ENGINEERING SURVEYING 1 VERTICAL CURVES

Transcription

1 RMI Univrsity ENGINEERING SURVEYING VERIA URVES Wnvr two gradints intrsct on a road or railway, it is ncssary to connct tm wit a vrtical curv to improv visibility (summit curvs), prvnt sock impacts or passngr discomfort (bot summit and sag curvs) and to improv visual apparanc Vrtical curvs allow vicls to pass smootly from on gradint to anotr In practic, road and railway gradints ar comparativly flat and it is oftn unimportant wat typ of vrtical curv is usd; t usual curvs ar circular or parabolic Howvr, it is bst to us a vrtical curv aving a constant rat of cang of gradint, i, a parabola and as it turns out, parabolic vrtical curvs ar vry asy to calculat and us For som frways (ig-spd roads) a vrtical curv wos rat of cang of gradint incrass or dcrass wit t lngt of t curv is somtims usd, g, a cubic parabola Howvr, sinc frways ar gnrally mad up of rlativly flat gradints, curvs of tis typ ar somtims rgardd as an unncssary rfinmnt cubic parabola is somtims usd as a sag vrtical curv, wr its proprtis allow a uniform rat of incras of cntrifugal forc (gratr passngr comfort) and lss filling in t vally is rquird Unlss indicatd otrwis, in ts nots, a vrtical curv is assumd to b a simpl parabola GRADIENS A gradint is a dimnsionlss numbr, gradint = tanα ris gradint = or run α run ris In road and railway dsign, gradints ar usually xprssd in prcntags; g, a road of +4% gradint riss 4 units vrtically in 00 units orizontally us, a gradint of p% is qual to p/00 Gradints rising from lft to rigt ar positiv and gradints falling lft to rigt ar ngativ Rprsnting gradints as prcntags as a usful connction wit calculus: if vrtical distancs ar masurd along t y-axis and orizontal distancs along t x-axis tn a 4% gradint is a matmatical gradint of 004 or dy / dx = 004 us t gradint (in %) dividd by 00 is t drivativ Vrtical urvsdoc (Marc 006)

2 RMI Univrsity Vrtical curvs connct two gradints and in sctional viw, t gradint to t lft of t vrtical curv will b dnotd by p% and t gradint to t rigt will b dnotd by q% Altrnativ notations ar g % or G % for t lft-and gradint and g % or G % for t rigt-and gradint p% vrtical curv q% Figur Gradints can also b xprssd as in x, i, vrtical in x orizontal A gradint of +4% is quivalnt to a gradint of in 5 If t gradint is xprssd as p%, tn it is quivalnt to a gradint of in x wr x = 00/p YPES OF VERIA URVES (i) Summit curvs: Vrtical curvs wr t total cang in gradint is ngativ +q% +p% (a) +p% -q% (b) -p% (c) -q% Figur Vrtical urvsdoc (Marc 006)

3 RMI Univrsity t A b t total cang in gradint, tn in prcnt A% = q% p% () In Figur (a) t gradints ar p% = +35% and q% = +4%, in Figur (b) t gradints ar p% = +35% and q% = -30% and in Figur (c) ty ar p% = -5% and q% = -50% total cang in gradint for ac curv is Figur (a) A % = + 4 ( + 35) = %, Figur (b) A % = 30 ( + 35) = 65% Figur (c) A % = 50 ( 5) = 5% In ac cas, A is a ngativ quantity and t vrtical curvs ar summit curvs (ii) Sag curvs: Vrtical curvs wr t total cang in gradint is positiv -p% +q% -p% (a) -q% (b) +q% +p% (c) Figur Vrtical urvsdoc (Marc 006) 3

4 RMI Univrsity In Figur (a) t gradints ar p% = -5% and q% = +5%, in Figur (b) t gradints ar p% = -45% and q% = -0% and in Figur (c) ty ar p% = +0% and q% = +40% total cang in gradint for ac curv is Figur (a) A % = + 5 ( 5) =+ 40%, Figur (b) A % = 0 ( 45) =+ 35% Figur (c) A % = + 40 ( + 0) =+ 30% In ac cas, A is a positiv quantity and t vrtical curvs ar sag curvs Sag vrtical curvs ar also known as vally curvs It sould b notd tat a summit curv will only av a "tru" ig point and a sag curv will only av tru low point wn tr is a cang of sign btwn t grads 3 EQUAION OF A VERIA URVE p q y x H datum Figur 3 In Figur 3 two gradints, p and q ar joind by a vrtical curv of lngt, ar tangnt points and t x-y coordinat origin is vrtically blow wit t x-axis bing t datum for rducd lvls y H is t rducd lvl of basic rquirmnt for t vrtical curv is tat t rat of cang of gradint (wit rspct to orizontal distanc) sall b constant is rquirmnt can b xprssd in two ways (i) Sinc t gradint can also b t drivativ dy/dx tn t rat of cang of gradint is a scond drivativ and w may writ d y K dx = (3) Vrtical urvsdoc (Marc 006) 4

