Problem. A simply supported uniformly loaded beam has a clear $span{l}_n=35ft.$ a width $b=12in.,$ and a total depth $h=20in.,d=17$ in$.,$ and $A_s=1\mathrm{.}32i{n}^2.$ It is subjected to a service dead$-$load moment $M_D=240,000in.1b$ and a service live$-$load moment $M_L=320,000in-lb.$ Solve the following steps to determine if the beam satisfies the various deflection criteria for short and long$-$term loading. It is assumed that $80\%$ of live load is continuously applied for $12$ months. Given: $f_c^{\text{'}}=3000$ psi, $f_y=50,000$ psi, normal weight concrete, $E_s=29{10}^6$ psi.

i$)$ From Table $8-1$ find Minimum Required Depth $(h_{\text{m}\text{i}\text{n}})$ and compare with $h.$ Check if the deflection calculations have to be made.

ii$)$ Determine: $($a$)$ modulus of elasticity of concrete, $($b$)$ modular ratio $n,($c$)$ rupture strength of concrete

iii$)$ If $fr=411.$ psi and $E_c=3,122,019$ psi, $($a$)$ Gross moment of Inertia $I_g,($b$)$ cracking moment $M_{cr}$

iv$)$ The cracked moment of inertia of the beam cross section, $I_{cr}.($assume $n=9\mathrm{.}29)$

v$)$ Convert length of the span to inches and find $\frac{5l^2}{48E_c},$