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


Published on

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

Published in: Engineering, Technology, Business
1 Like
  • Be the first to comment

No Downloads
Total views
On SlideShare
From Embeds
Number of Embeds
Embeds 0
No embeds

No notes for slide

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

  1. 1. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load Pierluigi Olmati a,⇑ , Francesco Petrini b , Konstantinos Gkoumas b a Faculty of Engineering and Physical Sciences, University of Surrey, GU2 7XH Guildford, UK b Sapienza University of Rome, Department of Structural and Geotechnical Engineering, Via Eudossiana 18, 00184 Rome, Italy a r t i c l e i n f o Article history: Available online xxxx Keywords: Performance-based blast engineering Fragility analysis Concrete cladding wall panels Cladding system a b s t r a c t This paper presents a probabilistic method to support the design of cladding wall systems subjected to blast loads. The proposed method is based on the broadly adopted fragility analysis method (conditional approach), widely used in Performance-Based Design procedures for structures subjected to natural hazards like earthquake and wind. The cladding wall system under investigation is composed by non-load bearing precast concrete wall panels. From the blast design point of view, these wall panels must protect people and equipment from external detonations. The aim of this research is to compute both the fragility curves and the limit states exceedance probability of a typical precast concrete cladding wall panel considering the detonations of vehicle borne improvised explosive devices. Moreover, the limit states exceedance probability of the cladding wall panel is estimated by Monte Carlo simulation (unconditional approach) in order to validate the proposed fragility curves. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Designing structures to withstand blast loads is common prac- tice for many government and commercial buildings. Generally in the design practice, a set of design scenarios are selected and the integrity of the blast-resistant structural members and of the pro- tective elements is assessed by using non-linear dynamic analyses with an equivalent single degree of freedom (SDOF) method. In such a way (adopting a deterministic approach for the hazard char- acterization), the probability of exceeding a particular limit state is not evaluated. In addition to the above, in designing a structural component subjected to blast loads, the current state of practice is to assume that the capacity is deterministic. The adoption of deterministic values for the demand is princi- pally due to a lack of knowledge of the hazard probability density function. This is common for Low Probability–High Consequence (LPHC) events [1]. As a partial consideration of the uncertainty affecting the blast load, simplified approaches are usually adopted. For example, in [2] the use of a magnification coefficient of 20% is applied on the assumed amount of explosive. However, this is limited to explosive storage facilities. In an antiterrorism design, the amount of explosive is characterized by elevated uncertainty depending on both technical and socioeconomic factors. These uncertainties lead the engineering community toward the implementation of probabilistic methods [3], something that is now crucial for both academics and practitioners. With specific reference to blast-resistant structures, some authors started carrying out investigations about the use of probabilistic methods for the assessment and design of structural components and structural systems. In [4], results of a parametric investigation on the reliability of reinforced concrete slabs under blast loading are presented, in order to establish appropriate probabilistic distributions of the resistant parameters. In [5], the extension of probabilistic approaches from the performance-based earthquake engineering to the blast design problems are provided, also by suggesting appropriate variables for the intensity measures IMs, the damage measures DMs, and the response parameters defi- nition. In [6], Monte Carlo simulations are performed in order to estimate the failure probability of windows subjected to a blast load made by a vehicle bomb. In [7], the fragility curves are presented for two kinds of glazing systems. In [8], the design in a probabilistic way of a sacrificial cladding for a blast wall is described, deployed to pro- tect vulnerable objects against an accidental explosion. Due to the above considerations, the definition of appropriate frameworks for the probabilistic design of blast resistant structures is an important objective for the engineering scientific community. To this regard, during the last decade Performance-Based Design (PBD) has been recognized as a powerful methodology for verifying the achievement of design performance objectives of structural systems during their design life [9]. Probabilistic approaches have been extensively implemented in the state of the art methods for 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: (P. Olmati). Engineering Structures xxx (2014) xxx–xxx Contents lists available at ScienceDirect Engineering Structures journal homepage: Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  2. 