Formula used
For a thick-walled cylinder under axisymmetric loading, the Lame stress solution gives radial and hoop stresses as functions of radius:
- Radial stress: σr(r) = A − B / r²
- Hoop stress: σθ(r) = A + B / r²
The constants are determined from boundary conditions σr(ri) = −Pi and σr(ro) = −Po:
- A = (Pi ri² − Po ro²) / (ro² − ri²)
- B = (ri² ro² (Pi − Po)) / (ro² − ri²)
If ends are closed, the uniform axial stress is σz = A. Von Mises stress is computed from σθ, σr, and σz.
How to use this calculator
- Enter inner radius ri and outer radius ro.
- Choose a length unit that matches your radii.
- Enter internal pressure Pi and external pressure Po.
- Set an evaluation radius within the wall thickness.
- Select end condition and output unit, then calculate.
- Use the profile table to inspect stress variation.
- Download a CSV or PDF report if needed.
Example data table
| # | ri (mm) | ro (mm) | Pi (MPa) | Po (MPa) | Max hoop at ri (MPa) |
|---|---|---|---|---|---|
| 1 | 50 | 100 | 10 | 0 | 16.67 |
| 2 | 30 | 60 | 8 | 0 | 13.33 |
| 3 | 40 | 90 | 12 | 2 | 18.94 |
| 4 | 25 | 75 | 6 | 1 | 8.60 |
| 5 | 60 | 120 | 15 | 0 | 25.00 |
Values are illustrative. Use the calculator for exact results.
Professional article
1) Why Lame stresses matter
Thick cylinders behave differently from thin shells because stress varies strongly through the wall. Lame equations capture that variation using radius-dependent terms. This calculator reports radial stress, hoop stress, axial stress, and Von Mises stress, letting you judge where yielding is most likely during pressure loading.
2) Key assumptions and scope
The model assumes linear elasticity, axisymmetric loading, and a long cylinder where end effects are negligible. Pressures act uniformly at the inner and outer surfaces. Temperature gradients, residual stresses, and plasticity are not included. For rapid transients, confirm whether dynamic effects change the effective pressure history.
3) Boundary conditions with real data
Radial stress equals the negative of applied pressure at each boundary: σr(ri)=−Pi and σr(ro)=−Po. This is a strong validation point. If you input Pi=10 MPa and Po=0 MPa, the calculator will show σr at the inner wall near −10 MPa and at the outer wall near 0 MPa.
4) Where the maximum hoop stress occurs
Hoop stress typically reaches its maximum magnitude at the inner radius when internal pressure dominates. For common radius ratios ro/ri between 1.5 and 3.0, the inner-wall hoop stress can be 1.2× to 2.5× higher than the mean stress. The “critical checks” panel reports this maximum directly.
5) Closed versus open ends
End condition changes axial stress. With closed ends, the pressure load on the end caps produces a uniform axial stress term (σz). With open ends, σz is set to zero. Many pressure vessel components are effectively closed-ended in service, so evaluating both cases helps bracket real operating conditions.
6) Using Von Mises for yielding
Design often compares Von Mises stress to material yield strength. For example, if a steel yield strength is 250 MPa, a computed Von Mises of 140 MPa suggests elastic margin, while 260 MPa indicates likely yielding. Always incorporate codes, temperature derating, and safety factors, not just a single criterion.
7) Interpreting the stress profile table
The profile table samples stresses from ri to ro. Radial stress increases toward the outer surface, while hoop stress decreases with radius for internal pressure cases. Increasing profile points improves resolution. Use the table to identify stress gradients near the bore, where cracking, corrosion, and fatigue often initiate.
8) Practical workflow for engineers
Start with geometry and pressure limits, then compute peak hoop and Von Mises stresses. Compare to allowable stress from your selected standard, and iterate on wall thickness. If external pressure is significant, include Po. Finally, export CSV/PDF for documentation and peer review. This structured workflow supports traceable design decisions.
FAQs
1) What is the difference between hoop and radial stress?
Hoop stress acts tangentially around the cylinder circumference, while radial stress acts through the wall thickness. Hoop stress usually governs yielding under internal pressure because it is larger near the inner wall.
2) Why is radial stress negative at the inner surface?
By convention, compressive stress is negative. Internal pressure pushes inward on the wall surface, creating compressive radial stress equal to −Pi at r=ri. The boundary condition confirms correct setup.
3) When should I select “closed ends”?
Select closed ends when end caps or closures transmit pressure thrust into the cylinder, producing axial stress. Many pipes with sealed ends, pressure vessels, and hydraulic cylinders behave this way in operation.
4) Can I use this for thin-walled cylinders?
You can, but thin-wall formulas may be simpler. As a rule of thumb, if thickness is less than about 10% of inner radius, thin-wall estimates become reasonable, though Lame remains accurate.
5) Does the calculator include stress concentrations?
No. It assumes a smooth, uniform cylinder without holes, grooves, threads, or notches. For features like ports or keyways, apply appropriate stress concentration factors or use finite element analysis.
6) Why does hoop stress reduce toward the outer radius?
In the Lame solution, the hoop term includes +B/r², which decreases as radius increases. This produces the common pattern of highest hoop stress at the bore and lower hoop stress near the outer wall.
7) What input checks help avoid invalid results?
Ensure ro>ri, evaluation radius lies within the wall, and pressures are non-negative. Also confirm consistent units. The calculator enforces these limits and will show errors instead of producing misleading outputs.