Minimum railway curve radius

90-foot (27.43 m) radii on the elevated 4 ft 8 ¹⁄₂ in (1,435 mm) Chicago 'L'. There is no room for longer radii at this street intersection

The minimum railway curve radius, the shortest design radius, 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. Superelevation is not a factor on tramway tracks. Minimum radius of curve is one parameter in the design of railway vehicles as well as trams.

History

The first proper railway was the Liverpool and Manchester Railway which opened in 1830. Like the trams that had preceded it over a hundred years, the L&M had gentle curves and gradients. Amongst 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 previous trains. The gentler the curves, the longer the visibility.

Planning

Railway curves are generally too large a radius to draw with a compass, so what used to be done was to have a set of "Set squares" with radii from 25mm to 1000mm in increments of 25mm. Amongst other things, the point of the compass would be off the page, if not off the table.

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 350-foot (106.7 m) radius (16 degrees 26 minutes) or sharper, but normally 410-foot (125.0 m) radius (14 degrees) 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 (8 degrees) is preferred.

The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost everything is proportionately smaller.

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 are unfriendly to sharp curves. Various type of articulated locomotives Mallet, Garratt, 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.

• The TGR K Class was
  • 610 mm (2 ft) gauge
  • 99 ft (30 m) radius curves
• Example Garratt
  • 1,000 mm (3 ft 3 ³⁄₈ in) gauge
  • 25 kg/m (50.40 lb/yd) rails
  • Main line radius - 175 metres (574 ft)
  • Siding radius - 84 metres (276 ft)
• 0-4-0
  • GER Class 209
  • 1,435 mm (4 ft 8 ¹⁄₂ in)

Couplings

Not all couplers can handle very sharp curves. This is particularly true of the European buffer and chain couplers, where the buffers get in the way.

Train lengths

A long heavy freight train, especially those with light and heavy waggons mixed up, may have problems going round very sharp curves, as the drawgear forces may pull intermediate waggons off the rails causing derailments. Solutions might include:

• marshall light waggons at rear of train
• intermediate locomotives
• ease curves.
• reduced speeds
• less cant (superelevation)
• More but shorter trains.
• problem is less bad with bulk say coal trains, where all waggons weigh the same.
• better driver training
• driving controls that display drawgear forces.
• New Electronically Controlled Pneumatic brakes have the potential to reduce errant drawgear forces, besides displaying even more information to the drive.

A similar problem occurs with harsh changes in gradients.

Speed

As a heavy train goes round a bend at speed, the centrifugal force the train exerts on the rails is sufficient to move the actual track, which is only held in place by ballast. To counter this, a cant is used, that is, a height difference between the outside and inside rails on the curve. Ideally the train should be tilted such that resultant (combined) force acts straight "down" through the bottom of the train, so the rails feel little or no sideways force. Some trains are capable of tilting to enhance this effect for passenger comfort.

The cant can't of course be ideal at the same time for both fast passenger trains and slow freight trains.

The relationship between speed and tilt can be calculated mathematically. Specifically, the gravitational and centripetal forces need to be in the ratio 1 : tan θ, where θ is the angle by which the train is tilted due to the cant:

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

Geometrically, tan θ can be expressed (approximately, for small angles) in terms of the track gauge and the cant:

$\tan\theta\approx\sin\theta=\frac{h_a+h_b}{G}$

Combining these gives

$g=\frac{v^2}{r}\frac{(h_a+h_b)}{G}$

which rearranges to give the formula for maximum speed on a curve:

$r=\frac{Gv^2}{g(h_a+h_b)}$

where G is the rail gauge, v is speed in m/s (1 m/s = 3.6 km/h), g is gravitational acceleration (9.8 m/s²), h_a is cant, and h_b is cant deficiency.

This table shows examples of curve radii. The values used when building high-speed railways varies, and depends on how much wear and safety desired.

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

Transition curves

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 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.

The super-elevation (aka cant) must also be transitioned.

The higher the speed, the longer the transition.

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 is given by

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

positive for troughs, negative for crests, where m is the mass of the train and v is the speed in m/s. 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 1 (section 2) in the UK has a minimum vertical curve radius of 10000m. High Speed 2, with the higher speed of 400 km/h, stipulates much larger 56000m radii. In both these cases the experienced change in 'weight' is less than 1%.

Problem curves

• The Australian Standard Garratt had flangeless leading driving wheels which tended to cause derailments on sharp curves.
• Sharp curves on the Port Augusta to Hawker line of the South Australian Railways caused derailment problems when bigger and heavier SAR X class locomotives were introduced, requiring deviations to ease the curves.
• 5-chain (101 m; 330 ft) curves on the Oberon railway line, New South Wales, limited steam locomotives to the 19 class.

List of minimum curve radii

Gauge	Radius	Location	Notes
1,435 mm (4 ft 8 1⁄2 in)	7,000 m (22,966 ft)	China	Typical 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 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 high-speed railways (300 km/h)
1,435 mm (4 ft 8 1⁄2 in)	3,500 m (11,483 ft)	China	Typical 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 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 Railway 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	Turkey
1,435 mm (4 ft 8 1⁄2 in)	160 m (525 ft)	NSW, Zig Zag	40 km/h
1,435 mm (4 ft 8 1⁄2 in)	100 m (328 ft)	NSW, 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
1,067 mm (3 ft 6 in)	95 m (311.68 ft)	Newmarket, New Zealand	Extra heavy concrete sleepers
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
1,435 mm (4 ft 8 1⁄2 in)	61 m (200 ft)	London Underground Central line	(between White City and Shepherd's Bush)
1,067 mm (3 ft 6 in)	60 m (197 ft)	Queensland Railways
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 (122.9 ft)	Kalka-Shimla Railway	or 48 degrees
1,435 mm (4 ft 8 1⁄2 in)	29.00 m (95.14 ft)	New York Subway
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)
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
1,495 mm (4 ft 10 7⁄8 in)	10.973 m (36.00 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)	10.058 m (33.00 ft)	Boston Green Line
610 mm (2 ft)	4.9 m (16 ft)	Chicago Tunnel Company	6.1 m (20 ft) in grand unions.
1,676 mm (5 ft 6 in)	175 m (574 ft)	Indian Railways	the sharpest curve permitted on Broad Gauge
5 ft 2 1⁄2 in (1,588 mm)	28 ft (8.534 m) in yard, 50 ft (15.240 m) elsewhere	Streetcars in New Orleans

See also

Category:Articulated locomotives Degree of curvature, civil engineering Heaviest trains

Lateral motion device Longest trains Matheran Hill Railway

Radius of curvature (applications) Railway systems engineering Track transition curve Turning radius