international journal of damage mechanics 2014 abdollahzadeh 3 24

Article

Risk assessment of structuressubjected 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 loadshave received considerable attention in recent years. The design and construction of public buildingswhich provides life safety in the face of explosions is receiving renewed attention from structuralengineers (Committee on Feasibility of Applying Blast Mitigating Technologies and DesignMethodologies from Military Facilities to Civilian Buildings, 1995; Elliot et al., 1992, 1994). Suchconcern 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 Methodologiesfrom Military Facilities to Civilian Buildings, 1995; Elliot et al., 1992, 1994). For many urbansettings, the unregulated traffic brings the terrorist threats within the perimeter of the building.

International Journal of Damage

Mechanics

2014, Vol 23(1) 3–24

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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: [email protected]

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For these structures, the modest goal is protection against damage in the immediate vicinity of theexplosion and the prevention of progressive collapse. In this sense, computer simulations could bevery valuable in testing a wide range of buildings types and structural details over a broad range ofhypothetical events (Committee on Feasibility of Applying Blast Mitigating Technologies andDesign 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 structurecould 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 certainlimit 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 alimit threshold for calculating the mean annual frequency of event. Term of structural collapse isintended to the loss of ability to withstand gravity loads. This approach considers the blast action inthe form of the blast fragility, defined as the probability of collapse when a given blast event hastaken place in the structure. Blast fragility is evaluated using an advance simulation method. It isassumed that a possible blast scenario is identified by quantity of the explosive mass and the locationof 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 andNafday, 1990). As a case study, the blast fragility of a generic four-storey steel building is calculatedand 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 couldpotentially cause significant damage and can be written as (Elliot et al., 1994):

P Cð Þ ¼XA

P CjAð ÞP Að Þ ð1Þ

where ‘A’ represents a critical event such as earthquake, blast, and so on. Formally, ‘A’ can bewritten as the logical union of the potential critical events, that is:

A � EQþWindþGas Explosionþ BlastþMISC ð2Þ

Equation (1) is written using the total probability theorem assuming that the critical event ‘A’ ismutually exclusive (i.e., they cannot happen simultaneously) and collectively exhaustive (i.e., all thepotential ‘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 ofeach problem, some of the terms in ‘A’ might be dominant in comparison to others. The deminimis riskvdm is in the order of 10�7/year (Pate-Cornell, 1994). Therefore, if the annual risk of occurrence of anycritical eventA is considerably less than the deminimis level, it could be omitted from the critical eventsconsidered in equation (2). Hence, the multi-hazard acceptance criteria can be written as following:

�C ¼X

PðCjAÞ�A � vdm ð3Þ

4 International Journal of Damage Mechanics 23(1)



The above-mentioned criteria could be used for both probability based design and assessments ofstructures for limit state collapse.

Considering a particular case in which the critical event is only blast, the design/assessmentcriterion 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 blastevents of significance, respectively. PðCjBlastÞ represents blast fragility. In this case, it is assumedthat 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 eventoccurred 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-politicalconsiderations 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 (Asproneet al., 2010).

Alternatively, in cases where vBlast cannot be identified, one could perform a scenario-basedcalculation of the probability of collapse and compare it against an acceptable threshold thatis larger than de minimis level (e.g., 10�2 is the conditional collapse probability necessary toachieve the de minimis level of less than 10�6/year, see Ellingwood, 2006). It should be notedthat employing the blast hazard formulation makes it possible to consider the rehabilitationstrategies with respect to blast. Risk reduction techniques for blast and earthquake can besimilar (i.e., composite wrapping of columns and steel bracing installations). In fact, suchcorrelation had been verified (Asprone et al., 2008), in which it has been demonstratedthat a seismic retrofit intervention (e.g., steel bracing installations) can lead to a reductionin 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 (Asproneet 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 ischaracterized by its speed, intensity, and duration. These are fundamental parameters in orderto evaluate the actions induced by an explosion in the vicinity of the structural elements. Thenumerical values of these parameters depend on several aspects, such as type and amount of theexploding mass, interest target distance from explosion, geometry of the target, and type ofreflecting surfaces (e.g., the ground in case of external explosions or walls or slabs in case ofclosed-in explosions). In the past decades, several investigations have been performed on suchaspects 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 trendover time similar to that shown in Figure 1, where a positive decaying phase is followed by aweaker negative phase which has a longer duration and a lower intensity. However, the phenom-enon is very quick and can last up to 10�2 s. Charges situated extremely close to a target structureimpose 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 ofenergy released by a detonation in the form of a blast wave. A universal normalized description ofthe blast effects can be given by scaling distance relative to ðE=PoÞ

