperial units V(,r = 0.6 lnl ';[;.,. + M,/(MIV - d/2) (6.10) where M, is the cracking moment and M and V are the. moment and shear force at the section under consideration. If this equation is transformed to S.I. units, it is assumed that fey] = 0.8 leu and a partial safety factor of 1.5 applied tofeu, then Ver = 0.037 bd liu + M,/(MIV - d/2) It is obviously conservative to ignore the d/2 term, in which case the following equation, which appears i11 the Code, is o~!ained. Vcr = 0.037 bd!feu + V(M/M) (6.11) It should be noted that, in equation (6.11), d is the dis- tance from the extreme compression fibre to the centroid of the tendons. If the modulus of rupture of the concrete is fro then the cracking moment is given by M, =if, +Ip,)l/y where /,)/ is the tensile stress due to prestress at an extr~me fibre, distance y from the centroid. ACI·ASCE Committee 323 [163] originally suggested that, in Imperial units,fr = 7.5 ./ley! and the Australian Code [160] subsequently reduced this to 6 !f..I' I to allow for shrinkage cracking. repeated loading and varia.tions in concrete quality. If the latter expression is converted to S.l. units, it is assumed that fcyl = 0.8 /1'11 and a partial safety factor of 1.5 is applied to fe". then the Code design "alue of 0.37 /f,." is obtained. Again, only SO% of the prestress should be taken to give a consistent partial safety factor. Hence, the fol- lowing design equation, which appears in the Code, is obtained. M, =(0.37 /1..'/1 + 0.8/p,) l/y (6.12) A minimum value of Vcr of 0.1 bd /l.u is stipulated in the Code. This value originated in the American Code as, in Imperial units, 1.7 bd .fl.y ]' The reason for this value is not apparent but if it is converted to S.l. units. it is assumed that In'1 :::: O.8fCII and a partial safety factor of 1.5 is applied to I;..;" then the Code value is obtained. The majority of the beams for which equation (6.10) was found to give a good lower bound fit had relatively high levels of prestress, with the ratio of effective prestress to tendon characteristic strength (f,,,/fpII) in excess of 0.5. These beams were thus representative of Class 1 or Class 2 beams. but not necessarily of Class 3 beams which can have much lower levels of prestress. A modified expres- sion for Vcr was thus derived for Class 3 beams which gives a linear transition from the reinforced concrete shear clauses (f,,,,//,1lI =0) to the Class 1 and Class 2 formula (equation (6.11)) when .f;,/(,,,, = 0.6. In view of the two terms of equation (6.11) it was proposed [1611 that for Class 3 members. Vcr = A + B (6.13) where A depends Oil material strength and is analogous to the shear force calculated from the I'e values of Table 6. I. and B is the shear force to f1exurally crack the beam. Both A and B are to be determined. The term, A, was written as a function of V,. and the effective prestress (f,,,) A = (I - nf"Jf"lI) v,.hd 73 .
- 44. where n isto be determined. This function was chosen for A because it reduces to the reinforced concrete equation whenfpe= O. . Equation (6.11) can be expanded to the following by using equation (6.12) _ !. V MoV (6 14) Vrr - 0.037 bd !/"u + 0.371!cu y M + M . where Mo =0.8fpl Ily is the moment to produce zero stress at the level of the steel centroid. It is thus convenient to write the term, B, of equation (6.13) as Mo VIM and Vcr becomes for reinforced and all classes of·.prestressed concrete (6.15) Forreinforcedconcrete.fp, =Mo =oand hence Vcr =vcbd, which agrees with the reinforced concrete clauses. In order that equations (6.14) and (6.15) for Classes 1 and 2 and Class 3 respectively agree for fp,lfpu =0.6. it is necessary that 0.037 bd I!c.. + 0.371!cu f~, = (1 - O.6n) vcbd A shear failure is unlikely to occur if MIV > 4h and thus it is conservative [161] to put MIV = 4h. It is further assumed that d == h, fly = bh2 /6 (the value for a rec- tangular section), feu = 50 N/mm2, Ve = 0.55 N/mm 2 (i.e. 0.5% steel) and thus n = 0.55. Hence. the fol- lowing equation, which appears in the Code. is obtained Vcr =(1...,.. 0.55fpelfplI) vcbd + MoVIM (6.16) In view of the large number of simplifications made in deriving this equation. Reynolds. Clarke and Taylor [161] compared it. with the partial safety factors removed. with observed Vcr values from 38 partially pre- stressed beams. The ratios of the experimental ultimate shear forces to V"r were. with the exception of one beam which had a ratio ·of 0.77. in the rang~ 0.97 to 1.40 with a mean of 1. 18. The exceptional beam had a cube strength of only 20 N/mm2 and a high amount of web reinforce- ment. The total area of both tensioned and untensioned steel in the tension zone should be used when assessing Ve from Table 6.1; and, in equation (6.16). d should be the dis- tance from the extreme compression fibre to the centroid of the steel in the tension zone. The total area of tension steel is used because the longitudinal steel contributes to the shear strength by acting as dowel reinforcement and by controlling crack widths. and thus indirectly influencing the amount of aggregate interlock. Thus any bonded steel can be considered. The Code also implies that unbonded tendons should be considered. but it could be argued that they should be excluded because they cannot develop dowel strength and are less effective in controlling crack widths. When both tensioned steel of area A.,(I) and characteristic strengthf,>u(/) and untensioned steel of areaA.,(u)and charac- teristic strength[Vl.(u)are present.fp,.lfpu should be taken as. by analogy. the ratio of the effective prestressingforce (PI) divided by the total ultimate force developed by both the tensioned and untensioned steels. i.e. 74 PAAs(/)fpu(t) + As(u)/yL(u»' It will be recalled that the d/2 term which appears in equation (6.10) was ignored in deriving equations (6.11) and (6.16). An examination of the bending moment and shear force diagrams of Fig. 6.9 reveals that the value of MIV at a particular section is equal to the value of (MIV-d/2) at a section distance dl2 from the particular sec- tion. It is thus reasonable to consider a value of Vcrcalcu- ,.,,,'tated from equations(6.11) and (6.16) to be applicable for a distance 0{'d72·in"'i£i?Uireation of increasing mo- ment from the particular section under consideration. Finally. contrary to the principles of statics, the Code does not permit the vertical component of the forces in inclined tendons to be added to Ver to give the total shear resistance. this requirement was based upon the results of tests on prestressed beams, with tendon drape angles of zero to 9.95°. reported by MacGregor. Sozen and Siess [164]. They concluded that the drape decreased the shear strength. However. since. except at the lowest point of a tendon. the effective depth of a draped tendon is less than that of a straight tendon. equations (6.11) and (6.16) do predict a reduction in shear strength for a draped. as com- pared with a straight. tendon. It is not clear whether the reduction in strength observed in the tests was due to the tendon inclination or the reduction in effective depth. If it is because of the latter. then the Code effectively allows for the reduction twice by adopting equations (6.11) and (6.16) and excluding the vertical component of the pre- stress. Hence. the Code, although conservative, does seem illogical in its treatment of inclined tendons. *shear reinforcement The shear force (Ve) which can be carried by the concrete alone is the lesser of V;~ and Vcr. If Vc exceeds the applied shear force (V) then. thioretically. no shear reinforcement is required. However, the Code requires nominal shear reinforcement to be provided, such that 0.87 fy"AsJbsv ;;:. 0.4 N/mm2, if V;;:. 0.5 Ve' These requirements were taken directly from the American Code. Thus shear rein- forcement need not be provided if V < 0.5 Ve' In addition the Code does not require shear reinforcement in members of minor importance nor where tests have shown that shear reinforcement is unnecessary. The CP 110 handbook [112] de'fines members of minor importance as slabs. footings. pile caps and walls. However. it is not clear whether such members should be considered to be minor in bridge situations and the interpretation of the Code . obviously involves 'engineering judgement'. If V exceeds Vc then shear reinforcement should be pro- vided in accordance with Asv V - Ve s: =0.87 [VI' d, (6.17) where dr is the distance from the extreme compression fibre to the centroid of the tendons or to any longitudinal bars placed in the comers of the links. whichever is the greater. The amount of shear reinforcement provided should exceed the minimum referred to in the last para- graph. Equation (6.17) can be derived in the same way as equation (6.7). The basic maximum link spacing is 0.75 d " which is the same as that for reinforced concrete beams. However. if V < > 1.8 Ve, the maximum spacing should be reduced to 0.5 d,: the reason forthis is not clear. In addition, for any value of V. the link spacing in flanged members should not exceed four times the web width: this requirement presum- ably follows from the ·CP 115 implication that special con- siderations should be given to beams in which the web depth to breadth ratio exceeds four. Maximum shear force ,.)/". In order to avoid premature crushing of the web concrete. it is necessary to impose aJ;l upper limit to the maximum shear force. The Code tabulates maximum design shear stresses which are derived from the same formula (0.75 ./Tcu) as those for reinforced concrete. The shear stress is considered to act over a nominal area of the web breadth. minus an allowance for ducts. times the distance from the extreme compression fibre to the centroid of aU (tensioned or untensioned) steel in the tension zone. Clarke and Taylor [154] have considered prestressed concrete beams which failed by web crushing and found that the ratios of observed web crushing stress to the Code value of 0.75 I1cIl were in the range 1.04 to 4.50 with.a ·meanof2.l3. It has been suggested by Bennett and Balasooriya [165] that beams with a web depth to breadth ratio in excess of ten could exhibit a tendency to buckle prior to crushing: such a ratio could be exceeded in a bridge. How- ever. the test data considered by Clarke and Taylor [154] induded specimens with ratios of up to 17; and Edwards [!66] has tested a prestressed box girder having webs with slenderness ratios of 33 and did not observe any instability problems. It thus appears that web instability should not be a problem in the vast majority of bridges. It is mentioned previously that a reduced web breadth, to allow for ducts, should be used when calculating shear stresses. The Code stipulates that the reduced breadth should be the actual breadth less either the duct diameter for ungrouted ducts or two-thirds the duct diameter for grouted ducts. These values were originally suggested by Leonhardt [167] and have subsequently been shown to be reasonable by tests carried out by Clarke- and Taylor [154] on prisms with ducts passing through them. It should be noted that when checking the maximum. she..ar.Jo.rccJ.D.Y-.Y-e.rticJtlj:..QIDPOOe1lLQLpr.e.s.tm£SJbould be___ considered o!'!l:x_fqL~~Jj!lll~Jm~.t@.£:k~(U.~Y!'~ This again defies statics but is consistent with the approach to calculating Veo and Vcr. Slabs The Code states that the flexural shear resistance of pre- stressed slabs should be calculated in exactly the same manner as that of prestressed beams. except that shear reinforcement is .not required in slabs when the applied shear force is less than Yr' This recommendation does not appear to be based upon test data and the author has the same reservations about the recommendation as those dis- cussed previously in connection with reinforced slabs. For design purposes, it would seem reasonable to aver- Ultimate limit state - shear and torsion age shear forces over a width of slab equal' to twice the effective depth, and to carry out.the shear design calcu- lation for the shear forces acting on planes normal to the tendons and untensioned reinforcement. A similar approach, for reinforced slabs. is discussed elsewhere in this chapter. *Punching shear· The Code specifiesa.Aifferent critical shear perimeter for prestressed concrete than for reinforced concrete. The perimeter for prestressed concrete is taken to be at a dis- tance of half of the overall slab depth from the load. The section should then be considered to be uncracked and Veo calculated as for flexural shear. In other words, the principal tensile stress at the centroidal axis around the critical perimeter should be limited to 0.24 l1cu. It should be noted.that Clause 7.4 of the Code refers to values of Veo in Table 32 of the Code whereas it should read Table 31. If shear reinforcement is required. it should be designed in the same way as that for flexural shear. The above design approach is. essentially. identical to that of the American Code [168] and was originally proposed by Hawkins. Crisswell and Roll [169]. They considered data from tests on slab-column specimens and slab systems. and found that the ratios of observed to cal- . culated shear strength (with material partial safety factors removed) were in the range 0.82 to 1.28 with a mean of 1.06. The data were mainly from reinforced concrete slabs but 32 of the slabs were prestressed. In addition. the specimens had concrete strengths and depths less than those which would occur in bridges. However. the author feels that the Code approach should be applicable to bridge structures. Finally, the reservations. expressed earlier when dis- cussing reinforced concrete slabs. regarding non-uniform shear distributions and the presence of voids are also applicable to prestressed slabs. Torsion - general Equilibrium and compatibility torsion In the introduction to this chapter it is stated that. accord- ing to the Code. torsion calculations have to be carried out only at the ultimate limit state. An implication of this fact is that it is necessary to think in terms of two types of torsion. Equilibrium torsion In a statically determinate structure. subjected to torsional loading. torsional stress resultants must be present in order to maintain equilibrium. Hence. such torsion is referred to as equilibrium torsion and torsional strength must be pro- vided to prevent collapse occurring. An example of equilibrium torsion is that which arises in a cantilever beam due to torsional loading. It is assumed in the Code that the torsion reinforcement provided to resist equilibrium torsion at the ultimate limit
- 45. state is adequateto control torsional cracking at the ser- viceability limit state. Compatibility torsion In a structure which is statically indeterminate, it .is theoretically possible to provide no torsional strength, and to prevent collapse occurring by designing more flexural and shear strength than would be necessary if torsional strength were provided. The explanation of this is that a stress resultant distribution, within the structure, with zero torsional stress resultants and which satisfies equilibrium can always be found. Since such a distribution satisfies equilibrium it leads to a safe lower bound design [27]. Although a safe design results from a stress resultant distribution with no twisting moments or torques, it is obviously necessary, from considerationsof compatibility, for various parts of the structure to displace by twisting. Hence, such torsion is refened to as compatibility torsion.. The torsion which occurs in bridge decks is, generally, compatibility torsion and it would be acceptable, in an elastic analysis, to assign zero torsional stiffness to a deck. This would result in zero twisting moments or torques throughout the deck and bending moments greater than those which would occur jf the full torsional stiffness were used. In the above discussion it is implied that either zero or the full torsional stiffness should be adopted. However, it is emphasised that any value of torsional stiffness could be adopted. As an example, Clark and West [170] have shown that it is reasonable, when considering the end diaphragms of beam and slab bridges, to adopt only SO% of the torsional stiffness obtained by multiplying the elastic shear modulus of concrete by Ji (see equation (2.43». Finally, although it is permissible to assume zero tor- sional stiffness at the ultimate limit state, which implies the provision of no torsion reinforcement, it is necessary to provide some torsion reinforcement to control any tor- sional cracks which could occur at the serviceability limit state. The Code assumes that the nominal flexural shear reinforcement discussed earlier in this chapter is sufficient for controlling any torsional cracks. Combined stress resultants The Code acknowledges the facl Ihat, at a particular point, the maximum bending moment, shear and torque do not generally occur under the same loading. Thus, when Fig. 6.11 Torsional cracks designing reinforcement to resist the maximum torque, reinforcement, which is present and is in excess of that required to resist the other stress resultants associated with the maximum torque, may be used for torsion reinforce- ment. Torsion of reinforced concrete Rect~ngular section Torsional shear stress Methods of calculating elastic and plastic distributions of torsional shear stress are available for homogeneous sec- tions having a variety of cross-sectional shapes, including rectangular. The calculation of an elastic distribution is generally complex, and that of a plastic distribution is gen- erally much simpler. However, neither distribution is cor- rect for non-homogeneous sections such as cracked struc- tural concrete. In order to simplify c~lculation' procedures, the Code adopts a plastic distributio'n of torsional shear stress over the entire cross-section. It is emphasised that such a dis- tribution is assumed not because it is correct but merely for convenience. The Code also gives allowable nominal tor- sional shear stresses with which to compare the calculated plastic shear stresses. The allowable values were obtained from test data. It can be seen that the above approach is similar to that adopted for flexural shear, in which allowable nominal flexural shear stresses, acting over a nominal area of breadth times effective depth, were chosen to give agree- ment with test data. The plastic torsional shear stress distribution is best cal- culated by making use of the sand-heap analogy [171] in which the constant plastic torsional shear stress (VI) is proportional to the constant slope ('lJ) of a heap of sand on the cross-section under consideration. In fact V, = T'lJIK (6.18) where T is the torque and K is twice the volume of the heap of sand. The plastic shear stress for a rectangular section can thus be evaluated from Fig. 6.10 as follows Volume of sand-heap= (hmi,,/2)(hmax){'lJhmi,,/2) - (2)(1/6)(h",i")(I1,,,i,,/2)('lJll",i,,/2) '(1/4)'1' h2 mi" (hmax - hmi,,/3) y, A.v--+-+- x, (a) Cross-section Fig. 6.12(a),(b) Space truss analogy Thus, from equation (6.18), 2T VI =h2 ml" (hmax - hm;" 13) (6.19) This equation is given in the Code. -'*.,orsion reinforcement Design If the applied torsional shear stress calculated from equation (6.19) exceeds a specified value (V,ml,,) it is necessary to provide torsion reinforcement. One might expect VI",;" to be taken as the stress to cause torsional cracking or that corresponding to the pure torsional strength of a member without web reinforcement. In fact, it is taken in the Code to be 2S% of the latter value. Such a vaiue was originally chosen by American Concrete Com- mittee 438 [172] because tests have shown [173, 174] that the presence of such a torque does not cause a significant reduction in the shear or flexural strength of a member. The tabulated Code values of Vlmi" are given by 0.067 /f.," (but not greater than 0.42 N/mm2 ) and are design values which include a partial safety factor of 1.S applied to feU' The formula is based upon that originally proposed by the American Concrete Institute but has been modified to (a) convert from cylinder to cu~ strength, (h) allow for partial safety factor and (c) allow for the fact Ihat the American Concrete Institute calculates VI from an equation based upon the skew bending theory of Hsu [17S] instead of the plastic theory. If VI exceeds Vlmi,,' torsion reinforcement has to be pro- vided in the form of longitudinal reinforcement plus closed links. The reason for requiring both types of reinforcement is that. under pure torsional loading, principal tensile stress- es are produced at 45° to the longitudinal axis of a beam. Hence. torsional cracks also occur at 4So and these tend to form continuous spiral cracks as shown in Fig. 6.11. It is necessary to have reinforcement, on each face, parallel and normal to the longitudinal axis in order that the torsional cracks can be controlled and adequate torsional strength developed. The amounts of reinforcement required are calculated by considering, at failure, a space truss. This is analogous to the plane truss considered for flexural shear earlier in this Chapter. In the space truss analogy. a spiral failure surface Compression strut . y, (b) Elevation on a larger face is considered which, theoretically, could form at any angle. However, for design purposes, it is considered to fonn at the same angle (4S0) as the initial cracks in order that the amount of stress redistribution· required prior to collapse may be minimised. The failure surface assumed by the Code is shown in Fig. 6.12 together with relevant dimensions. . If the two legs of a link have a total area of A.,v and' the links are spaced at sv. then y1ls" links cross a line parallel to a compression strut on a larger face of the member. If the characteristic strength of the link reinforcement is fy," then the steel force at failure in each larger face is Fy =fyv(A.,.,I2){y lls,,) Similarly, the steel force at failure in each smaller face is Fx =fyv(A.)2} (xlls,,) The total resisting torque is T =FyXl + FxYI =fy,A,,,,xlylls,, (6.20) At the ultimate limit state, fy" has to be divided by a partial safety factor of I.IS to give a design stress of 0.87 fyl" Hence, in the code, fyv in equation (6.20) is replaced by 0.87 fy," Furthermore, the Code introduces an efficiency factor, which is discussed later, of 0.8 in order to obtain good agreement between the space truss analogy and test results. Hence, in the Code, equation (6,20) is presented as A.,. ;, T Sv 0.8x)YI (0.87!v,') (6.21) The force Fy is considered to be the vertical component of a principal force (F) which acts perpendicular to the failure surface and thus F = F). ./2. The principal force also has a horizontal (longitudinal) component FI./2 which, from above, is equal to F,,, A force of this mag- nitude acts in each larger face and, similarly, a horizontal (longitudinal) force of magnitude F., acts in each smaller face. Thus the total horizontal (longitudinal) force. which tends to elongate the section, is 2(Fx + Fv). This elongat- ing force has to be resisted by the longitudinal reinforce- ment: if the total area of longitudinal reinforcement is Ad. 77
- 46. Diagonal thrust (b) Section .....Components of diagonal thrustsc ' Corners spall Fig. 6.13(a),(b) Diagonal torsional compressive stresses and it has a characteristic strength of/yL AsUyL =2(F.. + Fy) =Asv/YV(XI + YI)/s" Thus, in the Code, A.•L ;;;=~:v(f) (Xl + Yl) (6.22) Detailing The detailing of torsional reinforcement has been considered by Mitchell and Collins [176] and the fol- lowing Code rules are based very much on their work. Diagonal compressive stresses occur in the concrete between torsional .cracks, and such stresses near the edges of a section cause the comers to spall off as shown in Fig. 6.13. In order to prevent premature spalling it is necessary to restrict the link spacing and the flexibility of the portion of longitudinal bar between the links. Tests reported by Mitchell and Collins [176] indicate that the link spacing should not exceed (Xl + YI)/4 nor 16 times the longitudinal comer bar diameter. The Code specifies these spacings and also states that the link spacing should not exceed 300 mm. The latter limitation is intended to control cracking at .the serviceability limit state in large members where the two other limitations can result in large spacings. In addition, the longitudinal comer bar diameter should not be less than the link diameter. The characteristic strength of all torsional reinforcement is limited to 425 N/mm2 primarily because such a restric- tion exists for shear reinforcement. However, it is justified by the fact that some beams, which were tested by Swann [177] and reinforced with steel having yield stresses in excess of 430 N/mm2 , failed at ultimate torques slightly less than those predicted by the Code method of calcula- tion [178]. The reason for this was that the large concrete strains necessary to mobilise the yield stresS' of such high' strength steel resulted in a reduction in the efficiency factor . mentioned earlier and discussed in the next section of this chapter. Maximum torsional shear stress The space truss of·Fig. 6.12 consists of the torsion re-I inforcement acting as tensile ties pIllS concrete compressive struts. As is also the case for flexural shear, it is necessary, to limit the compressive stresses in the struts to prevent the struts crushing prior to the torsion reinforcement yielding" iu tension. This is achieved by limiting the nominal tor- sional shear stress. However, the derivation of the limiting values in the Code is connected with the choice of the efficiency factor applied to the reinforcement in equation (6.20). It is mentioned elsewhere in this chapter that an effi- ciency factor has to be introduced into equation (6.20) in order to obtain agreement between test results and the pre- dictions of the space truss analogy. Swann [178] found that the efficiency factor decreases with an increase in the nominal plastic torsional shear stress at collapse and also decreases with a decrease in specimen size. The dependence of the efficiency factor on stress is explained by the fact that, if the nominal shear stress is high, the stresses in the inclined compression struts be- tween the torsional cracks of Fig. 6.11 are also high. Thus, at high nominal stresses, a greater reliance is placed upon 'the ability of the concrete in compression to develop high stresses at high strains than is the case for 'low nominal stress. In view of the strain-softening exhibited by concrete in compression (see Fig. 4.1 (a» it is to be expected that' the efficiency factor spould decr~ase with an increase in nominal stress. The size effect is explained by the fact that spalling occurs at the comers <>fa member in torsion as shown in Fig. 6.13. This alters the path of the torsional shear flow and can be considered to reduce the area of concrete' resist-: ing the torque. As shown in Fig. 6.14, this effect is more significant for a small than for a large section. In order to formulate a simple design method which takes the above points into account, Swann [178] proposed that the efficiency factor should be chosen to be a value which could be considered to result in an acceptably high ' nominal stress level in large sections. The reason for con- sidering large sections was that these are least affected by comer spalling. Consideration of tests carried out on beams with a maxi- mum cross-sectional dimension of about 500 mm indi- cated that 0.8 was a reasonable value to take for the effi- ciency factor. The nominal stress which could be attained with this factor was 0.92 #Cu. If a partial safety factor of 1.5 is applied to te." a design maximum nominal torsional shear stress (v,,,) of 0.75 If.,u is obtained. This value is tabulated in the Code with an upper limit of 4.75 N/mm2 • It should be noted that the stress is the same as the design maximum nominal flexural shear stress. Thus, Swann [178] essentially chose an efficiency factor to give the same design maximum shear stress for torsional shear as for flexural shear. Having established a nominal maximum (a) Sections before spalling (b) Sections after spalling Fig. 6.14(a),(b) Torsional size effect [178] stress and an efficiency factor for large sections, it was decided to adopt the same efficiency factor ,<0.8) for .small sections and to determine reduced nommal maximum stresses for the latter by considering test data. It was f~und that the stress of 0.92 ./!c.. had to be modified by multiply- ing by (y /550) for sections where Yl < 550 mm. Hence the design st:ess of 0.75 !fell also has to be multiplied by this ratio. It ,is emphasised that the Code requi~s. that both of the following be satisfied: I. The total (flexural plus torsional) shear stress (v + v,) should not exceed 0.75 /fCII or 4.75 N/mm2 2. In the case of small sections (Yl < 550 mm} the tor- sional shear stress should not exceed 0.75 v'/cu(y1/550) or 4.75 (yI/550) N/mm2 • T-, L- and I-sections Torsional shear stress The plastic shear stress for a flanged section could be obtained from considerations of the appropriate sand-heap, such as that shown in Fig. 6.15(a) for a T-section. How- ever, the junction effects make the calculations ~a~her t~di. ous and the Code thus permits a section to be dIVided mto its component rectangles which are then considered indi- vidually. The manner in which the section is divided into rec- tangles should be such that the function r.(hm/l~h'il/) is maximised. This implies that, in general, the section should be divided so that the widest of the possible compo- nent rectangles is made as long as possible as shown in Fig. 6.15(b). and (c). The total torque (T) applied to the section is then apportioned among the component rec- tangles such that each rectangle is subjected to' a torque: , b h A (a) Actual h ~ ~ (b) ,Idealised h, > bw (c) Idealised h, < bw (d) R~inforcement Fig. 6.JS(a)-(d) Torsion of T-sectiQn T(hmqxh 3 min) r.(hmax h3 min) The above considerations imply that an anomalous situ- ation arises in the treatment of flanged sections for the following reason. An examination of equation (2.43) shows that the divi- sion into rectangles and the apportioning of the torque is carried out on the basis of the approximate elastic stiffnesses of the rectangles. However, the nominal torsional shear stress in each rectangle is calculated using equation (6.19) based upon plastic theory. ¥Torsion reinforcement If the nominal torsional shear stress in any rectangle is less than the appropriate V'mi" value discussed earlier in connection with rectangular sections, then no torsion reinforcement is required in that rectangle. Otherwise, reinforcement for each rectangle should be designed in accordance with equations (6.21) and (6.22), and should be detailed so that the individual rectangles are tied together as, for example, in Fig. 6.15(d). Maximum torsional shear stress The maximum nominal torsional shear stresses discussed 79 •
- 47. Median line x x lJ Shaded area =Ao Curvature l-11...L.._ _-'-'(L....l":I..,.~ (c) Section!bro~ mombrane at X-X -ll+-hwo/2 (a) Cross-section (b) Median line and enclosed area Ao Fig. 6.16(a)-(c) Torsion of box section L:___ Fig. 6.17 Box girder notation in connection with rectangular sections should not be exceeded in any individual rectangle; Box sections Torsional shear stress The elastic torsional shear stress distribution is easily cal- culated for a box section. Hence, the Code requires the nominal stress to be calculated from the standard formula for a thin-walled closed section [76]: VI = Tl2h,.,c0o where hw() is the wall thickness at the point where the shear stress V, is determined andAo is the area within the median line of the section as shown in Fig. 6.16. The above equation can be derived by applying the membrane analogy [76] in which the variable elastic tor- sional shear stress is proportional to the variable slope ('jJ) of a membrane inflated over the cross-section. The mathematical expression used is identical to that for the sand-heap analogy for plastic torsion (equation (6.18» butK is now twice the volume under the membrane. With refer- ence to Fig. 6.16 tlnd by ignoring the slight curvature of the membrane (so that a section through the membrane has straight edges), the slope at a particular point is 'i' = hlhwo ' where h = height of membrane, and the volume underthe membrane is ""hAo. Thus, from equation (6.18), ' VI = T'i'/2hAo = Tl2hwl~o (6.23) Equation (6.23) is, strictly, for thin-walled boxes' whereas it could be argued that many concrete box sec- tions are not thin. Maisel and Roll [41] have shown that the error (6 v,)in calculating VI for a thick-walled box from 80 equation (6.23) is, with the notation of Fig. 6.17, 6VI /t 2 I'o(b I b -1 2d) V, = 2Ao III,·· "hi, . "hOI' (6.24) 1frorsion reinforcement It can be seen from equation (6.23) that the nominal tor- sional shear stress at a point is dependent upon the wall thickness at that point and thus varies around a box with non-uniform wall thickness. If the nominal torsional shear stress at any point exceeds the' V"ntn values discussed eat'lier in connection with rectangular sections, then torsion reinforcement must be provided. The design of torsion reinforcement is complicated in the Code by the fact that two sets of equations may be used and the lesser of the amounts so calculated may be pro- vided. The two sets of equations are (a) equations (6.21) and (6.22) which were derived, for solid rectangular sec- tions, from the space truss analogy and (b) equations (6.25) and (6.26) below which were also derived from ,the space truss analogy, but with the reinforcement assumed to be concentrated along the median line of the box walls and with the efficiency factor taken to be 1.0. Asv T -~ Sv Ao (0.87fyv) (6.25) A Asv fyv (perimeter of Ao) sL ~--r Sv JyL 2 (6.26) The reason for having two sets of equations is that (6.21) and (6.22) were originally intended for beams of relatively small cross-section and they can be over- conservative for large thin-walled box sections in which Xl is greater that about 300 mm [179]. Swann and Williams [179] have tested model reinforced concrete box beams under pure torsional loading and found that the observed ultimate torques exceeded the ultimate torques calculated from either set of equations. It was also found that equations (6,.21) and (6.22) were more conser- vative than (6.25) and (6.26): this was to be expected since the, models were large in the sense that they were models of large prototype sections. The consideration of a box girder subjected only to tor- sion is rather academic sinc;e, in practice, flexural loading is also generally present. There is thus an essentially con- stant compressive stress (fco,') due to flexure over the cross-sectional area (Ac) of one flange. Thus a compre~sive in-plane force of AicQ" acts in conjunction with an in-plane shear force of Acv,. Hence, from equation (AI7), the Stress Tendon design curve ~.1----.;1~4-0.00185 (a) Author's criterion Fig. 6.18 Tendons as torsion reinforcement V •Strain (a) Stress - resultants on entire cross-section Fig. 6.19(a),(b) Combined stress-resultants amount (A) of longitudinal reinforcement required is given by (O. 87tvdA = -Arico' + IAcv11 or A = -A/ca,' + IAcv11 0.87fyl_ 0.87fvl. This implies that the amount of reinforceU;ent calculated from equation (6.22) or (6.26) may be reduced by Aic",'! O.87/yl.' The validity of this approach to design has been confinned by the tests of Swann and Williams [179] and is consequently permitted by the Code. It should be noted that the Code adopts for At' the area of section sub- jected to flexural compressive stresses instead of simply the flange area, and that the Code also refers to a stress fve ~h~W~k· . The reduction in 'longitudinal steel area due to the effect of flexural compressive stresses could, theoretically, be applied to sections other than boxes but the Code limits the reduction to box sections only. Maximum torsional shear stress The maximum nominal torsional shear stresses discussed in connection with rectangular sections have been shown to be reasonable for box sections by the tests of Swann and Williams [179]. Ultimate limit state - shear and torsi'on Stress Idealised tendon curve Idealised reinforcement curve Strain (b) Lampert's criterion (b) Stress - res41tants at a point *r . f 'orslon 0 prestressed concrete General The code essentially assumes that prestressed concrete members can be designed to resist torsion by ignoring the prestress and designing them as if they were reinforced. Thus values of VI' Vtm/n' Vtu, As,' and A.L should be calcu- lated in accordance with the equations 'for reinforced con- crete presented earlier in this chapter. The validity of such a~ approach has been checked, with bridges speci- fically tn mtnd, by Swann and Williams [179]. who carried out tests on model prestressed concrete box beams. It has been mentioned that the greatest pennissible characteristic strength for conventional torsion reinforce- ment is 425 N/mm 2 • This value was chosen because the large yield strains associated with higher strengths reduce the efficiency factor in equation (6.21) to less than 0.8. Thus, to achieve an efficiency factor of 0.8 when using tendons as torsion reinforcement, it is necessary to ensure that the additional tendon strain. required to mobilise the tendons' ultimate strength, does not exceed the yield strain of other reinforcement in the section having a character- istic strength of 425 N/mm2 • If the latter is conservatively assumed to be hot-rolled, the design yield strain is 0.87 x 425/200000 = 0.00185. Hence, the design ultimate stress
- 48. ,Predetermined position (yPzp) F / of centroid of tendons 1 / • ..i..____._..... __..___.__y ~ . .~ • z Fig. 6.20 Segmental box beam (fpd) for the tendons should be the lesser of 0.87!pu.and (see Fig. 6.18(a» !p~ plus the stress increment equivalent to a strain increment of 0.00185, where f~e is the effective pre- stress in the tendons. Although the above argument seems logical, the result- ing limiting stresses are not included in the Code. Instead, the design stress is limited to the· lesser of 460 N/mm2 and (O.87fpII - fp~). The specification of such stresses seems to indicate that a criterion suggested by Lampert [180] and by Maisel and Swann [181] was misinterpreted by the draft- ers. The suggested criterion was that tendons and conven- tional reinforcement should reach yield at about the same stage of failure of the section, in order that excessively large strains would not develop in one type of reinforce- ment prior to yield of the other. This implies that (see Fig.6.18(b» f,lIl - [". == fl' where [,,,1 is the tendon design stress at ultime.te. The draft- ers took tv to be 460 N/mm2 (the greatest value likely to be used)· although it exceeds the greatest value (425 N/mm2 ) permitted for torsion reinforcement. Hence, f~/(I ==f"e + 460 N/mm2 • However, fpd cannot exceed 0.87 fplI' thus, fpti should be taken as the lesser of 0.87 fpu and (fp~ +460 N/mm2 ). It thus appears that the limiting stresses given in the Code are not, as stated in the Code, design stresses but the stress increment necessary to raise the stress from the effective prestress to the design stress. It should also be mentioned that it is inconsistent to apply a partial safety factor to fplI but not to fy , and thus the second limit should be (fpe+ 400 N/mm2 ). Hence the stress increment in the Code should be 400 N/mm2 • In conclusion, the Code stress limits are stress illcre- ments and not design stresses; moreover, the author would suggest that the alternative criterion illustrated in Fig. 6.18(a) is more logical. Alternative design methods The Code method of design of a section subjected to a general loading is to consider the stress resultants acting on the entire cross-section, as shown in Fig. 6.19(a), and then to superpose the effects of any local actions. This method is probably very suitable to small sections but for large sections (and particularly for box beams) it could be consi- dered desirable to vary the reinforcement over, for exam- ple. the depth of a web. It is then necessary to consider the stress-resultants acting at various points of the cross- section al> shown in Fig. 6.19(b). Each point would gener- ally be subjected to both in-plane and bending stress resul- tants and thus the sandwich method discussed in Chapter 5 82 E E ~ II ~ ~-- •• •Fig. 6.al Example 6.1 6/26•• • would be appropriate for design. A simplified version of such an approach has been described by Swann and Wil- liams [179] and Maisel and Swann [181]. The Code refers to 'other design methods' without specifically mentioning any; however, the above approach of considering the stress resultants at points of the cross- section would be acceptable. Segmental construction In segmental construction it is not generally convenient to provide continuous longitudinal conventional reinforce- ment. Thus longitudinal tendons in excess of those needed for flexure have to be provideg. The Code states that the line of action of the longitudi- nal elongating force should coincide with the centroid of the steel actually provided. If the longitudinal steel capacities required in each flange and web of a segmental box beam are as shown in Fig. 6.20, then they can be replaced by a force F, situated at (y, i), where: F, = IFI Y = IFly/IFi i = IFiz/IFi and each summation is for i = 1 to 4. Thus the total ultimate capacity of the tendons needs to be F, and their centroid needs to be at {J, n. In practice, it is often simpler to calculate the necessary total ultimate capacity of tendons situated at predetermined positions such that their centroid is at, say, (yp. zp). In this case F, must be such that the capacity of each flange or web exceeds the appropriate value of F;. Hence, moments should be taken about each web and flange in tum to give four inequalities. For example. by taking moments about web 2 . (Y2 - y,,)F, ;!: F1(Y2 - YI) + F~(Y2 - Y~) + f4(Y2 - Y4) Slab ,..,..------.... '" "1/ , A '7 t .. ' ,/ 14600~ vY I I NO...·~ B I 7' A o I 4 ..... t i Pier 'Critical ~ , . Iperimeter. I I I I , I , '"" ,/ ...... _-_.. _-_..... (a) Plan /1%(d= 1060) I i -~I.... (b) Section A-A Flg.6.22(a),(b) Example 6.2 The largest value of F, obtained from the four inequalities is the total ultimate tendon capacity which must be pro- vided. Swann and Williams [179] give a numerical exam- ple of such a calculation, and they also present the results of tests on two model segmental box beams against which the above method of calculation was checked. Examples "*6.1 Flexural shear in reinforced concrete Design stirrups of 250 N/mm2 characteristic strength to resist an ultimate shear force of 540 kN applied to the sec- tion shown in Fig. 6.21. The concrete is of grade 40. Nominal applied shear stress = v = 540 x 103 /(300 x 600) = 3 N/mml From Table 6 of Code or 0.75 ./fc.u, maximum allowable shear stress = v" = 4.75 N/mm2 v" > v. o.k. Area of tension reinforcement A., = 2950 mml lOOA,/bd = (l00 x 2950)/(300 x 600) = 1.64 From Table 5 of Code. allowable shear stress without shear reinforcement v,. = 0.88 N/mm2 From equation (6.7), An· 300(3.00 - 0.88) ~ 2.92 mml/mm -s-" = 0.87 x 250 - The stirrup spacing should not exceed 0.75 d = 450 mm. 12 mmstirrups(2Iegs)at75 mmcentresgive3.02 mm2/mm. ~6.2 Punching shear in reinforced concrete Design the slab shown in Fig. 6.22 to resist punching shear if the characteristic strengths of the steel and con- crete are 425 and 40 N/mm2 respectively and the col~mn reaction at the ultimate limit state is 15 MN. The critical .,e"nme.ter is at 1.5h = (1.5)(1100) = 1650 mm The critical section is shown on Fig. 6.22. Perimeter =(2)(600) + (2)(1200) + (217)(1650) =14000 mm 'Average' effective depth, d =(3 x 980 + 1 x 1060)/4 = 1000 mm Nominal applied shear stress, Y =15 x 106 1(14 000 x 1000) =1.07 N/mm2 From Table 6 of Code or 0.75 lieu, maximum allowable shear stress =Yu = 4.75 N/mm"';' Only take half for slab, :. Yu = 2.38 NfmtnJ Vu > v, o.k. Average lOOAslbd = (3 + 1)/2 = 2 From Table 5 of the Code, allowable shear stress without shear reinforcement Vc = '0.95 N/mm 2 From Table 8 of Code, t. = 1.00 v - ;"v.. = 1.07 - 0.95 = 0.12 N/mm 2 . Thus shear reinforcement required, but must provide for at least 0.4 N/mm2 From equation (6.8) IAsv ;!:(0.4)(14 000)(1000)/(0.87)(425) = 15 100 mm2 This amount of reinforcement must be provided along a perimeter 1.5h from the loaded area and also along a perimeter 0.75h from the loaded area. Length of perimeter at 0.75h = (2)(600) + (2)(1200) + (2")(825) = 8780 mm Thus on 1.5h perimeter provide 15100/14 = 1080 mm2/m. and, on 0.75h perimeter, 15100/8.78 = 1720 mm2/m. Now check on perimeter at (1.5 + 0.75)/1 = 2.25h. Length of perimeter = (2)(600) + (2)(1200) + (211")(2475) = 19200 mm Nominal applied shear stress = v = 15 x 106 /(19200 x 1000) = 0.78 N/mm2 v < !;"vc• thus no need to provide more shear reinforce- ment. *6.3 Flexural shear in prestressed concrete The pretensioned box beam shown in Fig. 6.23 is sub- jected to a shear force of 0.9 MN with a co-existing moment of 3.15 MNm at the ultimate limit state. Design shear reinforcement with a characteristicstrimgth of 250 N/mm 2. The initial prestress is 70% of the characteris- tic strength and the losses amount to 30%. The concrete is of grade 50 and the section has been designed to be class I. From Table 32 of Code or 0.,75 /I.:". maximum. aJl()wabl~ shear stress VII = 5.3 N/mm 2 83
- 49. 34N" 15.2 mm lowrelaxation M"" Area = 47 8175 mm2 ' .' L !~. 1035 Bottom fibre modulus ... 126.43 x 10' mm3 Neutral a)(I&I& 610 mm from bottom fibre Fig. 6.23 Example 6.3 maximum allowable shear force Vu .. Vu bd where b' .. total web breadth .. 250 mm. d .. effective depth of steel in tension zone .. 945.5 mm V" .. (5.3)(250)(945.5)10"' =1.25 MN Vu>V, :. o.k. From Table 2l,ApJpu~ 227.0 kN Effective prestress" (34)(0.7)«).7)(227) =3780 kN Uncracked in flexure Compressive stress at centroidal axis = 3780 x 103 /478175 =7.91 N/mml Allowable principal tensile stress I, =0.24 ./leu =1.70 N/mml From equation 6.9, Vco .. (0.67)(250)(1035)x Cracked in flexure 1(1.70)1 + (0.8)(7.91)(1.70) = 641 x 103 N = 0.641 MN Distance of centroid of all tendons from bottom fibre = (16 x 64 + 16 x 115+ 2 x 984)/34 = 142 mm Eccentricity =510 - 142 =368 mm '" e, " Prestress at bottom fibre = 7.91 + (3780 x 103 '*'3681 125.4fx .108) '-.. = 19.0 N/niml From equation (6.12), cracking moment is M, = (0.37 /SO + €Jx 19.0) 125.43 x 108 x 10-9 = 2.23 MNm From equation (6.11), Vcr = (0.037)(250)(1035 - 142) /SO + (0.9 X 10') (2.23/3.15) = 696 X 103 N = 0.696 MN Vcr must be at least O.lbd /Tcu = 0.158 MN Thus use Vcr = 0.696 MN. Shear reinforcement Shear capacity without shear reinforcement Ve is lesser of Vcn and Vcr ,',:Vc " Veo = 0.641 MN 84 1- E E 3...... x, =530mm ~"-----4 .. • .. • b =600mm ~"---'-'-'--I FII.6.24 Example 6.4 From equation (6.17), E E ~...... .=-: A,vis"" (0.9-0.641)106 /(0.87 x 250 x 971) = 1.23 mma/mm V > 1.8Vc' maximum spacing is Jesser of 0.75 x 971 = 728mm and 4 x 125 =500 mm Provide 12 mm stirrups (2 legs) at 175 mm centres. ~ (0.87/,,1' ) .. 2 x 113 (0.87 x 250) s" . b 175 250 = 1.12 N/mml > 0.4 N/mm2 , O.K. "*6.4 Torsion in reinforced concrete An end diaphragm of a beam and slab bridge is, for design purposes, considered as a rectangular reinforced concrete beam, 600 mm wide and '1200 mm deep. The concrete is of grade 40 and the minirpum cover is 30 mm. Design torsion reinforcement, hayin'g a characteristic strength of 425 ,N/mm2 , to resist an ultimate torque of 290 kNm which co-exists with an ultimate shear force of 500 kN. From equation (6.19), the nominal torsional shear stress is _ 2 x 290 X 10 8 _ 1 61 NI 2 v, - 6002 (1200 _ 600/3) - . mm From Table 7 of Code or 0.067 /!cu, allowable torsional shear stress without torsion reinforcement Vtm/n = 0.42 N/mm2 V, > V,m/n' thus torsion reinforcement required. Assume d = 1144 mm, XI = 530 mm, Yt = 1130 mm (see Fig. '6.2~) Nominal flexural shear stress = V = 500 x 103 1(600 x 1144) = 0.73 N/mml v + V, = 2.34 N/mm2 From Table 7 of Code or 0.75 /!cll' maximum allowable shear stress V,U = 4.75 N/mm2 v," > v + V" thus section big enough. From equation (6.21), links should be provided such that A.•./sv" 290 x 108 /(0.8 x 530 x 1130 x 0.87 x 425) = 1.64 mm2 /mm f.-____ 900!!1.,.,,_____~ E E o N N ... 150mm ,.~.l25 mm ~ 1 -.-~ Effective depth (d) of 15.2 mm low relaxation 25 mm strands in tension zone'" 1145 mm 150mm -'... Fig. 6.25 Example 6.5 From equation (6.22), required area of longitudinal rein- forcement is A"" = 1.64 (425/425)(530 + 1130) = 2720 mm 2 4 No. 32 mm give 3220 mm2 (l bar in each comer) Stirrup spacing should not exceed the least of (a) (530 + 1130)/4 = 415 mm (b) 16 x 32 = 512 mm (c) 300 mm to mm stirrups (2 legs)at90 mm centl'es give 1.74 mm2 /mm If 40 mm cover to main steel then d, XI and YI are as assumed. The above reinforcement should be provided in addition to any flexural and shear reinforcement. Although the Code does not permit one to do so, it would seem reasonable to reduce the area of longitudinal torsion reinforcement in the flexural compression zone in a similar manner to that permitted for box sections. f6.5 Torsion in prestressed concrete The rectangular box section shown in Fig. 6.25 is sub- jected to an ultimate torque of 610 kNm and an ultimate vertical shear force 230 kN. A co-existing bending moment produces an average flexural compressive stress of 20 N/mm2 over the flexural compression zone which extends to a depth of 300 mm below the top <;>f the section. The concrete is of grade 50 and the characteri,stic strengths of the prestressing steel and the torsional reinforcement are 1637 N/mm2 and 425 N/mm 2 respectively. The effective prestress in the tendons is 860 N/mm2. The minimum cover is 25 mm. Design suitable torsion reinforcement. Area within median line Ao = (1220 - 150)(900 - 125) = 829250 mm2 From equation (6.23), the nominal torsional shear stress in a flange is vtf = 610 x 108 /(2 x 150 x 829250) = 2.45 N/mm2 and in a web is V,w = 610 x 106/(2 x 125 x 829250) = 2.94 N/mm 2 From Table 7 of Code allowable torsional, shear stress without torsion reinforcement V"ni,,== 0.42 N/mm2 v, >V,min, thus torsion reinforcemerit required. The Code does not, in this context, define band d to be used to calculate the flexural shear stress which acts only in the webs. Assume b = 2 x 125 == 250 mm, d = 1145 mm. Ulllfnate limit slttte - she,lr a/.J torsion Flexural shear stress in webs VI<' = 230 x 103 /(250 x 1145) = 0.80 N/mm 2 and flexural shear stress in flanges v,= O. For each flange, total shear stress = v, + v,, = 0 + 2.45 = 2.45 N/mm2 For one.web, total shear stress = v'" + V,w = 0.80 + 2.94 = 3.74 N/mm 2 From Table 7, maximum allowable shear stress V/ri = 4;7'5"N/-m1Jl2 v,.. > v + V" thus,section big enough. Assume XI = 82501m, Yl = 1140 mm. From equation (6.21), links should be provided such that As..!sv = 610 x 108 /(0.8 x 825 x 1140 x 0.87 x 425) = 2.19 mm2/mm . From equation (6.25), links should be provided such that Asisv = 610 x 106 /(829 250 x 0.87 x 425) =1.99 mm2/mm Provide links such that A,•..Isv == 1.99 mm2 /mm. The longitudinal torsion reinforcement could be provided by conventional reinforcement or by additional prestress- ing steel. If the latter were to be used; it would be logical to calculate a design stress as follows (see stress-strain urve in Fig. 5.10). fpe = 802 N/mm 2 Strain at Jpe == 0.00401 Allowable strain increment = 0.00185 (see text), thus total strain == 0.00586 at which the stress is (see Fig. 5.10) 1147 N/mm2 • The area of longitudinal conventional reinforcement would be, from equation (6.26), As1• =1.99(425/425)(2 x 1070 + 2 X 775)/2 = 3670 mm2 or, the area of longitudinal prestressing steel with an effec- tive prestress of 802 N/mm~ would be, from equation (6.26), A,d, = 1.99(0.87 X 425/1147)(2 x 1070 + 2 X 775)/2 = 1180 mm2 Thus provide either 1835 mm2 of conventional reinforce- ment or 590 mm2 of extra prestressing steel in each flange. The top flange areas may be reduced by, respectively 20(900 x 150 + 2 x 125 x 150)/(0.87 x 425) = 9330 mm2 and 20(900 x 150 + 2 x 125 x 150)/1147 = 301Omm2 Thus no torsion reinforcement is required in the top flange. The bottom flange could be reinforced with 4 No. 25 mm bars (giving an area of 1960 mm2) or 5 additional tendons (giving an area of 694 mm2 ). 12 mm diameter stirrups (2 legs) at 100 mm centres give 2.26 mm2/mm, which is adequate for the transverse torsion reinforcement. '. It is emphasised that, as written, the Code would not permit a design stress of 1147 N/rnm2 to be adopted for the tendons. The Code requires the lesser of 460 N/mm2 and (0.87 tim - Ipe) = 564 N/mm2 to be adopted, i.e. 460 N/mm2. Thus, in accordance with the Code, no advantage could be taken of the larger ultimate strength of the tendons.