2. the PBD of structures under different kind of hazards such as earth- quake [10,11], wind [12–14], and hurricanes [15]. The last case is an example of multi-hazard situation that is expected to be one of the main directions of PBD approaches in the future [16]. In the PBD context, a powerful tool is represented by the fragility analysis (see for example [17–19]). As it is well known, the structural fragility is expressed as the cumulative probability distribution of attaining a certain Damage Measure (DM) condi- tional to the Intensity Measure (IM) of the hazard. The efficiency of the fragility approach is strictly related to the appropriateness of the IM in terms of ‘‘sufficiency’’ and ‘‘efficiency’’, meaning that the IM must accurately describe all pertinent hazard sources (see [20]). Despite the above, a rigorous approach that is consistent with the well-established PBD frameworks adopted in presence of other hazards has not been defined for blast resistant structures. This paper is an effort in that direction with a specific focus on the fra- gility analysis. The fragility analysis is applied in order to compute the probability of exceeding a limit state (‘‘probability of exceed- ance’’) of a precast wall panel subjected to blast loads (in particular far-field surface-blast loads [2]) in a PBD perspective. As case study, a precast concrete cladding wall panel with the dimensions of 3500 mm in length and 1500 mm in width, with a cross sectional thickness of 150 mm is considered. The panel is subjected to a blast load generated by a vehicle borne improvised explosive device. The wall under investigation is a non-load bear- ing precast concrete wall panel used as exterior cladding for build- ings. Typically, the length and the width of these walls are subject to specific architecture requirements while their thickness is approximately 15 cm. The steel reinforcements are generally placed in the middle of the cross section. This kind of wall panels should be designed in order to protect occupants and equipment from external detonations. Non-linear dynamic analyses are carried out by the well-estab- lished method of the equivalent non-linear SDOF system, where the precast concrete wall panel is modeled by an equivalent non- linear SDOF on the basis of energetic considerations. Furthermore, both the fragility curves and the probabilities of exceedance are computed using Monte Carlo simulations. The fragility curves are evaluated for the case-study wall panel for each defined limit state called here Component Damage Level (CDL). Then the fragility curves are used in order to estimate the probability of exceedance of the cladding wall panel subjected to blast load scenarios (vehicle borne improvised explosive devices). Finally, the probability of exceedance of the wall panel subjected to the same scenarios is estimated by the unconditional approach (based on a single Monte Carlo simulation) in order to validate the obtained results (see [4,11]). In addition to the fragility analysis of the examined structural member typology (an innovative aspect of the paper), some preli- minary indications on the selection of a sufficient and efficient IM for PBD of blast-resistant structures are provided. 2. Fragility analysis As previously stated, the fragility of a structure under the action of a certain hazard is expressed as the cumulative probability dis- tribution of a certain DM conditional to the IM of the considered hazard. Probabilistic PBD approaches identify the generic struc- tural performance by means of acceptable occurrence frequencies for some threshold values (representing structural limit states) of an appropriate DM during a reference period of time [21,22]. The determination of such occurrences is affected by large amounts of uncertainty. The fragility approach allows the designer to express in a synthetic and efficient manner this uncertainty by making use of conditional probability relations and by highlighting the dependences of these occurrences from the IM. In earthquake engineering, the fragility approach has been mostly developed during last twenty years and applied for PBD purposes. The fragility curves have been developed also for struc- tures subjected to flood [23], fire [24], and windborne debris in hurricane prone regions [25]. The fragility curves are nowadays extensively used for the state of practice methods of structural risk evaluation for structures under natural hazards. Among other techniques proposed for the evaluation of the fra- gility curves, Monte Carlo analysis is extensively used [26]. Two main issues need to be addressed primarily in order to develop fragility curves under a single hazard by Monte Carlo approaches. These are due to the fact that: (i) the computational effort required in order to obtain the desired level of approxima- tion is often challenging [27]; and, (ii) the individuation of an effi- cient and sufficient scalar IM for fragility representation is needed. The last point is essential since, in case of a vectorial IM, the structural fragility needs to be represented in terms of surfaces, something that is required for example in the case of performance analysis under multiple hazards (see for example [28]). In this paper, this issue is discussed focusing the attention on the criticism of choosing a scalar IM. As a first step, the uncertainties characterizing blast-engineer- ing problems need to be properly individuated and addressed (Fig. 1). These uncertainties can be divided into three main groups: hazard uncertainties (e.g. explosive, stand-off distance); structure uncertainties (e.g. stiffness, dimensions, damping, material characteristics, damping, etc.); interaction mechanism uncertainties (e.g. the reflected pres- sure, pressure duration, etc.); This classification of the uncertainties in three groups (load, structure, interaction mechanisms) is generally valid for many engineering fields. The IM in general should be chosen among the first group of uncertain parameters or as a combination of those parameters, while the entity of the blast action given a certain IM is determined by the parameters characterizing the interaction between the IM and the structural parameters. In probabilistic terms, hazard and structural parameters can be characterized as unconditional with respect to parameters belonging to one of the other two groups, while parameters representing the interaction mechanisms must be usually characterized in conditional probabilistic terms with respect to the hazard and the structural parameters [12]. Fig. 1. Uncertaint parameters of vehicle borne improvised explosive device scenarios. 2 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  3. 3. 