Short Answer:

Hoop stress is the stress that acts around the circumference of a cylindrical or spherical vessel subjected to internal pressure. It is also known as circumferential stress. This stress acts tangentially to the surface and tends to split the cylinder along its length.

In a thin-walled pressure vessel, hoop stress is one of the major stresses developed due to internal pressure. It is generally greater than the longitudinal stress and is given by the formula:

where  = internal pressure,  = diameter of the cylinder, and  = wall thickness.

Detailed Explanation:

Hoop Stress

When a cylindrical or spherical vessel is subjected to internal pressure, it experiences forces acting in different directions on its wall. These forces generate two main types of stresses in the material — hoop (circumferential) stress and longitudinal (axial) stress.

The hoop stress acts tangentially around the circumference of the cylinder and tries to tear it apart along its length. In simple words, it is the stress developed in the wall of the vessel in the circumferential direction due to internal pressure trying to expand the cylinder radially.

Hoop stress is very important in the design of pressure vessels because it is generally twice the longitudinal stress and hence controls the wall thickness and safety of the vessel.

Derivation of Hoop Stress in a Thin Cylinder

To understand the concept of hoop stress, consider a thin-walled cylindrical vessel of

• internal diameter = ,
• internal radius = ,
• wall thickness = , and
• internal fluid pressure = .

Assume that the thickness of the wall is small compared to its diameter (), so that the stress can be considered uniform across the thickness.

Now, cut the cylinder vertically along its axis into two equal halves. The internal pressure acts on the inner surface, tending to separate the two halves.

Step 1: Force due to internal pressure
The internal pressure acts perpendicular to the inner surface, producing a total horizontal force that tries to push the two halves apart. The projected area on which pressure acts is equal to the internal diameter multiplied by the length of the cylinder :

Step 2: Resisting force by material
The resistance to this force is offered by the material of the cylinder wall. The resisting area on each side of the cut is equal to the wall thickness  multiplied by the length . Since there are two walls, the total resisting area is:

Let the hoop stress in the wall be . Therefore, the total resisting force developed in both walls is:

Step 3: Equilibrium condition
For equilibrium, the total pressure force is equal to the resisting force:

Simplifying,

Since , the expression can also be written as:

This is the standard formula for hoop stress in a thin cylinder.

Nature of Hoop Stress

1. Direction:
   It acts tangentially around the circumference of the cylinder.
2. Type:
   It is a tensile stress, since it tends to pull the material apart around the circumference.
3. Magnitude:
   The hoop stress is maximum at the inner surface and assumed constant across the thickness for thin cylinders.
4. Comparison:
   Hoop stress is twice the longitudinal stress, which acts along the axis of the cylinder.

Importance of Hoop Stress

1. Determines Wall Thickness:
   Hoop stress is the controlling factor in deciding the thickness of the vessel wall to ensure safety under internal pressure.
2. Prevention of Failure:
   Since hoop stress is maximum, failure due to internal pressure usually occurs along the length of the cylinder.
3. Design of Pressure Vessels:
   Engineers use the hoop stress formula to design boilers, pipes, gas cylinders, and hydraulic components to prevent bursting.
4. Safety in Operation:
   Understanding hoop stress helps maintain the structural integrity of vessels operating under pressure.

Applications of Hoop Stress

1. Boilers: Cylindrical shells of boilers experience hoop stress due to steam pressure.
2. Pipes: In water supply systems and oil refineries, pipes carry high-pressure fluids causing hoop stress in the pipe wall.
3. Gas Cylinders: Used for storing compressed gases such as oxygen or LPG, where internal pressure produces hoop stress.
4. Pressure Vessels: Industrial tanks designed for high pressure must resist hoop stress safely.
5. Aircraft and Submarines: The outer shell of these structures experiences hoop stress due to external and internal pressure differences.

Factors Affecting Hoop Stress

1. Internal Pressure (p):
   Hoop stress increases directly with internal pressure.
2. Diameter (d):
   Larger diameters increase the stress because a greater area is exposed to pressure.
3. Wall Thickness (t):
   Hoop stress is inversely proportional to thickness — thicker walls reduce stress.
4. Material Properties:
   Materials with higher tensile strength can safely withstand higher hoop stress.
5. Temperature:
   Elevated temperatures can reduce material strength, affecting the safe stress limit.

Failure Due to Hoop Stress

In most cylindrical pressure vessels, failure occurs when the hoop stress exceeds the tensile strength of the material. This failure appears as a longitudinal split along the cylinder because the circumferential direction experiences the maximum stress.

Thus, to prevent such failures, the hoop stress must be limited within the safe working stress of the material, and a suitable factor of safety must be used during design.

Conclusion

Hoop stress is the circumferential tensile stress produced in the wall of a thin cylinder or sphere subjected to internal pressure. It acts tangentially around the circumference and tends to split the cylinder along its length. The formula for hoop stress is

It plays a vital role in designing safe and efficient pressure vessels, as it determines the required wall thickness and ensures that the structure can withstand the internal pressure without bursting or failure.