Peak pressure p0
-Contact size
-Max shear stress
-Safety factor on load
-to first yield (Tresca)All results
| Effective radius R | - |
| Effective modulus E* | - |
| Contact radius a | - |
| Peak pressure p0 | - |
| Mean pressure | - |
| Maximum shear stress | - |
| Depth of maximum shear | - |
| Approach of the bodies (sphere only) | - |
| Peak pressure at first yield | - |
| Load at first yield | - |
| Softer body (lower yield strength) | - |
Method and formulas
These are the classical Hertz relations (Hertz 1882; see Johnson, Contact Mechanics, or Shigley, Budynas and Nisbett) for frictionless, elastic, smooth bodies, so every result can be checked by hand. Define the effective radius and modulus first.
| Quantity | Formula |
|---|---|
| Effective radius | 1/R = 1/R1 + 1/R2 (external); 1/R = 1/R1 − 1/R2 (socket or groove); R = R1 for a flat |
| Effective modulus | 1/E* = (1 − ν1²)/E1 + (1 − ν2²)/E2 |
| Sphere: contact radius | a = (3 F R / 4 E*)1/3 |
| Sphere: peak and mean pressure | p0 = 3 F / (2 π a²), pm = F / (π a²) = 2 p0 / 3 |
| Sphere: approach | δ = a² / R |
| Sphere: max shear and depth (ν = 0.3) | τmax ≈ 0.31 p0 at z ≈ 0.48 a |
| Cylinder: half-width | b = √(4 w R / π E*), w = F / L |
| Cylinder: peak and mean pressure | p0 = 2 w / (π b), pm = π p0 / 4 |
| Cylinder: max shear and depth (ν = 0.3) | τmax ≈ 0.30 p0 at z ≈ 0.78 b |
| First yield (Tresca) | 2 τmax = Sy of the softer body; load scales as p03 (sphere) or p02 (cylinder) |
The shear stress factors are the widely quoted values for a Poisson ratio of 0.3; they vary by a few percent for other materials, so treat the shear stress and depth as good estimates for steels. The Hertz solution assumes the contact patch is small compared with the radii and the size of the bodies, and that the load is normal. The calculator finds the softer body as the one with the lower yield strength.
Worked example: 10 mm steel ball on a steel flat, 1 000 N
Both bodies steel, E = 200 000 MPa, ν = 0.3, so E* = 1 / (2 × 0.91 / 200 000) = 109890 MPa. For a flat R = R1 = 10 mm. Contact radius a = (3 × 1000 × 10 / (4 × 109890))1/3 = 0.409 mm. Peak pressure p0 = 3 × 1000 / (2π × 0.409²) = 2859 MPa. Approach δ = a² / R = 0.0167 mm. Maximum shear stress ≈ 0.31 × 2859 = 886 MPa at a depth of about 0.48 × 0.409 = 0.196 mm.
Questions
What is Hertz contact stress?
Hertz contact stress is the local compressive stress where two curved elastic bodies, such as a ball on a flat, a roller on a race or two gear teeth, are pressed together. The contact is not a point but a small patch whose size follows from the load, the curvature of the surfaces and the elastic moduli. The peak pressure at the centre of the patch is much higher than the average nominal pressure, and the maximum shear stress sits below the surface, not on it.
How do you calculate the contact patch size and peak pressure for a sphere?
With effective radius R (1/R = 1/R1 + 1/R2) and effective modulus E* (1/E* = (1 − ν1²)/E1 + (1 − ν2²)/E2), the contact radius is a = (3 F R / 4 E*)^(1/3) and the peak pressure is p0 = 3 F / (2 π a²). Example: a 10 mm radius steel ball (E = 200 GPa, ν = 0.3) pressed on a steel flat with 1 000 N gives a = 0.409 mm and p0 = 2859 MPa.
How is line contact (cylinders) different from point contact?
Parallel cylinders touch along a strip of half-width b = √(4 w R / π E*), where w is the load per unit length, and the peak pressure is p0 = 2 w / (π b). Because the load is spread along the length, line contact carries far more load than point contact for the same peak pressure. The formula assumes a long contact without end effects: crowned rollers and edge loading are not modelled.
Where is the maximum shear stress and why does it matter?
For the common case of ν ≈ 0.3 the maximum shear stress is about 0.31 p0 at a depth of roughly 0.48 a below a sphere contact, and about 0.30 p0 at 0.78 b below a line contact. This subsurface peak is why bearings and gear teeth start to fail by subsurface fatigue and why case-hardened layers must be deeper than this depth, so the calculator reports both the stress and the depth.
When does the surface yield?
Using the maximum shear (Tresca) criterion, the softer body starts to yield when the peak pressure reaches about 1.6 times its yield strength for a sphere or about 1.67 times for a line contact. The calculator reports the safety factor on load for first yield, which is the load multiple at which this happens. Allowable pressures for rolling fatigue life, hardness and case depth are separate checks and are not covered here.
Does the calculator handle concave surfaces?
Yes. Choose a ball in a spherical socket or a cylinder in a cylindrical groove and enter the socket radius as a positive number larger than the ball radius. The effective radius is then 1 / (1/R1 − 1/R2). Conforming contact gives a larger patch and lower pressure, but the Hertz assumption that the patch is small compared with the radii gets weaker as R2 approaches R1.
What does this calculator not cover?
It covers frictionless, smooth, elastic contact under a static normal load, with the half-space assumption (the patch is small compared with the body size and radii). It ignores friction and tangential traction, surface roughness, plastic deformation, rolling fatigue life, lubrication, thin or case-hardened layers, and elliptical contacts between bodies with different principal curvatures (such as a ball in a bearing groove). It is an engineering aid; verify critical designs.
Engineering aid for guidance. It assumes frictionless, smooth, elastic contact under a static normal load and does not cover friction and tangential traction, roughness, plastic deformation, rolling fatigue life, lubrication, case-hardened layers or elliptical contacts. Material values are typical: use certified data and test the real assembly. Verify critical designs; you remain responsible for the design. Nothing you enter leaves your browser except through a share link you copy yourself.