# A New Formulation on Seismic Risk Assessment for Reinforced Concrete Structures with Both Random and Bounded Uncertainties

Authors: Xiao-Xiao Liu; Yuan-Sheng Wang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

A new formulation on seismic risk assessment for structures with both random and uncertain-but-bounded variables is investigated in this paper. Limit thresholds are regarded as random variables. The median of random variables is described through an improved multidimensional parallelepiped (IMP) convex model, in which the uncertain domain of the dependent bounded variables can be explicitly expressed. The corresponding Engineering Demand Parameters are taken to be dependent and follow a multidimensional lognormal distribution. Through matrix transformation, a given performance function is transformed into the regularized one. An effective method based on active learning Kriging model (ALK) is introduced to approximate the performance function in the region of interest rather than in the overall uncertain space. Based on ALK model, the failure probabilities for different limit states are calculated by using Monte Carlo Simulation (MCS). Further, the failure probabilities for different limit states in 50 years can be obtained through coupling the seismic failure probability with the ground motion hazard curve. A six-story reinforced concrete building subjected to ground motions is investigated to the efficiency and accuracy of the proposed method. The interstory drift and the acceleration as two responses of the case study are, respectively, obtained by utilizing Incremental Dynamic Analysis and nonlinear history analysis.


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## Body

## 1. Introduction

Recent earthquake hazards have caused serious economic and social loss [1, 2]. Currently, a number of academics [3–7] have emphasized the importance of performance-based seismic design (PBSD). A large amount of approaches for reliability analysis of structures has been proposed in recent years. Limit state fragility curves as considerable decision-making tools have been proposed to assess the reliability of RC structures [8–14]. The well-known Cornell’s three analytical seismic risk formulae have been widely used to carry out structural seismic reliability analysis [15, 16]. For instance, Eads et al. [17] used these analytical risk formulations to estimate collapse risk of a four-story office building. In Lu et al. [18, 19] the structural seismic hazard analysis could be studied by combining an improved cloud method and the analytical formulation of the damage hazard. Lü et al. [20] also applied Cornell’s three analytical seismic risk formulae to evaluate the seismic reliability of Chinese code-conforming buildings. In Wu et al. [21] the seismic risk assessment of RC buildings subjected to near-fault and far-fault ground motions was investigated using Cornell’s three analytical formulations. Moreover, the finite element reliability module based on the first-order reliability method (FORM) and MVFOSM has been proposed for seismic reliability problems [18, 19, 22, 23]. Song et al. [24] developed a component reliability method to identify the most probable failure members of RC buildings subjected to strong ground motions. Then the probability of a progressive collapse of the damaged structures could be calculated by the integral reliability method. Lü et al. [25] proposed a semi-analysis approach integrating the improved point method and moment method to analyze the nonlinear seismic reliability of a specific structure. The applicability of the reliability methodologies such as FORM, SORM, and HOMM was elaborated by Song [26] and Lu et al. [27]. Note that all of these reliability methods were based on the framework of performance-based seismic design (PBSD).However, the above-mentioned contributions are usually based on specific probability distributions of uncertain variables, which may be imprecise for some uncertain parameters because of insufficient experimental data. Unreasonable assumptions may cause misleading results in probabilistic reliability analysis [28, 29], if a probabilistic model is adopted. Therefore, the nonprobabilistic convex model was presented to describe the uncertain parameters with the limited available information. In the last several years, the structural reliability analysis methods based on the nonprobabilistic convex models have been intensively investigated [30–33] and they have provided effective supplements to traditional probabilistic reliability analysis. Moreover, the structural hybrid reliability analysis with both random uncertainty and bounded uncertainty has been proposed in recent years [34–36].Guo and Du [37] developed a unified reliability analysis framework to deal with both random and interval variables in multidisciplinary systems. Jiang et al. [38] constructed an algorithm with high efficiency and robust convergence performance to compute the hybrid reliability with both random and interval variables. Recently, an improved unified analysis approach [39] for structural hybrid reliability has been developed based on FORM. Moreover, sensitivity analysis for hybrid reliability with both probabilistic and convex variables was investigated in Guo and Du [34, 40], Wang et al. [41], and Zhang et al. [42, 43]. In Luo et al. [35], a probability and convex set mixed reliability model was proposed. Subsequently, the minimum reliability index was used as the constraint in reliability-based design optimization (RBDO) when both random and convex variables were considered. Yang et al. [44, 45] demonstrated that if performance functions were highly nonlinear or had multiple design points, the existing algorithms [38, 46] (Xiao et al., 2006) would be very inaccurate. To overcome the above problems, a number of scholars presented an active learning Kriging model (ALK) for hybrid reliability analysis [43–45, 47, 48]. When the Kriging model is constructed, the performance function need not be approximated throughout the uncertain space, but only in some region of interest.It is noteworthy that traditional convex models such as interval model and ellipsoidal model are found not capable of dealing with complex “multisource uncertainty” problems. Therefore, a more general convex model, namely, “multidimensional parallelepiped (MP) model”, was proposed in recent work [49–51]. This kind of convex model can take into account the independent and dependent uncertain parameters in a unified framework. To remedy the scarcity of the existing MP model, Ni et al. [52] presented an improved MP (IMP) model, in which the uncertainty domain of the interval variables could be explicitly expressed by a matrix inequality.In conclusion, based on the framework of PBSD, the reliability analysis of a given RC structure subjected to ground motions should be discussed with a combination of probability and IMP convex models. This means that a more general hybrid reliability analysis (MGHAR) for complex seismic engineering problems is developed in this paper. Limit thresholds are considered as random variables. The median of random variables is expressed by using the IMP model. The structural responses are taken to be dependent and follow a multidimensional lognormal distribution. Through matrix transformation, the performance function is mapped into the normalized performance function. A method based on ALK model named ALK-MGHAR is proposed. The reason is that a surrogate only rightly predicting the sign of the performance function is found capable of satisfying the precision requirement of MGHAR. Then Monte Carlo Simulation (MCS) is efficiently performed based on ALK-MGHAR. Further, the failure probabilities in 50 years can be computed by combining seismic failure probability and the ground motion hazard curve. This procedure is called ALK-MGHAR-MCS. A six-story RC building is used to demonstrate the efficiency and accuracy of ALK-MGHAR. The interstory drift and the acceleration are selected as two-dimensional Engineering Demand Parameters (EDPs), which are, respectively, calculated by Incremental Dynamic Analysis (IDA) and nonlinear history analysis (NHA).


## 2. MGHAR with MCS Method

### 2.1. Seismic Risk Formulation with Probabilistic Model

When only random variables are involved in an uncertain structure, the reliability in conjunction with the PBSD approach can be evaluated by traditional probabilistic reliability method. The limit state function or performance function is expressed asG(X), with the vector of random variables X={x1,x2,...,xm}. The reliability is denoted as the probability that the structural response exceeds the specified damage level under a given ground motion intensity. In probabilistic reliability theory, therefore, the failure probability of a structure for a specific limit state can be defined as follows:(1)Pf=PrGX<0=∫GX<0∫⋯∫fXdXwhere Pf is the structural failure probability under a damage state, Pr{•} represents the probability of an event, and f(X) is the joint probability density function (PDF) of random variables X. For normal random variables, X can be transformed into standard normal random variables u through a linear transformation, as follows:(2)ui=xi-μiσi,i=1,2,…mwhere μi and σi are mean and standard deviation of random variables xi, respectively.For nonnormal random variables, numerous available techniques, such as Nataf transformation [57] and Rosenblatt’s transformation [58], can transform the variables into approximately equivalent normal variables. Through such a treatment, the well-known FORM can be carried out for solving the structural reliability in (1).In order to evaluate the reliability of RC buildings, the familiar Cornell approach [16] is adopted in this paper. The mean annual frequency (MAF) of exceeding a specified limit state per year is normally defined as(3)λEDPLS=∫IMPEDP∣IMLS∣imdHIMimdimdimwhere PEDP∣IMLS∣im is the structural failure probability for a specific limit state and can be solved through (1). HIM(im)=P[IM≥im] is the ground motion hazard function and denotes the MAF of a specific earthquake event (IM≥im); IM is the intensity measure (peak ground acceleration, spectral acceleration, etc.).For specific site conditions, a simplified model proposed by Cornell et al. [15] can be used to carry out the ground motion hazard analysis, expressed as(4)HSax=PSa>x=k0·x-kwhere k0 is a constant depending on the ground motion characteristics,k is the slope of the seismic hazard curve in logarithmic coordinates, and Sa is spectral acceleration.When earthquake occurrences in time are assumed to be a Poisson process [59], the failure probabilities for different limit states in 50 years are calculated by using the following expression:(5)PEDP=1-1-λEDP50


### 2.2. Seismic Risk Formulation with Both Random and Bounded Variables

When both random variables and IMP convex variables appear in an uncertain structure, the performance function can be denoted asG(X,Y), where Y={y1,y2,...,ym} denotes the vector of marginal intervals and will be expounded in Section 3.2. The IMP convex variables are actually uncertain-but-bounded quantities. Due to the coexistence of random and bounded variables, the limit state G(X,Y)=0 produces a cluster of limit state surfaces in the stochastic space. The minimum limit state, minG(X,Y), which denotes the worst case of a given structure, is the most concerned in this study. A stringent reliability requirement can be satisfied only when the worst case is taken into account. For problems with both random and bounded variables, the failure probability of a structure for a specific limit state is defined as(6)PfU=PrminY⁡GX,Y⁡<0=∫minY⁡GX,Y<0∫⋯∫fXdXwhere PfU is the maximum failure probability when the minimum limit state is considered.The reliability analysis of RC structures with both random and IMP variables can be investigated using (6). When the interstory drift and the acceleration are simultaneously considered, the performance function further elaborated in Section 3 is expressed as G((Δ,δ),(D,A)). In this performance function, threshold capacity values corresponding to the two EDPs are described as random variables. The median of random variables is represented by the IMP variables. The failure probability of structures subjected to earthquakes is expressed as(7)PfU=∬minμα⁡GΔ,δ,D,AfD,A∣Sa=xdDdAwhere PfU is the maximum failure probability corresponding to the minimum limit state minG((Δ,δ),(D,A)), f(·) is the bivariate PDF,D is the interstory drift,A is the acceleration, and Δ andδ are threshold capacity values of the interstory drift and the acceleration, respectively.For convenience, this subsection usesM(w) to replace the minimum value of the performance function minG((Δ,δ),(D,A)). Then, the MAF of exceeding the two specific limit states per year is defined as(8)λD,AΔ,δ=∫∬MwfD,A∣Sa=xdDdAdHSaxdxdx=∫∬MwfD,A∣Sa=xdDdAhSaxdxwhere HSa(·) is the ground motion hazard function of the site derived from PHSA and Sa is the spectral acceleration. In this paper, MCS as the benchmark of ALK-MGHAR can be implemented to obtain an accurate result. Two steps required here are detailed as follows.Step 1.
A great deal of random samples involved in the performance functionG((Δ,δ),(D,A)) is generated.Step 2.
An optimization problem elaborated in Section4 is performed at each of the simulated samples. Subsequently, a failure indictor can be obtained at the corresponding sample.Based on the above-mentioned two steps, (8) can be rewritten as(9)λD,AΔ,δ=∫∬MwfD,A∣Sa=xdDdAhSaxdx=∫1NMCS∑j=1NIFwjhSaxdxwhere IFw is the failure indictor function and it can be expressed as(10)IFw=1Mw≥00Mw<0where M(w) is the minimum value of the performance function which can be elaborated in Section 3. In this paper, it should be noted that M(w)≥0 represents the failure region of a given structure and M(w)<0 denotes the safe region.Remark 1.
(i) The failure probabilities in 50 years can be calculated by (5). (ii) Since the uncertainties in both responses originated from the same source of uncertainties, the two EDPs are taken to be dependent and follow a bivariate lognormal distribution. The bivariate PDF is expressed as follows [7]:(11)fD,A∣Sa=x=12πDAσD∣xσA∣x1-ρ21/2∗exp⁡-121-ρ2∗lnD-μD∣xσD∣x2+lnA-μA∣xσA∣x2-2ρlnD-μD∣xσD∣xlnA-μA∣xσA∣xwhere μD∣x and σD∣x are log-mean and log-standard deviation of the maximum interstory drift, respectively; μA∣x and σA∣x are log-mean and the log-standard deviation of the acceleration, respectively; ρ is the correlation coefficient between lnD and lnA.The mean vectorμT and covariance Σ of the bivariate PDF are expressed as follows:(12)μD,AT=μD∣x,μA∣xΣ=σD∣x2ρσD∣xσA∣xρσD∣xσA∣xσA∣x2ρ can be estimated by the following expression:(13)ρ^=1/n∑i=1nlnDi·lnAi-μ^D∣im·μ^A∣imσ^D∣im·σ^A∣imwhere ρ^ is the estimator of the correlation coefficient ρ;n is the number of ground inputs; μ^D∣im and σ^D∣im are the estimators of log-mean and log-standard deviation of the maximum interstory drift, respectively; μ^A∣im and σ^A∣im are the estimators of log-mean and log-standard deviation of acceleration, respectively.


