$8\mathrm{.}8.($a$)$ Develop the torsion differential equation solution for the W section having torsionally fixed ends and an eccentrically applied concentrated load at midspan.

$($b$)$ Compute the torsion constant $J,$ the warping constant $C_w,$ and $($i$.$e$.,\frac{1}{a})$; then use them in parts $($c$)$ through $($f$).$

$($c$)$ Compute the combined bending stress, including warping torsion and ordinary flexure components, at $z=0,0\mathrm{.}3L,$ and $0\mathrm{.}5L$ unless otherwise instructed.

$($d$)$ Compute the maximum shear stress in the web, including Saint$-$Venant torsional shear and flexural shear, at the same locations indicated in $($c$).$

$($e$)$ Compute the flange shear force $V_f$ due to warping restraint at the same locations indicated in $($c$).$

$($f$)$ Compute the maximum shear stress in the flange, including Saint$-$Venant torsion, warping shear, and vertical flexure flange shear, at the same locations indicated in $($c$).$

$($g$)$ Give a tabular summary of all stresses.

Problem $8\mathrm{.}8$