Stability of Homogeneous Slopes: From Chart to Closed-Form Solutions and from Deterministic to Probabilistic Analysis

Slopes are rarely homogeneous. However, stability solutions for homogeneous slopes are still useful because simple assumptions on ground conditions have to be made when there is no or little site investigation, typically at the early design stage of a project. Many stability charts have been developed for slope stability analysis since the pioneer work of 1937, however, stability charts are a product of the 20th century and not very compatible with the digital design workflow in the 21st century. Instead, closed-form solutions (CFS), which are more compatible with modern digital design workflow, are developed in this study for dry, saturated, and unsaturated slopes. The CFS can directly be used for deterministic analysis. Limit state surface (LSS), which separates the failure and safe domains, is ultimately important for probabilistic analysis. With the proposed CFS, LSS can be defined analytically, therefore conducting a probabilistic analysis is as straightforward as a deterministic analysis. Four methods are considered in this study: first-order reliability method (FORM), second-order reliability method (SORM), direct integration method (DIM), and Monte Carlo simulation (MCS). Two examples are used to illustrate the application. In the first example, probability of failure obtained by FORM in this study (defining LSS by the proposed CFS) agrees well with those reported in the literature (defining LSS by the strength reduction finite-element method). It is shown that LSS in the uncorrelated normalized u-space can be highly nonlinear, in such a situation FORM is less accurate, while SORM still performs well compared with DIM and MCS. In the second example, it is demonstrated that with the proposed CFS, digital automation of slope design can readily be implemented in a computer spreadsheet using various design methods.

This study presents a series of simple analytical solutions for dry, saturated, and unsaturated slopes. These analytical solutions are more convenient to use than the conventional slope stability charts, as analytical solutions can be incorporated into a spreadsheet program and readily be used for parametric studies and design automation. With the proposed analytical solutions, carrying out a probabilistic analysis is as simple as a deterministic analysis. Therefore, slope stability can be evaluated through probability of failure, which better captures the uncertainty of the ground parameters. An Excel spreadsheet that contains automation scripts and practical examples is provided in the Supplemental Material, which can be directly used or adapted for specific design problems.