Adapted Design Of Experiments for Dimension Decomposition Based Meta Model in 2 Structural Reliability Analysis 3

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1 Title: 1 Adapted Design Of Experiments for Dimension Decomposition Based Meta Model in 2 Structural Reliability Analysis 3 4 Authors & Affiliations: 5 Mehralah RakhshaniMehr 1*, Assistant Professor, 6 1 University of AL-Zahra 7 m.rakhshianimehr@alzahra.ac.ir 8 POB: Hamed Mirkamali 2, M.Sc. of Earthquake Engineering, 11 2 Shiraz University of Technology 12 hamedmirkamali@gmail.com 13 Tel= Mohsen Rashki 3, Assistant Professor, 16 3 University of Sistan and Baluchestan 17 Mrashki@eng.usb.ac.ir 18 Tel= Abdolhamid Bahrpeyma 4, Assistant Professor, 21 2 University of Sistan and Baluchestan Tel=

2 39 Abstract 40 Reliability analysis of structures is often problematic for the structures with nonlinear and 41 complex limit state functions (LSF). For these cases, simulation methods often provide 42 accurate failure probability, but with high number of structure s LSF analysis. This paper 43 presents an efficient combination of Monte Carlo Simulation (MCS) method and Univariate 44 Dimension Reduction (UDR) based Meta-model to approximate the failure probability of 45 structures with few LSFs evaluation. For this purpose, the design of experiment used in the 46 Meta model is adapted such that the expected failure samples in MCS being approximated 47 with higher accuracy. Several numerical and engineering reliability problems are solved by 48 the proposed approach and the results are verified by MCS. Results show that the proposed 49 approach highly reduces the required number of structural analysis to provide proper results. 50 Keywords: Structural Reliability, Simulation Methods, Univariate Dimension Reduction, 51 Design of Experiment, Monte Carlo simulation Introduction 53 In structural reliability analysis, the failure probability P f is defined as [1]: 54 f G( X) x P f X dx (1) where f X (X) is the joint probability density function of the vector of basic random variables 55 X=[x 1,x 2,,x n ] T, which represents uncertain quantities such as material properties, loads, 56 boundary conditions and geometry. In Eq. (1), G(X) is the limit state function (LSF) in which 57 G(X)>0 represents the safety domain and G(X) <0 represents the failure domain. However, 58 the failure probability of a given problem by means of Eq. (1) is not a straight approach 59 because the joint probability density function F X (X) is not always available. In some cases, 60 Eq. (1) cannot also be integrated analytically even if the F X (X) is available, especially for the 61 2

3 complex structures with low failure probabilities and implicit LSFs. Therefore, in order to 62 avoid such calculation, various techniques have been proposed such as: a) approximation 63 methods (i.e. First Order Reliability Method (FORM), the Second Order Reliability Method 64 (SORM)) and b) simulation methods [2-5]. FORM and SORM are accurate for reliability 65 problems with linear and moderate LSFs, but they are inaccurate for high nonlinear LSFs and 66 difficult to solve when the actual implicit LSF usually cannot be easily expressed explicitly. 67 Besides, in some cases, FORM and SORM may suffer convergence problems [6]. Hence, 68 when an accurate reliability evaluation is required, simulation methods are often employed Monte Carlo Simulation 70 Monte Carlo Simulation (MCS) is considered as a most efficient and accurate simulation 71 method and is commonly used for the evaluation of the probability of failure for structures, 72 either for comparison with other methods or as a standalone reliability analysis tool [7, 8]. 73 This method involves sampling the design space based on the mean, variance and PDF values 74 of random variables. From a mathematical point of view, MCS allows to estimate the 75 expected value of a quantity of interest more specifically, suppose the goal is to evaluate 76 f hx ( ), that is, an expectation of a function h: x with respect to the Probability 77 Density Function (PDF) [9]: 78 h ( X ) f x( X ) dx h f ( X ) (2) X The idea behind MCS is a straight forward application of the law of large numbers that states 79 if X x...x 1, x2, n is independent and identically distributed from the PDF fx( X ), then the 80 N 1 empirical average hx ( i ) converges to the true value of f hx ( ) when N leads to. 81 N i 1 Therefore, if the number of samples N is large enough, then f hx ( ) and it can be 82 accurately estimated by the corresponding empirical average: 83 3

