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, ., ' ~{cfcnilig to your quen'y.on the ubove sUbjeot. it issuggested thnt tll~:tbllowln8'elem;,nts arc c()llsldetcd ill the calculation aftlle length ofthe transition ofllie spirulcurvlJ: .. · . i.. ,. \;.(1. neslgn Speed . .. '. : ':. ..; .• !', • • b. Radius ot'lhe circular curve .'. .. . . '.," 'I; .

.. c .. Superolevatioll ofUle c\I.t'Ve , . . , . J .. Road Width (mt) . I

I. e, Crowllofthe)und-(%) ......·'1·'·'····, 'Ii .:EXltlrlple:····'.·::.,j . . Ii X",:, ',6. Mme, SUIlCrclcitatioh 3,05 x.8.2% ""p.25Q .';', i,,'t:·:

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i , Tile type' Of spiral recommended is tile Euler, ,Spiral or Clpthoid, which by virtue of its flexibJlity and simplicity is the most commonly used worldwide!

Ir 5.2.1.1 The Clothoid and its applications

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Tile basic equation of tile clotlloid ;s .' .'

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where

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A = Parameter R = Radius at L = Length of

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cif the clothoid distance L .from thespi ral ,

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, the origin (R Fro )

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The parameter is in practice the element thatd;e'termines tile "size" of ,theClothoid( li,k,e the radius[:fort'lle" circular curves). This is,illustrated in Figure ,5.1 where the relationship between 'parameters, 'Radii and lengths may be 'nppr.eci at ed. ' ' , , l One can intuitively see the extreme flexibility of this type of spiral, 'which can tie adapted to any i,required condition .. Because of the: relatively complex i formulas' for its' calculation, the clo,t.h.oidwas, i,n the past, seldom used for highway design, howeve~ the diffusion in recent years of sophisticated ,inexpensive' ca~culating machine's "has made it possible the practical us~ ~"f "th'i;g

,type of, ',spiral. ' I ": ,,' .... !; ..

Foll'owing are the defini,tions and,formulas;'fo/the' cal c u 1 at i on 0 f the C lot hoi d s, the s e .s h 0 u 1 d be' r e !I din ",:",,' conjunct'ion with Figure' 5.2 where the curve's :'diagrams , are shown. I': " ::

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5.2.t.:~,C!qthoidsdefinil

A, = , ",: ,j

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Parameter of ,ne ctothotd "Radiu's" of the' 'ircular curve

Length of tran ition ' Abscissa ofth, ransit,ion en,d Ordinate, of th! ransit ion end Offset qf the I Gular curve Sh i ft o~ t he pc, c' of rt angency Long t'angent of "hEl: trans it ion short tangent 'oj'.'the transi." ,ion

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== Angle of the C10thoid at R '=;.I,)./ll1g16.,:II: t.118 r:el\t:er of tile cirr;ulnl" .curv(~ - ".Tangent 1e'ngtl') of the curve ·i . _. External'distance between middle of !?'urve and PI. == Length' of ·the c1 rcu'l ar 'curve" " . , Total 1 eng'th of! the cUl-ve" :' , '

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5,2.1,3 Clothoids formulas

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DR XM TL TS e

Lc; L T E

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(R x COS\1l + V): - R X-(R x Sin01 X-(Y/Tan0l Y/SiN0

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8(Radl y. R 2Lt

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,,"Lc + ! (R+DR 1 x.Tan (! . .t:::.. (2) (R+DR)/Cos ( .t:::.. /21 ~

~nsymmetrica1 c10thoid~:

+ XM R

.Formulas for each element of the A

2, Ltl & Lt 2 , 0 1 & O2 , xl & X2 ,

& XM2' TLf & TL2 andTS 1 & TS 2 I

'symmgtrical clothoids except: .

Lt'

two c 1 ot00 i ds, i. e. A 1 Y 1& Y 2;> DR 1 .& .pR2 ' are the 'sa~e ~s· for

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,'., ..... . ·8 Lc L Tl

,T2

:: 1:;.,-01 - 02 :: 8(Rad) x R :: Lc + L tl + L t2 . i :: (R+DR

1) .Jan (1:;.,/2) + XM 1 - (DRj-DR2

1l/coS(90- A ) _ (R+QR

2) ran (.t:::../2) + XM2 + (DR 1-DR2,)/cos(90-A)'

::.For unsymmetrical curves the ~ flf meaningless, E

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"<:,; .The formulas shown for ,the calculation of; X :and,Y" allow. , correct ,re'sults up to an: angle of clothoid (0') inthe range

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0 0 < !1l < 900 . ' ".. . l: . . For. cases: where :th,El. a:ngle·!il should beg,reater:, the. integration for XahdY,Shoulrl be .extended.

