Thin-walled cylindrical vessel under internal pressure: what is the expression for the longitudinal (axial) stress?

Introduction / Context:Design of thin-walled pressure vessels distinguishes between circumferential (hoop) stress and longitudinal (axial) stress. Correct formulas are critical for sizing wall thickness, selecting materials, and verifying code compliance. For thin cylinders (t ≪ D), membrane theory provides simple expressions used in preliminary design and quick checks.

Given Data / Assumptions:

• Internal pressure p (uniform).
• Cylinder mean diameter D and wall thickness t, with t ≪ D.
• Closed ends carrying internal pressure load.

Concept / Approach:The longitudinal stress arises from the pressure force on the end caps, balanced by the axial stress in the cylindrical wall across the cross-sectional metal area. Force balance yields σ_l = p D / (4 t), whereas the hoop stress from tangential force balance is σ_h = p D / (2 t). The longitudinal stress is thus half the hoop stress in a thin cylinder.

Step-by-Step Solution:

Verification / Alternative check:Compare with hoop stress formula σ_h = p D /(2 t); longitudinal is half of hoop for thin cylinders, matching classic results.

Why Other Options Are Wrong:

• p D /(2 t) is the hoop stress, not longitudinal.
• 2 p D / t and p t / D do not follow membrane equilibrium for thin shells.
• p / 2 lacks dimensional consistency.

Common Pitfalls:Using internal radius rather than mean diameter inconsistently; applying thin-wall formulas when t is not small relative to D.

Final Answer:σ_l = p D / (4 t)