3.full

Article
Risk assessment of structures
subjected to blast
Gholamreza Abdollahzadeh and Marzieh Nemati
Abstract
Attacking city centers with pack portable bombs has become one of the regular terrorist attacks around
the world. In these situations, life losses and injuries can be caused from various sources such as direct
blast effects, structural collapse, debris impact, fire, and smoke. Casualties could increase when indirect
effects are combined with closed exits or timely evacuation. So, calculating the annual risk of the struc-
tural collapses resulting from extreme loading conditions is subjected to many efforts. In this paper, the
annual risk of blast-induced progressive structural collapse is calculated. The blast fragility is also calcu-
lated by a simulation procedure which generate possible blast configuration, and finally kinematic plastic
limit analysis is used to verify the structural stability under gravity loading. As a case study, the blast
fragility and the annual risk of collapse of a four-storey steel building are calculated.
Keywords
Blast load, progressive collapse, annual risk, risk assessment, blast fragility
Introduction
Due to the accidental or intentional events occurred for structures all over the world, explosive loads
have received considerable attention in recent years. The design and construction of public buildings
which provides life safety in the face of explosions is receiving renewed attention from structural
engineers (Committee on Feasibility of Applying Blast Mitigating Technologies and Design
Methodologies from Military Facilities to Civilian Buildings, 1995; Elliot et al., 1992, 1994). Such
concern arose initially in response to air attacks during Second World War (Baker et al., 1983;
Jarrett, 1968; Smith and Hetheringtob, 1994), continued through the cold war (Al-Khaiat et al.,
1999), and more recently, this concern has grown with the increase of terrorism worldwide
(Committee on Feasibility of Applying Blast Mitigating Technologies and Design Methodologies
from Military Facilities to Civilian Buildings, 1995; Elliot et al., 1992, 1994). For many urban
settings, the unregulated traﬃc brings the terrorist threats within the perimeter of the building.
International Journal of Damage
Mechanics
2014, Vol 23(1) 3–24
! The Author(s) 2013
Reprints and permissions:
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DOI: 10.1177/1056789513482479
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Faculty of Civil Engineering, Babol University of Technology, Babol, Iran
Corresponding author:
Gholamreza Abdollahzadeh, Faculty of Civil Engineering, Babol University of Technology, Babol, Iran.
Email: abdollahzadeh@nit.ac.ir

For these structures, the modest goal is protection against damage in the immediate vicinity of the
explosion and the prevention of progressive collapse. In this sense, computer simulations could be
very valuable in testing a wide range of buildings types and structural details over a broad range of
hypothetical events (Committee on Feasibility of Applying Blast Mitigating Technologies and
Design Methodologies from Military Facilities to Civilian Buildings, 1995).
Moreover, a performance-based design aims to ensure the satisfactory performance of the struc-
ture during its lifetime. Therefore, it needs to consider all the possible critical actions the structure
could experience in the future. Considering the uncertainty involved in characterizing these elements,
it seems inevitable to address the probabilistic performance-based design. The target structure reli-
ability in such probabilistic framework is represented by the probability of failure. More specifically,
it is represented by the mean annual frequency of the structural response which exceeds a certain
limit threshold and identified based on the designed performance objectives (Asprone et al., 2010).
This study aims to evaluate the probability of failure. The structural collapse was considered as a
limit threshold for calculating the mean annual frequency of event. Term of structural collapse is
intended to the loss of ability to withstand gravity loads. This approach considers the blast action in
the form of the blast fragility, defined as the probability of collapse when a given blast event has
taken place in the structure. Blast fragility is evaluated using an advance simulation method. It is
assumed that a possible blast scenario is identified by quantity of the explosive mass and the location
of the blast within the structure. For each possible blast scenario generated by the simulation,
stability is verified by performing a plastic limit analysis on the damaged structure (Corotis and
Nafday, 1990). As a case study, the blast fragility of a generic four-storey steel building is calculated
and then the annual risk of collapse is evaluated.
Blast hazard assessment/design
For the limit state collapse, the probability of collapse is considered as all possible events that could
potentially cause significant damage and can be written as (Elliot et al., 1994):
P Cð Þ ¼
X
A
P CjAð ÞP Að Þ ð1Þ
where ‘A’ represents a critical event such as earthquake, blast, and so on. Formally, ‘A’ can be
written as the logical union of the potential critical events, that is:
A EQ þ Wind þ Gas Explosion þ Blast þ MISC ð2Þ
Equation (1) is written using the total probability theorem assuming that the critical event ‘A’ is
mutually exclusive (i.e., they cannot happen simultaneously) and collectively exhaustive (i.e., all the
potential ‘A’s are considered). Obviously, the events contributed to ‘A’ are varied based on the type,
location, and function of the structure to be designed or assessed. So depending on the particulars of
each problem, some of the terms in ‘A’ might be dominant in comparison to others. The de minimis risk
vdm is in the order of 10À7
/year (Pate-Cornell, 1994). Therefore, if the annual risk of occurrence of any
critical event A is considerably less than the de minimis level, it could be omitted from the critical events
considered in equation (2). Hence, the multi-hazard acceptance criteria can be written as following:
C ¼
X
PðCjAÞA vdm ð3Þ
4 International Journal of Damage Mechanics 23(1)

