Three-dimensional stability of slopes and excavations

Three-dimensional (3D) limit analysis of stability of slopes is presented. Such analyses are not common, because of the difficulties in constructing three-dimensional mechanisms of failure in frictional soils. A class of admissible rotational mechanisms is considered in this paper. The failure surface has the shape of a curvilinear cone (‘horn'), with upper and lower contours defined by log-spirals; all radial cross-sections of the surface are circular. In the special case of cohesive soils (undrained behaviour), the shape of the failure surface reduces to a torus. An alternative failure surface is generated when the axis of rotation intersects the circle that generates the surface. The 3D mechanism is further modified with a plane-strain central insert to ensure the transition to a plane-strain mechanism if no restraint is placed on the slope width. Also, the spherical failure surface considered in the literature is re-examined. The critical height of slopes with finite width is determined, and the results are presented in the form of graphs and tables for a practical range of parameters. A separate set of results is given for the critical depth of excavations, where the extent of the failure mechanism is defined by the geometry of the earthworks. An example illustrates the practical use of the results.