Visibility Problems in Crest Vertical Curves

Transcription

1 Viibility Problem in Cret Vertical Curve M. LIVNEH, J. PRASHKER, and J. UZAN, echnion, Irael Intitute of echnology he length of a cret vertical curve i governed by viibility conideration. he minimum length i baed on the topping ight ditance; the ma:ximum length i baed on the paing ight ditance, and overtaking i allowed throughuul il length. he object of the preent paper i theoretical determination of the zone of overtaking viibility in a curve deigned on a belowmaximwn bai. he analyi cover 2 cae: (a) overtaking vehicle inide and oncoming vehicle outide the curve and (b) both vehicle outide the curve. he correponding curve geometrie were alo conidered. he equation obtained were computer-olved for curve with lope difference ranging from 2 to 12 percent, paing ight ditance correponding to the deign peed range of 5 to 11 km/hr, and length limit correponding to the topping and paing ight ditance repectively. Reult were rendered in convenient graph form, permitting determination of Ute type of diviion line and the length of the no-overtaking zone to be marked on a 2-way 2- lane highway in the vicinity of the curve. he length of the no-overtaking zone increae with the overall length of the curve, up to the maximum (unretricted overtaking). he concluion i that, in order to reduce the no-overtaking zone in below-maximum cae, it hould preferably be a hort a poible within the requi1 ement limit of overtaking viibility and driving convenience. CONVENIONAL DESIGN of a c1 et vertical curve (hereinafter referred to imply a curve) in a 2-way 2-1ane highway involve 2 extreme cae: minimal, with overtaking prohibited altogether, and maximal, with overtaking allowed throughoutit length, baed repectively on the ight ditance required for topping and paing (ove1 taking) at a given peed. Even in a minimal curve a point exit beyond which viibility i in ufficient for overtaking. he deign i frequently baed on a criterion radiu that generally atifie thi minimal requirement, and overtaking i allowed along part of the curve in uch cae. A olid diviion line throughout the curve and beyond it would be unrealitic and make every driver a potential offender, a i bound to happe.n wherever overtaking i prohibited along obviouly afe tretche. Removal of the overtaking retriction i indicated on the highway by a broken line parallel to the olid line, and advance notice of the reduced ight ditance i indicated by a pecial cloely paced broken line (warning line) equal in length to the paing ight ditance and originating at the point beyond which there i no overtaking viibility. A vehicle that tart to overtake hort of thi point will be able to complete the move; after thi point, the warning line i equivalent to the olid line. he o.bject of thi paper i the theoretical determination of the different zone from the viewpoint of overtaking afety. hi wa done graphically or by mean of practical field tet in itu. Paper ponored by Committee on Operational Effect of Geometric. 76

2 77 y x Figure 1. Parabolic cret vertical curve. DEERMINAION OF VISIBILIY ZONES he deign i generally baed on a parabolic curve (Fig. 1) with the following equation: Ai 2. H y = - 4 x + ix + A (la) Ai = i - j (lb) where y = elevation of a point on the curve; x = ditance of the point from the origin of the coordinate ytem; Ai = algebraic difference of Ute tangential lope i and j; 2 = length of the curve in plan; and HA = elevation of point A, the initial point of the curve. In the range of longitudinal lope ued in road deign, it can be hown that the radiu of curvature i contant at cloe approximation: y " l/r = ---- ( 1 + y'2f/2 (2) Becaue the lope y' i relatively mall, y ' 2 in the denominator i negligible with repect to unity, and we have R :!:! 1/y" = 2 Ai An additional aumption i that the true length and projection of a loping line are equal. In thee circumtance, the reult would not be affected by light rotation of the ytem. Accordingly, the abolute value of lope i and j are immaterial in the following analyi; the only ignificant parameter are h, level of driver' eye above the ground, and the difference Ai. he ight ditance in both direction are ymmetric with repect to the interection point C of the 2 tangent. In determining the driver' ight ditance with repect to an oncoming vehicle, the 4 cae given in able 1 hould be ditinguihed. According to the German Code (1), h equal the height of the oncoming car, o that cae la and 2b are identical. - Cae la Cae la, in which the overtaking vehicle i inide and the oncoming vehicle i outide the curve, i hown in Figure 2. he parabola equation i ued to obtain the elevation (3)

