Thin-walled cylinder under internal pressure If p is the internal pressure in a thin cylinder of diameter d and wall thickness t, what is the circumferential (hoop) stress σ_h developed in the shell?

Introduction / Context:
Thin-cylinder formulas are used widely for boilers, pipes, and pressure vessels where wall thickness is small relative to diameter (t ≪ d). Two principal membrane stresses develop: hoop and longitudinal.

Given Data / Assumptions:

• Internal pressure p, cylinder diameter d, wall thickness t.
• Thin-wall assumption: stress is uniform through thickness; t/d is small.
• No end caps or fittings altering membrane stress locally.

Concept / Approach:
Equilibrium of half the cylinder under pressure gives the hoop stress. Cutting along a diametral plane and balancing pressure force with resisting stresses yields the standard expression.

Step-by-Step Solution:

Verification / Alternative check:
Longitudinal stress σ_l from end-cap equilibrium: σ_l = p * d / (4 * t), exactly half the hoop stress—consistent with thin-cylinder theory.

Why Other Options Are Wrong:

• p d / t and 2 p d / t overestimate; they ignore the factor 2 from two walls resisting.
• p d / (4 t) is the longitudinal stress, not hoop stress.
• p t / d is dimensionally inconsistent for stress.

Common Pitfalls:
Confusing hoop and longitudinal formulas or missing the factor of 2 in the hoop-stress derivation.

Final Answer:
σ_h = p * d / (2 * t)