This document presents a probabilistic fragility analysis method to support the performance-based design of precast concrete cladding wall panels subjected to blast loads. Non-linear dynamic analyses are used to model a case study wall panel as an equivalent nonlinear single-degree-of-freedom system. Fragility curves are developed for limit states of the wall panel and used to calculate the probability of exceeding those limit states under different blast scenarios. Monte Carlo simulation is also used to validate the fragility curve results. The method provides a rigorous probabilistic approach for designing blast-resistant structures consistent with performance-based design frameworks for other hazards.

1. Fragility analysis for the Performance-Based Design of cladding wall
panels subjected to blast load
Pierluigi Olmati a,⇑
, Francesco Petrini b
, Konstantinos Gkoumas b
a
Faculty of Engineering and Physical Sciences, University of Surrey, GU2 7XH Guildford, UK
b
Sapienza University of Rome, Department of Structural and Geotechnical Engineering, Via Eudossiana 18, 00184 Rome, Italy
a r t i c l e i n f o
Article history:
Available online xxxx
Keywords:
Performance-based blast engineering
Fragility analysis
Concrete cladding wall panels
Cladding system
a b s t r a c t
This paper presents a probabilistic method to support the design of cladding wall systems subjected to
blast loads. The proposed method is based on the broadly adopted fragility analysis method (conditional
approach), widely used in Performance-Based Design procedures for structures subjected to natural
hazards like earthquake and wind. The cladding wall system under investigation is composed by
non-load bearing precast concrete wall panels. From the blast design point of view, these wall panels
must protect people and equipment from external detonations. The aim of this research is to compute
both the fragility curves and the limit states exceedance probability of a typical precast concrete cladding
wall panel considering the detonations of vehicle borne improvised explosive devices. Moreover, the
limit states exceedance probability of the cladding wall panel is estimated by Monte Carlo simulation
(unconditional approach) in order to validate the proposed fragility curves.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Designing structures to withstand blast loads is common prac-
tice for many government and commercial buildings. Generally in
the design practice, a set of design scenarios are selected and the
integrity of the blast-resistant structural members and of the pro-
tective elements is assessed by using non-linear dynamic analyses
with an equivalent single degree of freedom (SDOF) method. In
such a way (adopting a deterministic approach for the hazard char-
acterization), the probability of exceeding a particular limit state is
not evaluated. In addition to the above, in designing a structural
component subjected to blast loads, the current state of practice
is to assume that the capacity is deterministic.
The adoption of deterministic values for the demand is princi-
pally due to a lack of knowledge of the hazard probability density
function. This is common for Low Probability–High Consequence
(LPHC) events [1]. As a partial consideration of the uncertainty
affecting the blast load, simplified approaches are usually adopted.
For example, in [2] the use of a magnification coefficient of 20% is
applied on the assumed amount of explosive. However, this is
limited to explosive storage facilities. In an antiterrorism design,
the amount of explosive is characterized by elevated uncertainty
depending on both technical and socioeconomic factors. These
uncertainties lead the engineering community toward the
implementation of probabilistic methods [3], something that is
now crucial for both academics and practitioners.
With specific reference to blast-resistant structures, some
authors started carrying out investigations about the use of
probabilistic methods for the assessment and design of structural
components and structural systems. In [4], results of a parametric
investigation on the reliability of reinforced concrete slabs under
blast loading are presented, in order to establish appropriate
probabilistic distributions of the resistant parameters. In [5], the
extension of probabilistic approaches from the performance-based
earthquake engineering to the blast design problems are provided,
also by suggesting appropriate variables for the intensity measures
IMs, the damage measures DMs, and the response parameters defi-
nition. In [6], Monte Carlo simulations are performed in order to
estimate the failure probability of windows subjected to a blast load
made by a vehicle bomb. In [7], the fragility curves are presented for
two kinds of glazing systems. In [8], the design in a probabilistic way
of a sacrificial cladding for a blast wall is described, deployed to pro-
tect vulnerable objects against an accidental explosion.
Due to the above considerations, the definition of appropriate
frameworks for the probabilistic design of blast resistant structures
is an important objective for the engineering scientific community.
To this regard, during the last decade Performance-Based Design
(PBD) has been recognized as a powerful methodology for verifying
the achievement of design performance objectives of structural
systems during their design life [9]. Probabilistic approaches have
been extensively implemented in the state of the art methods for
http://dx.doi.org/10.1016/j.engstruct.2014.06.004
0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
E-mail address: pierluigi.olmati@gmail.com (P. Olmati).
Engineering Structures xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

2. the PBD of structures under different kind of hazards such as earth-
quake [10,11], wind [12–14], and hurricanes [15]. The last case is
an example of multi-hazard situation that is expected to be one
of the main directions of PBD approaches in the future [16].
In the PBD context, a powerful tool is represented by the
fragility analysis (see for example [17–19]). As it is well known,
the structural fragility is expressed as the cumulative probability
distribution of attaining a certain Damage Measure (DM) condi-
tional to the Intensity Measure (IM) of the hazard. The efficiency
of the fragility approach is strictly related to the appropriateness
of the IM in terms of ‘‘sufficiency’’ and ‘‘efficiency’’, meaning
that the IM must accurately describe all pertinent hazard sources
(see [20]).
Despite the above, a rigorous approach that is consistent with
the well-established PBD frameworks adopted in presence of other
hazards has not been defined for blast resistant structures. This
paper is an effort in that direction with a specific focus on the fra-
gility analysis. The fragility analysis is applied in order to compute
the probability of exceeding a limit state (‘‘probability of exceed-
ance’’) of a precast wall panel subjected to blast loads (in particular
far-field surface-blast loads [2]) in a PBD perspective.
