When a hardened steel ball rests on a flat plate, the entire load passes through a contact patch smaller than a pinhead. Intuitively you might expect the stress there to be enormous — and it is. A 100 N load, roughly the weight of a 10 kg mass, can generate more than 1 GPa of pressure inside that tiny spot. That is several times the yield strength of mild steel, yet well-designed parts survive millions of cycles. Understanding why is the entire point of Hertz contact theory.
This article walks through how to estimate contact pressure and contact size for curved bodies, works a full numerical example, and explains the subsurface stress that engineers most often forget.
Why this calculation matters
Contact stress decides the fate of a surprising range of machine elements: ball and roller bearings, gear teeth, cam-and-follower pairs, wheel-rail interfaces, and press fits. None of these fail because the average stress is high. They fail because a concentrated, repeated contact load drives fatigue cracks just below the surface.
If you size these components using nominal stress — load divided by some projected area — you will be wildly optimistic. The real contact area is tiny and load-dependent: it grows only as the cube root of force. So the pressure does not scale the way bulk-stress intuition suggests. You need contact mechanics to get a number you can defend in a design review.
The core formula
Heinrich Hertz solved the elastic contact problem in 1881 for smooth, frictionless, non-conforming bodies. For a sphere of radius R pressed onto a flat surface — a flat is just a sphere of infinite radius — three results matter.
First, combine the two materials into an effective modulus E*:
1 / E* = (1 - v1^2) / E1 + (1 - v2^2) / E2
Then the contact radius a:
a = ( 3 F R / (4 E*) ) ^ (1/3)
And the maximum contact pressure p0, which sits at the center of the patch:
p0 = 3 F / (2 pi a^2)
Two features are worth internalizing. The contact radius grows with the cube root of load, so doubling the force enlarges the patch by only about 26 %. And the peak pressure is exactly 1.5 times the mean pressure F/(pi a squared) — the pressure distribution is a hemisphere, not a flat block.
There is a third result that experienced engineers care about most. The largest shear stress does not occur at the surface. It sits roughly 0.48a below the surface and reaches about 0.31 p0. That buried shear peak is where rolling-contact fatigue cracks are born.
A worked example
Take a steel ball of radius R = 10 mm pressed onto a flat steel plate with a force F = 100 N. Both bodies are steel: E = 210 GPa, Poisson's ratio v = 0.3.
Step 1 — effective modulus.
1/E* = 2 x (1 - 0.3^2) / 210e9 = 1.82 / 210e9
E* = 115.4 GPa
Step 2 — contact radius.
a = ( 3 x 100 x 0.010 / (4 x 115.4e9) ) ^ (1/3)
a = ( 6.50e-12 ) ^ (1/3) = 1.87e-4 m = 0.187 mm
Step 3 — maximum contact pressure.
p0 = 3 x 100 / (2 pi x (1.87e-4)^2) = 1.37e9 Pa = 1371 MPa
So a modest 100 N load produces about 1.37 GPa of peak contact pressure inside a contact circle just 0.37 mm across. The maximum shear stress is roughly 0.31 x 1371 = 425 MPa, located about 0.09 mm beneath the surface. That subsurface number — not the surface pressure — is what you compare against the material's shear fatigue strength.
Common mistakes
Treating contact pressure as linear in load. It is not. Pressure scales with F to the power 1/3, because the patch grows as the load grows. Tripling the load raises peak pressure by only about 44 %.
Using yield strength as the limit. Hertzian contact routinely runs above the uniaxial yield stress without permanent damage, because the material under the patch is in triaxial compression. The governing limit is usually subsurface shear fatigue, not first yield.
Ignoring the second body's curvature. For two curved bodies, R is an effective radius combining both. Two convex surfaces give a smaller effective radius and a higher pressure; a ball in a conforming groove gives a larger one and a lower pressure.
Forgetting the smooth-surface assumption. Hertz theory assumes ideal elastic, frictionless, smooth surfaces. Real roughness concentrates load on asperities, so local peaks can far exceed the Hertzian p0. Treat the Hertzian value as a clean baseline, not a ceiling.
Try the interactive NovaSolver calculator
Working the cube roots by hand is fine once, but for design iteration it helps to see the contact patch and pressure update live as you change load, radius, and material. The Hertz contact stress calculator on NovaSolver does exactly that — enter the geometry and load, and it returns contact radius, peak and mean pressure, and the depth of maximum shear.
Related calculators
- Line contact (cylinder on flat) — for rollers, pins, and cam followers, where the patch is a strip rather than a circle.
- Elliptical contact — for general curved bodies such as crowned rollers and gear teeth.
- Subsurface stress field — to see where the shear peak really sits beneath the surface.
You can browse the full set in the structural contact tools hub.
Closing note
Hertz contact theory is one of those rare pieces of classical mechanics that is both elegant and immediately useful. The takeaways are simple: contact patches are tiny, contact pressures are high, pressure scales only with the cube root of load, and the damage usually starts below the surface. Get those four ideas right and most bearing, gear, and cam problems become tractable. Run your own numbers, check the subsurface shear, and let the surface pressure be the start of the analysis rather than the end of it.
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