The above-mentioned criteria could be used for both probability based design and assessments of
structures for limit state collapse.
Considering a particular case in which the critical event is only blast, the design/assessment
criterion can be written as:
vC ¼ P CjBlastð ÞvBlast vdm ð4Þ
where vC and vBlast stand for the annual rate of collapse and annual rates of occurrence of blast
events of significance, respectively. PðCjBlastÞ represents blast fragility. In this case, it is assumed
that after blast event, there is enough time to repair the strategic structure back to its intact state.
Note that vC is rate of exceedance and not a probability; however, for very rare events, the prob-
ability is approximately equal to the annual rate. Estimation of the annual rate of a blast event
occurred by terrorist attack cannot be easily quantified and defined analytically. In other words,
the estimation of vBlast is not entirely an engineering problem since it depends on socio-political
considerations and how the structure is strategically vulnerable against such events. However,
in order to facilitate calculations, it is assumed here that vBlast t is a known quantity (Asprone
et al., 2010).
Alternatively, in cases where vBlast cannot be identified, one could perform a scenario-based
calculation of the probability of collapse and compare it against an acceptable threshold that
is larger than de minimis level (e.g., 10À2
is the conditional collapse probability necessary to
achieve the de minimis level of less than 10À6
/year, see Ellingwood, 2006). It should be noted
that employing the blast hazard formulation makes it possible to consider the rehabilitation
strategies with respect to blast. Risk reduction techniques for blast and earthquake can be
similar (i.e., composite wrapping of columns and steel bracing installations). In fact, such
correlation had been verified (Asprone et al., 2008), in which it has been demonstrated
that a seismic retrofit intervention (e.g., steel bracing installations) can lead to a reduction
in the risk of blast-induced progressive collapse. However, multi-hazard assessment of a gen-
eric RC frame structure, for both blast and earthquake events, had been performed (Asprone
et al., 2010).
Blast loading
An explosion mainly induces a quick and significant increase of pressure within the place it occurs,
i.e., air or water. Such overpressure propagates as a wave, the so-called blast wave, and is
characterized by its speed, intensity, and duration. These are fundamental parameters in order
to evaluate the actions induced by an explosion in the vicinity of the structural elements. The
numerical values of these parameters depend on several aspects, such as type and amount of the
exploding mass, interest target distance from explosion, geometry of the target, and type of
reflecting surfaces (e.g., the ground in case of external explosions or walls or slabs in case of
closed-in explosions). In the past decades, several investigations have been performed on such
aspects and they have provided reliable numerical procedures for the quantification of the over-
pressure time histories. In the case of blast explosion, the induced overpressure follows a trend
over time similar to that shown in Figure 1, where a positive decaying phase is followed by a
weaker negative phase which has a longer duration and a lower intensity. However, the phenom-
enon is very quick and can last up to 10À2
s. Charges situated extremely close to a target structure
impose a highly impulsive, high-intensity pressure load over a localized region of the structure
(Ngo et al., 2007).
Abdollahzadeh and Nemati 5

Blast wave scaling laws
All blast parameters are primarily dependent on the distance from the explosion and the amount of
energy released by a detonation in the form of a blast wave. A universal normalized description of
the blast effects can be given by scaling distance relative to ðE=PoÞ1=3
and scaling pressure relative
to Po, where E is the energy release (kJ) and Po the ambient pressure. For convenience, however, it is
a general practice to express the basic explosive input or charge weight (W) as an equivalent mass of
TNT. The results are then given as a function of the dimensional distance parameter (scaled dis-
tance) Z ¼ ðR=WÞ1=3
, where R is the actual effective distance from the explosion. W is generally
expressed in kilograms. Scaling laws provide parametric correlations between a particular explosion
and a standard charge of the same substance (Ngo et al., 2007).
Prediction of blast pressure
Blast wave parameter for conventional high-explosive materials have been the focus of a number of
studies during the 1950s and 1960s (Ngo et al., 2007).
As mentioned earlier, the blast action can be modeled by a quick decay pressure time–history
curve. This curve can be approximated by a triangular shape identified by two parameters, namely,
the initial peak pressure PSO and the duration tplus of positive phase. These parameters, which depend
on the amount of explosive and the distance from the charge, can be evaluated according to empirical
formulas available in literatures (Departments of the Army, the Navy and the Air Force – USA, 1990;
Henrych, 1979; Mills, 1987; Newmark and Hansen, 1961; Ngo et al., 2007; Department of Housing
and Urban Development, Iranian National Rules of Structures, 2010).
Peak overpressure. For the estimation of peak overpressure due to spherical blast, different rela-
tions are presented by researchers such as following ones.
Brode relations (Brode, 1955). Peak overpressure for near field (when PSO are greater than 10 kg/
cm2
) and middle or far fields (when PSO is between 0.1 and 10 kg/cm2
) are as:
PSO ¼
6:7
Z3
þ 1 PSO 4 10 kg=cm2
ð5Þ
Figure 1. Blast overpressure in air.

