Thin cylinders under internal pressure In a thin cylindrical shell carrying a flowing liquid under internal pressure p, what is the ratio of circumferential (hoop) stress to longitudinal stress in the wall?

Introduction / Context:
Design of thin-walled pressure vessels (pipes, tanks) requires distinguishing between hoop (circumferential) and longitudinal stresses. Knowing their ratio helps prioritize reinforcement and assess failure modes.

Given Data / Assumptions:

• Thin cylinder assumption (wall thickness t is small compared to diameter d).
• Internal pressure p from the contained fluid.
• Uniform stress distribution through thickness; end effects ignored for hoop stress derivation.

Concept / Approach:
Classical thin-cylinder formulas are:

• Hoop stress, sigma_h = pd / (2t)
• Longitudinal stress, sigma_l = pd / (4t)

Step-by-Step Solution:
Write the ratio R = sigma_h / sigma_lR = (pd/(2t)) / (pd/(4t))R = (1/2) / (1/4) = 2

Verification / Alternative check:
Use a quick numeric check: let p = 1, d = 1, t = 1. Then sigma_h = 0.5, sigma_l = 0.25, ratio = 2.

Why Other Options Are Wrong:
0.5 and 1.5 do not follow from the thin-cylinder relations; 1 would imply equal stresses which is not the case; 'None of these' is incorrect because the correct ratio is exactly 2.

Common Pitfalls:
Confusing radius with diameter in formulas; using thick-cylinder theory unnecessarily; forgetting that end plates halve the longitudinal stress relative to the hoop stress.

Final Answer:
2