Thin cylindrical shells with riveted joints: For a riveted cylindrical shell of diameter d and plate thickness t under internal pressure p, with joint efficiency η, the hoop (circumferential) stress is given by

Introduction / Context:Thin pressure vessels experience two principal membrane stresses: hoop (circumferential) and longitudinal. When a longitudinal riveted joint is present, its efficiency reduces the effective load-carrying thickness against hoop stress.

Given Data / Assumptions:

• Thin cylinder (t ≪ d), uniform internal pressure p.
• Riveted longitudinal joint with efficiency η (0 < η ≤ 1).
• Membrane theory (neglecting bending and stress concentrations).

Concept / Approach:For a seamless thin cylinder, hoop stress is σh = p * d / (2 * t). With a longitudinal joint of efficiency η, the effective resisting thickness is η * t, so the hoop stress increases to σh = p * d / (2 * η * t).

Step-by-Step Solution:

Verification / Alternative check:Longitudinal stress for a seamless thin cylinder is σL = p * d / (4 * t). The presence of a circumferential joint would affect σL instead; here the longitudinal joint affects hoop stress as shown.

Why Other Options Are Wrong:

• p * d / (4 * t * η) corresponds to longitudinal stress with joint efficiency, not hoop stress.
• p * d / (t * η) and 2 * p * d / (t * η) overstate the stress by missing the factor 2 or adding an extra 2.
• p / (2 * t * η * d) has wrong dependence on diameter.

Common Pitfalls:Mixing hoop and longitudinal stress formulas; always note the factor 2 difference.

Final Answer:σh = p * d / (2 * t * η)