Hertzian Cylinder Contact Stress Calculator (Roller Bearings)

Hertzian contact stress analysis evaluates the intense elastic contact stresses that develop when curved surfaces press against one another under mechanical load.

Unlike flat area contacts, line contact compresses into a narrow rectangular strip of width 2b, generating compressive stresses often exceeding 1,000 MPa.

The contact pressure distribution across the contact strip is semi-elliptical, peaking at p₀ along the center line.

The triaxial compressive state prevents immediate surface yielding; however, maximum shear stress peaks beneath the surface at depth ~0.78b, which initiates subsurface rolling-contact fatigue cracks.

p₀ is the peak normal compressive stress at the contact center.

τmax is the maximum subsurface shear stress governing subsurface fatigue pitting.

Contact width 2b indicates the physical footprint of the contact patch.

• Curvature: Conforming surfaces (e.g. roller inside a concave raceway) have a larger equivalent radius R*, dramatically reducing contact stress.
• Material modulus: Ceramic rollers (Si₃N₄, E=310 GPa) have higher stiffness, yielding higher peak stresses than steel under the same load.

• Sizing cylindrical and needle roller bearings.
• Checking contact fatigue on spur and helical gear teeth.
• Railway wheel-rail rolling contact fatigue calculations.

• Assumes frictionless elastic contact without plastic yield or tangential traction forces.