4 f N 1 h( X ) h( x i) N (3) i 1 The relevance of DMC to the reliability problem (1) follows from a simple observation 84 that the failure probability P f can be written as: 85 N 1 Pf f G ( X ) x X dx I X f X f x X f f hx N ( ) ( ) dx E I ( X ) ( i ), (4) i 1 where X x...x 1, x2, n are independent and identically distributed from the PDF fx( X ), I is 86 f a counting vector with values of zero and unity for samples in the failure and safe regions. As 87 it is seen a vast number of simulations have to be performed in order to achieve great 88 accuracy, especially for low values of failure probability. In an effort to reduce the excessive 89 computation cost of MCS using purely random sampling methodologies, which are 90 considered as the drawback of the method, various variance reduction techniques have been 91 proposed such as importance sampling [10-12], Direct sampling [13], line sampling [14,15], 92 Weighted Average Simulation Method (WASM) [16], Subset simulation [17], polynomial 93 chaos [18] and stochastic perturbation technique [19]. Unfortunately, most of these 94 techniques are not as generally applicable as MCS. For example, Importance Sampling 95 requires detailed information about the failure regions for being useful, and it faces 96 difficulties when applied to high-dimension problems [16, 20] First Order Reliability Method 98 First Order Reliability Method (FORM) is widely used to approximate the failure probability 99 of structures, which becomes a basic reliability analysis approach of reliability-based design 100 codes [21, 22]. In FORM, structural failure probability is estimated based on the reliability 101 index (β) by linearizing limit state function on the failure surface; i.e. P ( ), which 102 corresponds to minimum distance of the origin to the limit state function in the standard 103 normal area [21,23]. Generally, the main goal of FORM is the most probable point (MPP 104 f 4

5 i.e.u * ) search (b= U * ) [24, 25]. Hasofer Lind proposed a general iterative method for 105 computing reliability index [23] which was extended by Rackwitz and Fiessler to include 106 distribution information of random variables [26], which is called the HL RF method. This 107 method involves following steps to estimate the probability of failure [26] based on the HL 108 RF method as: 109 Step 1. Transform random variables in X-space into U-space by the following relation 110 e u x x e (5) x where u is the standard normal variable with the mean and standard deviation equal to zero 111 and one, respectively. and e x e are equivalent mean and standard deviation of the random 112 x e variable x, respectively that for normal random variable, x and e. The equivalent 113 x x x mean and standard deviation of non-normal random variables can be determined by the 114 following equations [26-28]: 115 e x 1 f (x) x 1 F (x) x, (6) e e x xx F x, (7) 1 (x) where F x(x) is cumulative distribution, f x(x) is probability distribution, 1 is inverse 116 standard normal cumulative distribution and is standard normal probability distribution 117 function. 118 Step 2. Find the reliability index. 119 The reliability index search is done based on an iterative process that can be reformulated 120 based on design point (U) as: 121 U k1 k 1k 1, (8) where 122 5

6 k 1 T gu ( k ) T gu ( ), (9) k k 1 T g( U ) g( U ) U k k k T gu ( ) k, (10) where ( ),,..., 1 2 n T g U g u g u g u is gradient vector of the limit state function at 123 the design point U k. 124 Step 3. Calculate the failure probability. 125 The probability of failure based on the FORM can be estimated as P ( ) [21, 29] Meta Models in structural reliability 127 When performance evaluation of a structure is computationally expensive, the number of 128 simulation-based function evaluations required for reliability analysis must be carefully 129 controlled. To that end, researchers have explored the use of Meta models, namely, simpler 130 approximate models calibrated to sample runs of the original simulation. The approximate 131 model or Meta model can replace the original one, thus reducing the computational burden of 132 evaluating numerous [30-35]. Bucher and Bourgund [36] proposed a quadratic polynomial 133 response surface without cross terms. The response surface represents the LSF along the 134 coordinate axes of the space of standard normal random variables. Nguyen et al. [37] 135 proposed an adaptive RSM based on a double weighted regression technique. For the first 136 iteration, a linear response surface is chosen, for the following iterations, a quadratic response 137 surface with cross terms is considered according to complementary points. Kang et al. [38] 138 proposed an efficient RSM applying a moving least squares approximation instead of the 139 traditional least squares approximation. Allaix and Carbone [39] discussed the locations of 140 the experimental points used for evaluating parameters of the response surface. Recently, 141 Dimension Reduction Method (DRM) as an efficient approach is used to reduce the analysis 142 computational costs [40-43]. 143 f 6

