If I want to find the smallest load magnitude that would cause a structure to buckle, I would pick the direction of the load such that the eigenvector of the "stress stiffness matrix" which has the most negative eigenvalue (most compressive load) is aligned with the eigenvector of the regular stiffness matrix which has the lowest eigenvalue (least stiff direction). Buckling load fraction is typically found by solving the generalised eigenvalue problem `K * v + s * Ksigma * v = 0` where K and Ksigma are the stiffness and stress stiffness matrices respectively.

It can be shown that i-th eigenvalue of the stiffness matrix is equal to duplicated strain energy stored in the element assemblage (when the corresponding displacement is i-th eigenvector) divided by the square of the Euclidian norm of i-th eigenvector (see p.727 of K.-J. Bathe. Finite element procedures. 1996).

In numerical analysis, one can gets the eigenvalues of any matrix A with

|A-lambda*I|=0 where lambda are the eigenvalues and I is the identity matrix.

The eigenvalues allows to calculate the conditioning number of A which correspond to the 2-norm of A and the 2-norm of its inverse. A can be the stiffness matrix.

In dynamics, I=M the mass matrix.

The conditioning number gives the information if A is well conditioned or ill-conditioned.

If A is ill-conditioned it is necessary to solve this problem before using any numerical solving algorithm because in some cases A becomes singular when finding a zero pivot. Several methods can be used to equilibrate a matrix.

The static stiffness matrix is the dynamic stiffness matrix of the structure for an excitation at zero frequency (static thus). All its eigenvalues should be positive if the structure is not free to move around (no rigid modes). Zero eigenvalues indicate rigid modes, up to six (translations and rotations) and even more than six when the structure has internal hinges. Negative eigenvalues (if not due to numerical errors) indicate internal instability.

Wei Hao Koh The eigenvalues of the element stiffness matrix in static finite element analysis are the natural frequencies of the element. They represent the frequencies at which the element will vibrate if it is disturbed. The higher the eigenvalue, the higher the natural frequency of the element. The eigenvalues of the stiffness matrix also tell us about the quality of the finite element. A good finite element will have eigenvalues that are well-separated. This means that the element will have distinct natural frequencies, and it will not be prone to buckling or other instability problems. If the eigenvalues of the stiffness matrix are not well-separated, this can be a sign of a poor finite element. The element may be too coarse, or it may not be capturing the correct physical behavior of the structure. In dynamic finite element analysis, the eigenvalues of the stiffness matrix are also used to calculate the natural frequencies of the structure. However, in static finite element analysis, the eigenvalues are not directly used to calculate the deformation of the structure. Instead, they are used to calculate the stiffness of the element.

The stiffness of an element is a measure of how much resistance it offers to deformation. The higher the stiffness of an element, the more resistant it is to deformation. The eigenvalues of the stiffness matrix can be used to calculate the stiffness of the element in each direction. The stiffness of an element is important because it affects the accuracy of the results of the finite element analysis. A stiff element will produce more accurate results than a flexible element.

I hope this

A regular finite element has a good conditioning number. The reference is the diagonal identity matrix which its conditioning number is equal to 1 . A degenerated finite element (surface nearly zero or volume nearly zero) has an ill-conditioned number very greater than 1 . The surface or the volume can be formulated in a matrix.

The explanations are in my published book:

Méthodes Numériques Appliquées - Cours - Exercices corrigés

Author: Mohammed Lamine MOUSSAOUI

• First Edition july 2022
• Publisher: Editions Universitaires Europeennes, Sciencia Scripts, Dodo Books Indian Ocean Ltd and Omniscriptum S.R.L. publishing group
• ISBN: 978-613-8-40666-2
• Editor: Sciencia Scripts
• Available at: https://MoreBooks.de

Unfortunately, many answers given above either given for the total stiffness matrix or on natural frequencies of the system.

First, in order to talk about natural frequencies there should also be a mass matrix. Wei Hao Koh's question does not consider the mass matrix.

Similarly, Wei Hao Koh's question is the relation of eigenvalues of the element stiffness matrix with the quality of the finite element method but not the eigenvalues of the total stiffness matrix.

The short answer is no information can be obtained from the eigenvalues of the element stiffness matrix. Eigenvalues of the total stiffness matrix gives more information on the finite element model some of them explained by Mohammed Lamine Moussaoui.

Best