Cant in a velodrome

The cant of a railway track or camber of a road (also referred to as superelevation, cross slope or cross fall) is the rate of change in elevation (height) between the two rails or edges. This is normally greater where the railway or road is curved; raising the outer rail or the outer edge of the road providing a banked turn, thus allowing vehicles to maneuver through the curve at higher speeds than would otherwise be possible if the surface is flat or level.

## Rail

The cant in a curve of the Nuremberg–Ingolstadt line
Track lubrication on a reverse curve in an area prone to movement due to wet beds

On railways, cant helps a train steer around a curve, keeping the wheel flanges from touching the rails, minimizing friction and wear.

The main functions of cant are the following:

• Improve distribution of the load across both rails
• Reduce wear on rails and wheels
• Neutralize the effect of lateral forces
• Improve passenger comfort
Railway superelevation at work

The necessary cant in a curve depends on the expected speed of the trains and the radius. However, it may be necessary to select a compromise value at design time, for example if slow-moving trains may occasionally use tracks intended for high-speed trains.

Generally the aim is for trains to run without flange contact, which also depends on the tire profile of the wheels. Allowance has to be made for the different speeds of trains. Slower trains will tend to make flange contact with the inner rail on curves, while faster trains will tend to ride outwards and make contact with the outer rail. Either contact causes wear and tear and may lead to derailment. Many high-speed lines do not permit slower freight trains, particularly with heavier axle loads. In some cases, the impact is reduced by the use of flange lubrication.

Ideally, the track should have sleepers (railroad ties) at a closer spacing and a greater depth of ballast to accommodate the increased forces exerted in the curve.

At the ends of a curve, the amount of cant cannot change from zero to its maximum immediately. It must change (ramp) gradually in a track transition curve. The length of the transition depends on the maximum allowable speed; the higher the speed, the greater length is required.

For the United States, standard maximum unbalanced superelevation of 75 mm (3 in), the formula is this:

${\displaystyle v_{max}={\sqrt {\frac {E_{a}+3}{0.00066d}}}}$

The maximum value of cant (the height of the outer rail above the inner rail) for a standard gauge railway is approximately 150 mm (6 in).[citation needed] For high-speed railways in Europe, maximum cant is 180 mm (7 in) when slow freight trains are not allowed.[1]

Track unbalanced superelevation (cant deficiency) in the United States is restricted to 75 mm (3 in), though 102 mm (4.0 in) is permissible by waiver. The maximum value for European railways varies by country, some of which have curves with over 280 mm (11 in) of unbalanced superelevation to permit high-speed transportation. The highest values are only for tilting trains, because it would be too uncomfortable for passengers in conventional train cars.[2]

### Physics of track cant

Ideally, the amount of cant ${\displaystyle E_{a}}$, given the speed ${\displaystyle v}$ of a train, the radius of curvature ${\displaystyle r}$ and the gauge ${\displaystyle w}$ of the track, the relation

${\displaystyle v^{2}={\frac {E_{a}rg}{\sqrt {w^{2}-E_{a}^{2}}}}\approx {\frac {E_{a}rg}{w}}}$

must be fulfilled, with ${\displaystyle g}$ the gravitational acceleration. This follows simply from a balance between weight, centrifugal force, and normal force. In the approximation it is assumed that the cant is small compared to the gauge of the track. It is often convenient to define the unbalanced cant ${\displaystyle E_{u}}$ as the maximum allowed additional amount of cant that would be required by a train moving faster than the speed for which the cant was designed, setting the maximum allowed speed ${\displaystyle v_{max}}$. In a formula this becomes

${\displaystyle v_{max}\approx {\sqrt {\frac {(E_{a}+E_{u})rg}{w}}}={\sqrt {\frac {(E_{a}+E_{u})g}{dw}}}}$

with ${\displaystyle d=1/r}$ the curvature of the track, which is also the turn in radians per unit length of track.

In the United States, maximum speed is subject to specific rules. When filling in ${\displaystyle g=32.17\,\mathrm {ft/s^{2}} }$, ${\displaystyle w=56.5\,\mathrm {in} }$ and the conversion factors for US customary units, the maximum speed of a train on curved track for a given cant deficiency or unbalanced superelevation is determined by the following formula:

${\displaystyle v_{max}\approx {\frac {3600}{63360}}{\sqrt {\frac {32.17\cdot 12(E_{a}+E_{u})}{56.5\cdot d{\frac {\pi }{1200\cdot 180}}}}}\approx {\sqrt {\frac {E_{a}+E_{u}}{0.00066d}}}}$

with ${\displaystyle E_{a}}$ and ${\displaystyle E_{u}}$ in inches, ${\displaystyle d}$ the degree of curvature in degrees per 100 feet and ${\displaystyle v_{max}}$ in MPH.

### Examples

In Australia, the Australian Rail Track Corporation is increasing speed around curves sharper than an 800-metre (2,625 ft) radius by replacing wooden sleepers with concrete ones so that the cant can be increased.[3]

### Rail cant

The rails themselves are now usually canted inwards by about 10 to 5 per cent.

