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Hertz Line Contact Simulator — Two Parallel Cylinders

Visualize Hertzian line contact between two parallel cylinders. Adjust line load, radii and Young's modulus to see the contact half-width and maximum contact pressure for roller bearings, cams and gear flanks.

Parameters
Line load P'
N/mm
Cylinder 1 radius R1
mm
Cylinder 2 radius R2
mm
Young's modulus E
GPa

Both bodies are assumed to be steel (Poisson's ratio nu = 0.3). For convex-concave contact, enter a large radius for R2.

Results
—
Equivalent radius Re
—
Contact half-width a
—
Max contact pressure pmax
—
Average pressure pavg
Cylinder Cross-Section & Hertz Pressure

Left: cross-section of the two cylinders with the contact strip of width 2a / Right: Hertz pressure p(x)=pmax·√(1-(x/a)²), peak at the centerline.

Theory & Key Formulas

Hertz line contact formulas for two parallel cylinders of radii $R_1$ and $R_2$ pressed together by a line load $P'$ per unit length.

Equivalent radius and equivalent modulus:

$$\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2}, \quad \frac{1}{E^*} = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2}$$

Contact half-width $a$ and maximum contact pressure $p_\text{max}$:

$$a = \sqrt{\frac{4\,P'\,R_e}{\pi\,E^*}}, \quad p_\text{max} = \sqrt{\frac{P'\,E^*}{\pi\,R_e}} = \frac{2\,P'}{\pi\,a}$$

Pressure distribution across the strip and average pressure:

$$p(x) = p_\text{max}\sqrt{1-\left(\frac{x}{a}\right)^2}, \quad p_\text{avg} = \frac{\pi}{4}\,p_\text{max} = \frac{P'}{2a}$$

For convex-concave contact (e.g. ball in an outer-race groove), substitute a negative radius for the concave side into $1/R_e$. Point contact (sphere) uses a different formula where $a$ scales with the cube root of the load.

What is the Hertz Line Contact Simulator

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In a roller bearing, rollers roll between the inner and outer races. How narrow is the strip that actually carries the load?
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That is exactly Hertzian line contact. Roughly speaking, two cylinders touch theoretically along a line, but when pressed together the contact spreads into a thin strip of width $2a$. With the simulator defaults (line load $P'=1000$ N/mm, $R_1=20$ mm, $R_2=30$ mm, steel-steel), the strip is only $2a \approx 0.73$ mm wide — about eight hair widths — and it carries around 1750 MPa, several times the yield stress of typical steel.
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Several times the yield stress? Why doesn't it just deform plastically?
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That is the interesting part — Hertz contact is a triaxial compressive state. Right under the contact, the material is confined by its surroundings, which keeps the shear stress small. Real damage is initiated by the maximum shear stress $\tau_\text{max} \approx 0.30\,p_\text{max}$ at a subsurface depth $z \approx 0.78\,a$. Fatigue cracks grow from there and produce flaking — this is what limits bearing life. In the simulator, increase $P'$ and watch $p_\text{max}$ — it grows with $\sqrt{P'}$, so quadrupling the load only doubles the stress. That is very different from point contact (sphere), which scales as the 2/3 power.
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When I change $R_1$ or $R_2$, which one has more impact?
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Good question. The equivalent radius is a harmonic-mean style combination: $1/R_e = 1/R_1 + 1/R_2$, so the smaller radius dominates. With $R_1=20$ and $R_2=30$ you get $R_e=12$ mm, very close to $R_1$. Doubling $R_2$ from 30 to 60 only raises $R_e$ to 15 mm and drops $p_\text{max}$ by about 10%. Doubling the smaller side $R_1$ from 20 to 40 raises $R_e$ to 24 mm and cuts $p_\text{max}$ by about 30%. Always grow the bottleneck radius first.
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Changing Young's modulus $E$ also moves the stress. Softer material means lower stress, right?
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Right — $p_\text{max} \propto \sqrt{E^*}$, so halving $E$ drops the stress by a factor $1/\sqrt{2}$. A plastic gear ($E \approx 3$ GPa) sees about 1/8 of the contact stress of a steel gear ($E=210$ GPa). The price is a wider contact strip and more deformation, which can hurt accuracy and add noise. Design is always a trade-off. For the point-contact (sphere) case see hertz-contact.html, for subsurface stress distribution see subsurface-stress-hertz.html, for elliptical contact see contact-ellipse-hertz.html, and for wheel-rail line contact see wheel-rail-contact.html.

