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Mechanical
Residual hoop stress at radius r after autofrettage
Hoop stress left in the wall after the autofrettage pressure is removed, found by subtracting the elastic Lamé response to that pressure from the peak stress.
- Ref
- MEC-74
- Rev
- v1
- Field
- Mechanical
- Result
- σθ,res [MPa]
After the pressure is removed the wall unloads elastically. The residual stress is the peak stress minus the elastic Lamé response of the whole cylinder to the same pressure, σθ,res = σθ,A − [PAa²/(b² − a²)](1 + b²/r²). The peak term is the yielded-zone or elastic-ring expression of the previous entry, chosen by where r lies relative to c. The result is valid only if unloading stays elastic everywhere; check the reverse-yielding screening ratio and the full Tresca measure, because the Bauschinger effect can cause reverse yielding earlier and reduce the retained compression. At the bore the formula reduces to σY(1 − PA/PY).
Hoop stress left in the wall after the autofrettage pressure is removed, found by subtracting the elastic Lamé response to that pressure from the peak stress. It is compressive near the bore, which is what delays fatigue cracking there.
Calculator
- Peak hoop stress at this radius, σθ,A177.3 MPa
- Elastic unloading term, PAa²/(b² − a²) × (1 + b²/r²)223.4 MPa
- Residual hoop stress at the bore−173.2 MPa
- Residual hoop stress at the outside surface28.71 MPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After a 177.4 MPa treatment the hoop stress at the bore (50 mm) is −173.2 MPa, compressive, while the outside surface carries 28.71 MPa in tension. The two balance through the wall.
- At the radius chosen here (60 mm) the residual hoop stress is −46.19 MPa. Compare it with the untreated case, where the stress at zero pressure is zero everywhere.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Outer radius | 60 to 300 mm | −415.8 | 20.25 | 436 | +10% | +55.6 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | −231 | −15.4 | 215.6 | +10% | +10 % |
| a Inner radius (bore) | 10 to 150 mm | −95.1 | 97.92 | 193 | +10% | +79.4 % |
| f Target plastic fraction of the wall | 0.05 to 0.95 | −114.7 | 14.23 | 128.9 | +10% | +20.5 % |
| r Radius where the stress is wanted | 10 to 300 mm | −64.16 | 38.15 | 102.3 | +10% | −124 % |
Sweep of Radius where the stress is wanted
| Radius where the stress is wanted (mm) | Residual hoop stress at radius r (MPa) |
|---|---|
| 10 | — |
| 46.25 | — |
| 82.5 | 35.44 |
| 118.8 | — |
| 155 | — |
| 191.3 | — |
| 227.5 | — |
| 263.8 | — |
| 300 | — |
Sweep one variable
Variables and how each one contributes
Residual hoop stress at radius r
What the equation gives you. Everything else feeds it.
Inner radius (bore)
The inner (bore) radius of the cylinder, in mm, in the same length unit as b. It must be smaller than b. Changes the residual stress through the pressure, the front and the Lamé response; the sign of the effect depends on r. Raising it by 5 mm (to 55 mm) moves the result by +79.4 %, from −46.19 to −82.85 MPa.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Changes the residual stress through the pressure, the front and the Lamé response; the sign of the effect depends on r. Raising it by 5 mm (to 105 mm) moves the result by +28.1 %, from −46.19 to −59.17 MPa.
Uniaxial yield strength
The uniaxial yield strength of the material at the autofrettage temperature, in MPa. The model treats the material as elastic and perfectly plastic and uses the Tresca criterion, so the wall yields where the hoop stress minus the radial stress reaches this value. Scales every term, so the residual stress changes in direct proportion at a fixed fraction f. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from −46.19 to −50.04 MPa.
Target plastic fraction of the wall
The fraction of the wall thickness (b − a) that is yielded at peak pressure, a plain number between 0 and 1. A value of 0 means yielding only just begins and 1 means the whole wall has yielded. The slider runs from 0.05 to 0.95. Raises the pressure and so deepens the compression near the bore, while pushing the tension outward in the wall. Raising it by 0.05 (to 0.45) moves the result by +25.4 %, from −46.19 to −57.9 MPa.
Radius where the stress is wanted
The radius at which the stress is wanted, in mm, anywhere in the wall from the bore (a) to the outside (b). Moves the point through the wall: the hoop stress is most compressive at the bore and rises to tension towards the outside. Raising it by 5 mm (to 65 mm) moves the result by −105 %, from −46.19 to 2.127 MPa.
Worked example with the values above
- Peak hoop stress at r = 60 mm (yielded zone, interface at 70 mm): σθ,A = 177.3 MPa.
- Elastic unloading term: PAa²/(b² − a²) = 59.15 MPa, multiplied by 1 + b²/r² = 3.778, gives 223.4 MPa.
- σθ,res = 177.3 − 223.4 = −46.19 MPa.
Assumptions and limits
- A long, uniform, single-piece thick cylinder with uniform internal pressure, no external pressure, small deformation and an initially stress-free isotropic material.
- Elastic, perfectly plastic behaviour with the Tresca criterion at constant temperature, and a hoop stress that is the largest principal stress (σθ ≥ σz ≥ σr). The axial stress must be checked separately: open ends, closed ends and axial restraint give different states.
- End regions, openings and attachments are ignored. This is a teaching and preliminary-analysis model, not a qualified manufacturing specification.
- Unloading is purely elastic. If reverse yielding occurs this formula overstates the compression.
Common mistakes
- Using the result without checking for reverse yielding on unloading.
- Forgetting to subtract the Lamé response: the peak stress alone is not the residual stress.
- Trusting the closed-form residual stress when the vessel is not a plain long cylinder, or when unloading may yield. Use an elastoplastic finite-element model instead, with these points in mind: model the actual geometry (axisymmetric where geometry and loading allow, otherwise 3D); do not impose plane strain merely because the cylinder is long, and represent the closed-end pressure thrust and the supports explicitly; use a plasticity law backed by material data, with kinematic or combined hardening where load reversal matters; apply the load in steps, 0 → trial PA → 0 → Ps, keeping the plastic history between steps; refine the mesh and pressure increments until the plastic front, residual stress and permanent deformation stop changing; and check signed hoop, radial and axial stresses as well as equivalent stress and plastic strain. A conventional von Mises model will not reproduce this Tresca result exactly, and no universal 2/√3 factor converts one into the other: that factor holds only when the axial stress sits midway between the radial and hoop stresses. Validate the stress against measurement before specifying a production treatment.
Statistics and uncertainty
The residual stress is the difference of two large terms, so small errors are amplified: a 1 % error in the pressure (1.77 MPa) shifts the residual hoop stress at the bore by 4.73 MPa, which is 1.58 % of the yield strength. This is why residual stress is verified by measurement, for example by diffraction or bore-slitting methods, and not assumed from the formula alone.
Where it is used
- Showing the compression at the bore that delays fatigue cracking in a pressure vessel or gun barrel.
- Checking the stress balance: the residual hoop stress must integrate to the equilibrium of the wall.
Maths you need
- Thick-walled cylinder: a wall too thick for the thin-wall formula
- Tresca criterion: yield when the largest stress difference reaches σY
- Superposition: adding elastic responses to a known state