PSO ¼
0:975
Z
þ
1:455
Z2
þ
5:85
Z3
À 0:019 0:1 5 PSO 5 10 kg=cm2
ð6Þ
where Z is scaled distance (as explained above).
Henrych relations[xv]. Here, important parameter for classifying the relation is scaled distance,
and relations are as below:
PSO ¼
14:072
Z
þ
5:54
Z2
þ
0:357
Z3
þ
0:00625
Z4
0:05 Z 5 0:3 ð7Þ
PSO ¼
6:194
Z
þ
0:326
Z2
þ
2:132
Z3
0:3 Z 5 1 ð8Þ
PSO ¼
0:662
Z
þ
4:05
Z2
þ
3:288
Z3
1 Z 10 ð9Þ
Brode relations for middle and far ﬁelds explosion show a better adoption with empirical formulas,
while Henrych relations show a better adoption with empirical formulas for near-ﬁeld explosion; for
this reason, for near distances ðZ 0:5Þ Henrych relations and for middle and far distances
ðZ 4 0:5Þ results of Brode relation were used in this study. Figure 2 shows peak overpressure due
to blast according to scaled distance (Department of Housing and Urban Development, Iranian
National Rules of Structures, 2010).
Time duration of positive phase. Time duration of positive phase tplus is the duration where
pressure due to blast is more than the environmental pressure. It is obvious that duration
of applying load is an important parameter in calculating the response of the structure.
Hence, in blast researches, negative phase was neglected and positive phase duration can
then be assumed as blast duration. There is a diagram in TM5-1300 standard for calculating
Figure 2. Peak overpressure due to blast according to scaled distance (Department of Housing and Urban
Development, Iranian National Rules of Structures, 2010).

the duration of positive phase which Izadifard and Maheri have simplified with this
equation:
log10 tplus=W1=3
À Á
¼ 2:5 log10 Zð Þ þ 0:28 Z 1 ð10Þ
log10 tplus=W1=3
À Á
¼ 0:31 log10 Zð Þ þ 0:28 Z ! 1 ð11Þ
Figure 3 shows comparison between this equation and TM5-1300 standard’s diagram. Other equa-
tions for the estimation of duration of positive phase are available in literature (Department of
Housing and Urban Development, Iranian National Rules of Structures, 2010).
Effects of blast explosion
Blast explosion has two kinds of effects on the civil structures. The primary effect of a blast explosion
on civil structures is caused by such a rapid and intense action that it is able to induce severe local
structural damages. In fact, the applied loads are so fast that they are unable to activate the global
vibration modes of the structure, since the inertia corresponding to such modes has no sufficient time
to react. Therefore, the blast-induced overpressures hit directly the single frame elements, which
behave as independent structures and can be modeled as fixed end elements (Departments of the
Army, the Navy and the Air Force – USA, 1990).
An indirect effect of blast explosion on civil structure is progressive collapse. The progressive
collapse can be defined as a mechanism involving a large part of a structure, triggered by local less
extensive damage in the structure. In fact, a blast explosion occurring within or near a building can
cause the loss of one or more single frame elements. Having lost some elements, the whole structure
can become unstable, failing under the present vertical loads. So, the structure can eventually
develop a global mechanism, which is widely referred to as the progressive collapse mechanism
(Allen and Schriever, 1972; ASCE/Structural Engineering Institute, 2005; General Services
Administration, 2003). Design and/or assessment of structures accounting for such failure
Figure 3. Comparison between Izadifard and Maheri equation with TM5-1300 diagram (Department of Housing and
Urban Development, Iranian National Rules of Structures, 2010).