7 In order to use the capabilities of MCS and simultaneously reduce the computational efforts, 144 this paper presents a framework that efficiently employs the DRM to evaluate the reliability 145 of structures. The proposed framework is presented after a brief review on DRM Dimension Reduction Method 147 The DRM is a newly developed technique to calculate statistical moments of the output 148 performance function [40-43]. There are several DRMs depending on the level of dimension 149 reduction: (1) Univariate Dimension Reduction Method (UDRM), which is an additive 150 decomposition of N-dimensional performance function into one-dimensional functions; (2) 151 Bivariate Dimension Reduction (BDR), which is an additive decomposition o N-dimensional 152 performance function into at most two-dimensional functions; (3) Multivariate Dimension 153 Reduction (MDR), which is an additive decomposition of N-dimensional performance 154 function into at most S-dimensional functions, where S N. 155 According to UDRM, any N-dimensional performance function h(x) can be additively 156 decomposed into one-dimensional functions as [44]: 157 hˆ X h X N h 1 i1 xi i1 N N h 1 N i1,,,,,, 1,,, (11) where is the mean value of a random variable and N is the number of design variables Proposed framework 159 This study employs the efficiency of the UDR-based Meta modeling in conjunction with the 160 accuracy of MCS to provide a suitable framework for structural reliability analysis. The idea 161 is to concentrate the experiments of the UDR-based Meta model in the region with high 162 failure probability to correctly approximate the performance function value for the samples 163 that are expected to be in failure set. The following steps could be conducted to provide the 164 desirable results: 165 7

8 5. 1. Axial Design Of Experiments (DOE) based on the desired reliability index 166 UDR-based Meta model requires axial DOE to approximate the LSF. Determination of the 167 location of experience samples in the proposed approach is based on the perception of 168 conducting the MCS sampling and excluding part of safe area. According to this approach 169 and Figure 1, which is presented in standard normal space (U), the space is divided into two 170 separate regions D 1 and D 2 and it is assumed that D 1 is selected such that no failure occurs in 171 this region. Here D 1 is chosen as the region inside a sphere with radius β [45]. 172 To reduce the computational cost, the location of DOEs could be considered such that the 173 corpus of experience samples in each axis being condensed in boundary of D 1 and D 2 regions 174 and most probable of failure regions. These samples should use for interpolation purposes of 175 DRM for the separated term in each dimension. By employing an anticipated reliability index 176 and mapping the proposed experience samples in the physical space (original design space), 177 the location of experiences for each variable is as follow: X DOE. t arg et.., (12) 2 where and are the mean value and standard deviation of random variable, respectively, 179 is the location coefficient Generation of random sample based on crude MCS 181 By conducting the proposed step for design of experiment, at the next step, the crude MCS 182 should perform to generate random samples to approximate the failure probability. Figure 2, 183 schematically shows the proposed DOEs and the generated samples for a two dimensional 184 problem. According to the proposed approach, the approximated performance function 185 correspond to each sample is achievable by employing the UDR-based Meta model LSF approximation by UDRM based-meta model and reliability evaluation 187 8

9 Eq. (11) should employ in this step to approximate the LSF. The following is the resulting 188 function for a limit state function with two random variables X 1 and X 2 : 189 ˆ,,, h X h X h x h x h (13) in which for the samples produced by MCS, through employing the experiments and a proper 190 interpolation technique, the values of hx, and h, x 1 2 for each dimension are 191 achievable thorough interpolation. Then, the value of performance function for each ample 192 could be approximate using Eq. (13). The failure probability then could be approximate by 193 (4). 194 In this study, the effectiveness of various interpolation techniques is also investigated to 195 suggest a proper technique to use in the proposed framework. These are several one- 196 dimensional interpolations implemented by MATLAB toolbox that are presented in Table The kriging method (method #3) as a new developed approximation method is also used in 198 the approach and compared with common interpolation techniques Kriging Method 200 Kriging Meta model is an interpolation technique based on statistical theory, which consists 201 of a parametric linear regression model and a nonparametric stochastic process. It needs a 202 design of experiments to define its stochastic parameters and then predictions of the response 203 can be completed on any unknown point. Give an initial DOE X x1, x2,..., x N, with n x R ( i 1,2,..., N ) the ith experiment, andg G( x1), G( x2),..., G( x N ),, with G( x i ) R 205 i 0 0 the corresponding response to X [46]. The approximate relationships between any experiment 206 X and the response Gx ( ) can be denoted as: 207 T G( x) F(, x) z( x) f ( x) z( x), (14) 9

