Circular curvesCircular curves

Horizontal curvesHorizontal curves

DefinitionDefinition

Horizontal, circular or simple curves are curves of constant radius required to connect two straights set out on the ground.

UsageUsage

roads, railways, kerb lines, pipe lines and may be set out in several ways, depending on their length and radius.

Circular Curve GeometryCircular Curve Geometry

FORMULAFORMULA

2tan

RT

2sin2

RC

2cos1

RM

1

2secRE

3602

RL

R

Cxree 60

9.1718)(deg1

R

Cxute

9.1718)(min1

Chainage T1 = Chainage I + tangent Chainage I + tangent lengthlengthChainage T2 = Chainage Chainage T1 – arc length – arc length

Setting out methodSetting out method

Offset From The Tangent Line

Offset From The Long Chord Line

Deflection Angle MethodSub Chords Line Method

OFFSET FROM TANGENT LINE

Given Radius, R 600m

Deflection angle ,

18024'

Offset 20m

Chainage intersection point, I

2140m

Draw the table from the formula:Draw the table from the formula:

Ofset

R R2 Y2 R2-Y2

formula 22 YRRX

22 YR 22 YRRX

IP ) Tangent line Y2 X2

X1

Y1

T1

circular arc T2

PROCEDURE PROCEDURE Tangent length = R tan θ/2

= 600 tan (1824/2) = 97.20m

Chainage T1 = chainage I – tangent length= 2140.00 - 97.20 = 2042.80m

Arc length = 2 x R x 360= 2 x 600 x 18o24’ = 192.68m 360

Chainage T2 = chainage T1 + arc length = 2042.80 + 192.68 = 2235.48m

Ofset R R2 Y2 R2-Y2

(1) (2) (3) = (2)2 (4) = (1)2 (5) = (3) – (4) (6) = √(5) (7) = (2) – (6)

0

6006002 = 360000

02 = 0360000 – 0 =

360000√ 360000 =

600.000600 – 600 =

0.000

20 202 = 40360000 – 40=

359600√ 359600 =

599.667600 – 599.667 =

0.333

40 402 = 1600360000 – 1600 =

358400√ 3584000 =

598.665600 – 598.665 =

1.335

60 3600 356400 596.992 3.008

80 6400 353600 594.643 5.357

97.20 9447.84 350969.109 592.426 7.574

22 YR 22 YRRX

OFFSET FROM LONG CHORD LINE

Given Radius, R 600m

Deflection angle ,

18024'

Offset 20m

Chainage intersection point, I

2140m

Draw the table from the formula:Draw the table from the formula:

Ofset R R2 Y2 R2-Y2

formula

22 YR 22 )2/(WR

22 )2/(22 WRYRX

22 )2/(22 WRYRX

PROCEDURE PROCEDURE Long chord length = 2R sin θ/2

= 2 x 600 sin (1824/2) w = 191.857m w/2 = 95.929 m

Tangent length = R tan θ/2 = 600 tan (1824/2) = 97.20m

Chainage T1 = chainage I – tangent length= 2140.00 - 97.20 = 2042.80m

Arc length = 2 x R x 360= 2 x 600 x 18o24’ = 192.68m 360

Chainage T2 = chainage T1 + arc length = 2042.80 + 192.68 = 2235.48m

Ofset

R R2 Y2 R2-Y2 w/2 2 x

(1) (2) (3) = (2)2 (4) = (1)2 (5) = (3) – (4) (6) = √(5) (7) (8) = √(3) – (7) (9) =(6) – (8)

0

6006002 = 360000

02 = 0360000 -0 =

360000√ 360000 =

600.000

95.9292 = 9202.277

√(360000 – 9202.277)=

592.282

600 - 692.282 = 7.785

20 202 = 40360000 – 40 =

359600√ 3596000 =

599.667

599.667 – 592.282 =

7.385

40 402 = 1600 358400 598.665 6.383

60 3600 356400 596.992 4.710

80 6400 353600 594.643 2.361

95.929 9202.277 350797.723 592.282 0.000

22 YR 22 )2/(wR

DEFLECTION ANGLE DEFLECTION ANGLE METHODMETHOD

Given

Radius, R 600m

Deflection angle ,

18024'

Offset 20m

Chainage intersection point, I

2140m

R

Cxree 60

9.1718)(deg1

formula

RCx

ute

9.1718)(min1

Draw the table form Draw the table form for deflection angle for deflection angle methodmethod

Stn.

