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Mechanical

First-yield pressure of a thick cylinder

Internal pressure at which the bore of a thick cylinder first reaches the Tresca yield condition, for an open or free-ended wall with the stress ordering σθ ≥ σz ≥ σr.

Ref: MEC-69
Rev: v1
Field: Mechanical
Result: P_Y [MPa]

P_{Y} = \frac{σ_{Y}}{2} (1 - \frac{a^{2}}{b^{2}})

P_Y is the internal pressure at which the inside surface of a thick cylinder first meets the Tresca yield condition, σ_θ − σ_r = σ_Y, with the hoop stress the largest and the radial stress (equal to minus the pressure) the smallest. Below it the wall is wholly elastic. The factor 1 − a²/b² is always below 1, so no thickness of wall can lift P_Y above σ_Y/2. It is the lower limit of the autofrettage window.

Internal pressure at which the bore of a thick cylinder first reaches the Tresca yield condition, for an open or free-ended wall with the stress ordering σθ ≥ σz ≥ σr. Below it the wall is wholly elastic.

Calculator

Solve for

P_YFirst-yield pressureComputed

MPa

aInner radius (bore)

10150

bOuter radius

60300

σ_YUniaxial yield strength

MPa

1001,500

Fully plastic limit for comparison, σ_Y ln(b/a)207.9 MPa
Ceiling that no wall thickness can exceed, σ_Y/2150 MPa
Largest autofrettage pressure for purely elastic unloading at the bore, 2P_Y225 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.

First-yield pressure (MPa) current value 112.5 MPa

σ_YUniaxial yield strengthKey driver

37.5563

aInner radius (bore)

3.97149

bOuter radius

45.8146

A cylinder of bore 50 mm and outer radius 100 mm in 300 MPa steel first yields at 112.5 MPa, which is 54.1 % of its fully plastic limit of 207.9 MPa.
Making the wall much thicker cannot help beyond a point: even an infinitely thick wall would first yield at σ_Y/2 = 150 MPa.

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	37.5	562.5	525	+10%	+10 %
a Inner radius (bore)	10 to 150 mm	3.973	148.5	144.5	+10%	−7 %
b Outer radius	60 to 300 mm	45.83	145.8	100	+10%	+5.79 %

Sweep of Outer radius

Outer radius (mm)	First-yield pressure (MPa)
60	45.83
90	103.7
120	124
150	133.3
180	138.4
210	141.5
240	143.5
270	144.9
300	145.8

Sweep one variable

Vary

First-yield pressure (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

P_Y

First-yield pressure

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. Lowers P_Y: a larger bore leaves a thinner wall. Raising it by 5 mm (to 55 mm) moves the result by −7 %, from 112.5 to 104.6 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. Raises P_Y, but with diminishing return, because the factor 1 − a²/b² levels off below 1. Raising it by 5 mm (to 105 mm) moves the result by +3.1 %, from 112.5 to 116 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 P_Y in direct proportion. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from 112.5 to 121.9 MPa.

Worked example with the values above

Radius ratio squared: a²/b² = 2,500 / 10,000 = 0.25, so 1 − a²/b² = 0.75.
Half the yield strength: σ_Y/2 = 150 MPa.
P_Y = 150 × 0.75 = 112.5 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

Using the formula for a closed-end wall without checking the axial stress ordering.
Using an ultimate strength instead of the yield strength.

Statistics and uncertainty

P_Y is proportional to the yield strength, so a ±5 % scatter in σ_Y gives a ±5 % scatter in P_Y. The bore radius and wall thickness enter through a ratio, so machining tolerances matter less than the material variation.

Where it is used

Setting the lower end of an autofrettage pressure window.
Finding the largest working pressure of an untreated thick cylinder before any yielding.

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

Related formulas

Practise this formula