1=3 and scaling pressure relativeto Po, where E is the energy release (kJ) and Po the ambient pressure. For convenience, however, it isa general practice to express the basic explosive input or charge weight (W) as an equivalent mass ofTNT. 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 generallyexpressed in kilograms. Scaling laws provide parametric correlations between a particular explosionand 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 ofstudies during the 1950s and 1960s (Ngo et al., 2007).

As mentioned earlier, the blast action can be modeled by a quick decay pressure time–historycurve. 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 dependon the amount of explosive and the distance from the charge, can be evaluated according to empiricalformulas 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 Housingand 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 ¼6:7

Z3þ 1 PSO 4 10 kg=cm2 ð5Þ

Figure 1. Blast overpressure in air.




PSO ¼0:975

Zþ1:455

Z2þ5:85

Z3� 0:019 0:15PSO 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 ¼14:072

Zþ5:54

Z2þ0:357

Z3þ0:00625

Z40:05 � Z5 0:3 ð7Þ

PSO ¼6:194

Zþ0:326

Z2þ2:132

Z30:3 � Z5 1 ð8Þ

PSO ¼0:662

Zþ4:05

Z2þ3:288

Z31 � 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; forthis reason, for near distances ðZ � 0:5Þ Henrych relations and for middle and far distancesðZ4 0:5Þ results of Brode relation were used in this study. Figure 2 shows peak overpressure dueto blast according to scaled distance (Department of Housing and Urban Development, IranianNational Rules of Structures, 2010).

Time duration of positive phase. Time duration of positive phase tplus is the duration wherepressure due to blast is more than the environmental pressure. It is obvious that durationof 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 canthen 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 thisequation:

log10 tplus=W1=3

� �¼ 2:5 log10 Zð Þ þ 0:28 Z � 1 ð10Þ

log10 tplus=W1=3

� �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 ofHousing 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 explosionon civil structures is caused by such a rapid and intense action that it is able to induce severe localstructural damages. In fact, the applied loads are so fast that they are unable to activate the globalvibration modes of the structure, since the inertia corresponding to such modes has no sufficient timeto react. Therefore, the blast-induced overpressures hit directly the single frame elements, whichbehave as independent structures and can be modeled as fixed end elements (Departments of theArmy, the Navy and the Air Force – USA, 1990).

An indirect effect of blast explosion on civil structure is progressive collapse. The progressivecollapse can be defined as a mechanism involving a large part of a structure, triggered by local lessextensive damage in the structure. In fact, a blast explosion occurring within or near a building cancause the loss of one or more single frame elements. Having lost some elements, the whole structurecan become unstable, failing under the present vertical loads. So, the structure can eventuallydevelop a global mechanism, which is widely referred to as the progressive collapse mechanism(Allen and Schriever, 1972; ASCE/Structural Engineering Institute, 2005; General ServicesAdministration, 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 minimumlevels of strength, continuity, and ductility, whereas in the direct approach, progressive collapsescenarios are directly analyzed. Actually, the progressive collapse mechanism is most often identifiedas the predominant mode of failure after a blast event (National Research Council, 2001), and it isalready the subject of wide research related to the protection of critical infrastructures (Agarwalet al., 2003; ASCE/Structural Engineering Institute, 2005; Bennett, 1988; General ServicesAdministration, 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 conditionalprobability for the event of progressive collapse given that a blast event takes place near or inside thestrategic structure in question.

