Lecture 7 stress distribution in soil
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The document provides information about stress distribution in soil due to self-weight and surface loads. It discusses Boussinesq's formula for calculating vertical stress in soil due to a concentrated surface load. The formula shows that vertical stress is directly proportional to the load, inversely proportional to depth squared, and depends on the ratio of radius to depth. A table of coefficient values used in the formula for different ratios of radius to depth is also provided.
![Vertical Stress in Soil
Stress Due to a Concentrated Load
According to Boussinesq’s analysis, the vertical stress
increase at point A caused by a point load of magnitude P
is given by
2 2 5/2
3 1
2 [1 ( / ) ]
z
P
z r z
σ
π
=
+
39Dr. Abdulmannan Orabi IUST
… … . 7 − 1
1
∆
.
Q
or
2z b
P
I
z
σ =](https://image.slidesharecdn.com/lecture7stressdistributioninsoil-161122164340/85/Lecture-7-stress-distribution-in-soil-39-320.jpg)
![Equation shows that the vertical stress is
Directly proportional to the load
Inversely proportional to the depth squared, and
Proportional to some function of the ratio ( r/z).
Vertical Stress in Soil
Stress Due to a Concentrated Load
where
2 5/2
3 1
2 [1 ( / ) ]
bI
r zπ
=
+
40Dr. Abdulmannan Orabi IUST
… … … … . 7 − 2](https://image.slidesharecdn.com/lecture7stressdistributioninsoil-161122164340/85/Lecture-7-stress-distribution-in-soil-40-320.jpg)
![Vertical Stress in Soil
Vertical Stress caused by a strip load
α
β
6
B
Vertical stress at point A can be determined by equation:
[ sin cos( 2 )]o
z
q
σ α α α β
π
= + +
P
56Dr. Abdulmannan Orabi IUST
… . 7 − 6](https://image.slidesharecdn.com/lecture7stressdistributioninsoil-161122164340/85/Lecture-7-stress-distribution-in-soil-56-320.jpg)
![B
[
0.25
0.5
]
^
0.5
0.25
Worked Examples
Example 4
Refer to figure below, The magnitude of the strip
load is 120 kPa. Calculate the vertical stress at
points, a , b, and c.
57Dr. Abdulmannan Orabi IUST](https://image.slidesharecdn.com/lecture7stressdistributioninsoil-161122164340/85/Lecture-7-stress-distribution-in-soil-57-320.jpg)
![_
6
4
"
+
_
1 2 2[( )( ) ( )]o
z
q a b b
a a
σ α α α
π
+
= + −
Vertical Stress Due to Embankment Loading
The vertical stress increase in the soil mass due to
an embankment of height H may be expressed as
Vertical Stress in Soil
" )where:
`4+ a`
) `4+ a`
58Dr. Abdulmannan Orabi IUST
… . 7 − 7](https://image.slidesharecdn.com/lecture7stressdistributioninsoil-161122164340/85/Lecture-7-stress-distribution-in-soil-58-320.jpg)
Lecture 7 stress distribution in soil
- 1. INTERNATIONAL UNIVERSITY FOR SCIENCE & TECHNOLOGY وا م ا و ا ا CIVIL ENGINEERING AND ENVIRONMENTAL DEPARTMENT 303322 - Soil Mechanics Stress Distribution in Soil Dr. Abdulmannan Orabi Lecture 2 Lecture 7
- 2. Dr. Abdulmannan Orabi IUST 2 Das, B., M. (2014), “ Principles of geotechnical Engineering ” Eighth Edition, CENGAGE Learning, ISBN-13: 978-0-495-41130-7. Knappett, J. A. and Craig R. F. (2012), “ Craig’s Soil Mechanics” Eighth Edition, Spon Press, ISBN: 978- 0-415-56125-9. References
- 3. Stress in soil due to self weight Stress Distribution in Soil Stress in soil due to surface load 3Dr. Abdulmannan Orabi IUST
- 4. Stress due to self weight The vertical stress on element A can be determined simply from the mass of the overlying material. If represents the unit weight of the soil, the vertical stress is Variation of stresses with depth A Ground surface zz ⋅= γσ 4Dr. Abdulmannan Orabi IUST
