Question #$3$

A pipe with closed ends has a wall thickness of $10$ mm and an inner diameter of $0\mathrm{.}60$ m $.$ It filled with a gas at $20$ MPa pressure and subjected to a torque about its long axis of $1200kN*m.$

a$.$ Determine the three principal normal stresses and the maximum shear stress. Recall from basic mechanics that:

$=\frac{T}{2r_{\text{a}\text{v}\text{g}}^{2}t}$

b$.$ If the pipe is constructed of oil$-$quenched and tempered $4340$ steel with a yield strength of $1620$ MPa $,$ what is the safety factor against yielding using the Tresca Maximum Shear Stress Criterion?

c$.$ Prior to loading, an inside crack is discovered through $25\%$ of the thickness of the sample. With this crack, what is the safety factor against fracture if the plane strain fracture toughness is $87\mathrm{.}4$MPa$\sqrt[\phantom{2}]{m}?$