From Problem to Failure – Insights from the Eigenvalue Problem of the Stiffness Matrix in Non-linear Structural Analysis

Stiffness characterizes the response behaviour of systems and links generalized displacements and forces of that system. Stiffness matrices thus contain plenty of information about system behaviour. In a finite element setting, the global stiffness matrix is assembled and used in the analysis, but it is rarely output and subjected to analysis itself. Here, we use the eigendecomposition of the stiffness matrix in continuous and discrete mechanical settings. Particularly, we observe eigenvalues and -vectors in relation to structural failure in illustrative examples for educational purposes. Aside from the classical case of buckling modes, we study a particular strength reduction technique which is used in geotechnical engineering practice for ultimate limit state analyses. We briefly touch upon links to model reduction and check whether failure loads can be estimated from pre-failure information in non-linear settings. The paper has educational character, drawing links between different fields of engineering and emphasizing visualization.