fragility analysis for the performance-based design of cladding wall panels subjected to blast load

Engineering Structures xxx (2014) xxx–xxx

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Engineering Structures

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Fragility analysis for the Performance-Based Design of cladding wallpanels subjected to blast load

http://dx.doi.org/10.1016/j.engstruct.2014.06.0040141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (P. Olmati).

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 loStruct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

Pierluigi Olmati a,⇑, Francesco Petrini b, Konstantinos Gkoumas b

a Faculty of Engineering and Physical Sciences, University of Surrey, GU2 7XH Guildford, UKb 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 engineeringFragility analysisConcrete cladding wall panelsCladding system

a b s t r a c t

This paper presents a probabilistic method to support the design of cladding wall systems subjected toblast loads. The proposed method is based on the broadly adopted fragility analysis method (conditionalapproach), widely used in Performance-Based Design procedures for structures subjected to naturalhazards like earthquake and wind. The cladding wall system under investigation is composed bynon-load bearing precast concrete wall panels. From the blast design point of view, these wall panelsmust protect people and equipment from external detonations. The aim of this research is to computeboth the fragility curves and the limit states exceedance probability of a typical precast concrete claddingwall panel considering the detonations of vehicle borne improvised explosive devices. Moreover, thelimit states exceedance probability of the cladding wall panel is estimated by Monte Carlo simulation(unconditional approach) in order to validate the proposed fragility curves.

� 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 inthe design practice, a set of design scenarios are selected and theintegrity of the blast-resistant structural members and of the pro-tective elements is assessed by using non-linear dynamic analyseswith an equivalent single degree of freedom (SDOF) method. Insuch a way (adopting a deterministic approach for the hazard char-acterization), the probability of exceeding a particular limit state isnot evaluated. In addition to the above, in designing a structuralcomponent subjected to blast loads, the current state of practiceis 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 densityfunction. This is common for Low Probability–High Consequence(LPHC) events [1]. As a partial consideration of the uncertaintyaffecting the blast load, simplified approaches are usually adopted.For example, in [2] the use of a magnification coefficient of 20% isapplied on the assumed amount of explosive. However, this islimited to explosive storage facilities. In an antiterrorism design,the amount of explosive is characterized by elevated uncertaintydepending on both technical and socioeconomic factors. Theseuncertainties lead the engineering community toward the

implementation of probabilistic methods [3], something that isnow crucial for both academics and practitioners.

With specific reference to blast-resistant structures, someauthors started carrying out investigations about the use ofprobabilistic methods for the assessment and design of structuralcomponents and structural systems. In [4], results of a parametricinvestigation on the reliability of reinforced concrete slabs underblast loading are presented, in order to establish appropriateprobabilistic distributions of the resistant parameters. In [5], theextension of probabilistic approaches from the performance-basedearthquake engineering to the blast design problems are provided,also by suggesting appropriate variables for the intensity measuresIMs, the damage measures DMs, and the response parameters defi-nition. In [6], Monte Carlo simulations are performed in order toestimate the failure probability of windows subjected to a blast loadmade by a vehicle bomb. In [7], the fragility curves are presented fortwo kinds of glazing systems. In [8], the design in a probabilistic wayof 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 appropriateframeworks for the probabilistic design of blast resistant structuresis 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 verifyingthe achievement of design performance objectives of structuralsystems during their design life [9]. Probabilistic approaches havebeen extensively implemented in the state of the art methods for

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the PBD of structures under different kind of hazards such as earth-quake [10,11], wind [12–14], and hurricanes [15]. The last case isan example of multi-hazard situation that is expected to be oneof the main directions of PBD approaches in the future [16].

In the PBD context, a powerful tool is represented by thefragility analysis (see for example [17–19]). As it is well known,the structural fragility is expressed as the cumulative probabilitydistribution of attaining a certain Damage Measure (DM) condi-tional to the Intensity Measure (IM) of the hazard. The efficiencyof the fragility approach is strictly related to the appropriatenessof the IM in terms of ‘‘sufficiency’’ and ‘‘efficiency’’, meaningthat the IM must accurately describe all pertinent hazard sources(see [20]).

