Optimization Under Longevity Constraint

Abstract

This chapter deals with probabilistic approaches to optimal design of structures made from quasibrittle material and loaded by cyclic forces [YCC97, PKK97, BIMS05a, BIMS05b, BIM07]. Special attention is devoted to different problem formulations and analytical solution methods. First we present some basic assumptions and relations. Then we formulate the optimal structural design problem based on a probabilistic approach. We must minimize the cost functional (volume of material) under constraints on the number of loading cycles before global fracture and on the probability of nondestructive behavior of the body. The original constraints are transformed to inequalities imposed on the stress in the uncracked body at the crack location. The resulting problem of optimal shape design consists of cost functional minimization under stress constraints, and can be solved by conventional methods. Several examples of structural design problems for statically determinate and indeterminate beams and frames are presented in the chapter. Then we use another probabilistic approach, based on the application of moment inequalities, for optimal structural design under a longevity constraint (constraint on the number of cycles). Here we require that the mathematical expectation (first moment) of the critical number of cycles must be greater than the given number of cycles, and the dispersion (second movement) of the critical number of cycles must be less than a given value. It is shown that this problem can be transformed to that of the structural volume minimization under a system of stress constraints. The presentation follows research results of [BRS03a].