“Performance-Based Hurricane Engineering (PBHE) framework”. Structural Safety, 45, 24-35. DOI: 10.1016/j.strusafe.2013.07.002

Structural Safety 45 (2013) 24–35

Contents lists available at ScienceDirect

Structural Safety
journal homepage: www.elsevier.com/locate/strusafe

Performance-Based Hurricane Engineering (PBHE) framework
Michele Barbatoa,* , Francesco Petrinib , Vipin U. Unnikrishnana , Marcello Ciampolib
a
b

Louisiana State University and A&M College, Civil & Env. Eng. Dept., 3418 PFTH, Baton Rouge, LA 70803, USA
Sapienza University of Rome, Struct. & Geotech. Eng. Dept., via Eudossiana 18, Rome 00184, Italy

a r t i c l e

i n f o

Article history:
Received 20 November 2012
Received in revised form 23 May 2013
Accepted 17 July 2013
Keywords:
Performance-Based Engineering
Hurricane engineering
Multiple hazards
Windborne debris
Hurricane wind
Residential buildings

a b s t r a c t
This paper presents an innovative fully-probabilistic Performance-Based Hurricane Engineering (PBHE)
framework for risk assessment of structural systems located in hurricane-prone regions. The proposed
methodology is based on the total probability theorem and disaggregates the risk assessment into elementary
components, namely hazard analysis, structural characterization, environment–structure interaction analysis, structural analysis, damage analysis, and loss analysis. This methodology accounts for the multi-hazard
nature of hurricane events by considering both the separate effects of and the interaction among hurricane
wind, flood, windborne debris, and rainfall hazards. A discussion on the different sources of hazard is provided, and vectors of intensity measures for hazard analyses are proposed. Suggestions on the selection of
appropriate parameters describing the interaction between the environmental actions and the structure, the
structural response, and the resulting damage are also provided. The proposed PBHE framework is illustrated
through an application example consisting of the performance assessment of a residential building subjected
to windborne debris and hurricane strong winds. The PBHE framework introduced in this paper represents
a step toward a rational methodology for probabilistic risk assessment and design of structures subjected to
multi-hazard scenarios.
c 2013 Elsevier Ltd. All rights reserved.

1. Introduction
Performance-Based Engineering (PBE) is a general methodology
that (1) defines the performance objectives for structural systems
during their design life, (2) provides criteria and methods for verifying the achievement of the performance objectives, and (3) offers
appropriate methodologies to improve the design of structural systems. In the last two decades, significant research efforts have been
devoted to the development of PBE in earthquake engineering [1,2],
and have led, e.g., to the Performance-Based Earthquake Engineering
(PBEE) framework implemented by the Pacific Earthquake Engineering Research (PEER) Center [2,3]. More recently, the civil engineering
community has shown significant interest toward the possible development and extension of PBE to other subfields of structural engineering [4]. In particular, Performance-Based Blast Engineering has
received considerable attention in the US after the terrorist attacks
of September 11, 2001 [5]. Other PBE examples are PerformanceBased Fire Engineering [6], Performance-Based Tsunami Engineering
[7], and Performance-Based Wind Engineering (PBWE) [8,9]. In earthquake engineering, modern design codes have gradually substituted
prescriptive approaches with PBEE procedures for the design of new
facilities and the retrofit of existing ones [10,11]. PBEE has been shown

* Corresponding author. Tel.: +1 225 578 8719; fax: +1 225 578 4945.
E-mail addresses: mbarbato@lsu.edu (M. Barbato) francesco.petrini@uniroma1.it
(F. Petrini) vunnik1@lsu.edu (V.U. Unnikrishnan) marcello.ciampoli@uniroma1.it (M.
Ciampoli).
0167-4730/$ - see front matter c 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.strusafe.2013.07.002

to facilitate design and construction of structural systems based on a
realistic and reliable assessment of the risk associated with seismic
hazard, thus leading to a more efficient use of resources for construction, maintenance, and retrofit of structures [12,13].
The advantages demonstrated by a PBE approach to civil engineering provide a strong motivation to develop a PBE methodology
for structures subjected to hurricanes. The need for assessing and
improving the resilience of the built environment subjected to hurricane hazard is widely recognized. Some initial interest in PBE has
been expressed in hurricane engineering [14–16], but a complete and
rigorous framework is still needed.
The development of a Performance-Based Hurricane Engineering (PBHE) methodology presents several additional challenges when
compared to other existing PBE methodologies. In fact, while other
PBE methodologies focus on single hazards, the landfall of a hurricane
involves different hazard sources (wind, windborne debris, flood, and
rain) that interact to generate the hazard scenario for a given structure and to determine its global risk. Thus, hurricanes can be viewed,
and must be analyzed, as multi-hazard scenarios. In addition, monetary losses due to structural and non-structural damage assume more
relevance for hurricane events than for other types of hazard (e.g.,
earthquakes) for which no (or very short) warning is available. Therefore, for hurricane hazard, performance levels related to limitation
of the monetary losses due to damage may be required for a large
portion of existing or newly designed structures.
During the last decade, significant attention has been also devoted to multi-hazard scenarios [17–19]. Multi-hazard scenarios raise

M. Barbato et al. / Structural Safety 45 (2013) 24–35

non-trivial issues mainly related to the following three problems: (1)
modeling the interaction among concurrent sources of hazard; (2)
calibrating design values having comparable occurrence rates for different hazards; and (3) balancing the design in order to attain similar
safety levels with regard to multi-hazard scenarios implying hazards
that, if taken separately, would drive design solutions in different
(and even opposite) directions (e.g., increasing the elevation of the
structure as a safe guard against flood may result in increased wind
loads).
In this paper, the PBE approach is formally extended to develop
a fully-probabilistic PBHE methodology. The interaction among the
multiple hazards is discussed, and a scheme for representing the uncertainties from all pertinent sources and their propagation through a
probabilistic performance assessment analysis is proposed. Analytical
models of the relevant environmental phenomena generated by hurricane events are briefly described. The paper includes suggestions
for candidate parameters for the probabilistic characterization of: (1)
the interaction between the structure and the hazard sources; (2)
the structural response; (3) the resulting structural damage; and (4)
the consequences of the structural damage. The proposed approach is
illustrated through an application focused on the performance assessment of a residential building subjected to both wind and windborne
debris hazard.
2. Proposed PBHE framework
In a PBE approach, the structural risk is conventionally measured
by the probability of exceeding (within a given reference period usually taken as one year) a specified value of a decision variable, DV, corresponding to a target performance. Each DV is a measurable attribute
of a specific structural performance and can be defined in terms of
cost/benefit for the users and/or the society (e.g., loss of human lives,
economic losses, exceedance of safety/serviceability limit states). An
assessment based on PBE provides a probabilistic description of the
appropriate DV for different design choices in order to allow a rational
decision among different design options.
A PBE procedure for structures subject to hurricane hazard can
be decomposed into elementary phases that must be carried out in
sequence. Perhaps the most important expected feature of the procedure is the qualitative independence of each phase from the others
(i.e., the choice of the parameters that are characteristic for a given
phase is independent from the parameters adopted in the previous
phases). The PBHE framework proposed in this paper is based on the
total probability theorem, similar to the PEER PBEE and the PBWE
frameworks. The structural risk is defined in terms of a given DV as
follows

G (DV) =

G (DV DM) · f (DM EDP)

· f (EDP IM, IP, SP) · f (IP IM, SP) · f (IM)
· f (SP) · dDM · dEDP · dIP · dIM · dSP

(1)

where G(·) = complementary cumulative distribution function, and
G(·|·) = conditional complementary cumulative distribution function;
f(·) = probability density function, and f(·|·) = conditional probability
density function; DM = damage measure (i.e., a parameter describing
the physical damage to the structure); EDP = engineering demand parameter (i.e., a parameter describing the structural response for the
performance evaluation); IM = vector of intensity measures (i.e., the
parameters characterizing the environmental hazard); SP = vector
of structural parameters (i.e., the parameters describing the relevant
properties of the structural system and non-environmental actions);
and IP = vector of interaction parameters (i.e., the parameters describing the interaction phenomena between the environment and
the structure). In Eq. (1), IM and SP are assumed as uncorrelated and
independent of IP, while IP is dependent on both IM and SP. The extension to the case of vectors of DM and EDP is straightforward.

25

By means of Eq. (1), the problem of risk assessment is disaggregated into the following tasks (see Fig. 1): (1) hazard analysis,
(2) structural characterization, (3) interaction analysis, (4) structural
analysis, (5) damage analysis, and (6) loss analysis. Detailed explanation of steps (1), (4), (5), and (6) can be found in the PBEE literature
[2], while steps (2) and (3) have been introduced in PBWE to rigorously model the effects on the structural response of the interaction
between the structural system and the environment (e.g., the aerodynamic effects, see [20]). In particular, the probabilistic hazard analysis
phase (i.e., the probabilistic characterization of IM) can be performed
by using the (joint) probability density function f(IM). The IM should
be chosen as strictly independent on the investigated structure. Thus,
the probabilistic information about IM should be provided by meteorologists, climatologists, and other experts in atmospheric sciences,
while the engineers have the task of clarifying what information is
needed.
3. Characterization of uncertainties
The PBHE framework described in the previous section requires
the identification of the uncertainties that affect the structural performance and the evaluation of the interaction phenomena occurring
among the different hazards and the structure. It is noted here that uncertainties can be classified into two different categories, i.e., aleatoric
uncertainties, which are due to natural variability of physical, geometrical, and mechanical properties, and epistemic uncertainties, which
are due to lack of knowledge, imprecise modeling, and limited statistical information [21]. Aleatoric uncertainties are inherent in nature
and, thus, are virtually irreducible. On the contrary, epistemic uncertainties can and should be reduced as much as possible, e.g., by
implementing more accurate and realistic models. While epistemic
uncertainties can significantly affect the confidence on the end results
of the PBHE framework proposed in this study, their detailed study is
beyond the scope of this paper.
This paper focuses on the characterization of the hazard for a single
structure located in a hurricane-prone region. Three different zones
can be distinguished [20] (see Fig. 2):
1. The environment, i.e., the space surrounding the structure but sufficiently far from it, where the parameters describing the wind
field and the other hurricane-related environmental actions are
not influenced by the presence of the structure itself.
2. The exchange zone, i.e., the space immediately surrounding the
structure, where the structural configuration and the environmental action are strongly correlated, and the interaction between the
structure and the environmental agents, as well as the presence
of adjacent structures, cannot be disregarded.
3. The structural system, which includes the structure (characterized
by a set of uncertain parameters collected in a vector S) as well
as the non-environmental actions and/or elements that can modify the structural behaviour (characterized by a set of uncertain
parameters collected in a vector A).
Hereinafter, the uncertain basic parameters of interest describing the environmental actions in the environment are collected in
the vector IM; the uncertain basic parameters describing the structural system and non-environmental actions or devices applied to the
structure are collected in the vector SP; and the uncertain (usually
derived) parameters of interest in the exchange zone are collected in
the vector IP (Fig. 2). Examples of IP are the aerodynamic and hydrodynamic coefficients, as well as the parameters defining the impact
energy of windborne debris. The uncertain parameters IP describing
the exchange zone can be chosen so that they do not affect directly
the uncertain parameters characterizing both the environment (IM)
and the structural system (SP). Instead, uncertainty propagation from
the structural system and the environment to the exchange zone is

26


Fig. 1. Probabilistic analysis components in the proposed PBHE framework.

Fig. 2. Identification of the uncertain parameters needed to describe the interaction between environment and structure in PBHE.

likely (e.g., the integrity of the building envelope affects the values of
the wind pressure acting on the building surfaces).
In Fig. 2, the different sources of uncertainties corresponding to
the environment, the structural system, and the exchange zone are
shown, and the different hazard sources and their interaction identified. The environmental hazard due to a hurricane event in a specified
geographic region is generated by the following four main sources of
hazard:

2. Water bodies (described by the uncertain vector F), which can
produce flood damage (flood hazard).
3. Sources of windborne debris (described by the uncertain vector D),
which can produce windborne debris damage (windborne debris
hazard).
4. Rainfall rates (described by the uncertain parameter vector RA),
which can induce flash flooding and direct damage to the interiors
of building when the building envelope has been breached (rainfall
hazard).

1. Hurricane strong winds (described by the uncertain vector W),
which can produce wind damage (wind hazard).

These various sources of hazard usually interact to produce the
actual hurricane hazard for a given structure. Typical examples are


the interaction between wind and waves in offshore sites, or the interaction between storm surge and wind in coastal regions. A set of
uncertain parameters included in vectors W, F, D, and RA must be
selected in order to describe the multiple hazards using IM. This set
must accurately describe all pertinent hazard sources and must be
as small as possible in order to be both “sufficient” and “efficient”
[22]. An analogous selection operation (for vectors A and S) is needed
to describe the structural behavior by SP. To derive the probabilistic
characterization of the hurricane actions on a structural system, the
proposed PBHE methodology also requires the identification of the
vector IP of the stochastic parameters describing the interaction between the environment and the structural system in the exchange
zone.
4. Multi-hazard characterization of hurricane events
Unlike other existing PBE engineering methodologies, in which
only a single hazard source is considered (e.g., PBEE and PBWE, which
consider earthquake and wind hazard only, respectively), the proposed PBHE framework innovatively accounts for concurrent and interacting hazard sources, i.e., storm surge and water bodies that can
cause flooding, windborne debris, rainfall, and strong winds. It also
accounts for the possible sequential effects of these distinct hazards.
The multi-hazard nature of the phenomena related to hurricanes and
their effects on the built environment can manifest in the following
three different modalities [23]:
1. Independent hazards, when different hazards affect the structure
independently. For example, windborne debris and flood hazard
can be considered as independent of each other because no mutual
interaction between the two hazards has the effect of modifying
the intensity of the corresponding actions. These hazards can occur
individually or simultaneously.
2. Interacting hazards, when the actions produced on a structure by
different hazards are interdependent (e.g., wind and windborne
debris hazards).
3. Hazard chains, when the effects of some hazards modify sequentially the effects of other hazards. For example, the actions on a
structure due to windborne debris can damage the structural envelope and increases the vulnerability of the structure to strong
winds.
In the proposed framework, the first two cases (i.e., independent
and interacting hazards) are treated within the hazard analysis, by
assuming proper interaction models between the hazards (e.g., by
using a proper joint probability distribution function to describe the
variability of the IM for different hazards [24,25]). The study of hazard chains requires modeling the structural system configuration and
properties as a function of the level of structural damage caused by
the different hazards. In particular, the presence of a hazard chain implies that the SP can change as a consequence of DM exceeding certain
thresholds. Thus, structural characterization, interaction analysis, and
structural analysis cannot be carried out without any information or
assumption on the values of DM. It is noteworthy that the proposed
probabilistic approach is consistent with the state-of-the-art in hurricane hazard and loss modeling, which can be identified with the
HAZUS® methodology [26].
However, the proposed PBHE framework presents the following
major differences when compared with the HAZUS methodology:

27

By contrast, the proposed PBHE framework is specifically developed to account for the characteristics of an individual building.
Thus, it has the potential to provide more accurate results.
2. HAZUS is not intended for use as a structural design tool. On
the contrary, the proposed PBHE framework is the first step toward a performance-based design methodology, which includes
the performance-based assessment procedure described in this
paper.
3. HAZUS approximates the multi-hazard nature of the hurricane
events as a simple superposition of various effects produced by
different sources of hazards, i.e., wind, windborne debris, flood
and rainfall. The proposed PBHE framework directly models the
multi-hazard nature of hurricanes by taking into account also the
effects due to the interaction between different hazard sources.
4. The proposed PBHE framework is significantly more flexible than
HAZUS, since it is based on the total probability theorem, which allows for independence of the different analysis components. This
property permits to take advantage of the state-of-the-art knowledge in the research subfields involved in the assessment and
design of structures located in hurricane prone regions, e.g., in
climatology, structural analysis, structural design, material technology, and loss analysis.

5. Performance expectations
In PBE, several performance expectation levels are defined based
on the severity of structural and non-structural damage and the corresponding losses (e.g., see [8,14,27]). Commonly, two main performance expectation levels with each level having different performance objectives are identified [8,20], i.e., a high level performance expectation (related to serviceability requirements) and a low
level performance expectation (related to structural safety and/or integrity). For the PBHE framework proposed in this paper, additional
considerations are needed to account for the fact that early warning of
the population is possible in case of hurricane hazard, in contrast with
other hazards (like the seismic hazard) for which warning is impossible or very limited. Thus, empty buildings during the hurricane transit
are not rare. In this situation, significant losses due to the damage to
non-structural components (e.g., building envelope, interiors of the
building) can occur without problems for people (because occupants
left the building) or for the structural integrity (because the structural parts do not suffer damages). In view of this consideration, an
additional intermediate performance level related to non-structural
damage is introduced.
The three performance expectation levels can be further subdivided in sub-levels or performance levels, e.g., the high performance
expectation level for a building can be related to occupant comfort (higher) and/or continued occupancy (lower). Moreover, different performance expectation levels need to be defined for different
structural typologies (e.g., buildings, bridges). A possible list of performance expectations for buildings and their short description are
provided in Table 1.
6. Description of the analysis steps
This section presents a brief description of the analysis steps of
the PBHE framework. Particular emphasis is given to the differences
between the PBHE framework and other existing PBE frameworks.
6.1. Hazard analysis

1. HAZUS is a GIS-based natural hazard assessment software used
for the regional risk and loss assessment of structures. Although it
is possible to use the HAZUS software for individual buildings, the
corresponding results can only estimate the average loss for the
class of buildings that are similar to the one under consideration.

The hazard analysis provides the probabilistic description of the
intensity measures IM. A comprehensive vector IM is obtained by
considering the components of the basic random parameter vectors
W, F, D, and RA that describe the different sources of hurricane hazard

28


Table 1
Classification of building performance expectation for PBHE.
Category

Level

Description

Damage level

High: comfort and safety of occupants

Occupant comfort

No or little discomfort to the building
occupants
No threat to safety of building
occupants, small economic losses
No threat to safety of building
occupants, some economic losses
Safety of building occupants is
jeopardized, significant economic
losses
Structural integrity is jeopardized,
reduced safety
Visible signs of structural distress,
structure is not safe

No damage

Continued occupancy
Intermediate: damage to
non-structural elements

Limited damage to envelope/content
Extensive damage to envelope/content

Low: structural integrity

Structural damage
Extensive structural damage

(Fig. 2). It is noteworthy that, for a specific structure s and a specific
performance objective p, the elements of IM that do not represent a
relevant hazard for s and/or have small influence on p can be neglected
or treated as deterministic. The reduced vector IM(s,p) (i.e., the vector
IM specialized for the structure s and the performance p) can be used
more efficiently than the vector IM at a negligible loss of sufficiency.
For example, the flooding hazard can be neglected in the case of
structures that are sufficiently far from water bodies.
The selection of the IM components strictly depends on the choice
of the (usually deterministic) models used to describe the environmental phenomena related to the various hazards, as illustrated in
the available technical literature. In general, the key parameters of
these models are treated as stochastic variables. In this paper, the
IM components are identified by selecting state-of-the-art models for
hurricane-related environmental phenomena. While the selected IM
depend on the specific models, the approach proposed here to identify
IM is general and can also be applied to different hazard models.
6.1.1. Wind field and wind hazard characterization
A model of the wind field associated with a hurricane is needed to
characterize the wind hazard. The following three methodologies (of
increasing complexity) can be adopted to define the hurricane wind
field and the corresponding hazard [26]:
1. The statistical description of the gust wind velocity, V, at the structural location is directly derived from existing data by fitting a
proper probability distribution [28].
2. The site specific statistics of some fundamental hurricane parameters are obtained, and a Monte Carlo approach is used to sample these parameters from the statistical information. Using the
sampled values, a mathematical representation of the hurricane
is obtained, and the statistics of the parameters that describe the
hurricane actions are evaluated for the structure of interest at its
specific location [29].
3. The full track of the hurricane is modelled, from its initiation over
the ocean until final dissipation [30]. Several tracks are simulated.
The statistics of the parameters describing the hurricane actions
are estimated from the parameter values in each simulated track
for the structure of interest at its specific location.
The different methodologies provide different vectors IM. In particular, the first methodology gives W = V, while the other two
methodologies give [15]:

Minor exterior damage, no interior damage
Exterior damage, minor interior damage
Significant exterior and interior damage

Structural components and/or connections
are damaged
Loss of integrity of structural components,
significant reduction or loss of bearing load
capacity

central pressure deficit; B = Holland parameter [31]; H* = atmospheric boundary layer height; z0 = terrain roughness length; and
the superscript T denotes the matrix transpose operator.
6.1.2. Flood hazard characterization
The flood hazard due to the presence of water bodies surrounding the structure depends on the total water surface elevation with
respect to the mean surface, ηtot , and on the flooding water velocity,
Vwater (i.e., the value of the component of the water velocity orthogonal to the flooding barriers). These two parameters allow for the computation of the volumetric rate of flow and can be used as synthetic
indicators of the flood intensity. The basic parameters characterizing
these indicators can be selected as the components of F.
Three main natural phenomena cause water level increase and
contribute to flood hazard: the astronomical tide (ηtide ), the waves
(ηwave ), and the storm surge (ηsurge ). The total water surface elevation
is the sum of the three contributions at the same instant of time, i.e.,
ηtot = ηtide + ηwave + ηsurge . The flooding water velocity, Vwater , can
be assumed, as a first approximation, equal to the highest velocity
for each of the three considered phenomena (Vtide , Vwave , and Vsurge ,
respectively). Specific basic parameters subvectors can be defined for
each of the three contribution (i.e., Ftide , Fwave , Fsurge ), and the vector
F obtained as the union of the three subvectors.
The flood hazard due to the astronomical tide can be characterized
by the two random variables ηtide and Vtide , i.e., Ftide = [ηtide Vtide ]T .
The individual characterizations of the flood hazard due to waves and
storm surge require more detailed considerations. The water level,
ηwave , and the wave speed, Vwave , can be directly related to: the water depth, d; the wave height, H; the wave length, L; and the wave
period, T. The last three quantities can be obtained by propagating
in space and time [32] the waves corresponding to a given wave energy density spectrum (e.g., JONSWAP [33]) valid for the sea waves
as determined by the wind field (i.e., RMW, Vc , Δpc , and B), as well
as by other parameters [32]. A storm surge is defined as the water
surface response to wind-induced surface shear stress and pressure
fields. Storm surges can produce considerable short-term increases
in water level. Current storm surge models are based on the depthaveraged momentum and continuity equations for steady long waves
under the hypothesis of incompressible water [34].
Based on the existing literature, the following vector F is suggested
for a suitable flood hazard characterization
F = [RMW Vc

pc B H ∗ z0 ηtide Vtide d U curr zb ]T

(3)

(2)

where Ucurr = current velocity, and zb = sea bottom friction roughness.

where RMW = radius of maximum wind (defined as the radial distance between the storm center and the maximum wind location);
Vc = translational speed of the center of the storm; Δpc = hurricane

6.1.3. Windborne debris hazard characterization
The vector D of intensity measures for windborne debris hazard
describes the intensity of the wind field (needed to determine the

W = [RMW

Vc

pc

B

H∗

z0 ]T


impact wind speed), and the characteristics of the windborne debris
that could affect the structure. The additional parameters needed to
describe the debris are: the density of upwind buildings with respect
to the investigated structure, nbuildings ; the properties of the different
(potential) debris types, e.g., Md = mass per unit area of the debris,
CD,d = drag coefficient of the debris (and/or other parameters describing the debris flight characteristics), and Ad = reference area of the
debris; and the resistance model for the missile sources (which contributes to determine the number of windborne debris). The following
vector D is suggested in this study:
D = RMW Vc

pc B H ∗ z0 nbuildings Md C D,d Ad

T

(4)

6.1.4. Rainfall hazard characterization
The high rainfall rate associated with hurricane events can induce
significant damage to the interior of buildings when the building envelope has been breached [35]. To the best of the authors’ knowledge,
no analytical rainfall hazard model is available in the technical literature. However, several models based on the interpolation of statistical
data define the correlation between the rainfall rate and other fundamental hurricane parameters. One of the more widely accepted and
used models is the one implemented in HAZUS® [26,36], which is
valid for tropical cyclones. The estimates of rainfall rates resulting
from this model are employed in HAZUS® to evaluate the amount of
water that enters the buildings through broken windows and glass
doors, while they are not used to assess the risk associated with inland flash flooding. Consistently with HAZUS® , this study does not
consider inland flash flooding hazard. The proposed vector RA of basic
random parameters is given by
RA = [RMW

Vc

pc

B

H∗

z0

pc ]T
˙

(5)

˙
where p c is the first time derivative of the hurricane central pressure.

6.2. Structural characterization
The step of structural characterization in the PBHE framework
provides the probabilistic description of the components of SP, which
define the geometrical and/or mechanical properties of the structure that characterize its response to environmental and man-made
loading. Uncertainties affecting SP are well-known and have been
extensively investigated in the past decades for ordinary buildings
[37,38]. They are usually identified as the parameters determining
the structural resistance and stiffness [39]. However, parameters describing shape, size, and orientation of structural components can
also be considered, since they can affect the load acting on the structure. In addition to the above parameters, robustness, connectivity,
and redundancy are also critical in the analysis of wind-induced effects on structures. Robustness implies the property of a structure
not to respond disproportionately to either abnormal events or initial
local failure [40]. A general framework, based on the total probability theorem, was proposed in the literature to assess probabilistically
the robustness of systems subject to structural damage [41,42]. This
framework is consistent with the proposed PBHE framework, e.g., by
using as DV the robustness index [41]. However, the computation of
the robustness index for structures subjected to hurricane hazard requires significant research and implementation work, and is outside
the scope of this paper.
Particular attention is needed when a hazard chain is possible or
likely. The probabilistic description of SP (e.g., the first- and secondorder statistics, as well as the distribution type) needs to be expressed
as a function of the damage parameter DM. A typical example of
this situation occurs when the behavior of buildings subjected to
windborne debris hazard is considered [43]. If windows or doors break
due to windborne debris impact, the characteristics of the building

29

envelope (described by SP) vary, causing a change in the internal
pressure coefficients and in the loads acting on the structure [44].
6.3. Interaction analysis
The interaction parameters IP describe the physical interaction
between structure and environment, which influences the structural
response and performance, as well as the intensity and distributions
of the environmental actions as a result of the interaction between
the structure and the environment. Typical examples of IP are the
aerodynamic pressure coefficients and the aerodynamic derivatives
for dynamic wind actions, the rate of water flow impacting the structure for flooding actions, the kinetic energy and linear momentum of
the impacting missile for windborne debris, the wind pressure on the
internal and external building surfaces for wind actions on a building
envelope, and the rate of water intrusion in a building under strong
rain for rainfall action [45].
In other words, the IP are parameters that influence the intensity
of the environmental actions on the structure, and that depend simultaneously on IM and SP, as well as on their variability (e.g., the
aerodynamic derivatives of a bridge depend both on wind direction
and velocity, which are components of IM, and on structural damping,
which is a component of SP). In deterministic terms, this dependency
is described by a mechanistic model of the IP as functions of the IM
and SP (e.g., see Fig. 3(a)).
In a probabilistic analysis, the uncertainty of both IM and SP must
be taken into account in order to obtain the probability distributions
of IP, which can be derived by using probability distributions conditional to IM and SP. The propagation of uncertainties from IM and SP to
IP can be performed, e.g., by characterizing the IP via parametric probabilistic distributions whose parameters are deterministic functions
of IM and SP (see [9]). The conceptual separation of the interaction
analysis from others analysis steps carried out in PBE approaches is an
aspect of novelty of the proposed PBHE framework with respect to the
original PEER approach. This clear separation between independent
parameters (IM and SP) and derived parameters (IP) has also the merit
of highlighting the correct direction of uncertainty propagation.
Examples of interaction analysis in structural engineering subfields other than hurricane engineering are: soil–structure interaction
analysis in earthquake engineering, fluid–structure interaction analysis in offshore engineering and wind engineering, and heat-transfer
analysis in fire engineering.
6.4. Structural analysis and damage analysis
The structural analysis phase provides the probabilistic description of a proper EDP, which concisely represents the essential aspects
of the structural response for damage and performance evaluation.
Examples of EDP are: axial force, shear force, bending moment, and
stress state in structural and non-structural elements; response quantities describing the structural motion (deflections, velocities, and accelerations of selected points); structural deformation indices (e.g.,
interstory or global drift ratio and beam chord rotation).
The damage analysis provides the probabilistic description of DM
conditional to the values of EDP. The results of a probabilistic damage analysis are commonly expressed in terms of fragility curves as
shown in many recent applications in hurricane engineering [28,46–
48]. For example, for low-rise wood residential construction, the damage states of interest relate to those components that are essential to
maintain the integrity of the building envelope, i.e., roofs, windows,
and doors, since the building envelope is the residential construction component that is most vulnerable to hurricane-induced damage
[46].
In some applications, it is convenient to assume DM = EDP, e.g.,
in the case of low rise gable roof structures, in which the number

30


Fig. 3. Different representations of fragility curves in case of interaction between two hazards: (a) relation between IP and IM, (b) fragility curve as a function of IP, and (c) fragility
curves as functions of IM.

of lost roof panels due to the uplift pressure generated by hurricane winds can be chosen as both DM and EDP. The hurricane wind
fragility is then expressed as the cumulative probability distribution
conditional to the uplift pressure (IP) or to the wind gust velocity (IM).
In case of hazard chains (e.g., if wind and windborne debris hazards
are considered), the representation of the fragility as the probability
of DM conditional to IM, namely P(DM|IM), can be used to assess the
effects of the interaction, e.g., based on the differences between the
functions P(DM|IM) obtained by considering undamaged or broken
windows (e.g., after a missile impact).
Fig. 3 shows two alternative representations of the fragility curve
for the roof panel uplift limit state, i.e., P(DM|IP) and P(DM|IM), in the
case of low-rise gable-roof buildings under the chain-effect of wind
and windborne debris hazards. In this example, the wind velocity
V is assumed as IM, the uplift pressure p is assumed as IP, and the
number of roof panels that are lost due to wind is assumed as DM.
The damage scenario corresponding to the loss of the ﬁrst roof panel
(DM = 1) is considered. The relation between uplift pressure and
wind velocity (i.e., IP = IP(IM)) is different for the cases “broken” and
“unbroken” windows (see, e.g., ASCE standard as shown in Fig. 3(a)).
In fact, this relation must take into account the internal pressurization
of the building caused, e.g., by the failure of a door/window due to
windborne debris impact (i.e., the chain effect). The fragility curves
obtained from the technical literature in the form P(DM|IP) are the
same for the two case of “broken” or “unbroken” windows, since
they depend only on the properties of the roof panels (see Fig. 3(b)).
Two different IP values (identiﬁed in Fig. 3(a) and (b) as p1 broken and
p1 unbroken ) correspond to a given IM value (V1 in Fig. 3(a) and (c)).
Thus, two different fragility curves P(DM|IM) can be built for the two
cases of “broken” or “unbroken” windows, which highlight the effects
of considering the interaction between wind and windborne debris
hazard. In the case of vector IM and IP, the fragility can be represented
through appropriate fragility surfaces [49].

