I mentioned in a previous article, The Cubic Parabola – a complicated simplification, that the curvature diagram of the Cubic Parabola increases linearly up to a peak point and then drops down. Only that first section of the parabolic curve can be used as an alignment transition.
The curvature diagram of the Clothoid is however linear to infinity.
This difference was not relevant to railway engineers for quite some time, and the Cubic Parabola was for a long time used as the default transition for railway track alignments, perhaps longer than the Clothoid.
For sure the cubic parabola can be found in all the pre-computer age track standards as the default transition. I was hoping to share here a few snapshots from old standards, but who would have thought that CD readers will disappear one day?
Here is a snapshot from Handbook no 3, the British track standard from 1973:
Even though it gives the cubic parabola formula, the handbook follows on with versines.
Interesting!
Trying to find out when the British Railways have switched to Clothoid, I noticed this section in 1993’s edition of British Railway Track:
The Hallade method sets out a clothoid spiral? Is that really true?
If you define the “spiral” using the Cubic Parabola formula, you’ll get a Parabola (duh!).
Do we really trace a Clothoid if we use linearly increasing versines measured relative to constant length chords?
I had to try this. Pen and paper first …
… and then, based on the question marks above, I drew this 2000m long curve, for versines measured on 20m chord, with a versine increment of 2mm. Not by hand.
Looks Clothoid enough to me!
The statement in the book is indeed correct. Hallade sets out a Clothoid.
However
On my hand drawn sketch, if I measure the versine at the mid points of each segment, I get versines vM which are very close to the average value of the adjacent versines, measured at the corners.
vM3 ≅ (v3 + v4)/2
Hence, even this broken line shape, which is not a clothoid, has a linear versine diagram, for chord lengths equal to the distance 2-4.
This is one of the main reasons why Hallade and other versine-based methods are not as used now as they were before for setting out or quality control of railway lines.
(To be continued)
References:
Cope, Geoffrey H. (1993) British railway track: design, construction and maintenance. Permanent Way Institution.
*** (1973) Handbook No 3. Civil Engineering Department.
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