Conditions for minimum stiffness of proportionally loaded structures

Abstract

During proportional loading of structures a significant re-arrangement of stresses may occur. It may result e.g. in the transition from a softening to a stiffening structure. This may even be the case for constant material stiffness. Since the terms stiffening structure and softening structure are lacking universally accepted definitions, it is not astonishing that the search for conditions for minimum stiffness of proportionally loaded multi degree of freedom structures has so far attracted very little attention. Presentation of such a condition, in the framework of the Finite Element Method (FEM), is the task of this publication. It is based on a linear eigenvalue problem that renders complex eigenvalues and eigenvectors possible. The point of inflection in the real part of the complex section of a specific eigenvalue function is shown to mark the load-level of minimum stiffness of a proportionally loaded structure. Complex eigenvalues require the indefiniteness of both real symmetric coefficient matrices of the underlying linear eigenvalue problem. In the given case, the first one depends on a dimensionless load parameter, whereas the second one is obtained by specialization of the first one for the onset of loading. These matrices are established with the help of a special hybrid finite element, available in a commercial FE program. The proposed condition for minimum stiffness of proportionally loaded structures is verified numerically. The results obtained in a parameter study of the investigated structure are found to agree well with the ones following from an alternative condition for the transition from a softening to a stiffening structure. This condition is based on a mechanically objective arc length in the form of a representative displacement of the structure.