A rectangular beam with width $b=12$ in$.$ and height $h=24$ in$.$ is simply supported with a span length of $30ft$ as shown below. The beam carries no live loads, but it supports its own weight $($unit weight of concrete is $150pcf)$ in addition to a uniform dead load of $0\mathrm{.}2ki\frac{p}{f}t$ and a concentrated load $P_D=20$ kips at its midspan. The compressive strength at $28$ days is $4,000$ psi $($normal$-$weight concrete$)$ and Grade $60$ reinforcement is used $($area of each #$10$ bar is $1\mathrm{.}27i{n}^2$ and diameter is $1\mathrm{.}27in.).$ The elastic modulus of steel is $29,000$ksi. Center$-$to$-$center spacing for the #$10$ bars is $3$ inches. Modulus of rupture and elastic modulus of normal$-$weight concrete are given by the equations below:

$f_r=7\mathrm{.}5*\sqrt[\phantom{2}]{f_c^{\text{'}}}$

$E_c=57,000*\sqrt[\phantom{2}]{f_c^{\text{'}}}.in$ psi

Do the following:

A$.$ At the critical section for bending $($mid$-$span$),$ check if the ACl code crack control requirement is satisfied using the actual stress in the rebars. Use the following equation for the maximum spacing allowed:

$s$min$(15*\frac{40,000}{f_s}-2\mathrm{.}5,12*\frac{40,000}{f_s})$

B$.$ Calculate the deflection at mid$-$span under the total dead load after $20$ years.