Track transition curve

A transition curve (also, spiral easement or, simply, spiral) is a

spiral-shaped length of highway or railroad track that is used
between sections having different profiles and radii, such as
between straightaways (tangents) and curves, or between two
different curves.[1]

Centripetal force on vehicles on roads

without and with a transition curve The red Euler spiral is an example of
an easement curve between a blue
straight line and a circular arc, shown
in green.

Straight sections of road connected directly

by a circular arc. |

This sign aside a railroad (between

Ghent and Bruges) indicates the
start of the transition curve. A
parabolic curve (POB) is used.

Straight section connected to a circular arc

via a Cornu spiral|

Comparison of a poorly designed road with

no transition curve with the sudden
application of centripetal force required to
move in a circle versus a well designed road
in which the centripetal acceleration builds
up gradually on a Cornu spiral before being
constant on the circular arc. The second
animation shows the increasing curvature of
the transition curve which is able to connect
to a circular arc of progressively smaller
radius.

In the horizontal plane, the radius of a transition curve varies continually over its length between the
disparate radii of the sections that it joins—for example, from infinite radius at a tangent to the nominal
radius of a smooth curve. The resulting spiral provides a gradual, eased transition, preventing undesirable
sudden, abrupt changes in lateral (centripetal) acceleration that would otherwise occur without a transition
curve. Similarly, on highways, transition curves allow drivers to change steering gradually when entering or
exiting curves.

Transition curves also serve as a transition in the vertical plane, whereby the elevation of the inside or
outside of the curve is lowered or raised to reach the nominal amount of bank for the curve.

History
On early railroads, because of the low speeds and wide-radius curves employed, the surveyors were able to
ignore any form of easement, but during the 19th century, as speeds increased, the need for a track curve
with gradually increasing curvature became apparent. Rankine's 1862 "Civil Engineering" [2] cites several
such curves, including an 1828 or 1829 proposal based on the "curve of sines" by William Gravatt, and the
curve of adjustment by William Froude around 1842 approximating the elastic curve. The actual equation
given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known
as a cubic parabola.

In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes
and tighter curves were necessary, were the principles beginning to be applied in practice.

The 'true spiral', whose curvature is exactly linear in arclength,

requires more sophisticated mathematics (in particular, the ability to
integrate its intrinsic equation) to compute than the proposals that
were cited by Rankine. Several late-19th century civil engineers
seem to have derived the equation for this curve independently (all
unaware of the original characterization of the curve by Leonhard
Euler in 1744). Charles Crandall[3] gives credit to one Ellis
Brusio spiral viaduct and railway
Holbrook, in the Railroad Gazette, Dec. 3, 1880, for the first (Switzerland, built 1908), from above
accurate description of the curve. Another early publication was
The Railway Transition Spiral by Arthur N. Talbot,[4] originally
published in 1890. Some early 20th century authors[5] call the curve "Glover's spiral" and attribute it to
James Glover's 1900 publication.[6]

The equivalence of the railroad transition spiral and the clothoid seems to have been first published in 1922
by Arthur Lovat Higgins.[5] Since then, "clothoid" is the most common name given the curve, but the
correct name (following standards of academic attribution) is 'the Euler spiral'.[7]

Geometry
While railroad track geometry is intrinsically three-dimensional, for practical purposes the vertical and
horizontal components of track geometry are usually treated separately.[8][9]

The overall design pattern for the vertical geometry is typically a sequence of constant grade segments
connected by vertical transition curves in which the local grade varies linearly with distance and in which
the elevation therefore varies quadratically with distance. Here grade refers to the tangent of the angle of
rise of the track. The design pattern for horizontal geometry is typically a sequence of straight line (i.e., a
tangent) and curve (i.e. a circular arc) segments connected by transition curves.

The degree of banking in railroad track is typically expressed as the difference in elevation of the two rails,
commonly quantified and referred to as the superelevation. Such difference in the elevation of the rails is
intended to compensate for the centripetal acceleration needed for an object to move along a curved path, so
that the lateral acceleration experienced by passengers/the cargo load will be minimized, which enhances
passenger comfort/reduces the chance of load shifting (movement of cargo during transit, causing accidents
and damage).

It is important to note that superelevation is not the same as the roll angle of the rail which is used to
describe the "tilting" of the individual rails instead of the banking of the entire track structure as reflected by
the elevation difference at the "top of rail". Regardless of the horizontal alignment and the superelevation of
the track, the individual rails are almost always designed to "roll"/"cant" towards gage side (the side where
the wheel is in contact with the rail) to compensate for the horizontal forces exerted by wheels under normal
rail traffic.

