,.^„.. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 lifBS KS II ^>^IJ£ L25 il.4 1.6 Photographic Sdences Corporation 33 WEST MAIN STREET WEBSTER, NY. MSSO (716) 872-4503 CIHM/ICMH Microfiche Series. CIHM/ICIVIH Collection de microfiches. Canadian Institute for Historical Microraproductions Institut Canadian da microraproductions historiquas 1980 Technical and Bibliographic Notes/Notes techniques et bibliographiques Th to The Institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which may significantly change the usual method of filming, are checked below. D D D D D D D Coloured covers/ Couverture de couleur I I Covers damaged/ Couverture endommagie Covers restored and/or laminated/ Couverture restaurde et/ou pellicuide □ Cover title missing/ Le titre de couverture manque I I Coloured maps/ D Cartes gdographiques en couleur Coloured Ink (i.e. other tiian blue or black)/ Encre de couleur (i.e. autre que bleue ou noire) Coloured plates and/or illustrations/ Planches et/ou illustrations en couleur Bound with other material/ Reli6 avec d'autres documents Tight binding may cause shadows or distortion along interior margin/ La reliure serr^e peut causer de I'ombre ou de la distortion le long de la marge intdrieure Blank leaves added during restoration may appear within the text. Whenever possible, these have been omitted from filming/ II se peut que certaines pages blanches ajouties lors d'une restauration apparaissent dans le texte, mais, lorsque ceia itait possible, ces pages n'ont pas M filmdes. Additional comments:/ Commentaires suppldmentaires: L'Institut a microfilm* le meilleur exemplaire qu'il lui a 6t6 possible de sn procurer. Les details de cet exemplaire qui sont peut-Atre uniques du point de vue bibliographique, qui peuvent modifier une image reproduite. ou qui peuvent exiger une modification dans la mithode normale de filmage sont indiquAs ci-dessous. I I Coloured pages/ D D Pages de couleur Pages damaged/ Pages endommagAes Pages restored and/oi Pages restaurdes et/ou peiliculdes Pages discoloured, stained or foxe( Pages dicolories, tachetdes ou piqu^es Pages detached/ Pages d6tach6es Shov.rthrough/ Transparence Quality of prir Quality indgale de I'impression Includes supplementary materi{ Comprend du materiel suppl^mentaire |~~| Pages damaged/ I I Pages restored and/or lami.iated/ I I Pages discoloured, stained or foxed/ I I Pages detached/ I I Shov.rthrough/ I I Quality of print varies/ I I Includes supplementary material/ Onlv edition available/ Seule Mition disponible Pages wholly or partially obscured by errata slips, tissues, etc., have been refilmed to ensure the best possible image/ Les pages totalement ou partiellement obscurcies par un feuillet d'errata, une pelure. etc., ont 6ti film^es A nouveau de fapon A obtenir la meilleure image possible. Th po of fill Oi be th si( ot fir si( or Til sh Tl wl M dij er be rij re m This item is filmed at the reduction ratio checked below/ Ce document est film* au taux de reduction indiquA ci-dessous. 10X 14X 18X 22X y 26X »X 12X 16X 20X 24X 28X 32X The copy filmed here hes been reproduced thenks to the generosity of: Nationel Library of Canada L'exemplaire film* fut reproduit grflce k la g6n6rosit6 de: Bibiiothdque nationale du Canada The Images appearing here are the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Les images suivantes ont M reproduites avec le plus grand soin. compte tenu de la condition at de la nettet6 de l'exemplaire film*, et en conformity avec les conditions du contrat de filmage. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres sion, and ending on the last page with a printed or illustrated impression. The last recorded frame on each microfiche shall contain the symbol -♦- (meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Maps, plates, charts, etc., » >ay b? filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les exemplaires originaux dont Sa couverture en papier est imprimie sont fiimis en commen9ant par le premier plat et en terminant soit par la derniAre page qui comporte une empreinte d'impression ou d'illustration, soit par le second plat, salon le cas. Tous les autres exemplaires originaux sont filmto en commen^ant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernlAre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbole •-^ signifie "A SUIVRE", le symbols V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent £tre film6s d des taux de reduction diffirents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de Tangle sup^rieur gauche, de gauche d droite. et de haut en bas, en prenant le nombre d'images n6nessaire. Les diagrammes suivants illustrent la m6thode. 