3. Blast load model The side-on blast pressure Ps0 (MPa), can be estimated by the formula of Mills [29] (Eq. (1)), while the side-on specific impulse is0 (Pa s), is estimated by the formula of Held [30] (Eq. (2)). Where, Z is the scaled distance (Eq. (3)), W is the explosive weight, and R is the stand-off distance. Ps0 ¼ 1:772 1 Z3 À 0:114 1 Z2 þ 0:108 1 Z ð1Þ is0 ¼ 300 1 2 ffiffiffiffiffiffi W 3 p ð2Þ Z ¼ R ffiffiffiffiffiffi W3 p ð3Þ The explosive amount is commonly expressed as an equivalent TNT charge [2] by the Equivalent Factor (EF), which multiplies the weight of the explosive charge utilized. Both Eqs. (1) and (2) are valid for free-air explosions. In this study detonations occurring on a surface (surface explosions) are considered, therefore the energy of the detonation is confined by the ground surface, creating a larger demand than that of the free-air explosion. The surface blast demand is calculated by using the same equations for the free-air explosions but with a charge weight (W) increased by 80% [31]. Then, the reflected pressure Pr (MPa) for a normal angle of incidence is computed using Eq. (4) [29]: Pr ¼ 2Ps0 7Patm þ 4Ps0 7Patm þ Ps0 ð4Þ where Patm is the atmospheric pressure (0.1 MPa). For simplicity, the negative pressure phase is neglected from the blast load time history [31]. And the duration of the blast load (td) is computed from the side-on pressure and the specific impulse, assuming a triangular impulse (Eq. (5)). td ¼ 2is0 Ps0 ð5Þ The variation in the pressure is assumed approximately to follow the Friedlander pulse shape shown in Eq. (6). Further details on the sensitivity of the structural response due to exponential or tri- angular blast loading are investigated in [32]. PðtÞ ¼ Pr 1 À t td e Àbt td ta t td ð6Þ In Eq. (6), ta (s) is the arrival time of the blast load, taken here as zero, and b is the decay coefficient. In this study a value of 1.8 for b is assumed [31]. The clearing effect is conservatively neglected in this study since the cladding wall panel is generally a single part of a large building façade, thus the conditions for clearing of the reflected shock wave are in general not satisfied. In Fig. 2 the blast load time histories computed for different values of the explosive weight W (kg) and stand-off distance R (m) with the above-men- tioned procedure are shown. The obtained curves are found to be in good agreement with the curves obtained by SBEDS [31]. The blast load is considered uniformly distributed on the cladding wall, which is typical for a scaled distance higher than approximately 2.0 m/kg1/3 [31]. 4. Cladding panel model A precast concrete cladding wall system has some advantages with respect to the traditional non-load bearing masonry claddings. Studies on improving the performances of traditional masonry claddings under blast loads are presented in [33,34], while in [35], the load bearing capacity of load bearing concrete walls subjected to a blast demand is investigated. The first advan- tage of precast concrete wall panels versus traditional masonry claddings is the greater resistance of the cladding system to a blast demand (see for example the advantage of using precast concrete wall panels for protecting steel stud constructions [36]). Precast concrete wall panels can be integrated with other mate- rials for improving the resistance to environmental attacks such as acid rains and chlorine ions exposure. In [37] the protection perfor- mance of cellulosic fiberboard panels from ballistic attacks is investigated. Furthermore, in [38] the behavior of concrete insu- lated panels is investigated, focusing on the shear ties connecting the two concrete wythes confining the insulation. The cladding panel taken as case study is 3500 mm long by 1500 mm wide, with a cross sectional thickness of 150 mm. The panel is connected along two of its edges to the external frame of a building, and in particular, it is assumed to be simply- supported by the beams of the frame. Length, width and cross sec- tional thickness of the panel are considered as stochastic variables, due to the construction tolerances used in the precast concrete industry (see for example [39]). The assumed mean values and Coefficients of Variations (COVs) are shown in Table 1. The longitu- dinal reinforcement consists of ten reinforcement bars of 10 mm diameter located in the mid-axis of the cross section. The mean value and the COVs of the reinforcement strength are provided in Table 1. The panel is assumed to not have shear reinforcement. 4.1. Concrete The concrete compressive strength fc is considered as a stochas- tic variable, while the Young’s modulus of the concrete Ec and the concrete density q are expressed as functions of fc. The mean value of fc is 28 MPa (4060 psi), with a COV of 0.18, as adopted in [40] for a lognormal probability density function (see Table 1). The Young’s Fig. 2. Blast loads by the adopted model (broken lines) and the SBEDS model (solid lines). Table 1 Input data. Symbol Description Mean COV Distribution fc Concrete strength 28 MPa 0.18 Lognormal fy Steel strength 495 MPa 0.12 Lognormal L Panel length 3500 mm 0.001 Lognormal H Panel height 150 mm 0.001 Lognormal b Panel width 1500 mm 0.001 Lognormal c Panel cover 75 mm 0.01 Lognormal W Explosive weight 227 kg 0.3 Lognormal R Stand-off distance 15 m 20 m 25 m 0.05 Lognormal P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 3 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  4. 4. modulus is computed by Eq. (7) [41] while the concrete density is computed by Eq. (8) [42,43]. Both Ec and fc are expressed in MPa while q is expressed in kg/m3 . Ec ¼ 22; 000 fc 10 0:3 ð7Þ q ¼ Ec 0:043ðf0:5 c Þ 1 1:5 ð8Þ The compressive strength increment of the concrete due to the high strain rate is accounted for in this study. This increase is taken into account by means of the Dynamic Increase Factor (DIF), a multipli- cative coefficient of the concrete static compressive resistance. Since the compressive strength enhancement of the concrete varies marginally for a ductile flexural response over the range of the strain velocity, the DIF can be assumed constant and equal to 1.19. This hypothesis is in accordance with the compressive strength enhancement proposed in [31], and it leads to an increase of the computational efficiency by avoiding cyclic iterations in the algo- rithm of the SDOF equation solver. However, cyclic iterations are necessary for computing the strength increase of the reinforcement. This increase is sensitive to the ductile flexural response, as opposed to the compressive strength increase of the concrete. 