Thin cylindrical shell under internal pressure — stress ratio: For a thin cylinder, what is the ratio of longitudinal stress to hoop (circumferential) stress under pressure p?

Introduction:
Thin-cylinder theory yields two primary membrane stresses: hoop and longitudinal. Remembering their magnitudes is essential for shell and pressure vessel design.

Given Data / Assumptions:

• Thin shell: t/D small, membrane action.
• Internal pressure p; diameter D; thickness t.

Concept / Approach:
Standard thin-cylinder formulas are: σ_hoop = pD/(2t) and σ_long = pD/(4t). The ratio σ_long/σ_hoop = (pD/(4t)) / (pD/(2t)) = 1/2.

Step-by-Step Solution:

Verification / Alternative check:
Equilibrium of a longitudinally cut cylinder and a transversely cut cylinder independently lead to the two expressions above, confirming the 1/2 ratio.

Why Other Options Are Wrong:

• 3/4, 1, 1.5, 2: Do not follow from thin-shell equilibrium relations.

Common Pitfalls:
Confusing which stress is larger; hoop stress is twice the longitudinal stress in thin cylinders.

Final Answer: