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 railways 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 railway companies is built to accommodate for a 2881NaN1 radius, but normally a 4101NaN1 radius is used as a minimum, as some freight carriages (freight cars) are handled by special agreement between railways that cannot take the sharper curvature. For the handling of long freight trains, a minimum 5741NaN1 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] 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 1001NaN1 curve radius.
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, Meyer & Fairlie) 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 60km/h, buffer-and-chain couplers increase the minimum radius to around 1500NaN0. As narrow-gauge railways, tramways, and rapid transit systems normally do not interchange with mainline railways, instances of these types of railway 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 reactive centrifugal force may cause negative effects: passengers and cargo may experience unpleasant forces, the inside and outside rails will wear unequally, and insufficiently anchored tracks may move. 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: 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:
| \tan\theta= | v2 |
| gr |
| r= | v2 |
| g\tan\theta |
| \tan\theta ≈ \sin\theta= | ha+hb |
| G |
| r= | v2 | = | |
|
| Gv2 | |
| g(ha+hb) |
| Curve radius | |||||||
|---|---|---|---|---|---|---|---|
| Cant 160mm, cant deficiency 100mm, no tilting trains | 630m (2,070feet) | 1800m (5,900feet) | 2800m (9,200feet) | 4000m (13,000feet) | 5400m (17,700feet) | 7000m (23,000feet) | |
| Cant 160mm, cant deficiency 200mm, with tilting trains | 450m (1,480feet) | 1300m (4,300feet) | 2000m (7,000feet) | no tilting trains planned for these speeds | |||
See main article: Track transition curve.
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, and the speed, is given by
| R=mg\plusmn | mv2 |
| r |
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 100000NaN0[6] and High Speed 2, with the higher speed of 400km/h, stipulates much larger 560000NaN0 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.
| Radius | Location | Gauge | Notes | ||
|---|---|---|---|---|---|
| NaN8000 | N/A (maglev) | Chūō Shinkansen | |||
| NaN7000 | data-sort-value=1435 | ||||
| NaN5500 | data-sort-value=1435 | ||||
| NaN4000 | data-sort-value=1435 | ||||
| NaN3500 | data-sort-value=1435 | ||||
| NaN2000 | data-sort-value=1435 | ||||
| NaN1200 | data-sort-value=1435 | Typical of medium-speed railways Passenger | |||
| data-sort-value=1435 | Typical of medium-speed railways Freight | ||||
| NaN800 | data-sort-value=1435 | Typical of medium-speed railways Passenger | |||
| NaN800 | data-sort-value=1435 | Typical of medium-speed railways Freight | |||
| NaN250 | data-sort-value="1067" | Deviated line. | |||
| NaN240 | data-sort-value="1435" | 5000abbr=onNaNabbr=on - 15000NaN0 | |||
| NaN200 | data-sort-value="1435" | ||||
| NaN200 | Homebush triangle | data-sort-value="1435" | 5000abbr=onNaNabbr=on - 15000NaN0 | ||
| NaN190 | data-sort-value="1435" | ||||
| NaN175 | data-sort-value="1676" | ||||
| North American rail network | data-sort-value="1435" | Preferred minimum on freight main lines | |||
| NaN160 | data-sort-value="1435" | 40 km/h | |||
| NaN125 | data-sort-value="1435" | Minimum radius for general service | |||
| NaNfeet120NaNfeetft (120ft)[9] | data-sort-value="1676" | ||||
| NaN100 | data-sort-value="1435" | Rolling stock limited to NaN500 and NaN300 - restricted to NSW Z19 class 0-6-0 steam locomotives | |||
| NaN95 | data-sort-value="1067" | Extra heavy concrete sleepers[10] | |||
| NaN87.8 | data-sort-value="1435" | Absolute minimum radius; not on lines for general service | |||
| NaN85 | Windberg Railway | data-sort-value="1435" | (between Freital-Birkigt and Dresden-Gittersee) - restrictions to wheelbase | ||
| NaN80 | Queensland Railways | data-sort-value="1067" | Central Line between Bogantungan and Hannam's Gap | ||
| NaNfeet70NaNfeetft (70ft) | JFK Airtrain | data-sort-value="1435" | |||
| NaN68.6 | Washington Metro[11] | data-sort-value="1429" | |||
| NaN61 | data-sort-value="1435" | (between White City and Shepherd's Bush) | |||
| NaNfeet50NaNfeetft (50ft) | data-sort-value="1435" | Cromford and High Peak Railway, Derbyshire, England until 1967 | |||
| Matadi-Kinshasa Railway | data-sort-value="762" | original line. | |||
| data-sort-value="600" | |||||
| NaN45 | data-sort-value="1000" | ||||
| NaN40 | data-sort-value="600" | on original line at Beddgelert | |||
| Victorian Narrow Gauge | data-sort-value="762" | NaN16 on curves (NaN32 on straightaways) | |||
| NaN37.47 (48°) | data-sort-value="762" | ||||
| NaN30 | N/A (monorail) | Rubber-tired, monorail-guided light rail downtown people mover system.[12] | |||
| NaNfeet29NaNfeetft (29ft) | data-sort-value="1435" | [13] | |||
| NaN27 | Chicago 'L' | data-sort-value="1435" | |||
| NaN25 | Sydney Steam Motor Tram 0-4-0 | data-sort-value="1435" | Hauling 3 trailers | ||
| NaN22 | data-sort-value="1435" | Depot tracks in Grodzisk Mazowiecki, Poland[14] | |||
| NaN21.2 | data-sort-value="610" | Sharpest curves were originally 13.7m (44.9feet)[15] | |||
| NaN18.25 | data-sort-value="610" | 1 in 20 (5%); 8km/h on curve; 20km/h on straight | |||
| NaN15.24 | Streetcars in New Orleans[16] | data-sort-value="1587.5" | Revenue service | ||
| NaN8.53 | data-sort-value="1587.5" | Yard tracks | |||
| NaN13.11 | data-sort-value="1435" | Light rail, former streetcar system | |||
| NaN10.973 | data-sort-value="1495" | ||||
| NaN10.67 | data-sort-value="1067" | ||||
| NaN10.058 | data-sort-value="1435" | ||||
| NaN10.06 | data-sort-value="1435" | ||||
| NaN4.9 | data-sort-value="610" | NaN6.1 in grand unions. Not in use. |