Track alignment is a mathematical/geometrical description of a railway track’s three-dimensional curvature horizontally and vertically as well as inclination to the horizontal plane and where the values are calculated taking into account the speed of the trains.
A major railway project requires to undergo various stages of the project cycle plan before the construction begins. Some important stages of the project are real estate procurement (that is, the purchase of land for the new railway), geotechnical engineering (that the ground really holds heavy trains at high speeds), dimensioning of track substructure (bank, cutting or tunnel) and designing of track superstructure (tracks, catenary, and signaling systems).
During the planning process, the track’s position must be determined in three directions (horizontal, vertical, and lateral), all with millimeter precision. In order to calculate the theoretical center lines of the track, mathematical calculations are required, which will limit the speed both vertically (“with and against”) and laterally (“curves”).
This track alignment article deals with some of these mathematical calculations, namely for horizontal curves and connecting transition curves (or in plain language “the curves of the track”).
Track width and track spacing
The track width is measured between the rail heads 14 mm below the top of the rail. In tight curves (less than 200 m radii), the track width is increased by 10 – 30 mm, the so-called track width supplement.
The track spacing is the c/c distance between two tracks. It is normally 4.5 m in Sweden but is increased to 6 m between every other track in railway yards and four-track, among other things to make room for signals and catenary poles. Track spacing is controlled by the vehicles’ permitted width (load profile).
The track gauges used are different around the world. In Germany, the minimum track distance is 4.0 m, but there the load profile is a little narrower than in Sweden. In the case of narrow tracks, track distances are usually a little smaller due to narrower vehicles. On high-speed lines, the track spacing is made slightly larger due to the draft around trains.
Horizontal curves
A horizontal curve is defined by a curve radius. The shorter the radius, the tighter the curve and the lower the maximum permitted speed (design speed).
The larger the radius of the curve, the more comfortable it is for passengers, the less fatigue and operational problems for the train, and the more secure environment is for the goods on the train. Thus, the radius of curvature is always reduced gradually from infinity to the final radius and at the end of the curve the radius is again increased to infinity (= straight track). These are called transition curves (clothoid) and are mathematical segments of spirals.
Rail cant(UK) or superelevation(US)
The rail cant or superelevation is part of the rail track which is equivalent to cant or superelevation in roadways. The rail cant is provided o reduce the impact of the centrifugal force in a curve by giving the track a lateral slope – rail elevation (English “cant”). This railway cant or elevation is not chosen to correspond to the maximum speed(design speed) as in roadways.
In roadways, the road is designed for the maximum design speed, however, the cars may travel at different speeds. The extra centrifugal force on the cars is accommodated by sliding further outwards from the center of the curvature. However, since trains run on rails there is no possibility of sliding and the trains can overturn if the superelevation is too high.
Thus, In the case of railways, the part of the centrifugal force is eliminated at the maximum permitted speed(design speed), but not so great that the train leans too much at low speed. For example, Normally it is less than 160 mm in Sweden.
The most normal thing in Sweden is to add 2/3 of the theoretical maximum value of the rail elevation in mixed traffic, however no more than 160 mm. The rail super elevation is also continuously increased from zero at the straight track to the maximum at the horizontal curve
Therefore, Since both slow and fast trains must run on the same track, the superelevation provided to the track may be different than the theoretically calculated superelevation. It is said that you get a rail elevation deficit for the fast trains and a surplus for the slow ones. One way to compensate for this is to use inclined trains.
Slow-moving freight trains on tracks with high rail elevation mean higher pressure on the inner rail in curves. This can lead to the flattening of the rail head. In addition, large rail super elevation affects the stability of the train negatively because of the lateral force that the locomotive’s traction provides when, for example, starting from a standstill in a tight curve with maximum rail elevation. This means that the rail super elevation must be limited to keep down the risk of derailment and overturning wagons inwards in curves.
However, for high-speed lines (over 250 km/h), where no slow freight trains run, the slope can be increased to 180 – 200 mm to avoid excessively long curve radii.
Rail Cant deficiency
As it is noted earlier, the railway cant that is provided is smaller than the cant corresponding to the maximum speed. That is to prevent the overturning or high rail side pressure for trains running at lower speeds. This will normally introduce a Rail cant deficiency which is the additional rail rise that would be needed to completely equalize the centrifugal force at the maximum permitted speed.
A formula is rail cant deficiency (hb)=gauge*curve acceleration/weight acceleration, ie rail cant deficiency is curve acceleration times a constant. Curve acceleration=curve force/mass and it is proportional to speed2/curve radius.
The theoretical value of required superelevation or cant for a certain speed of the train is calculated as 
Therefore,
The maximum permissible speed is chosen so that the rail elevation deficiency (curving force) does not exceed the permissible limit.
