Solve Lame stresses for thick cylinders under pressure. Evaluate hoop, radial, and axial behavior accurately. Use exports, graphs, and examples for faster engineering decisions.
Lamé constants
A = (pi ri2 - po ro2) / (ro2 - ri2)
B = ri2 ro2 (pi - po) / (ro2 - ri2)
σr(r) = A - B / r2
σθ(r) = A + B / r2
σz = A for closed ends, and 0 for open ends.
σvm = √{[(σθ-σr)² + (σr-σz)² + (σz-σθ)²] / 2}
This calculator applies Lamé theory for thick-walled cylinders under axisymmetric pressure loading. Internal and external pressures are entered as positive magnitudes, while radial stress is reported with compressive sign at loaded boundaries.
The hoop stress is usually largest at the inner wall during internal pressurization, which is why bore regions often govern design checks.
| Case | ri | ro | pi | po | r | Ends | σr(r) | σθ(r) |
|---|---|---|---|---|---|---|---|---|
| Case A | 50 mm | 100 mm | 12 MPa | 2 MPa | 75 mm | Closed | -4.5926 MPa | 7.2593 MPa |
| Case B | 40 mm | 90 mm | 15 MPa | 5 MPa | 60 mm | Open | -8.0769 MPa | 3.0000 MPa |
It calculates radial and hoop stresses in thick-walled cylinders subjected to internal and external pressure. This page also reports axial stress for closed ends, principal stresses, maximum shear stress, and von Mises equivalent stress at the selected radius.
Radial stress acts inward on a pressurized inner surface, so it is compressive by sign convention. For internal pressure pi, the radial stress at the bore becomes approximately -pi.
For cylinders under internal pressure, the hoop stress is typically highest at the inner radius. It decreases toward the outer wall, which makes the bore region critical for many strength checks.
Use closed ends when pressure loads are resisted by end caps or vessel heads, creating axial stress. Use open ends when the cylinder does not carry net axial pressure load through its wall.
No conversion is required if your values are already consistent. Enter both radii in one length unit and all pressures in one pressure unit. Output stresses stay in that pressure unit.
Thin-wall formulas assume nearly uniform hoop stress through thickness. Thick-wall analysis is needed when wall thickness is significant relative to radius, because stress varies strongly from inner to outer surface.
Von Mises stress combines the principal stress state into one equivalent value for ductile design checks. It helps compare multiaxial stresses against a yield strength or allowable limit.
The graph shows how radial, hoop, and axial stresses vary from inner radius to outer radius. It helps you locate peak tensile regions, pressure boundary effects, and thickness-driven stress gradients quickly.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.