Calculator Input
Example Data Table
Example case: inner pressure 120 MPa, outer pressure 15 MPa, inner radius 50 mm, outer radius 100 mm, closed ends.
| Radius (mm) | σr (MPa) | σθ (MPa) | σz (MPa) | von Mises (MPa) |
|---|---|---|---|---|
| 50.000 | -120.000 | 160.000 | 20.000 | 242.487 |
| 62.500 | -69.600 | 109.600 | 20.000 | 155.192 |
| 75.000 | -42.222 | 82.222 | 20.000 | 107.772 |
| 87.500 | -25.714 | 65.714 | 20.000 | 79.179 |
| 100.000 | -15.000 | 55.000 | 20.000 | 60.622 |
Formula Used
This calculator applies Lamé equations for a thick cylinder under internal and external pressure.
Radial stress:
σr = A − B / r2
Hoop stress:
σθ = A + B / r2
Lamé constants:
A = (Pi ri2 − Po ro2) / (ro2 − ri2)
B = ri2 ro2 (Pi − Po) / (ro2 − ri2)
Axial stress for closed ends:
σz = (Pi ri2 − Po ro2) / (ro2 − ri2)
Maximum shear stress:
τmax = |σθ − σr| / 2
von Mises stress:
√{[(σθ − σz)2 + (σz − σr)2 + (σr − σθ)2] / 2}
Compression is shown as negative radial stress near pressure boundaries. Use consistent units throughout the calculation.
How to Use This Calculator
- Enter inner and outer pressure using the same pressure unit.
- Enter inner and outer radius using the same length unit.
- Choose a radius location where you want the local stress result.
- Select open or closed ends based on your cylinder condition.
- Set graph points to control the stress distribution resolution.
- Optionally enter allowable stress for a quick design comparison.
- Click the calculate button to view results, graph, and distribution table.
- Use the CSV or PDF buttons to export the calculated output.
Frequently Asked Questions
1. What makes a cylinder thick?
A thick cylinder has a wall large enough that stress changes noticeably across the thickness. Thin-wall formulas no longer describe the radial and hoop stress distribution accurately.
2. Why is radial stress often negative?
Radial stress represents pressure acting inward on the wall. Under the usual sign convention, compressive stress is negative, so the radial stress at the pressure boundary appears negative.
3. Where is hoop stress highest?
For a cylinder under internal pressure, hoop stress is usually greatest at the inner surface. It decreases toward the outer surface as the radius increases.
4. When is axial stress included?
Axial stress is included for closed-end cylinders because pressure loads the end caps. For open ends, the idealized axial stress from pressure is taken as zero.
5. Can I use any units?
Yes. Use any pressure and length units you want, but stay consistent. The calculator does not convert units automatically between mixed systems.
6. What does von Mises stress tell me?
von Mises stress combines the principal stresses into one equivalent value. It is commonly used for ductile material yielding checks and quick design screening.
7. Can this handle external pressure too?
Yes. You can enter both internal and external pressure. The equations account for either case, provided the geometry and units are entered consistently.
8. Is the graph exact or approximate?
The graph plots exact equation values at selected radius points. Increasing the graph point count makes the displayed curve smoother and the table more detailed.