A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration. In plane (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 and so forms 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 bank is reached.
The red Euler spiral is an example of an easement curve between a blue straight line and a circular arc, shown in green.
Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an osculating circle (click on thumbnail to observe).
This sign aside a railroad (between Ghent and Bruges) indicates the start of the transition curve. A parabolic curve (POB) is used.
If such an easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point (the tangent point where the straight track meets the curve) with undesirable results. With a road vehicle, a transition curve allows the driver to alter the steering in a gradual manner.
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"[1] 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[2] 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,[3] originally published in 1890. Some early 20th century authors[4] call the curve "Glover's spiral" and attribute it to James Glover's 1900 publication.[5]
The equivalence of the railroad transition spiral and the clothoid seems to have been first published in 1922 by Arthur Lovat Higgins.[4] Since then, "clothoid" is the most common name given the curve, but the correct name (following standards of academic attribution) is 'the Euler spiral'.[6]
While railroad track geometry is intrinsically three-dimensional, for practical purposes the vertical and horizontal components of track geometry are usually treated separately.[7][8]
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.
^Rankine, William (1883). A Manual of Civil Engineering (17th ed.). Charles Griffin. pp. 651–653.
^Crandall, Charles (1893). The Transition Curve. Wiley.
^Talbot, Arthur (1901). The Railway Transition Spiral. Engineering News Publishing.
^ abHiggins, Arthur (1922). The Transition Spiral and Its Introduction to Railway Curves. Van Nostrand.
^Glover, James (1900). "Transition Curves for Railways". Minutes of Proceedings of the Institution of Civil Engineers. pp. 161–179.
^Archibald, Raymond Clare (June 1917). "Euler Integrals and Euler's Spiral--Sometimes called Fresnel Integrals and the Clothoide or Cornu's Spiral". American Mathematical Monthly. 25 (6): 276–282 – via Glassblower.Info.
^Lautala, Pasi; Dick, Tyler. "Railway Alignment Design and Geometry" (PDF).
Simmons, Jack; Biddle, Gordon (1997). The Oxford Companion to British Railway History. Oxford University Press. ISBN0-19-211697-5.
Biddle, Gordon (1990). The Railway Surveyors. Chertsey, UK: Ian Allan. ISBN0-7110-1954-1.
Hickerson, Thomas Felix (1967). Route Location and Design. New York: McGraw Hill. ISBN0-07-028680-9.
Cole, George M; and Harbin; Andrew L (2006). Surveyor Reference Manual. Belmont, CA: Professional Publications Inc. p. 16. ISBN1-59126-044-2.
Railway Track Design 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 (3rd ed.). New York: McGraw.
March 04, 2023
track, transition, curve, track, transition, curve, spiral, easement, mathematically, calculated, curve, section, highway, railroad, track, which, straight, section, changes, into, curve, designed, prevent, sudden, changes, lateral, centripetal, acceleration, . A track transition curve or spiral easement is a mathematically calculated curve on a section of highway or railroad track in which a straight section changes into a curve It is designed to prevent sudden changes in lateral or centripetal acceleration In plane 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 and so forms 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 bank is reached The red Euler spiral is an example of an easement curve between a blue straight line and a circular arc shown in green Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip also known as an osculating circle click on thumbnail to observe This sign aside a railroad between Ghent and Bruges indicates the start of the transition curve A parabolic curve POB is used If such an easement were not applied the lateral acceleration of a rail vehicle would change abruptly at one point the tangent point where the straight track meets the curve with undesirable results With a road vehicle a transition curve allows the driver to alter the steering in a gradual manner Contents 1 History 2 Geometry 3 See also 4 References 5 SourcesHistory EditOn 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 1 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 Brusio spiral viaduct and railway Switzerland built 1908 from above 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 2 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 3 originally published in 1890 Some early 20th century authors 4 call the curve Glover s spiral and attribute it to James Glover s 1900 publication 5 The equivalence of the railroad transition spiral and the clothoid seems to have been first published in 1922 by Arthur Lovat Higgins 4 Since then clothoid is the most common name given the curve but the correct name following standards of academic attribution is the Euler spiral 6 Geometry EditThis section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed January 2010 Learn how and when to remove this template message While railroad track geometry is intrinsically three dimensional for practical purposes the vertical and horizontal components of track geometry are usually treated separately 7 8 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 EditDegree of curvature Minimum railway curve radius Railway systems engineeringReferences Edit Rankine William 1883 A Manual of Civil Engineering 17th ed Charles Griffin pp 651 653 Crandall Charles 1893 The Transition Curve Wiley Talbot Arthur 1901 The Railway Transition Spiral Engineering News Publishing a b Higgins Arthur 1922 The Transition Spiral and Its Introduction to Railway Curves Van Nostrand Glover James 1900 Transition Curves for Railways Minutes of Proceedings of the Institution of Civil Engineers pp 161 179 Archibald Raymond Clare June 1917 Euler Integrals and Euler s Spiral Sometimes called Fresnel Integrals and the Clothoide or Cornu s Spiral American Mathematical Monthly 25 6 276 282 via Glassblower Info Lautala Pasi Dick Tyler Railway Alignment Design and Geometry PDF Lindamood Brian Strong James C McLeod James 2003 Railway Track Design PDF Practical Guide to Railway Engineering American Railway Engineering and Maintenance of Way Association Archived from the original PDF on November 30 2016 Sources EditSimmons 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 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 3rd ed New York McGraw Retrieved from https en wikipedia org w index php title Track transition curve amp oldid 1118166605, wikipedia, wiki, book, books, library,
