Problem $2$

The T$-$like structure ABCD $($with uneven arms BC and BD$)$ is subjected to forces $P$ and $2P$ at $C$ and $D,$ respectively. The structure has a circular cross$-$section with diameter $=40mm.G=80$GPa,$v=0\mathrm{.}3,S_y=300$MPa.

A strain gage oriented the $x-$axis is mounted at point $R$ in the cross$-$section a$-$a$.$

This strain gage reads ${}_R=5{10}^{-4}.$

a$)$ Determine the value of $P.$

b$)$ Under this loading condition, what would be the reading of a strain gage mounted at point $S$ along the direction shown $(40$ to the $x-$axis$).$

c$)$ Determine the factor of safety for the structure based on the von Mises criterion.

Cross$-$section $a-a$

Strain gauge orientation at $S$