Variational criteria for stiffness maxima and minima of proportionally loaded structures as solutions of an inverse problem

Abstract

Stiffness is a key term of structural mechanics. The same applies to the mechanical properties stiffening and softening of structures subjected to proportional loading. In the course of the loading process, originally stiffening (softening) structures may become softening (stiffening) structures. This occurs at the unknown load level at which the stiffness of the structure concerned attains a maximum (minimum) value. Extreme values of the stiffness of proportionally loaded structures are turning points of their mechanical behavior. Therefore, it is astonishing that analytical criteria for stiffness maxima and minima of such structures do not exist. In recent publications it has been claimed that points of inflection of eigenvalue functions of a special linear eigenvalue problem in the framework of the Finite Element Method (FEM) mark extreme values of the stiffness of proportionally loaded structures. The task of this work is to present the scientific foundation of this assertion in the form of criteria for stiffness maxima and minima based on variational calculus, termed variational criteria. This amounts to the solution of an inverse problem, with the mentioned numerical results as the observed effect and the sought variational criteria as the unknown cause. In general, analytical solutions for extreme values of the stiffness of proportionally loaded structures are inaccessible. Nevertheless, knowledge of their scientific basis is not only a fundamental scientific value in its own right, but also enhances the understanding of the intricacies of FE analysis for numerical determination of the load level at stiffness maxima and minima. The main motive for this work is to fill a significant scientific void of the literature in the area of the mechanics of engineering structures.