Problem $4(25\%)$

A singly reinforced rectangular beam must be designed to have a depth approximately $1\mathrm{.}5$ times its width $($see figure below$).$ In addition to the self weight $($assume ${}_{\text{c}\text{o}\text{n}\text{c}}=150l\frac{b}{f}{t}^3),$ the $24ft$ long simply supported beam supports a live load of $1\mathrm{.}5ki\frac{p}{f}t.$ Assuming $f_c^{\text{'}}=4,000$ and $f_y=40,000$ and the LRFD load combination: $1\mathrm{.}2$ DL $+1\mathrm{.}6$ LL for your design, determine the dimensions and reinforcement required for the following conditions:

$($a$)=0\mathrm{.}4*{}_b$

$($b$)={}_{\text{m}\text{a}\text{x}}$

Include a sketch of each cross$-$section, including bar spacing, clear cover, etc. Assume #$3$ stirrups are in use. Explain your results.

$w_u=1\mathrm{.}2w_{DL}+1\mathrm{.}6w_{LL}$