Short Answer:

Longitudinal stress is the type of stress developed along the length or axis of a thin cylindrical or spherical vessel when it is subjected to internal pressure. It acts in the axial direction and tends to pull the cylinder apart at its ends.

In a thin-walled cylinder, this stress is caused by the internal pressure acting on the circular end caps of the vessel. The magnitude of longitudinal stress is given by the formula:

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

Detailed Explanation:

Longitudinal Stress

When a cylindrical vessel is subjected to internal fluid pressure, the pressure acts not only on the curved surface but also on the circular end caps of the vessel. This internal pressure tends to push the end caps apart, which causes a tensile stress along the axis of the cylinder. This stress developed parallel to the length of the vessel is known as longitudinal stress.

In simple words, longitudinal stress is the stress that acts along the lengthwise direction of a cylinder or pressure vessel due to the force trying to separate its ends. It is also called axial stress because it acts along the central axis of the vessel.

The longitudinal stress is smaller than the hoop (circumferential) stress, but it still plays an important role in the design of pressure vessels, pipes, boilers, and other cylindrical components subjected to internal or external pressure.

Derivation of Longitudinal Stress in a Thin Cylinder

To understand longitudinal stress, let us consider a thin-walled cylindrical vessel with the following parameters:

• Internal pressure =
• Internal diameter =
• Radius =
• Wall thickness =

When the cylinder is pressurized internally, the pressure acts on the circular end caps. This pressure tends to blow the ends off the cylinder, and the wall of the cylinder must provide an equal and opposite resisting force to keep it in equilibrium.

Step 1: Force due to internal pressure on the end cap

The total pressure force acting on one circular end of the cylinder is equal to the pressure multiplied by the area of the end cap.

 

This force tends to push the end cap away from the cylinder.

Step 2: Resisting force due to the material of the cylinder

The resistance to this internal pressure is provided by the longitudinal stress () developed in the material of the cylindrical wall.
The resisting area is the total cross-sectional area of the cylinder wall acting in the longitudinal direction, which is the product of the circumference of the cylinder and the thickness of the wall.

 

Hence, the total resisting force developed in the wall is:

Step 3: Equilibrium condition

For equilibrium, the total resisting force must be equal to the total pressure force acting on the end cap:

Simplifying,

Since ,

This is the expression for longitudinal stress in a thin cylinder.

Nature of Longitudinal Stress

1. Direction:
   Longitudinal stress acts parallel to the axis (length) of the cylinder.
2. Type:
   It is a tensile stress, because the internal pressure tries to pull the ends of the cylinder apart.
3. Distribution:
   It is assumed to be uniform across the thickness of the wall for thin-walled cylinders (where ).
4. Magnitude:
   Longitudinal stress is half the hoop (circumferential) stress:

Comparison Between Hoop Stress and Longitudinal Stress

• Hoop Stress (): Acts circumferentially and tends to split the cylinder along its length.
• Longitudinal Stress (): Acts along the length and tends to pull the cylinder apart at its ends.
• Relation:

Hence, hoop stress is always greater than longitudinal stress.

Due to this, most failures in pressure vessels occur along the longitudinal direction, because the circumferential stress (hoop stress) is higher.

Applications of Longitudinal Stress

1. Boilers:
   Longitudinal stress acts along the length of cylindrical boiler shells due to internal steam pressure.
2. Pipes and Hydraulic Cylinders:
   Pressurized fluid inside pipes and hydraulic systems creates longitudinal stress that affects their design.
3. Gas Cylinders:
   Compressed gas cylinders experience longitudinal stress due to internal gas pressure.
4. Pressure Vessels:
   Used in chemical plants and refineries where internal pressure produces both hoop and longitudinal stresses.
5. Design of End Closures:
   Longitudinal stress helps in determining the strength and thickness of end caps or domes in pressure vessels.

Factors Affecting Longitudinal Stress

1. Internal Pressure (p):
   Increases directly with internal pressure.
2. Cylinder Diameter (d):
   Larger diameters cause higher longitudinal stress for the same pressure.
3. Wall Thickness (t):
   Thicker walls reduce longitudinal stress since stress is inversely proportional to thickness.
4. Material Strength:
   Stronger materials can resist higher stress without failure.
5. Temperature and Fatigue:
   High temperature or repeated pressure cycles can reduce material strength and increase stress effects.

Conclusion

Longitudinal stress is the tensile stress that acts along the axis of a thin cylindrical vessel when it is subjected to internal pressure. It occurs due to the pressure acting on the end caps of the cylinder and is given by the formula:

This stress plays a vital role in the safe design of pressure vessels, boilers, and pipelines. Although it is smaller than the hoop stress, it must be considered to ensure the cylinder can safely resist the internal pressure without deformation or failure.