Most finite element packages will offer you two types of eigenvalue buckling analysis.
The linear eigenvalue buckling analysis (called “Buckling load” in LUSAS), where, if the applied load is \(P\), each eigenvalue is a load factor \(\lambda\), such that the associated buckling load is \(\lambda P\).
The non-linear eigenvalue buckling analysis (called “Eigenvalues of stiffness matrix” in LUSAS), where the eigenvalues indicate whether the model is stable or not at the applied load. All eigenvalues positive indicates stable, whereas a negative eigenvalue indicates unstable (buckled). An eigenvalue equals zero at the buckling load for its associated mode. Non-zero eigenvalues also have associated modes, but these are not buckling modes.
The linear analysis is much simpler, quicker to run, and adequate in most cases but does not correctly model any of the following situations:
pre-stressing / post-tensioning or initial deformations
assymmetric buckling modes
geometric or material non-linearity
axial loads that cause lateral displacements or vice-versa
lateral loads
axial displacements that do not vary as the square of the lateral displacements (to a second order approximation)
The archetypal case where these effects are likely to be significant is in the analysis of a shallow arch, e.g. a Von Mises Truss.
The non-linear analysis works in all situations, but is much more complicated and more computationally expensive, as the analysis needs to use trial and error to extract eigenvalues at various different load factors to narrow down which ones give eigenvalues of zero.