## 2.1. Seismic Risk Formulation with Probabilistic Model

When only random variables are involved in an uncertain structure, the reliability in conjunction with the PBSD approach can be evaluated by traditional probabilistic reliability method. The limit state function or performance function is expressed asG(X), with the vector of random variables X={x1,x2,...,xm}. The reliability is denoted as the probability that the structural response exceeds the specified damage level under a given ground motion intensity. In probabilistic reliability theory, therefore, the failure probability of a structure for a specific limit state can be defined as follows:(1)Pf=PrGX<0=∫GX<0∫⋯∫fXdXwhere Pf is the structural failure probability under a damage state, Pr{•} represents the probability of an event, and f(X) is the joint probability density function (PDF) of random variables X. For normal random variables, X can be transformed into standard normal random variables u through a linear transformation, as follows:(2)ui=xi-μiσi,i=1,2,…mwhere μi and σi are mean and standard deviation of random variables xi, respectively.For nonnormal random variables, numerous available techniques, such as Nataf transformation [57] and Rosenblatt’s transformation [58], can transform the variables into approximately equivalent normal variables. Through such a treatment, the well-known FORM can be carried out for solving the structural reliability in (1).In order to evaluate the reliability of RC buildings, the familiar Cornell approach [16] is adopted in this paper. The mean annual frequency (MAF) of exceeding a specified limit state per year is normally defined as(3)λEDPLS=∫IMPEDP∣IMLS∣imdHIMimdimdimwhere PEDP∣IMLS∣im is the structural failure probability for a specific limit state and can be solved through (1). HIM(im)=P[IM≥im] is the ground motion hazard function and denotes the MAF of a specific earthquake event (IM≥im); IM is the intensity measure (peak ground acceleration, spectral acceleration, etc.).For specific site conditions, a simplified model proposed by Cornell et al. [15] can be used to carry out the ground motion hazard analysis, expressed as(4)HSax=PSa>x=k0·x-kwhere k0 is a constant depending on the ground motion characteristics,k is the slope of the seismic hazard curve in logarithmic coordinates, and Sa is spectral acceleration.When earthquake occurrences in time are assumed to be a Poisson process [59], the failure probabilities for different limit states in 50 years are calculated by using the following expression:(5)PEDP=1-1-λEDP50


## 2.2. Seismic Risk Formulation with Both Random and Bounded Variables

When both random variables and IMP convex variables appear in an uncertain structure, the performance function can be denoted asG(X,Y), where Y={y1,y2,...,ym} denotes the vector of marginal intervals and will be expounded in Section 3.2. The IMP convex variables are actually uncertain-but-bounded quantities. Due to the coexistence of random and bounded variables, the limit state G(X,Y)=0 produces a cluster of limit state surfaces in the stochastic space. The minimum limit state, minG(X,Y), which denotes the worst case of a given structure, is the most concerned in this study. A stringent reliability requirement can be satisfied only when the worst case is taken into account. For problems with both random and bounded variables, the failure probability of a structure for a specific limit state is defined as(6)PfU=PrminY⁡GX,Y⁡<0=∫minY⁡GX,Y<0∫⋯∫fXdXwhere PfU is the maximum failure probability when the minimum limit state is considered.The reliability analysis of RC structures with both random and IMP variables can be investigated using (6). When the interstory drift and the acceleration are simultaneously considered, the performance function further elaborated in Section 3 is expressed as G((Δ,δ),(D,A)). In this performance function, threshold capacity values corresponding to the two EDPs are described as random variables. The median of random variables is represented by the IMP variables. The failure probability of structures subjected to earthquakes is expressed as(7)PfU=∬minμα⁡GΔ,δ,D,AfD,A∣Sa=xdDdAwhere PfU is the maximum failure probability corresponding to the minimum limit state minG((Δ,δ),(D,A)), f(·) is the bivariate PDF,D is the interstory drift,A is the acceleration, and Δ andδ are threshold capacity values of the interstory drift and the acceleration, respectively.For convenience, this subsection usesM(w) to replace the minimum value of the performance function minG((Δ,δ),(D,A)). Then, the MAF of exceeding the two specific limit states per year is defined as(8)λD,AΔ,δ=∫∬MwfD,A∣Sa=xdDdAdHSaxdxdx=∫∬MwfD,A∣Sa=xdDdAhSaxdxwhere HSa(·) is the ground motion hazard function of the site derived from PHSA and Sa is the spectral acceleration. In this paper, MCS as the benchmark of ALK-MGHAR can be implemented to obtain an accurate result. Two steps required here are detailed as follows.Step 1.
A great deal of random samples involved in the performance functionG((Δ,δ),(D,A)) is generated.Step 2.
An optimization problem elaborated in Section4 is performed at each of the simulated samples. Subsequently, a failure indictor can be obtained at the corresponding sample.Based on the above-mentioned two steps, (8) can be rewritten as(9)λD,AΔ,δ=∫∬MwfD,A∣Sa=xdDdAhSaxdx=∫1NMCS∑j=1NIFwjhSaxdxwhere IFw is the failure indictor function and it can be expressed as(10)IFw=1Mw≥00Mw<0where M(w) is the minimum value of the performance function which can be elaborated in Section 3. In this paper, it should be noted that M(w)≥0 represents the failure region of a given structure and M(w)<0 denotes the safe region.Remark 1.
(i) The failure probabilities in 50 years can be calculated by (5). (ii) Since the uncertainties in both responses originated from the same source of uncertainties, the two EDPs are taken to be dependent and follow a bivariate lognormal distribution. The bivariate PDF is expressed as follows [7]:(11)fD,A∣Sa=x=12πDAσD∣xσA∣x1-ρ21/2∗exp⁡-121-ρ2∗lnD-μD∣xσD∣x2+lnA-μA∣xσA∣x2-2ρlnD-μD∣xσD∣xlnA-μA∣xσA∣xwhere μD∣x and σD∣x are log-mean and log-standard deviation of the maximum interstory drift, respectively; μA∣x and σA∣x are log-mean and the log-standard deviation of the acceleration, respectively; ρ is the correlation coefficient between lnD and lnA.The mean vectorμT and covariance Σ of the bivariate PDF are expressed as follows:(12)μD,AT=μD∣x,μA∣xΣ=σD∣x2ρσD∣xσA∣xρσD∣xσA∣xσA∣x2ρ can be estimated by the following expression:(13)ρ^=1/n∑i=1nlnDi·lnAi-μ^D∣im·μ^A∣imσ^D∣im·σ^A∣imwhere ρ^ is the estimator of the correlation coefficient ρ;n is the number of ground inputs; μ^D∣im and σ^D∣im are the estimators of log-mean and log-standard deviation of the maximum interstory drift, respectively; μ^A∣im and σ^A∣im are the estimators of log-mean and log-standard deviation of acceleration, respectively.


## 3. Two-Dimensional Performance Limit State Function

### 3.1. Performance Function of Both Limit States

The performance function of both limit states, which allows considering the relationship between EDPs and limit thresholds, is defined as follows [7, 60]:(14)GΔμΔ,σΔ,ZμZ,σZ,D,A=AZμZ,σZN1+DΔμZ,σZN2-1where D is the vector of the maximum interstory drift, Δ is the vector of the maximum interstory drift threshold, A is the vector of the acceleration, Z is the vector of the acceleration threshold, μΔ and σΔ are mean and standard deviation of the interstory drift threshold, respectively, and μZ and σZ are mean and standard deviation of the acceleration threshold, respectively.D andAare considered as random variables and follow the bivariate lognormal distribution. Δ and Z are assumed to be lognormally distributed. μΔ and μZ are described by the IMP variables.The desired performance function can guarantee that the two peak EDPs stay below their respective critical values over a specified duration. WhenN1=1, a sector/triangle acceptable region is generated to realize the equivalent between the notion of the performance function and treatment of joint probability density function (JPDF) of the two dependent EDPs. Therefore, (14) can be simplified as(15)GΔμΔ,σΔ,ZμZ,σZ,D,A=AZμZ,σZ+DΔμΔ,σΔN2-1When the simplest case withN2 = 1 is considered, the performance function is written as(16)GΔμΔ,σΔ,ZμZ,σZ,D,A=DZμZ,σZ+AΔμΔ,σΔΔμΔ,σΔZμZ,σZ-1


