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- 1. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load Pierluigi Olmati a,⇑ , Francesco Petrini b , Konstantinos Gkoumas b a Faculty of Engineering and Physical Sciences, University of Surrey, GU2 7XH Guildford, UK b Sapienza University of Rome, Department of Structural and Geotechnical Engineering, Via Eudossiana 18, 00184 Rome, Italy a r t i c l e i n f o Article history: Available online xxxx Keywords: Performance-based blast engineering Fragility analysis Concrete cladding wall panels Cladding system a b s t r a c t This paper presents a probabilistic method to support the design of cladding wall systems subjected to blast loads. The proposed method is based on the broadly adopted fragility analysis method (conditional approach), widely used in Performance-Based Design procedures for structures subjected to natural hazards like earthquake and wind. The cladding wall system under investigation is composed by non-load bearing precast concrete wall panels. From the blast design point of view, these wall panels must protect people and equipment from external detonations. The aim of this research is to compute both the fragility curves and the limit states exceedance probability of a typical precast concrete cladding wall panel considering the detonations of vehicle borne improvised explosive devices. Moreover, the limit states exceedance probability of the cladding wall panel is estimated by Monte Carlo simulation (unconditional approach) in order to validate the proposed fragility curves. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Designing structures to withstand blast loads is common prac- tice for many government and commercial buildings. Generally in the design practice, a set of design scenarios are selected and the integrity of the blast-resistant structural members and of the pro- tective elements is assessed by using non-linear dynamic analyses with an equivalent single degree of freedom (SDOF) method. In such a way (adopting a deterministic approach for the hazard char- acterization), the probability of exceeding a particular limit state is not evaluated. In addition to the above, in designing a structural component subjected to blast loads, the current state of practice is to assume that the capacity is deterministic. The adoption of deterministic values for the demand is princi- pally due to a lack of knowledge of the hazard probability density function. This is common for Low Probability–High Consequence (LPHC) events [1]. As a partial consideration of the uncertainty affecting the blast load, simpliﬁed approaches are usually adopted. For example, in [2] the use of a magniﬁcation coefﬁcient of 20% is applied on the assumed amount of explosive. However, this is limited to explosive storage facilities. In an antiterrorism design, the amount of explosive is characterized by elevated uncertainty depending on both technical and socioeconomic factors. These uncertainties lead the engineering community toward the implementation of probabilistic methods [3], something that is now crucial for both academics and practitioners. With speciﬁc reference to blast-resistant structures, some authors started carrying out investigations about the use of probabilistic methods for the assessment and design of structural components and structural systems. In [4], results of a parametric investigation on the reliability of reinforced concrete slabs under blast loading are presented, in order to establish appropriate probabilistic distributions of the resistant parameters. In [5], the extension of probabilistic approaches from the performance-based earthquake engineering to the blast design problems are provided, also by suggesting appropriate variables for the intensity measures IMs, the damage measures DMs, and the response parameters deﬁ- nition. In [6], Monte Carlo simulations are performed in order to estimate the failure probability of windows subjected to a blast load made by a vehicle bomb. In [7], the fragility curves are presented for two kinds of glazing systems. In [8], the design in a probabilistic way of a sacriﬁcial cladding for a blast wall is described, deployed to pro- tect vulnerable objects against an accidental explosion. Due to the above considerations, the deﬁnition of appropriate frameworks for the probabilistic design of blast resistant structures is an important objective for the engineering scientiﬁc community. To this regard, during the last decade Performance-Based Design (PBD) has been recognized as a powerful methodology for verifying the achievement of design performance objectives of structural systems during their design life [9]. Probabilistic approaches have been extensively implemented in the state of the art methods for http://dx.doi.org/10.1016/j.engstruct.2014.06.004 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: pierluigi.olmati@gmail.com (P. Olmati). Engineering Structures xxx (2014) xxx–xxx Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 2. the PBD of structures under different kind of hazards such as earth- quake [10,11], wind [12–14], and hurricanes [15]. The last case is an example of