Hertzian contact stress calculator

For point contact (sphere-sphere, sphere-plane) the bodies initially touch at a point; under load a circular contact area of radius a forms. For line contact (cylinder-cylinder, cylinder-plane) they touch along a line; under load a rectangle of width 2b forms over the effective length L. Line contact is therefore computed with the line load F' = F/L and the pressure rises more slowly with force than for point contact.

Select the concave switch for the second body. The radius R2 is then applied with a negative sign: 1/R = 1/R1 − 1/R2. The concave body conforms, the contact area grows and the pressure drops. The prerequisite is |R2| > R1; for nearly equal radii the area becomes elliptical and the simple theory underestimates it - the calculator warns in this case.

No. The negative radius changes the size of the contact area and the level of the pressure, not the shape of the distribution: semi-elliptical across the width 2b for line contact, semi-ellipsoidal across the circular area for point contact. The chart therefore shows the same curve shape in every case, only with different axis values. The reason lies in the half-space approximation underlying the Hertzian solution: there the geometry of both bodies enters solely through the gap, that is through the sum of the curvatures - a negative summand changes the value of that sum, not the form of the equation. Hütte (Das Ingenieurwissen, fig. 5-47) even uses the concave case to depict the distribution, while the convex case beside it shows only the contact area.

For sphere-sphere and cylinder-cylinder with convex bodies it does not: the reduced modulus 1/E* = (1−ν1²)/E1 + (1−ν2²)/E2 and the equivalent curvature 1/R = 1/R1 + 1/R2 are both sums, so swapping gives an identical result - even with dissimilar materials. In the cases involving a plane, body 2 is always the plane and R2 does not enter at all. With a concave counter body the convex one must be R1, because there 1/R = 1/R1 − 1/R2 applies. Swapping them does not produce a wrong number but the message that |R2| has to exceed R1.

Separately for both bodies: steel E = 210,000 N/mm² and ν = 0.3, grey cast iron E ≈ 110,000 N/mm² and ν = 0.25, bronze E ≈ 95,000 N/mm² and ν = 0.35, aluminium E ≈ 70,000 N/mm² and ν = 0.33. You can pick a material from the database that fills E and ν, or enter both values freely. ISO 6336 uses 206,000 N/mm² for steel; the difference affects p_max by only about 1.3 percent.

At the surface there is triaxial compression that does not cause yielding. Failure starts with the shear stress inside the material, whose maximum is about 0.31·p_max (point) or 0.300 to 0.304·p_max (line, depending on source: own numerics 0.3003, Niemann 0.304) at a depth of 0.48·a or 0.78 to 0.79·b. In addition there is the alternating shear stress τ_yz (0.25·p_max at 0.5·b for line, 0.215·p_max at 0.35·a for point contact, after Niemann), which reverses sign as the contact rolls over, lies closer to the surface and is regarded as the actual cause of pitting fatigue. This is exactly where the first cracks (pitting) form under rolling load. The hardening depth must therefore lie well below this zone.

The one with the lower allowable pressure - not the softer one. Young's modulus does not appear in the check at all; through E* it only lowers the pressure itself and thus helps both bodies. Both see the same p_max, but each has its own allowable level derived from its own yield strength, and the lower one governs. Usually that is the body with the lower yield strength; because the conversion factor for point contact depends on Poisson's ratio, it can be the other one in exceptional cases. The calculator therefore compares the allowable pressures and states below the field which body the governing value comes from. Two limits: with a brittle partner (ceramics, glass, grey cast iron) this check does not apply, because the tensile stress at the rim of the contact area fails first. And for case-hardened parts the strength at the depth of maximum shear stress counts, not the surface hardness.

The yield strength of standard materials depends on the product size: thicker parts reach lower values because they harden through more slowly during tempering. C45E is listed at 490 N/mm² for small dimensions and drops below that as the part gets thicker. Enter the governing dimension of your part and the calculator picks the matching step. If the field stays empty, the smallest step applies and thus the highest value of the material - for thick parts that is the unsafe side. If the entry lies beyond the standard table, the calculator says so instead of silently carrying the last value forward.

That depends on the application. For ductile steel under static load the yield onset is at 1.67·Rp0.2 (line) or 1.61·Rp0.2 (point). Hardened rolling pairs withstand 4000 to 4600 N/mm² statically (the level of the static load rating C0 with small permanent deformation), but only about 1500 N/mm² in continuous operation. Worm-wheel bronzes or grey cast iron have their own values. The calculator offers these guide values with context and source and shows only those matching the computed contact type - a ball-bearing value no longer appears next to a line contact. If a selected guide value contradicts the materials above, such as a hardened rolling pair next to POM, the calculator says so instead of silently colouring the check green. Do not mix the contexts.

Hertz theory assumes homogeneous, linearly elastic materials and the half-space: the contact dimensions must be small compared with the radii of curvature (a, b « R). Friction, tangential forces, lubricant film and residual stresses are not covered, nor is the edge stress at roller ends (which is why rollers are crowned). Generally curved bodies with an elliptical contact area (ball in a groove, crossed cylinders) require the elliptical theory with coefficients. For very soft materials such as plastics the values are only indicative because of viscoelasticity.