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Sec 40. r/Me amaiiiuiwn.) *2aiiaiiviii Sociisti^ of Ciuif 8ttijin«et». INCORPOKATED 1887. TRANSACTIONS. N.'B.— This Sdcioty, as a body, docs not hold itself responsible for tho facts and opinions stated in any of its publications. "TUB TRANSITION CUEVE." By Henry R. Lordly. To be read on Friday, 28th October, 1892. Although considerable has been written of lato on the subject of " Transition Curves," much of which has been very interesting and of service to the profession, yit the (juostion of deriving and laying out tho curve practically has not yet received e.xhaustive treatment. The necessity of such a curve and the great benefit to bo derived by its use has been w thoroughly discussed by different writers, that it will bo treated as foreign to this paper. For reforenco to these points we recommend Mr. Wicksteed's paper, in " Transactions " of this Socisty, "Vol. V, Part I. The transition curve to which our attention will be given here, is the curve which has been worked up and developed y Prof. C. L. Crandali, C.E., of Cornell University. (lUr. Ellis Holbrook," it is said, first proposed and used tho principles in laying out curves for the Pittsburg, Cincinnati & St. Louis R.R., 1882, and a gentleman at Lehigh University worked up a solution of it a few years ago, but to Prof. Crandali, the honor is due for having put it in its picsent state of efficiency.) This curve, as will bo seen later, is strictly mathematically correct, and it has now been tested suiHciently in the field to show that besides this, its easy manipuhition makes it invaluable to the engineer. In order to discuss it here, we will take it up under the following heading: (1.) Derivation of formula). (2.) Tables from the results of f 1). (3.) Examples and general con- clusions. (1.) In order to counteract centrifugal force upon a circular curve the outer rail must he elevated, tho change from the tangent being gradual to promote easy riding and to prevent twisting the trucks. Thoreforo, taking the centrifugal force proportional to the elevation at every point, the curvature, in passing from the circular curve to the tangent, must increase directly with tho distance. ' R. R. Gaz., Deo. 3, 1880. Suppose in figure 1, AB to be a circular cui-ve with centre C. Now if we begin at B to reduce the curvature directly witli distance, continuing this reduction until curvature is zero, main- taining the same central angle I. us in circular curve, the new curve will pass outside AB, having a tangent of/F(i, parallel to DE, the tangent of AB, and at a certain distance, KA, I'rom it. As is customary A is called the P.O. ; F the P.T.C., and B the P.C Again let H be the angle which the curve at any point L, makes wi-h the initiol tangent FG, s the length of arc FL. Then since by hypothesis the curvature at F is zero and increases with the distance from P; therefore the curvature of any point L, distant 8 from P, is equal to a constant multiplied by s. For con- venience to avoid fractions later, this curvature is expressed by 2ks, k being the constant depending on the rate of change of cur- vature. di The radius of the curvature equals ^^, H being any angle.and and curvature varies invei-sely as the radius of curvature, we have . = 'ihs, or, after integrating, H = ks'. Ify is the ordinate, we have dy = ds nin H = ds sin ki' (a), but sin a (a being any angle) in series is equal to ;|c a? , a' a' «-3! + Sl-f!'''"- + therefore in (a) dy = ds (ks' — .^j ki^ k's' therefore y = .j — „ .^-^ ks' + 5! • k's" 11.5.4.3.2 3 42 + - — (b) ^ 13'iO ^ ^ 1320 "^ 3 ! = 1.2.3; 5 ! = 1.2.3.4.5. and so on. In tbe same way, for the abscissa .c, we have dx = ds cos = ds cos ks' a' a' •>\ + A\— • • • and cos a in series = 1 / , k's' , A's" \ therefore (ix = (is I 1 — 2! ''' 4! / X = s — k^l 10 1^' + 216 ~ (c) From these two equations (b) and (c) the length s of the arc FL is expressed in terms of the co-ordinates of the curve, but before using them the investigation must be carried further. After passing this point two methods of procedure are open. First, reducing the curvature from the P. C, in the same ratio as it was increased. Equation (b) will then give the radial ordinates for this part of the curve, besides the ordinates — from the tangent— for the flret part. Also the curve and offset will bisect each other at P. If we should now place the second derivative for the first part at P equal to the second derivative of the second part at P, the K will be eliminated, leaving s to be ascertained for any given value of !