multi-hazard situation that is expected to be one of the main directions of PBD approaches in the future [16]. In the PBD context, a powerful tool is represented by the fragility analysis (see for example [17–19]). As it is well known, the structural fragility is expressed as the cumulative probability distribution of attaining a certain Damage Measure (DM) condi- tional to the Intensity Measure (IM) of the hazard. The efﬁciency of the fragility approach is strictly related to the appropriateness of the IM in terms of ‘‘sufﬁciency’’ and ‘‘efﬁciency’’, meaning that the IM must accurately describe all pertinent hazard sources (see [20]). Despite the above, a rigorous approach that is consistent with the well-established PBD frameworks adopted in presence of other hazards has not been deﬁned for blast resistant structures. This paper is an effort in that direction with a speciﬁc focus on the fra- gility analysis. The fragility analysis is applied in order to compute the probability of exceeding a limit state (‘‘probability of exceed- ance’’) of a precast wall panel subjected to blast loads (in particular far-ﬁeld surface-blast loads [2]) in a PBD perspective. As case study, a precast concrete cladding wall panel with the dimensions of 3500 mm in length and 1500 mm in width, with a cross sectional thickness of 150 mm is considered. The panel is subjected to a blast load generated by a vehicle borne improvised explosive device. The wall under investigation is a non-load bear- ing precast concrete wall panel used as exterior cladding for build- ings. Typically, the length and the width of these walls are subject to speciﬁc architecture requirements while their thickness is approximately 15 cm. The steel reinforcements are generally placed in the middle of the cross section. This kind of wall panels should be designed in order to protect occupants and equipment from external detonations. Non-linear dynamic analyses are carried out by the well-estab- lished method of the equivalent non-linear SDOF system, where the precast concrete wall panel is modeled by an equivalent non- linear SDOF on the basis of energetic considerations. Furthermore, both the fragility curves and the probabilities of exceedance are computed using Monte Carlo simulations. The fragility curves are evaluated for the case-study wall panel for each deﬁned limit state called here Component Damage Level (CDL). Then the fragility curves are used in order to estimate the probability of exceedance of the cladding wall panel subjected to blast load scenarios (vehicle borne improvised explosive devices). Finally, the probability of exceedance of the wall panel subjected to the same scenarios is estimated by the unconditional approach (based on a single Monte Carlo simulation) in order to validate the obtained results (see [4,11]). In addition to the fragility analysis of the examined structural member typology (an innovative aspect of the paper), some preli- minary indications on the selection of a sufﬁcient and efﬁcient IM for PBD of blast-resistant structures are provided. 2. Fragility analysis As previously stated, the fragility of a structure under the action of a certain hazard is expressed as the cumulative probability dis- tribution of a certain DM conditional to the IM of the considered hazard. Probabilistic PBD approaches identify the generic struc- tural performance by means of acceptable occurrence frequencies for some threshold values (representing structural limit states) of an appropriate DM during a reference period of time [21,22]. The determination of such occurrences is affected by large amounts of uncertainty. The fragility approach allows the designer to express in a synthetic and efﬁcient manner this uncertainty by making use of conditional probability relations and by highlighting the dependences of these occurrences from the IM. In earthquake engineering, the fragility approach has been mostly developed during last twenty years and applied for PBD purposes. The fragility curves have been developed also for struc- tures subjected to ﬂood [23], ﬁre [24], and windborne debris in hurricane prone regions [25]. The fragility curves are nowadays extensively used for the state of practice methods of structural risk evaluation for structures under natural hazards. Among other techniques proposed for the evaluation of the fra- gility curves, Monte Carlo analysis is extensively used [26]. Two main issues need to be addressed primarily in order to develop fragility curves under a single hazard by Monte Carlo approaches. These are due to the fact that: (i) the computational effort required in order to obtain the desired level of approxima- tion is often challenging [27]; and, (ii) the individuation of an efﬁ- cient and sufﬁcient scalar IM for fragility representation is needed. The last point is essential since, in case of a vectorial IM, the structural fragility needs to be represented in terms of surfaces, something that is required for example in the case of performance analysis under multiple hazards (see for example [28]). In this paper, this issue is discussed focusing the attention on the criticism of choosing a scalar IM. As a ﬁrst