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" The elements that normally are fixed for the design ofa spirnled c~~ye are' the radi~s of the circular portion (R) )lnd the ilengtllof the tr'1nsi,tlon (Lt). ..I'" The rad I Us 'i s fi xed' depend i ng on the, des I gn sp,eed a,nd environmenta:l characteristics, while thelength,j:of the transition (clothoid) should be fixed so thnt. illl the superelevation,run-off'(disGus'sed hereiqafter)i,can be accommodated:within its :length. ' ; , Sin c e t his w 0 u 1 d mo s t 0 f the t.i me s res u It 'i n ': b r 0 ken clothoid parameter (Al,. for 'practical purposes this is ,nortnallyr'ounded up 'for, making use of av~;i,1able tabulated values.l.~,

I , However, with the aid of modern calculators the: use of tables is no longer strictly necessary and the ~esigner has virtually unlimited choices,. '

For exampl e, a most equal ing, the length the circular'curve. of the clothoids (~)

fIl = 4

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'eye ple'cising effect is' obt~ined by of the clothoids and ,the /,le'ngth of This 'is'achieved by fixing t'tle angle as '1/4 the deflection angle (L'>.), i.e.

Another case may be when there is ',limited space :for the development of the superelevation run-off. , In such case the, curve is ,better formed b,y the two transitio~s only, meeting in one point of fixed radius. This would completely eliminate'thecircular portion with the,cQndition that:

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In modern design practice, the tendency is to eliminate short tangent lengths between curves"not only in 'brol~en"",:""" back curves but also in: case of curves of opposite di rection. It has been obs,erved that, between I,curves, ta,ngen~ le,n,gths sho.t:ter,than 100-150 rneters;create uneasiness of driv .. ing',-} because' the continuous; changes

"t~rom, straight ", co'urset,o curv,ed course' are ,'''- 'cause of' fragmented' drivin~ techniqUes which fatigue the operators." ;'In such' condft,ions a continuouslx curved

',(winding) al"ignment is preferable, ':.,

5':2.1.4 sJbsequent curv'es connected by clothoids "

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'Fixing ent ail found,

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an alignment that",achievesw'hat said abqve, may several 'trials ,be'f.o're' a satisfactory solution is, m~inly d~e to ~h~bffsets of t~e circula~ curv~s

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"is common practice, while e,st>;lblishing ,the ,prelim;inary (P) line, to f~x points of curves, as these are normally located in,the most constrained situations. lit is importantth~refore to fin~ a clothoid para~eter connecting the'two circular curves without moving! them from the originally set positi6n~ I

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The following' allows the solution of this problem. I There may be 2"('two) ,cases: i) the curves are, in opposite direction, and ii) ,the curves are in the same directiibn. In both cases the distance (d) between the two centres, (Cl & e2) of the:,curves must: be caJ,culated 'from,the survey ~~ta of the'p line. ;' ;

i ) Curves in opposite direction , ,

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'The clothoidto be,found ~ust ,satisfy condition (Refer to Figure;'5.3)·

the :fbl.lpwing

d = (R'1.+DR1+R2+Df\2)2 + (XM1'+XM2)2

Where

d = Distance between centres of curves Rl G Radius ,Of larger curve R2 :;: Radius of smaller curve DR1!XMl = Offset and shift of larger curve DR2!XM2,:;: Offset and shift of smaller ('''''''9

the distance, (D) between :the arches c must 'also be known, thus; ,

"0 := d:- (R1+R2)

The, parameter (A), of the clothoids ,qoni two curves t's obta::q,Q'",d by':

~; ';" "~;'( R 1 R2/R 1+R2) 0!::~5DO .:?5 (2. ~134+0 . 0843 (Rl . . . . , . '. . . ::.