mechanism can follow a direct approach or an indirect approach (Ellingwood and Leyendecker,
1978). In the indirect approach, resistance to progressive collapse is pursued guaranteeing minimum
levels of strength, continuity, and ductility, whereas in the direct approach, progressive collapse
scenarios are directly analyzed. Actually, the progressive collapse mechanism is most often identified
as the predominant mode of failure after a blast event (National Research Council, 2001), and it is
already the subject of wide research related to the protection of critical infrastructures (Agarwal
et al., 2003; ASCE/Structural Engineering Institute, 2005; Bennett, 1988; General Services
Administration, 2003; National Research Council, 2001).
Blast fragility
Using simulation-based reliability methods for risk assessment (Asprone et al., 2010)
The blast fragility denoted by PðCjBlastÞ, in the context of this work, is defined as the conditional
probability for the event of progressive collapse given that a blast event takes place near or inside the
strategic structure in question.
Consider that real vector represents the uncertain quantities of interest, related to structural
modeling and loading conditions. Let P ð Þ represent the probability density function (PDF) for the
vector . The PðCjBlastÞ can be written as follows:
PðCjBlastÞ ¼
Z
ICjBlast ð ÞP ð Þd ð12Þ
where ICjBlast ð Þ is an index function which is equal to unity in the case where leads to blast-induced
progressive collapse and otherwise, it is equal to zero. Here, the probability of progressive collapse
PðCjBlastÞ is calculated by generating Nsim samples i from PDF P ð Þ. The event of progressive
collapse is identified by the ratio index c ið Þ which is the factor that the gravity loads should be
multiplied in order to create a global collapse mechanism. In case it assumes a value less than unity,
the event of progressive collapse is actually activated, since the acting loads are sufficient to induce
instability in the structure. Moreover, the uncertain quantities of interest here is the position of
explosive mass with respect to the structure. Obviously, any other uncertain quantity such as those
related to structural modeling and amount of explosive can be added to vector of uncertain par-
ameters . For each simulation realization i, the following two steps are performed:
(1) A local dynamic analysis is performed on the column elements affected by the blast in order to
verify whether they can resist the explosion and keep their vertical load carrying capacity.
(2) After identifying the damaged columns to be removed, a kinematic plastic analysis is performed
on the damaged structure in order to evaluate the progressive collapse index c ið Þ and to control
whether the structure is able to carry the gravity loads in its post-explosion state.
Local dynamic analysis
Since the blast-induced action is very rapid, consequently the structural inertia does not have suf-
ficient time to respond, the individual elements react to it as if they were fixed-end elements.
Moreover, for the same reason, the structural damping can be ignored (Williams and Newell,
1991). For each simulation realization, the step 1 described above is conducted, performing the

dynamic analysis of an un-damped distributed-mass fixed-end beam subject to triangular impact
loading, for all the columns on the same floor as the explosion. Moreover, for the sake of simplicity
in calculations, it is assumed that the blast action is constant across the length of the columns
(Asprone et al., 2010).There are two ways to identify the damaged columns, closed-form solution
or computer-based analysis. Second one was chosen in this study in order to achieve more accurate
analysis because of considering nonlinear properties of materials and ignoring the assumption which
was accepted in closed-form solution for simplicity.
Closed-form solution. In closed-form solution, the period of first mode vibration of a fixed end
beam with constant EI, constant distributed mass m, and length L should be calculated at first. Then
it is assumed that the column is replaced with a single degree of freedom (SDOF) system with the
same period of vibration. By finding the equation of response of an un-damped SDOF system
subject to triangular impulse loading Y(t), the maximum response can be found (Clough and
Penzien, 1993). It can be shown that the maximum bending moment and shear will take place at the
fixed ends and will be calculated as follows:
Mmax ¼ 1:26
4:73
L
2
EI ð13Þ
Vmax ¼ 1:24
4:73
L
3
EI ð14Þ
In order to verify whether the individual column can resist the explosion, the maximum blast-
induced bending moment and shears, Mmax and Vmax, are compared against the ultimate bending
and shear capacity of elements at its ends.
In fact, the linear elastic analysis method incorporated for the local dynamic analysis of each
column arrives at a closed-form solution and makes it particularly easy to quickly check the affected
columns and identify those which needed to be removed for each blast scenario generated. The
accuracy of the checking phase could be improved by using the non-linear time-step methods in
order to solve the equation of motion under the blast impact loading (Asprone et al., 2010).
Using computer program. Computational methods in the area of blast effects mitigation are gen-
erally divided into those used for prediction of blast loads on the structure and those for calculations
of structural response to the loads. Computational programs for blast prediction and structural
response use both first principle and semi-empirical methods. Programs using the first principle
method can be categorized into uncoupled and coupled analyses. The uncoupled analysis calculates
blast load as if the structure were rigid and then apply these loads to a responding model of the
structure.
For a coupled analysis, the blast simulation module is linked with the structural response module.
In this type of analysis, the CFD (Computational Fluid Mechanics) model for blast load prediction
is solved simultaneously with the CSM (Computational Solid Mechanics) model for structure
response to account the motion of the structure while the blast calculation proceeds. The pressures
that arise due to motion and failure of the structure can be predicted more accurately. Examples of
this type of computer codes are AUTODYN, DYN3D, LS-DYNA, and ABAQUS. Table 1 sum-
marizes a listing of computer programs that are currently being used to model blast effects on
structures (Ngo et al., 2007).