In civil engineering, cant is often referred to as cross slope or camber. It helps rainwater drain from the road surface. Along straight or gently curved sections, the middle of the road is normally higher than the edges. This is called "normal crown" and helps shed rainwater off the sides of the road. During road works that involve lengths of temporary carriageway, the slope may be the opposite to normal – for example, with the outer edge higher – which causes vehicles to lean towards oncoming traffic. In the UK, this is indicated on warning signs as "adverse camber".

On more severe bends, the outside edge of the curve is raised, or superelevated, to help vehicles around the curve. The amount of superelevation increases with its design speed and with curve sharpness.

### Off-camber

An off-camber corner is described as the opposite of a banked turn, or a negative-bank turn, which is lower on the outside of a turn than on the inside.

On race courses, they are one of a handful of engineering factors at the disposal of a course designer in order to challenge and test drivers' skills.[5] Off-camber corners were described by a training guide for prospective racers as "the hardest corners you will encounter" on the track.[6] Many notable courses such as Riverside International Raceway combine off-camber corners with elevation and link corners for extra driver challenge.[7]

On the street, they are a feature of some of the world's most celebrated paved roads, such as "The Dragon" (US 129) through Deals Gap[8] and "The Diamondback" (NC 226A) in North Carolina,[9] Route 78 in Ohio,[10] Route 125 in Pennsylvania,[11] Route 33 in California,[12] and Betws-y-Coed Triangle at Snowdonia National Park in Wales.[13]

To mountain bikers and motorcyclists on trails and dirt tracks, off-camber corners are also challenging, and can be either an engineered course feature, or a natural feature of single-track trails.[14][15][16][17]

Camber in virtual race circuits is carefully controlled by video game race simulators to achieve the designer's desired level of difficulty.[5]

## References

1. ^ 2002/732/EC. *, Commission Decision of May 30, 2002 concerning the Technical Specification for Interoperability
2. ^ Zierke, Hans-Joachim. "Comparison of upgrades needs to recognize the difference in curve speeds". Retrieved April 10, 2008.
3. ^ "North South – strategy for growth Craven AU\$421.6 million Investment for Sydney Brisbane Corridor" (PDF). Links. Australian Rail Track Corporation Ltd. (11). August 2005. Archived from the original (PDF) on September 29, 2009. Retrieved November 22, 2012. Concrete re-sleepering of all curves of less than an 810-metre radius, using some 220,000 sleepers to increase cant deficiency and super-elevation, will be undertaken allowing for increased train speeds and further reducing transit times.
4. ^ ""KNOCK-KNEED" RAILS". The Queenslander. February 7, 1925. p. 9. Retrieved November 20, 2011 – via National Library of Australia.
5. ^ a b Luke McMillan (September 6, 2011), "A Rational Approach To Racing Game Track Design", Gamasutra
6. ^ Kenton Koch (2013), "Kenton Koch on Driving Technical Corners", Mazdaspeed Motorsports Development, Mazda North American Operations, Off camber corners: These corners are the hardest corners you will encounter...
7. ^ Van Valkenberg, Paul (October 1983), "What's It Really Like Out There?", Road & Track, 35: 67–69, Riverside International Raceway is a good example of a course with no isolated textbook turns: Every corner is either combined with another, or banked, off-camber, rising or falling.
8. ^ Darryl Cannon (September 25, 2012), "Deals Gap Revealed—Tail of the Dragon", Super Streetbike, Bonnier Group, [One of] the two worst corners [is] "Guardrail cliff", a sharp off-camber left ...
9. ^ Scot J. Marburger (2011), Top motorcycling roads: the Deep South, Gunsmoke Engineering
10. ^ Greg Harrison (July 2001), "Riding Roller-Coaster Roads on History's Trail", American Motorcyclist, American Motorcyclist Association: 31–32, [It] offers all types of curves—off-camber tight stuff, sweepers and esses that make me scramble from one side of the bike to the other while my foot stabs for the right gear.
11. ^ Miller, Robert H. (2010) [1997]. "PA125 – A Reptilian Tour on PA's Best Road". In Backroad Bob; Robert H. Miller. Motorcycle Road Trips (Vol. 14) Roads & Road Houses – Tour de Gastronomy. 14. p. 4. ISBN 9781452460512. Changing elevation a thousand feet at a time as it snakes over six mountain passes it offers no rest from decreasing radius, off-camber, blind and switchback curves.
12. ^ John Pearley Huffman (June 28, 2013), "The 10 Best Fourth of July Road Trips: Great Places and the Great Roads To Get You There", Edmunds.com, Route 33 has everything. It rolls across the Santa Ynez Mountains and plunges into the Cuyama Valley in relentlessly interesting ways. That includes midcorner elevation changes, off-camber hairpins, tightening-radius sweepers and straights long enough to hit terminal velocity. It's 72 miles of pure entertainment.
13. ^ The World's Best Motorcycle Routes, MCE Insurance
14. ^ "Riding Off-Camber Corners Over A Rise With Andrew Short – Pro Secrets – Dirt Rider Magazine", Dirt Rider, July 21, 2009
15. ^ Advanced Off Camber, MTB Techniques
16. ^ Steve Geall; Robin Kitchin; Greg Minaar (2001). The Ultimate Guide to Mountain Biking. Globe Pequot. p. 57. ISBN 9781585743032.
17. ^ Andrew Trevitt (October 3, 2011), "Riding skills series: Camber and Elevation—Using Both to Your Advantage", Sport Rider

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