For the track realignment methods (Hallade or similar) the existing alignment is surveyed by measuring the outer rail offsets (versines) to overlapping chords. One such setup is drawn below:

A’, B’, C’ are points on the outer rail of the track and for point B’ the versine B’D’ =  v_{B ex} is measured to the chord A’C’. We have here the triangle A’B’C’ defining an existing versine measurement setup. A similar triangle can be defined for the proposed (realignned) track and, by overlapping the two, results the following figure:

This figure defines the arrangement of the existing and proposed track (identified with the outer rail) for three consecutive points. To move the existing to the proposed location is required a lateral movement of the track, a slue (slew). In the above figure the slues of the three points are noted s_A , s_B  and s_C .

All the track rectification and realignment methods are in fact different calculation methods for providing these slues which are required to move the track from the existing to the proposed position.

In the Hallade method the slues are calculated based on the differences between existing and proposed versines.

This calculation procedure is based on the following main assumptions:

• the segments AA’, BB’, CC’ are considered parallel. The points are placed on circular arcs and the segments are practically radial, but this assumption is acceptable due to the great difference between the radius of the curve compared to the versines.
• The points B’, B, D, D’ are assumed co-linear. Again – the error implied by this assumption is not significant for the Hallade calculation.
• The start slues are null – in the calculation presented here the realignment starts at point 1. All the slues prior to this point are null.

The distance DD’ can be calculated from:

DD’ ≈ B’D – B’D’   (1)

DD’ ≈ B’B + BD – B’D’   (2)

B’B is the slue of point B’. BD is the proposed versine of B and B’D’ is the existing versine of B’. Considering this, (2) becomes:

DD’ ≈ s_B + v_{B ex} – v_{B prop}   (3)

DD’ is divided into two segments:

DD’ ≈ D’E + ED    (4)

Based on the mid-point theorem D’E is half of A’A (slue in A’ , s_A) and ED is half of C’C (slue in C’ , s_C):

Replacing (5) and (6) in (3) is producing:

s_A – 2 s_B + s_C = 2(v_{B ex} – v_{B prop} ) = 2δ_B  (8)

where δ_B is the difference between the existing and proposed versine for point B, δ_B = v_{B ex} – v_{B prop} .

The equation (8) is true throughout the entire alignment. We can write this equation for all the points of the alignment starting from point 0 to a current point i:

point 0:  s_-1 – 2 s₀ + s₁ = 2δ₀  (8.0)
point 1:  s₀ – 2 s₁ + s₂ = 2δ₁  (8.1)
point 2:  s₁ – 2 s₂ + s₃ = 2δ₂  (8.2)
………………………………………….
point i-1:  s_i-2 – 2 s_i-1 + s_i= 2δ_i-1  (8.i-1)
point i:  s_i-1 – 2 s_i + s_i+1 = 2δ_i  (8.i)
————————————– (+)

Summing all these into one we are getting (9):

One of the initial assumptions says the realignment starts at point 1 and all the slues before this point are null, therefore s_-1 = 0 and  s₀ = 0. Replacing these in (9) produces (10):

That is to say that the difference between the slues at any consecutive points (i) and (i+1) is equal to twice the sum of differences between proposed and existing versines to the point (i). This formula (10) can be used to define a Hallade computation formula for the slues, when the existing and proposed versines are known.

The formula (10) can be written for all the points starting from 0 to the current point i. Summing them into one we are getting formula (11):

 Therefore, by accumulating the effects, the slue at any point is the summation of the sums of the differences between existing and proposed versines, on all previous points.

The double sum ΣΣδ_i is called Hallade Moment at point i+1. This sum is calculated based on the measured, existing versines and on the known, proposed versines. The slue at any point is twice the value of the Hallade Moment.

Well, now, all this is based on knowing the proposed versines. How does the designer know the proposed versines?

That is, dear reader, another story. Part of it was presented in the article about the type of versines used for track realignment methods. The rest will follow soon …

References and further reading:

• 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
• Ellis, I. (1998). The Hallade Training Manual.
• 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.