Circular Curve

7/27/2019 Circular Curve

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PART V

S IMPLE CIRCULAR CURVES

v

F i gu r e 1

Assume t h a t AV and VD are two straight portions o f a proposed highway andt h a t t h e curve BC i s t o be used as a gradual change o f direction between them.

The curve has a constant radius R a n d i s called a simple c i r c u l a r curve .The curve s t ar t s a t B and ends a t D , ' . < A00 = < DCO = 90'Po i n t V i s referred t o as the vertex or Point o f Intersection ( P . I . ) .


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One of t h e va lues we m u s t always know, or be a b l e t a determine, for anysimple curve i s the d e f l e c t i o n ang l e a t t h e v e r t e x . This angle i s designatedby either I or A . I n th e case o f F i g u r e 1, the survey i s assumed t o be pro-gressing from A towards 0. I f this i s t h e case the d e f l e c t i o n ang l e a t thev e r t e x i s as shown. 0 v

0I n F ig u r e 2 t h e quadrilateral BOCV has c e r t a i n qualities o f syrrnnetry, as

fol 1ows :OB = OC because both are rad i i


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1 . Derivation of Formulae0 v

Figure 3

Abb.E.C.P.T.C.T.

BNameBeginning o f CurvePo i n t o f CurvatureTangent-Curve Po in t

I n A BOV


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0Figure 4The s t r a i g h t l i n e BC i s called the Long Chord, abbreviated L.C.In A BEO, < BEO = go0- A ABE = Sin - or BE = BO Sin TI30 2

L'C' and 00 = Rut BE = -The l i n e EF i s called t h e Middle Ordinate, abbreviated M .O .I n A BE0 and sector BFOM. 0 = OF - OE b u t OF = R and

- - AOE - Cos 7OB but OB = R


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Sub s t i t u t i - n g the values of OF and OE i n the o r i g i n a l expression f o r t h eM.O. we g e t ;

AM.O. = R - R CDS 7

The expression , (1-Cos $) i s known as t h e Versine o f $ and t a b l e s g i v i n gvalues o f t he Versine are publ~shed. This i s why we o f t e n see the above equat ioni n th e f o r m .

F i gu r e 5

i s referredr n a l and i s

In the right-triangle OBV and t h e Sector BFOE = OV - OF b ut - dOV - Sec - o r2 -


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6OV = OB Sec 2 b u t 0 0 ' R soA a l s o OF = RV = R Sec Z.

The o r i g i n a l expression was;E = OV - OF; i f we make the a b o v e s u t s t i t u t i o n s then

AE = R (set - 1 ) ------------------ aF romEgua t i o n ( 1 ) ; T = R T a n $ orR = .--- - Substitute t h i s va lue o f R i n (a ) aboveA 'Tan

T AE = --- A SecA (Sec - 1 ) E = TT a n Z-A L CCos 2 Cos 21 , - - -ACos 2 S i n 2 S i n 7

A 1 - CO S AThere i s an identity in t r igonometry o f the f o rm Tan = Sin AA

A 1 - Cos 2-I n the case o f ( b ) above i t would have t h e form Tan - = ASinFrom t h i s ;


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622. Radius of C u r v a t u r e and Degree of Cu r v e

There a r e three methods by means of which we can designatet h e Hsharpnesstq f curvature of a curve.

SHARP CURVE

FLAT CURVEa ) Radius

The sharpness of c u r v a t u r e is inversely proportional to theradius, ie. a reduction i n radius increases the sharpness.b) Degree of Curve, chord de f i n i t i o n - Dc

Degree of curve according t o the chord definition, Dc, isd e f i n e d as the angle subtended by a chord having a length of onefull s t a t i o n or 100 St. in the foot system and a 100 meters int he metric sys tem . This method i s followed i n railroad pract ice.The r a d i u s of such a curve may be computed by t h e followingexpression:


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Note t h a t t h e r a d i u s o f c u r v a t u r e v a r i e s i n v e r s e l y as the degreeo f c u r v e and t h e r a d i u s o f l o curve accord ing t o the chor dd e f i n i t i o n i s 5729.651 u n i t s o f measurements.

