Elliptical Hertz Contact Simulator — General Hertzian Contact
Real-time major and minor semi-axes (a, b) and peak pressure p_max for the elliptical contact patch that appears when two bodies meet with different principal curvatures, such as crossed cylinders. Steel-steel fixed (E=210 GPa, nu=0.3).
Parameters
Normal load F
N
Body 1 principal radius R_1x
mm
Body 2 principal radius R_2x
mm
Curvature ratio B/A
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Simplification: R_1y = R_1x (body 1 is a sphere); body 2's R_2y is derived from B/A. Both bodies are steel (E=210 GPa, nu=0.3) by default.
Results
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Major semi-axis a [mm]
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Minor semi-axis b [mm]
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Peak contact pressure p_max [MPa]
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Aspect ratio a/b
Contact ellipse and pressure profile
Left: top view of the contact ellipse (sized 2a x 2b). Right: pressure distribution along the major axis (half-ellipsoid, colored by pressure).
Equivalent radius $R_{eq} = 1/(2\sqrt{AB})$. The coefficients $m, n$ depend on the curvature ratio $k = B/A$ and are obtained here by linear interpolation between tabulated values.
Major and minor semi-axes of the contact ellipse:
$$a = m\left(\frac{3FR_{eq}}{E^*}\right)^{1/3},\qquad b = n\left(\frac{3FR_{eq}}{E^*}\right)^{1/3}$$
$$p_{max} = \frac{3F}{2\pi a b},\qquad p_{avg} = \frac{F}{\pi a b}$$
At $k = 1$ (ball contact) $m = n = 1$ and the formulas reduce to the classical Hertz sphere contact.
What is the Elliptical Hertz Contact Simulator?
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Hertz contact gives you a circular patch when a ball presses into a flat plate, right? Why is this one called "elliptical"?
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Good catch. A ball on a plate has the same curvature in every direction, so the patch comes out as a circle. But two crossed cylinders (axes at 90 degrees) have a tight curvature in one direction and a gentle one in the other — the patch stretches into an ellipse. Set the B/A slider above to 1 and you get a circle; push it up to 5 or 10 and the ellipse grows long and thin.
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I see — the pressure plot on the right keeps the same dome shape while the patch on the left gets squashed. Why do we need a lookup table of coefficients? Isn't there a clean formula?
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The exact solution involves complete elliptic integrals, so there is no closed-form algebraic expression. Engineering practice uses tabulated coefficients m and n from Roark or Johnson's textbook: a = m*(3FR_eq/E*)^(1/3), b = n*(3FR_eq/E*)^(1/3). This tool linearly interpolates the standard table points (k = 1, 2, 5, 10, 50), which is accurate enough for design first cuts and matches handbook values closely.
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Is the peak pressure formula the same as the ball case?
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Same form: p_max = 3F/(2*pi*a*b). The reason is that the pressure distribution is a half-ellipsoid whose volume equals the total load F. Once a and b are known, the peak pressure follows for any ellipse shape. The mean pressure is F/(pi*a*b), so peak is always 1.5 times the mean — that ratio holds regardless of how elongated the patch becomes.
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What about wheel on rail — circle or ellipse?
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Classic elliptical contact. The wheel tread and rail crown have different curvatures along and across the direction of travel, so the patch is an ellipse elongated along the rolling direction. Real wheel/rail analysis uses dedicated tools because the curvatures vary, but this simulator captures the right first-order trends. For a deeper dive see wheel-rail-contact.html.
FAQ
With k=1 the coefficients become m=n=1, the major and minor axes are equal, and the contact patch is a perfect circle. This recovers the standard ball contact case (see hertz-contact.html), confirming that this general formulation contains ball contact as a special case. Sweeping the load F from 1000 N to 8000 N you can also verify that a grows by roughly a factor of 2 (since 8^(1/3) is about 2).
This tool focuses on the geometric effect of the ellipse and on how the m, n coefficients change with curvature ratio, so material properties are fixed to keep the variation purely geometric. Steel-on-steel is the most common engineering case and gives E* = 115.4 GPa. If you need to vary materials or model dissimilar pairs, use hertz-contact.html in combination with this tool.
The limit B/A → infinity corresponds to line contact. This tool clamps to k = 50 (the upper tabulated point) and does not extrapolate beyond it. For very elongated contacts (k much greater than 50) treat the geometry as line contact instead — see hertz-line-contact.html. In real engineering practice (crossed cylinders, ball bearing grooves, etc.) k usually stays between 2 and 20.
This simulator only covers the surface pressure on the contact patch. For principal and shear stresses below the surface (the seeds of rolling-contact fatigue), see subsurface-stress-hertz.html. In elliptical contact the peak shear stress sits at a depth between roughly 0.5b and 0.5a, that is, between the ball-contact (0.48a) and line-contact (0.78a) limits.
Real-world applications
Ball-bearing groove contact: the ball and the inner/outer race groove have different curvature radii along and across the rolling direction. Race grooves are typically machined with a radius 0.515 to 0.52 times the ball diameter, giving an elliptical contact patch elongated in the circumferential direction. Tighter conformity lowers stress but increases rolling friction and heat, so groove design is a stress/life vs friction trade-off.
Crossed cylinders and perpendicular rollers: orthogonally crossed cylinders show up in needle-bearing chamfers, cam followers, and tensile-test specimen grips. For the same load, elliptical contact gives a smaller area and higher stress than line contact, so accurate evaluation requires a general Hertzian model like this one.
Helical and spiral-bevel gear tooth contact: in helical or spiral-bevel gears the tooth surfaces have different curvatures along the helix direction and across the tooth height, so the contact patch is elliptical. ISO 6336 pitting-life calculations account for this and the resulting stresses can be tens of percent higher than for pure Hertz line contact.
Wheel-rail contact: the most familiar elliptical Hertz application. On straight track the patch is close to circular or a short ellipse; in tight curves with flange contact the patch becomes very long and narrow with sharp pressure peaks. As the rail head wears, both the patch shape and the stress level evolve over time.
Common misconceptions and caveats
The most common misconception is to think that the formula for p_max is different for circular and elliptical patches. In reality both use p_max = 3F/(2*pi*a*b); the circular case is just the special case a = b. As you sweep the curvature ratio in this tool the formula for p_max never changes — only the values of a and b do — because the pressure distribution is always a half-ellipsoid.
Next, watch out for interpolation error in the m, n coefficients. This tool linearly interpolates five table points (k = 1, 2, 5, 10, 50). The true values come from complete elliptic integrals, and linear interpolation typically introduces a few percent error around k = 2 to 5. For research-grade work use the Hamrock-Brewe fit or direct integral evaluation, but for first-cut design and teaching this accuracy is more than enough.
Finally, never forget the Hertz assumptions. The theory presumes perfectly elastic, smooth, friction-free, small-deformation contact. Above the steel yield-equivalent pressure (around 4 GPa peak), with surface roughness comparable to contact width, or under thin boundary-lubrication films, the predicted values become first approximations only. For lifetime predictions of real components you also need to include frictional heating, subsurface shear, residual stress, and tribological history. For deeper analysis: ball contact in hertz-contact.html, line contact in hertz-line-contact.html, subsurface stress in subsurface-stress-hertz.html, wheel-rail in wheel-rail-contact.html.