A railway vehicle moving along a circular curve is subjected to an inertial centrifugal acceleration (ac), directly proportional with its speed and in reverse proportion with the curve radius. This lateral acceleration is perceived by the passengers as an uncomfortable sensation and, above a certain limit, endangers the lateral stability of the running vehicle. In order to compensate at least a part of its effect, when this centrifugal acceleration reaches a certain limit, the track is inclined towards the centre of the curve with the cross-level angle α (see figure 1).
The traditional way to measure this inclination is the cant, E, defined as “the vertical difference in heights of the two rails of a track, measured at centerline of the rail heads (S)” (NR/L2/TRK2049 – 2010, Track Design Handbook, B.1.1).
When this inclination is applied, the gravitational acceleration g will generate a parallel component with the plane of rail, compensating a part of the centrifugal acceleration.
Figure 1. The non-compensated lateral acceleration
In railway alignment design the complex dynamic behaviour of the vehicle is simplified to simple equations in order to define easy to understand and apply design parameters and standard limits. This simplification takes out the differences between the suspended and un-suspended mass, the suspension behaviour, the lateral and vertical thrust of the vehicle, the bogie attack angles, the vehicle acceleration or braking – to mention only a few main elements. The general standard rules for alignment design consider the vehicle a material point moving with constant speed at low rail level, along the centerline of the track.
The limits of the track alignment design parameters are in such a way defined to compensate for this simplified but easy to use approach. These limits are carrying safety factors, dynamic conditions and other constraints to ensure a safe and comfortable riding.
Taking into account all these simplifications accepted by the design standards, the final, non-compensated lateral acceleration of the vehicle, seen as a material point, can be considered:
The cross-level angle α is small, hence cos α is considered 1. The equation (1), with acceptable precision, becomes:
v is the speed of the vehicle (m/s)
R is the curve radius (m)
g is the gravitational acceleration (m/s²)
E is the applied cant (mm)
S is the cross-level standardised reference for rail heads centerline distance (mm) (see European Norm EN13848 – Track Geometry Quality). In UK this is conventionally considered to be 1502mm. (Cole, 1993, British Railway Track).
This equation allows to define the cant and all its related parameters, established to limit the non-compensated lateral acceleration.
Most of the track standards around the world (the British Track Design Handbook, TRK/2049 included) are using the concept of Cant Deficiency, D, instead of non-compensated lateral acceleration aq.
The Cant Deficiency is derived from (2) and defined as:
The factor 11.82 is defined for normal gauge and takes into account g, S and the speed unit conversion from m/s to km/h.
An important fact to understand and consider is that this cant constant value – 11.82 is a conventional figure. It is defined based on a presumed conventional rail centerline distance of 1502mm, for normal gauge track and it is related to the standardised way of measuring the cant, defined in the European Norm EN13848 – Track Geometry Quality.
It is established based on general equations and it’s not defined dependant of rail type or gauge variance (as long as the track has a normal track gauge).
That is why this constant should not be changed based on the variance of the normal gauge and kept the same for a gauge of 1432mm, 1435mm, 1438mm.
The cant constant carries with it all the simplifications mentioned above. Due to these simplifications, its around the world generalised value for normal gauge is 11.8 – an acknowledgement of the level of precision used to define it.
Also, as was demonstrated above, the cant constant is computed and based on the conventional rail centerline distance (considered in UK to be 1502mm) and not on gauge (1435mm). The difference between the two is significant and considering the later when defining the cross-angle in causing a significant error in cross level and cant.
This error is sumarised in table 1.
|cant (mm)||cant error (mm)|
Table 1. Cant error when is presumed computed based on gauge instead of rail centerline distance.
The error becomes even more significant when is propagated over sixfoot for siding geometry or for canted S&C design.