In order to model a structure, well knowing of the structure is necessary. One of the most sig-
nificant realizations of a structure model is material behaviors.
Blast loads typically produced very high strain rates in the range of 102
–104
sÀ1
. This high
straining rate would alter the dynamic mechanical properties of target structures and accordingly,
the expected damaged mechanisms for various structural elements. For steel structures subjected to
blast effects, the strength of steel can increase significantly due to strain rate effects (Maleki and
Rahmanieyan, 2011). Figure 4 shows the approximate ranges of the expected strain rates for dif-
ferent loading conditions (Ngo et al., 2007).
In this study, in order to verify whether the column can resist the blast load or not, the three-
dimensional (3D) model of column subject to triangular blast load was analyzed using nonlinear
explicit ABAQUS which takes into account both material nonlinearity and geometric nonlinearity.
Also it is assumed that the steel mechanical properties increase significantly due to the strain rate
effect. For considering the effect of strain rates, Cowper–Symonds model was used (Hibbot,
Karlsson and Sorensen Inc, 2006). The ratio of dynamic yield stress to static yield stress (R) for
plastic strain rate is defined in equation (15), where n and D are constants related to materials (Chen
and Liew, 2005; Maleki and Rahmanieyan, 2011; Saeed and Vahedi, 2009) and proposed amounts
for soft steel and were n ¼ 5 and D ¼ 40 (Saeed and Vahedi, 2009).
Table 1. Examples of computer programs used to simulate blast effects and structural response (Ngo et al., 2007).
Name Purpose and type of analysis Author/Vendor
BLASTX Blast prediction, CFD code SAIC
CTH Blast prediction, CFD code Sandia National Laboratories
FEFLO Blast prediction, CFD code SAIC
FOIL Blast prediction, CFD code Applied Research Associated, Waterways
Experiment Station
SHARC Blast prediction, CFD code Applied Research Associated,Inc.
DYNA3D Structural response þ CFD (coupled analysis) Lawrence Livermore National Laboratory (LLNL)
ALE3D Coupled analysis Lawrence Livermore National Laboratory (LLNL)
LS-DYNA Structural response þ CFD (coupled analysis) Livemore Software Technology Corporation
(LSTC)
Air3D Blast prediction, CFD code Royal Military of Science College, Cranfield
University
CONWEP Blast prediction (empirical) US Army Waterways Experiment Station
AUTO-DYN Structural response þ CFD (coupled analysis) Century Dynamics
ABAQUS Structural response þ CFD (coupled analysis) ABAQUS Inc.
CFD: Computational Fluid Mechanics.
Figure 4. Strain rates associated with different types of loading (Ngo et al., 2007).

The effect of increase in strain ratio when we consider the mechanical behavior of steel under
static load as a reference was shown in Figure 5.
_PL ¼ D R À 1ð Þn
ð15Þ
After modeling, applying blast load, and analyzing the column, we should find whether the
interested column went to plastic region or not, especially at its ends. To find this, equivalent plastic
strain and Mises stress are helpful criteria; in this study, Mises criterion is employed. If the stress in
any of the columns modeled for any of generated Nsim samples was more than Mises stress, we
consider that the column was failed. Figure 6 shows Mises stress contour in one column.
Figure 6. Modeled steel column and Mises stress contour.
Figure 5. Stress–strain relation by considering the effect of strain rate (Chen and Liew, 2005).