- 50. Chapter 7 Serviceability limits~ate -f . Introduction As explained in Chapter 4, the criteria which have to be satisfied at the serviceability limit state are those of per- mis~ible steel and con~rete stress, permissible crack width, interface shear in composite construction and vibratio~. In this chapter, methods of satisfying the criteria of permis- sible steel and concrete stress, and of cra£k width in both reinforced and prestressed concrete construction are pte- sented. The additional criteria, associated with interface shear and tensile stress in the in-situ concrete of composite construction, are presented in Chapter 8. Compliance with the vibration criterion is discussed, together with other aspects of the dynamic loading of bridges, in Chapter 12. Reinforced concrete stress limitations ""*General As discussed in Chapter 4, the concrete compressive stres- ses in reinforced concrete should not exceed 0.5feu and the reinforcement, tensile or compressive, stresses should not exceed O.8fy. Although it is not stated in the Code, the above limit- ations should be applied only to axial or flexural stresses. Thus it is not necessary to consider flexural shear or tor- sional shear stresses at the serviceability limit state. In practice, axial and flexural stress calculations will involve the application of conventional elastic modular ratio theory. However, a modular ratio is not explicitly stated in the Code, and it is necessary to calculate a value from the stated modulus of elasticity of the reinforcement (200 kN/mm2 ) and the modulus of elasticity of the con- crete. Short term values of the latter modulus are given in the Code and these vary from 25 kN/mml for a charac- teristic strength of 20 N/mm2 to 36 kN/mml for a charac- 86 teristic strength of 60 N/mml • The Code is not explicit as to whether these short term moduli shOuld be used when calculating stresses or whether they should be adjusted to give long-term moduli. However, the Code does state that a long-term modulus, equal to half of the short term value, should be used when carrying out crack width calculations. In view of this requirement, and the fact that all structural Codes of Practice have hitherto adopted a long-term mod- ulus, the author would suggest that, for stress calculations in accordance with the Code, a long-term modulus should be adopted. Since a relatiyely weak concrete having a characteristic strength of less than about 40 N/mm2 is gen- erally adopted for reinforced concrete, the long-term mod- ular ratio calculated from the Code would be between 13 and 16. These values are of the same order as the value of 15 which is general~y adopted f~r design purposes at present. Tension stiffening In the above discussion, conventional modular ratio theory is mentioned. This theory generally considers a cracked section and ignores the stiffening effect of the concrete in tension between cracks (tension stiffening); hence, it over- estimates stresses and strains as shown in Fig. 7.1. The difference between lines AB and OC of Fig. 7.1 is a measure of the tension stiffening effect and can be seen to decrease with an increase in load above the initial cracking load. This decrease results fr@m the development of further cracks and the gradual breakdown of bond between the reinforcement and the concrete. Hence the strain (E.) cal- culated ignoring tension stiffening should be reduced by an amount (E/S) to give the actual strain (Em). The Code permits tension stiffening to be taken into account in certain crack width calculations (as referred to later in this chapter), but it is not clear whether one is permitted to allow for it in permissible stress calculations. However, test results [183] indicate that, at stresses of the order of the steel stress limitation of 0.8fy, the tension stif- fening effect is negligible. Therefore, it seems to be reasonable to ignore tension stiffening when 'carrying out stress calculations. Load N Cracking load o --Actual ----Calculated on cracked stiffness -;1 . "Ncl ; I 1"/ I /1 I "/ I I / t f ts ~; N" (,. ;" 51 I / I I /" I I ; Fig. 7.1 Tension stiffening "*Slabs """" B /'c """ Stress Strain If the principal stresses in a slab do not coincide with the reinforcement directions, it is extremely difficult to calcu- late accurately the concrete and reinforcement stresses. As the Code gives no guidance, the author would suggest the following procedure: 1. Assume the section to be uncracked and calculate the four principal extreme fibre stresses caused by the stress resultants due to the applied loads. 2. Where a principal tensile stress exceeds the permis- sible design value, assume cracks to form perpendicu- lar to the direction of that principal stress. The per- missible design value could be taken to be the Class 2 prestressed concrete limiting stress of 0.45 !feu (see Chapter 4). 3. Consider each set of cracks in tum and calculate an equivalent area of reinforcement perpendicular to these cracks. 4. Using the equivalent area of reinforcement, calculate the stresses in the direction perpendicular. to the cracks by using modular ratio theory. 5. Compare the extreme fibre concrete compressive stress with the allowable value of 0.5 feu. 6. If the calculated stress in the equivalent area of rein- forcement is In, then calculate the stress in an i-th layer of reinforcement, inclined at an angle Q' i to the direction perpendicular to the cracks, from {; =bIn where b is discussed later. Compare /; with the per- missible value of O.8fY' The calculation of the equivalent area of reinforcement (step 3 above) is explained by considering a point in a cracked slab where the average direct and shear strains, referred to axes perpendicular and parallel to a crack (see Fig. 7.2), are En, E" Ynt. The strain in the direction of an i-th layer of reinforce- ment at an angle IXi to the n direction is, by Mohr's circle 2 • 2 • Ei = En COS 01.1 + Et sm 01.; - Ynt S1l1 IXi COS IX; The steel stress is thus li=E.• Ei Fig. 7.2 Cracked slab where Es is the elastic modulus of the steel. If the steel area per unit width is Ai' the steel force per unit width is FI=Atfl If N such layers are considered, the total resolved steel force in the II direction is N FII .. I FI cos2 IXI t .. t N == £, I At (Ell cos4 IXI + E, sin2 IXi cos2 IXI- 1.. 1 Ynt sin IXi cos3 IXi) The force FII can be considered in terms of an equivalent area (A'l) of reinforcement per unit width in the n direc- tion. Thus F,. =An Es En' Hence, by comparison with the previous equation, N E An = I Ai (COS 4 IXi + E: sin 2 IXi COS2IXi - i .. l Ynt • . ·3 £" sm IXi COS IXi) n It is reasonable, at the serviceability limit state, to· assume that the nand t directions will very nearly coincide with the principal strain directions. Thus Ynl ==0 and the third term in the brackets of the above equation can be ignored. There are now three cases to consider for a slab not subjected to significant tensile in-plane stress resultants: 1. If the slab is cracked on one face only and in one direction only, En» Et and the expression for An reduces- to N An =. ~ AI cos 4 IXI 1= 1 (7.1) 2. If the slab is cracked in two directions on .the same face, then Et will be of the same sign as En' If E, is again taken to be zero, the calculated value ofA"will be less than the true value. It is thus conservative to use equation (7.1). 3. If the slab is cracked in two directions on opposite faces, E, will be of opposite sign to En and could take
- 51. any value. Theprecise value of E/En to adopt is very difficult to determine, although some guidance is given by Jofriet and McNeice [184]. As an approxi- mation, it is reasonable to assume that, at the ser· viceability limit state, it is unlikely that En < /E,/; it is thus conservative to assume EnlE, = -1. In which case N ~ A/ cos2 (X/ (cos2 (X/ - sin2 (X/) ._ ""- (7.2) /= I It is implied in equation (7.2) that reinforcement in- clined at more than 45" to the n direction should be ignored. By implication the stress transformation factor, 6, refel'- red to earlier in this chapter should be taken as cos2 0:/ if equation (7.1) is used to evaluate An and as (C05 2 0:/- sin2 0:/) if equation (7.2) is used. Little error is involved in adopting the above approxima- tions for An when the reirtforcement is inclined at less than about 25° to the perpendicular to the cracks. If, when using the above approximations with reinforcement inclined at more than 25", it is found that the reinforce- ment stresses are excessive, it would be advisable toesti- mate a more accurate value of EI'.IE, (as opposed to, the above approximate values of zero and -1) as described in r184]. Crack control in reinforced concrete General Statistical approach The design crack widths given in Table 4.7, and discussed in Chapter 4, are design surface crack widths and are derived from considerations of appearance and durability. In order to calculate crack widths it is necessary to decide on an interpretation of the design crack width~ is it a maximum, a mean or some other value? It is not possible to think in terms of a maximum crack width but it is feas- ible to predict a crack width with a certain probability of exceedence. This can be illustrated by considering two nominally identical beams having zones of constant bend- ing moment. If each beam is subjected to the same loading and the widths of all of the cracks within the respective constant moment zones are measured, then distributions of crack widths can be plotted as shown in Fig. 7.3. It is found that, although the maximum width of crack measured on each of the two nominally identical beams may be very different the crack widths exceeded by a cer- tain percentage of the results are quite similar. For this reason, the design crack widths are defined as those having a certain proQability of exceedence. ' The probability of exceedence that has been chosen in the Code is 20% (i.e., 1 in 5 crack widths greater than the design value), which is the same percentage as that adopted in Lhe building code, CP 110. At first sight it may appear very liberal to permit t in 5 crack widths to exceed 88 Noof cracks -, -Beam 1 "'~,Beam2 Width exceeded by 11% of cracks I I I I I Crack width ."Ig. 7.3 Crack width distributions for nominally identical beams the design value; however, the formula used to calculate the '1 in 5' crack width is applicable only to the constant moment zQnes of specimens tested under laboratory con- ditions. The exceedence level in practice is much less than 20% and, for buildings, it has been estimated [182] that the chance of a specified width being exceeded by any single crack width will lie in the range'lQ-3 to 10-4• The main reasons for this reduction in probability are: 1. The specified design loading under which cracking should be checked is, ~ssentially, the full nominal loading. This is greater than the 'average' loading which occurs for a significant length of time. 2. In design, lower bound estimates of the material prop- erties are used; thus the probability of the stiffness being as low as is assumed in calculating the strains is low. 3. ,Structural members are not generally subjected to 'uniform bending over any great length, and thus the ,:anly cracks in a member which have any serious chance of being critical are those close to the critical sections of the memb~r. These will be few in number compared with the tptal population of ·cracks. The above reduction in probability will be less dramatic in bridges because the latter are subjected to repeated load- ings which cause crack widths to gradually increase during the life of the bridge. In addition, the design loading is more likely to be achieved on a bridge than on a building. However, considerably tess than 20% of the crack widths in a bridge should exceed the design value if the design is carried out in accordance with the Code formulae. Crack control in the Code General approach Although cracking due to such effects as the restraint of shrinkage and early thermal movement is a significant practical problem, it is cracking induced by applied loading that is used as a basis for design in the Code. However, the Code does require at least 0.3% of the gross concrete area of mild steel or 0.25$ of high yield steel to be provided to control restrained shrinkage and early thermal movement cracks. These percentages are rather less than those suggested by Hughes [185] and should be used with caution. The only crack widths that need to be calculated accord- ing to the Code are those due to axi~1 and flexural stress. resultants. Hence, flexural shear and torsional shear cracks do not have to be considered explicitly since, as discussed . in Chapter 6, it is assumed that the presence of nominal links control the widths of such cracks. Cracks due to applied loadings which cause axial and flexural stress resultants are controlled by limiting the spacing of the bars. This approach is identical to that adopted in BE 1173. The bar spacing should not generally exceed 300 mm, and should be such that the crack widths in Table 4.7 are not exceeded midway between the bars under the specified design loading. It should be noted that the Code gives a different method of ensurin,s",that the crack widths are not exceeded for each type of structural del1l~Jll. Loading Since the widths of cracks are controlled primar- ily for durability purposes, it is logical to define the design loading as that which can be considered to be virtually permanent rather than that of occasional but more severe loads. This is because, after the passage of occasional severe loads, the crack widths return to their values under permanent loading provided that the reinforcement reinained elastic during the application of the occasional load. Since the limiting reinforcement stress is 0.81", the reinforcement will remain elastic. Ai one time the drafters considered that 50% of HA loading should be taken to be permanent, together with pedestrian loading; dead load and superimposed dead load. When the appropriate partial safety factors were applied, the resulting design load was: dead load + 1.2 (superimposed dead load) + 0.6 (HA) + 1.0 (pedestrian loading) Subsequently, it, was decided to check crack widths under full HA loading but the partial safety factors were altered so that the design load became: dead load + 1.2 (superimposed dead load) + 1.0 (HA) + 1.0 (pedestrian loading) This design loading is given, in the general design section of Part 4 of the Code, with the requirement that the wheel load should be excluded except when considering top flanges and cantilever slabs. In addition, for spans less than 6.5 m, 25 units of HB loading with associated HA loading should be considered. This additional loading was introduced to comply with the requirement of Part 2 of the Code that all bridges should be checked for 25 units of HB loading. It is not clear whether top flanges and cantilever slabs should be loaded with 25 units of HB when they span less than 6.5 m. However, it would seem reasonable to design such slabs for the more severe of the local effects of the HA wheel load or 25 units of HB loading. The above loadings are very similar to those in BE 1173, with the exception that the latter document refers to 30 units of HB loading for loaded length less than 6.5 m. The implication in BE 1173 appears to be that the 30 units of HB loading should be applied to top flanges and cantilever slabs. Finally, the Code clause concerned with reinforced con- crete walls requires that any relevant earth pressure load- ings should be considered in addition to dead, superim- posed dead and highway loadings. Stiffnesses Although the design crack widths· and ioad- ings, referred to previously, are given in the general design section of Part 4 of the Code, the clauses concerned with crack width calculations appear much later in the Code under the heading: 'Spacing of reinforcement'. Guidance on the stiffnesses to be adopted in the calculations also appears under this heading. In all crack width calculations the Code requires _the elastic_~odulus of the· concrete to ,be a long-term v~lue .equal to-halfof the tabulated. short-term value. Hence a modular ratio Of about 13 to 16 would be used to calculate the strains ignoring tension stiffening. In certain situations which are discussed later in this chapter, the Code permit~ the strains ignoring tension stiffening to be reduced. It is thus necessary to determine the strain EIS in Fig. 7.1. This ~s best achieved by initially considering an axially re- lOforced cracked section (having a concrete area ofAc and a steel area ofAs) which is axially loaded. At a crack all of the applied force <tV) is carried by the steel: N =E" El As However, the average steel and concrete forces are given by: Ns =Es em As Nc =Aclcm where femis the average tensile stress in the concrete be- tween the cracks. But N= Ns + Ne :. E. El As =E" Em As + Aclem .'. Em =El - AelemlEs As Or Ets =AclemlEs As At the cracking load, fern is obviously equal to the tensile strength (f,) of the concrete. At higher loads, tests [183, 186, 187] indicate that lemreduces in accordance with lem =1,Iser 111 where fser is the steel stress at a crack at the cracking load and 11 is the steel stress at the load corresponding to the strain El' For an axially reinforced and loaded section it is obvious that Ae "'" bh where band h are the breadth and overall depth respectively. However, for a flexural member, it is necessary to define an effective area (Kbh) of concrete in tension over which the average stress lem acts. Hence, for flexure, and considering only surface strains E,•. = Kbh Iser f, 1E. As11 In order that strains at any depth (at) from the compres- sion face can be considered the above expression is modi- fied to Ets =Kbhf,erf, (at - x)IE"AsIl(h - x) where x is the neutral axis depth. In the Code, cracking is generally checked under HA loading only and the values of (YfL Yf3) at the ultimate limit state are 1.32 for dead load and 1.73 for HA loading, giving an average of 1.53; hence 11 "'" 0.871,,/1.53 =0.571". 89
- 52. However, implicit inthe tension stiffening fonnula in the Code is the assumption that I. =0.S8/yo This is because the fonnula was originally derived for CP ItO, for which I. =0.S8/y is correct. However, the difference between this value of 11 and the 'correct' value for the Code is negligible. Hence tts = Kbh Iserlt(a' - x)/Es A.(0.S8 /y) (h - x) Tests carried out by Stevens [188] indicated that, on average, KlserII0.S8E., = 1.2 X 10-3 N/mm2 • It is emphasised that this constant has the dimensions of N/mm2 • Thus Ets =1.2bh(a' - x) 1O-31As (h - x) fy Hence, the tension stiffening fonnula in the Code is obtained as Em = E1 - 1.2bh(a' - X) 10-3IA;(h - x)fy' ',-J (7.3) Beeby [189] has shown that equation (7.3) provides a reasonable lower bound fit to the instantaneous results of Stevens and a reasonable average fit to the latter's results obtained after long tenn loading of two years' duration. However, test results, under short tenn loading, reported by Rao and Subrahmanyan [186], Clark .and Speirs [183] and Clark and Cranston [187] show tension stiffening val- ues about one-third of those of Stevens and of those pre- dicted by equation (7.3). In addition, the latest CEB tem- sion stiffening equation [110] is in reasonable accord [187] with the data presented in [183], [186] and [187]. There is thus evidence to suggest that equation (1.3) overestimates tension stiffening. . Finally, equation (7.3) was originally derived with buildings in mind and thus the effects of repeated loading were not considered. Bridges are subjected to repeated loading and it is reasonable to assume that such loading reduces the tension stiffening. There is a lack of experi- mental data in this respect and, as an interim measure, it might be sensible to adopt the CEB recommendation [1 to] that tension stiffening under repeated loading should be taken as 50% of that under instantaneous loading. Tests on model solid [87, 126] lnd voided [71] slab bridges indicate that, for HB loading, the tension stiffening, as a proportion of the instantaneous value, is of the order of 60%, 50% and 40% after tOOO, 2000 and 4000 load applications respectively. These values are reasonably consistent with the CEB value. *Crack control calculations Beams Base, Read, Beeby and Taylor [190] have carried out tests on 133 reinforced concrete beams. They found that there was an average difference of only 13% between the crack control perfonnances of plain and defonned bars, and thus it is not necessary to have separate crack width fonnulae for the two types of bar. This point is particularly valid in 90 Load ~ / -Actual '/ ----Calculated on / cracked stiffness I I I I I I I I I I I I I I I I I I I I 'I Strain Fig. 7.4 Tension stiffening: high steel percentage view of the fact that crack widths are calculated only at positions mid-way between bars where the bar type has the least influence. Base et al. found that the distributions of crack width were Gaussian and that, for defonned bars, the mean crack width (wm) could be predicte.d by Wm =1.67 acr Em where aer is the perpendicular distance from the point where the crack width is to be predicted to the surface of the nearest reinforcing bar. The standard deviation of the crack width population at any strain was found to be 0.416 Wm. It is mentioned previously in this chapter that the proba- bility of exceedence adopted in the Code is 20%. In a Gaussian distribution, a. value of mean"plus 0.842 standard deviations is exceeded bf 20% of the population; thus the design crack width is g~ven by W = (1 + 0.842 x 0.416) 1.67acr Em = 2.3aer Em However, in the Code, tension stiffening is not taken into account for beams and it is thus assumed that Em = E1' Hence the following Code equation is obtained W =2.3aer El (7.4) The reason for ignoring tension stiffening in beams is that it is envisaged that reinforced concrete bridge beams would be heavily reinforced; in which case the load-strain relationship is as shown in Fig. 7.4. It can be seen that the tension stiffening effect is a small proportion of the actual strain and can be ignored. Equation (7.4) is very similar to the equation in BE 1/73; however, in the latter document the constant is not 2.3 but is 3.3 for defonned bars and 3.8 for plain bars. The value of 3.3 is appropriate to the 1% exceedence level [190], which is a much more severe criterion than the 20% level adopted in the Code. However, it should be remembered that the design crack widths specified in BE 1173 and the Code are also different. The BE 1173 value of 3.8 was obtained by increasing the value of 3.3 for defonned bars by 13% and rounding up [190]. Solid slab bridges .At one stage in the drafting of the Code, it was hoped to p~epare simple bar spacing rules which would obviate the need to carry out crack width calculations. However, it was found that, due to the large number of variables to be considered, it was possible to produce such rules only for . single sp!ln solid slab bridges. Consequently an exercise was carried out [191] in which a range of solid slab bridges was designed atthe ultimate limit state either by yield line" theory or in accordance with elastic momentfiellds. The bar spaCings required to control crack widths to the design values were then determined. Bar spacing tables based upon this exercise did appear in some drafts of the Code, but it Was eventually decided to simplify the bar spacing rules considerably. Consequently, the Code simply states that the longitudinal bar spacing should not exceed ISO mm and the transverse bar spacing should not exceed 300 mm. These values apply to continuous slabs in addi- tion to single span slabs and, in view of this, the author would suggest that 'longitudinal' be interpreted as primary and 'transverse' as secondary in order to avoid excessive cracking in certain situations. For example, the Code implies that the spacing of the transverse bars in the region of an interior support of a slab bridge continuous over dis- crete columns could be 300 mm; however, in such situ- ations the' transverse bending moments could be large enough to cause excessive cracking if the bar spacing were not considerably less than 300 mm. It is thus suggested that the bar spacing rules should be interpreted with 'engineering judgement'. From present design experience, it is known that, in some situations, a longitudinal bar spacing of 150 mm does not control the crack widths to the design values unless the reinforcement is used at a stress less than the maximum permitted in BE 1173 (= 0.56Iy)' The drafters anticipated that when designing in these situations in accordance with the Code, bar spacings less than 150 mm would be forced upon the designer because of the large amount of reinforcement that· would be. required to satisfy the ultimate limit state criterion. An examination of [191] shows that if bar spa~ings were to be calculated for single span slab bridges; they would generally be greater than the Code values of 150 and 3(}() mm, except for elastically designed slabs having skew angles greater than about 30° and yield line designed slabs having skew angles greater than about 15°. The Code val- ues ·should thus' be used with caution if the skew angle exceeds these values. Finally, it is worth mentioning that the reason for having bar spacing rules is to avoid can-ying out specific crack width calculations. However. since the Code requires stress calculations to be carried out at the serviceability limit state, albeit at a different design load to that for the crack width calculation, a large proportion of the data required for a crack width calculation would already be calculated. Thus, to a certain extent, the advantage of quoting bar spacing rulel'; is lost and litlle extra effort is involved in carrying out a complete crack width calcu- lation by, for example, applying the general procedure sug- gested by Clark [192, 193]. an =strass normal to line of symmetry .- - - . - _.1--.- I 0' = sttess tangential I t ofree surface . of void or soff.it (a) Elastic stresses In a quadrant (b) Crack pattern (194] Flg.7.S(a),(b) Cracks in voided slabs Sidefece Soffit Slab bridges with longitudinal circular voids Cracks due to longitudinal bending The spacing and -widths of cracks due to longitudinal bending are very simi- lar to those occurring, at the sam~ steel strain, in solid slabs. Thus, the Code limits the spacing of longitudinal reinforcement to 150 mm which is the same as the value for solid slabs. This spacing was not checked for voided slabs by either calculation or experiment; however, the author would suggest that, with the same precautions as discussed for solid slabs,it is a reasonable value to adopt in practice. Cracks due to transverse bending The stress raising • effects of the voids result in the response of a voided slab to transverse bending being very different to that of a solid slab. Cracking due to transverse bending in voided slabs has been described in some detail by Clark aod Elliott [194] and their findings can be summarised as follows: 1. Linear elastic analyses indicated that: (a) The peak stress occurs at the crown of the. void as shown in Fig. 7.5(a). It is here, on the inside ·of the void, that the first crack should initiate. (b) The peak stress on the outer face does not occur at the void centreline, but at approximately the quarter points of the void spacing as shown in Fig. 7.5(a). Thus cracks propagating from the outer face should initiate at the quarter points. 2. The theoretical predictions were confinned by tests on transverse strips of voided slabs. An actual crack pat- tern is shown in Fig. 7.5(b). 91
- 53. Reinforcement / / ... ... 'f (a'Pattern 1 ----:::-:.:---:: ----Crack pattern 1 FIR. 7.6(R),(b) Phm views of crock pntlcrnR in ~(llid slah .The tests indicated that in order to obtain a controlled nack pallern. such as th~lt shown in Fig. 7.s(b), with no cracks passing completely through the flange. it was necessary to hnve at least I% of transverse reinforcement in the fhmge. This reinforcement should he cnlculatcd as a percentage of the mi/limum !lange area. This miilimum reinforcement percentage requirement is given in the 'Code f~'r .predominantly tension flanges, together with an upper IIII1It of 1500 mm2 /m. The laller value was introduced to avoid excessive amounts of reinforcement in slabs ~ith very t!lick flange... Clark and Elliott I194J have suggested that, In such cases, it may be preferable to consider the minimum flange thickness as having two critical layers: one Inyer would be adjacent to the outer face and the other adjacent to the crown of the void. The thickness of each layer .would he equal to twice the relevant cover plus the ba.r .dlameter. and. each layer would he provided with a nlJllImum of 1% of reinforcement. This suggestion is simi- lar to that recommended by Holmberg lJ95J. In the case of predominantly compression flanges,· the Code requires that the area of transverse reinforcement should be the lesser of 1000 mm2 /m or 0.7% of the minimum !lange area. These values were chosen because the test~ reported by Clark and Elliott 1194) indicated that the moment at which su(:h steel would be stressed to 2.10 N/mm 2 would be greater than the cracking moment of the section. Thus. if cra(:king did occur due to an unex- pected severe loading situation. the reinforcement would not yield suddenly and a controlled crack pattern would occur. In addition tn the ahove limitations on the area of trans- verse .reint~)rcement. it is necessnry to limit the spacing of the rCll1fOrCelllenf. The Code states that the spacing should not exceed the solid slah value of 300· mm nor twice the minimum !lange thickness. The latter crit.erinn was intro- duced to discourage the use of large diameter hars if thin flanges were adopted. since Stich flanges are sutijected to p:n1icularly large stress concentrations. . ' Flanges In order to discuss crack control in the !langes of beam and ~Iah. l'cllular slah anll hox hcum construction it is first (b, Crack pattern 2superposed on crack pattern 1. Crack pattern 1 Crack pattern 2 / necessary to consider. in general terms. cracking in slabs.. Beeby [196] has investig~ted cracking In slabs spanning one ~~y and fu.und that there are two basic crackpatternll (see I'Ig. 7.6): I. A pattern controlled by the deformation imposed on - the section. . 2. A pattern controlled by the proximity of the re- inforcement. . Beeby [196] proposed a theory which adequately pre- dicts the properties of the two patterns and their interac- tion. In addition. formulae for predicting the widths of cracks at any point on a slab were derived. These formulae are too complicnted for design purposes nnd thull Beeby [182] reduced them to the following single design formula w= 1 + K (0", - Cmi';) 2 (h - x) . , where em/" is the minimum cover to the tension steel and KI and K2 are constants which depend upon the probahility . of exceedence of the design crack width. The appropriate values of KI and K2 are 3 and 2. respectively, for the 20% prohability of exceedence adopted in the Code. Hence, the following crack width equation, which isl given in the Code. is obtained (7.5) It should be noted that the strain. allowing for tension stiffening, is used because it is considered that slabs are relatively lightly reinforced and have load- strain relation- ships similar to that shown in Fig. 7.1 ..In such l~ases, the tension stiffening effect is significant and should be calcu- lated from equation (7.3): although. as mentioned earlier in this chapter, the tension stiffening could be reduced hy repeated loading. Equation 17_5) wa~ved from tests in which the re- inforcement was perpendicular to the cracks. The general problem of CflIt-k control wht'n the reinforcement is not perpendicular to the cracks has been wlIsidercd hy Clark ;% //////. A, F t= P ~I-'1-. ?: (a' Right bridge (b) Skew bridge, orthogonal transverse reinforcement (c' Skew bridge, skew transverse reinforcement .I ILongitudinal beams/webs A, • Transverse reinforcement Crack due to local transverse bending FiJ.l. 7.7(1I)-·(e) Cracking in fllInges 11921. This study showed that equation (7.5) is applicable provided that the crack width calculation is carried out in a direction perpendicular to the crack and that all of the re- inforcement should be resolved to an equivalent area of steel perpendicular to the crack. However, equation (7.3), which predicts the tension stiffening effect. has not been checked for reinforcement which is not perpendicular to .the cracks. In view of the complex stress state in the con- crete between the cracks. it would be advisable to ignore tension stiffening when considering such arrangements of reinforcement. r.lIl1gitudinal reinforcement in .!7anges Although it is not clear in the Code, it is unnecessary to calculate the spacing of the longitudinal reinforcement in a flange because a simple bar spacing rule is given. A bar spacing rule for tension flanges was derived by noting that, for HA loading, the design load at the ser- viceability limit state is about 70% of that at the ultimate limit state: thus EI = 0.7 x O.R7f..1200 x IO~ If it is assumed that f", """ 0.8 fl' then I'm"'" n.R x 0.7 x O.R7f,J200 X 10" Hence, a reasonahleupper limi.t to Em is 0.001. Since the strain is very nearly constant over the depth of a flange., the neutral axis depth 't) for the flange tends to infinity. Hence, equation (7.5) reduces to w =3acr E",and, forE".= o.onI and a design value of w of 0.2 mm (see Table 4.7). an = 67 mm, which results in a bar spacing of about 150 mm. This maximum har spacing is given in the Code forplJ!dominantly t~nsion flanges together with a value of 300 nllll for predominantly compression flanges. The latter value simply complies with the general maximum bar spacing given in the Code. Trall,~verse reiltforcement ill flanges When drafting the clauses for crack control in flanges. it was assumed that the effects of local bending in the flanges would generally dominate the transverse bending effects. Thus the major principal moment in a flange would act very nearly per- pendicular to the longitudinal beams or webs. Tests [R7, 126. 1921 on slab bridges and slab elements indicate that, lit the serviceability limit state. it is rellsonable to assume that the cracks are perpendicular.to the principal mome~t direction. There is no reason to assume that the cracks in a slab acting as a flange would not form in the same direc- tion and, thus, they would be very nearly parallel to the longitudinal beams or webs. It can be seen from Fig. 7.7 that, in the case of aright bridge or of a skew bridge in which the transverse rein- forcement is perpendicular to the beams or webs, the rein- forcement would be perpendicular to the service load cracks. Hence, equations (7.3) and (7.5) can be applied, to a unit width of slab perpendicular to the cracks, with A, equal to the llreaof transverse reinforcement per unit width. However. for a skew bridge in which the transverse reinforcement is parallel to the supports and such that it makes an angle (IX) to the perPendicular to the crack; as shown in Fig. 7.7(c), equations (7.3) and (7.5) cannot be applied directly. It is first necessary to calculate an effec- tive area of reinforcement perpendicular to the crack hy using either equation (7.1) or (7.2). IIHuet, equation (7.1) is adopted in the Code because the drafters had top flanges primarily in mind and it is most likely that these will be cracked on one face only and in one direction. Further- more, since the longitudinal reinforcement would generally
- 54. be parallel tothe longitudinal beam or webs, equation (7.1) reduces to An = At COS4~ whereAt is area of transverse reinforcement per unit width. HenceAs , in equation (7.3), should be replaced by An and the latter value used to calculate the neutral axis depth and the strain in a direction perpendicular to the cracks. Fi~aIly, the tension stiffening equation is not dependent upon bar spacing because it was derived from tests on beams; whereas, in the Code, it is used only for slabs. One would expect tension stiffening in slabs to depend upon the bar spacing and, indeed, test data indicate such a depen- dence for large bar spacings. However, tests reported by· Clark and Cranston [187] show that the influence of bar spacing is insignificant provided that the bar spacing does not exceed about 1.5 times the slab depth. Since such large spacings are unlikely to occur in a bridge, the tension stif- fening equation can be applied to flanges. However, the reservations expressed earlier, regarding skew reinforce- ment and repeated loading, should be considered. Columns If tensile. str~sse~I.!I~_ i~~o'luitttf;·llietT·~the column should be consIdered as a beam for crack control purposes and equation (7.4) used. Walls If tensile stresses occur in a reinforced concrete wall, then it is obviously reasonable that the wall should be consid- ered as a slab for crack control purposes. The Code takes suc~ an approach and also distinguishes between the two exp<:>~ure conditions (see Table 4.7) which are .applicable to a"Vall. Severe exposure This condition includes surfaces in con- tact with backfill and is thus appropriate to. the. back faces of retaining walls and wing walls. The design crack width is 0.2 mm which is also the value for soffits. Hence the Code permits the bar spacing rules for slab bridges, of 150 and 300 mm for longitudinal and transverse reinforcement, respectively, to be adopted for walls subjected to a severe exposure condition. However, it should be noted that calculations for walls were not carried out to check specifically that the spacings of 150 and 300 mm would be reasonable. Very severe exposure A leaf pier is an example of a wall subject to the effects of salt spray; hence, its exposure condition is classed by the Code as very severe. The bar spacing rules for slab bridges are appropriate only to the severe exposure condition and it is thus necessary to carry out crack width calculations for walls if the exposure con- dition is very severe. Thus equation (7.5) should be used, in conjunction with equation (7.3). Bases Except for the statement that 'reinforcement need not be provided in the side faces of bases to control cracking' , the 94 Code is rather vague regarding crack control in bases. The relevant clause states that the method of checking crack widths depends on the type of base and the design assump- tions. Before discussing this statement, it should be men- tioned that, although reinforcement is generally provided in the side faces of deep members to control cracks for aesthetic purposes, it is not necessary to do this in bases because they are generally buried. In drafting the Code clause on crack control in bases it was intended that the various components of a base should be checked for cracking in accordance with the most appropriate of the procedures given for other struc- tural elements. It was intended that 'beam components' should be checked by applying equation (7.4), and that 'slab components' (e.g., spread footings) should be con- sidered as follows: 1. If a moderate or severe exposure condition (see Table 4.7) is appropriate, then apply the bar spacing rules, of 150 and 300 mm, for slab bridges. 2. If a very severe exposure condition is appropriate, then apply equations (7.3) and (7.5). The reasons for these recommendations are the same as those discussed previously in connection with walls. It is obvious that 'engineering judgement' is required when checking crack widths in bases. Prestressed concrete stress limitations *Seams In Chapter 4, limiting ~alues of prestressing steel stresses and concrete compressive and tensile stresses are given. It is emphasised in Chapter 4 that, although prestressing steel stress criteria are given in the Code, another clause essentially states that these can be ignored because it is not necessary to calculate tendon stress changes due to the effects of applied loadings. Concrete stresses can be calculated by applying conven- tional elastic theory; but the calculation of tensile stresses in Class 3 members requires some comment. Although a Class 3 member is, by definition, cracked at the service- ability limit state, it is considered to be uncracked for the purposes of calculating stresses. It is permissible to do this because the hypothetical tensile stresses in Table 4.6(a) were calculated from test data by assuming elastic uncracked behaviour. Hence, for stress calculation pur- poses, a cracked Class 3 member is considered in exactly the same way as an uncracked Class 1 or 2 member. It should be noted that a disadvantage of the above approach to design is that, because the section is actually cracked in practice, the actual compressive stress exceeds the value calculated for an uncracked section. Thus the actual compressive stress could exceed the allowable com- pressive stress from Table 4.3(a) although the calculated compressive stress might not. Thus the author would sug- gest that, when designing Class 3 members, it would be good practice not to stress the concrete to its allowable limit in compression. Slabs In an uncracked slab (Class 1 or 2), conventional elastic theory can be applied in the usual way to calculate the principal concrete stresses. However, the approach adopted for cracked Class 3 beams, in which hypothetical tensile stresses appropriate to an uncracked beam are cal- culated, would not, in general, be correct for cracked Class 3 slabs. This is because the hypothetical tensile stresses in Table 4.6 were calculated from test data for beams and, although probably applicable to slabs in which the prestressed and non-prestressed steel are parallel to the principal stress direction, they would not be applicable to slabs in which these directions do not coincide. Test data are not available for such situations and the author would suggest the following interim measures which are based upon consideration of equation (7.1). 1. Beeby and Taylor [123] have studied, theoretically and experimentally, cracking in Class 3 members. Their theoretical expression for the hypothetical ten- sile stress is a function of the area of prestressing steel perpendicular to the crack. Equation (7.1) suggests that if the area of prestressing steel per unit width is A ps and it is at an angle ~ to the major principal stress direction. then the equivalent area of prestressing steel perpendicular to the crack is Aps COS4~. A study of Beeby and Taylor's work indicates that it is reason- able to assume that a reduction of steel area from Aps to A COS4~ results in a reduction of the hypotheticalps tensile stress from, say, !ht to !ht COS2~. Thus the author would suggest that, when the prestressing ten- dons are at an angle ~ to the major principal stress directions, the limiting hypothetical tensile stresses of Table 4.6(a) should be multiplied by COS2~. 2. The depth factors of Table 4.6(b) should not be modi- fied. 3. Any additional conventional reinforcement should be considered in terms of an equivalent area of re- inforcement perpendicular to the crack by using equa- tion (7.1). This equivalent area should be used to cal- culate the increase of hypothetical tensile stress which is permitted when additional reinforcement is present. 4. If the final limiting hypothetical tensile stress is less than the appropriate limiting tensile stress for a Class 2 member, the section will be uncracked and should he treated as a Class 2 member. Losses in prestressed concrete "*Initial prestress In order that excessive relaxation of the stress in the ten- dons will not occur, the normal jacking force should not exceed 70% of the characteristic tendon strength. How- ever. in order to overcome the effects offricfdon, the jack- ing force may be increased to 80% of the characteristic tendon strength, provided that the stress- strain curve of the tendon does not become significantly non-linear above a stress of 70% of the characteristic tendon strength. The above requirements are essentially identical to those of BE 2173. ~oss due to steel relaxation If experimental data are available, then the loss of pre- stress in the tendon should be taken as the relaxation. after 1000 hours duration, for an initial load equal to the jacking force at transfer. This value is taken because it is approxi- mately equal to the relaxation, after four years, for an ini- tial force of 60% of the tendon strength: this force is roughly the average tendon force over four years [197]. In the absence of experimental data, the relaxation loss should be taken as 8% for an initial prestress of 70% of the characteristic tendon strength, decreasing linearly to 0% for an initial prestress of 50% of the characteristic tendon strength. These values were based upon tests on plain cold drawn wire [112]. - The Code also refers to losses given in Part 8 of the Code, but this appears to be a mistake because no losses are given in Part 8. The Code losses are essentially the same as those of BE 2173 except for the reduced loss which may be adopted . if the initial jacking force is less than 70% of the charac- teristic tendon strength. Loss due to elastic deformation of the concrete The elastic loss may be calculated by the usual modular ratio procedure. For post-tensioned construction, the elastic loss may be calculated either, exactly, by considering the tensioning sequence, or, approximately. by multiplying the final stress in the concrete adjacent to the tendons by half of the modular ratio. The latter procedure is an approximate method of allowing for the progressive loss of prestress which occurs as the tendon forces are gradually transferred to the concrete. Elastic losses calculated in accordance with the Code will thus be identical to those calculated in accordance with BE 2173. Loss due to shrinkage of concrete For a normal exposure condition of 70% relative humidity the Code gives shrinkage strains of 200 and 300 micro- strains, for post-tensioning and pre-tensioning, respectively. These values are identical to the CP 115 values. However, for a humid exposure condition of 90% relative humidity, the Code also gives shrinkage strains of 70 and 100 microstrains, for post-tensioning and pre-tensioning. respectively. Loss due to creep of concrete According to Neville r198]. the relationship hetween c~ep of concrete and the stress to strength ratio is of the form shown in Fig. 7.8. It can be seen that, for a particular concrete. creep is directly proportional to stress for stress
- 55. Stress Strength ~0.7 ------------- ... 0.3--- Creep Fig. 7.8 Creep-stress/strength relationship to strength ratios less than approximately 0.3. The Code limit of one-third, below which creep can be considered to be proportional to stress, is thus reasonable. The Code gives values for the specific creep strain (creep strain per unit stress) of: 1. For pre-tensioning; the lesser of 48 x 10-8 and 48 x 10-0 x 4O!fc,per N/mm2 • 2. For post-tensioning; the lesser of 36 x 10-8 and 36 x 10-8 x 4O!fc; per N/mm2 .. In the above, tel is the cube strength at the time of transfer. The values of specific creep strain are identical to those in BE 2/73 and CP 115. If the compressive stress anywhere in the section exceeds one-third of the cube strength at transfer, the specific creep strain should be increased as indicated by Fig. 7.8. The Code gives a factor, by which the above specific creep strains should be multiplied, which varies linearly from unity at a stress-to-strength ratio of one-thi~d to 1.25 at a ratio ofone-half (the greatest allowable ratIo under any conditions). This factor is less than that indi- cated by tests [198] on uniaxial compression specimens because, generally, it is parts, rather than the whole, of a cross-section which are subjected to stresses in excess of one-third of the cube strength. The above requirements are identical to those of BE 2173 but it should be noted that the Code also states that half of the total creep may be assumed to take place in the first month after transfer and three-quarters in the first six months. Loss due to friction in duct In post-tensioning systems, losses occur due to friction in the duct caused by unintentional variations in the duct profile ('wobble') and by curvature of t~e duct. The Code adopts the conventional friction equation [199] Px = Po exp(- Kx + IJXlrps) where P" prestressing force at a distance x from the jack Po = prestressing force in the tendon at the jack radius of curvature of ductr".. K = wobble factor /.L = coefficient of friction of tendon. 96 In the absence of specific test data, the Code gives a general value of K of 33 x 10-4 per metre, but a value of 17x 10-4 may be used when the duct former is rigid or rigidly supported. The Code gives the following values for /.L: 1. 0.55 for steel moving on concrete. 2. 0.30 for steel moving on steel. 3. 0'.25 for steel moving on lead. The above values of K and fA. ~ identical to those in CP 115 and were originally based upon the test data of Cooley [199]. However, the Construction Industry Re- search and Information Association has now assessed all of the available experimental data on K and /.L values and has recommended [200]: 1. i.t values of 0.25 and 0.20 for steel moving on steel and lead, respectively. 2. For other than long continuous construction, the K values given in the Code. 3. A K value of 40 X 10-4 per metre for long continuous construction because it has been suggested [201] that the Code value of 33 x 10-4 underestimates friction losses in such situations. The author presumes that a value· of 20 x 10-4 would be adopted for rigidly sup- ported ducts. . Finally, where circumferential tendons are used, the fol- lowing Il values are recommended in the Code: 1. 0.45 for steel moving in smooth concrete. 2. 0.25 for steel moving on steel bearers fixed to the concrete. 3. 0.10 for steel moving on steel rollers. These values are the same as those in CP 115 and were originally suggested by Creasy [202].' Other losses The Code does not give specific data for calculating the losses due to steam curing nor, in post-tensioning systems, the losses due to friction in the jack and anchorages and to tendon movement at the anchorages during transfer. Instead reference is made, as in CP 115, to specialist advice. Deflections General It is explained in Chapter 1 that there is not a limit state of excessive deflection in the Code since a criterion, in the form of an allowable deflection or span-to-depth ratio, is not given. However, in practice, it is necessary to calcu- late deflections in order, for example,to calculate rotations in the design of bearings. The Code thus gives methods of calculating both short and long term curvatures in Appen- dix A of Part 4. The procedure is to calculate the curvature assuming, Ecfc Compressive stresses in concrete Tensile stresses- in cracked concrete Level of centroid of tension !.~ reinforcement 1N/mm2 (short-term) 0.55 N/mm2 (long-term) Fig. 7.9 Tension stiffening for curvature calculations first, an uncracked section and / second, a cracked section. The larger of the two curvatures is then~dopted. Lo~g term effects of creep are allowed for byustng an effective elastic modulus, for the concrete which is less than the short-term modulus. Shrinkage is allowed for by separately calculating the curvature due to shrinkage. Short-term curvature The short-term elastic moduli for concrete, which are tabu- lated in the Code, should be used to calculate the short- term curvature under imposed loading. The calculation for the uncracked section is straightfor- ward. However, the calculation for the cracked section is more complicated because of the need to allow for tension stiffening. It is mentioned elsewhere in this chapter, that when cal- culating crack widths in flanges, tension stiffening is allowed for by subtracting a 'tension stiffening strain' from the reinforcement strain calculated by ignoring tension stif· fening. However, when calculating deflections, a different approach is adopted: a triangular distribution of tens~le stress is assumed in the concrete below the neutral aXIS, with a stress of 1 N/mm2 at the centroid of the tension reinforcement, as shown in Fig. 7.9. The stress of 1 N/mm2 was derived from the test results of Stevens [188] which are referred to earlier in this chapter. . The CP 110 handbook [112] implies that the peutral aXIs should be calculated from the stress diagram of Fig. 7.9 by employing a trial-and-error approach; but, as suggested by Allen [203], it is simpler to adopt the following procedure, which involves little error: 1. Calculate the neutral axis depth (,x) ignoring tension stiffening. . 2. Calculate the extreme fibre concrete compressive stress due to the applied bending moment. 3. Calculate the extreme fibre strain (Fe) by dividing the stress by the elastic modulus of the concrete (EJ. 4. Calculate the curvature (E,JX). Allen gives equations which aid the above calculations for csctangular and T-sections. . Long-term curvature The curvature calculated above would increase under long-term loading due to creep of the concrete. Creep in compression is allowed for by dividing the short-term elas- tic modulus of the concrete by (1 + <1» where <I> is a creep coefficient. It is emphasised that <I> does not, in this chap- ter, refer to the same coefficient as it does in Chapter 8. In fact, the creep coefficient ~ referred to in Chapter 8 is iden- tical to the creep coefficient </> referred to above. An appropriate value of <I> can be determined from the data given in Appendix C of the Code, provided that sufficient prior knowledge of the concrete mix and curing conditions are known. Since <I> depends .upon many variables, the CP 110 handbook [112] gives a table of <I> values which may be used in the absence of more detailed information. As an aitc1'l1alive, a simplified method of obtaining <I> is given by Parrott [204]. Creep of the concrete in tension is allowed for by re- ducing the tensile stress in the concrete, at the level of the centroid of the tension reinforcement, from its short-term value of 1 N/mm2 to a long-term value of 0.55 N/mm 2• The latter value was again derived from the test results of Stevens [188]. The long-term curvature can be calculated by following the same procedure as that given earlier for the short-tenn curvature. Shrinkage curvature The Code gives the following expression for calculating the curvature ('IjIs) due to shrinkage. 'IjIs =Po Er./d (7.6) where d is the effective depth, Ec.. is the free shrinkage strain and Po is a coefficient which depends upon the per- centages of tension and compres~ion steel. The free shrinkage strain can be determined from the data given in Appendix C of the Code. However, as is also the case for <1>, the free shrinkage strain depends upon many variables. Thus the CP 110 handbook [112] suggests a value of 300 x 10-/ for a section less than 250 mm thick and a value of 250 x 10-/ for thicker sections. The handbook also indicates how the shrinkage develops with time. As an alternative, Parrott [204] gives a graph for estimating shrinkage. Values of the coefficient Po are given in a table in Appendix A of the Code. The tabulated values are based upon empirical equations derived by Branson [205] from two sets of tt~st data. Hobbs [206] has compared the Code values of Po with these data and also with an additional set of data. He found that the Code values are conservative and are particularly conservative, by a factor of about 2, for lightly reinforced and doubly reinforced sections. In view of this, Hobbs suggests an alternative procedure for calculating shrinkage curvatures, which is based upon theory rather than empirical equations, and which gives good agreement with test data [206]. General calculation procedure . The following general procedure is suggested in the Code for calculating the iong-telm curvature:
- 56. Moment Short term Shortterm Creep due to . • transient permanent permanent Shrmkage,_ ~4 .14 .,4 '" Total Permanent Curvature Fig. 7.10 Calculation of long term curvature I. Calculate the instantaneous (i.e., short·term) curva- tures under the total design load and under the peona- nent design load. " 2. Calculate the long4 term curvature under the perlllilnent design load. 3. Calculate the difference between the instantaneous curvatures under the total and permanent design loads. This difference is the curvature due to the transient design loads. Add this value to the long-term curva- ture under the permanent load. 4. Add the shrinkage curvature. The above procedure is logical and its net effect is ilIus- trated in Fig. 7.10. . Examples ~7.1 Reinforced concrete The top slab ofa beam and slab bridge has been designed at the ultimate limit state and is shown in Fig. 7.1 J. The Table 7.J Example 7.1. Nominal load effects la) Local moments (kNmlm) , ~ Load Dead Superimposed dead HA wheel 45 HB units 25 HB units (h) Glohal effects Load Transverse 0.45 0.22 10.8 11.7 7.56 Transverse moment (kNm/m) Longitudlna~ 0.0 0.0 7.20 7.65 5.26 Longitudinal compre~,ive stress in slab (N/mm2) Top Bottom Superimposed dead 0.0 0.33 0.14 1.60 0.25 2.55 1.48 HA 0.0 3.90 HA wheel 6.0 0.61 45 HB units 43.0 6.24 25 HB units 23.3 3.63 98 Transverse steel Y16-100 Longitudinal steel Y16-200 Fig. 7.11 Example 7.1 eharacteristic strengths of the reinforcement and concrete are 425 and 30 N/mm2 , respectively. . Mid~way between the beams, the local bending moments given in Table 7.1 (a) act together with the global ~ffects given in Table 7.1(b). The effects due to HB include those due to associated HA. It is required to check that the slab satisfies the service- ability limit state criteria under load combination 1. *Genera/ From Table 2 of the Code, short-term elastic modulus of concrete ::i::: 28 kN/mm2. Use a long-term value of 28/2 = 14 kN7mmB for both cracking and stress calculation (Clause 5.3.3.2). Modular ratio = 200/14 = 14.3. Each design load effect will be calculated as nominal load effect x Yfl. x Yf3. Details of these partial safety fac- tors are given in Chapters 3 and 4. '*Cracking Due to transverse bending HA design moment == (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) . + (l0.8)(l.2)(0:~3) + (6.0)(1.2)(0.83) = 17.5 kNmlm 2S HB design moment ~ (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) + (7.56)(1.1)(0.91) + (23.3)(1.1)(0.91) =31.6 kNm/m ~rit,((atwoment == 31.6 kNmlm Area of bottom ~teel = area of top steel = 2010 mm2/m . Ifelastic neutral axis is at depth x, then by taking first moments of area about the neutral axis: (1/2)(1000) Xl + (13.3)(2010) (x - 38) = (14.3)(2010)(152 - x) :. x:::: 62.2 mm .Second moment of area, I, about the neutral axis is given by: (113)(1000)(62.2)3 + (13.3)(2010)(24.2)2 + (14.3)(2010)(89.8)2 = 0.328 x lOP mm 4 /m Bottom steel stress = (14.3)(31.6 x 106 )(89.8)/0.328 x 109 =.124 N/mm2 Soffit strain ignoring tension stiffening is. . £1 = (124/200 x 10:1)(137.8/89.8) = 9.51 x 10-4 From equation (7.3), soffit strain allowing for tension stif- fening is Em = 9.51 X 10-4 - (1.2)(1000)(200)(10-3)/(2010 x 425) = 6.70 x 10-4 Maximum crack' width occurs mid-way between bars . where the distance (acr) to the nearest bar is given by "acr = /502 + 482 - 8 = 61.3 mm. From equation (7.5), crack width is (3)(61.3)(6.70 x 10- 4 ) = 0.094 mm w = 1 + (2)(61.3-40)/(200-62.2) From Table of the Code, allowable crack width is 0.20 mm. !.JttDue to longitudinal bending The longitudinal bars are in a region of predominantly compressive flexural stress and thus the Code maximum spacing is 300 mm. This exceeds the actual spacing of 200 mOl. *Stresses Limiting values The limiting stresses are 0.5 x 30 = 15 N/mm2 for concrete in compression, and 0.8 x 425 = 340 N/mm2 for steel in tension or compression. Due to transverse bending HA design moment = (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) + (10.8)(1.2)(1.0) + (6.0)(1.2)(1.0) = 20.9 kNm/m 45 HB design moment = (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) + (11.7)(1.1)(1.0) + (43.0)(1.1)(1.0) = 60.9 kNrnlm' Critical moment = 60.9 kNmlm From crack width calculations, x = 62.2 mm, I = 0.328 X 109 mm4 /m Maximum concrete stress == (60.9 x 108 )(62.2)/0.328 x 109 = 11.5 N/mm2 <15 N/mm2 Steel stresses are (14.3)(60.9 X 106 )(89.8)/0.328 x 109 = 238 N/mm2 tension and (14.3)(60.9 x 106 )(24.2)/0.328 x 109 = 64 N/mm2 compression Both < 340 N/mmI Due to longitudinal effects Maximum coml?ressive stresses occur under 45 units of HB loading. Assume section to be uncracked and ignore reinforcement. Extreme fibre stresses due to nominal local moment = ±(7.65 x 103 )(6)/(200)2 = ± 1.14 N/mm2 Net top fibre design stress is (0.33)(1.2)(1.0) + (6.24)(1.1)(1.0) + (1.14)(1.1)(1.0) = 8.51 N/mm2<15 N/mm2 1500 175 900 600 ~I Fig. 7.12 Example 7.2 Maximum tensile stress in reinforcement occurs under the HA wheel load. The longitudinal global stresses under the HA wheel load and superimposed dead load are: Top = 0.61 + 0.33 = 0.94 N/mml Bottom = 0.25 + 0.14 = 0.39 N/mm2 These are equivalent to an in-plane force of 200 (0.94 + 0.39)/2 = 133 kN/m, and a moment of [(0.94 - 0.39)/2] (2002 ) 10-3 /6 = 1.83 kNm/m. Total nominal moment = 7.2 + 1.83 = 9.03 kNrnlm. If the in-plane force is conserva~ively ignored, the design load effect is a moment of (9.03)(1.2)(1.0) = 10.8 kNm/m. Area of bottom steel = area of top steel = 1010 mm2 /m. If elastic neutral axis is at depth x and is above the top steel level (112)(1000) x 2 = (14.3)(1010)[(136 - x) + (54 - x)] :. x = 50.7 mm and is above the top steel. Second moment of area = (1/3)(1000)(50.7)3 + (14.3)(101Q)(85.32 + 3.32) = 0.149 X 109 mm4 /m Bottom steel stress = (14.3)(10.8 x 108 )(85.3)/0.149 x 109 =88 N/mm2 < 340 N/mm2 '*7.2. Prestressed concrete A bridge deck is constructed from the pre-tensioned I-beams shown in Fig. 7.12. Each beam is required to resist the moments, due to nominal loads, given in Table 7.2. Determine the prestressing force and eccentricity required to satisfy the serviceability limit state criteria, under load combination 1, for each of the three classes of prestressed concrete. Assume that the losses amount to Table 7.2 Example 7.2. Design data Moment (kNm) Stress (N/mm2 ) Load Nominal Design Top Bottom Dead 477 477 +3.67 - 5.24 Superimposed dead 135 162 + 1.25 - 1.78 HA .949 949 +7.30 - 10.42 HB + Associated HA 1060 1060 +8.15 -11.64
- 57. 11% at transferand finally amount to 34%. The concrete is of grade 50 and, at transfer, the concrete strength is 40 N/mm2 • General Area = 475250 mm2 Centroid is 529.2 mm from bottom fibre. Second moment of area = 4.819 x 1010 mm4 Bottom fibre section modulus = 9.11 x 107 mm3 Top fibre section modulus = 13.0 x 107 mm3 From Table 24 of the Code, the allowable compressive stress for any class of prestressed concrete is 0.33 x 50 = 16.5 N/mm2 • From Table 25 of the Code, the allowable compressive stress at transfer for any class of prestressed concrete is 0.5 x 40 = 20 N/mm2 • The usual sign convention of compressive stresses beb1g positive and ten'sHe stresses being negative is adopted: The design moments are calculated as nominal moments X 'tIL x 'tf3; ang are Dead load = (477)(1.0)(1.0) = 477 kNm . Superimposed dead load = (135)(1.2)(1.0) = 162 kNm HA load = (949)(1.2)(0.83) = 949 kNm HB + associated HA load = (1060)(1.1)(0.91) = 1060 kNm Hence HB loading is the critical live load. The design moments, together with the extreme fibre stresses which they induce, are given in Table 7.2. Class 1 The allowable tensile stresses in the concrete are zero under the design service load, and 1 N/mm2 at transfer or under a service load condition of dead load alone. The critical stresses, for the given section and loading, are the transfer stresses. If the prestressing force before any losses occur is P and its eccentricity is e, then at the top 0.89P 0.89Pe -1 = 475 250 - 13 x 107+ 3.67 :~ -2.494 x 106 = P(1 - 3.656 x 1O-3e) and at the bottom O.89P O.89Pe___ + _____ 5.24 = 20 475250 9.11 x 107 :. P(l + 5.217 X 10-3 e) = 13.478 X 106 Table 7.3. Example 7.2. Design stresses Stresses (N/mm2) Design load Class 1 Solving simultaneously P = 4087 kN and e = 440 mm (i.e., 89.2 mm from soffit). ' The resulting stresses under the various design loads are given in Table 7.3. Class 2 The allowable tensile stresses in the concrete are: 1. Under the design service load, 3.2 N/mm2 (from' Table 26 of the Code or 0145 Ifcu)' 2. Under dead plus superimposed dead load, zero. 3. At transfer, 2.9 N/mm2 (from Table 26 of the Code or 0.45 1Tct). The critical stress, for the given section and loading, is the tensile stress at the bottom fibre under the full design ser· vice load. Hence 0.66 P 0.66 Pe ' -3.2 475 250+ 9.11 x 107 - 5.24 - 1.78 - 11.64 ... P(l + 5.217 X 10-3 e) = 11.132 X 108 Assuming the same eccentricity (440 mm) as that adopted for Class 1: P ='3378 kN (i.e., 83% of that for Class 1) T~e resulting stresses under the various design loads are given in Table 7.3. Class 3 The allowable tensile stresses in the concrete are: 1. Under the design service load, the basic hypothetical tensile stress from Table 27 of the Code or Table 4.6(a) of this book is 5.8 N/mm2 for a design crack width Of 0.2 mm. The section is 900 mm deep and the stress of 5.8 N/mm2 'has to be multiplied by 0.75 (from Table 2~ of the Code or Table 4.6(b) this book) to give a, final stress of 4.35 N/mm2 • It will be assumed that no conventional reinforcement is present and thus the hypothetical stress cannot be increased. 2. Under dead plus superimposed dead load, zero. 3. At transfer, a Class 3 member must be treated as if it were Class 2 and thus the allowable stress is 2.9 N/mm2 • The critical stress, for the given section and loading, is Class 2 Class 3 Top Bottom Top Bottom Top Bottom Transfer PS+DL PS+DL+SDL PS+DL+SDL+LL PS final prestress DL dead load SDL superimposed dead load LL HB + associated HA load 100 -1.0 +0.2 +1.5 +9.6 +20.0 +13.5 +11.7 + 0.1 - 0.2 + 0.8 + 2.1 +10.2 +15.6 + 0.1 +14.1 +10.2 + 1.0 + 9.1 + 8.4 + 2.3 + 7.3 - 3.2 +10.4 - 4.4 again the tensile stress at the bottom fibre under the full design service load. Hence 435 = 0.66 P + 0.66 Pe -5.24 - 1.78 - 11.64 - . 475250 9.11 x 107 :. PO + 5.217 X 10-3 e) = 10.304 X 10 8 Assuming the same eccentricity (440 mm) as that adopted" for Classes 1 and 2: P = 3127 kN (i.e., 77% and93%'of those for Classes 1 and 2, respectively). The resulting stresses under the various design loads are given in Table 7.3. General comments In an actual design, the section should also be checked at the ultimate limit state. In addition other sections along the length of the beam would need to be checked, particularly with regard to transfer stresses. The losses have been assumed to be the same for each class but they will obviously be different because of the .4iffetenrprestressing forces. Thedifferenn:lasses of prestressed concrete have been catered for by merely altering the.prestress but, in practice, consideration would also be given to altering the cross· section. I I i
- 58. Chapter 8 Precast concreteand composite construction Precast concrete The design of precast members in general is based upon the design methods for reinforced or prestressed concrete which are discussed in other chapters. Bearings and joints for precast members are considered as part of this chapter. Bearings The Code gives design rules for two types of bearing: cor- bels and nibs. *Corbels A corbel is defined as a short cantilever bracket with a shear span to depth ratio less than 0.6 (see Fig. B.l(b». The design method proposed in the Code is based upon test data reviewed by Somerville [207]. The method assumes, in the spirit of a lower bound design method, the equilibrium strut and tie system shown in Fig. B.1(a). The calculations are carried out at the ultimate limit state. In order to assume such a strut and tie system it is first necessary to preclude a shear failure. This is done by proportioning the depth of the corbel, at the face of the supporting member, in accordance with the clauses cover- ing the shear strength of short reinforced concrete beains. The force (F,) to be resisted by the main tension re- inforcement can be determined by considering the equilib- rium of the strut and tie system as foHows (the notation is. in accordance with Fig. 8.1). V =Fe sin ~ P, = H + Fe cos ~ = H + V cot ~ (B.1) Somerville [207] suggests that ~ can be determined by assuming a depth of concrete x having a constant compres- sive stress of 0.4 feu and considering equilibrium at the face of the supporting member, as shown in Fig. 8.l(b) and (c). An iterative procedure is suggested in which x/d is first assumed to be 0.4; in which case cot ~ = a.,lO.&l and the compressive force in the concrete is given by Fe = 0.4 f,'u bx cos P 102 '''~'' , v H Steel F, .......-.ti~e~-". (a) Strut and tie v A F, ...-+---_~ d / x / I / x~/ (b) Deterrnination of ~ F, d-x/2 Fe cos ~ =O.4feu bx (c) Horizontal forces at A-A Fig. 8.I(a)-(c) Corbel strut and tie system L..----......-r'---,Corner shears d .£.d 3 Main reinforcement .', ··Horizontal links Bars to anchor links (a) Local bearing failure (b) Horizontal links Fig. 8.2(a),(b) Corbel detailing The horizontal component of this force is 0.4 feu bx COS2~ For equilibrium, the tensile force (F1) in the reinforcement must equal this horizontal component of the concrete com. pressive force; thus F, = 0.4 feu bx cos2 ~. A new value of ~can then be calculated from cot ~ =aj(d - x/2) This procedure can be continued iteratively. (8.2) (8.3) It should be noted that both Somerville [207] and the CP 110 handbook [112] give the last term of equation (8.2), incorrectly, as cos ~ instead of cos2 ~. The area of reinforcement provided should be, not less than '0.4% of the section at the face of the supporting member. This requirement was determined:empirically from the test data. It is important that the reinforcement is adequately anchored: at the front face of the corbel this can be achieved by welding to a transverse b~r or by bending the main bars to form a loop. In the latter cas€( the bearing area of the load should not project beyond the straight por- tion of the bars, otherwise shearing of the comer of the corbel could occur as shown in Fig. B.2(a). Theoretically no reinforcement, other than that referred to above, is required. However, the Code also requires horizontal links, having a total area equal to 50% of that of . the main reinforcement, to be provided as shown in Fig. B.2(b). Horizontal rather than vertical links are required because the tests, upon which the design method is based, showed that horizontal links were more efficient for values of ajd < 0.6. The Code states that the above design method is appli- cable for ajd < 0.6.The implication is thus that, if ajd ~ 0:6, the corbel should be designed as a flexural can~ tilever. However, the test data showed that the design l.ink In supporting~ member Fig. 8.3 Nib ~ . V(Line of action is at outer edge of loaded area) '---4-... Imaginary compressive strut method based upon a strut and tie ~ystem was applicable to avid values of up to 1.5. In view of this the CP 110 hand- book [112] suggests that, as a compromise, the method can be applied to corbels havingavldvalues of up to 1.0. Finally, in order to prevent a local failure under the load, the test data indicated that the depth of the corbel at the outer edge of the bearing should not be less than 50% of the depth at the face of the supporting member. *Nibs The Code requires nibs less than 300 mm deep to be designed as cantilever! slabs at the ultimate limit state to resist ,a bending moment of Vav (see Fig. B.3). Clarke [20B] has shown by tests that this method is safe, but that an equilibrium strut and tie system of design is more appropriate for nibs which project less than 1.5 times their depth. The distance av is taken to be from the outer edge of the loaded area (i.e. the most conservative position of the line of action of V) to the position of the nearest vert- ical leg of the links in the supporting member. The latter position was chosen from considerations .of a strut and tie system in which the inclined compressive strut is as shown in Fig. B.3. Detailing of the reinforcement is particularly important in small nibs and the Code gives specific rules which are· confirmed by the test results of Clarke [20B].
- 59. I,, ---Code -----Williams (205) 1.0 0.5 ,,, o 0.2 Fig.8.4 Bearing stresses Bearing stress ~~ 0.4 Nonnally the compresSive stress at the ultimate limit state between two contact surfaces should not exceed 40% of the characteristic strength of the concrete. If the bearing area is well-defined and binding re- inforcement is provided near to the contact area, then a tri- axial stress state is set up in the concrete under the bearing area and stresses much higher than 0.4 feu can be resisted. The Code gives the following equatio!l for the limiting bearing stress (fb) at the ultimate limit state = 1.5 feu 1 + 2yp,)Yo (8.4) where ypo and Yo are half the length of the side of the loaded area and of the resisting concrete block respec- tively. This equation was obtained from a recommendation of the Comite Europeen du Beton, but the original test data is not readily available. However, Williams [209] has reviewed all of the available data on bearing stresses and found that a good fit to the data is given by fb =0.78 feu (yp,)y?)-O.441 ·(8.5) In Fig. 8.4, the latter expression, with a partial safety factor of 1.5 applied to feu, is compared with the Code equation. It can be; seen that the latter is conservative for smal(Ioaded areas.' Th~ Code recogrises that extremely high bearing stress- es can be developed in certain situations: an example is in concrete hinges [210], the design of which will probably be covered in Part 9 of the Code. At the time of writing, Part 9 has not been published and thus the Department of 104 0.6 O.S 1.0 Transport's Technical Memorandum on Freyssinet hinges [211] will, presumably, be used in the interim period. This document pennits average compressive stresses in the throat of a hinge of up to 2 feu. -¥Halving joints Halving joints are quite common in bridge <,:onstruction and the Code gives two alternative design methods at the ulti- mate limit state: one involving inclined links (Fig. 8.5(a» and the other involving vertical links (Fig. 8.5(b». Each method is presented in the Code in tenns of reinforced concrete, but can also be applied to prestressed concrete. Inclined links When inclined links are used, the Code design method assumes the eqUilibrium strut and tie system, shown in Fig. 8.5(c), which is based upon tests carried out by ReynOlds [212]. The Code emphasises that, in order that the inclined links may contribute to the strength of the joint, they must cross the line of action of the reaction Fv' For eqUilibrium, the vertical component of the force in the links must equal the reaction (Fv). Hence Fv =Asv fyv cos 9 where Am fyv and 9 are the total area, characteristic strength and inclination of the links respectively. A partial safety factor of 1.15 has to be applied to fyv and hence the Code equation is obtained. ,Inclined links Horizontal reinforcement ----~~-r-------'''"--t,.. to resist moment at root of half end cantilever plus any horizontal forces Main tension reinforcement (a) Inclined links - 1--' TFv Minimum end cover to main tension reinforcement" (b) Vertical links Strut (concrete) Tie - .. (main reinforcement) L--_ _ _ _I (c) Strut and tie system Fig. 8.5(a)-(c) Halving joint F,. =As,. (0.87 fVl') cos e (8.6) It should be noted that, theoretically, any value of emay be chosen.: However, the crack which initiates failure forms at about 45° as shown in Fig. 8.5(a), and thus it is desirable to incline the links at 45°. The adoption of such a value of f) is implied in the Code. Reynolds suggested [212] that, althougl} his tests showed that it is possible to reinforce a joint ',so that the maximum allowable shear force for the full beam section could be carried, it would be prudent in 'practice to limit the reaction at the joint to the maximum allowable shear force for the reduced section (with an effective depth of do). This limit was suggested in order to prevent over- reinforcement of the joint and. hence, to ensure a ductile joint. One might expect that the above limit would imply Fv = vllbd,,: instead, the Code gives a value of Fv =4vrbdo. The reason for this is that, in a draft of CP 110, 4vc was the maximum allowable shear stress in a beam and was thus equivalent to VII' which occurs in the final version of CP 10 and the Code. However, the clause on halving joints was written when 4"" was in CP 110 and was not subse- quently altered when V" (= 0.75 It,,) was introduced (see Chapter 6). Finally, .the horizontal component of the tensile force in the inclined links has to be transferred to the bottom main tension reinforcement. Nonnally one would assume that such transfer occurs by bond, but the Code assumes that the transfer occurs in two ways: half by bond with the con- crete, and half by friction between the links and bars. For the latter to occur the links must be wired tightly to the main bars. The division of the force transfer into bond and friction was not based upon theory but was an interpreta- tion of the test results. Since only half of the force has to be t-rtlnsferred by bond, the anchorage length of the main tension reinforcement (lsb) given in the Code is only half of that which Would be calculated by applying the anchor- age bond stresses discussed in Chapter 10., Vertical links As an alternative to the inclined link system of Fig. 8.5(a), a halving joint may be reinforced with vertical links as shown in Fig. 8.6(b). The vertical links should be designed by the method described in Chapter 6 and anchored around longitudinal reinforcement which extends to the end of the beam as shown in Fig. 8.5(b) [212]. Horizontal reinforcement in half end The Code does not require flexural reinforcement to be designed in half ends. Howt(ver, it would seem prudent to design horizontal reinforcement to resist the moment at the root of the half end cantilever. Such reinforcement should also be designed to resist any horizontal forces. Composite construction General In the context of this book, composite construction refers to precast concrete acting compositely with in-situ con- crete. Very often in bridge construction, the precast con- crete is prestressed and the in-situ concrete is reinforced. The design of composite construction is complicated not only by the fact that, for example, shear and flexure cal- culations have to be carried out for both the precast unit and the composite member, but because additional calcu- lations, such as those for interface shear stresses, have'also to be carried out. The various calcula'tions are now discussed individually. Ultimate limit state Flexure The, flexural design of precast elements can be carried out in accordance with the methods of Chapter 5. In the case of a composite member, the methods of Chapter 5 may be applied to the entire composite section provided that hori- zontal shear can be transmitted, without 'excessive slip, across the interface of the precast and in-situ concretes. The criteria for interface shear stresses, discussed in Chap- ter 4, ensure that excessive slip does not occur.
