The tensile$-$reinforced rectangular beam shown below is made using steel with $$y $=60$ ksi and modulus of elasticity, ES$=29000$ ksi. Assume an elastic plastic curve for steel reinforcement. The concrete used has the stress$-$strain curve shown below with limit of elastic response at a strain of $0\mathrm{.}0005,$ maximum stress at $0\mathrm{.}002,$ and ultimate strain of $0\mathrm{.}003.($a$)$

$f_r=450$

$($b$)$

The concrete elastic modulus is Ec $=4000$ ksi, and modulus of rupture is $$r $=450$ psi.

Based on the above given information,

Determine the moment and unit curvature at a point corresponding to:

$1.$

Before and after cracking

$2.$

When fc $=0\mathrm{.}45$ f$$c

$3.$

When fs $=$ fy

$4.$

When $$c $=0\mathrm{.}003$