Problem $2(20$ Points$)$

An AISI $1018$ steel has a yield strength, S$\_($y$)=295$MPa. Under the plane stress condition the stresses at a critical point are

determined to be: $\backslash$sigma $\_($x$)=-30$MPa,$\backslash$sigma $\_($y$)=-65$MPa and $\backslash$tau $\_($xy$)=40$MPa

Determine the principle stresses at the critical point $(5$ points$)$

Determine the factor of safety, using the distortion$-$energy theory $(5$ points$)$

Determine the three principal shear stresses $($in$-$plane and out$-$of$-$plane$)$ and thus find the maximum shear stress $(5$ points$)$

Determine the factor of safety, using the maximum shear stress theory $(5$ points$)$