For the structure shown in Figure $2,$ with a uniformly distributed factored load $\backslash ($ w$\_\{$w$\}=4\backslash )$ kips $\backslash (/$ f t $\backslash )$ applied to beam $\backslash ($ B C $\backslash ),$ answer the following questions:

$($a$)$ Draw the bending moment, shear force, and normal force diagrams.

$($b$)$ Find the minimum area of steel required to serve as a cross section for member AB $.$ Assume welded connections and shear lag factor $\backslash (\backslash$mathrm$\{$U$\}=0\mathrm{.}75\backslash )$

$($c$)$ Find the minimum plastic modulus a standard rolled W section must have to adequately resist the bending moments in beam BC$.$ Assume that the top flange is continuously laterally braced. Neglect the effect of axial load

$($d$)$ Select the lightest W section for beam BC assuming no lateral bracing for the top flange. Neglect the effect of axial load. Check the adequacy of the cross section in shear.

$($e$)$ Using the section from part $($d$)$ and considering the effect of the axial load along BC$,$ determine the amplification factor $\backslash (\backslash$beta$\_\{1\}\backslash )$ and explain why $\backslash (\backslash$beta$\_\{2\}\backslash )$ is equal to zero. Use the appropriate interaction equation to check the safety of the cross section under combined bending and compression.

$($Assume $\backslash ($ F$\_\{$y$\}=36\backslash$mathrm$\{$ksi$\}\backslash )$ and $\backslash ($ F$\_\{0\}=58\backslash$mathrm$\{$ksi$\}\backslash )$ for member AB and $\backslash ($ F$\_\{$y$\}=50\backslash$mathrm$\{$ksi$\}\backslash )$ for member BC $)$

Figure $2$: The structure under consideration in Problem $2$