Kinematic plastic analysis on the damaged structure
After identifying and removing the damaged elements, it should be verified whether the damaged
structure can withstand the applied vertical loads. This is essentially a global stability analysis of the
damaged structure. A possible approach to performing such analysis would be to conduct a plastic
limit analysis. A plastic limit analysis (Corotis and Nafday, 1990; Watwood, 1979) involves finding
the load factor c on the applied loads for which the following effects occur:
(1) Equilibrium conditions are satisfied.
(2) A sufficient number of plastic hinges are formed in the structure in order to activate a collapse
mechanism in the whole structure or in a part of it.
It is assumed that the non-linear behavior in the structure is concentrated at the element ends and
mid height of them and these points are capable of developing their fully plastic moment (i.e., the
brittle failure modes such as axial and shear failure or the ultimate rotational failure do not take
place before the member has developed its plastic bending capacity). It has been shown (Grierson
and Gladwell, 1971) that the procedure for the plastic limit analysis can be defined as a linear
optimization programming with the objective of minimizing the load factor c. This linear program-
ming problem could be resolved by employing a simplex algorithm. For example, in the particular
case of a framed structure, the independent mechanisms are classified as follows (Grierson and
Gladwell, 1971) (Figure 7): (a) the soft-storey mechanisms in which the plastic hinges at both
ends of all the columns within a given storey are activated, (b) the beam mechanisms in which (at
least) three hinges are formed in given beam, and (c) the joint mechanisms in which the end hinges of
all the frame elements converging into a given joint are activated.
In static loading problems, a c less than or equal to unity indicates that the structure is already
unstable under the applied loads. On the other hand, the threshold for c in instantaneous dynamic
loading problem is equal to 2. In case of progressive collapse, it has been shown that a value 2 is
probably conservative and the actual value of c causing instability in the structure is between 1 and
2 (Ruth et al., 2006). It should be mentioned that the plastic limit analysis algorithm presented here
Figure 7. Principal Mechanisms (Asprone et al., 2010).

ignores some second-order non-linear actions that could prevent a mechanism from forming (e.g.,
the catenaries actions and the arch effects) (Asprone et al., 2010).
In this study, to find c, for any of generated Nsim samples, SAP2000 nonlinear program has
been run. Figure 8 shows one of the samples modeled in SAP2000 program. In this model, blast
occurred in storey 2 for bomb in p64 place, so it named c64s2. In this sample, these plastic
hinges were activated due to c ¼ 0:25; therefore, structure fails due to the soft storey
mechanism.
Calculating the blast fragility
As mentioned in the previous section, the blast fragility is defined as the probability of progressive
collapse when a blast event takes place inside the structure. The progressive collapse event can be
characterized by a Bernoulli-type variable that is equal to unity in the event of progressive collapse
and equal to zero otherwise (Asprone et al., 2010). Using the kinematic plastic limit analysis
described in the previous section, the Bernoulli collapse variable denoted by ICjBlast can be deter-
mined as a function of the collapse load factor c:
ICjBlast ¼ 0 if c 4 c,th ð16Þ
ICjBlast ¼ 1 if c c,th ð17Þ
where c,th is the threshold value for the load factor indicating the onset of progressive collapse
varying between 1 and 2.The MC procedure can be used to generate Nsim realizations of the uncer-
tain vector i according to its PDF p (Asprone et al., 2010). Finally, the conditional probability of
Figure 8. Sample c64s2 modeled in SAP2000.

progressive collapse in equation (5) can be solved numerically as the expected value of the Bernoulli
collapse index variable ICjBlast:
P CjBlastð Þ ¼
PNsim
i¼1 ICjBlast ið Þ
Nsim
ð18Þ
It can be shown that the coefficient variation of conditional progressive collapse probability can be
calculated as follows (Asprone et al., 2010):
COVP CjBlastð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À P CjBlastð Þ
Nsim:P CjBlastð Þ
s
ð19Þ
Numerical example
A possible application of the methodology described in the previous section can refer to the calcu-
lation of the mean annual risk for progressive collapse of a generic steel-framed building. A numer-
ical example is here presented; the characteristics of the case study structure are outlined in the
following.
Structural model description
The building studied here is a generic four-storey steel-framed structure designed according to the
American seismic provisions by using SAP2000 program. The structural model is illustrated in
Figure 8, presenting a plan of the generic storey; column sections are different on each floor, so
totally we have seven types of section (IPE180, IPE200, IPE220, IPE240, IPE270, IPE300, and
IPE330) for columns, whereas two types of beam are present, Type A and Type B, whose section
names are IPE180 and IPE220, respectively (Figure 9 shows sections of beams in plan); the floors are
supposed to be one-way joist slabs, with 0.3 m thick.
The soil was assumed to be type 2 and building was located in high seismic zone, and design dead
and live loads were listed in Table 2.
Figure 10 shows a 3D view of the model. Each storey is 3.00 m high. The non-linear behavior in
the sections is assumed to be only flexural and is modeled based on the concentrated plasticity
concept. It is assumed that the plastic moment in the hinge sections is equal to the ultimate
moment capacity. Materials parameters are outlined in Table 3.
Characterization of the uncertainties
As mentioned in this methodology, the uncertain quantity of interest in this study is the position of
explosive mass with respect to a fixed point within the structure, denoted by R. Formally, the vector
of uncertain parameter contains one uncertain quantity: ¼ Rf g. The following assumptions are
made in order to determine the possible values of :
. It is obvious that the explosion could take place inside or outside of the structure. Furthermore,
there are three types of bomb: back portable bomb for inside structure explosion, car bomb for
explosions which happened in parking level or outside of structure, and truck bomb for outside
explosion (Asprone et al., 2010). But in this study, it assumed that the explosion just could