4.2. Reinforcing steel A grade 450 MPa steel is used. Due to material standard requirements the average yield strength is higher than the speci- fied yield. For estimating the mean value of the yield strength, a static strength increase factor equal to 1.1 is adopted as indicated in [31]. A COV of 0.12 is used as proposed in [40] for a lognormal probability density function (see Table 1). Young’s modulus is taken as a deterministic value and assumed to be equal to 210 GPa. The steel strength increase due to the strain velocity is taken into account by the Cowper and Symonds model [44]. Thus, the DIF is provided by Eq. (9): DIF ¼ 1 þ d=dt q 1 n ð9Þ where de/dt is the strain-rate demand of the reinforcement, q is taken equal to 500 sÀ1 and n is taken equal to 6. Both q and n are estimated by fitting the strength increase versus the strain rate in [31]. By solving the SDOF equation of motion, the DIF is iteratively updated until a convergence threshold is reached. The strain rate of the steel reinforcement (de/dt) in Eq. (9) is calculated approximately by Eq. (10). de dt ¼ dS dt L 8 d 2EcJc ð10Þ where L is the length of the cladding panel, Ec is the Young’s mod- ulus of the concrete, Ic is the moment of inertia of the cracked cross section, d is the distance from the extreme compression fiber of the cross-section to the centroid of the tensile reinforcement, and dS/dt is the rate of the resistance force developed by the panel (S) when subjected to the demand. Eq. (10) is valid for simply supported ele- ments when the response is governed by the flexural behavior with- out shear failure. 5. Equivalent SDOF mechanical model of the panel In order to model a structural element subjected to a blast load with an equivalent SDOF, the latter is defined as a system that has the same energy of the original structural element (in terms of work energy, strain energy, and kinetic energy) when the structural element, if subjected to a blast load, deflects in a given deformed shape. The displacement field of the component can be expressed as u(x,t) = U(x)y(t), where U(x) is the assumed deformed shape of the component under the blast load and y is the maximum displacement of the component. Furthermore, the displacement of the component is obtained by the SDOF equation by assuming a flexural deformed shape by: KLMM€yðtÞ þ C _yðtÞ þ SðyðtÞÞ ¼ FðtÞ ð11Þ where y(t) is the displacement of the SDOF system and M is the total mass of the component, S(y(t)) is the resistance of the component as a function of the displacement expressed in unit force (see Fig. 3), F(t) is the blast pressure multiplied by the impacted area (A) expressed in force units, C is the damping (the percentage of the critical damping is assumed to be 1% in all the analyses), KLM is the load-mass transformation factor equal to the ratio of KM and KL (the mass transformation factor and the load transformation fac- tor respectively). The last two are evaluated by equating the energy of the two systems (in terms of work energy and kinetic energy respectively). KL ¼ RL 0 pðxÞUðxÞdx RL 0 pðxÞdx KM ¼ RL 0 mðxÞU2 ðxÞdx RL 0 mðxÞdx ð12Þ Referring to Eq. (12) and Fig. 3, p(x) is the blast load shape on the component, m(x) is the distributed mass, and r is the resistance of the element in terms of pressure. The load-mass transformation fac- tor KLM is different at each deformation stage of the component response; for a bilinear resistance function two values of the KLM can be defined: one for the elastic response and one for the plastic response. More details on the equivalent SDOF method are provided in [2,31]. In order to obtain the bilinear resistance function of the simply supported concrete cladding wall, it is necessary to compute the resisting moment (Mr) in the mid-span of the panel. The yielding point of the resistance function (Sy) is obtained by Eq. (13) where the loaded surface (A) is equal to the cladding wall length (L) times the cladding wall width (b) and ry is the pressure resistance function. Sy ¼ 8Mr L ¼ ryA ð13Þ The resisting moment is evaluated by Eq. (14) [31]: Mr ¼ Asfdy d À Asfdy 1:7bfdc ð14Þ where As is the reinforcements area, fdy is the dynamic yield strength of the reinforcing steel, fdc is the dynamic compressive strength of the concrete, b width of rectangular section. It is also necessary to evaluate the yielding displacement of the resistance function. For a simply supported component the yielding displacement (de) is given by Eq. (15). de ¼ 5 384 8MrL2 EcJ ð15Þ J ¼ 0:5ðJg þ JcÞ ð16Þ where J is the moment of inertia of the cross section as evaluated by Eq. (16), while Jc and Jg are computed by Eqs. (17) and (18) respectively. Jc ¼ Gbd 3 ð17Þ Jg ¼ bH 3 12 ð18Þ 4 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  5. 5. The coefficient G in Eq. (19) is evaluated starting by the design chart provided in [31]. In this study an analytical formula (see Eq. (19)) is determined by fitting the curves of the original chart [31]. G ¼ ð3320:3p3 À 181:98p2 þ 5:8624pÞ n 7 0:7 ð19Þ where p is the percentage of reinforcement in the cross section of the panel evaluated by neglecting the reinforcements cover, and n is the ratio between the steel Young’s modulus and the Ec. Eq. (19) is valid for single reinforced cross-sections only. Since Sy and de are evaluated, the resistance function of the simply supported cladding wall can be defined. Finally, the central difference method is then used to solve Eq. (11) which presents three non-linarites: one due to the bilinear shape of the resistance function, one due to the load mass transformation factors, and the last due to the dynamic strength enhancement of the reinforcing steel (which affects the resistance function). 