Hertz theory guide

Hertzian Contact Stress in Rolling Bearings

Understand effective material and curvature properties, contact-patch size, and maximum Hertz pressure in rolling contacts.

Prepared by BearingSolve9 min readUpdated 13 July 2026

Why rolling contacts develop high pressure

A rolling element and raceway appear to meet at a point or line before loading. Elastic deformation creates a small contact patch that carries the rolling-element load. Because the patch is small, local pressure can reach the gigapascal range even when the applied force seems modest.

Ball-bearing contacts are generally elliptical. Roller contacts are closer to line contacts but require edge and profile corrections. The circular point-contact model below is a useful introduction, not a complete bearing-contact model.

Three-dimensional rendering of a steel bearing ball pressed into a curved raceway with an orange elliptical contact patch — A normal rolling-element load creates a small elastic contact patch. The deformation is exaggerated; the footprint is elliptical, with principal semi-axes commonly denoted `a` and `b`.

Model boundary

Use the equations below only for smooth, frictionless, elastic bodies with a circular contact patch. A ball-to-raceway contact is normally elliptical and requires the two principal curvatures, elliptical semi-axes, and the actual raceway geometry.

Effective material and curvature properties

\frac{1}{E^{*}} = \frac{1 - {ν_{1}}^{2}}{E_{1}} + \frac{1 - {ν_{2}}^{2}}{E_{2}}

reduced elastic modulus

\frac{1}{R^{*}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}

effective radius for the simplified contact

Here, E is Young's modulus, ν is Poisson's ratio, and R describes surface curvature. The scalar R* equation applies to the circular-contact simplification. Convex and concave surfaces require opposite curvature signs; a real ball-to-raceway calculation resolves curvature independently in two principal directions.

Simplified circular contact

a = \sqrt[3]{\frac{3 F R^{*}}{4 E^{*}}}

contact radius

p_{0} = \frac{3 F}{2 π a^{2}}

maximum Hertz pressure at the contact centre

Three-dimensional Hertz pressure dome over an elliptical footprint, blue at the edge and red at the centre — For the simplified circular contact, pressure follows `p(r)` = `p₀`√(1 − `r`²/`a`²) and falls continuously to zero at `r` = ±`a`.

Pressure decreases from p₀ at the centre to zero at the contact edge. Subsurface shear stresses also develop below the surface, which is why a bearing analysis should consider more than the peak surface pressure alone.

Worked example

For two steel bodies with E = 210 GPa and ν = 0.30, the reduced modulus is approximately 115.4 GPa. With F = 1,000 N and R* = 10 mm:

Contact radius: a ≈ 0.402 mm
Maximum pressure: p₀ ≈ 2.95 GPa

Standards and further reading

K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985.
NASA TP-2473 — Hertzian Contact in Two and Three Dimensions
ISO 16281:2025 — Bearing-specific load distribution and rating-life context

This educational guide does not reproduce the standards. Its equations and examples have been checked against the cited public references, but the guide has not been independently certified or reviewed for your application. Use the current standard, manufacturer data, and an appropriate engineering review for final bearing selection.

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