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

Hint:

A simple way to remember the relationship is to think about how a cylinder would fail. It's more likely to split open along its length (due to hoop stress) than to be pulled apart at its ends (due to longitudinal stress). This implies that the hoop stress is the larger of the two. Specifically, Hoop Stress = 2 \(\times\) Longitudinal Stress. For a thin-walled sphere, the stress is uniform in all directions and equals the longitudinal stress of a cylinder (\(pD/4t\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the fundamental relationship between the two principal stresses—hoop stress and longitudinal stress—in a thin-walled cylindrical pressure vessel subjected to internal pressure.
Step 2: Key Formula or Approach:
We need to recall the standard formulas for hoop stress (\(\sigma_1\)) and longitudinal stress (\(\sigma_2\)) derived from the equilibrium of forces in a thin-walled cylinder. Let \(p\) be the internal pressure, \(D\) be the internal diameter, and \(t\) be the wall thickness.

• The formula for hoop (or circumferential) stress is: \(\sigma_1 = \frac{pD}{2t}\)
• The formula for longitudinal (or axial) stress is: \(\sigma_2 = \frac{pD}{4t}\)

Step 3: Detailed Explanation:
Let's compare the two formulas directly. \[ \sigma_1 = \frac{pD}{2t} \] \[ \sigma_2 = \frac{pD}{4t} \] We can rewrite the expression for \(\sigma_1\) in terms of \(\sigma_2\): \[ \sigma_1 = 2 \times \left( \frac{pD}{4t} \right) \] Since \(\sigma_2 = \frac{pD}{4t}\), we can substitute this into the equation: \[ \sigma_1 = 2\sigma_2 \] Step 4: Final Answer:
The hoop stress in a thin-walled cylindrical pressure vessel is exactly twice the longitudinal stress. This is a fundamental result in the mechanics of materials.