In Sweden, there are four train categories: Category A with hb max 100 mm, category B with hb max 150 mm, category C with hb max 180 mm, and category S (inclined train) with hb max 245 mm.[1]
Speeds in horizontal curves[edit | edit wikitext]
The following relationship applies to the vehicle’s maximum speed:
where
R is radius of curvature [m],
v maximum speed [km/h],
ha rail cant [mm],
hb rail cant deficiency [mm] and
The constant 11.8 comes from 1.05*s/3.6²/g where s is track width [mm], g is the gravity factor [m/s²], 3.6 is the conversion between m/s and km/h, and 1, 05 is an arbitrary extra margin that the Swedish Transport Administration has added.
Below is a guide for the horizontal radius required for a certain speed for non-inclined train design in Sweden. Three categories are entered; categories A, B, and C.
Cat A: rail rise 160 mm and rail rise deficiency 100 mm (e.g. rigid MD bogies)
Cat B: rail rise 160mm and rail rise deficiency 153mm (e.g. soft Asea bogies)
Cat C: rail rise 160 mm and rail rise deficiency 180 mm (X74, however, applied in Norway since 1982 and France 1967)
| Curve radius(m) | Cat A (Km/h) | Cat B(km/h) | Cat C(km/h) |
|---|---|---|---|
| 1000 | 148 | 162 | 169 |
| 1500 | 181 | 199 | 207 |
| 2000 | 209 | 230 | 240 |
| 2500 | 234 | 257 | 268 |
| 3000 | 257 | 282 | 294 |
| 3500 | 277 | 304 | 317 |
| 4000 | 296 | 325 | 339 |
Note that the values are theoretical; the X74 railcar train is only approved for 200 km/h and categories A and B (the “eighties’ wagons” from State companies) normally cannot exceed 160km/h
Rail Cant excess or surplus (E)
If the actual cant is larger than the theoretical value of the cant for the design speed, it is said that there is cant excess or surplus.
At low speeds,
The minimum radius that a train can turn at low speed depends on the type of train. The coupling and the associated equipment affect the ease of turning on sharp curves. All railway vehicles in Sweden are made so that they can handle a radius of 150 m. With standardized screw couplings, you cannot handle sharper curves as the associated buffers can touch each other. On narrow-gauge railways, you often have other couplings as the need for adaptation to a standard is less and narrow-gauge is often chosen to be able to have tighter curves. Trams are specially adapted to tight curves and can often take curves with a radius of 20 m, even with normal tracks.
Sharp curves can limit how heavy and long trains can be, especially on uphill gradients, as carriages can overturn curves from excessive traction forces. The problem is less if all wagons have the same load and care is taken to load all wagons roughly the same amount. In this way, very heavy trains can go in fairly sharp curves.
Vertical curves
Correspondingly, vertical curve radii are specified when a climb begins or ends and these too are given smooth transitions. At high speeds, a crest or dip can become uncomfortable for the passengers, so you don’t want too small a vertical radius. The transition is then called a ramp where the radius increases linearly.
The formula below applies, where v is the speed in km/h and a is permitted cornering acceleration, which can be around 0.3 m/s².
The minimum vertical radius is limited by the formula below in normal cases in Sweden.
Slope
Ascents are measured in parts per thousand (‰) or vertical meters per 1,000 horizontal meters.
The slope that is provided varies with the type of train as shown below in Sweden’s case.
| Track category | Slope-  |
| Track for freight transport | 10 |
| Track for passenger’s train | 25 |
| Track near platforms | 10 |
| Turnouts and parking areas | 2 |
For heavy freight trains, in Sweden, they try to stay below 10 per thousand (in Switzerland 27 per thousand since the 1880s, in the USA even steeper). For larger gradients, the weight of freight trains must be limited or heavier locomotives must be used (in Switzerland, e.g. Ae 6/6).
Since the 1950s, the subway in Stockholm has used 42 ‰ (Skanstullsbron), and trams even steeper, even up to 60 ‰
.
FAQ
Which type of track alignment is preferable?
Straight lines are the desirable line element
• No rail cant is needed
• The same rail super elevation (ha=0) fits all train speeds
• Easy to maintain
• Simple insertion of track switches (turnouts), it is possible to put switches in one horizontal curve but it is necessary to enforce speed restrictions and
maintenance problems are more frequent.
• Along the platform: Good visibility for staff when the door is closed
• Road planners prefer curves with large curve radii due to less risk of
glare, driver fatigue, and water runoff skew:
For further studies check the contact patch and railbaltica
Comparison of rail tracks and roadways
