Hertzian Contact Stress Calculator

Overview

When two curved machine elements are pressed together under a normal force, the theoretical point or line contact expands to a small contact area, creating Hertzian contact stresses. These highly localized stresses are critical in the design of gears, rolling-element bearings, cam–follower mechanisms, ball contacts and cylindrical line contacts. Typical failures appear as surface cracks, pitting or spalling.

This calculator is based on the formulations in Shigley’s Mechanical Engineering Design and can compute:

• Maximum contact pressure \(p_\text{max}\)
• Maximum shear stress in each body
• Depth of maximum shear stress below the surface
• Contact patch size: circular diameter (2a) or rectangular width (2b)
• Shear stress vs depth charts for both objects

Note: For more background on contact stresses, spherical and cylindrical contact, see Chapter 3.19 (Contact Stresses) of Shigley's Mechanical Engineering Design.

Definitions

Cylindrical contact: The contact of two cylindrical parts where the initial line contact becomes a narrow rectangular area under load. The contact area has width 2b and length l.

Modulus of elasticity (Young’s modulus): The ratio of uniaxial stress to strain in the linear (proportional) region. Typical values: Aluminum ≈ 69 GPa, Steel ≈ 200 GPa.

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

Proportional limit: The largest value of stress for which stress and strain remain linearly related (Hooke’s law).

Spherical contact: The contact of two spherical surfaces where a nominal point contact becomes a circular area of radius a under load. A maximum contact pressure \(p_{max}\) acts on this area.

Shear stress: Stress that acts tangential to a surface, tending to cause sliding between adjacent material layers.

Stress: Force per unit area, typically given in MPa (N/mm²) or psi.

List of Equations

Detailed equations and step-by-step derivations used in this Hertzian contact stress calculator are provided here:

List of equations and calculation steps for Hertzian contact stress calculations

Supplements

An additional calculation tool for Contact Calculations of Sphere in Circular Race has been developed to calculate contact stresses in ball bearings.

Example Calculation

Problem: A steel ball is pressed against a flat steel plate with a normal force of 500 N. The ball diameter is 20 mm. Estimate the Hertzian contact pressure and depth of maximum shear stress.

Given:

• Object-1: Steel sphere, diameter \(d_1 = 20\,mm\)
• Object-2: Flat steel plate
• Force \(F = 500\,N\)
• Elastic modulus \(E_1 = E_2 = 210\,GPa\)
• Poisson’s ratio \(ν_1 = ν_2 = 0.30\)

Inputs to the calculator:

1. Object-1 shape: Sphere
2. Object-2 shape: Plane
3. Poisson’s ratio: 0.30 for both objects
4. Elastic modulus: 210 / 210, unit: GPa
5. Diameter: 20 mm for Object-1 (Object-2 diameter hidden for plane)
6. Force: 500 N

Typical results (order of magnitude):

• Maximum contact pressure \(p_{max} \approx 1600 \, MPa\)
• Contact diameter \(2a \approx 0.50 \, mm\)
• Maximum shear stress \(\tau_{max} \approx 480 \, MPa\)
• Depth of maximum shear \(z \approx 0.20 \, mm\)

These values are typical for highly loaded Hertzian contacts such as ball bearings and gear tooth contacts, where contact areas are small and stresses are high.

Reference

• Budynas, R. G., Nisbett, J. K. (2014). Shigley's Mechanical Engineering Design , 10th edition. McGraw-Hill.