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Contact Calculations of a Ball in a Bearing Race

Hertzian contact of a sphere in a circular race
Hertzian contact tool

This calculator evaluates contact stresses of a sphere (ball) in a circular race. This configuration occurs in deep-groove ball bearings and similar rolling-element bearings.

The tool is an extension of the Hertzian Contact Stress Calculator. It computes maximum Hertzian contact pressure, maximum shear stress, rigid approach of the bodies, and the dimensions of the elliptical contact patch.

The underlying equations follow classical Hertz theory and the formulations summarized by Pilkey and Johnson; see the reference section below.

Contact Stress of Ball Calculator

INPUT PARAMETERS
Parameter Sphere Circular race Unit
Poisson's ratio [ ν1, ν2 ] —
Elastic modulus [ E1, E2 ]
Radius of objects R1, R2
R1, R3
Force [F]
 

Note: Use dot “.” as decimal separator.

RESULTS
Parameter Obj-1 Obj-2 Unit
Maximum Hertzian contact pressure [pmax] ---
Maximum shear stress [τmax] ---
Rigid distance of approach of contacting bodies [d] ---
Semimajor axis of contact ellipse [a] ---
Semiminor axis of contact ellipse [b] ---

Definitions

Ball bearing: A rolling bearing that uses balls between inner and outer races to reduce friction and support radial and/or axial loads.

Modulus of elasticity (Young’s modulus): Slope of the stress–strain curve in the linear elastic region under uniaxial loading. Typical values: aluminium ≈ 69 GPa, steel ≈ 200 GPa.

Poisson’s ratio: Ratio of lateral strain to longitudinal strain under uniaxial stress in the elastic range.

Shear stress: Stress component acting tangentially to a surface; it tends to cause sliding between material layers.

Hertzian contact: Localized contact between curved elastic bodies where the nominal point or line contact becomes a finite area and high compressive and shear stresses develop.

List of Equations

Key relationships used in the calculator (Hertzian contact of a sphere in a circular race):

a = 1.145 · na · (F · K · γ)1/3
b = 1.145 · nb · (F · K · γ)1/3
Pmax = 0.365 · nc · [ F / (K2 · γ2) ]1/3
τmax = σc (0.3906 k5 − 1.1198 k4 + 1.2448 k3 − 0.7177 k2 + 0.2121 k + 0.3)
d = 0.655 · nd · (F2 · γ2 / K)1/3
A = ½ (1/R1 − 1/R2)
B = ½ (1/R1 + 1/R3)
γ = (1 − ν12)/E1 + (1 − ν22)/E2
K = 1 / (2/R1 − 1/R2 + 1/R3)
E(e) = ∫0π/2 √(1 − e2 sin2φ) dφ
K(e) = ∫0π/2 dφ / √(1 − e2 sin2φ)
e = √(1 − (b / a)2)
k = b / a
A / B = [ K(e) − E(e) ] / [ (1/k2) · E(e) − K(e) ]
na = (1 / k) · [ 2 k E(e) / π ]1/3
nb = [ 2 k E(e) / π ]1/3
nc = (1 / E(e)) · [ (π2 k E(e) / 4) ]1/3

Note: The polynomial expression for τmax is obtained by curve fitting to Table 4.1 of Ref-2.

List of Parameters

Symbol Parameter
a Semimajor axis of contact ellipse
b Semiminor axis of contact ellipse
Pmax Maximum Hertzian contact pressure
τmax Maximum shear stress
d Rigid distance of approach of contacting bodies
A, B Coefficients in equation for locus of contacting points
E(e), K(e) Complete elliptic integrals of the second and first kind
k Ellipse aspect ratio (b / a)

Supplements

Link Usage
Hertzian Contact Calculator Calculates Hertzian contact parameters such as contact pressure, shear and von Mises stresses for spherical and cylindrical contacts.
Contact Stresses in a Steel Ball Bearing Worked example for contact stresses in a steel ball bearing using the sphere-in-race configuration.

Reference