Thin cylindrical boiler — ratio of hoop (circumferential) stress to longitudinal stress For a thin-walled cylindrical boiler under internal pressure, what is the ratio of the tensile (normal) stress in the circumferential (hoop) direction to the tensile stress in the longitudinal (axial) direction?
Mechanical Engineering
Strength of Materials
Difficulty: Easy
Choose an option
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A4
-
B3
-
C2
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D1
-
E1/2
Answer
Correct Answer: 2
Explanation
Introduction / Context:In thin pressure vessels, two principal membrane stresses arise: the circumferential (hoop) stress and the longitudinal (axial) stress. Correctly identifying their magnitudes is crucial for safe shell thickness design.
Given Data / Assumptions:
- Thin-wall assumption: wall thickness t is small compared to diameter D.
- Internal pressure p is uniform; end caps are integral.
Concept / Approach:Classical thin-cylinder formulas:sigma_hoop = p * D / (2 * t)sigma_longitudinal = p * D / (4 * t)Taking the ratio gives 2.
Step-by-Step Solution:
Write equilibrium of half-cylinder in the hoop direction to get sigma_hoop.Write equilibrium of the cylindrical shell end to get sigma_longitudinal.Compute ratio: (pD / 2t) / (pD / 4t) = 2.Verification / Alternative check:Dimensionally consistent; widely cited in pressure vessel design texts.
Why Other Options Are Wrong:
- 4 or 3: overestimates; not supported by thin-wall equilibrium.
- 1: would imply equal stresses, which is incorrect.
- 1/2: reversed ratio.
Common Pitfalls:Using diameter versus radius inconsistently; mixing thick-wall formulas with thin-wall assumptions.
Final Answer:2