The minimum railway curve radius is the shortest allowable design radius for the centre line of railway tracks under a particular set of conditions. It has an important bearing on constructions costs and operating costs and, in combination with superelevation (difference in elevation of the two rails) in the case of train tracks, determines the maximum safe speed of a curve. Minimum radius of curve is one parameter in the design of railway vehicles^{[1]} as well as trams.^{[2]} Monorails and guideways are also subject to minimum radii.
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. Among other reasons for the gentle curves were the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. There was no signalling at this time, so drivers had to be able to see ahead to avoid collisions with other trains on the line. The gentler the curves, the longer the visibility. The earliest rails were made in short lengths of wrought iron, which does not bend like later steel rails introduced in the 1850s.
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 350-foot (106.7 m) radius, but normally 410-foot (125.0 m) radius is used as a minimum, as some freight cars are handled by special agreement between railroads that cannot take the sharper curvature. For handling of long freight trains, a minimum 717-foot (218.5 m) radius is preferred.
The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost everything is proportionately smaller.^{[3]}
As the need for more powerful (steam) locomotives grew, the need for more driving wheels on a longer, fixed wheelbase grew too. But long wheel bases are unfriendly to sharp curves. Various types of articulated locomotives (e.g. Mallet, Garratt and Shay) were devised to avoid having to operate multiple locomotives with multiple crews.
More recent diesel and electric locomotives do not have a wheelbase problem and can easily be operated in multiple with a single crew.
Not all couplers can handle very sharp curves. This is particularly true of the European buffer and chain couplers, where the buffers extend the profile of the railcar body. For a line with maximum speed 60 km/h (37 mph), buffer-and-chain couplings increase the minimum radius to around 150 m. As narrow gauge railways, tramways and metros 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.
A long heavy freight train, especially those with wagons of mixed loading, may struggle on sharp curves, as the drawgear forces may pull intermediate wagons off the rails. Common solutions include:
A similar problem occurs with harsh changes in gradients (vertical curves).
As a heavy train goes round a bend at speed, the reactive centrifugal force can cause negative effects: passengers and cargo may feel unpleasant forces, the inside and outside rails will wear unequally, and insufficiently anchored track may move. To counter this, a cant (superelevation) is used. Ideally the train should be tilted such that resultant (combined) force acts straight "down" through the bottom of the train, so the wheels, track, train and passengers feel little or no sideways force ("down" and "sideways" are given with respect to the plane of the track and train). 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 9.80665 m/s²:
Rearranging for r gives:
Geometrically, tan θ can be expressed (using the Small-angle approximation) in terms of the track gauge G, the cant h_{a} and cant deficiency h_{b}, 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.
Curve radius | ≤ 33 m/s = 120 km/h |
≤ 56 m/s = 200 km/h |
≤ 69 m/s = 250 km/h |
≤ 83 m/s = 300 km/h |
≤ 97 m/s = 350 km/h |
≤ 111 m/s = 400 km/h |
---|---|---|---|---|---|---|
Cant 160 mm, cant deficiency 100 mm, no tilting trains |
630 m | 1800 m | 2800 m | 4000 m | 5400 m | 7000 m |
Cant 160 mm, cant deficiency 200 mm, with tilting trains |
450 m | 1300 m | 2000 m | no tilting trains planned for these speeds |
Tramways typically do not exhibit cant, due to the low speeds involved, instead use outer grooves of rails as a guide in tight curves.
A curve should not become a straight all at once, but should gradually increase in radius over time (a distance of around 40 m - 80 m for a line with a maximum speed of about 100 km/h). 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.
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 m, and the speed v, 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 v^{2}/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 (section 2) in the UK has a minimum vertical curve radius of 10,000 m (32,808 ft)^{[5]} and High Speed 2, with the higher speed of 400 km/h (250 mph), stipulates much larger 56,000 m (183,727 ft) radii.^{[6]} 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.
