Thin-walled cylindrical vessel under internal pressure: the circumferential (hoop) stress compared to the longitudinal stress is

Introduction / Context:
For thin cylindrical shells under internal pressure, two primary membrane stresses arise: hoop (circumferential) and longitudinal (axial). Understanding their relationship is essential for safe pressure vessel design and sizing of thickness.

Given Data / Assumptions:

• Thin-walled assumption: wall thickness t << radius r.
• Uniform internal pressure p; end caps integral.
• Elastic, isotropic material; negligible external loads.

Concept / Approach:
The classic thin-shell formulas are: hoop stress sigma_h = p * r / t, longitudinal stress sigma_l = p * r / (2 * t). These follow from free-body equilibrium of a longitudinal cut (for hoop) and a transverse cut (for longitudinal).

Step-by-Step Solution:
Write hoop stress: sigma_h = p * r / t.Write longitudinal stress: sigma_l = p * r / (2 * t).Take ratio: sigma_h / sigma_l = (p r / t) / (p r / (2 t)) = 2.Hence, hoop stress is twice the longitudinal stress.

Verification / Alternative check:
Textbook derivations and code background equations (e.g., for cylindrical shells) reproduce this 2:1 relationship under the thin-wall assumption.

Why Other Options Are Wrong:
Half or equal: contradicts the derived ratio.Eight times: far exceeds the thin-wall relationship and would imply grossly different mechanics.

Common Pitfalls:
Applying thin-wall formulas to thick shells (t not small vs r), or forgetting that external attachments can alter local stresses.

Final Answer:
Twice