1 2 3 1 2 3 4 5 6 K T I THE IRIIM TRANSITION CURVE BY E. S. M. Looelace, B.A.Sc. A.M. Can. Soc. C.E. Entered according to Act of Parliament, in tlie year 1893, by E S M Lovelace, in the office of the Minister ofAgricnUure and Sta- tistics at Ottawa. PREFACE. .M. Sta- 1 A» moHt engineers who have given the Bubject the slightest coDsidcratioD have ackDowledged the ndvuntages to be derived from the use of transition curves in the location of a line of Rail- way, the writer has no intention of discussing the question further than to say, that as sectionmen almost invariably ease off the ends of the circular curves as staked out (causing thereby either absolute kinks or el<>e portions of track of a much sharper degree of curvature than the main curve), it would seem to be the duty of the engineer to avoid such sources of danger to a train becom- ing derailed by locating the curve at once in the position which it will be made to take finally. The trouble hitherto has been that tho transition curves pro- posed have cither been of so complicated a nature as to render their location very troublesome, or else, mere approximations which engineers instinctively object to. The transition curve which the writer has undertaken to describe is mathematically exact, and its location requires very little more work, either mental or otherwise, than does that of an ordinary circular curve. In preparing the following, the writer received a good many suggestions from the papers on the subject read before the Can- adian Society of Civil Engineers, and also from the remarks of those gentlemen who took part in the subsequent discussions on such papers. E. S. M. L. 54 St. Matthew St., Montreal. < \ \ \ \ CHAPTER I. FORSIUkE. s \ -1-69.^ •<5|. \ Id Ft... 1 let AB and A'B be two tangents, intersecting at B, fafvlm ^ ^1, "r^" ^''' ^'^' ^»»^ -^" «f which vary from 7? at the points Pand P'to infinity at A and A'. 6 As most engineers are more oonoerncd with praodcal results *ihan with the theoretical transrormutions by which such results are obtained, all demonstrations, etc., have been purposely placed at the end of these notes, where all who choose may see for them- selves that the following six equations (giving all the necs^ssary information for the location of the proposed curves, and used in working up the table about to be described) are correct. Assuming R, radius of main curve, and the constant nuuibur m, as known, then will AP, the chord of the transition curve, be equal to the constant number m divided by three times the radius of the main curve, or letting the chord AP = c. (1) c = 111 Tr (^) sin 2« = m the co-ordinates of the point P arc = c cos c sin^ the co-ordinates of the point O are (3) ( Ax\ = { NP = (-i) AH= c /2-}-cos2'^N cos'^ _ C y^a + COS 2"\ 6 \ cos '^ / c /2-cos2'^> or An= AN- R sin ^ti HO = % C^-?^'f'\ or HO = NP^R cos 3« ^ b \ sm f' / the distance AB is given by i^) AB = HOUn- + An 2 Letting — equal the length of transition curve from A to P CO L = '"f .