Minimum railway curve radius


The minimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination with superelevation in the case of train tracks, determines the maximum safe speed of a curve. The minimum radius of a curve is one parameter in the design of railway vehicles as well as trams; monorails and automated guideways are also subject to a minimum radius.

History

The first proper railway was the Liverpool and Manchester Railway, which opened in 1830. Like the tram roads that had preceded it over a hundred years, the L&M had gentle curves and gradients. Reasons for these gentle curves include the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. The gentler the curves, the greater the visibility, thus boosting safety via increased situational awareness. The earliest rails were made in short lengths of wrought iron, which does not bend like later steel rails introduced in the 1850s.

Factors affecting the minimum curve radius

Minimum curve radii for railroads are governed by the speed operated and by the mechanical ability of the rolling stock to adjust to the curvature. In North America, equipment for unlimited interchange between railroad companies are built to accommodate for a radius, but normally a radius is used as a minimum, as some freight carriages are handled by special agreement between railroads that cannot take the sharper curvature. For the handling of long freight trains, a minimum radius is preferred.
The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost all the equipment is proportionately smaller. But standard gauge can also have tight curves, if rolling stocks are built for it, which however removes the standardisation benefit of standard gauge. Tramways can have below curve radius.

Steam locomotives

As the need for more powerful locomotives grew, the need for more driving wheels on a longer, fixed wheelbase grew too. But long wheel bases do not cope well with curves of a small radius. Various types of articulated locomotives were devised to avoid having to operate multiple locomotives with multiple crews.
More recent diesel and electric locomotives do not have a wheelbase problem, as they have flexible bogies, and also can easily be operated in multiple with a single crew.
Not all couplers can handle very short radii. This is particularly true of the European buffer and chain couplers, where the buffers extend the length of the rail car body. For a line with a maximum speed of, buffer-and-chain couplers increase the minimum radius to around. As narrow-gauge railways, tramways, and rapid transit systems normally do not interchange with mainline railroads, instances of these types of railroad in Europe often use bufferless central couplers and build to a tighter standard.

Train lengths

A long heavy freight train, especially those with wagons of mixed loading, may struggle on short radius curves, as the drawgear forces may pull intermediate wagons off the rails. Common solutions include:
A similar problem occurs with harsh changes in gradients.

Speed and cant

As a heavy train goes around a bend at speed, the centripetal force may cause negative effects: passengers and cargo may feel unpleasant forces, the inside and outside rails will wear unequally, and insufficiently anchored tracks may move. To counter this, a cant is used. Ideally, the train should be tilted such that resultant force acts vertically downwards through the bottom of the train, so the wheels, track, train and passengers feel little or no sideways force. Some trains are capable of tilting to enhance this effect for passenger comfort. Because freight and passenger trains tend to move at different speeds, a cant cannot be ideal for both types of rail traffic.
The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing centripetal force: θ is the angle by which the train is tilted due to the cant, r is the curve radius in meters, v is the speed in meters per second, and g is the standard gravity, approximately equal to 9.81 m/s²:
Rearranging for r gives:
Geometrically, tan θ can be expressed in terms of the track gauge G, the cant ha and cant deficiency hb, all in millimeters:
This approximation for tan θ gives:
This table shows examples of curve radii. The values used when building high-speed railways vary, and depend on desired wear and safety levels.
Tramways typically do not exhibit cant, due to the low speeds involved. Instead, they use the outer grooves of rails as a guide in tight curves.

Transition curves

A curve should not become a straight all at once, but should gradually increase in radius over time. Even worse than curves with no transition are reverse curves with no intervening straight track. The superelevation must also be transitioned. Higher speeds require longer transitions.

Vertical curves

As a train negotiates a curve, the force it exerts on the track changes. Too tight a 'crest' curve could result in the train leaving the track as it drops away beneath it; too tight a 'trough' and the train will plough downwards into the rails and damage them. More precisely, the support force R exerted by the track on a train as a function of the curve radius r, the train mass, and the speed, is given by
with the second term positive for troughs, negative for crests. For passenger comfort the ratio of the gravitational acceleration g to the centripetal acceleration v2/r needs to be kept as small as possible, else passengers will feel large changes in their weight.
As trains cannot climb steep slopes, they have little occasion to go over significant vertical curves. However, high-speed trains are sufficiently high-powered that steep slopes are preferable to the reduced speed necessary to navigate horizontal curves around obstacles, or the higher construction costs necessary to tunnel through or bridge over them. High Speed 1 in the UK has a minimum vertical curve radius of and High Speed 2, with the higher speed of, stipulates much larger radii. In both these cases the experienced change in weight is less than 7%.
Rail well cars also risk low clearance at the tops of tight crests.

Problem curves

GaugeRadiusLocationNotes
N/A JapanChūō Shinkansen
ChinaTypical of China's high-speed railway network
ChinaTypical of China's high-speed railway network
ChinaTypical of high-speed railways
ChinaTypical of China's high-speed railway network
ChinaTypical of high-speed railways
AfricaTypical of medium-speed railways Passenger
AfricaTypical of medium-speed railways Freight
AfricaTypical of medium-speed railways Passenger
AfricaTypical of medium-speed railways Freight
DRCongo Matadi-Kinshasa RailwayDeviated line.
Border Loop -
Wollstonecraft station, Sydney
Homebush triangle -
Turkey
Indian Railways
North American rail networkPreferred minimum on freight main lines
Lithgow Zig Zag40 km/h
North American rail networkMinimum radius for general service
Bay Area Rapid Transit
Batlow, New South WalesWeight limit: and
- restricted to NSW Z19 class 0-6-0 steam locomotives
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In reference to the Batlow Line, 5 x 66'-0" chains does not equal 300 metres, but rather 110.584 metres.
Source: - 1" = 25.4 mm
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Newmarket, New ZealandExtra heavy concrete sleepers
North American rail networkAbsolute minimum radius; not on lines for general service
Windberg Railway - restrictions to wheelbase
Queensland RailwaysCentral Line between Bogantungan and Hannam's Gap
JFK Airtrain-
Washington Metro
London Underground Central line
Gotham CurveCromford and High Peak Railway, Derbyshire, England until 1967
Matadi-Kinshasa Railwayoriginal line.
Welsh Highland Railway
Bernina Railway
Welsh Highland Railwayon original line at Beddgelert
Victorian Narrow Gauge on curves;
Kalka-Shimla Railway
N/A MetromoverRubber-tired, monorail-guided light rail downtown people mover system.
New York Subway
Chicago 'L'
Sydney steam tram
0-4-0		Hauling 3 trailers
Warsaw Commuter RailwaysSide track in Grodzisk Mazowiecki, Poland
Darjeeling Himalayan RailwayThe sharpest curves were originally
Matheran Hill Railway1 in 20 ; on curve; on straight
1,588 mm in revenue,
in yard		Streetcars in New Orleans
San Francisco Municipal RailwayLight rail, former streetcar system
Toronto Streetcar System
Taunton Tramway
Boston Green Line
Newark Light Rail
Chicago Tunnel Company in grand unions. Not in use.