Thin-walled cylindrical pressure vessel: what is the longitudinal stress (axial membrane stress) under internal pressure p, diameter d, and wall thickness t?

Introduction / Context:
Design of thin-walled cylinders relies on two primary stresses: circumferential (hoop) and longitudinal (axial). Correct formulas determine required thickness and verify code compliance.

Given Data / Assumptions:

• Thin-wall assumption: t ≪ d.
• Uniform internal pressure p; flat or hemispherical heads integrated so axial force is carried by shell.
• Elastic, isotropic behavior; no external loads.

Concept / Approach:
From static equilibrium of a free body cut normal to the axis, the pressure force on the projected area equals the membrane force in the shell wall. Using diameter d instead of radius r simplifies the standard relationship.

Step-by-Step Solution:
Hoop stress for reference: σh = p d / (2 t).Longitudinal stress: balance pressure on end area (π d^2 / 4) with shell force (σl * π d t) → σl = p d / (4 t).Select the matching option: p d / 4 t.

Verification / Alternative check:
Using radius r = d/2 yields σl = p r / (2 t); substituting r gives p d / 4 t, consistent with standard texts.

Why Other Options Are Wrong:
p d / 2 t: this is the hoop stress, not longitudinal.p d / t and p d / 8 t: over- and under-estimations not supported by equilibrium.

Common Pitfalls:
Mixing up hoop and longitudinal stresses; applying thin-wall formulas to thick shells where stress distributions are not uniform.

Final Answer:
p d / 4 t