10 T where,..., 1 p is a regression coefficient vector. Similar to the polynomial was built 208 T by response surface method, f ( x) f1( x), f2( x),..., f ( ) T p x makes a global simulation in 209 design space. In the ordinary Kriging, F(, x) is a scalar and always taken as F(, x). 210 Hence, the estimated Gxcan ( ) be simplified as: 211 G( x) F(, x) z( x) z( x), (15) here zxis ( ) a stationary Gaussian process [46]. The statistic characteristics can be denoted as: 212 E(z(x)) 0, (16) Var(z(x)), (17) 2 z 2 ( i), ( j ) z ( i, j ), (18) Cov Z x Z x R x x 2 where is the process variance, and z x, x are discretional points from the whole samples X. 213 i j R( x, x ) is the correlation function about x i and x j with a correlation parameter vector θ [46]. 214 i j 7. Numerical and engineering examples 215 Five numerical and engineering problems are investigated in this section. For each example, 216 the results obtained by using the proposed approach are compared with the FORM and the 217 accurate solution provided by using the MCS Example This example is presented to investigate the effect of the different interpolation and 220 prediction methods on the function approximation in the proposed approach. The 221 performance function presented as: 3 3 f x, x x x 18 and the distribution of random variables is: x 1 N (10,5), x 2 N (9.9,5) [47]. 223 The example is solved by three approaches and the results are presented in Table 2. The six 224 employed interpolation methods are also shown in this table based on their rank to provide

11 accurate solution. According to Table 2 and as it could be found using Figure 3, among 226 various interpolation methods were used in the proposed approach, the method #1 presented 227 suitable approximation in such a manner that by 13-time function evaluation, provided results 228 in good agreement with MCS with 10 4 function evaluation. Result shows that the FORM 229 requires few LSF evaluations to provide solution but the obtained solution is highly different 230 with the accurate result provided by the MCS and the proposed approach Example A nonlinear limit state with two independent standard normal variables is considered [48] cos. g X X X X X (19) In this example, the accuracy of the method for a nonlinear limit state function is 234 investigated. The results are presented in Table 3. According to Table 3 and as it could be 235 found by using Figure 4, the proposed method provides acceptable results when four 236 techniques are used to interpolate the results in step 2. Results presented in the table shows 237 that the proposed method provides accurate solution for the problem while the number of 238 function evaluation used by the method was even less than those required in FORM. It should 239 be noted due to the nonlinearity of the LSF, the reliability index determined by FORM is 240 higher than the correct reliability index Example An implicit reliability problem with highly nonlinear performance function is investigated in 243 this example. Figure 5 shows the problem of a four-story building excited by a single period 244 sinusoidal impulse of ground acceleration. The building contains isolated equipment on the 245 second floor. The motion of the lowest floor is resisted by a nonlinear hysteresis force due to 246 the building's base isolation bearings and an additional stiffness force, if its displacement 247 exceeds d c. Each floor has a mass of m f and between floors the stiffness and damping 248 coefficient are k f and c f, respectively. The statistical parameters of the basic random variables

12 are listed in Table 4. All variables are assumed to be lognormal and independent. The limit 250 state function is defined by [48]: 251 f 1 g m2 i fi g X max r t r t 0.5 max z t r t i2,3,4 f2 m max r t r t (20) 252 where refers to the displacement of floor and is the inter-story 253 displacement of two consecutive floors. The accelerations and are of the ground and 254 the smaller mass block, respectively. The displacement is of the larger mass block, and 255 represents the displacement of the equipment isolation system. The limit state function in Eq. 256 (20) is the sum of three expressions of failure modes. The first term describes damage of the 257 structural system due to excessive deformation. The second term represents damage of 258 equipment which is caused by excessive acceleration. 259 The last term represents the damage of the isolation system. They are multiplied by weighing 260 factors, which emphasize the three failure modes equally. Eq. (20) expresses it is desirable 261 that (1) none of the inter-story displacements exceeds from 0.04 m, (2) the peak acceleration 262 of the smaller mass block (the equipment) is less than 0.5 m/s 2, and (3) the displacement 263 across the equipment isolation system is less than 0.25 m. By Failing one or two of the 264 conditions does not necessarily lead to a failure in the limit state function, e.g., but 265 will decrease the value of the limit state function. In these simulations, the system fails 266 mainly because of the large acceleration of the smaller mass. The estimation result of by 267 direct full scale MCS of 10 5 sample size is [48]. 268 The problem is solved by three method and the results are presented in Table 5. Results 269 shows the convergence of FORM failed to a proper solution. The reason is the high- 270 nonlinearity of performance function and the dimension size of the problem. However, it 271 could be found that by employing the proposed method, an approximation of failure

