Thin cylinder under internal pressure – Ratio of longitudinal strain to hoop strain\nA 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)?

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)