As case study, a precast concrete cladding wall panel with the
dimensions of 3500 mm in length and 1500 mm in width, with a
cross sectional thickness of 150 mm is considered. The panel is
subjected to a blast load generated by a vehicle borne improvised
explosive device. The wall under investigation is a non-load bear-
ing precast concrete wall panel used as exterior cladding for build-
ings. Typically, the length and the width of these walls are subject
to specific architecture requirements while their thickness is
approximately 15 cm. The steel reinforcements are generally
placed in the middle of the cross section. This kind of wall panels
should be designed in order to protect occupants and equipment
from external detonations.
Non-linear dynamic analyses are carried out by the well-estab-
lished method of the equivalent non-linear SDOF system, where
the precast concrete wall panel is modeled by an equivalent non-
linear SDOF on the basis of energetic considerations. Furthermore,
both the fragility curves and the probabilities of exceedance are
computed using Monte Carlo simulations.
The fragility curves are evaluated for the case-study wall panel
for each defined limit state called here Component Damage Level
(CDL). Then the fragility curves are used in order to estimate the
probability of exceedance of the cladding wall panel subjected to
blast load scenarios (vehicle borne improvised explosive devices).
Finally, the probability of exceedance of the wall panel subjected
to the same scenarios is estimated by the unconditional approach
(based on a single Monte Carlo simulation) in order to validate the
obtained results (see [4,11]).
In addition to the fragility analysis of the examined structural
member typology (an innovative aspect of the paper), some preli-
minary indications on the selection of a sufficient and efficient IM
for PBD of blast-resistant structures are provided.
2. Fragility analysis
As previously stated, the fragility of a structure under the action
of a certain hazard is expressed as the cumulative probability dis-
tribution of a certain DM conditional to the IM of the considered
hazard. Probabilistic PBD approaches identify the generic struc-
tural performance by means of acceptable occurrence frequencies
for some threshold values (representing structural limit states) of
an appropriate DM during a reference period of time [21,22]. The
determination of such occurrences is affected by large amounts
of uncertainty. The fragility approach allows the designer to
express in a synthetic and efficient manner this uncertainty by
making use of conditional probability relations and by highlighting
the dependences of these occurrences from the IM.
In earthquake engineering, the fragility approach has been
mostly developed during last twenty years and applied for PBD
purposes. The fragility curves have been developed also for struc-
tures subjected to flood [23], fire [24], and windborne debris in
hurricane prone regions [25]. The fragility curves are nowadays
extensively used for the state of practice methods of structural risk
evaluation for structures under natural hazards.
Among other techniques proposed for the evaluation of the fra-
gility curves, Monte Carlo analysis is extensively used [26].
Two main issues need to be addressed primarily in order to
develop fragility curves under a single hazard by Monte Carlo
approaches. These are due to the fact that: (i) the computational
effort required in order to obtain the desired level of approxima-
tion is often challenging [27]; and, (ii) the individuation of an effi-
cient and sufficient scalar IM for fragility representation is needed.
The last point is essential since, in case of a vectorial IM, the
structural fragility needs to be represented in terms of surfaces,
something that is required for example in the case of performance
analysis under multiple hazards (see for example [28]). In this
paper, this issue is discussed focusing the attention on the criticism
of choosing a scalar IM.
As a first step, the uncertainties characterizing blast-engineer-
ing problems need to be properly individuated and addressed
(Fig. 1). These uncertainties can be divided into three main groups:
hazard uncertainties (e.g. explosive, stand-off distance);
structure uncertainties (e.g. stiffness, dimensions, damping,
material characteristics, damping, etc.);
interaction mechanism uncertainties (e.g. the reflected pres-
sure, pressure duration, etc.);
This classification of the uncertainties in three groups (load,
structure, interaction mechanisms) is generally valid for many
engineering fields.
The IM in general should be chosen among the first group of
uncertain parameters or as a combination of those parameters,
while the entity of the blast action given a certain IM is determined
by the parameters characterizing the interaction between the IM
and the structural parameters. In probabilistic terms, hazard and
structural parameters can be characterized as unconditional with
respect to parameters belonging to one of the other two groups,
while parameters representing the interaction mechanisms must
be usually characterized in conditional probabilistic terms with
respect to the hazard and the structural parameters [12].
Fig. 1. Uncertaint parameters of vehicle borne improvised explosive device
scenarios.
2 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

3. 3. Blast load model
The side-on blast pressure Ps0 (MPa), can be estimated by the
formula of Mills [29] (Eq. (1)), while the side-on specific impulse
is0 (Pa s), is estimated by the formula of Held [30] (Eq. (2)). Where,
Z is the scaled distance (Eq. (3)), W is the explosive weight, and R is
the stand-off distance.
Ps0 ¼ 1:772
1
Z3
À 0:114
1
Z2
þ 0:108
1
Z
ð1Þ
is0 ¼ 300
1
2
ffiffiffiffiffiffi
W
3
p
ð2Þ
Z ¼
R
ffiffiffiffiffiffi
W3
p ð3Þ
The explosive amount is commonly expressed as an equivalent TNT
charge [2] by the Equivalent Factor (EF), which multiplies the
weight of the explosive charge utilized.
Both Eqs. (1) and (2) are valid for free-air explosions. In this
study detonations occurring on a surface (surface explosions) are
considered, therefore the energy of the detonation is confined by
the ground surface, creating a larger demand than that of the
free-air explosion. The surface blast demand is calculated by using
the same equations for the free-air explosions but with a charge
weight (W) increased by 80% [31]. Then, the reflected pressure Pr
(MPa) for a normal angle of incidence is computed using Eq. (4)
[29]:
Pr ¼ 2Ps0
7Patm þ 4Ps0
7Patm þ Ps0
ð4Þ
where Patm is the atmospheric pressure (0.1 MPa). For simplicity,
the negative pressure phase is neglected from the blast load time
history [31]. And the duration of the blast load (td) is computed
from the side-on pressure and the specific impulse, assuming a
triangular impulse (Eq. (5)).
td ¼
2is0
Ps0
ð5Þ
The variation in the pressure is assumed approximately to follow
the Friedlander pulse shape shown in Eq. (6). Further details on
the sensitivity of the structural response due to exponential or tri-
angular blast loading are investigated in [32].
