Clinging to Old Ways: Spirals and Cant Deficiency

Northeastern railroad managers continue to make their jobs and riders’ experience harder by clinging to old track design standards. Many curves around the Northeastern rail system can support higher speeds than trains are allowed through them.

The major limitation on speed through a curve stems from centripetal acceleration. Centripetal acceleration redirects an object’s path off a straight line along a curve; passengers on a train feel it as a sensation of being pulled off one’s feet toward the center of a circle. To partially relieve this sensation, tracks are canted or banked so that the reaction force from the track both resists the train’s weight and provides some centripetal force. The centripetal acceleration still felt by the passenger, i. e. that not provided by the track, is expressed as the amount of additional bank needed for the track to supply all the centripetal force. The term for this is unbalanced bank or cant deficiency. Some residual centripetal acceleration acting on the passenger and vehicle is perfectly safe.

Centripetal acceleration a obeys the following formula, where F is force, m is mass, v is velocity (speed), and r is radius.

F = ma = mv²/r

Mass drops out, giving

a = v²/r

Railroads limit speed over curves based on the radius, the cant, and the cant deficiency, so that centripetal acceleration on the car and passenger remains under safe limits. Railroad documents express cant as the distance by which the outer rail is elevated relative to the inner rail. On Metro-North’s and other US companies’ track charts, the cant in inches is located below a line showing the location of curves as bulges. The number above the line is the curvature in degrees per 100 feet of chord. The cant deficiency is similarly expressed as the additional superelevation required for the track to supply the entire centripetal force. The sum of the cant and cant deficiency divided by the track gauge is the sine of the equivalent bank angle; centripetal acceleration a is proportional to its tangent.

a = g tan(sin^-1((E_A + E_U)/G))
g = gravitational acceleration
E_A = cant deficiency
E_U = cant deficiency
G = track gauge
E_A, E_U, and G must be in the same unit.

In Europe, railroads generally allow 150 mm (6”) of cant and another 150 mm (6”) of cant deficiency on non-tilting trains. Tilting trains typically permit a cant deficiency up to 225 mm (9”). The Northeast Corridor contains curves with up to 6” cant, but cant deficiency is limited to 3”. I have a theory as to why.

The length of a spiral is a secondary factor in allowable speeds over a curve. Spirals are the transitions from curves to straight track. They smooth the change in degree of curvature, tilt, and centripetal acceleration. It is not acceleration that causes discomfort; it is the change in acceleration, called jerk. Trains traveling over a transition from a straight section to a curve have to keep the centripetal jerk within set limits. The standard American Railway Engineering and Maintenance-of-Way Association (AREMA) formula, the de facto US standard, is the following:

L = 1.63 E_u V; L is the spiral length in feet, E_u is the cant deficiency, in inches, and V is the speed in mi/h.

In exceptional circumstances, the formula L = 1.22 E_u V is used.

The latter formula, and presumably the former as well, incorporates an arbitrary factor of 2 based on vague comfort standards. There is substantial English language literature substantiating the notion that the AREMA formulas are too conservative.

In Europe, formulas specify maximum jerk as the change in elevation per second; the standard depends on the country but typically lands around 50 — 55 mm/s change in cant or cant deficiency. Tilting trains are typically allowed 70 — 80 mm/s. Essentially, the standard AREMA formula allows 23 mm/s, less than half of what non-tilting trains are allowed in most of Europe; the latter one allows 31 mm/s. In other words, AREMA standards generally require a spiral of double the length as Europe does to handle a given speed. On closely spaced curves, it is likely that AREMA’s overly conservative jerk allowance depresses speeds.

For example, take the reverse curves at Port Chester, New York, on the New Haven Line. On the basis of their radii, they permit 75 mi/h at 12” equivalent cant (sum of cant and cant deficiency), yet trains are restricted to 45 mi/h. At that speed and at Metro-North’s current limit of 3” cant deficiency, the standard AREMA formula (23 mm/s) dictates a spiral of 280 feet is needed to transition into the tightest curve. In contrast, at the speed the curve actually permits at 6” cant deficiency, it prescribes almost a 750 foot spiral and roughly double that for the length needed to transition between the two halves of the reverse curve.

The European standard of 50 mm/s prescribes a 350 foot spiral, a minor increase in length, for 75 mi/h running at 12” equivalent cant. Alternatively, the currently prescribed spiral should support up to 60 mi/h at that much equivalent cant. That is, at a minimum, a 33% speed increase at one of the biggest speed restrictions on the line, which is worth ten seconds over the roughly half-mile slow zone. The braking and acceleration that is avoided saves yet more time. The slowest spots along the line offer many speedup opportunities that merely require adoption of modern operating standards. Since improved operations come a lot cheaper than new right-of-way, realization of these easy speedups behooves riders and the railroad.