adapted design of experiments for dimension decomposition...

Scientia Iranica A (2019) 26(6), 3060{3071

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Adapted design of experiments for dimensiondecomposition-based meta-model in structuralreliability analysis

M. Rakhshani Mehra;�, H. Mirkamalib, M. Rashkic, and A. Bahrpeymac

a. Department of Civil Engineering, Alzahra University, Tehran, P.O. Box 1993893973, Iran.b. Department of Civil Engineering, Shiraz University of Technology, Shiraz, Iran.c. Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran.

Received 10 May 2016; received in revised form 7 October 2017; accepted 5 March 2018

KEYWORDSStructural reliability;Simulation methods;Univariate dimensionreduction;Design of experiment;Monte Carlosimulation.

Abstract. Reliability analysis of structures is often problematic for the structures withnonlinear and complex Limit State Functions (LSFs). For these cases, simulation methodsoften provide accurate failure probability, but with a high number of LSFs in the analysisof the structure. This paper presents an e�cient combined meta-model of Monte CarloSimulation (MCS) and Univariate Dimension Reduction (UDR) to approximate the failureprobability of structures with evaluation of few LSFs. For this purpose, the design of theexperiment applied in the meta-model was adapted such that the expected failure samplesin MCS were approximated with higher accuracy. Several numerical and engineeringreliability problems were solved by the proposed approach and the results were veri�edby MCS. Results showed that the proposed approach highly reduced the required numberof structural analyses to provide proper results.© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

In structural reliability analysis, the failure probabilityPf is de�ned as [1]:

Pf =ZG(X)

fx (X) dX; (1)

where fX(X) is the joint probability density func-tion of the vector of basic random variables X =[x1; x2; :::; xn]T , which represents uncertain quantitiessuch as material properties, loads, boundary condi-tions, and geometry. In Eq. (1), G(X) is the Limit

*. Corresponding author.E-mail addresses: [email protected] (M.Rakhshani Mehr); [email protected] (H.Mirkamali); [email protected] (M. Rashki)

doi: 10.24200/sci.2018.20221

State Function (LSF) in which G(X) > 0 representsthe safety domain and G(X) < 0 represents the failuredomain. However, the failure probability of a givenproblem by means of Eq. (1) is not a straight approach,because the joint probability density function FX(X)is not always available. In some cases, Eq. (1) cannotbe integrated analytically, even if FX(X) is available,especially for the complex structures with low failureprobabilities and implicit LSFs. Therefore, in orderto avoid such calculation, various techniques havebeen proposed, e.g., a) approximation methods (i.e.,First Order Reliability Method (FORM) and SecondOrder Reliability Method (SORM)) and b) simulationmethods [2-5]. FORM and SORM are accurate forreliability problems with linear and moderate LSFs,but inaccurate for highly nonlinear LSFs and di�cultto solve when the actual implicit LSF cannot beexpressed explicitly. Besides, in some cases, FORM andSORM may su�er convergence problems [6]. Hence,

M. Rakhshani Mehr et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 3060{3071 3061

when an accurate reliability evaluation is required,simulation methods are often employed.

1.1. Monte Carlo simulationMonte Carlo Simulation (MCS) is considered as themost e�cient and accurate simulation method and iscommonly used for the evaluation of the probabilityof failure for structures, either for comparison withother methods or as a standalone reliability analysistool [7,8]. This method involves sampling the designspace based on the mean, variance, and PDF valuesof random variables. From a mathematical point ofview, MCS allows to estimate the expected value of aquantity of interest more speci�cally. Suppose the goalis to evaluate Ef [h(X)]; an expectation of the functionh : x ! R with respect to the Probability DensityFunction (PDF) [9] is:Z

Xh(X)fx(X)dx = Ef [h(X)] : (2)

The idea behind MCS is straight forward applicationof the law of large numbers, which states if X =[x1;x2;:::; xn] is independent from and distributed iden-tically to the PDF fx(X), then the empirical average1N

