THE HISTORY OF THE DERIVATION OF THE AREMA SPIRAL

Halsey G. Brown, P.E.a

ABSTRACT The paper traces the history and technical details of the AREMA transition spiral back to the initial use by railroads in 1880 and then to the earliest known discussion in 1694. The AREMA spiral equation, expressed intrinsically as d=ks, requires a parametric form to calculate cartesian coordinates. These equations are derived using the calculus and the infinite series. Prior to the current AREMA equations, an empirical curve based on the same theory was used. The derivation of this curve, known as the 10 chord spiral, includes a treatment of the 10 chord accuracy versus the “true” spiral. An understanding of this concept is helpful when using the modern equations. Earlier articles discussing the transition spiral were published by the ASCE, ICE, Engineering News Record, Railroad Gazette (Railway Age) and several Universities. Other transition curves were used or proposed during this period. These other curves are beyond the scope of the paper and are only mentioned briefly. Articles by Holbrook, Talbot, and Glover are discussed as the earliest applications of the subject curve for railroad use. The curve discussed in this paper is also referred to as the Clothoid, Cornu spiral and Euler spiral. The work of Cesaro, Cornu and Euler respectively are examined, revealing the same equations that are used today. Prior to these works, the mathematician Jacob Bernoulli discussed the same curve in 1694. The paper includes fragment illustrations of Holbrook’s 1880 railroad article and Euler’s 1743 publication. INTRODUCTION An internet search for the subject “railway transition curve” or “railway spiral” will yield many sites dealing with the subject. With respect to the history of this curve, some of the statements you will find are vague generalizations and some are actually incorrect. One statement explains that the spiral curve recommended by AREMA originated with the mathematician Leonhard Euler. Another gives the credit to the mathematician James Bernoulli. Who proposed that these equations be used for railroad curves ? The purpose of this paper is to trace the current AREMA spiral equations1 as far back as possibleb and look at several interesting aspects of the curve. The scope of the discussion is limited to AREMA Chapter 5, section 3.1.2-3.1.4, and does not cover the equations for the length of spiral or superelevation.

a Assistant Division Engineer Structures, CSX Transportation Inc. 1 Bell Crossing Rd. Selkirk, NY 12158 b Previous historical reports include AREA Proceedings Vol. 40 1939, pp 172-173 and Engineering News Record (May 1919) Vol. 82, No. 19, pp. 924-925

DISCUSSION OF CURRENT EQUATIONS We start with the spiral equation (EQ 4) found in AREMA chapter 5, Section 3.1: ksd = where: d = degree of curvature of the spiral at any point k = increase in degree of curvature per 100-foot station along the spiral s = length along spiral in 100-foot stations For this discussion, the important characteristic of equation EQ4 is that the relationship between the degree of curvature and the spiral length is linear. In the mathematics world, there are other spiral relationships that exist such as the Archimedian, Logarithmic and Hyperbolic spiralsc. In the past, there have been other designs that have been used or proposed as railroad easement curves. These curves include the Tapering Curve, Lemniscate, Parabola, Cubic Parabola, Cotes Spiral and spirals named after the engineer such as the Searles, Hood, and Stevens Spiral. The above equation is simple enough, although it is noted that plotting this equation is different than a normal x-y mathematical function. The calculation of cartesian coordinates for the AREMA spiral requires a parametric equation, meaning x and y are expressed by two separate equations. The formula d=ks is the intrinsic form of the curve, another math term that the reader can look up if desired. I have heard engineers state that we learn so much theory in college but never get a chance to use it in the real world. During the research for this paper, I found one text2 that states “the railroad engineer is rarely an expert mathematician” and later in the same text, the author states that the railroad engineer is “entirely ignorant of the Differential Calculus.” The development of the AREMA spiral is a practical application of calculus and the infinite series and results in a parametric equation; not bad! The full derivation of equations similar to those found in AREMA Chapter 5, section 3.1, can be found in the more comprehensive route surveying texts3,4. Before going further, let us consider the derivationd of equations 5, 6, 8 and 9. For convenience, fragments of the current AREMA Section 3.1 are shown in Figure 1.

