Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 88

This approach includes the following three steps: (1) the measured cohesions and friction
angles of soils are collected, (2) the best-fit marginal distributions are determined, and (3)
the copula that provides the best fit to measured dependence structure is identified. To
decouple steps (2) and (3), Kendall's tau is adopted to determine the copula parameters
underlying the four candidate copulas (Gaussian, Plackett, Frank, and No.16 copulas). From
hereon, X 1 and X 2 denote the cohesion c and friction angle ϕ, respectively.

2.3.1 Measured data of cohesion and friction angle
To construct the bivariate distribution of shear strength parameters, some measured data
of cohesion and friction angle reported in the literature (Li et al. 2000; Wu et al. 2005;
Zhang et al. 2013) are collected. Tables 2.2 through 2.4 summarize the measured cohe-
sions and friction angles of soils from Xiaolangdi Hydropower Station, silty clay in Taiyuan
area, and several typical strata in the Hangzhou area of China, respectively. On the basis of
these measured data, the coefficients of variation (COV) (see Equation 1.2 of Chapter 1 ),
Pearson's rho ( Equation 2.11 ) , and Kendall's tau ( Equation 2.14 ) can be obtained, which
are also listed in these tables. Significant variations underlying the measured shear strength
parameters can be observed. Furthermore, both Pearson's rho and Kendall's tau indicate
that there exists a strongly negative correlation between cohesion and friction angle. These
observations coincide with the observation reported in Tang et al. (2012, 2013a). Since
the data in Table 2.2 have the largest sample sizes, they are further employed to construct
the bivariate distributions of shear strength parameters. These data are obtained from
various tests, namely (1) consolidated-drained test (referred to as CD hereafter), (2) consol-
idated-undrained test (CU), and (3) unconsolidated-undrained test (UU). The sample sizes
( N ) are 63, 64, and 61, respectively. The resulting Pearson's rho are −0.544, −0.702, and
−0.623 for CD, CU, and UU datasets, respectively. The corresponding Kendall's tau are
−0.384, −0.544, and −0.4 47.

2.3.2 Identification of best-fit marginal distributions
To fit the marginal distributions of c and ϕ, four candidate distributions, namely Normal
truncated below zero (referred to as TruncNormal hereafter), Lognormal, Gumbel truncated
below zero (referred to as TruncGumbel hereafter), and Weibull distributions are examined
(Lumb 1970; Baecher and Christian 2003). These four distributions can guarantee that the
simulated shear strength data are positive, which satisfy the requirements of positive c and
ϕ. Table 2.5 summarizes the PDFs f ( x ; p , q ) and domains of distribution parameters ( p , q )
associated with the four candidate distributions. The relationships between ( p , q ) and (μ, σ)
for the four distributions are also provided in Table 2.5 .
There are many goodness-of-fit test methods for identifying the best-fit marginal distribu-
tions underlying measured data. For example, Section 1.2.3 of Chapter 1 introduced the com-
monly used Kolmogorov-Smirnov (K-S) test. In this chapter, both the Akaike Information
Criterion (AIC) (Akaike 1974) and Bayesian Information Criterion (BIC) (Schwarz 1978)
are adopted to identify the best-fit marginal distributions. A marginal distribution resulting
in the smallest AIC and BIC values is considered to be the best-fit marginal distribution. The
AIC and BIC are, respectively, defined as

N

∑

AIC =− 2 ln (;,) fx pq + 2 k (2.25)

i 1

i = 1