step, the uncertainties characterizing blast-engineer- ing problems need to be properly individuated and addressed (Fig. 1). These uncertainties can be divided into three main groups: hazard uncertainties (e.g. explosive, stand-off distance); structure uncertainties (e.g. stiffness, dimensions, damping, material characteristics, damping, etc.); interaction mechanism uncertainties (e.g. the reﬂected pres- sure, pressure duration, etc.); This classiﬁcation of the uncertainties in three groups (load, structure, interaction mechanisms) is generally valid for many engineering ﬁelds. The IM in general should be chosen among the ﬁrst group of uncertain parameters or as a combination of those parameters, while the entity of the blast action given a certain IM is determined by the parameters characterizing the interaction between the IM and the structural parameters. In probabilistic terms, hazard and structural parameters can be characterized as unconditional with respect to parameters belonging to one of the other two groups, while parameters representing the interaction mechanisms must be usually characterized in conditional probabilistic terms with respect to the hazard and the structural parameters [12]. Fig. 1. Uncertaint parameters of vehicle borne improvised explosive device scenarios. 2 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 3. 3. Blast load model The side-on blast pressure Ps0 (MPa), can be estimated by the formula of Mills [29] (Eq. (1)), while the side-on speciﬁc impulse is0 (Pa s), is estimated by the formula of Held [30] (Eq. (2)). Where, Z is the scaled distance (Eq. (3)), W is the explosive weight, and R is the stand-off distance. Ps0 ¼ 1:772 1 Z3 À 0:114 1 Z2 þ 0:108 1 Z ð1Þ is0 ¼ 300 1 2 ﬃﬃﬃﬃﬃﬃ W 3 p ð2Þ Z ¼ R ﬃﬃﬃﬃﬃﬃ W3 p ð3Þ The explosive amount is commonly expressed as an equivalent TNT charge [2] by the Equivalent Factor (EF), which multiplies the weight of the explosive charge utilized. Both Eqs. (1) and (2) are valid for free-air explosions. In this study detonations occurring on a surface (surface explosions) are considered, therefore the energy of the detonation is conﬁned by the ground surface, creating a larger demand than that of the free-air explosion. The surface blast demand is calculated by using the same equations for the free-air explosions but with a charge weight (W) increased by 80% [31]. Then, the reﬂected pressure Pr (MPa) for a normal angle of incidence is computed using Eq. (4) [29]: Pr ¼ 2Ps0 7Patm þ 4Ps0 7Patm þ Ps0 ð4Þ where Patm is the atmospheric pressure (0.1 MPa). For simplicity, the negative pressure phase is neglected from the blast load time history [31]. And the duration of the blast load (td) is computed from the side-on pressure and the speciﬁc impulse, assuming a triangular impulse (Eq. (5)). td ¼ 2is0 Ps0 ð5Þ The variation in the pressure is assumed approximately to follow the Friedlander pulse shape shown in Eq. (6). Further details on the sensitivity of the structural response due to exponential or tri- angular blast loading are investigated in [32]. PðtÞ ¼ Pr 1 À t td e Àbt td ta t td ð6Þ In Eq. (6), ta (s) is the arrival time of the blast load, taken here as zero, and b is the decay coefﬁcient. In this study a value of 1.8 for b is assumed [31]. The clearing effect is conservatively neglected in this study since the cladding wall panel is generally a single part of a large building façade, thus the conditions for clearing of the reﬂected shock wave are in general not satisﬁed. In Fig. 2 the blast load time histories computed for different values of the explosive weight W (kg) and stand-off distance R (m) with the above-men- tioned procedure are shown. The obtained curves are found to be in good agreement with the curves obtained by SBEDS [31]. The blast load is considered uniformly distributed on the cladding wall, which is typical for a scaled distance higher than approximately 2.0 m/kg1/3 [31]. 4. Cladding panel model A precast concrete cladding wall system has some advantages with respect to the traditional non-load bearing masonry claddings. Studies on improving the performances of traditional masonry claddings under blast loads are presented in [33,34], while in [35], the load bearing capacity of load bearing concrete walls subjected to a blast demand is investigated. The ﬁrst advan- tage of precast concrete wall panels versus traditional masonry claddings is the greater resistance of the cladding system to a blast demand (see for example the advantage of using precast concrete wall panels for protecting steel stud constructions [36]). Precast concrete wall panels can be integrated with other mate- rials for improving the resistance to environmental attacks such as acid rains and chlorine ions exposure. In [37] the protection perfor- mance of cellulosic ﬁberboard panels from ballistic attacks is investigated. Furthermore, in [38] the behavior of concrete insu- lated panels is investigated, focusing on the shear ties connecting the two concrete wythes conﬁning the insulation. The cladding panel taken as case study is 3500 mm long by 1500 mm wide, with a cross sectional thickness of 150 mm. The panel is connected along two of its edges to the external frame of a building, and in particular, it is assumed to be simply- supported by the beams of the frame. Length, width and cross sec- tional thickness of the panel are considered as stochastic variables, due to the construction tolerances used in the precast concrete industry (see for example [39]). The assumed mean values and Coefﬁcients of Variations (COVs) are shown in Table 1. The longitu- dinal reinforcement consists of ten reinforcement bars of 10 mm diameter located in the mid-axis of the cross section. The mean value and the COVs of the reinforcement strength are provided in Table 1. The panel is assumed to not have shear reinforcement. 