5 RMI Univrsity (ii) Sinc t rat of cang of gradint is a constant K, tn it is also qual to t total cang in gradint A dividd by t lngt q p A = = K (3) gradint at any point on t vrtical curv can b found by intgrating quation (3) dy d y = dx = K dx = Kx dx dx + is a constant of intgration and wn x = 0 (t tangnt point = p and quation for t gradint bcoms or ) t gradint is p us dy Kx p dx = + (33) dy q p = x + p dx rducd lvl y (igt abov datum) of any point on t vrtical curv can b found by intgrating (33) dy Kx y = dx = ( Kx + p) dx = + px dx + is a constant of intgration and wn x = 0 (t tangnt point ) t rducd lvl is H us = H and substituting t xprssion for K, t quation for t rducd lvl (yvalu) bcoms is is an quation of t form q p y = x + px+ H q p K of gradint i, a = =, b is t initial gradint at rducd lvl of ( i, c = H) (34) (35) y = ax + bx + c (a parabola) wr a is alf t rat of cang ( i, b p) = and c is t of If t gradints ar givn in prcntags tn quation (35) bcoms q% p% p% = y x x + H (36) Wn t sign of t cofficint of x is ngativ, t curv is a summit curv and wn t cofficint is positiv it is a sag curv is will b dmonstratd by numrical xampls for ac of t curvs in Figurs and Vrtical urvsdoc (Marc 006) 5

6 RMI Univrsity Summit vrtical curvs in Figur : Figur (a) Figur (b) Figur (c) p % =+ 35% q % =+ 4% p % =+ 35% q % = 30% p % = 5% q % = 50% tn 35 = y x x tn = y x x tn y = x + x + 4 ( + 35) K = = H 30 ( + 35) 65 K = = H 50 ( 5) 5 K = = H Sag vrtical curvs in Figur : Figur (a) Figur (b) Figur (c) p % = 5% q % =+ 5% p % = 45% q % = 0% p % =+ 0% q % =+ 40% tn y = x + x tn y = x + x tn y = x + x + 5 ( 5) 40 K = = H 0 ( 45) 35 K = = H + 40 ( + 0) 30 K = = H It sould b clar from t drivation of t quation of t vrtical curv and from t valuations abov, tat onc t gradints av bn fixd tn only on otr paramtr nds to b fixd to dfin t curv uniquly at is, itr t rat of cang of gradint K, or t lngt of t vrtical curv dtrmination of ts paramtrs is dalt wit in latr sctions Vrtical urvsdoc (Marc 006) 6

7 RMI Univrsity 4 PROPERIES OF HE PARABOI VERIA URVE H A y x p E F D datum q B igst point Figur 4 In Figur 4, E is a vrtical parabolic curv btwn two grads p and q wic intrsct at, ar tangnt points and t x-y coordinat origin is vrtically blow wit t x- axis bing t datum for rducd lvls y H is t rducd lvl of orizontal lngt of t vrtical curv is and t igst point of t curv is a distanc D from [] quation of t curv is q p Substituting for x, t rducd lvl of y = x + px+ H bcoms q p + + = ( + ) + p H q p H From Figur 4, t orizontal distanc from to is and w can us tis distanc to calculat t rducd lvl of as ( ) H + p + q Equating ts two xprssions for t rducd lvl of givs ( q+ p) + H = H + p + q( ) = H + ( p q) + q Vrtical urvsdoc (Marc 006) 7

8 RMI Univrsity anclling and r-arranging givs i, = ( p q) ( p q) = (4) is vry important rlationsip: t orizontal distancs from t tangnt points to t intrsction point ar qual is of considrabl us in solving vrtical curv problms [] In Figur 4, F is t mid-point of t lin and t rducd lvl (R) of F is t man of t rducd lvls of and ( + ) q p R F = + H 4 rducd lvl of E, a point on t curv at a orizontal distanc / from R rducd lvl of is E ( q+ 3 ) q p p = + p + H = + H 8 p R = + H vrtical distancs E and F ar t diffrncs btwn t R's of and E, and and F i, ( ) ( ) p q p q E =, F = 8 4 F E = (4) Hnc, t parabola biscts t vrtical from t intrsction point to t mid-point of t lin joining t tangnt points [3] gradint of t curv at E is givn by quation (34) as dy q p q p = + p = + dx E is is also t gradint of t lin joining t tangnt points, nc in Figur 4, AB is paralll to and is tangntial to t curv at E, a point midway btwn t tangnt points is Vrtical urvsdoc (Marc 006) 8