Frequently Asked Questions

The line contact formula is expressed in terms of the line load P' [N/mm]. For an effective contact length L [mm] and total load P [N], P' = P / L. For example, a 25 mm long roller carrying 25 000 N gives P' = 1000 N/mm. In practice the effective length is shorter than the nominal length because crowning (a barrel profile) is applied to the roller ends to avoid edge stress concentration.
For a convex roller in a concave groove (e.g. a ball in an outer-race groove), substitute a negative radius for the concave side into 1/R_e = 1/R1 + 1/R2. The closer the two radii are, the larger R_e becomes, the contact strip widens and the contact pressure drops. This is the "conformal contact" design strategy. This simulator only accepts positive values, so as an approximation enter a large value for R2 instead.
It depends strongly on the application and material, but as a guideline roller bearings made of high-carbon chromium bearing steel (SUJ2 / AISI 52100) allow p_max = 2000 to 4000 MPa. Gear flanks (case-hardened steel) allow 1500 to 2500 MPa, and shaft-cam contacts 1000 to 2000 MPa. Yield is not reached even when p_max exceeds the uniaxial yield stress because of the triaxial compressive state. Life prediction is done with cycle-based models such as the Lundberg-Palmgren or Harris formulas rather than from p_max alone.
For ideal parallel-cylinder contact use line contact. Cylindrical roller bearings, needle bearings, tapered roller bearings, gear meshing, cam-and-follower and wheel-rail are all line contact. Ball bearings, ball-on-flat and hemispherical tips are point contact. Intermediate cases (tapered or spherical rollers) produce elliptical contact. See hertz-contact.html for detailed point contact and contact-ellipse-hertz.html for elliptical contact.

Real-World Applications

Roller bearings (cylindrical, needle, tapered): This is the most direct application of this simulator. The contact between the roller and the inner/outer race is a classical parallel-cylinder line contact, and the design load is converted into a line load P' before evaluating p_max. For SUJ2 bearing steel, p_max = 4000 MPa is a typical limit for rolling-contact fatigue life. In practice the roller ends are crowned (barrel-shaped) to avoid edge stress concentration and to keep the pressure distribution centered.

Gear flank contact (Hertzian stress): Gear teeth in mesh see a line contact whose curvature varies along the line of action. Gear-design standards such as ISO 6336 and AGMA 2001 evaluate the surface contact stress sigma_H at the pitch point and compare it to an allowable value, with the underlying formula being exactly Hertzian line contact. It is the basis for pitting and flank damage (flaking) predictions.

Cam and follower contact: Cam profiles in contact with roller or flat followers are also evaluated as Hertzian line contact. Since the cam curvature changes continuously through the cycle, R1 (the cam radius) is computed at each angle and p_max at the worst angle is taken as the design-critical value. For automotive valve cams, p_max = 1500 to 2000 MPa is typical.

Wheel-rail contact: The contact between a railway wheel tread and the rail head is approximately a two-cylinder line contact, reaching p_max = 1000 to 1500 MPa. It is the basis for predicting shelling and squat defects on rail heads and the wear of wheel flanges under repeated rolling. See wheel-rail-contact.html for the full treatment.

Common Misconceptions and Cautions

The most common misconception is that "once contact pressure exceeds the yield stress, the material yields immediately". In Hertz contact the material right under the load is in a triaxial compressive state, confined by its surroundings, and the maximum shear stress is less than half of the principal stress. For steel, even when p_max reaches 3 to 5 times the uniaxial yield stress (200 to 800 MPa), a von Mises-based shakedown analysis often stays in the elastic regime. That is precisely why roller bearings can be operated at p_max = 4000 MPa. Do not carry the intuition from uniaxial tensile loading directly to Hertz contact.

The next most common error is to assume that "because the maximum stress is at the surface, surface treatment alone determines life". The peak pressure p_max in line contact does occur at the centerline of the contact strip, but the maximum shear stress tau_max about 0.30*p_max that drives rolling-contact fatigue (RCF) develops at a subsurface depth z about 0.78*a. With the default settings, a = 0.364 mm, so the critical depth is about 0.28 mm. Hardening must therefore cover not just the surface but also that depth — the effective case depth specified for carburizing or induction hardening is chosen with margin above this theoretical critical depth.

Finally, remember that "Hertz formulas ignore friction, lubrication and tangential force". They assume perfectly elastic, frictionless, normal-load-only contact. Real rolling contact has microslip and tangential forces, which move the maximum shear stress closer to the surface (McEwen's solution). Thin oil films can produce surface-initiated damage such as peeling and smudging. EHL (elastohydrodynamic lubrication) analysis and detailed depth-resolved stress evaluation (see subsurface-stress-hertz.html) extend the picture. Hertz formulas are a powerful first-cut design tool, but understanding the boundary of their assumptions is decisive in practice.

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