Optimal prestress of cracked unilateral structures: finite element analysis of an optimal control problem for variational inequalities

Abstract

The structural analysis problem concerning static stabilization of cracked structures through prestress reinforcement is studied in this paper. A linearly elastic structure with given crack interfaces along which frictionless no-tension no-penetration mechanism hold (the so called unilateral contact mechanism) is assumed. The prestressing control, which in most civil engineering applications is realized by externally placed prestressed cables, is modelled by means of prestressing (control) forces. The static analysis (state) problem turns out to be a variational inequality. The goal of the stabilization is taken to be the healing of the cracks in the structure with the minimum reinforcement among an available and technologically feasible set and this is expressed by means of a suitably written combined cost function. The arising optimal control problem is a nonsmooth and nonconvex one, thus appropriate techniques recently developed in the area of nonsmooth analysis are used for its solution. Numerical examples illustrate the theory and parametric numerical experimentation with various algorithms permit us discuss certain aspects of the computer application. Appropriate condensation techniques for the reduction of the problem dimensionality and the integration of the studied algorithms in a general purpose finite element software are given.