Improved analytical solution for toppling stability analysis of rock slopes

A solution is developed, based on a limit equilibrium approach, for analysing the toppling stability of rock slopes, which are characterized by blocks whose thickness is significantly smaller than the height of the block at the crest. The method is general in form and is applicable to a range of discontinuity geometries. In this study, the determination of the transition position from toppling to sliding is discussed in detail. It is clarified that the transition position from toppling to sliding should be determined by the frictional characteristics of the block base, for both cases of the friction angle along the dominant discontinuities being greater or less than the angle of the cut slope with the line normal to the dip of the dominant discontinuities. The effect of the angle of the block base with the normal to the dip of the dominant discontinuities, βbr, on the toppling stability is analysed. It is indicated that toppling stability greatly diminishes with increasing βbr, and that the ratio of the increment of the factor of safety to its initial value with βbr=0 will vary between 10% and 20% as βbr changes from 0° to ±5°, and 30% and 50% with βbr up to ±10°. A large discrepancy occurs between the calculated factor of safety against its actual value when βbr is considered equal to zero, in the case of the block base being non-normal to the dip of the dominant discontinuities. A spreadsheet procedure is presented for facilitating the method and by which several cases of toppling are analysed. The results indicate that the proposed solution represents the asymptotic value of the support force necessary to stabilize the slope against toppling as the slenderness ratio tends to infinity and that, when the slenderness ratio is greater than approximately 15–25, the support force calculated by the proposed solution provides an accurate estimate of the actual value.