13 probability is achievable by 65-times function evaluation, while the result is in agreement 273 with MCS with 10 5 function evaluation. Result shows that proposed approach provides 274 accurate solution when the Kriging (interpolation #3) method is employed Example Consider a roof structure, which is subjected to a uniformly distributed vertical load q, as 277 shown in Figure 6.The example is adopted from [49]. The top cords and the compression bars 278 are concrete, and the bottom cords and the tension bars are steel. In structural analysis, the 279 uniformly distributed load q transformed into three nodal loads, each is P ql. The serviceability limit state of the structure with respect to its maximum vertical displacement 281 was considered. The limit state function is given by: 282 g 2 ql ua 2 AcEc AsE s (21) where u a is the allowable displacement and is set to be 0.03 m, E and A denote the Modulus 283 of elasticity and cross-sectional area, and the subscripts s and c indicate the steel and concrete 284 material, respectively. Table 6 summarizes the statistical information of the random variables. 285 All random variables are assumed independent normal. The probability of failure is found to 286 be after direct Monte Carlo simulations. 287 Table 7 presents the results from MCS, FORM and proposed method. The results from the 288 proposed method agrees reasonably well with the Monte Carlo result. The relative error is 5% 289 after 49-times function evaluation Example In automobile engineering, the front axle beam is used to carry the weight of the front part of 292 the vehicle (Figure 7) [50]. As the entire front part of the automobile rests on the front axle 293 beam, it must be robust enough in construction to ensure its reliability. An I-beam is often

14 used in the design of front axle due to its high bend strength and light weight. As shown in 295 Figure.7, a dangerous cross section happens in the I-beam part. The maximum normal stress 296 and shear stress are M and T separately, where M and T are bending 297 W x W p moment and torque, respectively, W x and W P are section factor and polar section factor, 298 respectively, which are given as a( h 2 t ) b 3 3 W x h ( h 2 t ) 6h 6h (22) 2 3 W p 0.8bt 0.4 a ( h 2 t ) t (23) To test the static strength of the front axle, the limit-state function can be expressed as 300 g 2 2 s 3 (24) where σ S is limit-state stress of yielding. According to the material property of the front axle, 301 the limit stress of yielding σ S is 460 MPa. The geometry variables of I-beam a, b, t, h and the 302 local M and Tare independent normal variables with distribution parameters listed in Table Table 9 presents the estimated values of failure probability with different methods. The 304 number of samples used for each method are also listed in Table 9. It shows that the proposed 305 method can achieved a good result with the least number of samples. According to Table 9, 306 among various interpolation methods used in the proposed approach, the methods #1, #2, #5 307 and #6 presented suitable approximation in such a manner that by 49-time function 308 evaluation, provided acceptable results with MCS with 10 5 function evaluation. 309 Efficiently solving these examples involving highly nonlinear and implicit LSFs confirms the 310 high potential of method to be applicable in real world engineering problems Conclusions

15 In this study, an adapted DOE presented for decomposition based Meta models in structural 314 reliability. The idea of the proposed approach is based on simulation approaches that separate 315 the design space into two safe and unsafe regions. The employed DRM- based Meta model 316 requires a one-dimensional interpolation method to approximate the LSF; hence, this study 317 also investigated the efficiency and accuracy of various interpolation techniques on 318 applicability of the proposed method. Solving numerical and engineering problems with different interpolation techniques prove that among the investigated interpolation approaches, 320 using the spline and kriging method in the proposed approach provides results with 321 acceptable accuracy. By using the proposed framework, it could be found that the method 322 presents efficiency similar to FORM while the accuracy is close to MCS