PðtÞ ¼ Pr 1 À
t
td
e
Àbt
td ta t td ð6Þ
In Eq. (6), ta (s) is the arrival time of the blast load, taken here as
zero, and b is the decay coefficient. In this study a value of 1.8 for
b is assumed [31]. The clearing effect is conservatively neglected
in this study since the cladding wall panel is generally a single part
of a large building façade, thus the conditions for clearing of the
reflected shock wave are in general not satisfied. In Fig. 2 the blast
load time histories computed for different values of the explosive
weight W (kg) and stand-off distance R (m) with the above-men-
tioned procedure are shown. The obtained curves are found to be
in good agreement with the curves obtained by SBEDS [31]. The
blast load is considered uniformly distributed on the cladding wall,
which is typical for a scaled distance higher than approximately
2.0 m/kg1/3
[31].
4. Cladding panel model
A precast concrete cladding wall system has some advantages
with respect to the traditional non-load bearing masonry
claddings. Studies on improving the performances of traditional
masonry claddings under blast loads are presented in [33,34],
while in [35], the load bearing capacity of load bearing concrete
walls subjected to a blast demand is investigated. The first advan-
tage of precast concrete wall panels versus traditional masonry
claddings is the greater resistance of the cladding system to a blast
demand (see for example the advantage of using precast concrete
wall panels for protecting steel stud constructions [36]).
Precast concrete wall panels can be integrated with other mate-
rials for improving the resistance to environmental attacks such as
acid rains and chlorine ions exposure. In [37] the protection perfor-
mance of cellulosic fiberboard panels from ballistic attacks is
investigated. Furthermore, in [38] the behavior of concrete insu-
lated panels is investigated, focusing on the shear ties connecting
the two concrete wythes confining the insulation.
The cladding panel taken as case study is 3500 mm long by
1500 mm wide, with a cross sectional thickness of 150 mm. The
panel is connected along two of its edges to the external frame
of a building, and in particular, it is assumed to be simply-
supported by the beams of the frame. Length, width and cross sec-
tional thickness of the panel are considered as stochastic variables,
due to the construction tolerances used in the precast concrete
industry (see for example [39]). The assumed mean values and
Coefficients of Variations (COVs) are shown in Table 1. The longitu-
dinal reinforcement consists of ten reinforcement bars of 10 mm
diameter located in the mid-axis of the cross section. The mean
value and the COVs of the reinforcement strength are provided in
Table 1. The panel is assumed to not have shear reinforcement.
4.1. Concrete
The concrete compressive strength fc is considered as a stochas-
tic variable, while the Young’s modulus of the concrete Ec and the
concrete density q are expressed as functions of fc. The mean value
of fc is 28 MPa (4060 psi), with a COV of 0.18, as adopted in [40] for
a lognormal probability density function (see Table 1). The Young’s
Fig. 2. Blast loads by the adopted model (broken lines) and the SBEDS model (solid
lines).
Table 1
Input data.
Symbol Description Mean COV Distribution
fc Concrete strength 28 MPa 0.18 Lognormal
fy Steel strength 495 MPa 0.12 Lognormal
L Panel length 3500 mm 0.001 Lognormal
H Panel height 150 mm 0.001 Lognormal
b Panel width 1500 mm 0.001 Lognormal
c Panel cover 75 mm 0.01 Lognormal
W Explosive weight 227 kg 0.3 Lognormal
R Stand-off distance 15 m 20 m 25 m 0.05 Lognormal
P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 3
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

4. modulus is computed by Eq. (7) [41] while the concrete density is
computed by Eq. (8) [42,43]. Both Ec and fc are expressed in MPa
while q is expressed in kg/m3
.
Ec ¼ 22; 000
fc
10
0:3
ð7Þ
q ¼
Ec
0:043ðf0:5
c Þ
1
1:5
ð8Þ
The compressive strength increment of the concrete due to the high
strain rate is accounted for in this study. This increase is taken into
account by means of the Dynamic Increase Factor (DIF), a multipli-
cative coefficient of the concrete static compressive resistance. Since
the compressive strength enhancement of the concrete varies
marginally for a ductile flexural response over the range of the strain
velocity, the DIF can be assumed constant and equal to 1.19.
This hypothesis is in accordance with the compressive strength
enhancement proposed in [31], and it leads to an increase of the
computational efficiency by avoiding cyclic iterations in the algo-
rithm of the SDOF equation solver. However, cyclic iterations are
necessary for computing the strength increase of the reinforcement.
This increase is sensitive to the ductile flexural response, as opposed
to the compressive strength increase of the concrete.
4.2. Reinforcing steel
A grade 450 MPa steel is used. Due to material standard
requirements the average yield strength is higher than the speci-
fied yield. For estimating the mean value of the yield strength, a
static strength increase factor equal to 1.1 is adopted as indicated
in [31]. A COV of 0.12 is used as proposed in [40] for a lognormal
probability density function (see Table 1). Young’s modulus is
taken as a deterministic value and assumed to be equal to
210 GPa. The steel strength increase due to the strain velocity is
taken into account by the Cowper and Symonds model [44]. Thus,
the DIF is provided by Eq. (9):
DIF ¼ 1 þ
d=dt
q
1
n
ð9Þ
where de/dt is the strain-rate demand of the reinforcement, q is
taken equal to 500 sÀ1
and n is taken equal to 6. Both q and n are
estimated by fitting the strength increase versus the strain rate in
[31]. By solving the SDOF equation of motion, the DIF is iteratively
updated until a convergence threshold is reached. The strain rate of
the steel reinforcement (de/dt) in Eq. (9) is calculated approximately
by Eq. (10).
de
dt
¼
dS
dt
L
8
d
2EcJc
ð10Þ
where L is the length of the cladding panel, Ec is the Young’s mod-
ulus of the concrete, Ic is the moment of inertia of the cracked cross
section, d is the distance from the extreme compression fiber of the
cross-section to the centroid of the tensile reinforcement, and dS/dt
is the rate of the resistance force developed by the panel (S) when
subjected to the demand. Eq. (10) is valid for simply supported ele-
ments when the response is governed by the flexural behavior with-
out shear failure.