### 3.2. The IMP Model

Recent work [49, 50] indicated that the MP model could deal with the problems where correlated variables and independent variables coexist. However, Ni et al. [52] and Jiang et al. [51] stated the main deficiencies of the existing MP model and then proposed an IMP model. The correlation coefficient between uncertain-but-bounded variables in the IMP model can be easily calculated and the uncertainty domain of the bounded variables can be explicitly expressed through a matrix inequality. When a two-dimensional problem with bounded variables μΔ and μZ is considered, the IMP model degenerates into a parallelogram. The marginal intervals of the two variables μΔ and μZ are denoted as μΔI and μZI, respectively. The two marginal intervals μΔI and μZI are defined as(17)μΔI=μΔL,μΔR,μΔC=μΔL+μΔR2,μΔW=μΔR-μΔL2(18)μZI=μZL,μZR,μZC=μZL+μZR2,μZW=μZR-μZL2where μΔL and μΔR are the lower bound and upper bound of μΔI; μZL and μZR are the lower bound and upper bound of μZI; μΔC and μZC represent the midpoints of μΔI and μZI; μΔW and μZW represent interval radii of the two marginal intervals. If bounded variables μΔ and μZ are independent of each other, the uncertainty domain of the two variables will become a rectangular domain μΔL,μΔR×μZL,μZR, expressed as Ωr. If the two bounded variables are dependent, the uncertainty domain will form a parallelogram domain Ωp, as shown in Figure 1. According to the principles proposed by Ni et al. [52], the uncertainty domain in the IMP model can be constructed.Figure 1
The IMP convex model with a two-dimensional problem.It can be observed from Figure1 that the center of the parallelogram coincides with that of the rectangular domain. Moreover, the shape of the parallelogram can exhibit the degree of correlation between μΔI and μZI (Ni et al. [52] have completed the proof). In the IMP model, a quadrilateral side need not be set to be parallel to the abscissa axis. From Figure 1, it can be seen thatf represents the semi-axis length in the direction HD→ ande represents the semi-axis length in the direction HC→. A new correlation coefficient between uncertain-but-bounded variables can be defined as(19)ρΔZ=e-fe+fwhere ρΔZ has the range [-11], ρΔZ=ρZΔ and ρΔΔ=ρZZ=1. When m=0 and ρΔZ=1, the bounded variables μΔ and μZ are linearly and positively dependent. When e=0 and ρΔZ=-1, the two bounded variables are linearly and negatively dependent. When e=f and ρΔZ=0, the two bounded variables are independent. When 0<f<e and ρΔZ>0, the two bounded variables are positively dependent. When 0<e<f and ρΔZ<0, the two variables are negatively dependent. Then, the symmetrical correlation matrix ρ can be defined as(20)ρ=ρΔΔρΔZρZΔρZZ=1ρΔZρZΔ1Subsequently, the uncertainty domain established by the IMP model can be explicitly expressed by the following matrix inequality [51, 52]:(21)-e≤ρ-1T-1R-1μ-μC≤ewhere the diagonal matrices T, R and the vectors e, μ, μC are defined as(22)T=diagw1,w2,wi=1∑j=12ρi,j,i=1,2R=diagμΔW,μZW,e=11T,μ=μΔμZT,μC=μΔCμZCTin which diag (w1,w2) represents the diagonal elements w1 and w2 of the diagonal matrix and ρ(i,j) represents the element in theith row and thejth column of the correlation matrix ρ. Further, aShape Matrix C can be defined as [51, 52](23)C=RTρEquation (21) can be rewritten as follows:(24)C-1μ-μC≤eNote that when the interval radiusμΔW is equal to μZW, the parallelogram is a rhomb. The IMP domain can be expressed as(25)Ωp=μ∣C-1μ-μC≤eConsider a generaln-dimensional problem with marginal intervals μiI, i=1,2,⋯,n, of uncertain variables. In such circumstances, the uncertain domain is a multidimensional parallelepiped. Analogous to the definition of (20), the correlation between any two uncertain-but-bounded variables μi and μj can be expressed by the correlation coefficient ρij. Then, the correlation matrix ρ for all of bounded variables can be defined as [51, 52](26)ρ=1ρ12⋯ρ1nρ211⋯ρ2n⋯⋯⋱⋮ρn1ρn2⋯1where ρ is an m×m symmetric matrix. The analytical expression of the multidimensional parallelepiped uncertain domain can be derived in the similar way from the two-dimensional case, and the detailed process is omitted.Remark 2.
(i) When creating a new IMP model, we need the marginal intervals of all bounded variables and the correlation between any two variables. Assume that there aren-dimensional uncertain variables μi,i=1,2,…,n, andm test samples μ(r),r=1,2,…,m, then the detailed procedure of building the IMP convex model could be found in Jiang et al. [49, 50] and Ni et al. [52]. For multidimensional problems, the IMP model can be efficiently developed through decomposing the complicated n-dimensional problems into n(n+1)/2 sample two-dimensional problems.


### 3.3. Normalization of the Uncertainty DomainΩp

To transform the IMP model into a regular interval (bounded) model, a regularization method will be performed in this section. Then, the parallelepiped uncertainty domainΩp can become a multidimensional cube after the regularization method is adopted. Introduce the following transformation:(27)δ=C-1μ-μCThe uncertain-but-bounded variablesμ can be mapped into the δ space and then the uncertainty domain Ωp becomes ann-dimensional cube Ω∗, denoted as(28)Ω∗=δ∣δ≤ewhere δ={δ1,δ2,…,δn}.The semi-axis lengthe andf can be solved when only translation, scaling, and rotation are required in the above process. Thus, this regularization method can be categorized as affine transformation proposed by Jiang et al. [49, 51]. By using the transformation, the uncertainty domain Ωp becomes a standard cube center and the length of each side is 2 in δ space. Meanwhile, the bounded variables are independent of each other in δ space.


## 3.1. Performance Function of Both Limit States

The performance function of both limit states, which allows considering the relationship between EDPs and limit thresholds, is defined as follows [7, 60]:(14)GΔμΔ,σΔ,ZμZ,σZ,D,A=AZμZ,σZN1+DΔμZ,σZN2-1where D is the vector of the maximum interstory drift, Δ is the vector of the maximum interstory drift threshold, A is the vector of the acceleration, Z is the vector of the acceleration threshold, μΔ and σΔ are mean and standard deviation of the interstory drift threshold, respectively, and μZ and σZ are mean and standard deviation of the acceleration threshold, respectively.D andAare considered as random variables and follow the bivariate lognormal distribution. Δ and Z are assumed to be lognormally distributed. μΔ and μZ are described by the IMP variables.The desired performance function can guarantee that the two peak EDPs stay below their respective critical values over a specified duration. WhenN1=1, a sector/triangle acceptable region is generated to realize the equivalent between the notion of the performance function and treatment of joint probability density function (JPDF) of the two dependent EDPs. Therefore, (14) can be simplified as(15)GΔμΔ,σΔ,ZμZ,σZ,D,A=AZμZ,σZ+DΔμΔ,σΔN2-1When the simplest case withN2 = 1 is considered, the performance function is written as(16)GΔμΔ,σΔ,ZμZ,σZ,D,A=DZμZ,σZ+AΔμΔ,σΔΔμΔ,σΔZμZ,σZ-1


## 3.2. The IMP Model

Recent work [49, 50] indicated that the MP model could deal with the problems where correlated variables and independent variables coexist. However, Ni et al. [52] and Jiang et al. [51] stated the main deficiencies of the existing MP model and then proposed an IMP model. The correlation coefficient between uncertain-but-bounded variables in the IMP model can be easily calculated and the uncertainty domain of the bounded variables can be explicitly expressed through a matrix inequality. When a two-dimensional problem with bounded variables μΔ and μZ is considered, the IMP model degenerates into a parallelogram. The marginal intervals of the two variables μΔ and μZ are denoted as μΔI and μZI, respectively. The two marginal intervals μΔI and μZI are defined as(17)μΔI=μΔL,μΔR,μΔC=μΔL+μΔR2,μΔW=μΔR-μΔL2(18)μZI=μZL,μZR,μZC=μZL+μZR2,μZW=μZR-μZL2where μΔL and μΔR are the lower bound and upper bound of μΔI; μZL and μZR are the lower bound and upper bound of μZI; μΔC and μZC represent the midpoints of μΔI and μZI; μΔW and μZW represent interval radii of the two marginal intervals. If bounded variables μΔ and μZ are independent of each other, the uncertainty domain of the two variables will become a rectangular domain μΔL,μΔR×μZL,μZR, expressed as Ωr. If the two bounded variables are dependent, the uncertainty domain will form a parallelogram domain Ωp, as shown in Figure 1. According to the principles proposed by Ni et al. [52], the uncertainty domain in the IMP model can be constructed.Figure 1
The IMP convex model with a two-dimensional problem.It can be observed from Figure1 that the center of the parallelogram coincides with that of the rectangular domain. Moreover, the shape of the parallelogram can exhibit the degree of correlation between μΔI and μZI (Ni et al. [52] have completed the proof). In the IMP model, a quadrilateral side need not be set to be parallel to the abscissa axis. From Figure 1, it can be seen thatf represents the semi-axis length in the direction HD→ ande represents the semi-axis length in the direction HC→. A new correlation coefficient between uncertain-but-bounded variables can be defined as(19)ρΔZ=e-fe+fwhere ρΔZ has the range [-11], ρΔZ=ρZΔ and ρΔΔ=ρZZ=1. When m=0 and ρΔZ=1, the bounded variables μΔ and μZ are linearly and positively dependent. When e=0 and ρΔZ=-1, the two bounded variables are linearly and negatively dependent. When e=f and ρΔZ=0, the two bounded variables are independent. When 0<f<e and ρΔZ>0, the two bounded variables are positively dependent. When 0<e<f and ρΔZ<0, the two variables are negatively dependent. Then, the symmetrical correlation matrix ρ can be defined as(20)ρ=ρΔΔρΔZρZΔρZZ=1ρΔZρZΔ1Subsequently, the uncertainty domain established by the IMP model can be explicitly expressed by the following matrix inequality [51, 52]:(21)-e≤ρ-1T-1R-1μ-μC≤ewhere the diagonal matrices T, R and the vectors e, μ, μC are defined as(22)T=diagw1,w2,wi=1∑j=12ρi,j,i=1,2R=diagμΔW,μZW,e=11T,μ=μΔμZT,μC=μΔCμZCTin which diag (w1,w2) represents the diagonal elements w1 and w2 of the diagonal matrix and ρ(i,j) represents the element in theith row and thejth column of the correlation matrix ρ. Further, aShape Matrix C can be defined as [51, 52](23)C=RTρEquation (21) can be rewritten as follows:(24)C-1μ-μC≤eNote that when the interval radiusμΔW is equal to μZW, the parallelogram is a rhomb. The IMP domain can be expressed as(25)Ωp=μ∣C-1μ-μC≤eConsider a generaln-dimensional problem with marginal intervals μiI, i=1,2,⋯,n, of uncertain variables. In such circumstances, the uncertain domain is a multidimensional parallelepiped. Analogous to the definition of (20), the correlation between any two uncertain-but-bounded variables μi and μj can be expressed by the correlation coefficient ρij. Then, the correlation matrix ρ for all of bounded variables can be defined as [51, 52](26)ρ=1ρ12⋯ρ1nρ211⋯ρ2n⋯⋯⋱⋮ρn1ρn2⋯1where ρ is an m×m symmetric matrix. The analytical expression of the multidimensional parallelepiped uncertain domain can be derived in the similar way from the two-dimensional case, and the detailed process is omitted.Remark 2.
(i) When creating a new IMP model, we need the marginal intervals of all bounded variables and the correlation between any two variables. Assume that there aren-dimensional uncertain variables μi,i=1,2,…,n, andm test samples μ(r),r=1,2,…,m, then the detailed procedure of building the IMP convex model could be found in Jiang et al. [49, 50] and Ni et al. [52]. For multidimensional problems, the IMP model can be efficiently developed through decomposing the complicated n-dimensional problems into n(n+1)/2 sample two-dimensional problems.


