15. horizontal curve ranging.r1 student (1)
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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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3. Fundamental Geometrical Theorems
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?
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3. Fundamental Geometrical Theorems
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T1
T2
I
θ
Tangent
Angle of Deflection ,Angle of Deviation orAngle of Intersection.
O
3. Fundamental Geometrical Theorems
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O
T1
T2
I
θ
θ
Cyclic Quadrilateral(Q T1 I T2 )
3. Fundamental Geometrical Theorems
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OO
T1
T2
I
θ
θ
ΔOT1T2 is an isosceles triangle.
3. Fundamental Geometrical Theorems
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O
T1
T2
I
θ
θ
ΔO T1 I and ΔO I T2 are congruent triangles
O
3. Fundamental Geometrical Theorems
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O
T1
T2
I
θ
θβ
θ + β = 180 。
3. Fundamental Geometrical Theorems
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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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4. Fundamental Geometrical Theorems
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OO
T1
T2
I
θ
θ
The Straights mean the Tangents
Intersection point, i.e. I.P.
Radius of Curve
Long Chord
4. Fundamental Geometrical Theorems
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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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