5. Equivalent SDOF mechanical model of the panel
In order to model a structural element subjected to a blast load
with an equivalent SDOF, the latter is defined as a system that has
the same energy of the original structural element (in terms
of work energy, strain energy, and kinetic energy) when the
structural element, if subjected to a blast load, deflects in a given
deformed shape. The displacement field of the component can be
expressed as u(x,t) = U(x)y(t), where U(x) is the assumed deformed
shape of the component under the blast load and y is the maximum
displacement of the component. Furthermore, the displacement of
the component is obtained by the SDOF equation by assuming a
flexural deformed shape by:
KLMM€yðtÞ þ C _yðtÞ þ SðyðtÞÞ ¼ FðtÞ ð11Þ
where y(t) is the displacement of the SDOF system and M is the total
mass of the component, S(y(t)) is the resistance of the component as
a function of the displacement expressed in unit force (see Fig. 3),
F(t) is the blast pressure multiplied by the impacted area (A)
expressed in force units, C is the damping (the percentage of the
critical damping is assumed to be 1% in all the analyses), KLM is
the load-mass transformation factor equal to the ratio of KM and
KL (the mass transformation factor and the load transformation fac-
tor respectively). The last two are evaluated by equating the energy
of the two systems (in terms of work energy and kinetic energy
respectively).
KL ¼
RL
0
pðxÞUðxÞdx
RL
0
pðxÞdx
KM ¼
RL
0
mðxÞU2
ðxÞdx
RL
0
mðxÞdx
ð12Þ
Referring to Eq. (12) and Fig. 3, p(x) is the blast load shape on the
component, m(x) is the distributed mass, and r is the resistance of
the element in terms of pressure. The load-mass transformation fac-
tor KLM is different at each deformation stage of the component
response; for a bilinear resistance function two values of the KLM
can be defined: one for the elastic response and one for the plastic
response. More details on the equivalent SDOF method are provided
in [2,31].
In order to obtain the bilinear resistance function of the simply
supported concrete cladding wall, it is necessary to compute the
resisting moment (Mr) in the mid-span of the panel. The yielding
point of the resistance function (Sy) is obtained by Eq. (13) where
the loaded surface (A) is equal to the cladding wall length (L) times
the cladding wall width (b) and ry is the pressure resistance
function.
Sy ¼
8Mr
L
¼ ryA ð13Þ
The resisting moment is evaluated by Eq. (14) [31]:
Mr ¼ Asfdy d À
Asfdy
1:7bfdc
ð14Þ
where As is the reinforcements area, fdy is the dynamic yield
strength of the reinforcing steel, fdc is the dynamic compressive
strength of the concrete, b width of rectangular section.
It is also necessary to evaluate the yielding displacement of
the resistance function. For a simply supported component the
yielding displacement (de) is given by Eq. (15).
de ¼
5
384
8MrL2
EcJ
ð15Þ
J ¼ 0:5ðJg þ JcÞ ð16Þ
where J is the moment of inertia of the cross section as evaluated by
Eq. (16), while Jc and Jg are computed by Eqs. (17) and (18)
respectively.
Jc ¼ Gbd
3
ð17Þ
Jg ¼
bH
3
12
ð18Þ
4 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

5. The coefficient G in Eq. (19) is evaluated starting by the design
chart provided in [31]. In this study an analytical formula (see Eq.
(19)) is determined by fitting the curves of the original chart [31].
G ¼ ð3320:3p3
À 181:98p2
þ 5:8624pÞ
n
7
0:7
ð19Þ
where p is the percentage of reinforcement in the cross section of
the panel evaluated by neglecting the reinforcements cover, and n
is the ratio between the steel Young’s modulus and the Ec. Eq.
(19) is valid for single reinforced cross-sections only. Since Sy and
de are evaluated, the resistance function of the simply supported
cladding wall can be defined. Finally, the central difference method
is then used to solve Eq. (11) which presents three non-linarites:
one due to the bilinear shape of the resistance function, one due
to the load mass transformation factors, and the last due to the
dynamic strength enhancement of the reinforcing steel (which
affects the resistance function).
6. Performance definition
For structural components subjected to blast loads in a flexural
response regime, generally two response parameters are of inter-
est: the support rotation angle (h) and the ductility ratio (l). These
parameters are defined in Eqs. (20) and (21):
h ¼ arctg
2dmax
L
ð20Þ
l ¼
dmax
de
ð21Þ
where dmax is the maximum displacement of the component.
A structural component subjected to a blast load is generally
expected to yield (ductility greater than 1), as it is economically
impractical to design a component to remain in elastic range.
While other significant response parameters can be defined (for
example [4] considers the strain on reinforcements), this study
focuses on the above defined response parameters, since these
are usually adopted for antiterrorism design [31].
In a performance-based blast design prospective, five Compo-
nent Damage Levels (CDLs) [31] are considered: Blowout (BO),
Hazardous Failure (HF), Heavy Damage (HD), Moderate Damage
(MD), and Superficial Damage (SD). Following the [31], the above
mentioned levels are defined as follows:
Blowout (BO): the component is overwhelmed by the blast load
causing debris with significant velocities.
Hazardous Failure (HF): the component has failed, and debris
velocities range from insignificant to very significant.
Heavy Damage (HD): the component has not failed, but it has sig-
nificant permanent deflections causing it to be un-repairable.
Moderate Damage (MD): the component has some permanent
deflection. It is generally repairable, if necessary, although replace-
ment may be more economical and aesthetic.
Superficial Damage (SD): the component has no visible permanent
damage.
