Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 77

(e.g., McNeil et al. 2005; Genest and Favre 2007; Salvadori and De Michele 2007). Recently,
the copula theory has been applied to geotechnical engineering. For example, Uzielli and
Mayne (2011, 2012) investigated the dependence between load-displacement model param-
eters underlying vertically loaded shallow footings on sands using copula. Li et al. (2012a,
2013a) constructed the bivariate distribution of hyperbolic curve-fitting parameters under-
lying load-settlement curves of piles using copulas. Tang et al. (2013a) investigated the
impact of copula selection on geotechnical reliability. Wu (2013a) proposed a copula-based
sampling method for probabilistic slope stability analysis. Wu (2013b) further employed the
Gaussian and Frank copulas to model the trivariate distribution among cohesion, friction
angle, and unit weight of soils. It is evident that the applications of copulas to geotechnical
engineering are scarce, especially in geotechnical reliability analysis (Dutfoy and Lebrun
2009; Tang et al. 2013b, c). The potential applications of copulas for geotechnical reliability
analyses should be further explored.
This chapter aims to develop a copula-based approach for modeling and simulating the
bivariate distribution of shear strength parameters. In this approach, the aforementioned
three limitations are removed. This chapter is organized as follows. In Section 2.2, the
copula theory is briefly introduced. Thereafter, the procedure for step-by-step modeling of
the bivariate distribution of shear strength parameters is illustrated using measured data. In
Section 2.4, the simulation algorithms for copulas and bivariate distribution are presented
in detail. The effect of copulas on the probability of retaining wall overturning failure is
investigated in Section 2.5.

2.2 CoPula theorY

As mentioned in the introduction, one objective of this chapter is to model and simulate the
bivariate distribution of shear strength parameters using copulas. To facilitate the under-
standing of subsequent copula applications, the copula theory is first introduced. The def-
inition of copulas is presented in Section 2.2.1. Then, two commonly used dependence
measures, namely Pearson linear correlation coefficient and Kendall rank correlation coef-
ficient, are explained in Section 2.2.2. The adopted four bivariate copulas in this chapter
are provided in Section 2.2.3.

2.2.1 Definition of copulas
The word copula originated from a Latin word for “link” or “tie” that connects different
things. To define it (e.g., Nelsen 2006): Copulas are functions that join or couple mul-
tivariate distribution functions to their one-dimensional marginal distribution functions.
Alternatively, copulas are multivariate distribution functions whose one-dimensional mar-
ginal distributions are uniform in the interval of [0, 1]. Since Sklar's theorem is the founda-
tion of many applications of the copula theory, such a theorem is introduced first.
Sklar's theorem (e.g., Nelsen 2006). Let F ( x 1 , x 2 , …, x n ) be the joint CDF of a random
vector X = [ X 1 , X 2 , …, X n ]. Its marginal CDFs are F 1 ( x 1 ), F 2 ( x 2 ), …, F n ( x n ). The concept of a
marginal CDF has been introduced in Chapter 1 . A joint CDF is defined as the probability
that X is less than or equal to a specific numerical vector [ x 1 , x 2 , …, x n ]:

… …

Fx x (, , , x ) = PX ( ≤ xX , ≤ x , , Xx ≤ ) (2.1)

12 n 1 1 2 2 n n

where P (.) denotes the probability. Then there exists an n -dimensional copula C such that
for all real [ x 1 , x 2 , …, x n ],