Problem #$19$

A simple$-$span beam, $22$ in $.$ deep by $12$ in $.$ wide, is subjected to large concentrated loads at the thirdpoints of the span. The distributed dead load is negligible. The peak moment requires $3-$#$8,$ Grade $75$ longitudinal bars. Concrete cover is $1-\frac{1}{2}in.,f_c^{\text{'}}$ is $4000$ psi $,$ and #$3$ stirrups $($also Grade $75)$ spaced at $9$ in $.$ are provided over the entire length of the beam. The beam span is $21$ ft $.$

a$.$ If only two of the three longitudinal bars are to extend to the support, determine the total length of the short bar using Sections $25\mathrm{.}4\mathrm{.}2\mathrm{.}4$ and $9\mathrm{.}7\mathrm{.}3\mathrm{.}3$ of the ACI Code. Assume at least one of the provisions in Section $9\mathrm{.}7\mathrm{.}3\mathrm{.}5$ is satisfied.

b$.$ Do the two bars that continue to the supports meet development length requirements for continuing reinforcement $($Section $9\mathrm{.}7\mathrm{.}3\mathrm{.}4)?$

c$.$ Determine whether Sections $9\mathrm{.}7\mathrm{.}3\mathrm{.}8\mathrm{.}1$ and $9\mathrm{.}7\mathrm{.}3\mathrm{.}8\mathrm{.}3$ are satisfied. Assume that two longitudinal bars extend $6$ in $.$ past the center of each support, and $V_u$ is $34\mathrm{.}1$ kips at the support.