Divergence Instability Conditions in the Optimum Design of Nonlinear Elastic Systems Under Follower Forces

Abstract

Nonlinearly elastic discrete systems under nonconservative, compressive loading [1,2,3] of follower type, that may lose their stability through divergence, are considered. Using a general mathematical formulation a thorough parametric discussion of the critical, prebuckling and postbuckling, large displacement response, is comprehensively presented. The predominant effects on the nonlinear divergence instability of the material nonlinearity as well as of the loading parameters defining the degree of nonconservativeness, are completely revealed. Necessary and sufficient conditions for the existence of regions of devergence instability, are properly established. By means of these conditions the boundary between divergence (static) and flutter (dynamic) instability is found. The case of existence of a double critical point (coincidence of the first and second static buckling eigenmodes), obtained as a result of the linear stability analysis [4–6], is also discussed. At the aforementioned boundary, the (critical) buckling load corresponds to the maximum load-carrying capacity that can be determined by means of a nonlinear (static) stability analysis. Thus, a further insight into the role of certain parameters of paramount importance for the change of mechanism of instability from divergence to flutter, and vice-versa, is also gained.