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?
Mechanical Engineering
Strength of Materials
Difficulty: Easy
Choose an option
-
A1/2
-
B3/4
-
C1
-
D1.5
-
E2
Answer
Correct Answer: 1/2
Explanation
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:
Write hoop stress: σ_h = pD/(2t)Write longitudinal stress: σ_l = pD/(4t)Form ratio: σ_l/σ_h = (pD/(4t)) / (pD/(2t)) = 1/2Verification / 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:
1/2