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B- TRANSACTIONS. ■Thin Society, ns a bojy, does not hold itself reapODsible for the facti and oplaloni stated iu any uf its imblioationa. DISCUSSION ON TRANSITION CURVES. By M. W. HoifKiss^B.A.Sc., A.M.Can.Soc.C.E. To be i-eaJ Friday, 25th November, 1892. The much vexed question of transition curves is getting pretty nearly settled when such excellent inethous as that of Mr. Lord- ley's are given us and good tables made out which he tells us is being done. Hib method is very acfiriits uud not so difficult to use. 1 have employed a method which ib not quite so accurate witbout cor- rections, as the one in Mr. Lordley's paper, but it is much more simple to apply,. and is, to my mind, sufficiently accurate for 999 cases out of 1000, and with corrections given below can be made as accurate ^8 W'j please. It is composed of two separate curves, and is in fact two separate cubic parabola*, one of which is measured along the tangent from the F. T. C. to the middle of the offset opposite to the F. C. marked F in Mr. Lordley's tig. 1, which I will reliw to aa simply tig. 1 in my following remarks, and the otber is measured ailong the circular arc beginning at the F. 0} and cou- tinaing back to the point P, where it meets the first cubic parabola. The following is the formula employed for the first cubic parabola : — Let the carvaturo increase in proportion to x measured along the tangent and not along the curve as in Mr. Loi-dloy's paper. Then we have d'y dv dy dx -,= 2 kx= = lu'' kx' y^ J From fig. 1 it will be seen when x = FK, r = 2 R and x = a " ,c = FK, y = 2 '""^ •'-'=<' where o = the offset 1 D (1) (-') (3) •. 2Aa=r ka- 2Ji~2 X673U J) 2^920 ka and :; =-Tr=, Da H ~'d (22920) a = 185'4 Ji (4) (8) Now, this part of the transition is only used from the F. T. C. up to the jM>int F opposite to the F. (\ in the circular arc in fig. 1,'aiid iIlo difloreiice of length l>etweeii 'a' in this formula and the length <>!' llie curve is smaller tliuu is usually measured in th« fi«ld. Xo nhuu' Uuw to |Hit ill this part of the curve, let us take ac •iiample. Suppone we desire to put a traiititioa uq a 4" caiTe urid wo choose an ottaot of '.'. fl. Thoi-o numbers are taken as a simplo C'uso in ordci- to bo bolter uiiduistood. Fiom (5) wo boo that this gives us "a"=Ul ft. Now, tho ordinates from tlie tangent F Ji (lig. 1) arc prop, to x\ as will ho scon from (3), and 2 tlio ordinate at P will bo .7 =1 ft. Tho ordinatos then will bo, beginning at tho P. T. C, 1 ft. from tho P. T. C. 10 X 1 = 00 20 30 35 40 45 SO 125 lau (■Il)'xi = /20 \' (i) (i) (i3t) X» = /125\3 00 00 00 X 1 =01 X 1 =02 W^l Xl='0-i X 1 =-04 OH, etc. 87 X 1 =117 1 fl = 186 ft. This 11 it very O'irivoiiiorit length, 131 ft., of transition, which being doulilud gives in 2i)2 ft. of tr:iiisition eurve. If wo always make tho ollsei half tho degree of curvature, that is 7: =—, it will bo seen from (5) that wo will always get this length, and indeed it can bo nearly always used to good advantage. But as wo will soimtitnes require other long; lis or other oti'sels, wc can prepare, say, half a dozen such little tables as the above, and put them in the hack of the IJeld-hook. Wo can have, jierhaps, i) 7) o Jj D and tabulate tho urdiiiates as above fur tho transition curve for every 5 li., say, from the P. T. C. to the P. C. One table does for each longih of 'a.' Make it out for an offset of 2 ft., and then, if youi' oll'sot is any other multijjlo ot 2, multiply the ordin- atos BO tabulated by this inuliipio mentally when you are putting in tho transition curve to got tho proper ordiiiates. Sd far we have only given the curve for tho lirst half of the total, that is from the P T. G. to tho P. 0. I intend to show that ."