16 9. References 325 [1] Ditlevsen, O. and Madsen, H.O. Structural reliability methods, New York, Wiley (Vol ) (1996). 327 [2] Hasofer, A.M. and Lind, N.C. Exact and invariant second-moment code format, Journal of 328 the Engineering Mechanics division, 100(1), pp (1974). 329 [3] Rackwitz, R. and Flessler, B. Structural reliability under combined random load 330 sequences, Computers & Structures, 9(5), pp (1978). 331 [4] Der Kiureghian, A., Lin, H.Z. and Hwang, S.J. Second-order reliability 332 approximations, Journal of Engineering Mechanics, 113(8), pp (1987). 333 [5] Liu, P.L. and Der Kiureghian, A. Optimization algorithms for structural 334 reliability. Structural safety, 9(3), pp (1991). 335 [6] Wang, L. and Grandhi, R.V. Safety index calculation using intervening variables for 336 structural reliability analysis, Computers & structures, 59(6), pp (1996). 337 [7] Metropolis, N. and Ulam, S. The Monte Carlo method, Journal of the American statistical 338 association, 44(247), pp (1949). 339 [8] Yazdani, A., Nicknam, A., Khanzadi, M. and Motaghed, S. An artificial statistical method to 340 estimate seismicity parameter from incomplete earthquake catalogs A case study in 341 metropolitan Tehran, Iran, Scientia Iranica. Transaction A, Civil Engineering, 22(2), p (2015). 343 [9] Zuev, K. Subset Simulation Method for Rare Event Estimation, An Introduction. ArXiv 344 preprint arxiv: (2015). 345 [10] Wang, Z. and Song, J. Cross-entropy-based adaptive importance sampling using von Mises- 346 Fisher mixture for high dimensional reliability analysis, Struct Saf, 59: (2016). 347 [11] Fan, H. and Liang, R. Importance sampling based algorithm for efficient reliability analysis 348 of axially loaded piles, Computers and Geotechnics, 65, pp (2015). 349 [12] Ibrahim, Y. Observations on applications of importance sampling in structural reliability 350 analysis, Struct Safety, 9(4), (1991). 351 [13] Nie, J. and Ellingwood, B.R. Directional methods for structural reliability analysis, Struct 352 Safety, 22(3), (2000). 353 [14] Depina, I., Le, T.M.H., Fenton, G. and Erikson, G. Reliability analysis with Metamodel Line 354 Sampling, Struct Saf, 60, 1-15 (2016). 355 [15] Pradlwarter, H.J., Schueller, G.I., Koutsourelakis, P.S. and Charmpis, D.C. Application of 356 line sampling simulation method to reliability benchmark problems, Struct Safety, 29(3), (2007). 358 [16] Rashki, M., Miri, M. and Moghaddam, M.A. A new efficient simulation method to 359 approximate the probability of failure and most probable point, Structural Safety, 39, pp (2012)

17 [17] Au, S.K. and Beck, J.L. "Estimation of small failure probabilities in high dimensions by 362 subset simulation", Probabilistic engineering mechanics, 16(4), pp (2001). 363 [18] Ghanem, R. and Spanos. "Polynomial chaos in stochastic finite elements", Journal of Applied 364 Mechanics, 57(1), pp (1990). 365 [19] Kamiński, M. and Świta, P. "Structural stability and reliability of the underground steel tanks 366 with the Stochastic Finite Element Method". Archives of Civil and Mechanical 367 Engineering, 15(2), pp (2015). 368 [20] Ghavidel, A., Mousavi, S.R. and Rashki, M. The effect of FEM mesh density on the failure 369 probability analysis of structures, KSCE Journal of Civil Engineering, pp.1-13 (2017). 370 [21] Gong, J.X. and Yi, P. A robust iterative algorithm for structural reliability analysis, 371 Structural and Multidisciplinary Optimization, 43(4), pp (2011). 372 [22] Johari, A., Momeni, M. and Javadi, A.A. An analytical solution for reliability assessment of 373 pseudo-static stability of rock slopes using jointly distributed random variables method, 374 Iranian Journal of Science and Technology Transactions of Civil Engineering, 39, pp (2015). 376 [23] Hasofer, A.M. and Lind, N.C. Exact and invariant second-moment code format, Journal of 377 the Engineering Mechanics division, 100(1), pp (1974). 378 [24] Keshtegar, B. and Miri, M. Reliability analysis of corroded pipes using conjugate HL RF 379 algorithm based on average shear stress yield criterion, Engineering Failure Analysis, 46, 380 pp (2014). 381 [25] Johari, A., Mousavi, S. and Nejad, A.H. "A seismic slope stability probabilistic model based 382 on Bishop's method using analytical approach", Scientia Iranica. Transaction A, Civil 383 Engineering, 22(3), p.728 (2015). 384 [26] Rackwitz, R. and Flessler, B. Structural reliability under combined random load sequences, 385 Computers & Structures, 9(5), pp (1978). 386 [27] Santosh, T.V., Saraf, R.K., Ghosh, A.K. and Kushwaha, H.S. Optimum step length selection 387 rule in modified HL RF method for structural reliability, International journal of pressure 388 vessels and piping, 83(10), pp (2006). 389 [28] Yang, D. Chaos control for numerical instability of first order reliability method, 390 Communications in Nonlinear Science and Numerical Simulation, 15(10), pp (2010). 392 [29] Zhou, W., Gong, C. and Hong, H.P. "New Perspective on Application of First-Order 393 Reliability Method for Estimating System Reliability", Journal of Engineering 394 Mechanics, 143(9), p (2017). 395 [30] Yun, W., Lu, Z., Jiang, X. and Zhao, L.F. "Maximum probable life time analysis under the 396 required time-dependent failure probability constraint and its meta-model 397 estimation", Structural and Multidisciplinary Optimization, 55(4), pp (2017)