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~--------,-------- , e, : • " '. ~ . .".. I * "Jhis formUla, derives i from; a seri.es of mdhematical <regressions, therefore the, results are not absolutely ,precise. Tli'e error: is, "however, negligible, for'

practical ap~iication"{n' n'ormalconditioris.In cases,' where absolut~, precisjop,',ts required few, reiterations ·(trial\·and. error are necessary). . '

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'CURVESOF(fpPOSlTOIRECTION ... '-CONNECTED BY CLOTHOIDS

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,·~!.NAL ,~RAVER~E, =l~t.~.

XM2

0.

iRIG1NAL lP)' N' ': , ' IDR2· 7-' _£CJ.-rRAVERSE LINE' ..• ', . . --t.-:.::.:,~,' -? -:---:- ,' .. "

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-- tDRl _0:--, XM . . I .. .1 .. 1 ----. ,-,.' . . . . -

'( )0 7~ 0 '2~ ~ . (R R )-',0077 J A= RtR2 'xD' ~ 2.2154+0.0845x-R

T .2 xD Rt+R. ". ,I..., . t+R2

. D =V R,+,.l+ '0,,"'_ ~ (R t + R~) =V (R;+DR1+ R2+OR2)2.j.{XM1+XM:l,;{ Rt + R2)

c!:= ATN (ocr ) _ ATN( XM t+XM2 .) rRt+Rz \RI+DRI+R2+ DR 2

. ' R2 - . (R.+DR )- -eo -OM = R TANc!:+ - 2, so:.; • 2 . ,Sin cC

~: -'-

QM = RI~ANit!Rl\+DR~)::"CC'C~~ ;i:'" --"'·-~~-'--'C-'=--'·--·' Sin cC

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MSTiTS = XM1 -

Rl (Rf+ DR1) - ' COS-cC ,

-Ian c!: '~-

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The same may be, found graphically by chrirts included in Figure 5.5 ai follows

usingithe ,I' I I

,,1. Enter R1 and R2 in chart b) (curv~s in oppo:site direction).!

2. At the intersection read'the value of R1R~/~1+R2. 3. Enter this value and the' D (calculated) in 'chart

a) . ~ .: ...' i

4. At th'e',:,intersectjon read ,the value of!the parameter (A). ,,' ;' ,

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ii) Curves in the same direction

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In case of curves in ,the same direction (broken :back curves) the connection wit:h clothoid is poss~~le only if the smaller curve i~ inscribed in the l~rger on'e, because, un1 i ke the previou.s case where th.e:, two curve's arches, are connect,ed by 2 ,(two) c10thoids having the same parameter and flex point (R' =1 ') ot 8T=T8, in this cose the two'curve arc'hesl ure connecfed by a segment o'f ,clothoid huving the! two' ends id~ntified, by the two radii values. ,!

I, Should the two curVes not ,be 'on,e inscri,bed in the other, than an auxiliary arch circumscribing bofh of them must' be inserted and then the connections made.

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In this c~se the clothoid ~o be found must satisfy the follo'w'ing condition (Refer to Figure, 5.4)

d = (R1+DR1-R2-DR2J 2 + ~XM2-XM1)2

,the distance (D) between arbhes will' be , ,

, I D = R1-d-R2i . . ; ,

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The larger radius, iS~'a"ways' identified "~"1Il1~lle'r radius tiY'.:R2., ;,

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d ~ 1\ ... ,1 ••••• '.

, The 'parnll\eter, (A) of the clothoid, oLWhi:ch a .... segment., is, ,used' for, (':onnecti ng t he ,two curves, is

obtained by : ' ' " ':. ,

:..: (R 1 R2/R 1-R2 ),0 . 75 DO . 2. ( 2 . 2134+0.0843( R 1 R2/R1-R2IJ -1 :0077'() 'J'" ,r., .,' " 'I' ' , ;: • • , •• j ., - j....

