Ab initio estimates of stability limits on nonlinear load-displacement paths: potential and limitations

Abstract

In order to avoid a fully nonlinear prebuckling analysis by the finite element method for the mere purpose of btaining the stability limit in the form of a bifurcation or a snap-through point, this limit may be estimated by means of the solution of a suitable linear eigenvalue problem. What seems to be most suitable in this context, is a consistent linearization of the mathematical formulation of the static stability condition. It can be interpreted as the stability criterion for the tangent to the load-displacement diagram at a known equilibrium state in the prebuckling domain. Based on this linearization, higher-order estimates of the stability limit can be obtained from scalar postcalculations. Unfortunately, the order of such an estimate is only defined in an asymptotic sense. Nevertheless, for many engineering structures the geometric nonlinearity in the prebuckling domain is moderate. In this case, the general information from asymptotic analysis is frequently relevant for the entire prebuckling domain. This allows good ab initio estimates of stability limits based on nonlinear load-displacement paths. The nucleus of this article is the discussion of the potential and the limitations of determination of stability limits based on ab initio estimates of nonlinear load-displacement paths. The theoretical findings are corroborated by the results from a comprehensive numerical study.