PROBLEM $2.$

The state of plane strain on a differential element of material is given as $lo{n}_x=0,lo{n}_y=-225({10}^{-6}),$ and ${}_{xy}=120({10}^{-6}).$ Determine

a$.$ the equivalent strains on an element oriented $-40$ counterclockwise from the differential element.

b$.$ the in$-$plane principal strains on the oriented element.

c$.$ the maximum in$-$plane shear strain and the associated average normal strain on the oriented element.

d$.$ Use maximum shear$-$stress theory to determine the factor of safety of the design if the yield stress of the material is ${}_Y=95\mathrm{.}0$MPa. Recall that $F.S.=\frac{{}_Y}{},$ where $$ is determined from maximum shear$-$stress theory.

e$.$ Use maximum distortion$-$energy theory to determine the factor of safety of the design if the yield stress of the material is ${}_r=95\mathrm{.}0$MPa.