Question

The relationship between the hoop stress \( \sigma_1 \) and the longitudinal stress \( \sigma_2 \) of a closed cylindrical thin-walled pressure vessel is:

• A. \( \sigma_1 = 2\sigma_2 \)
• B. \( \sigma_1 = \sigma_2 \)
• C. \( \sigma_1 = \frac{\sigma_2}{2} \)
• D. \( \sigma_1 = \frac{1}{3}\sigma_2 \)

Hint:

In thin-walled pressure vessels, the hoop stress is always greater than the longitudinal stress by a factor of 2. This is a critical consideration for material strength and design safety.

Answer 1

The Correct Option is A

For a thin-walled pressure vessel, the relationship between the hoop stress \( \sigma_1 \) and the longitudinal stress \( \sigma_2 \) is derived based on the internal pressure and geometry of the vessel. The hoop stress (also known as the circumferential stress) acts around the circumference of the cylinder, while the longitudinal stress acts along the length of the cylinder. The general formula for the hoop stress and longitudinal stress in a thin-walled cylindrical pressure vessel under internal pressure is given by:
\[ \sigma_1 = \frac{pR}{t} \quad {(hoop stress)} \]
\[ \sigma_2 = \frac{pR}{2t} \quad {(longitudinal stress)} \] Where:
- \( p \) is the internal pressure,
- \( R \) is the radius of the vessel,
- \( t \) is the wall thickness.
From these equations, we can observe that the hoop stress is twice the longitudinal stress. Therefore, the relationship between the two stresses is:
\[ \sigma_1 = 2\sigma_2 \] Thus, the correct answer is (A).