\i exactly similar formula onn bo U8od for the ^ccond jiai t of the transiiion curve, only it will be mca- sured iVom the P. C backwards from the point B in fig. 1 to the P, C, at tho ])oint i\ But the curvaiuro will be supposed to vary as the distance from B measured atony the Circular arc to- wards A and uj) to tho P. C. Now, since the curvutuioof tho straight lino /f /(T is constant (and always zuro), its curvature can properly bo represented by '* • . a = 131 ft. 1 4 ■ a = 93 U. 2 a = 2G2 ft. 4 . a = 372 (t. u horizontftl straight line fw a basr of no curvaturo. Let A B (&g. 4) represent this. O t A B Fio. 4. Again, since the curvature of the transition varies as the dis- tance from the P. T. C, its curvature can be properly repre- sented by a straight line beginning at A and mnlcing an angle wiih A B depending on the constant 2 K in (1). Lot such a line ho AC and let A 5 = a in tig. 1 . Then the ordinate from 4 fl to il C at any point x from A will repiesent the curvaturo a; that point, and of courne will represent tlia difference of curva* ture between the tangent and the transition curve at that point. Now, tt similar diagram will apply to the second part of the transition curve. For since the curvature of the circular arc is constant, its cmvaturo can properly bo represented by a horison- tal straight lino at a distance of '5 above the base line A B of no cutvaluro, since the curvature of the cirilo is -p. Let this be re presont-jd by the -straight lino D E. Now make D E equal to tlio length of the circular arc between iho P. C. and the P. (7.» and also equal to the length of tangent from the P. T. C. to the P. C, and also, of course, equal to A fi in fig. 4. JD B "13? Fio. 6. Then, since at the P. C the curvature of the transition curve is equal to that of the circular aic, its curvature at this point can properly be represented by the ordinate from E to the base line of no curvature, the same an that of the circular arc. And since the decrease in the curvati!;e of the transition curve is pro- portionnto to the distance from the P. C measured backwards towaids the P. C, this decrease of curvature can properly be represented by a straight line drawn from the point E in (tig. 6) below D E, and m:iking an angle with it depending on the con- stant K in (1), and this angle will, of course, be equal to the angle P j4 C in (fig. 4) and the curvature of the transition at any point X from the P. C' will be represented by the ordinate be- tween this inclined line and the base line of no curvature. Then let the curvature of the transition curve be represented by the strait line O E. Then the difference of curvaturo between the circle and transition at any point x from E will be represented by the ordinate from I> E io E in fig. 5. Now, on examining figs. 4 and 5, it will at once be seen that they nii'oiy fit into one another, and that one is not complete without the other. Of course B in fig. 4 i> equal to B (J in fig. 5 J that is, the curvature of the two parts of the total transi- tion is the sumo at the P. C. or at point P in fig. 1, And as the second part of the transition curve reeedos from the circular are at any point with the same anglo as the first part of the tran- sition recedes from the tangent at the corresponding point, both depending equally on the constant k in (2), and as the length of the circular arc is equal to the length of the tangent from the P. T. C. to the P. C, it follows that the inclination of the first part at F to the tangent PAT is equal to the inclination of the BCuOiid part at P to the tangent or liiu circutui- arc at P. C, and consequently the first and second parts of the tiansition have a common tangent at the point P, and from (6) it is evident that the point P must bisect the offset at the P. C. It will be BeoD AY)m tho explanation above that the 8amo equa- tions will iipply to tho second part of the tranHition curve m for the first part, with thin diMtinetion : the formulio for tho Hucond part roprcHont nol tlju total curvatiiro, or tangent, or ordinate, na, in the formulro for tho HrBt part, but represent the difference between these functions of the tircular arc and the transition curve. Then the above equations are identical for tho two parts. Now, since the l6liigTl>"Cf tfre-fatigiSirfftTrm'the P. T. C. is so very nearly equal to that of the first part of the transition, and the length of the circular arc is so very nearly the same length au the tieoond part of the tiansilion curve, 1 am pretty suie in 4ssertjn^ that a passenger riding in tho train passing over tLi» part of tho track would not be able to say whether the variation in curvatui* was calctjiated according to the length of the ti-an- sitioD curve or of the combined lengths of tho tangent and the circular arc, . TJie second part of tho