c ) Deqree o f Curve, a r c d e f i n i t i o n - D,The c u r v a t u r e i s expressed by s t a t i n g the "degree of c u r v e "

Da w h i c h h a s t r a d i t i o n a l l y been d e f i n e d as th e angle subtendeda t the c e n t r e o f t h e curve by an a r c 100 f t . l o n g . I n t h e m e t r i csystem, D , s d e f i n e d as the a n g l e by a 100 m arc .a

S m a l l e r values of D, decreases the sharpness o f c u r v a t u r e .The a r c d e f i n i t i o n f o r degree o f curve i s m o s t frequentlyfo l lowed i n highway p r a c t i c e .

When d e s i g n i n g a c u r v e , Da , is u s u a l l y se l ec t ed on the b a s i sof design speed, superel e v a t i o n a n d road sur face f r i c t i o n f a c t o r .

Figure 7I f a 100 f t or meter arc subtends an a n g l e o f l o , the r a d i u s

o f c u r v e i s 5729.578 u n i t s . We refer to this as a "one degreecurve' .


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3 . Length o f Curvea ) Lenqth o f Curve Chord D e f i n i t i o n - LCThe l e n g t h o f c u r v e on the chord b a s i s , is t h e s u m a t i o n o f chordswhich approximate t h e curve and, i s t h e r e f o r e , a n i n e x a c t expression .The LC obtained w i l l always be l e s s t h a n th e t r u e a r c l e n g t h . Thed i f f e r e n c e w i 11 incr ease a s D, increases.

Where C = 100,A = Deflectoin a ng l e o f curve

FIGURE 8b ) Length o f Curve A r c D e f i n i t i o n - 1,The length o f cur ve or a r c l e n g t h f o r corresponding r a d i u s

v a r i e s d i r e c t l y t o t h e central a ng l e subtended by t h e a r c . SeeF i g u r e 7.

T h i s is an exact expression.Note t h a t t h e l e n g t h of curve ( o n t h e chords b a s i s ) L C , i s somewhatless t h a n th e a c t u a l a r c l e n g t h L a . T h i s d i f f e r e n c e w i l l i n c r easea s D or D i n c r e a s e s .a c


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6The Arc Basis is Used f o r t h e Calculations Presented i n Th i sChanter.

Actual ly both chords and arc methods a re used i n NorthAmerica. For long gradual curves, which are commor! i n railroadpractice, t h e chord basis (arc length considered to b e same aschords) is normally used. For highway curves and c u r v e d proper tyboundaries, t h e arc bas i s is more common.

In f i e l d p r a c t i c e , field measurements of t h e curve a re l a i do u t with t a p e , along the chord and n o t along t h e a r c , r e s u l t i n gi n er ror . Thi s er ro r can be r educed when the a rc b a s i s is usedby using short a r c ("chord' f ) length or applying t h e differencebetween t h e arc length and t h e chord length. For example, for a100 meter 2 c u r v e , the chord length is 99 .955 meters and for a10' curve t h e chord l e n g t h i s 99 .873 meters. The f o l l ow inggeneral rules a r e suggested in the me t r i c system:100 meter a r c s "chordsn u p t o l o curves

030 meter a r c s +*chordsuu p to 4 c u r v e s20 mete r arcs "chords" up to l oV curves

010 meter arcs "chordsw u p to 25 curves3 meter a r c s wcho r d s l tUp t o 100' curves

4 . Examp l e Problem f 1THO tangents i n t e r s e c t a t Station 3 + 16.770. The deflection

angle to t h e right is 40~00 ' 00 " . It is decided t o d e s i g n t h ehighway fo r a m a x i m u m speed of 9 0 km/hr, a nd u s i n g A A S H O *recommendation f o r supere levat ion and friction a mi irnum r ad i a sof 270 m e t e r s and a maxim~rn degree of curve, Da is to be 22'.Calcula te T, La, R and t h e stationing of th e P .C. and P .T . using20 meter a rc length.* American Association o f S t a t e Highway Organization


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66ExplanatlonWhen we speak of a station on a curve we a r e g iven a specific

location from t h e s t a r t of t h e survey. In t h e above problem, t h eP . I . is given as Stn. 3 + 16.770. What we are saying is t h a tP.1. l a 316.770 m from t h e s t a r t of t h e survey.