- 60. r In-situ 1 Composite :-- .~..- '.1-- _.-. - . - 'ceritroic:r-' ....- .._- ' •. . -_ .•.- .._- . -- ....- ... --1_-+#. - . - . - . - . _. -14----.1_ Precast (a) Section (b) Shear stross dueto Va1 Fig. 8.6(a)-(d) Shear in composite beam and slab section *Shear It is not necessary to consider interface' shear at the ulti- mate limit state because the interface shear criteria dis- cussed in Chapter 4, for the serviceability limit state, are intended to ensure adequate strength at the ultimate limit state in addition to full composite action at the service- ability limit state. Thus it is necessary to consider only vertical shear at the ultimate limit state. The fact that interfac'e shear is chec! at the serviceability limit state and vertical sh~ar at the ultimate limit state causes a minor problem in the organis- ation of the calculations and introduces the possibility of errors being made. It is understood that in the proposed amendments to CP 110, which are being drafted at the time of writing, the interface shear criteria are different to those in the Code, and the calculations will be carried out at the ultimate limit state. The design of a precast element to resist vertical shear can be carried out in accordance with the methods described in Chapter 6. However, the design of a com- posite section to resist vertical shear is more complicated and there does not seem to be an established method. The latter fact reflects the lack of appropriate test data. The appropriate Code clause merely states that the design rules for prestressed and reinforced concrete should be applied and, when in-situ concrete is placed between precast prestressed units, the principal tensile stress in the prestressed units should not anywhere exceed 0.24 !feu (see Chapter 6). It is thus best to consider the problem from first principles. In the folIowing, it will be assumed that the precast units are prestressed and thus the problem is one of determining the shear capacity of a prestressed-. reinforced composite section when it is flexurally uncracked (Vr~) and also when it is 'flexurally cracked (VeT)' The terminology and notation are the same as those of Chapter 6. It is emphasised that the suggested pro- cedures are tentative and that test data are required. There are two general cases to consider: 'I. Precast prestressed beams with an in-situ reinforced concrete top slab to form a composite beam and slab bridge. • 2. Precast prestressed beams with in-situ concrete placed between and overthe beams to form a composite slab. . .106 (c) Shear stress dueto Vc2 (d) Total shear stress *Composite beam and slab It is suggested by Reynolds, Clarke and Taylor [161] that Veo for a composite section should be detennined on the basis of a limiting principal tensile stress of 0.24 I!cu 9t the centroid of the composite .section. This approach is thus similar to that for pre- stressed concrete. The shear force (Vel) due to self weight and construc- tion l~ads produces a shear stress distribution in the precast member as shown in Fig. 8.6(b). The shear stress in the . precast member at the level of the composite centroid is!,. The additional shear force (VeZ) which acts on the com- posite section produces a shear stress distribution in the composite section as shown in Fig. 8.6(c). The shear stress at the level of the composite centroid, due to Vc2 , is fs. The total shear stress distribution is shown in Fig. 8.6(d) and the shear stress at the level of the com- posite centroid is (I. +f'.;~. ' Let the compressive stress at the level of the composite centroid due to self weight and construction loads plus 0.8 of the prestress befcp' (The factor of 0.8 is explained in Chapter 6.) Then the major principal stress at the c~m posite centroid is given by II =-f,./2 + 1(f'cpI2)2 + (I. + I's)2 This stress should not exceed the limiting value of f, = 0.24/!cu' Hence, on substitutingfl =f" and rearranging f .• = II? + f,.,J,- I. But 1'..=V,.zAy!Ib, where I, band Aji refer to the com- posite section. Hence Ib Vc2 = A)i [II/ + f,.,,!, - I.] (8.7) Finally, the total shear capacity (Veo).is given by Veo = Vet + Ve2 (8.8) The above calculation would be carried out at the junc~ tion of the flange and web of the composite section if the centroid were to occur in the flange (see Chapter 6). The above approach, suggested by Reynolds, Clarke and Taylor, seems reasonable .except, possibly, when the section is subjected to a hogging bending moment. In such ..... '-....~.. a situation the reinforced concrete in-situ flan~e might be racked and it could be argued that the in-Situ concrete ~hould ;hen be ignored. However, as explained in Chapter 6; a significant amount of shear c~n be transmitted by dowel action of reinforcement, which would be present .in the flange, and by aggregate interlock a~ro~s the cracks. It thus seems reasonable to include the m-sltu flange. as part of a homogeneous section, whether or not the section is subjected to hogging bending. , When calculating Vcr> which is the shear capacity of the member cracked in flexure, it is reasonable, if the member is subjected to sagging bending, to apply the prestress.ed concrete clauses to the composite section because the tn- situ flange would be in compression, However, when the .section is subjected to hogging bending, it would be con- servative to ignore the cracked in-situ flange and to carry out the calculations by applying the prestressed concrete clauses to the precast section alone. . When the prestressed concrete clauses are applied to the composite section, the author would suggest that the fol- lowing amendments be made. 1. Classes 1 and 2: (a) Replace equation (6.12) with M, =Mb(l - y"I,Jyc!") + (8.8) (0.37 /f.'"b + 0.8 [p,) I,.Iyc where the subscripts b andc refer to the precast beam and the composite sections respectively and Mb is the moment acting on the precast beam alone. Equation (8.8) is derived as follows. The design value of the compressive stress at the bot- tom fibre due to prestress and the moment acting on the precast section alone is 0.8 Ip ,- MbYb11b The additional stress to cause cracking is (see Chapter 6) 0.37 /t:.uh + O.81p , - MhYb11h Thus the additional moment, applied to the com- posite section to cause cracking, is: Ma = (0.37 /I."b + a.8lp, - MhYb1h)1e1Ye The total moment is the cracking moment: M, = Ma + Mb which on simplification gives equation (8.8) (b) Replace the term (£I !1c,,) in equation (6.11) with (d 17 b +d· If .) where the subscripts band ih vJC14 I ..;JCUI , refer to the precast beam and in-situ concrete respectively, and db and d; are defined in Fig. 8.7(a). In this context db is measured to the centroid of all of the tendons. 2. Class 3: (a) Calculate Mo from M" = Mb (1 - Ybl/v"/h) + O.8Ip ,I,./y,. (8.9) which can be d~rived in a similar manner to equa- tion (8.8). (b) Replace the term (I,.d) in equation (6.16) with dif db x y·'liQ·····~(, ..', ". X Centroid of tendons, or all steel in tension zone; as appropriate, (a) Beam and slab Fig. 8.7(a),(b) Composite sections. (vcbdb +vc/d;), where Vcb and Vc' are the nominal shear stresses appropriate to Icub and lell' respec- tively. db is measured to the centroid of all of the steel in the tension zone. Finally, it is suggested that the maximum allowable shear force should be calculated from V'4 =0.7Sb(dh/tllh + d, !f:.u,) (8.10) db is measured to the centroid of all of the steel in the tension zone. The above suggested approach is slightly different to that of BE 2173. :If::Composite slab In order to 'comply.strictly. with the ~od.e when calculating Veo for the composite sectIOn, the prmcl- pal tensile stress at each point in the precast beams should be checked. However, the author would suggest that it is adequate to check only the principal tensile s~ress at the centroid of the composite section. When carrytng out the V calculation no consideration is given to whether the0 0 ' • in-situ concrete between the beams is cracked. This IS because the adjacent prestressed concrete restrains the in- situ concrete and controls the cracking [113]. This effect is discussed in Chapter 4 in connection with the allowable flexural tensile stresses in the in-situ concrete. Provided that the latter stresses are not exceeded, the in-situ and precast concretes should act compositely. It could be argued that, when subjected to hogging bending, the in-situ concrete above the beams shoul,d be ignored. However, the author would suggest that It be included for the same reasons put forward for including it in beam and slab composite construction. The general Code approach differs from the approach.of BE 2173, in which ?,~eas of plai~ in-situ concrete. w.h.lch develop principal tensile stresses m excess of the hmltmg value are ignored. . When calculating Vcr, the in-situ concrete could be flex- urally cracked before the precast concrete cracks. It is not clear how to calculate Vcr in such a situation although tbe author'feels that the restraint to the in-situ concrete pro- vided by the precast heams should enab~e one to a~ply the prestressed concrete clauses to the entire composite sec- tion. However, in view of the lack of test data, the aut~or would suggest either of the following two conservative approaches. 1. Ignore all of the in-situ concrete and apply the pre- stressed concrete clauses to the precast beams alone. as proposed by Reynolds, Clarke and Taylor [161]. 107
- 61. 2. Apply thereillforced concrete clauses to the entire composite section. Finally. it is suggested that the maximum allowable shear force for a section (h" + hi) wide should be calcu- . lated from v" = O.75d(b,. /1,'"1> + hi /i.'"I) (8.11 ) where h" and b; are defined in Fig. 8.7(b). The above suggested approach is different to that of BE 2173 in which a modified form of equation (6.11) is adopted for Vcr' Serviceability limit state *General It is mentioned in Chapter 4 that the stresses which have to he checked in a composite member are the compressive and tensile stresses in the precast concrete, the com- pressive and tensile stresses in the in-situ concrete and the shear stress at the interface between the two con- cretes. For the usual case of a prestressed precast unit acting compositely with in-situ reinforced concrete. the stress calculations are complicated by the fact that different 10ld levels have to be adopted for checking the various stresses. This is because. as discussed in Chapter 4, different yp values have to be applied when carrying out the various stress calculations. It is explained in Chapter 4 thaI the value of Yp implied by the Code is unity for all stress calculation under any load except for HA and HB loading. For the latter loadings, YP is unity for all stresscalcu- lations except for the compressive and tensile stresses in the prestressed concrete. when it takes the values of 0.83 and 0.91 for HA and HB loading respectively. It is emphasised that the actual values to he adopted for the 'various stress calculations are not 'stated in the Code. The values quoted above can be deduced from the assumption that it was the drafters' intention that. in general. Yp should be unity at the serviceability limit state. However, there is an impor- tant implication of this intention, which may have been overlooked by the drafters. The tensile stresses in the in- situ concrete have to be checked under a design load of 1.2 HA or 1.1 HB (because Yfl. is 1.2 and 1.1 respectively and YP = 1.0) as compared with 1.0 HA or 1.0 HB when designing in accordance with BE 2173. despite the allow- able tensile stresses in the Code and BE 2173 being identi- cal. This suggests that. perhaps. the drafters intended th~ Code design load to be 1.0 HA or 1.0 HB and, hence, the Yrl values to be 0.83 and 0.91 respectively. This argument also throws some doubt on the actual intended values of YI~ to he adopted when checking interface shear stresses. In conclusion. although it is not entirely clear what value of Yf3 should he adopted for HA and HB loading, and it could be argued that (Yf', Yp) should always be taken to be unity. the following values of Yp will be assumed. I . HA - 1.0 except for stresses in prestressed concrete when it is 0.83. lOR o Average interface shear stress = v" = O.4fcubih;lbel 1--,-- Minimum moment section O.4fcub;h; Precast ,...,._-, .....L .......---...-----+1 Maximum moment . section Fig. 8.8 Interface shear at ultimate limit state 2. HB - 1.0 except for stresses in prestressed concrete when it is 0.91. t Compressive and tensile stresses Compressive and tensile stresses in either the precast or In-situ concrete should not exceed the values discussed in Chapter 4. Such stresses can be calculated by applying elastic theory to the precast section or to the cOIl)posite section as appropriate. The difference between the elastic moduli of the two concretes should be allowed for if their strengths differ by more than one grade. It is emphasised that the allowable flexural tensile stress- es of Table 4.4 for in-situ concrete are applicable only when the in-situ concrete is in direct contact with precast prestressed concrete. If the adjacent precast unit is not pre- stressed then' the flexural cracks in the in-situ concrete should be controlled by applying equation (7.4). The allowable flexural tensile stresses in the in-situ con- crete are interpreted differently in the Code and BE 2/73. In the Code it is explicit,ly stated that they are stresses at the contact surface; when~!ls in BE '2173 they are applicable to all of the in-situ concrete. but those parts of the latter in. which the allowable stress is exceeded are not included in the composite section. In neither the Code nor BE 2173 is it necessary to calcu- late the flexural tensile stresses in any in-situ concrete which is not considered, for the purposes of stress calcu- lations, to be part of the composite section. :¥Interface shear stresses In terms of limit state design, it is necessary to check inter- face shear stresses for two reasons. I. It is necessary to ensure that, at the serviceability limit state, the two concrete components act compositely. Since shear stress can only be transmitted across the interface after the in-situ concrete has hardened, the loads considered when calculating the interface shear stress at the serviceability limit state shoul~ consist only of those loads applied after the concrete has hard- ened. Thus the self weight of the precast unit and the in-situ concrete should be considered in propped but not in unpropped construction. In addition. at the ser- viceability limit state it is reas~ahle to calculate the interface shear stress by using elastic theory; hence Vir = VS,llh,. (8.12) CompreSSion Tension -------&- - - -... E; ~---------- ·1 -------------lI 1---e::~~~~~~~:..=.:t=-:.:::;-=============l -"""""Original I positions ~!/ 1 . "." .. ,..,." '"~''' "I I I I I.I I1--___-.£_---' __________________ ...1 I-----_____.!:.I?!!...-..____--.~ E; =free shrinkage of in-situ concrete Eb/ = creep plus shrinkage of top of precast concrete Ebb =creep plus shrinkage of bottom of precast concrete Strains Fig. 8.9 Differential shrinkage pIllS creep I = horizontal interface shear stress = shear force at point considered = first moment of area, about the neutral axis of the transformed composite section, of the concrete to one side of the interface = second moment of area of the transformed composite section be = width of interface. 2. It is necessary to ensure adequate horizontal shear strength at the ultimate limit state. The shear force per unit length which has to be transmitted across the interface is a function of the normal forces acting in the in-situ concrete at the ultimate limit state. The lat· ter forces result from the total desigll load at the ulti- mate limit state. If a constant flexural stress of 0.4 feu (see Chapter 5) is assumed at the point of maximum moment, then the maximum nonnal force is 0.4Jeu bi hi where bi is the effective breadth of in-situ concrete above the interface and hi is the depth of in- situ concrete or the depth to the neutral:8xis if the latter lies within the in-situ concrete. It is conservative to assume that the normal force is zero at the point of minimum moment, which will be considered to be dis- tance I from the point of maximum moment. Hence an interface shear force of 0.4feu bi hi must be transmit- ted over a distance I (see Fig. 8.8); thus the average interface shear stress is (8.13) A Technical Report of the Federation Internationale de la Precontrainte [213] suggests that the average shear stress should be distributed over the length I in pro- portion to the vertical shear force diagram. It can be seen from the above that, in order to be thorough, two calculations should be carr.ied out: I. An elastic calculation at the serviceability limit state .which considers only those loads which are applied after the in-situ concrete hardens. Stresses 2. A plastic calculation at the ultimate limit state which considers the total design load at the ultimate limit state. However, the Code does not require these two calcu- lations to be carried out: instead a single elastic calculation is carried out at the serviceability limit state which con- siders the total design load at the serviceability limit state. This is achieved by taking V in equation (8.12) to be the shear force due to the total design load at the serviceability limit state. This procedure, though illogical, is intended to ensure that both the correct serviceability and ultimate limit state criteria can be satisfied by means of' a single calculation. ' Finally, as mentioned previously in this chapter, the interface shear clauses in CP 110, which are essentially identical to those in the Code, are being redrafted at the time of writing. It is understood that the new clauses will require the calculation to be carried out only at the ultimate limit state by considering the total design load at this limit state. This procedure, if adopted, would be more logical than the existing procedure. ~Differential shrinkage "When in-situ concrete is cast on an older precast unit, much of the movement of the latter due to creep and shrink- age will already have taken place, whereas none of the shrinkage of the in-situ concrete will have occurred. Hence, at any time after casting the in-situ concrete, there will be a tendency for the in-situ concrete to shorten rela- tive.to the precast unit. Since the in-situ concrete acts compositely with the precast unit, the latter restrains the movement of the former but is itself strained as shown in Fig. If.9. Hence, stresses are developed in both the in-situ and precast concretes as shown in Fig. 8.9. It is possible to calculate the stresses from considerations of equili- brium, and the necessary equations are given by Kajfasz, . Somerville and Rowe [113].
- 62. - 'l Support I ituconcrete ~=ii$§~~~1Es22S£t:-Continuity reinforcement (a) Beams supported on pier 'l Support I, '95&S.-/ ity fill reinforcement ransversely pre-stressed in-situ concrete crosshead (b) Beams embedded in crosshead Fig. 8.10(a),(b) Continuity in composite construction it is emphasised that the above calculations need to ~e carried out only at the serviceability limit state since the stresses arise from restrained deformations and can thus be ignored at tl)e ultimate limit state. The explanation of this is given in Chapter 13 in connection with a discussion of thermal stresses. The Code does not give values of Yf'~ and Yt3 to be used when assessing the effects of differential shrinkage at the serviceability limit state; but it would seem to be reason- able to use 1.0 for each. The most difficult part of a differential shrinkage calcu- lation is the assessment of the shrinkage strains of the two concretes, and the creep strains of the precast unit. These strains depend upon many variables and, if data from tests on the concretes and precast units are not available, esti- mated values have to be used. For beam and slab bridges in a normal environment the Code gives a value of 100 x W'(' for the differential shrinkage strain, which is defined as the difference between the shrinkage strain of the in-situ concrete and the average shrinkage plus creep strain of the precast unit. This value was based upon the results of tests on composite T-beams reported by Kajfasz, Somerv'ilIe and Rowe [113]. The test results indicated dif- ferential shrinkage strains which varied greatly, but the value quoted in .the Code is a reasonable value to adopt for design purposes. Although it is not stated in the Code, it was intended that the current practice [6. 113J of ignoring differential shrinkage effects in composite slabs, consisting of preten- sioned beams with solid in-situ concrete infill, be con- tinued. Finally. the stresses induced by the restraint to differen- tial shrinkage are relieved by creep and the Code gives a reduction factor of 0.43. The derivation of this factor is discussed later in this chapter. 110 Continuity Introduction A multi-span bridge formed of precast beams can be made continuous by providing an in-situ concrete diaphragm at each support as shown in Fig. 8.10(a). An alternative form of connection in which the ends of the precast beams are not "supported directly on piers but instead are embedded in a transversely prestressed in-situ concrete crosshead, with some tendons passing through the ends of the beams, has been described by Pritchard [214] (see Fig. 8.10 (b». A bridge formed by either of the above methods is stati- cally determinate for dead load but statically indeterminate for live load; and thus the in-situ concrete di!lphragm or crosshead has to be designed to resist the hoggil'ng moments which will occur at the. supports. The design rules for reinforced concrete can be applied to the dia- phragm, but consideration should be given to the following points. Moment redistribution Tests have been carried out on half-scale models of con- tinuous girders composed. of precast I-sections with an in-situ concrete flange and support diaphragm, as in Fig. 8.IO(a), at the Portland Cement Association in America [215, 216]. It was found that, at collapse, moment redistributions causing a reduction of support moment of about 30% could be achieved. It thus seems to be reasonable to redistribute moments in composite bridges provided that the Code upper limit of 30% for reinforced concrete is not exceeded. '*Flexural strength The Code permits the effect of any compressive stresses due to prestress in the ends of the precast units to be ignored when calculating the ultimate flexural -strength of connections such as those in Fig. 8.10. This recommendation is based upon the results of tests carried out by Kaar, Kriz and Hognestad [215]. They car- ried out tests on continuous girders with three levels of prestress (zero, 0.42 fcy/ and 0.64 fey') and three percen- tages of continuity reinforcemeflt (0.83, 1.66 and 2.49). It was found to be safe to ignore the precompression in the precast concre(e except for the specimens with 2.49% re- inforcement. In addition, it was found that the difference between the flexural strengths calculated by, first, ignoring and. second, including the precompression was negligible except for the highest level of prestress. Kaar, Kriz and Hognestad thus proposed that the precompression be ignored provided that the reinforcement does not exceed 1.5%. and the stress due to prestress does not exceed 0.4 fc)'1 (i.e. about 0.32.r'II)' Although the Code does not quote these criteria, they will generally be met in prac- tice. In addition to the above tests. good agreement between calculated and observed flexural strengths of continuous connections involving inverted T-beams with ad.ded in-situ concrete has been reported by Beckett [217). Shear strength The shear strength of continuity connections of the type shown in Fig. 8.1O(a) has been investigated by Mattock and Kaar [218]. who tested fifteen half-scale models. The reinforced concrete connections contained no shear rein- forcement, but none of the specimens failed in shear in the connection. The actual failures were as follows: thirteen by shear in the precast beams, one by flexure of a precast beam and one by interface shear. When designing, to resist shear, the end of a precast beam which is to form part of a continuous composite bridge, it should be rcmcmbercu that the end of the beam will be subjected to hogging bending. Thus flexural cracks could form at the top of the beam; consequently this region should be given consideration in (he shear design. Sturrock [219] tested models of continuity joints which simulated the type shown in Fig.8.10(b). The tests showed that no difficulty should be experienced in re- inforcing the crosshead to ensure that a flexural failure, rather than a shear failure in either the joint or a precast beam away from the joint, would occur. Serviceability limit state Crack widths and stresses in the reinforced concrete dia- phragm can be checked by the method discussed for re- inforced concrete in Chapter 7. Since the section in the vicinity of the ends of the pre- cast beams is to be designed as reinforced concrete, it is almost certain that tensile stresses will be developed at the tops of the precast beams. If the beams are prestressed, the Code implies that these stresses should not exceed the allowable stresses, for the appropriate class of prestressed member, given in Chapter 4, This means that no tensile stresses are permitted in a Class 1 member: this seems rather severe in view of the fact that any cracks in the in-situ concrete will be remote from the tendons. Conse- quently, the CP 110 handbook [112] suggests that the ends of prestressed units, when used as shown in Fig. 8.10, should always be considered as Class 3 and, hence, crack- ing permitted at the serviceability limit state. : Shrinkage and creep , The deflection of a simply supported composite beam changes with time because of the effects of' differential shrinkage and of creep due to self weight and prestress. Hence, the ends of a simply supp0l1ed beam rotate as a function of time, as shown in Fig. 8.11 (a), and, since there is no restraint to the rotation, no bending moments are developed in the beam. But, in the case of a beam made continuous by providing an in-situ concrete dia- phragm, as shown in Fig. 8.10, the diaphragm restrains the rotation at the end of the beam and bending moments are developed as shown in Fig. 8. II(b). A positive rotation occurs at the end of the beam because of creep due to prestress and thus a sagging moment is developed at the connection. The sagging moment is relieved by the fact that negative rotations occur as a result of both differential shrinkage and creep due to self weight, which cause hogging moments to be (a) Simply supported (b) Continuous Fig. 8.11 Long-term effects developed at the connection. Nevertheless, a net sagging moment at the connection will generally be developed. Since this sagging moment exists when no imposed load- ing is on the bridge, bottom reinforcement, as shown in Fig. 8.IO(a), is often necessary, together with the usual top reinforcement needed to resist the hogging moment under imposed loading. The provision of bottom rein- forcement has been considered by Mattock [220]. In order to examine the influence of creep and shrinkage on connection behaviour, Mattock [220] tested two con- tinuous composite beams for a period of two years. One beam was provided with bottom reinforcement in the dia- phragm and could thus transmit a significant sagging moment, whereas the other beam had no bottom re- inforcement. The latter beam cracked at the bottom of the diaphragm after about one year and the behaviour of the beam under its design load was adversely affected. Mattock found that the observed variation due to creep and shrinkage of the centre support reactions could be pre- dicted by the 'rate of creep' approach [198]. He thus sug- gested that this approach should be used to predict the support moment, which is of particular interest in design. The 'rate of creep' approach, which is adopted in the Code, assumes that under variable stress the rate of creep at any time is independent of the stress history. Hence, if the ratio of creep strain to elastic strain at time t is ~ and the stress at time 1 is J, then the increment of creep strain ( bEe) in time &1 is given by (8.14) whereE is the elastic modulus. <In the continuity problem under consideration, it is more convenient to work in terms of moment (M) and curvature (t!); and thus, by analogy with stress and strain bt! = (MI El)b~ (8.15) The effects of creep and shrinkage are discussed in detail in reference [220]. In the following analyses, they are treated less rigorously but with sufficient detail to illus- trate the derivation of the relevant formulae in the Code. Creep due to prestress The general method is illustrated here by considering the two span continuous beam shown in Fig. 8. 12. The beam has constant flexural stiffness and III
- 63. Maximum curvature =~ / M ~ if--~--~--""tJir----~i L L..............._----....._._----_...-.j Fi~. 8.12 Effect of creep on continuous beam equal spans. The prestressing force is P and the maximum' eccentricity in each span is e. Hence the maximum pre- stressing moment in each span is Pe. At time t, let the curvatures associnted with tl;.;;:;e moments be jJ and the restraint moment at the internal support be M. In time Of, the curvature'I' will increase due to creep by 6'1', where ll~, =,,6~ =(Pe/El)o~ If the spans were freely supported, the change in end ro- tation of span 1 at the internal support would be k6,~ =k (Pe/El)o~ and of span 2 would be -ko~, = -k (Pe/EI)6B (8.16) (8.17) where k depends upon the tendon profile. For a straight profile k = L, and for a parabolic profile k = 2L/3. The rotation due to M if the spans were simply supported would be ± 2MLl3EI with the negative and positive signs being taken for spans 1 and 2 respectively. In time Of, these rotations would change due to creep by (-2MLl3E/) op and (+ 2MLl3E/)6P (8.18) (8.19) Also in time Of, the restraint moment would change by ~M. The rotation, due to this change, at the ends of spans I and 2 respectively would be (- 20MLl3EI) and (+ 20MLl3El) (8.20) (8.21) Since the two spans are joined at the support, their net changes of rotation must be equal. The net change of ro- tation for span 1 is obtained by summing expression (S.16), (8.18)' and (S.20); and, for span 2, by sU1'Jlming expres- sions (S. 17), (S.19) and (S.21). If the two net changes of rotation are equated and the resulting equation re- arranged, the following differential equation is obtained elM + M = 3kPe dl3 2 The right hand side of this equation is the sagging restraint moment which would result, in the absence of creep, if the beam were cast and prestressed as a monolithic continuous beam. This moment will be designated M". Hence dM dP + M =Mr The solution of this equation with the boundary condition 112 that, when t = 0, P= 0, M = 0 is M = Mp [l - exp (-B)] In the Code, the expression [1 - exp (- B)] is designated <1>1; thus M = M"CPI (S.22) Although equation (8.22) has been derived for a two-span beam, it is completely general and is applicable to any span arrangement, provided th~t the appropriate value of Mp is used. Hence, fol' any continuous beam, the restraint moments at any time can be obtained by calculating the restraint moments which would occur if the beam were cast and prestressed as a monolithic continuous beam and by then multiplying these moments by the creep factor <PI. The ahove analysis implies that the prestress moment (Pe) should be calculated by considering the prestressing forces applied to the entire composite section. Hence the eccentricity to be used should be that of the prestressing force relative to the centroid ofthe composite section. This is explained as follows. Consider the composite beam shown in Fig. S.13. The eccentricities of the prestressing force P are e'l and ec with respect to the precast beam and composite centroids respectively. In time t after casting the in-situ concrete, creep of the precast concrete will cause the axial strain of the precast beam to change by (P/A,,)~ and its curvature to change by (Pe,,!EbI,,) B, where the sub- script b refers to the precast beam. The in-situ concrete is initially unstressed and thus does not creep. In order to maintain compatibility between the two concretes it is necessary to apply an axial force of P~ and a moment of p..,,~ to the precast beam. However, since the composite section must be in a state of internal equilibrium. it is now necessary to apply a capcelling force of P~ and a cancel- ling moment of Peb~ t< the composite section. The net moment applied to the composite section and which pro- duces a curvature of the composite section is thus (PeI>B) + (PP) (ec - e,,) = PeeB Hence. the eccentricity used should be that relative to the centroid of the composite section. A more rigorous proof is given in reference [220]. The creep factor Bis dependent upon a great number of variables. Appendix C of Part 4 of the Code gives data for the assessment of the following effects on ~ : relative humidity, age at loading, cement content, water-cement ratio, thickness of member and time under load. Unfortu- nately, the basic data required to assess these effects are not generally known at the design stage. For design pur- poses, one is interested in 13ce, which is the value towards which ~ would eventually tend. In the absence of more precise data, Mattock [220] suggested that ~e(' should be taken as 2. This value implies that the creep factor CPI to be applied to the restraint moment due to creep is 0.S7. In practice the value of ~C'e is likely to be between 1.5 and 2.5. and the adoption of the average value of 2 for B,.,. implies a maximum error of 10% in the value of CPl. If ~ is calculated from the data in Appendix C of the Code, it should be remembered that its value should be based on the increase in creep strain from the time that the heam is made continuous by casting the in-situ concrete, and not from the time of prestressing. Precast concrete and composite construction Precast / / Potential p, ~p ~~ ~Composite centroid Precast centroid Tendon -- -- - - -- -- - --1-- --- / -----------r·---- / / ---------~------- ~trai~--E----Pp~--------Pjf-1------ .I'P ._ ~J-~~ - ~ -------. ------------ _.:1___ Pec~ centroid L-.._ _ _ _ _-L.._ _ _ __ Pebl3 Potential creep strains Stress resultants applied to precast section Stress resultants applied to composite section Net stress resultants applied at composite centroid . Fig. 8.13 Effect of creep on composite section Creep due to dead load By an analysis similar to that given abpve, it can be shown that the restraint moment due to creep under dead load is given by M =M.A>I (8.23) Md is the hogging restraint moment which would occur if the beam were cast as a monolithic continuous beam; i.e. the restraint moment due to the combined dead loads of the precast beam and in-situ concrete applied to the composite section. For the two-span beam considered in the last sec- tion, Md is a hogging moment of magnitude (WL2/8) where w is the total (precast plus in-situ) dead load per unit length. CP1 can again be taken as 0.87. Shrinkage Before considering the effects of shrinkage on the restraint moments of a continuous beam, the relief of shrinkage stresses, in general, due to creep is examined. In the analysis which follows, it is assumed that the relation- ships between creep strain per unit stress (specific creep) and time, and between shrinkage strain (E.) and time, have the similar forms shown in Fig_ 8.14, so that, E., = K(E)f) where K is a constant. Hence, using equation (8.14) E, = KP/E and OE. = KoBlE (8.24) Consider a piece of concrete which is restrained against shrinkage so that a tensile shrinkage stress is developed. At time t, let this stress bef. In time Ot, let.'the increase of shrinkage be OE.. and the change of the shrinkage stress be 6[. Also in time Ot, there would be a potential creep strain of (f/E)O~. Thus the net potential change of strain, which results in a change of shrinkage stress, is 6E., - (f/E)b~ Hence, the change of stress is or df f E dEs dff + = dB Hence, using equation (8.24) df d~ + f= K Shrinkage (~.) and specific creep (~/E) strain 'Off< --------i~~~~---- ~cc/E Time(t) Fig. 8.14 Specific creep-time and shrinkage-time curves The solution of this equation with the boundary condition that, when t = 0, ~ = 0, f = 0 is f =K [1 - exp (- B)1 =K<I>I Hence, using equation (S.24) f =EE 11• P In the Code, the expression CPl/~ is designated cP ; thus f= EE,cp (8.25) Hence, the shrinkage stress, at any time t, is the shrinkage stress (EE.,) which would occur in the absence of creep multiplied by the creep factor cp. If the limiting value (~,.c) of ~ is again taken as 2, then cP = 0.43. This factor is referred to earlier in this chapter in connection with the relief of differential shrinkage stresses. Differential shrinkage The general method is again illus- trated by considering a two-span continuous beam which is symmetrical about the internal support. Due to the differ- ential shrinkage between the precast beam and the in-situ flange, at any time t, there will be a constant curvature ('jJ) imposed throughout the length of the beam, and the change (o,,) of curvature in time Ot can be calculated by considering Fig. 8.15. In time Ot, the differential shrink- age strain will change by OE,. If it is required that the precast beam and the flange stay the same length, it is necessary to apply to the centroid of the flange a tensile force of of = OEs Ee! Ac!
- 64. In-situ flange centroid Precast :nn'nn!:'t••4-------------------,--- centroid 8Facent j---~ Potential shrinkage strainForce applied Force applied Net stress resultants applied at composite centroid •to in-situ flange to composite section lrig. 8.15 Differential shrinkage where Act and Eel are the area and elastic modulus respec- tively of the flange concrete. Since the composite section must be in a state of internal equilibrium, it is now neCes- sary to apply a cancelling compressive force of bF to the composite section. This force has an eccentricity of ace"t with respect to the centroid of the composite section. Hence a moment bFace~t is effectively applied to the com- posite section and this moment induces the curvature1jl. Thus 6~, = bFacentlEl = bEsEctActaee,,/EI where the EJ value is appropriate to the composite section. If at time t, the restraint moment at the internal support is M and it changes by bM in time bt, then an analysis, identical to that presented earlier in this chapter for creep due to prestress, results in the following differential equa- tion dM 3 dE dj3 + M =2"E'i A~f aUnt d; Using equation (8.24) dM 3 K dB +M = '2 EctActacmt E The solution of this equation with the boundary condition that, when t = 0, ~ = 0, E. = 0, M = 0 is M= tEctActacent ~ [1 -exp(-~)] Using equation (8.24) M - 1. [1 - exp(-~)] - 2 (EsEctActaw,,) ~ or (8.26) (8.27) where Me., is the hogging restraint moment which would occur in the absence of creep. Although equation (8.26) has been derived for a two-span beam, equation (8.27) is completely general and is applicable to any span arrange- ment. The Code again assumes a value of 0.43 for <1>. For design' purposes, one is interested in Ee/iff' which is the value towards which E.• would eventually tend. Appendix C of the Code gives data which enable the effects on 114 shrinkage of the following to be assessed: relative humid- ity, cement content, water-cement ratio, thickness of member and time. Hence, the differential shrinkage strain can be assessed: a typical value would be about 200 X 10-6 • This value is much greater than the recommended value of 100 X 10-6 quoted earlier in this chapter, when differential shrinkage calculations for beam and slab bridges were discussed. This is because the latter value allows for the creep strains of the precast beam whereas, when carrying out the restraint moment calculations, the creep and shrinkage effects' are treated independently (compare Figs. 8.9 and 8.15). Finally, it should be noted that the Code states that Mcs can be taken as Me. = Ediff EctAct acent <I> (8.28) for anilnternal support. This is appr.oximately correct for a large number of spans (5 qr more) but it will underestimate the restraint moment for, beams with fewer spans. In gen- eral the value of Me. calculated from equation (8.28) needs to be multiplied by a constant ~hich depends upon the ,span arrangement. Equation (8.26) shows that the constant . ':.'js 3/2 for a two-span beam with equal span lengths. :< Appropriate values for other numbers of equal length spans are: 1. Three spans: 1.2 for both internal supports. 2. Four spans: 1.29 for first internal supports, 0.86 for centre support. 3. Five or more spans: 1.27 for first internal supports, 1.0 for all other supports. Values for unequal spans would have to be calculated from first principles. Combined effects of creep and differential shrinkage The net sagging restraint moment due to creep under prestress and dead load and due to differential shrinkage can be obtained by summing equations (8.22), (8.23) and (8.27) tl,vith due account being taken of sign. Hence ' .~;~ -; M =(M" - Me/)<I>1 - Mc,,<I> (8.29) Examples of these calculations are given in [113], [216] and [219]. The calculations need to be carried out only at the serviceability limit state since the restraint moments are _____ .:1 I g N ... 1000 + + + + + + l'-_______+_______-/ J51 ~I ~ + + + + + + + + ... + + + Quarter span Fig. 8.16 Composite beam cross-sections due to imposed deformations and can be ignored at the ultimate limit state. Code flotation The notation adopted in the Code for the creep factors is confusing. Appendix C of Part 4 of the Code adopts the symbol <I> for the creep factor which, in this chapter, has been given the symbol ~ (or ~cc after a long period of time). However, the main body of Part 4 of the Code uses ~c'(" <I> and <1>1 in the same sense as they are used in this chapter. Thus care should be exercised when assess- ing a creep factor from Appendix C of the Code for use in composite construction calculations. The notation adopted for the creep factors is also refer- red to in Chapter 7 in connection with the calculation of long-term curvatures and deflections. '*Example - Shear in composite construction . A hridge deck consists of pretensioned precast standard M8 beams at 1 m centres acting compositely with a 160 mm thick in-situ concrete top slab. The characteristic strengths of the shear reinforcement to he designed and of the precast and in-situ concretes are 250 N/mm2, 50 N/mm 2 and 30 N/mm2 respectively. Foq.r tendons are deflected at the quarter points and the tendon patterns at mid-span and at a support are shown in Fig. 8.16. The total tendon force after all losses have occurred is 3450 kN. The span is 25 m, the overall beam length 25.5 m and the nominal values per beam of the critical shear forces and moments at the support and at quarter- span for load combination I are given in Table 8.1. Design reinforcement for both vertical and interface shear at the two sections. Support Table 8.1 Nominal values of stress resultants Support Quarter span Load Shear force Moment Shear force Moment (kN) (kNm) (kN) (kNm) Self weight 163 0 81 763 Parapet 27 0 , 14 132 Surfacing 29 0 15 135 HB + associated HA 332 0 196 1333 Section properties The modular ratio for the in-situ concrete is /30/50 = 0.775. Reference [36] gives section properties for the pre- cast and composite sections; these are summarised in the upper part of Table 8.2. The composite values are based upon a modular ratio of 0.8 which is only slightly differ- ent to the correct value of 0.775. Table 8.2 Section properties Property Area (mm2) Height of centroid above bottom fibre (mm) Second moment of area (mm4) First moment of a;ka above composite centroid (mm") First moment of area above interface (mm") Precast Composite· 393450 454 642 65.19 x 09 124.55>.< JOP 44.4 x OP 116.0 X lOR ,-----,----------
- 65. The first momentsof area, about the composite centroid, of the sections above the composite centroid and above the interface are also required. These have been calculated and are given in the lower part of Table 8.2. *Vertical shear .'*At support Centroid of tendons from soffit = (15 X 60 + 12 X 110 + 2 X 1080 + 2 X 1130)/31 = 214 mm There is no applied moment acting, thus the stress at the c composite centroid is due only to the prestress and is fop = (3.45 X 1Ot;/393 450)- (3.45 X 106 ) (454 - 214) (642 - 454)/(65.19 x 109) = 6.38 N/mm2 Allowable principal tensile stress (see Chapter 6) =f, = 0.24 /50 ,;, 1.70 N/mm2 • Design shear force at the ulti- mate limit state 'acting on the precast section alone is nom- inal value XYfL xYp Vel = 163·X 1.2 x 1~ 15 = 225 kN Inclination of the four deflected tendons = arctan (970/6500) = 8.490 Vertical component of inclined prestress = (4/31) (3.45) (sin 8.49)103 = 66 kN Net design shear force on precast section = 225 - (0.8) (66) = 172 kN = V where 0.8 is the partial safety factor applied to the prestress (see Chapter 6). Shear stress at composite centroid is f =V(A-)/lb = (172 x 10 3 ) (44.4 x 10 6 )= 073 N/In 2 ", y (65.19 X 109) (160) . m Additional shear force (vd which can be carried by the composite section before the principal tensile stress at the composite.centroid reaches 1.7 N/mm2 is, from equation (8.7), ' _ (124.55 X 109 ) (160) Vc2 - 116 X 106 x = 459 kN ( /(1.7)2 + (0.8) (6.38) (1.7) - 0.73) 10-3 V(o = Vc + Vc2 = 225 + 459 = 684 kN It is not necessary to consider the section cracked in flex- ure at the support and thus the ultimate shear resistance of the concrete alone is Ve =VcO =684 kN Design shear force is V =225 + (27 x 1.2 x 1.15) + (29 x 1.75 x l.l5) + (332 x 1.3 x 1.1) = 795 kN V> Ve,thus links are required such that A". 051 , 116 V- VI' 0.87 Ivl,d, d, = distance from extreme compression fibre of com- posite section to the centroid of the lowest tendon = 1270 mm A... _ (795 - 684)106 _ 2 s;- - (0.87)(250)(1270) - 402 mm /m V" 1.8 Ve (Clause 7.3.4.3), thus maximum link spacing is lesser of O.75d, = 952 mm and 4 x 160 = 640 mm. 10 mm links (2 legs) at 390 mm centres give 403 mm2 /m. Finally, in the above calculations, the full value of the pre- stress has, been taken at the supPort, although the support would lie 'within the transmission zone. BE 2/73 requires a reduced value of prestress to be adopted but the Code does not state that this should be done. However, in prac- tice, one would normally estimate the build;up of pre- stress within the transmission zone, and calculate the shear capacity in accordance with the estimated pre- stress. *At quarter span Centroid of tendons from soffit = (15 x 60 + 14 x 110 + 2 x 160)/31 = 89 mm Design moment at ultimate limit state acting on precast section alone = 763 x 1.2 x 1.15 = 1053 kNm Stress at composite centroid due to the moment = 1053 X 106 (642 - 454)/(65.19 x 109 ) = 3.04 N/mm2 (tension) Stress at composite centroid due to prestress ' = (3.45 x 106 /393450) - (3.45 x 106 ) (454':- 89) (642 - 454)/(65.19 x 109 ) = 5.14 N/mm2 , Total stress at composite, centroid to be used in equation (8.7) is f~p = (0.8) (5.14) - 3.04 = 1.07 N/mm2 Design shear force at the ultimate limit state acting on the precast section alone is Vel = 81 x 1.2 x 1.15 = 112 kN Shear stress at composite centroid is Is =(112 x 103 )(44.4 x 106 )/(65.19 x 109 ) (160) =0.48 N/mm2 Additional shear force (Vd which can be carried by the composite section before the principal tensile stress at the composite centroid reaches 1.70 N/mm2 is (124.55 X 109 ) (160) Vc2 = 116 X 106 X ()(1.7)2 + (1.07) (1.7)-0.48)10-3 = 290 kN VeO = Vet + Ve2 = 112 + 290 = 402 kN The section must now be considered to be cracked in flex- ure. Stress at extreme tension fibre due to prestress is fpl :: (3.45 x 106 /393 450) + (3.45 x 106 ) (454 - 89) (454)/(65.19 x 109 ) = 17.54 N/mm2 The cracking moment is given by equation (8.8) Mt = (1053 x 106) x (1 - 454 x 124.55 x 109 /642 x 65.19 x 109 ) + (0.37 v'5O + 0.8 x 17.54) 124.55 x 109 /642 2860 x 106 Nmm = 2860 kNmM_'~"''''--- . Design moment at the ultimate limit(state is M = 1053 + (132 x 1.2 x 1.15) + (135 x 1.75 X 1.15) + (1333 x 1.3 X 1.1) 3413 kNm Design shear force nt the ultimate limit state is V = 112 + (14 x 1.2 x 1.15) + (15 x 1.75 x 1.15) + (196 x 1.3 x 1.1) = 442 kN From the modified form of equation (6.11), suggested in this chapter Vcr = (0.037) (160) (1111 /50 + 130 /30)10-3 + 442 (2860/3413) = 421 kN Ultimate shear resistance (Ve) is the lesser of VeO aJ1d Vcr and thus V( = Vc() = 402 kN V> Ve, thus links are required such that A". = (442 - 402)106 _ 21 oS. 0.87 X 250 x 1270 - 145 mm In However, minimum links must be provided such that A,.., ( 0.87 fy ,' ) = 0.4 N/mm2 SI' b or AS!, _ (0.4) (160)10 3 = 294 mm2/m SI' - 0.87 x 250 V" 1.8Ve, thus the maximum link spacing is the:same as at the support (= 640 mm) 10 mm diameter links (2 legs) at 500 mm g,ive 314 mm2 /m Maximum allowable shear force At support, distance of centroid of tendons in tension zone from soffit is (15 x 60 + 12 x 110)/27 = 82 mm. Thus, in equation (8.10) d; = 130 mm db = 1200 - 82 = 1118 mm Vu = (0.75) (160) (1118 /50 + 130/30)10-3 = 1034 kN Allowing for vertical component of inclined prestress, because uncracked Vu = 1034 + (0.8) (66) = 1087 kN 795 kN, thus section is :¥ Interface shear' ~At sU,/Jport Total design shear force at serviceability limit state is (163 x L.ox 1.0) + (27 x 1.0 x 1.0) + (29 x 1.2 x 1.0) + (332 x 1.1 x 1.0) = 590 kN From equation (8.12), interface shear stress is = (590 x 10 3 ) (71.6 X 10 8 ) = 1 16 NI 2 Vh (124.55 x 109) (300) . mm Type 1 surface is not permftted for beam and slab bridge construction. The allowable shear stress for Type 2 surface is 0.38 N/mm2 (see Table 4.5) and this cannot be increased by providing links in excess of the required minimum of 0.15%. The allowable shear stress for Type 3 surface is 1.25 N/mmz (see Table 4.5); and thus the minimum amount of steel of 0.15% is all that is required, but thelaitence must be removed from the top surface of the beam. Thus pro- vide (0.15) (300) (1000)/100 = 450 mm2 /m This exceeds that provided for vertical shear. Thus use 10 mm links (2 legs) at 350 mm centres which give 449 mm2 /m *At quarter span Total design shear force at serviceability limit state is (81 x 1.0 x 1.0) + (14 x 1.0 x 1.0) + (15 x 1.~ x 1.0) + (196 x 1.1 x 1.0) = 329 kN Vh = 0.63 N/mm2 Again only Type 3 surface can be used and 10 mm links at 350 mm centres would be required. This amount of re- inforcement exceeds that required for vertical shear. 117
- 66. Chapter 9 Substructures and foundations Introduction TheCode does not give design rules which are specifically concerned with bridge substructures. Instead, design rules, which are based upon those of CP 110, are given for col- umns, walls (both reinforced and plain) and bases. In ad- dition, design rules for pile caps, which did not originate in CP 110, are given. The CP 110 clauses were derived for buildings and, thus, the column and wall clauses in the Code are also more relevant to buildings than to bridg~s. In view of this, the approach that is adopted in this chapter is, first, to give the background to the Code clauses and, then, to discuss them in connection with bridge piers, col- umns, abutments and wing walls. However, these struc- tural elements are treated in general terms only, and, for a full description of the various types of substructure and of their applications, the reader is referred to [221]. The author anticipates that the greatest differences in section sizes and reinforcement areas, between designs carried out in accordance with the current documents and with the Code, will be noticed in the design of substruc- tures and foundations. The design of the latter will also take longer because, as explained later in this chapter, more analyses are required for a design in accordance with the Code because of the requirement to check stresses and crack widths at the serviceability limit state in addition to . strength at the ultimate limit state. In CP 110, from which the s.ubstructure clauses of the Code were derived, it is only necessary to check the strength at the ultimate limit state, since compliance with the serviceability limit state criteria is assured by applying deemed to satisfy clauses. The column and wall clauses of CP 110 were derived with this approach in mind. The fact that the CP 110 clauses have been adopted in the Code without, apparently, all~w ing for the Code requirement that stresses and crack widths at the serviceability limit state should be checked, has led to a complicated design procedure. This complication is mainly due to the fact that it has not yet been established which limit state will govern the design under a particular set of load effects. Presumably, as experience in using the Code is gained, it will be possible to indicate the most likely critical limit state for a particular situation. 118 Columns r *General tJ{:..Definition A column is not defined in the Code; but a wall is defined as having an aspect ratio, on plan, greater than 4. Thus a column can be considered as a member with an aspect ratio not greater than 4. ..:J(;.Effective height The Code gives a table of effective heights (Ie), in terms of the clear height (10)' which is intended only to be a guide for various end conditions..The table, which is here sum- marised in Table 9.1, is based upon a similar table in CP 110 which, in turn, WilS based upon a table in CP 114. The effective heights have been derived mainly with framed buildings in miri'd and do not cover, specifically, the types of column which oCCur in bridge construction. Indeed, in view of the variety of different types of articula- tion which can occur in bridges, it would be difficult to produce a table covering all situation.s. It would thus appear necessary to consider each particular case individ- ually by examining the likely buckling mode. In doing this, consideration should be given to the way that movement can be accommodated by the bearings, the flexibility of the column base (and soil in which it is founded), and whether the articulation of the bridge is such that the col- umns are effectively braced or can sway. Some of these aspects are discussed by Lee [222]. Table 9.1 Effective heights of columns Column type Braced, restrained in direction at both ends Braced, partially restrained in direction at one or both ends ' Unbraced or partially braced, restrained in direction at one end, partially restrained at other Cantilever /,J/" 0.75 0.75-1.0 1.0-2.0 2.0-2.5 N Ultimate -- -------------------- Serviceability I 1--_ _ _--'-1.__________..,... 1 Fig. 9.1 Lateral deflection Reference is made in Table 9.1 to braced and unbraced columns: the Code states that a column is braced, in a particular plane, if lateral stability to the structure as a whole is provided in that plane. This can be achieved by designing bracing or bearings to resist all lateral forces. ~Slenderness limits A column is considered to be short, and thus the effects of its lateral deflection can be ignored, if the slenderness ratios appropriate to each principal axis are both less than 12. The slenderness ratio appropriate to a particular axis is defined as the effective height in respect to that axis divided by the overall depth in respect to that axis. The overall depth should be used irrespective of the cross- sectional shape of the column. If the slenderness ratios are not less than 12, the column is defined as slender and lateral deflection has to be con- sidered by using the additipnal moment concept which is explained later. The limiting slenderness ratio is taken as 12 because work carried out by Cranston [223] indicated that buckling is rarely a significant design consideration for slenderness ratios less than 12. This work formed the basis of theTP 110 clauses for slender columns. It is possiSle for 'slender columns to buckle bey combined lateral bending and twisting: Marshall [224] reviewed all of the available relevant test data and concluded that lateral torsional buckling will not influence collapse provided that, for simply supporteg ends, I" :s;;; 500 b2 /h where h is the depth in the plane under consideration and b is the width. Cranston [223] suggested that this limit should be reduced, for design purposes, to I,,:S;;; 250 b% (9.1) Cranston also suggested the following limit for columns for which one end is not restrained against twisting, and this limit has been adopted in the Code for cantilever col- umns. (9.2) In addition, the Code requires that I" should not exceed 60 times the minimum column thickness. This limit is stipulated because Cranston~s study did not include col- umns having a greater slenderness ratio. This. limit on 10 is, generally, more onerous than that implied by equation (9.1) and, thus, the latter equation does not appear in the Code. It should be noted that, for unbraced columns (which frequently occur in bridges), excessive lateral deflections can occur at the serviceability limit state for large slender- ness ratios. This is illustrated in Fig. 9.1. An analysis, which allows for lateral deflections, is not required at the serviceability limit state and, c9nsequently, CP 110sug- gests a slenderness ratio limit of 30 for unbraced columns. This limit does not appear in the Code, but it would seem prudent to apply a similar limit to bridge columns, unless it is intended to consider, by a non-linear analysis, lateral deflections at the se.rviceability limit state. In the above discussion of slenderness limits, it is implicitly assumed that the column has a constant cross- section throughout its length. However, many columns used for bridges are tapered: for such columns, data given by Timoshenko [225] indicate that, generally, it is con· servative to calculate the slenderness ratio using the aver· age depth of column. Ultimate limit state Short column '*Axial load Since lateral deflections can be neglected in. a short column, collapse of an axially loaded column occurs when all of the material attains the ultimate concrete com- pressive strain of0.0035 (see Chapter 4). The design stress- strain curve for the ultimate limit state (see Fig. 4.4) indicates that, at a strain of 0.0035, the compressive stress in the concrete is 0.45 ie,.. Similarly Fig. 4.4 indicates that, at this strain, the compressive stress in the reinforce- ment is the design stress which can ;'ary from 0.718 fy to 0.784Jy (see Chapter 4). The Code adopts an average value of 0.75 fy for the steel stress. It is not clear why the Code adopts the average value for columns, but a minimum value of O.72fy for beams (see Chapter 5), If the areas of concrete and steel are Ac and A.•r respec- tively, then the axial strength of the column is . N =0.45 fruAc + 0.75 ivA,,· (9.3) The Code recognises me fact that some eccentricity of load will occur in practice and, thus, the Code requires a minimum eccentricity of 5% of the section depth to be adopted. It is not necessary to calculate the moment due to this eccentricity, since allowance for it is made by r,e- ducing the ultimate strength, obtained from equation (9.3), by about 10%. This reduction leads to the Code formula for an axially loaded column: N =0.4 fr,,Ac + 0.67 i,A.,r (9.4) Axial load plus uniaxial bending When a bending moment is present. three possible methods of design are given in the Code: 1. For symmetrically reinforced rectangular or circular columns, the design charts of Parts 2 and 3 of CP 10
- 67. It N I -e= MIN 0.4fCI Elevation i I -IP'oo I I, I i ~ h,.~ Fig. 9.2 Plain column section [128. 130] may be used. These charts were prepared using the rectangular - parabolic stress-strain curve for concrete and the tri-linear stress-strain curve for reinforcement discussed in Chapter 4. Allen [203] gives useful advice on the use of the design charts. 2. A strain compatibility approach (see Chapter 5) can be adopted for any cross-section. An area of reinforce- ment is first proposed and then the neutral axis depth is guessed. Since the extreme fibre compressive strain is 0.0035, the strains at all levels are then defined. Hence, the stresses in the various steel layers can be determined from the stress-strain curve. The axial load and bending moment that can be resisted by the column can then be determined. These values can be compared with the design values and, if deficient, the area of reinforcement and/or neutral axis depth modi- fied. The procedure is obviously tedious and is best performed by computer. 3. The Code gives formulae for the design of rectangular columns only. The formulae, which are described in the next section. require a 'trial and error' design method which can be tedious. An example of their use is given by Allen [203]. Although the Code formulae are for rectangular sections only, similar formulae could be derived for other cross-sections. In conclusion, it can be seen that a computer, or a set of design charts, is required for the efficient design of col- umns subjected to an axial load and a bending moment. -fCode formulae Consider an unrein/orced section at col- lapse under the action of an axial load (N) and a bending moment (M). If the depth of concrete in compression is d,., as shown in Fig. 9.2, then by using the simplified reclan- 120 II ?M 1 III I • • 1A.2 A;, • • ~d2 __._ .--1£.......-...._._..~d' 'II···_··_···········-F---·---..----. I, I 0.0035 I ~t---..dc •..._ ..-.j Section elevation Section plan Strains 0.4fcu Stresses O.72fy , 1 Fe ~; I t. Fig. 9.3(a),(b) Reinforced "column section Stress resultants gular stress block for concrete with a constant stress of 0.4 feu: Eccentricity (e) = MIN = (hI2) - (d,./2) :. de =h-2e (9.5) For equilibrium N =0.4 feubdc =0.4 !c."b (h - 2e) (9.6) Hence, only nominal reinforcement is required if the axial load does not exceed the value of N given by equation (9.6). However, it should be noted from this equation that, when e > h12, N is negative. Hence, equation (9.6) should not be used for e > h12; however. the Code specifies the more conservative limit of e = h/2 - d I, where d I is the depth from the surface to the reinforcement in the more highly compressed face. When N exceeds the value given by equation (9.6), it is necessary to design reinforcement. At failure of a re- inforced concrete column, the strains, stresses and stress resultants are as shown in Fig. 9.3. It should be noted that the Code now takes the design stress of yielding compres- sion reinforcement to be its conservative value of 0.72 f}" •M. N i ._._..__..... My' (a) Biaxial interactioli diagram Fig. 9.4 Biaxial bending interaction''lilagraln For equilibrium N = 0.4 !c'l/hd,. + 0.72 t,A:,I +/,2A"'2 (9.7) and, by taking moments about the column centre line. M =0.2 !c,"bde (/1 - de) + O.72 fvA.~, (h/2 - d') -J.~2A.'2 (h/2 - d2) (9.8) These equations are difficult to apply because the depth (tic) of concrete in compression and the stress (/,2) in the reinforcement in the tension (or less highly compressed) face are unknown. The design procedure is thus to assume values of d,. and f,2' then calculate A.:, and A.,2 from equa- tion (9.7). and check that the value of M calculated from equation (9.8) is not less than the actual design moment. If M is less than the actual design moment. the assumed val- ues of d,. and /,2 should be modified and the procedure repeated. Guidance on applying this procedure is given by Allen [2031. However. the Code does not allow de to be taken to be less than 2d I. From Fig. 9.3 it can be shown that. at this limit, the strain in the more highly compressed reinforcement is 0.00175, which is less than the yield strain of 0.002 (see Chapter 4). However. serious errors in the required quantities of reinforcement should~not arise by assuming that the stress is always 0.72 fy. as in Fig. 9.3. When the eccentricity is large (e > h/2 - d2) and thus the reinforcement in one face is in tension, the Code permits a simplified design method to be used in which the axial load is ignored at first and the section designed as a beam. .The required design moment is obtained by taking .moments about the tension reinforcement. Thus. from ·Fig.9.3. ,M + N(d - h/2) = Fe (d - d,J2) + F,: (d - d') (9.9) :The right-hand side of this equation is the ultimate moment lof resistance (Mu) of the section when considered as a 'beam. Hence, the section can be designed. as a beam. to resist the increased moment (Ma) given by the left-hand side of equation (9.9). i.e. Ma = M + N(d - h/2) Now, d - h/2 = hl2 - d2• (9. to) in the lorm Mil = M + N(h12 - d2) «l~~~5r:c~;, and the Code gives equ~ti~n~ I C' ",(9~j:' ''''''~ Actual -'-"-Idealised (.~'!.)lln + (My )an= 1 Mu. Muy (b) Section ABC If force equilibrium is considered. N =Fc + F: - F" or F, =(Fc + F'.,) - N (9.12) In the absence of the axial force N, Fa =Fc + F,: =Fi, where Fb is the tension steel force required for the section considered as a beam. Hence. from equation (9.12). when the axial force is present F.r =Fb - N The area of tension reinforcement is obtained by dividing F" by the tensile design stress of 0.87 fy; hence A., =Ab - NIO.87 fy (9.13) wh~re A., and Ah are the required areas of tension re- inforcement for the column section-and a beam respectively. Hence. the section can be designed by, first. designing it as a beam to resist the moment Mn from equation (9.]]) and then reducing the area .of tension reinforcement by (!VIO.87 fy). Axial load plus biaxial bending If a .column of known dimensions and steel area is analysed rigorously. it is pos- sible to construct an interaction diagram which relates fail- ure values of axial load (N) and moments (Mx• My). about the major and minor axes respectively. Such a diagram is shown in Fig. 9.4(a),. where N"zrepresents the full axial load carrying capacity'given by equation (9.3). A section, parallel to the MxMy plane. through the diagram for a par- ticular value of NIN"z would have the shape shown by Fig. 9.4(b), where. assuming an axial load N. Mil.• and Mil}' are the maximum moment capacities for bending about the major and minor axes respectively. The shape of the diagram in Fig. 9.4(b) varies according to the value ofNINuz but can be represented approximately by (MxIMI/..>'''n + (MvlMlly)"'n = I (9.14) (¥n is a function ofNINllz. Appropriate values are tabulated in the Code. ·· ..When,designing a column subjected to biaxial bending.' it is first necessary to assume a reinforcement area. Values 4~~~.:'~~., '. :.~.~~-,~t~~'~l. ~, .,', .". ",~;j~~it ,fr~ 121
- 68. N I I I I I I e I -.!~ I I I I I I N Initial eccentricity M; =Ne; Initial moment Fig.9.5 Additional moment N Additional eccentricity of N"" Mil.• and Mlly can then be calculated from first prin- ciples or obtained from the design charts for uniaxial bend- ing. It is then necessary to check that the left-hand side of equation (9.14) does not exceed unity. "*Slender columns General approach When an eccentric load is applied to any column, lateral deflections occur. These deflections are small for a short column and can be ignored, but they can be significant in the design of slender columns. The deflections and their effects are illustrated in Fig. 9.5, where it can be seen that the lateral deflections increase the eccentricity of the load and thus produce a moment (M"dd) which is additional to the primary (or initial) moment (M;). Hence, the total design moment (M,) is given by M,= M;+ Madd where M;= Ne; M"dtl = Neatld (9.15) (9.16) (9.17) e,. and e"dd are the ipitial and additional eccentricities respectively. Since the section design is carried out at the ultimate limit state, it is necessary to assess the additional eccen- tricity at collapse. The additional eccentricity is the .lateral deflection. and the latter can be determined if the distri- hution of curvature along the length of the column can be calculated. The distribution of curvature for a column subjected to an axial load and end moments is shown in Fig. 9.6. where 'I'" is the maximum curvature at the centre of the column at collapse. The actual distribution ofcurva- ture depends upon the column cross-section, the extent of cracking and of plasticity in the concrete and reinforce- ment. However. it can-be seen. from Fig. 9.6. that it is unconservative to assume a triangular distribution, and conservative to assume a rectangular distribution. For these two distributions the central deflections are given by, respectively: ('"dd =1;'1',/12 ('",Id = 1;11',,/8 122 Additional moment M N '-V Total moment Fig. 9.6 Curvature distributions M,= M;+ Madd --Actual --- Conservative --- Unconservative , , Thus, Cranston [223] ~uggested that, for design pur- poses. a reasonable value to adopt would be: eadd =1;,p1l110 (9.18) 'jill can be determined if the strain distribution at col- lapse can be assessed. It is thus necessary to consider the mode of collapse. Unless the slenderness ratio is large, it is unlikely that a reinforced concrete column will fail due to instability, prior to material failure taking place [223]. Hence, instability is ignored initially and, for a balanced section in which the concrete crushes and the tension steel yields simultaneously, the strain distribution is as shown in Fig. 9.7. The additional moment concept used in the Code is based upon that of the C.E.B. [226] in which the short-term con- crete crushing strain (Eu) is taken as 0.0030. In order to allow for long-term effects under service conditions, the latter strain has to be multiplied by a creep factor which Cranston [223J suggests should be conservatively taken as l.25. Hence. Ell = 0.00375. The strain (E..) in the rein- forcement is that appropriate to the design stress at the ultimate limit state. Since the characteristic strength of the reinforcement is unlikely to exceed 460 N/mm2 • E.. can be conservatively taken as 0.87 x 460/200 x 103 = 0.002. Hence, the curvature is given by 'jill = (0.00375 + 0.002)/d = O.00575id (9.19) M N "'. d Fig. 9.7 Collapse strains for billanced section It is now necessary to consider the possibility of an insta- bility failure as opposed to a material failure. In such a situation, the strains are less than their ultimate values and, hence, the curvature is less than that given by equation (9.19). The C.B.B. Code [226] allows for this by reducing the curvature obtained from equation (9.19) by the follow- ing empirical amount le/50 000h2 It also assumed that d =11, so that the curvature is obtained finally as 'I'" = (0.00575 -1)50 OOOh)/h (9.20) If this curvature is substituted into equation (9.18), then the following expression for the lateral deflection (or ad- ditional eccentricity) is obtained e,l/Itt = (hI1750)(/~/h)2 (1- 0'()035 [)h) (9.21) It should be noted that for a slenderness ratio of 12 (i.e. just slender), e"dd = O.OSh, but for the maximum permitted slenderness ratio of 60, eadtl = 1.63h. Hence, for very slender columns, the additional eccentricity and, hence, the additional moment can be very significant in design terms. Minor axis bending If h is taken as the depth with respect to minor axis bending. then the additional eccentricity is given by equation (9.21). Hence. the total design moment, which is obtained from equations (9.15), (9.17) and (9.21). is given in the Code as M, = M; + (Nh/1750) (/,.Ih)2 (1- O.00351)h) (9.22) ~M, Loading Lateral Moments deflection Fig. 9.8 Effect of unequal end moments 1, is taken as the greater of the effective heights with respect to the major and minor axes. It should be noted that M/ should not be taken to be less than 0.05 Nh, in order to allow for the nominal minimum initial eccentricity of 0.05h. The column should then be designed, by anyone of the methods discussed earlier felr short columns, to resist the axial load N and the total moment M" In Fig. 9.5, the maximum additional moment occurs at the same location as the maximum initial moment. If these maxima do not coincide, equation (9.15) is obviously con- servative. Such a situation occurs when the moments at the ends of the column are different, as shown in Fig. 9.8. To be precise, one should determine the position where the maximum total moment occurs and then calculate the latter moment. However, in order to simplify the calcu- lation for a braced column, Cranston [223] has suggested that the initial moment, where the total moment is a maxi- mum, may be taken as M; =0.4 M + 0.6 M2 but M/ <t 0.4 M2 (9.23) where M1 and M2 are the smaller and larger of the initial end moments respectively. For a column bent in double curvature. M is taken to be negative. It is possible for the resulting total moment (M,) to be less than Mi. In such '3 situation, it is obviously necessary to design to resist M2 and thus M, should never be taken to be less than M2• For an unbraced column, the Code requires the total moment to be taken as the sum of the additional moment and the maximum initial moment. This can be very con- servative for certain bridge columns which are effectively fixed at both the base and the top, but which can sway under lateral load or imposed deformations (e.g. tempera- ture movement). Major axis bending A column which is loaded eccentric- ally with respect to its major axis can fail due to large additional moments developing about the minor axis. This· is because the slenderness ratio with respect to the minor axis is greater than that with resp~ct to' the major axis.
- 69. N ~+----------4---+Y b M; x h Fig.9.9 Major axis bending Hence, with reference to Fig. 9.9 the column should be designed for biaxial bending to resist the following moments M,x =M/ + (Nh/1750) (lexfh)2 (1- 0.0035Iexfh) M,.I' =(Nb/1750) (ley/b)2 (1- O.0035Ieyfb) (9.24) (9.2~) where M,.~ and M,y are the total moments about the major (x) axis and minor (y) axis respectively, and lex and ley are the effective heights with respect to these axes. Cranston [223] has shown that, for a braced column with II :t> 3b, it is conservative to design the column solely for bending about the major axis, but the slenderness ratio should then be calculated with respect to, the minor'axis. Hence, in such situations, the Code permits the column to be designed to resist the axial load N and the following total moment about the major axis. M, = Mi + (Nh/1750)(I)b)2 (1- 0.00351)b) (9.26) where Ie is the greater of lex and ley. Biaxial bending When subjected to biaxial bending, a column should be designed to resist the axial load Nand moments (Mx = MIX! My = M,y) such that equation (9.14) is satisfied. The total moments about the major and minor axes respectively are M,x = Mix + (Nh/1750) (lexfh? (1- 0.0035 lex/h) M,y = Miy + (Nb/1750) (ley/b)2 (1 - 0.0035 ley/b) (9.27) (9.28) where Mix and Miy are the initial moments with respect to the major and minor axes respectively. Serviceability limit state General The design of columns in accordance with the Code is complicated by the fact that it is necessary to check stress- es and crack widths at the serviceability limit state in addition to carrying out strength calculations·at the ulti- mate limit state. fStresses At the serviceability limit state, the compressive stress in the concrete has to be limited to 0.5 fm and the reinforce- ment stresses to 0.8 f).. Thus, for an axially loaded col- 124 J' ..... O.~ ~O.5fcu O.87fJ ' ~O.4fcu (a) Section ,(b) Limiting stresses at serviceability limit state (c) Limiting stresses at ultimate limit state Fig. 9.10(a)-(c) Stress comparison umn, the design resistance at the serviceability limit state is, usually, ' Ns =0.5 fcuAc + (0.5 fcuEs/Ec)Asc This value generally exceeds the design resistance at the . ul'timate limit state, as given by equation (9.4). Since the design load at the ultimate limit state exceeds that at the serviceability limit state, it can be seen that ultimate will generally be the critical limit state when the loading is pre- dominantly axial. When the loading is eccentric to the extent that one face is in tension, the stress conditions at the ultimate and ser- viceability limit states will be as shown in Fig. 9.10. Since the -average concrete stress at the serviceability limit state (0.25 feu) is much less than that at the ultimate limit state (0.4 feu), it is likely th,at, with regard to concrete stress, the serviceability limit state will be critical. *Crack widths The Code considers that if a column is designed for an ultimate axial load in excess of 0.2 fcuAn it is unlikely that flexural cracks will occur. For smaller axial loads, it is necessary to check crack widths by considering the column to be a beam and by applying equation (7.4). From Table 4.7, it can be seen that, since a column could be subjected to salt spray, the allowable design crack width could be as small as 0.1 mOl. Hence, equation (7.4) implies a maxi- mum steel strain of about 1000 X 10-6 , or a stress of about 200 N/mm2 • For high yield steel, this stress is equi- valent to about 0.48/1' Hence, when crack control is con- sidered, the reinforcement stress in Fig. 9.IO(b) is limited to much less than 0.81", and thus crack control could be the critical design criterion for columns with a large eccen- tricity of load. Summary Ultimate is likely to be the critical limit state for a column which is either axially loaded or has a small eccentricity of load. However, due to the fact that either the limiting compressive stress or crack width could be the critical design criterion, serviceability is likely to be the critical limit state for a column with a large eccentricity of load. It appears that, in order to simplify design, studies should be carried out with a view to establishing guidelines for iden- tifying the critical limit state in a particular situation. Reinforced concrete walls General Definitions Rctuining wulls, wing w!llIs and similar structures which, primm'i!y, are subjected to bending should be considered us slabs and designed in accordance with the methods of Chapters 5 and 7. The following discussion is concerned with walls subjected to significant axial loads. In terms of the Code, a reinforced concrete wnll is a vertical load-bearing member with an aspect ratio, on plan, greater than 4; the reinforcement is assumed to contribute to the strength, and has an area of at least 0.4% of the cross-sectional area of the wall. This definition thus covers reinforced concrete abutments. The limiting value of 0.4% is greater than that specified in CP 114 because tests have shown that the presence of reinforcement in walls reduces the in-situ strength of the concrete [227]. Hence,under axial loading, a plain concrete wall can be stronger than a wall with a small percentage of reinforcement. Slenderness The slenderness ratio is the ratio of the effective height to the thickness of the wall. A short wall has a slenderness ratio less than 12. Walls with greater slenderness ratios are considered to be slen- der. In general, the slenderness ratio of a braced wall should not exceed 40, but, if the area of reinforcement exceeds 1%, the slenderness ratio limit may be increased to 45. These values are more severe than those' for columns because walls are thinner than columns, and thus deflec- tions are more likely to lead to problem~. If lateral stability is not provided to the structure as a whole, then a wall is considered to be unbraced and its slenderne~s ratio should not exceed 30. This rule ensures that deflections will not he excessive. The above slenderness limits were obtained from CP 110 and were thus derived with shear walls and in-fill panels in framed structures in mind. They are thus not necessarily applicable to the types of wall which are used in bridge construction. However, the slenderness ratios should not result in any further design restrictions com- pared with existing practice. Analysis The Code requires that forces and moments in reinforced concrete walls should be determined by elastic analysis. When considering bending perpendicular to an axis in the plane of a wall, a nominal minimum eccentricity of 0.05h should be assumed. Thus a wall should be designed for a moment per unit length of at least 0.05 nwh where nw is the maximum load per unit length. Ultimate limit state Short walls '*Axialload An axially loaded wall should be designed in accordance with equution (9.4). '*Eccentric loads If the load is eccentric such that it pro- duces bending about an axis in the plane of a wall, the wall should be designed on a unit length basis to resist the combined effects of the axial loud per unit length and the bending moment per unit length. The design could be car. ried out either by considering the section of wall as an eccentrically loaded column of unit width or by using the 'sandwich' approach, described in Chapter 5, for design- ing against combined bending and in-plane forces. If the load is also eccentric in the plane of the wall, an elastic analysis should be carried out, in the plane of the wall, to determine the distribution of the in-plane forces per unit length of the wall. The Code states that this analysis may be carried out assuming no tension in the concrete. In fact, any distribution of tension and com. pression, which is in equilibrium with the applied loads could be adopted at the ultimate limit state since, as ex- plained in Chapter 2, a safe lower bound design would result. Each section along the length of.. the wall should then be designed to resist the combined effects of the moment per unit length at right angles to the wall and the compression, or tension, per unit length of the wall. The design could be carried out by considering each section of the wall as an eccentrically loaded column or tension member of unit width, or by using the 'sandwich' approach. Slender walls The forces and moments acting on a slender wall should be determined by the same methods previously described for short walls. The pol1ion of wall, subject to the highest intensity of axial load, should then be designed as a slen- der column of unit width. Serviceability limit state Stresses The comments made previously regarding stress calcu- lations for columns are also appropriate to walls. Crack widths Walls should be considered as slabs for the purposes of crack control calculations, and the details of the Code requirements are discussed in Chapter 7.
