Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 5 / a

Assume a thin cylindrical wall, uniform internal pressure and closed ends carrying the pressure load. The radial stress is small compared with the membrane stresses when $t\ll r$.

For an end cap, the outward pressure force is $P\pi r^2$. The wall's axial tensile force is ${\sigma }_z(2\pi rt)$. Equilibrium therefore givesFor a longitudinally split half-cylinder of length $L$, pressure has resultant $P(2rL)$ across the projected rectangular area. Two cut wall sections each carry ${\sigma }_{\theta }tL$, soThus hoop stress is twice the closed-end axial stress in this model. An open-ended tube without an external end load does not acquire the same pressure-induced axial membrane stress.