Problem $2(40$ points$)$

A prestressed concrete beam supports a live load of $0\mathrm{.}27$kip$\frac{s}{f}t$ over a simply supported span of $26ft.$ The beam has I$-$section with a depth of $15\mathrm{.}8in.$ The thickness of the flange and web are $2\mathrm{.}4$ and $3\mathrm{.}2in,$ respectively. The width of the flange is $7\mathrm{.}9in.$ The concrete density is $=150l\frac{b}{f}{t}^3.$ The beam is to be prestressed by an effective prestressing force of $53$kips at a suitable eccentricity such that the resultant stress at the bottommost fiber of the beam at the mid$-$span is zero.

a$)$ Find the eccentricity required for the force.

b$)$ If the tendon is concentric, find the magnitude of the prestressing force necessary to satisfy the same condition of zero resulting stress at the bottom fiber of the beam mid$-$span.