- 70. Plain concrete walls "*General Aplain concrete wall or abutment is defined as a vertical load-bearing member with an aspect ratio, on plan, greater than 4; any reinforcement is not assumed to contribute to the strength. If the aspect ratio is less than 4, the member should be considered as a plain concrete column.. The following. design rules for walls can also be applied to columns, but, as indicated later, certain design stresses need modifi- cation. The definitions of 'short', 'slender', 'braced' and 'unbraced', which are given earlier in this chapter for re- inforced concrete walls, are also applicable to plain con- crete walls. The clauses, concerned with slenderness and lateral support of plain walls, were taken directly from CP 110 which in tum were based upon those in CP 111 [228]. In order to preclude failure by buckling the slenderness ratio . of a plain wall should not exceed 30 [229]. The effective heights given in the Code are summarised in Table 9.2. Table 9.2 Effective heights of plain walls Wall type Unbraced. lateralty spanning structure at top Unbraced, no laterally spanning structure at top Braced against lateral movement and rotation Braced against lateral movement only * l., = distance between centres of support I 1" = distance between a support and a free edge 1.5 2.0 0.75* or 2.0t 1.0* or 2.5t In order to be effective, a lateral support to a braced wall must be capable of transmitting to the structural ele- ments, which provide lateral stability to the structure as a whole, the following forces: I. The static reactions to the applied horizontal forces. 2. 2.5% of the total ultimate vertical load that the wall has to carry. A lateral support COUld be a horizontal member (e.g., a deck) or a vertical member (e.g., other walls), and may be considered to provide rotational restraint if one of the fol- lowing is satisfied: I . The lateral support and the wall are detailed to provide bending restraint. 2. A deck has a bearing width of at least two-thirds of the wall thickness, or a deck is connected to the wall by means of a bearing which does not allow rotation to occur. ~. The wall supports, at the same level, a deck on each side of the wall. Forces Members, which transmit load to a plain wall, may be considered simply supported in order to calculate the re- action which they transmit to the wall. 126 I.. h ~I h-2e" Fig. 9•.11 Eccentrically loaded short wall'at collapse If the load is eccentric in the plane of the wall, the eccentricity and distribution of load along the wall should be calculated from statics. When calculating the distribu- tion of load (i.e., the axial load per unit length of wall), the concrete should be assumed to resist no tension. If a number of walls resist a horizontal force in their plane, the distributions of load between the walls should be in proportion to their relative stiffnesses. The Code clause concerning horizontal loading refers to shear con- nection between walls ahd was originally written for CP 110 with shear walls..in buildings in mind. However, the clause could be applied, for example, to connected semi-mass abutments. When considering eccentricity at right angles to the plane of a wall, the Code states that the vertical load transmitted from a deck may be assumed to act at one-third the depth of the bearing area back from the loaded face. It appears from the CP 110 handbook [112] that this requirement was originally intended for floors or roofs of buildings bearing directly on a wall. However, the inten- tion in the Code is, presumably, also to apply the require- ment to decks which transmit load to a wall through a mechanical or rubber bearing. Ultimate limit state Axial load plus bending normal to wall Short braced wall The effects of lateral deflections can be ignored in a short wall and thus failure is due solely to concrete crushing. The concrete is assumed to develop a constant compressive stress of "Awfcu at collapse, where "A... is a coefficient to be discussed. The concrete stress dis- tribution at collapse of an eccentrically loaded wall is as shown in Fig. 9. t t. The centroid of the stress block must design criterion, serviceability is likely to be the critical limit state for a column with a large eccentricity of load. It appears that, in order to simplify design, studies should be carried out with a view to establishing guidelines for iden- tifying the critical limit state in a particular situation. Reinforced concrete walls General Definitions Retaining wulls, wing walls and similar structures which, primarily, are subjected to bending should be considered as slabs and designed in accordance with the methods of Chapters 5 and 7. The following discussion is concerned with walls subjected to significant axial loads. In terms of the Code, a reinforced concrete wall is a vertical load-bearing member with an aspect ratio, on plan, greater than 4; the reinforcement is assumed to contribute to the strength, and has an area of at least 0.4% of the cross-sectional area of the wall. This definition thus covers reinforced concrete abutments. The limiting value of 0.4% is greater than that specified in CP 114 because tests have shown that the presence of reinforcement in walls reduces the in-situ strength of the concrete [227]. Hence,under axial loading, a plain concrete wall can be stronger than a wall with a small percentage of reinforcement. Slenderness The slenderness ratio is the ratio of the effective height to the thickness of the wall. A short wall has a slenderness ratio less than 12. Walls with greater slenderness ratios are considered to be slen- der. In general, the slenderness ratio of a braced wall should not exceed 40, but, if the area of reinforcement exceeds 1%, the slenderness ratio limit may be incre,ased to 45. These values are more severe than those for columns because walls are thinner than columns, and thus deflec- tions are more likely to lead to problems. If lateral stability is not provided to the structure as a whole, t~en a wall is considered to be unbraced and its slenderness ratio should not exceed 30. This rule ensures that deflections will not be excessive. The above slenderness limits were obtained from CP 110 and were thus derived with shear walls and in-fill panels in framed structures in mind. They are thus not necessarily applicable to the types of wall which are used in bridge construction. However, the slenderness ratios should not result in any further design restrictions com- pared with existing practice. Analysis The Code requires that forces and llIoments in reinforced concrete walls should be determined by elastic analysis. When considering bending perpendicular to an axis in the plane of a wall, a nominal minimum eccentricity of 0.05h should be assumed. Thus a wall should be designed for a moment per unit length of at least 0.05 nwh where nw is the maximum load per unit length. Ultimate limit state Short walls Axialload An axially loaded wall should be designed in accordance with equation (9.4). Eccentric loads If the load is eccentric such that it pro- duces bending about an axis in the plane of a wall, the wall should be designed on a unit length basis to resist the combined effects of the axial load per unit length and the bending moment per unit length. The design could be car- ried out either by considering the sec.tion of wall as an eccentrically loaded column of unit width or by using the 'sandwich' approach, described in Chapter 5, for design- ing against combined bending and in-plane forces. If the load is also eccentric in the plane of the wall, an elastic analysis should be carried out, in the plane of the wall, to determine the distribution of the in-plane forces per unit length of the wall. The Code states that this analysis may be carried out assuming no tension in the concrete. In fact, any distribution of tension and com- pression, which is in equilibrium with the applied loads could be adopted at the ultimate limit state since, as ex- plained in Chapter 2, a safe lower bound design would result. Each section along the length of.. the wall should then be designed to resist the combined effects of the moment per unit length at right angles to the wall and the compression, or tension, per unit length of the wall. The design could be carried out by considering each section of the wall as an eccentrically loaded column or tension member of unit width, or by using the 'sandwich' approach. Slender walls The forces and moments acting on a slender wall should be determined by the same methods previously described for short walls. The po~ion of wall, subject to the highest intensity of axial load, should then be designed as a slen- der column of unit width. Serviceability limit state Stresses The comments made previously regarding stress calcu- lations for columns are also appropriate to walls. Crack widths Walls should be considered as slabs for the purposes of crack control calculations, and the details of the Code requirements are discussed in Chapter 7.
- 71. Plain concrete walls General Aplain concrete wall or abutment is defined as a vertical load-bearing member with an aspect ratio, on plan, greater than 4; any reinforcement is not assumed to contribute to_..··· the strength. ",,' #..-'-" '"'~. If the aspect ratio is less than 4, the member should be considered as a plain concrete column. The following design rules for walls can also be applied to columns, but, as indicated later, certain design stresses need modifi· cation. The definitions of 'short'. •slender' , 'braced' Ilnd 'unbraced', which are given earlier in this chapter for reo inforced concrete walls, are also applicable to plain con· crete walls. The clauses, concerned with slenderness and lateral support of plain walls, were taken directly from CP 110 which in tum were based upon those in CP 111 [228]. In order to preclude failure by buckling the slenderness ratio of a plain wall should not exceed 30 [229]. The effective heights given in the Code are summarised in Table 9.2. . Table 9.2 Effective heights of plain walls Wall type Unbraced, laterally spanning structure at top Unbraced, no laterally spanning structure at top Braced against lateral movement and rotation Braced against lateral movement only * 10 = distance between centres of support . I I" = distance between a support and a free edge 1.5 2.0 0.75~ or Z.ot 1.0* or 2.5t In order to be effective, a lateral support to a braced wall must be capable of transmitting to the structural ele- ments, which provide lateral stability to the structure as a whole, the following forces: I. The static reactions to the applied horizontal forces. 2. 2.5% of the total ultimate vertical load that the wall has to carry. A lateral support COUld be a horizontal member (e.g., a deck) or a vertical member (e.g., other walls), and may be considered to provide rotational restraint if one of the fol- lowing is satisfied: 1. The lateral support and the wall are detailed to provide bending restraint. 2. A deck has a bearing width of at least two-thirds of the wall thickness, or a deck is connected to the wall by means of a bearing which does not allow rotation to occur. 3. The wall supports, at the same level, a deck on each side of the wall. Forces Members, which transmit load to a plain wall, may be considered simply supported in order to calculate the re- aciion which they transmit to the wall. 126 I .'~, I i nw , I, I ...' I.... I I, I f4---.---'-h'------""'i~1 ","~__h_-2_e"-,,'_.~ Fig. 9.11 Eccentrically loaded short wall at collapse .. If the load is eccentric in the plane of the wall, the eccentricity and distribution of load along the wall should be calculated from statics. When calculating the distribu- tion of load (Le., the axial load per unit length of wall), the concrete should be assumed to resist no tension. If a number of walls resist a horizontal force in their plane, the distributions of load between the walls should be in proportion to their relative stiffnesses. The Code clause concerning horizontal loading refers to shear con- nection between walls aqd was originally written for CP 110 with shear walls ,in buildings in mind. However, the clause could be applied, for example, to connected semi-mass abutments. When considering eccentricity at right angles to the plane of a wall, the Code states that the vertical load transmitted from a deck may be assumed to act at one-third the depth of the bearing area back from the loaded face. It appears from the CP 110 handbook [112] that this requirement was originally intended for floors or roofs of buildings bearing directly on a wall. However, the inten- tion in the Code is, presumably, also to apply the require- ment to decks which transmit load to a wall through. a mechanical or rubber bearing. Ultimate limit state Axial load plus bending normal to wall Short braced wall The effects of lateral deflections can be ignored in a short wall and thus failure is due solely to concrete crushing. The concrete is assumed to develop a constant compressive stress of "•.feu at collapse, where "w is a coefficient to be discussed. The concrete stress dis- tribution at collapse of an eccentrically loaded wall is as shown in Fig. 9.11. The centroid of the stress block must i' (a) Braced-code (b) Braced Fig. 9.12(a)-(c) Lateral deflection of a slender wall coincide with the line of action, of the axial load per unit length of wall (nw), which is at an eccentricity of ex. Hence the depth of concrete in compression is 2(h12 - ex) =h - 2ex Thus the maximum possible value of nw is given by (9.29) The coefficient "w varies from 0.28 to 0.5. It is tabulated in the Code and depends upon the following: I. Concrete strength. For concrete grades less than 25, lower values of "ware adopted than f<?r concrete grades 25 and above. This is because of the difficulty of controlling the quality of low grade concrete in a wall. Hence, essentially, a higher 'value of Ym is adopted for low grades than for high grades. 2. Ratio of clear height between supports to wall length. Tests reported by Seddon [229] have shown that the stress in a wall at failure increases as its height to length ratio decreases. This is because the base of the wall and the structural member(s) bearing on the wall restrain the wall against lengthwise expansion. Hence, a state of biaxial compression is induced in the wall which increases its apparent strength above its uni- axial value. The biaxial effect decreases with distance from the base or bearing member and, thus, the aver- age stress, which can be developed in a wall, increases as the height to length ratio decreases. CP III permits an increase in allowable stress which varies linearly from 0%, at a height to length ratio of 1.5, to 20%, at a ratio of 0.5 or less. Similar increases have been adopted in the Code. ea ex1 nw Lateral ~ load I I / I, I, I, I, (c) Unbraced 3. Ratio of wall length to thickness. It is not clear why the Code requires "'IV to be reduced, when the ratio of wall length to thickness is less than4 (Le., when the wall becomes a column). The reduction coefficient varies linearly from 1.0 to 0.8 as the length to thick- ness ratio reduces from 4 to I: The reason could be to ensure that the value of "w does not exceed 0.4 when the aspect ratio is 1, because 0.4 is the value adopted for reinforced concrete columns and beams. Slender braced wall At the base of a wall, the eccen- tricity of loading is assumed to be zero. Thus the eccentricity varies linearly from zero at the base to ex at the top. A slender wall deflects laterally under load in the same man- ner as a slender column. The lateral det1ection increases the eccentricity of the load and the Code takes the net maximum eccentricity, to be (0.6ex +eo), as shown in Fig. 9.12(a). The additional eccentricity (eo) is taken, empirically. to be 1;/2500h, where I, and h are the effec- tive height and thickness of the wall respectively. It should be noted that the Code mistakenly gives the additional eccentricity as 1./2500h. If ex in equation (9.29) is replaced by (O.6ex + eo). the following equation is obtained for the ultimate strength of a slender braced wall: (9.30) The above assumption of zero eccentricity at the base of a braced wall is based upon considerations of walls in buildings [II2}. In the case, for example, of an abutment an eccentricity could exist at the bottom of the wall as . shown in Fig. 9. 12(b). If the eccentricities at the top and bottom are ed and e,.2 respectively. the author would sug- 127
- 72. I+-h/2 N V I NI V )ii21 I. Fig. 9.13 Shear normal to wall gest that, by analogy with equation (9.23), the maximum net eccentricity should be taken as the greater of > (0.4 ext + 0.6 ex2 + eu) and (0.6 e.tl + 0.4 ex}. + ea) The. appropriate net eccentricity should then be substituted for e.r in equation 9.29. Slender unbraced wall The lateral deflection of a slender unbrl,lced wall is shown in Fig. 9.12(c). The net eccen- tricities, fro,!! the wall centre line. at the top and bottom of the wall are e.,! and (e.r 2 + ea) respectively. .The Code requires every section of the wall to be capable o.f resisting the load at each of these eccentricities. Hence, by repla- cing ex in equation (9.29) by each of these eccentricities, the ultimate strength of a slender unbraced wall is the lesser of: 11",.= (h - 2ex l) f...wt.." (9.31) and . II ... =(h - 2ex2 - 2ea) f...w/Cl( (9.32) -t.Shear In general the total shear force in a horizontal plane should not exceed one-quarter of the associated vertical load. The reuson for this requirement is not clear but, since the requirement was taken from CP 110. it was intended pre- sumably for shear walls bearing on a footing or a floor. Thus it appears that the design criterion was taken to be shear friction with a coefficient of friction of 0.25. A shear force at right angles to a wall arises from a change in bending moment .down the wall. The maximum moments at the ends of a wall occur when the load is at its greatest eccentricity of h12. The maximum change of i2R moment over the length of the wall occurs when the eccen- tricities at each end are of opposite sign, as shown in Fig. 9.13, and is given by N(h12 + hI2)= Nh. Hence, the constant shear force throughout the length of the wall is V =Nhll,. In order that V does not exceed 0.25N, it is necessary that l)h should exceed 4. In fact, the Code states that it is not necessary to consider shear forces.normal to the wall ifl,/h exceeds 6. The Code is thus conservative in this respect. When considering shear forces in the plane of the wall, it is necessary to check that the total shear force does not exceed 0.25 of the associated total vertical load, and that the average shear stress does not exceed 0.45 N/mm 2 for concrete of grade 25 01' above, or 0.3 N/mm2 for lower grades of concrete. The reason for assigning these allow- able stresses is not apparent. e(Searing The bearing stress under a localised load should not exceed the limiting value given by equation (8.4). Serviceability limit state "*Deflection The Code states that excessive deflections will not occur in a cantilever wall if its height-to-Iength ratio does not exceed 10. The basis of this criterion is not apparent, but the CP 110 handbook [112] adds that the ratio can be increased to 15 if tension does not develop in the wall under lateral loading. -tCrack control It is necessary to control cracking due to both applied load- ing (flexural cracks) and the effects of shrinkage and temperature. Flexural cracking Reinforcement, specifically to control flexural cracking, only has to be provided when tension occurs over at least 10% of the length of a wall, when subjected to bending in the plane of the wall. In such situ- . ations, at least 0.25% of high yield steel or 0.3% of mild steel should be provided in the area of wall in tension: the spacing should not exceed 300 mm. These percentages are identical to those discussed in the next section when con- sidering the control of cracking due to shrinkage and temperature effects. The spacing of 300 mm is in accor- dance with the maximum spacing discussed in Chapter 7. Shrinkage and temperature effects In order to control cracking .due to the restraint of shrinkage and temperature movements, at least 0.25% of high yield steel or 0.3% of mild steel should be provided both horizontally and verti- cally. These percentages are identical to those for water- retaining structures in CP 2007 [230], but it should be noted that they are much Jess than those given in the new standard for water retaining structures (BS 5337) [231], and are also much less than those. suggested by Hughes [1851. The author would thus suggest that the values of 0.25% and 0.3% should be used with caution. Bridge piers and columns Hurlier in this chapter, the Code clauses concerned with c()lumns and reinforced walls are presented and brief men- tion lTude of their application. In the following discussion the design of bridge piers and columns in accordance with the Code i$ considercd briefly and compared with present practice. *Effective heights The Code clauses concerning effective heights are intended. primarily, for buildings and are not necessarily applicahle to bridge pier!; and column!;. However, this criti- cism is equally applicable to the effective heights given in the existing design document (CP 114). Thus, there is flO difference in the a!;sessment of effective heights in accordance with the Code and with CP 114. Slender columns and piers ('r 114 defines a slender column as one with a slenderness ratio in excess of 15. whereas the Code critical !;lenderness ratio is 12. This means that some columns. which could be considered to he short at present, would have to be con- sidered as slender when designed i'n accordance with the C()de. ep 114 allows for slenderness by applying a reduction factor to the calculated permissible load for a short column. The redlll,tion factor is a function of the slenderness ratio. This upproach is simple. but does not reflect the true Ix~haviour of a slender column at collapse. Thus the reduc- tion fac!!',r approach has not been adopted in: the Code: inst('ad. the additional moment concept. which is described earlier in this chapter. is used. Use of the latter concept requires more lengthy calculations. and thus the design of slender {'olumns. in accordance with the Cod~, will take longer than their design in accordance with CP 114. Design procedure In accordance with CP I 14. ollly olle calculation has to be IIndertaken-- the permissible load has to he checked under working load ('onditions. However. in accordance with the Code;' three calculations. each under a different load con- dition. have to be carried out. These calculations are con- cerned with strength at the ultimate limit state. stresses at the serviceahility limit stilte and. if appropriate, crack width at the serviceability limit state. Hence, the design procedure will he much longer for II column designed in accordance with the Code. A further problem arises when applying the Code: it is not clear in advance which of the three design calculations will be critical. However, it is likely that ultimate will be the critical limit state for a column, which is either axially loaded or is· subjected to a relatively small moment. For columns subjected to a large moment, either the limiting concrete compressive stress at the serviceability limit state, or the limiting crack width at the serviceability limit state could be critical. If the latter criterion is critical then it may be necessary to specify columns with greater cross- sectional areas than are adopted at present. This is because a very large amount of reinforcement would be required to control the cracks. For a column size currently adopted, the required amount of reinforcement may exceed the maxi- mum amount permitted by the Code. This possibility is increased by the fact (see Chapter 10) that the maximum amount of reinforcement permitted in a vertically cast col- umn is 6% in the Code as compared with 8% in CP 114. Bridge abutments and wing walls The design of abutments and wing walls in accordance with the Code is very different to their design to current practice. A' major difference is the number of analyse!! which need to be carried out. At present a single analysis covers all aspects of design but, in accordance with the Code, five analyses. each under a different design load. have to be carried out for the following five design aspects: I. Strength at the ultimate limit state. 2. Stresses at the serviceability limit state. 3. Crack widths at the serviceability limit state; but deemed to satisfy rules for bar spacing are appropriate in some situations (see Chapter 7). 4. Overturning. The Code requires the least restoring moment due to unfactored nominal loads to exceed the greatest overturning moment due to the design loads (given by the effects of the nominal loads multiplied by their appropriate YfL values at· the ultimate limit state). 5. Factor of safety against sliding and soil pressures due to unfactored nominal loads in accordance with CP 2Q04 [92J. A further important difference in design procedures occurs when considering the effects of applied defor- mations described in the Code and in the present documents. In the latter. all design aspects are considered under work- ing load conditions. and thus the effects of applied defor- mations (creep. shrinkage and temperature) need to he considered for all aspects of design. However. as explained in Chapter 3, the- effects of applied defor- mations can be ignored under collapse conditions. Thus Part 4 of the Code pemlits creep, shrinkage and temperature effects to he ignored at the ultimate limit state. The impli- cation of this is that less main reinforcement would be required in an abutment designed to the Code than one designed to the existing documents. Although the effects of applied deformations can he 129
- 73. ignored at theultimate limit state, they have to be con- sidered at the serviceability limit state. The effects of applied deformations thus contribute to the stresses at the serviceability limit state. Since less reinforcement would be present in an abutment designed to the Code than one designed to the existing documents, the stresses at the ser· viceability limit state would be greater in the former abut· ment. However, it is unlikely that they would exceed the Code limiting stresses of 0.8t" and O.S /,." for reinforce~ men! and concrete respectively. The main bar spacings will generally be greater for abutments designed in accordance with the Code than for those designed in accordance with the present documents. This is because the Code maximum spacing of ISO mm (see Chapter 7) will generally be appropriate for abut· ments, whereas spacings of about 100 mm are often neces· sary at present. The Code should thus lead to less,conges- tion of main reinforcement. Finally, the Code does permit the use of plastic methods of analysis, and the design of abutments and wing walls is an area where plastic methods could usefully be applied. In particular, Lindsell [2321 has tested a model abutment with cantilever wing walls. and has shown that yield line theory gives reasonable estimates of the loads at collapse of the abutment and of the wing walls, An alternative plas- tic method of design is the Hillerborg strip method, which is applied to an abutment in Example 9.2 at the end of this chapter. Foundations General A foundation should be checked for sliding and soil bear- ingpressure in accordance with the principles of CP 2004 1921. The latter document Is written in terms of working stress design and thus unfactored nnminalloads should be used when checking sliding and soil bearing pressure. Unwever. when carrying out the structural design of a foundation. the design loads appropriate to the various limit stutes should be adopted. Uence. more calculations hav!.' to be cltrried out when designing foundations in Ilcclll'dance with the Code ihan for those designed in accordance with the current documents, III .thc absence of a more accurate method. the Code permits the u!lual assumption 01' a linear distribution of hearing stress under a foundation. Footings Ultimate limit state ,.'lou"(1 The critical secticlll for hending ill taken at the fal'e of the column or wall as shown in Fill. 9,14(a), Re- inf()rcement should he designed for the total moment at this l'riticul se<.'tion and. except for·the reinforcement parallel to the shorter side of a rectanllular footing. it should be l.lO I, I, I ---d-I I.) Flexure , I, I, ~-..-.-.-..-.t-., I, ~....,D I, I, I, Iotd~ Ibl Flexural shear Critical aactlona Crltlca' section. ( 1.Sh 'b-._-_..__l-- ~ectlon'-0--' .Critical I i ' :1 ............ /'. leI Punching shear Fig. 9.14(.)-(t) Crilical sections for footing spread uniformly across the base. Reinforcement parallel to the shorter' side should be distributed as !lhown in Fig, 9.1!1i. The latter requirement ill empirical and was based upon a lIimilar requirement in the ACI Code 116RI, The Code is thus more precise than CP 114 with regard to the distribution of reinforcement. Flexural shtar The total shear on a section. at a distance equal to the effective depth from the face of the column or wall (see Fig. 9.14(b». should be checked in accordance with the method given in Chapter 6 for flexural shear in beams. These requirements, when allowance is made for the different dellign loads. are very similar to those of BE 1/73. It is worth mentioning that the Code clause is identical to that in CP 110. except that the latter document require!! ~________._.!.L.____ ,.~ :I I I I I I I I I " I" 'I,' ," ~ • ~.............._-toJ-_.............................. (~;~~.~) ..~~ (r~1!l) A. h '" overall slab depth I I I I I I I I I I ... .'. . ,. Plreaifot"" ,,. .14 ~ reinforcement (~1 -1) A. ~-;Ti 2" A." total area of reinforcement parallel to shorter side 1~1" /1 .._ Central band width Fig. 9.15 Distribution of reinforcement in rectangular footing the critical section to be at Ilh times the effective depth from the face. This critical section was adopted in CP 110 because a critical section, at a distance equal to the effec- tive depth from the loaded face, would have resulted in much deeper foundations than those previously required in accordance with CP 114. Punching shear The critical perimeter and design method discussed for slabs in Chapter 6 should be used for foot- ings. The perimeter is shown in Fig, 9.14(c), '*serviceability limit state Stresses It is necessary to restrict the stresses to the limit- ing values of 0.81y and 0.5 fe" in the reinforcement and concrete respectively. Crack widths As discussed in Chapter 7, footings should be treated as slabs when considering crack control. Piles The Code does not give specific design rules, for piles. However, once the forces acting on a pile have been assess- ed, the pile can be designed as a column in accordance with CP 2004 and the Code. Pile caps Ultim~te limit stat~ The reinforcement in a pile cap may be designed either by bending theory or truss analogy. The shear strength then has to be checked. . . Bending theory When applying the bending theory [233], the pile cap is considered to act as a wide beam in each direction. The total bending moment at any section Concrete strut 1..oo"'=!--,I,..4~-Reinforcement ~~-i-"""_Q"" tie Fig. 9.16' Ttussanalogy for piJecap can be obtained at that section, and the total amount of reinforcement at the section determined from simple bend- ing theory as described in Chapter S. Such a design method is not correct because a pile cap acts as a deep. rather than a shallow, beam; however, the method has been shown by tests to result in adequate designs [234]. This is probably because most pile caps' fail in shear and the method of design of the main reinforcement is, largely, irrelevant [234]. The total I1mount of reinforcement cal- culated at a' section should be unifonnly distributed across the section. Truss analogy The truss analogy assumes a strut and tie system within the cap, and is in the spirit of a lower bound method of design. The strut· and tie system for a four-pile cap is shown in Fig. 9.16. Formulae for detennining the forces in the ties for various arrangements of piles are given by Allen [203] and Yan [235]. It can be seen from Fig. 9,16 that, because of the assumed structural action, the reinforcement, calculated from the tie forces, should be concentrated in strips over the piles. However, since it is considered good practice to have some reinforcement throughout the cap, the Code requires 80% of the rein- forcement to be concentrated in strips joining the piles and the remainder to be unifonnly distributed throughout the cap. Tests carried out by Clarke [234] have demonstrated the adequacy of the truss analogy, Flexural shear The Code requires flexural shear to be checked across the full width of a cap at a section at the face of the coluf!1n, as shown in Fig, 9.17(a). It should be noted that the critical section is not intended to coincide with the actual failure plane, but is chosen merely because it is convenient for design purposes. The question now arises as to what allowable design shear stress should be used in association with the above critical section. Tests carried out by Clarke [234] have indicated that the basic design stresses given in Table 6.1 should be used, except for those partsof the critical sec- tion which are crossed by flexural reinforcement which is fully anchored by passing over a pile. For the latter parts of the critical section, the basic design shear stresses should be enhanced to allow for the increased shear resis- tance due to the short shear span (see Chapter 6). The enhancement factor (2dfav) where d is the effective depth and av , is the shear span which, in the present context. is
- 74. C~~ical section r./___~_____?' /' q 1 ) . --- ~-::-- - - - Enhance Vc __ -T-::;-- -- over these I I lengths , /' p.---~---- / -, I " .... ,J 2600kN 280kN Bearing which permits only rotation d~ X _14 P _I . __,_.._ "_"" --<.,.."...._, d ;.. effective depth of cap (a) Flexural shear # ,,/..U' / Critical ~,> '/_...----+- section . / I ' '< I I , ,/ ~ ~--- (b) Punching shear-Code (c) Punching shear-actual Fig; 9.17(a)-(c) Shear in pile caps taken as the distance between the face of the column and the nearer edge of the piles, viewed in elevation, plus 20% of the pile diameter. The Code states that the reason for adding 20% of the pile diameter is to allow for driving tolerances. However, Clarke [234] has suggested the same additional distance in order to allow for the fact that the piles are circular, rather than rectangular, and thus the 'average' shear span is somewhat greater than the clear distance between pile and critical section. The value of 20% of the pile diameter, chosen by Clarke [234], is simi- lar to the absolute value of 150 mm suggested by Whittle and Beattie [233] to allow for dimensional errors. How- ever, the result is that the allowable design stress varies along the critical section, as shown in Fig. 9.17(a), and the total shear capacity at the section should be obtained by summing the shear capacities of the component' parts of the section. ' It is understood that, in a proposed amendment to CP 110, the critical section for flexural shear is located at 20% of the diameter of the pile inside the face of the pile. Thus the critical section is at the distance av , defined in Fig 9.17(a), from the face of the column. This criticalsec- tion is more logical than that defined in the Code, but the 132 8m O,6m H I,m D Fig. 9.18 Bridge column resulting shear force at the critical section will be only marginally different. Punching shear Clarke [:~34] suggests that punching of the column through the cap:' need only be considered if the pile spacing exceeds four times the pile diameter. which is unlikely; thus the Code only requires punching of a pile through the cap to be considered. The critical section given in the Code for punching of a comer pile is extremely difficult to interpret and originates from the 1970 CEB recommendations [226]. The relevant diagram in the latter document shows that the correct interpretation is as shown in Fig. 9. 17(b). The Code does not state what value of allowable design shear stress should be used with the critical section. In view of this, the author would suggest using the value from Table 6.1 which is appropriate to the average of the two areas of reinforcement which pass over the pile, This suggestion is not based upon considerations of the Code section of Fig. 9.17(b) but of the section which would actually occur as shown in Fig. 9.17(c). The basic shear stress, obtained from Table 6.1 should then be enhanced by (2dlav), where . av should be taken as the distance from the pile to the critical section (I.e. dI2). Thus, in all cases in which failure could occur along the Code critical section. the enhance- ment factor would be equal to 4. It is understood that, in a proposed amendment to CP 110, punching shear is checked by limiting the shear stress calculated on the perimeter of the column to 0.8/hu' This limiting shear stress is very similar to the maximum nominal shear stress of 0.75 JTcu which is specified in the Code (see Chapter 6). Serviceability limit state Realistic values of stresses and crack widths in a pile cap, at the serviceability limit state, could only be assessed by carrying out a proper analysis; such an analysis would probably need to be non-linear to allow for cracking. Since it is difficult to imagine serviceability problems arising in a cap which has been properly designed and detailed at the ultimate limit state, a sophisticated analysis at the ser- viceability limit state cannot be justified. Thus, the author would suggest ignoring the serviceability limit state criteria for pile caps. Examples '"*9.1 Slender column A reinforced concrete column is shown in Fig. 9.18. The loads indicated are design loads at the ultimate limit state. Design reinforcement for the column, at the ultimate limit state, if the characteristic strengths of the reinforce- ment and concrete are 42S N/mm2 and 40 N/mmD respec- tively. Assume that the articulation of the deck is (a) such that sidesway is prevented and (b) such that sidesway can occur. ·'f:.No sidesway With sidesway prevented, the column can be considered to be braced. Consider the column to be partially restrained in direction at both ends, and, from Table 9.1, take the effective height to be the same as the actual height, I.e.. Ie = 8 m. Slenderness ratio = 8/0.6 = 13.3. This exceeds 12, thus the column is slender. Assume a minimum eccentricity of O.OSh = 0.03 m for the vertical load. Initial moment at top of column = Ml = 0' Initial moment at bottom of column = M2 = 280 x 8 + 2600 x 0.03 = 2318 kNm Since the column is braced, the initial moment to be added to the additional moment is, from equation (9.23), Mi = (0.4)(0) + (0.6)(2318) = 1391 kNm From equation (9.22), the total moment is M, = 1391 + (2600 x 0.6/1750)(13.3)2(1 - 0.0035 x 13.3) = 1391 + 150 = 1541 kNm However,this moment is less than M2 , thus design to resist M, = M2 = 2318 kNm. M/bh2 = 2318 x 108 /(1200 x 6002) = 5.37 N/mm2 Nlhh = 2600 x 03/(1200 x 600) = 3.61 N/mm2 Assume 40 mm bllrs with 40 mm cover in each face, so that d/h = 540/600 = 0.9. Substructures and foundations Hence use Design Chart 84 of CP 110: Part 2 [128]: from which 100 As,!bh = 2.8 :. Asc = 2.8 x 600 x 1200/100 = 20160 mml Use 16 No. 40 mm bars (20160 mm2) with 8 bars in each face. :Jf.Sidesway Since sidesway can occur, consider the column to be a cantilever with effective height equal to twice the actual height, i.e. Ie = 16 m. SJr.ndenless ratio == 16/0.6 = 26.7 The initial and additional moments are both maximum at the base. Hence, M; = 23,18 kNm and, from equation (9.22), the total moment IS M, = 2318 + (2600 x 0.6/1750)(26.7)2 (1 - 0.0035 x 26,7) = 2318 + 576 =2894 kNm ! Mlbh2 =2894 x 106 /(1200 x 6002 ) = 6.70 N/mm2 Nlbh = 3.61 N/mm2 (as before) From Design Chart 84 of CP 110: Part 2, 100 As,!bh = 3.6 :. Asc = 3.6 x 600 x 1200/100 = 25 920 mml Use 22 No. 40 mm bars (27 '720 mml) with II bars in each face. . It should be noted that the above designs have been car- ried out only at the ultimate limit state. In an actual design, it' would be necessary to check the stresses and .crac~ widths at the serviceability limit state by carrying out elastic analyses of the sections. '¥ 9.2 Hillerborg strip method applied to an abutment .. A reinforced concrete abutment is 7 m high and 12 m wide. At each end of the abutment there is a wing wall which is structurally attached to the abutment. The lateral loads acting on the abutment are the earth pressure, which varies from zero at the top to 5H kN/m2 at a depth H; HA surcharge, the nominal value of which is 10 kN/m2 (see Chapter 3); and the HA braking load which acts at the top of the abutment and may be taken t~ have a nominal value of 30 kN/m width of abutment. The Hillerborg strip wethod will be used to obtain a lower bound moment field for the abutment. . At the ultimate limit state, the design loads (nominal load x YfL x YfJ) are: Earth pressure 5H X 1.5 x 1.15 = 8.625H kN/m2 HA surcharge = 10 x 1.5 x 1.10 = 16,5 kN/m2 HA braking = 30 x 1.25 x 1.10 = 41.25 kN/m The wing walls and abutment base are considered to provide fixity to the abutment, which will thus be designed as if it were fixed on three sides and free on the fourth. The load distribution is chosen to be as shown in Fig. 9.19 (see also Chapter 2). Thus at the top of the abutment all of the load is considered to be carried in the y direction; at the centre of the base, all of the load is considered to be car- ried in the x direction; in the bottom comers, the load is
- 75. ,....1..;,.6;.,;;.5,;.;,kN;..;.;/-{'m;.;...2__.... 41.25kN/m 2S.1kN/m2 4m 7m ---~-r--I ." 1~=11..1 .''_I~ f~1 2.5m tSm 51.0kN/m2 ---,--r-B~----~~~--------~~--~--4-~~B 68.3kN/m2 1m 1.5m 76.9kN/m2 t~-~: ~ Pressure distribution - - Free edge ~Flxededge -- - Load dispersion line ---- Zero moment line --- Typical strip' '(ii-'/'.(ii!: Strong edge band Fig. 9.19 Abutment "':467kNm~' . .~-467kNm ~+374kNm Fig. 9.20 Strip AA considered to be shared equally between the x and ydirec- tions. It is emphasised that any distribution of load could be choseiland that shown in Fig. 9.19 iSinerely one pos- sibility. . In order .that the resulting moments do not depart too much from the. elastic. moments. and thus serviceability problems do not arise, the zero moment lines shown in Fig. 9.19 are chosen. Typical strips AA, BB, CC, DD and EE of unit width are now considered. StripAA Tile loading and bending moments are shown in Fig. 9.20. Strip 88 The loading and bending moments ate shown in Fig. 9.21. Strip CC . ." . . Strip EE. which carries the braking load. earth pressure and surcharge at the top of the abutment,must also pro- 134 34.1SkN/m 34.15kN/m ~~~--~ 6m -137kNm~ , ~-137kNm ~------:::;;;> +17kNm Fig. 9.21 Strip BB 0.5 m+______ O.5m -¥------- 3m 51.0kN/m-¥-._--- 1.5m 1.5m 76.9kN/m Fig. 9.22 Strip CC .Ji=12:4kN -187kNm +43kNm vide a reaction to strip CC. Hence. the loading and bend- ings moments for strip CCare shown in Fig. 9~22; The reaction (R) can first be obtained by taking moments about the point of zero moment, and then the bending moment diagram can be calculated. "~..,.,, - -_._---_.__. O.5m R=6.2kN O.5m 3m 25.5kN/m 1.5 III 1.5m ....f. "........-....-.----..- ~m:'r77 38.5kN/m Fi;. 9.23 Strip DD 6.2kN/m 12.4kN/m 6.2kN/m e ooeocoo 25.1 +41.25 = 66.35kN/m ~ . .~ 2m1m 6m. 1m2m i4=-~----- "I' 'I. iI +22kNm Reactions from K direction strips Earth pressure, surcharge and braking .,~7~~~,~~~ /1-763kNm ". ~ '.' ..: +627*Nm . l<'lg. 9.24 Strip EE Strip DO Strip DD is similar to strip CC and its loading and bending moments are shown hi Fig. 9.23. Strip EE Strip EE acts as a strong edge band (1 m wide) which not only supports the surcharge, earth pressure and braking loads but also supports the ends of typical strips CC and DO. Thus the loading and bending moments are as shown in Fig. 9.24. The loading is taken, conservatively, as that' at a depth of 1 m. *9.3 Pile cap Design the four-pile cap shown in Fig. 9.25 if the charac· teristic strengths of the reinforcement and concrete are 425 N/mm2 and 30 N/mm2 respectively. The design load at the ultimate limit state is 5200 kN. Load ~r pile = 5200/4 = 1300 kN The ca will be designed by both bending theory and truss analog methods. "fBendingtheory Bending Total bending moment at column centre line ::: 2 x 1300 x 0.75 = 1950 kNm Assume effective depth = d = 980 mm From equation (5.7), the lever arm is O.4m O.76m O.76m O.4m Substructures and foundations f ~ 0.3m I ----~.3~ --l-~--r------~·-- "....,..........,~ / .,I I -I- I I , I ',_...-;' I ~ m___-tI .....--..... 0.3 m 0.3 mO.2 ml " I I I I+-' i /I ' .... ...-;1" I ~~~O.74m~+~-0~.7~6~m~·4+~-~O.~76~m--~+~O.4m~ 16200kN 1.1 m Fla. 9.25 Pile cap ( / 5 x 1950 X lOll ) Z =0.5 x 980 1 + vI - 30 x 2300 x 9802 =943 mm But maximum allowable z = 0.95d = 0.95 x 980 = 931 mm Thus z = 931 mm From equation (5.6), required reinforcement area is As = 1950 x 108 /(0.87 x 425 x 931) = 5665 mm' Use 19 No. 20 mm bars' (5970 mm2 ) Flexural shear 100 A/bd = (100 x 5970)/(2300 x 980) =0.26 From Table 5 of Code, allowable shear stress without shear reinforcement = Vi" = 0.36 N/mm2 • This stress may be enhanced by (2dla,,) for those parts of the critical sec- tion il1dicated in Fig. 9.17(a). a" = 200 + 0.2 x 500 = 300 mm Enhancement factor = 2 x 980/300 = 6.53 Enhanced Vc = 6.53 x 0.36 = 2.35 N/mm2 Shear capacity of critical section = «2)(2.35)(500) + (0.36)(2300 - 2 x 500)] 980 x lO-a = 2760 kN Actual shear force = 2 x 1300 = 2600 kN < 2760 kN :. O.K. 135
- 76. Punching shear Tile criticalsection shown in Fig. 9.17(b) would occur under the column in this example. Thustake critical section 11t' cotner of column, as shown in Fig. 9:25, at [(0.75 - 0.3)/2 - 0.25] = 0.386 m from pile. The latter value will be assumed for avo . .' From Fig. 9.17(b), length of perimeter is 980 + 500 = 1480 mm As for flexutal shear~. v,: = Q,36 N/mm 2 . Enhancement factor ~ (2 x 980/386) = 5.08 Enhanced Vc = 5.08 x 0.36 = 1.83 N/mm2 Shear capacity of critical sectiQn = 1;83 x 1480 x 980 x 1O~3 = 2650 leN Actual shear force = 1300 kN< 2650 kN :. O.K. . :¥. Truss analogy Truss FQr equilibrium, the force in each of the reinforcement ties of Fig. 9.16 is NIISd. = (5200 x 1500)/(8 x 980) = 995 kN Required reinforcement area is A" = 995 x lOa/(0.87 x 425) = 2690 mm2 Since there at two ties in each direction, the total re~ inforcement area in each direction is 2 x 2690 = 5380 mm 2 • It can be seen that the truss theory requires less reinforcement than the bending theory, and this is gener~ ally the case. 80% of the tie reinforcement should be provided over the piles, i.e. 0.8 x 2690 = 2150 mm2 Use 7 No. 20mm bars (2200 mm2 ) over the piles. The remaining 20% (540 mm2) should be placed between 136 the cap centre line and the piles, i.e. 2 x 540 ::::' 1080 mm2 should be placed between the piles. Use 4 No. 20 mm bars (1260 mm2 ) between the piles. '*Flexural shear Over a pile, 100 Aslbd = (100 x 2200)/(500 x 980) . = 0.45. From Table 5 of Code, Vc = 0.51 N/mm2 Enhancement factor = 6.53 (as for bending theory) Enhanced Vc = 6.53 x 0.51 = 3.33 N/mm2 Between piles, 100 AJbd = (100 x 1260)/(1000 x 980) =0.13. From Table 5 of Code, Vc = 0.35 N/mm2 Assume no reinforcement outside piles, thus Vc = 0.35 N/mm2 Shear capacity of critical section =[(2)(3.33)(500) + (0.35)(2300 - 2 x 500)] 980 X 10"a = 3710 kN' . Actual shear force = 2 x 1300 = 2600 kN < 3710 kN :. O.K. *Punching shear Length of critical section = 1480 mm (as for bending theory) As for flexural shear, Vc'= 0.51 N/mm2 Enhancement factor = 5.08 (as for bending theory) Enhanced Vc = 5.08 x 0.51 = 2.59 N/mm2 . Shear capacity of critical section = 2.59 x 1480 x 980 x lO-a = 3760 kN Actual shear force = 1300 kN < 3760 kN :. O.K. It c~m be seen from the above calculations that the truss theory design results in a greater shear capacity than does the bending theory design. Chapter 10 Detailing Introduction In this chapter, the Code clauses, concerned with con- siderations affecting design details for both reinforced and prestressed concrete, are discussed and compared with those in the existing design documents. Table 10.1 Nominal covers Conditions of exposure Nominal cover (mm) for concrete grade 25 30 40 ~50 Moderate Surfaces sheltered from severe 40 rain and against freezing whilst saturated with water, e.g. ( I) surfaces protected by a waterproof membrane; (2) internal surfaces whether subject to condensation or not; (3) buried concrete and concrete continuously under water Severe (I) Soffits (2) Surfaces exposed to driving rain, alternate wetting and drying; e.g. in contact Vith back-fill and to freezing whilst wet Very severe 30 25 20 30 25 (I) Surfaces subject to the NtA 50* 40* 25 effects of de-icing salts or salt spray, e.g. roadside structures and marine structures (2) Surfaces exposed to the NtA NtA 60 50 action of sea water with abrasion or moorland water having a pH of 4.5 or less. * Only applicable if the concrete has entrained air (see text) Reinforced concrete *'Cover The cover to a particular bar should be at least equal to the bar diameter, and is also depen.dent upon the exposure con- dition and the concrete grade as shown by Table 10.1. These values are very similar to those given in Amend- ment 1 to BE 1173: however, there are two important dif- ferences. First, the Code considers all soffits to be subjected to severe exposure conditions, whereas BE 1173 distinguishes between sheltered soffits and exposed ·soffits. Thus. for the soffit of a slab between precast beams, the Code would require, for grade 30 concrete, a minimum cover of 40 mm, whereas BE 1173 would require only 30 mm. Hence, top slabs in beam and slab construction may need to be thicker than they are at present. Second, for roadside structures subjected to salt spray and constructed with grade 30 or 40 concrete, the Code requires the concrete to have entrained air. BE 1173 does not have this requirement. The Code thus requires a dramatic change in current practice. The footnote to Table 10.1 appears in the Code with a reference to Part 7 of the Code. However, Part 7 refers only to the permitted vari- ation in specified air content without giving the latter, but Clause 3.5.6 of Part 8 of the Code does specify air contents for various maximum aggregate sizes. Bar spacing if·Minimum distance between bars For ease of placing and compacting concrete, the Code relates the minimum distance between bars to the maxi- mum aggregate size. The Code clauses were taken from CP 116 and are thus more detailed than those in CP 114, although they are very similar in implication. I.n addition to rules for single bars and pairs of bars, the Code -gives rules for bundled bars since the latter are allowable. 137
- 77. Maximum spacing ofbars in tension In order to 'cbntrol crack widths. to the 'val~es given in Table 4,7, thernaximul11 spacing of bars has to be Ihnited; . The procedures for cal~ulating maximum har spacings are discussed in Chapter 7. The Code also stipulates that, in no circumstances, should the spacing exceed 300,mm. This was considered a .reas()nablema~iinumspacingto ensure that, in all reinforcedconcrete bridge members, the . bars wouldbesufficientlyclbse together for them to be assumed ·to·fonn a'smeared' layer of reinforcement,.. rather than act as individual bars. Minimum reinforcement areas *Shrinkage andtemperatur~'reinfbrce~ent' In those parts of a structure where cr@cking could occur due ,to restraint to. shrinkage or thermal movements,at least 0.3% of mild. steel or 0.25% of high yield steel should be provided. These values are less than those sug- gested .by Hughes [185] and the author would suggest that they.be used with caution. The Code values originated in CP 2007. "*Beams and slabs .A minimum area of tension reinforcement is required in, a beam or slab in order to ensure that the cracked strength of the. section exceeds its uncracked strength; otherwise, any reinforcement would yield as soon as cracking occurred, and extremely wide cracks would result. The cracking moment of a rectangular concrete beam is given by M, =f,bh2 /6 where !tis the tensile strength of the concrete, and band h are the breadth and overall depth respectively. If the beam is reinforced with an area of reinforcement (As)at an effec- ,tive depth (d) and having a characteristic strength (fy), the ultimate moment ofresistance is given by M,. =:=fyA,.•z where z is the lever arm~ Since it is required that Mu~ M"then f,.A.z ~ f,hh2/6 .01' .' ( h2 ). '... f ,- ==16.7:IJ.. d. .. fy Beeby ri191has shownthatji"" 0.556)];,;: thus, for the maximum allowable value of Icuof' 50 N/mm2, ft = 3..9N/mm2. Hence, forfy =' .250 N/mm2 and 410 Nlmm2 re'spectively, the required minimum reinforcement percen- tages are 0.26 and 0.16. These values agree very well with . the Code values of 0.25 and 0.15 respectively. The latter .values cannot be compar~9directly with thos~ in CP 114 because the CP 114 values are expressed as a percentage of the gross section, rather than the effective section. However; the· Code will generally require greater minimum areas of reinforcement than does CP 114.. 138 load Column service ,.",~~.,..,.,~,..,..,..,..;".,..,....,_......,;.... load load Carried ~~~~~~~~~~~b';'com:ret& ' .......... '-... ·Time Fig. 10.1 Load transfer in column 'under service' load conditions The above minimum reinforcement areas are. given in " . the Code under the heading 'Minimum area of main i. reinforcement' '. but, since a minimum .area of secondary i re.inforcement is not speCified for solid slabs, it would seem prudent also to apply the Code values to secondary reinforcement. Voided slabs Although a minimum area of secondary reinforcement in 'i solid slabs is not specified, values are given for voided slabs. These values are discussed in detail in Chapter 7. Columns Under long-term service load conditions, load is transfer- red from the concrete to the reinforcement as shown in Fig. 10.1. The load transfer occurs because the concrete creeps and shrinks. If the area of reinforcement is very small. there is a danger of the reinforcement yielding under service load conditions. In order to prevent yield, ACI Committee 105 [236] proposed a minimum rein~ forcement area of 1%. This value is adopted in the Code and is a little greater than that (0:8%) in CP 114. How- ever, if a column is lightly loaded, the area of reinforce- mentis allowed' to be. less than 1% but not less than (0.15 Nlfy), where Nis the ultimate axial load and fy is the characteristic strength of the reinforcement. This require- ment is intended to cover a case where a column is made much larger than is necessary to carry the load. In order to ensure the stability of ,a reinforcement cage pdor to casting, the Code requires (as does CP114) the main bar diameter to be at least 12 mm. In addition, the Cbde require~ at least six main bars· for circular columns and four bars for rectangular columns.. t Walls It is explained in Chapter 9 that a reinforced concrete wall, , which carries a significant axial Imid, should have at least. . 0.4% vertical reinforcement. This requirement is necessary because !imaller amounts of reinforcement .canresult in ,a reinforced wall which is weaker than a plain concrete wall [227]. Links Links are generally present in a member for two reasons: to act as shear or torsion reinforcement, and to restrain main compression bars. The minimum requirements for links to act as shear reinforcement are discussed in detail in Chapter 6: In the following, the requirements for a link to restrain a com- pression bar are discussed. Beams and columns The link diameter should be at least one-quarter of the diameter of the largest compression bar, and the links should be spaced at a distance which is not greater than twelve times the diameter of the smallest compression bar. These requirements are the same as those in CP 114. except that the latter also requires the link spac- ing in columns not to exceed the least lateral dimension of the column nor 300 mm, and the link diameter not to be less than 5 mm. The latter requirement is automatically sulil'oli~J by the fad lltul the slllullest available bar has a diameter of 6 mm (although it is now difficult to obtain reinforcement of less than 8 mm diameter). Walls and slabs When the designed amount of compres- sion reinforcement exceeds 1%, links have to be provided. The link diameter should not be less than 6 mm nor one- quarter of the diameter of the largest compression bar. In the direction of the compressive force, the link spacing should not exceed 16 times the diameter of the compres- sion bar. In the cross-section of the member, the link spac- ing should not exceed twice the member thickness. These requirements were taken from the ACI Code [168] and are different to those ofCP 114. Maximum steel areas In order to ease the placing and compacting of concrete, the amount of reinforcement in a member must be restricted to a maximum value. The Code values are as follows. Beams and slabs Neither the area of tension reinforcement nor that of com- pression reinforcement should exceed 4%. CP 114 requires only that the area of compression reinforcement should not exceed 4%. Columns The amount of longitudinal reinforcement should not exceed 6% if vertically cast, 8% if horizontally cast nor 10% at laps. The CP 114 amount is always 8%. Hence, the Code is 1110re restrictive with regard to vertically cast columns, and this fact, coupled with the small allowable design crack width, could result in larger columns - as discussed in Chapter 9. Walls The area of vertical reinforcement should not exceed 4%. No limit is given in CP 114. Bond General All bond calculations in accordance with the Code are car- ried out at the ultimate limit state. t T+'OT ~______~'OX~______~~I Fig. 10.2 Local bond The Code, like Amendment 1 to BE 1/73, recognises two types of deformed bars: 1. Type 1, which are, generally. square twisted. 2. Type 2, which have, geneFaHy, transverse ribs. Type 2 bars have superior bond characteristics to type 1 bars. However, unless it is definitely known at the design stage which type of bar is to be used on site, it is necessary to assume type 1 for deSign purposes. Local bond Consider a beam of variable depth subjected to moments which increase in the same direction as the depth increases, as shown in Fig. 10.2. The tension steel force at any point (x) is T, where T= Mlz and M and z are the moment and lever arm respectively at x. The rate of change of T is dT _ z(dM/dx) - M(dz/dx) dx - Z2 But dMIdx is the shear force (V) at x and the Code assumes dzldx "'" tan e•. Hence dT V - M tan eslz dx z But dTldx is also equal to the bond force per unit length; which is fbs (l:us), where fb. is the local bond stress and (l:us) is the sum of the perimeters of the tension rein- forcement. Hence V - M tan fj,/z A. (l:us ) = z S or V - M tan fJ.lz A, = (l:us)z (10.1 ) The Code assumes that z"'" d (and adjusts the allowable values of fbs accordingly). If M increases in the opposite direction to which d increases, the negative sign in equa- tion (10.1) becomes positive. Hence, the following Code equation is obtained I _ V + M tan 0.,1d Jb.• - (l:us)d (10.2) In addition to the modification to allow for variable depth. this equation differs to that in CP 114 because the 139
- 78. CP 114 equationis written in terms of the lever arm rather than the effective depth. The allowable local bond stresses at the ultimate limit state depend on bar type and concrete strength: they are given in Table 10.2. The bond stresses for plain, type 1 deformed and type 2 deformed are in the approximate ratio 1 : 1.25 : 1.5. It is understood thl1t the tabulated values were obtained by considering the test data of Snowdon [237] and by scaling up the CP 114 values, for plain bars at working load conditions, to ultimate load conditions. Snowdon's tests on 150 mm lengths of various types of bar indicated that the bond stresses developed by plain, square twisted (type 1) and ribbed (type 2) bars were in the approximate ratio 1 : 1.3: 3.5. Hence the Code ratio is c reasonable for type 1 deformed bars but can be seen to be conservative for type 2 deformed bars. However, Snowdon found that the advantage of the latter bars over plain bars decreased with an increase in diameter, particu)a1'1y with low strength concrete. Table 10.2 Ultimate local bond stresses Local bond stress (N/mm2) for concrete grade Bar type 30 :s; 4020 25 Plain 1.7 2.0 2.2 2.7 Deformed Type 1 2.1 2.5 2.8 3.4 Deformed Type 2 2.6 2.9 3.3 4.0 The Code local bond stresses are about 1.5 to 1.6 times those in BE 1/73 if overstress is ignored. However, it should be remembered that the Code bond stresses are intended for use at the ultimate limit state with a steel stress of 0.87 f)., whilst the BE 1/73 bond stresses are intended to be used at working load with a steel stress of about 0.56 I,.. The ratio of these steel stresses is 1.55, and thus the result of carrying out local bond calculations in accordance with the Code and with BE 1173 should be about the same. The author understands that, in a proposed amendment to CP 110, local bond calculations are not required at all. If this proposal is adopted, then local bond calculations will, presumably, be omitted from the Code also. Anchorage bond The conventional expression for the anchorage length (L) of a bar, which is required to develop a certain stress (j,) can be obtained from any standard text on reinforced con- crete, and is (10.3) where rna is the average anchorage bond stress and cp is the bar diameter. Since bond calculations are carried out at the ultimate limit state, the reinforcement stress (j,) is the design stress at the ultimate limit state and is 0.87 f" for tension bars and O.72/y for compression bars. However, if more than the required amount of reinforcement is provided. then the 140 stress in the bar is less than the design stress and lower values ofis, than those given above, may be used. The allowable average anchorage bond stresses (fba) depend upon bar type, concrete strength and whether the bar is in tension or compression. Higher values are permit- ted for bars in compression because some force can be transmitted from the bars to the concrete by end bearing of the bar. The allowable anchorage bond stresses are given in Table 10.3. The stresses for plain, type 1 deformed and type 2 deformed bars are .in the approximate ratio 1 : 1.4 : 1.8, and those for bars in compression are about 25% greater than those for bars in tension. It is understood that the values for bars in tension were obtained by consid- ering the test data of Snowdon [237] and by scaling up the CP 114 values, for plain bars at working load conditions, to ultimate load conditions. Snowdon's tests indicated that the anchorage lengths for plain, square twisted (type 1) and ribbed (type 2) bars were in the approximate ratio 1 : 1.4 : 2. The Code ratio agrees very well with Snow- . don's results. The Increase of 25%, when bars are in com· pression, was taken from that implied in CP 114. Table 10.3 Ultimate anchorage bond stresses Bar type Plain, in tension Plain, in compression Deformed, type I, in tension Deformed, type I, in compression Deformed, type 2, in tension Defon:ned, type 2. in . compression Anchorage bond stress (N/mm2) for concrete grade 20 25 30 1.2 1.4 1.5 1.5 1.7 1.9 1.7 1.9 2.2 2.1 2.4 2.7 2.2 2.5 2.8 2.7 3.1 3.5 ;:!!: 40 1.9 2.3 2.6 3.2 3.3 4.1 The Code average bond stresses for plain bars are about 1.5 those in BE 1/73, if overstress is ignored. However, since the ratio of steel stresses is 1.55 (see previous dis- cussion of local bond stresses), anchorage lengths for plain bars will be about the same whether calculated in accor- dance with the Code or BE 1173. But anchorage lengths for deformed bars will be shorter by about 13% for type I bars and 29% for type 2 bars. This is because BE 1173 allows increases in bond stress. above the plain bar values, of only 25% for type I bars and 40% for type 2 bars. These percentages compare with 40% and 80%. respec- tively, in the Code. Bundled bars The Code permits bars to be bundled into groups of two, three or four bars. The effective perimeter of a group of hars is obtained by calculating the sum of the perimetcrs of the individual bars and, then, by multiplying by a reduc- tion factor of 0.8. 0.6, or 0.4 for groups of two, three or four bars respectively. The resulting perimeter, so calcu- lated, is less than the actual exposed perimeter of the group of bars to allow for difficulties in compacting con- crete around groups of bars in contact. Steel force T .. ~--~--+4--~-.--.-.I 1 ---~ 1 Mlz ' I -+-_--====-1 1 , 1 t. beam Fig. 10.3 Steel force diagram "JY:; Lap lengths In general, as in CP 114, a lap length should be not less than the anchorage length calculated from equation (10.3). However. for deformed bars in tension, the lap length should be 25% greater than the anchorage length. This requirement is to allow for the stress concentrations which occur at each end of a lap, and which result in splitting of the concrete along the bars at a lower load than would occur for a single bar in a pull-out test [238]. Such split- ting does not occur with plain bars, which fail in bond by pulling out of the concrete. In addition to the above requirements, the Code requires the followTng minimum lap lengths to be provided for a bar of diameter <p: I. Tension lap length 4: 25 <P + 150 mm 2. Compression lap length 4: 20 cp + J50 mm These minimum lengths are much ITIore conservative than those in CP 114 for small diameter bars, and slightly less conservative for large diameter bars. Bar curtailment and anchorage General curtailment As in CP 114, a bar should extend at least twelve dia- meters beyond the PQint at which it is no longer needed to . carry load. A bar should also be extended a minimum distance to allow for the fact that. in the presence of ~hear, a bar at a particll!ar section has to carry a force greater than that cal- culated by dividing the bending moment (M) at the section by the lever arm (z). A rigorous analysis [239] of the truss of Fig. 6.4(a), rather than the simplified analysis of Chap- tcr 6, shows that the total force, which has to be carried by the main tension reinforcement at a section where the moment and shear force are M and V, respectively, is T = Mlz + (VI2) (cot () - cot ex) The Code assumes 0 = 45°. thus T = Mlz + (VI2) (1 - cot ex) (10.4) In Fig. 10.3, the distributions of tension force due to , bending (Mlz) and total tension force (T) are plotted for a general case. It can be seen from Fig. 10.3 that the increase in steel force due to shear can be allowed for by de~igning the reinforcement at a section to resist only the moment at that section, and by extending, the reinforce- ment ~eyond that section by the distance6x in Fig. 10.3. The dIstance 6x can be found by equating the total steel force at a section at x to the steel force due to moment only at a section at (x +6 x). The maximum increase in steel force due to the shear force, and, hence, the maximum value of6x occurs when cot IX is zero (i.e. vertical stirrups) and equation (10.4) becomes T = Mlz + VI2 (l0.5) For a central point load (2W) 'on a beam of span I, the moment and shear force at x are: M.t = Wx Vx = W From equation (10.5) T.r =Wxlz + WI2 The moment at (x + 6X) is M~·+D.X =W(x + 6X) t::.x can be found from Tx = Mx~D.xlz Th'us WXlz + ~/2=W(x + 6Jc)/z From which I1x == il2 If this analysis is repeated for a uniformly distributed load. it can be shown that6x is, again, about z/2. Hence, if the longitudinal reinforcement is designed solely to resist the moment at a section, the reinforcement should be extended a distance zl2 beyond that section. However, the Code. conservatively, takes the extension length to be the effec- . tive depth. .Curtailment in·tensionzones In addition to the above general requirements, the Code requires anyone of the following conditions to be met before a bar is curtailed in a tension zone. . 1. In order to control the crack width at the curtailment point, a bar should extend at least an anchorage length, calculated from equation (10.3) with f, = 0.87 fy, beyond the point at which it is no longer required to resist bending. 2. Tests, such as those carried out by Ferguson and Mat- loob [2401, have shown that the shear capacity of a section with curtailed bars can be up to 33% less than that of a similar section in which the bars are not cur- tailed. In order to be conservative, the Code requires the shear capacity at a section. where a bar is curtailed, to be greater than twice the actual shear force. 3. In order to control the crack width at the curtailment point, at least double the amount of reinforcement required to resist the moment at that section should be provided. This requirement was taken from the ACI Code [168]. 141