The thresholds corresponding to these CDLs are defined in
terms of the response parameters h and l. For a non-structural
concrete cladding wall without shear reinforcement, neglecting
tension membrane effect, the CDL thresholds are those reported
in Table 2. The Fragility Curves are computed in the following for
each of the mentioned CDLs.
7. Algorithm for the computation of the fragility curves
As described in previous sections, the blast load on the panel is
a function of the peak pressure and of the impulse density (Eqs. (4)
and (2) respectively); the pressure is function of Z only (Eqs. (1)
and (4)), while the impulse density depends on both the Z and
the W (Eq. (2)). Consequently, two detonations with the same Z
can have different impulse density, depending on the amount of
explosive. Thus, the two explosions have the same peak pressure
but different duration.
Summarizing, since the blast load depends on both the Z and
the W, the choice of the IM for computing the fragility curves is a
crucial issue. In this study the Z is taken as the IM. Some aspects
related to this choice are discussed in the next section. Note that
for higher values of the Z the cladding wall has a lower structural
response than for lower values.
The Fragility Curves (FCs) are developed for each CDL. The algo-
rithm implemented in MATLABÒ
for the fragility curves evaluation
is shown in Fig. 4. With reference to the same figure, ‘‘i’’ indicates
the ith
point of the fragility curve, ‘‘j’’ indicates the ‘‘j’’th CDL, and
‘‘k’’ indicates the kth
stand-off distance (R) for which the fragility
curve is computed. ‘‘N’’ is the maximum value for ‘‘i’’ and therefore
the total number of points forming the fragility curve. ‘‘M’’ is the
maximum value for ‘‘j’’ and therefore the total number of the CDLs.
Finally ‘‘L’’ is the maximum value of ‘‘k’’ therefore the total number
Fig. 3. Resistance versus displacement relation of a component.
Table 2
Component damage levels and the associated thresholds in terms of response
parameters [31].
Component damage levels h (degree) l (–)
Blowout 10° None
Hazardous failure 610° None
Heavy damage 65° None
Moderate damage 62° None
Superficial damage None 1
P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 5
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

6. of the stand-off distances for which the fragility curve related to
the jth
CDL is computed.
The name ‘‘FC-CDL (j,k)’’ indicates the fragility curve computed
for the kth
R, the jth
CDL, by varying the W (and consequently the Z).
The ith
point of the fragility curve (named FC-CDL (i,j,k)) is com-
puted by considering the blast load at the kth
R and the ith
W.
The minimum and maximum amount of W should be enough for
computing the values of the FC-CDL (j,k) ranging from 0 to 1.
The FC-CDL (i,j,k) is obtained by a Monte Carlo simulation and
the complete (cyclic) procedure of Fig. 3 is hereby described.
first a kth
R is selected;
then the jth
CDL is selected;
consequently the ‘‘i’’ index is increased by solving the previ-
ously introduced equations for each ‘‘i’’ in order to evaluate
the ith
points of the FC-CDL (j,k) until tracking the complete
FC-CDL (j,k);
after that, a new jth
CDL is considered with the same value of
‘‘k’’. When j = M a different R is selected and the previous two
described cycles are repeated until k = L;
finally, the piecewise curves obtained point by point with the
above steps they are interpolated by a lognormal cumulative
function, see Fig. 6.
As said, the fragility curves describe the conditional probability
of exceedance (P(X x0|Z)) of the response parameter X (chosen
case-by-case as the most critical between the values of h and l,
see Table 2) with respect to the threshold x0 (identifying the
CDL). As expected, for a constant number of samples at each ith
point, the COV of P(X x0|Z) increases with the decreasing of
P(X x0|Z). In order to obtain an acceptable COV, the number of
samples adopted in the analysis is increased with the decreasing
of P(X x0|Z); this means that the number of samples increase with
increasing Z. In this work, an exponential law has been set for this
increasing trend. In Fig. 5 the number of the samples and the rela-
tive COVs are shown in function of P(X x0|Z) for the fragility curve
related to the heavy damage CDL and for R equal to 20 m.
8. Sufficiency of the intensity measure and results of the
fragility analysis
For better understanding the sufficiency of the adopted inten-
sity measure (the scaled distance Z), some considerations can be
made with reference to the pressure–impulse diagrams [45]
related to the case study panel.
For this purpose, reference is made to the mean values of both
materials and geometrical parameters (see Table 1), and the DIFs
for the concrete and steel are taken as constant and equal to 1.19
and 1.20 respectively. The pressure–impulse diagrams referred to
different values of h are shown in Fig. 7. Generally, three regions
can be individuated in the pressure–impulse diagrams, each
related with a different regime of structural response subjected
to a load time history. These are defied as: impulsive (I), dynamic
(D), and pressure (P) region, depending on the characteristics of
the load time history with respect to the dynamical proprieties
of the structure [2].
Two blast loads are taken into account. These loads can be cho-
sen in such a way that they are characterized by the same IM (and
consequently by the same peak pressure) but having different W
and R. As a matter of fact, the two blast loads are consistent with
two different demands on the pressure–impulse diagrams, having
the same peak pressure but different values of the impulse density.
As it can be observed in Fig. 7, the difference between the struc-
tural response of the panel subjected to the two above mentioned
blast demands (again, characterized by the same pressure peak but
by different impulse densities) depends on the position of these
demands in the pressure–impulse diagram. Thus, if these blast
demands are located in the impulsive region (I), a certain value
for this difference will be observed, while if blast demands are
located in the dynamic (D) or pressure (P) regions, then this differ-
ence will be lower than in the previous case.
Considering the above, it can be concluded that the sufficiency
of the chosen IM is greater in the D and P regions than in the I
region. In the I region a fragility surface made by considering both
R and W as independent elements of a vectorial IM would be more
appropriate.
For taking account this approximation, as explained in the pre-
vious section, the fragility curves are computed for different values
of the R (R ¼ Z
ffiffiffiffiffiffi
W3
p
). In what follows, when the fragility curves are
used for estimating the failure probability of a component damage
Fig. 4. Fragility curve computation flowchart.
