Problem #$5(35$ points$)$: A state of plane stress has components $\backslash$sigma $\_($x$)=-200$MPa,$\backslash$sigma $\_($y$)=400$MPa, and $\backslash$tau $\_($xy$)=-100$MPa, as shown on the element below. For this material, the yield strength S$\_($y$)=500$ MPa $.$

$($a$)$ Draw a complete Mohr's circle for this stress state. Clearly label all important points on the circle.

$($b$)$ Using Mohr's circle, determine the principal stresses and the maximum in$-$plane shear stress.

$($c$)$ Show all stresses from part $($b$)$ on properly oriented stress element$($s$).$ Be sure to include all stresses acting on the elements, as well as the angle$($s$)$ with respect to the original x$-$y coordinate system.

$($d$)$ Does the material yield, according to the Maximum Shear Stress $($Tresca$)$ theory? Justify your answer with calculations.

ANSWERS:

$\backslash$sigma x$=35\mathrm{.}40$ ksi $\backslash$sigma y$=124\mathrm{.}6$ ksi $\backslash$tau x$$y$=22\mathrm{.}60$ ksi

PLEASE SHOW EACH STEPTO PROVE THESE ANSWERS