- 79. o (a) Bearing force Bb= $ 14-14-"-~ o o f.---- $_.. ___+1 (b) Definitions of s" Fig. IO.4(a),(b) Bearing force at bend o The above requirements are far more complicated than those of CP 114, unless one chooses to apply option 1 and continue a bar for a full anchorage length. *Anchorage at a simply supported end The Code requires one of the following conditions to be satisfied. 1. As in CP 114,. a bar should extend for an anchorage length equivalent to twelve times the bar diameter; and no bend or hook should begin before the centre of the support, 2. If the support is wide and a bend or hook does not begin before dl2 from the face of the support (where d is the effective depth of the member), a bar should . extend from the face of a support for an anchorage length equivalent to (dI2' + 12 <1», where II> is the bar diameter. 3. Provided that the local bond stress at the face of a support is less than half the value in Table 10.2, a straight length ofbar should extend, beyond the centre line of the support. the greater of 30 mmor one-third of the support width. This clause was originally writ- ten. for CP 110, to covet small precast units [112]. and it is not clear whether the clause is applicable to bridges. Bearing stresses inside bends The bearing stress on the concrete inside a bend of a bat of diameter 11>, which is bent through an angle II> with a radius' r. should be calculated by assuming the resultant force (R) 142 on the concrete is uniformally spread over the length of the bend. Hence, with reference to Fig. 1O.4(a) the resultant force is R =2Fbt sin (8/2) where FbI is the tensile force in the bar at the ultimate limit state. The'bearing area is II> [2r sih (8/2)] Thus the bearing stress fb is given by fb ,= 2Fbl sin (8/2)/<1> [2r sin (8/2)] ... fb =.Pb/rep which is the equation given in the Code. (10.6) The bearing stress calculated from equation (10.6) should' not exceed the allowable value given by equation (8.4). In the present context, the bar diameter is the length of the loaded area and thus, in equation (8.4), Ypo =11>/2. Similarly the bar spacing (ab) is the length of the resisting concrete block and thus, in equation (8.4), Yo =abl2. On substituting into equation (8.4), the following Code ex- pression is obtained. 1.5 feu 1 +2(j)/ab However, for a bar adjacent to a face of a member, as shown in Fig. 1O.4(b), the length of the resisting concrete block is (c + II> + abI2), where c is the side cover. Thus Yo should be taken as(c + II> + abI2)/2; but the Code, by redefining ab as (c + 11», implies that Yo = (c + 11»/2 The Code thus seems. to be conservative in this situation. It appears that the Code requirements were based upon those of the CEB [226]. which,in fact, defines ab as (c + 11>/2) for a bar adjacent to a faGe of a member. The Code definitions of ab are summarised in Fig. lO.4(b). It is not necessary to carry outthese bearing stress caicu- lations if a bar is not assumed to be stressed beyond the bend. Hence, bearing stress calculations are not required for standard end hooks or bends. . Prestressed concrete The following points concerning detailing in prestressed concrete are intended to be additional to those discussed previously for reinforced concrete. Cover to tendons Bonded tendons As in BE 2/73, th~ Code requires the covers to bonded tendons to be the same as those to bars in reinforced con- crete. Hence. the comments made earlier in this chapter regarding the reinforced concrete covers are relevant. Tendons in ducts. Again. as in BE 2/73, the Code specifies a minimum cover of 50 mm to a duct. In addition a table, which is identical to that in BE 2/73. is provided (in Appendix D of Part 4 of the Code) for covers to curved tendons in ducts. 11:xternal tendons Part 4 of the Code refers to Clause 4.8.3 of Part 7 of:>the Code for the definition of an external tendon. However, the latter clause does not exist in Part 7, but exists in Part 8 of the Code as Clause 5.8.3. It defines an external ten- don as one which 'after stressing and incorporation in the work, but before protection, is outside the structure'. This definition is essentially the same as that in BE 2/73. As in BE 2/73, the Code requires that, when external tendons are to be protected by dense concrete, the cover to the tendons should be the same as if the tendons were internal. In addition, the protective concrete should be anchored, by reinforcement, to the prestressed member, and should be checked for cracking. The Code is not specific regarding how the latter check should be carried alit: BE 2/73 refers to the reinforced concrete crack width formula of BE 1/73. When using the Code, the author would suggest that equation (7.4) for beams should be used. Spacing of tendons Bonded tendons The minimum tendon spacing should comply with the .minimum spacings specified for reinforcing bars. The latter spacings are similar to those of CP 116 and, as explained earlier in this chapter, are similar to those specified in BE 2/73. In addition, BE 2/73 requires com- pliance with the maximum spacings specified in CP 116, whereas the Code does not refer to maximum spacings. Tendons in ducts The Code gives a number of requirements -for the clear distance between ducts; these requirem~nts are identical to those in BE 2/73. '*Transmission length in pre-tensioned members In both the Code and BE 2/73, the transmission length is defined as the length required to transmit the initial pre- stressing force in the tendon to the concrete. The transmission length depends upon a great number of variables (e.g. concrete compaction and strength, tendon type and size) and. ifpossible, should be determined from tests carried out under site or factory conditions, as appropriate. If such test data are not available, the Code gives recommended transmission lengths for wire and for strand. The Code implies values which are identical to those in BE 2/73. The suggested transmission lengths for wires are based i::::t ---------- --- ----!~~;:,:~~::re this length ---------- - h"._ Stress distribution Fig. 10.5 Splitting at end of prestressed member on data from tests carried out in the laboratory and on site by Base [242]; and those for strands are based on data from tests carried out only in the laboratory by Base [243]. The CP 110 handbook [112] warns that the transmission lengths for strands,· which were based upon laboratory data, could be exceeded on site; it also warns that they should not be used for compressed strands, for which transmission lengths can be nearly tw~ce those given in the Code. In members which have the tendons arranged vertically in widely spaced groups, the end section of the member acts like a deep beam when turned through 90° (see Fig. 10.5). This is due to tile fact that, towards the end of the member, concentrated loads are applied by the ten- dons, and, at some distance from the end, the prestress is fully distributed over the section. Hence, the end face of the member is in tension and a crack can form, as shown in Fig: 10.5. Vertical links should be provided to control the crack, and Green [241] suggests that, by analogy with a deep beam, the required area of the vertical reinforce- ment (As) should be calculated from: As =0.2hfebJfs < O.04Pklh where (10.7) h = vertical clear distance between tendon groups bw = width of web, or end block, at a distance h from the end of the member ie = average compressive stress between the tendons at a distance h from the end of the member . is = permissible reinforcement stress (O.87iy ) Pk = total initial prestressing force " In Fig. 10.5 and the above discussion, the prestressing forces have been considered to be applied at the end of the member, whereas, of course, they are transferred to the concrete over the transmission length, which is typically of the order of 400 mm. Thus the deep beam analogy tends to overestimate the tendency to crack, and equation (10.7) should be conservative. End blocks in post-tensioned members General In a post-tensioned member, the prestressing forces are applied directly to the ends of the member by means of relatively small anchorages. The forces then spread out 143
- 80. 1 End block Compressed wedge of concrete tt Splitting action t t t (a) Splitting action Fig. 10~6(a).(b) End block with symmetrical anchorage (1m 0.8 r::v: 0.6 0.4 0.2 Tensile stress (1m Co",P"~~::.Tstress ..-~: Distance from loaded face (b) Transverse stress distribution along block centre line oL-----~0~.1.---~0~.2r-----0~.~3----~0~.A4-----n0~.5-----n~----~,----,~----~0~~9----~1.6 ypo/Yo Fig. 10.7 Ma."(imum transverse tensile stress in end block [244] over the cross-section of the member and. in this region of spread (the end block), high local stresses occur. In par- ticular, large transverse bursting stresses occur: it is easiest to examine these st.resses by considering an end block subjected to a single symmetrically placed prestressing force. Single anchorage Single symmetricallY placed anchorages have been studied both theoretically and experimentally [244]. Qualitatively, the structural behaviour consists of a cone of compressed concrete being driven into the end block and, thus, causing splitting of the end block as shown in fig. 10.6(a). The splitting action cailses transverse bursting stresses which are greatest across a horizontal or vertical section through the axis o'fthe end block. The distribution of transverse stresses along such a section is of the form shown in Fig. 1O~6(b),where it can be seen that compressive stress- es exist near to the loaded face. but at a distance of about 0.2 y" from the loaded face (where 2 Yo is the length of the side of the end block) the stress becomes tensile. The ten- sile stress reaches a maximum of ltin at about 0.5 Yo from the loaded face, and then decreases to nearly zero at about 2yo from the loaded face (Le.at a distance equal to the 144 length of the side of the:, end block) [244]. It can be seen that it is reasonable to approximate the actual stress dis- tribution to the triangular stress distribution shown in Fig. 1O.6(b). The maximum stress is mainly dependent upon the ratio of the length of the side of the loaded area (2 Ypo) to that of the end block (2 Yo). The ratio of maximum transverse ten- sile stress (f,m) to the average compressive stress, over the total cross-sectional area of the end blo.ck (fal'e), is plotted againstYpd.vo in Fig. 10.7; the relationship was determined experimentally [244]. The bursting tensile stresses have to be resisted by re- inforcement, and thus the total bursting tensile force to be resisted is of primary interest. The bursting tensile force can be obtained by integrating a number of stress dia- grams, similar to that of Fig. 10.6(b). As one might expect, the bursting tensile force (Fbsl) is mainly dependent upon' YP(jyo' The ratio of Fbsl to the maximum prestressing ten- don force (Pk) is plotted against YpdYo in Fig. 10.8. In this figure, the range of values obtained from various theories [244], values determined from tests [244] and the Code values are given. It can be seen that the experimental val- ues exceed the theoretical values, and that the Code values have been chosen to lie between the theoretical and experimental values. Fbst 0.4 Pk - 0.3 0.2 0.1 I 0.3 I 0.2 I 0.40'---* Fig. 10.8 Bursting tensile force The tendon force for use in determining the bursting force should be the greatest load that the tendon will carry during its life. This will be the jacking load for a bonded tendon, and the greater of the jacking load .and the tendon load at the ultimate limit state for an unbonded tendon. The latter load may be assessed, as explained in Chapter 5, from Table 30 of the Code. However, the Code clause on end blocks refers to Tables 20 to 23 of the Code: these tables give characteristic strengths of tendons, and it is not clear whether the reference is an error, or whether it is intended that the load at the ultimate limit state should be taken as that equivalent to the characteristic strength of the tendon. It would seem more appropriate to use a load assessed from Table 30. The bursting tensile force, calculated from the Fh.•1Pk ratios given in the Code and plotted in Fig. 10.8, should be resisted by reinforcement. From Fig. 10.6(b) it can be seen that this reinforcement should be distributed in a region extending from 0.2)'" to 2),,, from the loaded face of the end block. The reinforcement should be designed at the ultimate limit state and, thus, its design stress is 0.87/Y' However, in order to control cracking, the reinforcement stress should be limited to a value correspond.ing to a strain of 0.001 (Le. 200 N/mm 2 ) if the cover to the reinforce- ment is less than 50 mm. If the end block is rectangular. the'value of ypjyo is different in the two principal directions. Hence, Fh..1 should be determined in each of the principal directions and rein- forcement proportional accordingly. But, for detailing pur- poses, it is generally more convenient to use the greater area of reinforcement in both directions. The above design method, in which all of the bursting tensile force is resisted by reinforcement, is the method given in the Code. However, the Code does permit the adoption of alternative design methods, in which some of the bursting tensile force is resisted by the concrete, to be adopted. One such method is that suggested by Zielinski and Rowe [244]; when using this method, the values of F".,/Pk given in the Code should not be used but, instead, the test values given in Fig. 10.8 should be used. This design procedure is. first for the appropriate value of YllI,!Y", to obtain the maximum tensile stress frqm Fig. 10.7 I 0.5 I 0.6 Tensile· stress (1m - .....--Tests -·-·-Code ')7"..-7.17)7 Range of ~U~.:::; theories I 0.8 I 0.7 ~~~~~~LU~~<~~~~_~~_--+ Distance ~ Provide reinforcement --.' in this region Fig. 10.9 Stresses resisted by concret~ end reinforcement from loaded face and the bursting tensile force from Fig. 10.8. The ideal- ised triangular stress distribution diagram of Fig. ]0.6(b) is then constructed and the pennissible tensile stress in the concrete (flp) superimposed on the diagram as shown in Fig. 10.9. Only those areas where the stress exceeds the permissible tensile stress of the concrete need to be re- inforced, as shown in Fig. 10.9. The bursting tensile force to be resisted by reinforcement (f.) as a fraction of the total bursting tensile force (Fhsl) is equal to the ratio of the area of the shaded part of the stress diagram to the total area; hence. (10.8) The permissible steel stress used to calculate the required area' of reinforcement is usually chosen to be 140 N/mm2 • The strain associated with this stress is gener- .ally too small to cause observable cracking of the concrete. Regarding the value to be taken for the permissible tensile stress in the concrete, BE 2173 states that it should be the cylinder splitting strength of the concrete divided by 1.25, and this document also gives values of cylinder splitting strength for various grades of concrete. The above two design methods lead to similar amounts of reinforcement because. although in the second method some of the bursting tensile force is resisted by the con- crete. the total bursting tensile force to be resisted is greater (see Fig. ]0.8). BE 2173 also permits the use of either of the two design methods. 145
- 81. Y01 Y01 Y02 Y02 Y03 Y03 .'ig. 10.10 Multipleanchorages Multiple anchorages Prism for ·-anchorage 1 .-Anchorage 1 .'_ Anchorage 2 Prism for -.- anchorage 2 ." Prism for anchorage 3 Anchorage 3 Very often the total prestressing force is applied to the end of a member by a numb~r of anchorages. Tests have been carried out on end blocks with mUltiple anchorages by Zielinski and Rowe [246], and the results indicate that each anchorage may be associated with a prism of con- crete, which acts like an end block for the particular anchorage. as shown in Fig. 10.10. Each prism is symmet- rically loaded by its anchorage, and its vertical dimension is the lesser of twice the distance from the centre line of its anchorage to the centre line of the nearer adjacent anchor- age, and twice the distance from the centre line of its anchorage to the edge of the concrete. The bursting tensile force and the required amount of reinforcement in each prism can be assessed by either of " the methods for single anchorages described earlier. ~n addition, the individu~1 pristn~ sh?uld ~ tied together .reinforcement. No gUIdance IS gIven In the Code on ~<tzsign such reinforcement, but a method has been '~rke [246]. which is based on the French .~ '<:ses ~ ~ .~~ ~ ~loaded face of an end block ~ §i ~. t.h.& .If-.*"'- ~ ~ con~ (I~ ~ h' splitt~~b~ .....~SI:::¢C SPlitti~J>'()~$ ~~($' ~~es arise for similar "'~in this chapter, in ~n pre-tensioned ~qy be used toaregreal .....t- ~ ;$ !$ • ~ <&-~1lJ o·~ the aXIs e ~ b~ ~v b stresses alt..1 0<;: ~~.,# .$ F· 10 6(b)·~ (IV '(). 1lJ'<i .....1lJ'<i Ig. ~ • C5 b ~ ~ es exist near tf'.# ~ ~o 0.2 y" from the'$' .s ~""... ,,'I,; side of the end b'r a- sile stress reaches .)iJ' .~' the loaded face, ana." 2y-o from the loaded 144 ~dbY dded / _,orage, .flmsverse Crack A--- 1-='--_. -- -_.- .-. - ·--A L, L Reinforcement ~~_-"..~"",",,___ Transverse stress across AA Compression Fig. 10.11 Spulling tcnsile stresses due to eccentric prestress stress distribution along a line parallel to the axis of an eccentric prestressing force is as shown in Fig. 10.11 [2481. Figure 10.11 also shows the crack which can occur at the loaded face. Green [241] suggests that vertical re- inforcement sufficient to resist a force of O.04Pk should be placed as near as possible to the loaded face of the member to control the crack. It would seem prudent to provide a minimum amount of such reinforcement in all end blocks whether or not the prestressing force is highly eccentric. Further guidance on resisting spalling tensile stresses, based upon the French C~de [247], is given by Clarke [246]. End block and beam of different shapes Prestressed concrete beams are generally of a non- rectangular shape with flanges; however, the end blocks are often made rectangular. As one might expect, stress concentrations occur where the section changes from rectangular to non-rectangular, and, at the junction, a second region of transverse burst~ng tensile stress occurs. Tests [246] indicate that; at the junction of end block and beam, reinforcement sho~ld be provided to resist a burst- ing tensile force equal to 70% of that calculated within the end block. Summary The steps in the design of a general end block can be summarised as follows. 1. Design reinforcement to prevent bursting of the indi- vidual prisms of concrete associated with each anchorage. 2. Design reinforcement to tie together the individual prisms. 3. Design reinforcement to resist the spalling tensile stresses between, and at a distance, from anchorages. 4. Design reinforcement to prevent bursting at the junc- tion of a rectangular end block and a non-rectangular beam. Clarke [246] gives a detailed worked example which illustrates the above design procedure. Chapter 11 Lightweight aggregate concrete ~Introduction The design recommendations which are discussed in pre- vious chapters are intended only for concretes made from normal weight aggregates. AU naturaUy occurring aggre- gates, with the exception of pumice, are of normal weight but, as such aggregates become scarce, it is likely that manufactured aggregates will become more popular than they are at present. The majority of manufactured aggre- gates (e.g., expanded shale and clay, foamed blast furnace slag and sintered pulverised fuel ash) are lightweight. In addition to the above consideration of the future availability of natural aggregates, the use of lightweight aggregate concrete has obvious advantages where ground conditions are poor, and there is a need to reduce, as much as possible, dead loads and, thus, foundation loads. Lightweight aggregate concrete has been used through- out the world for both reinforced and prestressed concrete construction, but it has been used far less in Great Britain than in some other countries. This is particularly true of bridge construction [280]. The first lightweight aggregate concrete road bridge built in Great Britain was the Redesdale Bridge, which was constructed by the Forestry Commission in North- umberland. This bridge has a single span of 16.8 m. It is constructed of prestressed precast invet;ted T-beams with in-situ concrete in-fill to form a composite slab. The beams were cast from a concrete composed of ,sintered pul- verised fuel ash (Lytag) coarse aggregate and natural sand fine aggregate, with a wet density of 1890 kg/m3, to give a minimum strength at transfer of 35 N/mm2 and a minimum 28-day strength of 48 N/mm 2. As explained later, greater losses of prestress occur with lightweight, as compared with normal weight, aggregate concrete; for this bridge, the losses were assumed to be 40% greater at transfer, and 30% greater finally. The .Transport and Road Research Laboratory carried out tests on beams identical to those used in the Redesdale Bridge and found that they behaved satisfactorily under both static and repeated loading [249]. Although the Redesdale Bridge was the first lightweight aggregate concrete road bridge in Great Britain, it does not form part of a public highway. The first lightweight aggre- gate concrete brid,ge to be built over a public highway. the Glasshouse Wood Footbridge at Kenilworth having a span of 31.S m. was designed by the Warwickshire Sub-Unit of the Midland Road Construction Unit. This bridge was opened in 1974. Lytag was used for both coarse and fine aggregates: the 28-day strength was 45.7 N/mm 2 and the design air-dry density was 1700 kg/m3. . Examples are to be found in Staffordshire of composite slab motorway bridges with spans of about II m, which were constructed with normal weight aggregate precast concrete inverted T-beams with lightweight aggregate (Lytag) in-situ in-fiU concrete. Lightweight aggregate con- crete as in-fiU for composite slabs has also been used else- where in Great Britain. Recently Kerensky, Robinson and Smith [250] have reported the successful completion', in 1979, of the Friar- ton Bridge over the River Tay at Perth. This is a steel- concrete composite bridge consisting of a steel box girder with a composite lightweight aggregate concrete deck slab. Lytag was used for both coarse and fine aggregates, and the design strength and air-dry density were 30 N/mm2 and 1680 kg/m3 respectively. In the following, the structural properties of lightweight aggregate concrete, and how these are dealt with by the Code, are discussed. At present, the requirements for the structural use of lightweight aggregate concrete in highway structures are covered by BE 11 [251]. However, this document limits the, j.lse of lightweight aggregate concrete to in-filling (such as between inverted T-beams), and only gives data on density, modulus of elasticity and allowable tensile stresses for in-fill concrete. *Durability The durability of lightweight aggregate concrete can be very good, as was demonstrated by one of the early con- crete ships - the Selma. This ship was constructed of a concrete with expanded shale aggregate and the reinforce- ment cover was only 10 mm. The reinforcement was 147
- 82. still in excellentcondition after forty years in service [96]. In lIon-marine environments, it is thought that, because of the greater porosity of lightweight aggregates, which permits the relatively easy diffusion of carbon dioxide through the concrete, carbonation of the concrete may . occur to a greater depth when lightweight aggregates are used. The Code thus requires the cover to the reinforce- ment to be 10 mm greater than the appropriate value obtained from Table 10.1 for normal weight concrete. However, this requirement may be conservative because tests, carried out by Grimer [252], in which specimens of five different lightweight aggregates and one normal weight aggregate were exposed to a polluted atmosphere" for six years, showed that the effect of the type of aggre- gate on the rate of penetration of the carbonation front was small in comparison with the effect of mix proportions. Strength 1:Compressive strength The minimum characteristic strengths permitted by. the Code when using lightweight aggregate concrete '~e 15 N/mml, 30 Nlmml and 40 N/mml for reinforced, post-tensioned and pre-tensioned construction respectively. It should be noted that BE 11 requires a minimum strength of 22.5 N/mm2 for in-fill concrete. These strengths can be attained readily with lightweight aggregates [96], and details of mixes suitable for prestressed concrete have been given by Swamy et al. [253]. One important difference between concretes made with lightweight and normal weight aggregates is that the gain of strength with age may be different. .In particular the gain in strength with certain lightweight aggregates may be very small for rich mixes [254]. Tensile strength The.tensile strength of any concrete is greatly influenced . by the moisture content of the concrete, because drying reduces the tensile strength. The flexural tensile strength tends to be reduced by drying more than the direct tensile strength. Curing conditions affect the tensile strength of light- weight aggregate concrete more than normal weight aggre- gate concrete. Although the tensile strengths are. similar for moist cured specimens, the tensile strength of lightweight aggregate concrete when cured in dry conditions can be up to 30% less than that of comparable normal weight con- crete [96]. The relatively reduced tensile strength does not influ- ence the design of reinforced concrete, but has to be allowed for in the design of prestressed concrete. No specific guidance is. given in the Code, but the CP 110 handbook [112] suggests that all allowable tensile stress- es, referred to in the prestressed concrete clauses for nor- 148 mal weight concrete, should be multiplied by 0.8. This value seems reasonable in view of the reduction in tensile strength referred to in the last paragraph. The implications of the factor of 0.8 are: 1. The allowable tensile stresses for Class 2 pre- tensioned and post-tensioned members are 0.36 !feu and 0.29 I!cu respectively, instead of 0.45 /!CU and 0.36 /Tcu, respectively, for normal weight aggregate concrete (see Chapter 4). 2. The basic allowable hypothetical tensile stresses for Class 3 prestressed members, which are given in Table 4.6(a), should be multiplied by 0.8. 3. The allowable concrete flexural tensile stresses for in-situ concrete, when used in composite construction, which are given in Table 4.4, should be multiplied by 0.8. The allowable stresses, so obtained, are very close to those given in BE 11. 4. When calculating the shear strength of a prestressed member uncracked in flexure, the design tensile strength of the concrete (fi) should be taken as 0.19/!cu, instead of 0.24 0.. for normal weight aggregate concrete (see Chapter 6). 5. When calculating the shear strength of a Class 1 or 2 'prestressed member cr,cked in flexure, the design flexural tensile strength of the concrete (fr) should be taken as 0.30 !feu, instead of 0.37 .ftcu for normal weight aggregate concrete (see Chapter 6). Hence, equation (6.12), for the cracking moment, becomes Mt = (0.3 /leu + 0.8 fpt)lly (11.1) "'t Shear strength The shear cracks, which develop in members of light- weight aggregate concrete, frequently pass through the aggregate rather than around the aggregate, as occurs in members of normal weight aggregate concrete. Hence, the surfaces of a shear crack tend to be smoother for light- weight aggregate concrete, and less shear force can be transmitted by aggregate interlock across the crack (see Chapter 6). Since aggregate interlock can contribute 33% to 50% of the total shear capacity of a member [152], the shear strength of a lightweight aggregate concrete member can be appreciably less than that of a comparable normal weight concrete member. Tests carried out by Hanson [255] and by Ivey and Buth [256] on beams without shear reinforcement have shown that, for a variety of lightweight aggregates, it is reasonable to calculate the shear strength of a lightweight aggregate concrete member by multiplying the shear strength of the comparable normal weight aggregate con- crete member by the following factors. 1. 0.75 if both coarse and fine aggregates are light- w e i g h t . / 2. 0.85 if the coarse aggregate is lightweight and the fine aggregate is natural sand. In the Code an average value of 0.8 has been adopted for any lightweight aggregate concrete, and, by analogy, the same value has been adopted for torsional strength. Hence: 1. The nominal allowable shear stresses for reinforced concrete (vc), which are given in Table 6.1, should be multiplied by 0.8. 2. The maximum nominal flexural,. or torsional, shear stress should be 0.6 ./f.,u (but not greater than 3.8 N/mm2 and 4.6 N/mm2 for reinforced and pre- stressed concrete respectively), instead of 0.75 lieu (but not greater than 4.75 N/mml and 5.8 N/mml for reinforced and prestressed concrete respectively) for normal weight aggregate concrete (see Chapter 6). Only half of these values should be used for slabs (see Chapter 6). 3. When calculating the shear strength of a Class 1 or 2 prestressed member cracked in flexure, although it is not stated explicitly in the Code, the first term of the right-hand side of equation (6.11) should be taken as 0.03 bd v'.fcu,instead of 0.037 bd./f.,u as used for nor- mal weight aggregate concrete. 4. The limiting torsional stress (Vtmln) above which tor- sion reinforcement has to be provided should be 0.054 ./!cu (but not greater than 0.34 N/mml), instead of 0.067 ./Tcu (but not greater than 0.42 N/mmD ) as used for normal weight aggregate concrete (see Chap- ter 6). 5. Although not stated in the Code, it would seem pru- dent to multiply the basic limiting interface shear stresses for composite construction, which· are given in Table 4.5, by 0.8. Bond strength Shideler [257] has carried out comparative pull-out tests on deformed bars embedded in eight different types of lightweight aggregate concrete and one normal weight aggregate concrete. The average ultimate bond stresses developed with the lightweight aggregate concretes were, with the exception of foamed slag, at least 76% of those developed with the normal weight aggregate concrete. The Code reduction factor of 0.8 to be applied, for deformed bars, to the allowable bond stresses of Tables 10.2 and 10.3 thus seems reasonable. Short and Kinniburgh [258] have reported the results of pull-out and 'bond beam' tests in which plain bars were embedded in three different types of lightw~ight aggregate concrete and in normal weight aggregate concrete. The average ultimate bond stresses developed with the light- weight aggregate concretes were 50% to 70% of those developed with the normal weight aggregate concrete. The Code reduction factor of 0.5 to be applied, for plain bars, to the allowable bond stresses of Tables 10.2 and 10.3 thus seems reasonable. Shideler's tests with foamed slag aggregate indicated that the average bond stress could be as low as 66% for horizontal bars due to water gain forming voids in the con- crete under the bars. The Code thus advises that allowable bond stresses should be reduced still further (than 20% and 50%) for horizontal reinforcement used with formed slag aggregate: appropriate reduction factors would seem to be 0.65 and 0.33 for deformed and plain bars respec- tively. The lower bond stresses developed with lightweight aggregate concrete imply that transmission lengths of pre- tensioned tendons are greater than the values discussed in Chapter 10 for use with normal weight aggregate concrete. The Code gives no specific advice on transmission lengths for lightweight aggregate concrete, but the CP 110 hand- book [112] suggests that they should be taken as 50% greater than those· for normal weight aggregate concrete. This increase seems reasonable, 'as an upper limit, when compared with test data collated by Swamy [259]. Bearing strength It can be seen from Fig. 10.6(a), which illustrates an end block of a post-tensioned member, that a bearing failure is, essentially, a tensile splitting failure. Hence the allowable bearing stresses for lightweight aggregate concretes should reflect their reduced tensile strength discussed earlier in this chapter. Since the tensile strength of lightweight aggregate concrete can be up to 30% less than that of comparable normal weight aggregate concrete [96], the Code requires the limiting t?earing stress for lightweight aggregate concrete to be two-thirds of that calculated from equation (8.4). The Code implies that the above reduction should be applied only when considering bearing stresses inside bends of reinforcing bars, but it would seem prudent to apply the reduction to all bearing stress calculations involving lightweight aggregate concrete. Movements Thermal properties Lightweight aggregate concrete has a cellular structure and, thus, its thermal conductivity can be as low as one- fifth of the typical value of 1.4 W/moC for normal weight aggregate concrete [96]. The reduced thermal conductivity is of great benefit in buildings, because it provides good thermal insulation. However, for bridges, it implies that the differential temperature distributions are more severe than those discussed in Chapters 3 and 13 for normal weight aggregate concrete. Although differential temperature distributions are more severe with lightweight aggregate concrete, their effects are mitigated by the fact that the coefficient of thermal expansion can be as low as 7 x lO-s/oC [96], as compared with approximately 12 x 1O-o/oC for normal weight aggre- gate concrete. The lower coefficient of thermal expansion also means that overall thennal movements of a bridge are less when lightweight aggregate concrete is used. This fact, coupled with the lower elastic modulus of lightweight aggregate concrete (see next section), means that thennal stresses, which result from restrained thermal movements, are less than for normal weight aggregate concrete. 149
- 83. Elastic modulus The elasticmodulus of lightweight aggregate concrete can range from 50% t075% of that of normal weight aggregate concrete of the same strength [254]. The higher values are associated with fGamed blast furnace slag aggregate and the lower values with expanded clay aggregate [254]. It is mentioned in Chapters 2 and 4 that the Code gives a table of short-term elastic moduli for normal weight aggre- gate concretes. These values are in good agreement with the following relationship, suggested by Teychenne, Parrot and Pomeroy [20] from considerations of test data. E - 9 1f, 0.33c - . cu where Ec is the elastic modulus in kN/mml and fCIl is the characteristic strength in N/mm2 • The latter authors further suggested, from considerations of test data, that the elastic modulus of a lightweight aggregate concrete with a density of Dr (kg/m3) could be predicted from (11.2) Equation (11.2) was based upon data from sixty mixes covering four different lightweight aggregates with con- crete densities in the range 1400 kg/ms to 2300 kg/m3. The Code states that the elastic modulus of a lightweight aggregate concrete, with a density in the above range, pan be obtained by multiplying the elastic modulus of a normal weight aggregate concrete by (D,/2300)2 • The resulting elas- tic modulus will thus agree closely with that predicted by equation 01.2). They will also be within 20% of those specified in BE 11. The reduced elastic modulus of lightweight aggregate concrete has the following design implications. 1. Stresses arising from restrained shrinkage or thermal movements are less than for normal weight aggregate concrete. 2. Elastic losses in a prestressed member can be up to double those in normal weight aggregate concrete members. 3. Lateral deflections of columns are greater than for normal weight aggregate concrete. Hence stability problems are more likely to occur, and additional moments (see Chapter 9) are greater. These factors are allowed for in the Code by: 150 (a) Defining a short column as one with a slenderness ratio ofnot greater than 10, as compared with the critical ratio of 12 for normal weight aggregate concrete columns (see Chapter 9); (b) Substituting the divisor of 1750, in the additional moment parts of equations (9.21) to (9.28), by a divisor of 1200. Hence, the additional moment is increased by nearly 50%. The requirement to reduce the divisor from.t 750 to 1200 implies that Creep the assumed extreme fibre concrete strain at fail- ure for lightweight aggregate concrete is about 0.00633 as compared with 0.00375 for normal weight aggregate concrete (see Chapter 9). This increase is reasonable in view of the reduced elas- tic modulus and the greater creep of lightweight aggregate concrete (see next section). The data on creep of lightweight aggregate concrete, as compared with that of normal weight aggregate concrete, are conflicting. Although creep of lightweight aggregate concrete can be up to twice that of normal weight aggre- gate concrete [96], it has also been observed [260] that less creep may occur with structural lightweight aggregate con- crete as compared with normal weight aggregate concrete. As is true of all other concretes, creep of lightweight aggregate concrete depends upon a great number of fac- tors, and it is desirable to obtain test data appropriate to the actual conditions under consideration. In lieu of such data, S.pratt [96] suggests that creep of lightweight aggre- gate concrete should be assumed to be between 1.3 and 1.6 times that of normal weight concrete under the same conditions. In similar circumstances, the CP ItO handbook [112] suggests that the loss of prestress due to concrete creep should be assumed to be 1.6 times that calculated for normal weight aggregate concrete. Shrinkage Great variations occur in the shrinkage values for light- weight aggregate concrete; values up to twice those for normal weight aggregate'foncrete ha~e been reported [96]. In the absence of data pe.rtaining to the actual conditions under consideration. Spratt [96] suggests the adoption of an unrestrained shrinkage strain of between 1.4 and 2.0 times that of normal weight aggregate concrete under the same conditions. In similar circumstances, the CP ItO handbook [112] suggests that the loss of prestress due to shrinkage should be assumed to be 1.6 times that calcu- lated for normal weight aggregate concrete. Losses in prestressed concrete From the previous considerations, it is apparent that total losses in prestressed lightweight aggregate concrete may be up to 50% greater than those in prestressed normal weight aggregate concrete. This is because of the smaller elastic modulus and the greater creep and shrinkage of lightweight aggregate concrete. Chapter 12 Vibratio'n and fatigue Introduction In this chapter, the dynanlic aspects of design are con- sidered in .terms of vibration and fatigue. Hence, reference is made to Parts 2, 4 and 10 of the Code. Vibration Design criterio~ It is explained in Chapter 3 that it is not necessary to con- sider vibrations of highway bridges, because the stress increments due to the dynamic effects are within the allowance made for impact in the nominal highway load- ings [t07]. In addition, vibrations of railway bridges are allowed for by multiplying the nominal static standard railway loadings by a dynamic factor. Hence, specific vibration calculations only have to be carried out for footbridges and cycle track bridges. It is explained in Chapter 4 that the appropriate design criterion for footbridges and cycle track bridges is that of discomfort to a user; this' is quantified i~ the Code as a maximum vertical acceleration of 0.5 110 mls·, where fo is the fundamental natural frequency in Hertz of,tpe unloaded bridge [107]. . Compliance Introduction The above design criterion is given in Appendix C of Part 20f the Code, which also gives methods of ensuring compliance with the criterion. It should be noted that the criterion and the methods of compliance are the same as those in BE 1177. The background to the compliance rules has been given by Blanchard, Davies and Smith [107]. They considered a pedestrian, with a static weight of 0.7 kN and a stride length of 0.9 m. walking in resonance , , 0.6 " ',9>. '0$ ', 0.2 , . , I ,, "I I I I. I o~----~~----~~__~~~____~ 1.0 1.1 1.2 ,,:: 1.3 1.4 Fig. 12.1 Attenuation factor t Natural freq~ency {pacing frequency with the natural frequency of a footbridge. It was found that it was possible to excite a bridge in this way if the natural frequency did not exceed 4 Hz, since the latter value is a reasonable,upperlirilit, of applied pacing fre- quency (frequencies above,3 Hz representing running). ,Thus, if the natural frequency exceeds 4 Hz, resonant vibrations do not occur; ,however,it is still necessary to calculate the amplitude of. the non-resonant vibrations which do occur when a pedestrian strides at the maximum possible frequency of 4 Hz. Analyses were carried out to determine an attenuation factor defmed as the ratio of the maximum acceleration when walking below the resonant frequency to that when walking at the resonant frequency. The results of such an analysis have the form shown in Fig. 12.1. It can be seen that the attenuation factor drops rapidly and, at a frequency ratio of 1.25 (for which the natural frequency is 5 Hz if the pacing frequency is 4 Hz), the attenuation factor is very small. It is thus considered 151
- 84. f t K=1.0 I- -I f I f t 14 -I- -I K=:0.7 f 11 f f 11 I- -I- _14 t MI K 1+--7-'----IoI+----=..,;.--:-+l4--~-+i-1 1.0 .. 0.6 0.8 0.8 ';;0.,6 0.9 F'g. rd Configuration factor (K).1" ' . , ' , ' ", .. , time Fig. 12.3 Decay due to damping very ,difficult to excite bridges with natural frequencies greater than 5 Hz, and their vibration may be ignored. Hence the C~de states tha,t, if the natural frequency of the unloaded bndge exceeds 5 Hz, the vibration criterion is deemed to, be satisfied. If the natural frequency lies between 4 Hz and 5 HZ," Blanchard, Davies and Smith [107] suggest that the maxi- mum bridge acceleration should first be calculated using the natural frequency. This maximum acceleration should then be multiplied by an attenuation factor which varies linearly from 1.0 at 4 Hz to 0.3 at 5 Hz, as shown in ~ig. 12.1, to give the maximum acceleration due to a pac- Ing frequency of 4 Hz. This approach has been adopted in the Code. , In th~ above discussion it is implied that it is necessary to conSider only a single pedestrian crossing a bridge. This requireme~t ,was proposed by Blanchard, Davies and Smith [107] by considering some existi~g bridges: for each bridge the number of pedestrians required to produce a maximum acceleration just equal to the allowable value of 0:5 If" was calculated. It was concluded that, in order that the more sensitive, of the existing bridges could be con- sidered to be just acceptable to the Code' vibration criterion, theappliep loading should be limited to a single pedestrian. ." . Calculat;on of natural frequency The Code requires that the natural frequency of a concrete bridge should be determined by ,{,:onsidering ihe uncracked secti?!l (neglectingthereinforceInent), ignoring shear lag, but Including the stiffness of parapets. The, Code also requires· the,shoft-term elastic modulus of concrete to be used. It would seem appropriate to use the dynamic modulus, and Appendix B of Part 4 of the Code tabulates such moduli for various concrete strengths. The tabulated values are in good agreement with data presented by Neville [108]. " Wills [2~1] has used the material and section properties, referred to In the last paragraph, to obtain good agreement 152 6 4 2 0;~--~1~O~--~20~--~3~0-----L----~ 4050 Main span (I) metres Fig. 12.4 Dynamic response factor ('IjI) for 6 = 0.05 between' predicted and observed natural 'frequencies of existing bridges. ' The natural frequency of a bridge should be ,calculated, by including superimposed dead load but excluding pedes~ trian live loading. For bridges of constant cross-section and up to four spans, the fundamental natural frequency can be obtained easily by using, for example, tables presented by Gorman [262]. The fundamental natural frequency ifo) is given by [262] f =o where !Ef /p:4 EI = flexural rigidity • A = cross-sectional area P = density . ,. i, L = length of bridge , (12.1) ~ parameter dependent· on' span arrangement' and lengths, support conditions and the vibration mode. For bridges of varying cross-section, it is necessary to us 7either a compu~er ~rogram, such as that adopted by WIlls [261], or a sImphfied analysis based on a uniform cross-section. In the latter approach, a bridge of varying' cross-section is replaced by a bridge having a constant ~ross-section with a mass per unit length and flexural rigid- Ity. equal to the weighted means of the actual masses per umt length and flexural rigidities of the bridge. Equation (12.1) can then be used. Wills [263] explains this pro- cedure and shows that it leads to satisfactory estimates ofthe . natural frequencies of bridges having cross-sections which vary significantly. Nine bridges were considered and the calculated frequencies were within 6% of the measured frequencies except for one bridge, which had an error of 15%. . " .,' Forcei1' ' , , ~----"'T-----i-"+~ ". . 25% of static load ," ,...- A" " . , . Static ~__'~U~.--_J_---1---4-'--':¥-~'='"--- , load .... Fig. 12.5 Moving pulsating force Calculation of acceleration Simplified method For a bridge of constant cross-section and up to three symmetric spans (as shown in Fig. 12.2), the Code gives the following formula for calculating the maximum vertical acceleration (a): a =4rrF0 Y.• k,, where fo = fundamental natural frequency (Hz) Y. = static deflection (m) K = configuration factor '' dynamic response factor (12.2) Iffo lies inthe range 4 Hz to 5 Hz, the acceleration calcu- lated from equation (12.2) should be reduced by applying the attenuation factor discussed earlier in this chapter. Equation (12.2) was derived by Blanchard, pavies and Smith [107] and represents, in a simple form, tbe results of a study of a number of bridges with different span arrangements. For each bridge, a numerical solution to its governing equation of motion was obtained. The static deflection (y.) should be calculated, at the midpoint of the main span, for a vertical load of O.7kN, which represents a single pedestrian. The configuration factor (K) depends upon the number and lengths of the spans; values are given in Fig. 12.2. Linear interpolation may be used for intermediate values of 1111 for three span bridges. The dynamic response factor ('I) depends upon the main span length and the damping characteristics of the bridge. If a bridge is excited, the amplitude of the vibration gradu- ally decreases due to damping, as shown in Fig. 12.3. The damping is expressed in terms of the logarithmic dec- rement «(), which is the natural logarithm of the ratio of the amplitude in one cycle to the amplitude in the following cycle. The Code suggests that, in the absence of more pre- cise data, the logarithmic decrement, of both reinforced -One foot ---Actual combined -·-Idealised combined Right Time, and prestressed concrete footbr~dges, should be assumed to be 0.05. In the past, a larger value has often been taken for reinforced concrete than for prestressed concrete, because it was felt that cracks in reinforced concrete dissipate energy and thus improve the damping. However, accord- ing to Tilly [264], this view is not supported by experi- mental data. Furthermore, when considering a bridge, it is the overall damping, due to energy dissipation at joints, etc, in addition to inherent material damping. which is of interest. Tests on existing concrete footbridges have indi- cated logarithmic decrements in the range 0.02 to 0.1, and thus the Code value of 0.05 seems reasonable [264]. The relationship between the dynamic response factor (,,) and the main span length (I) is given graphically in the Code. The relationship is shown in Fig. 12.4 for the logarithmic decrement of 0.05 suggested in the Code. General method For bridges with non-uniform cross- sections, and/or unequal side spans, and/or more than three spans, the above simplified method of determining the maximum vertical acceleration is inappropriate. In such situations, it is necessary to analyse the bridge under the action of an applied,.moving and pulsating point load, which represents a pedestrian crossing the bridge;. The amplitude of the point load was cho~n so that, when applied to a simply supported singli~ span bridge, it pro- duces the same response as that produced by a pedestrian walking across the bridge [107]. In Fig. 12.5, the force- time relationship is shown for a single foot. The relation- ship obtained by combining consecutive single foot rela- tionships is also shown. It can be seen that the combined effect can be represented by a sine wave with an amplitude of about 25% of the static single pedestrian load of 0.7 kN. In fact the Code takes the amplitude to be 180 N and, as discussed earlier in this chapter, the pacing fre- quency is taken -to be equal to the natural frequency of the bridge (fo). Hence the pulsating point load (Fin Newtons). given in the Code, is: 153
- 85. F == 180sin (211 it,1) (12,3) where T is the time in. seconds. It is mentioned earlier in this Chapter that the assumed stride length is 0.9 m. Thl,ls, ifthe.pacing frequency is bars, the implications of not considering concrete and pre- stressing tendons are discussed briefly in a simplistiC man- '. nero ,t;,Hz, the required velocity (v, in mls) of the pulsating point Concrete load is given by: . Concrete in compression can withstand,. for .2 million.' . V, =0.9 fo (12.4) cycles of repeated loading, a maximum stress' of about .Wills [261] discusses two metbadsofailalysing a bridge 60% of the static strength if the minimum stress in acycle .'under the above moving pulsating point.load. On~_Jllethod·-···· is zei'0-·t16~~ The maximum. stress which can be. tolerated requit(:sa large amount of computer stor~ge.<space and the increases as tfie minimum stress increases. Since Part 4 of other more approximate method requires much less storage the Code:: specifies a limiting compressive stress of 0.5 feu .. space'. Wills [261J shows that the approximate method is for concrete at the serviceability limit state (see Chapter adequate for many footbridges. 4), it is very unlikely that fatigue failure of concrete in compression would occur. It is thus reasonable for the Forced vibrations Up to now in this chapter, only those vibrations which . result from normal pedestrian use of a footbridge have . been considered. However, it is also necessary to consider the possibility of damage arising due to vandals deliber- ately causing resonant oscillations. It was not possible [107] to quantify a loading or a criterion for this action; thus the Code merely gi~es a warning that reversals of load effects can occur. However, the Code does suggest that, for prestressed concrete, the section should be provided with . unstressed reinforcement capable of resisting a reverse moment of 10% of the static live load moment. Fatigue 'General Code approach . . Although Part 1 of the Code refers to fatigt,le .under the heading 'ultimate limit state', it is the repeated apph- cation of working loads which cause deterioration to a stage where failure occurs; Alternatively, the working loads may cause minor fatigue damage; which could result in the bridge being considered unserviceable. Hence, fatigue cal- culations !lre.catriedout separately from the calculations to check compliance with the ultimate.and serviceability limit state criteria:' in Part 4 ()f the Code, fatigue is dealt with under 'Other considerations'. The design fatigue loading is specified in Part 10 of the Code and, since it is a design loading, partial safetyfactors (YjLalid Yf3) do not have to be applied. '. . . When.deterrniningth~response ora.bridge .tofatigue 10ading,Part lof the Code requires the use of a linear elastiC method with>the elastic modulus·of concrete equal to its short-term value. With regard to .concrete bridges, Part 4 .of the Code requires only the fl1tigue strength (or life) of reinforcing hars to be assessed. Thus concrete and prestressing ten- dons do not have to be considered in fatigue calculations. Befote presenting the Code requirements for reinforcing 154 Code not to require calculations for assessing the fatigue life of concrete in compression. It should be noted that 2 million cycles of loading during the specified design life of 120 years are equiv- . alent to about SO applications of the ·full design load per day. . . For concrete in tension, cracking occurs at a lower stress under repeated loading than under static loading. Cracking does not occur, in less than 2 million cycles of repeated loading, if the maximum t~nsile stress does not exceed about 60% of the static tensile strength, if the niinimum "stress is zero or compressive [265, 266]. The maximum tensile stress which can be tolerated without cracking . increases if the minimum stress is tensile. The reduced resistance to cracking of concrete subjected to repeated loading has two implications. . 1. Shear cracks may form at a lower load, with a pos- sible decrease in shear strength. 2. Flexural cracks may form at a .lower load, resulting in either cracks in Class 1 or 2 prestressed members, or cracks wider than thee. allowable values in reinforced concrete or Class 3 pre'stressed concrete members. With the partial safety factors for loads and for material' properties.that ha.ve been adopted in the Code at the ulti~ mate limit state, it is unlikely that principal tensile stresses under working load conditions.would be great enough to cause fatigue shear cracks. Thus a bridge, designed ,to resist static shear in accordance with the Code, should exhibit adequate shear resistance when subjected .to reo peated loading. The interface shear str~ih of' composite members should also be adequate under repeated loading. Badoux and Hulsbos [279] have tested composite beams under 2 million cycles of loading. The test specimens were essen- tially identical to those, tested under static loading. by Saemann and Washa [118], which.are discussed. in Chapter 4; It was found that, under repeated loading,. the interface shear strength was reduced. However, the allowable inter- face shear stresses, which were proposed in [279] for re- peated loading, exceed the allowable stresses given in the Code for static loading at the serviceability limit state (see Chapter 4). Since the latter stresses have to be checked under the full design load at the serviceability limit state (i.e. under dead plus imposed loads) it is very unlikely that interface shear fatigue failure would occur. The require- . ment to check interface shear stresses under the full design load is discussed fully in Chapter 8. Flexural cracking under repeated loading is now con- sidered. Reinforced concrete Repeated loading causes cracks to form at a lower load than under static loading and, subse- quently, the cracks are wider. However, the author would suggest that the breakdown of tension stiffening under repeated .loading has a greater influence. on crack widths than does the reduction in the load at which cracking occu~s. Tension stiffening under repeated loading is dis- c:.:::;sed in Chapter 7, where the author suggests that, as an interim measure, tension stiffening under repeated loading should be taken as 50% of that under static loading. Class I prestressed concrete Flexural tensile stresses are not permitted under service load conditions and thus re- peated loading cannot cause cracking. Class 2 prestressed concrete No flexural tensile stresses are permitted under dead plus superimposed dead loads (see Chapter 4), and thus the minimum flexural stress is always compressive. For repeated loading, the maximum tolerable tensile stress, appropriate to a compressive minimum stress, is about 60% of the static tensile strength. Since the flexural tensile stress permitted by the Code may be up to 80% of the static flexural tensile strength (see Chapter 4), it is possible that flexural fatigue cracking could occur in a Class 2 member designed in accordance with the Code. It is significant that, iIi CP 115, the allow- able tensile stresses for repeated loading are about 65% of those for non-repeated loading; the latter stresses are very similar to the Code values. It would thus seem prudent to take about 65% of the Code values when considering re- peated ioading. Class 3 prestressed concrete A Class 3 member is designed to be cracked under the serviceability limit state design load. Repeated loading may cause the cracks to be wider than under static loading. However, the permissible hypothetical tensile stresses in the Code (see Chapter 4) are conservative in comparisionwith test data [120, 122, 123], and thus excessive cracking under repeated loading should not occur. Prestressing tendons The allowable .concrete flexural, compressive and tensile stresses specified in the Code imply that the stress range, under service load conditions, of a prestressing tendon in a Class 1 or 2 member cannot exceed about 10% of the ulti- mate static strength. If it is assumed that the effective prestress in a tendon is about 45% to 60% of its ultimate strength, then test results indicate that it may be conservatively assumed that 2 mil- lion cycles of stress Clm be withstood by a tendon without failure, providing that the stress range does not exceed about 10% of the ultimate static strength [265]. It is thus unlikely that fatigue failure of a prestressing tendon, in a Class 1 or 2 member. would occur under service load con- ditions. In a Class 3 member, designed for the m/lXimum.allow- able hypothetical tensile stress of 0.25 feu (see Chapter 4), stress ranges in tendons could be. up to 15% of the ultimate static strength of a tendon. Fatigue failure of tendons are thus possible in some Class 3 members. However, it should be emphasised that, in the vast majority of Class 3 members, tendon stress ranges ",ill be . much less than the values quoted above and fatigue fail- ures would then be unlikely. Nevertheless, it would seem prudentto ensure that,. for all classes, tendon s~ss.ranges do not exceed 10% of the ultimate static strength of the tendon. It should be noted that the conservative tolerable stress range of 10% of the ultimate static strength, which is quoted in the last paragraph, is based upon tests on pre- stressing tendons in air. However, the work of Edwards [267, 268] has shown that tendon fatigue strength can be less when embedded in concrete than in air: for 7-wire strand, the stress range in concrete was about 8% of the ultimate static strength as compared with about 13% in air. Fatigue failure of anchorages and couplers, rather than of a tendon, should also be considered, since they can withstand smaller stress ranges [265]. Particular attention should be given to the possibility of an anchorage fatigue failure when unbonded tendons are used, because stress changes in the tendon are transmitted directly to the anchorages. Code fatigue highway loading A table in Part 10 of the Code gives the total number of commercial vehicles (above 15 kN unladen weight) per year which should be assumed to travel in each lane of various types of road. The number of vehicles varies from 0.5 >< 108 for a single two-lane all purpose road to 2 >< 108 for the slow lane of a dual three-lane motorway. Part 10 of the Code also gives a load spectrum for commercial vehicles showing the proportions of vehicles having various gross weights (from 30 kN to 3680 kN) and various axle arrangements. The load spectrum depicts actual traffic data in terms of twenty-five typical com- mercial vehicle groups. It is obviously tedious in design to have to consider a number of different axle arrangements, and thus it was decided to specify a standard fatigue vehicle. The intention was that each type <?f commercial vehicle in the load spec- trum would be represented by a vehicle having the same gross weight as the actual vehicle, but having the axle arrangement of the standard fatigue vehicle. The standard fatigue vehicle was chosen to give the same cumulative fatigue damage, for welded connections in steel bridges, as do the actual vehicles. However~ John- son and Buckby [24] have emphasised that equivalence of cumulative fatigue damage does not occur for shear con- nectors in composite (steel-concrete) construction. This is because fatigue damage in welded steel connections is proportional to the third or fourth po'wer of the stress range whereas, for a shear connector. it is proportional to the eighth power. Fatigue damage of unwelded reinforcing bars is proportional to stress range to the power 9.5 (see 155
