Define thick cylinder.

Short Answer:

A thick cylinder is a type of pressure vessel in which the wall thickness is large compared to its internal diameter. Generally, a cylinder is considered thick when the ratio of the inner diameter to wall thickness is less than 20, that is, .

In a thick cylinder, the stress across the wall is not uniform. The internal pressure causes a significant difference between the inner and outer stresses, and hence, both radial stress and hoop (circumferential) stress vary across the wall thickness. Thick cylinders are used for high-pressure applications such as hydraulic presses, gun barrels, and pressure vessels.

Detailed Explanation:

Thick Cylinder

A thick cylinder is a hollow circular vessel designed to resist high internal or external pressures where the wall thickness cannot be neglected compared to the internal radius. When the wall is thick, the variation in stress across the thickness becomes significant, and therefore, it must be considered in the design and analysis.

A cylinder is classified as thick if:

where,

  •  = wall thickness,
  •  = internal diameter.

Unlike thin cylinders, where stress is assumed to be uniform, thick cylinders experience non-uniform stress distribution due to the variation of internal pressure across the wall. These cylinders are widely used in applications involving high-pressure fluids, such as steam boilers, hydraulic presses, gun barrels, and high-pressure pipelines.

Stresses in a Thick Cylinder

When a thick cylinder is subjected to internal pressure, two main stresses are developed within its wall:

  1. Radial Stress (σr):
    • Acts perpendicular to the wall of the cylinder (along the radius).
    • It is maximum at the inner surface and minimum at the outer surface.
  2. Hoop or Circumferential Stress (σθ):
    • Acts tangentially along the circumference of the cylinder.
    • It is maximum at the inner surface and decreases towards the outer surface.

Because of the large wall thickness, the magnitude of these stresses varies considerably from the inner to the outer surface, and hence their distribution must be determined accurately.

Lame’s Equations for Thick Cylinder

The variation of hoop and radial stresses in a thick cylinder can be determined using Lame’s equations, which are derived from the conditions of equilibrium and compatibility.

Let,

  •  = internal radius,
  •  = outer radius,
  •  = internal pressure,
  •  = external pressure (usually zero for open cylinders).

Then, the general equations for stresses at any radius  are:

 

where  and  are constants determined from boundary conditions:

  • At  (compressive),
  • At .

From these,

 

By substituting these values into Lame’s equations, we can find the radial and hoop stresses at any point in the wall.

The maximum hoop stress always occurs at the inner surface and is given by:

Difference Between Thin and Thick Cylinders (Conceptually)

Although both types of cylinders carry internal pressure, their stress distribution differs greatly:

  • In thin cylinders, the wall is assumed thin enough for stress to remain uniform across thickness.
  • In thick cylinders, the stress varies non-uniformly across the wall, with maximum stress occurring at the inner surface.

Hence, the design of thick cylinders requires a more detailed analysis using Lame’s equations to ensure safe operation under high pressure.

Applications of Thick Cylinders

Thick cylinders are used where high internal or external pressures exist. Some typical applications include:

  1. Hydraulic Presses: Cylinders that operate under very high hydraulic fluid pressure.
  2. Gun Barrels: Designed to withstand the explosive pressure of expanding gases.
  3. Pressure Vessels: Used in chemical and power plants where high pressure is common.
  4. High-Pressure Pipes: Used in oil and gas industries for transporting pressurized fluids.
  5. Injection Molding Machines: Cylinders that contain molten materials under high pressure.

Advantages of Thick Cylinders

  1. High Pressure Resistance: They can safely handle high internal or external pressures.
  2. Better Strength: The thick walls provide greater mechanical strength and stability.
  3. Durability: More resistant to wear, fatigue, and failure compared to thin cylinders.
  4. Safe Operation: Suitable for applications where failure could lead to catastrophic results.

Limitations of Thick Cylinders

  1. Heavy Weight: Due to large wall thickness, they are bulky and heavy.
  2. Complex Stress Distribution: Requires detailed analysis and design calculations.
  3. Expensive: More material and machining increase manufacturing costs.
  4. Difficult Inspection: It is harder to detect defects or cracks deep within thick walls.

Design Considerations

While designing a thick cylinder, engineers must consider:

  1. Internal and External Pressures: To calculate the correct stresses using Lame’s equations.
  2. Material Strength: The material must have high yield strength and ductility.
  3. Factor of Safety: Sufficient margin must be provided to prevent bursting.
  4. Manufacturing Accuracy: Precision in dimensions ensures uniform stress distribution.
Conclusion

A thick cylinder is a pressure vessel whose wall thickness is large compared to its diameter, generally when . It is designed to carry high internal or external pressures safely. In such cylinders, both radial and hoop stresses vary across the thickness, and their values are determined using Lame’s equations. Thick cylinders are essential in applications like hydraulic presses, gun barrels, and pressure vessels where high pressure and safety are critical. Understanding their stress behavior ensures reliable and safe mechanical design.