## 3.3. Normalization of the Uncertainty DomainΩp

To transform the IMP model into a regular interval (bounded) model, a regularization method will be performed in this section. Then, the parallelepiped uncertainty domainΩp can become a multidimensional cube after the regularization method is adopted. Introduce the following transformation:(27)δ=C-1μ-μCThe uncertain-but-bounded variablesμ can be mapped into the δ space and then the uncertainty domain Ωp becomes ann-dimensional cube Ω∗, denoted as(28)Ω∗=δ∣δ≤ewhere δ={δ1,δ2,…,δn}.The semi-axis lengthe andf can be solved when only translation, scaling, and rotation are required in the above process. Thus, this regularization method can be categorized as affine transformation proposed by Jiang et al. [49, 51]. By using the transformation, the uncertainty domain Ωp becomes a standard cube center and the length of each side is 2 in δ space. Meanwhile, the bounded variables are independent of each other in δ space.


## 4. Application to Reliability Analysis

The performance limit state equation (15) defines the failure mode of the RC structure and it plays an important role in the reliability assessment. For convenience of expression, the vectorsD,A, Δ,Zare uniformly expressed asX=D, A, Δ,Z. Through the regularization of the uncertain variablesXand μ intouand δ elaborated in Section 3, the limit state functionG(X, μ) can be mapped into the regularized limit state functionG(u, δ). Since the random and bounded variables coexist, correspondingly, the limit stateG(u, δ) constitutes a cluster of limit state surfaces in the standardu space. In the wholeu-space Ωu, a safe region of thelimit state functionG(u,δ)is denoted as Ωs=u∣minδ⁡Gu,δ<0 and a critical region is expressed as Ωc=u∣G(u,δ)=0. Then a failure region can be denoted as Ωf=Ω∖(Ωs∪Ωc). Therefore, the failure probability with both random and bounded variables can be defined as(29)PfU=PrMu=minδ⁡Gu,δ>0∣δ∈Ω∗where M(u) is the minimization of the limit state function and can be solved by the following optimization problem:(30)minδGu,δs.t.δ≤eIn this paper, a global optimization algorithm called as DIRECT algorithm [61] is used to realize the global optimum. Then, (8) can be rewritten as(31)λ=∫PrMu=minδ⁡Gu,δ>0∣δ∈Ω∗hSaxdxNote that the failure probabilities in 50 years can be obtained by substituting (31) into (5).If the FORM method is used to solve (29), (29) can be obtained by using the following double-loop optimization:(32)β∗=minuuTus.t.minδ⁡Gu,δ=0δ≤ewhere β∗ is the minimum reliability index. A mathematical programming method (MPM) and a single-loop iterative (SLI) method [35] were proposed to solve the above double-loop optimization problem. Subsequently, the maximum failure probability of a structure can be approximately calculated by the following expression:(33)PfU=Φ-β∗=1-∫-∞β∗12πexp⁡-u22duGeometrically,β∗ is the shortest distance from the origin to the region Ωc in the standardu-space Ω. Some literatures [38, 62, 63] showed that the reliability of a structure with both random and convex variables could be evaluated using the minimum reliability index. However, β∗ could not accurately measure the reliability of a structure when the performance function was nonlinear or had multiple design points [64, 65]. Therefore, an efficient method based on active learning Kriging model (ALK) will be used to predict (30).


## 5. ALK-MGHAR for the Reliability Analysis

### 5.1. Fundamental Theory

To reduce the computational cost, an ALK method [43, 47, 48, 66–68] is introduced to approximate the performance in the region of interest rather than throughout the overall space. Based on the Kriging model, then, the failure probability PfU can be calculated through MCS method. When the Kriging model is built, the performance function need not be approximated throughout the whole uncertainty space, but only in the region of interest. An imitated Efficient Global Optimization (EGO) for global optimization is usually required in ALK models. When searching the minimum value within global optimization, we require selecting the points at which the value of the objective function G(u,δ) has the maximum expectation value to be smaller than the current least value. Therefore, a learning function called Expected Improvement Function (EIF) proposed by Jones et al. [69] was employed in EGO. For the problems with both random and bounded variables, a right predication for the sign of the objective function G(u,δ) rather than the specific value of G(u,δ) can be adopted in the Kriging model. In order to greatly enhance the prediction for the sign of G(u,δ), the point at which the sign of G(u,δ) has the maximum risk value to be wrongly predicted should be added into the DoE. Moreover, an active learning function termed as Expected Risk Function (ERF) [43, 44] is used to recognize this point. The reason why a Kriging model providing a right prediction for the sign of G(u,δ) can satisfy the precision requirement of MGHAR is given by the following corollary. Note that the proposed corollary is based on the existing properties [45, 64]. Therefore, the proof of the corollary is omitted.Corollary 3.
If a Kriging model can rightly predict the sign ofG(u,δ), then the sign of minδ⁡G(u,δ) is the same as that of minδ⁡G^(u,δ). Sequentially, the optimization problem defined by (30) can be replaced by the following optimization:(34)minδG^u,δs.t.δ≤e


### 5.2. Revisiting Kriging Model

For the sake of convenience, we use the vectorΘ to replace (A,D,Z,Δ). The Kriging model G(Θ) can be denoted as(35)GΘ=FΘ,β+zΘwhere F(Θ,β) is the polynomial regression part and β is the vector of regression coefficients. z(Θ) is a Gaussian random process whose mean and covariance are expressed as(36)EzΘ=0covza,zb=σ2Rθ,a,bwhere σ is the standard deviation of the Gaussian process, a and b are two arbitrary points, and R(·) is a correlation function with parameters θ, a, and b. Here, the Gaussian function is selected and it can be expressed as(37)Rθ,c,d=exp⁡-∑i=1nθici-di2where n is the number of sample points and θi, ci, and di are theith component of θ, c, and d, respectively.The Kriging model needs a DoE to define its statistical parameters and then predictions forG(Θ) at unknown points can be implemented. Given a DoE: Θ1,Θ2,⋯,ΘmT with Θ(i) theith training points and a vector of the performance function G=GΘ1,GΘ2,⋯,GΘmT with G(Θ(i)) theith function value. In this study, ordinary Kriging model [70] is utilized to simplify (35) which means that F(Θ,β) can be replaced by a constant β^. Therefore, the predicted value G^(Θ) and the predicted variance s2(Θ) are, respectively, represented by(38)G^Θ=β^+rTΘRG-1β^(39)s2Θ=σ21-rTΘRrΘ+1-1TR-1rΘ21TR-11In (38)-(39),(40)β^=1TR-11-11TR-1G(41)σ2=1mG-β^1TR-1G-β^1(42)rΘ=Rθ,Θ,Θ1,Rθ,Θ,Θ2,⋯Rθ,Θ,ΘmIn (38)-(39), R is an m×m matrix whose element is given by(43)Rij=Rθ,Θi,ΘjIn (38)-(42), 1 denotesm-dimensional unit vector and r(Θ) is the correlation vector between Θ andith training point.The parameterθ in (42)-(43) can be solved through the unconstrained optimization problem, expressed as(44)θopt=argminθ⁡R1/mσ2Because only the Kriging model with the global optimum parameterθopt can provide the most exact predictions for the performance function G(Θ), the optimal parameter θopt should be solved through global optimization strategy and the modified DIRECT algorithm [71] is adopted in this paper.The Kriging model provides a predicted valueG^(Θ) for the performance function G(Θ) at an unknown point Θ. Nevertheless, there exist some uncertainties in this prediction process because a Gaussian random variate G(Θ)~N(G^(Θ),s2(Θ)). It should be noted that G^(Θ) is only an approximate value rather than an actual value of G(Θ). Therefore, there exists a risk that the sign of G(Θ) is wrongly predicted. To implement the prediction for the sign of G(Θ), the point at which the sign of G(Θ) has the largest risk value to be wrongly predicted needs to be identified. Then this point should be added into DoE and the sign prediction of G(Θ) is significantly improved. To identify the aforesaid point and build an ALK model, a learning function named ERF is introduced to provide a precise sign prediction and it is expressed as(45)ERΘ=-signG^ΘG^ΘΦ-signG^ΘG^ΘsΘ+sΘϕG^ΘsΘin which sign(·) is the sign function and Φ(·) and ϕ(·) are the cumulative distribution function (CDF) and the PDF of the standard normal distribution, respectively. If a point is able to maximize ERF, the sign of G(Θ) at this point will have the largest risk being wrongly predicted. Subsequently, this point is added into DoE. The procedure of combining MCS and ALK-MGHAR named ALK-MGHAR-MCS will be presented in Section 5.3.


### 5.3. Summary of ALK-MGHAR-MCS for Reliability Analysis

Step 1 (selection of recorded ground motions).
Thirty real earthquake records (M=6.0-7.0,R=15-20km) are chosen whose response shapes are similar to the shape of the target response spectrum, according to the selection of ground motion inputs elaborated by Liu et al. [7].Step 2 (analysis of the structure subjected to each of the ground motions).
Analyze a given structure under each of the ground motions generated in Step 1 which is scaled from low hazard levels to higher hazard levels until a structural collapse occurs. The maximum interstory drift can be obtained using IDA and the maximum acceleration needs to be calculated through NHA. The present writer [7] illustrated the methodology of calculating the two EDPs.Step 3 (estimate of the mean and the standard deviation originated from (11)).
The mean and the standard deviation from the response samples obtained in Step 2 are required and then the bivariate PDF can be evaluated. Subsequently, a great deal of sampling points in the performance function (11) can be generated through the bivariate PDF.Step 4 (define the initial DoE).
(a) The number of training points included in the initial DOE is supposed to be small. The number of 12 is selected in this study, according to the existing investigations [47, 48].
(b) Latin hypercube sampling (LHS) is utilized to produce the samples, which are uniformly distributed in the uncertain space. For bivariate lognormal variablesDandA, Nataf transformation is used to transform the two variables into the approximately equivalent normal variables. The lower and upper bounds of equivalent normal variables are chosen as Fi-1(Φ(±5)) and then these variables are converted into the standardu-space. Here, Fi-1(·) is the inverse CDF of equivalent normal variables. Two steps are required for the IMP convex variables μΔ and μZ. Firstly, the IMP model needs to be constructed. Then LHS is used to generate a large amount of samples uniformly covering the space defined by (28) which is mapped into the original space defined in (25).
(c) These chosen sampling points are used to estimate the performance functionG(Θ) and then an initial Kriging model is constructed with the initial DoE.Step 5 (generate a large amount of samples as candidate points).
(a) Denote the set of candidate points asΩLHS. For bivariate lognormal random variables, MCS is adopted to generate samples. For IMP convex variables, LHS is employed to obtain the uniformly distributed sampling points as Step 4.
(b) Denote the number of sampling points inΩLHS as NΩLHS. To cover the overall uncertain space, the number of points NΩLHS (NΩLHS=105) is supposed to be sufficiently large. Note that the performance function is not calculated in this stage. All of the points are only considered as candidate points. The newly increased training points in the next few steps will be chosen among the points.Step 6 (identify the newly added training points in the set of the candidate points).
The point maximizing ERF among the candidates is selected as a new training point, which should be added into the DoE. The point is expressed as(X(∗),μ(∗)).Step 7 (stopping condition).
When the maximum value of ERF is small enough, the sign of the performance functionG(X,μ) in the uncertain space has little risk value to be wrongly predicted. Then, go to Step 9. The stopping condition adopted here is E[R(X(∗),μ(∗))]≤10-4.Step 8 (obtain an updated DoE and construct a new Kriging model).
If the stopping condition in Step 7 is not satisfied,G(X,μ) at the point (X(∗),μ(∗)) should be calculated. Add the point (X(∗),μ(∗)) into the DoE and construct a new Kriging model. Go back to Step 6.Step 9.
Implement MGHAR with MCS presented in Section2 based on the Kriging model.