T h e problem also s t a t ed that A is 40' t o t h e r i g h t , t h i s sayst h a t wh e n we stand at t h e P.C. and l o ok in the direction in whicht h e survey is progressing, t h e curve deflects t o the r i g h t .Solution

Given P . I . = S t n . 3 + 16 .770A 1 40

20 meter a r cFor safety Da may b e rounded down t o ZOO.Using a rc definitionRecompute R a n d c a l c u l a t e T and La

A 40T = R tan - - 2 8 6 . 4 7 9 t a n - 104.270

2 2

P . I . Station 3 + 16 .770

P .C . S t a t i o n 2 + 12.500La 2 + 00.000

P.T. Station 4 + 12.500


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Figure 9

5. Laying out a Curve by Deflection Ang l e sCurves are s t a k e d out usually b y the use of deflection angles

t u r n e d a t t h e P.C. from the tangent to sta t ions a l o n g the curvetogether with the use of cho rds measured from station to s t a t i o nalong the c u r v e .

I n the past 100 ft. chord l e n g t h c o u l d be l a i d out quiteaccurately, however, in t h e metric s y s t em 100 mete r chords w i l lr e su l t i n a large e r ro r . Most curves are presently laid out i n20 meter chords or less and t h e discussion p r e s e n t e d here w i l luse 20 meter chord l eng th .


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68First let us cons ider the first 20 m e t e r past the P.C.

N o t e that the a n g l e subtended by a 20 meter a rc "chord" w i l lbe propor t iona l to the degree of curve Da.

In i sosce le s A , AO


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Consider the second 20 metre chord BCMeasure 20 m from B t o C . J o i n AC . In isosceles t r i ang l e OCA angleVAC = D. To locate po in t C measure 20 m from point B. With the transita t the P.C., reading O0 on V, turn off angle D to a l i g n rape ande s t ab l i s h C.

/ FIGURE 11

FIGURE 12


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From: Figure 10: Note t h a t deflection a n g l e D/2 - 1 / 2 a n g l e AOB a tcentreFigure 11: Note t h a t deflection angle D 1 / 2 angle a t c e n t r eFfgure 12: Note that deflection angle 3D/2 = 1/2 a n g l e AOD a t

centre

From t h i s we can deduce a general r u l e .If we s e t a t r a n s i t up a t e i t h e r t h e P.C . or t h e P.T . , and

s i g h t on t h e v e r t e x with t h e plates set at zero, we can turn thecorrect a n g l e t o an y p o in t on a curve of constant radius bymerely s e t t i n g o n t h e p l a t e s an angle equa l to half the angle a tthe centre sub tended by t h e chord j o i n i n g t h e i n s t ru men t an d th edesired point.

PC . n V=PI.

Figure 13 - Figure of D e f l e c t i o n Angles on a Simple CurveOn th e above f i g u r e let p o i n t s a , b , c , d represent s t a t i o n

p o i n t s on a simple curve. Point a is a n odd distance from P.C.and distance dB i s also an odd increment. The deflection anglesare:


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U s u a l l y the P.C. and P.T. d o not f a l l a t even s t a t i o n s but weare usually required to pl a ce our stakes a t the even stations onthe curve . The f i rs t and l a s t odd length of arc or odd s t a t i o n ,therefore, is us ua l l y l e s s than 20 meters and the f i r s t and l a s tdeflection angles are less than D/2.

In the case of an arc of a circle, the angle subtended a t t h ecentre i s d i r e c t l y proportional t o the l e n g t h of t h e arc.

