B. Pichler:
"Zur Zuverlässigkeit initialer Abschätzungen von Stabilitätsgrenzen auf nichtlinearen Last-Verschiebungspfaden elastischer Strukturen mittels der Methode derFiniten Elemente"; Institute for Strength of Materials, Vienna University of Technology (TU Wien), 1999.
Frequently, loss of stability is the reason for the failure of thin-walled structures. Calculation of stability limits, i.e., of bifurcation or snap-through points on nonlinear load-displacement paths of elastic structures, requires the use of a geometrically nonlinear theory. For the computation of such stability limits the Finite Element Method is employed.Solving the resulting system of nonlinear algebraic equations by means of an incremental-iterative Newton procedure and an accompanying check of functions which are well suited for the identification of stability limits, may require excessive computer resources. Therefore, initial estimates of stability limits may be determined on the basis of linear eigenvalue analysis. The resulting error may be of the form of a significant overestimation of the load corresponding to snap-through.
In order to improve the quality of initial estimates of stability limits on nonlinear load-displacement paths, a special kind of an accompanying linear eigenproblem has been developed. It is based on the consistent linearization of the discretized stability criterion for static, conservative problems.
The usefulness of this method was demonstrated for different technical problems in the Dr.-techn.-thesis by Helnwein. Based on the characteristic properties of the consistently linearized eigenvalue problem, obtained from an asymptotic investigation, higer-order estimation functions were found.
For several engineering structures which are characterized by moderate displacements and, hence, by a moderate influence of nonlinearity in the prebuckling domain, the information obtained by means of asymptotic analysis remains relevant for an initial eigenvalue analysis. Thus, for such problems it is possible to make an initial assessment of stability limits. The higher-order estimates usually result in good approximations of the true solution of stability limits on nonlinear load-displacement paths.
On the other hand, for structures with a strong influence of the nonlinearity in the prebuckling domain, in the form of a strong redistribution of stresses, an asymptotic approach cannot be utilized for an estimate of the stability limit, solely based on information which is available at \la=0.
In order to further improve the quality of estimates of stability limits alternative estimation methods are defined and extensions of the existing higher-order estimation functions are presented. It is shown that indicators based on the first derivative of the eigenvector of the consistently linearized eigenproblem with respect to the loadparameter can lead to a conservative reliability-criterion for ab initio estimates of stability limits.
Keywords: diplomarbeit, diploma thesis