6.5. Loss analysis
The loss analysis step gives the estimate of the annual probability
of exceedance of DV, G(DV), where DV can be used as an indicator for
structural risk. Hurricanes are among the most costly natural hazards
to impact residential construction in the southeast coastal area of
the United States [46]; thus, DV is usually expressed in monetary
terms. It is noteworthy that, from a loss-based design perspective,
non-structural and structural damage are both losses; moreover, in
addition to direct losses, hurricanes can lead to social disruption for
extended periods of time, including the need to relocate building
inhabitants [25].
DV can be chosen as the repair cost related to the hurricane induced damage, or a percentage of the insured value, or the lifetime
cost of the structural system, evaluated by taking into account the
construction and maintenance costs, the repair costs after an event,
the economic losses due to damage (also to building contents), and
the loss of functionality [50]. Even in the simplest cases, repair costs
are highly uncertain, and updated data from insurance companies are
needed to obtain an appropriate probabilistic description of repair
costs.
In addition, both ethic and technical problems arise when the DV
is related to the loss of human life and/or to a life quality index for
the structure subjected to the hurricane. Further research is needed
to overcome the technical challenges related to the inclusion of these
aspects in evaluating the losses associated with the structural damages and failures due to the hurricanes. In addition, a constructive
dialogue is needed among different stakeholders to determine a consensus on when and how to consider life quality indices and costs
associated with loss of life into hurricane risk assessment.


31

Table 2
Probabilistic characterization of external and internal pressure coefficients.
Location/
condition
GCp
Roof (near
ridge)
Roof (away
from ridge)
Windward wall
Leeward wall
Side wall
GCpi
Enclosed
Breached

Mean

COV

Distribution

−0.855

0.12

Normal

−1.615

0.12

Normal

0.95
−0.76
−1.045

0.12
0.12
0.12

Normal
Normal
Normal

0.15
0.46

0.33
0.33

Normal
Normal

and windborne debris hazard. A study of sufficiency and efficiency of
different potential IM [22], albeit important, is out of the scope of this
study.
Among the wind occurrence models available in the literature
[46,52], the Weibull distribution is adopted here to describe the hurricane wind speed variability [28]. The two-parameter Weibull cumulative distribution function, F(V), is given by:
F (V ) = 1 − exp −
Fig. 4. Plan view of the benchmarck residential development.

7. Application example
The proposed PBHE framework is illustrated through the risk analysis for a building belonging to a hypothetical residential development, located near the coast in South Florida and composed by 30
identical concrete block gable roof structures (see Fig. 4). This application example seems sufficiently advanced to display some of the
specific critical issues of the PBHE framework, and highlights the importance of the interaction between different hazards in a hurricane
risk analysis. However, it is also simple enough to avoid the complexities of more realistic applications, thereby maintaining the focus of
this paper on the illustration of the PBHE framework.
The risk analysis is performed for the building identified as “Target” in Fig. 4. The interaction of wind and windborne debris hazards
is taken into account. Roof covers are considered as debris sources,
whereas the windows and glass doors are considered as debris impact
vulnerable components [51].
7.1. Hazard analysis and structural characterization
In the present study, the 3-second hurricane wind speed V
recorded at 10 m above the ground is used as the only component
of W for characterizing the wind hazard. For the sake of simplicity,
the wind direction variability has been neglected by assuming that
the maximum local winds generated by hurricanes act only in the
most unfavorable direction for windborne debris hazard (i.e., in the X
direction in Fig. 4).
For windborne debris hazard, the considered intensity measures
(IM) are: the wind speed, V; the debris area, Ad ; and the mass per unit
area of debris, Md . It is assumed that the buildings in the benchmark
residential development are the only windborne debris source affecting the target structure. Thus, the parameter nbuildings (i.e., the density
of upwind buildings with respect to the investigated structure) can
be excluded from the D vector. All windborne debris are assumed
of sheet type with flight characteristics described by deterministic
parameters. The choice of IM is based on damage analysis results
available in the literature, which show a strong correlation between
the selected parameters and the structural damage produced by wind

V
a

b

(6)

The two shape parameters a and b are site specific and are determined
by fitting the hurricane wind speed records provided by the National
Institute of Standards and Technology (NIST) to a Weibull distribution.
The NIST wind speed records contain data sets of simulated 1-minute
hurricane wind speeds at 10 m above the ground in an open terrain
near the coastline for locations ranging from milepost 150 (near Port
Isabel, TX) to milepost 2850 (near Portland, ME), spaced at 50 nautical
mile intervals (92,600 m). Considering South Florida as the location
for the case study, the dataset corresponding to milepost 1400 is used
for fitting the distribution. The 1-minute hurricane wind speed (V )
dataset is converted into 3-second wind speed as V = 1.77V [37,44].
The two parameter Weibull distribution function is fitted using the
converted wind speeds, and the parameters are a = 25.2447 m/s and
b = 1.6688, respectively. The area and mass per unit area of debris
are assumed to follow a uniform distribution in the range [0.108,
0.184] m2 and [10.97, 23.35] kg/m2 , respectively [27].
The considered structural parameters (SP) are: the wind pressure
exposure factor (evaluated at the height h of the roof of the target
building), Kh ; the external pressure coefficient, GCp ; and the internal
pressure coefficient, GCpi . The pressure coefficients include the effects
of the gust factor G. The topographic factor, Kzt , the wind directionality factor, Kd , are modeled as deterministic and assumed equal to
one; whereas the total vulnerable area on each side of each building
is assumed equal to 20% of the total wall area. The wind pressure
exposure factor Kh is assumed as normally distributed with a mean
value of 0.71 and a coefficient of variation (COV) of 0.19 [39]. The
characterization of the external and internal pressure coefficients is
given in Table 2 [46,52].
7.2. Interaction analysis
The choice of the interaction parameters (IP) is crucially dependent
on the performance levels of interest and the corresponding monitored responses of the structural and non-structural elements. The
main parameters controlling the effects of windborne debris impact
are the impact linear momentum, Ld , the impact kinetic energy, Ed ,
and the number of impacting debris, nd . It is known from the literature
that the impact linear momentum is well correlated with damage for
envelope components with a brittle behavior (e.g., doors, windows,
see [53,54]), whereas the impact kinetic energy is better correlated

