$4(25\%)$ A round steel bar having Sy $=900$MPa is subjected to loads producing calculated stresses of

$($P$)/($A$)=-25$MPa,T$($c$)/($J$)=65$MPa, and M$($c$)/($I$)=80$MPa for the $2$D element on the Side $($labeled S $)$:

a$.$ Determine the principle stresses and maximum shear stress.

b$.$ Determine the safety factor with respect to initial yielding according to the maximum$-$shear$-$stress

theory and according to the maximum$-$distortion$-$energy theory.

c$.$ Draw the failure theory diagram clearly showing the stress state for these theories.