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.
A stability analysis of slopes based on a translational mechanism of failure is presented. The collapse mechanism is assumed to be in the form of rigid blocks analogous to slices in traditional slice methods. The proposed analysis, although based on the kinematical approach of limit analysis, always satisfies the equilibrium of forces acting on all blocks in the selected mechanism. All slope stability analyses based on the limit equilibrium of slices can be interpreted in the context of their implicitly assumed collapse mechanisms. The static assumptions made are equivalent to assuming an arbitrary strength of the soil on interfaces between slices. Solutions to stability factor yH/c from all analyses based on the limit equilibrium of slices fall into a relatively narrow range bounded by the solutions using the proposed analysis for two extreme assumptions of soil strength between the blocks. Solutions beyond this range obtained by any method of slices indicate unreasonable consequences when interpreted in the context of the failure mechanism. A convenient way to include pore pressure effects is also presented and implemented in the analysis of both translational and rotational slope collapse.
While computational tools have made most graphical methods and charts obsolete, stability charts for slopes are still routinely used in practice. The charts presented here are based on the kinematic approach of limit analysis that leads to a strict lower bound on stability number c/␥H or an upper bound on the safety factor. An earlier suggestion is employed in this paper to produce charts that eliminate the necessity for iterations. Charts are presented for slopes subjected to pore water pressure and also for those exposed to seismic forces.
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