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Theminimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination withsuperelevation (difference in elevation of the two rails) in the case oftrain tracks, determines the maximum safe speed of a curve. The minimum radius of a curve is one parameter in the design ofrailway vehicles^[1] as well astrams;^[2]monorails andautomated guideways are also subject to a minimum radius.

The first proper railway was theLiverpool and Manchester Railway, which opened in 1830. Like the tram roads that had preceded it over a hundred years, the L&M had gentle curves andgradients. Reasons for these gentle curves include the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. The gentler the curves, the greater the visibility, thus boosting safety via increased situational awareness. The earliestrails were made in short lengths ofwrought iron,^{[citation needed]} which does not bend like latersteel rails introduced in the 1850s.

Minimum curve radii for railways 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 railway companies is built to accommodate for a 288-foot (88 m) radius, but normally a 410-foot (125 m) radius is used as a minimum, as some freight carriages (freight cars) are handled by special agreement between railways that cannot take the sharper curvature. For the handling of long freight trains, a minimum 574-foot (175 m) radius is preferred.^[3]

The sharpest curves tend to be on the narrowest ofnarrow gauge railways, where almost all the equipment is proportionately smaller.^[4] But standard gauge can also have tight curves, if rolling stocks are built for it, which however removes the standardisation benefit of standard gauge. Tramways can have below 100-foot (30 m) curve radius.

As the need for more powerful steam locomotives grew, the need for more driving wheels on a longer, fixed wheelbase grew too. However, long wheel bases do not cope well with curves of a small radius. Various types ofarticulated locomotives (e.g.,Mallet,Garratt,Meyer &Fairlie) 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 flexiblebogies, and also can easily be operated in multiple with a single crew.

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• TheTasmanian 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 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

Not allcouplers can handle very short radii. This is particularly true of the Europeanbuffers 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), buffers increase the minimum radius to around 150 m (164 yd; 492 ft). Asnarrow-gauge railways,tramways, andrapid transit systems normally do not interchange with mainline railways, in Europe these often use bufferless central couplers and build to a tighter standard.

A long heavy freight train, especially those with wagons of mixed loading, may struggle on short radius curves, as thedrawgear 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 onunit 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).

As a heavy train goes around a bend at speed, thereactive 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.^{[dubious –discuss]} To counter this, acant (superelevation) is used. Ideally, the train should be tilted such thatresultant 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 oftilting 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 balancingcentripetal 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, andg is thegravity of Earth, approximately 9.81 m/s²:$${\displaystyle \tan \theta =\frac{v^2}{gr}.}$$Rearranging forr gives:$${\displaystyle r=\frac{v^2}{g\tan \theta }.}$$Geometrically, tanθ can be expressed (using thesmall-angle approximation) in terms of thetrack gaugeG, the canth_a andcant deficiencyh_b, all in millimeters:$${\displaystyle \tan \theta \approx \sin \theta =\frac{h_a+h_b}{G}.}$$This approximation for tanθ gives:$${\displaystyle r=\frac{v^2}{g\frac{h_a+h_b}{G}}=\frac{G{v}^2}{g(h_a+h_b)}.}$$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 usethe outer grooves of rails as a guide in tight curves.

A curve should not become a straight all at once, but should gradually increase in radius over time (a distance of around 40–80 m (130–260 ft) for a line with a maximum speed of about 100 km/h (62 mph)). Even sharper than curves with no transition arereverse curves with no intervening straight track. Thesuperelevation must also be transitioned. Higher speeds require longer transitions.

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, thesupport forceR exerted by the track on a train as a function of the curve radiusr, the train massm, and the speedv, is given by$${\displaystyle R=mg\pm \frac{m{v}^2}{r},}$$with the second term positive for troughs, negative for crests. For passenger comfort the ratio of thegravitational accelerationg to thecentripetal accelerationv²/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 trains are sufficiently high-powered that steep slopes are preferable to the reduced speed necessary to navigate horizontal curves around obstacles, or the higher construction costs necessary to tunnel through or bridge over them.High Speed 1 (section 2) in the UK has a minimum vertical curve radius of 10,000 m (32,808 ft)^[6] andHigh Speed 2, with the higher speed of 400 km/h (250 mph), stipulates much larger 56,000 m (183,727 ft) radii.^[7] In both these cases the experienced change in weight is less than 7%.

Railwell cars also risklow clearance at the tops of tight crests.

• TheAustralian Standard Garratt hadflangeless leading driving wheels that tended to cause derailments on sharp curves.
• Sharp curves on thePort Augusta toHawker line of theSouth Australian Railways caused derailment problems when bigger and heavierX class locomotives were introduced, requiring realignments to ease the curves.^[8]
• 5-chain (101 m; 330 ft) curves on theOberon,Batlow, andDorrigo lines, New South Wales limited steam locomotives to the0-6-0Z19 class.

This sectionneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources in this section. Unsourced material may be challenged and removed.(June 2019) (Learn how and when to remove this message)

• Hilton, George W.; Due, John Fitzgerald (1 January 2000).The Electric Interurban Railways in America. Stanford University Press.ISBN 978-0-8047-4014-2. Retrieved10 June 2014.

• Track geometry
• Radii

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