90-foot (27.43 m) radii on the elevated 4 ft 8+12 in (1,435 mm) standard gauge Chicago 'L'. There is no room for longer radii above this street intersection at this cross junction at the Wells and Lake street intersection in the northwest corner of the loop

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.

## 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,[citation needed] which does not bend like later steel rails introduced in the 1850s.

## Factors affecting the minimum curve radius

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 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 railways 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] 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 100-foot (30.5 m) curve radius.

### Steam locomotives

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, as they have flexible bogies, and also can easily be operated in multiple with a single crew.

### Couplings

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; 492 ft). 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.

### 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:

• marshaling light and empty wagons at the rear of the train
• intermediate locomotives, including remotely controlled ones
• easing curves
• reduced speeds
• reduced cant (superelevation), at the expense of fast passenger trains
• more, shorter trains
• better driver training
• driving controls that display drawgear forces
• Electronically Controlled Pneumatic brakes

A similar problem occurs with harsh changes in gradients (vertical curves).

### Speed and cant

As a heavy train goes around a bend at speed, the centripetal 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.[dubious ] 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:

${\displaystyle \tan \theta ={\frac {v^{2}}{gr}}}$

Rearranging for r gives:

${\displaystyle r={\frac {v^{2}}{g\tan \theta }}}$

Geometrically, tan θ can be expressed (using the Small-angle approximation) in terms of the track gauge G, the cant ha and cant deficiency hb, all in millimeters:

${\displaystyle \tan \theta \approx \sin \theta ={\frac {h_{a}+h_{b}}{G}}}$

This approximation for tan θ gives:

${\displaystyle r={\frac {v^{2}}{g{\frac {h_{a}+h_{b}}{G}}}}={\frac {Gv^{2}}{g(h_{a}+h_{b})}}}$

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.

## Transition 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.

## 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 m, and the speed v, is given by

${\displaystyle R=mg\pm {\frac {mv^{2}}{r}}}$

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 (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.