Despite the above, a rigorous approach that is consistent withthe well-established PBD frameworks adopted in presence of otherhazards has not been defined for blast resistant structures. Thispaper is an effort in that direction with a specific focus on the fra-gility analysis. The fragility analysis is applied in order to computethe probability of exceeding a limit state (‘‘probability of exceed-ance’’) of a precast wall panel subjected to blast loads (in particularfar-field surface-blast loads [2]) in a PBD perspective.

As case study, a precast concrete cladding wall panel with thedimensions of 3500 mm in length and 1500 mm in width, with across sectional thickness of 150 mm is considered. The panel issubjected to a blast load generated by a vehicle borne improvisedexplosive 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 subjectto specific architecture requirements while their thickness isapproximately 15 cm. The steel reinforcements are generallyplaced in the middle of the cross section. This kind of wall panelsshould be designed in order to protect occupants and equipmentfrom external detonations.

Non-linear dynamic analyses are carried out by the well-estab-lished method of the equivalent non-linear SDOF system, wherethe 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 arecomputed using Monte Carlo simulations.

The fragility curves are evaluated for the case-study wall panelfor each defined limit state called here Component Damage Level(CDL). Then the fragility curves are used in order to estimate theprobability of exceedance of the cladding wall panel subjected toblast load scenarios (vehicle borne improvised explosive devices).Finally, the probability of exceedance of the wall panel subjectedto the same scenarios is estimated by the unconditional approach(based on a single Monte Carlo simulation) in order to validate theobtained results (see [4,11]).

In addition to the fragility analysis of the examined structuralmember typology (an innovative aspect of the paper), some preli-minary indications on the selection of a sufficient and efficient IMfor PBD of blast-resistant structures are provided.

Fig. 1. Uncertaint parameters of vehicle borne improvised explosive devicescenarios.

2. Fragility analysis

As previously stated, the fragility of a structure under the actionof a certain hazard is expressed as the cumulative probability dis-tribution of a certain DM conditional to the IM of the consideredhazard. Probabilistic PBD approaches identify the generic struc-tural performance by means of acceptable occurrence frequenciesfor some threshold values (representing structural limit states) ofan appropriate DM during a reference period of time [21,22]. Thedetermination of such occurrences is affected by large amountsof uncertainty. The fragility approach allows the designer toexpress in a synthetic and efficient manner this uncertainty by

Please cite this article in press as: Olmati P et al. Fragility analysis for the PerfoStruct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004

making use of conditional probability relations and by highlightingthe dependences of these occurrences from the IM.

In earthquake engineering, the fragility approach has beenmostly developed during last twenty years and applied for PBDpurposes. The fragility curves have been developed also for struc-tures subjected to flood [23], fire [24], and windborne debris inhurricane prone regions [25]. The fragility curves are nowadaysextensively used for the state of practice methods of structural riskevaluation 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 todevelop fragility curves under a single hazard by Monte Carloapproaches. These are due to the fact that: (i) the computationaleffort 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, thestructural fragility needs to be represented in terms of surfaces,something that is required for example in the case of performanceanalysis under multiple hazards (see for example [28]). In thispaper, this issue is discussed focusing the attention on the criticismof 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 manyengineering fields.

The IM in general should be chosen among the first group ofuncertain parameters or as a combination of those parameters,while the entity of the blast action given a certain IM is determinedby the parameters characterizing the interaction between the IMand the structural parameters. In probabilistic terms, hazard andstructural parameters can be characterized as unconditional withrespect to parameters belonging to one of the other two groups,while parameters representing the interaction mechanisms mustbe usually characterized in conditional probabilistic terms withrespect to the hazard and the structural parameters [12].

rmance-Based Design of cladding wall panels subjected to blast load. Eng


Fig. 2. Blast loads by the adopted model (broken lines) and the SBEDS model (solidlines).

Table 1Input data.

Symbol Description Mean COV Distribution

fc Concrete strength 28 MPa 0.18 Lognormalfy Steel strength 495 MPa 0.12 LognormalL Panel length 3500 mm 0.001 LognormalH Panel height 150 mm 0.001 Lognormalb Panel width 1500 mm 0.001 Lognormalc Panel cover 75 mm 0.01 Lognormal

W Explosive weight 227 kg 0.3 LognormalR Stand-off distance 15 m 20 m 25 m 0.05 Lognormal

P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 3

3. Blast load model

The side-on blast pressure Ps0 (MPa), can be estimated by theformula of Mills [29] (Eq. (1)), while the side-on specific impulseis0 (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 isthe stand-off distance.

