(Solved): 3. (15 points) Consider a thin-walled pressure vessel with closed ends of length L, radius R, and t ...

3. (15 points) Consider a thin-walled pressure vessel with closed ends of length $L$, radius $R$, and thickness $t$ subjected to an internal pressure $p$ and torque $T$ as pictured below: We consider a loading situation, in which the torque is always given in terms of the internal pressure by $T=\pi p{R}^3$. For a thin-walled tube you may approximate the polar moment of area as $J=2\pi R^{3}t$. (a) Determine all non-zero stress components in the wall of the pressure vessel, and sketch the state of stress on an element aligned with the axial and circumferential directions - i.e., the $x$ - and $\theta$-directions - as shown above. (b) Construct Mohr's circle for this state of plane stress. You may normalize the normal stress and shear stress axes by $pR/t$. (c) Using Mohr's circle, determine (i) the maximum in-plane shear stress ${\tau }_{\max }$, (ii) the principal stresses ${\sigma }_{\max }$ and ${\sigma }_{\min }$, and (iii) the two angles ${\theta }_{\mathrm{p}}$ that describe the orientations of the principal planes with respect to the $x$-direction.