Problem $2(60$ points$)$

It is known that plane stress condition exists at a point in the $x-y$ plane. The strain rosette attached to this point $($see figure$)$ gives the following readings for each strain gauge:

${}_a=450({10}^{-6}),{}_b=300({10}^{-6}),$ and ${}_c=150({10}^{-6})$

The gauge $a$ is orientated along $x-$axis. Futhremore, it is known that ${}_x=54\mathrm{.}4$MPa and ${}_y=31\mathrm{.}32$ MPa at this point

a$)$ Determine ${}_x,{}_y,$ and ${}_{xy}$ at this point $(20$ pts $).$

b$)$ Determine $E,G,$ and $U$ at this point, assuming it is made of an isotropic, linear elastic material $(20$ pts$).$

c$)$ Determine the complete stress and strain matrix at this point $(10$ pts$).$

d$)$ If this location is the critical point in the structure, what is the safety factor of the structure based on Tresca criterion? The yield stress $S_y$ for the structure is $60$ MPa $(10$ pts$).$