Problem $1(50$ pts $)$ For the reinforced concrete beam cross$-$section shown below, compute the

nominal bending moment capacity $M_n.$ The beam is reinforced with $3$ No$.7$ bars. Use $f_c^{\text{'}}=$

$4000$ psi and Grade $60$ bars $($i$.$e$.,f_y=60$ksi.$).$ Make sure to check if the reinforcement yields or

or not at the time of beam failure, i$.$e$.,$ check if ${}_s{}_y$ or not.

Use the following equations as needed.

$a=\frac{A_s{f}_y}{0\mathrm{.}85f_c^{\text{'}}b}$cdots$-$cdots$(1)$

$M_n=A_s{f}_y(d-\frac{a}{2})$

$-----(2)$

${}_s=\frac{d-c}{c}0\mathrm{.}003,$ where $c=\frac{a}{{}_1}$

${}_1=0\mathrm{.}85$ if $f_c^{\text{'}}4$ksi

${}_1=0\mathrm{.}85-0\mathrm{.}05(f_c^{\text{'}}-4)$ if $f_c^{\text{'}}{}_1=0\mathrm{.}65f_c^{\text{'}}8$ksi$4$ksi $[use$ ksi for $f_c^{\text{'}}]$

${}_1=0\mathrm{.}65if{f}_c^{\text{'}}8$ksi

For the calculation $ofAs,$ use the chart provided