Consider that real vector � represents the uncertain quantities of interest, related to structuralmodeling and loading conditions. Let P �ð Þ represent the probability density function (PDF) for thevector �. The PðCjBlastÞ can be written as follows:

PðCjBlastÞ ¼

ZICjBlast �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-inducedprogressive collapse and otherwise, it is equal to zero. Here, the probability of progressive collapsePðCjBlastÞ is calculated by generating Nsim samples �i from PDF P �ð Þ. The event of progressivecollapse is identified by the ratio index �c �ið Þ which is the factor that the gravity loads should bemultiplied 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 induceinstability in the structure. Moreover, the uncertain quantities of interest here is the position ofexplosive mass with respect to the structure. Obviously, any other uncertain quantity such as thoserelated 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 toverify 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 performedon the damaged structure in order to evaluate the progressive collapse index �c �ið Þ and to controlwhether 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 impactloading, for all the columns on the same floor as the explosion. Moreover, for the sake of simplicityin 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 solutionor computer-based analysis. Second one was chosen in this study in order to achieve more accurateanalysis because of considering nonlinear properties of materials and ignoring the assumption whichwas accepted in closed-form solution for simplicity.

Closed-form solution. In closed-form solution, the period of first mode vibration of a fixed endbeam with constant EI, constant distributed mass �m, and length L should be calculated at first. Thenit is assumed that the column is replaced with a single degree of freedom (SDOF) system with thesame period of vibration. By finding the equation of response of an un-damped SDOF systemsubject to triangular impulse loading Y(t), the maximum response � can be found (Clough andPenzien, 1993). It can be shown that the maximum bending moment and shear will take place at thefixed ends and will be calculated as follows:

Mmax ¼ 1:264:73

L

� �2

EI� ð13Þ

Vmax ¼ 1:244:73

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 bendingand shear capacity of elements at its ends.

In fact, the linear elastic analysis method incorporated for the local dynamic analysis of eachcolumn arrives at a closed-form solution and makes it particularly easy to quickly check the affectedcolumns and identify those which needed to be removed for each blast scenario generated. Theaccuracy of the checking phase could be improved by using the non-linear time-step methods inorder 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 calculationsof structural response to the loads. Computational programs for blast prediction and structuralresponse use both first principle and semi-empirical methods. Programs using the first principlemethod can be categorized into uncoupled and coupled analyses. The uncoupled analysis calculatesblast load as if the structure were rigid and then apply these loads to a responding model of thestructure.

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 predictionis solved simultaneously with the CSM (Computational Solid Mechanics) model for structureresponse to account the motion of the structure while the blast calculation proceeds. The pressuresthat arise due to motion and failure of the structure can be predicted more accurately. Examples ofthis 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 onstructures (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 s�1. This highstraining rate would alter the dynamic mechanical properties of target structures and accordingly,the expected damaged mechanisms for various structural elements. For steel structures subjected toblast effects, the strength of steel can increase significantly due to strain rate effects (Maleki andRahmanieyan, 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 nonlinearexplicit 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 rateeffect. 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) forplastic strain rate is defined in equation (15), where n and D are constants related to materials (Chenand Liew, 2005; Maleki and Rahmanieyan, 2011; Saeed and Vahedi, 2009) and proposed amountsfor 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 responseþCFD (coupled analysis) Lawrence Livermore National Laboratory (LLNL)

ALE3D Coupled analysis Lawrence Livermore National Laboratory (LLNL)

LS-DYNA Structural responseþCFD (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 responseþCFD (coupled analysis) Century Dynamics

ABAQUS Structural responseþCFD (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 understatic load as a reference was shown in Figure 5.