3 78 of point E ', driver' eye level in the overtaking car, which i (4) he lope of the line of ight, y ', i y' ai ( ). -- x+m (5a) and m equal at leat the toppi ng ight ditance in the minimal cae. he elevation of point D ', level of r oof of oncoming vehicle, obtained from the lope of the ight line from Eq. 5a i given by (5b) (6) wllere Sp i the paing ight ditance. he ame elevation obtained from the parabola equation i Hn, =HA+ i +(Sp + x - ) j + h (7) ABLE 1 CASES USED IN DEERMINING SIGH DISANCE Cae Vehicle Poition in Curve Cae Vehicle Poition In Curve la Overtaking Inide 2a Overtaking Outide Oncoming Outide Oncoming Outide lba Overtaking Inide 2b Overtaking Outide Oncoming Inide Oncoming Inide 8 For a curve not baed on paing ight ditance, cae lb i irrelevant. y.. u :c..... > u :c x Figure 2. Sight line-one vehicle inide and one vehicle outide curve.

4 79 Equating both expreion, we have Ai 2 (Ai ) 4 x + 2 Sp - A1 x + (8) For a given curve with known Ai and 2 and for Sp correponding to a given deign peed, Eq. 8 yield x, the coordinate of the point beyond which there i ufficient viibility for overtaking. Of the 2 root of the quadratic equation, only the one atifying the following 3 condition i relevant: Cae 2a xo (x+m);2 (x +Sp) 2 Cae 2a, in which both vehicle are outide the curve, i hown in Figure 3. ituation exit only for Sp > 2. A the diagram how, hi (9a) (9b) (9c) where Sp = v + m + n + w = v + + w (1) v = the ditance from the vehicle ituated outide the curve to the interection point F of the line of ight of the vehicle with the grade line of the beginning of the curve (Fig. 3 ), and w = the correponding ditance when the vehicle i on the other ide of the curve. he lope of the ight line i (11) he angle between tangent and ight line are (i - y') and (-j + y ') repectively, and we have h h (12) v = i - y' x/r y m w Figure 3. Sight line-both vehicle outide curve.

5 8 h h w = -j +y' = -x/r + i (13) Where R i the curve radiu. Subtituting thee in Eq. 1, we have (2Sp - RAi)x 2 - RAi.(2Sp - RAi)x + 2R 2 hi = (14) x1 and x2, the root of Eq. 14, are ymmetrical with repect to the vertical through C, namely, he poition of the vehicle in the coordinate ytem are X1 = 2 - X2 (15) for the overtaking vehicle, meaured from the initial point of the curve, and (16) b = Sp - a - 2 (17) for the oncoming vehicle, meaured from the end point of the curve. Identification of Relevant Cae Cae la, one vehicle inide and one vehicle outide the curve, i repreented by Eq. 8. For thi cae, one of the olution mut obey condition in Eq. 9a, 9b, and 9c. Cae 2a, both vehicle outide the curve, i repreented by Eq. 14. For thi cae, both a and b, obtained from Eq. 16 and 17, mut be poitive. Excluion of one cae indicate validity of the other u u :c :c e;.. y.. u u :c :c 1 t b-a a f' Li 2 2 ection 2 x plane warning line ---- diviion line - - interrupted line Figure 4. Determination of no-overtaking zone-both vehicle outide curve.

