What this calculator does
Every curve on a railway has a speed above which it cannot be run — set not by the traction available, but by how much unbalanced lateral acceleration the track, the vehicles and the passengers will tolerate. This calculator works that speed out from three numbers you already have on the plan: curve radius, applied cant, and the cant deficiency limit in your network standard. It also gives you the posted speed once you round down to a 5 km/h board, and the slowest speed the curve can carry before cant excess starts punishing the low rail.
The formula
It is the equilibrium cant relation, rearranged. Equilibrium cant — the cant at which the resultant of gravity and centrifugal force sits square to the plane of the rails — is:
E_eq = (G_eff × V²) / (127 × R)
Solve for speed and you get the form this tool uses:
V = √(E × 127 × R / G_eff)
where V is speed in km/h, R is curve radius in metres, G_eff is the effective gauge in millimetres (the spacing between rail-head centres — about 1500 mm on standard-gauge track), and E is the cant available in millimetres.
The whole trick is what you substitute for E:
- Maximum speed uses
E_a + D_lim. The train is allowed to run above equilibrium by the permitted cant deficiency, so the applied cant and the deficiency allowance both count toward the budget. - Equilibrium speed uses
E_aalone. This is the speed at which the applied cant is exactly right and nothing is left over in either direction. - Minimum speed uses
E_a − X_lim. Below equilibrium the curve is over-canted for the traffic; once that gap exceeds the permitted cant excess you are outside the standard. If the applied cant is less than the excess limit, there is no lower bound at all.
Reading the result
The raw maximum is rarely the number that goes on the board. Speed boards are posted in 5 km/h steps, and you always round down — 124.98 km/h becomes a 120 km/h board, not 125. The tool reports the actual cant deficiency at that rounded speed too, which is the figure worth quoting in a design report: it shows how much margin the posted speed leaves against the limit you nominated.
The minimum-speed output matters on mixed-traffic lines. A curve canted generously for a fast passenger service can be genuinely poor track for a slow loaded freight — the vehicle leans on the low rail, and low-rail head wear, gauge-corner fatigue and geometry deterioration follow. If the tool reports a minimum speed above the speed your slowest regular traffic actually runs, the applied cant is too high for the traffic mix and the compromise cant needs revisiting.
Typical limits
Limits belong to your network, not to this tool. As a guide to the ranges you will see in practice: cant deficiency limits for conventional rolling stock on plain track commonly fall between 75 and 110 mm, with higher values (150 mm and above) permitted only for approved tilting or high-deficiency stock. Cant excess limits are typically in the same 90–110 mm region. Applied cant is usually capped around 150 mm on passenger-dominant track and often 110 mm or less where heavy freight dominates. Confirm the values that apply to you — for example RISSB AS 7635 (Australia), TfNSW ESC 210 (NSW), or EN 13803 (Europe).
Note also that cant deficiency is only one of several checks that can govern a curve speed. Sharp curves are frequently limited instead by turnout geometry, vehicle curving performance, rail wear rates, or a signalling constraint. Treat this calculator as the cant check, not the whole speed assessment.
Worked example
A 900 m radius curve on standard gauge carries 95 mm of applied cant, and the applicable standard permits 110 mm of cant deficiency and 110 mm of cant excess.
The cant budget at maximum speed is 95 + 110 = 205 mm. With 127 × 900 / 1500 = 76.2:
V_max = √(205 × 76.2) = √15 621 ≈ 124.98 km/h
Round down to the next 5 km/h board and you post 120 km/h. At 120 km/h the equilibrium cant is 1500 × 120² / (127 × 900) ≈ 188.98 mm, so the actual deficiency at the posted speed is about 93.98 mm — roughly 16 mm of margin against the 110 mm limit.
The equilibrium speed for 95 mm of cant is √(95 × 76.2) ≈ 85.08 km/h, so traffic running near 85 km/h feels no lateral push at all. Because the applied cant (95 mm) is below the excess limit (110 mm), there is no minimum speed — even a stationary train is within the excess limit on this curve.
FAQ
Why round down to 5 km/h? Because the posted speed is a limit, not a target. Rounding up would sanction operation beyond the deficiency limit you nominated. Some networks post in 10 km/h steps, or in mph — round down in whatever increment your signage standard uses.
Can I use this for a transition curve? No. This is the check for the circular portion of the curve. The transition has its own governing constraints — cant gradient, rate of change of cant and rate of change of cant deficiency — which the transition curve length calculator covers.
What if the answer is higher than line speed? Then the curve is not the constraint, and the line speed stands. That is a useful result: it tells you cant is not what you need to change to go faster here.
Does effective gauge really equal 1500 mm?
It is the conventional value for 1435 mm standard gauge — the nominal gauge plus roughly half a rail head each side, rounded. Using it gives the familiar E ≈ 11.8 V²/R. Narrow and broad gauge options in the tool use 1120 mm and 1670 mm on the same basis.
This tool provides indicative figures for preliminary design and checking. Final speeds must be verified against the standards, rolling-stock approvals and operating rules applicable to your network.