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Peak hoop stress at radius r during autofrettage

Hoop stress through the wall at the moment of peak autofrettage pressure: a logarithmic profile in the yielded zone, a Lamé profile in the elastic ring outside it.

Ref: MEC-73
Rev: v1
Field: Mechanical
Result: σ_θ,A [MPa]

σ_{θ,A} (r) = σ_{Y} (1 + \ln \frac{r}{a}) - P_{A} (r \leq c) σ_{θ,A} (r) = \frac{σ_{Y} c^{2}}{2 b^{2}} (1 + \frac{b^{2}}{r^{2}}) (r > c)

σ_θ,A(r) is the hoop stress at the moment of peak autofrettage pressure. The yielded zone (a ≤ r ≤ c) and the elastic ring (c < r ≤ b) are treated separately. In the yielded zone, radial equilibrium with σ_θ − σ_r = σ_Y gives σ_r = −P_A + σ_Y ln(r/a) and σ_θ = σ_r + σ_Y. In the ring the stresses are Lamé stresses with the constant σ_Yc²/(2b²). The pressure P_A is found from the target fraction f as in the autofrettage-pressure entry. The solution assumes the hoop stress is the largest principal stress, so the axial condition must be checked separately.

Hoop stress through the wall at the moment of peak autofrettage pressure: a logarithmic profile in the yielded zone, a Lamé profile in the elastic ring outside it. The two meet at the interface radius.

Calculator

Solve for

σ_θ,APeak hoop stress at radius rComputed

MPa

aInner radius (bore)

10150

bOuter radius

60300

σ_YUniaxial yield strength

MPa

1001,500

fTarget plastic fraction of the wall

0.050.95

rRadius where the stress is wanted

10300

Interface radius c70 mm
Hoop stress at the interface radius c223.5 MPa
Peak hoop stress at the bore, σ_Y − P_A122.6 MPa
Peak radial stress at this radius−122.7 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.

Peak hoop stress at radius r (MPa) current value 177.3 MPa

σ_YUniaxial yield strengthKey driver

59.1886

bOuter radius

−87.4243

aInner radius (bore)

120204

rRadius where the stress is wanted

152215

fTarget plastic fraction of the wall

147204

At peak pressure (177.4 MPa) the hoop stress is 122.6 MPa at the bore and rises to 223.5 MPa at the interface radius 70 mm, where the two profiles meet. Beyond the interface it falls again to the outside.
The hoop stress at the interface from both sides is the same, which is a check on the calculation: 223.5 MPa from the yielded side and 223.5 MPa from the elastic side.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ_Y Uniaxial yield strength	100 to 1,500 MPa	59.08	886.3	827.2	+10%	+10 %
b Outer radius	60 to 300 mm	−87.39	243	330.4	+10%	−12.6 %
a Inner radius (bore)	10 to 150 mm	119.9	204	84.09	+10%	−3.47 %
r Radius where the stress is wanted	10 to 300 mm	151.6	214.8	63.19	+10%	+16.1 %
f Target plastic fraction of the wall	0.05 to 0.95	146.9	204	57.06	+10%	−2.36 %

Sweep of Radius where the stress is wanted

Radius where the stress is wanted (mm)	Peak hoop stress at radius r (MPa)
10	—
46.25	—
82.5	181.5
118.8	—
155	—
191.3	—
227.5	—
263.8	—
300	—

Sweep one variable

Vary

Peak hoop stress at radius r (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_θ,A

Peak hoop stress at radius r

Result · MPa

What the equation gives you. Everything else feeds it.

Inner radius (bore)

mm · typical 10 to 150

The inner (bore) radius of the cylinder, in mm, in the same length unit as b. It must be smaller than b. Changes the stress through the position of the front and the pressure; the net effect depends on where r lies relative to c. Raising it by 5 mm (to 55 mm) moves the result by −3.47 %, from 177.3 to 171.1 MPa.

Outer radius

mm · typical 60 to 300

The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Changes the stress through the position of the front and the elastic ring; the net effect depends on where r lies relative to c. Raising it by 5 mm (to 105 mm) moves the result by −6.44 %, from 177.3 to 165.8 MPa.

σ_Y

Uniaxial yield strength

MPa · typical 100 to 1,500

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 hoop stress in proportion at a fixed fraction f, because every term scales with the yield strength. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from 177.3 to 192 MPa.

Target plastic fraction of the wall

· typical 0.05 to 0.95

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. Changes the stress through the pressure and the position of the front; where r is in the yielded zone the stress falls as the pressure rises, and where r is in the ring it rises. Raising it by 0.05 (to 0.45) moves the result by −2.92 %, from 177.3 to 172.1 MPa.

Radius where the stress is wanted

mm · typical 10 to 300

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 along the profile: the hoop stress rises in the yielded zone and falls in the elastic ring as r increases. Raising it by 5 mm (to 65 mm) moves the result by +13.5 %, from 177.3 to 201.3 MPa.

Worked example with the values above

Interface radius c = 70 mm and autofrettage pressure P_A = 177.4 MPa; r = 60 mm lies in the yielded zone.
Yielded zone: σ_r = −177.4 + 300 ln(60/50) = −122.7 MPa, and σ_θ = σ_r + σ_Y.
σ_θ,A = 177.3 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 (σ_θ ≥ σ_z ≥ σ_r). 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.

Common mistakes

Reading this as the stress in service: it is the stress at peak autofrettage pressure, before unloading.
Using a radius outside the wall.

Statistics and uncertainty

Both profiles scale with the yield strength, so the scatter in σ_Y passes directly into the stress. The location of the interface also depends on the pressure actually reached, so measured profiles show a smooth transition rather than the sharp change of the ideal model.

Where it is used

Drawing the loaded stress profile before the residual stress is found.
Checking that σ_θ ≥ σ_z ≥ σ_r holds, which the closed-form solution assumes.

Maths you need

Thick-walled cylinder: a wall too thick for the thin-wall formula
Tresca criterion: yield when the largest stress difference reaches σ_Y
Lamé equations: elastic stresses in a thick cylinder

Related formulas

Practise this formula