Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 114

2.6 SuMMarY anD ConCluSIonS
This chapter has employed the copula approach to model and simulate the bivariate dis-
tribution of shear strength parameters. Some measured data of shear strength parameters
reported in the literature are collected. Four copulas, namely Gaussian, Plackett, Frank, and
No.16 copulas, are selected to construct the joint probability distribution of cohesion and
friction angle. The impact of copulas on retaining wall reliability is investigated. Several
conclusions can be drawn:
1. Copulas can effectively decouple the determinations of marginal distributions and joint
probability distributions of shear strength parameters. Applying the copula theory, it
is not necessary that the shear strength parameters follow the same marginal distribu-
tions or must share a dependence structure characterized by the Gaussian copula. In
short, the copula theory provides a general and convenient approach for modeling and
simulating the bivariate distributions of shear strength parameters, which should be
highlighted in practical geotechnical applications.
2. The copula selection has a significant impact on geotechnical reliability. The fail-
ure probabilities produced by different copulas differ considerably. This difference
increases with decreasing probability of failure or increasing nominal factor of safety.
Significant difference in probability of failure could be observed for relatively small
COVs of shear strength parameters or a strongly negative correlation between cohe-
sion and friction angle.
3. The Gaussian copula may not capture the dependence structure between cohesion
and friction angle properly. When the Gaussian copula is applied to reliability analy-
ses, the probabilities of failure for the geotechnical structures may be underestimated
significantly. Thus, it is of importance to select the appropriate copula to characterize
the dependence structure underlying the shear strength parameters when enough data
are available.
4. The applications of copulas to geotechnical engineering are mostly concerned with
bivariate data. One reason for this is that relatively few copulas have practical
n -dimensional generalizations. One well-known example is the elliptical copulas such
as the Gaussian and t copulas. They can be easily extended to model and simulate
the multivariate distributions of multiple geotechnical parameters. However, the multi-
variate t copula is applicable only when complete multivariate data are available. With
incomplete multivariate data at hand, only the multivariate Gaussian copula can be
adopted.
aCknoWleDgMentS
This work was supported by the National Science Fund for Distinguished Young Scholars
(Project No. 51225903) and the National Basic Research Program of China (973 Program)
(Project No. 2011CB013506) and the National Natural Science Foundation of China (Project
No. 51329901).
aPPenDIX 2a: Matlab ® CoDeS
This appendix presents MATLAB codes stored in M-files for modeling and simulating the
bivariate distribution of shear strength parameters using copulas: (1) MATLAB codes for