Part $2(75$ marks$)$

$25$ Q$2\mathrm{.}1$ A machine frame is made of steel having $S_y=400$MPa and $S_{yy}=250$MPa. When marks loaded in a test fixture, the stresses were found to vary linearly with load. Two points on the surface were found to be most critical. With a $4-kN$ test load, stresses at these points were:

Point $b,{}_x=150$MPa,${}_y=-100$MPa.

Determine the factor of safety according to the

$($a$)$ maximum$-$normal$-$stress theory,

$($b$)$ maximum$-$shear$-$stress theory, and

$($c$)$ maximum$-$distortion$-$energy theory.

Discuss briefly the relative validity of each theory for this application. Considering all the information available, what is your best judgment of the value of test load at which yielding would be expected to begin $($see Figure below$)?$

Compute test load at which the frame will experience initial yielding. $($Bonus$)$