Thin-walled cylindrical vessel under internal pressure: what is the relationship between circumferential (hoop) stress and longitudinal stress?

Introduction / Context:
For thin-walled cylinders under internal pressure, two principal membrane stresses arise: the hoop (circumferential) stress around the circumference and the longitudinal stress along the axis. Understanding their ratio is essential for safe shell design, nozzle reinforcement, and head selection.

Given Data / Assumptions:

• Thin-wall assumption: wall thickness t ≪ diameter d.
• Uniform internal pressure p; negligible external loads.
• Linear elastic, isotropic material; small deformation.

Concept / Approach:
Balance forces on free-body sections. For hoop stress, cut the cylinder axially: the pressure acting on the projected area balances the tensile forces in the two walls. For longitudinal stress, cut a transverse section: pressure on the head balances the tensile ring in the shell. These classical force balances yield closed-form expressions.

Step-by-Step Solution:
Hoop stress: σh = p d / (2 t).Longitudinal stress: σL = p d / (4 t).Therefore, σh = 2 σL.This ratio guides which failure mode is critical; typically hoop governs thickness sizing for long cylinders.

Verification / Alternative check:
Dimensional checks and energy methods (e.g., strain energy minimization) confirm the same results under thin-shell theory. Codes and handbooks list these identities as baseline shell formulas.

Why Other Options Are Wrong:
(b) and (c) contradict the derived ratio; (d) is false because a fixed ratio exists for thin shells; (e) has no theoretical basis in this context.

Common Pitfalls:
Applying thin-wall equations to thick shells; omitting external loads or bending from supports; misusing diameter (inside vs. outside); ignoring joint efficiency in code calculations.

Final Answer:
σh = 2 σL