NPi=1

h(xi) converges to the true value of Ef [h(X)]

when N approaches +1. Therefore, if the number ofsamples N is large enough, then Ef [h(X)] and it canbe accurately estimated by the corresponding empiricalaverage:

Ef [h(X)] � 1N

NXi=1

h(xi): (3)

The relevance of DMC to the reliability problem (1)follows a simple observation that the failure probabilityPf can be written as:

Pf =ZG(X)

fx (X) dX =ZXIf (X)fx(X)dX

= Ef [If (X)] =1N

NXi=1

h(xi); (4)

where X = [x1;x2;:::; xn] is independent from anddistributed identically to the PDF fx(X), and If isa counting vector with values of zero and unity forsamples in the failure and safe regions, respectively.As it is seen, a vast number of simulations haveto be performed in order to achieve great accuracy,especially for low values of failure probability. In thee�orts to reduce the excessive computation cost of MCSusing purely random sampling methodologies, whichare considered as the drawback of the method, variousvariance reduction techniques have been proposed,e.g., importance sampling [10-12], direct sampling [13],line sampling [14,15], Weighted Average Simulation

Method (WASM) [16], subset simulation [17], polyno-mial chaos [18], and stochastic perturbation technique[19]. Unfortunately, most of these techniques are notas generally applicable as MCS. For example, impor-tance sampling requires detailed information about thefailure regions for being useful, and it faces di�cultieswhen applied to high-dimension problems [16,20].

1.2. First order reliability methodFirst Order Reliability Method (FORM) is widely usedto approximate the failure probability of structuresand has become a basic reliability analysis approachfor reliability-based design codes [21,22]. In FORM,structural failure probability is estimated based on thereliability index (�) by linearizing limit state functionon the failure surface, i.e., Pf � �(��), whichcorresponds to minimum distance of the origin from thelimit state function in the standard normal area [21,23].Generally, the main goal of FORM is the search for themost probable point (MPP), i.e., U�(b = jjU�jj) [24,25].Hasofer Lind proposed a general iterative method forcomputing reliability index [23], which was extendedby Rackwitz and Flessler to include distribution in-formation of random variables [26], called the HL-RFmethod. This method involves the following steps toestimate the probability of failure [26] based on theHL-RF method:

Step 1. Transform random variables in X-space intoU -space by the following relation:

u =x� �ex�ex

; (5)

where u is the standard normal variable with themean and standard deviations equal to zero and one,respectively, and �ex and �ex are equivalent meanand standard deviations of the random variable x,respectively; for normal random variable, �ex = �xand �ex = �x. The equivalent mean and standarddeviations of non-normal random variables can bedetermined by the following equations [26-28]:

�ex =1

fx(x)��1 fFx(x)g� ; (6)

�ex = x� �ex��1 [Fx(x)] ; (7)

where Fx(x) is cumulative distribution, fx(x) is prob-ability distribution, ��1 is inverse standard normalcumulative distribution, and � is standard normalprobability distribution function.Step 2. Find the reliability index.

The reliability index search is done based on aniterative process that can be reformulated based ondesign point (U) as:

Uk+1 = �k+1�k+1; (8)

where:

3062 M. Rakhshani Mehr et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 3060{3071

�k+1 = � rT g(Uk)krT g(Uk)k ; (9)

�k+1 =g(Uk)�rT g(Uk)UkkrT g(Uk)k ; (10)

where rg(U) = [@g/@u1;@g/@u2;:::; @g/@un]T is gra-dient vector of the limit state function at the designpoint Uk.Step 3. Calculate the failure probability.

The probability of failure based on FORM canbe estimated as Pf � �(��) [21,29].

