Hoop Stress in Thin Cylinders

In the study of mechanics of materials, understanding the stresses developed in pressure vessels like cylinders is crucial. A thin cylinder is a cylinder where the wall thickness is small compared to its diameter, typically when the ratio of diameter to thickness is greater than or equal to 20.

Understanding Hoop Stress

When a thin cylinder is subjected to internal pressure, stresses are developed in the walls of the cylinder. These stresses are primarily categorized into three types:

• Hoop stress (or Circumferential stress)
• Longitudinal stress (or Axial stress)
• Radial stress

Hoop stress acts along the circumference of the cylinder. Imagine cutting the cylinder across its diameter. The internal pressure pushes outwards on the curved surface, and the hoop stress is the stress resisting this outward bursting force. It is distributed uniformly through the thickness of the thin wall.

Nature of Hoop Stress

Consider a small element of the cylinder wall. The internal pressure acts radially outwards. To resist this outward force and maintain equilibrium, the cylinder wall experiences tension acting circumferentially. This tension is the hoop stress.

For a thin cylinder of diameter $d$, thickness $t$, and subjected to internal pressure $p$, the hoop stress ($\sigma_h$) is given by the formula:

$\sigma_h = \frac{pd}{2t}$

Since pressure $p$, diameter $d$, and thickness $t$ are positive values, the resulting hoop stress $\sigma_h$ is always positive. In stress analysis, positive stress indicates tensile stress, and negative stress indicates compressive stress.

Therefore, the hoop stress in a thin cylinder subjected to internal pressure is always a tensile stress, acting in the circumferential direction.

Comparison with Other Stresses

• Longitudinal Stress: This stress acts along the axis of the cylinder. It resists the force tending to separate the ends of the cylinder. For a closed thin cylinder, the longitudinal stress ($\sigma_l$) is given by $\sigma_l = \frac{pd}{4t}$. It is also a tensile stress, but its magnitude is typically half of the hoop stress.
• Radial Stress: This stress acts radially, through the wall thickness. On the inner surface, it is equal to the internal pressure $p$ (compressive, $\sigma_r = -p$), and on the outer surface, it is zero (atmospheric pressure). In thin cylinders, the maximum radial stress ($p$) is usually very small compared to the hoop and longitudinal stresses because the wall thickness $t$ is small. Often, the radial stress distribution across the thin wall is considered negligible or averaged to zero for simplicity in thin cylinder analysis.

Based on this analysis, the hoop stress is primarily circumferential and tensile in nature.

The question asks for the nature of hoop stress in a thin cylinder. As derived and explained, it is a tensile stress acting circumferentially.