$($a$)$ Discuss three different ways by which a structural component may fail.

$($b$)$ Starting with the expression for the von Mises equivalent stress find the

shear stress required for plastic flow in a stress field of pure shear. What

value does the Tresca criterion give for this situation?

$($c$)$ A solid shaft made from a material with a uniaxial yield stress of $Y=120$

$\frac{N}{m}{m}^2$ is to be subjected to a torque of $100$ Nm $.$ What diameter should be

used if yielding according to $($i$)$ Tresca yield criterion, and $($ii$)$ von Mises yield

criterion, is not to occur.

$[$Hint: The shear stress $$ at the surface of a solid circular shaft subjected to a torque $T$

is given by $=16\frac{T}{}{d}^3,$ where $d$ is the diameter of the shaft.$]$