Track transition curve: Difference between revisions

Revision as of 19:08, 10 April 2016

Latest revision as of 21:52, 12 June 2025

(34 intermediate revisions by 23 users not shown)

Line 1:

{{Short description|Mathematically-calculated curve in which a straight section changes into a curve}}

[[Image:Easement curve.svg|240px|thumb|The red [[Euler spiral]] is an example of an easement curve between a blue straight line and a circular arc, shown in green.]]

[[File:CornuSprialAnimation.gif|thumb|Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as its tip (click on thumbnail to observe).]]

[[Image:Parabolic transition curve.JPG|thumb|240px|This sign aside a railroad (between [[Ghent]] and [[Bruges]]) indicates the start of the transition curve. A parabolic curve (''POB'') is used.]]

A '''track transition curve''', or '''spiral easement''', is a mathematically calculated curve on a section of highway, or [[Track (rail transport)|railroad track]], where a straight section changes into a curve. It is designed to prevent sudden changes in [[centripetal acceleration|lateral (or centripetal) acceleration]]. In plan (i.e., viewed from above) the start of the transition of the horizontal curve is at infinite radius and at the end of the transition it has the same radius as the curve itself, thus forming a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of [[Cant (road/rail)|bank]] is reached.

[[Image:Parabolic transition curve.JPG|thumb|240px|This sign aside a railroad (between [[Ghent]] and [[Bruges]]) indicates the start of the transition curve. A [[parabolic curve]] (''POB'') is used.]]

If such easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point – the [[tangent|tangent point]] where the straight track meets the curve – with undesirable results. With a road vehicle the driver naturally applies the steering alteration in a gradual manner and the curve is designed to permit this, using the same principle.

A '''transition curve''' (also, '''spiral easement''' or, simply, '''spiral''') is a spiral-shaped length of highway or [[track (rail transport)|railroad track]] that is used between sections having different profiles and radii, such as between straightaways ([[tangent]]s) and curves, or between two different curves.<ref name="pwayblog">{{Cite web |last=Constantin |date=2016-07-03 |title=The Clothoid |url=https://pwayblog.com/2016/07/03/the-clothoid/ |access-date=2023-06-07 |website=Pwayblog}}</ref>

{| class="wikitable" style=width:200px

|-

! Centripetal force on vehicles on roads without and with a transition curve

|-

| [[File:Straight-circle.gif|frameless|upright=1.2|center]] Straight sections of road connected directly by a circular arc. |

|-

| [[File:Straight-cornu.gif|frameless|upright=1.2|center]] 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 [[centripetal acceleration|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 [[cant (road/rail)|bank]] for the curve.

==History==

Line 10 ⟶ 24:

| last = Rankine

| first = William

| authorlinkauthor-link = William John Macquorn Rankine

| title = A Manual of Civil Engineering

| year = 1883

| publisher = Charles Griffin

| edition = 17th

| pages = [https://archive.org/details/amanualcivileng02rankgoog/page/n656 651]–653

| pages = 651–653

| url = httphttps://booksarchive.google.comorg/details/books?id=VSIJAAAAIAAJamanualcivileng02rankgoog

}}</ref> cites several such curves, including an 1828 or 1829 proposal based on the "[[Sinesine wave|curve of sines]]" by [[William Gravatt]], and the ''curve of adjustment'' by [[William Froude]] around 1842 approximating the [[Elasticaelastica theory|elastic curve]]. The actual equation given in Rankine is that of a [[Polynomial#Graphs|cubic curve]], which is a polynomial curve of degree 3., Thisat wasthe time also known as a cubic parabola at that time.

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, didwere the principles startbeginning to be applied in practice.

[[File:Brusio Viaduct (158241421).jpeg|thumb|[[Brusio spiral viaduct]] and railway (Switzerland, built 1908), from above]]

The "'true spiral"', where thewhose 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 thisthe curve by [[Leonhard Euler]] in 1744). Charles Crandall<ref>{{cite book

| last = Crandall

| first = Charles

Line 27 ⟶ 42:

| year = 1893

| publisher = Wiley

| url = httphttps://booksarchive.google.comorg/details/books?id=DnA5AAAAMAAJ&pg=PAtransitioncurve02crangoog

}}</ref> gives credit to one Ellis Holbrook, in the Railroad Gazette, Dec. 3, 1880, for the first accurate description of the curve. Another early publication was ''The Railway Transition Spiral'' by [[Arthur N. Talbot]],<ref>{{cite book

| last = Talbot

Line 34 ⟶ 49:

| year = 1901

| publisher = Engineering News Publishing

| url = httphttps://booksarchive.google.comorg/details/books?id=k4WCNKMdOZ0C&pg=PP7railwaytransiti01talbgoog

