General Eigenvalue Buckling
In an eigenvalue buckling problem we look for the loads for which the model stiffness matrix becomes singular, so that the problem
has nontrivial solutions. KMN is the tangent stiffness matrix when the loads are applied, and the vM are nontrivial displacement solutions. The applied loads can consist of pressures, concentrated forces, nonzero prescribed displacements, and/or thermal loading.
Eigenvalue buckling is generally used to estimate the critical buckling loads of stiff structures (classical eigenvalue buckling). Stiff structures carry their design loads primarily by axial or membrane action, rather than by bending action. Their response usually involves very little deformation prior to buckling. A simple example of a stiff structure is the Euler column, which responds very stiffly to a compressive axial load until a critical load is reached, when it bends suddenly and exhibits a much lower stiffness. However, even when the response of a structure is nonlinear before collapse, a general eigenvalue buckling analysis can provide useful estimates of collapse mode shapes.
The Base State
The buckling loads are calculated relative to the base state of the structure. If the eigenvalue buckling procedure is the first step in an analysis, the initial conditions form the base state; otherwise, the base state is the current state of the model at the end of the last general analysis step (see General and Perturbation Procedures). Therefore, the base state can include preloads (“dead” loads), PN . The preloads are often zero in classical eigenvalue buckling problems.
If geometric nonlinearity was included in the general analysis steps prior to the eigenvalue buckling analysis (see General and Perturbation Procedures), the base state geometry is the deformed geometry at the end of the last general analysis step. If geometric nonlinearity was omitted, the base state geometry is the original configuration of the body.