Surveying and leveling

1. PREPARED BY:
Asst. Prof. GAURANG PRAJAPATI
CIVIL DEPARTMENT
CURVES
Mahatma Gandhi Institute Of
Technical Education
& Research Centre, Navsari (396450)
SURVEYING
4TH SEMESTER
CIVIL ENGINEERING

2. INTRODUCTION
• Curves are generally used on highways and railways where it is necessary to change the
direction of motion.
• A curve connect two straight lines.
• A curve is always tangential to the two straight directions.
• The two straight lines connected by a curve are called tangents.
 TYPE OF CURVES:
CURVES
SURVEYING
3140601
2
1) Horizontal curve 1) Vertical curve
A) Circular curve B) Transition curve i) Summit curve
i) Simple curve i) cubic parabola ii) Valley curve
ii) Compound curve ii) Spiral curve
iii) Reverse curve iii) Lemniscate

3. TYPES OF CIRCULAR CURVES
• There are three types of circular curves.
1. Simple curve
2. Compound curve
3. Reverse curve
CURVES
SURVEYING
3140601
3

4. TYPES OF CIRCULAR CURVES
1. Simple Curve:
• A simple circular curve consists of a single arc of the circle.
• It is tangential to both the straight lines.
2. Compound Curve:
• A compound curve consists of two or more simple arcs.
• The simple arcs turn in the same direction with their centers of curvature on the
same side of the common tangent.
• In figure, an arc of radius R1 has center O1 and the arc of radius R2 has center O2.
3. Reverse Curve:
• A reverse curve consists of two circular arcs which have their centers of
curvature on the opposite side of the common tangent.
• The two arcs turn in the opposite direction.
• Reverse curves are provided for low speed roads and railways.
CURVES
SURVEYING
3140601
4

5. TYPES OF CIRCULAR CURVES
CURVES
SURVEYING
3140601
5

6. DEFINITIONS AND NOTATIONS FOR SIMPLE CURVE
CURVES
SURVEYING
3140601
6

7. DEFINITIONS AND NOTATIONS FOR SIMPLE CURVE
 Back tangent:
The tangent (AT1) previous to the curve is called the back tangent or first tangent.
 Forward tangent:
The tangent (T2B) following the curve is called the forward tangent or second tangent.
 Point of Intersection (P.I.):
If the tangents AT1 and AT2 are produced they will meet in a point, called the point of
intersection. It is also called vertex (V).
 Point of curve (P.C.):
It is the beginning point T1 of a curve. At this point the alignment changes from a tangent to
a curve.
 Point of tangency (P.T.):
The end point of a curve (T2) is called the point of tangency.
 Intersection angle (ф):
The angle AVB between tangent AV and tangent VB is called intersection angle.
 Deflection angle (Δ):
The angle at P.I. between the tangent AV produced and VB is called the deflection angle.
CURVES
SURVEYING
3140601
7

8. DEFINITIONS AND NOTATIONS FOR SIMPLE CURVE
 Tangent distance:
It is the distance between P.C. to P.I. It is also the distance between P.I. to P.T.
 External distance (E):
It is the distance from the mid. point of the curve to P.I.
It is also called the apex distance.
 Length of curve (l):
It is total length of curve from P.C. to P.T.
 Long chord:
It is chord joining P.C. to P.T. T1 T2 is a long chord.
 Normal chord:
A chord between two successive regular stations on a curve is called normal chord.
Normally, the length of normal chord is 1 chain (20 m).
 Sub chord:
The chord shorter than normal chord (shorter than 20 m ) is called sub chord.
CURVES
SURVEYING
3140601
8

9. DEFINITIONS AND NOTATIONS FOR SIMPLE CURVE
 Versed sine:
The distance between mid. point of long chord (D) and the apex point C. is called
versed sine.
It is also called mid-ordinate (M).
 Right hand curve:
If the curve deflects to the right of the direction of the progress of survey, it is called
right-hand curve.
 Left hand curve:
If the curve deflects to the left of the direction of the progress of survey, it is called left
hand curve.
CURVES
SURVEYING
3140601
9

10. DESIGNATION OF CURVE
The sharpness of the curve is designated by two ways.
1. By radius (R)
2. By degree of curvature (D)
1. By radius (R):
• In this method the curve is known by the length of its radius ®.
For example,
• 200 m curve means the curve having radius 200 m.
• 6 chain curve means the curve having radius equal to 6 chain.
• This method is used in England.
CURVES
SURVEYING
3140601
10