2. Meta-models in structural reliability

When performance evaluation of a structure is compu-tationally expensive, the number of simulation-basedfunction evaluations required for reliability analysismust be carefully controlled. To that end, researchershave explored the use of meta-models, which aresimpler approximate models calibrated to sample runsof the original simulation. The approximate model ormeta-model can replace the original one, thus reduc-ing the computational burden of evaluating numerousproblem [30-35]. Bucher and Bourgund [36] proposeda quadratic polynomial response surface without crossterms. In their model, the response surface representedthe LSF along the coordinate axes of the space ofstandard normal random variables. Nguyen et al. [37]proposed an adaptive RSM based on a double weightedregression technique. For the �rst iteration, a linearresponse surface was chosen; for the following itera-tions, a quadratic response surface with cross terms wasconsidered based on the complementary points. Kanget al. [38] proposed an e�cient RSM applying a movingleast squares approximation instead of the traditionalleast squares approximation. Allaix and Carbone [39]discussed the locations of the experimental pointsused for evaluating parameters of the response surface.Recently, Dimension Reduction Method (DRM) asan e�cient approach has been used to reduce thecomputational costs of the analysis [40-43].

In order to use the capabilities of MCS and simul-taneously reduce the computational e�orts, this paperpresents a framework that e�ciently employs the DRMto evaluate the reliability of structures. The proposedframework is presented after a brief review of DRM.

3. Dimension reduction method

DRM is a newly developed technique to calculatestatistical moments of the output performance func-tion [40-43]. There are several DRMs dependingon the level of dimension reduction: (1) UnivariateDimension Reduction Method (UDRM), which is anadditive decomposition of N -dimensional performance

function into one-dimensional functions; (2) BivariateDimension Reduction (BDR), which is an additivedecomposition of N -dimensional performance functioninto at most two-dimensional functions; (3) Multivari-ate Dimension Reduction (MDR), which is an additivedecomposition of N -dimensional performance functioninto at most S-dimensional functions, where S � N .

According to UDRM, any N -dimensional perfor-mance function h(X) can be additively decomposedinto one-dimensional functions as [44]:

h (X)�= h (X)�NXi=1

h (�1; : : : ; �i�1; xi; �i+1; : : : ; �N )

� (N � 1)h (�1; : : : ; �N ) ; (11)

where �i is the mean value of a random variable Xiand N is the number of design variables.

4. Proposed framework

This study employs the e�ciency of the UDR-basedmeta-modeling in conjunction with the accuracy ofMCS to provide a suitable framework for structuralreliability analysis. The idea is to concentrate the ex-periments of the UDR-based meta-model on the regionwith high failure probability to correctly approximatethe performance function value for the samples that areexpected to be in the failure set. The following stepscould be conducted to provide the desirable results.

4.1. Axial Design Of Experiments (DOE)based on the desired reliability index

UDR-based meta-model requires axial DOE to ap-proximate the LSF. Determination of the location ofexperience samples in the proposed approach is basedon the perception created by conducting the MCSsampling and excluding the safe area part. In thisapproach, as shown in Figure 1, which is presented instandard normal space (U), the space is divided intotwo separate regions D1 and D2 and it is assumed thatD1 is selected such that no failure occurs in this region.Here, D1 is chosen as the region inside a sphere withradius � [45].

Figure 1. Excluding the safe area part.


To reduce the computational cost, the locationof DOEs could be considered such that the corpus ofexperience samples in each axis is condensed within theboundaries of D1 and D2 regions and regions with thehighest possibilities of failure. These samples shouldbe used with the aim of interpolating DRM for theseparated term in each dimension. By employing ananticipated reliability index and mapping the proposedexperience samples in the physical space (originaldesign space), the location of experiences for eachvariable is as follows:

XDOE = ��p

22:�target:�:�; (12)

where � and � are the mean value and standarddeviation of random variable, respectively, and � is thelocation coe�cient.

4.2. Generation of random sample based oncrude MCS

After conducting the proposed step in the design of theexperiment, in the next step, crude MCS should be per-formed to generate random samples for approximatingthe failure probability. Figure 2 schematically showsthe proposed DOEs and the generated samples for atwo-dimensional problem. According to the proposedapproach, the approximated performance function cor-responding to each sample is achievable by employingthe UDR-based meta-model.

