UPPER BOUND LIMIT ANALYSIS USING DISCONTINUOUS VELOCITY-FIELDS

A new method for computing rigorous upper bounds under plane strain co nditions 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 de fine a kinematically admissible velocity field as the solution of a li near programming problem. Unlike existing formulations, which permit o nly a limited number of velocity discontinuities whose directions of s hearing must be specified a priori, the new formulation permits veloci ty discontinuities at all edges shared by adjacent triangles and the d irections of shearing are found automatically. The variation of the ve locity 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 condition s. The objective function corresponds to the dissipated power, or some related load parameter of interest, and is minimised to yield the des ired 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-fric tional, purely cohesive and uniform, or purely cohesive and linearly v arying, and thus, quite general. The new procedure is very efficient a nd always requires fewer elements than existing methods to obtain usef ul upper bound solutions. Moreover, because of the extra degrees of fr eedom introduced by the discontinuities, the linear elements no longer need to be arranged in a special pattern to model incompressible beha viour accurately.