Th'is formula der:ives from,'a: ,series" of, mathemat:ical regressions, the'reforel',the results are not absolutelY preci 5,e;.",T.he eTror, is', however, neg1 i gib1 e 'fol" practical;'appl ication' in normal conditions. In 'cases where' absolute precision is required few reitera'ti'ons (trial and error are necessa~y). I

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," "'tcF i GU'RE:S.4':> , ,CURVES ,IN'SAMEDIRECTION* c6~fNEC"tED ,8YCLOTHOID

,: ( Rl R",\O.7= 0 0.25 X[2.21~~~.0843J.Rl R2,,-1.C077x OJ ? '

Rr':R;) .' .. '\,Rl-RJ . \

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I11III

, OF THECLOTHOID PARAMETER (A) FOR CONNECTING CIRCULAR CURVES IN THE SAME OR OFPOSIT DIRECTION

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c) FOR USE WITH ,CURVES IN THE - SAME ',DIRECTION

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parameter may be . , 'sume ,procedure.': (C) will be used

5.2.2 Superelevation in CUrVe ! .

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For horizontal curve design, it is nece,ssary tOi:determine superelev:at ion rates as app1 i cabl e overthei range of curvature used for eac'h': design speed. Earl ier' we have seen that for the calculation of 'the minimum radius, for" a given design speed, the,'maximum values of 'e (superelevation rate) and' f ,(si-de friction' factor) are used. ". .: However if the design speed is fixed, when the; radius of curvature varies, also (e+.o varies. . " . i In order to'determine the super,elevationrate,. as a function of ~he radius at 'Il giv.en' speed, the/d,i:~tribution . of e and 'f should be determined, •. That is,' itl'should be. identified how much of th~ centrifugal forced~veloped,on di~ferent curvnturecondftionl'ls counterac~ed by .the, side friction, 'and how much of it by the super~;levntion. ,:

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The method 'recommended "fbr t,hedistribution9f e and f 'is the AASHTO Method' 5 iri whichf is distributed along a curved 1 i,ne of par:abol ic fiorm.," ', ... ' .: . ' Based 'on :this method' the ,distribution ,of e,an'd fin all conditions of design. speed envisaged, ,conSidering max

. sUper,e 1 eyat,.; on ,rat es'( e), iof, i 10· pe rcent,', 'and ~8 percent,' are'" shown in Tables 5.2, and 5. 3,respecti ye ly. . F.or

. practjcal design purposes ;the val~es ,il'Jd{cate~ 'i,n'column (e) may be, rounded to the ,neurestO.5 percent, The superelevation rates ;presented,.,in 'Figures 5.6 and 5.7 ,were derived by plon ingthe .. evalues from Tables

'5.2 and 5.3 and,may be:used fo'r intermediate' Rudi'i"",', values. .j ~ ,;·fY,~.:· I supe'rel evat i on::."run~Q~f (I.,). denotes t he,.l engt h : of . road

".\i"needed to .. a'cco-iTiplis/l the change'·.in"cross:~lope from normal crown, section. t,o't:he fu.1,lY,superelevated ,section .. The minimum length required to.:uccomplish:thisl" cons-iders the maximum relative slope betWeen profile of the edges

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. Seurca : The Cansalt!l,t. Oerived irem :\Asnto :: =.;d f distribution S} ··R : iadius· ~f"curY'a.tl1ra [n) e :,Suiieraian!::~ ra.te (i1iJ) f: Side irlctioi'i factor RC : ReIGle -ldws! :r:ln HC -: Har;iiai Cr~va

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When simple'circular curves are used between tang;ents, to achi.eve the 'max1lnum superelevation, the supere;levation· run-off is split. in accordance to normal international practice, between the curved portion and the Itang'ent port ion in the rat io 1/3L inside the curve and i 2/3L 'on the tangent', the break even point being the itangent point of the curve. In'sonie c·ase,. especially; where ~S . curve are, introduced in the: alignment, the va111es shown' apove may :l),~,.e)\ceeded •. however ney.er beyond 1%. ,In such cases, the·';·distribution.· .. of rune-6ff 1engthsm·.ay also differ· froin the ratios stated. above .. it.mayj',cbe said:":"'· that although not desirable!, these relaxations' do not,: significantly affect the t:r:avelli",g characterfstics of·· the road from the safety vi~wp6int. .. . , ,

