Two contacting cylinders elastically deform into a narrow pressure band instead of touching along a geometric line; pressure within it is many times higher than a mean pressure based on the apparent contact area.
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Two contacting cylinders elastically deform into a narrow pressure band instead of touching along a geometric line; pressure within it is many times higher than a mean pressure based on the apparent contact area.
5,000 N over a 50 mm contact length between 40 mm and 80 mm steel cylinders (E* ≈ 115.4 GPa) give pH ≈ 525 MPa.
Classical Hertz theory: purely elastic behavior, smooth surfaces, contact width small compared with the cylinder diameters, and no plastic deformation.
This calculator finds the maximum Hertz contact pressure between two parallel cylinders from normal force, contact length, both diameters and the reduced elastic modulus.
Two parallel cylinders touch geometrically along a line. Under load, both deform elastically into a narrow pressure band of half-width b, with maximum pressure pH at its center: pH = √[(FN/L)·E*/(π·R)], with reduced radius R from 1/R = 2/D1 + 2/D2.
Picture two rubber rollers pressed together: instead of touching along an infinitely thin line, they flatten into a visible, narrow strip. For steel this deformation is far smaller and invisible, but the principle is identical — and pressure at the center of that strip is many times higher than a rough force-over-area estimate would suggest.
pH = √[(FN/L)·E* / (π·R)], with 1/R = 2/D1 + 2/D2
pH = √[(FN/L)·E* / (π·R)]1/R = 2/D1 + 2/D2| Symbol / input | Meaning |
|---|---|
| Maximum Hertz pressure pH | Peak pressure at the center of the contact band. |
| Normal force FN | Contact force acting normal to the contact surface. |
| Contact length L | Length of the line of contact between the two cylinders. |
| Cylinder 1 diameter D1 | Diameter of the first cylinder. |
| Cylinder 2 diameter D2 | Diameter of the second cylinder. |
| Reduced elastic modulus E* | Effective elastic modulus of the material pair, see the linked calculator. |
Normal force FN, contact length L, both cylinder diameters D1 and D2, and the reduced elastic modulus E* (from the linked calculator) set maximum pressure.
First compute E* with the reduced elastic modulus calculator if not already known. Then enter force, contact length and both diameters to find the maximum Hertz pressure.
5,000 N over a 50 mm contact length between 40 mm and 80 mm steel cylinders (E* ≈ 115.4 GPa) give pH ≈ 525 MPa.
A pressure of 525 MPa is already in a range that hardened bearing steel (allowable fatigue strength often above 1,000 MPa) tolerates without issue, but that softer structural steels can already exceed yield strength at. For comparison, the apparent mean pressure force/(contact length × contact width) would sit markedly below this actual peak value.
Force is given in N, lengths and diameters in mm, the reduced elastic modulus in GPa, and the resulting pressure in MPa.
Geometrically, two rigid cylinders touch only along an infinitely thin line, which would imply infinite pressure. Since real materials yield elastically, both cylinders deform locally, so the contact line widens into a narrow but finite pressure band. Within that band, pressure is not evenly distributed but follows a semi-elliptical profile, maximum pH at the center and zero at the band edges.
The calculation is used to design rolling bearings, gear tooth flanks (line contact at the tooth flanks), cams, rollers and sprockets — anywhere two curved surfaces roll or slide against each other under load.
Classical Hertz theory: purely elastic material behavior, ideally smooth surfaces, contact width small compared with the cylinder diameters, and no tangential friction forces in the contact area. Real surface roughness, lubricant film formation (elastohydrodynamics) and incipient plastic deformation at very high pressures are not captured.
Common mistake: A common mistake is confusing the apparent mean bearing pressure (force divided by contact area) with the actual Hertzian peak pressure — the latter is always markedly higher. It is also easy to use outer diameter instead of the diameter actually governing curvature at the contact.
Line contact arises from two cylinders or a cylinder and a flat touching; point contact arises from two spheres or a sphere and a flat, see the corresponding linked calculators.
Because the actual contact area under load is extremely small and pressure within it is additionally unevenly distributed, with a marked peak at the center — a rough force-over-area calculation assuming a larger area substantially underestimates this peak.
When the material's yield strength is exceeded locally (plastic rather than purely elastic deformation), for very rough surfaces, or when tangential friction forces in the contact area become significant.
The reduced radius is dominated by the smaller diameter; a much larger second diameter approaches the cylinder-on-flat special case, see that calculator.
No, it gives only static contact pressure; a complete design additionally needs fatigue strength, lubrication and dynamic load cycling.