18 [31] Bhadra, A. and Ionides, E.L. Adaptive particle allocation in iterated sequential Monte Carlo 399 via approximating meta-models, Statistics and Computing, 26(1-2), pp (2016). 400 [32] Dengiz, B., İç, Y.T. and Belgin, O. A meta-model based simulation optimization using 401 hybrid simulation-analytical modeling to increase the productivity in automotive industry, 402 Mathematics and Computers in Simulation, 120, pp (2016). 403 [33] Sudret, B. Meta-models for structural reliability and uncertainty quantification, arxiv 404 preprint arxiv: (2012). 405 [34] Cadini, F., Santos, F. and Zio, E. An improved adaptive kriging-based importance technique 406 for sampling multiple failure regions of low probability, Reliability Engineering & System 407 Safety, 131, pp (2014). 408 [35] Echard, B., Gayton, N. and Lemaire, M. AK-MCS: an active learning reliability method 409 combining Kriging and Monte Carlo simulation, Structural Safety, 33(2), pp (2011). 410 [36] Bucher, C.G. and Bourgund, U. A fast and efficient response surface approach for structural 411 reliability problems, Structural safety, 7(1), pp (1990). 412 [37] Nguyen, X.S., Sellier, A., Duprat, F. and Pons, G. Adaptive response surface method based 413 on a double weighted regression technique, Probabilistic Engineering Mechanics, 24(2), 414 pp (2009). 415 [38] Kang, S.C., Koh, H.M. and Choo, J.F. An efficient response surface method using moving 416 least squares approximation for structural reliability analysis, Probabilistic Engineering 417 Mechanics, 25(4), pp (2010). 418 [39] Allaix, D.L. and Carbone, V.I. An improvement of the response surface method, Structural 419 Safety, 33(2), pp (2011). 420 [40] Cai, W., "A dimension reduction algorithm preserving both global and local clustering 421 structure". Knowledge-Based Systems, 118, pp (2017). 422 [41] Xu, H. and Rahman, S. A moment-based stochastic method for response moment and 423 reliability analysis, In Proceedings of 2nd MIT Conference on Computational Fluid and 424 Solid Mechanics, pp (2003). 425 [42] Xu, H. and Rahman, S. A generalized dimension reduction method for multidimensional 426 integration in stochastic mechanics, International Journal for Numerical Methods in 427 Engineering, 61(12), pp (2004). 428 [43] Huang, B. and Du, X. Uncertainty analysis by dimension reduction integration and saddle 429 point approximations, Journal of Mechanical Design, 128(1), pp (2006). 430 [44] Lee, I., Choi, K.K. and Du, L. Alternative methods for reliability-based robust design 431 optimization including dimension reduction method, In ASME 2006 International Design 432 Engineering Technical Conferences and Computers and Information in Engineering 433 Conference, pp (2006). 434 [45] GhohaniArab, H. and Ghasemi, M.R. A fast and robust method for estimating the failure 435 probability of structures, P I Civil Eng Str B, 168(4): (2015)

19 [46] Lv, Z., Lu, Z. and Wang, P. A new learning function for Kriging and its applications to solve 437 reliability problems in engineering, Computers & Mathematics with Applications, 70(5), 438 pp (2015). 439 [47] Zou, T., Mahadevan, S., Mourelatos, Z. and Meernik, P. Reliability analysis of automotive 440 body-door subsystem, Reliability Engineering & System Safety, 78(3), pp (2002). 441 [48] Gavin, H.P. and Yau, S.C. High-order limit state functions in the response surface method 442 for structural reliability analysis, Structural safety, 30(2), pp (2008). 443 [49] Dubourg, V. Sudret, B. "Meta-model-based importance sampling for reliability Sensitivity", 444 Structural safety, pp. 49:27 36 (2014). 445 [50] Wang, P., Lu, Z. and Tang, Z. "An application of the Kriging method in global sensitivity 446 analysis with parameter uncertainty". Applied Mathematical Modelling, 37, pp (2013)

20 a t i o Interpolation Method Table 468 Table 1. The employed interpolation techniques Table 2. Reliability results for Example 1. Table 3. Reliability results for Example 2 Table 4. Statistical properties of random variables for Example 3 Table 5. Reliability results for Example 3 Table 6. Random variables for Example 4 Table 7. Reliability results for Example 4 Table 8. Random variables for Example 5 Table 9. Reliability results for Example Table 1. The employed interpolation techniques 470 Sign Interpolation Method #1 Spline #2 PCHIP #3 Kriging #4 #5 #6 Linear Cubic V5cubic Method Table 2. Reliability results for Example P f Reliability index No. of function evaluations FORM MCS # # UDRM # # # # Method Table 3. Reliability results for Example P f Reliability index No. of function evaluations FORM UDRM MCS #