11. DESIGNATION OF CURVE
2. By degree of curvature (D)
In this method the curve is designated by degree. The degree of curvature can be
defined by two ways:
A. Chord definition:
• The angle subtended at the center of curve by a chord of 20 m is called degree of
curvature.
e.g.
• If an angle subtended at the center of curve by a chord of 20 m is 5˚, the curve is
called 5˚ curve.
B. Arc definition:
• The angle subtended at the center of curve by an arc of 20 m length, is called
degree of curve.
• This system is used in America, Canada, India, etc.
CURVES
SURVEYING
3140601
11

12. RELATION BETWEEN RADIUS AND DEGREE OF CURVE
1. By chord definition:
• The angle subtended at the center of curve by a chord of 20 m is called degree of curve.
R = radius of curve
D = degree of curvature
PQ = 20 m = length of chord
From triangle PCO.
sin
𝐷
2
=
10
𝑅
∴ 𝑅 =
10
sin
𝐷
2
• When D is small, sin
𝐷
2
may be taken equal to
𝐷
2
.
∴ sin
𝐷
2
=
𝐷
2
CURVES
SURVEYING
3140601
12

13. RELATION BETWEEN RADIUS AND DEGREE OF CURVE
∴ 𝑅 =
10
𝐷
2
×
𝜋
180
(where, D is in degree)
=
10 × 360
𝐷 × 𝜋
=
1145.92
𝐷
R =
𝟏𝟏𝟒𝟔
𝑫
CURVES
SURVEYING
3140601
13

14. RELATION BETWEEN RADIUS AND DEGREE OF CURVE
2. By Arc definition:
The angle subtended at the center of curve by an arc of 20 m length is called degree of
curve.
2𝜋𝑅
360
=
20
𝐷
(where, D is in degree)
∴ 𝑅 =
20 × 360
2𝜋𝐷
=
1145.92
𝐷
R =
𝟏𝟏𝟒𝟔
𝑫
CURVES
SURVEYING
3140601
14

15. RELATION BETWEEN RADIUS AND DEGREE OF CURVE
• For 30 m arc:
2𝜋𝑅
360
=
30
𝐷
∴ 𝐷 =
1718.9
𝑅
OR
∴ 𝑹 =
𝟏𝟕𝟏𝟖. 𝟗
𝑫
CURVES
SURVEYING
3140601
15

16. ELEMENTS OF SIMPLE CIRCULAR CURVE:
CURVES
SURVEYING
3140601
16

17. ELEMENTS OF SIMPLE CIRCULAR CURVE:
In the figure,
T1 = P.C. = first point of tangency = Point of curve
T2 = P.T. = second point of tangency
V = P.I. = Point of Intersection
Δ = Deflection Angle
Ф = Intersection Angle
R = Radius of curve
CD = Mid ordinate (M)
CURVES
SURVEYING
3140601
17

18. ELEMENTS OF SIMPLE CIRCULAR CURVE:
1. LENGTH OF CURVE:
• If curve is designated by Radius:
l = length of arc T1C T2
= R × Δ (where Δ is in radian)
=
𝑹 𝚫 𝛑
𝟏𝟖𝟎
(where Δ is in degree)
• If curve is designated by Degree:
Length of arc = 20 m
∴ Length of curve = l =
𝟐𝟎 𝚫
𝟏𝟖𝟎
m (where D = Degree of curve for 20 m arc)
CURVES
SURVEYING
3140601
18

19. ELEMENTS OF SIMPLE CIRCULAR CURVE:
2. TANGENT LENGTH (T):
VT1 and VT2 are the tangent lengths.
T = VT1 = VT2 = tangent length
From Δ VT1O,
tan
Δ
2
=
V
𝑇
1
O
𝑇
1
=
T
𝑅
(∠VT1O and ∠VT2O are the right angles)
∴ T = R tan
𝚫
𝟐
•
3. LENGTH OF CHORD (L):
In the figure T1T2 is a long chord.
Length of long chord = L = T1T2 = 2 T1D
From Δ T1DO,
sin
Δ
2
= 𝑇
1
𝐷
𝑇
1
𝑂
= 𝑇
1
𝐷
𝑅
∴ L = 2 T1D
∴ L = 2R sin
𝚫
𝟐
CURVES
SURVEYING
3140601
19

20. ELEMENTS OF SIMPLE CIRCULAR CURVE:
4. EXTERNAL DISTANCE (E):
In the figure, VC is an external distance.
External distance = E = VC = OV – OC
From Δ VT1O,
cos
Δ
2
=
O
𝑇
1
OV
=
R
OV
∴ OV =
R
cos
Δ
2
= R sec
Δ
2
∴ E = OV – OC
= R sec
Δ
2
– R
∴ E = R (sec
𝚫
𝟐
– 1)
CURVES
SURVEYING
3140601
20