Equation No. 4, ksd = can also be expressed as 100

5730 klR

=

Where R = the radius of curvature at any point on the spirale. At the end of the spiral, the

radius will be equal to that of the connecting simple curve cR , thus 100

5730 kLRc

=

Combining these 2 equations results in lLRR c

=

c r=aΘ, r=eaΘ, r=a/θ respectively, with θ being the angle with respect to the x axis d A similar derivation of the equations is also found in the 1890 Talbot paper discussed later in this article. e It is customary (and necessary) to use the non-railroad definition of d in this derivation.

D = degree of circular curve d = degree of curvature of the spiral at any point l = Length from the T.S. or S.T., to any point on the spiral having coordinates x and y s = length l in 100-foot stations L = total length of spiral S = length L in 100-foot stations δ = central angle of the spiral from the T.S. or S.T. to any point on the spiral Δ = central angle of the whole spiral a = deflection angle from the tangent at the T.S. or S.T. to any point on the spiral b = orientation angle from the tangent at any point on the spiral to the T.S. or S.T. k = increase in degree of curvature per 100-foot station along the spiral Figure 1 – Fragment from AREMA Chapter 5, Section 3.1 (All functions are in feet or degrees)

Figure 2

Referring to figure (2), now consider a differential sector of the spiral which can be

treated as a simple curve and the relationship Rdld =δ applied, which when combined

with the above relation results in:

∫ ==LR

ldlLR

l

cc 2

2

δ (δ in radians)

Finally, with the relations previously used, k & s are restored and radians are converted to degrees, resulting in AREMA EQ 5:

2

21 ks=δ AREMA EQ 5

For EQ 8 and 9, referring to figure (2), the following relationships are developed: )(sinδdldy = )(cosδdldx = The sine and cosine terms are removed by applying the Maclaurin Series. Details of the infinite series expansion of sin(x) and cos(x) are outlined in most calculus texts. It is noted that when these equations were created, the sine and cosine values were not available at the push of a calculator button. It was desirable to remove the functions from the equations.

.....)!7!5!3

(753 δδδδ −+−= dldy

......)!6!4!2

1(642 δδδ

−+−= dldx

The previous relationship LR

l

c2

2

=δ is applied at the end of the spiral giving cR

L2

=Δ .

Combining the equations for δ and Δ, results in 2)(Ll

Δ=δ which is inserted into the

infinite series:

dlLl

Ll

Lly .....])(

!5)(

!3)([ 10

56

32 Δ

+Δ

−Δ= ∫

dlLl

Llx .....])(

!4)(

!21[ 8

44

2 Δ+

Δ−= ∫

Once this simple integration is accomplished, the Δ term is backed out the same way it was substituted in. A hint for anyone trying this is to separate out one of the l terms. For example, if l5 is expressed (l4l), all will work out right. Note that only the first two terms are required to obtain practical accuracy, and further terms are dropped. Finally, applying a constant so δ will be in degrees and using the relation

100/ls = gives: ssy 300001264.0582.0 δδ −= AREMA EQ 8 slx 2003048.0 δ−= AREMA EQ 9 A similar derivation exists starting instead with the radius of curvature ρ = ds/dδ. It is still necessary to use the arc definition of curvature to complete the derivation. In some articles, it has been customary to note that EQ 8 and EQ 9 result in nearly the same plot as the cubic parabola y= ax3. This is demonstrated by substituting EQ 5 into EQ 8 and only using the first term of the infinite series. It is only approximate and it is noted that the typical cubic equation would contain an x-y relationship not an s-y relationship. In some cases, if the constant (a) is chosen properly, the resulting coordinates fall very close to the path of the spiral. As previously mentioned, the cubic parabola has been used in the past by some US railroads and can be found in several older field manuals. The cubic parabola is still used in Europe.f The derivation of AREMA equation EQ 6 is based on the simple concept that the tangent of the angle “a” will be y/x. However, the simplicity disappears once equations EQ 8 and f See AREMA Practical Guide, Chapter 12 – European Curve and Turnout Mechanics. The equivalent organizations to ASCE and AREMA in the UK are the I.C.E. and the Permanent Way Institution (PWI).

