$2\mathrm{.}7.$ For a solid circular section with diameter $d,$ we know $Z=\frac{d^3}{6}$ and $S=\frac{d^3}{32},$

and for solid rectangular section with width $b$ and depth $d,$ we know $Z=b\frac{d^2}{4}$

Problems

and $S=b\frac{d^2}{6}.$ Using these relationships, derive shape factor expressions for the

following two shapes:

a$.$ channel section bent about its strong axis, and

$f=\frac{3d}{2}[\frac{b{d}^2-(b-t_w)(d-2t_f{)}^2}{b{d}^3-(b-t_w)(d-2t_f{)}^3}]$

b$.$ circular hollow tube

$f=\frac{16D}{}\frac{(D^3-d^3)}{(D^4-d^4)}.$