Slope stability analysis using rigid elements

Abstract

Slope stability problems are among the most commonly addressed problems in geotechnical engineering. In the past several decades, limit equilibrium methods have been most commonly used and widely accepted by engineers for slope stability analyses due to their relative simplicity and rich experience accumulated. However, it is well known that the solution obtained by a limit equilibrium method is not rigorous because neither static nor kinematic admissibility conditions are satisfied. Recently, many efforts have been made to utilize limit methods based on the upper bound or lower bound limit theorems in classical plasticity to assess the slope stability. The main objective of this thesis is to develop and provide a novel approach to the upper bound limit analysis of slope stability using rigid elements. A new rigid element formulation of the upper bound theorem is derived and presented. An efficient solution algorithm is adopted to obtain the factor of safety for the resulting optimization problem. A new three-dimensional (3-D) mesh generation scheme with the power of digital elevation model (DEM) in Geographical Information System (GIS) has been put forward as related to 3-D mesh generation for 3-D slope stability analysis. The major part of this thesis describes a new upper bound formulation using rigid elements for the limit analysis of two-dimensional (2-D) and 3-D slope stability problems. Rigid elements are used to construct a kinematically admissible velocity field, which makes it possible to perform the limit analysis of stability problems with complex geometries, soil profiles, groundwater conditions, and complicated loadings. The velocity discontinuities are permitted to occur at all inter-element boundaries. The task of finding the minimum value of the factor of safety can be formulated as a nonlinear programming problem with linear and nonlinear equality constraints by expressing mathematically the Mohr-Coulomb failure criterion, a flow rule, velocity boundary conditions, and the energy-work balance equation. A special feasible sequential quadratic programming algorithm (FSQP) has been applied, for the first time, to obtain the solutions for such nonlinear optimization problems. In FSQP, the nonlinear equality constraints are turned into inequality constraints and the objective function is replaced by an exact penalty function which penalizes nonlinear equality constraint violations only. It has been shown that the formulation of the proposed method can be easily reduced to the formulations of the upper bound limit analysis methods proposed by Michalowski (1995) and Donald and Chen (1997) if the same slice techniques with a translational failure mechanism are used. In other words, the upper bound limit analysis using slices may be viewed as a special case of the present method. The effects of pore water pressure are considered and incorporated into the rigid element formulations. The pore water pressure is treated as an external force, similar to gravity and surface fractions, and included through work terms in the energy-work balance equation. In addition, the slope stability problems with a nonlinear Mohr-Coulomb failure criterion have been analyzed by taking advantages of rigid elements in conjunction with a tangential technique, using a series of linear failure (or yield) surfaces tangent to the actual nonlinear failure/yield surface. The approximation is done on an interface-by-interface basis such that each discontinuity in the mesh has its own tangential friction angle and cohesion. This means, the friction angle and cohesion of the soil is different from interface to interface, depending on the location of the interface. Utilizing the DEM in GIS, a multi-layer DEM technique for modeling the slope geometry, geological stratifications, potential slip surface, and pore-water pressure conditions has been put forward in the thesis. A strategy for 3-D mesh generation, including the establishment of topological relationships between spatial entities, judgment of correlative relationships between spatial surfaces, implementation of strata separations, and reconstruction of spatial model, is then developed. The 3-D numerical mesh suitable for upper bound computations is subsequently generated. A range of 2-D and 3-D stability problems is used to examine the validation and capability of the proposed method. A few actual cases are studied to show the application of the present method in solving practical slope stability problems.