9 RMI Univrsity [4] In Figur 4, D is t orizontal distanc to t igst point of t curv (or t lowst point if t curv was a tru sag curv) gradint will b zro at t igst point (or lowst) and sinc t gradint is also t drivativ, tn w can st quation (34) qual to zro and solv x dy q p = x + p = 0 dx giving t orizontal distanc to t ig (or low) point of a parabolic vrtical curv as p D = q p (43) As mntiond prviously, a tru summit curv (aving a igst point) and a tru sag curv (aving a lowst point) will only occur wn tr is a cang of sign btwn q and p Not tat t trm q p in quation (43) is t numrical sum of t two gradints and tat D will always b a positiv quantity g, Summit p % = + 40% q % = 30% 4 4 D = = g, Sag p % = 40% q % = + 30% 4 4 D = = 3 ( 4) 7 Vrtical urvsdoc (Marc 006) 9

10 RMI Univrsity 5 EXAMPES OF PARABOI VERIA URVE OMPUAIONS In ts xampl computations, it is assumd tat vrtical curvs ar parabolic and tat t lngt of t curv as bn fixd according to crtain dsign principls s dsign principls (for dtrmining t lngt of vrtical curvs) ar covrd in subsqunt sctions of ts nots 5 ocating tangnt points of vrtical curvs Exampl A p = +35% d q = -4% B c: R: 5740 c: R: Figur 5 Figur 5 sows a rising gradint of +35% followd by a falling gradint of 4% connctd by a vrtical parabolic curv of orizontal lngt 0 m A, a point on t rising grad, as a cainag of m and a rducd lvl (R) of 5740 m and B, a point on t falling grad, as a cainag of m and R m alculat t cainag and R of t tangnts and and t intrsction point t t orizontal distanc from A to t intrsction point b d, tn sinc AB = m R 35 = R + 00 A d 4 4 RB = R + 50 or R = RB Equating t xprssions for R ( d) ( d) givs 35 4 R A + d = RB ( d) and d = 733 m Vrtical urvsdoc (Marc 006) 0

11 RMI Univrsity cainag of t intrsction point is = 733 m and t cainags of t tangnt points ar 60 m itr sid of t intrsction (du to t symmtry of t curv) Having dtrmind t cainags, t R's follow from t grads, giving Point ainag R A B abl 5 Exampl E A B D c: R: 7340 c: R: c: R: 780 c: R: 7570 Figur 5 In Figur 5 points A, B, D and E li on intrscting grads wic ar to b connctd by a parabolic vrtical curv of orizontal lngt 50 m cainags and R's ar sown on t diagram and t grads intrsct at alculat t cainag and R of t tangnts and and t intrsction point Gradint AB = = = 0050 = 50% Gradint DE = = = = + 345% Vrtical urvsdoc (Marc 006)

12 RMI Univrsity t t orizontal distanc from B to t intrsction point b d, tn R 50 = R + 00 B d RD = R + 00 or R = RD Equating t xprssions for and R ( d) ( d) givs RB d = RD d = 453 m ( d) cainag of t intrsction point is = m and t cainags of t tangnt points ar 75 m itr sid of t intrsction (du to t symmtry of t curv) Having dtrmind t cainags, t R's follow from t grads, giving Point ainag R A B D E A p = -50% abl B D R: 7390 q = +345% E c: R: 7340 c: R: 7677 c: R: c: R: 6980 c: R: 780 c: c: R: 7570 Figur 53 Vrtical urvsdoc (Marc 006)

13 RMI Univrsity 5 omputation of Rducd vls of points on vrtical curvs Onc t cainag and rducd lvl (R) of t tangnt points av bn dtrmind, it rmains to comput t R's of points along t vrtical curv is can b acivd by using quation (36) q% p% p% = y x x + H (36) wr t gradints p and q ar givn in prcntags, x is t orizontal distanc from t tangnt point to t point on t curv, H is t R of and y is t R of t point on t curv Exampl 3 onsidr t parabolic vrtical curv dtrmind in Exampl : = 0 m, p = +350%, q = 40% c = 7633 m, R = m c = 7833 m, R = m omput t R's of points on t vrtical curv at vn 0 m cainags quation, s (36) abov is q% p% p% = + + = R x x H x 00350x tabulatd rsults ar Point ainag x R abl 53 Vrtical urvsdoc (Marc 006) 3

