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?

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:

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