transition is i)ut in exactly as the tii-at part excepting, that you coromence at tho P. (7.' and measure backwards towards the P. G. alomj t/ie circular arc, and mea»mro the, ordinatoN outward from tho circular arc in tig. 1. But in practice the circular arc will be the distance of tho otfsot "o" outside of the cii'cular arc (which wo will call the inner circular Afc), and'wili be a continuation' of the tangent F K insto'ad of the lino D A, and Will have its P. C. at tho point K insloud of at A. Con8e(juontlyrit will be necessary to measure the orditiatos tor the transition curve t'riuards the distance of the offset, minus the tabulated ordinates, or, if "y" is the tabulated ordinate from the ipaer ciiciilar arc to the transition curve, it will be necessary.to measure tho distance (o—y) inwards instead of the distance ^'. y''- outwards. I always run in the regular circular curve in location just a.s if there wor* to be- no tiTjirottianrandthen, just before con- struction, go along tho~~rrflo and, vvith an a.-sistant, pull up tho stakes in the circular curve and movo themjn tho distance (o — y) for the second part of tho transition. But for tho first part of tho transition the stakes are movBd in onlj' the distance "y." Two men can do this as last as they can walk along, and one pull up the stakes and move them in the proper distance, wl.ilo the other gives that distance from tlio little tables. If the stakes are eet 50 ft. apart in a transition of 2(i2 tt., only five stakes will have to be moved on each end of the circular curve, which will give an idea of the rapidity with which this can be done. A number of miles can be done in a few honre even where there is considerable curvature, as in a hillj' country. It mujt be remembered that this transition is composed of two distinct parts which happen to fit in together nicely. But one is measured along a circular arc backwards, while the first k mea- sured along the tangent forwards. I always keep the same hubs as were used for the ordinary cir- oular curve, and when the transition is put in no other hubs are reqijired, but the old ones must b« prcsorvodj and if it is nece^- iary to rtm in the line again tho circular arc is run In first as if ho transition were going to be put in, and at any time convenient the •takes moved as directed above. In the transit book it is aniy necessary to note down the ofiset chosen or " a," as tho dogi'eo of carvatur,. will have been already noted for the ordinary circular carve. Then the transition curve can bo put in any time after- wards. And then we have to consider that by this method wo can lay down the transition curve from lines already ostablishod, and without any calculation more than can easily bo done mentally by any one who can do ordinary multiplication. As said above, this method, without using tho corrections, i» only an approximation, but a very close approximation, so cloM, in fact, that as far as making the trains rido easily over it, it is as good as if the variation.; of curvature were calculated to vary with the 'listance from tho P. n. along tho ti'ansition curve itself. But as there may be times when it will be neeessary to know tb« exact difTerence between the length of th« total transition »ad "i n'. tinil as lliis in a very oany miitiei' In lirul. mid llio icniiilMiK loi'iiiillu is very -tiuiplo. tlu' rtoliilioii U j^ivoi; in wliiit li»lli)W!<. Kill- llic lii'Nt pai t ot'llio Iruimilioii let s = ilisliiiiiH) of any iiiiiut I'ldni lliu /'. /. ''. lnl^■lSlll•o^i aiotif^ Iho liainilioii ciii'vo, and let X = tlio i:fin'os|)iindin!; dittancii mtmsiii'cd aliiiit; tin) laiiUi'iil. Tiien wu liavn ,1s = ^dr' f- ,1,/' - ^/ 1 + /■' v'd,- I'l'.Mii C') / A' X* It* X /.■■-'' VVliiin (■ = ((, .s — 1," — loiii;lli 1)1' tir-i liii IT mI ilm i rnmd imi cMlur / /,-',(' li it I: It'' \ /»•<«' ; "*' -fn /» .( WfitiDg /■ i;.r ^^;7;JV|;) -'" li.iv'.: -.to ''" -II-.- - " f '' OK III .., + \ ■ I 111 'I'lni.t wr Mi'i' ihal llio r:iliii ' d''|irii.K .ndy ■mi lln; |irii liirl i* (, It' we ninkr lliis |iindiiil l.'ilil, liieii li II.J715,') ' ^1 f- MOii,'i'.'Ni;i;;i — ■ooounDlis;: -f.Vi'. - I noiiii'JhTc: I'l Now il -I -l.'il II :iii(l U •-- in' .