The change of superelevation from zero in a tangent segment to the value selected for the body of a
following curve occurs over the length of a transition curve that connects the tangent and the curve proper.
Over the length of the transition the curvature of the track will also vary from zero at the end abutting the
tangent segment to the value of curvature of the curve body, which is numerically equal to one over the
radius of the curve body.

The simplest and most commonly used form of transition curve is that in which the superelevation and
horizontal curvature both vary linearly with distance along the track. Cartesian coordinates of points along
this spiral are given by the Fresnel integrals. The resulting shape matches a portion of an Euler spiral, which
is also commonly referred to as a "clothoid", and sometimes "Cornu spiral".

A transition curve can connect a track segment of constant non-zero curvature to another segment with
constant curvature that is zero or non-zero of either sign. Successive curves in the same direction are
sometimes called progressive curves and successive curves in opposite directions are called reverse curves.

The Euler spiral provides the shortest transition subject to a given limit on the rate of change of the track
superelevation (i.e. the twist of the track). However, as has been recognized for a long time, it has
undesirable dynamic characteristics due to the large (conceptually infinite) roll acceleration and rate of
change of centripetal acceleration at each end. Because of the capabilities of personal computers it is now
practical to employ spirals that have dynamics better than those of the Euler spiral.

See also
Degree of curvature
Minimum railway curve radius
 Railway systems engineering

References
1. Constantin (2016-07-03). "The Clothoid" (https://pwayblog.com/2016/07/03/the-clothoid/).
Pwayblog. Retrieved 2023-06-07.
2. Rankine, William (1883). A Manual of Civil Engineering (https://archive.org/details/amanualci
vileng02rankgoog) (17th ed.). Charles Griffin. pp. 651 (https://archive.org/details/amanualcivi
leng02rankgoog/page/n656)–653.
3. Crandall, Charles (1893). The Transition Curve (https://archive.org/details/transitioncurve02c
rangoog). Wiley.
4. Talbot, Arthur (1901). The Railway Transition Spiral (https://archive.org/details/railwaytransiti
01talbgoog). Engineering News Publishing.
5. Higgins, Arthur (1922). The Transition Spiral and Its Introduction to Railway Curves (https://ar
chive.org/details/cu31924031215142). Van Nostrand.
6. Glover, James (1900). "Transition Curves for Railways" (https://books.google.com/books?id=
aQsAAAAAMAAJ&pg=PA161). Minutes of Proceedings of the Institution of Civil Engineers.
pp. 161–179.
7. Archibald, Raymond Clare (June 1917). "Euler Integrals and Euler's Spiral--Sometimes
called Fresnel Integrals and the Clothoide or Cornu's Spiral" (http://www.glassblower.info/Eu
ler-Spiral/AMM/AMM-1918.HTML). American Mathematical Monthly. 25 (6): 276–282 – via
Glassblower.Info.
8. Lautala, Pasi; Dick, Tyler. "Railway Alignment Design and Geometry" (http://www.engr.uky.e
du/~jrose/RailwayIntro/Modules/Module%206%20Railway%20Alignment%20Design%20an
d%20Geometry%20REES%202010.pdf) (PDF).
9. Lindamood, Brian; Strong, James C.; McLeod, James (2003). "Railway Track Design" (http
s://web.archive.org/web/20161130162616/http://www.engsoc.org:80/~josh/AREMA/chapter
6%20-%20Railway%20Track%20Design.pdf) (PDF). Practical Guide to Railway
Engineering. American Railway Engineering and Maintenance-of-Way Association. Archived
from the original (http://www.engsoc.org/~josh/AREMA/chapter6%20-%20Railway%20Trac
k%20Design.pdf) (PDF) on November 30, 2016.

Sources
Simmons, Jack; Biddle, Gordon (1997). The Oxford Companion to British Railway History.
Oxford University Press. ISBN 0-19-211697-5.
Biddle, Gordon (1990). The Railway Surveyors. Chertsey, UK: Ian Allan. ISBN 0-7110-1954-
1.
Hickerson, Thomas Felix (1967). Route Location and Design. New York: McGraw Hill.
ISBN 0-07-028680-9.
Cole, George M; and Harbin; Andrew L (2006). Surveyor Reference Manual. Belmont, CA:
Professional Publications Inc. p. 16. ISBN 1-59126-044-2.
Railway Track Design (http://www.arema.org/eseries/scriptcontent/custom/e_arema/Practical
_Guide/PGChapter6.pdf) pdf from The American Railway Engineering and Maintenance of
Way Association, accessed 4 December 2006.
Kellogg, Norman Benjamin (1907). The Transition Curve or Curve of Adjustment (https://arch
ive.org/details/transitioncurve01nrgoog) (3rd ed.). New York: McGraw.
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