4.3. LSF approximation by the UDRM-basedmeta-model and reliability evaluation

Eq. (11) is employed at this step to approximate theLSF. The following is the resulting function for a limitstate function with two random variables X1 and X2:

h (X)�= h (X)�h (x1; �2)+h (�1; x2)�h (�1; �2) ; (13)

in which for the samples produced by MCS, throughemploying the experiments and a proper interpolation

Figure 2. DOE and the generated samples based onMCS.

Table 1. The employed interpolation techniques.

Sign Interpolation method

#1 Spline#2 PCHIP#3 Kriging#4 Linear#5 Cubic#6 V5cubic

technique, the values of h(x1; �2) and h(�1; x2) foreach dimension are achievable thorough interpolation.Then, the value of performance function for eachsample could be approximated using Eq. (13). Then,the failure probability could be approximate by Eq. (4).

In this study, the e�ectiveness of various inter-polation techniques is also investigated to suggest aproper technique for use in the proposed framework.It consists in several one-dimensional interpolationsimplemented by MATLAB toolbox, which are pre-sented in Table 1. The kriging method (method #3)as a newly developed approximation method is alsoused in the approach and compared with commoninterpolation techniques.

5. Kriging method

Kriging meta-model is an interpolation technique basedon statistical theory, which consists in a parametriclinear regression model and a non-parametric stochas-tic process. It needs a design of experiments to de�nethe stochastic parameters and then, predictions of theresponse can be completed on any unknown point.An initial DOE X = [x1; x2; :::; xN0 ] is given withxi 2 Rn(i = 1; 2; :::; N0) as the ith experiment andG = [G(x1); G(x2); :::; G(xN0);] with G(xi) 2 R as thecorresponding response to X [46]. The approximaterelationships between any experiment X and the re-sponse G(x) can be denoted as:bG(x) = F (�; x) + z(x) = fT (x)� + z(x); (14)

where �T = [�1; :::; �p] is a regression coe�cientvector. Built by response surface method similar to thepolynomial, fT (x) = [f1(x); f2(x); :::; fp(x)]T makes aglobal simulation in the design space. In the ordinarykriging, F (�; x) is a scalar always taken as F (�; x) = �.Hence, the estimated bG(x) can be simpli�ed as:bG(x) = F (�; x) + z(x) = � + z(x): (15)

Here, z(x) is a stationary Gaussian process [46]. Thestatistic characteristics can be denoted as:

E(z(x)) = 0; (16)

V ar(z(x)) = �2z ; (17)


Cov [Z(xi); Z(xj)] = �2zR(xi; xj); (18)

where �2z is the process variance; xi; and xj are

discretional points from the whole samples X; andR(xi; xj) is the correlation function about xi and xjwith a correlation parameter vector � [46].

6. Numerical and engineering examples

Five numerical and engineering problems are investi-gated in this section. For each example, the resultsobtained by using the proposed approach are comparedwith those by FORM and the accurate solution pro-vided by using the MCS.

6.1. Example 1This example is presented to investigate the e�ect ofthe di�erent interpolation and prediction methods onthe function approximation in the proposed approach.The performance function is presented as f (x1; x2) =x3

1 + x32 � 18 and the distribution of random variables

are x1 = N(10; 5); and x2 = N(9:9; 5) [47].The example is solved by three approaches and

the results are presented in Table 2. The six employedinterpolation methods are also shown in this tablebased on their ranks to provide an accurate solution.According to Table 2 and as shown in Figure 3, amongvarious interpolation methods used in the proposed ap-proach, method #1 presents a suitable approximationin such a manner that by 13 times function evaluation,the provided results are in good agreement with MCSwith 104 function evaluations. Result shows thatthe FORM requires few LSF evaluations to providesolution, but the obtained solution is highly di�erentfrom the accurate result provided by the MCS and theproposed approach.

6.2. Example 2A nonlinear limit state with two independent standardnormal variables is considered [48].

g (X)=�0:16(X1 � 1)3�X2+4�0:04cos(X1:X2) : (19)

In this example, the accuracy of the method for a non-linear limit state function is investigated. The resultsare presented in Table 3. According to Table 3 andas shown in Figure 4, the proposed method providesacceptable results when four techniques are used tointerpolate the results in Step 2. Results presentedin the table show that the proposed method hasprovided accurate solution to the problem, althoughthe number of its function evaluations is even lessthan that required by FORM. It should be noted thatdue to the nonlinearity of the LSF, the reliabilityindex determined by FORM is higher than the correctreliability index.