14 RMI Univrsity p = +350% 3 4 q = -40% 5 Ṭ 6 c:7633 R: c:780 R: 5838 c:700 R: c:70 R: c:740 R: c:760 R: 587 R: 5760 c:780 c:7833 R: Figur 54 Exampl 4 onsidr t parabolic vrtical curv dtrmind in Exampl : = 50 m, p = 50%, q = +345% c = m, R = 7677 m c = m, R = 7390 m omput t R's of points on t vrtical curv at vn 0 m cainags quation, s (36) abov is q% p% p% = + + = R x x H x 0050x 7677 tabulatd rsults ar Point ainag x R abl 54 Vrtical urvsdoc (Marc 006) 4

15 RMI Univrsity p = -50% R: R: 7390 q = +345% c: c: 580 R: 7376 c: 5300 R: 706 c: 530 R: c: 5340 R: 709 c: 5360 R: 7073 c: 5380 R: 7394 c: 5400 R: 7874 c: Figur 55 Exrcis A p = +385% q = -450% B c: R: 8380 c: R: 8365 Figur 56 Figur 56 sows a rising gradint of +385% followd by a falling gradint of 450% connctd by a vrtical parabolic curv of orizontal lngt 30 m A, a point on t rising grad, as a cainag of m and a rducd lvl (R) of 8380 m and B, a point on t falling grad, as a cainag of m and R 8365 m alculat t following: (i) cainags and R's of t tangnts and and t intrsction point (ii) R's of points on t curv at vn 0 m cainags (iii) cainag and R of t mid-point of t curv (iv) cainag and R of t ig-point of t curv Vrtical urvsdoc (Marc 006) 5

16 RMI Univrsity Exrcis A falling grad of 4% mts a rising grad of 5% at cainag m and R m At cainag m, t undrsid of a bridg as a R of 750 m two gradints ar to b joind by a parabolic vrtical curv of maximum lngt (roundd down to narst 0 m) to giv at last 4 mtrs claranc undr t bridg alculat t following: (i) lngt of t vrtical curv is lngt sould tn b roundd down to t narst 0 m for us in t following calculations (ii) cainags and R's of t tangnt points (iii) claranc btwn t curv and t undrsid of t bridg Not tat tis sould b at last 4 m (iv) R's of points on t curv at vn 0 m cainags (v) cainag and R of t mid-point of t curv (vi) cainag and R of t low-point of t curv 6 DESIGN ONSIDERAIONS FOR PARABOI VERIA URVES In prvious sctions, it as bn sown tat a vrtical parabolic curv is compltly dfind if t following lmnts ar known (i) Intrscting gradints p and q, (ii) ainag and R of t intrsction point (or t tangnt point) and (iii) Horizontal lngt dtrmination of a suitabl lngt is normally t rsponsibility of t traffic nginr or dsignr For any dsign spd, t minimum lngt of a vrtical curv will dpnd on on of two factors, namly t limitation of vrtical acclration or t allowabl minimum sigt distanc 6 ngt of Vrtical Parabolic urv dtrmind by imitation of Vrtical Acclration is is a mtod of computing taking into account t dsign spd of t road v (m/s) or mor commonly V (kp) and t allowabl vrtical acclration a dsign spd is usually takn to b t 85 prcntil spd, i, t spd tat is not xcdd by 85% of t vicls using t road vrtical acclration a, is t rsult of t vicl travrsing t curv at a constant spd; its vrtical vlocity componnt canging as t grad cangs from p at to q at is cang in t vrtical componnt of vlocity mans tat t vicl is subjct to an acclration a, (cntriptal acclration) givn by v a = (6) r r is t radius of circular curv approximating t vrtical parabolic curv and / r is t curvatur κ gnral quation for t curvatur of a curv y = f( x) is givn by Vrtical urvsdoc (Marc 006) 6

17 RMI Univrsity d y κ =± dx dy + dx 3/ (6) For a vrtical curv, dy dx is t gradint, wic is usually small and dy will b dx xcdingly small and may b nglctd, giving t curvatur κ (as an approximation) For parabolic vrtical curvs, quations (3) and (3) Now, sinc / r = κ d y/ dx Using t rlationsip t vrtical curv as or for grads in prcntags d y dx d y κ (63) dx (t rat of cang of gradint) is a constant K, and from d y q p dx = K = (64) and using quations (6) and (64) t vrtical acclration is ( q p) v a = (65) V (kp) v (m/s) = and r-arranging quation (65) givs t lngt of 36 ( q p) V = (66) 96a ( q% p% ) V = (67) 96a For dsign purposs, t maximum valu of vrtical acclration a, sould not xcd 0g (g is t acclration du to gravity 98 m/s) and otrwis com witin t rang 00g to 005g, dpnding on t importanc of t road On major igways, a maximum valu of a = 005g is considrd satisfactory Vrtical urvsdoc (Marc 006) 7

