Understanding Stresses in Thin-Walled Cylindrical Pressure Vessels

A thin-walled cylindrical pressure vessel is commonly used to store fluids or gases under pressure. Due to the internal pressure, stresses are developed within the wall of the vessel. The two primary stresses are the hoop stress and the longitudinal stress.

What is Hoop Stress?

Hoop stress, also known as circumferential stress, acts along the circumference of the cylinder. It is caused by the internal pressure pushing outward on the curved walls, trying to burst the cylinder along its length. Imagine cutting the cylinder across its diameter; the hoop stress resists this bursting force.

The formula for hoop stress ($\sigma_h$) in a thin-walled cylinder is:

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

Where:

• P is the internal pressure
• D is the internal diameter of the vessel
• t is the wall thickness

What is Longitudinal Stress?

Longitudinal stress, also known as axial stress, acts along the length of the cylinder. It is caused by the internal pressure pushing on the ends of the cylinder, trying to pull the cylinder apart along its axis. Imagine cutting the cylinder through a cross-section perpendicular to its axis; the longitudinal stress resists this pulling force.

The formula for longitudinal stress ($\sigma_l$) in a thin-walled cylinder is:

$\sigma_l = \frac{PD}{4t}$

Where:

• P is the internal pressure
• D is the internal diameter of the vessel
• t is the wall thickness

Comparing Hoop Stress and Longitudinal Stress

Now let's compare the formulas for hoop stress and longitudinal stress:

• Hoop stress ($\sigma_h$) = $\frac{PD}{2t}$
• Longitudinal stress ($\sigma_l$) = $\frac{PD}{4t}$

We can see that the term $\frac{PD}{4t}$ appears in both formulas. Substituting the longitudinal stress formula into the hoop stress formula:

$\sigma_h = \frac{1}{2} \times \frac{PD}{t}$

$\sigma_h = 2 \times \frac{PD}{4t}$

Therefore, the relationship between hoop stress and longitudinal stress is:

$\sigma_h = 2 \times \sigma_l$

This shows that the hoop stress is twice the longitudinal stress in a thin-walled cylindrical pressure vessel. This is why hoop stress is the critical stress and often governs the design thickness of the vessel wall, as it is the larger of the two stresses.

Based on the comparison of the formulas, hoop stress is twice the longitudinal stress.