Limit analysis of soil slope stability using finite elements and linear programming

Stability and deformation problems in geotechnical engineering are boundary value problems: differential equations must be solved for given boundary conditions. Solutions are found by solving differential equations derived from conditions of equilibrium, compatibility, and the soil constitutive relations, subjected to boundary conditions. For stability problems, the theory of plasticity is used to set up the differential equations. The limit equilibrium method is commonly used for slope stability analysis. However, it is well known that the solution obtained from the limit equilibrium method is not rigorous, because neither static nor kinematic admissibility conditions are satisfied. Limit analysis takes advantage of lower and upper bound theorems to provide relatively simple, but rigorous bounds on the true solution. Therefore, limit analysis is an efficient benchmarking method to check the validity of solutions obtained by the limit equilibrium methods. In this study, three-noded linear triangular finite elements are used to construct both statically admissible stress fields for lower bound analysis and kinematically admissible velocity fields for upper bound analysis. By assuming linear variation of nodal variables, the determination of the best lower and upper bound solution may be set up as a linear programming problem with constraints based on the satisfaction of static and kinematic admissibility. The effects of porewater pressures are considered and incorporated into the finite element formulations so that the effective stress analysis of saturated slopes may be done. Results obtained from limit analysis of homogeneous simple slopes with different ground water patterns are presented in the form of stability charts, and compared with those obtained from the limit equilibrium method. For more general problem—slopes with inhomogeneous soil profiles, arbitrary ground surface and water table, the lower and upper bounds to the true factor of safety are calculated directly using the numerical procedure. Various examples of slopes are selected from the literature and the results of limit analysis and limit equilibrium analysis of these slopes are compared.