Thin Cylinder Hoop Stress Calculation

Understanding the stress distribution in thin-walled cylinders, like pressure vessels or pipes, is a fundamental concept in mechanics of materials. When a thin cylinder is subjected to internal pressure, two primary types of stresses are developed: circumferential (or hoop) stress and longitudinal (or axial) stress. This problem focuses on calculating the average circumferential (hoop) stress.

Given Parameters for Thin Cylinder Stress

Let's identify the given values for the thin cylinder from the problem statement:

• Inner radius of the cylinder, \(r = 500 \text{ mm}\)
• Thickness of the cylinder wall, \(t = 10 \text{ mm}\)
• Internal pressure applied to the cylinder, \(p = 5 \text{ MPa}\)

Formula for Circumferential (Hoop) Stress

For a thin-walled cylinder, where the ratio of the inner radius to the wall thickness (\(r/t\)) is typically greater than 10, the average circumferential or hoop stress (\(\sigma_h\)) can be calculated using a simplified formula. This formula assumes uniform stress distribution across the wall thickness, which is a reasonable approximation for thin cylinders.

The formula for the average circumferential (hoop) stress in a thin cylinder is:

\[\sigma_h = \frac{pr}{t}\]

Where:

• \(\sigma_h\) is the circumferential (hoop) stress
• \(p\) is the internal pressure
• \(r\) is the inner radius
• \(t\) is the wall thickness

Step-by-Step Calculation of Hoop Stress

Now, let's substitute the given values into the formula to find the average circumferential (hoop) stress. It's important to ensure that all units are consistent. In this case, pressure is in MPa, and dimensions are in mm, which will yield stress in MPa.

Substitute these values into the formula:

\[\sigma_h = \frac{(5 \text{ MPa}) \times (500 \text{ mm})}{10 \text{ mm}}\]

First, multiply the pressure by the inner radius:

\[\sigma_h = \frac{2500 \text{ MPa} \cdot \text{mm}}{10 \text{ mm}}\]

Now, divide by the thickness:

\[\sigma_h = 250 \text{ MPa}\]

Final Result for Circumferential Stress

The calculated average circumferential (hoop) stress in the thin cylinder subjected to an internal pressure of 5 MPa is 250 MPa. This value represents the tensile stress acting along the circumference of the cylinder due to the internal pressure.