6.3. Example 3An implicit reliability problem with highly nonlinearperformance function is investigated in this example.Figure 5 shows the problem of a four-story buildingexcited by a single-period sinusoidal impulse of groundacceleration. The building contains isolated equipmenton the second oor. The motion of the lowest oor isresisted by a nonlinear hysteresis force in base isolationbearings of the building and an additional sti�nessforce, if its displacement exceeds dc. Each oor hasa mass of mf and between oors, the sti�ness anddamping coe�cients are kf and cf , respectively. Thestatistical parameters of the basic random variablesare listed in Table 4. All variables are assumed to belognormal and independent. The limit state functionis de�ned by [48]:

Table 2. Reliability results for Example 1.

Method Pf Reliability index No. of function evaluations

FORM 0.0832 1.384 13

MCS 0.0057 2.530 104

UDRM

Inte

rpol

atio

nm

ethod

#1 0.0057 2.530 13

#5 0.0043 2.628 13

#2 0.0042 2.636 13

#3 0.0026 2.794 13

#6 0.0024 2.820 13

#4 0.0020 2.878 13


Figure 3. Failure region in the MCS and the proposedapproach by using: (a) Spline, (b) PCHIP, and (c) kriginginterpolation for Example 1.

g (X) = 12:5(0:04�max jrfi (t)� rfi�1 (t)j)i=2;3;4

+ (0:5�max jzg (t)� rm2 (t)j)+ 2 (0:25�max jrf2 (t)� rm1 (t)j) ; (20)

where rfi refers to the displacement of the ith oor andrfi(t)� rfi�1(t) is the inter-story displacement of twoconsecutive oors. The accelerations �zg and �rm2 areof the ground and the smaller mass block, respectively.The displacement rm1 is of the larger mass block andrepresents the displacement of the equipment isolation

Figure 4. Failure region in the MCS and the proposedapproach by using: (a) Spline, (b) PCHIP, and (c) kriginginterpolation for Example 2.

system. The limit state function in Eq. (20) is thesum of three expressions of failure modes. The �rstterm describes damage to the structural system dueto excessive deformation. The second term representsdamage to equipment, which is caused by excessiveacceleration.

The last term represents the damage to the isola-tion system. They are multiplied by weighing factors,which emphasize the three failure modes equally. AsEq. (20) states, it is desirable that:

1. None of the inter-story displacements exceed0.04 m;


Figure 5. Base-isolated structure with an equipment isolation system on the 2nd oor, including the e�ects of isolationdisplacement limits [47].



FORM 7:4883:105 3.791 17

MCS 1:1900:104 3.675 106

UDRM

Inte

rpol

atio

nM

ethod

#1 1:1900:104 3.675 13

#3 1:1400:104 3.686 13

#2 1:0800:104 3.700 13

#5 1:3400:104 3.644 13

#4 6:2000:105 3.838 13

#6 4:3000:105 3.927 13

Table 4. Statistical properties of random variables for Example 3.

Variable Description Units Mean value COV

mf Floor mass Kg 6000 0.1kf Floor sti�ness N/m 30000000 0.1cf Floor damping coe�cient N/m/s 60000 0.2dy Isolation yield displacement m 0.05 0.2fy Isolation yield force N 20000 0.2dc Isolation contact displacement m 0.5 0kc Isolation contact sti�ness N/m 30000000 0.3m1 Mass of block 1 Kg 500 0m2 Mass of block 2 Kg 100 0k1 Sti�ness of spring 1 N/m 2500 0k2 Sti�ness of spring 2 N/m 100000 0c1 Damping coe�cient of damper 1 N/m/s 350 0c2 Damping coe�cient of damper 2 N/m/s 200 0T Pulse excitation period s 1.0 0.2A Pulse amplitude m/s/s 1.0 0.5


2. The peak acceleration of the smaller mass block(the equipment) be less than 0.5 m/s2;

3. The displacement across the equipment isolationsystem be less than 0.25 m.

Failing to meet one or two of the conditions doesnot necessarily lead to a failure in the limit statefunction, e.g., g(X < 0), but will decrease the valueof the limit state function. In these simulations, thesystem fails mainly because of the large accelerationof the smaller mass. The estimation of Pf with directfull-scale MCS of 105 sample size is 0.196 [48].

