Thin cylinder under internal pressure – Ratio of longitudinal strain to hoop strain A thin cylindrical shell of diameter d, thickness t, and length l is subjected to internal pressure p. If ν is Poisson’s ratio, what is the ratio of longitudinal strain to hoop strain (ε_longitudinal / ε_hoop)?
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A(0.5 − ν) / (1 − 0.5 ν)
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B(1 − 0.5 ν) / (0.5 − ν)
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C(1 + ν) / (1 − ν)
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D(1 − 2 ν) / (1 − ν)
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EIndependent of ν, equal to 0.5
Answer
Correct Answer: (0.5 − ν) / (1 − 0.5 ν)
Explanation
Introduction / Context:Thin-cylinder membrane theory gives closed-form expressions for stresses and strains under internal pressure. The ratio of longitudinal to hoop strain is important for estimating changes in length and diameter.
Given Data / Assumptions:
- Thin shell: t/d ≤ about 1/20, radial stress negligible compared to membrane stresses.
- Hoop stress σ_h = p d / (2 t); longitudinal stress σ_l = p d / (4 t).
- Linear elastic isotropic material with Young’s modulus E and Poisson’s ratio ν.
Concept / Approach:Strains considering Poisson coupling are ε_h = (σ_h/E) − ν (σ_l/E) and ε_l = (σ_l/E) − ν (σ_h/E). Form the ratio ε_l / ε_h.
Step-by-Step Solution:Let σ_l = 0.5 σ_h (from thin-cylinder stresses).ε_l = (σ_l − ν σ_h)/E = (0.5 σ_h − ν σ_h)/E = σ_h (0.5 − ν)/E.ε_h = (σ_h − ν σ_l)/E = (σ_h − ν * 0.5 σ_h)/E = σ_h (1 − 0.5 ν)/E.Therefore, ε_l / ε_h = (0.5 − ν) / (1 − 0.5 ν).
Verification / Alternative check:For ν = 0.3, ε_l / ε_h = (0.2) / (0.85) ≈ 0.235, consistent with the fact that hoop strain exceeds longitudinal strain.
Why Other Options Are Wrong:Inversions or ν-independent values contradict the derived dependence; expressions with (1 ± ν) are for other relationships (e.g., plane stress/plane strain conversions).
Common Pitfalls:Ignoring Poisson’s effect and assuming ε_l/ε_h = σ_l/σ_h = 0.5; mixing thin and thick cylinder assumptions.
Final Answer:(0.5 − ν) / (1 − 0.5 ν)