/ or AK. (The work of the above has been omitted hero; it is simple application of tlio calculus, and, if interested, the reader may readily follow the steps through himself.) Second—By continuini; the curve to B, and placing the curve so that it shall be the circle of curvature, we will get for flat curves the same result as in the last case. When the offsets are larger, AK and the curve will rot bisect each other, and the ordinates from the circle will dirt'er slightly from the correspond- in" ones of the tangent. The author claims this to be theoret- ically the correct method, particularly if the curve is to be run with a transit. It also gives simpler forraute. Now from equation (1) (anyarc)S=5^=2^- _ s_ •*• -2H' Bui at its limits S=R, and at the same time 2 6=2 I ,.„,,; the s then being s", the distance from the P. T. C. to the P. C and k= e Then r-^ 8' 2 E Sin r i=1.86v/FDr The latter value is obtained by asauming the 8ine=aic and thoTl g^jj putting 81. v/63y"=v/2iX6730 F-t-'D. <'=v^i„ jo" From e o (J° Left. 3° 00'=J I /1,=:'7° 28' 47+22.7 !• 55'. 2 D= 6° +62.7 1° 04'.6 T= 9° 46+02.7 Oo 28'. 8 T=234.44 46+42.7 44+82.7 Offset 3.92 P. T. C. O N. 20 W. 0° 07'. 2 1"=0° 03'. 92 • . 1)° 28' Hero we have taken S"=300. . • . F=3.92, and s ov x = 149.9. T'=234.44. Wo divide 300 by 5, whicli in GO ft. ibr chord length, which is reasonable length. Then as central angles are as ^quare of distance : (iyx9°=21° 0(i' .■. O7'.2=de(loction. (^)»x9°= 1° 26. 4' .-. 0° 28'. 8= (f)'x9°= 3° 14.4' .-. 1° 04'.8= " (|)'x9°=r 5° 45.6 .-.1° 55'. 2= (f) x9°=: 9° 00' .-. 3° 00' = Since a=:27°28' and 18° is usod up for T. curve remains for simple curve, which is (9° 28-r-U°) long= 1+57.8. (The ueflections from P. C to P. T. C are taken from Table No. 2, a description of which has been omitted for want of space. It is constructed from the tangent, in sories and an ctjuatioii, ''=i ■ ^(«'+ s'+ ss)— This table should be very valuable to the engineer. Without it we begin from the P. T. C and lan to the P. C with the defleetioris first found. If wo wish to simply put in the ott'sots and lun the curve latei', we place the stakes iis follows : P. I. (point of intersection)=48+67 T'= 2+34.4 . • . P. C. =46 + 32. + 4+57.8 P. T. .')0+90.4 ThenatSts. 46+32.6 and 50+90.4 oflsets are placed. (4+57.8):= A -.27° 28' D 6° As will be seen, it is better to work forward instead of from the P. T. C. and P. T. C to the circular curve. * Set up truuit and aat to 6° for backsight. • " " " 4» 44' + 1 = J8» 44' for baokalght, «tc. ^ii^'. // . . / ' */ / / / / If the front tangent bas not boon located, in beginning the cii-- cularciii'vo procood iw foilowH. Sot up transit at otlaet diatunce insido tho tungont, or ut the P. C. ; backsigiit to a point similarly oflfset ; thon run the curve as usual. At tho P. T. the operation will have to bo reversed. If, on tho contrarv, tlio front tangent has boon tixod, T' ami not T, must bo moasurod from tho vertex to locate a point from which to lay oil' F from the P. C, T'= \ \ - - - (R+ V) tan Y' and is found, from tig. 2, as follows : T'=HB+HO -EUnf+FUnf =(R+F) Un ^ or =T+F tan ^ Here we have A, D and F given. As the flguro indicates, the circular curve is moved parallel to itself to a distance F, from ils former position, in order to make room for the transition curve. The now cur^o then has an external distance, with referenco to the oid tangent, equal to or slightly less than the old, tho offset being small. Thus = F E' < E — ^^, Frcra Soarles' Table "VI we may take the E for a 1° curve ; divide this value by E' for D', and then change the latter value enough to avoid fractional minutes, before finding the length of tho curve and T. In case tho new curve should tit tho roadbed better by extend- ing as far outside tho old curve at centre as inside at the P. C. we would have : F F E'=F -—\- cos ^ * 2 Another important case ovises where a transition curve is to be put in on old track, tho no./ track being same length as the old. This is to prevent cutting the rails. In tig. 3 let BC=T= »R tan -a and BK=T'=(R'fF) tan ^■ The arc AC=length of old track = Ra'' arc r andarcGL=U'A°arc 1°. Now, tho length of now track from A to C, the transition curve boiug put in, is eijual to (Tli-|-2 (BC— HK)4-2 (e+e"), therefore by substitution wo got RA" arc 1°=R'A° arc l°+2 R tan '^ —'2. (R'+F) tai) .^4-2 («+«')) therefore R'= RA° arc 1°— 2 (R— F) tan '\ 2 (e+e') A° arc 1°— 2 tan > 2 The following will show the use of the above equation. Find tho data for a transition curve where the track is already laid on a G° curve, 800 ft. long. Taking 2-5 ft of tmnsition curve per degree wo have c=:2 X 150 =300. Then from the tables wo get F=3.92 ft.; Ce+e')= (.02+.04)=0.06; A=48'';R=-'f®=966. SubHtitutiiiK in the eijuation R'=799.{)0— 84«.e9— 0.12 .=887.43. 0.8;n«0— (1.89048 _— 4fi.91 ~— ().(l528tr therefore from Searles' Tablo VI wo get r)'=B''28'=(i°4fit}. L'=48^-6.460=742.35 ft. 'r'=»(887.4;J+a,02 tan -j =396,86 ft., whicli ii the ilutii required. Othor problems rai^lit bo taken, but wo buiievu enough bus buun given to Hhuw thu worthing of the curve. Some onginoors, it might bo romarlrautice demands the bo8t, and a properly qui