Stochastic Limit Load Analysis of~Elasto-Plastic Plane Frames

Abstract

Problems from plastic analysis and optimal plastic design are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the stress (state) vector. In practice, one has to take into account stochastic variations of several model parameters, such as material strength parameters, external load factors, cost coefficients, etc.. Hence, in order to get robust maximum load factors, i.e., load factors being insensitive with respect to stochastic parameter variations, the structural analysis problem under stochastic uncertainty must be replaced by an appropriate deterministic substitute problem. Here, a direct approach is proposed based on the primary costs for missing carrying capacity and the recourse costs (e.g. costs for repair, compensation for weakness within the structure, damage, failure, etc.). Based on the mechanical survival conditions of plasticity theory, a quadratic error/loss criterion is developed. The minimum recourse costs can be determined then by solving an optimization problem having a quadratic objective function and linear constraints. For each configuration of the structure, i.e., each vector of model parameters and each design vector, one has then an explicit representation of the “best” internal load distribution. Moreover, also the expected recourse costs can be determined explicitly. Consequently, an explicit stochastic non-linear program results for finding a robust maximum limit load/shakedown factor. The deterministic substitute problems are based on (i) minimizing the expected total costs and (ii) minimizing (e.g.) the weight of the structure subject to an expected recourse cost constraint. Some numerical examples are given.