0.00
0.02
0.04
0.06
0.08
0.10
0
20000
40000
60000
80000
100000
0.1 0.9 3.3 9.0 22.4 40.4 59.5 77.9 90.1 96.6 98.8 100.0
C.O.V.
N°ofsamples
P(Xx0|Z) [%]
N° of samples C.O.V.
Fig. 5. N° of samples and COV for the fragility curve representative of the HF
component damage level, for R equal to 20 m.
6 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

7. level, the specific fragility curve corresponding to the mean value
of R is used for this purpose. This increases considerably the suffi-
ciency of the chosen IM.
8.1. Results
This section presents the results regarding: (i) the fragility
curves of the case study cladding panel, and, (ii) the probability
of the limit state exceedance of such cladding panel estimated by
both the conditional and unconditional approaches. The last point
is important in order to validate the computed fragility curves by a
comparison of the exceedances obtained by the two approaches.
In Fig. 8 the computed fragility curves are shown for different
values of R. Focusing on the considered CDLs, from Fig. 8 can be
observed that the fragility curves of the SD level have a different
0
20
40
60
80
100
3.7 3.9 4.1 4.3 4.5
P(Xx0|Z)[%]
Z
Interpolated FC
Numerical FC
Fig. 6. Numerical and lognormal interpolated fragility curves.
Fig. 7. Pressure–impulse diagrams.
0
20
40
60
80
100
2.4 2.6 2.8 3.0 3.2 3.4
P(Xx0|Z)[%]
Z
Hazardous Failure
0
20
40
60
80
100
2.8 3.0 3.2 3.4 3.6 3.8 4.0
Heavy Damage
P(Xx0|Z)[%]
Z
0
20
40
60
80
100
3.0 3.5 4.0 4.5 5.0
P(Xx0|Z)[%]
Z
Moderate Damage
0
20
40
60
80
100
5 6 7 8 9 10 11
P(Xx0|Z)[%]
Z
Superficial Damage
Fig. 8. From top left clockwise, fragility curves for the HF, HD, SD, MD component damage levels.
Street
Level 2
Level 3
Level 1
Target
Fig. 9. Lines of defence.
P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 7
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

8. slope compared to that of the other three CDLs (HF, HD, and MD). It
should be noted that the SD level is based on the ductility (l) of the
component while HF, HD, and MD levels are based on the support
rotation (h). The SD level for a concrete cladding panel prescribes
the elastic response of the component, and for the case study panel
it appears to be more sensitive to the considered uncertainties
compared to the HF, HD, and MD levels. By varying the number
of samples the maximum obtained COV for the lower probability
of failure (close to zero), is about 9% (Fig. 5). This value is consid-
ered acceptable for the specific case, and it is consistent with other
studies on blast applications (see [7]).
For computing the limit state probability of exceedance by the
conditional approach it is necessary to develop a hazard analysis
for the stochastic characterization of the blast scenario and to solve
Eq. (23). In this study, a vehicle borne improvised explosive device
is considered. The amount of explosive (W) in the vehicle depends,
among else, on the security measures in place. These security mea-
sures can be structured in different lines (see Fig. 9) and for each
line of security a different mean value of W is expected. The
expected value of W decrease with the decreasing of R from the
target, since the line of security system reduces progressively the
severity of the possible attacks.
In the example of Fig. 9, level 1 prevents trucks entering the tar-
get zone, so no truck bomb should be expected. Level 2 in Fig. 9 (for
example a fence barrier) prevents vehicles entering. Finally, Level 3
prevents pedestrians approaching the target.
With this in mind, in the specific application a scenario con-
cerning a truck bomb (with about 4000–27,000 kg of TNT or equiv-
alent) is unreasonable (e.g. by assuming that the intelligence
service is able to prevent this threat). Therefore, a vehicle bomb
(with about 27–454 kg of TNT or equivalent) is considered. The
mean amount of TNT or equivalent in the vehicle is assumed equal
to 227 kg with a COV equal to 0.3 (see Table 1); this assumption is
in line with [46]. A set of stand-off distances are considered (15, 20
and 25 m) each with a coefficient of variation equal to 0.05, assum-
ing that the vehicle could impact a fence barrier but move no
further.
The conditional probability of exceedance of the CDL (P(X x0))
is evaluated by Eq. (23). As previously stated, X is the most critical
between the response parameters h and l, assumed here as uncor-
related. Consequently P(X x0) is the union of the two failure prob-
abilities evaluated by considering separately the two response
parameters characterizing the component damage level (see
Table 2 and Eq. (22)). The probability density function of the Z
(p(Z)) is computed by fitting the samples of both W and R with a
lognormal distribution. As mentioned above, the fragility curve
(P(X x0|Z)) used for evaluating Eq. (23) is the one corresponding
to the mean value of the R (Table 1).
PðX x0Þ ¼ PðH h0Þ [ PðM l0Þ ð22Þ
PðX x0Þ ¼
Z þ1
À1
PðX x0jZÞpðZÞdz
ffi
X1
i¼0
PðX x0jZÞipðZÞiDZi ð23Þ
The obtained results are shown in Table 3. The first column reports
the CDLs, while in the second and third columns report the P(X x0)
for each blast scenario obtained by Eq. (23) and by the uncondi-
tional approach respectively, the last one considered for comparison
purposes as ‘‘exact value’’ of the exceedance probability.
From these results the maximum percentage difference
between the P(X x0) computed by the conditional and uncondi-
tional approaches is 11%. Further studies are necessary to confirm
whether this percentage difference is acceptable or not.
However, it is also necessary to consider that the W in the vehi-
cle has an elevated dispersion, something that amplifies this differ-
ence due to the dependence of the impulse density to both R and
W. Thus, the difference between the P(X x0) computed by the con-
ditional and unconditional approaches increases with the increase
in the difference between the R with which the fragility curve is
computed (mean value of R) and the R of the Monte Carlo samples
of the unconditional approach.