- 5. ∑= ⋅=⋅++⋅+⋅= n i iinnz hhhh 1 2211 ...... γγγγσ Stress due to self weight Stresses in a Layered Deposit The stresses in a deposit consisting of layers of soil having different densities may be determined as Vertical stress at depth z1 Vertical stress at depth z2 Vertical stress at depth z3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 5Dr. Abdulmannan Orabi IUST
- 6. With uniform surcharge on infinite land surface Stress due to self weight Original land surface Conversion land surface ∗ 6Dr. Abdulmannan Orabi IUST
- 7. Stress due to self weight ∗ Vertical stresses due to self weight increase with depth, There are 3 types of geostatic stresses: a. Total Stress, σtotal b. Effective Stress, σ' c. Pore Water Pressure, u Vertical Stresses 7Dr. Abdulmannan Orabi IUST
- 8. Stress due to self weight Consider a soil mass having a horizontal surface and with the water table at surface level. The total vertical stress at depth z is equal to the weight of all material (solids + water) per unit area above that depth ,i.e Total vertical stress !"!#$ %#! ∗ 8Dr. Abdulmannan Orabi IUST
- 9. Stress due to self weight The pore water pressure at any depth will be hydrostatic since the void space between the solid particles is continuous, therefore at depth z: Pore water pressure & ∗ If the pores of a soil mass are filled with water and if a pressure induced into the pore water, tries to separate the grains, this pressure is termed as pore water pressure 9Dr. Abdulmannan Orabi IUST
- 10. Stress due to self weight Effective vertical stress due to self weight of soil The difference between the total stress ( !"!#$) and the pore pressure (u) in a saturated soil has been defined by Terzaghi as the effective stress ( ).' ' !"!#$ − The pressure transmitted through grain to grain at the contact points through a soil mass is termed as effective pressure. 10Dr. Abdulmannan Orabi IUST
- 11. Stress due to self weight Stresses in Saturated Soil If water is seeping, the effective stress at any point in a soil mass will differ from that in the static case. It will increase or decrease, depending on the direction of seepage. The increasing in effective pressure due to the flow of water through the pores of the soil is known as seepage pressure. 11Dr. Abdulmannan Orabi IUST
- 12. A column of saturated soil mass with no seepage of water in any direction. The total stress at the elevation of point A can be obtained from the saturated unit weight of the soil and the unit weight of water above it. Thus, Stress due to self weight Stresses in Saturated Soil without Seepage 0 A Solid particle Pore water )* )& + + 12Dr. Abdulmannan Orabi IUST
- 13. 0 A Solid particle Pore water )* )& + + + + Forces acting at the points of contact of soil particles at the level of point A Stress due to self weight Stresses in Saturated Soil without Seepage & ) ,)* − )- %#! where + . + /+ 0 1 %#! + .+ 2 )* 2 + 3 4 0 1 + 2 + . +4 13Dr. Abdulmannan Orabi IUST
- 14. Stress due to self weight Stresses in Saturated Soil without Seepage ) ) 5 6 7 8 Valve (closed) Stress at point A, • Total stress: • Pore water pressure: • Effective stress: * & ) * & ) * ' * − * 0 Stress at point B, • Total stress: • Pore water pressure • Effective stress: : & ) ) ∗ %#! : ,) ) - & : ' : − : : ' ) %;< 14Dr. Abdulmannan Orabi IUST
- 15. Stress due to self weight Stresses in Saturated Soil without Seepage Stress at point C, • Total stress: = & ) ∗ %#! > ,) - & > ' > − > > ' %;< • Pore water pressure: • Effective stress: Total stress Pore water Pressure, u Effective stress DepthDepth Depth 15Dr. Abdulmannan Orabi IUST