6 81 CL )( = =--= -----_, i-a: warning line ---- diviion line interrupted line plane Figure 5. Determination of no-overtaking zone-one vehicle inide and one vehicle outide curve. Mar king of Diviion Line Figure 4 how the line for cae 2a, both vehicle outide the curve. For reaon of ymmetry, a and b determine the length of the no-overtaking zone on both ide of the curve. hi cae i confined to a mall angle i. Figure 5 how the line for cae la, one vehicle outide and one vehicle inide the curve. he ditance x i obtained from Eq. 8, and the initial point of the diviion line i given by c = Sp+ x - 2 (18) wher e c i the poition of the oncoming vehicle, meaured from the end point of the curve. Deign Speed (km/ hr) ABLE 2 SOPPING AND PASSING SIGH DISANCE ACCORDING O DESIGN SPEED Paing Sight Ditance (m) Stopping Sight Ditance (m) COMPUAION DAA Equation 8 and 14 were olved on an Elliott 53 computer for the following i value:.2,.4,.6,.8,.1, and.12. Sight ditance or S-value age given in able 2 (2 ). he curve length correponding to each i and S wa determined for h = 1. 2 m, the elevation of an object on the road being taken a zero. 2 value ranged from 2 for topping to 2p for paing and are hown at 1-m interval in Figure 6 through 11. For given i, 2, and Sp, the correponding x i found ubj ect to the relevant condition. Reult of Eq. 14 were not plotted becaue of the low frequency of the cae involved.

7 o L.L......_.._.._ Numerical Example 1 Given Ai=. 8, 2 = 1,6 m, R = 2, m, deign peed v = 95 km/hr (correponding to S = 145 m, Sp = 64 m). he length of the curve baed on topping ight ditance i 2 = 7 m; and the length baed on permitting overtaking throughout the curve i 2p = 3,413 m. 2, the relevant cae i that of Eq. 8. he data hown in Figure 9 yield x = 1,32 m, c = 1, ,6 = 36 m, and the reulting marking cheme i hown 18 A Sp< M 1!!..!'.:!., 14 I. 1! m z- "' " 1! a v.... i v->' ::: : OC>:: BOO i, i.-i n, h) - t ). 1l?1 IH(l11 6 "-l-4-'1--+.ij'l*-l--+--l--l--l--+-+-l--+-l--i ". t-t-t-t-..r_=j,: : : = -J---J---J---+--J---+--J---1,..,,. ft....,. lenoth of cret vertical curve, meter Figure 6. Determination of x value for Sp ditance, variou length of vertical curve 2, and for Ai=.2. I' IL.11 t O ! <.J tt. " "' 4 6 IZOO ,, lenoth of cr11t 11ertical curve 1 meter Figure 7. Determination of x value for variou Sp ditance, variou length of vertical curve 2, and for A i = l-l-l-l-l-l-l-l-l-+h-F-_. - v I1 B!!!!I CD 24 > ,_...,li"...,...rt, i - r g l-4-A-v--1-_._ <.. 64m flu 1-J-j-j-j-j-j-j-+l?'-h«-!: :!!!o_..,_,_.._.._..._, l6oo l--l-1-1-l-l-l-+-,#'-.4<' i 'E 12 l--1-l-l il lhfll-..l:::=:. I A n...o i 1..., l < 1-1-t-M... i<-l--t :!:::=: !--!--!--!--!--!--!--!-.f--t 4 f-- 1-1,.uo I _. I.e u. IOM '" -: <, i" 'i length of cret vertical curve, meter L-l-l-l-l-l-l-1-l-U'< <, l"i n. """' "' w 4 1! l-l-l-l-l-l-l-l-f-,,#l-2p 4 to:j«..._ 35 J 3 '-'-'-'-'-'-,,_,,,,,._, g' l W.. J " m 2oo l-+ -If.'- jr,.,...,,.-i zooo l---! ;rn-,,m,." !--!-.f I-I-I-I- IV-: ;; : _...._,,,...._, 2 <OOO 6 2 lenoth of cre1 vertical curve, me1er Figure 8. Determination of x value for variou Sp ditance, variou length of vertical curve 2, and for Ai=.6. Figure 9. Determination of x value for variou Sp ditance, variou length of vertical curve 2, and for l:j,, i =.8.

