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Stability of a column

Problem description: Compute the stability limit of the column shown. The flexural stiffness EI_y<EI_z.

Mesh: Use the following element types:
ITE=56--see appendix B.1, four hexahedra.
ITE=61--see appendix B.2, four semi-loofs.

Material properties: $E=2\times 10^5$ MPa, $\nu=0.3$.

Support: Two point support. Statically determinate.

Solution: The minimum flexural stiffness EI_y can be computed as

$$EI_y=2\times10^{11}\,\frac{0.02\times0.01^3}{12}=333.3\,\mbox{[Nm$^2$ ]}$$

and the critical force

$$F_{\mbox{{\it crit}}}=\left(\frac{\pi}{l}\right)^2 EI_y=3290\,\mbox{[N]}$$

In the FEM method we constitute the initial stress (geometric) matrix ${\bf K}_\sigma$ for some reference loading ${\bf R}_0$. Subsequently, the generalized eigenproblem is solved

$$\det\vert{\bf K}_0 + \lambda {\bf K}_\sigma\vert= 0$$

where $\lambda$ is the load parameter such that the crititical loading

$${\bf R}_{\mbox{{\it crit}}} = \lambda {\bf R}_0$$

Therefore, it is convenient to choose a unit reference force in the $F_{\mbox{{\it crit}}}$ direction so that the load parameter directly represents the magnitude of the critical force.

Numerical solutions are shown below.

$F_{\mbox{{\it crit}}}$	ITE=56	ITE=61
3290	3490	3293