6. Performance definition For structural components subjected to blast loads in a flexural response regime, generally two response parameters are of inter- est: the support rotation angle (h) and the ductility ratio (l). These parameters are defined in Eqs. (20) and (21): h ¼ arctg 2dmax L ð20Þ l ¼ dmax de ð21Þ where dmax is the maximum displacement of the component. A structural component subjected to a blast load is generally expected to yield (ductility greater than 1), as it is economically impractical to design a component to remain in elastic range. While other significant response parameters can be defined (for example [4] considers the strain on reinforcements), this study focuses on the above defined response parameters, since these are usually adopted for antiterrorism design [31]. In a performance-based blast design prospective, five Compo- nent Damage Levels (CDLs) [31] are considered: Blowout (BO), Hazardous Failure (HF), Heavy Damage (HD), Moderate Damage (MD), and Superficial Damage (SD). Following the [31], the above mentioned levels are defined as follows: Blowout (BO): the component is overwhelmed by the blast load causing debris with significant velocities. Hazardous Failure (HF): the component has failed, and debris velocities range from insignificant to very significant. Heavy Damage (HD): the component has not failed, but it has sig- nificant permanent deflections causing it to be un-repairable. Moderate Damage (MD): the component has some permanent deflection. It is generally repairable, if necessary, although replace- ment may be more economical and aesthetic. Superficial Damage (SD): the component has no visible permanent damage. The thresholds corresponding to these CDLs are defined in terms of the response parameters h and l. For a non-structural concrete cladding wall without shear reinforcement, neglecting tension membrane effect, the CDL thresholds are those reported in Table 2. The Fragility Curves are computed in the following for each of the mentioned CDLs. 7. Algorithm for the computation of the fragility curves As described in previous sections, the blast load on the panel is a function of the peak pressure and of the impulse density (Eqs. (4) and (2) respectively); the pressure is function of Z only (Eqs. (1) and (4)), while the impulse density depends on both the Z and the W (Eq. (2)). Consequently, two detonations with the same Z can have different impulse density, depending on the amount of explosive. Thus, the two explosions have the same peak pressure but different duration. Summarizing, since the blast load depends on both the Z and the W, the choice of the IM for computing the fragility curves is a crucial issue. In this study the Z is taken as the IM. Some aspects related to this choice are discussed in the next section. Note that for higher values of the Z the cladding wall has a lower structural response than for lower values. The Fragility Curves (FCs) are developed for each CDL. The algo- rithm implemented in MATLABÒ for the fragility curves evaluation is shown in Fig. 4. With reference to the same figure, ‘‘i’’ indicates the ith point of the fragility curve, ‘‘j’’ indicates the ‘‘j’’th CDL, and ‘‘k’’ indicates the kth stand-off distance (R) for which the fragility curve is computed. ‘‘N’’ is the maximum value for ‘‘i’’ and therefore the total number of points forming the fragility curve. ‘‘M’’ is the maximum value for ‘‘j’’ and therefore the total number of the CDLs. Finally ‘‘L’’ is the maximum value of ‘‘k’’ therefore the total number Fig. 3. Resistance versus displacement relation of a component. Table 2 Component damage levels and the associated thresholds in terms of response parameters [31]. Component damage levels h (degree) l (–) Blowout 10° None Hazardous failure 610° None Heavy damage 65° None Moderate damage 62° None Superficial damage None 1 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 5 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  6. 6. of the stand-off distances for which the fragility curve related to the jth CDL is computed. The name ‘‘FC-CDL (j,k)’’ indicates the fragility curve computed for the kth R, the jth CDL, by varying the W (and consequently the Z). The ith point of the fragility curve (named FC-CDL (i,j,k)) is com- puted by considering the blast load at the kth R and the ith W. The minimum and maximum amount of W should be enough for computing the values of the FC-CDL (j,k) ranging from 0 to 1. The FC-CDL (i,j,k) is obtained by a Monte Carlo simulation and the complete (cyclic) procedure of Fig. 3 is hereby described. first a kth R is selected; then the jth CDL is selected; consequently the ‘‘i’’ index is increased by solving the previ- ously introduced equations for each ‘‘i’’ in order to evaluate the ith points of the FC-CDL (j,k) until tracking the complete FC-CDL (j,k); after that, a new jth CDL is considered with the same value of ‘‘k’’. When j = M a different R is selected and the previous two described cycles are repeated until k = L; finally, the piecewise curves obtained point by point with the above steps they are interpolated by a lognormal cumulative function, see Fig. 6. As said, the fragility curves describe the conditional probability of exceedance (P(X x0|Z)) of the response parameter X (chosen case-by-case as the most critical between the values of h and l, see Table 2) with respect to the threshold x0 (identifying the CDL). As expected, for a constant number of samples at each ith point, the COV of P(X x0|Z) increases with the decreasing of P(X x0|Z). In order to obtain an acceptable COV, the number of samples adopted in the analysis is increased with the decreasing of P(X x0|Z); this means that the number of samples increase with increasing Z. In this work, an exponential law has been set for this increasing trend. In Fig. 5 the number of the samples and the rela- tive COVs are shown in function of P(X x0|Z) for the fragility curve related to the heavy damage CDL and for R equal to 20 m. 8. Sufficiency of the intensity measure and results of the fragility analysis For better understanding the sufficiency of the adopted inten- sity measure (the scaled distance Z), some considerations can be made with reference to the pressure–impulse diagrams [45] related to the case study panel. For this purpose, reference is made to the mean values of both materials and geometrical parameters (see Table 1), and the DIFs for the concrete and steel are taken as constant and equal to 1.19 and 1.20 respectively. The pressure–impulse diagrams referred to different values of h are shown in Fig. 7. Generally, three regions can be individuated in the pressure–impulse diagrams, each related with a different regime of structural response subjected to a load time history. These are defied as: impulsive (I), dynamic (D), and pressure (P) region, depending on the characteristics of the load time