The problem has been solved by the three meth-ods and the results are presented in Table 5. Resultsshow the fail of FORM to converge to a proper solution.The reason is the high nonlinearity of performancefunction and the dimension size of the problem. How-ever, it is noteworthy that by employing the proposedmethod, an approximation of failure probability isachievable by 65 times function evaluation with the

results in agreement with MCS with 105 functionevaluations. Result shows that the proposed approachprovides accurate solution when the kriging (interpola-tion #3) method is employed.

6.4. Example 4Consider a roof structure subjected to a uniformlydistributed vertical load q, as shown in Figure 6. Theexample is adapted from [49]. The top cords and thecompression bars are concrete, and the bottom cordsand the tension bars are steel. In structural analysis,the uniformly distributed load q is transformed intothree nodal loads with each being P = ql/4. Theserviceability limit state of the structure with respect toits maximum vertical displacement is considered. Thelimit state function is given by:

g = ua � ql2

2

�3:81AcEc

+1:13AsEs

�; (21)

where ua is the allowable displacement and set to

Figure 6. A roof structure (redrawn from [48]).



FORM { { {

MCS 0.196 0.856 105

UDRM

Inte

rpol

atio

nm

ethod #3 0.1848 0.897 65

#2 0.2243 0.758 65

#5 0.2399 0.707 65

#1 0.2402 0.706 65

#6 8:3800:104 3.142 65

#4 7:6900:104 3.167 65


Table 6. Random variables for Example 4.

Variable Mean COVq (N/m) 20,000 0.07l (m) 12 0.01As (m2) 9:82� 10�4 0.06Ac (m2) 400� 10�4 0.12Es (N/m2) 1� 1011 0.06Ec (N/m2) 2� 1010 0.06

0.03 m, E and A denote the modulus of elasticity andcross-sectional area, respectively, and the subscripts sand c indicate the steel and concrete material, respec-tively. Table 6 summarizes the statistical informationof the random variables. All random variables areassumed to be independent normal. The probabilityof failure is found to be 9:37�10�3 after direct 5�107

Monte Carlo simulations.Table 7 presents the results of the MCS, FORM,

and proposed method. The results of the proposedmethod agree reasonably well with the Monte Carloresults. The relative error is 5% after 49 times functionevaluation.

6.5. Example 5In automobile engineering, the front axle beam isused to carry the weight of the front part of thevehicle (Figure 7) [50]. As the entire front part of theautomobile rests on the front axle beam, it must berobust enough in construction to ensure its reliability.An I-beam is often used in the design of front axledue to its high bend strength and light weight. Inthis example, as shown in Figure 7, a dangerous crosssection happens in the I-beam part. The maximumnormal stress and shear stress are � = M/Wx and� = T/Wp, respectively, in which M and T are bendingmoment and torque, respectively, and Wx and WP aresection factor and polar section factor, respectively,given as:

Figure 7. Schematic diagram of automobile frontaxle [48].

Table 8. Random variables for Example 5.

Variable Mean COVa (mm) 12 0.06b (mm) 65 0.325t (mm) 14 0.07h (mm) 85 0.425

M (N.mm) 3:5� 106 1:75� 105

T (N.mm) 3:1� 106 1:55� 105

Wx =a(h� 2t)3

6h+

b6h

hh3 � (h� 2t)3

i; (22)

Wp = 0:8bt2 + 0:4�a3(h� 2t)

�t�: (23)

To test the static strength of the front axle, the limit-state function can be expressed as:

g = �s �p�2 + 3�2; (24)

where �s is limit-state stress of yielding. Consideringthe characteristic of material in the front axle, thelimit stress of yielding �s is 460 MPa. The geometryvariables of I-beam, namely a, b, t, h, the local M; andT , are independent normal; they are listed with theirdistribution parameters in Table 8.



