Problem: A machine component is made from ductile material, with yield strengths in tension and

compression of $100$ ksi and Ultimate Strength of $125$ ksi $.$ The loading results in a $2$D element with

$2=-1,$ and $3=0.$ Consider $($a$)$ the maximum$-$normal$-$stress theory, $($b$)$ the maximum$-$shear$-$stress

theory, and $($c$)$ the maximum$-$distortion$-$energy theory. Which theory would be expected to agree most

closely with an actual test and why? Draw all three failure theory lines on a $1-2$ plot and clearly label

the point of failure for this stress element. Estimate the factors of safety for the other two theories for

that point.