4.1. Concrete The concrete compressive strength fc is considered as a stochas- tic variable, while the Young’s modulus of the concrete Ec and the concrete density q are expressed as functions of fc. The mean value of fc is 28 MPa (4060 psi), with a COV of 0.18, as adopted in [40] for a lognormal probability density function (see Table 1). The Young’s Fig. 2. Blast loads by the adopted model (broken lines) and the SBEDS model (solid lines). Table 1 Input data. Symbol Description Mean COV Distribution fc Concrete strength 28 MPa 0.18 Lognormal fy Steel strength 495 MPa 0.12 Lognormal L Panel length 3500 mm 0.001 Lognormal H Panel height 150 mm 0.001 Lognormal b Panel width 1500 mm 0.001 Lognormal c Panel cover 75 mm 0.01 Lognormal W Explosive weight 227 kg 0.3 Lognormal R Stand-off distance 15 m 20 m 25 m 0.05 Lognormal P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 3 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 4. modulus is computed by Eq. (7) [41] while the concrete density is computed by Eq. (8) [42,43]. Both Ec and fc are expressed in MPa while q is expressed in kg/m3 . Ec ¼ 22; 000 fc 10 0:3 ð7Þ q ¼ Ec 0:043ðf0:5 c Þ 1 1:5 ð8Þ The compressive strength increment of the concrete due to the high strain rate is accounted for in this study. This increase is taken into account by means of the Dynamic Increase Factor (DIF), a multipli- cative coefﬁcient of the concrete static compressive resistance. Since the compressive strength enhancement of the concrete varies marginally for a ductile ﬂexural response over the range of the strain velocity, the DIF can be assumed constant and equal to 1.19. This hypothesis is in accordance with the compressive strength enhancement proposed in [31], and it leads to an increase of the computational efﬁciency by avoiding cyclic iterations in the algo- rithm of the SDOF equation solver. However, cyclic iterations are necessary for computing the strength increase of the reinforcement. This increase is sensitive to the ductile ﬂexural response, as opposed to the compressive strength increase of the concrete. 4.2. Reinforcing steel A grade 450 MPa steel is used. Due to material standard requirements the average yield strength is higher than the speci- ﬁed yield. For estimating the mean value of the yield strength, a static strength increase factor equal to 1.1 is adopted as indicated in [31]. A COV of 0.12 is used as proposed in [40] for a lognormal probability density function (see Table 1). Young’s modulus is taken as a deterministic value and assumed to be equal to 210 GPa. The steel strength increase due to the strain velocity is taken into account by the Cowper and Symonds model [44]. Thus, the DIF is provided by Eq. (9): DIF ¼ 1 þ d=dt q 1 n ð9Þ where de/dt is the strain-rate demand of the reinforcement, q is taken equal to 500 sÀ1 and n is taken equal to 6. Both q and n are estimated by ﬁtting the strength increase versus the strain rate in [31]. By solving the SDOF equation of motion, the DIF is iteratively updated until a convergence threshold is reached. The strain rate of the steel reinforcement (de/dt) in Eq. (9) is calculated approximately by Eq. (10). de dt ¼ dS dt L 8 d 2EcJc ð10Þ where L is the length of the cladding panel, Ec is the Young’s mod- ulus of the concrete, Ic is the moment of inertia of the cracked cross section, d is the distance from the extreme compression ﬁber of the cross-section to the centroid of the tensile reinforcement, and dS/dt is the rate of the resistance force developed by the panel (S) when subjected to the demand. Eq. (10) is valid for simply supported ele- ments when the response is governed by the ﬂexural behavior with- out shear failure. 5. Equivalent SDOF mechanical model of the panel In order to model a structural element subjected to a blast load with an equivalent SDOF, the latter is deﬁned as a system that has the same energy of the original structural element (in terms of work energy, strain energy, and kinetic energy) when the structural element, if subjected to a blast load, deﬂects in a given deformed shape. The displacement ﬁeld of the component can be expressed as u(x,t) = U(x)y(t), where U(x) is the assumed deformed shape of the component under the blast load and y is the maximum displacement of the component. Furthermore, the displacement of the component is obtained by the SDOF equation by assuming a ﬂexural deformed shape by: KLMM€yðtÞ þ C _yðtÞ þ SðyðtÞÞ ¼ FðtÞ ð11Þ where y(t) is the displacement of the SDOF system and M is the total mass of