32


position), and is given by [51]
√
ud = V · 1 − exp − 2 · C · K · x

(8)

where x = g·X = dimensionless horizontal flight distance of the deV2
bris. The debris is assumed to hit the target building if the debris flight
distance is larger than the distance between the source and the target
building and, at the same time, the landing position falls within the
angle of hit (see Fig. 4).
Finally, the debris impact model uses the horizontal component of
the missile velocity (obtained from the debris trajectory model) and
data related to the missile size and mass (obtained from the debris
generation model) to compute the impact linear momentum of the
missile (i.e., the linear momentum corresponding to the windborne
debris velocity component orthogonal to the impacted surface, conditional to the event of at least one impact on windows). In this study,
the debris impact model gives the impact linear momentum as
L d = Md · Ad · ud

Fig. 5. Interaction analysis for windborne debris hazard.

with damage to envelope components with a ductile behavior (e.g.,
aluminum storm panels, see [48]). Hereinafter, it is assumed that the
glass windows and doors are unprotected and have a brittle behavior.
Based on this assumption, the IP selected in this study are: (1) the linear momentum of the debris at impact, Ld , for the windborne debris
hazard; and (2) the wind pressure acting on the walls and roof, pw ,
for the wind hazard.
The procedure proposed here for the interaction analysis corresponding to windborne debris hazard is summarized in the flowchart
shown in Fig. 5. The input of the interaction analysis for windborne
debris hazard is obtained from the hazard analysis and the structural
characterization steps. A debris generation model provides the number and type of windborne debris that can affect the structure under
consideration. The debris generation model used in this study is that
employed by the Florida Public Hurricane Loss Model (FPHLM), in
which the mean percentage of damage to roof covers is based on the
simulation results from a component-based pressure induced damage model, and is expressed as a function of the wind speed [27]. The
number of debris generated from each source house is calculated considering the percentage of roof cover damage at a given wind speed
and the geometry of the house [27,51].
The results of the debris generation model, derived according to
the geometry of the considered example case (i.e., density and relative position of debris sources with respect to the target structure),
are taken as input for the debris trajectory model [55–57]. The debris
trajectory model is used to assess if and at which impact velocity a
given windborne debris hits the building. In this study, the debris
trajectory model provides the landing position of the debris, which is
identified by the random variables X = along-wind flight distance and
Y = across-wind flight distance. The random variables are modeled
using a two-dimensional Gaussian distribution described by the following parameters: μX = mean along-wind flight distance; μY = mean
across-wind flight distance = 0 m; σ X = σ Y = 0.35μX = standard deviation of the along-wind and across-wind flight distances, respectively
[52]. The parameter μX is computed as:
μX =

2Md
1
˜
C · K ·T
·
ρa
2

2

˜
+ c1 · K · T

3

˜
+ c2 · K · T

where ρ a = 1.225 kg/m3 = air density; K =

4

˜
+ c3 · K · T

ρa ·V 2
2Md ·g

5

(7)

= Tachikawa num-

˜
ber; T = g·T = normalized time; g = gravity constant; T = flight time
V
in seconds; and C, c1 , c2 , and c3 = non-dimensional coefficients that
depend on the shape of the debris.
The flight time is assumed to follow a uniform distribution in the
interval [1, 2.5] s. For the sheet-type debris considered in this study,
C = 0.91, c1 = −0.148, c2 = 0.024, and c3 = −0.0014. The debris
horizontal velocity at impact, ud , is a function of the wind velocity
and the distance travelled by the debris (determined by its landing

(9)

The interaction analysis for the wind hazard provides the statistical characterization of the wind pressure, pw . For the sake of simplicity, in this study, the wind pressure is computed as
pw = qh · (G C p − G C pi )

(10)

where the wind velocity pressure at a quote h, qh , is given by
qh = 0.613 · K h · K zt · K d · V 2 · I

(11)

I = importance factor (assumed here equal to 1). In Eq. (11), V is
measured in m/s and qh is measured in N/m2 . It is noted here that
the simplified approach used in this study to perform the interaction
analysis for the wind hazard is appropriate for the simple and small
structures considered in the example application. However, larger
and more complex structures may require a more rigorous approach
based on the use of stochastic processes, random fields, and computational fluid dynamics to evaluate the wind effects on the structure.
7.3. Structural analysis and damage analysis
In this example application, the structural analysis step is not
needed explicitly because the engineering demand parameters (EDP)
can be assumed to coincide with the IP. The strengths of glass windows, glass doors, and roof (which are assumed to be the only components that can be damaged) are obtained from empirical relations
available in the literature and directly compared to the corresponding IP. Following a procedure commonly used in PBEE, the physical
damage conditions are represented using a limit state function g for
each damage limit state, i.e.,
g = DM − IP

(12)

where the demand measure (DM) corresponds to the limit state capacity for the given damage limit state. The damage limit states considered here are (1) the breaking of annealed glass windows/doors,
and (2) the uplift of the roof sheathings. The statistics of the damage
limit state capacities for different components and limit states are
provided in Table 3 [52,54].
7.4. Loss analysis
In this study, the decision variable (DV) is taken as the repair cost
of the building, RC, expressed as a percentage of the total cost of the
building. The complementary cumulative distribution of DV can be
used for informed risk-management decision [58] and is computed as
the convolution integral of the conditional probability of DV given DM
and the derivative of the complementary cumulative density function
of DM [4,59]. Since the repair costs associated with the different component limit states are not independent, the computation of G(DV)


33

Table 3
Statistics of limit state capacities.
Component

Limit state

Parameter

Mean (unit)

COV

Distribution

Roof
Windows
Windows

Uplift
Pressure
Impact

Rroof
RM
Rwindow

2762.7 (N/m2 )
4998.7 (N/m2 )
4.72 (kg m/s)

0.20
0.20
0.23

Normal
Normal
Lognormal

Table 4
Summary of parameters used in the risk assessment analysis.
Analysis step

Parameters

Symbol

Definition

Hazard analysis

IM

V

3-Second gust wind
speed
Area of debris
Mass of debris per
unit area
Wind pressure
exposure factor
External pressure
coefficient
Internal pressure
coefficient
Impact linear
momentum
Wind pressure on
the surface
Strength for
pressure (window)
Strength for uplift
(roof)
Strength for impact
(window)
Repair cost (% of
total cost)