Ps0 ¼ 1:7721Z3

� �� 0:114

1Z2

� �þ 0:108

1Z

� �ð1Þ

is0 ¼ 30012

ffiffiffiffiffiffiW3p� �

ð2Þ

Z ¼ RffiffiffiffiffiffiW3p ð3Þ

The explosive amount is commonly expressed as an equivalent TNTcharge [2] by the Equivalent Factor (EF), which multiplies theweight of the explosive charge utilized.

Both Eqs. (1) and (2) are valid for free-air explosions. In thisstudy detonations occurring on a surface (surface explosions) areconsidered, therefore the energy of the detonation is confined bythe ground surface, creating a larger demand than that of thefree-air explosion. The surface blast demand is calculated by usingthe same equations for the free-air explosions but with a chargeweight (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 2Ps07Patm 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 timehistory [31]. And the duration of the blast load (td) is computedfrom the side-on pressure and the specific impulse, assuming atriangular impulse (Eq. (5)).

td ¼2is0

Ps0ð5Þ

The variation in the pressure is assumed approximately to followthe Friedlander pulse shape shown in Eq. (6). Further details onthe sensitivity of the structural response due to exponential or tri-angular blast loading are investigated in [32].

PðtÞ ¼ Pr 1� ttd

� �e�bttd ta � t � td ð6Þ

In Eq. (6), ta (s) is the arrival time of the blast load, taken here aszero, and b is the decay coefficient. In this study a value of 1.8 forb is assumed [31]. The clearing effect is conservatively neglectedin this study since the cladding wall panel is generally a single partof a large building façade, thus the conditions for clearing of thereflected shock wave are in general not satisfied. In Fig. 2 the blastload time histories computed for different values of the explosiveweight W (kg) and stand-off distance R (m) with the above-men-tioned procedure are shown. The obtained curves are found to bein good agreement with the curves obtained by SBEDS [31]. Theblast load is considered uniformly distributed on the cladding wall,which is typical for a scaled distance higher than approximately2.0 m/kg1/3 [31].

4. Cladding panel model

A precast concrete cladding wall system has some advantageswith respect to the traditional non-load bearing masonrycladdings. 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 concretewalls subjected to a blast demand is investigated. The first advan-tage of precast concrete wall panels versus traditional masonrycladdings is the greater resistance of the cladding system to a blastdemand (see for example the advantage of using precast concretewall 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 asacid rains and chlorine ions exposure. In [37] the protection perfor-mance of cellulosic fiberboard panels from ballistic attacks isinvestigated. Furthermore, in [38] the behavior of concrete insu-lated panels is investigated, focusing on the shear ties connectingthe two concrete wythes confining the insulation.

The cladding panel taken as case study is 3500 mm long by1500 mm wide, with a cross sectional thickness of 150 mm. Thepanel is connected along two of its edges to the external frameof 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 concreteindustry (see for example [39]). The assumed mean values andCoefficients of Variations (COVs) are shown in Table 1. The longitu-dinal reinforcement consists of ten reinforcement bars of 10 mmdiameter located in the mid-axis of the cross section. The meanvalue and the COVs of the reinforcement strength are provided inTable 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 theconcrete density q are expressed as functions of fc. The mean valueof fc is 28 MPa (4060 psi), with a COV of 0.18, as adopted in [40] fora lognormal probability density function (see Table 1). The Young’s




modulus is computed by Eq. (7) [41] while the concrete density iscomputed by Eq. (8) [42,43]. Both Ec and fc are expressed in MPawhile q is expressed in kg/m3.

Ec ¼ 22;000fc

10

� �0:3

ð7Þ

q ¼ Ec

0:043ðf 0:5c Þ

� � 11:5

ð8Þ

The compressive strength increment of the concrete due to the highstrain rate is accounted for in this study. This increase is taken intoaccount by means of the Dynamic Increase Factor (DIF), a multipli-cative coefficient of the concrete static compressive resistance. Sincethe compressive strength enhancement of the concrete variesmarginally for a ductile flexural response over the range of the strainvelocity, the DIF can be assumed constant and equal to 1.19.This hypothesis is in accordance with the compressive strengthenhancement proposed in [31], and it leads to an increase of thecomputational efficiency by avoiding cyclic iterations in the algo-rithm of the SDOF equation solver. However, cyclic iterations arenecessary for computing the strength increase of the reinforcement.This increase is sensitive to the ductile flexural response, as opposedto the compressive strength increase of the concrete.

