The kinematic theorem of limit analysis is used to evaluate the amount of reinforcement necessary to prevent collapse of slopes. The results are also applicable to some failure modes of reinforced walls. Calculations were performed assuming uniform and linearly increasing distributions of reinforcement strength through the slope height. The computational results are presented in charts, which can be used in design. The seismic influence is substituted with a quasi-static horizontal force. While such an approach ignores the acceleration history and does not allow any insight into the behavior of a structure, it is being routinely used in practice, and the charts are o€ffered as a design aid to determine the amount or strength of reinforcement. The length of reinforcement was also calculated, based on collapse mechanisms which include rupture in some layers and pull-out in others. It was found that the distribution of reinforcement with variable spacing, to match the triangular distribution of ``smeared'' strength, is more economical than a uniform spacing. Uniform spacing requires longer reinforcement and larger strength.

The traditional approach to stability analysis of granular slopes leads to the safety factor that is associated with a planar failure surface approaching the slope face, whether the slope is "dry" or submerged. However, for partially submerged slopes, a more critical, nonplanar failure surface can be formed. A family of geometrically similar surfaces can be found that is characterized by the same safety factor. If the safety factor drops down to unity and the slope becomes unstable, then a mechanism of an size can form.

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 force acting on all blocks in the selected mechanics, all slope stability analyses based on the limit equilibrium of slices can be interpreted in the context of their implicitly assumed collapse mechanisms.

A limit analysis approach is applied to determine the amount of reinforcement necessary to prevent collapse of slopes due to reinforcement rupture, pullout, or direct sliding. The reinforcement is uniformly distributed over the height of the slope, and each layer of the primary reinforcement has the same length. A rigorous lower bound to the required reinforcement strength is calculated.

A three dimensional slope stability analysis for drained frictional cohesive material based on the upper bound technique of limit analysis is presented. A rigid-block toe or above the toe collapse mechanism is considered, with energy dissipation taking place along planar velocity discontinuities. The technique is appropriate for slope analysis when question arises as to the level of permissible loads on slopes where the load is confined to limited area (local load).

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 kinematic approach of limit analysis is explored in three-dimensional
3D stability analysis of slopes. A formal derivation is first shown indicating that, in a general case, the approach yields an upper bound to the critical height of the slope or an upper bound on the safety factor. A 3D failure mechanism is used to produce stability charts for slopes. The slope safety factor can be read from the charts without the need for iterations. While two-dimensional
2D analyses of uniform slopes lead to lower safety factors than 3D analyses do, a 3D calculation is justified in cases where the width of the collapse mechanism has physical limitations, for instance, in the case of excavation slopes, or when the analysis is carried out to back-calculate the properties of the soil from 3D failure case histories. Also, a 3D failure can be triggered by a load on a portion of the surface area of the slope. Calculations indicate that for the 3D safety factor of the loaded slope to become lower than the 2D factor for the same slope
but with a load-free surface, the load has to be very significant and equal to the weight of a soil column of the order 10−1 of the slope height.