Project ForgeA thin-walled cylinder of radius r and thickness t is open at both ends, and fits snugly between two rigid walls under ambient conditions. The material has Young's modulus E, Poisson's ratio ν, and coefficient of thermal expansion α. What is the minimum rise in temperature ΔT of the cylinder (uniform temperature, no buckling) required to prevent gas leakage if the cylinder has to store gas at an internal pressure p above atmospheric?
- AΔT = 3νpr/(2αtE)
- BΔT = (ν−1/4)pr/(αtE)
- ΔT = νpr/(αtE)
- DΔT = (ν+1/2)pr/(αtE)
Explanation
Since the two walls are independently rigid (externally fixed), the open-ended cylinder itself carries no axial stress from the pressure (σz=0) — only hoop stress σθ=pr/t. This hoop stress produces an axial Poisson contraction εz=−νσθ/E=−νpr/(tE). To avoid the cylinder pulling away from the walls (creating a leak path), the free thermal expansion must exactly offset this contraction: αΔT = νpr/(tE), so ΔT=νpr/(αtE).
Source: GATE ME 2020 Session 2 (ME-2) (2020)
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