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Bore hoop stress in service after autofrettage
Hoop stress at the bore when a service pressure is applied to an autofrettaged cylinder, valid while reloading stays elastic: the residual stress plus the elastic Lamé hoop stress of the service pressure.
- Ref
- MEC-80
- Rev
- v1
- Field
- Mechanical
- Result
- σθ,S(a) [MPa]
If reloading to the service pressure Ps is elastic, the stress in service is the residual stress plus the elastic Lamé stress from Ps. At the bore the hoop stress is σθ,S(a) = σθ,res(a) + Ps(b² + a²)/(b² − a²) and the radial stress is −Ps. The result is the hoop stress at one point; it does not by itself establish the maximum equivalent stress, the fatigue life or the allowable service pressure. Axial stress must be added consistently and the full Tresca measure checked through the wall.
Hoop stress at the bore when a service pressure is applied to an autofrettaged cylinder, valid while reloading stays elastic: the residual stress plus the elastic Lamé hoop stress of the service pressure.
Calculator
- Residual hoop stress at the bore (zero service pressure)−173.2 MPa
- Hoop stress at the bore of an untreated cylinder at the same pressure, Ps(b² + a²)/(b² − a²)166.7 MPa
- Radial–hoop Tresca measure at the bore, σθ − σr93.49 MPa
- Reduction in bore tensile stress by autofrettage173.2 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.
- At 100 MPa service pressure an untreated cylinder would carry 166.7 MPa of hoop tension at the bore. After a 177.4 MPa autofrettage it carries −6.511 MPa: autofrettage has taken 173.2 MPa off the bore tension.
- The radial stress at the bore is −100 MPa, so the radial–hoop Tresca measure there is 93.49 MPa, compared with a yield strength of 300 MPa. This one point does not give the allowable service pressure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Inner radius (bore) | 10 to 150 mm | −5,647 | 43.55 | 5,691 | +10% | +239 % |
| PA Peak autofrettage pressure | 50 to 600 MPa | −1,133 | 333.3 | 1,467 | +10% | +727 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | −206.5 | 1,193 | 1,400 | +10% | −461 % |
| Ps Service pressure | 0 to 300 MPa | −173.2 | 326.8 | 500 | +10% | −256 % |
| b Outer radius | 60 to 300 mm | −306.9 | 40.69 | 347.6 | +10% | −173 % |
Sweep of Service pressure
| Service pressure (MPa) | Bore hoop stress in service (MPa) |
|---|---|
| 0 | −173.2 |
| 37.5 | −110.7 |
| 75 | −48.18 |
| 112.5 | 14.32 |
| 150 | 76.82 |
| 187.5 | 139.3 |
| 225 | 201.8 |
| 262.5 | 264.3 |
| 300 | 326.8 |
Sweep one variable
Variables and how each one contributes
Bore hoop stress in service
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. Acts on both parts: a wider bore deepens the residual compression and also raises the elastic tension from the service pressure. The net effect depends on the two pressures. Raising it by 5 mm (to 55 mm) moves the result by +239 %, from −6.511 to −22.06 MPa.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Acts on both parts: a thicker wall reduces the residual compression and also reduces the elastic tension from the service pressure. The net effect depends on the two pressures. Raising it by 5 mm (to 105 mm) moves the result by −95.3 %, from −6.511 to −0.3038 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. Raises the service hoop stress one for one at a fixed pressure, because the residual stress starts from σY. Raising it by 25 MPa (to 325 MPa) moves the result by −384 %, from −6.511 to 18.49 MPa.
Peak autofrettage pressure
The peak internal pressure applied during autofrettage, in MPa. For a partial treatment it must lie between the first-yield pressure and the fully plastic limit. Lowers the service hoop stress: more autofrettage pressure leaves more compression at the bore. Raising it by 5 MPa (to 182.4 MPa) moves the result by +205 %, from −6.511 to −19.84 MPa.
Service pressure
The internal pressure in service, in MPa, applied after the autofrettage pressure has been removed. Raises the service hoop stress in direct proportion, by (b² + a²)/(b² − a²) for each MPa. Raising it by 10 MPa (to 110 MPa) moves the result by −256 %, from −6.511 to 10.16 MPa.
Worked example with the values above
- Residual hoop stress at the bore: σY − 2PAb²/(b² − a²) = −173.2 MPa.
- Elastic increment from the service pressure: Ps(b² + a²)/(b² − a²) = 100 × 12,500 / 7,500 = 166.7 MPa.
- σθ,S(a) = −173.2 + 166.7 = −6.511 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.
- Elastic reloading to the service pressure; the superposition fails if the service pressure reopens yielding.
Common mistakes
- Using this single point as the allowable service pressure or fatigue life.
- Forgetting to add the axial stress when the full Tresca measure is checked.
- Using these cylinder formulas for a sphere, a non-uniform wall, nozzles, heads, welded or layered construction, significant axial restraint or large deformation. Those need an elastoplastic finite-element model of the real geometry and load path.
Statistics and uncertainty
The service stress is the sum of a residual stress (uncertain, see the residual-stress entries) and an elastic stress that is known accurately from the pressure and geometry. The uncertainty is therefore mostly in the residual part, which is why a treated vessel is validated by measurement. The percentage changes quoted for the inputs look large because the result sits close to zero, so a small shift in MPa is a large fraction of it.
Where it is used
- Showing how much autofrettage lowers the tensile hoop stress at the bore under working pressure.
- Checking that the bore stays well inside yield in service.
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