8 " r-.-.,,.,._... _,_._,,,_t--1-t u... ' -.bot " 1 2 A' Z, l MI' i-.--r--i-4-l'--+-l--t-+-t-/f...Y.+ I Zp +t'i(ll'ja - > t-+-<-t-- - -<- -; i :.':: t--!--l--t-+-r--+-l--t-+-4-1' , -+ J t., :ruo Q) I/ 5 f-+-+-i-l--l-+-i-l--f--.ri/ i-+-f-i-l--f--! "' / 4 l-+-t--+-t llf" - : =: v U.- 1Jtt11111-t--t-t--1-i--+-+-i 83? 1,,om 3 t--< <ir -- : ::,,,._,,_.,_ t--<.-1.L..... '-i--l- -l- 1 1,.S l'oa O / H r I t--+ -f--! f--I length of cret vertical curve, meter Figure 1. Determination of x value for variou Sp ditance, variou length of vertical curve 2, and for ll. i = _... l"'..,.1 : : ;:: +--t--tr t-+-t--+--f--i- -!'.\. u r t l-of-l--+-f l--t 1,.._, r--+-l t--t c -- I- : ;: :! J" ;-;-t-t--f--l--i-4-lr--+-l--t-+-I / _ 1 _ ,c c---'----'---'-'-'--'--'--' length of cret vertical curve, meter Figure 11. Determination of x value for variou Sp ditance, variou length of vertical curve 2, and forll. i = ,.28 S !8lc 36 S = - =-c:.._i ,-:...:;- -...=.--:... Figure 12. Example of propoed marking. in Figure 12. For the ame data and R = 25, m, we have 2 = 2, m and, accordingly, x = 1,7 m, c = 1, , = 34 m. he correponding length of the diviion line i 1,4 m, compared with 1,4 m for R = 2, m. he radiu permitting unretricted overtaking under thee condition i 42,667 m. Numerical Example 2 Given ll.i =.4, 2 = 8 m, R = 2, m, deign peed v = 8 km/hr (correponding to S = 11 m, Sp = 55 m). he length of the curve baed on topping ight ditance i 2 = 22 m; and the length baed on permitting unretricted overtaking throughout the curve i 2p = 1,26 m. A Sp < 2, the relevant cae i again that of Eq. 8. he data hown in Figure 7 yield x = 1,32 m, c = = 36 m. For the ame data and a maller radiu R = 1, m, we have 2 = 4 m. Now Sp::> 2, and Eq. 14 i r elevant; it root are x1 = 38 m and X2 = 362 m. For x1, Eq. 16 and 17 yield a= 297 m and b = -187 m, a negativevalue, wherea it i required to be poitive. Hence, the cae of both vehicle outide the curve doe not apply here, and the diviion line i marked a for the cae with one vehicle outide and the other inide the curve. SUMMARY AND CONCLUSIONS hi analyi permit determination of the zone of overtaking viibility for a cret vertical curve in a 2-way 2-lane highway. he propoed marking (Fig. 4 and 5) comprie a warning line, equal in length to the paing ight ditance, indicating nearne of a zone of reduced overtaking viibility. In thi zone the driver i allowed to complete an overtaking move begun earlier but not allowed to attempt a new one once he ha paed the initial point of the warning line in the right lane. Analyi of the reult

9 84 how that the length of the no-overtaking zone increae with that of the curve, up to the maximum where overtaking i unretricted. he udden vanihing of the diviion line in the diagram i explained a follow: A 2 tend to 2p, x tend to 2 - Sp. In other word, the olution tend to the cae of the oncoming vehicle near the end point of the curve and the other within the curve at ditance x, At 2 = 2 8, x "' 2 - Sp; in other word, one vehicle i at the end point of the curve and the other at a ditance Sp from it hi correpond to the cae of both vehicle inide the curve with overtaking viibility throughout it length. Both x and 2 - x increae a 2 increae; c"' Sp - (2 - x) decreae accordingly; but a the rate of increae of x i teeper than that of c, the length of the diviion line increae with 2. he concluion i that, in order to reduce the no-overtaking zone where the deign cannot be baed on the paing ight ditance, the curve hould be a hort a poible but till comply with the requirement of topping ight ditance and driving convenience. he curve length complying with the latter apect i obtainable, for example, from R - 1 ft/ec 2 (19) REFERENCES 1. Ueberarbeiteter Entwurf der Richtlininen fuer die Anlagen von Landtraen (RAL). II eil, Linienfuhrung (RAL-L), A Policy on Geometric Deign of Rural Highway. American Aociation of State Highway Official, 1965.