Question

Based on the limiting factor to prevent bursting in thin cylinder, the design is primarily made based on which stress

• A. Maximum volumetric stress
• B. Maximum longitudinal stress
• C. Maximum shear stress
• D. Maximum hoop stress

Hint:

Since hoop stress is twice as large as longitudinal stress (\( \sigma_h = 2\sigma_l \)), thin-walled pipes and boiling sausages always split open along their lengthwise seams when overpressurized, never around their circumferences!

Answer 1

The Correct Option is D

Concept: A thin-walled cylindrical pressure vessel with internal diameter \( d \), wall thickness \( t \), and internal gauge fluid pressure \( p \) develops two primary types of tensile stress in its walls:

• Hoop Stress / Circumferential Stress (\( \sigma_h \)): The stress acting along the circumference of the cylinder wall, which resists the tendency of the cylinder to split open along a longitudinal line. Its value is given by: \[ \sigma_h = \frac{pd}{2t} \]

• Longitudinal Stress (\( \sigma_l \)): The stress acting parallel to the longitudinal axis of the cylinder, which resists the tendency of the cylinder to burst apart into two separate tubes. Its value is given by: \[ \sigma_l = \frac{pd}{4t} \]

Step 1: Comparing the magnitudes of the stress components.
Let us analyze the ratio between the two stress equations for a cylinder under identical internal pressure \( p \), diameter \( d \), and wall thickness \( t \): \[ \sigma_h = \frac{pd}{2t} \] \[ \sigma_l = \frac{pd}{4t} = \frac{1}{2}\left(\frac{pd}{2t}\right) = \frac{1}{2}\sigma_h \] This comparison shows that the hoop stress is exactly twice as large as the longitudinal stress: \[ \sigma_h = 2\sigma_l \]

Step 2: Identifying the structural failure mode.
Because hoop stress is twice as large as longitudinal stress, as the internal fluid pressure \( p \) inside the cylinder increases, the hoop stress will always reach the yield strength limit of the material first. \[ \sigma_h \rightarrow \sigma_{\text{allowable}} \quad \text{while} \quad \sigma_l = \frac{1}{2}\sigma_{\text{allowable}} \] Therefore, the cylinder will fail by splitting longitudinally due to excessive circumferential tension long before it can fail along a transverse cross-section due to longitudinal stress.

Step 3: Concluding the primary design criteria.
To prevent catastrophic bursting structural failure, engineers must design the vessel wall thickness based on the maximum stress component. Since hoop stress is the larger and more critical stress component, Maximum hoop stress serves as the primary limiting factor in thin cylinder design. This matches Option (D).