_"PL ¼ D R� 1ð Þn

ð15Þ

After modeling, applying blast load, and analyzing the column, we should find whether theinterested column went to plastic region or not, especially at its ends. To find this, equivalent plasticstrain and Mises stress are helpful criteria; in this study, Mises criterion is employed. If the stress inany of the columns modeled for any of generated Nsim samples was more than Mises stress, weconsider 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 damagedstructure can withstand the applied vertical loads. This is essentially a global stability analysis of thedamaged structure. A possible approach to performing such analysis would be to conduct a plasticlimit analysis. A plastic limit analysis (Corotis and Nafday, 1990; Watwood, 1979) involves findingthe 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 andmid height of them and these points are capable of developing their fully plastic moment (i.e., thebrittle failure modes such as axial and shear failure or the ultimate rotational failure do not takeplace before the member has developed its plastic bending capacity). It has been shown (Griersonand Gladwell, 1971) that the procedure for the plastic limit analysis can be defined as a linearoptimization 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 particularcase of a framed structure, the independent mechanisms are classified as follows (Grierson andGladwell, 1971) (Figure 7): (a) the soft-storey mechanisms in which the plastic hinges at bothends of all the columns within a given storey are activated, (b) the beam mechanisms in which (atleast) three hinges are formed in given beam, and (c) the joint mechanisms in which the end hinges ofall 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 alreadyunstable under the applied loads. On the other hand, the threshold for �c in instantaneous dynamicloading problem is equal to 2. In case of progressive collapse, it has been shown that a value 2 isprobably conservative and the actual value of �c causing instability in the structure is between 1 and2 (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 hasbeen run. Figure 8 shows one of the samples modeled in SAP2000 program. In this model, blastoccurred in storey 2 for bomb in p64 place, so it named c64s2. In this sample, these plastichinges were activated due to �c ¼ 0:25; therefore, structure fails due to the soft storeymechanism.

Calculating the blast fragility

As mentioned in the previous section, the blast fragility is defined as the probability of progressivecollapse when a blast event takes place inside the structure. The progressive collapse event can becharacterized by a Bernoulli-type variable that is equal to unity in the event of progressive collapseand equal to zero otherwise (Asprone et al., 2010). Using the kinematic plastic limit analysisdescribed 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 �c4 �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 collapsevarying 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 Bernoullicollapse index variable ICjBlast:

P CjBlastð Þ ¼

PNsim

i¼1 ICjBlast �ið Þ

Nsimð18Þ

It can be shown that the coefficient variation of conditional progressive collapse probability can becalculated as follows (Asprone et al., 2010):

COVP CjBlastð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� P CjBlastð Þ

Nsim:P CjBlastð Þ

sð19Þ

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 thefollowing.

Structural model description

The building studied here is a generic four-storey steel-framed structure designed according to theAmerican seismic provisions by using SAP2000 program. The structural model is illustrated inFigure 8, presenting a plan of the generic storey; column sections are different on each floor, sototally we have seven types of section (IPE180, IPE200, IPE220, IPE240, IPE270, IPE300, andIPE330) for columns, whereas two types of beam are present, Type A and Type B, whose sectionnames are IPE180 and IPE220, respectively (Figure 9 shows sections of beams in plan); the floors aresupposed to be one-way joist slabs, with 0.3m thick.

The soil was assumed to be type 2 and building was located in high seismic zone, and design deadand live loads were listed in Table 2.

Figure 10 shows a 3D view of the model. Each storey is 3.00m high. The non-linear behavior inthe sections is assumed to be only flexural and is modeled based on the concentrated plasticityconcept. It is assumed that the plastic moment in the hinge sections is equal to the ultimatemoment 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 ofexplosive mass with respect to a fixed point within the structure, denoted by R. Formally, the vectorof uncertain parameter contains one uncertain quantity: � ¼ Rf g. The following assumptions aremade 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 forexplosions which happened in parking level or outside of structure, and truck bomb for outsideexplosion (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 structureis allowed to people at each floor; consequently, a back portable bomb can explode from the firstto the fourth floor of the structure as shown in Figure 11.

. For each simulation realization, the center of explosion is determined. The explosion scenariooccurs with the same probability at each of the four floors of the building. Then the amount ofexplosive is defined as 35 kg of equivalent TNT (simulating a back portable bomb). All uncertainquantities are assumed to be uniformly distributed (i.e., the possible values for the uncertainquantity are all equally likely).

The process in determining the realization of � vector is clarified in Figure 11. Also, Figure 12 showsrealization of bomb place inside each floor.

