An analytical solution for analysis of toppling-slumping failure in rock slopes

When a slope consists of a horizontal weak rock or soil overlaid by some vertical strong slender rock columns, the slope is prone to a secondary toppling failure which is known as "toppling-slumping" failure. In the slope, rock columns impose pressure on the lower continuous weak rock or soil mass which leads to a differential settlement at the base of each rock column or a circular shear failure in the continuous mass. If the differential settlements reach a special threshold, rock column overturn and toppling-slumping will happen. However, the circular shear failure of the soil or weak rock mass which is a well-recognized failure mechanism will occur when shear stress in the slope exceeds its shear strength. In this paper, an attempt has been made to better understanding of the "toppling-slumping" failure mechanism through developing an analytical solution. For this purpose, first an Airy stress function is acquired for the lower continuous soil mass and its stress distribution and displacement are accordingly determined analytically due to the overlaid rock columns loads. Then, differential settlements at the base of all rock columns and shear stresses in the soil or weak rock mass are computed and compared with the toppling and shear failure criteria to assess slope failure. Finally, several design-charts are developed that determine the stability of a slope against toppling-slumping failure mechanism based on geometric characteristics of the slope and shear strength of its constituent material. Also, a Mathematica package code was developed to evaluate the stress distribution within the slope, and the safety factor. The results show that the vertical stress distribution plays a significant role in the settlement of the rock columns. Results also manifested that at a certain distance from the slope crest, the set of rock columns will act like a single column on a half-plane. The shear or toppling failure of the slope not only depends on the material's strength but also on the geometry and the normalized column distance.