Curves

All the basic deatils of curves

1. PREPARED BY: Asst. Prof. GAURANG PRAJAPATI CIVIL DEPARTMENT CURVES Mahatma Gandhi Institute Of Technical Education & Research Centre, Navsari (396450) SURVEYING 4 TH 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 Horizontal curve 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 R 1 has center O 1 and the arc of radius R 2 has center O 2 . 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 (AT 1 ) previous to the curve is called the back tangent or first tangent. Forward tangent: The tangent (T 2 B) following the curve is called the forward tangent or second tangent. Point of Intersection (P.I.): If the tangents AT 1 and AT 2 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 T 1 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 (T 2 ) 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. T 1 T 2 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. By radius (R) 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 .   When D is small, may be taken equal to .   CURVES SURVEYING 3140601 12

13. RELATION BETWEEN RADIUS AND DEGREE OF CURVE (where, D is in degree)     R   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. ( where, D is in degree)     R   CURVES SURVEYING 3140601 14

15. RELATION BETWEEN RADIUS AND DEGREE OF CURVE For 30 m arc :   OR   CURVES SURVEYING 3140601 15

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

17. ELEMENTS OF SIMPLE CIRCULAR CURVE: In the figure, T 1 = P.C. = first point of tangency = Point of curve T 2 = 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: LENGTH OF CURVE : If curve is designated by Radius : l = length of arc T 1 C T 2 = R × Δ (where Δ is in radian) = (where Δ is in degree)   If curve is designated by Degree : Length of arc = 20 m Length of curve = l = m (where D = Degree of curve for 20 m arc)   CURVES SURVEYING 3140601 18

19. ELEMENTS OF SIMPLE CIRCULAR CURVE: 2. TANGENT LENGTH (T): VT 1 and VT 2 are the tangent lengths. T = VT 1 = VT 2 = tangent length From Δ VT 1 O, tan = = ( VT 1 O and VT 2 O are the right angles) T = R tan   3. LENGTH OF CHORD (L): In the figure T 1 T 2 is a long chord. Length of long chord = L = T 1 T 2 = 2 T 1 D From Δ T 1 DO, sin = = L = 2 T 1 D L = 2R sin   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 VT 1 O, cos = = OV = = R sec E = OV – OC = R sec – R E = R ( sec – 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 T 1 DO , cos = = OD = R cos M = OC – OD = R - R cos M = R (1 - cos )   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: Linear method 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: By offsets from the long chord. By successive bisection of arcs or chords: By offsets from the tangents. CURVES SURVEYING 3140601 22

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

24.   METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE R= Radius of the curve o = Mid ordinate x = ordinate at distance x from the mid point of the chord T 1 and T 2 = Tangent points L = Length of long chord To obtain equation for O : From triangle OT 1 D: OD = O O = R – OD O O = R –   CURVES SURVEYING 3140601 24

25.   METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE In order to calculate ordinate O x to any point E, draw the line EE 1 , parallel to the long chord T 1 T 2 . Join EO to cut the long chord in G . O x = EF = E 1 D = E 1 O – DO (DO = R - O O ) = R – - (R - O O ) = R – - (R - O O )   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 T 1 T 2 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 ) OR CD = O O - R – Join T 1 C and T 2 C and bisect them at D 1 and D 2 respectively. At D 1 & D 2 set out perpendicular offsets C 1 D 1 = C 2 D 2 = (1- cos ) and obtain points C 1 and C 2 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: Radial offsets Perpendicular offsets Radial offsets: Let, O x = Radial offset DE at any distance x from T 1 along the tangent. T 1 D = x From ∆ T 1 DO, R + O x = O x = – R   CURVES SURVEYING 3140601 28

29.   METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE ii. Perpendicular offsets: O x = offset perpendicular to the tangent DE = O x T 1 D = x, measured along the tangent. From ∆ EE 1 O, E 1 O 2 = EO 2 - E 1 E 2 (T 1 O - T 1 E 1 ) 2 = EO 2 - E 1 E 2 (R - O x ) 2 = R 2 – x 2 (R - O x ) = O x = R –   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: Rankine method of tangential angles Two theodolite method Tacheometric method CURVES SURVEYING 3140601 30

31.   METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE 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 𝑇 1 and 𝑇 2 . Set the theodolite at P.C. 𝑇 1 . 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 T 1 A. With zero end of the tape pointed at T 1 and an narrow held at a distance T 1 A=C 1 , swing the tape around T 1 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 T 1 B. With the zero end of the tape pinned at A and an arrow held at a distance AB = C 2 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 T 2 is reached. Join the points T 1 ,A,B,C….T 2 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. V𝑇 1 A = ∆ 1 = Deflection angle for A. A𝑇 2 T is the angle subtended by the chord T 1 A in the opposite segment. A𝑇 2 𝑇 2 = VT 1 A = ∆ 1 V𝑇 1 B = ∆ 2 = 𝑇 1 𝑇 2 B   CURVES SURVEYING 3140601 35

36.   METHODS OF SETTING OUT SIMPLE CIRCULAR CURVE PROCEDURE: Setup one theodolite at P.C. (𝑇 1 )and the other at P.T. (𝑇 2 ) Clamp both plates of each transit to zero reading. With zero reading, direct the line of sight of the transit at 𝑇 1 towards V. Similarly direct the line of sight of the other transit at 𝑇 2 towards 𝑇 1 . 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 𝑇 1 will be along 𝑇 1 A and the line of sight of theodolite at 𝑇 2 will be along 𝑇 2 A. 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. (𝑇 1 ) 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 𝑇 1 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 𝑇 1 A. Direct the staff man to move in the direction 𝑇 1 A till the calculated staff intercept 𝑆 1 is obtained. The staff is generally held vertical. First point A is fixed. Set the deflection angle ∆ 2 directing the line of sight along 𝑇 1 B. 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: Weight of the vehicle (W) Centrifugal force (P) The centrifugal force is given by, 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/sec 2 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 = 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 = 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 Where, p = Polar distance of any point = 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 (+g 1 ) followed by a down grade (-g 2 ) An upgrade (+g 1 ) followed by another upgrade (+g 2 ). (g 1 >g 2 ) A down grade (-g 1 ) followed by another down grade (-g 2 ). (g 2 >g 1 ) A plane surface followed by down grade (-g 1 ) CURVES SURVEYING 3140601 48

49. TYPES OF VERTICAL CURVES 2. Valley Curve (Concave curve): It is provided in following situations: A down grade (-g 1 ) followed by a upgrade (+g 2 ). A down grade (-g 1 ) followed by another down grade (-g 2 ). (g 1 >g 2 ) An upgrade (+g 1 ) followed by another upgrade (+g 2 ). (g 2 >g 1 ) A plane surface followed by upgrade (+g 1 ) 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) = = Where g 1 , g 2 = Grades in % r = Rate of change of grade (%)   CURVES SURVEYING 3140601 50