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Railway Engineering Simulator

Wheel-Rail Contact Simulator — Elliptical Hertz Contact and Adhesion Limit

Quick answer
Wheel–rail contact is an elliptical Hertzian contact: the semi-axes are a = m·(3F·R_eq/E*)^(1/3) and b = n·(3F·R_eq/E*)^(1/3) (steel-on-steel equivalent modulus E* ≈ 115.4 GPa), and the peak contact stress is p_max = 3F/(2π·a·b). The adhesion limit of tangential force follows Coulomb friction, F_t,max = μ·P.

Visualize elliptical Hertz contact between a railway wheel and rail head. Change the axle load, wheel radius, rail crown radius and friction coefficient to learn about the contact ellipse, maximum contact pressure and the adhesion-limit tangential force.

Parameters
Axle load P
kN
Wheel radius R_w
mm
Rail crown radius R_r
mm
Friction coefficient μ
—

Steel-on-steel contact (E = 210 GPa, ν = 0.3) is assumed, giving an equivalent modulus E* ≈ 115.4 GPa. Typical gauges are 1067 mm (conventional rail) and 1435 mm (Shinkansen). Typical friction coefficients are μ ≈ 0.30 when dry, μ ≈ 0.10 when wet, and μ ≈ 0.05 with oil contamination.

While paused, moving a slider updates the results immediately.

Results
—
Contact ellipse semi-major a
—
Contact ellipse semi-minor b
—
Max contact pressure p_max
—
Max tangential F_t = μP (adhesion)
—
Maximum shear stress τ_max ≈ 0.30 p_max
—
Depth of τ_max z ≈ 0.40 a
Wheel-Rail Contact and Contact Ellipse

Left = wheel rolling on a rail and contact ellipse (Hertzian pressure) / Right = subsurface shear-stress field τ(z) and depth of τ_max. Changing the wheel load, wheel radius, or rail curvature updates the display every frame.

Contact Pressure Distribution p(x,y)

Left = wheel/rail cross section / right = top view of contact ellipse (major diameter 2a × minor diameter 2b)

Theory & Key Formulas

Hertz semi-elliptical pressure distribution over the ellipse: p_max at the center and zero at the boundary. Contours show 25/50/75% of the maximum.

Theory & Key Formulas

Wheel-rail contact forms an elliptical Hertzian contact because the curvatures differ in two orthogonal directions. Equivalent curvatures A and B and curvature ratio k:

$$A = \tfrac{1}{2}\!\left(\tfrac{1}{R_{wx}}+\tfrac{1}{R_{rx}}\right),\ B = \tfrac{1}{2}\!\left(\tfrac{1}{R_{wy}}+\tfrac{1}{R_{ry}}\right),\ k=\tfrac{B}{A}$$

Equivalent radius of curvature R_eq and equivalent modulus E* (steel-on-steel: E* ≈ 115.4 GPa):

$$R_{eq} = \tfrac{1}{2\sqrt{A B}},\quad \tfrac{1}{E^*} = \tfrac{1-\nu_1^2}{E_1}+\tfrac{1-\nu_2^2}{E_2}$$

Contact-ellipse semi-major axis a and semi-minor axis b (m and n are coefficients dependent on curvature ratio k), and maximum contact stress:

$$a = m\!\left(\tfrac{3F R_{eq}}{E^* }\right)^{1/3},\quad b = n\!\left(\tfrac{3F R_{eq}}{E^* }\right)^{1/3},\quad p_{max} = \tfrac{3F}{2\pi a b}$$

Maximum tangential force at the adhesion limit (Coulomb's law of friction):

$$F_{t,\max} = \mu\, P$$

The values of m and n depend on curvature ratio k. At k=1, m=n=1 (circular contact); as k increases, the ellipse becomes flatter. This tool evaluates k=1, 1.5, 2, 5, and 10 by linear interpolation of tabulated values.

