Stability of a column

Compute the stability limit of the column shown. The flexural stiffness $EI_y<EI_z.$

$E=2\times10^5\text{ MPa},$ $\nu=0.3.$

Two-point support. Statically determinate.

• Hexahedra, BEAM56G1
• Semi-loofs, BEAM61G1

The minimum flexural stiffness $EI_y$ can be computed as $$EI_y=2\times10^{11}\frac{0.02\times0.01^3}{12}=333.3\text{ Nm$^2$}$$ and the critical force $$F_\text{crit}=\left(\frac{\pi}{l}\right)^2 EI_y=3290\text{ N}.$$

In the FEM method we constitute the initial stress (geometric) matrix $\mathbf{K}_\sigma$ for some reference loading $\mathbf{R}_0.$ Subsequently, the generalized eigenproblem is solved $$\det|\mathbf{K}_0+\lambda\mathbf{K}_\sigma|=0$$ where $\lambda$ is the load parameter such that the crititical loading $$\mathbf{R}_\text{crit}=\lambda\mathbf{R}_0.$$

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

Numerical solutions are shown below.