Prob. $2(60$ pts $)$ A cylindrical helium tank with the outside diameter of $8"$ and inside diameter of ${6\mathrm{.}4}^{\text{'}\text{'}}$ is made of SAE $4140$ OQT $1000$ steel with $S_y=152$ksi and $S_u=168$ksi. During its lifetime, the cylinder is pressurized repeatedly between $5,000$ and $10,000$ psi.

$($a$)$ Determine the mean diameter $D_m$ and wall thickness $t,$ then show that it qualifies as thick wall cylinder.

$($b$)$ Calculate the longitudinal, hoop and radial stresses to the inside of the cylinder.

$($c$)$ Calculate the equivalent von Mises stresses to the inside of the cylinder for the two limit pressure values, and the corresponding mean stress ${}_m$ and alternating stresses ${}_a.$

$($d$)$ Calculate the true endurance limit $S_n\text{'}$ of the cylinder material for $90\%$ reliability, knowing that to the inside of the cylinder the surface finish is cold drawn $($show how you extract $S_n$ from the chart above$).$

$($e$)$ Calculate the design factor to the inside of the cylinder for infinite fatigue life using Goodman criteria.

$\backslash$table$[[$Stresses in thick$-$walled cylinders$*],[,$Stress at position $r,$Maximum stress$],[\backslash$table$[[$Hoop$],[($tangential$)]],\frac{p{a}^2(b^2+r^2)}{r^2(b^2-a^2)},\frac{p(b^2+a^2)}{b^2-a^2}($at inner surface$)],[$Longitudinal$,\frac{p{a}^2}{b^2-a^2},($uniform throughout wall$),],[$Radial$,\frac{-p{a}^2(b^2-r^2)}{r^2(b^2-a^2)},-p($at inner surface$)]]$

$a=$ inner radius; $b=$ outer radius; $r=$ any radius between a and $b$; $p=$ internal pressure,