history with respect to the dynamical proprieties of the structure [2]. Two blast loads are taken into account. These loads can be cho- sen in such a way that they are characterized by the same IM (and consequently by the same peak pressure) but having different W and R. As a matter of fact, the two blast loads are consistent with two different demands on the pressure–impulse diagrams, having the same peak pressure but different values of the impulse density. As it can be observed in Fig. 7, the difference between the struc- tural response of the panel subjected to the two above mentioned blast demands (again, characterized by the same pressure peak but by different impulse densities) depends on the position of these demands in the pressure–impulse diagram. Thus, if these blast demands are located in the impulsive region (I), a certain value for this difference will be observed, while if blast demands are located in the dynamic (D) or pressure (P) regions, then this differ- ence will be lower than in the previous case. Considering the above, it can be concluded that the sufficiency of the chosen IM is greater in the D and P regions than in the I region. In the I region a fragility surface made by considering both R and W as independent elements of a vectorial IM would be more appropriate. For taking account this approximation, as explained in the pre- vious section, the fragility curves are computed for different values of the R (R ¼ Z ffiffiffiffiffiffi W3 p ). In what follows, when the fragility curves are used for estimating the failure probability of a component damage Fig. 4. Fragility curve computation flowchart. 0.00 0.02 0.04 0.06 0.08 0.10 0 20000 40000 60000 80000 100000 0.1 0.9 3.3 9.0 22.4 40.4 59.5 77.9 90.1 96.6 98.8 100.0 C.O.V. N°ofsamples P(Xx0|Z) [%] N° of samples C.O.V. Fig. 5. N° of samples and COV for the fragility curve representative of the HF component damage level, for R equal to 20 m. 6 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  7. 7. level, the specific fragility curve corresponding to the mean value of R is used for this purpose. This increases considerably the suffi- ciency of the chosen IM. 8.1. Results This section presents the results regarding: (i) the fragility curves of the case study cladding panel, and, (ii) the probability of the limit state exceedance of such cladding panel estimated by both the conditional and unconditional approaches. The last point is important in order to validate the computed fragility curves by a comparison of the exceedances obtained by the two approaches. In Fig. 8 the computed fragility curves are shown for different values of R. Focusing on the considered CDLs, from Fig. 8 can be observed that the fragility curves of the SD level have a different 0 20 40 60 80 100 3.7 3.9 4.1 4.3 4.5 P(Xx0|Z)[%] Z Interpolated FC Numerical FC Fig. 6. Numerical and lognormal interpolated fragility curves. Fig. 7. Pressure–impulse diagrams. 0 20 40 60 80 100 2.4 2.6 2.8 3.0 3.2 3.4 P(Xx0|Z)[%] Z Hazardous Failure 0 20 40 60 80 100 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Heavy Damage P(Xx0|Z)[%] Z 0 20 40 60 80 100 3.0 3.5 4.0 4.5 5.0 P(Xx0|Z)[%] Z Moderate Damage 0 20 40 60 80 100 5 6 7 8 9 10 11 P(Xx0|Z)[%] Z Superficial Damage Fig. 8. From top left clockwise, fragility curves for the HF, HD, SD, MD component damage levels. Street Level 2 Level 3 Level 1 Target Fig. 9. Lines of defence. P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 7 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  8. 8. slope compared to that of the other three CDLs (HF, HD, and MD). It should be noted that the SD level is based on the ductility (l) of the component while HF, HD, and MD levels are based on the support rotation (h). The SD level for a concrete cladding panel prescribes the elastic response of the component, and for the case study panel it appears to be more sensitive to the considered uncertainties compared to the HF, HD, and MD levels. By varying the number of samples the maximum obtained COV for the lower probability of failure (close to zero), is about 9% (Fig. 5). This value is consid- ered acceptable for the specific case, and it is consistent with other studies on blast applications (see [7]). For computing the limit state probability of exceedance by the conditional approach it is necessary to develop a hazard analysis for the stochastic characterization of the blast scenario and to solve Eq. (23). In this study, a vehicle borne improvised explosive device is considered. The amount of explosive (W) in the vehicle depends, among else, on the security measures in place. These security mea- sures can be structured in different lines (see Fig. 9) and for each line of security a different mean value of W is expected. The expected value of W decrease with the decreasing of R from the target, since the line of security system reduces progressively the severity of the possible attacks. In the example of Fig. 9, level 1 prevents trucks entering the tar- get zone, so no truck bomb should be expected. Level 2 in Fig. 9 (for example a fence barrier) prevents vehicles entering. Finally, Level 3 prevents pedestrians approaching the target. With this in mind, in the specific application a scenario con- cerning a truck bomb (with about 4000–27,000 kg of TNT or equiv- alent) is unreasonable (e.g. by assuming that the intelligence service is able to prevent this threat). Therefore, a vehicle bomb (with about 27–454 kg of TNT or equivalent) is considered. The mean amount of TNT or equivalent in the vehicle is assumed equal to 227 kg with a COV equal to 0.3 (see Table 1); this assumption is in line with [46]. A set of stand-off distances are considered (15, 20 and 25 m) each with a coefficient of variation equal to 0.05, assum- ing that the vehicle could impact a fence barrier but move no further. The conditional probability of exceedance of the CDL (P(X x0)) is evaluated by Eq. (23). As previously stated, X is the most critical between the response parameters h and l, assumed here as uncor- related. Consequently P(X x0) is the union of the two failure prob- abilities evaluated by considering separately the two response parameters characterizing the component damage level (see Table 2 and Eq. (22)). The