When transition curves!are inserted between tangents and circular curves,the supere'levntion run-off, is ientire1y developed within the ;transition which .1engtl;1 should a'lways be determined accord,ngly; A method bf''!,app1ying. superelevation ;s shown 'in Figure 5.8. Ve'ry, fla't horizontal :curves require 1itt1e·.or 'nosupere!levation be.cause,as said, the amount of cent,rifugal' forces ,to' be" counteracted" .decreases 'as the radius of c;urvature·: .. increases. 'Th~ limit a~ which, to counteract the centrifugal force, it is sufficient that .both l/ln\,s have .. the normal crown slope va1ue'in the directio:n 'of the center of the curve, is indicated ;n Tab1,es 5.2 i and 5.3 with RC (Remove adverse Crown).· ,j . The limit (minimum radiusi for whicb thefe is no need of ,supere1~vation is indicated with.NC· (Normal cro~n) •

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Widening of Lanes in Curve "' ' " ,

On modern highWays, wit;h lanes 3.65m." Wi.de. ane! .' relatively straig~t, alignments, the' need for widenirig hai'"

. lessened considerab]y, but! for narrower Janes' and some conditions of curvllJ!iure:,especially ;n h;i1'ly and

,~ountainous ter~~in and on roads with high percentage of commercial t·raftic., it. may. be necessa'r.)i to widen the pavement to make vehi·cle operating characteristics" cOl]lparable:'to tangents sect;ions. ' ! . . The amounCof Widening, as .shown· in Tab]es 5.4 and 5.5 derived from. AASHTO, is applied to the insid,e edge of

",:t h epa vern e n tin cor res po n den ceo f t:h e ,,f u.] ,1 y supereleva(ed curve,:,s portion' and the tapering is. attained 1jri~ar1y over the superelevation run~off.

"Th'e val ues application t raf,f'i'c 'and

'i' . t,· '. , ! indicate,d,in T.ab.le 5 .. 4 'are ,for general and are 'based on standard compdsition of. on the S.ingle:.Unit"tru.ck (SU) vehicle type.

i·

,.

j--, '- ",- '- ',- -L:,-': I-- ~ ~ _ ~ 1_:;--~ I--- I--- , ~ I--

. .J'T~~.:.~,,~ _."

I--- '- .fL-, ';-C"'~:0f;;!

_ " - ,1,: ' . ,,' " " _' - -- -,:I. -': ; : " 'FIGURE5 8 --co: :,,-,L __

METHOO-:OFAPPLYING __ SU'PERE:LEVXTION-- '-' .-' - '-~:':'=":::' ~.:

NORMAL CROWN- , SUPERELEVATIONRUNOFF = L MAX. SUPERELEVATIOO ,

2/:3L _, 1/:3L

" q~TEREOGE OF +MENTJ " 5 =- SL9iE RELATIVE 0 CENTERLlN '\. 'I - e· LW

, -, NC.:"t,W' , - ROAD i"""t -~ .. r.N:-;-;C'- . == '-,~ ± ; __ NC'L~' _ 'jPC/PT SCI S ---

~S/ES TS/ST---_,-:---O .. ---~.:.:~-:-__ __ e'LW

,_-, i ROUNDING --_ --:~ -"----. - . . ........ . ~-. r-o. ...... _ -Rour; DING _ ' - _ - -- -0-----

'. '-. 0:; I :> '

~

(,)

z ~

w ~ !!i~'

~ en ~ ~

_ <i ~. ~~-_ ~> wz

C::: 0' 6~ , _ *C:::' :::;: 0; - c:: _____ z () ______ - WC::: _ • ILl ______ ...:----------- C:::(,) '_ xc.. ---- ---- - - ,-«;:) I ~I - _____ --"".:..-~=.o----~~- '~en , --~--.;.;;-..;. -----~~.7.

,-

.------.--.. -.. " •.. __ ._-- ._-

SSiES = BEGINNING/END OF SUPERELEVATION , ROUNDING: NC· LW/S

-~

MS '" MAXIMUM SUPEREtEVATION ; PCiPT .. POINT OF CURVE/TANGENT

e = SUPERaEVATION RATE LW = LANE WIDTH "

W = WIDENING TS/ST : TANGENT ISPlRAl./TANGENT SC/CS = SPIRAL/CIRClE/sPIRAL

;

- - '-.--. . ...• --_.-,-- - .