About the Wheel-Rail Contact Simulator

🙋
How exactly does a train wheel touch the rail? I only have a vague image of it rolling on top of the track.
🎓
Good question. The wheel and the rail actually meet on a small ellipse, not a point or a line. Look at the contact ellipse on the right of the simulator: about 11 mm long and 9 mm wide, so an area of only about 300 mm squared carries the entire 100 kN axle load. That is more than 3 tonnes per square centimeter. The reason it is elliptical is that the wheel has curvature in the rolling direction while the rail has curvature across its crown — the two perpendicular directions have different curvatures.
🙋
So everything is squeezed into that tiny area! The stress must be huge, right?
🎓
Yes. With the defaults (100 kN axle load, 460 mm wheel, 300 mm rail crown) the maximum contact pressure is about 480 MPa, comparable to the tensile strength of mild structural steel (400 to 500 MPa). In practice rails use high-carbon steel (C 0.7 to 0.8 percent) heat-treated to a yield stress of 800 to 1000 MPa, so the surface stays elastic. The real worry is the maximum shear stress at a depth of about 0.3 to 0.5 times a below the surface (around 4 mm). Fatigue cracks nucleate there and grow into the surface flaking we call shelling.
🙋
I am also curious about the maximum tangential force. What is this adhesion limit?
🎓
Trains accelerate by spinning their wheels with motors, but there is an upper limit on the traction that can be transmitted without slipping. It is given by F_t,max = mu times P: with mu around 0.3 (dry rail) and a 100 kN axle load, you can deliver at most 30 kN — exactly what the "Max tangential" card shows. Above that the wheel spins. When rain or wet leaves drop mu to 0.05, you can pull only 5 kN. That is why trains around Tokyo sometimes slip and run late in autumn.
🙋
When I move the wheel radius slider, the ellipse changes size. Does a bigger wheel reduce the stress?
🎓
Yes. In Hertz contact the stress scales as the 2/3 power of the load and the -2/3 power of the equivalent radius. So a larger wheel raises R_eq and lowers the stress. Shinkansen wheels are 910 mm, conventional intercity wheels 860 mm, and freight wheels also 860 mm — they are all kept large to spread the load. The same is true for the rail: increasing R_r from 300 mm to 600 mm enlarges the contact area and lowers the stress. Try the R_r slider and see.

Frequently Asked Questions

The wheel has a radius R_w (typically 460 mm) in the rolling direction but a nearly flat tread in the transverse direction. The rail is straight along the track but its crown has a radius R_r (typically 300 mm) in the transverse direction. With different curvatures in two perpendicular directions, Hertz theory gives an elliptical patch rather than a circular one. The further the curvature ratio k = B/A is from 1, the more elongated the ellipse becomes; at the standard condition k = 1.53 the semi-major/semi-minor ratio is about 1.30.
During traction the wheel spins (slip), and during braking it locks and slides. Spinning causes intense friction at the contact, melting and roughening the tread and locally hardening the rail. Sliding wears a wheel flat on a single spot, producing impact noise and large dynamic loads. Modern electric trains detect slip and slide and automatically reduce torque using re-adhesion control, maintaining stable acceleration and braking even in wet weather.
Water enters the contact and prevents the metal-to-metal adhesion that normally provides friction, so the apparent coefficient falls. mu drops from about 0.30 dry to about 0.10 wet, and to about 0.05 with oil or leaf sap contamination. The leaf problem in autumn causes major delays in the UK, Germany and Japan. Countermeasures include sanders, ceramic particle injection, and on-board friction modifier (FM) application.
In Hertz contact the material is under triaxial compression, so the maximum principal stress can exceed the uniaxial yield stress without producing yield. Plastic flow starts when the maximum shear stress at depth 0.3 to 0.5 times a (approximately 0.3 times p_max) exceeds the uniaxial shear yield stress (about half the yield stress). For railway high-carbon steel (yield 800 to 1000 MPa) the elastic limit is around p_max approx 1500 MPa, but repeated contact still drives rolling contact fatigue (RCF). See hertz-contact.html for spherical Hertz contact, and rolling-contact-fatigue.html and rolling-contact-stress.html for RCF.

Applications in Practice

Tread profile design for rolling stock: For Shinkansen, conventional intercity trains, metros and trams, the combination of wheel tread curvature (conicity) and rail head curvature determines contact stress and running stability. The Shinkansen uses a modified conicity of 1/40 with the 300 mm crown of a 60 kg rail to balance track following and contact stress.