the component, S(y(t)) is the resistance of the component as a function of the displacement expressed in unit force (see Fig. 3), F(t) is the blast pressure multiplied by the impacted area (A) expressed in force units, C is the damping (the percentage of the critical damping is assumed to be 1% in all the analyses), KLM is the load-mass transformation factor equal to the ratio of KM and KL (the mass transformation factor and the load transformation fac- tor respectively). The last two are evaluated by equating the energy of the two systems (in terms of work energy and kinetic energy respectively). KL ¼ RL 0 pðxÞUðxÞdx RL 0 pðxÞdx KM ¼ RL 0 mðxÞU2 ðxÞdx RL 0 mðxÞdx ð12Þ Referring to Eq. (12) and Fig. 3, p(x) is the blast load shape on the component, m(x) is the distributed mass, and r is the resistance of the element in terms of pressure. The load-mass transformation fac- tor KLM is different at each deformation stage of the component response; for a bilinear resistance function two values of the KLM can be deﬁned: one for the elastic response and one for the plastic response. More details on the equivalent SDOF method are provided in [2,31]. In order to obtain the bilinear resistance function of the simply supported concrete cladding wall, it is necessary to compute the resisting moment (Mr) in the mid-span of the panel. The yielding point of the resistance function (Sy) is obtained by Eq. (13) where the loaded surface (A) is equal to the cladding wall length (L) times the cladding wall width (b) and ry is the pressure resistance function. Sy ¼ 8Mr L ¼ ryA ð13Þ The resisting moment is evaluated by Eq. (14) [31]: Mr ¼ Asfdy d À Asfdy 1:7bfdc ð14Þ where As is the reinforcements area, fdy is the dynamic yield strength of the reinforcing steel, fdc is the dynamic compressive strength of the concrete, b width of rectangular section. It is also necessary to evaluate the yielding displacement of the resistance function. For a simply supported component the yielding displacement (de) is given by Eq. (15). de ¼ 5 384 8MrL2 EcJ ð15Þ J ¼ 0:5ðJg þ JcÞ ð16Þ where J is the moment of inertia of the cross section as evaluated by Eq. (16), while Jc and Jg are computed by Eqs. (17) and (18) respectively. Jc ¼ Gbd 3 ð17Þ Jg ¼ bH 3 12 ð18Þ 4 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 5. The coefﬁcient G in Eq. (19) is evaluated starting by the design chart provided in [31]. In this study an analytical formula (see Eq. (19)) is determined by ﬁtting the curves of the original chart [31]. G ¼ ð3320:3p3 À 181:98p2 þ 5:8624pÞ n 7 0:7 ð19Þ where p is the percentage of reinforcement in the cross section of the panel evaluated by neglecting the reinforcements cover, and n is the ratio between the steel Young’s modulus and the Ec. Eq. (19) is valid for single reinforced cross-sections only. Since Sy and de are evaluated, the resistance function of the simply supported cladding wall can be deﬁned. Finally, the central difference method is then used to solve Eq. (11) which presents three non-linarites: one due to the bilinear shape of the resistance function, one due to the load mass transformation factors, and the last due to the dynamic strength enhancement of the reinforcing steel (which affects the resistance function). 6. Performance deﬁnition For structural components subjected to blast loads in a ﬂexural response regime, generally two response parameters are of inter- est: the support rotation angle (h) and the ductility ratio (l). These parameters are deﬁned in Eqs. (20) and (21): h ¼ arctg 2dmax L ð20Þ l ¼ dmax de ð21Þ where dmax is the maximum displacement of the component. A structural component subjected to a blast load is generally expected to yield (ductility greater than 1), as it is economically impractical to design a component to remain in elastic range. While other signiﬁcant response parameters can be deﬁned (for example [4] considers the strain on reinforcements), this study focuses on the above deﬁned response parameters, since these are usually adopted for antiterrorism design [31]. In a performance-based blast design prospective, ﬁve Compo- nent Damage Levels (CDLs) [31] are considered: Blowout (BO), Hazardous Failure (HF), Heavy Damage (HD), Moderate Damage (MD), and Superﬁcial Damage (SD). Following the [31], the above mentioned levels are deﬁned as follows: Blowout (BO): the component is overwhelmed by the blast load causing debris with signiﬁcant velocities. Hazardous Failure (HF): the component has failed, and debris velocities range from insigniﬁcant to very signiﬁcant. Heavy Damage (HD): the component has not failed, but it has sig- niﬁcant permanent deﬂections causing it to be un-repairable. Moderate Damage (MD): the component has some permanent deﬂection. It is generally repairable, if necessary, although replace- ment may be more economical and aesthetic. Superﬁcial Damage (SD): the component has no visible permanent damage. The thresholds corresponding to these CDLs are deﬁned in terms of the response parameters h and l. For a non-structural concrete cladding wall without shear reinforcement, neglecting tension membrane effect, the CDL thresholds are those reported in Table 2. The Fragility