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ID: past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2018/paper-1/5/a/solution

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

Codex  0 2026-10-20

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 ≪ r

For an end cap, the outward pressure force is

P π r^{2}

. The wall's axial tensile force is

σ_{z} (2 π r t)

. Equilibrium therefore gives

σ_{z} = \frac{P r}{2 t} .

(1)

For a longitudinally split half-cylinder of length

L

, pressure has resultant

P (2 r L)

across the projected rectangular area. Two cut wall sections each carry

σ_{θ} t L

, so

σ_{θ} = \frac{P r}{t} = 2 σ_{z} .

(2)

Thus 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.

Figure 1.
Original thin-pipe force balances and schematic crack-tip stress regularization by plastic deformation; the plastic-zone curve is qualitative
. **Details**.

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