9. Conclusions
The probabilistic analysis of a precast concrete cladding wall
panel subjected to blast load has been presented. The blast load
model has been adopted on the basis of empirical laws, and both
the geometry and mechanical properties of the panel are assumed
as stochastic. A mechanical model equivalent to a single degree of
freedom has been adopted for describing the motion of the panel
under the blast load. The Monte Carlo simulation has been used
for computing: (i) the fragility curves of the cladding wall panel
subjected to blast load for several limit states (component damage
levels), (ii) the probability of exceedance of limit states of the clad-
ding wall panel by means of both the unconditional and condi-
tional approach for comparison purposes.
A discussion about the effectiveness of the intensity measure
chosen for the fragility analysis has been presented and analyses
have been carried out for different values of the stand-off distance.
It is expected that the scaled distance Z adopted in this paper, is a
sufficient intensity measure especially for blast demands belong-
ing to the dynamic and pressure region of the pressure–impulse
diagrams.
This study highlights the feasibility and effectiveness of the fra-
gility approach in the design of cladding wall panels and, generally,
of protective structures. This is one of the fundamental steps nec-
essary for developing a fully probabilistic Performance-Based
Design for blast resistant structures, something already done for
structures subjected to other hazards (e.g. earthquake and wind).
One of the main issues related to the completion of a probabilistic
performance-based blast engineering procedure consists in deter-
mining the hazard function; this issue is mainly due to the fact that
an explosion event (e.g. terroristic vehicle bomb attack) is a low
probability event, as described in [47–49].
Additional studies could focus on: (i) improving the intensity
measure sufficiency and efficiency, (ii) considering the degradation
of the cladding panel during the life-cycle, (iii) improving the
mechanical model that describes the response of the panel
Table 3
Comparisons of the probability of exceedance using the conditional and unconditional
approaches.
CDL Mean W = 227 kg, COV = 0.3 lognormal
Mean R, COV = 0.05 lognormal
Conditional
approach (%)
Unconditional
approach (%)
Percentage
difference D%
R = 20 m
SD 100.0 100.0 0.0
MD 96.6 97.5 0.9
HD 55.7 55.5 0.3
HF 13.6 12.1 11.0
R = 25 m
SD 100.0 100.0 0.0
MD 74.6 77.3 3.5
HD 14.2 12.6 11.2
HF 1.02 1.02 0.0
R = 15 m
SD 100.0 100.0 0.0
MD 97.9 99.9 2.0
HD 93.6 96.9 3.4
HF 67.8 72.6 6.6
8 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

9. subjected to a blast load, and (iv) developing appropriate methods
for complex system design [50] starting from the element-based
analysis provided in this paper.
Acknowledgments
The authors gratefully acknowledge the scientific contribution
of Prof. Franco Bontempi of the Sapienza University of Rome.
This work was partially supported by StroNGER s.r.l. from the
fund ‘‘FILAS – POR FESR LAZIO 2007/2013 – Support for the
research spin-off’’.
References
[1] Starossek U. Progressive collapse of structures. London, United
Kingdom: Thomas Telford Publishing; 2009.
[2] UFC 3-340-02. Structures to resist the effects of accidental explosions. United
States of America: Department of Defense; 2008.
[3] Netherton MD, Stewart MG. Blast load variability and accuracy of blast load
prediction models. Int J Protect Struct 2010;1(4):543–70.
[4] Low HY, Hao H. Reliability analysis of reinforced concrete slabs under
explosive loading. Struct Saf 2001;23:157–78.
[5] Whittaker AS, Hamburger RO, Mahoney M. Performance-based engineering of
buildings and infrastructure for extreme loadings. In: Proceedings of the AISC-
SINY symposium on resisting blast and progressive collapse. New York, USA;
December 4–5, 2003.
[6] Chang DB, Young CS. Probabilistic estimates of vulnerability to explosive
overpressures and impulses. J Phys Secur 2010;4(2):10–29.
[7] Stewart MG, Netherton MD. Security risks and probabilistic risk assessment of
glazing subject to explosive blast loading. Reliab Eng Syst Safety
2008;93:627–38.
[8] Linkute˙ L, Juocevicˇius V, Vaidogas ER. A probabilistic design of sacrificial
cladding for a blast wall using limited statistical information on blast loading.
Mechanika 2013;19(1):58–66.
[9] Augusti G, Ciampoli M. Performance-based design in risk assessment and
reduction. Probab Eng Mech 2008;23(4):496–508.
[10] Cornell CA, Krawinkler H. Progress and challenges in seismic performance
assessment. PEER Center News 2000;3(2).
[11] Vamvatsikos D, Dolšek M. Equivalent constant rates for performance-based
seismic assessment of ageing structures. Struct Saf 2011;33(1):8–18.
[12] Petrini F, Ciampoli M. Performance-based wind design of tall buildings. Struct
Infrastruct Eng – Maint Manage Life-Cycle Des Perform 2012;8(10):954–66.
[13] Spence SMJ, Gioffrè M. Large scale reliability-based design optimization of
wind excited tall buildings. Probab Eng Mech 2012;28:206–15.
[14] Spence SMJ, Gioffrè M. Efficient algorithms for the reliability optimization of
tall buildings. J Wind Eng Ind Aerodyn 2011;99(6–7):691–9.
[15] Barbato M, Petrini F, Unnikrishnan VU, Ciampoli M. Probabilistic performance-
based hurricane engineering (PBHE) framework. Struct Saf 2013;45:24–35.
[16] Decò A, Frangopol DM. Risk assessment of highway bridges under multiple
hazards. J Risk Res 2011;14(9):1057–89.
[17] Olmati P. Monte Carlo analysis for the blast resistance design and assessment
of a reinforced concrete wall. In: ECCOMAS thematic conference – COMPDYN
2013: 4th international conference on computational methods in structural
dynamics and earthquake engineering. Proceedings – an IACM special interest
conference, Kos Island, Greece. Code 104617; 12 June 2013–14 June 2013. p.
