Hertzian Contact Stress Calculator (Line and Point Contact)
Hertzian contact pressure and contact patch size for two parallel cylinders in line contact or two spheres in point contact, from the radii, the elastic constants and the normal force. Results include the contact half width or patch radius, the peak and mean pressure, and the maximum subsurface shear stress with the depth at which it occurs.
How to use this calculator
- Choose line contact for two parallel cylinders or a roller on a plane, and point contact for two spheres or a ball on a plane.
- Enter the radius of each body, using 0 for a flat surface and a negative value for a concave one such as the inside of a raceway.
- Enter the contact length for line contact, the normal force, and the Young's modulus and Poisson's ratio of both bodies.
- Read the contact patch size, the peak pressure and the maximum subsurface shear stress, which is what governs rolling contact fatigue rather than the surface pressure.
Frequently asked questions
Why is the maximum shear stress below the surface rather than at the point of highest pressure?
At the surface the stress state is close to hydrostatic compression directly under the load, and hydrostatic pressure does not shear material. Moving down, the principal stresses diverge and the shear stress grows, reaching a maximum at a depth of about 0.79 times the contact half width for line contact and 0.48 times the patch radius for point contact, at roughly 30 percent of the peak pressure. This is why rolling bearings fatigue from cracks that start below the surface and work upwards.
What is a reasonable peak contact pressure for hardened steel?
Hardened 52100 bearing steel runs at 1500 to 2000 MPa for a long life, and case hardened gears at 1500 to 1800 MPa at the pitch point. These are far above the yield strength of the material, and they are tolerable only because the material is harder than the pressure and the contact keeps moving so no single point sees many cycles. Above the shakedown limit the surfaces flatten plastically and the elastic solution stops describing what is happening.
Why does a concave surface reduce the contact pressure so much?
The effective radius is one over the sum of the reciprocals of the radii, with a concave surface entering as a negative number. A roller in a closely conforming raceway has a large effective radius, so the same load spreads over a wider band at a much lower pressure. It is the reason a small conformity error in a roller bearing changes its fatigue life out of all proportion to its size.
Why are rollers crowned at the ends?
The two dimensional solution assumes the two cylinders run parallel with a uniform load along the contact. A real roller ends abruptly, and the stiffness of the material either side of the end produces a pressure spike far above the average, which spalls the raceway edge. Crowning removes material at the ends so the load tapers off, at the cost of a slightly higher pressure in the middle.