4.2. Reinforcing steel

A grade 450 MPa steel is used. Due to material standardrequirements the average yield strength is higher than the speci-fied yield. For estimating the mean value of the yield strength, astatic strength increase factor equal to 1.1 is adopted as indicatedin [31]. A COV of 0.12 is used as proposed in [40] for a lognormalprobability density function (see Table 1). Young’s modulus istaken as a deterministic value and assumed to be equal to210 GPa. The steel strength increase due to the strain velocity istaken into account by the Cowper and Symonds model [44]. Thus,the DIF is provided by Eq. (9):

DIF ¼ 1þ d�=dtq

� �1n

ð9Þ

where de/dt is the strain-rate demand of the reinforcement, q istaken equal to 500 s�1 and n is taken equal to 6. Both q and n areestimated by fitting the strength increase versus the strain rate in[31]. By solving the SDOF equation of motion, the DIF is iterativelyupdated until a convergence threshold is reached. The strain rate ofthe steel reinforcement (de/dt) in Eq. (9) is calculated approximatelyby Eq. (10).

dedt¼ dS

dtL8

d2EcJc

� �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 crosssection, d is the distance from the extreme compression fiber of thecross-section to the centroid of the tensile reinforcement, and dS/dtis the rate of the resistance force developed by the panel (S) whensubjected 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 loadwith an equivalent SDOF, the latter is defined as a system that hasthe same energy of the original structural element (in termsof work energy, strain energy, and kinetic energy) when the


structural element, if subjected to a blast load, deflects in a givendeformed shape. The displacement field of the component can beexpressed as u(x, t) = U(x)y(t), where U(x) is the assumed deformedshape of the component under the blast load and y is the maximumdisplacement of the component. Furthermore, the displacement ofthe component is obtained by the SDOF equation by assuming aflexural 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 totalmass of the component, S(y(t)) is the resistance of the component asa 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 thecritical damping is assumed to be 1% in all the analyses), KLM isthe load-mass transformation factor equal to the ratio of KM andKL (the mass transformation factor and the load transformation fac-tor respectively). The last two are evaluated by equating the energyof the two systems (in terms of work energy and kinetic energyrespectively).

KL ¼R L

0 pðxÞUðxÞdxR L0 pðxÞdx

KM ¼R L

0 mðxÞU2ðxÞdxR L0 mðxÞdx

ð12Þ

Referring to Eq. (12) and Fig. 3, p(x) is the blast load shape on thecomponent, m(x) is the distributed mass, and r is the resistance ofthe element in terms of pressure. The load-mass transformation fac-tor KLM is different at each deformation stage of the componentresponse; for a bilinear resistance function two values of the KLM

can be defined: one for the elastic response and one for the plasticresponse. More details on the equivalent SDOF method are providedin [2,31].

In order to obtain the bilinear resistance function of the simplysupported concrete cladding wall, it is necessary to compute theresisting moment (Mr) in the mid-span of the panel. The yieldingpoint of the resistance function (Sy) is obtained by Eq. (13) wherethe loaded surface (A) is equal to the cladding wall length (L) timesthe cladding wall width (b) and ry is the pressure resistancefunction.

Sy ¼8Mr

L¼ ryA ð13Þ

The resisting moment is evaluated by Eq. (14) [31]:

Mr ¼ Asfdy d� Asfdy

1:7bfdc

� �ð14Þ

where As is the reinforcements area, fdy is the dynamic yieldstrength of the reinforcing steel, fdc is the dynamic compressivestrength of the concrete, b width of rectangular section.

It is also necessary to evaluate the yielding displacement ofthe resistance function. For a simply supported component theyielding displacement (de) is given by Eq. (15).

de ¼5

3848MrL

2

EcJð15Þ

J ¼ 0:5ðJg þ JcÞ ð16Þ

where J is the moment of inertia of the cross section as evaluated byEq. (16), while Jc and Jg are computed by Eqs. (17) and (18)respectively.

