Slide 1

AN EDUSAT LECTURE ON

CURVESAN INTRODUCTION LECTURE ONCURVEST1ABCT2OFig. 1 . A CURVECURVESCurves are regular bends provided in the lines of communication like roads, railways and canals etc. to bring about gradual change of direction. 2CURVEST1ABCT2OFig. 2. A CURVECURVESThey enable the vehicle to pass from one path on to another when the two paths meet at an angle. They are also used in the vertical plane at all changes of grade to avoid the abrupt change of grade at the apex.

3CURVESHORIZONTAL CURVESCurves provided in the horizontal plane to have the gradual change in direction are known as horizontal curves.VERTICAL CURVESCurves provided in the vertical plane to obtain the gradual change in grade are called as vertical curves. Curves may be circular or parabolic. Curves are generally arcs of parabolas. 4CURVESNEED OF PROVIDING CURVESCurves are needed on Highways, railways and canals for bringing about gradual change of direction of motion. They are provided for following reasons:-i) To bring about gradual change in direction of motion.ii) To bring about gradual change in grade and for good visibility.5CURVESNEED OF PROVIDING CURVES Contdiii) To alert the driver so that he may not fall asleep.

iv) To layout Canal alignment.

v) To control erosion of canal banks by the thrust of flowing water in a canal.6CURVES CLASSIFICATION OF CIRCULAR CURVESHorizental curves are classified as : Simple Curves.

Compound Curves.

Reverse Curves.

Transition curves.

Lemniscate curves.7CURVEST1ABCT2OFig. 3. A SIMPLE CURVEi) Simple Curve:

A simple curve Consists of a single arc of circle connecting two straights. It has radius of the same magnitude throughout.8RRCURVESii) COMPOUND CURVE A compound Curve consists of two or more simple curves having different radii bending in the same direction and lying on the same side of the common tangent. Their centres lie on the same side of the curve.AT1M P NCO1O2Fig.4 Compound Curve9R2R1CURVESiii) REVERSE OR SERPENTINE CURVEA reverse or serpentine curve is made up of two arcs having equal or different radii bending in opposite direction with a common tangent at their junction .Fig. 5. A Reverse or Serpentine Curve. MTheir centres lie on opposite sides of the curve. NO2O1AT1T2pB10R1R2R2R1CURVESREVERSE OR SERPENTINE CURVE Fig.6 A Reverse or Serpentine Curve. They are commonly used in railway sidings and sometimes on railway tracks and roads meant for low speeds. They should be avoided as far as possible on main lines and highways where speeds are necessarily high.AT1T2O2O1MNBP11CURVESiv) Transition CurveA curve of variable radius is known as transition curve. It is also called a spiral curve or easement curve. In railways, such a curve is provided on both sides of a circular curve to minimize super elevation. Excessive super elevator may cause wear and tear of the rail section and discomfort to passengers.12

CURVESv) Lemniscate CurveA curve similar to a transition curve and is generally adopted in city roads where the deflection angle is large. In figure below OPD shows the shape of such a curve. The curve is designed by taking a major axis OD, minor axis PP, with the region of O, and axis OA and OB. OP is known as the polar ray, and the polar angle.13

CURVESBBT1T2ORCAEFI/2Fig. 8 SIMPLE CIRCULAR CURVE14` AwesazASW32wqaa1Q2CURVESNAMES OF VARIOUS PARTS OF CURVEThe two straight lines AB and BC which are connected by the curve are called the tangents or straights to the curve. The point of intersection of the two straights (B) is called the intersection point or the vertex. When the curve deflects to the right side of the progress of survey ,it is termed as right handed curve and when to the left , it is termed as left handed curve.15CURVESNAMES OF VARIOUS PARTS OF CURVE(iv) The lines AB and BC are tangents to the curve. AB is called the first tangent or the rear tangent . BC is called the second tangent or the forward tangent. (v) The points ( T1 and T2 ) at which the curve touches the tangents are called the tangent points. The beginning of the curve ( T1) is called the tangent curve point and the end of the curve (T2) is called the curve tangent point.

16CURVESNAMES OF VARIOUS PARTS OF CURVE(vi) The angle between the lines AB and BC (ABC) is called the angle of intersection (I).(vii) The angle by which the forward tangent deflects from the rear tangent (BBC) is called the deflection angle () of the curve.(viii) The distance from the point of intersection to the tangent point is called tangent length ( BT1 and BT2).(ix) The line joining the two tangent points (T1 and T2) is known as the long chord.