- 86. Each wheel load20kN on 200 square or 225 diameter contact area ~/:: :t:3'~0"~0".. '.j:; + +f 1.1800 ~I_' . 600g' '.. ~I. 1800.1 conditions, .and hence the stress range in a. link is always very small. Welded bars General Part 4 of the Code permits the use of welded bars, Pl'Qyi~ed that the· following Jour requirements' ate.:· FI~.iiI6;~jit!lclar~ fatig~evehide . m e t : · next.section), .and thus the, standard f~tigue vehicle would 1. Welding must be carried out in accordance with Parts, no~give'e.~Il'yalenceof·.fatigue•.damaioforQJlwelded..b~~....'.....~~-,~·ind:·~i·~f.·tbe· C~.T~k)'feldin3· iJ·.··not.~nnitted . Howevel",.:it is·appropriate for welded bats. ..",..."""-' 'sinceite~h-reduce fatigue sttengthconsiderably; for The standard fatigue vehicle Spe9ified .in Part 10 of the example, tack· welding of stirrups to main bars can Codeha$ agrossweig~t of 320kN, and COrlsi,$ts of four reduce .the fatigue strength of the latter by about 35% ..' wheel~ on each Of four axles. The vehicle is similar to the (or fatigue life by 75%) [270].. shortest HB vehicle (see Fig. 3.6) but the transverse .spac- c 2. 'The welded bar is not part of a top slab. This is, pre- .. . f th .. hi' d'f~ ". h . F' 1:Z 6 '.' sumably, because such bars are subjected to the loeal 109 ~•..... ew ee 8. IS .' I erent, aSs ownm Ig. ". ...•. efflec·t·s' 'o'f' concentrated' wheel loads I'n ad'dl'tl'on to"th"e' ;'Part 10 ·of ihe Code also gives 1:1 simplified load, spec- trum for use with the standard fatigue veHicle, The sim- global effects of the standard fatigue vehicle. p1lfied spectrum gives the proportions of total 2.ommercial 3. The stress range is within that permitted by Part 10 of vehicles for various multiples of the standard fatigue the Code. This pOint is discu~sed in the following sec- vehicle gross weight of 320 kN. . tions of this chapter. Compliance clauses for reinforcing bars Unwelded bars Part 10 of the Code does nllt give a compliance clause for unwelded bars: instead maximum allowable stress ranges under the normal 'static' design loads at the serviceability limit state are specified in Part 4 of the Code. These ranges are 325 N/mm2 for high yield bars and 265 N/mm Z for mild steel bars, and are taken from BE 1173. T~eauthor is. unaware of the origin of these stress ranges,but their implications are now considered in a sim- plistic manner. Moss [269] has reported that the stress range (or) - cycles to failure (IV) relationship for a variety ofbigh yield bars, when the bars are always in tension, is No/ 5 =1.8x 1029 (12.5) Thus,,for a stress ra:hge of 325 N/mrh 2, the fatigue life (IV) is 0.2~X 108 cyclefis. However, a stress range as large as 325 N/mm2 is only likely to occur when there is a reversal of stress during the' stress '.cycle (I.e. the stress changes from compression to tension). In sjJch cirCllmstances the stress range fora particular number of cycles'may be up to one-third greater th~nthat predicted by equation (12;5). Hence, the upper limit to equation 02.5) can be expressed as Nor 9 :!'> =.~.8 ..x 1 ( l 3 o ( 1 2 . 6 ) . Thus,lor .....~. ,',stress!an~~' of ·325 N/tltm2, .. ihe{atigue ,life from. equatlOll (12.'5) IS 3.8 X 108 cycles. Hence, the fatigue life ofa bar, designed to have the maximum stress range ofPart 4 of the Code, is likely to be of the order of 0.25 x 108 to 3.8 X 108 cycles. Thus, the specified stress ranges seem. reasonable.' However, it shQuld be remem- bered that bending a bar can reduce its fatigue strength by up to 50% [265) ~ This fact should not affect sheilr links, since shear cracks are unlikely to occur under service load 156 4. Lap welding should not be used, because adequate control cannot be exercised over the profile of the root beads. Or - N relationships Part 10 of the Code gives Or -N relationships for the following cases: ' . 1. Welded intersections in fabric, or between hot rolled bars. 2. Butt weld between the ends of hot rolled bars. 3. Butt weld between the end of a hot rolled bar and the surface of a plate.. 4. Fillet weld between the end of a hot rolled bar .and the surface ofa plate. , . The appropriate design Or .,. N relationship, which corre- sponds to a 2.3% probabilit~ of failureis: 02.7) . where K is 1.52 for cases 1 and 2, 0.63 for case 3 and 0.43 for case 4. These relationships were derived, princi- pally, for structural steelwork connections and their use is thus restricted in the Code to hot rolled bars. ijowever, Moss [269] has tested butt welded connections in both hot rolled and cold worked bars, and he found that the Code relationship is a satisfactory lower bound fit to his test data. The Code gives three methods of using equation 02.7) to assess fatigue life. The methods are now described briefly in ascending order of accuracy and complexity. Full descriptions are 'given in Part 10 of the Code.. 1" . , " , i, 1 '" " . :, >', .' '," Assessment without damage calculation Equation (12.7) . has been used in conjunction with the simplified load spec- trum, referred to earlier in this chapter, to produce graphs of limiting stress range (OH) against loaded length (L) for various. categories of road. The loaded length is defined as the base length of the loop of the point load influence line which contains the greatest ordinate, as shown in Fig. 12.7. A full description of the derivation of the • Greatest + ve or - ve ordinate Loaded length (L) Fig. 12:7 Loaded length for fatigue c'alculation graphs is given by Johnson and Buckby [24], and the graphs are given in Part 10 of the Code. The graph of 0H against L for a butt weld between the ends of two bars is shown in Fig. 12.8 for a dual three- lane motorway. The stresses due to .the standard (320 kN) fatigue vehicle as it crosses the bridge in each slow lane and each adjacent lane in tum should be calculated. At a particular design point on the bridge, the. stress range is the greatest algebraic difference between these calculated stresses, irrespective ,of whether the maximum and minimum stres- ses result from the vehicle being in the same lane. The stres~ range, so calculated, should not exceed the appro- priate limiting stress range (OH)' In some situations (e.g. at an expansion joint) the stresses calculated due to the passage of the standard fatigue vehicle should be increased by an impact factor, which can be up to 25%. It is emphasised that a load spectrum is not used with this simplified method of compliance. Damage calculation. single vehicle method As with the previous method, a load spectrum is no't used: instead, the standard fatigue vehicle is applied to each slow and each adjacent lane in tum. However, instead of just considering the greatest stress range (as is done in the previous method), the fatigue damage due to each stress cycle is assessed by means of a 'damage chart' given in Part 10 of the Code. The total damage is assumed to be the sum of the damages contributed by each individual stress cycle. The derivation of the damage charts and an example of 20r-~--__~~~~~~~~~~ 1 2 5 10 20 50 100 200 Loaded length L (m) Fig. 12.8 Limiting stress range for butt welded bars for dual three-lane motorway their use are given in Appendices C and D, respectively, of Part 10 of the Code. Damage calculation. vehicle spectrum method. This method requires each vehicle in the load spectrum to be traversed along each lane, and consideration should be given to vehicles occurring simultaneously in one or more lanes. From the resulting stress histories, a stress spec- trum, which gives the number of occurrences of various stress ranges, can be produced by the methods described in Part 10 of the Code. The fatigue damage due to each stress range can then be calculated from'the Or - N curves given in the Code. The cumulative damage is assessed by using Miner's rule, which states that the cumulative damage is equal to the sum of the individual damages [271]. This summation should not exceed unity if the fatigue life is to be considered acceptable. 157
- 87. Chapter 13 Terylperature loading: "ntroduction.', . '~ ;, " .,: ~ .' A~ explained inChapter 3, it is necessary to considertwo aspects of temperature loading: overall temperature move- ments, and differential temperature effects.' The overall movements ate discussed first. .Having locllted the point on the structure which does not move (the stagnant paint), it is simple to calculate the overall movement at any other point of the structure. If the articulation of the bridge is not complicated, it is possible to locate the stagnant point by inspection (e.g. at a fixed . bearing). However, in .general, it is necessary to consider the relative stiffnesses of the deck, piers'and foundations in order to calculate the stagnant point. This calculation is discussed in detail by Zederbaum [272]. If any of the overall movements are restrained then stress resultants are induced in the structure. These stress resultants can be simply calculated from a knowledge of the restrained movements. Since these overall movement calculations are well known and are not contentious, they are not discussed further in this chapter. In contrast, many bridge engineers are uncertain as to /' how to include differential temperature effects into the design procedure. In particular, it is not clear how to deal with cracked Sections, and there are differing views ori the method of calculation of differential temperature effects at the ultimate limit state. The Code gives no guidance what- soever on these matters. In view of this, the remainder of this, chapter is concerned with differential .temperature effects, and design procedures are suggested. These pro- cedur~s are; to a certain extent, based upon current un- completed research at the University of Birmingham and, thus, they should be considered as interim measures until' the research is completed. . , . " . ' . Servic~abiHty limit state Compohent effects Introduction In the following a general non-linear differential temper- ature distribution is considered to be applied to a general section, as shown inFig; 13. L 158 It is.important to realise that, when a temperature dis- tribution isappJied to a stnicture, temperatute~iriduced strains occur, but temp'rature-induced stresses result only if such strains are restrained, either externally or internally. .' . . There are two basic approaches to the determination of the effects of a non-linear temperature distribution, such as that shown in Fig. 13.1: the strain method or the stress method. The two methods are based upon the same assumptions of structural behaviour, and their'end results are identical. In view of the fact that temperature loading is, in structural terms, an applied deformation, the author prefers the strain method; however, many bridge engineers prefer the stress method because they are used to working in terms of stresses, and because it is computationally more convenient for statically indeterminate structures. Each method is now presented. In·the following all stresses, strains and stress resultants are positive when tensile. Strain method Consider the general section and temperature· distribution of Fig. 13.1.'At a distance z above the bottom of the sec- tion, the temperature istz " the section breadth is bz, the elastic modulus is Ez and the coefficient of expansion is ~z; each is a function of z. The coefficient of expansion may vary with depth due to different materials existing in the section. The elastic modulus may vary for two reasons: 1. Different materials existing in the section. 2. Sttesses, due to co-existing force loading, varying through the depth ofthe section and, thus, being on different points of the material stress-strain curve. The potential thermal strain (t;) at Z, in the absence of any restraint, is (13.1) If it is assumed. that plane sections remain plane, then the section must take up the strains indicated by the dashed line of Fig. 13.2(a). The latter strains can be defined in terms of a strain (Eo) at Z = 0, and a curvature ('11). At any level the difference betwe.en the potential thermal strain and the strain that actually occurs is a strain which induces a thermal stress. Since. no external: force is applied to the . sections, these thermal stresses m~t be self-equilibrating. '. z General section Fig. 13.1 Temperature distribution h z (a) . Potential and final strains h Potentla.1 Fig. 13.2(a)-(d) Thermal strain method Temperature distribution (b) Strain Hence, in the absence of any external restraint, the section experiences an axial strain, a curvature amI a set of self- equilibrating stresses, as shown in Fig. 13.2. At distance z above the bottom of the section, the stress- inducing strain is , Eo + 'IIz - ~ztz Thus, the stress at z is f= Ez (Eo + 'IjI z - ~ztz) (13.2) For force and moment eqilibrium respectively: (13.3) S: fbzzdz =0 (13.4) If equation (13.2) is substituted into equations (13.3) and (13.4), Eo and '11 can be obtained as Eo =(F3F4 - F2FS)/FIF3 - Fl) '11 =(FIFs - F2F4)/(FIF3 - F22) (13.5) (13.6) . '~,:. (e) Curvature where F4 = Soh '" E b t d"z z z z Z Potential thermal strains (d) Self-equilibrating stresses (13.7) Having obtained Eo and '11, the self-equilibrating stresses can be obtained from equation (13.2). Uncracked section For the case of a section which is uncracked, and in which the concrete stresses are sufficiently low for the elastic modulus to be considered constant throughout the section, equations (13.5) and (13.6) reduce to 159
- 88. 130 Elevation Fig. J,3;3~ta~k~ds~ction 200 2or4Y16 Section £O:::;~I [If:bztzdz - A.iS: bztz(z - i) dZ] (13.8) 'v= ~cr: bztz (z - z) dz (13.9) where Ais the cross-sectional area, Z is the distance of the section "centroid from' the bottom of the section I is the secotldmoment of area about the, centroid and 'c¥c is the ~oefficient of expansion of concrete. The strain at the cen- troid of the section (e) is given by £';" £" + 'IjI i ¥' :. ~;= C¥c,Sh b'};tzdz (13.10) A 0, ,," .,' , Cracked section Tests carried out by Church [273] at the Transport and Road Research Laboratory, on both cracked and uncracked sections under the thermal loading shown in Fig. 13.3, have shown that the free thermal curvature of a cracked section is, typically, 20% greater than that of an uncracked section. Furthermore, the thermal curvature can be, calculatedfrom equation (13.6) by assuming the elastic modulus of the concrete, within the hpight of the crack, to be, zero. If it is assumed that the crack extends to the neutral ~xis of the cracked transformed section, and the thermal stresses dQ not affect' the position of the neutral axis, then equations (13.9) and (13.10) become 'IjI ==E:I[~..E.rAl,'(h :"'d- z) + , c¥cEc S~ bztz (z "" i)dZJ (13.11) "E = E~[C¥".E.rA.'t.,+ c¥cEc S:bztzdZ] whe,re . "'J:::: ," ~,; ,""" ,coefficieIltof eXpansion of reinforcement' E.. = elastic modulus of reinforcement A" = area ofreinf()ccement t.. , = temperature at level of reinforcement d:::;effective depth ofreinforcement 160 "I 25°C A :::; area of cracked transformedsectio~' I = second moment of area of crllckedtransformed section about its centroid ' Z = ~istance of centroid of crackedtra~sfo~ed ~." hon from the bottom of the section. " " .' " , . T~sts have not yet been ca~ried out under temperature distrIbutions which cause the cracked partofthesectiooto' be hotter than the uncracked part. However, _nsuch cit" cumstances, the first of equations (13.11) predicts for temperature differences of the shape shown in Fig. i3.3, that the free thermal curvature is, typically, 20% less than that of an uncracked section. . , . However, when the Code temperature distribution, with It~ non-ze~o tem~rature at the bottom of the secti()n.'(see Fig. 3.2) IS considered, it is found' that the differences between the calculated free, thermal curvatures 'for the uncracked and cracked section are much less than 20% (see E~ample 13.1). Thus the :'author w6uld' suggest that, for deSign purposes, the free thermal curvature could 00 . calc~lated from equation (13.9) (which is foran uncracked sectIOn), irrespective of whether the section is cracked or uncracked. Stress method In the stress method,the section is, at first, assumed to be fully re~trai~ed, so that" no, displacements. take place, as shown In Fig. 13.4(a). The streSs fa, at t, due to the restraint is LJ f2. '" L· I::,.t . iX. fa =- Ez c¥ztz fo -;: ~.z, E '" t, t- cI - E Hence, the restraining forceis F =fh "b dz. }o Z a If no longitudinal restraint is,present,iherestrainitlg' force must be released by application of a releasing force Fr = -F. The stress (f,) at z due to the releasing force is , f, = -E.z J:fobzdz/f: Ezbzdz ,(13.14) The net stress at z is nOW ifo + jJ). Potential + z + h (a) Potential and restrained strains (ti) Restrained strlilises (c) Stresses dueto relaxing force (d) Stresses dueto relaxing moment (e) Self-equilibrating stresses Fig. 13.4(8)-(e) Thermal stress method The restraining moment (M) can be obtained by taking moment~ about z :::; 0; if a positive moment is sagging: M :::; - J: (fa + ft)bzzdz (13.15) If no moment restraint is. present, the restraining moment tnust be released by application of a releasing moment M, = -M. The stress (f2) at z due to the releasing moment is Ii =-Eiz - i) (-M) / f: Ezbz(z - z)2dz (13.16) If the section is uncracked, equations (13.12) to (13.16) become (13.17) Ii == -(z:'" i)(-M)II , . If the section is cracked, the elastic modulus of the con- crete, within the height of the crack, should be assumed to be zero. The equations for F and M then become: rh' F:::; -<XsE,.Asts -C¥cEc bztzdz• f M =c¥sE,.Asts (h - d - z) + c¥cEc .I ~ bztz(z - z)dz 1 (13.18) This approach is very similar to that adopted by Hambly [274] for a cracked section. .... It can be seen from Fig. 13.4 that the self-equilibrating stresses can be calculated from (13.19) The stresses calculated from equation (13.19) are identi- cal to those calculated, by the strain method, from equa- tion (13.2). Externally unrestrained structure In a structure which is externally unrestrained, such as a simply supported beam, the strain £0 and the curvature 'IjI can occur freely. Thus the only stresses in the structure ar~ the self-equilibrating stresses calculated from equation (13.2) or (13.9). However, these stresses do not occur at the free ends of·, the structure, where plane sections distort and do not reinain plane. Hence the stresseS build-up from zero to the values given by equations (13.2) and (13.19). Such a build-up of stress implies that, in order to maintain equilibrium, longitudinal shear stresses occur near to the ends of a member. The calculation of these shear stresses is demon- strated in Example 13.2. Externally restrained structure If the strain £"or the curvature 'IjI is prevented from occur- ring by the presence of external restraints, then secondary stresses occur in addition to the self-equilibrating stresses. The secondary stresses can be calculated by using either a compatibility method or an eqUilibrium method. The author feels that the compatibility method 'explains the actual behaviour better, but the equilibrium method is gen- erally preferred because it is computationally more con- venient. The t~o methods are compared in the following section by considering a two-span beam. It should be emphasised that bridge decks are two- dimensional in plan, and thus the transverse effects of tem- perature loading should also be considered. However, the same principles, which are illusttated for a one-dimen- sional structure, can also be applied to two-dimen- sional structures (see reference [99]). Compatibility method The supports of the two-span beam shown in Fig. 13.5 are assumed to permit longitudinal movement, so that the strain £0 can occur. However, the free thermal curvature 'IjI is prevented from occurring by the centre support. If the centre support were absent, the beam would take up the 16f
- 89. (a) Fr~eDefl.ected shape t ·lR=3£ItI;// J21 (b)Force applied to give zeJO displacement atcehtre support <~.. ,.sel' . ~.... .. (e) .Thermal moments . Fig.. 13.S(a)-(c) Compatibility method ~r~e thermal curvature, and the displacement at the pos- ~~on .of the centre support would be [2~/2, as shown in Ig. 13.5(a). It is now necessary to apply a vertical force R.at the centre support; as shown in Fig. I3.5(b), to ~estore the beam at this point to the level of the centre support. Hence, for the two-span beam, R== 48 ErW~/2) / (2/)3 =3EI~/l T~is force induces the thermal moments shown in Fig. 13.5«:); the maximum moment, at the support, is M.i=.R(2/)/4 == 1.5 EN (13.20) un~f the sect~on is uncracked, the EI value of the th crack~d ~ectlon should be used in equaribn (13.20). If ,e sechon IS cracked, the author would suggest that the EI v<llue.of the cracked transformed section should be used in equatlon~I3.20). In addition, this value of Eland the neut.ral aXIs depth appropriate to the cracked transformed section should be used to calculate the secondary stresses due to the thermal moment. However, although~,could b~ calculated .from equations (13,6) and (13 II) 't' ·b bl ffi···· . ,I IS pro - a . y Sl.! ,Clently. accurate to calculate ~ from equation ~13.?) usmg .the second moment of area of the uncracked st:ctlOn,~s discussed earlier in this chapter. . , Equilibrium method At e~chsup~rt,the beam is first assumed to be fully r~stramed ,agamst rot~ti?n but not against longitudinal ~ovement. Thus restral?mg moments (M), given by equa~ .tlon (~3: 15)and shown In Fig. 13.6(a); are set up. The. net restra~n!ng moments are shown in Fig. I3.6(b), and the r~st:ammgmoment diagram is shown in Fig. I3.6(c). Sm~~n~extemaJmoments are applied,itis necessary, for ~qulll~num. to. cancel the end restraining moments by ::Plymg rel~a~mg moments which are equal and opposite i tlte re~~~~mmgmOmeflts~ as .shown in Fig. 13.6(d). If be beam IS analysed under the effects of the teleaslng 1: . I,: 162 M ~) (a) Restraining moments ) M (b) Net restrilining moments (c) ReStraining moment diagram (d) Releasing moments (e) Releasing moment diagram ~ (f) Net thermal moment = (c) + (e) Fig. 13.6(a)-(f) Equilibrium method ~oment~, the moment distribut.ion s.hown i.n F..ig.. 1~.6(e).. IS obtamed. The final thermal moments·h . F' 13 ' S ow In· Ig. .6(f), are obtained by summing the restraint ._ ments and the distributed releasing moments. The maxi- mum thermal moment is 1.5M;· by using equations (l3.12?, (1~.I4) and (13.15), this moment can be shown to be ~dentlcal to that given,by the compatibility method (equation (13.20».' If the secti?n is uncr~cked, the properties •of the uncracked section should be used to calculate the s·e·· _ d . . . . . con ary stresseS due to the thermal moments. It is m~ntioned earlier in this chapter that, if a section is cracked, ItS response to thermal loading Catl be calculated by assuming. the elastic modulus of the concrete, within the crack heIght, to be zero. Thus, the restraint moment sho.Uld be caIc~lated f~om the second of equations (13.18) which was denved usmg thisassumptton. Thepropei1ies . Of. the cracked transformed section shOUld be used to I u- 1 t . h· ca c a e t e secondary stresses due to the thernial mame t. . . . n s. Ultimate limit state . . .. ' . ~~en c~~sidering thermal effects under ultimate load con- . ?1t1~ns It IS essen~ial to bearin mind that the tIiermalload- mg I~ a deformatIOn rather thari a force. T~e significarice of this can be seen froin Fig. 13.7, whichcompaces the respon~e of a material ,to an applied St~ssanp an applied ..Attain at lioth tM servIceability and ultimate limit states. I' D=D~ad load L=Live load T=Temperature load (a) Serviceability (b) Ultimate Fig. 13.7(a),(b) Effects of applied strains E Although Fig. 13.7.is presented in terms of stress-strain curves,the following discussion is equally applicable to load-deflection or moment-rotation relationships. It can be seen from Fig. 13.7 that, at the serviceability limit state, the applied thermal strain results in a relatively large thermal stress, but, at the ultimate limit state, only a small thermal stress arises. In view of the comments made in the last paragraph, it is essential to consider carefully what the IQads are, and what load effects result (see also Chapter 3). The author would suggest that the· applied thermal strains or displace- ments should be iriterpreted as being nominal loads. These strains or displacements should be multiplied by the appropriate '{tL values to give the design loads. The design load effects are then the final strains.or displacements, and the stresses or stress resultants which arise from any restrain'ts. At the ultimate limit state, the stress or stress resultant design load effects arte very small and, if full plasticity is assumed, are zero. However, due consider- ation should be taken at the ultimate limit state of the magnitudes of the thermal strains or displacements. Thus, whereas at the serviceability limit state, it is necessary to limit the .totaI (dead + live + thermal) stress so that it is less than the specified permissible stress; at the ultimate limit state, one is more concerned about strain capacity (i.e. ductility) and it is only necessary to limit the total (dead + live + thermal) strain so that it is less than the strain capacity of the material. To summarise, the author would suggest that thermal stresses or stress resultants can be ignored at the ultimate limit state provided that it can be demonstrated that the structure is sufficiently ductile to absorb the thermal strains. The strains associated with the self-equilibrating stresseb are, typically, of the order.of 0.0001. Such strains are·.very small compared with .the strain capacity of concrete in compression, which the Code assumes to be 0.0035 (see Chapter 4). It thus seems reasonable to ignore, at the ulti- mate limit state, the strains associated with the self- equilibrating stresses. One would expect structural concrete sections to possess adequate ductility. in terms of rotation capacity, so that thermal moments could be ignored at the ultimate limit state. However, it is not clear whether they are also sufficiently ductile in terms of shear behaviour. Tests,· designed to examine these problems, are in progress at the University of Birmingham. To date, tests have been carried out on simply . supported beams under various combi- nations of force and thermal 'loading. It has been found that temperature differences as large as 30c C. through the depth of a beam, with peak temperatures of up to 50"C do not affect the moment of resistance or rotation capacity [273]. Tests on statically indeterminate beams are about to com- mence to ascertain whether adequate ductility, in terms of bending and shear, is available to redistribute completely the thermal moments and shear forces, which arise from the continuity. Design procedure General The logical way to allow for temperature effects in the design procedure is to check ductility at the ultimate limit state, and provide nominal reinforcement to control crack- ing, which may occur due to the temperature effects, at the serviceability limit state [275, 276]. However, the Code does not permit such an approach; thus, the following design procedure is suggested by the author. Ultimate limit state 1. Calculate the free thermal curvature and self- equilibrating stresses using the uncracked section. 2. Because of material plasticity, ignore the self- equilibrating stresses. 3. Calculate, from the free thermal curvature, the ro- tation required, assuming full plasticity of the section, such that no thermal continuity moments occur. 4. If the required rotation is less than the rotation cap- acity, ignore the thermal continuity moments. 5. If the required rotation exceeds the rotation capacity, add the thermal continuity moments to the moments due to the other loads and design accordingly. 163
- 90. 1000 1000 .100 surfacing ioooomm2 000 0 (a)Section (b) Nominal temperatures (OC) (c) Design temperatures (OC) (d) Free thermal strains (x 108) Fig. 13.8(~)':'(d) Example 13.1 It should be noted that step 4 assumes adequate ductility in shear in addition to adequate rotation capacity. How" ever, experimental evidence of adequate ductility in shear is not available at present. Serviceability limit'state I . Calculate the free thermal curvature (or the restraining" moment) and the self"equilibrating stresses, using the uncracked or cracked section as appropriate. 2. Calculate the secondary stresses due to any external restraints. 3. Compare the total stresses with the allowable values. In the case of a prestressed member designed as Class 1 for imposed force loadings, it would seem reason" . able to adopt the Class 2 allowable stresses when con" sidering thermal loading in addition to the other load" ings. Similarly, it would seem reasonable to adopt the Class 3 allowable stresses, under thermal loading, for a member designed to Class 2 for imposed force load" ings. It is worth mentioning that, for a reinforced concrete section, the effects of thermal loading at the serviceability limit state'are less onerous when deSigning to the Code than to the present documents. This is because the Code does not require crack widths to be checked under thermal loading. and the Code allowable stresses (0.5 feu for con" crete and 0.8/y for reinforcement) are greater than those specified in BE 1/73 (see Chapter 4). Examples 13.1 Uncracked and cracked rectangular section It is required to determine the response at the serviceability limit state of the section shown in Fig. 13.8(a) to the application of a differential temperature distribution. The 164 section is identical to that considered by Hambly [274]. The nominal positive temperature difference distribution, obtained from Figure 9 of Part 2 of the Code, is shown in Fig. 13.8(b). The concrete is assumed to be grade 30; thus, from Table 2 of Part 4 of the Code, the short"term elastic modulus of the concrete is 28 kN/mml • (It is not yet clear what value of elastic modulus to adopt, but the short"term value seems more appropriate than the long" term value.) The coefficients of thermal expansion of steel and concrete are each assumed to be 12 x 1O"8fC. The nominal temperature differences in Fig. 13.8(b) first have to be multiplied by a partial safety factor (Y,L) of 0.8 (see Chapter 3) to give the design temperature differ" ences of Fig. 13.8(c). The free thermal strains appropriate to the design temperature differences are shown in Fig. 13.8(d); these strains are the design loads. In the following analysis; the section is considered to be both uncracked and cracked}, and both the strain and stress methods are demonstrated.,' Uncracked Cross-sectional area = A = 1000 x 1000 = 1 X 108 mml Height of neutral axis = z.= 500 mm Second moment of area = I = 1000 x 10003 /12 =83.33 x 10' mm4 f: bztzdz = 1000 [(6.6) (150) + (1.2) (250) + (1) (200)] = 1.49 x 108 = 1000[(6.6) (150) (440.9) + (1.2) (250) (266.7) - (1) (200) (433.3)] = 0.43 x 10' Strain method From equation (13.10), the axial strain is £ = (12 X 10"8) (1.49 x 108)11 x 108 = 17.9 X 10"8 150 250 ·400 200 -1.0 -2.7 50 (e) Uncrecked (b) Cracked Fli. 13.9(a),(b) ~lf"equilibrating stresses (N/mm') Prom equation'(13.9), the curvature is 11' .. (12 x 10·') (0043 x 10')183.33 x 10" III 61.9 x 10·' mm"l Bottom fibre strain ::: Eo" 8 ... 'IjIt = 17.9 x 10·'- (61.9 x 10"") (500) ::: -13.1 x 10"' 696 'the self-equilibrating stresses, shown in Fig. 13.9(a), can now be obtained from equation (13.2). Slress method From equation (13.17), the restraining force and moment are: F =-(12 x 10"') (28 x 10') (1.49 x 10') = -0.501 x lOIN M = (12 x 10·')(28 x 10') (0.43 X 10") = 0.144 x 10' Nmm The self"equilibrating stresses can now be obtained, from equation (13.19): they are identical to those, calculated by the strain method, in Fig. 13.9(a). Cracked With an ela"tic modulus of the reinforcement of 200 ~/mm', 'the neutral axis depth, is found to be .304 mm: Thus! =696 mm. Area of transformed section .. A .. (1000) (304) + (200128) (10 000) '. 0:3754 'x 10' mm' Second moment uf area of transformed section .. 1 .. (1000) (304)·/3 + (200128) (10 000) (646)' .. 39.17 x 10' mm4 f: E,b,t,d: -1£,.4;1. +Ii E,.bt,d: .= (200 X 103 ) (10 000) 0.5) + (28 x 103)(1000)[(6.6) (150) + (1.661) (154)] = 37.88 x 10' ~ E,As/6 (h - d - 2) + S~ Ej:Jtzzdz =(200 x 103) (10 000) (1.5) (-646) + (28 x 103) (1000) [(6.6) (150) (245) + (1.661) (154) (88.4)] = 5.487 X 1011 Strain method From equations (13.11), the axial strain and curvature are (12 x 10"8) (37.88 x 101) ! =(28 x 103) (0.3754 X 108) = 43.2 X 10"8 (12 x 10-6)(5.487 >(1011 ) 'IjI = (28 x 103) (39.17 X 10") =60.0 x 10"9 mm" I Thus the cracked free thermal curvature is only 3% less than the uncracked free thermal curvature. Bottom fibre strain = Eo =i! - 'IjI i ::: 43.2 X 10"8 - (60 X 10"1) (696) = 1.4 x 10"8 The self"equilibratlng stresses, shown in Fig. 13.9(b). can now be obtained from equation (13.2). It can be seen that, after cracking, the extreme fibre compressive stress is reduced by 17% but the peak tensile stress is increased by 29%; however, the tensile stress is small. Stress method From equations (13. 18), the restraining forge and moment are: F =-(12 X 10"8) (37.88 x 10') = -0.454 x lOG N M =(12 x 10"6)(5.487 x 1011) =65.8 X 108 Nmm The self"equilibrating stresses can noW be obt~ll1ed from equation (13.19): they are identical to those calculated by the strain method. in Fig. 13.9(b). Design load effects If no extemal restraints are applied, the load effects are the axial strain, the curvature and the self"equilibrating stres· ses. These should be multiplied by the appropriate value of .Y/3to give the 'design load effects: In fact, ypls unity at the . serviceability limit state (see Chapter 4). Thus the stresses shown in Pig. 13.9 are the stresses which should be added to the dead and live load stresIes, aM the net' stresses coml,ared with the allowable values. 16~
- 91. 2.2 0.75 Fig. 13.10(a):-(c) Example13.2 ',' '", . 13.2B6)(:girder )i A prestresSttdconcrete co~tinuous .viaJ~ct~~ith spans of 45 m, ha~ the cross"section shown in Pig. 13.10(a). The concrete IS of grade SO. It is required tOdetermfue the longitudinal effe9ts of a positive temperature,distribution at both the serviceability and ultimate limit stat~s.', :i General ;1 ;. • Since the sectionis prestressed, it isassumedtobe ~ncracked. Cross"sectional area = A = (12) (0.25) + (5)(0;4) + (2) (0.75) (1.5S) = 7.325 ml '1 First moments of area about ~offit to determine i: Z = [(12) (0.25) (2.075) + (5) (0.4) (012) +, (2) (0.75) (1.55) (1.175)]/7.3'25 • = 1.277 m Second mo.m~nt of area =; I = (12) (0:25)~/I2 + (12) (0.25) (0.798)2 + (5)(0.4)3/12 + (5) (0.4) (1.077)2 + (2)(0.75) (1.55)3/12 + = 4.762 m4 ,,(2) (0.75) (1.55) (0; 102)2 Prom Table 2 of ,Part 4 of the Code, elastic modulus of concre~e =: E == 34 X 103 N/mm2 • The, coefficient of expansion JS 12 x W-8/°C. (: ~., Serviceability limit state ", ' T~e norri~nal" tempera~~re ',difference ~istribution, from Figure 9 ?f Part 2 of ttt;e Cdde, is shown in Pig. d.l0(b). These te~peratures h~vetObe multiplied by 0.8 (see Exa~ple 13.1) to obtain the design temperature differences of Fig. I3.W(c). 166 f ~ (bl Nominal (el Design 'temperatures temperatures , ~) ~" I~' .. ~t. . II: (12) (<t'~1?5) (6~6) +:(12) (0.1) (L92) + (2) (0.75~ (0:15) (0.72) + (5) (0.2) (1.0) = 15.35 'y: = (12) (0.15) (6.6) (~.8639) + (12) (0.1) (1.92) (0.7276) + (2) (0.75) (0.15) (0.72) (0.623) + (5) (0.2) (1.0) (- 1.21) = 10.83 Using equations (13.9) and (13.10) (for the strain method) the curvature and axial strain are: • ' ", = (12 X 10"8) (10.83)/4.762' = 27.3 X 10"8 mOl f: = (12 X 10"8) (15.35)17.325 J. , '= 25.1 )f10"8 The soffit strain is Eo = £ -'IIi =25.1 X 10-6 _' , (27.3 X 10"8) (1.277) =-9.7 x 10"' The self"equilibrating stresses, shown in Pig. 13.11(a), can now beobtained from equation (13.2). If it is assumed that the articulation is such that the axial strain can occur . freely, then the only secondary stresses to occur are those due to the restraint to the curvature. This restraint produces the thermal be~ding moment diagram,' shown. in Pig. 13.12. The maximum thermal moment occurs at the first interior support and is ' , ' , Ms = 1.27EI '" == (1.27)(34 x lQ6) (4.762) x (27.3 x 10-6) =5614 kNm The secondary stresses due to this moment are shown in Pig. I3.1l(b). The net stresses, which are obtained by adding the secondary. stresses to the self"equilibrating stresses, are shown in Pig. I3.11(c): ' -2.7 , 1.1 (a) Self-equilibrating 0.15 0.25 1.6 0.2 FIR. 13.11(a),(c) Thermal stresses (N/mm') • (bl Flexural secondary 0.2~r 1.3 -3.8 -0.3 ; (c) Net 0.4 1.0 1.2 Free thermal curvature =til 1 1 r r r r I EItII 1.27EltII 1.27EIw Fig. 13.12 Thermal moments in mUlti-span viaduct Free thermal curvature . PI..t;,~K<>'~hinges ~~~ r Failing span " 1/2 _I_ 1/2 ~I F.g. 13.13 Thermal rotations in multi-span vi~duct Longitudinal shear stresses At the free end of the via- duct, the self-equilibrating stresses do not occur, but longitudinal shear stresses occur in the zone within which the self-equilibrating stresses build-up. The shear stress is greatest at the web-cantilever junction. The average com- pressive stress in the cantilever, away from the free end, is, from Pig. I3.11(a), (2.7 - 0.6)/2 = 1.1 N/mm2 • In accordance with St. Venant's principle this stress will be assumed to build-up over a length, along the span, equal to the breadth of the cantilever (i.e. 3.5 m). Then the average longitudinal shear stress at the web-cantilever junction is equal to the average compressive stress in the cantilever (i.e. 1.1 N/mm2 ). Transverse effects In practice, the transverse effects of the temPerature distribution should also be investigated. Ultimate limit state The partial safety factor YfL at the ultimate limit state is 1.0 (see Chapter 3). ,Hence the free thermal curvature at the ultimate limit state is (27.3 x 10-8) (1.0(0.8) = 34.1 x 10-6 mOl. The worst effect that this pOsitive curva- ture can have on the structure at tIle ultimatelimit~ state is to cause rotation in a plastic hinge at the centre of Ii failing span, as shown in Pig. 13.13. This is because such a thermal rotation is of the same sign as the rotation due to force loading. (If a negative temperature difference, dis- 167 .
- 92. tribution were beingconsidered, the thermal rotation at a support hinge would be calculated.) The total thermal rotation (a,) is given by a, ='11112 where I is the distance between hinges. Thus a, = (34.1 x 10-8 )(22.5) = 0.77 X 10-8 radians. To obtain the design load effect this rotation must be multiplied by' . Yp, which is L 15 at the ultimate limit state. Thus the design rotation is 1.15 x 0.77 x 10-3 = 0.88 X 10-3 radians. 168 The Code does not give permissible rotations, but the CEB Model Code [110] gives a relationship between per- missible rotation and the ratio of neutral axis depth to effective depth. It is unlikely that the permissible rotation would be less than 5 x 10-3 radians. This value is much greater than the design rotation. Thus,' unless the moment due to the applied forces is considerably redistributed away from mid-span, the section has sufficient ductility to enable the thermal moments at the ultimate limit state to be ignored. Appendix A Equations for plate design Sign conventions The positive directions of the applied st,ress resultants ~r unit length and the reinforcement directtonex are shown 10 Fig. AI. Stress resultants with the superscript* are the required resistive stress resultants per unit length in the reinforce- ment directions x. y for orthogonal reinforcement and x, ex for skew reinforcement. The principal concrete force per unit length (Fe) which appears in equations (AI9), (A21), (A23), (A26), (A28) and (A30) is tensile when positive. Bending brthogonal x I y reinforcement Bottom Generally M; = Mx + IMxyl M: = My + IMxyl If M; < () M;=O M* = M + M~y y y Mx If M; < 0 M; =0 M; =Mx+I~~1 Top Generally M; = Mx -IMxyl (AI) (A2) (A3) (A4) (AS) Fig. A.I Slab element M; = My -IMxyl If M:> 0 M;= 0 M; = My If M;> 0 M;=O -1~M~x' M;. = Mx -I~ Skew x I ex reinforcement Bottom Generally M: = Mx + 2Mxy cot ex + My cot 2 ex + Mxy.+ My cot ex Ism ex . M Mxy + My cot ex M*ex =sin! ex + sin ex If M:< 0 M;=O (A6) (A7) (AS) (A9) (AlO) 169
- 93. * :::_1_.(M +1Mx + M cot (X)Z ) (All) Me'( sinz (X .y (Mx + 2Mxy cot (X + My cot (X If M;< 0 M~ =0 M: = Mx + 2Mxy cot (X + My cot 2(X + Top Generally I(MXY +M~y cot(Xfl M; = Mx + 2Mxy cot (X + My cot (X - IMxy +.My cot (X Ism (X M~ = My. -IMxy +,My cot (X Isin2 (X sm (X If M;> 0 M;=O (A12) ¥(A13) (A14) M~ = 1 (M _/ (Mxy + My cot (X)Z I)sin2(X y (Mx + 2Mxy cot (X + My cot2(X) (A15) If M*" > 0 M~=O M: = Mx + 2Mxy cot (X + My cot2(X - I (Mxy + My cot (X)Z/ My (A16) In-plane forces Orthogonal X, Y reinforcement Generally N: = Nx + INxyl N; = Nv + IN.•y I /.;, = -2 INxyl If N. < 0 Ni =0 170 (AI7) (AIS) (A19) Skew X, ~ reinforcement Generally N: = Nx + 2Nxy cot (X + Ny cotZ (X + INxy + Ny cot (X Ism (X N' = .Ny . + INxy +,Nycot (X I0< 5m2(X sm (X Fc =-2 (Nxy + Ny cot (X) (cot(X ± cosec (X) (A20) (A2l) (A22) (A23) (A24) (A25) (A26) In equation (A26), the sign in the last bracket is the same as the sign of (Nxy + Nycot (X). If N; < 0 N;=O N; =_1_ (N +1 (Nxy +Ny cot (X)Z sinz (X y Nx + 2Nxy cot (X t Ny cot Fc =(Nx + Nxy cot (X)Z + (Nxy + N~ cot (X)Z Nx + 2Nxy cot (X + Ny cot (X If N~ < 0 N:=O N; = Nx + 2Nxy cot (X + Ny coe ~ + I (NXY + Ny cot (X)21 Ny (X ) (A27) (A28) (A29) (A30) ....-~---.- ..-,.~ ...~.- -_.._--_._--_.•.. _.....-...--..-....... . Appendix B Transverse shear in cellular and voided slabs Introduction It is mentioned in Chapter 6 that no rules are given in the Code for the design of cellular or voided slabs to resist transverse shear. In this Appendix, the author suggests design approaches at the ultimate limit state. All 'stress resultants in the following are per unit length. Cellular slabs General The effect of a transverse shear force (Qy) is to deform the webs and flanges of a cellular slab, as shown in Fig. B.l(b). Such deformation is generally referred to as Vierendeel truss action. The suggested design procedure is initially to consider the Vierendeel effects separately from those of global transverse bending, and then to combine the global and Vierendeel effects. Analysis of Vierendeel tru~s Points of contraflexure may be assumed ,at the mid-points of the flanges and webs. Assuming the point of contraflex- ure in the web to be always at its mid-point implies that the. stiffnesses of the two flanges are always equal, irres- pective of their thicknesses and amounts of reinforcement. However, a more precise idealisation is probably not justified. The shear forces are assumed to be divided equally between the two flanges to give the loading, bend- ing moment and shear force diagrams of Fig. B.l. Design , Webs A(;eb can be de~igned as a slab, in accordance with the methods described in Chapters 5 and 6, to resist the bend- ing moments and shear forces of Figs. B.l(c) and (d), respectively. Flanges The flanges can be designed as slabs, in accordance with the method descri~ed in Chapter 6, to resist the shear' forces of Fig. B.l(d). . In addition to the Vierendeel bending moments, the global transverse bending moment (My) induces a force of Mylhe, where he is the lever arm shown in Fig. B.1(a), in both the compression and tension flanges. Thus, each flange should be designed as a slab eccentrically loaded by a moment Q"s/4 (from Fig. B.l(c» and either a compres- sive or tensile force, as appropriate, of Mylz. Voided slabs General The effect of a transverse shear force is to deform the webs and flanges of a voided slab in a similar manner to those of a cellular slab. However, since the web and flange thick- nesses of a voided slab vary throughout their lengths, analysis of the Vj~rendeel effects is not readily carried out. In view of this, a method of design is suggested which is based on considerations of elastic analyses of voided slabs and the actual behaviour of transverse strips of reinforced concrete voided slabs subjected to shear [277]. The sug- gested ultimate limit state method is virtually identical to an unpublished working stress method proposed by Elliott [278] which, in tum, is based upon the test data and design recommendations of Aster [277J. Although Aster's tests to ' failure were conducted on transverse strips of voided slabs, a similar failure mode has been observed in a test on a ' model voided slab bridge deck by Elliott, Clark and ~ Symmons [71]. The design procedure considers, independently, possible<! cracks initiating on the outside and inside of a void due to L the Vierendeel effects of the transverse shear. The latter ) " 171
- 94. Idealised Vierendeel truss 0.)2 ~ ~ I~ 0 r;- ..,I~ Oyl2 / ~ s (a) Section OyS/2 - ___ 0.)2 i..... :~ -/ , Oyl2 ""-- ---_ OyS/2 (e) Bending moments Fig. B.I(a)-(d) Cellular slab Oyl2 d Possible Oy/2j crack ~ Oyl2 ~ ~~~~--------~ o Compression ;.:..:..:5 he ( ",,~ ..,~., AI 0.)2 Oyl2 00 (b) Loading ----.--~ ------~~~--~ (d) Shear forces CDI '. I I :, I I, I.Htdl4 I I I I I I I I I I 0: Elastic bottom~____-I-_~ Bottom reinforcement strain I I fibre stresses ~~ (a) Elastic stresses Fig. B.2(a),(b) Vierendeel stresses in voided slab [277] effects and the global transverse bending effects are then combined. .In the following, the global transverse moment (My), co-existing with the transverse shear force (Qy) , is assumed to be sagging. 172 (b) Measured reinforcement strains Bottom flange design Elastic analysis of the uncracked section shows that the distribution of extreme fibre stress, due to Vierendeel action, is as shown in Fig. B.2(a) [277]: the peak stress Oyl2 000.)2 Fig. B.3 Bottom flange of voided slab 00 Fig. B.4 Top flange of voided slab occurs at about the quarter-point of the void (i.e., at d/4 from the void centre line, where d is the void diameter). Thus a crack may initiate, from the bottom face of the slab, at this critical section. It has also been observed that peak bottom flange rein- forcement strains, in cracked concrete slab strips, occur at about d/4 from the void centre line. This is illustrated by Fig. B.2(b), which shows some of Aster's. measured bot- tom reinforcement strains in a reinforced 'concrete trans- verse strip. Fig. B.2(a) shows that the Vierendeel bending stress at the centre line of the void is zero; hence, only a shear force acts at this section, as shown in Fig.' B.3. It is con~ servative, with regard to the design of the reinforcement in the bottom flange, to assume that the shear force (Qy) is shared equally between the two flanges. In fact, less than Q,J2 is carried by the bottom flange because it is cracked. Thus the Vierendeel bending moment at the critical sec- tion, d/4 from the void centre line, is: Mv =(Q,J2) (d/4) =Qyd/8 (B.l) The bottom flange reinforcement is also subjected to a tensile force of (Mylz), where My in this case is the maxi- mum 'global transverse moment and z is the lever arm for global bending shown in Fig. B.3. The resultant compres- sive concrete force (C) in the top flange is considered to act at mid-depth of the minimum flange thickness (t), because the design is being carried out at the ultimate limit i Reinforcement Critical section Critical secfion O.5t z state and the concrete can be considered to be in a plastic condition. • The bottom flange reinforcement should be designed for the combined effects of the force M,Jz and the Vierendeel momentMv• The section depth should be that at the critical section. Top flange design The extreme top fibre stress distribution, due to Vierendeel action, is similar in form to that, shown in Fig. B.2(a), for the bottom fibre. Thus, due to Vierendeel action, a crack may initiate, from the top face of the slab, at the critical section (distance d/4 from the void centre line). The Vierendeel bending stress is again zero at the centre line of the void, but it is now conservative, with regard to the design of the reinforcement in the top flange, to assume that all of the shear force is carried by the top flange. This assump- tion implies that the bottom flange is severely cracked due to global transverse flexure and cannot transmit any shear by aggregate interlock or dowel action. The Vierendeel bending moment at the critical section is (see Fig. B.4): (B.2) The top flange is also subjected to a compressive force of (M,Jz) which counteracts the tension induced in the top 173
- 95. Maximum tensile stress'"KQylh d/h = 0.800 d/h =0.775 d/h =0;750 d/h = 0.725 d/h =0.650 200)-----~----~~----~----_+·----~1 2 3 4 5 ( ) M . My/Qyh a aXlmum tensile stress at face of void h (c) Section Fig. B.S(a)-(e) Maximum tensile stress at face of void flange reinforcement by the Vierendeel moment My. Hence, the greatest tension in the reinforcement is obtained when My is a minimum. The top flange should be designed as an eccentrically loaded column (see Chapter 9) to resist the compressive force (Mylz), which acts at t/2 from the ~QP face, and the moment Mv. The depth of the column should be taken as the flange thickness at the critical section. Detailing of flange reinforcement The areas of flange reinforcement provided should exceed the Code minimum values discussed in Chapter to, and the bar spacings should be less than the Code maximum values discussed in Chapters 7 and to. Web design It is desirable to design the section so that the occurrence of cracks initiating from the inside of a void is prevented, because it is difficult to detail reinforcement to control such cracks. Elliott [278] has produced graphs which give the maxi- mum tensile stress on the inside of a void due to com- bined transverse bending and shear: it is conservatively ass~m~d that all of the· shear is carried by the top flange. Elltott s graphs are. reproduced in Fig. B.S. The maximum tensile stress obtained from Fig. B.5(a) should be compared with an allowable tensile stress. The author would suggest that the latter stress should be taken as 0.45 /Tcu: the derivation of this value, which is the 174 20~____~____~.-____*-____-+____~ c 0 2. 3 4 5 (b) Location of maximum tensile stress MylQyh Effective depth 14 s ~I Fig. B.6 Vertical web reinforcement in voided slab Code allowable flexural tensile stress for a Class 2 pre- tensioned member, is given in Chapter 4. Tensile stresses less than and greater than the allowable stress now have to be considered. Tensile stress less than allowable Cracking at the inside of a void would not occur in this situation, and vertical reinforcement in the webs should be provided. The work of Aster [277] indicates that the design can be carried out by considering the Vierendeel truss of Fig. B.l(b), for which the horizontal shear force at the point of contraflexure in the web is Q"slhe. The critical section for Vierendeel bending of a web is considered to be at d/4 above the centre line of the void, .as shown in Fig. B.6. The Vierendeel bending moment at this critical section is: -------_.- • _ _ _0 __- " _ _ • • • • _ _- - - h" C=Concrete strut T =Reinforcement tie .lg. B.7 lncli~ed web reinforcement in voided slab (B.3) Reinforcement at the critical section, with the effective depth shown in Fig. B.6, can be designed to resist the momentMv· The vertical reinforcement in the web is most con- veniently provided in the form of vertical links, as shown in Fig. B.6-; however, only one leg of such a link may be considered to contribute to the required area of reinforce- ment. This area should be added to that required to resist the longitudinal shear to give the total required area of link reinforcement. Tensile stress greater than allowable If the tensile stress obtained from Fig. B.5(a) is greater than the allowable stress, cracking will occur on the inside of the void. In this situation, it is preferable to reduce the size of the voids, so as to reduce the tensile stress, or to alter the positions of the voids in the deck, so that they are not in areas of high transverse shear. If cracking is not precluded by either of these means,· it is necessary to design the voided slab so that reinforcement crosses the crack, which initiates on the inside of the void. This can be done either by providing inclined reinforcement in the webs, or by providing a second layer of horizontal re- inforcement in the flange, close to the void. Inclined reinforcement The forces acting in -a web are shown in Fig. B.7. The horizontal shear force at the point of contraflexure of the web is Q,slhe (see discussion of vertical web reinforcement). For horizontal equilibrium , (T + C) cos IX =Q,sllze But T = C, from vertical equilibrium; thus T =Q,sI2he cos IX (B.4) Critical section for bottom laver of top flange reinforcement Fig. B.8 Additional horizontal reinforcement in voided slab Inclined reinforcement should be designed to resist this force. The reinforcement could take the form of. for example, inclined links or bars: the latter should be anchored by lapping with the top arid bottom flange re- inforcement. Additiona/horizontal reinforcement As an alternative to inclined reinforcement an additional layer of horizontal reinforcement may be provided as shoWn in Fig. B.S. The critical section for designing this reinforcement should be taken as the position of maximum tensile stress, obtained from Fig. B.5(b). The latter figure gives the position in terms of the angular displac~ment (cp): its horizontal dis- tance from the void centre line is thus d sin cj) 12. It is conservative to assmne that all of the transverse shear force is carried by the top flange and thus, from Fig. B.S, the Vierendeel bending .moment at the critical section is: My =Q,.d sin cj)/2 (B.S) The top flange is also subjected to a compressive force of (Mlz), which counteracts the tension induced in the reinforcement by the Vierendeel moment My. Hence, the greatest tension in the reinforcement is obtained when My is a minimum. The critical section should be designed as an eccen- trically loaded colm (see Chapter 9) to resist the com- pressive force (Mylz),which acts at tl2 from the top face; and the moment My. The depth of the column should be taken as the flange thickness at the critical section. Effect of global twisting moment A global twisting moment induces forces in the flanges; these forces can be taken into account in the suggested design methods by replacing My throughout by M; (obtained {lom the appropriate equation of Appendix A). 175
- 96. References 1. British StandardsInstitution, BS 153 Specification for steel girder bridges, Parts I & 2, pp. 16; Part 3A, pp. 40; Parts 3B & 4, pp. 64; 1972. 2. Kerensky, O. A., Major British Standard covers design and con- struction of bridges' , BSI News, July 1978, pp. 10-11. 3. Department of Transport, Technical Memorandum (Bridges) BE 1177 (lst amendment). Standard highway loadings, May 1979, p. 47. 4. British Standards Institution, BS 153. Specification for steei girder bridges. Part 3A, 1972. p. 39.' 5. Department of Transport. Technical Memorandum (Bridges) BE 117)' (lst revision, 1st amendment). Reinforced cOllcrete for highway struc- tures. July 1979. p. 19. 6. Department of Transport. Technical Memorandum (Bridges) BE 2/73 (3rd amendment). Prestressed concrete for highway structures, June 1975. p. 22. 7. British Standards Institution. The structural use of reinforced con- crete in buildings. CP 114: Part 2: 1969, p. 94. 8. British Standards Institution. The structurai use of prestressed con- crete in buildings. CP 115: Pan 2: 1969. p. 44. 9. British Standards Institution. The structural use ofprecast concrete. CP 116: Pan 2: 1969. p. 153. 10. Department of Transport. Specification for road and bridge works. 1976. p. 194. 11. Department of Transport. Technical Memorandum (Bridges) BE 1/ 76 (3rd amendmelll). Design requiremelllsfor elastomeric bridge bear- ings. September 1978. p: 14. 12. Department of Transport. Interim Memorandum ·(Bridges) 1MII (fst addendum). PTFE in bridge bearings. January 1972. p. 3. 13. Department of Transport. Departmental Standard BD 1178. Implementation of BS 5400 - The design and specification of steel, concrete and composite bridges. September 1978. p. 3. 14. Ferguson. H.• 'New code introduces Iimit·state approach to bridge design'. New Civil Engineer. 6 July 1978. pp. 13-14. . 15. British Standards Institution. CP llO: Part I: 1972. The structural lise of concrete. Part I. Design, materials and workmanship. p. ISS. 16. International Organization for Standardization. General principles for the verification of the safety of structure.v. International Standard ISO 2394. 1973; p. 5. 17. Beeby. A. W. and Taylor. H. P. J.. Accuracy ill analysis - influ- ence of Y/3. unpublished note submitted to a meeting of Comite Euro- peen du Beton Commission XI. Pavia. June 1977. pp. 12. 1R. Henderson. W.. Burt. M. E. and Goodearl. K. A.• 'Bridge load- ing', Proceedillgs of an .International Conference on Sieel Box Girder Rridges. London. Institution of Civil Engineers. 1973. pp. 193-201. 19. Cope. R. J. and Rao. P. V;. 'Non·linear finite element analysis of concrete slab structures'. Proceedings of the Institution of Civil Engineers. Part 2. Vol. 63. March 1977. pp. 159-179. 20. Teychenne. D. C.• Parrot. L. J. and Pomeroy. C. D.• 'The esti- ,mation of the elastic modulus of concrete for the design of structures'. Ruildillg Research Establishmelll Current Paper CP 23/78. March 197R. p. 11. . 176 21. British Standards Institution, Specification for hot rolled and hot rolled and processed high tensile alloy steel barsfor the prestressing of cOllcrete. BS 4486: 1980, p. 4. 22, British Standards Institution, Specification for nineteen-wire steel strand for prestressed concrete. BS 4757: 1971, p. 14. 23. Roik, K. 'H. and Sedlacek, G., 'Extension of engineets' bending and torsion theory considering shear deformation', Die Bautechnik, January 1970, pp. 20-32. . 24. JohnSon, R. P. and Buckby, R. J., CompoSite structures of steel and concrete. Vol. 2. Bridges. with a commelllary on BS S400: Pari S. Crosby Lockwood Staples. London, 1979, p. 524. 25. Moffatt, K. R. and Dowling, P. J.• 'Shear lag in steel box girder bridges'. The Structural Engineer. Vol. 53, October 1975. pp. 439- 448. 26. Moffatt. K. R. and Dowling, P. J.• 'British shear lag rules for composite girders', Proceedings of the American Society of Civil Engineers (Structural Division). Vol. 104, No. ST7, July 1978, pp.1123-1130. 27. Clark. L. A.• 'Blastic analysis of concrete bridges and the ultimate limit state'. The Highway Engineer. October 1977, pp. 22-24. 28. Pucher. A.• Influence Surfaces of Elastic Plates. Springer Verlag. Wien and New York. 1964. '; 29. Westergaard. H. M.• 'Comp~t~tions of stresses in bridge slabs due to wheel loads' •Public Roads. Vol. 2. No. I. March 1930. pp. 1-23. 30. Holland. A. D., and Deuce. T. L. G.• 'A review of small span highway bridge design and standardisation'. Journal of the Institution ofHighway Engineers. August 1970. pp. 3-29. 31. Bergg. J. A.• 'Eighty highway bridges in Kent'. Proceedings of the Institution of Civil Engineers. Part I. Vol. 54. November 1973, pp. 571-603. 32. Woolley. M. V.• 'Economic road bridge design in concrete for the medium span range 15 m-45 m·. The Structural Engineer, Vol. 52. No.4. April 1974. pp. 119-128. 33. Pennells. E.• Concrete Bridge Designer's Manual. Viewpoint Pub- lications. 1978. p. 164. 34. Somerville. G. and Tiller. R. M.• Standard bridge beams for spans from 7 m to 36 m, Cement and Concrete Association, Publication 32.005. November 1970. p. 32. 35. Dow-Mac Concrete Ltd.. Prestressed concrete bridge beams. 36. Manton. B. H. and Wilson. C. B. MoT/e & CA standard bridge beams. Cement and Concrete Association. Publication 32.012. March 1971. p. 36. 37. Somerville. G.• 'Three years' experience with M-beams': Sur- veyor. 27 July 1973, pp. 58-60. 38. Hook. D. M. A. and Richmond. B.• 'Western Avenue Extension- precast concrete box beams in cellular bridge decks'. The Structural Engineer. Vol. 48. No.3. March 1970. pp. 120-128. 39. Chaplin. E. C.. Garrett. R. J.• Gordon, J. A. and Sharpe. D. J.• 'The development of a design for a precast concrete bridge beam of U-section·. The Structural Engifleer. Vol. 51. No. 10. October 1971. pp. 383-3R8. 40. Swann, R. A., 'A feature survey of concrete box spine-beam bridges', Cement and COllcrete Association. Technical Repon 42.469. June 1972. p. 76. 41. Maisel, B, I. and Roll, F., 'Methods of analysis and design of con- crete box beams with side cantilevers', Cement and Concrete Associ- ation. Technical Repon 42.494. November 1974, p. 176. 42. Timoshenko, S. P. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill. 1959, p. 580. 43. Plantema. F. J. Sandwich construction, John Wiley and Sons, 1966, p. 246. 44. Libove, C. and Batdorf. S. B., 'A general 8I1IIl1 deflection theory for flat sandwich plates', U.S. National Advisory Committee for Aeronautics, Report No. 899, 1948, p. 17. e ' " 45. Guyon, Y., 'Calcul des ponts dalles', Annales des Ponts It Chaus- stes, Vol. 119, No. 29, pp. 555-589, ·No. 36, pp. 683-718, 1949. 46. Massonnet, C., Methode de calcul des ponts apoutres multiple ten- ant compte de leur rtsistance a la torsion, IABSB Publications, Vol. 10, 1950, pp. 147-182. 47. Morice, P. B. and Little, G., 'The analysis of right bridge decks , subjected to abnormal loading', Cement and Concrete Association, Publication 46.002, 1956, p. 43. 48, Rowe, R. E., Concrete Bridge Desigll, C. R. Books, 1962, p. 336. 49. Cusens, A, R. and Panta, R. P., Bridge Deck Analysis, John Wiley and Sons,' 1975, p, 278. 50. Department of Transport, Technical Memorandwn (JJridges) BE 6/ 74, Suite 0/bridge design and analysis programs. Program HECBIBI IS (ORTHOP)and Design Charts HECB/BS, October 1974, p. 3, 51. Morley, C. T., 'Allowing for shear deformation in orthotroplc plate theory', Proceedings of the Institution of Civil Engineers, Supplement paper 7367, 1971, p. 20. 52. Elliott, G., 'Partial loading on orthotropic plates', Cement and Cont;rete Association. Technical Report 42.519, October 1978, p. 60. 53. Goldberg, J. E. and Leve, H. L.• 'Theory of prismatic folded plate structures' ,IABSE Publication, Vol. 17, 1957, pp. 59-86, 54. De Fries-Skene, A; and Scordelis, A. C., 'Direct stiffness solution for folded plates'. Proceedings of the American Society of Civil Engineers (Structural Division), Vol. 90, No. ST 4, August 1964, pp. 15-48 55. Department of Transport, Technical Memorandum (Bridges) BE 3/ 74. Suite of bridge design and analysis programs. Program HECB/BI II (JtIUPDI), August 1974. p. 2. 56. Department of Transport, Technical Memorandum (JJridges) BE 41 76. Suite of bridge design and analysis programs. Program HECB/BI 13 (STRAND 2). March 1976. p. 5. 57. Department of Transport, Technical Memorandum (Bridges) BE 3/ 75. Suite of bridge desigll and analysis programs. Program HECB/B{ 14 (QUEST), February 1975. p. 2. 58. ,Department of Transport. Technical Memorandum (Bridges) BE 4/ 7S. Suite of bridge design and analysis programs. program HECB/B/ 16 (CASKET), February 1975. p. 3. 59. Cheung. Y. K.• 'The finite strip method in the analysis of elastic plates with two opposite simply supported ends', Proceedings of the Institution ofCivil Engineers. Vol. 40. May 1968, pp. 1-7. 60. Loa, Y. C. and Cusens. A. R.• 'A refined fini!e strip method for the analysis of orthotropic plates'. Proceedings of the Institution of Civil Engineers, Vol. 48. January 1971. pp. 85-91. 61. Hambly. E. C.• Bridge Deck Behaviour, Chapman and Hall Ltd, 1976. p. 272. 62, West, R., 'C &. CAICIRIA recommendations on the use of grillage analysis for slab and pseudo-slab bridge decks', Cement and Concrete Association. Publication 46.017. 1973, p. 24. 63. Robinson. K. E., 'The behaviour of simply supported skew bridge slabs under concentrated loads'. Cement and Concrete Associatioll. Research Repon 8, 1959, p. 184. 64. Rusch, H. and Hergenroder. A.• Influence surfaces for moments in skew slabs, Technological University of Munich, 1961, available as Cement and Concrete Association translation, p. 21 + 174 charts. 65. Balas, J. and Hanuska, A.• Influence Surfaces of Skew Plotes, Vydaratelstro Slovenskej Akademic. Czechoslavakia, 1964. 66. Naruoka, M. and Ohmura. H.• 'On the analysis of a skew girder bridge by the theory of orthotropic plates', IABSE Proceedings, Vol. 19. 1959, pp. 231-256. 67. YeginobaJi, A., 'Continuous skew slabs', Ohio State University Engineering Experimental Station. Bulletin 178, November 1959. 68. Yeginobali. A., 'Analysis of continuous skewed slab bridge decks', Ohio State University Engineering Experimental Station, Report EES 170-2, August 1962. 69. Schleicher, C:;. and Wegener, B., Continuous Skew Slabs. Tables, for Statical Analysis, Verlag fur Bauwesen, Berlin,1968. 70. Elliott, G. and Clark, L. A., Circular voided concrete slab, stiff- nesses in preparation. 71. Elliott, G., Clark. L. A. and Symmons. R. M., 'Test ora quarter- sq~ COlIC........... ~.C,_1tt '.,Concrete "'"A88eciat/on:-Technlcal Report 42.S27, December 1979, p, 40. . 72. "'Slliott, G. (Private communication), 73. Timoshenko, S. P. and Gere, J, M., Mechanics of Materials, Van Nostrand Reinhold, 1973. p. 608. 74. Holmberg, A., 'Shear-weak beams on elastic foundation,', IABSE . Publication, Vol. 10, 1960, pp. 69-85. 75. Clark, L. A., 'Comparison of various methods of calculating the torsional inertia of right voided slab bridges', Cement and Concrete Assoc/ation. Technical Report 42.508, June 1975, p. 34. 76. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hili, 1951, p. 506. 77, Jackson, N., 'The torsional rigidity of concrete bridge decks', Con- crete, Vol. 2, No. 11, November 1968, pp. 469-471. 78. Hillerborg, A., 'Strip metl,lod of design:, Viewpoint, 1975, p. 256. 79. Wood, R, H. and Armer, G. S, T., 'The theory of the strip method for design of slabs' , Proceedings of the Institution ofCivil Engineers, Vol. 41, October 1968, pp. 285-311. 80. Kemp, K, 0., 'A strip method of slab design with concentrated loads or supports'. The St~uctural Engineer, Vol. 49, No. 12, December 1971, pp. 543-548. 81. Fernando, J. S. and Kemp, K. 0., .'A generalized strip deftexion .. method of reinforced concrete slab design'. Proceedings of the Illstitu- tion ofCivil Engilleers, Part 2. Vol. 65. March 1978, pp. 163-174. 82. Spence, R, J. S. and Morley, C. T., 'The strength of single-cell concrete box girders of deformable cross-section' , Proceedings of the Institution of Civil Engineers, Part 2, Vol. 59. December 1975, pp.743-761. 83. Beeby, A. W., 'The analysis of beams in plane frames according to CP 110', Cement and Concrete Association. Development Report 44.001, October 1978, p. 34. 84. Jones, L. L. and Wood, R. H., 'Yield-line Analysis of Slabs. Thantes and Hudson, Chatto and Windus, 1967, p. 401. 85. Jones, L. L., Ultimate Load Analysis of Concrete Structures, Chatto and Windus, 1962. p. 248. 86. Granholm, C. A. and Rowe, R. E., 'The ultimate load of simply supported skew slab bridges', Cement and Concrete Association. Research Repon 12. June 1961, p. 16. 87. Clark, L. A.. 'The service load response of short-span skew slab bridges designed by yield line theory', Cement and Concrete Associ- ation. Technical Repon 42.464. May 1972, p. 40. . 88. Nagaraja, R, and Lash. S. D., 'Ultimate load capacity of reinforced concrete beam-and-slab highway bridges', Proceedings of the Ameri- can Concrete Institute, Vol. 67, December 1970, pp. 1003-1009. 89. Morley. C. T. and Spence, R. J. S.• 'Ultimate strength of cellular concrete box-beams', Proceedings of a Symposium on structural analYSis, lIOn-linear behaviour and techniques. Transpon and Road Research Laboratory Supplementary Repon 164 UC, 1975, pp. 193- 197. 90. Cookson, P. J., 'Collapse of concrete box gilders involving distor- tion of the cross-section', PhD Thesis. University of Cambridge, 1976, p.227. 91. Best. B. C. and Rowe. R. E~. 'Abnormal loading on composite slab bridges'. Cement and Concrete Association. Research Report 7. October 1959. p. 2~. 92. British Standards Institution, Code of Practice for foundations. CP 2004: 1972. p. 158. " ,.' 93. Hay. J. S., 'Wind forces on bridges - drag and lift coefficients', Transpon and Road Research Laboratory Report 6/3, 1974, p. 8. 94. Emerson. M.• 'Extreme values of bridge temperatures for design purposes'. Transpon and Road Research Laboratory Report 744, 1976. p. 25. 95. Emerson. M., 'Temperature differences in bridges: basis of design 177