EQ 9 are substituted into the relationship. The solution is a=δ/3 – constant, yet AREMA EQ 6 neglects the constant. EQ 10-EQ 13 are working equations not directly related to this discussion and are not derived here. Before discussing the chronological history of this curve, note the paragraph found in AREMA chapter 5, section 3.1.5e, repeated here: In staking by deflection, it is sometimes convenient to divide the spiral into a number of equal chords. The first or initial deflection a1 may be calculated for the first chord point. The deflections for the following chord points are a1 times the chord number squared. This paragraph is what is left of the AREA 10 Chord spiral concept. A term first appearing in the 1911 AREA proceedings and dropped in 1947. TEN CHORD SPIRAL CONCEPT Working backward in time, the current equations go back to the 1947 proceedings where it was proposed to delete all previous spiral information. The manuals between 1911 and 1947 recommended use of the ten chord spiral, an empirical equation providing the main chord length “C” and expressing X & Y in terms of this value (fig 3).

Fig 3 – AREA 10 Chord Spiral was used until 1947.

Angle “f” is the angle between a given chord and the X – axis.

The 1911 AREA convention report5 explained that the committee considered seven different types of spirals used by railroads including “a curve whose radius is inversely proportional to the length of arc, as developed by Crandall and Talbot.” C. L. Crandall was a Civil Engineering Professor at Cornell University and published a surveying text6. Arthur Newell Talbot, whose name shows up in other railroad discussions such as The Talbot Reports, was a professor at the University of Illinois. Professor Talbot also published a text on spiral curves7. The 1911 committee developed a “practical adaption” of the Talbot/Crandall spiral and the report includes a full derivation of the resulting curve described as the ten chord spiral. A limited discussion of the ten chord derivation and accuracy is helpful when using the modern equations The derivation of the ten chord spiral begins with the d=ks relationship. Referring to Figure 3, the spiral is divided into 10 equal chords. Using the relations a= δ/3 and s=10s/(n-1), where n is one of the ten points on the spiral, the following equation for the angle “f” between any chord and the X axis is developed:

Δ+−

= )300

133(2 nnf

Evaluating f at each of the ten chord points (n=1,2,3,4…), it is found that f will be 1,7,19,37,61,91,127,169,217 and 271 300ths of Δ. Now the opposite and adjacent sides of each chord triangle are calculated and summed to get the total X and Y for a given spiral:

⎥⎦⎤

⎢⎣⎡ Δ

+Δ

+Δ

+Δ

=300

271cos....30019cos

3007cos

300cos

10LX

⎥⎦⎤

⎢⎣⎡ Δ

+Δ

+Δ

+Δ

=300

271sin....30019sin

3007sin

300sin

10LY

The total spiral angle is: XYA =tan and also

AXC

cos=

With these equations evaluated at different values of Δ, the empirical equations shown in figure 3 were developed and appeared in the AREA manual until 1947. See the 1911 AREA report for further details and assumptions regarding this derivation. EQUATION ACCURACY The 1911 report clearly states that the resulting layout data will not agree exactly with the theoretical equations and then demonstrates in detail that the error is minor. This is a good point to discuss further.

The quantity “s” is defined in the AREMA Manual as the length to a point on the spiral in 100 ft. stations. A review of the full spiral curve derivation makes it clear that theoretically, “s” is the dimension measured along the arc of the curve. This measurement would be most difficult to accomplish in the field, and thus; a steel tape is used to form chords when laying out the curve g. The good news is that unless extreme curvature and length are used, the error is minorh. For such cases, a correction can be made to the calculations. Theoretically, one could calculate the length of the chord for each arc portion and use these measurements for the layout. The entire issue is eliminated if the theoretical equations for x and y are used and measured from the tangent. The 1911 report states: if closer adherence to the theoretical curve is desired, the entire spiral may be staked from the tangent by use of the coordinates x and y. This level of accuracy is generally not warranted, especially with the modern surfacing equipment in use today. Consider one additional interesting property of the spiral curve referred to in the current AREMA manual. The spiral deflects from a circular curve having the same degree as the spiral at that point at the same rate as it does from the tangent. A little thought by the reader is required to prove this. One last term related to this concept that appears in older texts and articles (another math term) is osculating circle. The first dictionary term for osculate is to kiss, however in mathematics, it refers to a circle that shares a common tangent to a given curve and is centered on the concave side of the curve. An osculating circle to a spiral would have a radius matching the degree of curvature of the spiral at the given point. Prior to the 1911 publication, AREA did not recommend a specific transition curve. The following statement is in the first AREA manual8 (1905): Any transition curve of the type of the Searles, Crandall, Holbrook, Talbot or cubic parabola, which shall be susceptible of being run in by deflection or offset is recommended. The 1901 AREA proceedings 9 included the results of a survey showing what type of spirals the different railroads were using (if any) and is interesting reading. The 1901 proceedings also mention a curve referred to as the Holbrook Spiral and includes the full derivation. The derivation begins with the same linear relationship between curvature and arc length with only the symbols differing (d=nL). BEFORE AREA Prior to 1899, there was no AREMA or AREAi so we look to other publications. In the Proceedings of the American Society of Civil Engineers, there are several articles prior to 1899 including an 1892 article10 by William Cain entitled: The Transition Curve whose