21 Interpolation Method # # # # # Table 4. Statistical properties of random variables for Example Variable Description Units Mean value C.O.V. m f Floor Mass Kg k f Floor Stiffness N/m c f Floor Damping Coefficient N/m/s d y Isolation Yield Displacement m f y Isolation Yield Force N d c Isolation Contact Displacement m k c Isolation Contact Stiffness N/m m 1 Mass of Block 1 Kg m 2 Mass of Block 2 Kg k 1 Stiffness of Spring 1 N/m k 2 Stiffness of Spring 2 N/m c 1 Damping Coefficient of Damper 1 N/m/s c 2 Damping Coefficient of Damper 2 N/m/s T Pulse Excitation Period s A Pulse Amplitude m/s/s Method Table 5. Reliability results for Example P f Reliability index No. of function evaluations FORM MCS # # UDRM # # # # Table 6. Random variables for Example Variable Mean COV 21

22 Interpolation Method Interpolation Method T(N.mm) q(n/m) 20, l(m) A s (m 2 ) A c (m 2 ) E s (N/m 2 ) E c (N/m 2 ) Method Table 7. Reliability results for Example P f Reliability index No. of function evaluations FORM MCS # # UDRM # # # # Table 8. Random variables for Example Variable Mean COV a(mm) b(mm) t(mm) h(mm) M(N.mm) Method Table 9. Reliability results for Example P f Reliability index No. of function evaluations FORM MCS # UDRM # # # #

23 # Figure 1. Excluding part of safe area Figures 490 Figure 2. DOE and the generated samples based on MCS Figure 3. Failure region in the MCS and proposed approach by using: (A) spline (B) PCHIP (C) Kriging interpolation for Example 1. Figure 4. Failure region in the MCS and proposed approach by using: (A) spline (B) PCHIP (C) Kriging interpolation for Example 2. Figure 5. Base isolated structure with an equipment isolation system on the 2 nd floor, and including the effects of isolation displacement limits [47] Figure 6. A roof structure (redrawn from [48]). Figure 7. Schematic diagram of automobile front axle [48]

24 492 Figure 1. Excluding part of safe area Figure 2. DOE and the generated samples based on MCS

25 501 Figure 3. Failure region in the MCS and proposed approach by using: 503 (A) Spline (B) PCHIP (C) Kriging interpolation for Example

26 Figure 4. Failure region in the MCS and proposed approach by using: 509 (A) Spline (B) PCHIP (C) Kriging interpolation for Example Figure 5. Base isolated structure with an equipment isolation system on the 2 nd floor, 513 and including the effects of isolation displacement limits [47]

27 516 Figure 6. A roof structure (redrawn from [48]) Figure 7. Schematic diagram of automobile front axle [48]

28 Biographies 546 Mehralah RakhshaniMehr received his BSc in Civil engineering from university of Sistan and 547 Baluchestan (USB), Iran, in 2007, MSc and PhD in Structural engineering from USB in 2009 and , respectively. He is currently Assistant Professor in AL-Zahra University, Tehran, Iran. His 549 main research areas are concrete structures analysis and rehabilitation, dynamic of structures and 550 structural reliability analysis. 551 Hamed Mirkamali received his BSc in Civil Engineering from Islamic Azad Universiry, Zabol, Iran, 552 in 2012, MSc in Earthquake Engineering from Shiraz University of Technology, Shiraz, Iran, in He carried out his research thesis titled Vulnerability Assessment of Gas Network Using the First 554 Order and Simulation Reliability Method under supervision of Dr. Hossain Rahnema and acquired 555 excellent grade of 19.4 out of 20. His research achievements have been quite significant and has 556 already published two conference papers. 557 Mohsen Rashki received his BSc in Civil engineering from University of Sistan and Baluchestan 558 (USB), Iran, in 2007, MSc and PhD in Structural engineering from USB in 2009 and 2014, 559 respectively. He is currently Assistant Professor in Architecture Department at USB and published 560 several research papers. His main research areas and interests are structural reliability analysis, 561 reliability based design optimization and Bayesian probabilistic approaches. 562 Abdolhamid Bahrpeyma received his MSc in Architectural Engineering from University of Science 563 and Industry, Tehran, Iran, in 1994, and PhD in Urban Planning Engineering from The University of 564 Paris 10, France, in He is currently Assistant Professor in Civil Engineering Department at 565 Sistan and Baluchestan University