Convergence aspect of limit equilibrium methods for slopes

Abstract

This note is concerned with the multiplicity of solutions for the factor of safety that may be obtained on the basis of the method of slices. Discontinuities in the function for the factor of safety are discussed and the reasons for false convergence in any iterative solution process are explored, with particular reference to the well-known Bishop simplified method (circular slip surfaces) and Janbu simplified or generalized method (slip surfaces of arbitrary shape). The note emphasizes that both the solution method and the method of searching for the critical slip surface must be considered in assessing the potential for numerical difficulties and false convergence. Direct search methods for optimization (e.g., the simplex reflection method) appear to be superior to the grid search or repeated trial methods in this respect. To avoid false convergence, the initially assumed value of factor of safety F0 should be greater than β1(=−tan α1 tan [Formula: see text]) where α1 and [Formula: see text] are respectively the base inclination and internal friction angle of the first slice near the toe of a slope, the slice with the largest negative reverse inclination. A value of F0 = 1 + β1, is recommended on the basis of experience. If there is no slice with a negative slope for any of the slip surfaces generated in the automatic, search process, then any positive value of F0 will lead to true convergence for F. It is necessary to emphasize that no slip surface needs to be rejected for computational reasons except for Sarma's methods and similarly no artificial changes need to be made to the value of [Formula: see text] except for Sarma's methods. Key words: slope stability, convergence, limit equilibrium, analysis, optimization, slip surfaces, geological discontinuity, simplex reflection technique.