-u .. ( Tho constant number m, which determines the len.rth of AP ^the chord of the transition curve) can of course be chosen :.t twlT ??'""'"' '*' '^' point- P, si.ht to ^and turn off twice the deflection angle to /', that is turn off 2o The central angle /'O// is always H'N.r three times the de- flection angle to the point I*. The above equations also apply to any other point on the transition curve between A and 1\ CHAPTEK II. TABLES. As8uinin<; that the chord distance to the point P is nixty feet fur each degree of curvature at that point (which assumption makes the value of the constant nambor m ur|Uil to 1031337), the following tabic gives the deflection angles and nil necessary data for a transition curve that can be applied to any uinin curve (from an 0° to a 10® one), the degree of curvature of which is Bonic multiple of 5 minutes. When the degrco of curvature of main curve is not a multiple of 5 minutes, all the necessary data (for the point where the transition curve meets the main curve) can be found by inter- polation in the table with the exception of the distance HO, which varies so rapidly that it must be calculated from equation (4). The deflection angles for points five feet apart on tran- sition curve can still be taken direct from the table. To use the table : — Look in 1st column for quantity corresponding to degree of curvature of main curve, and on the same line with it under the several headings will be found the deflection angle and distance to point where transition curve meets the main curve and also the quantities to be used in calculating the distance AB. In laying out the curve by means of offsets from the tangent the necessary distances will be found in columns 6 and 7. In laying out the curve by means of deflection angles from the point J, take from the 3rd column the chord distance of the lance also int trom iDCe from i4 to a 8° 25' 30' 35' 40/ 45/ 50/ 65/ 90 0/ 5/ 10/ 15/ 20/ 25/ 30/ 35' 40/ 45/ 50' 55' 10" 0' 14 TABLE. us of atur ransi urve •3 5 ^HO ffi L B c 2 d o a> Q Co-ordinates of Point P. ANorx NPory 680. H 674.1 667.5 661.1 654.8 648.6 642.6 636.6 630.8 625.1 4 9 619. 613. 608.6 603.1 597.9 592.7 687.7 682.7 577.8 673.0 505 510 515 520 525 630 535 640 545 560 555 560 565 570 575 580 585 590 595 600 608.2 7° 9'.49 501.1 62.93 510.0 12.63 602.8 63.61 513.3 18.23 606.9 64.84 518.5 27.06 510.7 66 78 523.7 3u.99 515.4 68.77 628.9 45.02 620.2 70.80 6.34.1 54.15 525.0 72.87 539.3 8® 3.38 629.7 74.98 540.0 4.68 530.4 76.28 544.5 12.71 5.34.5 77.13 549.7 22.15 539.2 79.33 560.0 22.72 539.5 79.46 564.9 31.68 543.9 81.56 560.1 41.33 548.6 83.84 665.4 61.07 553.3 86.17 670.0 69.73 667.6 b3.25 570.6 90 0.92 558.0 88.54 576. y 10.88 562.7 90.96 581.2 20.94 667.4 93.41 586.4 31.11 572.0 95.91 591.7 41.. 39 576.7 98.46 597.0 51.78 581.3 101.06 600.0 57.65 583.9 102.64 602.4 10° 2.28 585.9 103.71 607.7 12.90 590.5 106.41 Co-ordinates of Point O. AH 261.8 254.3 266.8 259.2 261.7 264.2 266.6 269.1 271.5 274.0 276.5 278.9 281.4 283.6 286.2 288.7 291.1 293.6 296.0 298.4 HO 696.4 690 684 678 672 666.8 661.2 656.8 660.6 645.3 640.3 635.3 630.5 625.7 621.1 616.6 612.1 607.8 603.6 699.4 i I U i I CHAPTER III. PFiOBLCMS IN LOCATION OP TRANSITION CDEVE. \ \ \ FtG. a Althou^li theoretically both methods are equally good, prac- tically it is better to run hi the second piece of transition curve from A' before locating the main curve PP ; but should it be considered necessary to reverse this order by putting in the main curve first, and then the second piece of transition curve, pro- ceed as follows : — In Fig 2. Suppose that one piece of transition and the main curve have been located, that the transit is at P and sighting along PT the tangent to both transition and main curves. ommcnoing at A call the points to be located P^ P^ P^, etc. 16 their rectangular co-ordinates Xj yj, x^ y^,, x^ y,, etc., and the deflection angles from A to these points d^^ ^"^j ft°^ etc. The exterior angle FSIi = SPP^ + RPj P therefore SPP^ = PSR - RP^P but tangent of angle RP^ P is equal to PR NPNR y- y, RPj~ AN-AN~~x-x, also the angle PSR is equal to the angle PTN or 3^; therefore the angle T/'P^, the deflection angle from the tangent PT, to P, the first point to be located, is given by angle TPPj = 3«-tan-i ^"^^ 35 — X_ Similarly angle TPP, = S'^- tan-i IZll aJ-Xfl u « « « (( (( (( TPP. =3"-tan-i IZlj x-x^ TPP^ =3'^-tan-i^-Zl* TPP^ =3^-tatri -^LJ^ x-aj^ ^'i^P, =3^-tan-i'Llll ^^7^1 =3'^-tan-i ^"-^i cc — cc 1 As the distances from A to points P^ P.