g This statement refers to the traditional layout method. Modern survey equipment provides other options. h In addition to the 1911 AREA accuracy discussion, this fact is stated in many survey texts and curve manuals. A concise discussion appears in the 1949 Searles, Chapin manual “Field Engineering” page 240. i AREMA was formed in 1899, then shortened to AREA. In 1997 AREA was renamed back to AREMA.

Curvature Varies Directly as its Length from the P.C. or Point where it Connects with the Tangent. It is a long title, but it states the use of the linear k-s relationship. In this article, there are familiar equations and a reference to Wellington as the first to propose such a transition curve. Arthur Wellington is a famed railroad Civil Engineer who produced a text book entitled: The Economic Theory of the Location of Railways. The 1901 edition is 980 pages, the original version was published in 1887. Wellington was also the editor for the Railroad Gazette and published an article11 on transition curves. The Wellington text and article do not mention the specific curve discussed in this article. Upon further research, it is revealed that Professor Talbot produced an 1890 article entitled The Railway Transition Spiral12 which appeared in the University of Illinois newsletter The Technograph. This article clearly uses the same d=ks relationship and the parametric equations for x and y. It is in this article we find nearly the exact list of equations that appear in the current AREMA Manual.j The following statement is found on page 96 of Professor Talbot’s Technograph article: The transition spiral was probably first used on the Pan Handle Railroad in 1881, by Mr. Elliot Holbrook. The principal part of the treatment here given was made before the writer’s attention was called to Mr. Holbrook’s use of the curve, and it is believed that most of the formulas and methods appear here for the first time. Professor Crandall’s text states that transition curves came into use in the United States approximately 1880 and mentions Mr. Ellis Holbrook. Another text by Tratman13 states that Hobrook was the first to use this curve in the United States. In a contribution14 to the Dec 3, 1880 edition of the Railroad Gazette (now Railway Age), Holbrook provides a derivation resulting in the same general equations that have been discussed. The discussion clearly starts with the equation Rl=A, with R= radius, l=distance from the tangent along the curve and A=constant. Holbrook worked for the Panhandle Railroad (Pennsylvania Railroad between Columbus and Pittsburgh) and later was the Chief Engineer for the Kansas City Southern. An articlek in the 1901 Engineering News15, written by Holbrook states: About 21 years ago the writer found occasion to use transition curves on railroads, and found that the problem had not been worked out. Further into the article Holbrook states: Having worked out the properties of such a spiral in connection with its relation to the adjacent tangent and circle and put it into use, the writer distributed among various assistant engineers of the Pan Handle road copies of the demonstration and tables, and from them it went to other roads. So according to several sources including Holbrookl, the use of d=ks for railway transition curves started with him. j Except for symbol use and minor algebraic manipulation, the equations are identical. k This article also appears in the 1906 text by F. Lavis entitled “Railroad Location Surveys and Estimates”.

