Points of inflection of special eigenvalue functions as indicators of stiffness maxima/minima of proportionally loaded structures

Abstract

The stiffness of a proportionally loaded structure may continuously increase or decrease. As a special exception, it may be constant. On the other hand, an initially stiffening (softening) structure may turn into a softening (stiffening) structure. At the load level of such a change the stiffness of the structure attains an extreme value. The task of this work is to present mathematical conditions for these load levels. Lack of them represents a void in the pertinent literature. The practical significance of the aforementioned changes is the one of indicators of the mechanical behavior to be expected after their occurrence. The second and the last author have recently presented a condition for the load level at which the stiffness of a proportionally loaded structure becomes a minimum value. It is given as d²(ℜ(χ₁(λ)))/dλ²=0, representing the condition for a point of inflection of the real part of a complex eigenvalue function χ₁(λ), where λ denotes a dimensionless load parameter. The underlying linear eigenvalue problem has two indefinite coefficient matrices, which is a necessary condition for complex regions of eigenvalue functions. These matrices are established with hybrid elements, available in a commercial finite element program. In the present work, d²(χ₁(λ))/dλ²=0 is shown to be the condition for the load level at which the stiffness of a proportionally loaded structure attains a maximum value. The eigenvalue function concerned has no complex region. It is also shown that the displacement elements, which are the basis for their extension to the employed hybrid elements, are unable to indicate the load level at an extreme value of the stiffness.