A rectangular beam with width $b=12in.$ 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 $4,000($normal$-$weight concrete$)$ and

Grade $60$ reinforcement is used $($area of each

#$10$ bar is $1\mathrm{.}27$ in ${?}^2$ and diameter is $1\mathrm{.}27$ in$.).$

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.