Hoop Stress Variation in a Thick Cylinder

The hoop stress is the circumferential stress that is exerted on a cylindrical object, such as a pipe or a cylinder. When the object is thick-walled, the hoop stress varies across the thickness of the object.

Hyperbolic Variation

The variation of the hoop stress across the thickness of a thick cylinder is hyperbolic. This means that the stress is highest at the inner surface of the cylinder and decreases as you move towards the outer surface.

Reasons for Hyperbolic Variation

The hyperbolic variation of hoop stress in a thick-walled cylinder can be explained by the following factors:

1. Poisson's Ratio: The hoop stress is related to the radial stress by Poisson's ratio. When a thick-walled cylinder is subjected to an internal pressure, the radial stress is compressive and causes the cylinder to contract radially. This results in an increase in the hoop stress at the inner surface of the cylinder.

2. Bending Stress: A thick-walled cylinder can also experience bending stress due to its own weight or external loads. The bending stress is highest at the inner surface of the cylinder, which contributes to the hyperbolic variation of the hoop stress.

3. Material Properties: The material properties of the cylinder, such as its modulus of elasticity and yield strength, can also affect the variation of the hoop stress across the thickness of the cylinder.

Conclusion

In summary, the hoop stress in a thick-walled cylinder varies across the thickness of the cylinder in a hyperbolic manner. This variation is influenced by Poisson's ratio, bending stress, and material properties.