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Table 5.4

WIDENING (M) OF TWO-LANES PAVEMENT ON CURVES (Based on Single-Unit Truck-SU-Vehicle Type)

.. , ------------.---------~----- .. --.----------.--------.--------.--------~--------------.-----

: DESIGN SPEED (1<011 h). : R ; -----------------;-------------"-..:--------------:..:.-------.:.-.-----~---'"7-,,------: ;

, ,: 30 ": 40 : 50 : 60 : 70 : 80 :. 90 100:,: . , ______________________________________ :...: ______________________________________ :... ___ , t

30 , 2.40

, ,. , ' , , , . 40

, 1. 95 , , , " . ,

50 , 1. 65 LBO

, , ,. , , 60 1.50' :,1 . 1 ;65 , , , , 70 1.35 1. 59 , , , , , 80 1. 20 1. 35

, 1. 50

, , , 90 1.05 1. 20

, 1.35 '.' ,

100 1 ;05, 1. 20 , ..

1.20 -. , , 1"

125 0.90 1. 05 " 1.05 1. 20 I;. , , . 150 0.90 0.90

, 1. 05 '1.05 . ' 1.20 ' i

I. , ' 175

, 0.75 / 0.90

, 0.90 ,1.05, 1.05 ' ' , , / / 11

200 ,: 0.75 , 0.75 , '0.90 0.90 , .L05, ' , , , , ' 225 0.75 , 0.75 , 0.90 '0.90 0.90 1. 05 "1 , ,

", I

250 0.60 , 0.75 , 0.75 0 .. 90 0.90 1.05 I! . , ,. Ij"

275 0.60 0.75 , 0.75 , 0;90 ", 0.90 0.90 I; , , " 300 0.60 0.60 0.75 0.75 0.90 0.90 ,I 1. as . " , , 350 0.60 0.60 0.75 0.75 0.75 0.90 " 0.90 , .

. , , , ,

" I," , .: , ./

400 0.60 450 500 600 700 BOO , , 900

, , 1000

, , 1500

, , ,

0.60 " 0.60 '0.75 ·0.75 0.60 0.60 '0.75 0.75

0.60 ;0.60 0,75 "

0.60 :0.60 0.60 , ,

0.60 0.60 , 0.60 0.60

" 0.75 " 0.90 0.90 'i 0.75 " 0.90 0.90 ,.

" " 0.75 ,.

0.75 0.90 ,

,I , 0.75 :1. 0.75 0.75 /

" , .0.,60 ·Ii 0.75 0.75 ,

" ,

0.60 , 0.60. 0.75 , " ,

0.60. ,.

. 0.60 0.75 , , , 0.60 r 0.60 0.60

, , . I .. _~ / , 0.60 , 0.60 , , , , , / , , 'I I'

!::: I I' !:, i I. ,I • ---------------------------------------------------~-----------~--------------------

2000 / , , /

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NOTES: ,The above is based on Norma.l Pavement .Width (w) = 6.'10 m. .For w = 6.70;ln. reduce the. above',Jalues by .0.30'm. For W '~ .. 7 .. 30 m. reduce the above vaJ ues' by" 0.60 /1)."

Values 'less than 0.60 m. lIlay be disregarded.'· .

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70

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USil1~':;;h;o, chnrts in r,lgure li.n tho par"mot!lJr may be ,. (OUlltl, ,g·raphlcl.ll1y following tha /,lame Pt:ocedUre.

clElflCr1 bed! en,.1 i e r : except. that.' chart (,C) w.I ll:! be i used'

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to f,ind· ~lR2/Rl-A2. j' ii SUlJerelevaHoll: in .curve! 'f! 'T--~------r-+--~-----i '. '. r i. ' .. " :;. ";, For hori.zqt.l~I)·l !;cUr"vede;sjgni:!'ft:f,fs,'.'ne,cessary to'det~rmine' '.r suporo 1 Bv~t'i'on/,,'rllt os lis.: ap!?]! cab1.e . OV~J,·t he,,'png<) or,.'·,",: ..... 'curvature. used for e.l1chdesji 9n, spef3d. . Eflr,) ier Iwe. /Jav.e.;,· seel1that for the calculation of the.tnfnimum·:radlus 'for' ? IJ'!/; Ve 11 d ~s i.,9 /1, .. s "a'e d .t.he., mil xi m.um·);fY~4l,1E1 ~;,;.,o'fej . '(sup",relevlltl011 ratEd and f!(siodefrlction:.;f/lct'or). are. Use.d'. .' : !. '. ..' i' .' ' ...... :: •. "' .• ,,. "'1·:"" Hqwover if t.ha desigl1speed ~s fixed, w.heh.theradius of curvatureya,ries, nlso(e.H)iY~ries. , .. ,l."lii . .' In,,ol'dor to determine the sUpeC,elevntiol1 'rJlt,~"Il&.a '. functionof the radiUS at a divanspaed. thGdfstfit~ttol1

.. of e and ·.f sho,!ld .be detormi,ned.Thllt is, ,it'. s~.cit;1d be .. . ident·Hhd how tnuchoft.he· cel1tdftlg'al forc~ deve1<:>ped on.

