Hertzian Contact Stress Calculator (Sphere Contact)
Use this tool to estimate contact patch radius and maximum contact pressure for two elastic bodies in normal contact. This implementation uses classical Hertzian point-contact equations for sphere-on-sphere or sphere-on-flat loading.
Equivalent radius: 1/R' = 1/R1 + 1/R2 (for flat surface, 1/R2 = 0)
Equivalent modulus: 1/E' = (1-ν1²)/E1 + (1-ν2²)/E2
Contact radius: a = [3FR'/(4E')]1/3
Max pressure: p0 = 3F/(2πa²)
What is Hertzian stress?
Hertzian stress is the localized contact stress that appears when two curved elastic bodies are pressed together. You see it in gears, rolling bearings, cam-follower systems, ball screws, rail-wheel interfaces, and even orthopedic implant contacts. Instead of spreading load over a large area, contact occurs over a tiny patch, so pressure rises quickly.
Because contact fatigue often starts at or near this high-pressure zone, estimating Hertzian pressure early in design helps prevent pitting, spalling, brinelling, and premature wear.
What this calculator computes
This page calculates key outputs for frictionless normal contact between isotropic elastic solids:
- Equivalent radius (R') from the two radii of curvature
- Equivalent elastic modulus (E') from both materials and Poisson ratios
- Contact patch radius (a) for circular point contact
- Maximum contact pressure (p0) at the center of the contact area
- Mean contact pressure and a quick estimate of maximum subsurface shear stress
Input guidance
1) Normal load (N)
Use the static or effective dynamic load that actually acts at the contact point. If load varies with time, analyze several load levels or use a duty-cycle method.
2) Radii of curvature (mm)
For sphere-on-sphere, enter both radii. For sphere-on-flat, check the flat-surface box; the second radius is treated as infinite. Smaller radii generally increase stress because the contact area shrinks.
3) Material properties (GPa and ν)
The elastic modulus controls stiffness. Higher modulus reduces contact size and raises pressure for the same force. Poisson's ratio usually ranges from 0.25 to 0.35 for many metals.
How to read the results
- Large a, lower p0: Better load spreading, usually improved contact durability.
- Small a, high p0: Higher risk of surface distress and rolling contact fatigue.
- Subsurface shear estimate: Useful for quick screening of crack-initiation risk below the surface.
Worked example
Suppose two steel components (E = 210 GPa, ν = 0.30) carry a 1000 N load with R1 = 10 mm and R2 = 20 mm. The calculator gives a small contact radius and a high pressure in the hundreds of MPa range. That is common for concentrated mechanical contacts and explains why hardening and surface finish matter so much.
Design tips to reduce Hertzian contact stress
- Increase contact radius where possible.
- Increase the number of simultaneous load-sharing contacts.
- Reduce peak load using better load paths, preload control, or damping.
- Use materials and heat treatments with strong contact-fatigue performance.
- Improve lubrication to reduce frictional heating and surface damage progression.
Assumptions and limitations
This calculator is based on classical Hertz theory. It assumes:
- Elastic deformation (no gross plasticity)
- Smooth, non-conforming contact geometry
- Frictionless normal loading in the basic equations
- Homogeneous, isotropic materials
Real systems may deviate because of roughness, coatings, residual stress, misalignment, edge loading, temperature, lubrication regime, tangential traction, and impact. For safety-critical designs, use FEA and validation testing.
Quick FAQ
Is this useful for bearings and gears?
Yes, as a first-pass estimate. Final rating should still follow relevant standards and detailed contact-fatigue methods.
Can I use plastic materials?
You can enter their elastic properties, but if stress approaches yield, linear Hertz assumptions become less accurate.
What unit system is used?
Inputs are N, mm, GPa; outputs are primarily mm and MPa.