Da Angle subtended by a f u l l 100 meter stationD = Angle subtended by a f u l l s t a t i o n (ie. may t o 100 or less)d = Angle subtended by an o d d l e n g t h stationR = Length of arc of an odd s t a t i o nS = Length of arc of a f u l l s t a t i o nC = Chord distance

The d e f l e c t i o n angle for an odd s t a t i o n = d/2OR in more general form for an arc length, S

d a D- -. D and d i n degrees2 2s


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The degree of curve, Da is defined f o r a 100 rn a rc l e n g t h ,since we are using 20 rn a rcs t h e angle , D subtended by t h e 20 rna rc is 1/5 of Da namely 4.

The deflection angle, D/2 for a f u l l 20.000 m station is thus2 The distance from P.C. to the f i r s t f u l l station is ( 2 +20.000) - ( 2 + 12.500 = 7.50 m ) , and t h e d i s t a nc e from the l a s tfull station on t h e curve to P.T, is ( 4 + 12.500) - (4 + 00,000)= 12.500 rn.

The deflection angles f o r t h e odd increments a t t h e beginningand t h e end of t h e curve are:

*I- 0 45 ' ( a t beginning o r curve)2- 11 5 ' (at end o f c u r v e )

Chord distances for t h e i n i t i a l and f i n a l o d d increments ofarc and the full station are:C 1 = 2R S i n d , / Z = 2 r 286.479 S i n 0.75' = 7 . 5 0 0 meterC2 = 2R S i n d 2 / 2 = 2 x 286.479 S i n 1.25'= 12.499meterC20 = 2R S i n D l 2 = 2 x 286.479 S in 2' = 19.996 meters

It can be seen t h a t the chords l e n g t h are near ly equal t o thea rc length and no correction would have to b e a p p l i e d t o accountf o r t h e d i f f e r e nc e .

Let us now make up a s e t of field notes fo r t h e curve. It


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73should b e emphasized t h a t t h e form of field notes given is n o tnecessarily THE form of f i e l d notes though it s a t i s f i e s ou r needsquite well.

T h e notes a r e s e t up on t h e assumption t h a t t h e transit iss e t u p a t t h e P .C , (Stn 2 + 12.500) and t h a t t h e plates read0 00' 00'' w i t h t h e i n s t r u m e n t sighted on t h e P.I. ( S t n 3 +16 . 7 70 ) .

T h e notes are designed to be absolutely complete s o t h a t t h eman who is going to l a y o u t t h e curve does n o t h a v e t o r e f e r t oa n y o t h e r source f o r any information he may require with regardto t h e curve.

T h e bottom and t op lines i n t h e notes relate our particularc u r v e to the preceding and following c u r v e s .

T h e co lumn headed "Bearingt' orients the c u r v e i n relation t ot h e rest of t he pro j ec t as does t h e sketch.


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LEFT HAND PAGE OF FIELD BOOKFigure 14Note t h a t t h e s t a t i o n i n g increases f rom the b o t t o m o f t h e pagetowards the t o p so t h a t the ins - t rument -man, standfng a t P - C , andl o o k i n g a t t h e notes, sees t h e notes going away from h im as doest h e curve.

STAT ON9 + 18.570

4 + 12.5004 + 003 + 803 + 603 + 403 + 203 + 002 + 802 + 602 t 402 + 202 + 12.500

0 + 11.380

CURVE

BRG .$ 2 5 ' 1 8 ' ~

= 6 / 2

CURVE DATA

A = 40'00 ' 00"

Da = 20'

La = 200.000

T = 104.270

D /2 = 200100"d1/2 = 0 ~ 4 5 ' 0 0 "d2/2 = 115 '00"

PO I N T

P . C.

P.T.

P.C.

P.T.

LOCATION OF SIMPLE1

DEFLECTIONANGLE

2 0 ~ 0 000"1a045 00"16'45 ' 00"1445'00"12'45 '00"10'45' 00 "

8'45 '00"6 '00"

4O45 ' 00"2'45 ' 00"0'45 ' 00"oOOO ' 00"


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NORTH

OAKHURS T ROAD PAGE 2 0L -TRANS I T - SOKK I SHA A - I 2 CLOUDY,2QC30m. T E E L TAPE N0.8 ROHLF ING ,H

KOZ LOW, A TAPESTOf T, D ROD

P. 1 ,

R IGHT HAND PAGE

I? .

