Upper bound limit analysis using discontinuous velocity fields

Abstract

A new method for computing rigorous upper bounds under plane strain conditions is described. It is based on a linear three-noded triangular element, which has six unknown nodal velocities and a fixed number of unknown plastic multiplier rates, and uses the kinematic theorem to define a kinematically admissible velocity field as the solution of a linear programming problem. Unlike existing formulations, which permit only a limited number of velocity discontinuities whose directions of shearing must be specified a priori, the new formulation permits velocity discontinuities at all edges shared by adjacent triangles and the directions of shearing are found automatically. The variation of the velocity jump along each discontinuity is described by an additional set of four unknowns. All of the unknowns are subject to the constraints imposed by an associated flow rule and the velocity boundary conditions. The objective function corresponds to the dissipated power, or some related load parameter of interest, and is minimised to yield the desired upper bound. Since plastic deformation may occur not only in the discontinuities, but also throughout the triangular elements as well, the method is capable of modelling complex velocity fields accurately and typically produces tight upper bounds on the true limit load. The formulation is applicable to materials whose strength is cohesive-frictional, purely cohesive and uniform, or purely cohesive and linearly varying, and thus, quite general. The new procedure is very efficient and always requires fewer elements than existing methods to obtain useful upper bound solutions. Moreover, because of the extra degrees of freedom introduced by the discontinuities, the linear elements no longer need to be arranged in a special pattern to model incompressible behaviour accurately.