Maximum Speed on a Railway Curve (49 CFR 213.57)

Also known as railway curve speed limit · FRA curve speed · maximum timetable speed on a curve · cant and cant deficiency speed · track elevation and speed · Vmax = sqrt((Ea+Eu)/(0.0007 D)) · superelevation of railway track speed · unbalanced elevation curve speed · rail curve speed from degree of curvature · FRA track safety standards curves · equilibrium speed railway curve

V_{max} = \sqrt{\frac{E_a + E_u}{0.0007\,D}}\qquad (V\text{ in mph},\ E\text{ in inches},\ D\text{ in degrees})

sig figs

Worked example: Table 1: 1 degree, 0 in elevation, 3 in cant deficiency → 65 mph. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Change any unit and the answer follows.

Maximum Speed on a Railway Curve (49 CFR 213.57) explained

This is the formula in the United States Federal Railroad Administration's Track Safety Standards (49 CFR 213.57(b)) for the highest posted timetable speed on a curve. A train rounding a curve wants to go straight on. Raising the outside rail above the inside one, the cant, balances part of that sideways force. The train is also allowed to run with some of it left unbalanced, and the amount, stated as a height in inches like the elevation, is the cant deficiency. Both lift the permitted speed.

The terms: $V_{max}$ in miles per hour, $E_a$ the actual elevation of the outside rail in inches, $E_u$ the qualified cant deficiency in inches, and $D$ the degree of curvature in degrees, counted on a 100-foot chord (1 degree is a radius of about 5 730 ft). Because $E_a$ and $E_u$ enter as a sum, more cant and more allowed deficiency are interchangeable in the formula, while a sharper curve, a larger $D$ , lowers the speed. The regulation sets $E_u$ at 3 inches for every vehicle and allows more for vehicles the FRA approves, and it caps the elevation at 8 inches on the lowest two classes of track and 7 inches on classes 3 to 5. The constant 0.0007 belongs to these units of miles per hour, inches and degrees.

The regulation's Appendix A, Table 1 (three inches of unbalance) gives the published checks, in whole miles per hour: 65 mph on a 1 degree curve with no elevation, 93 mph with 3 inches of elevation, 93 mph on a half-degree curve with no elevation, 160 mph on that half-degree curve with 6 inches, and 46 mph on a 2 degree curve with no elevation. The table is rounded, so the allowance is half a mile an hour and a little more. A case worked by hand: 3 inches of elevation and 3 inches of cant deficiency on a 2 degree curve give $\sqrt{6/0.0014}=65.4654$ mph. The page also solves for the elevation, the cant deficiency or the curvature.

It is a regulatory formula for the posted timetable speed, in the units the regulation uses; the degree-of-curve-radius page relates the degree of curvature to the radius.

Maximum Speed on a Railway Curve (49 CFR 213.57) formula

V_{max} = \sqrt{\frac{E_a + E_u}{0.0007\,D}}\qquad (V\text{ in mph},\ E\text{ in inches},\ D\text{ in degrees})

Where

$V_{max}$ = Maximum allowable operating speed (mph)
$E_a$ = Actual elevation of the outside rail (in)
$E_u$ = Qualified cant deficiency (in)
$D$ = Degree of curvature (°)

Where this comes from

U.S. Code of Federal Regulations, 49 CFR 213.57, 'Curves; elevation and speed limitations' (Federal Railroad Administration, Track Safety Standards; US federal work, public domain), read 23 Oct 2026 at the Legal Information Institute: '(b) The maximum allowable posted timetable operating speed for each curve is determined by the following formula: V_max = sqrt((E_a + E_u)/(0.0007 D))', with V_max in m.p.h., E_a the actual elevation of the outside rail in inches, E_u the qualified cant deficiency in inches and D the degree of curvature in degrees; '(a) the maximum elevation ... may not be more than 8 inches on track Classes 1 and 2, and 7 inches on track Classes 3 through 5'; '(c) All vehicles are considered qualified ... with a cant deficiency, E_u, not exceeding 3 inches'; and Appendix A, Table 1, 'Three Inches Unbalance', the speeds in whole m.p.h. by elevation of the outer rail and degree of curvature: 0 degrees 30 minutes 93 mph at no elevation and 160 at 6 inches; 1 degree 65 at no elevation and 93 at 3 inches; 2 degrees 46 at no elevation. Read it