^ P^ etc., will have boon already decided on when running in the first piece of tran- and the angcnt have tran- 11 Bition curve, it is only necessary to take from the table the values of Xj yj, ajjy,, X3 1/3, etc., corresponding to these distances, and insert them in the above equations : For instance : in the example given in the last chapter the point P^ is 200 feet from A, therefore from table x^ = 200.0 and y^ = 3.88 and P is 395.9 feet from A, therefore from table at = 393. 9 and y = 29. 97 6 is also 4° 21'. 04, therefore 3« is 13° 3'. Consequently the angle TPP, = 130 3' - tan-i ^^J\- ^J^ = 13° 3' - 7"^ 40' = 5° 23' ij\fiS . 9 — JOO A similar problem is that in which it becomes necessary to put in an intermediate hub at any point, say P,, for then the transit being moved to P^ and sighted along the tangent UP^ the deflection angle P^P^U= 3^5 - tan-i ^g "^^ a;, -a;. « i( « i( (( lie (( (( (i P3/'5^=3^3-tan-iy5_iyL »j — OSg " P,P,U= 3^3 - tan-i y«~y« .r.-xj (( AP,U=3e^^e^^2e^ or if running the curve forward to P 18 the deflection angle P,/*, F= tan* 1^L_J^ _3«, (I (( * I\P.V=taLn-^ ^ — ^-3^ 7* 5 »7-a^5 (( « " PP,F=tani ^-J^-SfJ. X-X3 If there be more than one intermediate hub, proceed in an ex- actly similar way. Referring- again to Fig. 1 it will be seen that OH - Ris the offset distance which a circular curve of radius 0^ would have to be moved towards the centre to make room for the tran- sition curves, so that if an external distance corresponding to a radius equal to Off be; taken from a volume on circular curves, this offset will have to bo added to it to give the external dis tance EB in Fig. 1. also since EB = OH sec — R 2 a value of R may be chosen which will (with the corresponding value of OH taken from the table) make EB equal to any re- quired distance. For a given intersection angle /, EB is as small as possible when 3^ = -, or when in the table '^ = _, at the same time the 2' 6 main curve PP' reduces to zero, the points P and P' come together at E, and R is the minimum radius which can be used so long as »i = 1031337, the value chosen in the construction of the table. Conversely, for a given minimum radius R, EB is as small as possible when m has a value equal to 9^^ sin - found by com- 19 bining equations (1) and (2) ; theoretically this would be the correct curve to adopt in every case, but to use it one would have to be content to do without the aid of tables, as such would require to be infinite in extent. dis In special cases when the values given in the table are not suitable, and it bec(^0es necessary to depend entirely on the for- mulae given in the Ist Chapter, proceed as follows : — Assume a convenient length for the chord distance AP or c ; then, as R is supposed given ; by equation (1) rn = 3/?, c. Knowing m and c, ^ is given by equation (2). By equation (4) AH and HO can now be easily calculated, and their values substituted in equation (5) will give the tan- gent distance AB. Finally find the length of the transition curve from equation (6). To locate the curve ; — Call the chord distances from A to the points to be located a, 2a, 3a, 4a, etc., and the deflection angles to these points, ^1^ ^2^ ^,^ ^^ etc., then by equation (2) sin 2^, = — (( (( (( (( <( sin 2«., = 4 - = 4 sin 2^, m sin 2^3^ 9^ = 9sin20j m thus it is only necessary to calculate the value of sin 2^,, for this, multiplied by 4, 9, 16, 25, etc., will at once give the values of sin 2^2, sin 2% sin 2f?4, sin 2^^ , etc. 0°, 6^°, ^3°, etc., can now be found from a table of natural sines by inspection. 20 When « i, not g^.,„ ,h„„ ,,y g„ ,^ ^^^1^ 1^^ ^^^^^^^^ ^_^^^^^ bo „„Uip,io<, b, 4, 9. 16, 26, etc.. eo «;« J ,,„.^ ,„ .^ )o given. onoe enoupfh pe H'e° it onoe CHAPTER IV. DERIYATION OF FORMULJE AND GENERAL CONCLUSIONS RELATIYE TO CURVE. Let AKP be the proposed transition curve, and assume that the radius of curvature at any point P of this curve varies in- versely as the chord distance from A to that point, that is. 09 JiCt r m m '6AP and therefore AP= ^ = equation (1) Assume a system of polar co ordinates in which AP is the radms vector, and the angle which it makes with the tangent AN the vectorial anisic. " PST is tangent to the transition curve at Pand .4 7^ is the polar distance hQiAP=C " AT^2' The formula for the n.diiis of curvature at any point P is dp .