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Mechanical

Lamé radial stress in a thick cylinder

Radial stress at radius r in a thick-walled cylinder under internal and external pressure.

Ref: MEC-42
Rev: v1
Field: Mechanical
Result: σ_r [MPa]

σ_{r} = A - \frac{B}{r^{2}} A = \frac{p_{i} a^{2} - p_{o} b^{2}}{b^{2} - a^{2}} B = \frac{(p_{i} - p_{o}) a^{2} b^{2}}{b^{2} - a^{2}}

a and b are the internal (bore) and external radii, r is the radius where the stress is wanted (a ≤ r ≤ b), and p_i and p_o are the internal and external pressures entered as positive values. Normal stress is positive in tension, so pressure gives a negative (compressive) radial stress at a loaded surface. The equations hold for a homogeneous, isotropic, linearly elastic cylinder with small deformation, away from the ends. The constants A and B come from the two boundary conditions σ_r(a) = −p_i and σ_r(b) = −p_o. The defaults are the worked example from the reference sheet: a 50 mm bore, 75 mm outer radius and 10 MPa internal pressure.

Radial stress at radius r in a thick-walled cylinder under internal and external pressure. It equals minus the pressure at each surface and is always compressive for pressure loading.

Calculator

Solve for

σ_rRadial stressComputed

MPa

p_iInternal pressure

MPa

1100

p_oExternal pressure

MPa

0100

aInternal radius

10100

bExternal radius

60400

rRadial position in the wall

5075

Lamé constant A8 MPa
Lamé constant B45,000 MPa·mm²
Hoop stress at the same radius26 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.

Radial stress (MPa) current value −10 MPa

p_iInternal pressureKey driver

−100−1

rRadial position in the wall

aInternal radius

−0.226

p_oExternal pressure

−10−10

bExternal radius

−10−10

The radial stress runs from −10 MPa at the bore to 0 at the outside surface, which are exactly the boundary conditions (−p_i and −p_o). It is a good first check on any working.
Under pressure the radial stress is compressive throughout and much smaller than the hoop stress (−10 against +26 MPa at the bore), so it is the hoop stress that governs failure.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
p_i Internal pressure	1 to 100 MPa	−100	−1	99	+10%	+10 %
r Radial position in the wall	50 to 75 mm	−10	0	10	+10%	−31.2 %
a Internal radius	10 to 100 mm	−10	−0.2262	9.774	+10%	—
p_o External pressure	0 to 100 MPa	−10	−10	5.684e-14	+10%	0 %
b External radius	60 to 400 mm	−10	−10	3.553e-15	+10%	0 %

Sweep of Radial position in the wall

Radial position in the wall (mm)	Radial stress (MPa)
50	−10
53.13	−7.945
56.25	−6.222
59.38	−4.765
62.5	−3.52
65.63	−2.449
68.75	−1.521
71.88	−0.7108
75	0

Sweep one variable

Vary

Radial stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_r

Radial stress

Result · MPa

What the equation gives you. Everything else feeds it.

p_i

Internal pressure

MPa · typical 1 to 100

The pressure acting on the inside (bore) surface, in MPa, entered as a positive value. At the default radius (the bore) the radial stress is exactly −p_i, so a 10 % rise in p_i makes σ_r 10 % more negative.

p_o

External pressure

MPa · typical 0 to 100

The pressure acting on the outside surface, in MPa, entered as a positive value. Zero by default, which makes the radial stress 0 at the outside. External pressure makes it −p_o there and leaves the bore value at −p_i: with p_o = 2 MPa the stress at mid-wall (62.5 mm) becomes −4.8 MPa instead of −3.5 MPa.

Internal radius

mm · typical 10 to 100

The internal radius (bore radius), in mm. Sets where the wall starts. The default r equals a, so raising a above r puts the point inside the bore, where there is no material and no result is shown: raise r with it.

External radius

mm · typical 60 to 400

The external radius, in mm. The wall thickness is b − a. The radial stress is still −p_i at the bore and −p_o at the outside, but a thicker wall spreads the change between them more gradually. Thickening the wall helps the hoop stress far more than the radial stress.

Radial position in the wall

mm · typical 50 to 75

The radius at which the stress is wanted, in mm. It must lie in the wall, between a and b. Moves the point through the wall: at the defaults the radial stress rises from −10 MPa at the bore (50 mm) to −3.5 MPa at mid-wall (62.5 mm) and 0 at the outside (75 mm). The sweep chart shows the whole distribution.

Worked example with the values above

Wall term: b² − a² = 75² − 50² = 3,125 mm².
Lamé constants: A = (10 × 50² − 0 × 75²) / 3,125 = 8 MPa, and B = (10 − 0) × 50² × 75² / 3,125 = 45,000 MPa·mm².
σ_r = A − B/r² = 8 − 45,000 / 50² = −10 MPa.

Assumptions and limits

A homogeneous, isotropic, linearly elastic material with small deformation, so the stress stays below yield.
Uniform internal and external pressure on a long cylinder, at a section away from the ends, nozzles and other discontinuities.

Common mistakes

Entering a radius r that lies outside the wall. The result is only defined for a ≤ r ≤ b.
Entering the pressures with a minus sign. Enter p_i and p_o as positive values; the formula already makes the radial stress negative.

Statistics and uncertainty

σ_r is linear in the pressures, so a 1 % error in p_i is a 1 % error here, and the values at the two surfaces depend on the pressures alone. Between the surfaces the 1/r² term makes the answer sensitive to the position: a 1 % error in r gives about a 2 % error in the B/r² part. In service the pressure is rarely known better than a few per cent and often has peaks above the nominal value, so treat the result as a nominal elastic stress, not a failure prediction.

Where it is used

Checking the stress through the wall of a gun barrel, hydraulic cylinder or pressure vessel.
Radial stress at the interface in shrink-fit and compound cylinder calculations.

Maths you need

Radial and hoop stress: perpendicular normal stresses in a cylinder wall
Algebra: substituting into A − B/r²

Related formulas

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