Question $4(15$ points$).$ A simply supported beam is loaded as shown. The cross section at location C of the beam is as shown below right, where C is somewhere between the two applied loads P$.$ Point O on the cross section is on the neutral axis of the beam.

a$)(5$pts$)$ Derive the shear force and bending moment along the $x$ axis.

b$)(3$pts$)$ Determine the distribution of normal stress on the cross section of the beam as a function of $y($at location C$).$

c$)(2$pts$)$ Determine the maximum $($magnitude$)$ of the normal stress on the cross$-$section C$.$

d$)(5$pts$)$ Using Castigliano's theorem to determine the deflection at middle point of the beam.