## 5.1. Fundamental Theory

To reduce the computational cost, an ALK method [43, 47, 48, 66–68] is introduced to approximate the performance in the region of interest rather than throughout the overall space. Based on the Kriging model, then, the failure probability PfU can be calculated through MCS method. When the Kriging model is built, the performance function need not be approximated throughout the whole uncertainty space, but only in the region of interest. An imitated Efficient Global Optimization (EGO) for global optimization is usually required in ALK models. When searching the minimum value within global optimization, we require selecting the points at which the value of the objective function G(u,δ) has the maximum expectation value to be smaller than the current least value. Therefore, a learning function called Expected Improvement Function (EIF) proposed by Jones et al. [69] was employed in EGO. For the problems with both random and bounded variables, a right predication for the sign of the objective function G(u,δ) rather than the specific value of G(u,δ) can be adopted in the Kriging model. In order to greatly enhance the prediction for the sign of G(u,δ), the point at which the sign of G(u,δ) has the maximum risk value to be wrongly predicted should be added into the DoE. Moreover, an active learning function termed as Expected Risk Function (ERF) [43, 44] is used to recognize this point. The reason why a Kriging model providing a right prediction for the sign of G(u,δ) can satisfy the precision requirement of MGHAR is given by the following corollary. Note that the proposed corollary is based on the existing properties [45, 64]. Therefore, the proof of the corollary is omitted.Corollary 3.
If a Kriging model can rightly predict the sign ofG(u,δ), then the sign of minδ⁡G(u,δ) is the same as that of minδ⁡G^(u,δ). Sequentially, the optimization problem defined by (30) can be replaced by the following optimization:(34)minδG^u,δs.t.δ≤e


## 5.2. Revisiting Kriging Model

For the sake of convenience, we use the vectorΘ to replace (A,D,Z,Δ). The Kriging model G(Θ) can be denoted as(35)GΘ=FΘ,β+zΘwhere F(Θ,β) is the polynomial regression part and β is the vector of regression coefficients. z(Θ) is a Gaussian random process whose mean and covariance are expressed as(36)EzΘ=0covza,zb=σ2Rθ,a,bwhere σ is the standard deviation of the Gaussian process, a and b are two arbitrary points, and R(·) is a correlation function with parameters θ, a, and b. Here, the Gaussian function is selected and it can be expressed as(37)Rθ,c,d=exp⁡-∑i=1nθici-di2where n is the number of sample points and θi, ci, and di are theith component of θ, c, and d, respectively.The Kriging model needs a DoE to define its statistical parameters and then predictions forG(Θ) at unknown points can be implemented. Given a DoE: Θ1,Θ2,⋯,ΘmT with Θ(i) theith training points and a vector of the performance function G=GΘ1,GΘ2,⋯,GΘmT with G(Θ(i)) theith function value. In this study, ordinary Kriging model [70] is utilized to simplify (35) which means that F(Θ,β) can be replaced by a constant β^. Therefore, the predicted value G^(Θ) and the predicted variance s2(Θ) are, respectively, represented by(38)G^Θ=β^+rTΘRG-1β^(39)s2Θ=σ21-rTΘRrΘ+1-1TR-1rΘ21TR-11In (38)-(39),(40)β^=1TR-11-11TR-1G(41)σ2=1mG-β^1TR-1G-β^1(42)rΘ=Rθ,Θ,Θ1,Rθ,Θ,Θ2,⋯Rθ,Θ,ΘmIn (38)-(39), R is an m×m matrix whose element is given by(43)Rij=Rθ,Θi,ΘjIn (38)-(42), 1 denotesm-dimensional unit vector and r(Θ) is the correlation vector between Θ andith training point.The parameterθ in (42)-(43) can be solved through the unconstrained optimization problem, expressed as(44)θopt=argminθ⁡R1/mσ2Because only the Kriging model with the global optimum parameterθopt can provide the most exact predictions for the performance function G(Θ), the optimal parameter θopt should be solved through global optimization strategy and the modified DIRECT algorithm [71] is adopted in this paper.The Kriging model provides a predicted valueG^(Θ) for the performance function G(Θ) at an unknown point Θ. Nevertheless, there exist some uncertainties in this prediction process because a Gaussian random variate G(Θ)~N(G^(Θ),s2(Θ)). It should be noted that G^(Θ) is only an approximate value rather than an actual value of G(Θ). Therefore, there exists a risk that the sign of G(Θ) is wrongly predicted. To implement the prediction for the sign of G(Θ), the point at which the sign of G(Θ) has the largest risk value to be wrongly predicted needs to be identified. Then this point should be added into DoE and the sign prediction of G(Θ) is significantly improved. To identify the aforesaid point and build an ALK model, a learning function named ERF is introduced to provide a precise sign prediction and it is expressed as(45)ERΘ=-signG^ΘG^ΘΦ-signG^ΘG^ΘsΘ+sΘϕG^ΘsΘin which sign(·) is the sign function and Φ(·) and ϕ(·) are the cumulative distribution function (CDF) and the PDF of the standard normal distribution, respectively. If a point is able to maximize ERF, the sign of G(Θ) at this point will have the largest risk being wrongly predicted. Subsequently, this point is added into DoE. The procedure of combining MCS and ALK-MGHAR named ALK-MGHAR-MCS will be presented in Section 5.3.


## 5.3. Summary of ALK-MGHAR-MCS for Reliability Analysis

Step 1 (selection of recorded ground motions).
Thirty real earthquake records (M=6.0-7.0,R=15-20km) are chosen whose response shapes are similar to the shape of the target response spectrum, according to the selection of ground motion inputs elaborated by Liu et al. [7].Step 2 (analysis of the structure subjected to each of the ground motions).
Analyze a given structure under each of the ground motions generated in Step 1 which is scaled from low hazard levels to higher hazard levels until a structural collapse occurs. The maximum interstory drift can be obtained using IDA and the maximum acceleration needs to be calculated through NHA. The present writer [7] illustrated the methodology of calculating the two EDPs.Step 3 (estimate of the mean and the standard deviation originated from (11)).
The mean and the standard deviation from the response samples obtained in Step 2 are required and then the bivariate PDF can be evaluated. Subsequently, a great deal of sampling points in the performance function (11) can be generated through the bivariate PDF.Step 4 (define the initial DoE).
(a) The number of training points included in the initial DOE is supposed to be small. The number of 12 is selected in this study, according to the existing investigations [47, 48].
(b) Latin hypercube sampling (LHS) is utilized to produce the samples, which are uniformly distributed in the uncertain space. For bivariate lognormal variablesDandA, Nataf transformation is used to transform the two variables into the approximately equivalent normal variables. The lower and upper bounds of equivalent normal variables are chosen as Fi-1(Φ(±5)) and then these variables are converted into the standardu-space. Here, Fi-1(·) is the inverse CDF of equivalent normal variables. Two steps are required for the IMP convex variables μΔ and μZ. Firstly, the IMP model needs to be constructed. Then LHS is used to generate a large amount of samples uniformly covering the space defined by (28) which is mapped into the original space defined in (25).
(c) These chosen sampling points are used to estimate the performance functionG(Θ) and then an initial Kriging model is constructed with the initial DoE.Step 5 (generate a large amount of samples as candidate points).
(a) Denote the set of candidate points asΩLHS. For bivariate lognormal random variables, MCS is adopted to generate samples. For IMP convex variables, LHS is employed to obtain the uniformly distributed sampling points as Step 4.
(b) Denote the number of sampling points inΩLHS as NΩLHS. To cover the overall uncertain space, the number of points NΩLHS (NΩLHS=105) is supposed to be sufficiently large. Note that the performance function is not calculated in this stage. All of the points are only considered as candidate points. The newly increased training points in the next few steps will be chosen among the points.Step 6 (identify the newly added training points in the set of the candidate points).
The point maximizing ERF among the candidates is selected as a new training point, which should be added into the DoE. The point is expressed as(X(∗),μ(∗)).Step 7 (stopping condition).
When the maximum value of ERF is small enough, the sign of the performance functionG(X,μ) in the uncertain space has little risk value to be wrongly predicted. Then, go to Step 9. The stopping condition adopted here is E[R(X(∗),μ(∗))]≤10-4.Step 8 (obtain an updated DoE and construct a new Kriging model).
If the stopping condition in Step 7 is not satisfied,G(X,μ) at the point (X(∗),μ(∗)) should be calculated. Add the point (X(∗),μ(∗)) into the DoE and construct a new Kriging model. Go back to Step 6.Step 9.
Implement MGHAR with MCS presented in Section2 based on the Kriging model.


## 6. Case Study: A Six-Story RC Building

### 6.1. Design of the Case Study according to Chinese Codes

To verify the efficiency and accuracy of the proposed method, a sample six-story reinforced concrete (RC) frame building is taken for case study in this paper. The example structure is designed according to Chinese codes [72], which represents the typical mid-rise RC buildings in China. The geographical location of the modelled building is designated to be western China. This is a Class II site, with eight times the intensity of an earthquake and the designing ground acceleration is set as 0.20. The seismic grade of the example building is level 2 [73]. The plane and elevation views of the designed frame structure are shown in Figure 2. The total height of building is 19.8 m. The lengths of longitudinal spans and transversal spans are 36 m and 14.4 m, respectively. The columns and beams’ cross sections are 600 mm×600 mm and 300 mm×500 mm, respectively. The distributed steel in beams and columns is designed by using Chinese codes [72], as shown in Figure 3.Figure 2
The plane and elevation views of the designed frame structure.Figure 3
The distributed steel in columns and beams.


### 6.2. Modeling of the Designed RC Building

By using the finite element platform OpenSees [74], a three-dimensional model of the given structure (Figure 4) can be developed in this section. A rigid diaphragm MPC (multipoint constraints) is employed to model the floor concrete slabs. Nonlinear beam-column elements are used to model column and beam members. The Kent-Scott-Park model with no tension stiffening is adopted to model the concrete material. Column and beam cross sections are discretized to fibers of confined concrete, unconfined concrete, and steel reinforcement. The structure is considered as deterministic and other parameters are not regarded as uncertain in the present analysis. The parameters for confined concrete (identified by the subscript “core”) are listed asfc,core =34.5MPa,fu,core =24.1MPa, ε0,core =0.005, εu,core =0.020. The parameters for unconfined concrete (identified by the subscript “cover”) are described as follows:fc,cover=27.6MPa,fu,cover=0MPa, ε0,cover=0.002, εu,cover=0.006. A bilinear material called Steel01 material is used to model the reinforcing steel with parametersE=206GPa, fy=400MPa, andb=0.01. The structural responses such as interstory drift and acceleration can be, respectively, obtained through IDA and NHA. The fundamental mode for the given RC structure is set at 1.0018 sec by modal analysis.Figure 4
DoE obtained by ALK-MGHAR.