- 97. I."l requirements', Transport andRoad Research Laboratory Report 765, 1977, p. 39. 96. Spratt, B. H., 'The structural use of lightweight aggregate con- crete', Cement and Concrete Association, Publication 45.023, December 1974, p. 68. 97. Black, W., 'Notes on bridge bearings', Transport and Road Research Laboratory Report 382, 1971, p. 10. 98. Emersoit~ M., 'The calculation of the distribution of temperature in bridges, Transport and Road Research Laboratory Report 561, 1973, p.20. 99. Blythe, D. W. R. and Lunniss, R. C., 'Temperature difference effects in concrete bridges', Proceedings of a Symposium on bridge temperature. TranSport and Road Research Laboratory Supplementary Report 442, 1978, pp. 68-80. 100. Jones, M. R., 'Bridge temperatures calculated by a computer program', Transport and Road Research Laboratory Report 702, 1976, p. 21. 101. Mortlock, J. D., 'The instrumentation of bridges for the measure- ment of temperature and movement', Transport and Road Research Laboratory Report 641, 1974, p. 14. 102. Henderson, W., 'British highway bridge loading', Proceedings of the Institution ofCivil Engineers, Part 2, June 1954, pp. 325-373. 103. Flint, A. R. and Edwards, L. S., 'Limit state design of highway bridges', The Structural Engineer, Vol. 48, No.3, March 1970, pp.93-108. 104. Rutley, K. S., 'Drivers' and Passengers' maximum tolerance to lateral acceletation', Transport and Road Research Laboratory ~ech nical Note 455, 1970. 105. Burt, M. E., 'Forces on bridge due to braking vehicles', Trans- port and Road Research Lalloratory Technical Note 401, 1969, p..5. 106. Department of Transport, Technical Memorandum QJrldges) BE5 (1st amendment). The design of highway bridge parapets, August 1978, p. 15. 107.· Blanchard, J., Davies, B. L. and Smith, J. W., 'Design criteria and analysis for dynamic loading of footbridges', Proceedings of a Symposium 011 dynamic behaviour of bridges, Transport and Road Research Laboratory Supplementary Report 275, 1977, pp. 90-100. 108. Neville, A. M., Properties of Concrete, Sir Isaac Pilman& Sons Ltd., London, 1965, p. 532. 109. Rowe, R. E.• 'Current European views on structural safety', Pro- ceedings of the Amerlcim Society of Civil Engineers (Structural Div- ision), Vol. 96, No. ST3, March 1970, pp. 461-467.. 110. Comite Euro-International du Beton, CEB-FIP model code for concrete structures, 1978. III. Comi.e Europeen du Beton, Recommendation.s for an Inter- national code of practice for reinforced concrete, 1964, p. 156. 112. Bate, S. C. C., et al., Handbook on the unified code for structural concrete, Cement and Concrete Association, 1972, p. 153. 113. Kajfasz, S., SOinerville, G. and Rowe, R. E., An Investigation of the behaviour of composite beams, Cement and Concrete Association, Research Report IS, November 1963, p. 44. 114. Mikhailov, O. V., 'Recent research on the action of unstressed concrete in composite structures (precast monolithic structures)', Pro- ceedings of the Third Congress of the Federation Internationale de la Precontrainte. Berlin. 1958, pp. 51-65. liS. American Concrete Institute - American Society of Civil Engineers Committee 323, 'Tentative recommendations for prestressed concrete', Proceedings of tile American Concrete Institute, Vol. 54, No.7, January 1958, pp. 545-578. 116. British Standards Institution, Composite construction in structural steeland concrete. CP 117: Part 2: beams for bridges: 1967, p. 32. 117. Hanson, N. W., 'Precast·prestressed concrete bridges. 2. Hori- zontal shear connections', Journal of the Research and Development Laboratories of the Portland Cement Association, Vol. 2, No.2, May 1960, pp. 38-58. 118; Seamann, J. C. and Washa, G. W., 'Horizontal shear connections '?etween precast beams and cast·in-place slabs', Proceedings of the American Concrete Inslitute, Vol. 61, No. II, November 1964, pp. 1383-1410. 119. Beeby, A.W., 'Short term deformations of reinforced concrete members'. Cement and Concrete Association. Technical Report 42.408, March 1968, p. 32. 120:' . Bate, S. C. C., 'Some experimental data relating to the design of. 178 prestressed concrete', Civil Engineering and Public Works Review Vol. 53, 1958, pp. 1010-1012, 1158-1161,1280-1284. . , 121: Concrete Society, 'Flat slabs in post-tensioned concrete with par_ ticular regard to the use of unbonded tendons - design recommen- dations', Concrete Society Technical Report 17, July 1979, p. 16. 122. Abeles, P. W., 'Design of partially prestressed concrete beams' . Proceedings of ,the American Concrete Institute, Vol. 64, OctObe; 1967, pp. 669-677. 123. Beeby A. W. and Taylor, H. P. J., 'Cracking in partially pre_. stressed members', Sixth Congress of the Federation Internationale de Precontralnte, Prague, 1970, p. 12. . 124. Schiessl, P., '~dmissible crack ,::vidth inreinforc~ ~~Iprete)truc tures', Contribution It 3 -17, Inter-Association Colloquium on the behaviour in service of concrete structures, Preliminary Report, Vol. 2, Liege, 1975. 125. Beeby, A. W., 'Corrosion of reinforcing steel in concrete and its relation to cracking', The Structural Engineer, Vol. 56A, March 1978, pp. 77-81. 126. Clark, L. A., 'Tests on slab elements and skew slab bridges deSigned in accordance with the factored elastic moment field', Cement and Concrete ASSOCiation, Technical Report 42.474, September 1972, p.47. 127. Beeby, A. W., .'The design of sections for flexure and axial load according to CP 110', Cement and Concrete Association, Develop";ent Report 44.002, December 1978, p. 31. 128. British Standards Institution, CP 110: Part 2: 1972. The struc- tural use of concrete. Part 2. Design charts for singly reinforced beams. doubly reinforced beams and rectangular columns, p. 90. 129. Pannell, F. N., 'The ultimate moment of resistance of unbonded prestressed concrete beams', Magazine ofConcrete Research, Vol. 21, No. 66, March 1969, pp. 43-54. 130. British Standards Institution, CP 110: Part 3: 1972. The Struc- tural use of concrete. Part 3. Design charts for circular c~/umns and prestressed beams, p. 174. 131. Taylor, H. P. J. and Clarke, J. L., 'Design charts for prestressed concrete T-sections', Cement and Concrete Association. Departmental Note DNI301 I, August 1973, p. 19. 132. Kemp, K. 0., 'The yield criterion for orthotropically. reinforced concrete slabs', International Journal ofMechanical Sciences, Vol. 7, 1965, pp. 737-746. 133. Wood, R. H., 'The reinforcement of slabs in accordance with a pre-determined field of moments', Con~rete, Vol. 2, No.2, February 1968, pp. 69-76. 134. Hillerborg, A.., 'Reinfo~~ement of slabs and shells designed according to the. theory of elasticity', Belong., Vol.38, No.2, 1953, pp.l01-109. . 135. Armer, G. S. T.; 'Discussion of reference 133', Concrete, Vol. 2, No.8, August 1968, pp. 319-320. . 136. Uppenberg, M., Skew concrete bridge slabs on line supports, Institutionen for Brobyggnad Kungl, Tekniska Hogskolan, Stockholm, 1968. 137. Hallbjorn, L., 'Reinff!rced concrete skew bridge slabs supported on columns', Division of Structural Engineering and Bridge Building, Royal Institute of Technology, Stockholm, 1968. 138. Clements, S. W., Cranston, W. B. and Symmons, M. G., 'The influence of section breadth on rotation capacity', Cement and Con- crete Association. Technical Report 42.533, September 1980, p. 27. 139. Morley, C. T., 'Skew reinforcement ~f concrete slabs against bending and torsional moments' , Proceedings of the Institution ofCivil , Engineers, Vol. 42, January 1969, pp. 57-74. 140. Kemp, K. 0., Optimum reinforcement in (J concrete slab sub- jected to multiple loadings, International Association for Bridge and Structural Engineering, 1971, pp. 93-105. . 141. Nielson, M. P., 'On the strength of reinforced concrete discs', Acta Polytechnica Scandinavica, Civil Engineering and Building Con- struction Series No. 70, 1969, p. 254. 142. Clark, L. A., 'The provision of·tension and compression re- inforcement to resist in·plane forces'. Magazine ofConcrete Research, Vol. 28, No. 94, March 1976, pp. 3-12. 143. Peter, J., 'Zur Bewehrung von Schieben und Schalen fur Haupt- spannungen schiefwinklig zur Bewehrungsrichtung', Doctoral Thesis, Technical University of Stuttgart. 1964, p. 213. 144. Morley, C. T. and Gulvanessian, H.• 'Optimum reinforcement of ....:...------."- .-- concrete slab elements', Proceedings of the Institution of Civil Engineers. Part 2, Vol. 63, June 1977, pp. 441-454. 145. Morley, C. T., 'Optimum reinforcement of concrete slab elements against combinations of moments and membrane forces', Magazine of Concrete Research, Vol. 22, No. 72, September 1970, pp. IS5-162. 146. Clark, L. A. and West, R., 'The behaviour of solid skew slab bridges under longitudinal prestress', Cement and Concrete Associ- ation. Technical Report 42.501, December 1974, p. 40. 147. Clark, L. A. and West, R., 'The behaviour of skew solid and voided slab bridges under longitudinal prestress;, Seventh COf}gress of the Federation Internationale de la precontrainte". New 'York, May 1974, p. 8. . 148. Shear Study Group. The shear strength of reinforced concrete beams, Institution of Structural Engineers, London, 1969, p. 170. 149. Baker, A. L. L.,Yu, C. W. and Regan, P. E., 'Bxplanatory note on the proposed Unified Code Clause on shear in reinforced concrete beams with special reference to the Report of the Shear Study Group' , The Structural Engineer, Vol. 47, No.7, July 1969, pp. 285-293. ISO. Regan, P. B., 'Safety in shear: CP 114 and CP 110', Concrete, Vol. 10, No. 10, October 1976, pp. 31-33. lSI. Taylor, H. P. J., 'Investigation of the dowel shear forces carried by the tensile steel in reinforced concrete beams', Cement and Con- crete Association. Technical Report 42.431, February 1968, p. 23. IS2. Taylor, H. P. J., 'Investigation of the forces carried across cracks in reinforced concrete beams in shear by Interlock of aggregate', Cement and Concrete Association, Technical Report 42.447, November 1970, p. 22. IS3. Pederson, C., 'Shear in beams with bent-up barsi, Final Report of the International Association for Bridge a,1d Structural Engineering Colloquium on Plasticity In Reinforced Concrete, Copenhagen, 1979, pp.79-86. 154. Clarke, J. L. and Taylor, H. P. J., 'Web crushing - a review of research', Cement and Concrete Association, Technical Report 42.509, August 1975, p. 16. 155. TJlylor, H. P. J., 'Shear strength of large beams', Proceedings of the Amerlcan Society ofCivil Engineers (Structural Division) Vol. 98, No. STH, November 1972, pp. 2473-2490. 156. Regan, P. E., 'Design for punching shear', The Structural Engineer, Vol. 52, No.6, June 1974, pp. 197-207. 157. Nylander, H. and Sundquist, H., 'Punching of bridge slabs', Pro- ceedings ofa Conference on Developments in Bridge Design and Con- .rtruction, Cardiff, 1971, pp. 94-99. 158. Moe, J., 'Shearing strength of reinforced concrete slabs and foot- ings under concentrated loads', Portland Cement Association. Development Department Bulletin D47, April 1961, p. 135. 159. Hanson,.I. M., 'Influence of embedded service ducts on strengths of flat plate structures', Portland Cement Association. Research and Development Bulletin RD OD5.01D, 1970. 160. Hawkins, N. M., 'The shear provision of AS CA35-SAA Code for prestressed concrete', Civil Engineering Traniactions, Institute of Engineers. Australia, Vol. CE6, September 1964, pp. 103-116. 161. Reynolds, G. C., Clarke, J. L. and Taylor, H. P. J. 'Shear pro- visions for prestressed concrete in the Unifted Code, CP 110: 1972', Cement and Concrete Association. Technical Report 42.500, October 1974, p. 16. 162. Sozen, M. A. and Hawkins, N. M. 'Discussion of a paper by ACI-ASCE Committee 326. Shear and diagonal tension', Proceedings of the American Concrete Institute, Vol. 59, No.9, September 1962, pp.1341-1347. 163. MacGregor, J. G. and Hanson, J. M. 'Proposed changes in shear provisions for reinforced and prestressed concrete beams' , Proceedings of the American Concrete Institute, Vol. 66, No.4, April 1969, pp. 276-288. 164. MacGregor, J. G., Sozen, M. A. and Siess, C. P., 'Effect of draped reinforcement on behaviour of prestressed concrete·. beams', Proceedings of t~e Arrzericall Concrete Institute, (,Vol. 57, No.6, qecember 1960, pp. 649-677. ' , ' 16~i ~ennett, E. VJ.. nC! Balasooriya,: B. M. A., 'Shear st~ngth of prestressed beam~.with thin webs failing in inclined compression', Proceedings of the Amel 'an Concrete Institute, Vol. 68, No.3, March 1971, pp. 204-212. 166. Edwards, A. D., 'The structural behaviour of a prestressed box beam with thin webs under combined bending and shear' , Proceedings of the Institution of Civil Engineers, Part 2, Vol. 63, March 1977, pp. 123-135. 167. Leo~hardt, F., '~bm.inderung der Tragfahigkeit des Betons infolge stabforml~er, ~htwmkhg zur Druckrichtung angedrahte Einlagen', Included m: KDlttel, G. amd Kupfer, M. (Eds), Stahlbetonbau: Berichie aw Forschung and Praxis. Festschrift Rusch, Wilhelm Ernst & Sohn, Berlin, 1969, pp. 71-78. 168. American Concrete Institute, Building code requirements lor re- . iIIforced concrete lACI ~I8-77), 1977; p. 103. : 169."._H;a~~s, N. M., Crisswell, M. E.and Roll, F., 'Shear strength of slabs wlt~,t shear reinforcement', American Concrete Inslltute, Special PublIcation SP-42 , Vol. 2, 1974, pp. 677-720. 170: Clark, L. A. and West, R.,· 'The torsional stiffness of support diaphragms in beam and slab bridges' , Cement and Concrete Associa- tion, Technical Report 42.510, August 1975, pp. 20. 171. Nadal, A., Theory of Flow and Fracture of Solids, McGraw-HiI:, New York, 1950. . 172. American Concrete Institute Committee 438, 'Tentative recom- mendations for the design .of reinforced concrete members to resist tor- sion', Proceedings ofthe American Concrete Institute, Vol. 66, No. I, January 1969, pp. 1....8. 173. Cowan, H. J. and Armstrong, S., 'Experiments on the strength of reinforced and prestressed concrete beams and of concrete encased steel joists in combined bending and torsion', Magazine of Concrete Research, Vol. 7, No. 19, March 1955, pp. 3-20. 174. Brown, B. L., 'Strength of reinforced concrete T-beams under combined direct shear and torsion' , Proceedings of the American Con- crete Institute, Vol. 51, May 1955, pp. 889-902. 17S. Hsu, T. T. C., 'Torsion of structural concrete - plain concrete rectangular sections', Americ.an Concrete Institute Special Publication SP-18, 1968, pp. 203-238. 176. Mitchell, D. and Collins, M. P., 'Detailing for torsion', Proceed: ings of the American Concrete Institute, Vol. 7, September 1976, pp.506-5l1. 177. Swann, R. A., 'Experimental basis for a design method for rectangular reinforced concrete beams in torsion', Cement and Con- crete Association, Technical Report 42.452, December 1970, p. 38. 178. Swann, R. A., 'The effect of size on the torsional strength of rectangular reinforced concrete. beams', Cement and Concrete Associ- ation, Technical Report 42.453, March 1971, p. 8. 179. Swann, R. A. and Williams, t-., 'Combined loading tests on model prestressed concrete box beams with steel mesh', Cement and Concrete Association. Technical Report 42.485, October 1973, p. 43. 180. Lampert, P., 'Torsion and bending in reinforced and prestressed concrete members', Proceedings of the Institution of Civil Engineers, Vol. 50, December 1971, pp. 487-505. 181. Maisel, B. I. and Swann, R. A., 'The design of concrete box . spine-beam bridges', Construction Industry Research and Information Association. Report 52, November 1974, p. 52. 182. Beeby, A. W., 'The prediction and control of flexural cracking in reinforced concrete members', American Concrete Institute Special Publication 30 (Cracking. deflection and ultimate load ofconcrete slab systems), 1971, p. 55. 183. Clark, L. A. and Speirs, D. M., 'Tension stiffening in reinforced concrete beams and slabs under short-term load' , Cement and Concrete Association. Technical Report 41.52/, July 1978, p. 19. 184. Jofriet, J. C. and McNeice, G. M., 'Finite element analy~is of reinforced concrete slabs', Proceedings ~f the American Society of C)vil Engineers (Structural Division), Vol. 97, No. ST3, March 1971, pp. 785-806. 185. Hughes, B. P., 'Early thermal movement and cracking of con- crete', Concrete, Vol. 7, No.5, May· 1973, pp. 43,44. 186. Rao, P. S. and Subrahmallyan, B. V. 'Trisegmental moment- curvature relations for reinforced concrete members', Proceedings of I the Americqn Concrete Institute, Vol. 70, No.6, May 1973, I pp. 346-351. . 187 Clark,~. A. and Cranston, W. ·8. 'The influence of bar spacing on tension stiffening in reinforced concrete slabs', Proceedings of International Conference 01' Concrete Slabs, Dundee, 1979, pp; 118-128. 188. Stevens, R. F., 'Deflections of reinforced concrete beams', Pro- ceedings of the Institution of Civil Engineers, Part 2, Vol. 53, Sep- tember 1972, pp. 207-224. 179
- 98. 189. Beeby, A.W., 'Cracking and corrosion'. Concrete in the Oceans, Technical Report No. 1,1978, p. 77. 190. Base, G. D., Read, J. B., Beeby, A. W. and Taylor, H. P. J., •An investigation of the crack control characteristics ofvarious types of bar in reinforced concrete beams', Cement and Concrete Association, Research Report 18, December 1966; Part I, p. 44; Part 2, p. 31. 191. Clark, L, A.• 'Crack control in slab bridges', Cement and Con- crete ASSociation. Departmental Note 3009, October 1972, p. 24. 192... Clark, L. A., 'Flexural cracking in slab bridges', Cement and Concrete Association, Technical Report 42.479. May 1973, p. 12. 193. Clark. L. A.• 'Strength and serviceability considerations' in the arrangement of the reinforcement in concrete skew~lab bridges', Con- struction Industry Research and Information Association, .ReI!JIFI.Jr, October 1974, p. IS. ' . ,,,,~ 194. Clark, L. A. and Elliott, G.. 'Crack control in concrete bridges', The. Structural Engineer, Vol. 58A, No.5, May 1980, pp, 157-162. 195, Holnsberg, A., 'Crack width prediction and minimum reinforce- ment for crack. control', Soertlyk af Bygnillgsstatiske Meddelelser. VoI.44, No.2, 1973. 196. Beeby, A. W., 'An investigation of cracking in slabs spanning one way', Cement and Concrete 1ssociation, Technical. Report 42.433, April 1970, p. 31. . . 197. Walley, F. and Bate, S. C. C., A guide to the B.S, Code 0/Prac- tice/or prestressed concrete, Concrete Publications Ltd., 1961, p. 95. 198. Neville; A. M., Creep of Concrete: Plain, reinforced and pre- stressed, North-HolIand Publishing Co., 1970, p. 622. 199. Cooley, E. H., 'Friction in post-tensioned prestressing systems', Cement and Concrete Association, Research Report No.1, October 1953, p. 37. 200. Construction Industry Research and Information Association, Pre- stressed concrete - friction losses during stressing, Report 74, Febru- ary 1978, p. 52. .' 201. Longbottom, K. W. and Mallett, G. P., 'Prestressing steels', The Structural Engineer, Vol. 51, No. 12, December 1973, pp. 455-471. . 202. Creasy, L. R., 'Prestressed concrete cylindrical tanks', Proceed- , ings of the Institution of Civil Engineers, Vol. 9, January 1958, pp. 87-114. 203. Allen, A. H.• 'Reinforced concrete design to CP 110 - simply explained', Cement and Concrete Association. Publication 12.062, 1974, p. 227. 204. Parrott, L. J., 'Simplified methods of predicting the deformation of structural concrete', Cement alld COllcrete Association, Development Report 3, October 1979, p. 11. 205. Branson, D. E., Deformation of Concrete Structures, McGraw- Hill International Book Company, New York. 1977, p..546. 206. Hobbs, D. W., 'Shrinkage-induced curvature of reinforced con- crete members'. Cement and Concrete Association, Development Report 4,November 1979. p. 19. . 207. Somerville, G.• 'The behaviour and design of reinforced concrete corbels', Cement and Concrete Association. Technical Report 42.472, August 1972, p. 12. 208. Clarke, J. L.• 'Behaviour and design of small nibs', Cement and Concrete Association. Technical Report 42.512, March 1976, p.8. 209. Williams, A., 'The bearing capacity of concrete loaded over a limited area', Cement and Concrete Associatioll. Technical Report 42.526, August 1979, p. 70. 210. Base, G. D., 'Tests on four prototype reinforced concrete hinges', Cement and Concrete Association. Research Report 17, May 1965, p.28. 211. Department of Transport, 'Technical Memorandum (J3ridges) BE 5175. Rules for the design and use ofFreyssinet concrete hinges in highway structures', March 1975, p. 11. 212. Reynolds, G. C., 'The strength of half;joints in reinforced con- crete beams', Cemelll and Concrete Association. Techllical Report 42.415, June 1969, p. 9. 213. Federation .Internationale de la Precontrainte. 'Shear at the inter- face of precast and in situ concrete', Techllical Report FIP1914. August 1978. p. IS. 214. Pritchard, B. P... 'The use of continuous precast beam decks for the MI,I Woodford Interchange .viaducts', The Structural Engineer, Vol. 54; No.. 10, October 1976, pp. 377-382. 215. Kaar, P. H.• King. L. B. and Hogriestad, E., 'Precast-prestressed 180 concrete bridges. I. .Pilot tests of continuous girders', Journal o/tM Research and Development LaboratorieS 0/ the Port/ond Cement Association, VoL 2, No.2, May 1960, pp. 21-37. 216. Mattock, A. H. and Kaar, P.H., 'Precast-prestressed' concrete bridges. 3. Further tests of continuous girders', Jou~l 0/ the Research and Development Laboratories 0/ the Port/ond Cement . Association, Vol. 2, No.2, May 1960, pp. 3S-58.· . 217." Beckett, D., An Introduction to Structural Design (I) Concrete bridges; Surrey University Press, Henley-on-Thames, 1973, pp. 93- '96. 21S. Mattock, A. H. and Kau, P. H., 'Precast-prestressed concrete ..bridges. 4. Shear tests of continuous girders'" Journal 0/the Research iiillJDevelopment Laboratories 0/ the Portland Cement AlSociation, Vol. 3,No. 1, January 1961, pp. 19-46. 219. Sturrock, R. D., 'Tests 011 model bridge beams In precast to In . situ concrete con,s~ction', Cement and Concrete AlSociation, Techni- cal Report 42.488, January 1974, p. 17. c 220. Mattock, A. H., 'Precast-prestressed concrete bridges. 5. Creep and shrinkage studies', Journal 0/ ,!he Research and Development Laboratories of the Portland Cement Association, Vol. 3, No.3, Sepo , tember 1961, pp. 30-70. 221. Building Research Establishment, BritIge Foundations and Sub- structures, HMSO, 1979. 222. Lee, D. J., 'The theory and practice of bearings and expansion joints for bridges', Cement and Concrete Association, Publication 12.036, 1971, p. 65. 223. Cranston, W. B., 'Analysis and design of reinforced concrete col- umns', Cement and Concrete Association, Research Report 41.020, 1972, p. 54. 224. MarshalI, W. T., 'A survey of the problem of lateral instability In reinforced concrete beams', Proceedings 0/ the Institution of Civil Engineers, Vol. 43, July 1969, pp. 397-406. . 225. Timoshenko, S. P., Theory of Elastic Stability, McGraw-Hili Book Co., New York and London, 1936, p. SIS. 226. CoiniteS Europeen du Beton, 'International recommendations for the design and construction of concrete structures', 1970, p. SO. 227. Larsson, L. E., 'Bearing capacity of plain and reinforced concrete walls', Doctoral Thesis, Chalmers Technical University, Ooteborg, Sweden, 1959, p. 248.. 228. British Standards Institution, Structural recommendations for loadbearing walls. CP 110: Part 2: 197()', p. 40. 229. Seddon, A. E., 'The strength of concrete walIs under axial and eccentric loads', Proceedings ofa Symposium on. The Strength o/Con- crete Structures, London, May ~956, pp. 445..,..473. 230. British Standards' Institution, Design and construction of re- inforcedand prestressed concrete structures for the storage of water and other aqueous liquids. CP 2007: Part 2: 1970, p. SO. 23!. British Standards Institution, The structural use 0/ concrete for retaining aqueous liquids. BS 5337: 1976, p. 16. 232. Lindsell, P., 'Model analysis of a bridge abutment', The Struc- tural Engilleer, Vol. 57A. No.6, June 1979, pp. 183-19t'. 233. Whittle. R. T. and Beattie. D., 'Standard pile caps', Concrete, Vol. 6, No. I, January 1972, pp. 34-36 and No.2, February 1972, pp.29-31.. 234. Clarke, J. L., 'Behaviour and design of pile caps with four piles', Cement and Concrete Association. Technical Report· 42.489, November 1973, p. 19. 235. Yan, H. T.• 'Bloom base allowable in the design of pile caps', Civil Engineering and Public Works Review, Vol. 49, No. 575, May 1954, pp. 493-495 and No. 576, June 1954, pp. 622-623. 236. American Concrete Institute Committee lOS, 'Reinforced concrete column investigation - tentative final report of Committee 105', Pro- ceedings of the American COllcretelnstitute. Vol. 29, No~5, February 1933, pp. 275-282. 237. Snowdon. L. C., 'Classifying reinforcing bars for bond strength', Buildillg Research Station. Current Paper 36170. November 1970. 238. Ferguson, P. M. and Breen. J. E.• 'Lapped splices for high- strength reinforcing bars'. Proceedings ofthe American Concrete Insti- tute,Vol. 62. September 1965, pp. 1063-1078. 239. Nielsen. M. P. and Braestrup, M. W. 'Plastic shear strength of reinforced concrete beams', Bygningsstatiske Meddelelser, Vol. 46, No.3, 1975, pp. 61-99. 240. Fergulson, P. M. and MatJoob, F. N., 'Effect of bar cut-off on bond and shear strength of reinforced concrete beams', Proceedings 0/ the American Concrete Institute, Vol. 56, No. I, July 1959, pp: 5- 24. 241. Green, j, K., 'Detailing for standard prestressed concrete bridge beams', Cement and Concrete Association, Publication 48.018, December 1973, p. 21. 242. Base, G. D., 'An investigation of transmission length in pre- tensioned concrete', Cement and Concrete Association, Research ,~, Report 5, August.l95S, p. 29. . . 243. Base, G. D., 'An investigation of the use of stnne! in pre- tensioned prestressed concrete beams', Cement and Concrete~Associ- ation; Research Report 11, January 1961, p. 12. ' 244. Zielinski, J. and Rowe, R. E., 'An investigation of the stress dis- tribution in the anchorage zones of post-tensioned concrete members' , Cement and Concrtte Association, Research Report 9, September 1960, p. 32. 245. Zielinsld, I. and Rowe, R. E., 'The stress distribution associated with groups of anchonges In post-tensioned concrete members', Cement and Concrete Association, Res.earch Report 13, October 1962, p.39. 246. Clarke, J. L., 'A guide to the design of anchor blocks for post- tensioned prestressed concrete members', Construction Industry Re.fearch and Information AssOciation, Guide I, June 1976, p. 34. 247. Prench Oovemment, 'Conceptlonet calcul du beton precontraint', Publication 73-64 BIS. 24S. Lenschow, R. I. and Sozen, M. A., 'Practical analysis of the anchorage zone problem in prestressed beams', Proceeding, 0/ the American Concrete Institute, Vol. 62, No. 11, November 1965, pp. 1421-1439. 249. Howells, H. and Raithby, K. D., 'Static and repeated loading tests on lightweight prestressed concrete bridge beams', Transport and Road Research Laboratory, Report 804, 1977, p. 9. 250. Kerensky, O. A., Robinson, J. and Smith B. L., 'The dellgn and construction of Friarton Bridge', The Structural Enginetr, Vol. 5SA, No. 12, December 1980, pp. 395-404. 251. Department of Transport, 'Lightweight aggregate concrete for use in highway structures', Technical Memorandum (JIridges) No. BE II, April 1969, p. 4. 252. Grimer, P. J., 'The durability of steel embedded In lightweight concrete', Building Research Station, Current Paper (Eng/ntering Series) 49, 1967, p. 17. 253. Swamy, R. N., Sittampalm, K., Theodonkopoulos, D., Ajibade, A. O. and Winata, R., 'Use of lightweight aggregate concrete for structunl applications', in Advances in Concrete Slab' Technology, Dhir, R. K. and. Munday, J. G. L. (Eels), Pergamon Press,1979, pp.4O-4S. 254. Teychenne, D. C., 'Structunl concrete made with lightweight aggregate', Concrete, Vol. I, No.4, April 1967, pp. 111-122. 255. Hanson, J. A., 'Tensile strength and diagonal tension resistance of structural lightweight concrete', Proceedings o/the American Concrete Institute, Vol. 5S, No. I, July 1961, pp. 1-40. 256. lvey, D. L. and Buth, E., 'Shear capacity of lightweight concrete be!lms', Proceedings of the American Concrete Institute, Vol. 64, No. 10, October 1967, pp. 634-643. 257. Shideler, J. J., 'Lightweight-aggregate concrete for structural use', Proceedings ofthe American Concrete Institute, Vol. 54, No.4, October 1957, pp. 299-328. 25S. Short, A. and Kinniburgh, W., Ughtweight Concrete, Applied Science Publishers (third ed), 1975, p. 464. 259. Swamy, R. N., 'Prestressed lightweight concrete', in Develop- ments in Prestressed Concrete - I; Sawko, F. (Ed.), Applied Science Publishers, 1975, pp. 149-191. 260. Evans, R. H. and Paterson, W. S., 'Long-term deformation characteristics of Lytag lightweight-aggregate concrete', The Str!ICtural Engineer, Vol. 45, No. I, January 1967, pp. 13-21. 261. Wills, J., 'A simplified method to calculate dynamic behaviour of footbridges' '.Transport and Road Research Laboratory. Supplemen- tary Report 432, 1975, p. 10. 262. Gorman~ D. J., Free Vibration of Beams and Shafts, lohn Wiley and Sons, 1975, p. 386. . . 263. Wills, J., 'Correlation of calculated IIId measured dyn~ic be6ivio.ur of bridges', Proceedings oj 'aSympoStUift on dynamic behaviour of-bridges. Transport and Road Research Laboratory. SUp- plementary Report 275, 1977, pp. 70-89. 264. Tilly, G. P., 'Damping of highway bridges: a review', Proceed- ings 0/a Symposium on dynamic behaviour 0/bridges, Transport and Rood Research Laboratory, Supplementary Report 275, 1977, pp. 1-9. 265. American Concrete Institute Committee 215·, 'Consldentions for design of concrete structures subjected to fatigue loading', Proceedings 0/ tile American Concrete. lnstitute, Vol. 71, No.3, March 1974, pp.97-121. 266. Bennett, E. W., 'Fatigue in concrete', Concrete, Vol. S, No.5, May 1974, pp. 43-45. • 267. Price, K. M. and Edwards, A. D., 'Fatigue strength of prestressed concrete flexural ..members', Proceedings o/Jhe Institution of Civil Engineers, Vol. 47, October 1970, pp. 205-226. 268. Edwards, A. D. and Picard, A., 'Fatigue characteristics of pre- stressing strand', Proceedings of the InSlitution 0/ Civil Engineers, Vol. 53, Part 2, September 1972, pp. 323-336. 269. Moss, p. S., 'Axial fatigue of high-yield reinforcing bars in air' , Transport and Road Research Laboratory, Supplementary Report 622, 19S0, p. 29. 270. Burton, K. T. and Hognestad, E., 'Fatigue tests of reinforcing bars - tack welding of stirrups', Proceedings ofthe American Concrete Institute, Vol. 64, No.5, May 1967, pp. 244-252. 271. Miner, M. A., 'Cumulative damage in fatigue', Transactions 0/ the American Society of Mechanical Engineers, Vol. 67, Series E. September 1945, pp. AI59-AI64. 272.. Zederbaum, J., 'Factors influencing the longitudinal movement of a concrete bridge system with special reference to deck contraction', American Concrete Institute Special Publication 23 (First International Symposium on Concrete Bridge Design),-1967, pp. 75-95. 273. Church, J. G. 'The effects of diurnal thermal loading on cracked, concrete bridges', MSc Thesis, University of Birmingham, 19SI. 274. Hambly, E. C., 'Temperature distributions and stresses in con- crete bridges', The Structural Engineer, Vol. 56A, No.5, May 1978, pp. 143-14S. 275. Clark, L. A., 'Discussion of reference 274', The Structural Engineer, Vol. 56A, No.9, September 1975, p. 244. 276. Hambly, E. C., 'Reply to reference 275', The Structural Engineer, Vol. 57A, No. I, January 1979, pp. 28-29. 277. Aster, H., 'The analysis of rectangular hollow reinforCed concrete slabs supported on four sides', PhD Thesis, Technological University of Stuttgart, April 1968, p. 111. 278. Elliott, G. E., 'Designing for transverSe shear in voided slabs', Cement and Concrete Association, unpublished, p. 4. 279. Badoux, J. C. and Hu1sbos. C. L., 'Horiwntal shear connection in composite concrete beams under repeated loads', Proceedings o/the American Concrete Institute, Vol. 64, No. 12. December 1967, pp. 811-819. 280. The Concrete Society. 'A review of the international use of light- weight concrete in highway bridges', Concrete Society Technical Report No. 20. August 1981, p. IS. 181
- 99. Index abutment, 129-30, 133-S analysis,9-31 elastic, S, 9, 10, 13-16 finite element, IS finite strip, 1S folded plate, IS grillage, 13, IS-16 limit, 19-20 local effect, 10 lower bound, 10, 19, 20-3 model,27 non-linear, 9 plastic, 10, 19-27,27-31 plate theory, 13- IS serviceability limit state, 9-10 ultimate limit state, 10 upper bound, 19, 20, 22-7 beam cracking, 90 flexure, SS-S shear, 65-70, 72-S torsion, 76-S3 beam and slab, 13, 17, 19,26 bearings, 36 bearing stress, 104, 142 bond, 139-42 anchorage, 140 bundled bars, 140 local, 139-40 box beam, 13, SO-I, SS box girder, 13, 20, 26-7 braking load. 32, 41 carriageway, 34 cellular slab, 12-13, 17, 19,70,171 central reserve loading, 41 centrifugal load, 32, 41 characteristic load, 4 characteristic strength, 4-5, 4S-6 concrete, 4S prestressing tendon, 46 .reinforcement, 45 coefficient of friction bearing, 36 duct, 96 skidding, 41 coefficient of thennal expansion, 36, 4S, 149 collision load parapet, 32, 41 support, 32, 42 column, 11S-2S additional moment, 122-3 axial load, 119 biaxial bending, 121-2, 124 bridge, 129 cracking, 94, 124 effective height, liS, 129 reinforcement, 13S, 139 short, 119-22 slender, 122-4, 133 slenderness, 119, 129 stresses, 124 uniaxial bending 119-21, 123--4 composite construction, S4, 10S-17 beam and slab, 106-7 continuity, 110-1S differential shrinkage, 109-10, III flexure, lOS interface shear, SO-I, 10S-9, 117 slab, 11, 12-13,27-9, SI, 107-S, 110 stresses, 49- 51 vertical shear, 106-S, 116-17 concrete characteristic strength, 4S elastic modulus, 9, 4S, 150 fatigue, 154-S lightweight, 147-S0 stress limitations, 49-51, 10S-9 stress-strain, 4S, 49 . consequence factor, 7 corbel, 102-3 cracking, S8-94 base, 94 beam, 90 column, 94, 124 II
- 100. design crack width,51-2 early thennal, 88 flange, 92-4, 98-9 footing, 131 prestressed conc~te, 51 reinforced concrete, 88-94, 98-9 slab, 48, 91, 92-3 torsional, 76, 77, 78 wall, 94, 125, 128-9 < voided slab, 91-2 creep column, 122 composite construction, 110, 111-15 data, 45 deflection, 97 lightweight concrete, 150 load, 32, 35 loss, 95-6 curtailment, 141-2 cycle track loading, 32, 42 damping, 153 dead load, 32, 34-5 deflection, 52, 96-8 design criteria, 4S-52 design life, 4 design load, 5 design resistance, 7 design strength, 6 detailing, 137-46 detenninistic design, 2, 3 differential settlement, 32, 35 dispersal of load, 40 durability, 52, 147-8 dynamic loading, 42, 151-7 early thennal movement, 88 earthquake, 38 effective flange width, 9-10, 57 effective wheel pressure, 40 elastic modulus concrekl, 9, 45, 150 steel,9 end-block, 143-6 erection, 32, 35, 36,38 exceptional load, 32, 38 fatigue, 4, 42, 52, 154-7 concrete, 47, 154-5 prestressing tendon, 155 reinforcement, 156-7 flexural shear, 65-70, 72-5, 83-4, 130-1 finite element, 15 finite strip, 15 folded plate, 15 footing, 130-1 footway loading, 32, 42 foundation, 34, 130-3 gap factor, 6 global load factor, 6 184 grillage analysis, 15-16, 18-19 HA loading, 38, 39-40 halving joint, 104-5 HB loading, 38, 40-1 highway loading, 38-42 application, 40-1 brlJking, 32, 41 ',~centrifugal, 32, 41 collision, 32, 41-2 , HA, 38, 39-40 HB, 38, 40-1 primary, 38-41 secondary, 41-2 skidding, 32, 41 verge, 41 Hillerborg strip method, 10, 20, 130, 133-5 hypothetical flexural tensile stress, 51, 94-5 I·beam, 13 ice, 38 impact, 38, 42 implementation of Code, 2 influence line, 6, 34 influence surface, 16 interface' shear, SO-I, 10S-9, 117 inverted T-beam, 51 lap length, 141 lightweight aggregate concrete, 147-50 limit analysis, 19-20 limit state serviceability, 4, 9, 48-52, 86-101 ultimate, 4, 10, 48 limit state design, I, 3- 4 load,32-44 application, 34 braking, 32, 41 centrifugal, 32, 41 collision, 32, 41-2 combination, 32-3 dead, 32, 34-5 differential settlel1lent,32~ 35 dynamic, 42, 151-7 effect, 5-6 erection, 32, 35, 36, 38 exceptional, 32, 38 fatigue, 4, 42, 52, 154-7 fill, 32, 35 footway and cycle track, 32, 42 friction at bearing, 33 HA, 38, 39-40 HB, 38, 40-1 highway, 32, 38-42 lurching, 32 nosing, 32 pennanent, 32, 34-5 railway, 32, 42 skidding, 32, 41 superimposed dead, 32, 35 temperature, 32, 36-8, 158-68 traction, 32 transient, 32, 35-42 wind, 32, 35-6 loaded length, 38-9 load factor design, 3 -----_..._.. .-_..----.' local effects, 10, 15, 26, 31,93,98-9 losses; 95-6, 150 lower bound method, 10, 19, 20-2 M-beam, 12 membrane action, 26 model analysis, 27 modular ratio, 86, 89 moment redistribution, 10, 20-2, 53, 110 natural frequency, 151-2 nib, 103 nominal load, 4 .. notional lane, 34 partial safety factor load,S, 33-4 load effect,S, 52-"3 material, 6-7, 46-8 pier, 129 pile, 131 pile cap, 131-3, 135-6 plate bending, 59-60, 169-70 in-plane forces, 60-1, .17() orthotropic, 13-14 shear defonnable, 14-15 theory, 13-15 .' Poisson's ratio, 9, 15-16, 18/45 precast concrete, 102-5 prestressed concrete beam, 57-8,.61-3, 94-5 cracking, 51,~ design criteria, 48-51 end~block, 143-6 fatigue, 155 J losses, 35, 95-6, 150 serviceability; limit state, 94~5. 99-101" shear, 72-5, 83-4 slab, 61, 95 : torsion, 81-3, 85 prestressing tendon characteristic strength, 46 cover, 142-3 elastic modulus, 9 fatigue, 155 initial stress, 95 spacing, 143 stress limitation, 48 stress~strain, 46 transmission length, 143 railway loading, 32, 42 rectangular stress block, 55 reinforcement anchorage, 140, 142 bond, 139-42 characteristic strength, 45 cover, 137, 148 curtailment, 141-2 elastic modulus, 9 fatigue, 156-7 maximum, 139 minimum, 138-9 ·"~ing, 137-8 stress limitation, 48, 86-8, 98-9 stress-strain, 46 retaining wall, 94 segmental construction, 82-3 serviceability limit state, 4, 86-101 analysis, 9-10 base, 94 beam, 90, 94-5 column, 94, 128-9 composite construction, 108-9, 111 cracking, 88-94 deflection, 96-8 design criteria, 48-52, 53 flange, 92-4 footing, 131 pile cap, 133 slab, 87-8, 91-2, 95 stress limitation, 86-8, 94-5 temperature effects, 158-62, 164-7 Wall, 94, 125, 128-9 shear, 65-75 at points of contraflexure, 68-9 beam, 65-70, 72-5 cellular slab, 70, 17i composite construction, 106..!9, 115-17 defonnation, 12, 14-15, 16, 171 flexural, 65-70, 72-5, 83-4,130-1 interface, 50-1, 108-9, 117 lag, 9-10 modulus, 16 prestressed concrete, 72-5, 83-4 punching, 70-2, 75, 83, 131, 132 reinforced concrete, 65-72,83 reinforcement, 66-8, 69-70, 74-5 short members, 68 slab, 69-72, 75 transverse, 70, 171-5 voided slab, 70, 72, 171-5 shrinkage curvature, 97 data, 45 differential, 109-10, 111 lightweight concrete, 150 load, 32, 35 loss, 95 skew slabI prestressed, 61 reinforced, 63 yield line theory, 23, 24, 29-30 skidding load, 32, 41 slab IS5
- 101. bending, 59-60, 169-70 bridge,11 cellular, 12-13, 17, 19,70, 171 composite, 11, 12-13,27-9,51, 107-8, 110 cracking, 48, 91, 92-3 Hillerborg strip method, 20 in~planeforces, 6O~1, 170 membrane action, 26 prestressed, 61, 95 shear, 69-72, 75 skew, 23, 24, 29-30, 61, 63 stiffness, 16, 18-19 stresses,' 87~8 voided, 11-12, 16-17, 19,70,72,91-2, 171-5 yield line theory; 22-6, 27-31, 130 snow load, 38 stiffness, 9, 10 axial,9 . flexural, 9, 14, 16-19 for grillage analysis, 18-19 for plate analysis, 16-18 of beam and slab, 17, 19 of cellular slab, 17, 19 of discrete boxes, 17, 19 of slab, 16, 18-19 of voided slab, 16-17, 19 shear, 9, 12, 14, 16-19 torsional, 9, 10, 13, 14, 16-19 stress limitation, 48-51 concrete in compression, 49, 86:"'8, 94-S, 98-101, 108 concrete in tension, 49-50, 51, 94-5, 98-101, 108 interface shear, 50-1, 108-9, 117 prestressing tendon, 48 reinforCement, 48, 86-8, 98-9 stress-strain concrete, 45, 49 design, 48 prestressing tendon, 46 reinforcement, 46 superimposed dead load,.32, 35 temperature, 32, 36-8, 158-68 combination of range and difference, 38 difference, 36-8, 158-68 frictional bearing restraint, 36 range, 36 seJ:Viceability limit state, 158.,..62, 164-7 ultimate limit state, 162-3, 163-4, 167-8 . tension stiffening, 86, 89-90, 97 . beam, 90 .flange, 92-3,94 top hat beam, 13 torsion, 75-.83 186 box section, 80-1, 85 compatibility, 76 cracking, 76, 77, 78 equilibrium, 75-6 flanged beam; 79-80 prestf!;lssed concrete, 81-3, 85 ./ rectangular section, 76-9, 84-5 reinforced concrete, 76-81, 84...85 reinforcement, 77~8, 79,80-1 segmental construction, 82-3 traffic lane, 34 U-beam,13 ultimate limit state, 4 analysis, 10 beam.55-8,65~9,7S-83 column, 119-24 .... ' composite construction, 105'-8 design criteria, 48 fleXUre, 5'-60, 61"':3 footing, 130-1 in-plane forces, 60-1, 63-4 pile cap, 131..,.3 plate, 58-61 shear, 65-15, 171-5 slab, 58-61, 69-75 . temperature effects, 162-3, 163-4, 167-8 torsion, 75-83 . upper bound method, 19,20, 22-7 verge loading, 41 verification, 7 vibration, 4, 42, 48, 52, IS1-4 voided slab, 11-12 cracking, 91-2 shear, 70, 72, 171-S stiffness, 16-17, 19 wall, 125-9 cracking, 94, 125, 128-9 effective height, 126 plain, 126-9 reinforced, 125 retaining, 94 shear, 128 t. short, 125, 126-7 slender, 125, 127-8 slenderness, 125, 126 stresses, 125 wing, 94, 129-30 welding, 156 wheel load dispersal, 40 effective pressure, 40 HA, 38,40 HB,40 wind load, 32, 35-6 wing wall, 94, 129-30 working lane, 34 working stress design, 2-3 yield line theory, 10,22-6, 53 abutment, 130 composite slab, 27-9 slab bridge, 23-6, 27-30 top slab, 26, 31 wing wall, 130 • ~ ...•..-.----- ..-...•~-.-~--..._.'. . .._---_......_._...._..__._------_....•........_..._......._...._--------- Chapter 1 Introduction Code format (p.1) All ten parts of the Code have now been published. However. Part 9 on bearings has been published in two sections: 9.1 Code of Practice for design of bridge bearings; and 9.2 Specification for materials, manufac- ture and installation of bridge bearings. Since Purt 9 has now been published. Appendix F of Part 2. which covered bearings. has been deleted. Highway bridges (p.2) The Department of Transport (DTp) has published an implementation document for each Part of the Code except Parts 7 and 8, for which the DTp requires its own specification document PO) to be used. However. at the time of writing, the latter document is being revised. The relevant DTp implementation documents are: BD 15/82. General principles for the design and con- struction of bridges. Use of BS 5400 : Part 1 : 1978 [281]. BD 14/82. Loads for highway bridges. Use of BS 5400 : Part 2 : 1978. (Including amend- ment No.1) [282]. ' BD 24/84. Design of concrete bridges. Use of BS 5400 : Part 4 : 1984 [2H3J. BD 20/83. Bridge. bearings. Use of BS 540() : Part 9 : 1983 (284). BD 9/81. Implementation of BS )400 : Part to : 1980. Code of Practice for fatigue [285]. BA 9/81. Usc of BS 5400 : Part 10: 19XO: Code of Practice for fatigue. (Induding Amendment No.1) (286). Characteristic strengths (p.4) In addition to characteristic strengths, characteristic stresses have now been introduced. These are defined as the stresses at which material stress-strain curves become non-linear. Design load effects (p.5) Values of Yj3 in Part 4 are no longer dependent on the type of loading. 1
- 102. Chapter 2 Analysis General requirements(p.S) Part 4 now states that the ef(ects of shear lag on structural ari~lysis may be neglected. except for cable- stayed superstructures. at both the serviceability and ultimate limit states. Serviceability limit state (p.S) Axial. torsional and shearing stiffnesses may now be based on the concrete section ignoring the presence of reinforcement (as before). or the gross section includ- ing the reinforcement on a m~dular ratio basis. The tabulated short'-term elastic modulus of concrete shOLJld be used for analysing a structure under applied forces. However: when analysing a structure under applied deformations. a modulus intermediate between the tabulated value and half that value should now be used. The; value adopted should reflect the proportions of permanent and short-term deformations. Ultimate limit state (p.10) In BO JS/S2. the DTp seems to imply that plastic methods of analysis arc those based only on considera- tions of collapse mechanisms. In addition, Part 4 permits the usc of plastic methods only with the agreement of the relevant bridge authority. The stiffnesses used in an elastic analysis at the ultimate limit state should be based on the same section properties as those used at the serviceability li'mit state. The load effects due to restraint of tors,ional and distortional warping may be neglected for longitudinal memhers. Local effects (p.1 0) The statement that the worst loading case occurs in the regions of sagging moment under load combination 1 2 has been removed..It is now necessary to determine the most adverse combination of global and local effects. Hillerborg strip method (p.20) This method is no longer referred to in the Code. Moment redistribution (p.20) The clauses on moment redistribution in Part 4 : 1984 are very much different to those in Part 4 : 1978. The following four conditions now have to be met. 1. Adequate rotation capacity has to be verified at sections where moments are reduced either by reference to test data or by calculating the rotation capacity as the lesser of: . 0.008 + 0.035 (0.5 - dcld,,) (2.68a) or, for reinforced concrete 0.6 t1> /(d - de) ' ( 2 . 6 8 b ) or. for prestressed concrete to/(d - de) (2.68c) but {O, and *0.015 where de' = calculated depth of concrete in com- pression at the ultimate limit state d,. =effective depth of a rectangular section. or the overall depth of a compression flange d ~ effective depth to tension reinforcement <1> = diameter of smallest tensile reinforcing bar. It should be noted that rota,tion capacity can be exhausted either due to crushing of concrete or due to fracture of reinforcemeht or prestressihg steel. Expression (2.68a) is' a linear approximat,ion to the relationship given in the CEB Model Code (1 lO). which was based. on a review of t~I',t J~at(,l carried out by Siviero [287]. The expression results in a conservative estimate of the rotation which can occur prior to crushing of the concrete in the compression zone. Expressions (2.68b) and (2.68c) generally give, conservative estimates of the rotation which can~ develop prior to fracture of reinforcings~c:cJor " , i ...........~.-....~.......;.........-..........~'-...'.. ;1/0111/1',.11'1'. ':"J"'.'i prestressing steel respectively. It is understood that ml == 822 kN m/m ' the expre$sions were derived from considerations y .. 1.251 m of the ultimate elongations and gauge lengths spe- "'2 =35.7 kN mlm cified in the British Standards for the various types It should also be noted that in accordance with of reinforcing and prestressing steels, and from BOll4/82 the OTp would require full HA to be applied data provided by manufacturers of prestressing to the third lane instead of the HAl3 loading required s t e e l s . , . .. ' " .~, , ", by Part 2 of the Code (see Fig. 2.28). In this case it is 2. The effects of th"~ r"distributlonof't()flgitudinal '!oundJhat: moments on the transverse Inress resultants should"" ......"",. , be assessed by means of It non-linear analysis. The ml .. 884 kN'~/m author would suggest that a linear analysis could y II1II 0.861 m ' also be used all indicated on page 22. h12:= 18.9 kN m/m 3. The greater of the shearllund reactions before and The required"transvers(;!- momcnt of resistance has after redistribution should be used for design. as therefore decreas'ed substantially because the change in suggested on page 22. loading affects me~anilim (b) but not (c). and the 4. The overall depth of the member should not ex- longitudinal moment of resistance has increased. This ceed 1.2 m. A depth limit has been introduced illustrates the extreme sensitivity of yield line d~sign to because the available t"ltt data (287) on rotation changes in loading. , '" capacity do not cover dtH'!p memberA, and thore is The unltll for the parapet loading in Table 2.2 should evidence to sugcst Chill rotation tmpaeity reduces be correctad to kN/m. with an Inerea!!e In depih. 2.2 Vleld line design of skew slab bridge (p.29) Part '* h6W requlr@s Vtt bo Hike" ali 1.U for all imlds when ulilng Ii plil§tie method of linalyslli at th@ ultimate! limit litat@, Heneo; eDeh f:fltfy 1ft the !UHiOi'lti ~ofuffin of Tllhl~ 2,2 lihould ftbW be t, IS, und tht! Idiliiln' lOads in tht! fiHUll:blumH lil'iould.b6 multiplied by (1, 1~/t.l) fot HAl HI lind footway loadings, Ali Ii r"!lull of this Inef@ii!l&! in Ibattlngl It ill found thai: • ". 2.3 Vield line dellgn of top Ilab (p,31) As for E:xample 2.2,'tht% value of Yp.. upplled to the HA wh@@lload shoUld floW b@ L15 ifiliiOlld gf LL Htifice, tlu~ d"lIign load Is HOW i7~,5 kHI As a ft!sult of this hu~r@a§e in loading, it is found that: y == 0.231 m (m, "f hi;!) == 14.4 kN m/Ift , •
- 103. Chapter 3 . Loadings General(p.32) ~mcndment No.1 to Part 2 of the Code was published in March 1983. The amendment corrected and revised. the text. but the technical content· was un~ltered: Hence. Ch~pter 3 of the m,ain text still' refle'cts .the current Purt 2. However. the DTp has issued its own umendments to Part 2'in the form of BD 14/82 [282]. In the following sections, these amendments are discus- sed. It is emphasised that the amendments are those of . the DTp and not BSt Nevertheless, Part 2 is undqr revision at the time of writing, and the DTp's amend- ments will probably be incorporated in the revision. Partial safety factors (p.33) The DTp has amended some values of Y/I. in Table 3.1. The changes are discussed later. Superimposed dead load (p.35) The partial safety factors of ].75 and 1.2 given in Table 3.1 for superimposed dead load are applied only to the deck surfacing load. Smaller partial safety factors of 1.2 and ].0 at ultimate and serviceability limit states re- spectively are applied to other superimposed dead loads. Differential settlement (p.35) The nominal. value of differential settlement assumed should have a 95% probability of not being exceeded (juring the design life of the structure. The partial safety factors arc now specified as1.2 and].O at ultimate and serviceability limit statesr~s~ectively.. 4 Temperature'range (p.36) Coefficients of thermal expansion are given for con- cretes made with various natural aggregates. The values. which allow for the presence of reinforcement" . vary from 9'>< lO-·lIrC for limestone to 13.5 x lO-I,/oC for chert. Erection loads (p.38) A partial safety factor of 1.0 is specified at the ser- viceabilitylimit state. HA loading (p.38) Type HA loading is uncJer review at the time of writing, and its value has to be agreed with the DTp for loaded lengths in excess of 40 rri. HB loading (p.40) The number of units of HB loading is specified as: 45 for motorways and trunk roads; 37.5 for principal roads; 30 for other public roads; and 25 for accom- modation roads and byways. Applica{;on (p.40) HA loading The full uniformly distributed and knife- edge loads arc applied to two notional lanes and O.h times these loads to all other notional lanes. This loading is more severe than the one-third HA in the' other lanes specified in the Code. /IB loading The HB vehicle can occupy any notional lane or can straddle any two notional lanes. Hence. for a bridge with four notional lanes. it is necessary to consider HB loading in one lane. full HA loading in two other lanes. and 0.6 HA loading in the remaining lane. Hence, this loading can be much more severe than that specified in Part 2 of the Code. :! '"' , ~ Collision with parapets (p,41) The moment and shear transmilled to the member supporting the parapet should be taken as the moment and shear resistance of the parapet at the ultimate limit state calculated in accordance with the Code. The partial safety factors have been increased from 1.25 and 1.0 to 1.5 and 1.2 at the ultimate and serviceability limit states respectively. Collision with supports (p.42) The normal and parallel components of the loads now have to be applied concurrently. as was the case in BE 1/77. In addition the partial safety factor at the ultimate limit state h:ts been increased from. 1.25 to 1.5. It is no longer necessary to consider the. serviceability limit state. These change~ imply that the loading is now very similar to the BE )n7 loading. For superstructures with a headroom clearance of less than 5.7 m above a carriageway. a single nominal collision load of 50 kN, in any direction between horizontal and vertical. has to be applied to the super- structure soffit. 5
- 104. Chapter 4 Material propertiesand design criteria Concrete (p.45) Characteristic strengths (p.45) The lowest grade that can be used for reinforced concrete has been increased from 20 to 25. Stress-strain curve (p.45) The strain at which the parabola joins the horizontal line in Fig. 4.1(b) is how defined as 2.44 x 10-4 VTcu. Other properties (p.45) The characteristic stresses in compression and tension are 0.5 feu and 0.56 V'1:u respectively. Reinforcement (p.45) Characteristic strengths (p.45) In accordance with the revisions to the British Stan- dards for reinforcement, the quoted characteristic strength of both hot rolled and cold worked reinforcing bars is 460 N/mm2. Stress-strain curve (p.46) The characteristic stress in tension or compression is 0.75 fyo The characteristic stress is intended to be the stress at which the stress-strain curve becomes non- linear. Figure 4.2 indicates that this stress should be 0.8/y. This anomaly has probably arisen because, it is understood, the value of 0.75 /y was obtained by applying the BE In3 overstress factors to the BE In3 allowable tensile stresses. The first kink in the stress-strain curve of Fig. 4.4(b) should be labelled 0.7 fy. Characteristic strengths (p.46) A table is given for the characteristic strength of compacted strand for which a British Standard is not available. For other types of prestressing steel, the relevant British Standards are quoted. 6 Values (p.46) The values of the partial safety factors at the ultimate limit state are still 1.5 and 1.15 for concrete and steel respectively. However, new values which are applied to the characteristic stresses have been introduced at the serviceability limit state. The value for reinforcement is 1.0, and the values for concrete are given in Table 4.1A. The higher values arise for prestressed concrete and for uniform stress distributions because more crf;ep occurs under these conditions. The reason for the larger value in tension for post-tensioned, as opposed to pre-tensioned, construction is explained on page·51. Table 4.1A Ym values for concrete at serviceability Type ofstress Triangular compression Uniform compression Tension • Pre-tensioned + Post-tensioned Type ofconstruction Reinfon:ed Prestressed 1.00 1.33 1.25 1.67 1.25* 1.55+ Reinforcement (p.48) The limiting stress is equal to the characteristic stress of 0.75 fy because the partial. safety factor is 1.0. Due to other changes, the complica- tions referred to in the main text no longer exist. Prestressing steel (p.48) No reference is now made to a stress limitation. Compressive stresses in reinforced concrete (p.49) When the partial safety factors in Table 4.1A are applied to the characteristic stress of 0.5 fe", limit- ing stresses of 0.5 feu and 0.38 feu are obtained for triangular and uniform stress distributions respectively. Compressive stresses in prestressed concrete (p.49) When the partial safety factors in Table 4.1A are applied to the characteristic stress of 0.5 feu, Iimit- ing stresses of 0.4 j~u and 0.3 fnl arc obtained for triangular and uniformstress distributions respectively. - These stresses are greater than those in Table 43(a). However, as explained later. the loads under which the stresses are checked are also greater. In addition it should be. noted that a higher stress at a support is no longer permitted; The transfer stresses are the same as those in Table 4.3(b), except that upper limits of 0.4 fe/l and 0.3 fm are imposed for triangular and utfiT'orm stress distributions respectively. Thi,si it to prevent transfer stresses in excess of the limiti'll'gservice stresses occurring, when a member is post-tensioned at such an age that its concrete strength exceeds about three- quarters ~f the characteristic strength. Compressive stresses in composite construction (p.49) The permitted increase in limiting compressive stress is reduced from 50% to 25%, as was the case in BE 2173. .. Tensile stresses in composite construction (p.49) The values given in Table 4.4 are only applicable when tension is induced by sagging moments due to imposed service loading. Interface shear in composite construction (p.50) The relevant clauses have changed almost entirely. In par- ticular, shear calculations are now carrid out at the ultimate limit state, which is far more logical than the 1978 procedure. Two types of surface are now defined: 1. Roughened by wet brushing or subsequent tooling. 2. Laitance removed by jetting with air and/or water. 'Rough as cast' surfaces are classified as type 2. The design approach is now similar to the CP 117 approach, which was adapted for BE 2/73, and isoow compatible with that in BS 5400 : Part 5 : 1979 Jor steel/concrete composite construction. The new Part 4 method is based on the shear friction approach of Mattock and Hawkins [288]. Ftom test data they found that the maximum shear stress (vuL that could be transferred across a pre-existing crack was given by: Vu =1.38 N/mm2 + 0.8 pfy *0.3 f; (4.18) where p is the area of reinforcement per unit area with yield stress h. crossing the shear plane~ If the cylinder strength f: is assumed to be 0.8 feu, then the upper limit of Vu becomes 0.24 feu. If the material partial safety factors of 1.15, 1.5 and 1.5 are applied toIy, feu and 1.38 N/mm2(since the latter stress is that which can be transmitted by concrete alone)' respectively, then the. design shear strength (Vdu) is obtained as: Vdu =0.92 N/mm2 + 0.7 pfy *0.16 feu (4:1b) From considerations of tests on composite beams, Johnson [289] proposed that the upper limit of Vdu should be reduced to 0.15 feu, and this value is adopted in the Code. The stress of 0.92 N/mm2 represents the shear stress which can be transmitted by the concrete alo.ne. This stress should be a function of concrete strength. and the F1P 1213] has proposed that it should he taken as 0.025/..". The values so far discussed arc appropriate to a type 1 surface (or monolithic concrete) and have. essentially, been adopted in the Code. For a type 2 surface, the FIP has proposed that the shear stress transmitted by the concrete alone should be taken as 0.015 fC/I' This stress is 60% of that for a type 1 surface. and the same'percentagehas been adopted for the upper limit of shear stress to give 0.09/"/1' . . The Code actually refers to the shear force per unit length (V,). and gives the following design equation: V, =v, L.• +0.7 Aefy ;}kdeu Ls (4.1c) where La is the length of the shear plane, and A(. is the arell of steel per unit length crossing the shear plane (excluding co-existent bending steel). Values of V, are given in Tabl~ 4.5A. k t is 0.15 for monolithic concrete or a type 1 surface, and 0.09 for a type 2 surface. Table 4.SA V,values (N/mm2) Surface 1 2 Conc:rete grade 0.50 0.30 2S 0.63 0.38 30 0.75 0.45 iii!: 40 0.80 0.50 It can be seen that now the shear capacity can always be increased by providing additional reinforcement. Hence, the problems discussed on page 51 no longer arise. The modified stresses also give a better lower bound to the test data of Saemann and Washa [118] shown in Fig. 4.8(b). It is now stated in the Code that interface shear calculations do not have to be carried out for a compo- site slab formed from precast inverted T-beams with solid infill. Cracking ofprestressed concrete (p.51) . The flexural tensile stress limitations for a Class 2 member are now obtained by dividing the characteristic tensile stress of 0.56 VTcu by the appropriate partial safety factor given in Table 4.1A (i.e. 1.25 for pre- tensioned construction, and 1.55 for post-tensioned construction). The resulting limiting stresses are 0.45 VTcu and 0.36 VTcu, which are identical to those in Part 4 : 1978. It is now stated that members have to be designed for Class 1 under load combination 1, and Class 2 or 3 under load combinations 2 to·S. However, only 25 units of HB loading need to .be considered in combination 1. In addition, all live loading may be ignored for Iightiy trafficked highway bridges and railway bridges where the live loading is controlled. The DTp [283] has defined lightly trafficked struCtures to be accommoda- tion bridges, bridleway bridges, and foot/cycle track bridges. The DTp also requires that Class 3 prestressed concrete should not be used. At transfer, the flexural tensile stress is limited to 1 N/mm2for all classes. Limiting stresses based on the FIP recommendations
- 105. [290] ate nowgiven for joints in post-tensioned seg- ment,11 construction. For cement mortar joints. the net strt!ssl~s should be compressive and not less than 1.5 Nt mm2 at the serviceability limit stale. For resin mortar joints, the net stresses should be compressive at the serviceability limit state. I!1 addition, for resin, joints, during the jointing operation, the average stress should be between 0.2 N/mOl2 and 0.3 Nln~!m2, but nowhere· less than 0.15 Ntmm2,' and' ll~~ di,fEe{encc I(lctweeOl stresses across the section sl!toouldi oot cxcc<:(,I; @,'.5 Nt mOl2• ' Cracking of reinforced concrete (p,51) The design cfat'k widths in Table 4.7 have been altered slightly. The value for 'Severe,' exposure has been increased to 0.25 mm. The exposurc condition referred to as 'Very severe (2)' is now referred to as 'Extreme' and still has a design crack width of 0.1 mm. However, the uesign crack width for the 'Very severe (1)' expo- sure has been increased to 0.15 mm. The Hbove increases in design surtace crack widths should not be viewed as relaxations of standards. because the covers have also been increased (see Chapter lO). , It should also be noted that buried concrete is now considered to he in a severe. rather than a moderate, environment. Ultimate limit state (p.S3) The vHlucs of Yf3 have now been simplified consider- ably, and are no longer dependent on load type. A value of 1.1 should be used for all loads unless a plastic method of analysis is adopted, when 1.15 should be used. Serviceability limit state (p.53) The values hHVC again been simplified considerably and are now always 1.0. This implies that the design highway or railway loading under load combination 1 is 8 greater th:H1 the nominal load for al des!g,11 triter,ia; e.g. the design HA load is now 1.2 HA. l.htsexplHlIls why the limiting concrete compressive stresses in l,re.. stressed concrete of Table 4.3 have. been increased by 20%. In Part 4 : 1978. stresses in prestrt:.ssed c~mcrete were checked under 1.0 HA, whereas now they are checked under 1.2 HA. By increasing the limiting stress. by 20'),;, (the same increase as that applicu to the HA loading), the drafters appear to have assumcd that 4lead load stresses are always balanced by stresscs·due to prestress. It should be noted that under load con~ binations 2 and 3, the designHA load is 1.0 HA rather thHn 1.2 HA. Hence. larger compressive stresses are now permitted under load comrinations 2 anu 3 than . was the case in Part 4 : 1978. Although the limiting compressive st'resses for pre- stressed concrete have been increased to allow for the increased loading under 10Hd combination I, it seems anomHlous that the limiting tensile stresses have not also been increased. Reinforced concrete (p.54) Only two load levels now need consideration, since stresses and cmck widths arc now checked at the same design load, However, in lo~d combination I, only a maximum of 25 units of HB loading have to be considered for crack widths, whereas up to 45 units could be required for stresses. Prestressed concrete (p.S4) In load combination 1, only a maximum of 25 units of HB loading have to be ~onsidered for tensile stresses, wherea~ up to 45 units co~ld be required for compress- ive stresses. Composite construction (p.S4) Interface shear is now considered at the ultimate limit state. This simplifies the calculations considerably. Chspter5 Ultimate limit state - flexure and in-plane forces Assumptions (p.55) In order to ensure a ductile failure, the strain at the centroid of the tension reinforcement must now exceed the value given by equation (5.1), However. if the ultimate moment of resistance of the section is at least 1.15 times the required value it is not necessary to satisfy the above requirement. The reason for this is given latcr in the discussion of the prestressed concrete assumptions of page 57. Simplified concrete stress block (p.55) The simplified rectangular stress block can only be used for a rectangular section or a flanged section with the neutral axis within the flange. Singlyreinforced rectangular beam (p.56) In equation (5.8), correct d to d2 • Flanged beams (p.57) At the ultimate limit state, it is now permitted to take the full flange width as the effective wid~h. This allows for plastic redistribution of stresses at coilapse. Assumptions (p.57) 2. The Code no longer covers unbonded tendons. Hence, the table of failure stresses has been re- moved. 3. It is now necessary to ensure a ductile failur~ by checking that the strain in the outermost tendon exceeds the following value: fpl4 £ = 0.005 +-- E.,Ym (5.14a) If the outermost layer of tendons provides Jess than 25% of the total tendon area, th.e above require- ment should also be met at the centroid of the outermost 25% of tendon area. Equation (5.14a) . .", ',-........ ,!'"' ..... " can be derived from considerations of the strcss- strain curves given in Figs 4.4(c) and (d) in the same manner as equation (5.1) was derived. This ductility requirement can be very restrictive be- cause it is satisfied by few' currently designed standard bridge beams. As a result it is permissible to ignore the requirement provided that the ulti- mate moment of resistance of the section is at least 1.15 times the required value, The 1.'15 factor ensures that the vast majority of current designs would be acceptable to the new Code, Orthogonal reinforcement (p.59) Insert = before M;yin equation (5.18), 5.1 Prestressed beam section strength (p.61 ) In this example, the characteristic tendon strength would now be obtained from the relevant British Standard rather than a Code table. The outermost tendon strain is 0.014 which exceeds tbe value of 0.0121 obtained from equation (5.14a), However, it is not now permissible to use the simplified rectangular stress block because the neutral axis is not in the flange. Hence, a computer program would be needed to analyse the section in practice, In the strain compatibility approach the neutral axiS' depth is incorrectly stated as 330 mm instead of the correct value of 339 mm which was used in the subse- quent calculations. Table 29 in Part 4 : 1978 has been renumbered to Table 27 in Part 4 : 1984, 5.2 Slab (p.63) Three corrections in right-hand column: (i) Line 4: change M; > 0 to M; > 0 (ii) Line 6: change (0.9) to (-0.9) (iii) Last two lines: transpose the words 'top' and 'bottom'. 9
