Q$3$ A very stiff vertical column $AB$ subjected to axial force $P$ and a flexible beam $BC$ are connected by a rigid connection, as shown in Figure $2.$ Consider global buckling and local $($Euler$)$ buckling. At point $B,$ beams $AB$ and $BC$ are rigidly connected.

$($a$)$ Derive the equilibrium equation for the deformed configuration for stiff upstand $AB.$ Derive the equation for critical buckling force assuming small deformations. Sketch post$-$buckling deformation.

$[20]$

$($b$)$ Give equation for local $($Euler$)$ buckling of beam BC$.$

$[7]$

$($c$)$ Give critical buckling for structure. What will buckle first, upstand $AB$ or beam $BC?$ Assume that $L=a.$

$[6]$