Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6976301, 12 pages

https://doi.org/10.1155/2017/6976301

## The Method of Solving Structural Reliability with Multiparameter Correlation Problem

College of Science, Inner Mongolia University of Technology, Hohhot 010051, China

Correspondence should be addressed to Haibin Li

Received 1 July 2017; Revised 12 November 2017; Accepted 16 November 2017; Published 11 December 2017

Academic Editor: Fiorenzo A. Fazzolari

Copyright © 2017 Juan Du et al. 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

Correlation among variables must be considered to accurately reflect the level of structure reliability. This problem has referential value to engineering practice and has attracted attention from relevant scholars and industries. In this paper, Copula function was used to build the joint probability density function among all variables. The key is to describe the correlation among variables, solve the correlation parameter of Copula function, and select the type of correlation structure among variables. The correlation parameter of Copula function was solved using Pearson linear correlation coefficient and maximum likelihood estimation. Based on the Akaike information criteria (AIC) and Bayesian information criteria (BIC), the optimal Copula function was selected, and the correlation structure among variables was determined. Monte Carlo method, which is based on Nataf inverse transformation, was introduced and used to evaluate the reliability of the correlated variable. Finally, this paper proposed the reliability calculation method based on dual neural network and direct integration by establishing the dual neural network of original and integrand functions. Compared with the Monte Carlo method, the proposed method can be utilized to efficiently and precisely calculate the structure reliability of multiple correlated random variables.

#### 1. Introduction

In structural reliability analysis, variables are often correlated, for instance, the positive correlation between the seismic peak displacement and permanent displacement [1] and among the stresses of the weld fatigue damage [2] and the significant negative correlation between the shear strength parameters of rock and soil and the parameters of pile-load-displacement curve [3, 4]. Therefore, the correlation among variables must be considered to rationally analyze the structural reliability. The traditional calculation method for structural reliability includes first-order [5] and second-order moments [6], which are only limited to the linear correlation of variables. A 2D or multidimensional distribution model is often used to characterize the correlation among variables, such as 2D lognormal distribution [7] and multidimensional normal distribution [8]. However, one of the drawbacks of these models is the requirement of same-edge distribution variables, which greatly limit their application in reliability analysis.

The emergence and development of Copula theory solve the above problems and provide a new way to construct joint probability density function of correlated variables [9]. Copula theory was first proposed by Sklar in 1959 [10]. Under the framework of Copula function, variables could follow arbitrary edge distributions, and the linear and nonlinear correlations among variables could be defined [11]. Copula functions were introduced in computational structural reliability because of its capacity to handle the arbitrary correlations of variables. Li et al. used the 2D Frank Copula function to construct the joint probability distribution function of two seismic attenuation models [12]. Based on the Copula function, Tang et al. presented the load-displacement hyperbolic probabilistic analysis of pile foundation and used the Copula function to establish the joint probability distribution function of hyperbolic parameters [13]. Huang et al. established a Copula function model of rock mass shear strength parameters based on measured data [14]. Liu et al. used Copulas to develop a reliability model for systems with s-dependent degradation processes. The proposed model accommodates the assumptions of s-dependence among the degradation processes and allows for different marginal distributions [15]. For two-component and multiple-component systems with multiple failure modes, Liu and Fan established the mixed Copula models for time-independent reliability analysis of series systems, parallel systems, series-parallel systems, and parallel-series systems [16]. Based on the Copula function, Xu et al. constructed a joint probability density function, which was integrated on the failure field to calculate the failure probability of the structure [17]. In reliability-based design optimization (RBDO) of the structure, the exact joint probability density function of the input relevant variables is necessary to obtain the optimal design [18]. Lee et al. used the Copula function to consider the correlation among variables and solve the problem of RBDO [19, 20] and found that the correlation significantly affected RBDO results [21–25]. They also analyzed the influence of the confidence level for Copula function on RBDO results [26] and used the Bayesian and Markov chain Monte Carlo methods to select the Copula function [27]. For RBDO with varying standard deviations (STDs), Cho et al. used the Copula function for the design of sensitivity and then improved the efficiency of the calculation [28].

The calculation methods for structural reliability are classified in three categories. The first method solves the structural response or the probability feature quantity of performance function [29–31]; however, the computational accuracy in this method is heavily dependent on the form of performance functions. The second method was the Monte Carlo method, which directly applies sampling statistics [32–34]. The disadvantage of this method is the large computational complexity. The third method was direct integration [35], which has encountered mathematical difficulties in the multiple integral and quantitative description of computational accuracy.

