B. Pichler:
"Estimation of Stability Limits on Nonlinear Load-Displacement Paths";
Talk: 11th Inter-Institute Seminar on Youth and Computational Mechanics, Janowice, Poland; 1999-10-07 - 1999-10-10.
Loss of stability is a frequent reason for failure of thin-walled structures. Computation of static stability limits, i.e., bifurcation or limit points, on nonlinear load-displacement paths of elastic structures, requires the use of a geometrically nonlinear theory. A powerful method for determination of such points is the Finite Element Method (FEM). One of several possible bases of the FEM is the principle of virtual displacements. With the help of this principle a system of nonlinear partial differential equations is converted into a system of nonlinear algebraic equations.In order to avoid a fully nonlinear prebuckling analysis for the mere purpose of obtaining the stability limit, estimates of this limits, based on the solution of linear eigenvalue problems, have frequently been used. Helnwein, e.g., has suggested a consistent linearization of the mathematical formulation of the static stability condition. It can be interpreted as the stability criterion for the tangents to the nonlinear load-displacement diagrams at a known state of equilibrium in the stable prebuckling domain.
Based on the results of the investigation of the asymptotic properties of the consistently linearized eigenvalue problem, alternative functions can be defined for the purpose of reducing the estimation error. Those estimation functions, obtained from scalar post-calculations, can be identified as estimation functions of higher order. Since the order of the estimation error is not necessarily equal for bifurcation points and snap-through points, the investigation of the asymptotic properties must be performed independently for both types of loss of stability.
Different extensions of the higher-order estimation functions can be derived. They should even further increase the reliability of estimates of stability limits, which may be obtained. Unfortunately, the order of such estimates is only defined in an asymptotic sense. Nevertheless, for many engineering structures the effect of the geometric nonlinearity in the prebuckling domain is moderate. In this case, the general information from asymptotic analysis is not only relevant in the immediate vicinity of the stability limit, but in an extended part of the entire prebuckling domain.
Based on the angle between the eigenvector of the consistently linearized eigenvalue problem and the reference-load vector, an a-priori criterion to distinguish bifurcation modes from snap-through modes can be derived. Therefore, appropriate buckling-mode-specific estimation functions of higher order can be chosen for an initial estimation of the stability limit.
Keywords: stability limits, estimation, ab-initio, reliability