18 RMI Univrsity 6 Minimum Sigt Distanc Sigt distanc is dfind as t xtnt of a drivrs clar viw a road, sufficint to nabl t driv to ract to an mrgncy or to prmit saf ovrtaking of a vicl travlling at lss tan dsign spd y igt objct cut off igt sigt distanc Figur 6 Drivr's viw of an objct on a summit vrtical curv y igt bridg objct cut off igt sigt distanc Figur 6 Drivr's viw of an objct on a sag vrtical curv onstraints assumd for computation of sigt distancs: Higt of y of drivr ( ) passngr vicl commrcial vicl 5 m 80 m Objct cut-off igt ( ) drivr is assumd to ract to tir viw of tat portion of t objct ovr tis igt approacing vicl 5 m stationary objct on road 00 m vicl tail or stop ligt 060 m Vrtical urvsdoc (Marc 006) 8

19 RMI Univrsity 63 Stopping Distanc D S A tortical stopping distanc D S can b drivd from t quation Stopping Distanc = Raction Distanc + Braking Distanc (68) 63 Raction Distanc DR A drivr confrontd wit an mrgncy as firstly to prciv and scondly to ract to a situation bfor ty apply foot to brak tim laps btwn initial prcption and t instant wn t vicl braks act is calld t total raction tim ( R ) It can vary btwn 05 and 3 sconds dpnding upon circumstancs (drivrs prcption, atmospric conditions, tc) For dsign purposs a figur of R = 5 sc is usually adoptd wr R V Raction Distanc DR = R v = R (69) 36 is raction tim in sconds v is vlocity in mtrs pr scond (m/s) V is vlocity in kilomtrs pr our (kp) 63 Braking Distanc D B Having applid t braks, t vicl still as to stop Braking distanc is dfind as t lngt of roadway travlld from t tim t braks start to act until t vicl is brougt to a alt o stablis a formula, considr t kintic nrgy of t vicl: Kintic nrgy W K is t nrgy possssd by a body bcaus of its motion; it is masurd by t work don by t body as it is brougt to rst onsidr a body of mass m dclrating to rst From t 3rd quation of motion wr v u a = + s v is final vlocity u is initial vlocity a is acclration s is displacmnt F m motion s m Sinc t final vlocity will b zro ( v = 0) and t acclration will b ngativ tn 0= u + u s = a as Vrtical urvsdoc (Marc 006) 9

20 RMI Univrsity Now Work = Forc trfor u Work = ma = mu a u Distanc = F s = F and Forc = mass acclration = ma a and sinc kintic nrgy is a masur of t work don tn Wk ( ) = mass vlocity = mv (60) For a body of mass m, dclrating to rst on a orizontal surfac, t forc F= f N causing t dclration is a function of t normal raction forc N normal forc is qual to t wigt W, a forc wos magnitud is W = mg and f is t officint of ongitudinal Friction magnitud of t dclrating forc is F = f N N W m motion F = f N = f W = fmg Sinc t kintic nrgy W k is t work don (Forc Distanc) as t body is brougt to rst, tn giving t Braking Distanc D B as Forc Distanc = W k fmg d = mv Braking Distanc D B v = (6) gf v Using a valu of t acclration du to gravity of g = 98 m/s and wit V in kp ( V = ) 36 tn Substituting quations (69) and (6) into quation (68) givs Using a valu for t raction tim distancs as D B V = (6) 54 f V V Stopping Distanc DS = DR + DB = R + (63) f R = 5sc givs a common formula for t stopping D S V = 07V + (64) 54 f Vrtical urvsdoc (Marc 006) 0

