$($Q$4)$

The two $500-lb$ forces are vertical and the force $P$ is parallel to the $z$ axis. Knowing that ${}_{\text{a}\text{l}\text{l}}=8$ksi, do the following:

a$)$ Calculate the magnitude of P and the reactions.

b$)$ Draw the loading diagrams.

c$)$ Define the critical section and its loads.

d$)$ Draw stresses distribution across the cross$-$section.

e$)$ Extract the state of stress at the critical point.

Calculate the principal and maximum shear stresses and their planes orientation. $($Draw Mohr's Circle$)$

g$)$ Use Maximum shear failure theory to determine the smallest permissible outer diameter of the shaft.