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Contact Stresses in a Steel Ball Bearing

Worked example

This example illustrates how to evaluate Hertzian contact stresses in a steel ball bearing. The bearing consists of an inner race, an outer race and 12 balls. The balls and races are made of steel with elastic modulus E = 210 GPa and Poisson's ratio ν = 0.29.

The manufacturer specifies a rated radial load of P0 = 3600 N at 2500 rpm. An empirical relation P = 5 P0 / n is used to estimate the load on the topmost ball, where n is the number of balls:

P = 5 P0 / n = 5 × 3600 / 12 = 1500 N

The goal is to determine at the inner race–ball contact:

  • Maximum Hertzian contact pressure pmax
  • Maximum shear stress τmax
  • Contact ellipse dimensions (semimajor axis a, semiminor axis b)
Geometry and loading for contact stresses in a steel ball bearing

Figure – Geometry of the steel ball bearing contact problem.

Step 1 – Input data for the Hertzian contact analysis

The following material and geometric data are taken from the example and are used as inputs in the contact calculator.

INPUT PROPERTIES SUMMARY
Parameter Symbol Value Unit
Poisson's ratio of ball ν1 0.29 —
Poisson's ratio of race ν2 0.29 —
Elastic modulus of ball E1 210 GPa
Elastic modulus of race E2 210 GPa
Radius of ball R1 4.90 mm
Inner race groove radius R2 5 mm
Pitch radius of ball centers R3 25 mm
Maximum load on one bearing ball P 1500 N

How to reproduce these inputs in AmesWeb

  1. Open the Contact Calculations of Sphere in Circular Race calculator.
  2. Enter the material properties E1, E2, ν1, ν2 and the radii R1, R2, R3 as listed in the table.
  3. Set the load on the ball to P = 1500 N.
  4. Run the calculation to obtain contact pressure, shear stress and contact ellipse dimensions as shown below.

Step 2 – Hertzian contact results for the bearing

Using the input data above in the sphere–race contact calculator, the following results are obtained for the inner race–ball contact.

RESULTS
Parameter Symbol Obj-1 Obj-2 Unit
Maximum Hertzian contact pressure pmax 2193.6 MPa
Maximum shear stress τmax 684.9
Rigid distance of approach
of contacting bodies 		d 0.012 mm
Semimajor axis of contact ellipse a 2.101
Semiminor axis of contact ellipse b 0.156

Engineering interpretation

  • The maximum Hertzian contact pressure of 2193.6 MPa is high but typical for rolling-element bearings with very small contact areas and hardened steel races and balls.
  • The maximum shear stress, τmax = 684.9 MPa, occurs below the surface and is important for subsurface fatigue, pitting and spalling.
  • The contact patch is an ellipse with semiaxes a = 2.101 mm and b = 0.156 mm, showing that the contact is narrow in the rolling direction.

Conclusion

For the specified steel ball bearing and operating load, the Hertzian contact analysis gives:

  • Maximum contact pressure: pmax = 2193.6 MPa
  • Maximum shear stress: τmax = 684.9 MPa
  • Contact ellipse semiaxes: a = 2.101 mm, b = 0.156 mm

FAQ – Ball bearing contact stresses

What are Hertzian contact stresses in ball bearings?

Hertzian contact stresses are localized compressive and shear stresses that develop where the curved ball and raceway are pressed together. They are used to evaluate surface durability and fatigue life of rolling-element bearings.

Why is the contact area elliptical?

In a real bearing, curvature in two perpendicular directions leads to an elliptical contact patch instead of a point. The semimajor axis a and semiminor axis b of the ellipse are obtained from the Hertz equations and depend on geometry and load.

How can I check my own bearing design?

You can use AmesWeb’s Hertzian contact tools such as the Hertzian Contact Stress Calculator and Sphere in Circular Race to evaluate contact pressure and subsurface shear stresses for your own geometry, materials and loading.

Reference