FORM { { {

MCS 0.00479 2.59 105

UDRM

Inte

rpol

atio

nm

ethod #1 0.00307 2.74 49

#2 0.00297 2.75 49

#6 0.00297 2.75 49

#5 0.00280 2.77 49

#3 0.00225 2.84 49

#4 0.00002605 4.046 49




FORM 0.0194 2.06 49

MCS 0.0195 2.05 105

UDRM

Inte

rpol

atio

nm

ethod #1 0.0195 2.05 49

#2 0.0195 2.05 49

#5 0.0195 2.05 49

#6 0.0195 2.05 49

#3 0.0192 2.07 49

#4 0.0000145 4.18 49

Table 9 presents the estimated values of failureprobability with di�erent methods. The number ofsamples used for each method is also listed in Ta-ble 9. The table shows that the proposed methodcan achieved good results with the lowest numberof samples. According to Table 9, among variousinterpolation methods used in the proposed approach,the methods #1, #2, #5, and #6 present suitableapproximations in such a manner that by 49 timesfunction evaluation, they provide acceptable results incomparison with MCS with 105 function evaluations.

E�ciently solving these examples involving highlynonlinear and implicit LSFs con�rms the high potentialof the method to be applied to the real-world engineer-ing problems.

7. Conclusions

In this study, an adapted DOE was presented fordecomposition-based meta-models in structural relia-bility. The idea of the proposed approach was basedon simulation approaches that separated the designspace into two safe and unsafe regions. The employedDRM-based meta-model required a one-dimensionalinterpolation method to approximate the LSF; hence,this study also investigated the e�ciency and accuracyof various interpolation techniques in applicability ofthe proposed method. Solving numerical and engineer-ing problems with 6 di�erent interpolation techniquesproved that among the investigated interpolation ap-proaches, using the spline and kriging method in theproposed approach provided results with acceptableaccuracy. By using the proposed framework, we foundout that e�ciency of the method was similar to that ofFORM, while its accuracy was close to that of MCS.

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Biographies

Mehralah Rakhshani Mehr received his BSc inCivil Engineering from University of Sistan andBaluchestan (USB), Iran, in 2007, and MSc and PhDin Structural Engineering from USB in 2009 and 2014,respectively. He is currently Assistant Professor atAL-Zahra University, Tehran, Iran. His main researchareas are concrete structures analysis and rehabilita-tion, dynamics of structures, and structural reliabilityanalysis.

Hamed Mirkamali received his BSc in Civil Engi-neering from Islamic Azad University, Zabol, Iran, in2012, and MSc in Earthquake Engineering from ShirazUniversity of Technology, Shiraz, Iran, in 2015. Hecarried out his research thesis entitled \VulnerabilityAssessment of Gas Network Using the First Order andSimulation Reliability Methods" under supervision ofDr. Hossain Rahnema and acquired excellent gradeof 19.4 out of 20. His research achievements havebeen quite signi�cant and he has already published twoconference papers.

Mohsen Rashki received his BSc in Civil Engineeringfrom University of Sistan and Baluchestan (USB), Iran,in 2007, and MSc and PhD in Structural Engineer-ing from USB in 2009 and 2014, respectively. Heis currently Assistant Professor in the ArchitectureDepartment of USB. He has published several researchpapers. His main research areas and interests arestructural reliability analysis, reliability-based designoptimization, and Bayesian probabilistic approaches.

Abdolhamid Bahrpeyma received his MSc in Ar-chitectural Engineering from University of Science andIndustry, Tehran, Iran, in 1994, and PhD in UrbanPlanning Engineering from Paris Nanterre University,France, in 2007. He is currently Assistant Professor inCivil Engineering Department at Sistan and Baluches-tan University.

adapted design of experiments for dimension decomposition...

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