Chainage

Chord length

Deflection angle,(0 ‘ “)

Setting out angle, (0 ‘ “)

Tangent length = R tan θ/2 = 600 tan (1824/2)= 97.20m

Chainage T1 = chainage I – tangent length= 2140.00 - 97.20 = 2042.80m

Arc length = R x x 2 360= 600 x 18o24’ x 2 = 192.684m 360

Chainage T2 = chainage T1 + arc length = 2042.80 + 192.68= 2235.48m

PROCEDURE PROCEDURE

Stn. Chainage Chord length, C

Deflection angle,(0 ‘ “) – use formula

Setting out angle, (0 ‘ “) – cumulative deflection angle

T1 2042.821 0 00 0’ 0” 00 0’ 0”

1 2060 17.179 00 49’ 12” 00 49’ 12”

2 2080 20.000 00 57’ 18” 10 46’ 30”

3 2100 20.000 00 57’ 18” 20 43’ 48”

4 2120 20.000 00 57’ 18” 30 41’ 6”

5 2140 20.000 00 57’ 18” 40 38’ 24”

6 2160 20.000 00 57’ 18” 50 35’ 42”

7 2180 20.000 00 57’ 18” 60 33’ 0”

8 2200 20.000 00 57’ 18” 70 30’ 18”

9 2220 20.000 00 57’ 18” 80 27’ 36”

T2 2235.506 15.506 00 44’ 25” 90 12’ 1”

= 192.684 = 90 12’ 1” θ / 2 = 180 24’ / 2 = 90 12’ 1”

6006017.1799.1718

xx

60060000.209.1718

xx

60060506.159.1718

xx

+=

+=

Sub chords line Sub chords line methodmethod

• Given Radius, R 600m

Deflection angle ,

18024'

Offset 20m

Chainage intersection point, I

2140m

formula

Draw the table form Draw the table form for sub chord line for sub chord line methodmethodStn.

Chainage

Chord length

Offset

Ra

Ofset2

2

1

Rabb

Ofset2

)(2

Rb

Ofsetn

2

Rbcc

Ofsetn 2

)(1

Tangent length = R tan θ/2 = 600 tan (1824/2)= 97.20m

Chainage T1 = chainage I – tangent length= 2140.00 - 97.20 = 2042.80m

Arc length = R x x 2 360= 600 x 18o24’ x 2 = 192.684m 360

Chainage T2 = chainage T1 + arc length = 2042.80 + 192.68= 2235.48m

PROCEDURE PROCEDURE

Stn. Chainage Chord length

Offset

T1 2042.821 0 -1 2060 a = 17.179 0.2462 2080 b = 20.000 0.6203 2100 20.000 0.6674 2120 20.000 0.6675 2140 20.000 0.6676 2160 20.000 0.667

7 2180 20.000 0.6678 2200 20.000 0.6679 2220 20.000 0.667T2 2235.506 c =15.506 0.459

= 192.684 -

6002179.17 2

1 xOfset

6002)179.1720(20

2 xOfset

600202

n

Ofset

6002)20506.15(506.15

1 xOfset

n

First OffsetSecond Offset

Other Offset

Last Offset

PROCEDURE SETTING PROCEDURE SETTING OUT OUT

Circular curves may be set out in a variety of ways, depending on the accuracy required, its radius of curvature and obstructions on site.

Methods of setting out are as follows:

•Using one theodolite and a tape by the tangent angle method. This method can be used on all curves, but is necessary for long curves of radius unless they are set out by coordinates.

•Using two theodolites. This method can be used on smaller curves where the whole length is visible from both tangent points and where two instruments are available.

•Using tapes only by the method of offsets from the tangent. This method is used for minor curves only.

•Using tapes only by the method of offsets from the long chord. This method is used for short radius curves.

OPTICAL SQUAREOptical squares are simple sighting instruments used to set out right angles. They can be provided either with mirrors or with one or two prisms. Because of practical difficulties in using squares with mirrors, they have been replaced by squares with prisms: "prismatic squares". There are two major types of prismatic squares: single prismatic squares and double prismatic squares; both will be dealt with in the sections which follow.

FIELD WORKFIELD WORK

OFFSET LINE FROM BASELINE USED OFFSET LINE FROM BASELINE USED OPTICAL SQUAREOPTICAL SQUARE

STEP 1STEP 1STEP 2STEP 2

OFFSET LINE FROM BASELINEOFFSET LINE FROM BASELINE

STEP 1STEP 1 STEP 2STEP 2

STATION AT BASELINESTATION AT BASELINE

End of sub -topic• ExerciseExercise

• Practical 6 & 7 Practical 6 & 7