21. ELEMENTS OF SIMPLE CIRCULAR CURVE:
5. MID ORDINATE (M):
In the figure, CD is the mid ordinate.
It is also known as versed sine.
Mid ordinate = M = CD = OC – OD
From Δ T1DO,
cos
Δ
2
=
OD
O
𝑇
1
=
OD
𝑅
∴ OD = R cos
Δ
2
∴ M = OC – OD
= R - R cos
Δ
2
∴ M = R (1 - cos
𝚫
𝟐
)
CURVES
SURVEYING
3140601
21

22. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
Based on the instruments used in setting out the curves on the ground there are two
methods:
1. Linear method
2. Angular method
1. Linear methods:
• In these methods only tape or chain is used for setting out the curve. Angle
measuring instrument are not used.
• These methods are used where a high degree of accuracy is not required and the
curve is not short.
• Main linear methods are:
A. By offsets from the long chord.
B. By successive bisection of arcs or chords:
C. By offsets from the tangents.
CURVES
SURVEYING
3140601
22

23. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
A. By offsets from the long chord:
CURVES
SURVEYING
3140601
23

24. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
R= Radius of the curve
0o = Mid ordinate
0x = ordinate at distance x from the mid point of the chord
T1 and T2 = Tangent points
L = Length of long chord
To obtain equation for 0O:
From triangle OT1D:
OD = 𝑅2 −
𝐿
2
2
OO = R – OD
OO = R – 𝑹𝟐 −
𝑳
𝟐
𝟐
CURVES
SURVEYING
3140601
24

25. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
In order to calculate ordinate Ox to any point E, draw the line EE1, parallel to the long
chord T1 T2. Join EO to cut the long chord in G.
Ox = EF = E1D
= E1O – DO (DO = R - OO)
= R – 𝐸𝑂2 − 𝐸𝐸1
2
- (R - OO)
= R – 𝑹𝟐 − 𝒙𝟐 - (R - OO)
CURVES
SURVEYING
3140601
25

26. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
B. By successive bisection of arcs or chords:
CURVES
SURVEYING
3140601
26

27. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
• Join the tangent points T1 T2 and bisect the long chord at D.
• Erect perpendicular DC at D equal to the mid ordinate (M)
Mid ordinate ,
M = CD = R (1 - cos
𝜟
𝟐
)
OR
CD = OO - R – 𝑹𝟐 −
𝑳
𝟐
𝟐
• Join T1C and T2C and bisect them at D1 and D2 respectively.
• At D1 & D2 set out perpendicular offsets C1D1 = C2D2 = (1- cos
∆
4
) and obtain points C1
and C2 on the curve.
• By the successive bisection of these chords more points may be obtained.
CURVES
SURVEYING
3140601
27

28. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
C. By offsets from the tangents:
• The offsets from the tangents can be of two types:
i. Radial offsets
ii.Perpendicular offsets
i. Radial offsets:
Let, Ox = Radial offset DE at
any distance x from T1 along
the tangent.
T1D = x
From ∆ T1DO,
R + Ox = 𝑅2 + 𝑥2
Ox = 𝑹𝟐 + 𝒙𝟐 – R
CURVES
SURVEYING
3140601
28

29. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
ii. Perpendicular offsets:
Ox = offset perpendicular to the tangent
DE = Ox
T1D = x, measured along the tangent.
From ∆ EE1O,
E1O2 = EO2 - E1 E2
(T1O - T1 E1)2 = EO2 - E1 E2
(R - Ox)2 = R2 – x2
(R - Ox) = 𝑅2 − 𝑥2
Ox = R – 𝑹𝟐 − 𝒙𝟐
CURVES
SURVEYING
3140601
29

30. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
2. Angular method:
• In this method instruments like theodolite are used for setting out the curves.
Sometimes chain or tape is also used with the theodolite.
• These methods are used when the length of curve is large.
• These methods are more accurate than Linear methods.
• The Angular methods are:
1. Rankine method of tangential angles
2. Two theodolite method
3. Tacheometric method
CURVES
SURVEYING
3140601
30

31. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
1. Rankine method of tangential angles:
• This method is most frequently used for setting out the circular curves of large
radius and considerable length.
• This method is useful for setting out the curves for Railway, Highway and
Expressway with more accuracy.
• In this method, only one theodolite is used, hence it is called One Theodolite
Method.
• Rankine’s Principle:
“A deflection angle to any point on the curve is the angle at P.C. between the back
tangent and the chord from P.C. to that point.”
CURVES
SURVEYING
3140601
31

32. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
CURVES
SURVEYING
3140601
32

33. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
PROCEDURE:
• Set out 𝑇1 and 𝑇2.
• Set the theodolite at P.C. 𝑇1.
• With both the plates clamped to zero, direct the theodolite to bisect the point of
intersection (V).
• Release the upper clamp screw and set angle ∆1 on the vernier. The line of sight is thus
directed along the chord T1A.
• With zero end of the tape pointed at T1 and an narrow held at a distance T1A=C1, swing
the tape around T1 till the arrow is bisected by the cross hairs. Thus, the first point A is
fixed.
• Now Release the upper plate and set the second deflection angle ∆2 on the vernier so
that the line of sight is directed along T1B.
• With the zero end of the tape pinned at A and an arrow held at a distance AB = C2
swing the tape around A till the narrow is bisected by the cross hairs. Thus, the second
point B is fixed.
• Repeat the steps 6,7 till the last point T2 is reached.
• Join the points T1,A,B,C….T2 to obtain the required curve.
CURVES
SURVEYING
3140601
33

34. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
2. Two theodolite method:
CURVES
SURVEYING
3140601
34

35. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
• In this method, two theodolites are used, one at P.C. and other at P.T.
• In this method, tape/chain is not required.
• This method is used when the ground is unsuitable for chaining.
• In this method two theodolites are used one at P.C and the other at P.T.
• In this method tape/chain is not required. This method used when the ground is
unsuitable for chaining.
∠V𝑇1 A = ∆1= Deflection angle for A.
∠A𝑇2T is the angle subtended by the chord T1A in the opposite segment.
∠A𝑇2𝑇2 = ∠VT1A = ∆1
∠V𝑇1B = ∆2 = ∠𝑇1𝑇2 B
CURVES
SURVEYING
3140601
35

36. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
PROCEDURE:
• Setup one theodolite at P.C. (𝑇1)and the other at P.T. (𝑇2)
• Clamp both plates of each transit to zero reading.
• With zero reading, direct the line of sight of the transit at 𝑇1 towards V. Similarly
direct the line of sight of the other transit at 𝑇2 towards 𝑇1. Vernier A of both the
theodolites will show zero reading.
• Set the reading of each of the transits to the deflection angle for the first point A
equal to ∆1. The line of sight of theodolite at 𝑇1 will be along 𝑇1A and the line of
sight of theodolite at 𝑇2 will be along 𝑇2A.
• Move a ranging rod or an arrow in such a way that it is bisected simultaneously by
cross hairs of both the instruments. Thus, point A is fixed.
• Now, to fix the second point B, set reading ∆2 on both the instruments and bisect
the ranging rod.
• Repeat the above steps to obtain other points.
CURVES
SURVEYING
3140601
36

37. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
3. Tacheometric method:
• In this method, the angular and linear measurements are made by using a
tacheometer.
• This method is less accurate than Rankine’s method but the advantage is that,
chaining is completely eliminated.
• In this method, a point on the curve is fixed by the deflection angle from, the rear
tangent and by using tacheometrically, the distance of that point from P.C. (𝑇1) and
not from the preceding point as in Rankine’s method.
• Thus, each point is fixed independently and the error in setting out is not carried
forward.
CURVES
SURVEYING
3140601
37

38. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
CURVES
SURVEYING
3140601
38

39. METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE
PROCEDURE:
• Set the tacheometer at 𝑇1 and sight the point of intersection (V) when the reading
is zero.
• Set the deflection angle ∆1 on the vernier, thus directing the line of sight along 𝑇1A.
• Direct the staff man to move in the direction 𝑇1A till the calculated staff intercept
𝑆1 is obtained. The staff is generally held vertical. First point A is fixed.
• Set the deflection angle ∆2 directing the line of sight along 𝑇1B. Move the staff
backward or forward untill the staff intercept 𝑆2 is obtained thus fixing the point
B.
• Similarly, other points are fixed.
CURVES
SURVEYING
3140601
39

40. TRANSITION CURVES
• When a vehicle moves on a curve, there are two forces acting:
1. Weight of the vehicle (W)
2. Centrifugal force (P)
The centrifugal force is given by,
𝑃 =
𝑊 𝑣2
𝑔𝑅
Where, P = Centrifugal force in Kg or N
W = Weight of the vehicle Kg or N
V = Speed of the vehicle, m/sec
g = Acceleration due to gravity, m/sec2
R = Radius of the curve
CURVES
SURVEYING
3140601
40

