Hertz Contact Stress Calculator: Sphere and Cylinder Contact Pressure, Patch Size and Subsurface Shear

How this calculator works

Choose point contact (a sphere) or line contact (a cylinder). Enter the force, the radii of the two bodies (0 for a flat plane), the length of a cylinder and the modulus and Poisson's ratio of each material.

The calculator finds the effective modulus E* and radius R, then for a sphere the contact radius a = (3FR/4E*) to the one third, the peak pressure 3F over 2 pi a squared and the approach.

It gives the greatest shear stress, which is below the surface, where fatigue cracks begin.

Formulas used

• E^* = \left(\frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2}\right)^{-1}
• a = \left(\frac{3FR}{4E^*}\right)^{1/3},\quad p_0 = \frac{3F}{2\pi a^2}
• b = \sqrt{\frac{4FR}{\pi LE^*}},\quad p_0 = \frac{2F}{\pi bL}

Worked example

With the default numbers in the calculator above:

1. Effective (contact) modulus: 115384.62 MPa
2. Effective radius: 10.00 mm
3. Radius of the circular contact patch: 0.40 mm
4. Peak contact pressure at the centre: 2953.47 MPa
5. Approach of the two bodies (the combined flattening): 16.17 micrometres
6. Greatest shear stress, below the surface: 915.58 MPa at 0.19 mm below the surface
7. Average pressure over the patch: 1968.98 MPa (the peak is 1.5 times this)

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Contact Cylinder on a cylinder or plane (line contact).

1. Effective (contact) modulus: 115384.62 MPa
2. Effective radius: 10.00 mm
3. Half width of the contact strip: 0.0743 mm (a strip 0.15 mm wide)
4. Peak contact pressure at the centre of the strip: 428.53 MPa
5. Greatest shear stress, below the surface: 128.56 MPa at 0.0579 mm below the surface
6. Average pressure over the strip: 336.57 MPa (the peak is 4/pi, 1.27, times this)

Common questions

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Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Hertz theory of contact (Roark's Formulas for Stress and Strain, Chapter 14; Johnson, Contact Mechanics); ISO 281 Rolling bearings: dynamic load ratings and rating life; ISO 76 Rolling bearings: static load ratings.

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