What is Hertz contact stress?

Hertz contact stress is the localized stress that develops when two curved bodies are pressed together. The contact area is very small and the maximum pressure occurs at the center. This theory applies to elastic, non-conforming contacts like ball bearings, gears, and wheel-rail interfaces.

What is the maximum contact pressure formula for spheres?

For two spheres: p_max = (1/π)·(6F·E*²/R*²)^(1/3), where E* is the combined elastic modulus and R* is the combined radius. The contact area is circular with radius a = (3FR*/(4E*))^(1/3).

What is the combined elastic modulus E*?

The combined (effective) elastic modulus accounts for both bodies: 1/E* = (1-ν₁²)/E₁ + (1-ν₂²)/E₂. It represents the combined elastic compliance of both contacting surfaces.

When does Hertz theory apply?

Hertz theory assumes: elastic material behavior, contact area much smaller than body dimensions, frictionless contact, smooth surfaces, and non-conforming geometries. It does not apply to plastic deformation, rough surfaces, or conforming contacts.

Where is Hertz contact stress important?

Rolling element bearings, gear teeth, cam-follower systems, wheel-rail contact, indentation testing, and any machine element where curved surfaces press against each other. Contact fatigue (pitting) is often governed by Hertzian stress.

Free Engineering Tool

Hertz Contact Stress Calculator

Calculate maximum Hertzian contact pressure, contact area, and approach for sphere-on-sphere and cylinder-on-cylinder elastic contact.

Sphere ContactCylinder Contact#171

Contact Type

Radius R₁ (mm) — ∞ for flat

Radius R₂ (mm) — ∞ for flat, 0 = flat

Force F (N)

E₁ (GPa)

ν₁ (Poisson’s ratio)

E₂ (GPa)

ν₂ (Poisson’s ratio)

Quick presets

Results

Maximum Contact Pressure p₀

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Contact Radius / Half-Width a

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Combined E*

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Combined R*

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Combined Elastic Modulus

Sphere on Sphere

Cylinder on Cylinder (parallel)

⚠️ Note: Enter R₂ = 0 for a flat surface (R₂ → ∞). Hertz theory assumes elastic contact, small contact area relative to body size, and frictionless surfaces.

Hertz Contact Stress Calculator | Sphere & Cylinder | Free Online Tool

Published by Nikolai Shelkovenko on February 15, 2026 February 15, 2026

Results

Combined Elastic Modulus

Sphere on Sphere

Cylinder on Cylinder (parallel)