P. Helnwein, H.A. Mang, B. Pichler:
"Ab Initio Estimates of Stability Limits on Nonlinear Load-Displacement Paths: Potential and Limitations";
Computer Assisted Mechanics and Engineering Sciences, 6 (1999), 3/4; 345 - 360.
Determination of stability limits (bifurcation or limit points) on nonlinear load-displacement paths of elastic structures requires use of a geometrically nonlinear theory. With the help of the principle of virtual displacements and the Finite Element Method, the respective system of nonlinear differential equations is converted to a system of nonlinear algebraic equations. The incremental-iterative solution of this system of equations must include checks for stability limits.
In order to avoid a fully nonlinear prebuckling analysis for the mere purpose of obtaining the stability limit, estimates of this limit, 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 tangent to the load-displacement diagram at a known equilibrium state in the stable prebuckling domain. Based on this linearization, higher-order estimates of the stability limit can be obtained from a scalar postcalculation.
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 on nonlinear load-displacement paths.
The nuclei of this keynote lecture is the discussion of the feasibility of good ab initio estimates on nonlinear load-displacement paths.
The theoretical findings are corroborated by the results from a comprehensive numerical study.
Keywords: shell, stability, estimates, stability limits, buckling, derivative of eigenvector, indicator function, reliability-indicator