Contact of Cylinders and Rollers

Contact between two cylinders with parallel axes is the classical Hertzian case of line contact. Before loading, ideal surfaces touch along a line; elastic deformation spreads the load over a narrow contact strip of length \(L\) and total width \(2b\).

Assumptions of the Hertz line-contact model

The relations below apply to smooth, isotropic, linearly elastic bodies with small deformation and a contact width much smaller than the cylinder radii and length. The surfaces are assumed to be long enough that end effects do not dominate. Plastic indentation, roughness, misalignment, coatings, and lubrication effects require additional models.

Reduced properties

For external contact between two convex cylinders, the reduced radius is

\[\frac{1}{R^*}=\frac{1}{R_1}+\frac{1}{R_2}.\]

For a cylinder against a concave cylindrical surface, use the signed-curvature convention consistently. The reduced elastic modulus is

\[\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}.\]

Here \(R_1,R_2\) are radii of curvature, \(E_1,E_2\) are Young's moduli, and \(\nu_1,\nu_2\) are Poisson's ratios.

Contact-strip width

Let \(w=F/L\) be the normal load per unit contact length. For classical Hertzian line contact, the contact half-width is

\[b=\sqrt{\frac{4wR^*}{\pi E^*}}.\]

The total strip width is \(2b\). Use one consistent unit system: for example, \(w\) in N/mm, \(R^*\) in mm, and \(E^*\) in MPa gives \(b\) in mm.

Pressure distribution

Across the strip, the contact pressure follows a semi-elliptical distribution:

\[p(x)=p_0\sqrt{1-\frac{x^2}{b^2}},\qquad |x|\le b,\]

with the maximum pressure at the centre:

\[p_0=\frac{2w}{\pi b}=\sqrt{\frac{wE^*}{\pi R^*}}.\]

The integral of \(p(x)\) across the strip equals \(w\), so \(p_0\) is not an average pressure. At \(x=\pm b\), the ideal Hertz pressure falls to zero.

Worked example

Take two steel rollers with \(R_1=R_2=50\) mm, \(E_1=E_2=210\) GPa, \(\nu_1=\nu_2=0.30\), and \(w=100\) N/mm. Then \(R^*=25\) mm and \(E^*\approx115\) GPa. The calculated half-width is \(b\approx0.17\) mm and the maximum pressure is \(p_0\approx375\) MPa. The small width relative to the radii illustrates why contact stresses are much higher than nominal stresses in the rollers.

Physical interpretation

Increasing \(w\) increases both \(b\) and \(p_0\). A larger reduced radius broadens the contact and reduces pressure concentration, while a larger \(E^*\) narrows the strip and raises the maximum pressure. The maximum shear stress occurs below the surface, which helps explain rolling-contact fatigue and pitting.

Engineering use and limits

The model is used for cylindrical roller bearings, roller supports and simplified wheel–rail contact. Rollers of finite length may have high edge stresses; crowning, profiling or a more detailed numerical model is then used. A Hertz pressure is not by itself a complete design criterion: fatigue, hardness, lubrication, contamination and load cycles must also be checked.

Calculation procedure

1. Identify the local radii and whether contact is external or internal.
2. Find \(w=F/L\), \(R^*\), and \(E^*\).
3. Calculate \(b\), then \(p_0\).
4. Check that the contact width is small and that elastic assumptions are credible.
5. Assess edge effects, cyclic loading, lubrication, and the applicable contact-strength criterion.