Problem C $(25$ points$)$

For each of the beams loaded as shown below, $($a$)$ draw the shear force and bending moment diagrams, and $($b$)$ determine the magnitude and location of the maximum absolute value of the bending moment. Provide the general mathematical expressions for the shear and bending moments for all segments of the diagrams. Assume that the weight of the beam is negligible.

a$)$

b$)$