Bivariate probabilistic slope stability analysis using copulas and back analysis by Markov chain Monte Carlo

Copula functions have been prevalently used to take into account the correlation of random variables in probabilistic analyses. In this paper, copula functions of different families were used in probabilistic slope stability analysis along with a back-analysis to reduce the uncertainty of random input variables. In this regard, the cohesion (c) and internal friction angle (& oslash;) of the soil were incorporated as random variables into a computer code developed in MATLAB. The c-& oslash; pair values generated using the best copulas were exported to GeoStudio software to estimate the distribution of the factor of safety (FS) by various limit equilibrium (LE) methods. Additionally, the Markov chain Monte Carlo (MCMC) method using Metropolis algorithm was employed to adjust the values of random input variables in a back analysis of slope stability, so as to modify the FS values. It was shown that the results of different copulas are affected by both the back analysis and c-& oslash; sample size, where the effect of back analysis on reducing the uncertainty is much more than increasing the sample size. Frank copula showed the strongest performance and the least uncertainty in probabilistic slope stability analysis, while Gaussian showed the weakest performance. Finally, it was found that the Morgenstern-Price, Spencer, and Bishop methods are the most effective LE methods for probabilistic slope stability analysis when the correlation of random variables are addressed.