A General Algorithm for Limit Solutions of Plane Stress Problems

A computational approach to limit solutions is considered most challenging for two major reasons. A limit solution is likely to be non-smooth such that certain non-differentiable functions are perfectly admissible and make physical and mathematical sense. Moreover, the possibility of non-unique solutions makes it difficult to analyze the convergence of an iterative algorithm or even to define a criterion of convergence. In this paper. we use two mathematical tools to resolve these difficulties. A duality theorem defines convergence from above and from below the exact solution. A combined smoothing and successive approximation applied to the upper bound formulation perturbs the original problem into a smooth one by a small parameter E. As E + 0. the solution of the original problem is recovered. This general computational algorithm is robust such that from any initial trial solution. the first iteration falls into a convex hull that contains the exact solution(s) of the problem. Unlike an incremental method that invariably renders the limit problem ill-conditioned. the algorithm is numerically stable. Limit analysis itself is a highly efficient concept which bypasses the tedium of the intermediate elastic-plastic deformation and seeks the most important information directly. With the said algorithm. WC have produced many limit solutions of plane stress problems. Certain non-smooth characters of the limit solutions arc shown in the examplrz presented. Two well-known as well as one parametric family of yield functions are used to allow comparison with some classical solutions.