Q$1.(15$ Marks$)$

For the beam shown in figure $1$a$,$

a$)$ Derive equations for the shear force and bending moment for each section along the span of the beam.

b$)$ Clearly draw and label with proper units the shear and bending moment diagrams for the beam.

c$)$ If the beam's cross section is as shown in fig $\backslash (1$ b $\backslash ),$ determine the required spacing $"$ s $"$ if $4$ mm diameter steel bolts are equally spaced on both sides of the cross section and a factor of safety of $3$ is required. The bolts have a shear yield strength of $30$ MPa $.$