$($b$)$ The solid steel rectangular column is fixed at both ends having dimensions as illustrate in Figure $2($b$).$ The column is braced at its midpoint along $x-x$ axis using struts that are pin$-$connected. Given that the modulus of elasticity is $200$GPa, factor of safety is $2\mathrm{.}2$ and the yield strength is $250$MPa.

i$.$ Sketch the buckling shapes of the column

$[3$ Marks$]$

ii$.$ Determine the largest allowable length of the column

$[5$ Marks$]$

iii. Calculate the slenderness ratio of the column

$[3$ Marks$]$

iv$.$ Determine the ratio of axial stress corresponding to the buckling load and elastic strength of the materials

$[4$ Marks$]$