Hertzian Contact Stress: Ball-on-Flat Contact Radius & Max Pressure

Why doubling ball-bearing load increases contact stress by only 26% (2^(1/3))!

At age 23, Heinrich Hertz (1882) solved elastic deformation where a curved ball bearing, cam follower, or train wheel of radius R (m) presses against a flat raceway under normal force F (N)! If balls were truly rigid points, contact area would be zero and stress infinite! In reality, elastic flattening creates a circular contact patch of radius a = ((3 * F * R) / (4 * E_star)) ^ (1 / 3), where Reduced Contact Modulus satisfies 1 / E_star = ((1 - nu_1^2) / E_1) + ((1 - nu_2^2) / E_2)!

Peak contact pressure at the center of the patch equals 1.5x mean pressure: p_max = (3 * F) / (2 * PI() * (a ^ 2))! Because contact area grows with F^(2/3), Peak Hertzian Stress scales with the CUBE ROOT of load (F ^ (1/3))—increasing load 8x (from 1,000 N to 8,000 N on a R = 10 mm steel ball) only DOUBLES p_max from 2,953.47 MPa to 5,906.94 MPa!

Peak Sphere-on-Flat Hertzian Contact Pressure p_max (MPa)

Uses LET to evaluate E_star (Pa) and patch radius a (m) before computing peak central contact stress in MPa.

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Load, Radius, or Modulus", LET(Estar, 1 / ((2 * (1 - (B5 ^ 2))) / B4), a, ((3 * B2 * B3) / (4 * Estar)) ^ (1 / 3), ROUND(((3 * B2) / (2 * PI() * (a ^ 2))) / 1E6, 2)))

Hertzian Contact Patch Radius a (in Millimeters)

Computes elastic circular footprint radius in mm for identical mating materials (E = B4, Poisson nu = B5).

=LET(Estar, 1 / ((2 * (1 - (B5 ^ 2))) / B4), ROUND((((3 * B2 * B3) / (4 * Estar)) ^ (1 / 3)) * 1000, 4))

Further reading

• Microsoft Support — POWER function