K-equivalence

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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^[1] as well as trams.^[2]

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. The earliest rails were made in short lengths of wrought iron, which does not bend like later steel rails that were introduced in the 1850s.

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 Template:Convert radius (16 degrees 26 minutes) or sharper, but normally Template:Convert 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 Template:Convert radius (8 degrees) is preferred.

The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost everything is proportionately smaller.^[3]

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
- Template:Railgauge gauge
- Template:Convert radius curves
Example Garratt
- Template:RailGauge gauge
- Template:Convert rails
- Main line radius - Template:Convert
- Siding radius - Template:Convert ^[4]
0-4-0
- GER Class 209
- Template:Railgauge

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:

marshal light and empty waggons at rear of train
intermediate locomotives, including remotely controlled ones.
ease curves or deviate line.
reduced speeds
less cant (superelevation), but this is unfriendly to fast passenger trains.
More but shorter trains.
problem is less severe with bulk say coal trains, where all waggons weigh the same.
better driver training
driving controls that display drawgear forces.
New c2013 Electronically Controlled Pneumatic brakes have the potential to reduce errant drawgear forces, besides displaying even more information to the driver.

A similar problems occur with harsh changes in gradients (vertical curves).

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 θ

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

\tan θ \approx \sin θ = \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{G v^{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. The cant deficiency is the amount of additional cant that would be needed to neutralise the centrifugal force, and it is proportional to the curve force.

This table shows examples of curve radii. The values used when building high-speed railways vary, 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

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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 = m g \pm \frac{m v^{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.^[5] High Speed 2, with the higher speed of 400 km/h, stipulates much larger 56000m radii.^[6] In both these cases the experienced change in 'weight' is less than 7%.

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.^[7]
Template:Convert curves on the Oberon railway line, Batlow railway line, and Dorrigo railway line, New South Wales, limited steam locomotives to the 19 class.

List of minimum curve radii

Gauge	Radius	Location	Notes
Template:Railgauge	Template:Convert	China	Typical China's high-speed railway network (350 km/h)
Template:Railgauge	Template:Convert	China	Typical China's high-speed railway network (250 km/h~300 km/h)
Template:Railgauge	Template:Convert	China	Typical high-speed railways (300 km/h)
Template:Railgauge	Template:Convert	China	Typical China's high-speed railway network (200~250 km/h)
Template:Railgauge	Template:Convert	China	Typical high-speed railways (200 km/h)
Template:Railgauge	Template:Convert	DRCongo Matadi-Kinshasa Railway	Deviated Template:Railgauge line.
Template:Railgauge	Template:Convert	Border Loop	Template:Convert - Template:Convert
Template:Railgauge	Template:Convert	Wollstonecraft Railway Station, Sydney
Template:Railgauge	Template:Convert	Homebush triangle	Template:Convert - Template:Convert
Template:Railgauge	Template:Convert	Turkey	Turkey^[3]
Template:Railgauge	Template:Convert	NSW, Zig Zag	40 km/h
Template:Railgauge	Template:Convert	NSW, Batlow, New South Wales	Weight limit: Template:Convert and Template:Convert - restricted to NSW Z19 class 0-6-0 steam locomotives
Template:Railgauge	Template:Convert	Newmarket, New Zealand	Extra heavy concrete sleepers ^[8]
Template:Railgauge	Template:Convert	Windberg Railway (de:Windbergbahn)	(between Freital-Birkigt and Dresden-Gittersee) - restrictions to wheelbase
Template:Railgauge	Template:Convert	London Underground Central line	(between White City and Shepherd's Bush)
Template:Railgauge	Template:Convert	Queensland Railways
Template:Railgauge	Template:Convert	Matadi-Kinshasa Railway	original Template:Railgauge line.
Template:Railgauge	Template:Convert	Welsh Highland Railway
Template:Railgauge	Template:Convert	Bernina Railway
Template:Railgauge	Template:Convert	Welsh Highland Railway	on original line at Beddgelert
Template:Railgauge	Template:Convert	Victorian Narrow Gauge	Template:Convert on curves; (Template:Convert on straight)
Template:Railgauge	Template:Convert	Kalka-Shimla Railway	or 48 degrees
Template:Railgauge	Template:Convert	New York Subway	^[9]
Template:Railgauge	Template:Convert	Chicago 'L'
Template:Railgauge	Template:Convert	Sydney steam tram 0-4-0	Hauling 3 trailers
Template:Railgauge	Template:Convert	Darjeeling Himalayan Railway	The sharpest curves were originally Template:Convert ^[10]
Template:Railgauge	Template:Convert	Matheran Hill Railway	1 in 20 (5%); Template:Convert on curve; Template:Convert on straight
Template:Railgauge	Template:Convert	Toronto Streetcar System
Template:Railgauge	Template:Convert	Taunton Tramway
Template:Railgauge	Template:Convert	Boston Green Line
Template:Railgauge	Template:Convert	Chicago Tunnel Company	Template:Convert in grand unions.
Template:Railgauge	Template:Convert	Indian Railways	the sharpest curve permitted on Broad Gauge
Template:Railgauge	Template:Convert in yard, Template:Convert elsewhere^[11]	Streetcars in New Orleans

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References

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Template:Rail tracks Template:Railway track layouts

↑ Guide to Railcars, showing the minimum radii that each freight car is able to negotiate
↑ Canadian Light Rail Vehicle able to negotiate a Template:Convert radius curve
↑ ^3.0 ^3.1 Jane's World Railways 1995-1996 p728
↑ http://www.garrattmaker.com/history.html
↑ http://www.whatdotheyknow.com/request/24986/response/79568/attach/3/HS1%20Section%202%20Register%20of%20Infrastructure.pdf - page 19
↑ http://highspeedrail.dft.gov.uk/sites/highspeedrail.dft.gov.uk/files/hs2-route-engineering.pdf - page 4
↑ Australian Railway History September 2008, p291.
↑ Railway Gazette International March, 2012, page 23
↑ Railway Gazette International, July 2012, p18
↑ Trains: The Early Years, page 51, H. F. Ullmann,Getty Images, ISBN 978-3833-16183-4
↑ Lightrail now New Orleans RTA/Brookville streetcar

[1] Guide to Railcars, showing the minimum radii that each freight car is able to negotiate

[2] Canadian Light Rail Vehicle able to negotiate a Template:Convert radius curve

[Jane-3] 3.0 ^3.1 Jane's World Railways 1995-1996 p728

[4] ttp://www.garrattmaker.com/history.html

[5] ttp://www.whatdotheyknow.com/request/24986/response/79568/attach/3/HS1%20Section%202%20Register%20of%20Infrastructure.pdf - page 19

[6] ttp://highspeedrail.dft.gov.uk/sites/highspeedrail.dft.gov.uk/files/hs2-route-engineering.pdf - page 4

[7] Australian Railway History September 2008, p291.

[8] Railway Gazette International March, 2012, page 23

[9] Railway Gazette International, July 2012, p18

[10] Trains: The Early Years, page 51, H. F. Ullmann,Getty Images, ISBN 978-3833-16183-4

[11] Lightrail now New Orleans RTA/Brookville streetcar

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K-equivalence

Contents

History

Factors affecting the minimum curve radius

Steam locomotives

Couplings

Train lengths

Speed

Transition curves

Vertical Curves

Problem curves

List of minimum curve radii

See also

References

Navigation menu

K-equivalence

History

Factors affecting the minimum curve radius

Steam locomotives

Couplings

Train lengths

Speed

Transition curves

Vertical Curves

Problem curves

List of minimum curve radii

See also

References

Navigation menu

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