It should be noted that the vector � ideally needs to also include the uncertainties in the structuralmodeling 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 theuncertainties in blast loading parameters (for further discussion of the effect on blast risk, see Lowand Hao, 2001). Hence, the uncertainties in structural modeling and component capacity andamount 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. Thisassumption is supported by the fact that the columns on the other floors and the floor beams aresheltered from the blast wave by the floor slab system (Departments of the Army, the Navy and theAir 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

E¼ 210� 109(Pa) 240� 109 0 Multiplier 40 �¼ 7800 kg/m2

� ¼ 0.3 270� 109 0.025 Exponent 5

285� 109 0.1

297� 109 0.2

300� 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 bombw=15000 kg -

25000kg

car bombw=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=

ffiffiffiffiw3p

is calculated. Then, a triangularimpulse 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 isfurther 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 thecolumn, it is divided into two components and both of them act to the column simultaneously andused to verify whether the column fails.

For modeling the columns in ABAQUS, FRAME3D elements were used, and both the ends ofthe columns were fixed in all degrees. (As mentioned before) The column was meshed sweep withhex-dominated elements. Moreover, the blast load was applied only on one face of the column whichwas straightly affected by blast. Furthermore, this load was divided into two components in x andy 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 wascompared 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 finiteelements for each column member without affecting the accuracy.

With regard to limitation in experimental studies in blast field, for validating the modeling inABAQUS software, first we modeled a plate under blast loading according to Maleki andRahmanieyan (2011) and compared the results. The results were similar and hence we concludedthat 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 blastscenario were removed and plastic hinges assigned to the rest of columns and all the beams in threepositions (start, middle, and end of the elements). SAP2000 provides default-hinge properties andrecommends PMM hinges for columns and M3 hinges for beams. Default hinges are assigned to theelements (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 damagedstructure with SAP2000 software. The cumulative distribution function of the load factor denotedby P � � �cjBlastð Þ is plotted for possible values of �c in Figure 14 (this curve is drawn by usingMicrosoft Office Excel 2007 program). The threshold value identifying progressive collapse region is�C,th ¼ 12½ �, as marked in Figure 15. However, by considering a conservative value equal to 2, it canbe observed that P CjBlastð Þ, probability that a blast event leads to progressive collapse of the casestudy 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(λ<

λ C|Bl

ast)

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 tothe original structure. This explains why the probability that a blast event leads to a collapse loadfactor 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 thestorey in which the explosion takes place. This kind of plot is very helpful for identifying the criticalzones within which an explosion could most likely lead to progressive collapse. It can be observedthat 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 unstableand 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 fromequation (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 observedfrom 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 anengineering quantity and it depends more on the socio-political circumstances and the strategicimportance of the structure. For instance, in case of a non-strategic structure, vBlast can be in the

Figure 16. The critical zone.




order of 10�7 (Ellingwood, 2006), making annual risk of blast collapse negligible. Alternatively, incase of a strategic structure, vBlast can be as large as 10�4; in such case, blast hazard dominantlyincrease 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 yieldthe mean annual risk of collapse, the probability of progressive collapse needs to be multiplied bythe annual rate of significant blast event taking place.

Conclusions

A simple, useful, and applicable methodology for calculating the annual risk of a strategic structurecollapse is presented in the progressive collapse assessment framework. In this methodology, a blastevent of interest takes place and the probability of progressive collapse is calculated by realizing 324blast 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 verifywhether progressive collapse mechanisms under the vertical service loads on the damaged structureare activated (using SAP2000 program). As a numerical example, a case study is presented, in whichthe generic steel frame building’s annual rate of collapse is discussed. The following observationsand outcomes can be made:

. The probability of progressive collapse is found to be around 98%. The results of the presentedcase study seem to justify the 324 realization of blast scenario for calculating the probability ofprogressive collapse.

. This study exploits the particular characteristics of the blast action and its effect on the structurein order to achieve maximum efficiency in the calculations. More specifically, the use of acommon 3D finite element analysis renders the calculations significantly more rapid and therebyfeasible for implementation within a simulation procedure.

. The outcome of the realizations can be used to mark the location of critical blast scenarios on thestructural geometry and identify the risk-prone areas. An example of a simple and effectiveprevention 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 fragilityP 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 steelstructure can be extended in order to evaluate the vulnerability of a class of structures against theblast-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-profitsectors.

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