Jc ¼ Gbd3 ð17Þ

Jg ¼bH3

12ð18Þ



Fig. 3. Resistance versus displacement relation of a component.

Table 2Component damage levels and the associated thresholds in terms of responseparameters [31].

Component damage levels h (degree) l (–)

Blowout >10� NoneHazardous failure 610� NoneHeavy damage 65� NoneModerate damage 62� NoneSuperficial damage None 1


The coefficient G in Eq. (19) is evaluated starting by the designchart 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 � 181:98p2 þ 5:8624pÞ n7

� �0:7ð19Þ

where p is the percentage of reinforcement in the cross section ofthe panel evaluated by neglecting the reinforcements cover, and nis the ratio between the steel Young’s modulus and the Ec. Eq.(19) is valid for single reinforced cross-sections only. Since Sy andde are evaluated, the resistance function of the simply supportedcladding wall can be defined. Finally, the central difference methodis then used to solve Eq. (11) which presents three non-linarites:one due to the bilinear shape of the resistance function, one dueto the load mass transformation factors, and the last due to thedynamic strength enhancement of the reinforcing steel (whichaffects the resistance function).

6. Performance definition

For structural components subjected to blast loads in a flexuralresponse regime, generally two response parameters are of inter-est: the support rotation angle (h) and the ductility ratio (l). Theseparameters are defined in Eqs. (20) and (21):

h ¼ arctg2dmax

L

� �ð20Þ

l ¼ dmax

deð21Þ

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 economicallyimpractical to design a component to remain in elastic range.

While other significant response parameters can be defined (forexample [4] considers the strain on reinforcements), this studyfocuses on the above defined response parameters, since theseare 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 abovementioned levels are defined as follows:

� Blowout (BO): the component is overwhelmed by the blast loadcausing 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 permanentdeflection. 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 interms of the response parameters h and l. For a non-structuralconcrete cladding wall without shear reinforcement, neglectingtension membrane effect, the CDL thresholds are those reportedin Table 2. The Fragility Curves are computed in the following foreach of the mentioned CDLs.

7. Algorithm for the computation of the fragility curves

As described in previous sections, the blast load on the panel isa 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 andthe W (Eq. (2)). Consequently, two detonations with the same Zcan have different impulse density, depending on the amount ofexplosive. Thus, the two explosions have the same peak pressurebut different duration.

Summarizing, since the blast load depends on both the Z andthe W, the choice of the IM for computing the fragility curves is acrucial issue. In this study the Z is taken as the IM. Some aspectsrelated to this choice are discussed in the next section. Note thatfor higher values of the Z the cladding wall has a lower structuralresponse than for lower values.

The Fragility Curves (FCs) are developed for each CDL. The algo-rithm implemented in MATLAB� for the fragility curves evaluationis shown in Fig. 4. With reference to the same figure, ‘‘i’’ indicatesthe ith point of the fragility curve, ‘‘j’’ indicates the ‘‘j’’th CDL, and‘‘k’’ indicates the kth stand-off distance (R) for which the fragilitycurve is computed. ‘‘N’’ is the maximum value for ‘‘i’’ and thereforethe total number of points forming the fragility curve. ‘‘M’’ is themaximum value for ‘‘j’’ and therefore the total number of the CDLs.Finally ‘‘L’’ is the maximum value of ‘‘k’’ therefore the total number



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°

of s

ampl

es

P(X>x0|Z) [%]

N° of samples C.O.V.

Fig. 5. N� of samples and COV for the fragility curve representative of the HFcomponent damage level, for R equal to 20 m.


of the stand-off distances for which the fragility curve related tothe jth CDL is computed.

The name ‘‘FC-CDL (j,k)’’ indicates the fragility curve computedfor 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 forcomputing 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 andthe 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 evaluatethe ith points of the FC-CDL (j,k) until tracking the completeFC-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 twodescribed cycles are repeated until k = L;� finally, the piecewise curves obtained point by point with the

above steps they are interpolated by a lognormal cumulativefunction, see Fig. 6.

