Thin riveted cylindrical shell under internal pressure: For a shell of diameter d and thickness t subjected to internal pressure p, what is the longitudinal (axial) membrane stress in the shell plate (neglecting joint efficiency)?

Introduction / Context:
Thin-walled cylindrical pressure vessels develop two principal membrane stresses: circumferential (hoop) and longitudinal (axial). Correct formulas are crucial for safe design of boilers, tanks, and pipelines, including riveted or welded shells when joint efficiency is not specifically introduced.

Given Data / Assumptions:

• Thin shell: t ≪ d (thin-wall assumption).
• Uniform internal pressure p.
• Homogeneous, isotropic plate; joint efficiency effects neglected unless stated.

Concept / Approach:
Equilibrium of half the cylinder by a longitudinal cut gives the hoop stress; equilibrium of a segment cut by a transverse plane across the cylinder gives the longitudinal stress. Standard results: σ_hoop = p d / (2 t), and σ_long = p d / (4 t). The longitudinal stress is half the hoop stress in a thin cylinder.

Step-by-Step Solution:

Verification / Alternative check:
Compare with hoop stress derivation; ratio σ_hoop : σ_long = 2 : 1 for thin cylinders, consistent with classic results.

Why Other Options Are Wrong:

• p d / (2 t) is the hoop (circumferential) stress, not longitudinal.
• p t / d and 2 p t / d invert variables and give wrong trends.
• p d^2 / (8 t) has incorrect dimensions for stress.

Common Pitfalls:
Mixing hoop and longitudinal formulas; introducing joint efficiency without being asked; applying thin-wall formulas outside their validity range (t not small compared to d).

Final Answer:
σ_long = p d / (4 t)