Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5027958, 15 pages

https://doi.org/10.1155/2018/5027958

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

^{1}State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China^{2}Department of Mechanical and Civil Engineering, Northwestern Polytechnical University, Xi’an 710129, China

Correspondence should be addressed to Xiao-Xiao Liu; nc.ude.utjx@9891uilxx and Yuan-Sheng Wang; nc.ude.upwn@gnehsnauygnaw

Received 13 July 2018; Accepted 18 September 2018; Published 1 November 2018

Academic Editor: Allan C. Peterson

Copyright © 2018 Xiao-Xiao Liu and Yuan-Sheng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 as , with the vector of random variables . 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:where is the structural failure probability under a damage state, represents the probability of an event, and is the joint probability density function (PDF) of random variables . For normal random variables, can be transformed into standard normal random variables through a linear transformation, as follows:where and are mean and standard deviation of random variables , 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 aswhere is the structural failure probability for a specific limit state and can be solved through (1). is the ground motion hazard function and denotes the MAF of a specific earthquake event (); 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 aswhere is a constant depending on the ground motion characteristics,* k* is the slope of the seismic hazard curve in logarithmic coordinates, and 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:

##### 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 as , where 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 produces a cluster of limit state surfaces in the stochastic space. The minimum limit state, , 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 aswhere 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 . 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 aswhere is the maximum failure probability corresponding to the minimum limit state , is the bivariate PDF,* D* is the interstory drift,* A* is the acceleration, and Δ and* δ* are threshold capacity values of the interstory drift and the acceleration, respectively.

For convenience, this subsection uses to replace the minimum value of the performance function . Then, the MAF of exceeding the two specific limit states per year is defined aswhere is the ground motion hazard function of the site derived from PHSA and 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 function is generated.

*Step 2. *An optimization problem elaborated in Section 4 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 aswhere is the failure indictor function and it can be expressed aswhere is the minimum value of the performance function which can be elaborated in Section 3. In this paper, it should be noted that represents the failure region of a given structure and 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]:where and are log-mean and log-standard deviation of the maximum interstory drift, respectively; and are log-mean and the log-standard deviation of the acceleration, respectively; is the correlation coefficient between and .

The mean vector and covariance of the bivariate PDF are expressed as follows:

can be estimated by the following expression:where is the estimator of the correlation coefficient *ρ*;* n* is the number of ground inputs; and are the estimators of log-mean and log-standard deviation of the maximum interstory drift, respectively; and 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]:where is the vector of the maximum interstory drift, is the vector of the maximum interstory drift threshold, is the vector of the acceleration, is the vector of the acceleration threshold, and are mean and standard deviation of the interstory drift threshold, respectively, and and are mean and standard deviation of the acceleration threshold, respectively.** D** and** A **are considered as random variables and follow the bivariate lognormal distribution. and are assumed to be lognormally distributed. and 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. When , 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

When the simplest case with* N*_{2} = 1 is considered, the performance function is written as

##### 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 and is considered, the IMP model degenerates into a parallelogram. The marginal intervals of the two variables and are denoted as and , respectively. The two marginal intervals and are defined aswhere and are the lower bound and upper bound of ; and are the lower bound and upper bound of ; and represent the midpoints of and ; and represent interval radii of the two marginal intervals. If bounded variables and are independent of each other, the uncertainty domain of the two variables will become a rectangular domain , expressed as . If the two bounded variables are dependent, the uncertainty domain will form a parallelogram domain , 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.