15. Horizontal Curve Ranging.r1 Student (1)

SITE SURVEYING

Curve Ranging

1

Curve Ranging Scope of Coverage

1. Objectives

2. Examples of Curves

3. Fundamental Geometrical Theorems

4. Curve Elements

5. Designation of Curves

6. Setting Up Procedures (Calculations)2

1. Objectives

After studying this Chapter, the students should be able to make the necessary calculations to fix the positions of points forming a Horizontal Curve.

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2. Examples of Curves

In construction surveying, curves have to be set out on the ground for a variety of purposes:

1. Curve may form the major part of a roadway,

2. Curve may form a kerb line at a junction, or3. Curve may form the shape of an ornamental

rose bed in a town centre.

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What are Tangents to a Circle?What are Angle of Deflection?What are Angle of Curvature?Cyclic QuadrilateralIsosceles TrianglesCongruent TrianglesWhat are the angles encountered & what are their

relationship?

5


6

T1

T2

I

θ

Tangent

Angle of Deflection ,Angle of Deviation orAngle of Intersection.

O


7

O

T1

T2

I

θ

θ

Cyclic Quadrilateral(Q T1 I T2 )


8

OO

T1

T2

I

θ

θ

ΔOT1T2 is an isosceles triangle.


9

O

T1

T2

I

θ

θ

ΔO T1 I and ΔO I T2 are congruent triangles

O


10

O

T1

T2

I

θ

θβ

θ + β = 180 。


11

O

T1

T2

I

θ

θ/2O

θ/2

4. Curve Elements

1. Straights: What are the Straights?2. Intersection Point, I.P.?3. Angle of Deviation (Angle of Deflection, or

Angle of Intersection).4. Radius of Curve

Usually a multiple of 50 m.5. Tangent Length6. Long Chord7. Major Offset

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13

OO

T1

T2

I

θ

θ

The Straights mean the Tangents

Intersection point, i.e. I.P.

Radius of Curve

Long Chord


14

OO

T1

T2

I

θ

θ

Tangent Length

Major Offset

5. Designation of CurvesIn UK, curves are designated by the length of the

radius.The radius is usually in multiples of 50 m.

Curves can also be designated by the degrees subtended at the centre by an arc 100 m long.The Degree of Curvature is given as a No. of whole

degrees.The Degree of Curvature may be measured in Degrees

or Radians.

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6. Setting Up Procedures (Calculations)

1. Small Radius Curves:

(a) Finding the Centre

(b) Offset from the tangent

2. Large Radius Curves:

(a) Setting by Tangential Angles

(b) Using 2 Theodolites

(c) Setting Out by Co-ordinates16

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O

T1 I

T2

O

I T1

T2

θ αRR

R

Minor Road

Major Road

C

CCH 0 m

(of minor road)

Fig. 12.10Small Radius Curveby finding the centre.

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I

T1T2

BC

D

c1 c2

α1 α2

c3

c4

α3

α4θ/2

θ

θ

O

Fig. 12.15(a)

Large Radius Curve: Setting by Tangential Angles

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T1

B

O

X

α1

α1

c1

Fig. 12.15 (b)

Large Radius Curve: Method 2(b)- Using Two Theodolites

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AB

T1 T2

CID

α1

α2 α2

α1

Fig. 12.17

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A

I

O

T1

T2 CH 754.5°

B

CH 0105.260 E352.150 N

CH 20

CH 30

CH 40 (X)

Tangent Length22.510 m

CH 60 (Y) CH 80 CH 100

SSurvey Station.

148.500 E370.010 N

R = 572.960 m

Fig. 12.18

WCB 40° 00’ 00”

WCB44° 30’ 00”

1°

2°1.5°

4.5°

Large radius Curve: by Co-ordinates

6.1- Small Radius Curves• Method 1: Finding the Centre.

In Fig. 12.10, kerbs have to be laid at the roadway junction.

Consider the right-hand curve. The deviation angle α is measured from the plan

and the tangent lengths I T1 and I T2 (= R tan α/2) calculated.

The procedure for setting the curve is then as follows: -------

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O

T1 I

T2

O

I T1

T2

θ αRR

R

Minor Road

Major Road

C

CCH 0 m(of minor road)

Fig. 12.10

Procedure for setting out:

1. From I, measure back along the straights the distance I T1 and I T2.

2. Hammer in pegs at those points & mark the exact positions of T1 and T2 by nails.

3. Hook a steel tape over each nail and mark the centre O at the point where the tapes intersect when reading R. Hammer in a peg and mark the centre exactly with a nail.

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Procedure for setting out:

4. Any point on the curve is established by hooking the tape over the peg O and swinging the radius.

This method is widely used where the radius of curvature is less than 30 m.