Gauge | Radius | Location | Notes |
---|---|---|---|
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 7,000 m (22,966 ft) | China | Typical of China's high-speed railway network (350 km/h) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 5,500 m (18,045 ft) | China | Typical of China's high-speed railway network (250 km/h-300 km/h) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 4,000 m (13,123 ft) | China | Typical of high-speed railways (300 km/h) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 3,500 m (11,483 ft) | China | Typical of China's high-speed railway network (200-250 km/h) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 2,000 m (6,562 ft) | China | Typical of high-speed railways (200 km/h) |
1,067 mm (3 ft 6 in) | 250 m (820 ft) | DRCongo Matadi-Kinshasa Railway | Deviated 1,067 mm (3 ft 6 in) line. |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 240 m (787 ft) | Border Loop | 5,000 long tons (5,100 t; 5,600 short tons) - 1,500 m (4,921 ft) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 200 m (656 ft) | Wollstonecraft station, Sydney | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 200 m (656 ft) | Homebush triangle | 5,000 long tons (5,100 t; 5,600 short tons) - 1,500 m (4,921 ft) |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 190 m (623 ft) | Turkey^{[3]} | |
1,676 mm (5 ft 6 in) | 175 m (574 ft) | Indian Railways | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 160 m (525 ft) | Lithgow Zig Zag | 40 km/h |
1,676 mm (5 ft 6 in) | 120 m (390 ft)^{[8]} | Bay Area Rapid Transit | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 100 m (328 ft) | Batlow, New South Wales | Weight limit: 500 long tons (510 t; 560 short tons) and 300 m (984 ft) - restricted to NSW Z19 class 0-6-0 steam locomotives
___________________________________________________________________________ In reference to the Batlow Line (NSWGR), 5 x 66'-0" chains does not equal 300 metres, but rather 110.584 metres. Source: - 1" = 25.4 mm (generally accepted) ___________________________________________________________________________ |
1,067 mm (3 ft 6 in) | 95 m (312 ft) | Newmarket, New Zealand | Extra heavy concrete sleepers^{[9]} |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 85 m (279 ft) | Windberg Railway (de:Windbergbahn) | (between Freital-Birkigt and Dresden-Gittersee) - restrictions to wheelbase |
1,067 mm (3 ft 6 in) | 80 m (262 ft) | Queensland Railways | Central Line between Bogantungan and Hannam's Gap |
1,435 mm (4 ft 8½ in) | 70 m (230 ft) | JFK Airtrain | |
1,429 mm (4 ft 8 ^{1}⁄_{4} in) | 68.6 m (225 ft) | Washington Metro^{[10]} | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 61 m (200 ft) | London Underground Central line | (between White City and Shepherd's Bush) |
1,435 mm (4 ft 8½ in) | 50 m (160 ft) | Gotham Curve | Cromford and High Peak Railway, Derbyshire, England until 1967 |
762 mm (2 ft 6 in) | 50 m (164 ft) | Matadi-Kinshasa Railway | original 762 mm (2 ft 6 in) line. |
600 mm (1 ft 11 ^{5}⁄_{8} in) | 50 m (164 ft) | Welsh Highland Railway | |
1,000 mm (3 ft 3 ^{3}⁄_{8} in) | 45 m (148 ft) | Bernina Railway | |
600 mm (1 ft 11 ^{5}⁄_{8} in) | 40 m (131 ft) | Welsh Highland Railway | on original line at Beddgelert |
762 mm (2 ft 6 in) | 40 m (131 ft) | Victorian Narrow Gauge | 16 km/h or 10 mph on curves; (32 km/h or 20 mph on straight) |
762 mm (2 ft 6 in) | 37.47 m or 122.9 ft (48°) | Kalka-Shimla Railway | |
N/A (monorail) | 30 m (98 ft) | Metromover | Rubber-tired, monorail-guided light rail downtown people mover system.^{[11]} |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 29 m (95 ft) | New York Subway | ^{[12]} |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 27.43 m (90 ft) | Chicago 'L' | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 25 m (82 ft) | Sydney steam tram 0-4-0 |
Hauling 3 trailers |
610 mm (2 ft) | 21.2 m (70 ft) | Darjeeling Himalayan Railway | The sharpest curves were originally 13.7 m (45 ft) ^{[13]} |
610 mm (2 ft) | 18.25 m (59.9 ft) | Matheran Hill Railway | 1 in 20 (5%); 8 km/h or 5 mph on curve; 20 km/h or 12 mph on straight |
1,588 mm (5 ft 2 1⁄2 in) | 50 ft (15.24 m) in revenue, 28 ft (8.53 m) in yard^{[14]} |
Streetcars in New Orleans | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 43 ft (13.11 m) | San Francisco Municipal Railway | Light rail, former streetcar system |
1,495 mm (4 ft 10 ^{7}⁄_{8} in) | 10.973 m (36 ft) | Toronto Streetcar System | |
1,067 mm (3 ft 6 in) | 10.67 m (35 ft) | Taunton Tramway | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 33 ft (10.058 m) | Boston Green Line | |
1,435 mm (4 ft 8 ^{1}⁄_{2} in) | 33 ft (10.058 m) | Newark Light Rail | |
610 mm (2 ft) | 4.9 m (16 ft) | Chicago Tunnel Company | 6.1 m (20 ft) in grand unions. Not in use. |
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