Lame's Equation for Thick Pressure Vessels: Calculator & Formulas

In mechanical and civil engineering, not all cylinders are created equal. When the wall thickness of a cylinder is more than 1/20th of its diameter, it is classified as a "Thick Cylinder." For these components—like hydraulic rams, high-pressure pipes, and gun barrels—simple thin-shell formulas fail. This is where Lame's Equation becomes the industry standard for safety and precision.

As a Design Engineer, I often work with high-pressure systems where thin-cylinder formulas simply don't apply. When the wall thickness is significant, we must use Lame's Equation to ensure safety. Below is the original mathematical representation and a guide on how to solve it using my online scientific calculator.

What is Lame's Equation?

If you are in the field of engineering or if you are unfamiliar with it you should learn about Lame's Equation. Lame’s Equation is a crucial formula used to determine the stresses acting within thick cylinders or pressure vessels. When a pipe or cylinder possesses significant wall thickness, the stress distribution across its inner and outer sections is non-uniform—the stress is highest at the inner surface and progressively decreases as one moves outward. In such scenarios, the standard formulas applicable to thin cylinders are inadequate; therefore, Lame’s Equation is employed. This equation enables us to calculate the radial stress (σᵣ) and hoop stress (σθ) at any given radius (r) within the cylinder, thereby facilitating safe design practices—particularly for high-pressure equipment such as boilers, valve bodies, and pressure vessels.

Lame’s theory assumes that the material is homogeneous and that the longitudinal stress is constant. It helps us find two critical types of stresses:

Hoop Stress (σh): The tangential stress acting along the circumference.
Radial Stress (σr): The stress acting towards or away from the center.

Lame's Equation

For a thick cylinder subjected to internal and external pressure, the stresses at any radius r are given by:

Radial Stress (σ_r) = (b / r²) - a

Hoop Stress (σ_h) = (b / r²) + a

Where a and b are Lame's constants. In most engineering problems where external pressure is zero, these constants are calculated as:

b = (P × r₁² × r₂²) / (r₂² - r₁²)

a = b / r₂²

Step-by-Step Solving on Our Calculator

Lame's equations involve multiple squares and divisions. Follow this button sequence on our tool for an internal pressure P = 50 MPa, internal radius r₁ = 100mm, and external radius r₂ = 150mm.

1. Calculating Constant 'b'

Formula: b = (50 × 100² × 150²) / (150² - 100²)

Press these buttons:

👉 Press (
👉 Type 50 × 100 x² × 150 x² then press )
👉 Press ÷ then press (
👉 Type 150 x² - 100 x²
👉 Press ) then =

Your screen should show: 900000

2. Calculating Maximum Hoop Stress (at r = 100)

Formula: σ_h = (b / 100²) + a

Press these buttons:

👉 Press ( 900000 ÷ 100 x² )
👉 Press + 40 then =

Result: 130 MPa

Pro Tips for Engineering Students

Use the 'x²' Button: Don't type 150 × 150. Using the orange x² button on my calculator ensures the logic stays clean.
Don't Skip Brackets: In Lame's formula, the denominator subtraction must happen before division. Always use the ( ) keys.
Unit Consistency: Ensure your pressure is in N/mm² (MPa) and dimensions are in mm for consistent results.

Conclusion

Lame’s equation is a vital part of Strength of Materials (SOM). By using the original formulas and following the structured button sequence on our scientific calculator, you can ensure that your structural designs are safe and professional.

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