1803–15.
[18] Olmati P, Trasborg P, Naito CJ, Bontempi F. Blast resistant design of precast
reinforced concrete walls for strategic infrastructures under uncertainty. Int J
Crit Infrastruct; ISSN online: 1741-8038.
[19] Bazzurro P, Cornell CA, Shome N, Carballo JE. Three proposal for characterizing
MDOF nonlinear seismic response. J Struct Eng 1998;124(11):1281–9.
[20] Luco N, Cornell CA. Structure-specific scalar intensity measures for near-
source and ordinary earthquake ground motions. Earthq Spectra
2000;23:357–92.
[21] Pozzuoli C, Bartoli G, Peil U, Clobes M. Serviceability wind risk assessment of
tall buildings including aeroelastic effects. J Wind Eng Ind Aerodynam
2013;123:325–38.
[22] Ciampoli M, Petrini F. Performance-based Aeolian risk assessment and
reduction for tall buildings. Probab Eng Mech 2012;28:75–84.
[23] Van De Lindt J, Taggart M. Fragility analysis methodology for performance-
based analysis of wood-frame buildings for flood. Science 2009;10(3):113–23.
[24] Lange D, Devaney S, Usmani A. An application of the PEER performance based
earthquake engineering framework to structures in fire. Eng Struct
2014;66:100–15.
[25] Herbin AH, Barbato M. Fragility curves for building envelope components
subject to windborne debris impact. J Wind Eng Ind Aerodyn 2012;107–
108:285–98.
[26] Smith MA, Caracoglia L. A Monte Carlo based method for the dynamic ‘‘fragility
analysis’’ of tall buildings under turbulent wind loading. Eng Struct
2011;33:410–20.
[27] Taflanidis A, Beck JL. Stochastic subset optimization for reliability optimization
and sensitivity analysis in system design. Comput Struct 2009;87(5–
6):318–31.
[28] Lee KH, Rosowsky D. Fragility analysis of woodframe buildings considering
combined snow and earthquake loading. Struct Saf 2006;28:289–303.
[29] Mills CA. The design of concrete structures to resist explosions and weapon
effects. In: Proceedings of the 1st international conference for hazard
protection, Edinburgh; 1987.
[30] Held M. Blast waves in free air. Propellants, Explos, Pyrotech 1983;8(1):1–7.
[31] US Army Corps of Engineers. Methodology manual for the single-degree-of-
freedom blast effects design spreadsheets (SBEDS); 2008.
[32] Gantes CJ, Pnevmatikos NG. Elastic–plastic response spectra for exponential
blast loading. Int J Impact Eng 2004;30:323–43.
[33] Davidson JS, Fisher JW, Hammons MI, Porter JR, Dinan J. Failure mechanisms of
polymer-reinforced concrete masonry walls subjected to blast. J Struct Eng
2005;131(8):1194–205.
[34] Moradi LG, Davidson JS, Dinan RJ. Resistance of membrane retrofit concrete
masonry walls to lateral pressure. J Perform Constr Facil 2008;22(3):131–42.
[35] Naito CJ, Wheaton KP. Blast assessment of load-bearing reinforced concrete
shear walls. Pract Periodical Struct Des Constr 2006;11(2):112–21.
[36] Naito CJ, Dinan R, Bewick B. Use of precast concrete walls for blast protection
of steel stud construction. J Perform Constr Facil 2011;25(5):454–63.
[37] Jordan JB, Naito CJ. Calculating fragment impact velocity from penetration
data. Int J Impact Eng 2010;37:530–6.
[38] Naito C, Hoemann J, Beacraft M, Bewick B. Performance and characterization of
shear ties for use in insulated precast concrete sandwich wall panels. J Struct
Eng 2012;138(1):1–11.
[39] Precast Prestressed Concrete Institute (PCI). PCI design handbook, 7th ed.;
2010.
[40] Enright MP, Frangopol DM. Probabilistic analysis of resistance degradation of
reinforced concrete bridge beams under corrosion. Eng Struct
1998;20(11):960–71.
[41] Eurocode 2 – design of concrete structures – Part 1–1: general rules and rules
for buildings. European Committee for Standardization; 2005.
[42] Australian standard 3600 concrete structures. North Sydney: Standards
Association of Australia; 1988.
[43] American Concrete Institute (ACI): Building Code Requirements for Reinforced
Concrete; 2011.
[44] Cowper GR, Symonds PS. Strain hardening and strain rate effects in the impact
loading of cantilever beams. Applied mathematics report no. 28. Providence,
Rhode Island, USA: Brown University; 1957.
[45] Krauthammer T, Astarlioglu S, Blasko J, Soh TB, Ng PH. Pressure–impulse
diagrams for the behavior assessment of structural components. Int J Impact
Eng 2008;35:771–83.
[46] The Federal Emergency Management Agency (FEMA). Risk assessment a how-
to guide to mitigate potential terrorist attacks against buildings, providing
protection to people and buildings. FEMA 452; 2005.
[47] Olmati P, Petrini F, Bontempi F. Numerical analyses for the structural
assessment of steel buildings under explosions. Struct Eng Mech
2013;45(6):803–19.
[48] Olmati P, Gkoumas K, Brando F, Cao L. Consequence-based robustness
assessment of a steel truss bridge. Steel Compos Struct 2013;14(4):379–95.
[49] JalaliLarijani R, Celikag M, Aghayan I, Kazemi M. Progressive collapse analysis
of two existing steel buildings using a linear static procedure. Struct Eng Mech
2013;48(2):207–20.
[50] Sgambi L, Gkoumas K, Bontempi F. Genetic algorithms for the dependability
assurance in the design of a long-span suspension bridge. Comput-Aided Civ
Infrastruct Eng 2012;27(9):655–75.
P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 9
Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng
Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004