- 16. ) ) 5 6 7 8 Valve (open) ? ( @ AB - Stress due to self weight Stresses in Saturated Soil with Upward Seepage Stress at point A, • Total stress: • Pore water pressure: • Effective stress: * & ) * & ) * ' * − * 0 16Dr. Abdulmannan Orabi IUST
- 17. Stresses in Saturated Soil with Upward Seepage Stress due to self weight Stress at point B, • Total stress: • Pore water pressure • Effective stress: : & ) ) ∗ %#! : ,) ) - & : ' : − : : ' ) %;< − & 17Dr. Abdulmannan Orabi IUST
- 18. Stresses in Saturated Soil with Upward Seepage Stress due to self weight Stress at point C, • Total stress: • Pore water pressure: • Effective stress: = & ) ∗ %#! : ,) ) - & > ' > − > > ' %;< − ) & > ' %;< − & Note that h/H2 is the hydraulic gradient i caused by the flow, and therefore 18Dr. Abdulmannan Orabi IUST
- 19. Total stress Pore water Pressure, u Effective stress DepthDepth Depth Stress due to self weight Stresses in Saturated Soil with Upward Seepage 19Dr. Abdulmannan Orabi IUST
- 20. Stress due to self weight Stresses in Saturated Soil with Upward Seepage At any depth z, is the pressure of the submerged soil acting downward and is the seepage pressure acting upward. The effective pressure reduces to zero when these two pressures balance. This situation generally is referred to as boiling. > ' %;< − >C & 0 >C %;< & . >C 3. 3+ D2.+ 3 .+2 For most soils, the value of >C varies from 0.9 to 1.1 %;< & > ' 20Dr. Abdulmannan Orabi IUST
- 21. ) ) 5 6 7 8 Valve (open) ? ( @ AB - Stress due to self weight Stresses in Saturated Soil with Downward Seepage Stress at point A, • Total stress: • Pore water pressure: • Effective stress: * & ) * & ) * ' * − * 0 21Dr. Abdulmannan Orabi IUST
- 22. Stress at point B, • Total stress: • Pore water pressure • Effective stress: : & ) ) ∗ %#! : ,) ) − - & : ' : − : : ' ) %;< & Stress due to self weight Stresses in Saturated Soil with Downward Seepage 22Dr. Abdulmannan Orabi IUST
- 23. Stress due to self weight Stress at point C, • Total stress: • Pore water pressure: • Effective stress: = & ) ∗ %#! : ,) − ) - & > ' > − > > ' %;< ) & > ' %;< & Stresses in Saturated Soil with Downward Seepage 23Dr. Abdulmannan Orabi IUST
- 24. Pore water Pressure, uTotal stress Effective stress DepthDepth Depth Stress due to self weight Stresses in Saturated Soil with Downward Seepage 24Dr. Abdulmannan Orabi IUST
- 25. Worked Examples Example 1 A soil profile is shown in figure below. Calculate total stress, pore water pressure, and effective stress at A, B, C, and D. D C B A Ground surface G.W.T Sand Clay Sandγ = 16.3 kN/m^3 γ = 15.1 kN/m^3 γ = 19.8 kN/m^3 1.8 m 1.6 m 2.9 m 25Dr. Abdulmannan Orabi IUST
- 26. Stress due to self weight Total stress Effective stress Pore water pressure DepthDepthDepth γ1 X H1 γ1 X H1 + γ2 X H2 γ1 X H1 + γ2 X H2 + γ3 X H3 γ1 X H1 + γ2 X H2 + γsub X H3 γw X Hw 26Dr. Abdulmannan Orabi IUST
- 27. To analyze problems such as compressibility of soils, bearing capacity of foundations, stability of embankments, and lateral pressure on earth retaining structures, we need to know the nature of the distribution of stress along a given cross section of the soil profile. Stress due to surface load Introduction 27Dr. Abdulmannan Orabi IUST
- 28. When a load is applied to the soil surface, it increases the vertical stresses within the soil mass. The increased stresses are greatest directly under the loaded area, but extend indefinitely in all directions. Introduction 28Dr. Abdulmannan Orabi IUST Stress due to surface load
