Short Answer:

A thin cylinder is a type of pressure vessel in which the thickness of the wall is very small compared to its internal diameter. Generally, a cylinder is considered thin when the wall thickness is less than one-twentieth (1/20) of its internal diameter.

In a thin cylinder, the stress distribution across the wall thickness is assumed to be uniform. Such cylinders are commonly used to store or carry fluids under pressure, like boilers, gas tanks, pipes, and air receivers, where the internal pressure is relatively moderate and the wall thickness is small.

Detailed Explanation:

Thin Cylinder

A thin cylinder is a hollow circular vessel or pipe designed to contain fluid under internal or external pressure. The term thin refers to the condition when the thickness of the wall (t) is very small in comparison to the internal diameter (d) of the cylinder.

Mathematically, a cylinder is classified as thin-walled when:

In such cases, it is assumed that the pressure inside the cylinder acts uniformly on the inner surface and that the stress developed across the thickness of the wall is uniform. This simplification allows for easier analysis and design of the cylinder.

The main purpose of designing thin cylinders is to resist internal pressure safely without bursting or deforming excessively. These are used in various applications such as boilers, gas storage tanks, pipes, hydraulic cylinders, and pressure vessels in mechanical and civil engineering fields.

Assumptions in Thin Cylinder Theory

To simplify the analysis of a thin cylinder under internal pressure, the following assumptions are made:

1. The wall thickness is small compared to the diameter of the cylinder, i.e., .
2. The stress is uniformly distributed across the wall thickness.
3. The plane sections before loading remain plane even after the application of pressure.
4. The internal pressure acts equally in all directions.
5. The material of the cylinder is homogeneous, isotropic, and obeys Hooke’s law (linearly elastic).
6. The effect of radial stress is very small compared to the other stresses and hence neglected.

Under these assumptions, two main stresses are considered in a thin cylinder — hoop (circumferential) stress and longitudinal stress.

Types of Stresses in a Thin Cylinder

When a thin cylinder is subjected to internal pressure, two major stresses are developed:

1. Hoop or Circumferential Stress (σh):
   • It acts tangentially to the circumference of the cylinder.
   • It tends to split the cylinder along its length.
   • Formula:

where,
= internal pressure,
= internal diameter,
= wall thickness.

1. Longitudinal Stress (σl):
   • It acts along the length of the cylinder.
   • It tends to burst the cylinder open along its circular ends.
   • Formula:

From these expressions, it can be seen that hoop stress is twice the longitudinal stress, meaning the cylinder is more likely to fail along its length than at its ends.

Behavior of Thin Cylinders Under Pressure

When the fluid pressure acts inside the thin cylinder:

• The circumference of the cylinder increases slightly due to hoop stress.
• The length of the cylinder increases slightly due to longitudinal stress.
• The volume of the cylinder also increases.

These deformations are usually very small, and within the elastic limit of the material. However, they are important for the safe design of pressure vessels to avoid excessive strain or bursting.

Applications of Thin Cylinders

Thin cylinders are widely used in engineering applications where fluids are stored or transmitted under pressure. Some common examples include:

1. Boilers: Cylindrical shells of boilers operate under high internal steam pressure.
2. Pipes: Used in water supply systems, hydraulic lines, and oil refineries.
3. Compressed Gas Cylinders: Used for storing gases like oxygen, nitrogen, or LPG.
4. Pressure Vessels: Tanks and reservoirs used in industries to hold pressurized fluids.
5. Hydraulic and Pneumatic Cylinders: Used in actuators and machinery for power transmission.

Advantages of Thin Cylinders

1. Lightweight: Due to small wall thickness, thin cylinders are light and easy to handle.
2. Simple Stress Distribution: Stress analysis is straightforward because stress is assumed to be uniform.
3. Efficient Material Use: Less material is required, reducing cost.
4. Ease of Manufacturing: Fabrication and inspection are easier compared to thick cylinders.

Limitations of Thin Cylinders

1. Limited Pressure Capacity: Thin cylinders can only withstand moderate internal pressures.
2. Neglect of Radial Stress: The assumption of uniform stress distribution becomes inaccurate under very high pressures.
3. Elastic Deformation: Slight deformation under pressure may affect precise applications.
4. Unsuitable for High-Pressure Systems: For very high pressures, thick cylinders are preferred to ensure safety.

Difference Between Thin and Thick Cylinders (Conceptual Explanation)

• Thin Cylinder:
  Wall thickness  of internal diameter.
  Stress is assumed uniform.
• Thick Cylinder:
  Wall thickness  of internal diameter.
  Stress varies across thickness and must be calculated using Lame’s equations.

This distinction helps in selecting the correct design method based on the working pressure and the dimensions of the vessel.

Conclusion

A thin cylinder is a pressure vessel whose wall thickness is very small compared to its diameter, usually less than one-twentieth of the diameter. It is used to store or carry fluids under moderate pressure, assuming uniform stress distribution across its wall. Two main stresses develop in such a cylinder — hoop stress and longitudinal stress. The design and analysis of thin cylinders are fundamental in mechanical engineering, ensuring safe and efficient operation of pressure systems like boilers, tanks, and pipelines.