Curves are computed in the following for each of the mentioned CDLs. 7. Algorithm for the computation of the fragility curves As described in previous sections, the blast load on the panel is a function of the peak pressure and of the impulse density (Eqs. (4) and (2) respectively); the pressure is function of Z only (Eqs. (1) and (4)), while the impulse density depends on both the Z and the W (Eq. (2)). Consequently, two detonations with the same Z can have different impulse density, depending on the amount of explosive. Thus, the two explosions have the same peak pressure but different duration. Summarizing, since the blast load depends on both the Z and the W, the choice of the IM for computing the fragility curves is a crucial issue. In this study the Z is taken as the IM. Some aspects related to this choice are discussed in the next section. Note that for higher values of the Z the cladding wall has a lower structural response than for lower values. The Fragility Curves (FCs) are developed for each CDL. The algo- rithm implemented in MATLABÒ for the fragility curves evaluation is shown in Fig. 4. With reference to the same ﬁgure, ‘‘i’’ indicates the ith point of the fragility curve, ‘‘j’’ indicates the ‘‘j’’th CDL, and ‘‘k’’ indicates the kth stand-off distance (R) for which the fragility curve is computed. ‘‘N’’ is the maximum value for ‘‘i’’ and therefore the total number of points forming the fragility curve. ‘‘M’’ is the maximum value for ‘‘j’’ and therefore the total number of the CDLs. Finally ‘‘L’’ is the maximum value of ‘‘k’’ therefore the total number Fig. 3. Resistance versus displacement relation of a component. Table 2 Component damage levels and the associated thresholds in terms of response parameters [31]. Component damage levels h (degree) l (–) Blowout 10° None Hazardous failure 610° None Heavy damage 65° None Moderate damage 62° None Superﬁcial damage None 1 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 5 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 6. of the stand-off distances for which the fragility curve related to the jth CDL is computed. The name ‘‘FC-CDL (j,k)’’ indicates the fragility curve computed for the kth R, the jth CDL, by varying the W (and consequently the Z). The ith point of the fragility curve (named FC-CDL (i,j,k)) is com- puted by considering the blast load at the kth R and the ith W. The minimum and maximum amount of W should be enough for computing the values of the FC-CDL (j,k) ranging from 0 to 1. The FC-CDL (i,j,k) is obtained by a Monte Carlo simulation and the complete (cyclic) procedure of Fig. 3 is hereby described. ﬁrst a kth R is selected; then the jth CDL is selected; consequently the ‘‘i’’ index is increased by solving the previ- ously introduced equations for each ‘‘i’’ in order to evaluate the ith points of the FC-CDL (j,k) until tracking the complete FC-CDL (j,k); after that, a new jth CDL is considered with the same value of ‘‘k’’. When j = M a different R is selected and the previous two described cycles are repeated until k = L; ﬁnally, the piecewise curves obtained point by point with the above steps they are interpolated by a lognormal cumulative function, see Fig. 6. As said, the fragility curves describe the conditional probability of exceedance (P(X x0|Z)) of the response parameter X (chosen case-by-case as the most critical between the values of h and l, see Table 2) with respect to the threshold x0 (identifying the CDL). As expected, for a constant number of samples at each ith point, the COV of P(X x0|Z) increases with the decreasing of P(X x0|Z). In order to obtain an acceptable COV, the number of samples adopted in the analysis is increased with the decreasing of P(X x0|Z); this means that the number of samples increase with increasing Z. In this work, an exponential law has been set for this increasing trend. In Fig. 5 the number of the samples and the rela- tive COVs are shown in function of P(X x0|Z) for the fragility curve related to the heavy damage CDL and for R equal to 20 m. 8. Sufﬁciency of the intensity measure and results of the fragility analysis For better understanding the sufﬁciency of the adopted inten- sity measure (the scaled distance Z), some considerations can be made with reference to the pressure–impulse diagrams [45] related to the case study panel. For this purpose, reference is made to the mean values of both materials and geometrical parameters (see Table 1), and the DIFs for the concrete and steel are taken as constant and equal to 1.19 and 1.20 respectively. The pressure–impulse diagrams referred to different values of h are shown in Fig. 7. Generally, three regions can be individuated in the pressure–impulse diagrams, each related with a different regime of structural response subjected to a load time history. These are deﬁed as: impulsive (I), dynamic (D), and pressure (P) region, depending on the characteristics of the load time history with respect to the dynamical proprieties of the structure [2]. Two blast loads are taken into account. These loads can be cho- sen in such a way that they are characterized by the same IM (and consequently by the same peak pressure) but having