21 RMI Univrsity Rcnt tsts by t Australian Road Rsarc Board (ARRB) av found tat, on good dry pavmnts modrn passngr cars can consistntly aciv dclration rats in xcss of 0g Howvr, t valus usd for dsign purposs sould allow for dgradation of pavmnts skid rsistanc wn wt and for a rasonabl amount of surfac polising valus for t cofficint of longitudinal friction givn in abl 6 blow ar takn from Rural Road Dsign Guid to t Gomtric Dsign of Rural Roads, AUSROADS, Sydny, 993, p8 lowr valus assumd for t igr spds rflct t rduction in wt pavmnt skid rsistanc wit incrasing spd and t nd for latral vicl control ovr t longr braking distancs Initial Spd V (kp) officint of ongitudinal Friction f abl 6 Valus of officint of ongitudinal Friction f 64 ngt of Summit Vrtical urv for Stopping Sigt Distanc D Stopping Sigt Distanc D (Non-Ovrtaking Sigt Distanc) is usd to dtrmin t minimum lngt of a vrtical curv D sall b qual to t Stopping Distanc D S of a vicl travlling at dsign spd V wn an unobstructd viw is providd btwn a point (y igt) abov road pavmnt and a stationary objct of igt (objct cut off igt) in t lan of travl y igt objct cut off igt Stopping Sigt Distanc D = V V = R f D S Figur 63 Samuls, SE and Jarvis, JR, 978, Acclration and Dclration of Modrn Vicls, ARRB Rsarc Rport, ARR No86 from Rural Road Dsign Guid to t Gomtric Dsign of Rural Roads, AUSROADS, Sydny, 993, p8 Vrtical urvsdoc (Marc 006)

22 RMI Univrsity o dtrmin a minimum vrtical curv lngt, using Stopping Sigt Distanc D, tr cass aris (i) > D: lngt of vrtical curv gratr tan t stopping sigt distanc In tis cas, t vicl and t objct ar bot on t vrtical curv (ii) = D: lngt of vrtical curv qual to t stopping sigt distanc In tis cas, t vicl is at t bginning of t vrtical curv and t objct is at t nd of t curv (iii) < D: lngt of vrtical curv sortr tan t stopping sigt distanc In tis cas, t vicl and t objct ar bot on t grads joind by t vrtical curv 64 > D: ngt of Summit Vrtical urv Gratr tan Stopping Sigt Distanc stopping sigt distanc D S D S A p E F q B H y x Figur 64 In Figur 64 t following is known (s Sction 4 Proprtis of t Parabolic Vrtical urv) lin AB is tangntial to t curv at E (t mid-point of t curv) and is paralll to t lin btwn t tangnt points distanc = A= E = B is and t sign of is givn by t sign of ( q p) ( q p) F = E = = (65) 8 Vrtical urvsdoc (Marc 006)

23 RMI Univrsity = q p x H y x p x px datum q Figur 65 Equation (65) can b vrifid by considring Figur 65 and t quation of t parabolic vrtical curv q p y = x + px+ H For a point on t curv at a orizontal distanc x from t tangnt point t following tr componnts of quation (66) ar sown on Figur 65 q p (i) t vrtical distanc from t grad to t curv is = x, (ii) t vrtical distanc from t grad to t orizontal lin passing troug is px and (iii) t vrtical distanc from to t datum is H sum of t tr componnts is t y-coordinat of t point on t curv For t mid-point of t curv, t vrtical distanc is givn by t first componnt of quation (66) ( q p) q p = = 8 is dmonstrats a usful proprty of a parabolic vrtical curv, i,; t vrtical distanc from a tangnt lin to a point on t curv is givn by q p = (66) x (67) wr x is t orizontal distanc from t tangnt point Noting, in Figur 64, tat AB is a tangnt and E is a tangnt point, quation (67) can b usd to driv xprssions for t distancs and, noting tat q p in quation (67) as bn rvrsd to mak and positiv quantitis p q = S (68) Vrtical urvsdoc (Marc 006) 3

24 RMI Univrsity p q = ( S) D (69) Using quations (65), (68) and (69) t following manipulations yild an xprssion for t lngt of a vrtical curv By taking squar roots of bot sids of quations (68) and (69) w av Adding t quations liminats S p q = S ( ) = D S p q + = D Squaring bot sids and r-arranging givs = ( ) D p q p q ( + ) If t gradints ar givn in prcntags tn quation (60) bcoms = 00 ( % %) D p q ( + ) (60) (6) 64 = D: ngt of Summit Vrtical urv qual to t Stopping Sigt Distanc In tis cas, ltting D = in quations (60) and (6) givs or, if gradints ar givn in prcntags ( ) + = p q ( ) + 00 = p% q% (6) (63) Vrtical urvsdoc (Marc 006) 4