probability density function of the Z (p(Z)) is computed by fitting the samples of both W and R with a lognormal distribution. As mentioned above, the fragility curve (P(X x0|Z)) used for evaluating Eq. (23) is the one corresponding to the mean value of the R (Table 1). PðX x0Þ ¼ PðH h0Þ [ PðM l0Þ ð22Þ PðX x0Þ ¼ Z þ1 À1 PðX x0jZÞpðZÞdz ffi X1 i¼0 PðX x0jZÞipðZÞiDZi ð23Þ The obtained results are shown in Table 3. The first column reports the CDLs, while in the second and third columns report the P(X x0) for each blast scenario obtained by Eq. (23) and by the uncondi- tional approach respectively, the last one considered for comparison purposes as ‘‘exact value’’ of the exceedance probability. From these results the maximum percentage difference between the P(X x0) computed by the conditional and uncondi- tional approaches is 11%. Further studies are necessary to confirm whether this percentage difference is acceptable or not. However, it is also necessary to consider that the W in the vehi- cle has an elevated dispersion, something that amplifies this differ- ence due to the dependence of the impulse density to both R and W. Thus, the difference between the P(X x0) computed by the con- ditional and unconditional approaches increases with the increase in the difference between the R with which the fragility curve is computed (mean value of R) and the R of the Monte Carlo samples of the unconditional approach. 9. Conclusions The probabilistic analysis of a precast concrete cladding wall panel subjected to blast load has been presented. The blast load model has been adopted on the basis of empirical laws, and both the geometry and mechanical properties of the panel are assumed as stochastic. A mechanical model equivalent to a single degree of freedom has been adopted for describing the motion of the panel under the blast load. The Monte Carlo simulation has been used for computing: (i) the fragility curves of the cladding wall panel subjected to blast load for several limit states (component damage levels), (ii) the probability of exceedance of limit states of the clad- ding wall panel by means of both the unconditional and condi- tional approach for comparison purposes. A discussion about the effectiveness of the intensity measure chosen for the fragility analysis has been presented and analyses have been carried out for different values of the stand-off distance. It is expected that the scaled distance Z adopted in this paper, is a sufficient intensity measure especially for blast demands belong- ing to the dynamic and pressure region of the pressure–impulse diagrams. This study highlights the feasibility and effectiveness of the fra- gility approach in the design of cladding wall panels and, generally, of protective structures. This is one of the fundamental steps nec- essary for developing a fully probabilistic Performance-Based Design for blast resistant structures, something already done for structures subjected to other hazards (e.g. earthquake and wind). One of the main issues related to the completion of a probabilistic performance-based blast engineering procedure consists in deter- mining the hazard function; this issue is mainly due to the fact that an explosion event (e.g. terroristic vehicle bomb attack) is a low probability event, as described in [47–49]. Additional studies could focus on: (i) improving the intensity measure sufficiency and efficiency, (ii) considering the degradation of the cladding panel during the life-cycle, (iii) improving the mechanical model that describes the response of the panel Table 3 Comparisons of the probability of exceedance using the conditional and unconditional approaches. CDL Mean W = 227 kg, COV = 0.3 lognormal Mean R, COV = 0.05 lognormal Conditional approach (%) Unconditional approach (%) Percentage difference D% R = 20 m SD 100.0 100.0 0.0 MD 96.6 97.5 0.9 HD 55.7 55.5 0.3 HF 13.6 12.1 11.0 R = 25 m SD 100.0 100.0 0.0 MD 74.6 77.3 3.5 HD 14.2 12.6 11.2 HF 1.02 1.02 0.0 R = 15 m SD 100.0 100.0 0.0 MD 97.9 99.9 2.0 HD 93.6 96.9 3.4 HF 67.8 72.6 6.6 8 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),
  9. 9. subjected to a blast load, and (iv) developing appropriate methods for complex system design [50] starting from the element-based analysis provided in this paper. Acknowledgments The authors gratefully acknowledge the scientific contribution of Prof. Franco Bontempi of the Sapienza University of Rome. This work was partially supported by StroNGER s.r.l. from the fund ‘‘FILAS – POR FESR LAZIO 2007/2013 – Support for the research spin-off’’. References [1] Starossek U. Progressive collapse of structures. London, United Kingdom: Thomas Telford Publishing; 2009. [2] UFC 3-340-02. Structures to resist the effects of accidental explosions. United States of America: Department of Defense; 2008. [3] Netherton MD, Stewart MG. Blast load variability and accuracy of blast load prediction models. Int J Protect Struct 2010;1(4):543–70. [4] Low HY, Hao H. Reliability analysis of reinforced concrete slabs under explosive loading. Struct Saf 2001;23:157–78. [5] Whittaker AS, Hamburger RO, Mahoney M. Performance-based engineering of buildings and infrastructure for extreme loadings. In: Proceedings of the AISC- SINY symposium on resisting blast and progressive collapse. New York, USA; December 4–5, 2003. [6] Chang DB, Young CS. Probabilistic estimates of vulnerability to explosive overpressures and impulses. J Phys Secur 2010;4(2):10–29. [7] Stewart MG, Netherton MD. Security risks and probabilistic risk assessment of glazing subject to explosive blast loading. Reliab Eng Syst Safety 2008;93:627–38. [8] Linkute˙ L, Juocevicˇius V, Vaidogas ER. A probabilistic design of sacrificial cladding for a blast wall using limited statistical information on blast loading. Mechanika 2013;19(1):58–66. [9] Augusti G, Ciampoli M. Performance-based design in risk assessment and reduction. Probab Eng Mech 2008;23(4):496–508. [10] Cornell CA, Krawinkler H. Progress and challenges in seismic performance assessment. PEER Center News 2000;3(2). [11] Vamvatsikos D, Dolšek M. Equivalent constant rates for performance-based seismic assessment of ageing structures. Struct Saf 2011;33(1):8–18. [12] Petrini F, Ciampoli M. Performance-based wind design