Model point and line contacts using Hertz theory. Switch units, materials, and geometry instantly here. Get peak pressure, contact size, and safety insight now.
| Case | Mode | F | R1 | R2 | E1 / E2 | ν1 / ν2 | Typical output |
|---|---|---|---|---|---|---|---|
| A | Point | 1000 N | 10 mm | 20 mm | 210 / 210 GPa | 0.30 / 0.30 | p0 in MPa, a in mm |
| B | Point | 500 N | 12 mm | Flat | 70 / 210 GPa | 0.33 / 0.30 | Higher a for softer material |
| C | Line | 2000 N | 25 mm | Flat | 210 / 210 GPa | 0.30 / 0.30 | p0 in MPa, width in mm |
Hertz theory combines geometry and elastic compliance through two reduced quantities:
E* (reduced modulus) and R* (reduced radius).
1/E* = (1-ν1²)/E1 + (1-ν2²)/E2
1/R* = 1/R1 + 1/R2 (use 1/R2 = 0 for a flat surface)
a = [(3 F R*) / (4 E*)]^(1/3)
p0 = 3F / (2π a²)
pm = F / (π a²)
δ ≈ a² / R*
Convert total load to load per length: F' = F / L
b = sqrt[(4 F' R*) / (π E*)]
p0 = 2F' / (π b)
pm = F' / (2b)
Localized stresses at touching surfaces often control failure before bulk stresses do. Hertzian contact pressure estimates the peak compressive stress where two curved bodies meet. This is essential for sizing bearings, cam followers, gears, rollers, seals, and test fixtures.
Rolling-element bearings may see peak pressures in the hundreds to thousands of MPa, depending on load, curvature, and material stiffness. Tribology studies use contact size and mean pressure to select lubricants and predict film thickness. Metrology and indentation tests also use Hertz relations to interpret force–displacement data.
Load increases pressure strongly, but geometry and elastic compliance set how fast it rises. Smaller radii concentrate load into a smaller area, increasing peak pressure. Lower modulus materials spread the contact, increasing contact size and lowering peak pressure. Poisson ratio adjusts compliance through the reduced modulus term.
Point contact represents spheres or a sphere on a flat. The contact patch is circular and defined by radius a. Line contact represents cylinders or a cylinder on a flat. The patch becomes a rectangle-like strip with half-width b. The calculator switches equations and, for line contact, converts total load into load per length.
The maximum pressure p0 occurs at the center of the patch. Mean pressure pm is the load divided by contact area (point) or by strip area (line). Use p0 for yield and fatigue screening, and pm for lubrication and wear comparisons.
Hertz theory assumes smooth, elastic bodies and frictionless contact. Real parts have roughness, coatings, residual stress, and temperature-dependent properties. Surface roughness can raise true peak stresses, while coatings may change compliance and shift stresses deeper. If plastic deformation is expected, Hertz results become a lower-bound estimate.
After computing pressure and contact size, compare to allowable contact stress or hardness-based limits. For cyclic loading, assess rolling contact fatigue and pitting risk. Consider subsurface shear stress, edge loading, misalignment, and stress concentrations from grooves or chamfers. If you use a finite element model, these Hertz results provide a reliable starting validation point.
Start with realistic loads and geometry, then explore sensitivity by varying one input at a time. Keep units consistent; output in MPa for pressure and mm for sizes in most machine design contexts. Leave R2 blank for a flat counterface and document assumptions in your report exports. When results approach yield, redesign the curvature or distribute load across more contacts.
A flat counterface is modeled as an infinite radius. Leave R2 blank and the calculator uses 1/R2 = 0, which increases the reduced radius compared with two curved bodies.
Use maximum pressure p0 for quick yield and contact-fatigue screening. Mean pressure pm is better for comparing wear or lubrication conditions because it represents average loading over the contact patch.
Choose line contact for long cylinders or rollers where contact length is large compared with contact width, such as roller bearings or cam rollers. Provide the effective loaded length L.
Softer or lower-modulus materials deform more, increasing contact size. A larger patch distributes the same load, so peak pressure often drops, even though indentation and compliance increase.
The classic Hertz solution assumes frictionless contact and purely normal loading. Significant tangential forces introduce additional shear stresses and may alter damage risk, but the normal pressure estimate is still a useful baseline.
Use it as a first approximation only. Coatings and layered systems change compliance and stress distribution. For thin coatings or large modulus mismatch, specialized contact models or finite element analysis are recommended.
Recheck units, radii, and whether the load should be shared by multiple contacts. Very small radii or very short line-contact length can inflate pressure. If p0 approaches hardness or yield, redesign the contact.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.