25 RMI Univrsity 643 < D: ngt of Summit Vrtical urv lss tan t Stopping Sigt Distanc In Figur 66, t vicl (y igt ) and objct (objct cut off igt ) ar situatd on t gradints p and q rspctivly and ar at a distanc D apart D is t stopping sigt distanc and xcds t lngt of t parabolic vrtical curv and ar tangnt points and t lins and AB ar paralll AB is tangntial to t curv at E, wic is t midpoint of t curv following manipulations yild an quation for t lngt of t vrtical curv gradint of t lin ( /) ( /) ris p + q p + q = = = (64) run A d p E F q B d d stopping sigt distanc D Figur 66 An xprssion for t vrtical distanc t gradint of t lin at a distanc from d In Figur 66, t distanc can b obtaind by considring t grad p and is also sown as a dottd lin giving p+ q = ( ) p p q = d d d p d d p q = d (65) Vrtical urvsdoc (Marc 006) 5

26 RMI Univrsity Similarly Adding quations (65) and (66) givs but d+ d = D so w may writ p q = d (66) p q + = d + d ( ) p q + = ( D ) (67) From t prvious sction, t vrtical distanc is givn by quation (66) Howvr, for a summit curv, q p positiv w writ Substituting quation (68) into (67) givs Multiplying bot sids by or, if gradints ar givn in prcntags q p = 8 will b a ngativ quantity, making ngativ, tus, for p q = 4 (68) p q p q + = D 4 ( ) 4 p q and r-arranging givs and xprssion for ( + ) 4 = D p q ( + ) 400 = D p% q% (69) (630) 65 ngt of Sag Vrtical urv for Stopping Sigt Distanc D and laranc Higt H Stopping Sigt Distanc D (Non-Ovrtaking Sigt Distanc) is usd to dtrmin t minimum lngt of a vrtical curv D sall b qual to t Stopping Distanc D S of a vicl travlling at dsign spd V wn an unobstructd viw is providd btwn a point (y igt) abov road pavmnt and a stationary objct of igt (objct cut off igt) in t lan of travl Ovrad obstructions, suc as road or railway ovrpasss, sign or tollway gantris may limit t sigt distanc availabl on sag vrtical curvs Vrtical urvsdoc (Marc 006) 6

27 RMI Univrsity In Figur 67, t ovrad obstruction (bridg) is at igt H is t claranc igt H abov t road pavmnt y igt bridg H objct cut off igt Stopping Sigt Distanc D = D S V V = R f Figur 67 o dtrmin a minimum vrtical curv lngt, using Stopping Sigt Distanc D and laranc Higt tr cass aris H (i) > D: lngt of vrtical curv gratr tan t stopping sigt distanc In tis cas, t vicl and t objct ar bot on t vrtical curv (ii) = D: lngt of vrtical curv qual to t stopping sigt distanc In tis cas, t vicl is at t bginning of t vrtical curv and t objct is at t nd of t curv (iii) < D: lngt of vrtical curv sortr tan t stopping sigt distanc In tis cas, t vicl and t objct ar bot on t grads joind by t vrtical curv Vrtical urvsdoc (Marc 006) 7

28 RMI Univrsity 65 > D: ngt of Sag Vrtical urv Gratr tan Stopping Sigt Distanc A stopping distanc D p S D S F E q B Figur 68 In a similar mannr to Sction 64 (summit vrtical curvs), two quations can b writtn and manipulatd to yild xprssions for t minimum lngt of a vrtical curv q p = S q p = ( D S By taking squar roots of bot sids of quations (63) and (63)w av (63) ) (63) Adding t quations liminats S = S q p ( ) = D S q p Squaring bot sids and r-arranging givs + = D = ( ) D q p q p ( + ) H In quation (633) can b rplacd by t laranc Higt, t vrtical distanc btwn t point of tangncy E and t obstruction to t lin of sigt btwn objcts of igt and (633) Vrtical urvsdoc (Marc 006) 8

29 RMI Univrsity = ( ) D q p ( H + H ) If t gradints ar givn in prcntags tn quation (634) bcoms ( % %) D q p = 00 ( H + H ) (634) (635) Figurs 69 and 60 sow obstructions locatd vrtically abov t mid point of t curv and at a point to t lft of t mid point lin of sigt btwn objcts of igt and is paralll to t tangnt to t curv point of tangncy is vrtically blow t obstruction -p% obstruction D H +q% Figur 69 Obstruction at vrtical igt abov mid point of vrtical curv H -p% H D +q% Obstruction at vrtical igt H Figur 60 abov point to t lft of mid point of vrtical curv Vrtical urvsdoc (Marc 006) 9