happen inside the structure with back portable bomb. It assumed that the access to the structure
is allowed to people at each floor; consequently, a back portable bomb can explode from the first
to the fourth floor of the structure as shown in Figure 11.
. For each simulation realization, the center of explosion is determined. The explosion scenario
occurs with the same probability at each of the four floors of the building. Then the amount of
explosive is defined as 35 kg of equivalent TNT (simulating a back portable bomb). All uncertain
quantities are assumed to be uniformly distributed (i.e., the possible values for the uncertain
quantity are all equally likely).
The process in determining the realization of vector is clarified in Figure 11. Also, Figure 12 shows
realization of bomb place inside each floor.
It should be noted that the vector ideally needs to also include the uncertainties in the structural
modeling parameters and the structural component capacities. However, the overall effect of these
Figure 9. Storey view.
Table 2. Design dead and live load.
Tip stories (kg/m2
) Roof storey (kg/m2
) Stairs (kg/m2
) Side walls (kg/m2
)
Dead load 620 550 700 250
Live load 200 150 350 0

sources of uncertainty seems not to drastically affect the overall structural risk compared to the
uncertainties in blast loading parameters (for further discussion of the effect on blast risk, see Low
and Hao, 2001). Hence, the uncertainties in structural modeling and component capacity and
amount of explosive charge have not been considered in the present work.
Characterization of the parameters defining the local dynamic analysis
It is assumed that only the columns on the same floor as the explosion are affected by it. This
assumption is supported by the fact that the columns on the other floors and the floor beams are
sheltered from the blast wave by the floor slab system (Departments of the Army, the Navy and the
Air Force – USA, 1990).
Figure 10. Three-dimensional model view.
Table 3. Material properties.
Elastic properties
Plastic properties Rate dependent
General propertyTrue stress (Pa) True plastic strain Hardening Power law
E ¼ 210 Â 109
(Pa) 240 Â 109
0 Multiplier 40 ¼ 7800 kg/m2
¼ 0.3 270 Â 109
0.025 Exponent 5
285 Â 109
0.1
297 Â 109
0.2
300 Â 109
0.35

Figure 12. Bomb place realization.
Blast
Senario
Explosion takes
place inside the
structure
1st floor:
backpack bomb
w=35 kg
2nd floor:
backpack bomb
w=35 kg
3rd floor:
backpack bomb
w=35 kg
4th floor:
backpack bomb
w=35 kg
Explosion take
place outside
from the
structure
truck bomb
w=15000 kg -
25000kg
car bomb
w=200kg - 500kg
100 % 0 %
0 %25 % 0 %25 % 25 % 25 %
Figure 11. Blast realization logic tree.

Then, for each of the columns hit by the explosion at the distance R from the center of the charge,
given the amount of explosive w, the reduced distance Z ¼ R=
ffiffiffiffi
w3
p
is calculated. Then, a triangular
impulse loading is considered to be acting on the columns (Figure 13), whose parameters p0 (max-
imum initial pressure) and tplus (duration of the impulse) were illustrated in previous section. It is
further assumed that the intensity of the impact loading is uniform across the column height.
Furthermore, since such load generally acts in a direction that is not parallel to local axes of the
column, it is divided into two components and both of them act to the column simultaneously and
used to verify whether the column fails.
For modeling the columns in ABAQUS, FRAME3D elements were used, and both the ends of
the columns were fixed in all degrees. (As mentioned before) The column was meshed sweep with
hex-dominated elements. Moreover, the blast load was applied only on one face of the column which
was straightly affected by blast. Furthermore, this load was divided into two components in x and
y directions depend on the angle between bomb place and the column.
In this study, in order to check the accuracy of models, after-blast situation of first model was
compared with closed-form formulas. Since the model showed similar behavior in both methods,
modeling was confirmed.
In the interest of reducing computational time, it is important to use the smallest number of finite
elements for each column member without affecting the accuracy.
With regard to limitation in experimental studies in blast field, for validating the modeling in
ABAQUS software, first we modeled a plate under blast loading according to Maleki and
Rahmanieyan (2011) and compared the results. The results were similar and hence we concluded
that the modeling was fine. Therefore, we modeled all the samples in the same way.
Blast fragility
A simulation technique is used to generate 324 blast scenario realizations, assuming that the struc-
ture is subjected to its gravity loads and 30% of live loads. Also all the columns that failed in blast
scenario were removed and plastic hinges assigned to the rest of columns and all the beams in three
positions (start, middle, and end of the elements). SAP2000 provides default-hinge properties and
recommends PMM hinges for columns and M3 hinges for beams. Default hinges are assigned to the
elements (PMM for columns and M3 for beams). There is no extensive calculation for each member.
Figure 13. Blast impulse loading (Asprone et al., 2010).

