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 (difference in elevation of the two rails) 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^{[1]} as well as trams;^{[2]}monorails and automated guideways are also subject to a minimum radius.
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.
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 288-foot (87.8 m) radius, but normally a 410-foot (125.0 m) radius is used as a minimum, as some freight carriages (freight cars) are handled by special agreement between railroads that cannot take the sharper curvature. For the handling of long freight trains, a minimum 574-foot (175.0 m) radius is preferred.^{[3]}
The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost all the equipment is proportionately smaller.^{[4]}
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 do not cope well with curves of a small radius. 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 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 60 km/h (37 mph), buffer-and-chain couplers increase the minimum radius to around 150 m (164 yd). 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.
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 (vertical curves).
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.^{[dubious – discuss]} To counter this, a cant (superelevation) 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 ("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: ? 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 ? 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 ? 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 | 120 km/h; 74 mph (33 m/s) |
200 km/h; 130 mph (56 m/s) |
250 km/h; 150 mph (69 m/s) |
300 km/h; 190 mph (83 m/s) |
350 km/h; 220 mph (97 m/s) |
400 km/h; 250 mph (111 m/s) |
---|---|---|---|---|---|---|
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, they use the 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 40m-80m 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)^{[6]} and High Speed 2, with the higher speed of 400 km/h (250 mph), stipulates much larger 56,000 m (183,727 ft) radii.^{[7]} 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 |
---|---|---|---|
N/A (maglev) | 8,000 m (26,247 ft) | Japan | Ch Shinkansen (505 km/h [314 mph]) |
7,000 m (22,966 ft) | China | Typical of China's high-speed railway network (350 km/h [220 mph]) | |
5,500 m (18,045 ft) | China | Typical of China's high-speed railway network (250-300 km/h [160-190 mph]) | |
4,000 m (13,123 ft) | China | Typical of high-speed railways (300 km/h [190 mph]) | |
3,500 m (11,483 ft) | China | Typical of China's high-speed railway network (200-250 km/h [120-160 mph]) | |
2,000 m (6,562 ft) | China | Typical of high-speed railways (200 km/h [120 mph]) | |
250 m (820 ft) | DRCongo Matadi-Kinshasa Railway | Deviated line. | |
240 m (787 ft) | Border Loop | 5,000 long tons (5,100 t; 5,600 short tons) - 1,500 m (4,921 ft) | |
200 m (656 ft) | Wollstonecraft station, Sydney | ||
200 m (656 ft) | Homebush triangle | 5,000 long tons (5,100 t; 5,600 short tons) - 1,500 m (4,921 ft) | |
190 m (623 ft) | Turkey^{[4]} | ||
175 m (574 ft) | Indian Railways | ||
574 ft (175.0 m) | North American rail network | Preferred minimum on freight main lines | |
160 m (525 ft) | Lithgow Zig Zag | 40 km/h | |
410 ft (125.0 m) | North American rail network | Minimum radius for general service | |
120 m (390 ft)^{[9]} | Bay Area Rapid Transit | ||
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) ___________________________________________________________________________ | |
95 m (312 ft) | Newmarket, New Zealand | Extra heavy concrete sleepers^{[10]} | |
288 ft (87.8 m) | North American rail network | Absolute minimum radius; not on lines for general service | |
85 m (279 ft) | Windberg Railway (de:Windbergbahn) | (between Freital-Birkigt and Dresden-Gittersee) - restrictions to wheelbase | |
80 m (262 ft) | Queensland Railways | Central Line between Bogantungan and Hannam's Gap | |
70 m (230 ft) | JFK Airtrain | ||
() | 68.6 m (225 ft) | Washington Metro^{[11]} | |
61 m (200 ft) | London Underground Central line | (between White City and Shepherd's Bush) | |
50 m (160 ft) | Gotham Curve | Cromford and High Peak Railway, Derbyshire, England until 1967 | |
50 m (164 ft) | Matadi-Kinshasa Railway | original line. | |
50 m (164 ft) | Welsh Highland Railway | ||
45 m (148 ft) | Bernina Railway | ||
40 m (131 ft) | Welsh Highland Railway | on original line at Beddgelert | |
40 m (131 ft) | Victorian Narrow Gauge | 16 km/h or 10 mph on curves; (32 km/h or 20 mph on straight) | |
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.^{[12]} |
29 m (95 ft) | New York Subway | ^{[13]} | |
90 ft (27 m) | Chicago 'L' | ||
25 m (82 ft) | Sydney steam tram 0-4-0 |
Hauling 3 trailers | |
22 m (72 ft) | Warsaw Commuter Railways | Side track in Grodzisk Mazowiecki, Poland | |
21.2 m (70 ft) | Darjeeling Himalayan Railway | The sharpest curves were originally 13.7 m (45 ft) ^{[14]} | |
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^{[15]} |
Streetcars in New Orleans | |
43 ft (13.11 m) | San Francisco Municipal Railway | Light rail, former streetcar system | |
10.973 m (36 ft) | Toronto Streetcar System | ||
10.67 m (35 ft) | Taunton Tramway | ||
33 ft (10.058 m) | Boston Green Line | ||
33 ft (10.058 m) | Newark Light Rail | ||
4.9 m (16 ft) | Chicago Tunnel Company | 6.1 m (20 ft) in grand unions. Not in use. |