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 5 to 10 percent.

In 1925 about 15 of 36 major American railways had adopted this practice.^[4]

Off-camber

Off-camber bend to the left (UK road sign)

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.^[5][6] Off-camber corners are both feared and celebrated by skilled drivers.^[7][8] Handling them is a major factor in skilled vehicle control, both single-track and automotive; both engine-powered and human-powered vehicles; both on and off closed courses; and both on and off paved surfaces.

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.^[9] Off-camber corners were described by a training guide for prospective racers as "the hardest corners you will encounter" on the track.^[10] Many notable courses such as Riverside International Raceway combine off-camber corners with elevation and link corners for extra driver challenge.^[11]

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^[12] and the "Diamondback" (NC 226A) in North Carolina,^[13] Route 78 in Ohio,^[14] Route 125 in Pennsylvania,^[15] Route 33 in California,^[16] and Betws-y-Coed Triangle at Snowdonia National Park in Wales.^[17]

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.^{[18][19][20][21]} In cyclocross, off-camber sections are very common as the courses snake around ridges, adding difficulty.

Camber in virtual race circuits is carefully controlled by video game race simulators to achieve the designer's desired level of difficulty.^[9]