different W and R. As a matter of fact, the two blast loads are consistent with two different demands on the pressure–impulse diagrams, having the same peak pressure but different values of the impulse density. As it can be observed in Fig. 7, the difference between the struc- tural response of the panel subjected to the two above mentioned blast demands (again, characterized by the same pressure peak but by different impulse densities) depends on the position of these demands in the pressure–impulse diagram. Thus, if these blast demands are located in the impulsive region (I), a certain value for this difference will be observed, while if blast demands are located in the dynamic (D) or pressure (P) regions, then this differ- ence will be lower than in the previous case. Considering the above, it can be concluded that the sufﬁciency of the chosen IM is greater in the D and P regions than in the I region. In the I region a fragility surface made by considering both R and W as independent elements of a vectorial IM would be more appropriate. For taking account this approximation, as explained in the pre- vious section, the fragility curves are computed for different values of the R (R ¼ Z ﬃﬃﬃﬃﬃﬃ W3 p ). In what follows, when the fragility curves are used for estimating the failure probability of a component damage Fig. 4. Fragility curve computation ﬂowchart. 0.00 0.02 0.04 0.06 0.08 0.10 0 20000 40000 60000 80000 100000 0.1 0.9 3.3 9.0 22.4 40.4 59.5 77.9 90.1 96.6 98.8 100.0 C.O.V. N°ofsamples P(Xx0|Z) [%] N° of samples C.O.V. Fig. 5. N° of samples and COV for the fragility curve representative of the HF component damage level, for R equal to 20 m. 6 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 7. level, the speciﬁc fragility curve corresponding to the mean value of R is used for this purpose. This increases considerably the sufﬁ- ciency of the chosen IM. 8.1. Results This section presents the results regarding: (i) the fragility curves of the case study cladding panel, and, (ii) the probability of the limit state exceedance of such cladding panel estimated by both the conditional and unconditional approaches. The last point is important in order to validate the computed fragility curves by a comparison of the exceedances obtained by the two approaches. In Fig. 8 the computed fragility curves are shown for different values of R. Focusing on the considered CDLs, from Fig. 8 can be observed that the fragility curves of the SD level have a different 0 20 40 60 80 100 3.7 3.9 4.1 4.3 4.5 P(Xx0|Z)[%] Z Interpolated FC Numerical FC Fig. 6. Numerical and lognormal interpolated fragility curves. Fig. 7. Pressure–impulse diagrams. 0 20 40 60 80 100 2.4 2.6 2.8 3.0 3.2 3.4 P(Xx0|Z)[%] Z Hazardous Failure 0 20 40 60 80 100 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Heavy Damage P(Xx0|Z)[%] Z 0 20 40 60 80 100 3.0 3.5 4.0 4.5 5.0 P(Xx0|Z)[%] Z Moderate Damage 0 20 40 60 80 100 5 6 7 8 9 10 11 P(Xx0|Z)[%] Z Superficial Damage Fig. 8. From top left clockwise, fragility curves for the HF, HD, SD, MD component damage levels. Street Level 2 Level 3 Level 1 Target Fig. 9. Lines of defence. P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx 7 Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
- 8. slope compared to that of the other three CDLs (HF, HD, and MD). It should be noted that the SD level is based on the ductility (l) of the component while HF, HD, and MD levels are based on the support rotation (h). The SD level for a concrete cladding panel prescribes the elastic response of the component, and for the case study panel it appears to be more sensitive to the considered uncertainties compared to the HF, HD, and MD levels. By varying the number of samples the maximum obtained COV for the lower probability of failure (close to zero), is about 9% (Fig. 5). This value is consid- ered acceptable for the speciﬁc case, and it is consistent with other studies on blast applications (see [7]). For computing the limit state probability of exceedance by the conditional approach it is necessary to develop a hazard analysis for the stochastic characterization of the blast scenario and to solve Eq. (23). In this study, a vehicle borne improvised explosive device is considered. The amount of explosive (W) in the vehicle depends, among else, on the security measures in place. These security mea- sures can be structured in different lines (see Fig. 9) and for each line of security a different mean value of W is expected. The expected value of W decrease with the decreasing of R from the target, since the line of security system reduces progressively the severity of the possible attacks. In the example of Fig. 9, level 1 prevents trucks entering the tar- get zone, so no truck bomb should be expected. Level 2 in Fig. 9 (for example a fence barrier) prevents vehicles entering. Finally, Level 3 prevents pedestrians approaching the target. With this in mind, in the speciﬁc application a scenario con- cerning a truck bomb (with about 4000–27,000 kg of TNT or equiv- alent) is unreasonable (e.g. by assuming that the intelligence service is able to prevent this threat). Therefore, a vehicle bomb (with about 27–454 kg