- 29. •Allowable settlement, usually set by building codes, may control the allowable bearing capacity. •The vertical stress increase with depth must be determined to calculate the amount of settlement that a foundation may undergo Introduction 29Dr. Abdulmannan Orabi IUST Stress due to surface load
- 30. Introduction Foundations and structures placed on the surface of the earth will produce stresses in the soil These stresses will decrease with the distance from the load How these stresses decrease depends upon the nature of the soil bearing the load 30Dr. Abdulmannan Orabi IUST Stress due to surface load
- 31. Individual column footings or wheel loads may be replaced by equivalent point loads provided that the stresses are to be calculated at points sufficiently far from the point of application of the point load. Stress Due to a Concentrated Load 31Dr. Abdulmannan Orabi IUST Stress due to surface load
- 32. Stresses in soil due to surface load Vertical stress due to a concentrated load • Boussinesq’s Formula • Wastergaard Formula Stress Due to a Concentrated Load 32Dr. Abdulmannan Orabi IUST
- 33. Stress Due to a Concentrated Load Boussinesq’s Formula for Point Loads Joseph Valentin Boussinesq (13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. 33Dr. Abdulmannan Orabi IUST Stresses in soil due to surface load
- 34. In 1885, Boussinesq developed the mathematical relationships for determining the normal and shear stresses at any point inside a homogenous, elastic and isotropic mediums due to a concentrated point loads located at the surface Vertical Stress in Soil Stress Due to a Concentrated Load 34Dr. Abdulmannan Orabi IUST
- 35. The soil mass is elastic, isotropic (having identical properties in all direction throughout), homogeneous (identical elastic properties) and semi-infinite depth. The soil is weightless. Stress Due to a Concentrated Load Assumption: 35Dr. Abdulmannan Orabi IUST Vertical Stress in Soil
- 36. The distribution of σz in the elastic medium is apparently radially symmetrical. The stress is infinite at the surface directly beneath the point load and decreases with the square of the depth. Vertical Stress in Soil Stress Due to a Concentrated Load 36Dr. Abdulmannan Orabi IUST
- 37. At any given non-zero radius, r, from the point of load application, the vertical stress is zero at the surface, increases to a maximum value at a depth where , approximately, and then decreases with depth. E 39.25° Vertical Stress in Soil Stress Due to a Concentrated Load 37Dr. Abdulmannan Orabi IUST
- 38. Vertical Stress in Soil According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by Stress Due to a Concentrated Load D ∆ ∆ M ∆ N O P Q . P D 1 38Dr. Abdulmannan Orabi IUST
- 39. Vertical Stress in Soil Stress Due to a Concentrated Load According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by 2 2 5/2 3 1 2 [1 ( / ) ] z P z r z σ π = + 39Dr. Abdulmannan Orabi IUST … … . 7 − 1 1 ∆ . Q or 2z b P I z σ =
- 40. Equation shows that the vertical stress is Directly proportional to the load Inversely proportional to the depth squared, and Proportional to some function of the ratio ( r/z). Vertical Stress in Soil Stress Due to a Concentrated Load where 2 5/2 3 1 2 [1 ( / ) ] bI r zπ = + 40Dr. Abdulmannan Orabi IUST … … … … . 7 − 2
- 41. It should be noted that the expression for z is independent of elastic modulus (E) and Poisson’s ratio (µ), i.e. stress increase with depth is a function of geometry only. Vertical Stress in Soil Stress Due to a Concentrated Load 41Dr. Abdulmannan Orabi IUST
