Problem $2(50$ points$)$

The simply supported beam represented in the figure below $)=(377i{n}^2$ sustains its own weight $=393plf,$ superimposed dead loads $=100$ plf$,$ and live loads $=1,100$ plf$.$ The tendons are harped at midspan and subject to an effective prestressing force $F=308,255lb($with $f_{pe}>0\mathrm{.}4f_{pu}).$ Normal$-$weight concrete is used with strength $f_c^{\text{'}}=6,000.$

a$)$ Determine the value of the web$-$shear cracking stress, $v_{cw}$ at the section $\frac{h}{2}.$

b$)$ You are told that the flexure$-$shear cracking stress at the same critical section is $v_{ci}=1280.$ Determine if the shear reinforcement is needed at the section $\frac{h}{2}$ or not according to the ACI provisions.