of TNT or equivalent) is considered. The mean amount of TNT or equivalent in the vehicle is assumed equal to 227 kg with a COV equal to 0.3 (see Table 1); this assumption is in line with [46]. A set of stand-off distances are considered (15, 20 and 25 m) each with a coefﬁcient of variation equal to 0.05, assum- ing that the vehicle could impact a fence barrier but move no further. The conditional probability of exceedance of the CDL (P(X x0)) is evaluated by Eq. (23). As previously stated, X is the most critical between the response parameters h and l, assumed here as uncor- related. Consequently P(X x0) is the union of the two failure prob- abilities evaluated by considering separately the two response parameters characterizing the component damage level (see Table 2 and Eq. (22)). The probability density function of the Z (p(Z)) is computed by ﬁtting the samples of both W and R with a lognormal distribution. As mentioned above, the fragility curve (P(X x0|Z)) used for evaluating Eq. (23) is the one corresponding to the mean value of the R (Table 1). PðX x0Þ ¼ PðH h0Þ [ PðM l0Þ ð22Þ PðX x0Þ ¼ Z þ1 À1 PðX x0jZÞpðZÞdz ﬃ X1 i¼0 PðX x0jZÞipðZÞiDZi ð23Þ The obtained results are shown in Table 3. The ﬁrst column reports the CDLs, while in the second and third columns report the P(X x0) for each blast scenario obtained by Eq. (23) and by the uncondi- tional approach respectively, the last one considered for comparison purposes as ‘‘exact value’’ of the exceedance probability. From these results the maximum percentage difference between the P(X x0) computed by the conditional and uncondi- tional approaches is 11%. Further studies are necessary to conﬁrm whether this percentage difference is acceptable or not. However, it is also necessary to consider that the W in the vehi- cle has an elevated dispersion, something that ampliﬁes this differ- ence due to the dependence of the impulse density to both R and W. Thus, the difference between the P(X x0) computed by the con- ditional and unconditional approaches increases with the increase in the difference between the R with which the fragility curve is computed (mean value of R) and the R of the Monte Carlo samples of the unconditional approach. 9. Conclusions The probabilistic analysis of a precast concrete cladding wall panel subjected to blast load has been presented. The blast load model has been adopted on the basis of empirical laws, and both the geometry and mechanical properties of the panel are assumed as stochastic. A mechanical model equivalent to a single degree of freedom has been adopted for describing the motion of the panel under the blast load. The Monte Carlo simulation has been used for computing: (i) the fragility curves of the cladding wall panel subjected to blast load for several limit states (component damage levels), (ii) the probability of exceedance of limit states of the clad- ding wall panel by means of both the unconditional and condi- tional approach for comparison purposes. A discussion about the effectiveness of the intensity measure chosen for the fragility analysis has been presented and analyses have been carried out for different values of the stand-off distance. It is expected that the scaled distance Z adopted in this paper, is a sufﬁcient intensity measure especially for blast demands belong- ing to the dynamic and pressure region of the pressure–impulse diagrams. This study highlights the feasibility and effectiveness of the fra- gility approach in the design of cladding wall panels and, generally, of protective structures. This is one of the fundamental steps nec- essary for developing a fully probabilistic Performance-Based Design for blast resistant structures, something already done for structures subjected to other hazards (e.g. earthquake and wind). One of the main issues related to the completion of a probabilistic performance-based blast engineering procedure consists in deter- mining the hazard function; this issue is mainly due to the fact that an explosion event (e.g. terroristic vehicle bomb attack) is a low probability event, as described in [47–49]. Additional studies could focus on: (i) improving the intensity measure sufﬁciency and efﬁciency, (ii) considering the degradation of the cladding panel during the life-cycle, (iii) improving the mechanical model that describes the response of the panel Table 3 Comparisons of the probability of exceedance using the conditional and unconditional approaches. CDL Mean W = 227 kg, COV = 0.3 lognormal Mean R, COV = 0.05 lognormal Conditional approach (%) Unconditional approach (%) Percentage difference D% R = 20 m SD 100.0 100.0 0.0 MD 96.6 97.5 0.9 HD 55.7 55.5 0.3 HF 13.6 12.1 11.0 R = 25 m SD 100.0 100.0 0.0 MD 74.6 77.3 3.5 HD 14.2 12.6 11.2 HF 1.02 1.02 0.0 R = 15 m SD 100.0 100.0 0.0 MD 97.9 99.9 2.0 HD 93.6 96.9 3.4 HF 67.8 72.6 6.6 8 P. Olmati et al. / Engineering Structures xxx (2014) xxx–xxx Please cite this article in press as: Olmati P et al. Fragility analysis for the Performance-Based Design of cladding wall panels subjected to blast load. Eng Struct (2014), http://dx.doi.org/10.1016/j.engstruct.2014.06.004
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