Thin cylindrical shell under internal pressure: For a thin cylinder of diameter d, thickness t, and internal pressure p, what is the circumferential (hoop) stress in the shell?

Introduction / Context:Thin-walled pressure vessels experience two principal membrane stresses: hoop (circumferential) and longitudinal. Correctly computing these stresses is critical for safe vessel and pipe design.

Given Data / Assumptions:

• Thin cylinder: t ≪ d (thin-wall assumption).
• Internal pressure p uniformly distributed.
• Membrane theory (neglect bending through thickness).

Concept / Approach:Balancing forces on a diametral cut of the cylinder gives the hoop stress expression. For longitudinal stress, a different free-body leads to σ_long = pd/(4t). Here we specifically need the circumferential (hoop) stress σ_hoop = pd/(2t).

Step-by-Step Solution:

Verification / Alternative check:Textbook relation: σ_hoop = p r / t with r = d/2, giving σ_hoop = p d / (2 t). Longitudinal stress check: σ_long = p d / (4 t) is half of hoop stress, consistent with thin-cylinder theory.

Why Other Options Are Wrong:

• pd/t and pd/6t overestimate; pd/4t corresponds to longitudinal stress, not hoop.
• 2pt/d has wrong dependence and units for stress.

Common Pitfalls:Confusing hoop and longitudinal formulas; using diameter vs. radius incorrectly; applying thin-wall formulas when t is not small compared to d.

Final Answer:pd/2t