Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 76

Modeling and simulation of bivariate
distribution of shear strength
parameters using copulas
Dian-Qing Li and Xiao-Song Tang
2.1 IntroDuCtIon
It is well known that the shear strength parameters [cohesion ( c ) and friction angle (ϕ)]
are important parameters for evaluating deformation and stability of geotechnical struc-
tures, such as slope stability, bearing capacity of foundations, and earth pressure of retain-
ing walls. As far as the reliability analysis of these geotechnical structures is concerned,
the shear strength parameters are typically treated as uncertain parameters (Griffiths et al.
2011; Cherubini 2000; Abd Alghaffar and Dymiotis-Wellington 2007). Furthermore, it is
widely accepted that c and ϕ are negatively correlated in the literature (e.g., Low 2007; Li
et al. 2011; Tang et al. 2012, 2013a). To evaluate the reliability of geotechnical structures
exactly, the joint cumulative distribution function (CDF) or probability density function
(PDF) of shear strength parameters should be known. It is concluded by the previous stud-
ies (e.g., Low 2007; Li et al. 2011; Tang et al. 2012, 2013a) that the negative correlation
between cohesion and friction angle has a significant effect on geotechnical reliability and
ignoring such a correlation would lead to an overestimate of the probability of failure.
Many researchers studied the geotechnical reliability considering the dependence between
the shear strength parameters. In these studies, one or more of the three fundamental
assumptions have been made. First, the marginal distributions of the shear strength parame-
ters are normal distribution or have been transformed into normal distribution. Second, the
shear strength parameters have the same type of marginal distributions. Third, the depen-
dence structure between shear strength parameters is characterized by a Gaussian copula.
With regard to the third assumption, the well-known examples include the commonly used
Nataf model (Nataf 1962) and translation approach (Lebrun and Dutfoy 2009a, b; Li et al.
2012b; 2013b,c). In geotechnical practice, however, the shear strength parameters do not
always follow normal distribution and the same marginal distributions. Furthermore, the
Gaussian copula may not be adequate for characterizing the dependence structure between
c and ϕ as demonstrated by Tang et al. (2013a). Hence, it is of practical interest to develop a
more general and flexible approach for modeling the bivariate distribution of shear strength
parameters associated with geotechnical reliability problems.
To overcome the aforementioned three shortcomings, the past couple of years have wit-
nessed a growing interest in applying copulas for modeling the joint probability distribution
of multivariate data, particularly bivariate data (e.g., McNeil et al. 2005; Nelsen 2006).
Copulas are functions that join multivariate distribution functions to their one-dimensional
marginal distribution functions. There are many copulas in literature such as Gaussian, t ,
Frank, Clayton, Gumbel, and Plackett copulas. Each copula has its own dependence struc-
ture. Copulas provide a fairly general way for constructing multivariate distributions that
satisfy some nonparametric measure of dependence and prescribed marginal distributions.
The copula theory has been extensively used for financial and hydrological applications
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