F i g u r e 15

2+ 12.500

Show: Page No, (The right-hand and left-hand sheets have the samepage no.)

Project (If it is separate from t h a t on preceding and followingpages)

Party; Date; Weather Cond i t i ons .Sketch of curve (not necessary to sca le)Approximate o r i e n t a t i o n of curve (or correct values if known)


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. 76.He can check our curve by using latitudes and departures tocalculate t h e theoretical closing er ror a t t h e E.C. i f a seriesof chords are l a i d out from t h e B.C,

t ASSUMED NORTHA T=104.270

Figure 16 ( n o t t o scale)


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TraverseS i d e Dist .

Function Lat. Dep t

L'E A S T ACTUAL E.C.

IGNORING ANY 3ERROR I N STAKING

The ac tua l , stacked E.C. {neglecting errors i n l a y ou t ) falls0.012 rn outside t he forward t a n g e n t and 0.035 m beyond thetheoretical E.C. This error is n o t l a rge and greater accuracywould be aquired by having shorter chords lengths.


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786. Intermediate Set-up f o r Simple Circular Curve

S o m e t i m e s , because of the l e n g t h of a c u r v e or because o fphysical obstructions, t h e whole c u r v e cannot be r u n i n f r om e i t h e rthe P.C. o r t h e P.T. If t h i s is the case it becomes necessary t o s e tt h e t r a n s i t u p at some station on t h e curve and to orient i t i n such away t h a t t h e f i e l d no te s can be used without modification or f u r t h e rc a l c ~ l a t i o n .

With reference t o Figure 1 7 and us ing the f i e l d notes shownp rev i o u s l y we w i l l assume t h a t , w i t h t h e instrument se t up a t theP.C., we have staked t he curve t o Station 3 + 00.000 b u t are ~ n a b l e osee beyond t h a t p o i n t . We move t h e transit to Station 3 + 00.000.

Consider t he curve AOB (Fig. 1 7 ) a3 being a complete c u r v e i ni t s e l f , The c e n t r a l a ng l e AOB = 1 7 ~ 3 0 ~

Angle BAO = Angle ABO = 81 1 5 'T r i a n g l e AV,B is an isosecles d because t h e t w o sub tangen t

dis tances (TI * A V and V, 3 are equal.1Angle BAV , = Angle V, BA = 8'45'With t h e instrument set up a t B, with the plates clamped a t zero,

s i g h t back on A and clamp both mo t i o n s . Unclarnp t h e u p p e r motion andt u r n a clockwise angle of 8'45'. The line of s i g h t now lies along t h el i n e V BY a n d is t angent t o t h e c u r v e a t B. If B was t h e beg inn ing1 2of a curve, t h e deflection a n g l e t o Stn 3 + 20 wou l d be 2'00' If weplange t h e t e lescope we w i l l be sighting along t h e line BV and i f we2add 2'00" t o the 8'45' already s e t on the p l a t e s we w i l l b e sighted onStn 3 + 20 with 1 0 ~ 4 5 ' s e t o n t h e plates which is the v a l u e of th en o t e s f o r Stn 3 + 20.


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79Let us now assume t h a t we are a b l e t o s take to Stn 3 + 80 but

cannot see beyond Point C. We move our transit to Point C.Let us cons ide r curve BOC (Fig. 17 ) to be complete in itself.

