Upper bound theorem

Abstract: This paper describes a technique for computing rigorous upper bounds on limit loads under conditions of plane strain. The method assumes a perfectly plastic soil model, which is either purely cohesive or cohesive-frictional, and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory. The computational procedure uses three-noded triangular elements with the unknown velocities as the nodal variables. An additional set of unknowns, the plastic multiplier rates, is associated with each element. Kinematically admissible velocity discontinuities are permitted along specified planes within the grid. The finite element formulation of the upper bound theorem leads to a classical linear programming problem where the objective function, which is to be minimized, corresponds to the dissipated power and is expressed in terms of the velocities and plastic multiplier rates. The unknowns are subject to a set of linear constraints arising from the imposition of the flow rule and velocity boundary conditions. It is shown that the upper bound optimization problem may be solved efficiently by applying an active set algorithm to the dual linear programming problem. Since the computed velocity field satisfies all the conditions of the upper bound theorem, the corresponding limit load is a strict upper bound on the true limit load. Other advantages include the ability to deal with complicated loading, complex geometry and a variety of boundary conditions. Several examples are given to illustrate the effectiveness of the procedure.

Abstract: A new method for stability analysis in soils and rocks is presented, based on the upper bound theorem of classical plasticity. The sliding mass is divided into a small number of discrete blocks, with linear interfaces between blocks and either linear or curved bases to individual blocks. By equating the work done by external loads and body forces to the energy dissipated in shearing, either a safety factor or a disturbance factor may be calculated. The rigorous theoretical background is established, from which it may be demonstrated that for several well-defined classical slope problems the equations for the multi-block solution reduce to the published closed-form solutions. Powerful optimization routines are provided in the computer program EMU to search for the critical failure mechanism giving the lowest factor of safety. Several examples are given to demonstrate that, for problems where the exact answers are known, the new method produces accurate values of safety factor and predictions of failure mechanism. Applications to practical problems have shown that the new method is as simple as the conventional limit equilibrium methods for practitioners.