41. TRANSITION CURVES
• The centrifugal force (P) is inversely proportional to the radius of the curve (R).
• As the radius decrease, centrifugal force increases. Straight road has infinite radius
of curvature. Hence, centrifugal force on vehicles moving on straight road is zero.
• When a vehicle enters from straight road to the curve, its radius changes rom
infinite to R, resulting in sudden centrifugal force P on the vehicle. It causes the
vehicle to sway outwards. If this exceeds a certain value the vehicle may overturn.
• To avoid these effects, a curve of changing radius must be introduced between the
straight and the circular curve. Such a curve, is known as transition curve.
CURVES
SURVEYING
3140601
41

42. REQUIREMENTS OF A TRANSITION CURVE
• It should be tangential to the straight.
• It should meet the circular curve tangentially.
• Its curvature should be zero at the origin on straight.
• Its curvature at the junction with the circular curve should be the same as that of
the circular curve.
• The rate of increase of curvature along the transition should be the same as that of
increases of super elevation.
• The length should be such that full super-elevation (Cant) is attained at the
junction with the circular curve.
CURVES
SURVEYING
3140601
42

43. PURPOSES OF PROVIDING TRANSITION CURVE
The objects of providing transition curve are:
• To accomplish gradually the transition from the straight to the circular so that the
curvature is increased gradually from zero to a specified value.
• To provide a medium for the gradual introduction of super elevation.
• To provide extra widening on the circular curve gradually.
CURVES
SURVEYING
3140601
43

44. TYPES OF TRANSITION CURVES
There are mainly three types of transition curves:
1. Cubic Spiral
2. Cubic Parabola
3. The lemniscates Curve
1. Cubic Spiral
• The Cubic Spiral is best suited on Railways.
• The equation of a cubic spiral is,
Y =
𝒍𝟑
𝟔𝑹𝑳
Where,
y = perpendicular offset from the tangent
l = distance measured along the curve
R = Radius of the circular curve
L = Length of the transition curve
CURVES
SURVEYING
3140601
44

45. TYPES OF TRANSITION CURVES
2. Cubic Parabola
• This type of curve is used in railway line construction.
• The equation of a cubic parabola is,
Y =
𝒙𝟑
𝟔𝑹𝑳
Where,
y = perpendicular offset from the tangent
x = distance measured along the tangent
R = Radius of the circular curve
L = Length of the transition curve
CURVES
SURVEYING
3140601
45

46. TYPES OF TRANSITION CURVES
3. The lemniscates Curve
• Instead of providing intermediate circular curve, when entire curve is provided in
the form of transition curve, it is known as Lemniscate.
• This type of curve is used on highways.
• The equation of Bernouli’s lemniscates curve is,
P =k 𝒔𝒊𝒏 𝟐𝜶
Where,
p = Polar distance of any point
𝛼 = Polar deflection angle for any point
k = constant
CURVES
SURVEYING
3140601
46

47. VERTICAL CURVES
• It Is provided when there is a sudden change in gradient of a highway or a railway.
• It is provided when a highway or railway crosses a ridge or a valley.
• It smoothens the change in gradient so that there is no discomfort to the
passengers travelling in vehicles.
 Advantages:
• Due to vertical curve, change in gradient is gradual.
• It improves the appearance of roads.
• Road and railway journey becomes comfortable.
CURVES
SURVEYING
3140601
47

48. TYPES OF VERTICAL CURVES
1. Summit Curve (Convex curve)
2. Valley Curve (Concave curve)
1. Summit Curve (Convex curve):
It is provided in following situations:
• An upgrade (+g1) followed by a down grade (-g2)
• An upgrade (+g1) followed by another upgrade (+g2). (g1>g2)
• A down grade (-g1) followed by another down grade (-g2). (g2>g1)
• A plane surface followed by down grade (-g1)
CURVES
SURVEYING
3140601
48

49. TYPES OF VERTICAL CURVES
2. Valley Curve (Concave curve):
It is provided in following situations:
• A down grade (-g1) followed by a upgrade (+g2).
• A down grade (-g1) followed by another down grade (-g2). (g1>g2)
• An upgrade (+g1) followed by another upgrade (+g2). (g2>g1)
• A plane surface followed by upgrade (+g1)
CURVES
SURVEYING
3140601
49

50. TYPES OF VERTICAL CURVES
 Length of a Vertical Curve:
• The length of vertical curve can be obtained by dividing the algebraic difference of
the two grades by the rate of change of grade.
• Length of curve (L) =
𝑇𝑜𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑔𝑟𝑎𝑑𝑒
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑔𝑟𝑎𝑑𝑒
=
𝑔2− 𝑔1
𝑟
Where g1 , g2 = Grades in %
r = Rate of change of grade (%)
CURVES
SURVEYING
3140601
50