- 106. Chapter 6 Ultimate limitstate - shear and torsion Introduction (p.65) Interface shear calculations in composite construction are now carried out at the ultimate limit state. Beams without shear reinforcement (p.65) The design shear stresses given in Table 6.1 have been altered slightly. The new values are derived from the following equation: ' _ 0.27 (lOOA)Ih '( .IIIVc - - - ----.J vcu) Ym bd (6.18) The partial safety factor Ym is 1.25, and IC'u should not be taken as greater than 40 N/mm2 • The depth factor for slabs, discussed on page 69, is now also applicable to beams. Thus, for both slabs and beams the shear stress which can now be resisted in the absence of shear reinforcement is ~ v," Beams with shear reinforcement (p.66) When the nominal applied shear stress exceeds ~ vC" it is now necessary to provide shear reinforcement to resist the shear force in excess of (~ v(' - 0.4) bd rather than Ve b d. Hence, equations (6.6) and (6.7) become: b(v + 0.4 - ;'' v,.) A.I ·,. =--'-----=:::......!.<~ 0.87 fy ,' (sin a + cos a) A.". = b(v + 0.4 - !;.. ve) 0.87!,.v (6.6A) (6.7A) It is understood [291J that the reason for introducing the stress of 0.4 N/mm is to allow for the reduction of aggregate interlock under repeated loading. It appears that the stress of 0.4 N/mm2 was not based on test data, but was chosen because it is approximately the section- al shear stress which can be resisted by the minimum shear reinforcement required by the Code (see page 68). This requirement to overdesign shear reinforce- ment was originally introduced by British Rail (292)' In equation (6.7A),lvv is now limited to a maximum 10 "" r value of 460 N/mm2 instead of 425 N/mm2 • This value is still less than 480 N/mm2 and is thus conservative (sec page 67). There is now a requirement that the area of longitu- dinal reinforcement in the tensile zone should be at least: v A.f =2 (0.87 /y) (6.78) The derivation of this equation is given in full e1se- . where [293]. However, it can be seen from equa~ion (10.5) that the longitudinal reinforcement should really resist a force of VI2 in excess of the force due to bending alone. When designed shear reinforcement is necessary in a flanged beam, the longitudinal shear resistance at the flange/web junction has to be checked in accordance with the method explained in the amendments to Chapter 8. Minimum shear reinforcerrent (p.68)., Minimum shear reinforcement now has to be provided in all beams as follows: A.•v 0.4 b -;.:- =0.87fyv (6.7b) The values given by this equation are very similar to those in Part 4 : 1978.. Shear at point of contraflexure (p.68) The empirical design rule of Part 4 : 1978, which is discussed on page 69 has been omitted from Part 4 : 1984. Enhanced Ve values (p.69) The enhancement factor (;,,) is now given as a function of effective depth. rather than overall depth: 1/4 J- 7;,. =(500/d) .,.0. (6.7~r . . <. A fourth root relationship was proposed by Reg~I1" (150). Equation (6.7c) fits the test data of Fig. 6.5 better than the Part 4 : 1978 relationship. ~ 1-5dx~ ~ 1-5dx~ "5dI~ -------~ r I I : ~ I I ~ I I: :1'Sdy :.. _______ ~ 'tSdyI~ -------~~~i:ii~~ter I ® I I :''''-''-+-support "5d I I or load Y I I '-_.___ •.~ ..... _.J (a) (entre ~ 1'5dK~ ~ "5dx~ ex Shortest ,- straight line 1'1' touching ,- lSdI; -------- Y I I~ 1- sa ri -------- r I :~ Unsupported edge loaded ",'- area~/ /~ '~I~________ / (b) Edge (c) (ornerO) (d) (orner Oil 6.6A New punching shear perimeters An important point to note is that the enhancement factor from equation (6.7c) is less than unity for slabs with an effective depth greater than 500 mm. This fact, coupled with the requirement to resist an additional nominal stress of 0.4 N/mm2, will result in an increase of shear reinforcement in slab bridges. Shear reinforcement (p.69) Equation (6.7A) is now applicable to slabs. Maximum shear stress (p.70) The m,!ximum shear stress in slabs at least 200 mm thick is now the same as that for beams. Voided or cellular slabs (p.70) It is now,stated that the longitudinal ribs of voided slabs should be designed as beams. Transverse shear effects should be resisted by transverse flexural reinforcement: the method sug- gested in Appendix B of the main text could be used. However, it is now also stated that the top and bottom flanges should be designed as solid slabs, each to carry a part of the global transverse shear force and any shear forces due to torsional effects, proportional to the flange thicknesses. The author does not see the need for designing the flanges of a circular voided slab as individual solid slabs if the design method suggested in Appendix B is adopted. In addition, it is not clear how to include the shear forces due to torsion in the shear design of the flange since the torsional shear flow in the flange is perpendicular to the flexural shear flow. BS 5400 clause (p.72) The critical perimeter is now situated at 1.5 times the effective depth from the face of the load or column. Since the effective depth is different in the two steel directions, the critical perimeter is also at different distances in these two directions. It is also stated that the perimeter is rectangular whatever the shape of the loaded area (see Fig. 6.6A(a». There is little differ- ence in the lengths of the 1978 and 1984 perimeters for rectangular loaded areas; but, for a circular loaded area, the new perimeter is longer and this results in a greater punching shear strength. These changes origin- ated in the revisions to the building code [294]. The total shear capacity of the concrete is taken as the sum of the shear capacities of each straight portion of the critical perimeter. The flexural reinforcement perpendicular to each portion should be used to evalu- ate the appropriate value of Vc for each portion. The effective area of flexural reinforcement should now include all of the reinforcement within the width of the loaded area and a width extending to within three times the effective depth on each side of the loaded area. For the design of shear reinforcement. equation (6.8) has now been arranged in the form: 0.4 ~ b d :::; (~ A".) (0.87f,.,.) ~ V - V, (6.8A) where ~ b d is the area of the critical section. V is the applied shear force at the ultimate limit state. and V<, is the total shear capacity in the absence of shear rein- forcement. 11
- 107. All important additionin Part 4 : 19t-i4 is the pro- vision of design rules for thc punching of loads ncar to frec edges. The code perimeters and the layers of flexural reinforcement which arc considered to be in tension arc shown in Fig.6.6A(b - d). The author has discussed the question of which layers of steel arc in tension elsewhere [2')5], and is of the opinion that the stccl in the unloaded face is always in tension in case (b), and also in caSl' (c) eXCl~rt when both e. and c.I' in Fig. h.6A(c) arc less than the slab depth. In the latter case, all steel layers are in tension, and therefore the lesser of the top and bottom reinforcement areas in a particular direction should be used to determine tht:~ allowable shear stress approprime to that direction. For cases (b) and (c), the total shear capacity of the concrete is taken as 80%) of the sum of the shear capacities of the three or two portions, respectively, of the critical perimeter. It is understood that the reduc- tion of 2()(X) is to allow for the fact that it is not certain whether the design method is strictly applicable to supports ncar free edges. However, it is of interest to note that in thl' building codes [15, 294], a similar reduction of 20% allows for moment transfer at edge and corner columns [296]. Since slab bridges arc gener- ally seated on bearings which do not permit moment transfer, it is doubtful whether the 20% reduction is appropriate. Case (d), essentially, depicts a flexural shear failure, and the layers of tension reinforcement are shown correctly. The shear capacity of the concrete is calcu- lated for the critical section (of length b), with the effective depth and area of flexural steel, used to determine v,,, taken as the average of the values in the two reinforcement directions. If shear reinforcement is required for this case, then it is illogical to place the shear reinforcen:tent as required by the Code. i.e. on the critical perimeter and at a distance of 0.75 d inside it, since the latter reinforcement would make no con- tribution to the shear strength. The author would suggest that shear reinforcement should be designed in accordance with equation (6.8A), and placed within a distance from the load equal to an effective depth. as shown in Fig. 6.6a. It should be noted that the Code now states that a group of concentrated loads should be considered both singly and in combination. Finally, it is not necessary to consider punching through the flange of a circular voided slab, because the relative dimensions of wheel loads and flange thick- nesses make it highly unlikely that such a failure would occur. However, it is necessary to consider punching through the slab as a whole. No guidance is given in the Code as to how to carry out such a calculation. However, research is being carried out at the Univer- sity of Birmingham with a view to proposing a method of calculation. Shear in prestressed concrete (p.72) Additional clauses are now given on shear in trans- 12 ,I ,I / / ,I ,I / Links !n this I' I' ,1,1 region "'--7-/~,I / ,I ,I / / / J ,I / ( ( 6.6a Shear reinforccmentfor perimeter (d) Critical perimeter mission lengths. !ind on segmental construction Within tht.' transmission length of a pre-tensioned member, the shear capacity should be taken as the greater of the values obtained by assuming that: (a) thc section is reinforced and all tendons ignored: and (b) the section tS prestressed, with the appropriate value of . prestress at the considered section determined by lIsing , ' a linear variation of prestress over the transmission length. For a post-tensioned member of segmental construc- tion, the shear force at the ultimate limit state should not exceed: v =0.7 "IlL Ph tan 02 (6.8a) .' where Ph is the horizontal component of the prestres- sing force after all losses and,. YII. is the partial safety factor of 0.87 which is appliedto this force. Equation (6.8a) is based on the shear-friction theory [297], and U2 is the angle of friction of the'joint between segments. Tan 02 can vary from 0.7 for a smooth unprepared joint to 1.4 for a castellated joint. These values were prop- osed by Mast [297] from considerations of test data. Th~ 0.7 factor in equation (6.8a) is merely an additional factor of safety. Sections uncracked in flexure (p.72) There is now a general requirement that when calcu- lating the shear strength of a prestressed member, the prestressing force after losses should be multiplied by a partial safety factor of 1.15 where it adversely affects the shear strength and 0.87 in other cases. It is not stated whether a single value o(the partial safety factor should be used throughout a calculation, or whether the factor can take different values. A single value would seem more logical, and the author understands that this was the intention of the drafters. Because of the va!iable partial safety factor, equa- tion (6.8) on page 73 rather than equation (6.9) appe'ars in the Code, but fcp is calculated from the prestressing force multiplied by the appropriate partial safety factor (0.87). - ------ --- -- - -----_.-.<---"..---------- ------ .1 ....".,. Sections c~acked In flexure (1'.73), '" .,Sttlte it is now neCeSS<lfY to api;ly a partial safety 'f~ctor to the prctsttessingfQrce•.the following changes /have hl!len made III the Code: '• ...._' ,", .. I l. In e4u3tion(6.12r MI has changed to Me; and appears as Mrr ='(0.37 VJ:., +J~,) II)' (6.12A) J~" is calcula,ted fronlthe factmc:d value of the. prestressing forc~, (n~rtiul safety.fm:tor of 0.81). 2. In equatiOn (6'tt~Jl KIn =!"II'y. an~ ~Otlt/", and!,,(. are calculated '''()m the factored value of the pre- stressing forcl!. tit'should be noted that the minimum vltiue of V(., ~ould he obtained with either value (O.~7 or 1.15) ~)fthe partial safety factor. 3. Although not sUIted, one should. pr~sumahly~ also take P, to he the factored prestressing force :when , considering comhined tensioned and unt¢nsl()ned steel. Shear reinforcement (p. 74) It is now necessllry to 'provide at .·Ieastthe minimum amount of s~car reinforcement in ~II members. As is also the case for reinforced concrete. designed shear reinforcement 'has to resist an excess' nominal shear stress of 0.4 N/mm2 • 'Thus equation (6.17) appears as: }sv V + 0.4 b d, - Vc s:- =' d.87/yv a, (().17A) The area of longitudinal steel (reinforcement plus tendons) should exceed the value given by equation (6.78). However, the author suggests that this equation should be modified to allow for the effects of the prestress, which counteract the longitudinal tension due to shear. Thus, the area of longitudinal reinforce- ment required is less than that given by equation (6~7a). Consequently, a more realistic equation for prestressed concrete is: Iv pe(.) 1 (6.16a) A.• =2 -d; 0.H7J;. where P is the axial. component of t~e resultant effec- tive prestressing force (with the partial safely factor of 0.87 applied), and e(. is the distance of theJine of action of this force from the extreme compression fibre at the ultimate limit state. The force Pe,!d, in equation (6.16a) acts at depth d" and is one of a pair of forces (the other acts at the extreme compression fibre) which are stati- cally equivalent to the eccentric prestressing force P. When designed shear reinforcement is necessary in a flanged beam. the longitudinal shear resistance at the flange/Veb junction has to be checked in accordance with the method explained in the amendments to Chapter 8. Punching shear (p.75) The Code refers to values of Veo given in Table 28 of , II the Code. ht fact. iable 28 gives values of maximum' shettt sHess. Ti,e Code should refet· tu "hhlEt31 of Part ' 4 : 1978. but this table has not been included in't'art ' 4 : 1984. torsion reinforcement (p.77) Th~ notation has now changed: ASI ::: area of one leg of a closed link AsL :::: atell of one bar of longitudinal reiuforcel1lent sL =spacing .01' longitudinal rein(ot'ccll1ent ,i, Hence, equations (6.21) and (6.22) appear'in th~ Code as: ': As, ~ T (6.21 A) , Tv- !1,.6xlJ'1 (0..8.71;-:> ASL~,_A,SI (:flY) (6.22A) SL 'Sv fyL. Huwever, it is not clear what value should he taken for sl.in the situation shown in Fig. 6.12(a). Presumably. one "takes the average of the horrzontal and vertical spacirlgs. If thi!iis done. equation (6.22A) is. essen- : tially"".iQ~ntical to equation (6.22). Delai/ill8·.(p.78) Thl~ charactellistk strength of torsion I reinforcement should, noll be· assumed greater than 46() N/mm2 • This value stighdy exceeds the critical ' value of 430 N/mm2 found by Swann P·77.1 (s.ee page 78). .The area of links or tongitudtnal torsion reinforce- ment may be reduced by up to 2()'%. provided that the following product remain~ uncha;'1ged: A.vv Asl• --Sy 5L The implication of this detailing rule is that, the spiral failure surface (see Fig. 6. 12{b» would form at an angle other than 45°, which is the valuc assumed in the dcrivatio~ of equations (6.21A) and (6.22A). Torsionreinforcement (p.79) Equations (6.21A) and (6.22A) should be used for each individual rectangular section requiring torsion rein- forcement. Torsion reinforcement (p.80) The equations for rectangull;r sections are no longer included in the Code. Hence, only equations. (6.25) and (6.26). suitably modified to allow for the previously mentioned change of notation. have to he c<lOsidered. The required area of links is given by: }s" ~ 1" S; .... 2 Au (0.87 fYI') (6.25A) and the required area·of longitudinal steel is given by equation (6.22A). It is now permitted to reduce the amount of longitu- dinal reinforcement in flexural compressive zones, as discussed on page 81, for all sections (not just box 13
- 108. sections). The depthof the compn.'ssion zone should be taken as twice the cover to the closed links, since it is essentially, this depth of concrete which is considered effective in resisting torsion [177,180j. Torsion of prestressed concrete (p.81 ) It is now stated that compressive stresses in the con- crete due to prestress should be taken into account separately in the same manner as discussed on page 81 for flexural compressive stresses in reinforced concrete. 6.1 Flexural shear in reinforced concrete (p.83) The allowable shear stress without shear reinforcement is calculated from equation (6.1a): VI' = ~:;~ (l.64)'/.' (40)1/.1 = 0.871 N/mm2 From equation (6.7c), the depth factor is: ;,. = (5()()/600)1/4 =0.955 From equation (6.7A): A.... 300 (3.00 + 0.4 - 0.955 x 0.871) -s-,. = 0.87 x 250 =3.54 mm2 /mm This is an increase of 21% over that required by Part 4 : 1978: 16 mm stirrups (2 legs) at 100 mm centres give 4.02 mm2 /mm. 6.2 Punching shear in reinforced concrete (p.83) 460 N/mm2 steel will now be used. The critical perimeters in the longitudinal and trans- verse directions are, respectively: 1.5 x 980 = 1470 mm and 1.5 x 1060 = 1590 mm The critical perimeter is shown on Fig. 6.22A. For sides 1 and 3: Length = 2 x 1590 + 1200 =4380 mm . From equation (6.7c), depth factor;. =: (500/980)V4 = 0.845 From equation (6.la), allowable shear stress without shear reinforcement is: v,. =0.27 (3)'/.' (40)'/'/l.25 = 1.065 N/mm2 Shear capacity = VI =0.845 x 1.065 x 4380 x 980 x 10-:1 =3863 kN 14 Slab ~ 1470 +600+ 1470 ~ ------------l: . ~ I I I o I I ~ I I I I I I I B I :® Pier CD: I I I II IICritical $ permeter .... I I ~ ____._® ____ _._1 6.22A· New perimeter for Example 6.2 For sides 2 and 4: Length = 2 x 1470 + 600 = 3540 mm From equation (6.7c), depth factor~, :;;;: (500/1060)'/' := 0.829 From equation (6.1a), allowable shear stress without shear reinforcement is: v,. =0.27 (1)1/, (40)1/1/1.25 == 0.739 N/mm2 Shear capacity = V2 =0;829 x 0.739 x 3540 x 1060 x ':, 10-3 = 2298 kN Total shear capacity = Vc = 2(V1 + V2) = 2(3863 + 2298) x 10-:1 = 12.3 MN < 15 MN, thus shear reinforcement .needed. Left hand side of equation (6.8A) governs. Th~s, l:A". ~ 0.4 x 2 (4380 x 980 + 3540 x I060)/(O.87)(460) = 16,100 mm2 This is an increase of 7% over that required by Part 4 : 1978. This amount of reinforcement must he pro-' vided along a perimeter 1.5d from the loaded area, and also along a perimeter O.75d from the loaded area. The calculation should now he repeated ata perimeter 2.25d from the loaded area. 6.3 Flexural shear in prestressed concrete (p.83) A partial safety factor of either 0.1-17 or 1.15 has to be applied to the prestressing force when calculating V,," and Vcr' It is obvious from equation (6.8) on page 73 that Veo is a minimum when "'f' is a minimum: hence, the partial safety factor is taken as 0.87. It is also ~ , I---,~ G) ~ 0 r--- ~0 ........ .... ....II .. II III.... ~ ..c: Ia • 1--' ..... ~X, =520 b=600 6.24A Example 6.4 obvious from equations (6. 11) and (6.12A) that VI" is a. minimum when /pl is a minimum: hence, again, the partial safety factor is taken as 0.87. Compressive stress at centroidal axis =j;./, = 6.8H N/ ~ mm- V,," =(0.67)(250)(1035) Y(1.70)2+ (6.88)(1.70) = 0.662 x 101> N Prestress at bottom fibre =1,,, = 16.53 N/mm~ M.., = (0.37V5f) + 16.53) 125.43 x tol> x to-V =2.40 MNm V(., = (0.037)(250)( 1035-142)Y50 + (0.9 x 101»(2.40/3.15) =0.744 x 101> N Hence V, = V(." = 0.662 MN From equation (6.17A) A.,.. =(0.9 X 101> + 0.4 x 250 x 971 - 0.662 x 106 ) s.. 0.87 x 250 x 971 = 1.59 mm2 /mm This is an increase of 29% over that required by Part 4 : 1978. Provide 12 mm stirrups (2 legs) at 125 mm centres (1.81 mm2 /mm). It is emphasised that, for a Class 2 or 3 structure, the minimum value of V.., could be calculated from equa- tion (6.16) with either value (0.87 or 1.15) of the partial safety factor applied to the prestres~. Hence, both values would have to be tried. 6.4 Torsion in reinforced concrete (p.84) The characteristic steel strength would now be 460 NI mm2, and the cover would he 35 mm. As a result: d = 1139 mm, XI =520 mm,)'1 = 1120 mm. From equation (6.2IA): A.•ls,. =290 x Hl'/(1.6 x 520 x 1120 x 0.87 x 460) = O.77H mm:!/mm 10 mm at HX) mm centres give 0.785 mm2 /mm : thus use 2-legged 10 mm diameter stirrups. ,. • N In In ~ N In In .. -. l---'l~ ,0 •••• ; ',. ;.~ ... ',.. '.' ~ 784 ~ 6.25A Example 6.5 With 45 mm cover to main steel of 32 mm diameter, and using the average of the two SI. values shown in Fig. 6.24A: .', =(478 + 1078)/2 =778 mm From equation (6.22A): A.•, :; (778)(0.778)(460)/460 =605 mm2 Use 32 mm bar in each corner (804 mm2) The Code would now permit the area of longitudinal steel to be reduced in the flexural compression zone. 6.5 Torsion in prestressed concrete (p.85) The characteristic reinforcement strength would now be 460 N/mm2• and the cover would be 30 mm. Assume XI =815 mm~ YI =1130 mm. From equation (6.25A): Axis" === 610 x H1'1(2 x 829250 x 0.87 X460) =0.919 mm2 /mm 12 mm at 100 mm centres give 1.13 mm2 /mm: thus use 2-legged 12 mm diameter stirrups. Assume longitudinal reinforcement provided by a 32 mm bar in each corner, and one at mid-depth of each web. Use the maximum SL value of 784 mm shown in Fig. 6.25A. From equation (6.22A) A.•I. =(784)(0.919)(460)/460 =720 mm2 Use 32 mm bars (804 mm2 ) Cover to closed links =30 mm Thus, effective compression zone depth =2 x 30 =60mm Average stress in compression zone =20N/mm2 Permissible reduction of steel area =(20)(90() x 60)/(0.87 x 460) = 2699 mm2 Net area of each top bar =720 - 2699/2 <0 Thus, no torsion reinforcement is required in the top flange. 15
- 109. Chapter 7 Serviceability limitstate Introduction (p.86) Interface shear calculations in composite construction are now carried out at the ultimate limit state, General (p.86) It is emphasised that' stresses in reinforced concrete need to be checked only in the following situations: .. 1. Where the effects of applied deformations arc not considered at the ultimate limit state. The author assumes that the drafters had in mind the situation in which the full elastic effects of applied deforma- tions, with no relaxation due to the plastic be- haviour of materials, are considered at the ultimate limit state. Hence, if applied deformations are considered elastically at the ultimate limit state, less calculations have to be carried out. However, the true nature of the effects of applied defor- mations are confused by this design approach, since they are actually more significant at the serviceability rather than the ultimate limit state. 2. Where a plastic method of analysis has been used at the ultimate limit state, because such methods imply redistribution of elastic stresses. 3. Where combined global and local effects are con- sidered separately at the ultimate limit state. Such a situation would arise if a top slab were designed to resist local wheel load· effects using yield line theory. The concrete compressive stresses should not exceed 0.5fcu and 0.38fcu for triangular and uniform stress distributions respectively. Reinforcement. tensile or compressive stresses should not exceed 0.75fy. The value assumed for the elastic modulus of the concrete should be an appropriate value between the long-term and short-term values depending on the ratio of permanent to short-term effects. This implies a different modular ratio for each load case and load combination. 16 Slabs (p.87) At the top of page g8. I!."h-" should be corrected to l',11'1/ in two places: (a) immediately ahove equation (7.2): and (b) towards the end of the last paragraph. General approach (p.SS) It is now necessary to pro vide at least the following amount of reinforcement to control cracks due to the restraint of shrinkage and early thermal movements wherever restraint occurs: A.v =k, (A,. _. 0.5 A C{1,f (7.2a) where k, = 0.005 or 0.006 for high-strength or mild steel respectively, AI" is the gross cross-sectional area. and Am, is the area of the concrete core section more than 250 mm away from all surfaces. Equation (7.2a) results in up to twice as much reinforcement as that required by Part 4 : 1978. The reason for not taking As as a proportion of the gross cross-sectional area is that the cracks develop progressively from the surface. Hence, it is only necessary to reinforce the 'surface zones', which are each:assumed to be 250 mm thick [231]. Thus, equation (7.2a) is conservative since only half of the core area is subtracted from the total area. The spacing of the reinforcement should not exceed 150 mm. Loading (p.89) The design highway loading is now increased to 1.2 HA and 1.1 (25 units of HB) irrespec- tive of the span. Stiffnesses (p.89) The value assumed for the clastic modulus of the concrete should be an appropriate value between the long-term and short-term values. depend· ing on the ratio of permanent to short-term effects. This implies a different modular ratio for each load casc and load combination. The tension stiffening formula has been amended to: E",=EI- [3.8bl!(a l -d(.)] [1-.;t IO'~ *£1 (7.3A) E.,AAh-d..) I( J Equation (7.3A) differs from equation (7.3) in the following ways: I:. I. Notation: hi replaces h, and d,. replaces x. 2. lnstead of the service load stcel stress being assumed equal to O.5g/I ., the actual calculated value (E.•E.,) is used. Hence, E,~ rather than fl" appears in the equation. . 3. The second term in square brackets, in which Mq and MI( arc the live and permanent load moments respectively. has been introduced. to allow for the breakdown of tension stiffening due to .repeated live loads. For zero live loads. the two equations give the same result for a service load stress of 0.63/).. .. Crack control calculations (p.90) Crack width calculations now have to be carried out for all structural clements. including solid slab bridges. A single crack width equation is now given. This is equation (7.5) with the notation changed such that C"om replaces e",ill and de· replaces x. C"(I'" is the required nominal cover. It is emphasised that crack width cal- culations arc now carried out at an imaginary surface at a distance from the outermost bars equal to the re- quired nominal cover. Hence, with respect to calcu- lated crack widths a designer is no longer penalised for providing cover greater than that specified in the Code. It is not clear ·from the Code what value should be taken for C"om when calculating the width of a crack perpendicular to an inner layer of bars. Presumably it is the nominal cover plus the diameter of the outer layer of bars. For a section entirely in tension (e.g. a flange of a box girder). equation (7.5) reduces to (see page 93): w =3acr E", (7.Sa) This equation is now given in the Code. Specific guidance is now given on the combination of global and local effects for crack width calculations. The strain Em may be taken as the sum of the strains calculated separately for the global and local effects. Equation (7.Sa) may then be used to calculate the crack width. However, if the global effect induc~s compress- ion in a slah or flange, the crack width can be calculated from equation (7.5) with the depth of concrete in compression calculated by considering only the local bending moment. These approaches to calculating crack widths ar,. more conservative than that based on the strain due to combined glohal and local effects. Beams (p.94) Limiting values of prestressing steel stresses are no longer stated. Under HB loading. compressive and tensile stresses may have to be checked under different load levels in load combination 1 (see the discussion to the amendments of Chapter 4). Initial prestress (p.95) Immediately after anchoring, the tendon force should not exceed 70% or 75% of the characteristic tendon strength for pOSt-tensioned and pre-tensioned tendons respectively. Losses due to steel relaxation (p.95) The reference to Part 8 of the Code and the quoted loss in the range zero to 8% has been removed. Instead reference is made to the appropriate British Standard. 7.1 Reinforced concrete (p.98) The characteristic strengths are changed to 460 N/mm2 and 40 N/mm2 • If grade 30 concrete were used then it . would have to be air-entrained (see Chapter 10) unless the slab were supported on permanent formwork. General (p.98) From Table 3 of the Code, short-term elastic modulus of concrete =31 kN/mm2 . Hence, long term value := 3112"" IS.5 kN/mm2 . . The value for design depends on the ratio of perma- nent to short-term effects. It is not clear how this ratio can be calculated when both in-plane and bending effects have to be considered. However, for a top slab, where the dominant effects are due to wheel loads. it would seem reasonable to use the short-term elastic modulus. Hence, modular ratio = 200/31 =6.45, The values of yp for cracking are now 1.0 for HA and HB loading. Hence, the design loads are increased. Cracking (p.98) Due to transverse bending (p.98) HA design moment =(0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) + (10.8)(1.2)(1.0) + (6.0)(1.2)(1.0) =20.9 kNmlm 2SHB design moment ::: (0.45)(1.0)(1.0) + (0.22)(1.2)(1.0) + (7.56)(1.1)(1.0) + (23.3)(1.1)(1.0) = 34.7 kNmlm Critical moment = 34.7 kN mlm With modular ratio of 6.4S: de ::; 49.2 mm 1= 0.178 X 109 mm4 /m Required nominal cover = 35 mm (see Chapter 10 amendments). Thus crack width calculation carried out at a surface 35 mm from transverse steel, rather than at the soffit. Ratio of live to permanent moments> 1, thus equation (7.3A) reduces to Em = EI Strain at 35 mm below transverse steel is: £1 =(34.7 x 106 )(145.8)/(31 x 103 )(0.178 x 109 ) = 9.17 X 10-4 ... E", =9.17 X 10-4 acr = YS02 + 432 - 8 = 57.9 mm From eq~tion (~:~), crack width is " C,'" " 1., , t ' . 17V.l I ,
- 110. W =. _---.:.(3--'.)..:...(5_7_.9!...!)(,:....9._J7_x_I:...:..()_-oI..:...)_ ::::; (11'"'''mm 1 + (2)(57.9 - 35)/(200 - 49.2) . -- f,'rom Table 1 Part 4 : 1984. allowable crack width is O.'2S'mm. . F!clf .I,? {ongit~dinal bending (p.99) .;:;.~1~9'i~~S;: ,I~JI,~ wheell~~~: :,1::." .. ' ;lbrefcreslgn~stress due toglobal effects is: :;~:·(0.3~)(1.2)(1.0) + (0.61)(1.2)(1.0)::;: 1.13 NI mm2 :1' Bottom fibre design stress due to global effects is: (0.14)(1.2)(1.0) + (0.25)(1.2)(1.0) ::;: 0.47 NI mm2 Design stress 35 mm below transverse steel is: 0.47 + (l.i3 ::- 0.47)(5/200) ::;: 0.49 NI mm2 Strain == 0.49/31 x 1()3 =1.57 x 10-5 (compression) Design local moment =(7.20)(1.2)(1.0) =8.64 kN mlm With modular ratio of 6.45: dc·::;: 38.4 mm J ::;: 82.5 X 106 mm4/m Strain at 35 mm below transverse steel is: EI ::;: (8.64 x Hf)(156.6)/(31 x 103)(82.5 )( 106 ) =5.29 X 10-4 Ratio of live to permanent moments> 1. thus Em ::;: E Nett Em ::;: 5.29 )( 10-4 - 1.57 X 10-5 ::;: 5.13 X 10--4 acr = Vl()()2 + 592 - 8::;: 108.1 mm 2.trv/2.- ""/0 0 cnnm ::;: 35 + 16::;: 51 111m ? t /6 /(;::. ff From equation (7.5), crack width is f-., I~ W :: (3)(108.1)(5.13 x ~Q=~)___._ "" 0.097 mm 1 + (2)(108.1 - 51)/(200 - 38.4) . From Table 1 Part 4 : 1984, allowable crack width is 0.25 mm. Stresses (p.99) Limiting .values (p.99) The limiting stresses are now ?5 x 40 ::;: 20 N/mm2for concrete in flexural compress- Ion. 0.38 x 40 = 15.2 N/mm2 for concrete in axial compression, and 0.75 x 460 ::;: 345 N/mm2 for steel in tension or compression. Due to transverse bending (p.99) Critical moment is ,60.9 kN m/m (as in the main text) ::From crack width calculations, de =49.2 mm J =0.178 X 101) mm4/m Maximum concrete stress = (60.9 x 106 )(49.2)/0.178 x . 101) =16.8 N/mm2 < 20 N/mm2 Steel stresses are (6.45)(60.9 x 106 )(102.8)/0.178 x 101) = 227 N/mm2 tension .. a,nd .j6.45)(00.9 x 1<1')(11.2)/0.178 x 109 = 25 N/mm2 "compression Both < 345 N/mm2 Due to longitudinal effects (p.99) Compressive stress in concrete is (as in the main text) 8.51 N/mm2 < 15.2 NI mm2 • 18 Maximum tensile stress in reinforcement occurs· under the HA wheel load. due to an equivalent design moment of 10.8 kN m/m. From crack width calcu- lations: de = 38.4 mm J = 82.5 X 106 mm4 /m Bottom steel stress =(6.45)(10.8 x 106 )(97.6)/82,.5 x Ht' =82 N/mm2 < 345 N/mm2 7.2 Pre$tressed concrete (p.99) According to Part 4 : 1984 the section shown in Fig. 7.12 would be inadequate due to excessive tension .occuring at the bottom fibre under design HA loading. The reason for this is that the design load has been increased in Part 4 : 1984 and. although{(t~e."higher allowable compressive stresses compensate·' for this increase, the allowable tensile stresses have not been adjusted. Consequently the overall section depth must be increased to 950 mm. All other dimensions have been kept the same as in Fig. 7.12. The nominal and design dead load moments are now 486 kN m. It is necessary to consider only 25 units of HB loading when calculating tensile stresses under load combination I. The nominal moment for this loading is 660 kN m. All bridges now have to be designed as Class 1 under load combination 1. General (p.100) Area = 484 000 mm2 Centroid is 557 mm from bottom fibre Second moment of area =5.56 x 1010 mm4 Bottom fibre section modulus =9.98 x 107 mm3 Top fibre section modulus =14.13 X 107 mm3 From Table 22 of the Code, the allowable compressive stress for any class of preStressed concrete is 0.4 x 50 = 20 N/mm2 • ' From Table 23 of the Code. the allowable compressive stress at transfer for any class of prestressed concrete is 0.5 x 40 =20 N/mm2 • The imposed load design moments are increased be- cause Y[3 is now unity: HA = (949)(1.2)(1.0) = 1139 kN m HB = (1060)(1.1)(1.0) = 1166 kN m HB (25 units) = (660)(1.1)(1.0) =726 kN m The extreme fibre stresses induced by these, and the other. design moments are given in Table 7.2A. Table7.2A Example 7.2 Design data Moment (;kniim) Stress CINLmm1 ) Load Nominal Design Top 'Bot.tom· Oead 486 486 +3.44 -4.87 Superimposed dead 135 162 +.15 -1.62 HA 949 1139 +8.06 -11.41 HB 1060 1166 +8.25 -11.68 25HB 660 726 +5.14 -7.27 Class 1 (p.100) The allowable tensile stresses arc zero u,nder design service load (which for HB loading is only 25 units) and 1 N/mm2 at transfer. It is found that the top fibre at transfer and the bottom fibre under HA loading are critical. The required prestressing force and eccentric- ity are 4022 kN and 467 mm respectively. The resulting stresses are given in Table 7.3A. It can be seen that tension occurs under full HB loading but not under 25 units. Table 7.3A Example 7.2. Design Mresl>eS (N/mm2) Design load Top Transfer -1.0 PS + OL +0.2 PS + DL + SOL +1.3 PS + OL + SOL + HA +9.4 PS + OL + SDL + HB +9.6 PS + DL + SOL + 25HB +6.4 PS = final prestress DL = dead load SOL =superimposed dead load Bottom +19.3 +13.0 +11.4 0.0 -0.3 +4.2 19
- 111. Main te.~t pp.j02-J08 «; Chapter 8 Precast cOncrete and composite construction Corbels (p.102) The main reinforcement can no longer be anchored at the front face of a corbel by welding to a transverse bar, because of the possible reduction in fatigue capacity. It is now necessary to check a corbel at the ser- viceability limit state in addition to the ultimate limit state. However, no guidance is given regarding the nature of the serviceability check. At the time of writing this problem is under investigation at the Uni- versity of Birmingham. Nibs (p.103) Nibs are no longer referred to in the Code because they are not relevant to bridge construction. Bearing stress (p.104) Equation (8.4) is now replaced with: Ib = . 1.5fcu· :tleu (8.4 A) 1 + 2 VA"o,/A,up where Aeon and A,up are the contact and supp~rting area respectively. Equations (8.4) and (8.4A) are Iden- tical for a square contact area. Halving joints (p.104) It is now stated that halving joints should only be used where it is absolutely essentiaL This is because of the difficulty of providing access to such joints for inspec- tion and maintenance. It is now necessary to check a halving joint at the serviceability limit state in addition to the ultimate limit state. However. no guidance is given regarding the nature of the serviceability check. At the time of writing this problem is under inv~stigation at the Uni- versity of Birmingham. Shear (p.106) Interface shear is now considered at the ultimate limit state. The interface shear stress is calculated elastically from equation (8.12), and compared with the allowable 20 stresses discussed in the amendments to Chapter 4. Additional guidance is now givcn on the vertical shear strength of composite members. Composite beam and slab (p.106) The shcar c.apacity should be assessed by applying the prestressed concrete clauses to either the precast prestressed section acting alone, or the composite section. The first alternative is obviously the simpler and is conservative. Calculations for the second alternative could be carried out as described on pages 106 and 107. However, it isempha- sised that the partial safety factor applied. to the prestressing force can now be either 0.87 or 1.15 rather than 0.8. The last equation on page 106 should not be numbered (8.8). ., Composite slab (p.107) , The shear capacity should be assessed by either of the following methods: 1. The prestressedconcrete clauses are applied to the precast prestressed section acting alone. 2. The shear force. due to ultimate loads, is first apportioned between the infill concrete and precast units on the basis of their cross-sectional areas. The shear capacities of the infill and precast compo- nents life then assessed separately on the basis of their web breadths (the breadth of the infill is taken as the distance between precast webs) using the reinforced and prestressed concrete clauses respec- tively. The capacity of each component should exceed the shear force apportioned to it. Option 2 does not appear to be based on any test data. It seems illogical to apportion the total shear force on the basis of cross-sectional areas and to compare the resulting shear forces with capacities based on web widths. For a section with a large area of infill concrete. such as an edge beam. option 2 can result in links being required in the infill concrete. Thus, the author would recommend either the adoption of option 1 wherever possible. or the approach suggested on pages 107 and 108. .:X':~l, . . ';.:f!i'eneral (p. 108) ~<~..",...~ Interface shear is now checked at the ultimate limit ' state. Yfj is now unity for all serviceability calculation!!. Compressive and tensile stresses (p. 108) Allowance for the difference between the elastic mod- uli of the two concretes should be made if their characteristic strengths differ by more than to N/mm~. The allowable flexural tensile stresses of Table 4.4 for in-situ concrete are now applicable only when tension is induced by sagging moments due to imposed service loading. Interface shear stresses (p.108) Interface shear is now considered at the ultirt:late limit state. Differential shrinkage (p.109) In the absence of more exact data, shrinkage strains should be taken as those given for estimating the shrinkage loss in prestressed concrete (see page 95). An approximate value of the differential shrinkage strain is no longer given. Differential shrinkage effects can be ignored in com- posite slabs provided that the strengths of the precast and infill concretes do not differ by more than 10 NI mm:!. Flexural st~ength (p.110) The compressive stresses due to prestress in the ends of pre-tensioned units should now be assumed to vary linearly over the transmission length of the tendons. Creep due to prestress (p.111) The following correc- tions should be made to the left-hand column of page 112. 1. Eaeh side of equations (8.16) and (8.17) should be divided ~y 2. 2. The 2 should be removed from the term (2MLl3El) in expressions (8.18) to (8.21) and three lines above expression (8.18). . 3. The divisor of the right hand side of the penulti- mate equation should be 2L. Example - Shear in composite construction (p.115) It will be assumed that the shear force and moments in Table 8.1 have been obtained from an elastic analysis. so that Yp = 1.1 for all loads at the ultiPlate limit state Mllinlextpp. ~08-1l2. 115-117 ~., ~ i "" ": VerrtLcal s~ear (p.116):, ~/t.~. . .~. . l~.: At SfJ/JporiJp.116)',.." d., ... . ". If 15.2 mmstrands are used'and th~ concre·te strength at transfer is 40 N/mm2, then from equation (10.68). with k, =240 for strand. the transmission length is: I, = (240)(15.2)/v'4fl =577 mm For a linear development of stress within the transmis- sion length, the stress at the support =(250/577) of the fuJI value. Hence.f,." =(2501577)(6.38) -:= 2.76 N/mm2 Design shear force at the ultimate limit state acting on the precast section alone is: Vel = 163 X 1.2 X 1.1 =215 kN Vertical component of inclined prestress == (250/577)(66) = 28.6 kN With the partial safety factor of 0.87 applied to the prestress. the net design shear force on the p~ecast section is: V =215 - (0.87)(28.6) ;:: 190 kN Shear stress at composite centroid is f, = ~190 x 10 3 )(44.4 x 10 6 ) ;; 0.81 N/mm2 .• (65.19 x 109)(160) Additional shear force (Ve2) which can be carried by the composite section before the principal stress at the composite centroid reaches 1.7 N/mm2 is, from equa- tion (8.7) with the partial safety factor of 0.8 changed to 0.87 V _ (124.55 x lo9)(160} ("2 - x 116 X 106 (V(1.7)2 + (0.87)(2.76)(1.7) - 0.81)10-3 =314 kN VI' == V("() = Vel + Vc2 == 215 + 314:= 529 kN Design shear force is V =215 + (27 x 1.2 x 1.1) + (29 x 1.75 x 1.1) + (332 x 1.3 x 1.1) =781 kN V > V", thus links are required such that & = (781 - 529) lOb = 912 mm2 /m s,. (0.87)(250)(1270) 10 mm links (2 legs) at 150 mm centres give 1050 mm2 /m. At quarter span (p. 116) Design moment at ultimate limit state acting on precast section alone = 763 x 1.2 x 1.1 =1007 kN m Stress at composite section due to moment 21
- 112. = t007 X10" (642 - 454)/(65.19 x }(}'1) = 2.90 N/mm2 (tension) With the partial safety factor of 0.87 applied" to the stress due to prestress, the total stress at the composite centroid to be used in equation (8.7) is: f cp =(0.87)(5.14) - 2.90 = 1.57 N/m.m2 Design shear force at the ultimate limit state acting on the precast section alone is Vel = 81 x 1.2 x 1.1 =107 kN Shear stress at composite centroid is Is = (107 x 103 )(44.4 x 106 )/(65.19 x 1()9)(160) = 0.46N/mm2 Additional shear force (Vc-2) which can be carried by the composite section before the principal tensile stress at the composite centroid reaches 1.70 N/mm2is: V =(124.55 x 109)(160) x c-2 116 X 106 (Y(1.7)2 + (1.57)(1.7) - 0.46) 10--' =326 kN Vrll = Vd + Vc-2 =107 + 326 =433 kN The section is next co~sidered cracked in flexure. The cracking moment is now calculated from equation (8.8) on page 107 with the partial safety factor of 0.87 (rather than 0.8) applied to the extreme tension fibre stress due to the prestress (Jpt) of 17.54 N/mm2• The notation for cracking moment is now Mc-,;.hence Mc-, = (1007 x 106 ) x (1 - 454 x 124.55 x 109/642 x 65.19 x 109 ) + (0.37 V3() + 0.87 x 17.54) 124.55 x 109 /642 = 3114 x t<1' N mm =3114 kN m Design nioment at the ultimate limit state is M = 1007 + (132 x 1.2 XLI) + (135 x 1.75 x 1.1) + (1333 x 1.3 x 1.1) =3347 kN m Design shear force at the ultimate limit state is V= 107 + (14 x 1.2 x 1.1) + (15 x 1.75 xLI) + (196 x 1.3 XLI) =435 kN From the modified form of equation (6.11), suggested on page 107 Vc-, = (0.037)(160)(1111 V50 + 130 V30) 10--' +. 435 (3114/3347) =455 kN Vc =Vto =433 kN V > Ve, but minimum link requirement governs. Hence, provide minimum links, as on page 117, of 10 mm (2 legs) at 500 mm centres (314 mm2/m). 22 Interface shear (p.117) At support (p. 117) Design shear force at the ultimate limit state which acts after the top slab hardens is (27 x 1.2 x 1.1) + (29 x 1.75 x 1.1)+ (332 x 1.3 x 1.1) . =566 kN From equation (8.12), interface shear stress is =(566 :x: 10")(71.6. x Ht')= 1.08 N/mm2 v" (124.55 x 109)(300) Longitudinal shear force per unit length is V, = 1.08 x 300 :::; 325 N/mm For a rough as cast surface (i:e: Type 2),k, from Table 31 of Code is 0.09 and. for grade 30 concrete, V, =: 0.45 N/mm2 (see also Table 4.5A). From equation (4.lc), longitudinal shear force per unit length should not exceed 0.09 x 30 x 300 =810 N/mm > V, .'. O.K. From equation (4.lc) rearranged, required area of reinf<?rcement per unit length is AI' =(325 - 0.45 x jOO)/(0.7 x 250) = 1.09 mm2 /mm = 1090 mm2/m Required minimum =0.15% of c9ntact area =(0.151100)(300) =0.45 mm2/mm < 1.09 mm2 /mm 10 mm links (2 legs) at 150 mm centres (1050 mm2/m) provided .lor vertical shear is not sufficient for interface shear which requires 1090 mm2 /m. Hence. use 10 mm links (2 legs) at 125 mm ~ntres (1260 mm2/m). At quarter span (p. 117) De~ign shear force at the ultimate I.imit state which acts after the top slab hardens is (14 x 1.2 x 1.1) + (15 x 1.75 x 1.1) + (196 x 1.3 xLI) =328 kN For which V, = 189 N/mm, and A,. =(189 - 0.45 x 3(0)/(0.7 x 250) =0.309 mm2 /mm Hence, minimum of 0.15% (0.45 mm2/mm) governs for interface shear. This amount exceeds that provided for vertical shear (314 mm2 /m). Hence. provide 10 mm links (2 legs) at 350 mm centres which gives 449 mm2 /m. Chapter 9 Substructures and foundations General (p.118) A clause on the shear resistance of columns has been introduced into Part 4 : 1984. A column should be designed to resist shear as if it were a beam except that the ultimate shear stress (!;..v,.) resisted by the concrete can be multiplied by (1 + 0.05 N1Ac). N is the ultimate axial load in newtons and AI' is the cross-sectional area in mm2 • The multiplier is a more conservative version of that used in North American practice (168]. A column subjected to biaxial shear should be de- signed such that: (9.0a) where Vx and Vy are the applied shear forces, and V/I.t and VIIV are the corresponding shear capacities. There appearS to be no experimental or theoretical evidence to support this interaction formula, but it would seem to be conservative. No guidance is given in the Code regarding the effective shear area of a circular column nor the area of reinforcement in a circular column which should be used to determine the allowable shear stress from equation (6.1a). The American code [168] considers only that tension reinforcement which is in the half of the member opposite the extreme compression fibre, and takes the effective depth as the distance from the extreme compression fibre to the centroid of this reinforcement. The web width is taken as the diameter of the column. These recommendations are based on tests by Farodji and Diaz de Cassio [298J who found that the ACI equations for rectangular sections could be applied to circular sections if the external diameter were used for the effective depth, and the gross section area used for the product bd [299] Definition (p.118) A column is now defined as a member with an aspect ratio not greater than 4. Effective height (p.118) A more precise table of effective heights is now given, which replaces Table 9.1. The new table is summarised in Fig. 9.0a. It should be noted that the rotational restraint at a rotationally restrained end is assumed to be at least four times the column stiffness for cases 1,2 and 4 to 6, and eight times for case 7. The former value, which is generally conservative, was chosen to give the same effective heights for cases 1 and 2 as those used hitherto. However, -the effective height of a cantilever column is extremely sensitive to the rotational res- traint; hence, the restraint value was doubled for case 7 to give a more realistic value [300]. The effective heights can be derived analytically [225]. For case 4, it has been assumed that (a) under rotation the axial force remains perpendicular to the mid-plane of the elas- tomeric bearing, and (b) under lateral displacement the axial force acts through the centre line of the bearing [300]. Slenderness limits (p.119) The upper limit of slenderness ratio (l.Jh) is generally now 40, since this is considered a practical upper limit. However, when sway can occur, the upper limit.is 30, as suggested on page 119, and equation (9.2) is no longer used. Axial load (p.119) Equation (9.4) is now omitted from the Code. Code formulae (p.120) Equations (9.5) to (9.13) are still appropriate, if the following modifications are applied: 1. In equation (9.6), the term 0.4f('l/ should be re- placed twice by 0.45,t;·u' The latter value is correct for zero eccentricity and zero reinforcement, and is a conservative value for larger eccentricities when the minimum reinforcement, discussed on page 138, is provided. 2. In equations (9.7) and (9.8), the term O.72fy should be replaced with fy,' which is equal to expression 23
- 113. CaseJ 1,,:::0'7/0 Case 2 .Ie =.0.8510 Case6 11'=1'5/0 9.00 Encctivl~ hcight~ (4.1) on page 4X. Hence, the conservative value of 0.72/1' is no longer lIsed for all steels: instead, for current steel strengths; the appropriate values offl" arc O.7X4J; for mild steel and (I.725fy for high- .' strength steel. Axial load pillS biaxial bending (p.121) Nil: is now c"lculat~d from equation (9.3) with the term 0.75fl' replaced with fv(' (see the last paragraph f~r the ex- planation). Slender columns (p. 122) The nomin~1 minimum initial eccentricity in .Part 4 : 1978 of O.05h to allow for construction tolerances has been found unnecessarily large for the size of columns usually used in bridges. Hence, it now has an upper limit of 20 mm. In addition for biaxial bending, the nominal eccentricity is now 0.3 times the overall depth of the cross-section in the appropriate. plane of bending (but ::}20 mm). The notation used fo), the dimensions of a column caused confusion in Part 4: 1978. Hence. in Part 4 : 1984 the notation .shown in Fig. 9.9aisuscdconsis- tcntly. As a result: I. In l'quation (9.22). It should be replaced with hr' 2. If 11I'::}3 h.. equation (lJ.26) may be used with band " replaced with ht' and hI' respectively 3~ In equations (9~27).and·(9.28),band h should oc replaced with h, and hI' respectively. Stresses (p. 124) The concrete stress is now limited tqO.38J;/i instead of 0.5/;/1 under axial loading. Thus, the design resistance at serviceability n() longer exceeds that at ultimate.. .However. the ultimate limit state is still likely to be critical when the loading is predominantly axial. 24 I Case 7 Ie =2-3 /0 I I ~ No restraint to sideswayin cases 6 and 7, hy x- - +- - - -x I 1 9.9. Column section notation Crack widths (p. 124) The design crack width for very severe exposures has been increased to 0,15 mm and a different crack width formula introduced. with the result that crack control is now less likely to be the critical design criterion for a column with a large eccentricity of load. Axialloati (p.125) No reference is. now made to walls subjected only to axial loading. Eccl'Il(r;c load (p.125) The reference to 'no tension' when. determining the distribution of in-plane forces per unit length has now beenomitted, General (p.126) In order to reflect current practice, the Code clauses arc only appropriate to walls having a height to average thickness ratio not exceeding five. In view of this restriction. all references to effective height and to shori and slender walls arc now omitted. Unbraced walls are now referred to as walls unrestrained in ,'losition. ';':'., Axial load plus bending normlll to wall (p.126) The coefficient All' has been considerably simplified and Call now take one of two values: 0.35 for grade 15 or 20 concrete, and 0.4 for grades 25 and above. Slender.walls are no longer covered by the Code. Shear (p.128) No reference is now made to shear forces at right angles to a wall. However. it is understood that the clauses for shear forces in the plane of a wall. which arc discussed on page 128. should also be applied to shear forces'at right-angles to a wall. Bearing (p.128) Equation (8.4A) should be used Deflection (p.128) Since the height to thickness ratio is now limited to five, deflections need not be considered, Crack control (p. 128) Reinforcement has to be provided only to control shrinkage and temperature cracks. The required amount of reinforcement is given by equation (7.2a). • and its spacing should not exceed 150 mm. Effective heights (p.129) The new effective heights have been derived specifi- cally for bridges. Bridge abutments and wing walls (p.129) There are no longer deemed to satisfy ·rules for the bar spacings. Flexure (p. UO) Reinforcement should be distributed evenly across the width of the footing unless the width exceeds 1.5 (h('ol + 3d). where bcol is the width of the column. In such a case. two-thirds of the reinforcement should be concentrated in the width (bcol + 3d) centred on the column. Hence. Fig. 9.15 is no longer appro- priate. and Part 4 : 19H4 requires a greater concen- tration of reinforcement around the column than did Part 4 : 1Y78. Pllnching shear (p.l31) The appropriate perimeters arc now those shown in Fig. 6.6A. Serviceability limit state (p. 131) The limiting steel stress is now O.75J~. NeXlifal shear (p.DI) Any section across the full width riow has to be checked, as opposed to just a section at the face of the column, Punching shear (p.132) The perimeter shown in Fig. 9.17(b) is no longer used: instead. the perimeters shown in Fig. 6.6A sholiid he ustld for the punching bf both the columns and the piles. 9.1 Slender column (p.133) A characteristic strength of 425 N/mm2 is retained for the rC,inforcemcnt, because, at the time of vriting, column design charts are n()t generally available for 460 N/mm2 reinforcement. Calculations are presented only for the ultimate limit state. No sidesway (p.133) Case 2 of Fig. 9.0a is appropriate. Hem:e Ie .-:.::. 0.85 tv ;:: 0.85 X 8 =: 6.8 m Slenderness ratio = 6.8/0.6 :: 11.3. This do.es not exceed 12. thus the column is short. .The minimum eccentricity is the lesser of 0.05 x 0.6 = 0.03 m and 20 mm (i.e. 20. mm). Moment at base == 280 x 8 + 2600 x 0.02= 2292 kN m Mlbh'1. := Mlhyhx 2 = 2292 x 106 /(1200 x 60(2) =5.31 N/mm2 Nlbh =- N1hvhx := 2600 x m1 /(l200 x 6(0) . =3.61 N/mm2 . Assume 40 mrn main bars, 10 mm links and 50 mm cover for very severe exposure (see amendments to Chapter 10), so that dlhx = 520/600 = 0.867. Hence. use Design Chart 85 of CPll0 : Part 2 112g]: from which 100 A.,lhyh.t == 3.1 ... AS( = 3.1 x 1200 x flOo/I00 .::: 22.320 mm2 .Use 18 No. 40 mm bars (22680 mm2 ) with 9 bars in each face. Sidesway (p.133) Case 7 of Fig. 9.0a is appropriate. Hence Ie == 2.3/" :=:: 2.3 X 8 =18.4 m Slenderness ratio = 18.4/0.6 == 30.7. It is emphasised that this ratio exceeds the Part 4 : 1984 limitation of 30; however. in order to provide a design comparative with that of Part 4: 1978. the section size will not oe increased. The column is slender, and the initial and additional moments are both maxima at the base. l.i~ence . ¥l == 280 x 8::= 2240 kN m and, from equation (9.22). t'he total moment is Ntl == 2240 + (2600 x 0,(11750)(30.7)2 (I -·0.0035 x 30.7) == 2240 + 750 := 2990 kN m 25
- 114. M/hh2 =M/hyh.~ ";'~99(}XlOh 1(1200X 6()02) = 6.92 N/mm2 NIbil =Nllzyhx= 3.61 N/mm2 (as before) From DesigltChart 85 of CPllO : Part 2, 100 A.,,lhvhx =4.4 .. .. An =' 4.4 X 1200 X 6001100 ~ 31680 mm2 .. llse. 26 No. 40.mmbars{32700mm2Ywith 13 bars in . ..' eac~ fa~e. .... .... .........•.... ,.' ...... ... ' ::........ .... ...•...• . . . . Sheaf ~hear reinforcement will be designed only fol' the no . ,sl~esway case, w~ich has the.le.ast amOUnt oftension .remforccrnentofthe two cas¢~ c<)nsidered,.. . The t.ensionsteel consists' of 9 No. 40 mm bars (11 300 mm2 ). 100 As,lbd = 100 x 11300/(1200 x 520) = 1.81 From equation(6.1al-- '" Vc: =(O.27/L2S)(1.81)1~ (40)IIJ .. . =0.9 N/mm2 '. . . From equation (6.7c) ~' =(SO(}/S20)1i4 =0.99 Axial load enhancement factor = 1 + 0.05 x 2600 x 103/(600 x 12(0) =1.18 Thus, enhanced value of t-v(. = 1.18 x 0.99 x 0.9 = 1.05 N/mm2 Applied s~ear stress"'; v'= 280 x 103/(1200 x 520) =0.449 N/mm2 < 1.0S N/mm2, thus only nominal links required. 9.2 Hillerborg strip method applied to an abutment (p.133) Since the Hillerborg strip method is a plastic method, the value ofYf3 should be 1.15 for all loads. Hence, the HA surcharge and braking design loads shoul.d be . d 2 mcrease t~ 17.25 kN/m and 43.1 kN/m respectively. The me.thod of calculation thereafter is unchanged. In FIg. 9.19, the x and y axes are vertical and horizontal respectively. 9.3 Pile cap (p.135) The reinforcement characteristic strength is increased !o 460 N/mm 2 . The concrete. characteristic strength .is Increased to 40 N/mm2 to avoid the use of air-entrained concrete (see amendments to Chapter 10). Bending theory (p.135) Bending (p.135) Because of the increased characteris- tic.strengthof the steel, only 17 No; 20 mlTi bars (5340 ,mm2) are required. Flexural shear (p.135) 100 A,Ibd =(100 x 5340)/(2300 x 980) =0.237 From equation (6.18) v(' =(0.27/1.25)(O.237)1fJ (40)11.l =0.457 N/mm2 From equation (6.7c) ' . 26 . t . cap 9.258 Punching perimete.rJor Example 9,~ ~! =(500/980)114 =0.845 Cr~tical sect~on is where a" is aslargc as possible; thus, as In the maIO text, a" =300 mm, and the enhancement factor is 6.53. Thus, the shear capacity of the critical section = 0.845·x 0.457 [(6.53)(2 x 500) + (2300 - 2 x500)1 . x 980 x 10-' =2960 kN Actual shear force =2 x 1300 =2600 kN < 2960 kN .·.O.K. Punching shear (p.136) Only perimeter (d) of Fig. 6.6A can be accommodated. The dimensions of the perimeter are shown in Fig. 9:25a. In one direction, d = 980 mm and 100 A,Ibd = 0.237. If the same rein- forcement is provided in the orthogonal direction with an effective depth of 1000 nlp1, then 100 A,Ibd =0.232. Average 100 A,Ibd = 0.235 From equation (6.18) Vc =(0.27/1.25)(0.235)1/1(40)IA = 0.456 N/mm2 Average d = 990 mm From equation (6.7c) !;.. =(500/990)1/4 =0.843 The perimeter is 386 mm from the corner of the column (see pag,e 136). Hence, the shortshear span enhance- ment factor = 2 x 990/386 =5.13. Length of perimeter 'crossed by reinforcement anchored over pile = 1.63 - 2(0.15) V2== 1.21 m Shear capacity =0.843 x 0.456 [5.13 x 1210 + (1630 - 1210)] 990 x 10-3 == 2520kN Actual shear force = 1300 kN < 2520 kN .·.O.K. Truss analogy (p.136) Truss (p.136) In spite of the increased characteristic strength of the reinforcement, the main steel provided is the same as in the main text. . e "'1 Flexural shear (p.136) Over a pile, v,. from equation (6.1a) is Vi' =.: (0.27/1.25)(0.45)1/1 (40)'11 =0.566 N/mm 2 As before, the short shear span enhancement factor is 6.53. Between and outside piles. v.. = 0.39 N/mm 2 (Tahle 8 of Code). Depth factor. as in the bending theory calculations. = 0.845 Shear capacity = 0.845 [(6.53)(0.566)(2 x 5(0) + (0.39)(2300 - 2 x 500)]980 xlO~:.3 =3480 kN Act~al shear force = 2 x 1300 =2600 kN < 3480 kN .·.O.K. j Punchillg shear (p.136) The critical perimeter is shown in Fig. 9.25a. Average d =990 mm. There will be 9 bars (2830 mm2 ) crossing the perimeter in each of the orthogonal direc- tions, thus 100 A/bd = (100 x 2830)/(1150 x 990) =0.249 From equation (6.1a) Vc =(0.27/1.25)(0.249)'11 (40)1;., == 0.465 N/mm2 For the average d. the depth factor = 0.843. and the short shear span enhancement factor == 5.13. Shear capacity -'''·''.'''J..,JL~43 x 0.465 [5.13 x 1210 + (1630 - 1210)] 990 ",..... x 10-3 :;; 2570 kN Actual shear force == 1300 kN < 2570 kN ,'. O.K. 27
- 115. Chapter 10 Detailing Cover (p.137) Itis now emphasised in the Code that, although ttH' nominal cover is used for dcsign and is indkuted on the dnlwings, thc actual cover can be up to 5 mm less, The nominal covers in Purt 4: 197H have hecn increased by at least 5 mm. and air-entrained concrete is now required in morc'situations, Thc Part 4 : 19~4 values arc given in Table 1O.lA which sup(:rsedes Tublc 10.1, However, the nominal cover should also I'll.' not less than the bar size or maximum aggregute size plus 5 mill. The mure l)iI~lOull ~overs, etc, rcnc~t the durabil· ity problems which have bcen exp~~ricnced in the UK in recent years. Table IO.IA Nominal covers Nominal cover (mm) Envlronm~nt ror concrete grade 15 30 40 ~SO Extreme Exposed to abrasive action by l-ea water, or to water with a pH ~4.5 6.'i • 55 Very severe Directly afft'ctcd by dc-icing ',I, salts. or sea water spray ;. 50" 40 Severc Exposed to driving ran, or altcrnatc wetting and drying 45' 35 30 Moderate Shcltered. orpermancntly saturated with water with a pH >4.5 45 35 30 25 • Air-entrained concrete needed if liahle to freezing whilst wet. t Grade 30 permitted only for parapetsif air entrained and 60 mmcover. Minimum distance between bars (p. 137) Thc minimum distance betwecn rows of bars in in-situ .members has been increased to the maximum aggre- gate size, 28 Shrinkage and temperature reinforcement (p, 138) The arcas ()f reinforcement rcqulrell to t:ontrol shink· agc and temperature cracks havl~ heen, essentially. doubled to the values givl~n by equation (7.2a), • Beams and slabs (p, 138) Correcl the term (/t 2 /d)oin the last equation on page 13H to (h/df, Minimum areas of secondary reinforcement in pre- (10mmantly tensile areas of solid slahs are now given LIS O.12cyo of high strength stcel or 0.15'Yc, of mild steel. These percentages are numerically the same as those in the building codes (7.15). but are expressed in terms of the effective area rather than the gross area, Thus smaller minimum areas of secondary reinforcement are required than in accordan~ with CP114. The percentages of secO,ndary reinforcement men- tioned in the last paragr~ph should also be PfQ,vided where the main reinforcement resists compressfon. The diameter of the secondary reinforcement should not he less than one-quarter of that of the main bars, and the spacing should not exceed 300 mm, However, as ex- plained on page 139, links should be provided if the area of compression reinforcement exceeds 1%. The clauses concerned with the restraint of compression hars in s.labs (and as mentioned later, walls) are consis- tent with those in the new building. code (BSHI!O) 1294]. It is now necessary to provide longitudinal reinforce- ment to control cracking at the side of heams where thc depth of the side face exceeds 600 mm. Steel having an area of 0.05% of the effective section area should he provided in each face with a spacing not greater than 300 mm. Walls (p.138) The above requirements for secondary reinforccment in slabs. where the main reinforcement resists com- pression, are also applicable to walls. • Lap lengths (p.141) It is no longer necessary to increase the lap length by 25% for deformed bars in tension. However. it is known that the bond strength of top cast bars is less than that of bottom cast bars [301], and that the mode of failure of a lap changes from slip to splitting of the concr~te cover when the cover is less than about 2.5 times the bar diamcters [302, 303}, Hence, longer lap lengths are required in such situ- ations. Thus. the Code requires the lap length to be increased by 40% if any of the following conditions apply: 1. The bars are in the top of the section as cast. and the nominal cover is less than twice the bar diam- eter. 2, The clear distance between the lap and another pair of lapped bars is less than 150 mm. 3. The lap is in a corner, and the nominal cover to either face is less than twice the bar diameter. The lap lengthshould be increascd by 1000t'c, if either conditions 1 and 2 or 1 and 3 occur together, Anchorage at a simply supported end (p.142) Condition 3 on page 142 has heen removed from the Code. External tendons (p. 143) The Code no longer deals with external tendons. Transmission length in pre-tensioned members (p. 143) The following formula is now given for the transmission length (I,) wh(~re the initial stress does not exceed 75'X, of the characteristic strength of the tendon. and the concrete strength at transfer is at least 30 N/mm2: I, ::: k, <l>tv'r., (10.6a) where <t> is th,~ nominal diameter of the tendon in mm. and k/ is a coeffident (in the range 240 to 6(0) which depends on the type of tendon. This formula results in transmission length!. very similar to thos(~ in Part 4 : 1978 for wires, but thc transmission lengths for strand arc ahout 40(Yc, greater than those in Part 4 : 1978. This increase reflects currt:!nt test data. 29
- 116. • Chapter 1.1 . ".. .. ' Lightweight aggregate conCrete Introduction (p.147) ,The 'Cadena longer covers prestressed lightweight ,;ggregate concrete. Durability (p.147) Covers .should be 10 mm greater than those given in Table 1O.IA. Appendix A Equations for plate design In equation (AI3)on page 170 correct the third term on the right-hand side to My cot2 a. 30 f}j........ Compressive strength (p.148) Lightweight aggregate concrete for teinf()rced concrete must have a characteristic strength of at least 25 N/mm2 . Shear strength (p.148) The average value of O)~ for the reduction factor has bet:n retained. This factor should be applied to the normal weight aggregate concrete values of 11(, VI/' V,m in and V//I' However, the interface shear stresses for normal weight aggregate concrete have to be multiplied by 0.75 to give the equivalent lightweight aggregate concrete values. ,