Solve using the mentioned methods only $($Energy and Virtual Work Methods ONLY$)$

Consider a beam of length $2L$ which is pinned at its root, mid$-$span, and end as shown in Fig. $3.$ You may assume that $H_{23}^c=0$ and $H_{33}^c$ is known. The bean is subjected to a point moment $M_3=Q$ at its right$-$end.

a$)$ Solve for the exact internal moment $M_3(x_1)$ in the beam.

b$)$ Compute an approximate solution for the transverse displacement hat$(u{)}_2$ using Rayleigh$-$Ritz. You may chose any valid shape functions.

c$)$ Using your answer to $($b$)$ above, compute the corresponding approximate internal moment hat$(M{)}_3.$

d$)$ Compare your exact, $M_3,$ and approximate, hat$(M{)}_3$ internal moment. Comment on the accuracy of your approximate internal moment hat$(M{)}_3,$ as well as your expected accuracy of the approximate transverse displacement, hat$(u{)}_2.$

Figure $3$: Beam with three pins under a point moment.