17CURVES The arc T1FT2 is called the length of curve. The mid point(F) of the arc (T1FT2) is called the summit or apex of the curve.(xii) The distance from the point of intersection to the apex of the curve BF is called the apex distance.(xiii) The distance between the apex of the curve and the mid point of the long chord (EF) is called versed sine of the curve.(xiv) The angle subtended at the centre of the curve by the arc T1FT2 is known as central angle and is equal to the deflection angle () .

18CURVESELEMENTS of a Simple Circular CurveAngle of intersection +Deflection angle = 1800. or I + = 1800

(ii) T1OT2 = 1800 - I = i.e the central angle = deflection angle.

Tangent length = BT1 =BT2= OT1 tan /2 = R tan /2

19CURVESELEMENTS of a Simple Circular Curve(iv) Length of long chord =2T1E =2R sin /2Length of curve = Length of arc T1FT2

= R X (in radians) = R /1800 (vi) Apex distance = BF = BO OF = R sec. /2 - R = R (1 cos /2 )=R versine /2

20CURVES A curve may be designated either by the radius or by the angle subtended at the centre by a chord of particular length. In India, a curve is designated by the angle (in degrees) subtended at the centre by a chord of 30 metres (100 ft.) length. This angle is called the degree of curve (D). The degree of the curve indicates the sharpness of the curve.DESIGNATION OF CURVE21CURVESDESIGNATION OF CURVES. In English practice , a curve is defined by the radius of the curve in terms of chains, such as a six chain curve means a curve having radius equal to six full chains, chain being 30 metres unless otherwise specified. In America,Canada,India and some other countries a curve is designated by the degree of the curve. For example a 40 curve means a curve having angle of 90 degrees at the centre subtended by a chord of 30m length unless otherwise specified. 22The angle a unit chord of length 30 m subtends at the center of the circle formed by the curve is known as degree of the curve. It is designated as D. (Fig 10.2).A curve may be designated according to either the radius or the degree of the curve.When the unit chord subtends an angle of 1 degree, it is called a one-degree curve , when the angle is two-degree curve, and so on.It may be calculated that the radius of one-degree curve is 1,719 m.23

Degree of curve Degree of curve 24

When a particle move in a circular path, then a force (known as centrifugal force) acts upon it, and tends to push it away from the center. Similarly, when a vehicle suddenly moves from a straight to a curved path, the centrifugal force tends to push the vehicle away from the road or track.To counter balance the centrifugal force, the outer edge of the road or rail is raised to some height (with respect to the inner edge), so that the sine component of the weight of the vehicle (w sin ) may counter balance the over- turning force.25

Superelevations The height through which the outer edge of the road or rail is raised is known as superelevation or cant.In figure 10.4, p is the centrifugal force, w sin is the component of the weight of the vehicle, and h is the superelevation given to the road or rail.

26

Superelevations

27

Superelevations

28

CURVESRELATION between the Radius of curve and Degree of Curve.The relation between the radius and the degree of the curve may be determined as follows:-

Let R = the radius of the curve in metres. D = the degree of the curve. MN = the chord, 30m long. P = the mid-point of the chord.

In OMP,OM=R, MP= MN =15m MOP=D/2Then, sin D/2=MP/OM= 15/R

MNODD/2R R Fig.9 Degree of CurvePPTO29CURVESRELATION between the Radius of curve and Degree of Curve.Then,sin D/2=MP/OM= 15/ROr R = 15 sin D/2But when D is small, sin D/2 may be assumed approximately equal to D/2 in radians.Therefore: R = 15 X 360 D = 1718.87 D Or say , R = 1719 D

MNODD/2R R Fig. 10 Degree of CurveP This relation holds good up to 50 curves.For higher degree curves the exact relation should be used.(Exact)(Approximate)30Centrifugal force31