## List of selected minimum curve radii

N/A (maglev) 8,000 m (26,247 ft) Japan Chūō Shinkansen (505 km/h [314 mph])
1,435 mm (4 ft 8+12 in) 7,000 m (22,966 ft) Chinese high speed railway network 350 km/h [220 mph]
5,500 m (18,045 ft) 250–300 km/h [160–190 mph]
4,000 m (13,123 ft) 300 km/h [190 mph]
3,500 m (11,483 ft) 200–250 km/h [120–160 mph]
2,000 m (6,562 ft) 200 km/h [120 mph]
1,200 m (3,937 ft) Africa Typical of medium-speed railways (120 km/h [75 mph]) Passenger
Typical of medium-speed railways (80 km/h [50 mph]) Freight
800 m (2,625 ft) Typical of medium-speed railways (120 km/h [75 mph]) Passenger
800 m (2,625 ft) Typical of medium-speed railways (80 km/h [50 mph]) Freight
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+12 in) 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]
1,676 mm (5 ft 6 in) 175 m (574 ft 1+34 in) Indian Railways
1,435 mm (4 ft 8+12 in) North American rail network Preferred minimum on freight main lines
160 m (525 ft) Lithgow Zig Zag 40 km/h
125 m (410 ft 1+14 in) North American rail network Minimum radius for general service
1,676 mm (5 ft 6 in) 120 m (390 ft)[9] Bay Area Rapid Transit
1,435 mm (4 ft 8+12 in) 100 m (328 ft) Batlow, New South Wales Rolling stock limited to 500 long tons (510 t; 560 short tons) and 300 m (984 ft) - restricted to NSW Z19 class 0-6-0 steam locomotives
1,067 mm (3 ft 6 in) 95 m (312 ft) Newmarket, New Zealand Extra heavy concrete sleepers[10]
1,435 mm (4 ft 8+12 in) 87.8 m (288 ft 1116 in) 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
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 8+12 in) 70 m (230 ft) JFK Airtrain
1,435 mm (4 ft 8+12 in) 68.6 m (225 ft 1316 in) Washington Metro[11]
1,435 mm (4 ft 8+12 in) 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
762 mm (2 ft 6 in) Matadi-Kinshasa Railway original 762 mm (2 ft 6 in) line.
600 mm (1 ft 11+58 in) Welsh Highland Railway
1,000 mm (3 ft 3+38 in) 45 m (148 ft) Bernina Railway
600 mm (1 ft 11+58 in) 40 m (131 ft) Welsh Highland Railway on original line at Beddgelert
762 mm (2 ft 6 in) Victorian Narrow Gauge 16 km/h or 10 mph on curves (32 km/h or 20 mph on straightaways)
37.47 m or 122 ft 11+316 in (48°) Kalka-Shimla Railway
N/A (monorail) 30 m (98 ft) Metromover Rubber-tired, monorail-guided light rail downtown people mover system.[12]
1,435 mm (4 ft 8+12 in) 29 m (95 ft) New York Subway [13]
27 m (89 ft) Chicago 'L'
25 m (82 ft) Sydney steam tram
0-4-0
Hauling 3 trailers
22 m (72 ft) Warsaw Commuter Railway Depot tracks in Grodzisk Mazowiecki, Poland[14]
610 mm (2 ft) 21.2 m (69 ft 6+58 in) Darjeeling Himalayan Railway Sharpest curves were originally 13.7 m (44 ft 11+38 in)[15]
18.25 m (59 ft 10+12 in) 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+12 in) 15.24 m (50 ft 0 in) Streetcars in New Orleans[16] Revenue service
8.53 m (27 ft 11+1316 in) Yard tracks
1,435 mm (4 ft 8+12 in) 13.11 m (43 ft 18 in) San Francisco Municipal Railway Light rail, former streetcar system
1,495 mm (4 ft 10+78 in) 10.973 m (36 ft 0 in) Toronto Streetcar System
1,067 mm (3 ft 6 in) 10.67 m (35 ft 116 in) Taunton Tramway
1,435 mm (4 ft 8+12 in) 10.058 m (33 ft 0 in) Boston Green Line
10.06 m (33 ft 116 in) Newark Light Rail
610 mm (2 ft) 4.9 m (16 ft 1516 in) Chicago Tunnel Company 6.1 m (20 ft 316 in) in grand unions. Not in use.

## References

1. ^ "Guide to Railcars". World Trade Ref - Your One Stop Resource for Trade Information. Archived from the original on 2011-10-31.
2. ^ "The Canadian Light Rail Vehicles (The CLRVs) - Transit Toronto - Content". transittoronto.ca.
3. ^ Ziegler, Hans-Joachim (2005-10-28). "Results of railway history". The Shasta Route: Connecting Oregon and California by Passenger Rail. p. 13. Retrieved 5 December 2018.
4. ^ a b Jane's World Railways 1995-1996 p728
5. ^ "Metre-Gauge Beyer-Garratt 4-8-4 + 4-8-4". www.garrattmaker.com.
6. ^
7. ^
8. ^ Australian Railway History September 2008, p291.
9. ^ Paul Garbutt (1997). "Facts and Figures". World Metro Systems. Capital Transport. pp. 130–131. ISBN 1-85414-191-0.
10. ^ Railway Gazette International March, 2012, page 23
11. ^ "WMATA Summary – Level Rail Car Performance For Design And Simulation" (PDF). WMATA. 2013-10-13. Archived from the original (PDF) on January 14, 2016. Retrieved October 15, 2014.
12. ^ "Metromover System Expansion Study" (PDF). Miami-Dade MPO. September 2014. Archived from the original (PDF) on February 14, 2015. Retrieved February 13, 2015.
13. ^ Railway Gazette International, July 2012, p18
14. ^ "Charakterystyka linii" [Line parameters]. WKD (in Polish).
15. ^ Trains: The Early Years, page 51, H. F. Ullmann,Getty Images, ISBN 978-3833-16183-4
16. ^ Lightrail now New Orleans RTA/Brookville streetcar
17. ^ Superevevation