For each of these realizations, the collapse load factor c was calculated by modeling damaged
structure with SAP2000 software. The cumulative distribution function of the load factor denoted
by P cjBlastð Þ is plotted for possible values of c in Figure 14 (this curve is drawn by using
Microsoft Oﬃce Excel 2007 program). The threshold value identifying progressive collapse region is
C,th ¼ 12½ Š, as marked in Figure 15. However, by considering a conservative value equal to 2, it can
be observed that P CjBlastð Þ, probability that a blast event leads to progressive collapse of the case
study structure, is around 0.98. On the contrary, the value c ¼ 4 corresponds to the case that none
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
P(λλC|Blast)
λc
Figure 14. Blast fragility.
0
5
10
15
20
25
storey1 storey2 storey3 storey4
Figure 15. The blast scenarios that led to progressive collapse in the structure.

of the columns is eliminated due to the blast; in other words, it is the load factor corresponding to
the original structure. This explains why the probability that a blast event leads to a collapse load
factor load less than c 4 is equal to unity.
In order to gain further insight about the simulation results, the blast scenarios leading to pro-
gressive collapse, identified by 1 c 2, are plotted in Figure 15 illustrates the histogram for the
storey in which the explosion takes place. This kind of plot is very helpful for identifying the critical
zones within which an explosion could most likely lead to progressive collapse. It can be observed
that the collapse scenarios take place predominantly on second storey.
According to local dynamic analysis, it was recognized that in some positions such as 32, 35, 38,
41, 44, and 45, all the columns were failed due to blast, and so whole of the structure was unstable
and total collapse occurred. So if access to those areas were limited (middle span in y direction),
security of the building will increase. The critical zone was shown in Figure 16.
Discussion on case study
The annual risk of collapse, to compare with the de minimis threshold, can be calculated from
equation (20) as follows:
vc ¼ 0:98:vBlast ð20Þ
where 0.98 is the value of P CjBlastð Þ evaluated with the presented procedure. As can be observed
from equation (19), the blast fragility needs to be multiplied by the annual rate vBlast when a sig-
nificant blast event takes place. However, as mentioned before, this rate is difficult to evaluate as an
engineering quantity and it depends more on the socio-political circumstances and the strategic
importance of the structure. For instance, in case of a non-strategic structure, vBlast can be in the
Figure 16. The critical zone.

order of 10À7
(Ellingwood, 2006), making annual risk of blast collapse negligible. Alternatively, in
case of a strategic structure, vBlast can be as large as 10À4
; in such case, blast hazard dominantly
increase the annual risk of collapse. It should be noted that blast fragility is defined as the prob-
ability of progressive collapse, given that a significant blast event has taken place. In order to yield
the mean annual risk of collapse, the probability of progressive collapse needs to be multiplied by
the annual rate of significant blast event taking place.
Conclusions
A simple, useful, and applicable methodology for calculating the annual risk of a strategic structure
collapse is presented in the progressive collapse assessment framework. In this methodology, a blast
event of interest takes place and the probability of progressive collapse is calculated by realizing 324
blast scenario. In order to analyze the structural elements subjected to impulsive blast induced loads,
ABAQUS program is employed. An efficient limit state analysis is also implemented to verify
whether progressive collapse mechanisms under the vertical service loads on the damaged structure
are activated (using SAP2000 program). As a numerical example, a case study is presented, in which
the generic steel frame building’s annual rate of collapse is discussed. The following observations
and outcomes can be made:
. The probability of progressive collapse is found to be around 98%. The results of the presented
case study seem to justify the 324 realization of blast scenario for calculating the probability of
progressive collapse.
. This study exploits the particular characteristics of the blast action and its effect on the structure
in order to achieve maximum efficiency in the calculations. More specifically, the use of a
common 3D finite element analysis renders the calculations significantly more rapid and thereby
feasible for implementation within a simulation procedure.
. The outcome of the realizations can be used to mark the location of critical blast scenarios on the
structural geometry and identify the risk-prone areas. An example of a simple and effective
prevention strategy would be to limit or to deny the access to critical zones within the structure,
when they are identified by the presented procedure.
. This study determines the annual risk of collapse vc (equation (4)) according to the blast fragility
P CjBlastð Þ evaluated herewith the known annual rate of blast vBlast.
. Moreover, it should be noted that the methodology presented here for the assessment of a steel
structure can be extended in order to evaluate the vulnerability of a class of structures against the
blast-induced progressive collapse (i.e., masonry buildings, RC frame buildings, RC bridges).
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit
sectors.
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