QUESTION $4(140$ marks$)$

The bending moment diagram for a portal steel frame column BDI is shown in Fig. $4.$ The moment is about the major axis $(x$ axis$)$ of the column in the plane of the frame. The column also carries an ultimate compressive load of $135$ kN $.$ There are restraints at the bottom, point D and top of the column, so that the effective length for out$-$of$-$plane buckling is $L_{ey}=0\mathrm{.}545L$ where $L=6\mathrm{.}6m.$ The effective length for $x-$ axis column buckling is $L_{ex}=0\mathrm{.}85L.$ The column is $250$ UB $31\mathrm{.}4$ in grade $300$PLUS. Calculate the following and determine whether the column can carry the loads safely.

$($a$)$ The section moment capacity of the column under combined actions $($axial compression and bending$).(20$ marks$)$

$($b$)$ The in$-$plane member moment capacity $(M_{ix})$ of the column under combined actions.$(60$ marks$)$

$($c$)$ The out$-$of$-$plane member moment capacity $(M_{ox})$ of the column under combined actions. $($assume $k_t=k_l=k_r=1)(60$ marks$)$

The section properties of $250$UB$31\mathrm{.}4$ are given below $($from OneSteel Tables$)$

$d=252mm,t_f=8\mathrm{.}6mm,t_w=6\mathrm{.}1mm,A_g=4010m{m}^2$

$I_x=44\mathrm{.}5{10}^{6}m{m}^4,r_x=105mm,Z_{ex}=395{10}^{3}m{m}^3$

$I_y=4\mathrm{.}47{10}^{6}m{m}^4,r_y=33\mathrm{.}4mm,Z_{ey}=91\mathrm{.}4{10}^{3}m{m}^3,$

$I_w=65\mathrm{.}9{10}^{9}m{m}^6,J=89\mathrm{.}3{10}^{3}m{m}^4$

$f_y=320$MPa,$k_f=1\mathrm{.}0(form$ factor$),E=200000$MPa,$G=80000$MPa

Figure $4.$ Bending moment diagram for the column.