$($a$)$

$($b$)$

T$13\mathrm{.}1$ For the two columns subjected to concentric axial load $($see Figure T$13\mathrm{.}1($a$)$ and $($b$)),$

$($a$)$ Derive the governing differential equation for combined bending and axial behaviour.

$($b$)$ Obtain the critical buckling load and the effective length, $L_e,$ of the column.

Figure T$13\mathrm{.}1$: $($a$)$ Clamped$-$pinned, $($b$)$ Clamped$-$clamped column with concentric load.