$(2\mathrm{.}5-24)$

Problem $3\mathrm{.}5$ Prestressed concrete beams are sometimes manufactured in the following manner. High$-$strength steel wires are stretched by a jacking mechanism that applies a force $Q,$ as represented schematically in part $($a$)$ of the figure. Concrete is then poured around the wires to form a beam, as shown in part $($b$).$

After the concrete sets properly, the jacks are released and the force $Q$ is removed $[$see part $($c$)$ of the figure$].$ Thus, the beam is left in a prestressed condition, with the wires in tension and the concrete in compression.

Let us assume that the prestressing force $Q$ produces in the steel wires an initial stress ${}_0=620$MPa. If the moduli of elasticity of the steel and concrete are in the ratio $12$:$1$ and the cross$-$sectional areas are in the ratio $1$:$50,$ what are the final stresses ${}_s$ and ${}_c$ in the two materials?