As said, the fragility curves describe the conditional probabilityof exceedance (P(X > x0|Z)) of the response parameter X (chosencase-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 ofP(X > x0|Z). In order to obtain an acceptable COV, the number ofsamples adopted in the analysis is increased with the decreasingof P(X > x0|Z); this means that the number of samples increase withincreasing Z. In this work, an exponential law has been set for thisincreasing 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 curverelated to the heavy damage CDL and for R equal to 20 m.

8. Sufficiency of the intensity measure and results of thefragility analysis

For better understanding the sufficiency of the adopted inten-sity measure (the scaled distance Z), some considerations can bemade 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 bothmaterials and geometrical parameters (see Table 1), and the DIFsfor the concrete and steel are taken as constant and equal to 1.19and 1.20 respectively. The pressure–impulse diagrams referred todifferent values of h are shown in Fig. 7. Generally, three regionscan be individuated in the pressure–impulse diagrams, eachrelated with a different regime of structural response subjectedto a load time history. These are defied as: impulsive (I), dynamic(D), and pressure (P) region, depending on the characteristics ofthe load time history with respect to the dynamical proprietiesof 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 (andconsequently by the same peak pressure) but having different Wand R. As a matter of fact, the two blast loads are consistent withtwo different demands on the pressure–impulse diagrams, havingthe 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 mentionedblast demands (again, characterized by the same pressure peak butby different impulse densities) depends on the position of thesedemands in the pressure–impulse diagram. Thus, if these blastdemands are located in the impulsive region (I), a certain valuefor this difference will be observed, while if blast demands arelocated 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 sufficiencyof the chosen IM is greater in the D and P regions than in the Iregion. In the I region a fragility surface made by considering bothR and W as independent elements of a vectorial IM would be moreappropriate.

For taking account this approximation, as explained in the pre-vious section, the fragility curves are computed for different valuesof the R (R ¼ Z

ffiffiffiffiffiffiW3p

). In what follows, when the fragility curves areused for estimating the failure probability of a component damage



0

20

40

60

80

100

3.7 3.9 4.1 4.3 4.5

P (X

> x

0|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(X

> x

0|Z)

[%]

Z

Hazardous Failure

0

20

40

60

80

100

3.0 3.5 4.0 4.5 5.0

P(X

> x

0|Z)

[%]

Z

Moderate Damage

Fig. 8. From top left clockwise, fragility curves for t

Str

eet

Level 2

Level 3

Level 1

Target

Fig. 9. Lines of defence.



level, the specific fragility curve corresponding to the mean valueof 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 fragilitycurves of the case study cladding panel, and, (ii) the probabilityof the limit state exceedance of such cladding panel estimated byboth the conditional and unconditional approaches. The last pointis important in order to validate the computed fragility curves by acomparison of the exceedances obtained by the two approaches.

In Fig. 8 the computed fragility curves are shown for differentvalues of R. Focusing on the considered CDLs, from Fig. 8 can beobserved that the fragility curves of the SD level have a different

0

20

40

60

80

100

2.8 3.0 3.2 3.4 3.6 3.8 4.0

Heavy Damage

P(X

> x

0|Z)

[%]

Z

0

20

40

60

80

100

5 6 7 8 9 10 11

P(X

> x

0|Z)

[%]

Z

Superficial Damage

he HF, HD, SD, MD component damage levels.



Table 3Comparisons of the probability of exceedance using the conditional and unconditionalapproaches.

CDL Mean W = 227 kg, COV = 0.3 lognormalMean R, COV = 0.05 lognormal

Conditionalapproach (%)

Unconditionalapproach (%)

Percentagedifference D%

R = 20 mSD 100.0 100.0 0.0MD 96.6 97.5 0.9HD 55.7 55.5 0.3HF 13.6 12.1 11.0

R = 25 mSD 100.0 100.0 0.0MD 74.6 77.3 3.5HD 14.2 12.6 11.2HF 1.02 1.02 0.0

R = 15 mSD 100.0 100.0 0.0MD 97.9 99.9 2.0HD 93.6 96.9 3.4HF 67.8 72.6 6.6


slope compared to that of the other three CDLs (HF, HD, and MD). Itshould be noted that the SD level is based on the ductility (l) of thecomponent while HF, HD, and MD levels are based on the supportrotation (h). The SD level for a concrete cladding panel prescribesthe elastic response of the component, and for the case study panelit appears to be more sensitive to the considered uncertaintiescompared to the HF, HD, and MD levels. By varying the numberof samples the maximum obtained COV for the lower probabilityof failure (close to zero), is about 9% (Fig. 5). This value is consid-ered acceptable for the specific case, and it is consistent with otherstudies on blast applications (see [7]).

