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 do not cope well with curves of a small radius. Various types of articulated locomotives (e.g., Mallet, Garratt & Meyer) were devised to avoid having to operate multiple locomotives with multiple crews.

More recent diesel and electric locomotives do not have a wheelbase problem, as they have flexible bogies, and also can easily be operated in multiple with a single crew.

• The Tasmanian Government Railways 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) metre gauge
  • 25 kg/m (50.40 lb/yd) rails
  • Main line radius - 175 m (574 ft)
  • Siding radius - 84 m (276 ft) ^[5]
• 0-4-0
  • GER Class 209
  • 1,435 mm (4 ft 8+1⁄2 in) standard gauge

Couplings

Not all couplers can handle very short radii. This is particularly true of the European buffer and chain couplers, where the buffers extend the length of the rail car body. For a line with a maximum speed of 60 km/h (37 mph), buffer-and-chain couplers increase the minimum radius to around 150 m (164 yd; 492 ft). As narrow-gauge railways, tramways, and rapid transit systems normally do not interchange with mainline railways, instances of these types of railway in Europe often use bufferless central couplers and build to a tighter standard.

Train lengths

A long heavy freight train, especially those with wagons of mixed loading, may struggle on short radius curves, as the drawgear forces may pull intermediate wagons off the rails. Common solutions include:

• marshaling light and empty wagons at the rear of the train
• intermediate locomotives, including remotely controlled ones
• easing curves
• reduced speeds
• reduced cant (superelevation), at the expense of fast passenger trains
• more, shorter trains
• equalizing wagon loading (often employed on unit trains)
• better driver training
• driving controls that display drawgear forces
• Electronically Controlled Pneumatic brakes

A similar problem occurs with harsh changes in gradients (vertical curves).

Speed and cant

As a heavy train goes around a bend at speed, the reactive centrifugal force may cause negative effects: passengers and cargo may experience unpleasant forces, the inside and outside rails will wear unequally, and insufficiently anchored tracks may move. To counter this, a cant (superelevation) is used. Ideally, the train should be tilted such that resultant force acts vertically downwards through the bottom of the train, so the wheels, track, train and passengers feel little or no sideways force ("down" and "sideways" are given with respect to the plane of the track and train). Some trains are capable of tilting to enhance this effect for passenger comfort. Because freight and passenger trains tend to move at different speeds, a cant cannot be ideal for both types of rail traffic.

The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing centripetal force: θ is the angle by which the train is tilted due to the cant, r is the curve radius in meters, v is the speed in meters per second, and g is the standard gravity, approximately equal to 9.81 m/s²:

${\displaystyle \tan \theta ={\frac {v^{2}}{gr}}}$

Rearranging for r gives:

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

Geometrically, tan θ can be expressed (using the Small-angle approximation) in terms of the track gauge G, the cant h_a and cant deficiency h_b, all in millimeters:

${\displaystyle \tan \theta \approx \sin \theta ={\frac {h_{a}+h_{b}}{G}}}$

This approximation for tan θ gives:

${\displaystyle r={\frac {v^{2}}{g{\frac {h_{a}+h_{b}}{G}}}}={\frac {Gv^{2}}{g(h_{a}+h_{b})}}}$

This table shows examples of curve radii. The values used when building high-speed railways vary, and depend on desired wear and safety levels.

Tramways typically do not exhibit cant, due to the low speeds involved. Instead, they use the outer grooves of rails as a guide in tight curves.