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Mechanical

Lamé hoop stress in a thick cylinder

Circumferential (hoop) stress at radius r in a thick-walled cylinder under internal and external pressure.

Ref: MEC-43
Rev: v1
Field: Mechanical
Result: σ_θ [MPa]

σ_{θ} = 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.

Circumferential (hoop) stress at radius r in a thick-walled cylinder under internal and external pressure. Under internal pressure it is tensile and largest at the bore.

Calculator

Solve for

σ_θHoop stressComputed

MPa

p_iInternal pressure

MPa

1100

p_oExternal pressure

MPa

0100

aInternal radius

10100

bExternal radius

60400

rRadial position in the wall

5075

Radial stress at the same radius−10 MPa
Sum σ_θ + σ_r, equal to 2A at every radius16 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.

Hoop stress (MPa) current value 26 MPa

p_oExternal pressureKey driver

−334

p_iInternal pressure

2.6260

bExternal radius

10.355.5

aInternal radius

0.588

rRadial position in the wall

The hoop stress is highest at the bore (26 MPa) and falls to 16 MPa at the outside surface, a ratio of about 1.6. The wall does not share the load evenly, so a thick cylinder is designed for the bore.
At every radius σ_θ + σ_r = 2A, so through the wall the hoop and radial stresses change by equal and opposite amounts.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
p_o External pressure	0 to 100 MPa	−334	26	360	+10%	0 %
p_i Internal pressure	1 to 100 MPa	2.6	260	257.4	+10%	+10 %
b External radius	60 to 400 mm	10.32	55.45	45.14	+10%	−16.9 %
a Internal radius	10 to 100 mm	0.5882	26	25.41	+10%	—
r Radial position in the wall	50 to 75 mm	16	26	10	+10%	−12 %

Sweep of Radial position in the wall

Radial position in the wall (mm)	Hoop stress (MPa)
50	26
53.13	23.94
56.25	22.22
59.38	20.76
62.5	19.52
65.63	18.45
68.75	17.52
71.88	16.71
75	16

Sweep one variable

Vary

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

Variables and how each one contributes

σ_θ

Hoop 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. Directly proportional while the external pressure is zero: a 10 % rise in p_i raises every stress in the wall by 10 %.

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. External pressure squeezes the wall and lowers the tensile hoop stress: with p_o = 2 MPa the bore hoop stress falls from 26 to 18.8 MPa.

Internal radius

mm · typical 10 to 100

The internal radius (bore radius), in mm. Strong. A thinner wall carries the same pressure at higher stress: raising a by 10 % (to 55 mm, keeping r at the bore) raises the bore hoop stress by about 28 %, to 33.3 MPa.

External radius

mm · typical 60 to 400

The external radius, in mm. The wall thickness is b − a. A thicker wall lowers the stress, with diminishing returns: raising b by 10 % (to 82.5 mm) lowers the bore hoop stress by about 17 %, to 21.6 MPa.

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: the hoop stress falls from 26 MPa at the bore (50 mm) to 19.5 MPa at mid-wall (62.5 mm) and 16 MPa 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 = 8 MPa and B = 45,000 MPa·mm².
σ_θ = A + B/r² = 8 + 45,000 / 50² = 26 MPa.

Assumptions and limits

A homogeneous, isotropic, linearly elastic material with small deformation, below yield.
Uniform internal and external pressure on a long cylinder, away from the ends and any discontinuities.

Common mistakes

Using the thin-wall formula pa/t for a thick wall. At the default it gives 20 MPa where the true bore stress is 26 MPa.
Taking the average of the bore and outside stress as the working stress. Failure starts at the bore, where the stress is greatest.

Statistics and uncertainty

σ_θ is linear in the pressures. The radii act through b² − a², which is a difference of squares, so errors are amplified: at these proportions a 1 % error in either radius gives about a 2 % error in the bore hoop stress. This matters for a worn gun bore or a cylinder machined to a tolerance, where a small change in a changes the peak stress noticeably.

Where it is used

Checking the whole wall of a thick tube against yield, starting with the bore.
Comparing the hoop stress measured by a strain gauge on the outside with the value inside.

Maths you need

Radial and hoop stress: perpendicular normal stresses in a cylinder wall
Squares and fractions: B/r² and b² − a²

Related formulas

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