}}</ref> originally published in 1890. Some early 20th century authors<ref name="higgins">{{cite book

| last = Higgins

| first = Arthur

| title = The Transition Spiral and Its Introduction to Railway Curves

| url = https://archive.org/details/cu31924031215142

| year = 1922

| publisher = Van Nostrand

}}</ref> call the curve "Glover's spiral", attributingand attribute it to James Glover's 1900 publication.<ref>{{cite conference

| last = Glover

| first = James

| year = 1900

| title = Transition Curves for Railways

| booktitlebook-title = Minutes of Proceedings of the Institution of Civil Engineers

| pages = 161–179

| url = httphttps://books.google.com/books?id=aQsAAAAAMAAJ&pg=PA161

}}</ref>

The equivalence of the railroad transition spiral and the [[clothoid]] seems to have been first published in 1922 by Arthur Lovat Higgins.<ref name="higgins"/> Since then, "clothoid" is the most common name given the curve, even thoughbut the correct name (following standards of academic attribution) is "'the [[Euler spiral"]]'.<ref>[http://www.glassblower.info/Euler{{Cite journal |last=Archibald |first=Raymond Clare |author-Spiral/AMM/AMM-1918.HTMLlink=Raymond Clare Archibald |date=June 1917 |title=Euler Integrals and Euler's Spiral--Sometimes called Fresnel Integrals and the Clothoide or Cornu's Spiral |url=http://www.]glassblower.info/Euler-Spiral/AMM/AMM-1918.HTML |journal=American Mathematical Monthly, Volume |volume=25 (1918),|issue=6 pp.|pages=276–282 276 - 282|via=Glassblower. Raymond Clare ArchibaldInfo}}</ref>

==Geometry==

{{unreferenced section|date=January 2010}}

While railroad [[track geometry]] is intrinsically [[Threethree-dimensional space|three-dimensional]], for practical purposes the vertical and horizontal components of track geometry are usually treated separately.<ref>{{Cite web |url=http://www.engr.uky.edu/~jrose/RailwayIntro/Modules/Module%206%20Railway%20Alignment%20Design%20and%20Geometry%20REES%202010.pdf |title=Railway Alignment Design and Geometry |last=Lautala |first=Pasi |last2=Dick |first2=Tyler}}</ref><ref>{{Cite book |title=Practical Guide to Railway Engineering |last=Lindamood |first=Brian |last2=Strong |first2=James C. |last3=McLeod |first3=James |publisher=[[American Railway Engineering and Maintenance-of-Way Association]] |year=2003 |chapter=Railway Track Design |chapter-url=http://www.engsoc.org/~josh/AREMA/chapter6%20-%20Railway%20Track%20Design.pdf |archive-url=https://web.archive.org/web/20161130162616/http://www.engsoc.org:80/~josh/AREMA/chapter6%20-%20Railway%20Track%20Design.pdf |archive-date=November 30, 2016}}</ref>

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 [[Quadraticquadratic equation|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 [[Cant (road/rail)|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 (also referred to as ''cant'' or ''camber''), which is used to describeddescribe 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.

Line 69 ⟶ 85:

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 has two advantages. One is that it is easy for surveyors because the coordinates can be looked up in [[Fresnel integral]] tables. The other is that it 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]]

*[[Euler spiral]]

* [[Minimum railway curve radius]]

* [[Railway systems engineering]]

==NotesReferences==

{{reflist}}

==ReferencesSources==

{{refbegin}}

*{{cite book

| last =Simmons

| first =Jack

| authorlink =

|author2=Biddle, Gordon

| title =The Oxford Companion to British Railway History

| publisher =Oxford University Press

| year =1997

| ___location =

| pages =

| url =

| doi =

| isbn = 0-19-211697-5}}

*{{cite book

| last =Biddle

| first =Gordon

| authorlink =

| title =The Railway Surveyors

| publisher =Ian AllenAllan

| year =1990

| ___location = Chertsey, UK

| pages =

| url =

| doi =

| isbn = 0-7110-1954-1}}

*{{cite book

| last =Hickerson

| first =Thomas Felix

| authorlink =

| title =Route Location and Design

| publisher =McGraw Hill

| year =1967

| ___location =New York

| pages =

| url =

| doi =

| isbn = 0-07-028680-9}}

*{{cite book

| last =Cole

| first =George M

| authorlink =

|author2=and Harbin |author3=Andrew L

| title =Surveyor Reference Manual

Line 128 ⟶ 129:

| year =2006

| ___location =Belmont, CA

| url =

| doi =

| isbn = 1-59126-044-2

| page =16}}

Line 141 ⟶ 140:

| year = 1907

| edition = 3rd

| url = httphttps://booksarchive.google.comorg/details/books?id=ZVZCzW2codgCtransitioncurve01nrgoog}}

{{refend}}

{{Rail tracks}}