$(60$ pts $)$ Given the compound beam and its cross$-$section shown in the figures below:

a$.$ Draw the overall free body diagram for beam.

b$.$ Draw the free body diagram for each section of the compound beam

c$.$ Find the reaction forces at points $A,B,C,D,$ and E $.$

d$.$ Draw the shear and moment diagrams for the entire beam.

e$.$ If the beams cross$-$section has the dimensions shown, find the location of the centroid $\frac{?}{\text{b}\text{a}\text{r}}(y).$

f$.$ Find the Moment of Inertia about the centroidal axis $($I$.$e$.I_x).$ For a rectangle $I=\frac{1}{12}b{h}^3.$