# Model Stability Analysis

When a structural model is ill-conditioned (as reported by the condition number estimate) it could be a result modelling errors in the model. These errors could be of two types:

• Some elements may not be well connected or could be badly restrained, e.g. beam elements spinning about their axis.
• Some elements very stiff compared with all other elements in the model, e.g. a beam element of short length but a large section.

To detect such errors, model stability analysis, which is a qualitative analysis intended to reveal the causes of ill conditioning in models, can be useful. The analysis calculates the smallest and largest eigenvalues and corresponding eigenvectors of the stiffness matrix, i.e. it solves the problem

$\mathbf{Ku} = \lambda\mathbf{u}$

for eigenpairs $\left\{ \lambda,\mathbf{u} \right\}$. For each mode that is requested, element virtual energies are calculated for each element in the model. These are defined as follows.

The virtual strain energy $s_{e}$for large eigenpairs where

$s_{e} = {\mathbf{u}_{e}}^{T}\mathbf{K}_{e}$

and virtual kinetic energy $v_{e}$for small eigenpairs, defined as

$v_{e} = {\mathbf{u}_{e}}^{T}\mathbf{u}_{e}$

The virtual energies can be plotted onto elements as contours. Typically, for an ill conditioned model, a handful of elements will have large relative values of virtual energies.

• Where the ill conditioning is caused from badly restrained elements, such elements will have large relative virtual kinetic energies.

• If the ill conditioning is from the presence of elements with disproportionately large stiffnesses, then these elements will have large virtual strain.

The analysis also reports, in increasing order, the eigenvalues computed. For the case of badly restrained elements, there is usually a gap in the smallest eigenvalues. The number of smallest eigenpairs to be examined is given by the number of eigenvalues between zero and the gap.

## P-delta model stability analysis​

Note that we can also use p-delta analysis with model stability. As with static analysis, we replace $\mathbf{K}$ with $\mathbf{K} + \mathbf{K}_{g}$ in order to capture how loading on the structure affects its conditioning.