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*Curve Composition*

In setting out large radius curves, or in some cases small radius curves, pegs are set at regular intervals around the curve.

The interval is commonly 10 or 20 m & is measured as a RUNNING CHAINAGE, from the zero chainage point (CH 0 m) of the road system.

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Curve Composition

It would be very unlikely that either tangent point of the curve would coincide with a chainage which is at an exact tape length!

So what shall we do then ?

Refer to Fig. 12.14.27

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T1 T2

I

A

B

100120140 160

180 200 220

CH 126.000 CH 216.757

400 m Radius

CH 171.574

…to CH 0 point

13°

Initial sub-

chord

StandardSub-

chordsFinal Sub-chord

Fig. 12.14

Fig. 12.14• The straights AI & IB deviate by 13° at I, the I.P.

where the chainage is 171.574 m. Tangent lenghts IT1 & IT2

So chainage T1

Curve length

So chainage T2

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Fig. 12.14

• The last peg on the straight, measured at 20 m intervals from A, occurs at CH 120 m.

• So the 1st peg on the curve, at CH 140 m, lies at a distance of:

140 (-) 126 = 14 m from tangent point T1 . This short chord is called the initial sub-

chord.30

Fig. 12.14

• Thereafter, pegs are placed at standard chord intervals of 20 m occur at CH 160, 180 & 200 m.

• The final tangent point T2 is reached at 216.757 m; So the final chord is:

216.757 (-) 200.000 = 16.757 m This short chord is called the final sub-

chord. 31

Fig. 12.14: Summary

• Summarizing, the chord composition is derived as follows:

1) Chainage T1

2) CH at 1st peg on curve 3) So initial sub-chord4) CH at last peg on curve 5) So No. of standard chords 6) Chainage T2 7) So final sub-chord

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Fig. 12.14: Summary

• In setting out large radius curve, the chords must be almost equal to the arcs that they subtend.

• An accuracy of about 1 part in 10,000 is obtainable, provided the chord length does not exceed 1/20th of the length of the radius, i.e.

< R/20. 33

Method of Setting out Large Radius Curve

Method 2 (a)- Setting by Tangential Angles:

This is the common method of setting out large

radius curves when accuracy is required.

It uses tape and theodolite.

In Fig. 12.15, the tangent point T1 at the

beginning of the curve has been established.

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Setting by Tangential Angles…

BC and CD are equal standard chords, c2 and c3

chosen such that their length is < R/20.

TB is the Initial Sub-Chord, c1 is shorter than c2 &

c3 because the CH of T1 is irregular.

c4 is the Final Sub-Chord & is shorter than c2 & c3

too. 35

Setting by Tangential Angles:

• Tangential Angles: In Fig. 12.15, angles α1, α2, α3 & α4 are the angles

by which the curve deflects to the right or left. They are the tangential angles which are also

known as chord angles or deflection angles. They are more commonly known as the

Deflection Angles. Their values must be calculated in order to set

out the curve.36

Calculation of Deflection Angles

• In Fig. 12.15 (a), angle IT1B is the angle between T1I

& chord T1B.

• Angle T1OB is the angle at the centre subtended by

chord T1B.

So angle IT1B = ½ angle T1OB = α1

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I

T1 T2

BC

Dc1 c2

α1 α2

c3

c4

α3

α4

θ/2

θ

θ

O

Fig. 12.15(a)

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T1

B

O

X

α1

α1

c1

Fig. 12.15 (b)

Refer to Fig. 12.15(b):

• OX is the perpendicular bisector of chord T1B.

• So, angle T1OX = angle XOB = α1

• In triangle T1OX,

sin T1OX = T1X / T1O

= {c1 / 2} / R

= c1 / 2R40

Refer to Fig. 12.15(b)…..

• The value of any deflection angle (α1, α2, α3 &

α4) can similarly be found & the formula can

be written in general terms as:

sin α = c/2R ……(1)

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Eg. 9 (setting by tangential angles):

1. Two straights AI and IB have bearings of 80° & 110° respectively.

2. They are to be joined by a circular curve of 300 m radius.

3. The chainage of intersection point I is 872.485 m (Fig. 12.16)

4. Calculate the data for setting out the curve by 20 m standard chords.

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Fig. 12.16

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30°

O

N

AB

I

T1 T2

N 80° E S 70° ERadius= 300 m

30°

Table 12.2

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Chord Chord No.No.

LengthLength

(m)(m)ChainagChainag

ee

(m)(m)

DeflectioDeflectionn

AngleAngle

TangentiTangential Angleal Angle

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15. Horizontal Curve Ranging.r1 Student (1)