Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6141838, 11 pages

http://dx.doi.org/10.1155/2016/6141838

## Copula-Based Slope Reliability Analysis Using the Failure Domain Defined by the -Line

^{1}Key Laboratory of Geomechanics and Embankment Engineering, Ministry of Education, Hohai University, 1 Xikang Road, Nanjing 210098, China^{2}Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, 8 Daxue Road, Yichang 443002, China

Received 15 June 2016; Revised 21 August 2016; Accepted 29 August 2016

Academic Editor: Renata Archetti

Copyright © 2016 Xiaoliang Xu 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

The estimation of the cross-correlation of shear strength parameters (i.e., cohesion and internal friction angle) and the subsequent determination of the probability of failure have long been challenges in slope reliability analysis. Here, a copula-based approach is proposed to calculate the probability of failure by integrating the copula-based joint probability density function (PDF) on the slope failure domain delimited with the -line. Here, copulas are used to construct the joint PDF of shear strength parameters with specific marginal distributions and correlation structure. In the paper a failure (limit state) function approach is applied to investigate a system characterized by a homogeneous slope. The results show that the values obtained by using the failure function approach are similar to those calculated by means of conventional methods, such as the first-order reliability method (FORM) and Monte Carlo simulations (MC). In addition, an entropy weight (EW) copula is proposed to address the discrepancies of the results calculated by different copulas to avoid over- or underestimating the slope reliability.

#### 1. Introduction

Numerous problems regarding uncertainty in geotechnical engineering exist currently, such as soil and rock properties, environmental conditions, and theoretical models [1–3]. The application of reliability analysis for addressing uncertainty in geotechnical design is rather common [4–6]. To determine the reliability of a slope with high accuracy, it is imperative to investigate the relevance of parameters and establish their joint cumulative distribution function (CDF) or joint probability density function (PDF). However, establishing an accurate joint CDF or PDF remains a practical challenge in geotechnical engineering due to the limited available datasets [7–9]. Copulas may be used to address this problem [10, 11]. The use of copulas has the distinct advantage of enabling the construction of a joint CDF of random variables with any form of marginal distribution, overcoming the limitation of conventional approaches [12, 13] in which nonnormal variables must be converted into normal variables [9, 14].

In recent years, copulas have been applied to different disciplines, including finance [10, 15], biomedicine [16, 17], hydrology [18–20], and coastal engineering [21, 22]. In geotechnical engineering, Li et al. [23] used copulas to simulate the bivariate probability distribution of two curve-fitting parameters underlying the load-displacement models of piles. Tang et al. [9, 14] calculated the failure probabilities for an infinite slope and a retaining wall using the direct integration method and investigated the impact of different types of copulas on reliability. Marchant et al. [24] introduced a more general multivariate function for the spatial prediction of soil properties based on copulas. Wu [25, 26] determined the failure probability of a slope using a copula-based sampling method. Huang et al. [27] used a copula-based method to estimate the shear strength parameters of rock mass. Motamedi and Liang [28] proposed a probabilistic methodology for landslide hazard assessment at regional scale based on the copula modelling technique considering the possible dependence between the hazard elements.

Although copulas have been well developed and widely applied to determine the relevance of material properties and to establish the CDF or PDF, it is difficult to determine the performance function of a slope system in the critical state. Analysing the reliability of a slope remains a significant challenge in practice.

With the help of the -line [29], we propose a reliability analysis method based on copulas, which can be applied to any given homogeneous slopes. The remainder of this paper is organized as follows. Section 2 briefly introduces the copula functions and presents the procedure used in the copula-based slope reliability analysis. In Section 3, after finishing the detailed introductions of -line, we present an approach to identify the failure domain of a slope of any shape. The range of shear strength parameters, which can represent the slope failure domain, is also deduced. Section 4 explains how to calculate the failure probability by directly integrating the joint PDF of the considered variables on the failure domain identified by the -line. Section 5 presents a demonstrative application with evaluation of the failure probability for an example slope. Finally, the results calculated from different copulas are compared and discussed for an entropy weight (EW) copula in Section 6.

#### 2. Copulas for the Shear Strength Parameters

##### 2.1. Copula Function

A copula function is a multivariate distribution function with uniform marginal distributions which are all over . For an -dimensional uniform random vector , a copula is expressed as follows [30]:where and are the th uniform random variables on the interval and the corresponding values and represents the probability value.

Using Sklar’s theorem [31], the joint CDF of an -dimensional random variable can be established via the copula function aswhere is a copula function, is the marginal distribution function of the th random variable and are the copula parameters. Copulas offer the possibility of separately modelling the marginal distributions and the dependence structure among the random variables.

Using a one parameter copula function, the joint CDF of the shear strength parameters and , corresponding to cohesion and internal friction angle , can be expressed as follows:in which the parameter characterizes the dependence structure between the variables and .

The corresponding joint PDF iswhere and are the PDFs of and and is the copula density function given by

##### 2.2. Slope Reliability Assessment Using Copulas

The establishment of the joint CDF or PDF in terms of copulas for the implementation of the slope reliability analysis involves the following 4 steps.

