Define thick cylinder equation (Lame’s theorem).

Short Answer:

The thick cylinder equation, also known as Lame’s theorem, is used to determine the distribution of stresses (radial and hoop stresses) in the wall of a thick-walled cylinder subjected to internal or external pressure. Unlike a thin cylinder, where stress is assumed uniform, a thick cylinder experiences a variation of stress across its wall thickness.

According to Lame’s theorem, the general equations for radial and hoop (circumferential) stresses at any radius  within the wall of a thick cylinder are:

where  and  are constants determined by the internal and external boundary conditions.

Detailed Explanation:

Thick Cylinder Equation (Lame’s Theorem)

When a hollow cylinder or spherical shell is subjected to high internal or external pressure, the wall thickness cannot be neglected compared to its diameter. In such cases, the stresses within the wall are not uniform and vary along the radial direction.
This condition is analyzed using Lame’s theorem, developed by the French mathematician Gabriel Lame.

The theorem provides the mathematical relations for radial stress () and hoop or circumferential stress () at any point within the wall of a thick cylinder. These stresses vary from maximum at the inner surface to minimum at the outer surface.

Assumptions of Lame’s Theorem

  1. The cylinder is thick-walled, and the ratio of inner radius to outer radius is not negligible.
  2. The material of the cylinder is homogeneous, isotropic, and linearly elastic.
  3. The internal and external pressures are uniformly distributed.
  4. The cylinder is subjected only to internal and/or external pressure (no axial loading).
  5. The stress state is axisymmetric — it is the same at every point around the circumference.

Derivation of Lame’s Equations

Consider a thick cylindrical shell having:

  • Internal radius =
  • External radius =
  • Internal pressure =
  • External pressure =

Let  be the radius of an element inside the cylinder wall, and  be the small thickness of that element.

Step 1: Equilibrium of a small element

Consider a small cylindrical element of radius  and thickness .
The radial stress acting on the inner face is , and on the outer face is .

The hoop stress  acts tangentially around the circumference, producing circumferential strain but no net force in the radial direction.

For equilibrium in the radial direction,

Simplifying this expression,

Step 2: Relation between Hoop and Radial Stress

Rearranging,

This is a differential relation between hoop and radial stresses.

Step 3: General Solution

Differentiating the equation again and simplifying gives the general second-order differential equation for radial stress:

The solution of this equation is:

where  and  are constants.

Substituting this value of  in the earlier relation gives:

Hence, these are the Lame’s equations for the stresses in a thick cylinder:

Step 4: Determination of Constants (A and B)

The constants  and  are found using the boundary conditions at the inner and outer surfaces of the cylinder.

At the inner surface ():

At the outer surface ():

Substitute these conditions in the general equation of :

From :

Subtracting the second equation from the first:

 

Now, substitute  in one of the equations to get :

Thus,

Substitute these values into the general equations to find the stresses at any radius .

Final Form of Lame’s Equations

  1. Radial stress:
  1. Hoop stress:

Important Observations

  1. The hoop stress is always tensile and maximum at the inner surface of the cylinder.
  2. The radial stress is compressive, equal to the internal pressure at the inner surface, and decreases toward the outer surface.
  3. The stress distribution is non-linear, unlike in thin cylinders.
  4. For very thin walls (), the Lame’s equations simplify to thin-cylinder formulas.

Applications of Lame’s Theorem

  1. Design of high-pressure pipes and pressure vessels.
  2. Calculation of stresses in gun barrels, boiler drums, and hydraulic cylinders.
  3. Determination of bursting pressure and safe wall thickness.
  4. Analysis of internal pressure in steam or gas containers.
  5. Used in autoclaves and nuclear reactor pressure vessels.
Conclusion

The thick cylinder equation (Lame’s theorem) gives the relationship between radial and hoop stresses in a thick-walled cylinder under internal or external pressure. The stresses are not uniform but vary across the thickness. The equations,

accurately describe this variation. By applying appropriate boundary conditions, the constants  and  are determined, enabling engineers to design safe and efficient high-pressure components like pipes, cylinders, and pressure vessels.