\dp = ^J^ r »>y(i) dp^ 3cVc m integrating p = 1 (a) in there is no constant of integration as p and c vanish together. The formula for the Polar distance is c'dB p^ ^{dcy+c\dt>y Squaring both sides, transposing and simplifyin C '\/c'—p* by («) d^ cdc m m if>) 1 c* Integrating W= _ sin ^ — 2 m .'. sId 2^^ = — = equation (2) m thore is no constant of integration as c and ^ vanish together. The sine of APT, the angle which the tangent f*^*7' makes with the chord AF is — ^ = P AP c but by (a) :^ = ^ c w and by (2) ^ = sin2fl m :.sm APT =8{n2ft .'.APT =26 Also the central angle POII = PSN= SAP+ APS = fl + 2« = 3» The rectangular co ordinates of P are AN = c cos '^ and NP = c sin ■:)■ equation (3) The rectangular co-ordinates of are AH = AN- HN= AN-VP= AN- r sin 'de no = HV^- VO = NP+VO = NP+r cos S'^ Substituting in the above o cos for AN, c sin ^ for NP^ I equation (4) 24 3-^^^ (found by combining (1) and (2) for r, simplifying and reducing, AH becomes = 1 (?±oos^ ^ 6 COB ^ and HO becomes = ^ (^Z^os 2^) 6 sin^ J •- = equation (4) Keferring to Fig. (1) AB = AH+HB:. An+ OEi^nU equation (5) To find the length of transition curve itself:— the formula for a diflFerential of the curve is *. . dL = dL = ^J{dcy + c? (doy but by (ft) (dt^y = jLj^ m dc m^-c* slni' »C , „ -1 4 1 == = m (m^ - c*) ^ dc^ /I -i:\~2 rfc expanding by the Binomial Theorem V 2m^^8m -) rfc integrating. ^ = c . ^^ , ^^ .,,„,«„„ (,^ wheom = 1031337, (the number ohcen in makingupehe table) ; jj^ the second term in the above series = i_ of a foot when' 25 ring and ic when c 1 c = 254. 3 feet, and ^-- — . the third term in the above series = — 24?»* 10 of a foot when c = 518.6 feet, therefore it is only nocessb.y to use the second term when Z) > 4^ 10', and the third when D > 8° 35'. In constructing the table, a value for the constant number m was chosen wliich would give a reasonable length of transition curve for at least the great majority of cases (viz., those in which the degree of curvature of main curve varies from say 3° to 7*^) ; and which would also give to the table a convenient form for comparison with such tables on circular curves as may be found in the works of Searle, Shunk, etc. ; it was found as follows : — Let the chord distance in feet to any point on transition curve be numerically equal to the number of uinutes contained in the degree of curvature at that point, or if D = degree of curvature at the point, then 60Z) will equal the number of minutes in the degree of curvature at the same point that is. Let c = 60Z) giving about 60 feet of transition curve for each degree of cur- vature of main curve by (1) 60i> = !!i but when Z> = 1" r = 5729.65 and .-. m = 3, 5729.65, 60 = 1031337 In bending the rails it is also a convenience to know that if the chord length to a point on the transition curve be divided by sixty, the degre.; of curvature at that point is at once given. When necessary, other values of m may be found similarly. The curve is symmetrical with respect to a line, making an angle of 45° with the initial line AN, for since sin (90 - 2) = sin (90 + 2) c has the same value in equation (2) when ^ = 45-^, as it has when <^ = 45 + 0. When V» is 0, « = 45°, 2« = 90°, and sin 2^ = 1 .*. c = Vwi = a maximum. If a few points be plotted, it will be seen that the curve takes the form of a loop, the point of the loop being at the origin A. If the negative values of c be also taken, a second loop is obtained, so that the complete curve is in the form of the figure eight, and is, in fact, identical with a well-known curve called the Lemniscata. If it be assumed that sin 2f^ = 2fi c^ 1 equation (2) becomes 2^ = — or f) = kc^, where k ~ — 911 2m This is the formula that a member of the Canadian Society of Civil Engineers proposed using ; but looking at Figure 1 it will be seen that the length of the main curve depends upon the value of 6^, and therefore if 6^ is greater than say 6° or greater then 1°, a correct length for the main curve will not be obtained if such a formula has been used to calculate the value of ^. As has been already pointed out, however, the assumption may be used to advantage in the mere location of points on the tran- sition curve.