Thin cylindrical shell under internal pressure p: What is the maximum shear stress in terms of diameter d and wall thickness t?

Introduction / Context:
Thin-walled cylinders (like boilers and pipes) are commonly analyzed using thin-shell formulas. Besides hoop and longitudinal stresses, maximum shear stress is also important, particularly for yield criteria and fatigue checks.

Given Data / Assumptions:

• Thin cylinder: t ≤ D/20 so thin-wall theory applies.
• Internal pressure p, diameter d, wall thickness t.
• Plane stress at the wall; negligible radial stress compared to hoop and longitudinal normal stresses.

Concept / Approach:
Principal stresses in a thin cylinder are:Hoop stress: σ_h = p d / (2 t)Longitudinal stress: σ_l = p d / (4 t)Maximum in-plane shear stress equals half the difference of the principal stresses.

Step-by-Step Solution:

Verification / Alternative check:
Mohr's circle for plane stress with σ1 = σ_h and σ2 = σ_l gives radius R = (σ1 − σ2)/2 = p d / 8 t, matching the result.

Why Other Options Are Wrong:
p d / t, p d / 2 t, p d / 4 t: these correspond to hoop or longitudinal magnitudes or their full difference, not the maximum shear (which is half the difference).p t / d is dimensionally incorrect for stress.

Common Pitfalls:
Using hoop stress directly as shear; forgetting that maximum shear equals half the principal stress difference; mixing thin- and thick-wall formulas.

Final Answer:

p d / 8 t