Problem $\backslash$#$1$: $(30$ points$)$ The solid circular shaft shown in Figure $1$ is made of steel with an E value of $190$ GPa $.$ The shaft has a diameter of $80$ mm $.$

a$)$ Draw a free body diagram of the shaft and determine the support reactions.

b$)$ Draw a free body diagram of the beam's cut segment and determine the bending moment at any cross$-$section as a function of $\backslash ($ x $\backslash ).$

c$)$ Use the equations of slope and elastic curve; and the boundary conditions to determine the maximum defection of the shaft.

d$)$ Draw a sketch of the defection versus $\backslash ($ x $\backslash )$