There are three types of versines used for the track realignment and rectification methods (Radu, 2003):

– the versine measured with a real chord on the outer rail of the real existing track. This is the result of the versine (Hallade) survey and based on it are computed the track offsets (slues or slews) required to correct the track geometry.*Measured (existing) versine*– the versine not measured but computed using the formula v*Theoretical versineª*_{i}= C²/8R_{i }, where R_{i}is the centreline radius at point i. If this point is on a circular curve R is the radius of the curve. If it is on a transition curve R_{i }= RL/d , where d is the distance from the point i to the beginning of the transition curve and L is the total transition length.

The theoretical versine on a transition curve is:

The versine change *dv* between consecutive points along the linear transition curve is:

As the name suggest, this type of versine is theoretical, computed presuming a certain chord length but without making any chord offset measurement on the ideal, nondeformed track.

– the versine measured as offset to a chord on the ideal track, without any irregularities.*Ideal versineª*

This versine is the same as the *theoretical versine* if both ends of the chord are* on the same geometrical element (straight, circular curve or transition curve)*.

In the vicinity of the points where the curvature variation changes, the *ideal versine* is different from the *theoretical versine*. In the figure below, when measuring the versines for points 2 and 3, the chord is placed on two different alignment elements.

To measure the versine for point 2 the chord is placed between point 1 – on a straight – and point 3 – on the curve. Although not clearly visible in the figure, there is a versine V2 despite the fact that point 2 is actually on a straight, hence the theoretical versine is null.

Similarly for point 3, which requires the chord to be based on point 2 – on the straight – and on point 4 – on the curve? The versine V3, measured to the chord 2-4 is not equal to the circular arc theoretical versine.

Below is another example for a compound curve.

The versine v3 is an ideal one because is measured using the chord 2-4. But if we extend back the curve of radius R2 and find the point 2’ to define the chord length 2’-4, the versine v’3 measured using this chord is equal to the theoretical versine for the curve of radius R2.

When doing a *realignment or rectification* design using *Hallade* or other methods, the resulted versines are checked against the* ideal versines* and not the *theoretical* ones. If the track lining is checked on site, the measured versine should always be checked against the ideal versine. If the versine of a point on a straight is checked by placing the chord on the adjacent curve it is normal to get, even for an ideal track, a non-null versine.

In the *Hallade realignment method* (Ellis, 1998; Noblet, 2003) the ** ideal versines** at the start or end of a transition are computed by the following rules:

- When a transition starts or finishes
*at a half-chord point*, the first/last versines are modified by adding/subtracting 1/6 times the full transition rise, dv. - When the transition starts and finishes
*exactly midway between two half-chord points*the adjustment is equal to 1/48 of the full transition rise, dv. Unless the transition rise is very large, is generally accepted that in this case the rounding off, in Hallade realignment terms, is null.

The justification for these rules will be presented in a future post together with the equations to be considered when the transition curve has an arbitrary position relative to the half-chord points.

*Noteª: In some railway literature the labelling of the “theoretical” and “ideal” versines is the other way around – the versines measured on the ideal track are called “theoretical versines”. In fact that’s the way I’ve learned them also. But I chose to use the above way of defining them because it seems now more logical to me. *

*The reader familiarised with the other way of defining them is politely asked to excuse this labelling change which does not affect the essence of the subject: three distinct types of versines – one which is only the result of a mathematical formula and the other two measured on track – either ideal and without irregularities or real, with irregularities and implying also measuring tolerances and errors.*

**References:**

- Cope, G. (1993)
*British Railway Track – Design, Construction and Maintenance*. Permanent Way Institution, Echo Press, Loughborough. - Noblet, Y. (2003).
*Épures Hallades. Modification de trace par la méthode des flèches.*Available online: http://bazar.perso.free.fr/Files/Other/DOCUMENTATION/Trace/Fleches.pdf - Ellis, I. (1998).
*The Hallade Training Manual.*Available online: http://www.iainellis.com/hallade.pdf - Radu, C. (2003)
*Rectificarea si retrasarea curbelor de cale ferata (Rectification and realignment of the railway track curves)*. Course notes. Technical University of Civil Engineering Bucharest.

Hi Con. Thank you for your comment. Sorry, I did not have the time to prepare this – in the last year I have been busy with times tables practice and similar maths things (my boy is at that age) …

The calculation requires an introduction into a theory of what I can only translate as ‘evolving curves’.

It is strange that this concept and other related to Hallade theory doesn’t seem to be present at all in the technical literature written in English and available online.

What exactly are you after?

The formulas or the explanation of this?

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Hi Constantin,

You mention you intention to provide justification for the rules at the start and end of a transition in a future post together with the equations to be considered when the transition curve has an arbitrary position relative to the half-chord points. Have you had any progress with this?

I am struggling to find any explanation on this.

Thanks

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Once again another interesting article!!!!

In terms of cord measurement and versines it doesn’t really matter as long as the cord length is always the same and you adapt your equation to suit your method,

This link below gives the dimensions of a ruler that has been made to measure a few different radius of curves without having to do all the math. this works on a principal of an 8m cord length.

http://www.asa.transport.nsw.gov.au/sites/default/files/asa/railcorp-legacy/disciplines/civil/spc-201.pdf

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I was referring to the number of survey points required to produce a representative versine curve or Hallade.

We used to survey with a step of 20 m but that is not enough when it is about tight curves or short transitions.

It is important to find the right step for the surveyors to be more efficient and less expensive. Survey line is usually long.

Thanks for the article.

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I’m not sure I understand the question, Mehdi. Step like interval between consecutive points, half-chord? That’s usually 10 m (chord of 20 m) for metric and 33 ft (chord of 66 ft) for the imperial measuring system. For tighter curves shorter chords are used.

Or are you referring to the number of survey points required to produce a realignment design?

In that case the best is to measure from straight to straight – covering the entire curve and with enough points measured on the straight to pass over any potential transition curve and allow the correct definition of the straight bearing. If the curve is too long or just a rectification is required then … it depends on the particularities of the site and on the experience of the designer.

Does this make sense?

Thank you for your comment.

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Interesting…

Versines calculation is the method adopted by a number of software designers to assist engineers on realignment studies.

Now, the question is: what is the minimum step of survey points required to plot a representative versine for an optimal realignment study?

Regards,

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