The figure shows a fluid$-$pressure linear actuator $($hydraulic cylinder$)$ in which $D=10in,t=\frac{5}{8}$ in$,L=12in,$ and $w=1\mathrm{.}5in.$ Both brackets as well as the cylinder are of steel. The actuator has been designed for a working pressure of $2200$ psi. Eight $\frac{3}{8}-$in SAE grade $5$ coarse$-$thread bolts are used, tightened to $75$ percent of proof load.

Note; $K_b=\frac{A_{b}E}{L}$

$K_m=\frac{A_{m}E}{L}$

$($a$)$ Find the stiffnesses of the bolts and members, assuming that the entire cylinder is compressed uniformly and that the end brackets are perfectly rigid.

$20$ Points

$($b$)$ Using the Goodman fatigue criterion only, find the factor of safety guarding against a fatigue failure.

$20$ Points

$($c$)$ What pressure would be required to cause total joint separation?

$20$ Points