Sufficient conditions for existence of physically significant solutions in limiting equilibrium slope stability analysis

Engineering assessment of slope stability is usually performed using limiting equilibrium analysis. This framework includes a process of minimization which identifies the critical slip surface and its associated minimal safety factor. The approach makes sense only if a minimum safety factor exists, i.e. if there is a slip surface for which the safety factor is smaller than safety factors associated with all other slip surfaces. The present work establishes conditions which guarantee that slope stability problems have a physically significant minimum. The question of existence of a minimum is relevant to all slope stability formulations which satisfy equilibrium conditions without a priori assumptions with respect to the shape of potential slip surfaces. The main purpose of the present work is to "legitimatize" the approximate, but practically useful, limiting equilibrium technique by placing it on secure foundations. The present work shows that the restrictions required in order to ensure the existence of a minimum include three, well motivated, physical elements: (a) Two integral inequality constraints restricting legitimate forms of slip surfaces, and normal stress functions acting on them. These constraints represent the obvious observation that under usual conditions slopes fail by moving down and away from the main body of the slope. (b) The strength model (Mohr envelope), should imply a finite tensile strength. (c) A "cracking criterion" which specifies the consequences (crack formation) occurring when the soil’s tensile strength is fully mobilized.