0Angle BOC = 1 6 00'. Angle OBC = Angle OCB = 82'00'Triangle BV C is an isosceles triangle because the two subtangent2

distances (T2) BV2 and V C are equal.2Angle V p B C = Angle V2CB = 8 Y ~ ~ *W i t h t h e i n s t r um e n t s e t up a t C, with t h e plates clamped a t

8'451, sight b a c k on B and clamp both motlona. Unclarnp t h e uppermotion and turn the instrument clock~ise n additional 8O00', so t h a tt he plates read 16'45' and the line of s i g h t lies along the l i n e V C V2 3an d is tangent t o t h e c u r v e a t P o i n t C. If C was the beginning of acurve t h e deflection ang le t o Stn 4 + 00 would b e 2000tt, If we planget h e telescope the l i n e of sight will be along the subtangent CV and3i f we add 2O00" to t h e 16'45' a l r eady s e t on t h e pla tes we w i l l besighted on Stn 4 + 00 with 18'45' s e t on t h e plates which is the valuein the notes for S tn 4 + 00.

Again, we can proceed to s t a ke the next portion of the curvewithout changing the field notes.

The t w o statements allow us to formulate a general rule forintermediate setups on a simple circular curve.

When we occupy an in termediate s t a t i o n on a simple c f r c u l a r curveand sight b a c k on some previously established station, for t h e p u r p o s eof orienting the transit, the plates of the transit must be clamped att h e v a l u e of t h e deflection ang l e fo r t h e s t a t i o n being s i g h t e d on.After Sacksighting and clamping the lower motion, i f we change t h evalue of t h e angle on t h e plates t o t h a t fo r t h e de f l e c t i on ang l e of


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80the station that is occupied by the instrument, the line of sight willbe tangent to the curve and we can plunge t h e telescope and continuestaking the curve without making any change in the field notes.

Figure 17 Intermediate backsighting setup on a Simple Circular Cu rve(not too scale)


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81On Fig. 18 there is a s l i g h t l y different application of the ru l e .

In this case, the curve has been staked from t h e P.C. to Stn 3 + 00but the instrument is oriented by sighting ahead to t h e P.T. Thep r i n c i p l e is the same as that ou t l i n e d above.

Figure 18, Intermediate fores i ght ing s e t up on a Simple C i r c u l a rCurve (not too scale).


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82The curve has been s t a k e d from A t o B and t h e instrument h a s been

s e t u p a t 8. You will orient i t b y sighting ahead on the P.T.We will treat the sector BOC as an e n t i t y i n itself. He deduce

0t he a ng le BOC to be 22 30".0Angle OBC - Angle OCB = 78 45 'In a B V , C , B V l = V I C - T So A is isosceles1

0Angle V,BC = Angle V I C B = 1 1 15 'Following t h e rule s e t ou t on t h e previous example with t h e

instrument set u p at B, s e t t h e deflection angle for C (20~00') n theplates. Sight on C and clamp the lower motion. Unclamp the uppermotion and set the plates the deflection angle for point B.

Note t h a t 2g000' - 8'45' = 1 1 1 5 ' an d t h e l i n e of sight now l i e sa long the line. BV and we can continue staking without changing o u r1notes.

7 . F i e l d Procedurea ) Set up a t P.I. and l a y o u t tangents to establish P .C . and

P.T.b) Set up a t P . C . and sight back to P . I . Se t plates to

O O O O ' o O " .

c ) Turn off f i r s t deflection angle on transit and measwe outthe first subchord d is tance , pound in hub and writestationing on it.

d ) Continue l a y i n g ou t chords and placing hubs. When the last


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83station ( P . T . Is reached t h e last s u b c h o r d measured o u tshould be q u i t e close t o the P.T. hub establfshed earlier.If i t is not, a n er ror has been made and t h e c u r v e ha s t o ber u n i n a l l over again.

e ) It is advisable to use intermediate s e t u p s when s e t t i n g outthe c u r v e . They p r e v e n t communication d i f f i c u l t i e s and tendt o reduce t h e number of errors.


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PROBLEMS1. Calculate the radius of t h e curve t h a t w i l l pass t h rough Po in t P ,using 1a t t udes and departures. Determine th e stationing o f

t h e P . C . and P , T , and o f Point P on the curve.

2 . So lve for l e ng t h o f the curve i n t e r sec t i on dis tance AB, based on t hefo1 lowing diagram.


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3 . Calculate BC, area ABC (shaded) and ang l e d :

RADIUS 391.10 rn

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Circular Curve