### 6.3. Determination of Performance Limit Levels and the Uncertainty Domain

In this study, structural performance limit states are partitioned into four levels, i.e., normal operation (NO), immediate occupancy (IO), life safety (LS), and collapse prevention (CP). Limit states of the interstory drift and the acceleration are simultaneously considered in this study. The threshold values of both EDPs under each performance level are described as random variables, listed in Table1. The median of random variables under each performance level is treated as the IMP variables. The marginal intervals of the uncertain-but-bounded variables are shown in Table 2. The correlation coefficients between the bounded variables defined by (19) are ρΔZ=ρZΔ=0.33. The correlation matrices under four performance levels can be expressed as ρNO, ρIO, ρLS, and ρCP. Then all of the correlation matrices can be built:(46)ρNO=ρIO=ρLS=ρCP=10.330.331Table 1
Threshold capacity values of interstory drift and peak floor acceleration.
Threshold capacities of  both the EDPs NO level IO level LS level CP level Mean SD1 Mean SD1 Mean SD1 Mean SD1 Interstory drift threshold μ Δ 1 0.002 μ Δ 2 0.005 μ Δ 3 0.015 μ Δ 4 0.025 Acceleration threshold μ Z 1 0.4 μ Z 2 0.6 μ Z 3 0.8 μ Z 4 1.2 1: standard deviation of the interstory drift threshold and acceleration originated from Sun et al. [53].Table 2
The marginal intervals of IMP convex variables.
IMP convex variables Marginal intervals  ofμΔ μ Δ 1 μ Δ 2 μ Δ 3 μ Δ 4 0.001 8 1 , 0.002 0 2 0.004 8 1 , 0.005 0 2 0.014 8 1 , 0.015 0 2 0.002 4 1 , 0.002 5 2  Marginal intervals  ofμZ μ Z 1 μ Z 2 μ Z 3 μ Z 4 0 . 4 1 , 0 . 5 2 0 . 6 1 , 0 . 7 2 0 . 8 1 , 1 . 1 2 1 . 2 1 , 1 . 5 2 1: lower bound; 2: upper bound. 1 and 2 originated from FEMA 273 [54], FEMA 356 [55], and FEMA 445 [56].According to (22), the uncertainty domain Ωp can be explicitly expressed as(47)Ωp=μ0.0029-0.4670-0.00091.4788μΔ-μΔCμZ-μZCμ-μC≤111Then, the IMP model is transformed into the standardδ-space. Equation (14) will be a highly nonlinear performance function in the random space if N1≥2 and/or N2≥2.


## 6.1. Design of the Case Study according to Chinese Codes

To verify the efficiency and accuracy of the proposed method, a sample six-story reinforced concrete (RC) frame building is taken for case study in this paper. The example structure is designed according to Chinese codes [72], which represents the typical mid-rise RC buildings in China. The geographical location of the modelled building is designated to be western China. This is a Class II site, with eight times the intensity of an earthquake and the designing ground acceleration is set as 0.20. The seismic grade of the example building is level 2 [73]. The plane and elevation views of the designed frame structure are shown in Figure 2. The total height of building is 19.8 m. The lengths of longitudinal spans and transversal spans are 36 m and 14.4 m, respectively. The columns and beams’ cross sections are 600 mm×600 mm and 300 mm×500 mm, respectively. The distributed steel in beams and columns is designed by using Chinese codes [72], as shown in Figure 3.Figure 2
The plane and elevation views of the designed frame structure.Figure 3
The distributed steel in columns and beams.


## 6.2. Modeling of the Designed RC Building

By using the finite element platform OpenSees [74], a three-dimensional model of the given structure (Figure 4) can be developed in this section. A rigid diaphragm MPC (multipoint constraints) is employed to model the floor concrete slabs. Nonlinear beam-column elements are used to model column and beam members. The Kent-Scott-Park model with no tension stiffening is adopted to model the concrete material. Column and beam cross sections are discretized to fibers of confined concrete, unconfined concrete, and steel reinforcement. The structure is considered as deterministic and other parameters are not regarded as uncertain in the present analysis. The parameters for confined concrete (identified by the subscript “core”) are listed asfc,core =34.5MPa,fu,core =24.1MPa, ε0,core =0.005, εu,core =0.020. The parameters for unconfined concrete (identified by the subscript “cover”) are described as follows:fc,cover=27.6MPa,fu,cover=0MPa, ε0,cover=0.002, εu,cover=0.006. A bilinear material called Steel01 material is used to model the reinforcing steel with parametersE=206GPa, fy=400MPa, andb=0.01. The structural responses such as interstory drift and acceleration can be, respectively, obtained through IDA and NHA. The fundamental mode for the given RC structure is set at 1.0018 sec by modal analysis.Figure 4
DoE obtained by ALK-MGHAR.


## 6.3. Determination of Performance Limit Levels and the Uncertainty Domain

In this study, structural performance limit states are partitioned into four levels, i.e., normal operation (NO), immediate occupancy (IO), life safety (LS), and collapse prevention (CP). Limit states of the interstory drift and the acceleration are simultaneously considered in this study. The threshold values of both EDPs under each performance level are described as random variables, listed in Table1. The median of random variables under each performance level is treated as the IMP variables. The marginal intervals of the uncertain-but-bounded variables are shown in Table 2. The correlation coefficients between the bounded variables defined by (19) are ρΔZ=ρZΔ=0.33. The correlation matrices under four performance levels can be expressed as ρNO, ρIO, ρLS, and ρCP. Then all of the correlation matrices can be built:(46)ρNO=ρIO=ρLS=ρCP=10.330.331Table 1
Threshold capacity values of interstory drift and peak floor acceleration.
Threshold capacities of  both the EDPs NO level IO level LS level CP level Mean SD1 Mean SD1 Mean SD1 Mean SD1 Interstory drift threshold μ Δ 1 0.002 μ Δ 2 0.005 μ Δ 3 0.015 μ Δ 4 0.025 Acceleration threshold μ Z 1 0.4 μ Z 2 0.6 μ Z 3 0.8 μ Z 4 1.2 1: standard deviation of the interstory drift threshold and acceleration originated from Sun et al. [53].Table 2
The marginal intervals of IMP convex variables.
IMP convex variables Marginal intervals  ofμΔ μ Δ 1 μ Δ 2 μ Δ 3 μ Δ 4 0.001 8 1 , 0.002 0 2 0.004 8 1 , 0.005 0 2 0.014 8 1 , 0.015 0 2 0.002 4 1 , 0.002 5 2  Marginal intervals  ofμZ μ Z 1 μ Z 2 μ Z 3 μ Z 4 0 . 4 1 , 0 . 5 2 0 . 6 1 , 0 . 7 2 0 . 8 1 , 1 . 1 2 1 . 2 1 , 1 . 5 2 1: lower bound; 2: upper bound. 1 and 2 originated from FEMA 273 [54], FEMA 356 [55], and FEMA 445 [56].According to (22), the uncertainty domain Ωp can be explicitly expressed as(47)Ωp=μ0.0029-0.4670-0.00091.4788μΔ-μΔCμZ-μZCμ-μC≤111Then, the IMP model is transformed into the standardδ-space. Equation (14) will be a highly nonlinear performance function in the random space if N1≥2 and/or N2≥2.


## 7. Discussion

### 7.1. Reliability Analysis with Performance Function (16)

#### 7.1.1. Application of ALK-MGHAR to Performance Function (16)

Before performing ALK-MGHAR, the ground motion hazard analysis defined by (4) needs to be investigated. Since the site conditions defined in Section 6.1 are known, the two unknown parameters k0 and k in (4) are calculated through the interpolation method. Here, k0=7.9203×10-5 and k=2.3814. After the maximum seismic failure probability is estimated by ALK-MGHAR, the failure probability for different limit states in 50 years can be obtained by combining the calculated seismic failure probability with ground motion hazard curve. A Kriging model should be constructed to have a right prediction for the sign of the performance function G(Θ). Based on the procedure elaborated in Section 5.3, 12 training points are chosen to build an initial Kriging model and then the iterative process begins. After 51 iterations, the stopping condition is satisfied and 51 training points are totally added into DoE. Moreover, the Kriging model is updated for 51 times. Taking CP performance level as an example, the DoE including initial and added training points obtained by ALK-MGHAR is depicted in Figure 4. It can be observed that a plenty of the added training points focus on the vicinity of the limit state G(X,μ)=0. This demonstrates that ALK-MGHAR is found capable of only locally approximating the performance function in the region of interest rather than in the overall uncertain space.After the Kriging model is built, the optimization problem (30) can be replaced by the optimization (34). This means that there exists a risk that the sign of the actual performance function is wrongly predicted. Then MCS can be carried out based on the Kriging model. 105 sampling points are used and the global optimization algorithm is utilized to search the minimum value of the Kriging model at each simulated sample. The sign of the minimal function value at each sample is predicted through ALK-MGHAR, as shown in Figure 5. It can be seen that only one sign is wrongly predicted. This indicates the accuracy of the proposed method.Figure 5
Sign of the minimum performance function withN2=1 predicted by ALK-MGHAR.


#### 7.1.2. Different Methods for Reliability Analysis

When the performance function (16) is adopted in this subsection, the maximum failure probability defined by (7) can be calculated using different numerical methods, such as MCS, MPM, SLI, and the proposed ALK-MGHAR. Then the failure probabilities under different performance levels in 50 years can be obtained by substituting (9) into (5). Note that MPM and SLI originated from Luo et al. [35] are generally employed to solve a double-loop optimization problem. Therefore, both MPM and SLI based on FORM can be used to solve the optimization problem (2). The failure probabilities in 50 years obtained by different methods are shown in Table 3. It can be observed that both MPM and SLI yield the satisfactory results. This is because the performance function (16) is linear at this moment. Therefore, the accuracy of MPM and SLI has been less affected. Moreover, the computation cost of ALK-MGHAR is much less than that of other different methods. For example, MCS requires 2×100×105 function computations whereas the advocated method only requires 63 calls to the performance function. However, it is also found that the computational results obtained by the proposed method are in good coincidence with those obtained by MCS. These findings illustrate the efficiency and accuracy of ALK-MGHAR.Table 3
Results withN2=1 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 10.260% 1.653% 0.170% 0.0739% 2 × 100 × 10 5 - ALK-MGHAR 10.246% 1.647% 0.169% 0.0738% 63 0.135% MPM 9.951% 1.599% 0.165% 0.0716% 283 3.11% SLI 9.922% 1.598% 0.164% 0.0713% 147 3.52% ∗: calculated relative error under CP level.