For computing the limit state probability of exceedance by theconditional approach it is necessary to develop a hazard analysisfor the stochastic characterization of the blast scenario and to solveEq. (23). In this study, a vehicle borne improvised explosive deviceis 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 eachline of security a different mean value of W is expected. Theexpected value of W decrease with the decreasing of R from thetarget, since the line of security system reduces progressively theseverity 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 (forexample a fence barrier) prevents vehicles entering. Finally, Level 3prevents 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 intelligenceservice is able to prevent this threat). Therefore, a vehicle bomb(with about 27–454 kg of TNT or equivalent) is considered. Themean amount of TNT or equivalent in the vehicle is assumed equalto 227 kg with a COV equal to 0.3 (see Table 1); this assumption isin line with [46]. A set of stand-off distances are considered (15, 20and 25 m) each with a coefficient of variation equal to 0.05, assum-ing that the vehicle could impact a fence barrier but move nofurther.

The conditional probability of exceedance of the CDL (P(X > x0))is evaluated by Eq. (23). As previously stated, X is the most criticalbetween 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 responseparameters characterizing the component damage level (seeTable 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 alognormal distribution. As mentioned above, the fragility curve(P(X > x0|Z)) used for evaluating Eq. (23) is the one correspondingto the mean value of the R (Table 1).

PðX > x0Þ ¼ PðH > h0Þ [ PðM > l0Þ ð22Þ

PðX > x0Þ ¼Z þ1

�1PðX > x0jZÞpðZÞdz

ffiX1i¼0

PðX > x0jZÞipðZÞiDZi ð23Þ

The obtained results are shown in Table 3. The first column reportsthe 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 comparisonpurposes as ‘‘exact value’’ of the exceedance probability.

From these results the maximum percentage differencebetween the P(X > x0) computed by the conditional and uncondi-tional approaches is 11%. Further studies are necessary to confirmwhether 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 andW. Thus, the difference between the P(X > x0) computed by the con-ditional and unconditional approaches increases with the increasein the difference between the R with which the fragility curve iscomputed (mean value of R) and the R of the Monte Carlo samplesof the unconditional approach.

9. Conclusions

The probabilistic analysis of a precast concrete cladding wallpanel subjected to blast load has been presented. The blast loadmodel has been adopted on the basis of empirical laws, and boththe geometry and mechanical properties of the panel are assumedas stochastic. A mechanical model equivalent to a single degree offreedom has been adopted for describing the motion of the panelunder the blast load. The Monte Carlo simulation has been usedfor computing: (i) the fragility curves of the cladding wall panelsubjected to blast load for several limit states (component damagelevels), (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 measurechosen for the fragility analysis has been presented and analyseshave been carried out for different values of the stand-off distance.It is expected that the scaled distance Z adopted in this paper, is asufficient intensity measure especially for blast demands belong-ing to the dynamic and pressure region of the pressure–impulsediagrams.

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-BasedDesign for blast resistant structures, something already done forstructures subjected to other hazards (e.g. earthquake and wind).One of the main issues related to the completion of a probabilisticperformance-based blast engineering procedure consists in deter-mining the hazard function; this issue is mainly due to the fact thatan explosion event (e.g. terroristic vehicle bomb attack) is a lowprobability event, as described in [47–49].

Additional studies could focus on: (i) improving the intensitymeasure sufficiency and efficiency, (ii) considering the degradationof the cladding panel during the life-cycle, (iii) improving themechanical model that describes the response of the panel




subjected to a blast load, and (iv) developing appropriate methodsfor complex system design [50] starting from the element-basedanalysis provided in this paper.

Acknowledgments

The authors gratefully acknowledge the scientific contributionof Prof. Franco Bontempi of the Sapienza University of Rome.

This work was partially supported by StroNGER s.r.l. from thefund ‘‘FILAS – POR FESR LAZIO 2007/2013 – Support for theresearch spin-off’’.

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