Ad
Md
Structural
characterization

SP

Kh
GCp
GCpi

Interaction
analysis

IP

Ld
pw

Damage
analysis

DM

Rwindow
Rroof
RM

Loss analysis

DV

RC

requires the joint probability density function of the repair costs of all
component limit states, which is very difficult to obtain. To overcome
this difficulty, a very efficient multilayered Monte Carlo simulation
(MCS) approach is used in this study to estimate the loss hazard curve
[60]. The multilayered MCS approach is able to account for the uncertainty in the various parameters involved in the risk assessment
methodology (i.e., IM, IP, DM, and DV), which are summarized in Table
4.
The probabilistic hurricane loss analysis is performed for three different scenarios: (1) considering only the losses due to windborne debris hazard (the debris-only scenario); (2) considering only the losses
due to wind hazard (the wind-only scenario); and (3) considering the
losses due to windborne debris and wind hazards, and the effects
of their interaction (the interaction scenario). In the debris-only scenario, the repair cost is associated to the failure of a glass door or a
window due to the windborne debris impact (i.e., Ld ≥ RM ). No chain
reaction is considered, because the failure of a door or a window does
not affect the impact linear momentum of the other missiles. In the
wind-only scenario, the repair cost is associated to the failure of a
glass door and/or a window, as well as to the uplift of the roof due
to the wind pressure (i.e., pw ≥ Rwindow and/or pw ≥ Rroof ). In this
case, a chain reaction is possible because the failure of a glass door
or a window produces an internal pressurization of the building and
modifies the wind pressure acting on the other doors and windows
and on the roof (through the modification of the GCpi parameter from
enclosed to breached building, see Table 2). The interaction scenario
considers the failure of glass doors and windows due to both debris
impact and wind pressure, as well as the roof uplift due to the wind
pressure. In this case, two types of hazard chains are possible, corresponding to the internal pressurization of the building caused by
the failure of a door/window due to the windborne debris impact or
to the wind pressure. Thus, the two scenarios considering wind-only
and debris-only can be obtained as particular cases of the interaction

case by neglecting the wind pressure damage on the doors/windows
and roof for the debris-only scenario, and the damage on the doors/
windows due to windborne debris impact for the wind-only scenario.
Fig. 6 shows the flowchart of the multilayered MCS approach [60]
used for considering the interaction between wind and windborne
debris hazards. The number of hurricanes in each year is simulated
according to a Poisson random occurrence model with annual occurrence rate obtained from the NIST database. For each generated hurricane, a peak wind speed, V, is generated according to the Weibull
distribution. For each value of V, the value of the wind pressure on
the doors/windows and the roof is simulated using the pressure coefficients corresponding to the condition of enclosed buildings. The
linear momentum is also computed for each debris impact. If the
impact linear momentum and/or the wind pressure assume values
larger than the corresponding limit state capacity of the glass on any
of the four walls, the building envelope is considered to be breached
and the internal pressure is modified. The undamaged building components (doors/windows and roof) are checked for further damage
due to the modified pressure. A repair cost is then generated for each
damaged component according to an appropriate probability distribution. For the sake of simplicity, it is assumed that the repair cost
for the breakage of the windows on any side of the building or the
uplift of the roof can be represented by a lognormal random variable
with mean equal to 20% of the total cost of the building and COV
equal to 20%. The total repair cost for the single hurricane simulation
is equal to the sum of all the simulated component repair costs, with
a maximum value of 100% (total failure of the building). It is also assumed that the building is fully repaired after each hurricane event.
The single-year simulation is repeated a large number of times (in the
example, 10,000 samples are used) to estimate the annual probability
of exceedance (which coincides with the complementary cumulative
distribution function of DV) of the total repair cost.
The annual probabilities of exceedance of the repair cost for the
target building for the three different scenarios are shown in Fig. 7,
using a semi-logarithmic scale. A strong interaction is observed between the wind and the windborne debris hazards. This observation
suggests that the multi-hazard nature of hurricane must be taken into
account for accurate probabilistic loss analyses.

8. Conclusions
In this paper, a Performance-Based Hurricane Engineering (PBHE)
framework is proposed and illustrated. The methodology, that can be
used to evaluate the structural risk associated with facilities located
in hurricane-prone regions, is based on the total probability theorem
and builds on techniques already developed and used in other civil
engineering subfields. The problem of risk assessment is disaggregated into the following basic probabilistic components: (1) hazard
analysis, (2) structural characterization, (3) interaction analysis, (4)
structural analysis, (5) damage analysis, and (6) loss analysis. Each
of the analysis steps is briefly discussed in this paper. Particular emphasis is given to the differences between PBHE and other existing
performance-based engineering frameworks, e.g., the multi-hazard
nature of hurricane events, the presence of interacting hazard, and
the focus on high, intermediate, and low performance levels.
The feasibility of the proposed framework is demonstrated
through an application example consisting in the risk assessment

34


Fig. 6. Multilayered MCS approach for probabilistic hurricane loss estimation.

Annual mean upcrossing rate

10

10

10

10

-2

-3

Debris-only scenario
Wind-only scenario
Interaction scenario

-4

-5

0

10

20

30

40

50

60

70

80

90

100

Repair cost (% of building cost)
Fig. 7. Annual probability of exceedance of repair cost for different hazard scenarios.

for a target building in a hypothetical residential development under
three different hazard scenarios. It is observed that the interaction
between wind and windborne debris hazard can affect significantly
the value of the annual probability of exceedance of repair cost. This
observation suggests the need to consider the multi-hazard nature of
hurricane events for accurate probabilistic loss analysis.

Acknowledgements
The authors gratefully acknowledge support by: (1) the Louisiana
Board of Regents through the Pilot Funding for New Research Program
under Award No. NSF(2008)-PFUND-86; (2) the LSU Council on Research through the 2009–2010 Faculty Research Grant Program; (3)
the Longwell’s Family Foundation through the Fund for Innovation in
Engineering Research (FIER) Program; and (4) the Italian Ministry for
Education, University and Research – MIUR, which partially financed
the Wi-POD Research Project (2008–2010). Fruitful discussions with
Dr. Sauro Manenti (University of Pavia) are gratefully acknowledged.
Any opinions, findings, conclusions or recommendations expressed
in this publication are those of the authors and do not necessarily
reflect the views of the sponsors.

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“Performance-Based Hurricane Engineering (PBHE) framework”. Structural Safety, 45, 24-35. DOI: 10.1016/j.strusafe.2013.07.002

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“Performance-Based Hurricane Engineering (PBHE) framework”. Structural Safety, 45, 24-35. DOI: 10.1016/j.strusafe.2013.07.002