### 7.2. Reliability Analysis with Performance Function (15)

#### 7.2.1. Case 1:N2=2

If the parameterN2 in (15) is greater than or equal to 2, the performance function will be nonlinear. In this section, (15) withN2=2 is considered. Analogous to the performance function (16), 105 sampling points are generated and the minimal value of the Kriging model at each of the simulated samples is repeatedly explored to obtain the sign of the minimum function value. The sign of the extreme values predicted by ALK-MGHAR is shown in Figure 6. It can be observed that only 2 signs are wrongly predicted.Figure 6
Sign of the minimum performance function withN2=2 predicted by ALK-MGHAR.Based on different methods, the results are listed in Table4. It is seen that both MPM and SLI obtain the precise results. Qin et al. (2007) revealed that the nonlinearity of the performance function could make FORM produce large errors in reliability analysis. In general, both MPM and SLI based on FORM may generate large errors for nonlinear problems. However, the performance function (15) withN2=2 is not highly nonlinear and is monotonic with respect to the IMP convex variables defined in Section 3.2. Moreover, SLI is an effective method based on the assumption that the performance function is monotonic with respect to convex variables [35]. Therefore, the two methods and the proposed method behave very well in accuracy except in efficiency. ALK-MGHAR only requires 65 performance function computations that are less than 1/4 of MPM. This testifies the efficiency of the proposed method.Table 4
Results withN2=2 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 6.339% 1.140% 0.120% 0.0596% 2 × 100 × 10 5 - ALK-MGHAR 6.310% 1.136% 0.119% 0.0594% 65 0.34% MPM 6.105% 1.099% 0.116% 0.0575% 301 3.55% SLI 6.097% 1.097% 0.115% 0.0573% 159 3.83% ∗: calculated relative error under CP level.


#### 7.2.2. Case 2:N2≥2

The parameterN2 in (15) represents the nonlinearity of the performance function which reflects the limit state G(X,μ)=0. IfN2 is greater than or equal to 3, (15) will become a nonlinear function. Based on different methods, the results with different values ofN2 are listed in Table 5. It can be observed that both MPM and SLI provide results with a large error for this problem. The reason is that the nonlinearity of (15) makes MPM generate a large error as stated in Section 7.2.1. Because the performance function (15) for different limit states defined by Section 6.3 is monotonic within a specified interval, the performance of SLI is similar to that of MPM. From Table 5, it is also observed that ALK-MGHAR only needs 68 calls to the performance function (15) related toN2=3 and requires 73 function calls related toN2=6. However, both MPM and SLI still need many calls to the performance function. This demonstrates that no matter in accuracy or efficiency, ALK-MGHAR exhibits much better than other different methods.Table 5
Results with different values ofN2 by different methods.
N 2 Numerical method Failure probabilities in 50 years under four levels Function calls Relative error∗ NO level IO level LS level CP level N 2=3 MCS 5.67% 1.03% 0.098% 0.0424% 2 × 100 × 10 5 - ALK-MGHAR 5.652% 1.026% 0.0976% 0.0423% 68 0.338% MPM 4.145% 0.763% 0.0749% 0.0329% 327 22.34% SLI 4.347% 0.806% 0.0758% 0.0331% 161 22.03%  N 2=6 MCS 3.27% 0.71% 0.080% 0.0290% 2 × 100 × 10 5 - ALK-MGHAR 3.259% 0.708% 0.0797% 0.0288% 73 0.359% MPM 2.613% 0.559% 0.0623% 0.0226% 328 22.17% SLI 2.577% 0.546% 0.0624% 0.0223% 165 22.99% ∗: calculated relative error under CP level.


### 7.3. Reliability Analysis with Performance Function (14)

WhenN1 is equal to 3 andN2 is equal to 4, (14) becomes a more highly nonlinear function. After the Kriging model is constructed, the optimization problem for the true function (14) is replaced by the optimization for the Kriging model. Then MCS can be performed based on the Kriging model. Likewise, 105 samples are generated and the optimization is implemented for each of the simulated samples. The sign of the minimum function value at each sample predicted by ALK-MGHAR is shown in Figure 7. It is observed that only 2 signs are wrongly predicted. That illustrates the accuracy of the Kriging model and verifies the applicability of the property provided in Section 5.1 to MGHAR.Figure 7
Sign of the minimum performance function withN1=3 andN2=4 predicted by ALK-MGHAR.Based on different methods, the results for this problem are listed in Table6. It can be seen that both MPM and SLI obtain results with a large error. This is because the high nonlinearity of the performance function (14) withN1=3 andN2=4 makes FORM produce large errors. Both MPM and SLI proposed by Luo et al. [35] are based on FORM. The performance of SLI is still similar to that of MPM. From Table 6, it is also observed that ALK-MGHAR offers very precise results. This testifies that the nonlinearity of the performance function cannot affect the accuracy of the proposed method. Moreover, the computational cost of the proposed method is much less than that of both MPM and SLI. These findings demonstrate that ALK-MGHAR is suitable to deal with such complicated seismic engineering problems with strongly nonlinear performance functions.Table 6
Results withN1=3 and N2=4 by different methods.
Numericalmethods Failure probabilities in 50 years for different limit states Functioncalls Relativeerror∗ NO level IO level LS level CP level MCS 4.39% 0.95% 0.0892% 0.0355% 2 × 100 × 10 5 - ALK-MGHAR 4.376% 0.946% 0.0880% 0.0354% 65 0.342% MPM 2.858% 0.618% 0.0633% 0.0230% 321 34.89% SLI 2.898% 0.627% 0.0636% 0.0234% 163 33.97% ∗: calculated relative error under CP level.


## 7.1. Reliability Analysis with Performance Function (16)

### 7.1.1. Application of ALK-MGHAR to Performance Function (16)

Before performing ALK-MGHAR, the ground motion hazard analysis defined by (4) needs to be investigated. Since the site conditions defined in Section 6.1 are known, the two unknown parameters k0 and k in (4) are calculated through the interpolation method. Here, k0=7.9203×10-5 and k=2.3814. After the maximum seismic failure probability is estimated by ALK-MGHAR, the failure probability for different limit states in 50 years can be obtained by combining the calculated seismic failure probability with ground motion hazard curve. A Kriging model should be constructed to have a right prediction for the sign of the performance function G(Θ). Based on the procedure elaborated in Section 5.3, 12 training points are chosen to build an initial Kriging model and then the iterative process begins. After 51 iterations, the stopping condition is satisfied and 51 training points are totally added into DoE. Moreover, the Kriging model is updated for 51 times. Taking CP performance level as an example, the DoE including initial and added training points obtained by ALK-MGHAR is depicted in Figure 4. It can be observed that a plenty of the added training points focus on the vicinity of the limit state G(X,μ)=0. This demonstrates that ALK-MGHAR is found capable of only locally approximating the performance function in the region of interest rather than in the overall uncertain space.After the Kriging model is built, the optimization problem (30) can be replaced by the optimization (34). This means that there exists a risk that the sign of the actual performance function is wrongly predicted. Then MCS can be carried out based on the Kriging model. 105 sampling points are used and the global optimization algorithm is utilized to search the minimum value of the Kriging model at each simulated sample. The sign of the minimal function value at each sample is predicted through ALK-MGHAR, as shown in Figure 5. It can be seen that only one sign is wrongly predicted. This indicates the accuracy of the proposed method.Figure 5
Sign of the minimum performance function withN2=1 predicted by ALK-MGHAR.


### 7.1.2. Different Methods for Reliability Analysis

When the performance function (16) is adopted in this subsection, the maximum failure probability defined by (7) can be calculated using different numerical methods, such as MCS, MPM, SLI, and the proposed ALK-MGHAR. Then the failure probabilities under different performance levels in 50 years can be obtained by substituting (9) into (5). Note that MPM and SLI originated from Luo et al. [35] are generally employed to solve a double-loop optimization problem. Therefore, both MPM and SLI based on FORM can be used to solve the optimization problem (2). The failure probabilities in 50 years obtained by different methods are shown in Table 3. It can be observed that both MPM and SLI yield the satisfactory results. This is because the performance function (16) is linear at this moment. Therefore, the accuracy of MPM and SLI has been less affected. Moreover, the computation cost of ALK-MGHAR is much less than that of other different methods. For example, MCS requires 2×100×105 function computations whereas the advocated method only requires 63 calls to the performance function. However, it is also found that the computational results obtained by the proposed method are in good coincidence with those obtained by MCS. These findings illustrate the efficiency and accuracy of ALK-MGHAR.Table 3
Results withN2=1 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 10.260% 1.653% 0.170% 0.0739% 2 × 100 × 10 5 - ALK-MGHAR 10.246% 1.647% 0.169% 0.0738% 63 0.135% MPM 9.951% 1.599% 0.165% 0.0716% 283 3.11% SLI 9.922% 1.598% 0.164% 0.0713% 147 3.52% ∗: calculated relative error under CP level.


## 7.1.1. Application of ALK-MGHAR to Performance Function (16)

Before performing ALK-MGHAR, the ground motion hazard analysis defined by (4) needs to be investigated. Since the site conditions defined in Section 6.1 are known, the two unknown parameters k0 and k in (4) are calculated through the interpolation method. Here, k0=7.9203×10-5 and k=2.3814. After the maximum seismic failure probability is estimated by ALK-MGHAR, the failure probability for different limit states in 50 years can be obtained by combining the calculated seismic failure probability with ground motion hazard curve. A Kriging model should be constructed to have a right prediction for the sign of the performance function G(Θ). Based on the procedure elaborated in Section 5.3, 12 training points are chosen to build an initial Kriging model and then the iterative process begins. After 51 iterations, the stopping condition is satisfied and 51 training points are totally added into DoE. Moreover, the Kriging model is updated for 51 times. Taking CP performance level as an example, the DoE including initial and added training points obtained by ALK-MGHAR is depicted in Figure 4. It can be observed that a plenty of the added training points focus on the vicinity of the limit state G(X,μ)=0. This demonstrates that ALK-MGHAR is found capable of only locally approximating the performance function in the region of interest rather than in the overall uncertain space.After the Kriging model is built, the optimization problem (30) can be replaced by the optimization (34). This means that there exists a risk that the sign of the actual performance function is wrongly predicted. Then MCS can be carried out based on the Kriging model. 105 sampling points are used and the global optimization algorithm is utilized to search the minimum value of the Kriging model at each simulated sample. The sign of the minimal function value at each sample is predicted through ALK-MGHAR, as shown in Figure 5. It can be seen that only one sign is wrongly predicted. This indicates the accuracy of the proposed method.Figure 5
Sign of the minimum performance function withN2=1 predicted by ALK-MGHAR.


## 7.1.2. Different Methods for Reliability Analysis

When the performance function (16) is adopted in this subsection, the maximum failure probability defined by (7) can be calculated using different numerical methods, such as MCS, MPM, SLI, and the proposed ALK-MGHAR. Then the failure probabilities under different performance levels in 50 years can be obtained by substituting (9) into (5). Note that MPM and SLI originated from Luo et al. [35] are generally employed to solve a double-loop optimization problem. Therefore, both MPM and SLI based on FORM can be used to solve the optimization problem (2). The failure probabilities in 50 years obtained by different methods are shown in Table 3. It can be observed that both MPM and SLI yield the satisfactory results. This is because the performance function (16) is linear at this moment. Therefore, the accuracy of MPM and SLI has been less affected. Moreover, the computation cost of ALK-MGHAR is much less than that of other different methods. For example, MCS requires 2×100×105 function computations whereas the advocated method only requires 63 calls to the performance function. However, it is also found that the computational results obtained by the proposed method are in good coincidence with those obtained by MCS. These findings illustrate the efficiency and accuracy of ALK-MGHAR.Table 3
Results withN2=1 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 10.260% 1.653% 0.170% 0.0739% 2 × 100 × 10 5 - ALK-MGHAR 10.246% 1.647% 0.169% 0.0738% 63 0.135% MPM 9.951% 1.599% 0.165% 0.0716% 283 3.11% SLI 9.922% 1.598% 0.164% 0.0713% 147 3.52% ∗: calculated relative error under CP level.


