Track Superelevation Calculator
Calculate rail cant (superelevation) from curve radius and design speed using the AREMA formula e = 0.0007 × D × V².
About this calculator
Equilibrium superelevation is the amount of rail cant that makes a train's net lateral force at the wheel-rail interface zero at a given design speed — in other words, the banking angle where gravity alone balances the outward pull of the curve. This calculator uses the standard formula published in Chapter 5 (Track) of AREMA's Manual for Railway Engineering, e = 0.0007 x D x V^2, where D is the degree of curve (the 100-ft-chord definition, D = 5729.578 divided by radius) and V is design speed in mph; note the formula multiplies by D rather than dividing by radius directly, since D already folds that conversion in. Because railroads rarely build to the full equilibrium value — passenger comfort and freight stability both suffer at extremes — the applied (actual) cant is capped at whatever maximum you set (AREMA guidance: 6 in for freight track, up to 8 in for passenger).
The gap between equilibrium and applied cant is the cant deficiency, or unbalanced superelevation, which produces the sideways lean passengers feel; it's checked against your maximum allowed deficiency (commonly 3 in for freight). From there the calculator inverts the formula to find the maximum safe speed for the applied cant plus deficiency limit, and estimates the spiral transition (run-off) length needed to ramp cant smoothly into the curve, using AREMA's 31x-cant-length rule below 60 mph and 62x above it. Mixing up applied cant with equilibrium cant is the most common error — equilibrium is a theoretical target, not what track should actually be built to.
Inputs
Results
Equilibrium cant
7.22 in
≈ 2 credit cards
Figures current as of 2026. Source: American Railway Engineering and Maintenance-of-Way Association (AREMA), Manual for Railway Engineering, Chapter 5: Track
How to Use This Calculator
- Enter Design speed, Curve radius, and Max actual cant.
- Set Max cant deficiency.
- Review the Equilibrium cant (in) result.
- Use Applied cant (in) and Cant deficiency (in) to inform your decision.
How the result changes with Design speed
| Design speed | Equilibrium cant |
|---|---|
| 30 | 1.81 in |
| 45 | 4.06 in |
| 90 | 16.24 in |
| 150 | 45.12 in |
What each input means
- Design speed
- Maximum operating speed through the curve.
- Curve radius
- Radius of the horizontal curve in feet.
- Max actual cant
- Maximum allowed superelevation (6 in freight, 8 in passenger per AREMA).
- Max cant deficiency
- Maximum allowed unbalanced superelevation (typically 3 in for freight).
What each result means
- Equilibrium cant
- Theoretical superelevation for zero lateral force at design speed.
- Applied cant
- Actual superelevation limited by the maximum allowed.
- Cant deficiency
- Unbalanced superelevation (equilibrium minus actual).
- Deficiency within limit
- 1 = within limit, 0 = exceeds maximum allowed cant deficiency.
- Degree of curve
- Standard chord definition: D = 5729.578 / R.
- Max safe speed
- Maximum speed for the given actual cant plus maximum unbalanced.
- Spiral run-off length
- Minimum transition length to ramp from zero to full cant (AREMA).
- Lateral acceleration
- Uncompensated lateral acceleration experienced by passengers/freight.
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersDesign speed = 60, Curve radius = 2000, Max actual cant = 6, Max cant deficiency = 3 = 4 input(s) provided
- Calculate Equilibrium cantEquilibrium cant = 0.0007 * degreeOfCurve * speedMph * speedMph7.219 = 7.219
- Calculate Applied cantApplied cant = min(equilibriumCant, maxCantIn)6 = 6
- Calculate Cant deficiencyCant deficiency = equilibriumCant - actualCant1.219 = 1.219
Figures and sources
- AREMA equilibrium superelevation formula and maximum cant / run-off design standards (2026) — American Railway Engineering and Maintenance-of-Way Association (AREMA), Manual for Railway Engineering, Chapter 5: Track
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does the calculator multiply by degree of curve instead of dividing by radius?
The AREMA Manual for Railway Engineering's formula e = 0.0007 × D × V² already has the radius-to-curvature conversion baked into D, since degree of curve is defined as D = 5729.578 / R. Multiplying by D produces the same relationship as dividing by radius directly, just expressed in the units railroaders traditionally use for curve sharpness — sharper curves (smaller radius) have a larger D and therefore a larger equilibrium cant, exactly as physical intuition predicts.
Why can applied cant be lower than equilibrium cant, and what does that cost?
Equilibrium cant is a theoretical target where lateral force is exactly zero at design speed, but AREMA caps actual construction at 6 in for freight or 8 in for passenger track because extreme banking is uncomfortable and can be unstable for slow-moving or stopped trains. The calculator applies that cap directly (actualCant = min(equilibriumCant, maxCantIn)), and whatever equilibrium cant it couldn't deliver becomes cant deficiency — the sideways lean passengers and freight feel through the curve.
What does the maximum safe speed output actually tell me?
It inverts the equilibrium formula to solve for the highest speed at which your applied cant plus your maximum allowed cant deficiency together still satisfy e = 0.0007 × D × V² for this curve's degree of curve. In practice it answers: given how much cant is actually built into this curve and how much unbalanced lean you're willing to tolerate, how fast can a train go through it?
Why does the spiral run-off length formula change above 60 mph?
AREMA's transition-length guidance uses a 31×-applied-cant multiplier at or below 60 mph and doubles it to 62× above that speed, reflecting that faster trains need a gentler, longer ramp from zero cant into full cant to avoid an abrupt jerk as the vehicle enters the curve. The calculator picks whichever multiplier matches your design speed and applies it to the applied (not equilibrium) cant value.
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