| Abstract: | A 3-D mathematical approach to slope stability, which is based on limiting equilibrium and variational analysis, is presented. In the initial formulation there are three unknown functions: the slip surface, the normal stress and the shear stress direction over this surface. The minimum factor of safety is sought through variational extremization. The analysis indicates that the factor of safety is independent of the normal stress distribution over the critical slip surface. It also indicates that the direction of the elementary shear force over the slip surface depends on the slip surface function, but not on the normal stress function. The analysis yields a non-linear first order partial differential equation, relating the slip surface and its first partial derivatives. By limiting the analysis to symmetrical problems an ordinary differential equation, governing the slip surface path on the plane of symmetry, is derived. This equation enables the development of a numerical procedure to determine the minimal factor of safety of symmetrical 3-D slopes. Two possible failure modes are determined for homogeneous slopes. One mode consists of finite 3-D sliding body and the second represents cylindrical failure. Numerical analyses for some simple cases, of homogencous slopes are presented. |
