Ductile Design $-$ Equations $3\mathrm{.}12$ & $3\mathrm{.}13$

Equations $3\mathrm{.}12$ and $3\mathrm{.}13$ are:

When $y^{*}$ is located in the flange:

$\frac{M}{M_y}=\frac{}{{}_y}(1-\frac{b{d}^2}{6S})+\frac{b{h}^2}{4S}[1-\frac{1}{3}(\frac{}{{}_y}{)}^2]$

When $y^{*}$ is located in the web:

$\frac{M}{M_y}=\frac{M_p}{M_y}-(\frac{w{d}^2}{12S})(\frac{{}_y}{}{)}^2$

$($a$)$ Derive the analytical expressions for normalized moment$-$curvature response of a a wide$-$flange section. $($Note that Equation $3\mathrm{.}12$ contains two errors.$)$

$($b$)$ For a W $14$ x $90,$ plot the derived normalized moment$-$curvature response.

$($c$)$ On the same figure as $($b$),$ plot the normalized moment$-$curvature response of a rectangular section and qualitatively explain the differences.