# Cant and Cant Deficiency. Where is the 11.82 coming from?

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:

where:
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) 0 0.0 25 1.2 50 2.4 75 3.7 100 4.9 125 6.1 150 7.3

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.

## 18 thoughts on “Cant and Cant Deficiency. Where is the 11.82 coming from?”

1. Nii says:

Very detailed analysis, thanks Constantin

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2. Varun says:

Please publish paper if possible on these things with importance of cant deficiency , why the equilibrium cant cannot be applied and its difficulties etc…

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3. Varun says:

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4. The cant is limited to certain values (150, 180 etc) to ensure stable/safe/comfortable position of the running vehicle with maximum deficiency. Or has defined comfort limits for the stopped vehicle on canted curve – within a platform section or in open line. These limits are not universal, obviously depend on the track gauge and on the vehicle types and from here different limiting figures.

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5. Varun says:

Why the applied cant is limited to certain limits like 165 mm or something like that.

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6. Installing the equilibrium cant provides, in theory, equal vertical wear on both rails. The main advantage of providing cant deficiency is it allows tighter curves, flexible alignment layout on the ground topology. Cant deficiency also helps the vehicle steering.

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7. Varun says:

Hi Cosci,

When we apply cant to particular track why don’t we apply calculated equilibrium cant as applied cant without deficiency or excess considering train speed remains same and doesn’t have mixed traffic condition.What are the implications of providing equilibrium cant as applied cant.

Thanks

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9. cosci says:

Hi Uday. Thank you for your comment.
The cant – even though is a value expressed in mm – is measured as an angle and converted to a conventional height over a base of 1500mm. From this point of view even the 1505mm centerline distance is not entirely right.
I will update the post to show how the cant is measures and refer to the European Norm that defines the measurement method both for maintenance machines and hand held devices.
That will make the things clear enough.
You can check on a track gauge device – keep the inclination (cant) canstant, vary the gauge and check what happens. You will see – the cant doesn’t change.

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10. Hello Constantin Ciobanu, nicely summarized article giving the clarity from where 11.82 is computed.
Regarding computation of cant constant, since wheels are resting on rails, according to me, consideration of rail center gives more accurate values.

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