Consider the I$-$shaped cross$-$section shown below $($web$-$flange fillets are neglected$)$

subjected to axial force $P$ and bending moment $M.($Strong axis $=x-$axis; Weak axis $=y-$

axis$)$

I$-$Shaped Cross$-$Section

$A=$ total area

$A_w=$ web area $=(d-2t_f)t_w$

$M_{px}=x-$axis plastic moment $=Z_x{F}_y$

$M_{py}=y-$axis plastic moment $=Z_y{F}_y$

$Z_x=x-$axis plastic modulus

$Z_y=y-$axis plastic modulus

$P_y=$ yield force $=A{F}_y$

$F_y=$ yield stress

a$)$ The exact plastic $P-M$ interaction equations for flexure and axial compression are

given below. Note that the interaction equations consist of two regimes,

depending on the location of the plastic neutral axis. Derive equation S$-1$b$.$

$($Consider axial compression to be positive.$)$

i$.$ Case $1-$strong$-$axis flexure and axial compression

$\frac{A_w}{A}\frac{P}{P_y}1\mathrm{.}0,\frac{M}{M_{px}}=(1-\frac{P}{P_y})[1-(1-\frac{P}{P_y})\frac{A}{2b_{f}d}]\frac{Ad}{2Z_x}$

$0\frac{P}{P_y}\frac{A_w}{A},\frac{M}{M_{px}}+(\frac{P}{P_y}{)}^2\frac{A^2}{4t_w{Z}_x}=1\mathrm{.}0$

ii$.$ Case $2-$ weak$-$axis flexure and axial compression

$\frac{d{t}_w}{A}\frac{P}{P_y}1\mathrm{.}0,\frac{M}{M_{py}}=(1-\frac{P}{P_y})[\frac{4b_f{t}_f}{A}-(1-\frac{P}{P_y})]\frac{A^2}{8t_f{Z}_y}$

$0\frac{P}{P_y}\frac{d{t}_w}{A},\frac{M}{M_{py}}+(\frac{P}{P_y}{)}^2\frac{A^2}{4d{Z}_y}=1\mathrm{.}0$

b$)$ For a column with the following properties in$,t_w=0\mathrm{.}755in,b_f=10\mathrm{.}4in,$

$t_f=1\mathrm{.}25in),$ plot the interaction equations from a$)$ along with the interaction

equations based on the AISC Specification equations $($shown below$)$ and the

interaction equation for a rectangular section $($derived in class$).$ For all cases plot

normalized interaction equations vs$.\{$:$\frac{M}{M_p}).$