- 42. r/Z IB r/Z IB r/Z IB r/Z IB 0.00 0.4775 0.18 0.4409 0.36 0.3521 0.55 0.2466 0.01 0.4773 0.19 0.4370 0.37 0.3465 0.56 0.2414 0.02 0.477 0.20 0.4329 0.38 0.3408 0.57 0.2363 0.03 0.4764 0.21 0.4286 0.39 0.3351 0.58 0.2313 0.04 0.4756 0.22 0.4242 0.40 0.3294 0.59 0.2263 0.05 0.4745 0.23 0.4197 0.41 0.3238 0.60 0.2214 0.06 0.472 0.24 0.4151 0.42 0.3181 0.61 0.2165 0.07 0.4717 0.25 0.4103 0.43 0.3124 0.62 0.2117 0.08 0.4699 0.26 0.4054 0.44 0.3068 0.63 0.2070 0.09 0.4679 0.27 0.4004 0.45 0.3011 0.64 0.2024 0.1 0.4657 0.28 0.3954 0.46 0.2955 0.65 0.1978 0.11 0.4633 0.29 0.3902 0.47 0.2899 0.66 0.1934 0.12 0.4607 0.30 0.3849 0.48 0.2843 0.67 0.1889 0.13 0.4579 0.31 0.3796 0.49 0.2788 0.68 0.1846 0.14 0.4548 0.32 0.3742 0.50 0.2733 0.69 0.1804 0.15 0.4516 0.33 0.3687 0.51 0.2679 0.70 0.1762 0.16 0.4482 0.34 0.3632 0.52 0.2625 0.71 0.1721 0.17 0.4446 0.35 0.3577 0.53 0.2571 0.72 0.1681 0.54 0.2518 0.73 0.1641 Influence Factor Ib 42Dr. Abdulmannan Orabi IUST
- 43. r/Z IB r/Z IB r/Z IB r/Z IB 0.74 0.1603 0.94 0.0981 1.14 0.0595 1.34 0.0365 0.75 0.1565 0.95 0.0956 1.15 0.0581 1.35 0.0357 0.76 0.1527 0.96 0.0933 1.16 0.0567 1.36 0.0348 0.77 0.1491 0.97 0.0910 1.17 0.0553 1.37 0.0340 0.78 0.1455 0.98 0.0887 1.18 0.0539 1.38 0.0332 0.79 0.1420 0.99 0.0865 1.19 0.0526 1.39 0.0324 0.80 0.1386 1.0 0.0844 1.20 0.0513 1.40 0.0317 0.81 0.1353 1.01 0.0823 1.21 0.0501 1.41 0.0309 0.82 0.1320 1.02 0.0803 1.22 0.0489 1.42 0.0302 0.83 0.1288 1.03 0.0783 1.23 0.0477 1.43 0.0295 0.84 0.1257 1.04 0.0764 1.24 0.0466 1.44 0.0283 0.85 0.1226 1.05 0.0744 1.25 0.0454 1.45 0.0282 0.86 0.1196 1.06 0.0727 1.26 0.0443 1.46 0.0275 0.87 0.1166 1.07 0.0709 1.27 0.0433 1.47 0.0269 0.88 0.1138 1.08 0.0691 1.28 0.0422 1.48 0.0263 0.89 0.1110 1.09 0.0674 1.29 0.0412 1.49 0.0257 0.90 0.1083 1.10 0.0658 1.30 0.0402 1.50 0.0251 0.91 0.1057 1.11 0.0641 1.31 0.0393 1.51 0.0245 0.92 0.1031 1.12 0.0626 1.32 0.0384 1.52 0.0240 0.93 0.1005 1.13 0.0610 1.33 0.0374 1.53 0.0234 43Dr. Abdulmannan Orabi IUST Influence Factor Ib
- 44. r/Z IB r/Z IB r/Z IB r/Z IB 1.54 0.0229 1.66 0.0175 1.86 0.0114 2.5 0.0034 1.55 0.0224 1.67 0.0171 1.88 0.0109 2.6 0.0029 1.56 0.0219 1.68 0.0167 1.90 0.0105 2.7 0.0024 1.57 0.0214 1.69 0.0163 1.92 0.0101 2.8 0.0021 1.58 0.0209 1.70 0.0160 1.94 0.0097 2.9 0.0017 1.59 0.0204 1.72 0.0153 1.96 0.0093 3.0 0.0015 1.60 0.0200 1.74 0.0147 1.98 0.0089 3.5 0.0007 1.61 0.0195 1.76 0.0141 2.0 0.0085 4.0 0.0004 1.62 0.0191 1.78 0.0135 2.1 0.0070 4.5 0.0002 1.63 0.0187 1.80 0.0129 2.2 0.0058 5.0 0.0001 1.64 0.0183 1.82 0.0124 2.3 0.0048 1.65 0.0179 1.84 0.0119 2.4 0.0040 44Dr. Abdulmannan Orabi IUST Influence Factor Ib
- 45. Equation may be used to draw three types of pressure distribution diagram. They are: The vertical stress distribution on a horizontal plane at depth of z below the ground surface The vertical stress distribution on a vertical plane at a distance of r from the load point, and The stress isobar. Vertical Stress in Soil Pressure Distribution Diagram 45Dr. Abdulmannan Orabi IUST
- 46. The vertical stress distribution on a horizontal plane at depth of z below the ground surface U 5 5 Vertical Stress in Soil Distribution on a horizontal plane 46Dr. Abdulmannan Orabi IUST
- 47. The vertical stress distribution on a vertical plane at a distance of r from the point load . Vertical Stress in Soil Distribution on a vertical plane O 47Dr. Abdulmannan Orabi IUST
- 48. U Vertical Stress in Soil Stress isobars An isobar is a line which connects all points of equal stress below the ground surface. In other words, an isobar is a stress contour. 48Dr. Abdulmannan Orabi IUST