*(1) Estimation of the Marginal Distributions of Each Random Variable*. As for fitting copulas for shear strength, the first-line procedure is to determine the appropriate marginal distribution of each random variable. In various previous studies the normal, lognormal, and gamma distributions were selected as the best-fitting marginal distributions for the shear strength parameters ( and ) [25, 32, 33]. The lognormal distribution is defined only for nonnegative values [32]. Kolmogorov-Smirnov (K-S) test commonly used to check whether a set of data follows a certain distribution is adopted to examine marginal distribution [27]. The key procedure of K-S test is computing the test statistic , which can be expressed aswhere is the established marginal distribution functions and is the cumulative frequency distribution; it is defined asin which is the indicator function, equal to 1 if and equal to 0 otherwise. will be accepted if ,where is the critical value of a given significance level . can be obtained by using the established table [34] when and have been determined, and is the number of observed datasets.

*(2) Copula Parameter Estimation*. Two-dimensional copulas commonly used in geotechnical engineering belong to two different classes. The elliptical class includes the Gaussian and copulas. The Archimedean class includes the Frank, Gumbel, and Clayton copulas.

Most analyses performed on the shear strength parameters report a negative correlation coefficient between and , with value between −0.24 and −0.70. However, a positive correlation was also found in several cases [9, 25]. The shear strength parameters of rocks and soils typically have symmetric structures [9] and asymmetric structures [35, 36]. Considering the correlation characteristics and symmetry between shear strength parameters, the capability of the Gaussian, Frank, Clayton, Gumbel, and Plackett copulas is tested to model the dependence structure of the data [9, 25]. The considered copulas are symmetric copulas, except for the Clayton and Gumbel copulas. In addition, negative correlations may not be allowed in some of those copulas; nonetheless, a simple conversion which negates the values of one variable can achieve a positive value for the correlation.

Copula parameters can be obtained by using maximum likelihood estimation (MLE) [30], the Pearson linear correlation coefficient , and the Kendall or Spearman rank correlation coefficient [11]. In this paper, the Kendall rank correlation coefficient is chosen to estimate the copula parameter, . The parameter is independent of the marginal distribution. For the elliptical class of copulas, the relationship between and is given byFor the Archimedean class of copulas, Kendall’s is given bywhere is the generator function. Kendall’s may be computed by where is the total number of samples, , and is the sign function, which will be 1 for or in all other cases.

Then, the corresponding value is determined by using (8) or (9) depending on the copula class. The parameter of the Plackett copula was determined with the maximum likelihood method [30].

*(3) Identification of the Best-Fitting Copula*. There are various methods to identify the best-fitting copula model, among these candidate copulas, such as the Akaike information criterion (AIC) [37], root mean square error (RMSE), bias [38], and Cramér-von Mises statistical method [39, 40]. The AIC is the method most commonly used in engineering practice, and it can be defined as follows:In this study, AICc [41] takes a correction into consideration for finite sample sizes, which can be determined by the AIC:where is the number of estimated parameters and denotes the number of observations (for a single parameter, ). The copula with the smallest AICc value is preferred. The Cramér-von Mises statistics, , can be expressed as the sum of the squared differences between the true and empirical copulas for any considered point [39, 40]:where is the copula with the estimated value of the copula parameter, , and is the empirical copula given byin which and are empirical CDFs, is the indicator function, if , , and otherwise. A parametric bootstrapping approach is used to determine values [40]. The copula with the minimum and maximum can be considered as optimal. The RMSE and bias can be determined by the following equations [38]:where is the th observed value and is the th estimated value. The best-fitting model is the one that has the smallest RMSE or bias.

*(4) Slope Reliability Analysis*. Following three successive steps, the joint CDF or PDF of the random variable can be obtained. Subsequently, the probability of failure can be calculated by integrating the joint PDF established by copulas on the slope failure domain; this approach is called the copula-based direct integration method (CDIM). The determination of the slope failure domain and the integral solution of the probability of failure will be discussed in Sections 3 and 4, respectively.

#### 3. -Line and Slope Failure Domain

##### 3.1. -Line

According to Klar et al. [29], the factor of safety of a slope system can be expressed as follows based on the Mohr-Coulomb criterion:where and are the actual shear strength parameters of the soil slope, and are the mobilized parameters required for the limit equilibrium state, denotes the shear strength and denotes the mobilized shear strength with a factor of safety , and is normal stress. Klar et al. [29] proposed the concept of the -line based on (17) to determine the factor of safety geometrically. The -line (Figure 1) represents the combinations of and at which the slope will exist in limit equilibrium (). The combinations of and above the -line imply a stable condition (), whereas the combinations falling below the -line (namely, in the failure domain symbolized by ) are impossible and represent nonphysical conditions since the system cannot mobilize more than its capacity. The geometrical expression of can be written asin which denotes the distance between point and the origin and the distance between the origin and point is designated by . is an arbitrary point in space -, and is the intersection of the -line and .