30 RMI Univrsity 65 = D: ngt of Sag Vrtical urv qual to t Stopping Sigt Distanc In tis cas, ltting D = in quations (634) and (635) givs = or, if gradints ar givn in prcntags 00 = ( H ) + H q p ( H ) + H q% p% (636) (637) 653 < D: ngt of Sag Vrtical urv lss tan t Stopping Sigt Distanc d A H ( + ) p stopping distanc D obstruction H q H ( + ) B d Figur 6 In Figur 6, t vicl (y igt ) and objct (objct cut off igt ) ar situatd on t gradints p and q rspctivly and ar at a distanc D apart D is t stopping sigt distanc and xcds t lngt of t parabolic vrtical curv and ar tangnt points and t lins and AB ar paralll AB is tangntial to t curv at t mid-point following manipulations yild an quation for t lngt of t vrtical curv Equation (64) givs t gradint of t lin gradint = An xprssion for t vrtical distanc H ( ) grad p and t gradint of t lin p + q + can b obtaind by considring t Vrtical urvsdoc (Marc 006) 30

31 RMI Univrsity giving Similarly p+ q + = p q p = d+ d d ( ) ( ) H d p d ( ) H Adding quations (638) and (639) givs + = q p d (638) q p H ( + ) = d (639) q p H + = d + d ( ) ( ) q p but d+ d = D and = (s quation (68) noting tat t for a sag vrtical 4 curv q p is a positiv quantity) Substituting ts xprssions givs q p q p H + = D 4 4 ( ) ( ) 4 Multiplying bot sids by p q and r-arranging givs and xprssion for 4 = D H ( + ) p q or, if gradints ar givn in prcntags 400 = D H ( + ) p% q% (640) (64) Vrtical urvsdoc (Marc 006) 3

32 RMI Univrsity 66 ngt of Sag Vrtical urv for Hadligt Sigt Distanc S Figur 6 sows a vicl on a sag vrtical curv wos adligts ar at a vrtical igt abov t road pavmnt at t mid point of t curv longitudinal axis of t vicl is paralll to t tangnt to t vrtical curv at t mid point E usful portion of t adligt bam divrgs (upwards) at an angl θ and striks t road pavmnt at a orizontal distanc S (t Hadligt Sigt Distanc) from t vicl Hadligt Sigt Distanc = S = D S A p E θ q Sθ B Figur 6 o dtrmin a minimum vrtical curv lngt, using Hadligt Sigt Distanc S, tr cass aris (i) > S: lngt of vrtical curv gratr tan t adligt sigt distanc In tis cas, t vicl and t limit of t ligt bam ar bot on t vrtical curv (ii) = S: lngt of vrtical curv qual to t adligt sigt distanc In tis cas, t vicl is at t bginning of t vrtical curv and t limit of t ligt bam is at t nd of t curv (iii) < S: lngt of vrtical curv sortr tan t adligt sigt distanc In tis cas, t vicl and t limit of t ligt bam ar bot on t grads joind by t vrtical curv 66 > S: ngt of Sag Vrtical urv Gratr tan Hadligt Sigt Distanc In a similar mannr to prvious sctions, t proprty of t vrtical distanc btwn t tangnt and t parabolic curv can b mployd to giv t following quation q p + Sθ = S (64) In quation (64) it is assumd tat t vrtical distanc, sown as Sθ, is qual to t small arc of a larg circl of radius S subtnding t angl θ at its cntr (t adligt of t vicl) is is a rasonabl assumption, sinc in practic, t angl θ is small (usually º) and t grads ar also small Vrtical urvsdoc (Marc 006) 3

33 RMI Univrsity Rarranging quation (64) givs S ( q p) = + Sθ or, if gradints ar givn in prcntags = ( ) ( % %) ( + Sθ ) S q p 00 (643) (644) For a ligt bam wit θ = ( radians) and = ( % %) S q p S = 0750 m, quation (644) bcoms (645) 66 S: ngt of Sag Vrtical urv lss tan or qual to t Hadligt Sigt Distanc Hadligt Sigt Distanc = S = D S θ p q Figur 63 Figur 63 sows a vicl on t grad p (bfor t tangnt point) wos usful portion of t adligt bam intrscts t grad q (aftr t tangnt point) on t otr sid of t curv In tis cas < S Sinc w ar intrstd in dtrmining minimum lngts of vrtical curvs, w may considr t cas wn t vicl's adlamp is at and t adligt bam striks t road pavmnt at, i, t cas wn = S In suc a cas, substituting for S in quation (643) givs t formula to b usd for S or, if gradints ar givn in prcntags ( + θ ) S = q p 00( + Sθ ) = q% p% (646) (647) Vrtical urvsdoc (Marc 006) 33

34 RMI Univrsity For a ligt bam wit θ = ( radians) and S = q% p% = 0750 m, quation (647) bcoms (648) Vrtical urvsdoc (Marc 006) 34