of tall buildings. Struct Infrastruct Eng – Maint Manage Life-Cycle Des Perform 2012;8(10):954–66. [13] Spence SMJ, Gioffrè M. Large scale reliability-based design optimization of wind excited tall buildings. Probab Eng Mech 2012;28:206–15. [14] Spence SMJ, Gioffrè M. Efficient algorithms for the reliability optimization of tall buildings. J Wind Eng Ind Aerodyn 2011;99(6–7):691–9. [15] Barbato M, Petrini F, Unnikrishnan VU, Ciampoli M. Probabilistic performance- based hurricane engineering (PBHE) framework. Struct Saf 2013;45:24–35. [16] Decò A, Frangopol DM. Risk assessment of highway bridges under multiple hazards. J Risk Res 2011;14(9):1057–89. [17] Olmati P. Monte Carlo analysis for the blast resistance design and assessment of a reinforced concrete wall. In: ECCOMAS thematic conference – COMPDYN 2013: 4th international conference on computational methods in structural dynamics and earthquake engineering. Proceedings – an IACM special interest conference, Kos Island, Greece. Code 104617; 12 June 2013–14 June 2013. p. 1803–15. [18] Olmati P, Trasborg P, Naito CJ, Bontempi F. Blast resistant design of precast reinforced concrete walls for strategic infrastructures under uncertainty. Int J Crit Infrastruct; ISSN online: 1741-8038. [19] Bazzurro P, Cornell CA, Shome N, Carballo JE. Three proposal for characterizing MDOF nonlinear seismic response. J Struct Eng 1998;124(11):1281–9. [20] Luco N, Cornell CA. Structure-specific scalar intensity measures for near- source and ordinary earthquake ground motions. Earthq Spectra 2000;23:357–92. [21] Pozzuoli C, Bartoli G, Peil U, Clobes M. Serviceability wind risk assessment of tall buildings including aeroelastic effects. J Wind Eng Ind Aerodynam 2013;123:325–38. [22] Ciampoli M, Petrini F. Performance-based Aeolian risk assessment and reduction for tall buildings. Probab Eng Mech 2012;28:75–84. [23] Van De Lindt J, Taggart M. Fragility analysis methodology for performance- based analysis of wood-frame buildings for flood. Science 2009;10(3):113–23. [24] Lange D, Devaney S, Usmani A. An application of the PEER performance based earthquake engineering framework to structures in fire. Eng Struct 2014;66:100–15. [25] Herbin AH, Barbato M. Fragility curves for building envelope components subject to windborne debris impact. J Wind Eng Ind Aerodyn 2012;107– 108:285–98. [26] Smith MA, Caracoglia L. A Monte Carlo based method for the dynamic ‘‘fragility analysis’’ of tall buildings under turbulent wind loading. Eng Struct 2011;33:410–20. [27] Taflanidis A, Beck JL. Stochastic subset optimization for reliability optimization and sensitivity analysis in system design. Comput Struct 2009;87(5– 6):318–31. [28] Lee KH, Rosowsky D. Fragility analysis of woodframe buildings considering combined snow and earthquake loading. Struct Saf 2006;28:289–303. [29] Mills CA. The design of concrete structures to resist explosions and weapon effects. In: Proceedings of the 1st international conference for hazard protection, Edinburgh; 1987. [30] Held M. Blast waves in free air. Propellants, Explos, Pyrotech 1983;8(1):1–7. [31] US Army Corps of Engineers. Methodology manual for the single-degree-of- freedom blast effects design spreadsheets (SBEDS); 2008. [32] Gantes CJ, Pnevmatikos NG. Elastic–plastic response spectra for exponential blast loading. Int J Impact Eng 2004;30:323–43. [33] Davidson JS, Fisher JW, Hammons MI, Porter JR, Dinan J. Failure mechanisms of polymer-reinforced concrete masonry walls subjected to blast. J Struct Eng 2005;131(8):1194–205. [34] Moradi LG, Davidson JS, Dinan RJ. Resistance of membrane retrofit concrete masonry walls to lateral pressure. J Perform Constr Facil 2008;22(3):131–42. [35] Naito CJ, Wheaton KP. Blast assessment of load-bearing reinforced concrete shear walls. Pract Periodical Struct Des Constr 2006;11(2):112–21. [36] Naito CJ, Dinan R, Bewick B. Use of precast concrete walls for blast protection of steel stud construction. J Perform Constr Facil 2011;25(5):454–63. [37] Jordan JB, Naito CJ. Calculating fragment impact velocity from penetration data. Int J Impact Eng 2010;37:530–6. [38] Naito C, Hoemann J, Beacraft M, Bewick B. Performance and characterization of shear ties for use in insulated precast concrete sandwich wall panels. J Struct Eng 2012;138(1):1–11. [39] Precast Prestressed Concrete Institute (PCI). PCI design handbook, 7th ed.; 2010. [40] Enright MP, Frangopol DM. Probabilistic analysis of resistance degradation of reinforced concrete bridge beams under corrosion. Eng Struct 1998;20(11):960–71. [41] Eurocode 2 – design of concrete structures – Part 1–1: general rules and rules for buildings. European Committee for Standardization; 2005. [42] Australian standard 3600 concrete structures. North Sydney: Standards Association of Australia; 1988. [43] American Concrete Institute (ACI): Building Code Requirements for Reinforced Concrete; 2011. [44] Cowper GR, Symonds PS. Strain hardening and strain rate effects in the impact loading of cantilever beams. Applied mathematics report no. 28. Providence, Rhode Island, USA: Brown University; 1957. [45] Krauthammer T, Astarlioglu S, Blasko J, Soh TB, Ng PH. Pressure–impulse diagrams for the behavior assessment of structural components. Int J Impact Eng 2008;35:771–83. [46] The Federal Emergency Management Agency (FEMA). Risk assessment a how- to guide to mitigate potential terrorist attacks against buildings, providing protection to people and buildings. FEMA 452; 2005. [47] Olmati P, Petrini F, Bontempi F. Numerical analyses for the structural assessment of steel buildings under explosions. Struct Eng Mech 2013;45(6):803–19. [48] Olmati P, Gkoumas K, Brando F, Cao L. Consequence-based robustness assessment of a steel truss bridge. Steel Compos Struct 2013;14(4):379–95. [49] JalaliLarijani R, Celikag M, Aghayan I, Kazemi M. Progressive collapse analysis of two existing steel buildings using a linear static procedure. Struct Eng Mech 2013;48(2):207–20. [50] Sgambi L, Gkoumas K, Bontempi F. Genetic algorithms for the dependability assurance in the design of a long-span suspension bridge. Comput-Aided Civ Infrastruct Eng 2012;27(9):655–75. P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 9 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014),