$($Maximum $20$ points$)$ Consider the frame shown in figure $2($left$).$ The ends A and $D$ are fixed while the joints $B$ and $C$ are rigid. It is given that $L=4m$ and a uniformly distributed load of magnitude $w=1k\frac{N}{m}$ acts on $BC.$ EI is constant throughout. The fixed$-$end moments are given as$.M_{BC}=-w\frac{L^2}{12}$ and $M_{CB}=w\frac{L^2}{12}.$ For simplicity, assume that the joints $B$ and $C$ pre$-$loading and post$-$loading are at the same vertical level. Answer the following:

$($a$)(2$ points$)$ Does the frame undergo any sidesway? Explain by drawing the deflection diagram.

$($b$)(6$ points$)$ Find the moment at each joint using slope deflection method.

$($c$)(6$ points$)$ Find the moment at each joint using moment distribution method.

$($d$)(6$ points$)$ Draw the bending moment diagram for the frame.

Figure $2$: $($Left$)$ Frame for Question $2.$