A new model for rock slope stability analysis is proposed in which the distribution of normal forces on the contact planes is chosen to minimize the potential energy of the system, allowing calculation of the factor of safety against sliding. The classic wedge and plane are shown to be special cases of this more general model, which allows determination of the safety factor for any shape of prismatic contact surface. One of the practical implications of this work is that if three or more contain planes are involved the safety factor may be significantly lower than that calculated from the wedge model (which provides an upper limit on stability).
Wedge sliding and plane sliding stability analyses have been widely used for the last thirty years in the determination of rock slope stability. Some of the early work on the subject is due to Londe et al. (1969, 1970), John (1968), Wittim (1965) and Goodman (1976, 1989). The main results have been compiled by Hoek and Bray (1981) and a useful computer program has been prepared by Watts (1994). The requirements for sliding may be summarized as follows. Plane failures can occur when the strike of a discontinuity plane such as bedding is parallel or nearly parallel to the strike of the slope face and the weak plane daylights in the free face at a dip angle greater than the friction angle. Wedge failures can occur for a block defined by two planes whose line of intersection daylights in the free face and plunges sufficiently steeply that the destabilizing forces exceed the shear resistance. These classic methods, based on limiting equilibrium and typically implemented either through graphical methods such as stereographic projection, or via computer programs, assume that all shear stress in the contact plane(s) directly opposes motion. Given this assumption, the distribution of normal forces on the contact planes is statically determinate (for one or two contact planes), and hence the magnitude of the frictional resistance to sliding, and the factor of safety, can be calculated directly.
The fundamental problem, as stated by Warburton (1993), is to predict how the fixed surroundings and the forces acting on the block will affect its movement. Here, the important factors in the solution of stability problems are the shear strength, dip and dip direction of the discontinuity planes, the geometry of the slope, and the loading conditions. Certain geologic environments, however, produce blocks which cannot be adequately modeled as either wedges or plane slides. An example is blocks forming in cylindrically folded sedimentary rocks, where the surface of sliding is neither a single plane nor a double plane, but is curved. This type of block may be idealized as a prismatic block with multiple sliding planes, all with parallel lines of intersection. If the sliding planes number three or more the distribution of normal forces, and hence the factor of safety, is indeterminate. In this paper we introduce a new analytical model for sliding stability analysis of cylindrical sliding surfaces, which include plane and wedge slides as special cases. We show that if three or more contact planes are involved the safety factor may be significantly lower than that calculated from the wedge model.