The optimal profile of a rock slope under given geometry constraint is investigated by means of the upper bound method of limit analysis. The optimal condition corresponds to maximizing stability factor. It is assumed that rock failure condition is suitably described by the Hoek-Brown criterion. Genetic algorithm is applied for locating the optimal profile. The optimal profiles turn out to be concave profiles but there are heaps around the toes of the slopes. Compared with plane profiles, the percentage of increase in stability factor can reach up to nearly 50%. The key geometry parameters of the optimal profiles are provided for practical use. The new upper bound formulation is capable of computing the stability factors of slopes of any shape, thus the stability status of a slope under man-made excavation can be monitored during the whole construction period.


In mining industry, man-made slopes are excavated to create working surfaces. It is important to optimize the slope profile in order to maximize slope stability under certain geometry constraint. For instance, when the height and width of an excavation are assigned, a plane profile AC (see in Fig. 1) is widely applied in engineering practice, but it has been proved not the best one in terms of safety (Utili & Nova, 2007; Cala, 2007). A vertical profile ABC is the one with least stability. Thus, it can be deduced that there is a profile within the domain of the rectangle ABCD providing the highest stability.

Cala (2007) compared the difference of convex and concave profiles in stability using strength reduction technique. According to Cala (2007), circular concave slopes can increase the maximum slope inclination under given factor of safety. Besides, based on upper bound theory of limit analysis, Utili & Nova (2007) assumed a logarithmic spiral profile to prove it is more stable than a plane profile in terms of the stability factor. Although the above studies attempted to seek the best circular or logarithmic spiral profiles, none of them considered the arbitrary shape of the optimal profile and its contribution to enhance slope stability.

As reported in Hudson & Harrison (1997), there are two slope failure mechanisms for rock slopes:

  1. a curvilinear one when the rock behaves as an equivalent continuum and

  2. a linear one when it behaves as a discontinuum.

This paper deals with the first type of failure mechanisms. The rock behaves as a continuum when it is intact or when the scale of discontinuities is small compared with the scale of the slope, for instance, in the case of closely fractured or highly weathered (Wyllie & Mah, 2004). In this paper, the optimal profile of a rock slope is investigated by means of the upper bound method of limit analysis. The rock mass is treated as a continuum following Hoek-Brown failure criteria. The stability factor corresponding to the non-linear Hoek-Brown failure criterion is obtained by revising the solution following Mohr-Coulomb failure criterion. Genetic algorithm is applied to search for the optimal profile.

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