Thin cylindrical pressure vessel: For a cylinder of diameter d and thickness t under internal pressure p, what is the longitudinal (axial) membrane stress in the shell?

Introduction / Context:
Thin-walled pressure vessels develop two principal membrane stresses: circumferential (hoop) and longitudinal (axial). Correct formulas are vital for sizing thickness and checking allowable stress compliance in boilers, tanks, and pipelines.

Given Data / Assumptions:

• Thin cylinder, t ≪ d.
• Internal pressure p; ends closed.
• Uniform membrane stresses; neglect radial stress.

Concept / Approach:
Equilibrium on a free-body of half the vessel gives hoop stress sigma_h = p d / (2 t). Equilibrium on the end cap gives longitudinal stress sigma_L = p d / (4 t). The longitudinal stress is half the hoop stress in a thin cylinder with closed ends.

Step-by-Step Solution:

Verification / Alternative check:
Check ratio: sigma_h / sigma_L = 2, consistent with thin cylinder theory.

Why Other Options Are Wrong:

• p d / t and 2 p d / t are excessive (do not match equilibrium).
• p d / (2 t) is the hoop, not longitudinal, stress.
• p d / (6 t) has no basis in thin-cylinder derivation.

Common Pitfalls:
Using hoop formula for both directions; forgetting that end closures create axial stress; applying thick-cylinder relations to thin shells.

Final Answer:
p d / (4 t)