Computational Investigation of Structural Stability

Estimation of stability limits on nonlinear load-displacement paths from the prebuckling path has attracted several researchers in computational structural mechanics. The vehicle for such estimations are linear eigenvalue problems. The mathematical basis for the estimation functions to be used in this project is the consistent linearization of the static stability criterion for systems discretized by the Finite Element Method. Consistent linearization leads to a special form of a linear eigenvalue problem. The resulting eigenvalues provide estimation (eigenvalue) functions, which may contain important information about the mechanical behavior of the structure. Hence, the eigenvalue functions are investigated to obtain insight in the behavior of the structure. The main goals of this research project are the following: * investigation of the mechanical meaning of the sign of the curvature of the estimation functions at bifurcation points * investigation of the mechanical meaning of different kinds of cusps of the eigenvalue curves at snap-through points (provided different kinds of cusps are existing) * analysis of the influence of imperfections on the behavior of estimation functions According to the listed goals, in the first part, bifurcation problems will be analyzed. The sign of the curvature of the eigenvalue curves at bifurcation points may be positive or negative. As a special case, this sign may be zero. The mechanical meaning of the mentioned sign is yet unknown. There is numerical evidence that, with regards to symmetric bifurcation, the sign of the curvature of the eigenvalue curve at the stability limit is not correlated with imperfection sensitivity or insensitivity of the structure. However, the aforementioned special case, for which the eigenvalue curve exhibits a saddle point at the stability limit, represents a "border-line case" between imperfection sensitivity and insensitivity. One of the goals of the proposed project is to explore the described situation and present the respective mathematical proof for the border-line case. The goal of the second part is to obtain further insight in structures exhibiting a snap-through point. Therefore, the behavior of eigenvalue curves at snap-through points is analyzed. In all numerical examples analyzed so far, the second derivative of the eigenvalue curve (based on a load parameterization) tends to infinity at snap-through points. Hence, the eigenvalue curve has a cusp of first kind at this point. It will be one of the tasks of the project to investigate whether eigenvalue curves with a cusp of second kind (characterized by a finite value of the curvature) exist and to study the mechanical significance of such cusps provided they exist. A singularity-free parameterization using the arc-length as the parameter was proposed to overcome the singularity at snap-through points. From numerical evidence it appears that the eigenvalue curves based on this parameterization are approaching the stability limit monotonically. A mathematical proof of this monotonicity will be attempted in this project. In the third part, the influence of imperfections on the behavior of eigenvalue functions will be investigated. From the viewpoint of technical applications this is a very important task. Civil engineering structures nearly always have small imperfections, i.e. deviations from the theoretical shape or loading. These imperfections may have a significant influence on the structural behavior. This influence is reflected by the eigenvalue curves obtained by means of the consistently linearized eigenproblem. It is expected that this information can be exploited for an "early" assessment of the significence of the imperfections.