$($b$)$ A steel rectangle column is pinned at top and fixed at the bottom having dimension as shown in Figure $4.$ The load carrying capacity is increased by bracing the column along $xx$ axis using struts that are assumed to be pin connected at the mid$-$height. Given that the factor of safety of this column is $2\mathrm{.}5,$ modulus of elasticity is $200$GPa and yield strength is $250$MPa.

i$)$ Sketch the buckling shapes and calculate the effective length of the column.

ii$)$ Determine the greatest thickness, $b$ if its width is $3b.$

iii$)$ Calculate the slenderness ratio of the column and ratio of axial stress corresponding to the buckling load and elastic strength of the materials.

Figure $4$