A composite column with a configuration shown in Figure Q$4$ is fixed at the base. The outer steel tube diameter is $\backslash (\backslash$mathrm$\{$D$\}\_\{\backslash$circ$\}\backslash$mathrm$\{$mm$\}\backslash )$ and the hollow ratio $($ratio of inner tube diameter to outer tube diameter$)$ is $0\mathrm{.}7.$ Both steel tubes are having a similar thickness of $5$ mm $.$ An axial load of $\backslash (\backslash$mathbf$\{$P$\}\backslash$mathrm$\{$kN$\}\backslash )$ is applied at the top of the column through a rigid plate. Given the elastic modulus of concrete and steel, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$c$\}\}=32\backslash$mathrm$\{$GPa$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$s$\}\}=205\backslash$mathrm$\{$GPa$\}\backslash ),$ concrete compressive strength, $\backslash (\backslash$mathrm$\{$f$\}\_\{\backslash$mathrm$\{$cu$\}\}=\backslash$mathrm$\{$Y$\}\backslash )$ MPa and steel yield strength, $\backslash (\backslash$mathrm$\{$f$\}\_\{\backslash$mathrm$\{$y$\}\}=275\backslash$mathrm$\{$MPa$\}\backslash ).$ Solve the followings:

a$)$ Generate the equilibrium and compatibility equations for this condition. $(2\backslash \%)$

b$)$ If the failure of the column is governed by concrete crushing, find the maximum axial load $\backslash ($ P $\backslash ).(15\backslash \%)$

c$)$ At the maximum load in $($b$),$ calculate the axial forces and normal stresses in the outer and inner steel tubes. $\backslash ((8\backslash \%)\backslash )$

$($a$)$ Side view

Figure Q$4$