- 49. What is the vertical stress at point A of figure below for the two loads, P1 and P2 ? P1 = 350 kNP2 = 470 kN Z=2.5m 2.3 m1.1 m A Worked Examples Example 2 49Dr. Abdulmannan Orabi IUST
- 50. A four concentrated forces are located at corners of a rectangular area with dimensions 8 m by 6 m as shown in figure in the next slide. Compute the vertical stress at points A and B, which are located on the lines A – A’ , B – B’ at depth of 4 m below the ground surface. Worked Examples Example 3 50Dr. Abdulmannan Orabi IUST
- 51. 700 kN700 kN 700 kN700 kN 4 m 4 m 8 m B A’ A B’ Worked Examples Example 3 Dr. Abdulmannan Orabi IUST 51
- 52. Vertical Stress in Soil Westergaard Formula Westergaard proposed a formula for the computation of vertical stress by a point load, P at the surface as O + 2V + . / In which µ is Poisson’s ratio + 1 − 2X /,2 − 2X- 52Dr. Abdulmannan Orabi IUST … . 7 − 3
- 53. Vertical Stress in Soil Stress below a Line Load The vertical stress increase due to line load , , inside the soil mass can be determined by using the principles of the theory of elasticity, or 2 V P This equation can be rewritten as / 2 V 1 P 1P P 53Dr. Abdulmannan Orabi IUST … . 7 − 4
- 54. Vertical Stress in Soil Vertical Stress caused by a horizontal line load The vertical stress increase ( ) at point A in the soil mass caused by a horizontal line load can be given as : 2 P V P 1 / P 54Dr. Abdulmannan Orabi IUST … . 7 − 5
- 55. Vertical Stress in Soil Vertical Stress caused by a strip load The fundamental equation for the vertical stress increase at a point in a soil mass as the result of a line load can be used to determine the vertical stress at a point caused by a flexible strip load of width B. The term strip loading will be used to indicate a loading that has a finite width along the x axis but an infinite length along the y axis. 55Dr. Abdulmannan Orabi IUST
- 56. Vertical Stress in Soil Vertical Stress caused by a strip load α β 6 B Vertical stress at point A can be determined by equation: [ sin cos( 2 )]o z q σ α α α β π = + + P 56Dr. Abdulmannan Orabi IUST … . 7 − 6
- 57. B [ 0.25 0.5 ] ^ 0.5 0.25 Worked Examples Example 4 Refer to figure below, The magnitude of the strip load is 120 kPa. Calculate the vertical stress at points, a , b, and c. 57Dr. Abdulmannan Orabi IUST
- 58. _ 6 4 " + _ 1 2 2[( )( ) ( )]o z q a b b a a σ α α α π + = + − Vertical Stress Due to Embankment Loading The vertical stress increase in the soil mass due to an embankment of height H may be expressed as Vertical Stress in Soil " )where: `4+ a` ) `4+ a` 58Dr. Abdulmannan Orabi IUST … . 7 − 7
- 59. ^ 7 6 2 ` 120 aO+ 3 ` 2 ` Refer to figure below. The magnitude of the load is 120 kPa. Calculate the vertical stress at points, A , B, and C. Worked Examples Example 4 59Dr. Abdulmannan Orabi IUST
- 60. 1- Under the center: The increase in the vertical stress ( ) at depth z ( point A)under the center of a circular area of diameter D = 2R carrying a uniform pressure q is given by Vertical Stress in Soil Vertical Stress due to a uniformly loaded circular area 1 − 1 Q/ 1 / 60Dr. Abdulmannan Orabi IUST … . 7 − 8
- 61. Vertical Stress in Soil Vertical Stress due to a uniformly loaded circular area 6 6' Q 6' 6 61Dr. Abdulmannan Orabi IUST
- 62. Vertical Stress in Soil 2- At any point: The increase in the vertical stress ( ) at any point located at a depth z at any distance r from the center of the loaded area can be given Vertical Stress due to a uniformly loaded circular area where and are functions of z/R and r/R. 1' ' 1' ' 62Dr. Abdulmannan Orabi IUST … . 7 − 9