«,"---,( -- ,iii:i II. = liiiir an incli. II will bi' >ciMi llial il i^ :i very niniiilo ni.illi'i- to liiid tiii> valnn oT till' latio I'liiiji till.' aliovi! iDinmla. (i> willi a laMe (if lin^ uiitLliia line lo^uiillini will dii luc :ill tlie IniniM, and when tliis lo^arillnn i- multiplieil liy the lo pe.tive ox|)i(iioiit') dI' " '< " all ihe vulueij ot li'. b', &c , ran In; lound ul uuce, But it is ovideni that 6* and i* arc all ilio lenns iliai iioeU to bo oousiderod lu secure the groutosl accuracy over required. Ir jti also eady to find the dirt'erouL-e iu luiiglL between ibe •t-cond pari of ihi- IransiliDU rurvi- and ibal id llu- ciroulai ur'j, troui wliicli il i^ nicusuiod nil, in llii^ fidlowiiin- niannor ; Li't i bc! tin' leiiglli 111' any pmnL liiim lliu /'. ''.' nioa«ui od iikin.';' till' Iransiliiin i-iiJvo, and i' llio luMf^lli ul llic lui'reMpoiid n^' jiuiul inoasuied alunt; I lie iiiiiur I'irndar aic Ah bollire. |el 1/ = oi'dinalu nieasmod alinij< lliu inuei I'iri'ill.ii- arr, and lei H, — radiim ul' lurvalurn of ihf inner cin'ulai' arc, and l>^ itM du>;ree ot iiirvatun', and if, lU leni;lli lueasured along llie inner circular arc. Then, k x> il = •, A', W, +j/ _ Ky + T" " - ' "''i:^ ft. A x' ft, -' + :{«, when X — <(. s = s" = 1 + ■iW.rJayi- ■ ■ ■ «/ -«. = « o . . ■ (84) 3 (22920)'' D' a' \ (•.'2!fiO;7 But UK in practice tlio Itwt half longtli "f tho triiiiHilioii curvo U moinurod iilotiK tho outer ciirultti- mr, wliicli lies Jiml tlio iIih- tO' .eof the ntlai't "»" nuliiilly outward Irom tiio inner ciiculm- *i " s " : , what we roiilly wiint in not tlio ratio ■- l)»t the latio ^ . If .•let/), (t and II staml forHimiliir InDClions of lliin oiilor cirtujur i-c, a« follows, wo Clin oiisiiy traiisfonn lliu uliovo cijuiilion to give UK tiiiM ro.ii;ircmonl, for D, a, = l> n winco itii' factors in oac)i vnvy inverMoly as one another, ft_o 5730 — «/) ,„ui«, ^ ^ ft "- r.-:i(i '^ s," 57:10 — l> I , •■• rt °" 57;t() V '"+";((•: Da cr tiHing "i" to Htand for ^,^,,,^„ wo liavo a - 57^0 V "•" :i / It will bo noticed thai, nllliouifh in oui- Ibi niiila tor -^ lor llio lirst imrt of ihc transition inrvo. wo liiivo lal \ For the second jia-t, „ , i oD / b'\ h-i a — s., = a -l I 1 4. - I > 15730 V' +3/ 3f . • . total ditl'erencc = y to put in on ih.^ ground that, no mat- tor how indolent or ill-informed an ongincoi' might bo, ho would in all probability i)Ul it in. And then, after it is in, it is as good as the bof t. I can'l help but think that it is the one that will always bo used when all ongiiiccrs como to understand the importance of transition cuivo^ and to seo li(tw easy it is to put this one in on the ground. 1 1 is simplicity itsolf, is as good as if it wore better, and can bo accurately caU uliitod as to lor.gih, &c., from tbo fortnuliu given above, when such calculation is necessary, which will not bo onco in a thousand limes. Enough is us good as a feast and imicb more wbnie-on.o. And last but not least, it can be put in at any limo aflor location by simply moving the stakes as directed above. If as might happen sometimes, it is necessary to have a result ii.tin'itely cor.ect, the combi.n.tion of (6), (7.) and (8) will give it, and then we have D / t'\ Ir I /'''' b' b"' \ ■(573o(l + 3)-a}-"iro-72- + 2(r8-'^''; roDt b'\ b- {( I 57:w' A' 10 \\brM — oD' } Ti + &<•• ( (9) With this correction tbo method is i)iactically correct, and when it is re(iuired to find the dilloronco in length between the total length of the liansition cuive and '2a' this last formula (9) should always bo used, unless tbo offset ' o ' is small. For accuracy il should be I'uii out the same number of terms as is done above. If il should be desired to run in new tangents from an already established circular curve, this circular curve would then become the innoi' circulai arc, and the second part of the transition woidd be meiiaured along it. If it should bo required to know the ditlerence in length between tbo whole transition and '2 a,' this can bo found by combining (6) and (6J), and wo get \ 30 72 ^ ) 13 ft= = a :io The transition should bo iiut in as in the other case, of courae, except the orilinate ' y ' for the ti ansition curve is measured out- ward frotn the circular ore, and (o— ;/) outward from the oW tangent or ' !/ ' inward from the new tangent in all cases in same direction as tho other case. 7 d — a 72^ f +4 very approximately.