Rationale for axle load limits: Axle loads are limited by regulation in every country to protect rails and the subgrade. Japanese conventional lines allow about 16 to 17 tonnes, the Shinkansen 16 tonnes, European freight 22.5 tonnes, and US freight up to 32 tonnes. Higher axle loads raise contact stress and shorten the life of rails, wheels, bridges and ballast — they are set by trading maintenance cost against capacity.

Predictive maintenance of shelling and spalling: Surface flaking on rails (shelling) and damage near the wheel flange (head checks) are fatigue failures driven by repeated Hertz contact stress. JR companies in Japan run periodic ultrasonic and image-based inspection cars to forecast damage growth and grind the surface preventively. In the US, cab-mounted laser sensors provide continuous monitoring.

Curve negotiation and wear management: On curves the outer rail sees flange contact and the inner rail sees tread contact, so the contact state changes dramatically. On tight curves with radius below 300 m, flange or gauge-face lubrication lowers the friction coefficient to control flange wear and gauge-corner cracks. Modern light-rail vehicles also use independently rotating wheels and steering bogies to optimize the contact state.

Common Misconceptions and Pitfalls

The most common misconception is to assume that contact stress is proportional to axle load. In Hertz contact, stress actually scales as the 1/3 power of the load: doubling the axle load raises the contact pressure by only about 1.26 times, because the contact area itself grows as the 2/3 power of the load. Try increasing the axle load from 50 kN to 200 kN (4 times) in the simulator and you will see that p_max grows by only about 1.59 times. Conversely, reducing the axle load gives only limited relief — improving the rail crown curvature or enlarging the wheel diameter is much more effective.

The next common error is to think that raising the friction coefficient endlessly improves traction. The adhesion limit F_t = mu times P is indeed proportional to mu, but pushing mu too high accelerates wear on both the wheel and the rail and shortens rollout distance, hurting energy efficiency. In practice, friction modifiers are applied differently in different zones so that drivers see good adhesion under traction, while coasting and braking zones keep moderate rolling resistance. A single Shinkansen consist may consume tens of liters of friction modifier per service.

Finally, remember that this simulator is an idealized static Hertz analysis. In reality wheel-rail contact is also subject to dynamic loads from train motion (track irregularities, joint impacts), thermal stresses (summer rail buckling), impact loads from wheel flats, and creep forces (tangential forces under micro-slip). The p_max approx 480 MPa shown here is the static Hertz stress; designers multiply it by a dynamic amplification factor (DAF) of 1.5 to 2.0. Rolling contact fatigue, corrugation, hunting and other phenomena outside the scope of this tool also matter. Detailed evaluation uses dedicated CAE software (CONTACT, VAMPIRE, etc.) together with full-scale testing.

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How to Use

  1. Set wheel radius (mm) using the slider: typical values range 400–500 mm for freight cars, 350–380 mm for commuter rail after wear.
  2. Set rail radius (mm) using the slider: standard rail crowning gives ~300 mm nominal radius of curvature.
  3. Adjust normal load P (kN) with the slider: freight wagon axles typically apply 80–120 kN; light rail 40–60 kN.
  4. Set coefficient of friction μ using the slider: dry steel-on-steel ~0.4–0.5; wet or oiled surfaces drop to 0.2–0.3.
  5. Read contact ellipse semi-major axis a, semi-minor axis b (mm), peak Hertzian pressure p_max (GPa), and maximum tangential adhesion force F_t (kN).

Worked Example

Freight wagon: wheel radius 460 mm, rail radius 310 mm, axle load 100 kN, μ = 0.35 (light surface moisture). Simulator returns: a = 18.2 mm, b = 12.6 mm, p_max = 1.24 GPa, F_t = 35 kN adhesion limit. This means the 100 kN vertical load creates an elliptical contact patch 36.4 mm long × 25.2 mm wide with peak pressure exceeding typical rail yield (~1.0 GPa), and traction is capped at 35 kN before wheel slip occurs.

Practical Notes

  1. Worn wheels (increased radius) and flange contact (rail radius change) shift contact ellipse geometry; monitor b/a ratio—ratios <0.5 indicate narrow, high-pressure strips prone to shelling.
  2. On grades or curves, add lateral load; μ-values drop 15–25% in wet or sandy conditions, reducing F_t and increasing slip risk in traction drives.
  3. p_max >1.3 GPa in standard rail typically triggers accelerated rolling-contact fatigue; compare your result against rail grade (R260, R350) yield limits.