- 63. Vertical Stress in Soil Vertical Stress due to a uniformly loaded circular area 7 7' . Q 7 7'. 63Dr. Abdulmannan Orabi IUST
- 64. Vertical Stress in Soil Variation of with z/R and r/R.1' 64Dr. Abdulmannan Orabi IUST
- 65. Vertical Stress in Soil Variation of with z/R and r/R.1' 65Dr. Abdulmannan Orabi IUST
- 66. Variation of with z/R and r/R.' Vertical Stress in Soil 66Dr. Abdulmannan Orabi IUST
- 67. Vertical Stress in Soil Variation of with z/R and r/R.' 67Dr. Abdulmannan Orabi IUST
- 68. Vertical Stress in Soil Vertical Stress Caused by a Rectangular loaded area The increase in the vertical stress ( ) at depth z under a corner of a rectangular area of dimensions B = m z and L = n z carrying a uniform pressure q is given by: z o zq Iσ = c 3 +3 . 2 0 2 .+ d + 2 where : 68Dr. Abdulmannan Orabi IUST … . 7 − 10
- 69. Vertical Stress in Soil Vertical Stress Caused by a Rectangular loaded area c 1 4V 2 ` ` 1 ` ` 1 ` 2 ` 1 + e 2 ` ` 1 ` − ` 1 The influence factor can be expressed as ` d + 2 where : 69Dr. Abdulmannan Orabi IUST … . 7 − 11
- 70. The increase in the stress at any point below a rectangular loaded area can be found by dividing the area into four rectangles. The point A’ is the corner common to all four rectangles. Vertical Stress in Soil Vertical Stress Caused by a Rectangular loaded area 1 2 34 6' * f g c g 70Dr. Abdulmannan Orabi IUST
- 71. Vertical Stress in Soil Vertical Stress Caused by a Rectangular loaded area 71Dr. Abdulmannan Orabi IUST 4 6' 6' 6' 6' − h5 h5i − h5 + h5 h5 4 3 4 2 9 1 1 2 3 5 5 1 8 7 9 87 4 7 3 * − − f
- 72. Variation of with m and nc n m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.0047 0.0092 0.0132 0.0168 0.0198 0.0222 0.0242 0.0258 0.2 0.0092 0.0179 0.0259 0.0328 0.0387 0.0435 0.0474 0.0504 0.3 0.0132 0.0259 0.0374 0.0474 0.0559 0.0629 0.0686 0.0731 0.4 0.0168 0.0328 0.0474 0.0602 0.0711 0.0801 0.0873 0.0931 0.5 0.0198 0.0387 0.0559 0.0711 0.0840 0.0947 0.1034 0.1104 0.6 0.0222 0.0435 0.0629 0.0801 0.0947 0.1069 0.1168 0.1247 0.7 0.0242 0.0474 0.0686 0.0873 0.1034 0.1169 0.1277 0.1365 0.8 0.0258 0.0504 0.0731 0.0931 0.1104 0.1247 0.1365 0.1461 0.9 0.0270 0.0528 0.0766 0.0977 0.1158 0.1311 0.1436 0.1537 1.0 0.0279 0.0547 0.0794 0.1013 0.1202 0.1361 0.1491 0.1598 72Dr. Abdulmannan Orabi IUST
- 73. Variation of with m and nc n m 0.9 1 1.2 1.4 1.6 1.8 2.0 2.5 0.1 0.0270 0.0279 0.0293 0.0301 0.0306 0.0309 0.0311 0.0314 0.2 0.0528 0.0547 0.0573 0.0589 0.0599 0.0606 0.0610 0.0616 0.3 0.0766 0.0794 0.0832 0.0856 0.0871 0.0880 0.0887 0.0895 0.4 0.0977 0.1013 0.1063 0.1094 0.1114 0.1126 0.1134 0.1145 0.5 0.1158 0.1202 0.1263 0.1300 0.1324 0.1340 0.1350 0.1363 0.6 0.1311 0.1361 0.1431 0.1475 0.1503 0.1521 0.1533 0.1548 0.7 0.1436 0.1491 0.1570 0.1620 0.1652 0.1672 0.1686 0.1704 0.8 0.1537 0.1598 0.1684 0.1739 0.1774 0.1797 0.1812 0.1832 0.9 0.1619 0.1684 0.1777 0.1836 0.1875 0.1899 0.1915 0.1938 1.0 0.1684 0.1752 0.1851 0.1914 0.1955 0.1981 0.1999 0.2024 73Dr. Abdulmannan Orabi IUST
- 74. Variation of with m and nc n m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.2 0.0293 0.0573 0.0832 0.1063 0.1263 0.1431 0.1570 0.1684 1.4 0.0301 0.0589 0.0856 0.1094 0.1300 0.1475 0.1620 0.1739 1.6 0.0306 0.0599 0.0871 0.1114 0.1324 0.1503 0.1652 0.1774 1.8 0.0309 0.0606 0.0880 0.1126 0.1340 0.1521 0.1672 0.1797 2.0 0.0311 0.0610 0.0887 0.1134 0.1350 0.1533 0.1686 0.1812 2.5 0.0314 0.0616 0.0895 0.1145 0.1363 0.1548 0.1704 0.1832 3.0 0.0315 0.0618 0.0898 0.1150 0.1368 0.1555 0.1711 0.1841 4.0 0.0316 0.0619 0.0901 0.1153 0.1372 0.1560 0.1717 0.1847 5.0 0.0316 0.0620 0.0901 0.1154 0.1374 0.1561 0.1719 0.1849 6.0 0.0316 0.0620 0.0902 0.1154 0.1374 0.1562 0.1719 0.1850 74Dr. Abdulmannan Orabi IUST