## 7.2. Reliability Analysis with Performance Function (15)

### 7.2.1. Case 1:N2=2

If the parameterN2 in (15) is greater than or equal to 2, the performance function will be nonlinear. In this section, (15) withN2=2 is considered. Analogous to the performance function (16), 105 sampling points are generated and the minimal value of the Kriging model at each of the simulated samples is repeatedly explored to obtain the sign of the minimum function value. The sign of the extreme values predicted by ALK-MGHAR is shown in Figure 6. It can be observed that only 2 signs are wrongly predicted.Figure 6
Sign of the minimum performance function withN2=2 predicted by ALK-MGHAR.Based on different methods, the results are listed in Table4. It is seen that both MPM and SLI obtain the precise results. Qin et al. (2007) revealed that the nonlinearity of the performance function could make FORM produce large errors in reliability analysis. In general, both MPM and SLI based on FORM may generate large errors for nonlinear problems. However, the performance function (15) withN2=2 is not highly nonlinear and is monotonic with respect to the IMP convex variables defined in Section 3.2. Moreover, SLI is an effective method based on the assumption that the performance function is monotonic with respect to convex variables [35]. Therefore, the two methods and the proposed method behave very well in accuracy except in efficiency. ALK-MGHAR only requires 65 performance function computations that are less than 1/4 of MPM. This testifies the efficiency of the proposed method.Table 4
Results withN2=2 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 6.339% 1.140% 0.120% 0.0596% 2 × 100 × 10 5 - ALK-MGHAR 6.310% 1.136% 0.119% 0.0594% 65 0.34% MPM 6.105% 1.099% 0.116% 0.0575% 301 3.55% SLI 6.097% 1.097% 0.115% 0.0573% 159 3.83% ∗: calculated relative error under CP level.


### 7.2.2. Case 2:N2≥2

The parameterN2 in (15) represents the nonlinearity of the performance function which reflects the limit state G(X,μ)=0. IfN2 is greater than or equal to 3, (15) will become a nonlinear function. Based on different methods, the results with different values ofN2 are listed in Table 5. It can be observed that both MPM and SLI provide results with a large error for this problem. The reason is that the nonlinearity of (15) makes MPM generate a large error as stated in Section 7.2.1. Because the performance function (15) for different limit states defined by Section 6.3 is monotonic within a specified interval, the performance of SLI is similar to that of MPM. From Table 5, it is also observed that ALK-MGHAR only needs 68 calls to the performance function (15) related toN2=3 and requires 73 function calls related toN2=6. However, both MPM and SLI still need many calls to the performance function. This demonstrates that no matter in accuracy or efficiency, ALK-MGHAR exhibits much better than other different methods.Table 5
Results with different values ofN2 by different methods.
N 2 Numerical method Failure probabilities in 50 years under four levels Function calls Relative error∗ NO level IO level LS level CP level N 2=3 MCS 5.67% 1.03% 0.098% 0.0424% 2 × 100 × 10 5 - ALK-MGHAR 5.652% 1.026% 0.0976% 0.0423% 68 0.338% MPM 4.145% 0.763% 0.0749% 0.0329% 327 22.34% SLI 4.347% 0.806% 0.0758% 0.0331% 161 22.03%  N 2=6 MCS 3.27% 0.71% 0.080% 0.0290% 2 × 100 × 10 5 - ALK-MGHAR 3.259% 0.708% 0.0797% 0.0288% 73 0.359% MPM 2.613% 0.559% 0.0623% 0.0226% 328 22.17% SLI 2.577% 0.546% 0.0624% 0.0223% 165 22.99% ∗: calculated relative error under CP level.


## 7.2.1. Case 1:N2=2

If the parameterN2 in (15) is greater than or equal to 2, the performance function will be nonlinear. In this section, (15) withN2=2 is considered. Analogous to the performance function (16), 105 sampling points are generated and the minimal value of the Kriging model at each of the simulated samples is repeatedly explored to obtain the sign of the minimum function value. The sign of the extreme values predicted by ALK-MGHAR is shown in Figure 6. It can be observed that only 2 signs are wrongly predicted.Figure 6
Sign of the minimum performance function withN2=2 predicted by ALK-MGHAR.Based on different methods, the results are listed in Table4. It is seen that both MPM and SLI obtain the precise results. Qin et al. (2007) revealed that the nonlinearity of the performance function could make FORM produce large errors in reliability analysis. In general, both MPM and SLI based on FORM may generate large errors for nonlinear problems. However, the performance function (15) withN2=2 is not highly nonlinear and is monotonic with respect to the IMP convex variables defined in Section 3.2. Moreover, SLI is an effective method based on the assumption that the performance function is monotonic with respect to convex variables [35]. Therefore, the two methods and the proposed method behave very well in accuracy except in efficiency. ALK-MGHAR only requires 65 performance function computations that are less than 1/4 of MPM. This testifies the efficiency of the proposed method.Table 4
Results withN2=2 by different methods.
Numerical methods Failure probabilities in 50 years for different limit states Function calls Relative error∗ NO level IO level LS level CP level MCS 6.339% 1.140% 0.120% 0.0596% 2 × 100 × 10 5 - ALK-MGHAR 6.310% 1.136% 0.119% 0.0594% 65 0.34% MPM 6.105% 1.099% 0.116% 0.0575% 301 3.55% SLI 6.097% 1.097% 0.115% 0.0573% 159 3.83% ∗: calculated relative error under CP level.


## 7.2.2. Case 2:N2≥2

The parameterN2 in (15) represents the nonlinearity of the performance function which reflects the limit state G(X,μ)=0. IfN2 is greater than or equal to 3, (15) will become a nonlinear function. Based on different methods, the results with different values ofN2 are listed in Table 5. It can be observed that both MPM and SLI provide results with a large error for this problem. The reason is that the nonlinearity of (15) makes MPM generate a large error as stated in Section 7.2.1. Because the performance function (15) for different limit states defined by Section 6.3 is monotonic within a specified interval, the performance of SLI is similar to that of MPM. From Table 5, it is also observed that ALK-MGHAR only needs 68 calls to the performance function (15) related toN2=3 and requires 73 function calls related toN2=6. However, both MPM and SLI still need many calls to the performance function. This demonstrates that no matter in accuracy or efficiency, ALK-MGHAR exhibits much better than other different methods.Table 5
Results with different values ofN2 by different methods.
N 2 Numerical method Failure probabilities in 50 years under four levels Function calls Relative error∗ NO level IO level LS level CP level N 2=3 MCS 5.67% 1.03% 0.098% 0.0424% 2 × 100 × 10 5 - ALK-MGHAR 5.652% 1.026% 0.0976% 0.0423% 68 0.338% MPM 4.145% 0.763% 0.0749% 0.0329% 327 22.34% SLI 4.347% 0.806% 0.0758% 0.0331% 161 22.03%  N 2=6 MCS 3.27% 0.71% 0.080% 0.0290% 2 × 100 × 10 5 - ALK-MGHAR 3.259% 0.708% 0.0797% 0.0288% 73 0.359% MPM 2.613% 0.559% 0.0623% 0.0226% 328 22.17% SLI 2.577% 0.546% 0.0624% 0.0223% 165 22.99% ∗: calculated relative error under CP level.


## 7.3. Reliability Analysis with Performance Function (14)

WhenN1 is equal to 3 andN2 is equal to 4, (14) becomes a more highly nonlinear function. After the Kriging model is constructed, the optimization problem for the true function (14) is replaced by the optimization for the Kriging model. Then MCS can be performed based on the Kriging model. Likewise, 105 samples are generated and the optimization is implemented for each of the simulated samples. The sign of the minimum function value at each sample predicted by ALK-MGHAR is shown in Figure 7. It is observed that only 2 signs are wrongly predicted. That illustrates the accuracy of the Kriging model and verifies the applicability of the property provided in Section 5.1 to MGHAR.Figure 7
Sign of the minimum performance function withN1=3 andN2=4 predicted by ALK-MGHAR.Based on different methods, the results for this problem are listed in Table6. It can be seen that both MPM and SLI obtain results with a large error. This is because the high nonlinearity of the performance function (14) withN1=3 andN2=4 makes FORM produce large errors. Both MPM and SLI proposed by Luo et al. [35] are based on FORM. The performance of SLI is still similar to that of MPM. From Table 6, it is also observed that ALK-MGHAR offers very precise results. This testifies that the nonlinearity of the performance function cannot affect the accuracy of the proposed method. Moreover, the computational cost of the proposed method is much less than that of both MPM and SLI. These findings demonstrate that ALK-MGHAR is suitable to deal with such complicated seismic engineering problems with strongly nonlinear performance functions.Table 6
Results withN1=3 and N2=4 by different methods.
Numericalmethods Failure probabilities in 50 years for different limit states Functioncalls Relativeerror∗ NO level IO level LS level CP level MCS 4.39% 0.95% 0.0892% 0.0355% 2 × 100 × 10 5 - ALK-MGHAR 4.376% 0.946% 0.0880% 0.0354% 65 0.342% MPM 2.858% 0.618% 0.0633% 0.0230% 321 34.89% SLI 2.898% 0.627% 0.0636% 0.0234% 163 33.97% ∗: calculated relative error under CP level.


## 8. Conclusions

This paper provides an ALK-MGHAR method. MGHAR investigated here represents the reliability problem of a given RC structure with both random and IMP variables. Limit thresholds of different types of components are treated as random variables. The median of random variables is represented by an improved MP convex model, in which the uncertain domain of correlated bounded (interval) variables can be explicitly expressed by the matrix inequality. The corresponding EDPs are considered as dependent random variables and follow a multidimensional lognormal distribution. Through matrix transformation, the limit state function is mapped into the corresponding regularized one. The Kriging model is used to approximate the performance function in the region of interest rather than in the overall uncertain space. ALK-MGHAR starts its iterative process with constructing an initial Kriging model by defining an initial DoE with a handful of training points. The point at which the sign of the performance function has the maximum risk to be wrongly predicted should be added into DoE. Sequentially, the added point from candidate points is used to update the Kriging model. After the ALK model is constructed, a large amount of samples is generated and the optimization is implemented at each sample. Then, the maximum seismic failure probability for different limit states can be performed by using MCS. Further, the failure probabilities for different limit states in 50 years can be obtained through coupling the seismic failure probability and the ground motion hazard curve. To demonstrate the efficiency and accuracy of ALK-MGHAR, a six-story RC building is used as a case study. The interstory drift and the acceleration can be, respectively, obtained through IDA and NHA.From this numerical example, it is demonstrated that the proposed method exhibits much better than other different methods such as MPM and SLI no matter in accuracy or in efficiency. Meanwhile, ALK-MGHAR is found capable of providing very accurate results for MGHAR and requires only a few of calls to the performance function. The phenomenon testifies that ALK-MGHAR only rightly predicting the sign of the performance function can meet the accurate requirement of MGHAR. It is also illustrated that ALK-MGHAR is able to deal with complex seismic engineering problems with strongly nonlinear performance functions.



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*Source: 5027958-2018-11-01.xml*