Thick-Walled Cylinder Stress Calculator

Find hoop and radial stress in a thick-walled cylinder under internal and external pressure using Lame's equations, at both the inner and outer wall.

🔘 Thick-Walled Cylinder Stress Calculator
MPa
MPa
mm
mm
Hoop stress, inner wall (max)
—
Hoop stress, outer wall
—
Radial stress, inner wall
—
Step-by-step working

🔘 What is Thick-Walled Cylinder Stress?

Thick-walled cylinder stress is the exact elastic stress distribution inside a cylinder wall that is too thick, relative to its diameter, for the simpler thin-wall membrane formula to remain accurate. Lame's equations give the exact solution, hoop stress sigma_theta(r) = A + B/r^2 and radial stress sigma_r(r) = A - B/r^2, where A and B are two constants found from the internal pressure, external pressure, and the inner and outer radii.

Engineers reach for Lame's equations for gun barrels, hydraulic and pneumatic cylinders, high-pressure chemical reactor walls, and thick pipe fittings, anywhere the diameter-to-thickness ratio drops below about 20, the point where the thin-wall assumption stops being reliable. Unlike the thin-wall formula, which treats stress as uniform through the wall, Lame's solution shows stress actually varies continuously between the inner and outer surface.

A key result from this solution is that hoop stress is always highest at the inner wall and lowest at the outer wall for a cylinder under internal pressure, this inner-wall value is what is normally compared against the material's allowable stress. Radial stress runs in the opposite pattern, exactly matching the applied pressure at each free surface (-Pi at the inner wall, -Po at the outer wall), which serves as a useful built-in check on the formula itself.

This calculator solves for both Lame constants, reports hoop and radial stress at the critical inner and outer surfaces, and plots the full stress distribution across the wall, so you can see exactly how stress decays from the bore outward.

📐 Formula

σθ(r) = A + B/r²      σr(r) = A − B/r²
A = (Piri² − Poro²) / (ro² − ri²)
B = (Pi − Po) ri² ro² / (ro² − ri²)
Pi = internal pressure (MPa), Po = external pressure (MPa)
ri = internal radius (mm), ro = external radius (mm)
Example: Pi = 50 MPa, Po = 0, ri = 50 mm, ro = 100 mm → σθ(inner) ≈ 83.33 MPa, σθ(outer) ≈ 33.33 MPa.

📖 How to Use This Calculator

Steps

1
Enter the internal pressure. Type Pi, the internal pressure acting on the bore, in megapascals.
2
Enter the external pressure. Type Po, the external pressure acting on the outside surface, in megapascals (use 0 if none).
3
Enter the internal radius. Type ri, the internal (bore) radius, in millimeters.
4
Enter the external radius. Type ro, the external (outside) radius, in millimeters.

💡 Example Calculations

Example 1 — Hydraulic Cylinder Under Internal Pressure

A cylinder with Pi = 50 MPa, Po = 0, ri = 50 mm, ro = 100 mm

1
A = (50 × 50² − 0) / (100² − 50²) = 16.67 MPa
2
B = 50 × 50² × 100² / (100² − 50²) = 1,666,667 MPa·mm²
3
σθ(inner) = 16.67 + 1,666,667/2,500 = 83.33 MPa
σθ(inner) = 83.33 MPa
Try this example →

Example 2 — Thick-Wall Gas Cylinder

A cylinder with Pi = 30 MPa, Po = 0, ri = 80 mm, ro = 120 mm

1
A = (30 × 80² − 0) / (120² − 80²) = 24.00 MPa
2
B = 30 × 80² × 120² / (120² − 80²) = 3,456,000 MPa·mm²
3
σθ(inner) = 24.00 + 3,456,000/6,400 = 78.00 MPa
σθ(inner) = 78.00 MPa
Try this example →

Example 3 — Submerged Cylinder With External Pressure

A cylinder with Pi = 20 MPa, Po = 5 MPa, ri = 60 mm, ro = 100 mm

1
A = (20 × 60² − 5 × 100²) / (100² − 60²) = 6.88 MPa
2
B = (20 − 5) × 60² × 100² / (100² − 60²) = 843,750 MPa·mm²
3
σθ(inner) = 6.88 + 843,750/3,600 = 26.88 MPa
σθ(inner) = 26.88 MPa
Try this example →

❓ Frequently Asked Questions

What are Lame's equations?+
Lame's equations are the exact elastic stress solution for a thick-walled cylinder under internal and/or external pressure, giving hoop stress sigma_theta(r) = A + B/r^2 and radial stress sigma_r(r) = A - B/r^2 at any radius r between the inner and outer wall.
What are the Lame constants A and B?+
A = (Pi ri^2 - Po ro^2) / (ro^2 - ri^2) and B = (Pi - Po) ri^2 ro^2 / (ro^2 - ri^2), where Pi and Po are internal and external pressure and ri and ro are the internal and external radii. Both hoop and radial stress at any radius follow directly once A and B are known.
Where is hoop stress highest in a thick-walled cylinder?+
For a cylinder under internal pressure, hoop stress is always highest at the inner wall (r = ri) and lowest at the outer wall (r = ro). This inner-wall value is the one normally checked against the material's allowable stress.
Why does radial stress equal negative Pi at the inner wall?+
Radial stress at any free surface must equal the negative of the pressure acting on it, by definition of the boundary condition. At the inner wall the internal pressure Pi pushes inward on the material, giving sigma_r(ri) = -Pi, at the outer wall sigma_r(ro) = -Po.
When should I use thick-wall (Lame) equations instead of the thin-wall formula?+
Use Lame's equations whenever the diameter-to-thickness ratio D/t is below about 20, common design rule of thumb. Below that ratio, stress varies enough through the wall that the simpler thin-wall formula PD/2t noticeably underestimates the true peak stress at the inner surface.
Can this calculator handle a cylinder under external pressure only?+
Yes, set the internal pressure Pi to 0 and enter the external pressure Po. The same Lame equations apply, though a cylinder under external pressure alone should also be checked separately for buckling, which this formula does not cover.
What happens to hoop stress as the wall gets very thick?+
As the outer radius grows much larger than the inner radius, the inner-wall hoop stress approaches Pi for internal pressure alone, its theoretical minimum floor, no amount of added wall thickness can reduce hoop stress below the applied internal pressure itself.
Does this formula account for yielding or failure criteria?+
No, this calculator returns the elastic stress state only. Checking against material failure requires comparing the computed hoop, radial, and any axial stress using an appropriate yield criterion, such as Tresca (maximum shear stress) or von Mises, against the material's yield strength.
What units does this calculator use?+
Pressure (internal and external) is entered in megapascals, radii in millimeters, and both hoop and radial stress results are shown in megapascals.
How is this different from the thin-wall hoop stress calculator?+
The thin-wall formula sigma = PD/2t assumes uniform stress through the wall and is only accurate when D/t is 20 or greater. This calculator uses the exact Lame solution, valid at any D/t ratio, and reports how stress actually varies between the inner and outer wall rather than a single averaged value.
Tags: Thick Walled Cylinder Calculator Lame Equations Calculator Hoop Stress Thick Wall Calculator Radial Stress Calculator Cylinder Stress Distribution Calculator Formula Free Online

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What are Lame's equations?

Lame's equations are the exact elastic stress solution for a thick-walled cylinder under internal and/or external pressure, giving hoop stress sigma_theta(r) = A + B/r^2 and radial stress sigma_r(r) = A - B/r^2 at any radius r between the inner and outer wall.

What are the Lame constants A and B?

A = (Pi ri^2 - Po ro^2) / (ro^2 - ri^2) and B = (Pi - Po) ri^2 ro^2 / (ro^2 - ri^2), where Pi and Po are internal and external pressure and ri and ro are the internal and external radii. Both hoop and radial stress at any radius follow directly once A and B are known.

Where is hoop stress highest in a thick-walled cylinder?

For a cylinder under internal pressure, hoop stress is always highest at the inner wall (r = ri) and lowest at the outer wall (r = ro). This inner-wall value is the one normally checked against the material's allowable stress.

Why does radial stress equal negative Pi at the inner wall?

Radial stress at any free surface must equal the negative of the pressure acting on it, by definition of the boundary condition. At the inner wall the internal pressure Pi pushes inward on the material, giving sigma_r(ri) = -Pi, at the outer wall sigma_r(ro) = -Po.

When should I use thick-wall (Lame) equations instead of the thin-wall formula?

Use Lame's equations whenever the diameter-to-thickness ratio D/t is below about 20, common design rule of thumb. Below that ratio, stress varies enough through the wall that the simpler thin-wall formula PD/2t noticeably underestimates the true peak stress at the inner surface.

Can this calculator handle a cylinder under external pressure only?

Yes, set the internal pressure Pi to 0 and enter the external pressure Po. The same Lame equations apply, though a cylinder under external pressure alone should also be checked separately for buckling, which this formula does not cover.

What happens to hoop stress as the wall gets very thick?

As the outer radius grows much larger than the inner radius, the inner-wall hoop stress approaches Pi for internal pressure alone, its theoretical minimum floor, no amount of added wall thickness can reduce hoop stress below the applied internal pressure itself.

Does this formula account for yielding or failure criteria?

No, this calculator returns the elastic stress state only. Checking against material failure requires comparing the computed hoop, radial, and any axial stress using an appropriate yield criterion, such as Tresca (maximum shear stress) or von Mises, against the material's yield strength.

What units does this calculator use?

Pressure (internal and external) is entered in megapascals, radii in millimeters, and both hoop and radial stress results are shown in megapascals.

How is this different from the thin-wall hoop stress calculator?

The thin-wall formula sigma = PD/2t assumes uniform stress through the wall and is only accurate when D/t is 20 or greater. This calculator uses the exact Lame solution, valid at any D/t ratio, and reports how stress actually varies between the inner and outer wall rather than a single averaged value.

📌 Quick Tips

💡Use this calculator instead of the thin-wall formula whenever the diameter-to-thickness ratio is below about 20, gun barrels, hydraulic cylinders, and high-pressure fittings are common thick-wall cases.
💡Hoop stress is always highest at the inner wall and lowest at the outer wall for a cylinder under internal pressure alone, radial stress runs the opposite way, from -Pi at the inner wall to -Po at the outer wall.
💡Set the external pressure to 0 for the common case of a vessel exposed only to atmospheric pressure outside, or a positive value for a submerged or externally pressurized cylinder.
💡The two Lame constants A and B fully describe the stress state at any radius between the inner and outer wall, once found, both hoop and radial stress at any point follow directly.
💡As the wall gets thin relative to the diameter, this calculator's inner-wall hoop stress result converges toward the simpler thin-wall formula PD/2t, this is a useful way to sanity-check both formulas against each other.