Neural networks can approximate any functions and thus could be properly applied to structural reliability. Papadrakakis et al. combined neural network with Monte Carlo simulation to analyze the reliability of elastoplastic structure; the time-consuming calculation of the Monte Carlo simulation was reduced, and the efficiency of the calculation was improved [36]. Lopes et al. analyzed reliability using neural network instead of finite element analysis. The neural network had advantages in computational efficiency when compared with Monte Carlo method [37]. Zuo et al. used neural network to fit the performance function of the structure. The values of performance function and partial derivatives were obtained at the point of mean values. Hence, the moments of performance function were calculated based on the moments of random variables [38]. Cheng and Li used the neural network to simulate the limit state equation of long-span bridge. The genetic algorithm was used to train the network, and the failure probability of the structure was obtained [39]. Meng et al. utilized BP neural network for nonlinear mapping function trains to obtain the explicit expression of stress in response to the random variable. A study analyzed reliability and the sensitivity of metal structure by combining random perturbation theory and first-order second-moment method [40]. Elhewy et al. proposed a response surface method based on artificial neural networks and consequently reduced the computational complexity for reliability analysis [41]. Li et al. proposed a structural reliability method, which is based on neural network and direct integration, to solve the reliability of independent variables [42]. Dai and Cao developed a new neural network model based on wavelet support vector machine for reliability analysis; this model has extended the application of wavelet neural network to a high dimension [43].

This paper proposed a direct integration method that is based on dual neural network and could be applied for reliability calculation. The proposed method is composed of two neural networks with similar structure, multiple inputs, single output, and single hidden. By designing the function relation between the weights of two neural networks, one neural network is able to approximate integrands, whereas the other approximates original function. Therefore, the above networks were called integrand and original function neural networks. We only need to train the integrand neural network. Thus, the weights of original function neural network were provided directly by the function relation between the weights of the two neural networks. We subsequently used the original function neural network to calculate the multiple integral. In the proposed method, the integrand could easily obtain the sample data that were directly trained, and the integral computational accuracy would be greatly improved. Therefore, the proposed method can efficiently and accurately solve structural reliability problems.

First, we introduced the method by constructing the joint probability density function of correlation variable based on Copula function. This method includes the solution of correlation parameter and the selection of the structural type of the Copula function. Second, we introduced Nataf-Monte Carlo method. Third, we proposed direct integration based on dual neural network, which is in turn based on the integral form of reliability computation and the normalization method of the integral area. Fourth, we compared the proposed technique with Monte Carlo method and verified its effectiveness in simulation. Finally, the full-text conclusion and future prospects were presented.

#### 2. Construction of Joint Probability Density Function Based on Copula Function for Correlated Variables

##### 2.1. Copula Function

Copula theory was first proposed by Sklar in 1959. Sklar pointed out that any multidimensional joint distribution function could be decomposed into a corresponding edge distribution function and a Copula function. The Copula function determines the correlation among variables, including the size of the correlation coefficient and the type of the correlation structure [10]. According to its strict definition stated by Nelsen, the Copula function is a function that associates the joint distribution function of the variable with its edge distribution function. In essence, Copula function is also a joint distribution function. For -dimensional cases, the Copula function was defined as the -dimensional joint distribution function. The edge distribution was in the ^{n} space [9].

According to Sklar theorem [10], the joint distribution function of variables , could be expressed aswhere is the edge distribution function of the variable : . is the Copula function, and is the correlation parameter of the Copula function.

From (1), the joint probability density function could be obtained aswhere is the edge probability density function of variables and is the density function for .

If the edge distribution function of variables and the Copula function were known, then the multidimensional distribution model of the variables could be established using (1) and (2).

##### 2.2. Correlation Parameter of Copula Function

The correlation parameter of the Copula function characterizes the correlation among variables. The method of solving the correlation parameter is different for different types of Copula functions.

When the Copula function of multidimensional variable was an Ellipse Copula function, the number of correlation parameters was the same as that of the correlation coefficients among variables and had one-to-one correspondence. The correlation parameter of any two variables and was . The relationship between and Pearson linear correlation coefficient was [9]

When the Copula function of the multidimensional variable was the Archimedean Copula function, the multidimensional Archimedean Copula function had only one correlation parameter because only one generator existed. This parameter describes the overall correlation between the multidimensional variables . Maximum likelihood estimation is generally used to solve the correlation parameter [44].

##### 2.3. Selection of the Optimal Copula Function

Different Copula functions could describe different correlations among variables. Table 1 lists various Copula function types [9]. Therefore, selecting the optimal Copula function, which could best fit the correlation among variables, is necessary when constructing the joint probability density function among variables. In this paper, we used Akaike information criterion (AIC) and Bayesian information criterion (BIC) to select the optimal Copula function. The best Copula function has the smallest AIC or BIC values and is well fitted for the correlated structure among variables. For multidimensional distribution models, the AIC and BIC values were expressed as follows [45, 46]: where is the number of correlation parameters . In the Ellipse Copula function, , whereas in the Archimedean Copula function, .