. different cvrvllturo condlf.ipns is countllrilctedFby the '.' side frlctton,.llnd how much· 6f it ·by the supejele~lltlon. :

, .. '.'

" , .".." , ". . . . ... .: ;-' , '. ,~. -,',. :" "; ... '-. . :- ~ : ." .. - ,"" ." , .. - . : f :". ,,' ". . . .' .

The /Tiet /Jod recommended :' for. It he .•. ·:d istr ibut iOI;\, 0 f iB' Ilnd f,'·. is the AASHiO Method /). in'which' t' is distl"ibut<>dlalong a . curved 1 ine of pal"nbol ic form.. . . . . . i: .. ,

;1 "

( :~

).~. " ' '., i ,.

l3/1Sed on this method the distribution .of e and f,; in 11.11. conditiohS of design speed ,envisAged,. aonr;idering max .'

.·f .superelevl:ltion rates (e) of :10 pel"cent .and8,pel,,;;ent, .• ,: are "shown' i 11 Tab lea 5'. 2andlL 3 . 1"09\)901'; ve 1 y ;·"'··1'01"··'" prnpiioalde,sigl1 purposes th~ ValUes indicnfedjlI 'column (g') tnlly'bo· rolmdod . t 0 the he~"'estO. 5' poroent'; id l" ': , .. rhe superelevation rates .presentedirl'(o'Figl.lres 518. and

. 5~" War.G- dorived by plotting the e·~'valul!g froln' 'rabIes,,·,,··,,·,·'· , i" !5.2and 5.3 !Ind' /nI1Y be used fOI" intermediate' Radii ," i '.' ' ! .... I . _ ~.. . r .

' ... ;;, values. '. . .1'.,'. ' ••• I'

.• "f-. :,. ." . j ,: '(,n, SUper-elevation l'utl~Q~f. (l) denotes the 1ength .. , ,of f"ond . . j,': ..... , ',:.'.:,:., ',:'''.11 nO'e'ded to aocomplish the changei n cross 9101;1(1 f r~m

normal Cf"oWn,~ect ion to tho.flfllysUperaleV<ated ll!J.ot~Oti. ;~-'The rrd.nlmuin •. ',i;etl9th requ.il'ed t'o 'aooomplish th'ie;cohsHlers I rhe maximum, I'elut lve slope between profi llii'of,.the ;gdlles .. . ' '~~ ___ ~~ __ ~.~_~~~~~~~~_~_~~~~...'::~~:~~~~_i_~~_~S __ ~~;_l_,:~~_.: ____ ..;~.,_, ___ .,i~~_~~~~_;.~:-;,;';·'

:1 : " I ~', I. I : I 'C~~) . I I . 1 !i "', '" I .:' . ,..' I _I '. I 'I' .. ' I I' 1.._.. I I I. j I' 1 '

~.!:D~~i9n speed (11m/h) :,:30! 40:!, 50:! 00 ! '10! ,.' ~o !' i: 90 ! 100 : " .. "1' '-'-' "t·)·" , I ". I , '_1:.,

. ,- _. . ' .' , I ... . , r .. '-. ~.J. ., f", ',. _! -', , : r -,

!MIl~. 'relntive 1>1ope:,(%)' i 0':.7!';\ 0.70! '.·0'.65!0.eo: 0.551. .. 0;50,".:O:41l\ 0:4~! . t' ;'~-':"";--""~""-""--":"":'-~~7.'~·7;~~""'::;;""~:-:~---'":'~7--"'~":"1-~~~;-"-::~':--- .... -~~·~.::"~t~"I .... ~' .. IT7:---.... --- : . ~ I,·'i,.

": ::;i 1;1 ~ ':.! I .. ',:; ,

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