From this data, it appears that the concept of using the d=ks relationship began with Holbrook and was developed into a more convenient form by Talbot which we use today. American railroad survey texts prior to 1881 do not mention any type of transition curves (just simple and compound curves). The 1878 ASCE article by S. Whitney entitled The Theoretical Resistance of Railroad Curves16 only mentions the parabola as a transition curve. The 1872 New York Central Track Maintenance Specification does not mention any transition curve. Holbrook’s 1880 article is reprinted here with the kind permission of Railway Age Magazine. Due to the unusual size and configuration of the article, the print is very small but legible. If you are viewing this article in ADOBE, set the zoom to 200 % and it can be easily read. Forms of transition curves were being used in Europe much earlier than 1880 by Gravatt, Froude and others (see Crandall text), however this was not the d=ks spiral. There are articles in the Proceedings of the Institution of Civil Engineers developing the d=ks relationship, however, not until at least 190017,18. In the 1909 ICE proceedings19, a discussion between engineers had the following statement suggesting that the progress of transition curve use developed faster in the United States than in Europe: All English railways, he believed and certainly all railways in the United States had realized the importance of providing in a suitable methodical way spiral curves at the approaches to all circular curves on which high speeds obtained, and of keeping roads in the best possible condition for running. In addition, the following statement is found in a 1908 (PWI) article20 by H. E. Robarts: It is a somewhat singular fact that although the use of these curves has for many years been the common practice on many parts of the continent, and although they have been adopted in the majority of the principal American Railways, yet it is only within a comparatively recent time that they have been introduced to any extent at all by Railway Engineers in this country (England). There is a possibility that the spiral discussed in this article was previously used in another part of Europe or elsewhere in the world prior to use in America, however, I did not find any evidence to suggest this occurred.

l According to reference sources, Holbrook did write a text entitled “The Holbrook Spiral” however; I was unable to locate this text. It may have been his bound article. Note there is also an article by W.D. Taylor entitled “The Holbrook Spiral Curve” in the February 1906 Univ of Wisconsin Newsletter, The Wisconsin Engineer.

BEFORE RAIL USE In some parts of Europe, different terms are used for railroading: superelevation is cant, underbalance is cant deficiency and joint bars are fishplates. The spiral is typically referred to as the transition curve, but another term used is the Clothoid. The name Clothoid (other spellings exist) was given to the curve by the mathematician E. Cesaro in 188621. Note that this date is after Holbrook’s paper. The article by Cesaro reveals similar equations, but is not railroad related and builds on the work of previous mathematicians. Thus the curve did not start with Cesaro. The curve discussed in this paper is also commonly referred to as Euler’s Spiral or Cornu’s Spiral. Marie Alfred Cornu (1841-1902) was a French physicist and a professor at the Ecole Polytechnique in Paris. Cornu utilized the curve for optics and knowledge of optics and Fresnel diffraction is required to understand his work22. Now consider the work of the Swiss mathematician Leonhard Euler (1707-1783), one of the giants in math history. The article23 to be discussed here is “Methodus Inveniendi Lineas Curves Maxime Minimive Gaudentes.” Dated 1743 and consisting of 320 pages, it is an incredible work which covers elastic curves, oscillating motions and the Euler column buckling formula. The entire paper is in Latin and is very difficult to interpret unless, of course, you are fluent in Latin. Two extremely helpful publications24,25 were used to understand the Euler work. The derivation of the elastic curve formula is not related to surveying or directly to geometry, but begins more as a strength of materials problem. A rigorous treatment of the Euler problem is beyond the practical application of this paper, however the basic idea is as follows. The specific portion of the article is paragraph No. 47 “The Curvature of Elastic Ribbons which in their Natural State are not Straight”.

FIG 4 (Euler Figure 17) Diagram from Euler’s 1743 Paper

Referring to figure 4 (Euler figure 17), a weight p is placed on the end of the elastic ribbon forcing it into a straight line A-M. The distance between point A and M is defined as “s”. The ribbon has a constant stiffness EI but is referred to in this argument as Ek2. Euler then assumes the distance A-M is equal to the distance a-m. It is then stated that the moment about point m is ps. The quantity M/EI and the curvature of the ribbon are related by strength of materials. Also noting that the radius of curvature is the reciprocal of the curvature, the following equation is formed:

EIM

r=

1 or REIM =

Balancing forces, r

EIps = or utilizing the 1743 symbols, r

Ekps2

=

Rearranging terms gives rs= Ek2/p and noting that Ek2/p is a constant and calling it a2, results in the Euler equation : 2ars = (radius x arc length = constant) This is the same relationship as d=ks since d=constant/r Further into Euler’s article, the familiar parametric equations for x and y are created using the same infinite series method. Pages 276, 277 and figure 17 of Euler’s original 1743 paper are reprinted here with the kind permission of The Posner Memorial Collection, located at the Carnegie Mellon University Libraries, Pittsburgh, PA. At the time of this writing, a fine scanned copy of the entire Euler paper is available for viewing at www.posner.library.cmu.edu. To go back one step further, refer to the work of James Bernouilli (1654-1705) also known as Jacob or Jacque in some articles. To create further confusion, there was also the mathematician John Bernoulli (brother) and Daniel Bernoulli (nephew) who both made important math and physics contibutions. The initial study of the elastic curve prior to Euler was indeed done by James Bernoulli prior to 1700 where the equation rs=a2 appearsm. Euler gives credit to Bernoulli in his article stating that the nature of the elastic curve “has been done already in a most excellent fashion by that very great man Jacob Bernoulli.” CONCLUSION We have gone back over 300 years and are indeed still looking at the same basic equation. It is interesting to note that the symbol for length along the arc “s” was used in the 1600’s and is still being used in math texts today and by AREMA. m See reference for 1918 American Mathematical Monthly article on Euler, pg 279

In my research, I had hoped to find a definitive statement from Holbrook or Talbot explaining that they were building on the past work of the math community, but unfortunately, no absolute statement was discovered. The fact that “s” is used for arc length suggests that this did occur. The terminology appeared in some math books of the time26. In closing, there are publications available with sample spiral design problems, including the design of spirals for new alignments, design of a spiral for a compound curve and methods of inserting spirals into existing track. The AREMA Practical Guide has excellent worked examples of the problems listed above and more. The two texts previously referenced (Hickerson, Meyers) also have examples. In addition, some of the railroads produced their own design guides with equations and worked examples. These are difficult to obtain, but are worth the search.

References 1 AREMA 2008 Manual for Railway Engineering, Chapter 5, Part 3- Curves 2 H. C. Goodwin, Railroad Engineers Field Book, 1890, (John Wiley & sons) 3 Hickerson, Route Location and Design, 1967, (McGraw Hill) 4 Carl Meyer, David Gibson, Route Surveying and Design, 1980 (Harper & Row) 5 Report of Committee on Spirals, AREA Proceedings 1911, pp 417-427 6 Charles Crandall and Fred Barnes, Field Book for Railroad Surveying, 1910 (John Wiley & Sons) 7 A.N. Talbot, The Railway Transition Spiral 1927 (Mcgraw-Hill) 8 Report of Committee No. 5 Track, AREA Proceedings, 1905, pp 37 9 AREA Proceedings, 1901, pp 251-257 10 W. Cain, The Transition Curve whose Curvature varies Directly as it Length, ASCE Proc XXVI (1892) 11 A. M. Wellington, Railway Curves, Railroad Gazette 12 A. N. Talbot, The Railway Transition Spiral, University of Illinois Technograph, Vol. 5, 1890-1891 13 E.E. Russell Tratman, Railway Track and Maintenance, 1926 14 Ellis Holbrook, Railway Gazette (now Railway Age) December 3, 1880, pp 639 15 Ellis Holbrook, Spiral Curves, Engineering News, June 13, 1901 16 S. Whinery, On the Theoretical Resistance of Railroad Curves, AISC Proceedings VII, 79 (1878) 17 Arthur L. Higgins, The Transition Spiral and it Introduction to Railway Curves, 1922, (Van Nostrand) 18 James Glover, Transition Curves for Railways, ICE Proceedings, 1900, vol CXL 19 Spiller, Shortt, Discussion on Railway Curves, ICE Proceedings, 1909 pp 127 20 H. E. Robarts, Transition Curves on Railways, PWI Journal Abstracts, Vol. XXVI, part 3, 1908, pp334 21 E. Cesaro, Les Lignes Barycentriques, Nouvelles annals de mathematique, 1886, (3) tome 5 22 M. A. Cornu, Method nouvelle pour la discussion des problemes de diffraction dar les cas d’une onde cylindique. Journal de physique theorique et appliqué, Paris, Vol 3, 1874. 23 L. Euler, Methodus inveniendi lineas curves maxime minimive proprietate gaudentes, 1743 24 Topics for Club Programs, Euler Integrals and Euler’s Spiral – American Mathematical Monthly (AMM) Vol 25, 1918 25 W.A. Oldfather, C.A. Ellis, D. M. Brown, Leonhard Euler’s Elastic Curves –, ISIS Vol 20, Nov 1933 26 B. Williamson, An Elementary Treatise of the Differential Calculus, 1899, pp 286-287, Longman, Green