Assessing the applicability of local and global sensitivity approaches and their practical utility for probabilistic analysis of rock slope stability problems: comparisons and implications

Abstract

Characterization of properties governing the stability of rock slopes is essential for their design and analysis. Importance ranking of these properties can be obtained by the sensitivity indices that quantifies the extent to which different properties influence the stability of the slopes. This helps the designers to divert major laboratory/in situ investigation resources to evaluate highly ranked properties. Another usage is to perform the probabilistic stability analysis of slopes efficiently by treating low-ranked properties deterministically in the estimation of probability of failure (${P}_{\mathrm{f}}$). In this paper, the importance ranking of rock properties affecting the ${P}_{\mathrm{f}}$ of rock slopes prone to different failure mechanisms is performed using local/global sensitivity approaches (L/GSAs) and the accuracy of these approaches was assessed quantitatively. Four slope case studies indicating different structurally and stress-controlled failures were considered, and sensitivity analyses were performed using six different L/GSAs. Accuracy of the approaches was assessed by comparing the importance ranking of properties based on sensitivity approaches to that of the normalized errors, i.e., ${\varepsilon }_{i}$ invoked in the ${P}_{\mathrm{f}}$ by neglecting the uncertainties in these properties. Results indicated the superior accuracy of GSAs as compared to LSAs. Importance ranking was dependent upon the considered slope (failure mechanisms), with some slopes showing higher sensitivities to external parameters and others to inherent rock properties. An important guideline based on the analysis is suggested to consider the properties as deterministic/random variables in the probabilistic analysis. For the slopes with the minimum interaction effects in their sensitivity (planar, wedge, and stress controlled), uncertainties in multiple properties can be neglected based on the allowable error in the ${P}_{\mathrm{f}}$. Further, a dependence of ${\varepsilon }_{i}$ and corresponding importance ranking was observed on the selected value of property of interest (assumed as deterministic) across its domain in the analysis.

Appendix

1.1 Importance sampling

Importance sampling method is used to obtain a better estimate of ${P}_{f}$ of the system by sampling from a different distribution other than ${f}_{{\varvec{X}}}({\varvec{x}})$ that ensures preferred sampling from the failure region of the output domain leading to faster convergence and reduction in computational costs. The different distribution is known as importance sampling function $\psi ({\varvec{x}})$.

$${P}_{\mathrm{f},\mathrm{IS}}=\frac{1}{N}\sum_{k=1}^{N}{I}_{\left(Y\le 0\right)}({{\varvec{x}}}^{(k)})\frac{{f}_{{\varvec{X}}}({{\varvec{x}}}^{(k)})}{\psi ({{\varvec{x}}}^{(k)})}$$

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where {${{\varvec{x}}}^{(1)}, {{\varvec{x}}}^{(2)},\dots ,{{\varvec{x}}}^{(N)}$} is drawn from $\psi ({\varvec{x}})$.

In this article, first a FORM analysis is conducted in the standard normal space (${\varvec{U}}$) for the rock slopes and design point ${{\varvec{U}}}^{*}$ is obtained. Then, to obtain an efficient importance sampling function, mean of $n$-dimensional standard normal distribution (${\phi }_{n}({\varvec{u}})$) is shifted to ${{\varvec{U}}}^{*}$ as suggested by Melchers [29]. Thus,

$$\psi \left({\varvec{u}}\right)= {\phi }_{n}\left({\varvec{u}}-{{\varvec{U}}}^{*}\right).$$

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Sampling from $\psi \left({\varvec{u}}\right)$ will ensure greater sampling density in the failure region. Additionally, approximately 50% of these samples fall under the failure region of the system [29].