Minimum railway curve radius

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. 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 350 ft radius (16 degrees 26 minutes) or sharper, but normally 410 ft 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 ft 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
 * gauge
 * 99 ft radius curves
 * Example Garratt
 * gauge
 * 25 kg/m rails
 * Main line radius - 175 m
 * Siding radius - 84 m
 * 0-4-0
 * GER Class 209

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

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), ha is cant, and hb 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.

Transition curves
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 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.
 * 5 chain curves on the Oberon railway line, Batlow railway line, and Dorrigo railway line, New South Wales, limited steam locomotives to the 19 class.