Though sliding and toppling conditions of rock blocks on an inclined plane are easily established from static or dynamic equations, safety assessment of natural rock slopes against toppling failure is quite difficult. The paper presents the equations, which distinguish the static and dynamic approaches of this type of problems, followed by a numeric procedure implemented on a worksheet for the safety evaluation of a toppling slope. A commercial program (UDEC) was used to validate the numerical procedure, after being tested against the dynamic equations of a rigid block on an inclined plane. A probabilistic model considering the Spacing variability of the discontinuity spacing was developed.
It is commonly referred that the failure modes in rock slopes are sliding and toppling. Sliding occurs due to a translation movement of a rock block or a rock mass along a plane failure surface. Toppling involves rotation of rock blocks or columns around a fixed point on its base. The simplest assumption for the study of rock slopes is to consider a single rigid block on an inclined plane (Figure 1). This figure allows to establish the regions where the block is stable, where it slides or where it tilts. However, it can be seen that there is a zone where the block can slide and tilt at the same time. In this region, the correct instability mode cannot be deduced from the limit equilibrium equations (1) and (2) because they are not formally valid. In fact, the zone where both sliding and tilting may occur does not correspond to the intersection of both equations, since it is required that the force acting at the toe is large enough to provide block fixity (Bray & Goodman). As Sagaseta (1986) states, to define correctly the boundaries between the regions where sliding, tilting and sliding plus tilting occur it is necessary to establish the limit equilibrium equations starting by the dynamic motion equations of the block and then by imposing the particular conditions associated with the different modes of failure. The stability analysis of rock blocks on a slope was performed using a limit equilibrium analysis (Goodman & Bray, 1976; Hoek & Bray, 1977; Bobet, 1999) implemented in a common worksheet (Excel). The relative errors were found to be very small (less than 0.25%). Finally, the UDEC single block model was used to define the boundaries between the sliding, toppling and mixed sliding and toppling failure modes. Again, values of friction angle ranging from to 20° to 45° were used. The results showed that equations (5.2) or (7.1) - boundary between sliding and sliding+toppling - and equations (6.2) or (7.2) - boundary between sliding+toppling and toppling - were well modelled by UDEC, as the relative errors were small (in the range of the values already shown).
A 2D problem involving the toppling of a group of rock blocks on an excavation slope was analysed.