- 75. Variation of with m and nc n m 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 1.2 0.1777 0.1851 0.1958 0.2028 0.2073 0.2103 0.2124 0.2151 1.4 0.1836 0.1914 0.2028 0.2102 0.2151 0.2184 0.2206 0.2236 1.6 0.1874 0.1955 0.2073 0.2151 0.2203 0.2237 0.2261 0.2294 1.8 0.1899 0.1981 0.2103 0.2183 0.2237 0.2274 0.2299 0.2333 2.0 0.1915 0.1999 0.2124 0.2206 0.2261 0.2299 0.2325 0.2361 2.5 0.1938 0.2024 0.2151 0.2236 0.2294 0.2333 0.2361 0.2401 3.0 0.1947 0.2034 0.2163 0.2250 0.2309 0.2350 0.2378 0.2420 4.0 0.1954 0.2042 0.2172 0.2260 0.2320 0.2362 0.2391 0.2434 5.0 0.1956 0.2044 0.2175 0.2263 0.2324 0.2366 0.2395 0.2439 6.0 0.1957 0.2045 0.2176 0.2264 0.2325 0.2367 0.2397 0.2441 75Dr. Abdulmannan Orabi IUST
- 76. Approximate Method B B + z 2 1 z " O 76Dr. Abdulmannan Orabi IUST 2V:1H method A simple but approximate method is sometimes used for calculating the stress change at various depths as a result of the application of a pressure at the ground surface. The transmission of stress is assumed to follow outward fanning lines at a slope of 1 horizontal to 2 vertical.
- 77. Approximate Method For uniform footing (B x L) we can estimate the change in vertical stress with depth using the Boston Rule. Assumes stress at depth is constant below foundation influence area B B + z 2 1 z " d ,d - , - " O " O d 77Dr. Abdulmannan Orabi IUST … . 7 − 12 2V:1H method
- 78. Approximate Method B + z L B z Stress on this plane " j d ∗ Stress on this plane at depth z, " d ,d - , - Rectangular footing B B + z 2 1 78Dr. Abdulmannan Orabi IUST 2V:1H method
- 79. Newmark Method 79Dr. Abdulmannan Orabi IUST • Stresses due to foundation loads of arbitrary shape applied at the ground surface • Newmark’s chart provides a graphical method for calculating the stress increase due to a uniformly loaded region, of arbitrary shape resting on a deep homogeneous isotropic elastic region.
- 80. Newmark Method • The Newmark’s Influence Chart method consists of concentric circles drawn to scale, each square contributes a fraction of the stress. • In most charts each square contributes 1/200 (or 0.005) units of stress. (influence value, I) 80Dr. Abdulmannan Orabi IUST
- 81. Newmark Method 81Dr. Abdulmannan Orabi IUST The use of the chart is based on a factor termed the influence value, determined from the number of units into which the chart is subdivided. Influence value 0.005 A B 1 unit
- 82. Newmark Method A B Influence value = 0.005 Total number of block on chart = 200 and influence value = 1/200
- 83. The influence chart may be used to compute the pressure on an element of soil beneath a footing, or from pattern of footings, and for any depth z below the footing. It is only necessary to draw the footing pattern to a scale of z = length AB of the chart. (If z= 6m and AB = 30mm, the scale is 1/200). Newmark Method 83Dr. Abdulmannan Orabi IUST
- 84. The footing plan will be placed on the influence chart with the point for which the stress is desired at the center of the circles. Newmark Method The units (segments or partial segments) enclosed by the footing are counted, and the increase in stress at the depth z is computed as " c j Where I is the influence factor of the chart. " +00 2 0. . +. + 2+ 3 +3 0. . j `4 . 3 2 , 0+. + +. `+ 2- 84Dr. Abdulmannan Orabi IUST … . 7 − 13
- 85. Newmark Method 85Dr. Abdulmannan Orabi IUST