Hertzian Contact Pressure Calculator
p_mean is exactly 2/3 of p_max for every load, radius and material. Values are shown to 4 significant figures.
Reference case: a 10 mm steel ball (R1 = 5 mm, E = 210 GPa, v = 0.3) on a steel flat at 1000 N gives a contact radius of 0.3191 mm and a peak pressure of 4688 MPa. The same ball over a concave R2 = −10 mm socket drops to 2953 MPa at a larger 0.4021 mm contact.
1/R = 1/R1 + 1/R2 (1/R2 left out for a flat) 1/E* = (1 − v1²)/E1 + (1 − v2²)/E2 a = (3·F·R / (4·E*))^(1/3) p_max = 3·F / (2·π·a²) p_mean = F / (π·a²) approach = a² / R Units: mm, N, MPa (E in GPa × 1000). Source: H. Hertz, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985, chapter 4.
Valid for elastic, frictionless, non-conforming contacts only. The subsurface shear stress is not given. For preliminary estimation only, not a substitute for review by a qualified/licensed engineer.
Peak contact pressure of a ball on a flat, a ball or a socket
Ball bearings, cam followers, detent balls and ball-on-flat bolted joints all carry their load on a contact patch far smaller than the parts themselves. Before picking a steel grade or a hardness specification, a mechanical designer needs a defensible peak pressure in MPa. The classical Hertz solution gives it from just the load, the radii and the elastic constants.
This calculator takes the sphere radius R1 and a second radius R2: leave it at 0 or empty for a flat, enter a positive value for a second sphere, or a negative value for a concave surface of that radius. It combines the two bodies into an equivalent radius R and a reduced modulus E*, then returns the contact radius, the maximum and mean pressure and the elastic approach.
A worked case shows the scale: a 10 mm steel ball (R1 = 5 mm, E = 210 GPa, v = 0.3) pressed onto a steel flat with 1000 N gives a contact radius of 0.3191 mm and a peak Hertzian pressure of 4688 MPa. The same ball over a concave R2 = −10 mm socket drops to 2953 MPa at a larger 0.4021 mm contact. That is why conforming geometry is the cheapest way to cut contact stress.
How to use
- Enter the normal load F in newtons.
- Enter the radius R1 of the first body in mm. Set R2 to 0 or leave it empty for a flat, a positive value for a second sphere, or a negative value for a concave surface.
- Enter the Young's moduli E1 and E2 in GPa for the two materials.
- Enter the Poisson ratios v1 and v2 (0 to 0.5).
- Read the contact radius, p_max, p_mean, equivalent radius, E* and approach from the result cards, which refresh as you type.
FAQ
What is the maximum Hertzian contact pressure of a 10 mm steel ball pressed onto a steel flat with 1000 N?
For R1 = 5 mm, E1 = E2 = 210 GPa and v1 = v2 = 0.3, the reduced modulus E* is 115.38 GPa, the contact radius a is 0.3191 mm, p_max is 4688 MPa (4.688 GPa), the mean pressure p_mean is 3126 MPa and the approach is 0.0204 mm.
Why is the mean contact pressure two-thirds of the maximum Hertz pressure?
Hertz gives p_max = 3F/(2πa²) and p_mean = F/(πa²), so p_mean is exactly 2/3 of p_max whatever the load, radii or materials — a 4688 MPa peak always pairs with a 3126 MPa average.
What happens if the second surface is concave instead of flat?
A negative R2 is a concave surface of that radius; the equivalent radius 1/R = 1/R1 + 1/R2 becomes larger, so the contact is bigger and the pressure lower — R1 = 5 mm on a concave R2 = −10 mm at 1000 N with steel gives R = 10 mm, a = 0.4021 mm and p_max = 2953 MPa, versus 4688 MPa on a flat.
When does the Hertz point-contact formula stop being valid?
When 1/R = 1/R1 + 1/R2 is zero or negative — i.e. a concave R2 no larger in magnitude than R1 — there is no Hertz contact and the tool reports that instead of a number; the solution also assumes elastic, frictionless, non-conforming contacts and does not give the subsurface shear stress.