CURVESMETHODS OF CURVE RANGING A curve may be set out (1) By linear Methods, where chain and tape are used or (2) By Angular or instrumental methods, where a theodolite with or without a chain is used. Before starting setting out a curve by any method, the exact positions of the tangents points between which the curve lies ,must be determined. Following procedure is adopted:- 32CURVESMETHODS OF SETTING OUT A CURVEProcedure :-After fixing the directions of the straights, produce them to meet in point (B)Set up the Theodolite at the intersection point (B) and measure the angle of intersection (I) .Then find the deflection angle ( ) by subtracting (I) from 1800 i.e =1800 I.iii) Calculate the tangent length from the following equation Tangent length = R tan/2

33CURVESMETHODS OF SETTING OUT A CURVEProcedure :-

iv) Measure the tangent length (BT1) backward along the rear tangent BA from the intersection point B, thus locating the position of T1.

vi) Similarly, locate the position of T2 by measuring the same distance forward along the forward tangent BC from B.

34CURVESMETHODS OF SETTING OUT A CURVE Procedure (contd) :- After locating the positions of the tangent points T1 and T2 ,their chainages may be determined. The chainage of T1 is obtained by subtracting the tangent length from the known chainage of the intersection point B. And the chainage of T2 is found by adding the length of curve to the chainage of T1. Then the pegs are fixed at equal intervals on the curve.The interval between pegs is usually 30m or one chain length. ...............

35CURVESMETHODS OF SETTING OUT A CURVE Procedure (contd) :-This distance should actually be measured along the arc ,but in practice it is measured along the chord ,as the difference between the chord and the corresponding arc is small and hence negligible. In order that this difference is always small and negligible ,the length of the chord should not be more than 1/20th of the radius of the curve. The curve is then obtained by joining all these pegs. ...............

36CURVESMETHODS OF SETTING OUT A CURVE Procedure (contd) :- The distances along the centre line of the curve are continuously measured from the point of beginning of the line up to the end .i.e the pegs along the centre line of the work should be at equal interval from the beginning of the line up to the end. There should be no break in the regularity of their spacing in passing from a tangent to a curve or from a curve to the tangent. For this reason ,the first peg on the curve is fixed .

37CURVESMETHODS OF SETTING OUT A CURVE Procedure (contd) :- at such a distance from the first tangent point (T1) that its chainage becomes the whole number of chains i.e the whole number of peg interval. The length of the first sub chord is thus less than the peg interval and it is called a sub-chord. Similarly there will be a sub-chord at the end of the curve. Thus a curve usually consists of two sub-chords and a no. of full chords. 38CURVESExample : A simple circular curve is to have a radius of 573 m .the tangents intersect at chainage 1060 m and the angle of intersection is 1200. Find, Tangent Distance. Chainage at beginning and end of the curve. Length of the long chord. Degree of the curve. Number of full and sub chords.

Solution: Please see fig.11Given,The deflection angle, = 1800 1200 =600 Radius of curve = R = 573 m

PTO39CURVESFig.111060 mO729.151329.15T1T26001200330.85R=573m= L=600m40CURVESWe know ,tangent length = R tan /2 = 573 x tan 300 = 573x 0.5774 = 330.85 m (Ans.)(ii) Length of curve is given by: R 1800 = x 573x600 1800 = 600 m (Ans.)Chainage of first tangent point (T1) = Chainage of intersection point tangent length. = 1060 330.85 = 729.15 m (Ans.)

PTO41CURVES(iii) The length of long chord is given by: L = 2R sin /2 = 2 x 573 x sin 300 = 573 m ( Ans.)

Degree of CurveWe know the relation , R= 1719 D or D = 1719 R =30 Therefore , degree of curve is =30 (Ans.)

PTO42CURVES(v) Number of Full and sub chords: Assuming peg interval =30mChainage of T1 = 729.15 m = 729.15 30 = 24 full chain lengths + 9.15 mChainage of Ist peg on the curve should be 25 full chain lengths.The length of Ist sub chord= (25+00) (24 + 9.15) = 20.85 mChainage of T2 = 1329.15 Chain lengths. 30 = 44 full chain lengths + 9.15 m.Chainage of last peg on the curve =44 full chains. Therefore length of last sub chord = (44+9.15) (44+00) = 9.15m

PTOChain lengths.43CURVESNo. Of full chords = chainage of last peg chainage of Ist peg = 44 25 = 19So, there will be 19 full chords and two sub chords.Check:Length of full chords = 19x30 =570.00m Ist sub chord = 20.85m last sub chord = 9.15m

Total length of all chords = 600.00m

PTO(Same as length of curve)44CURVESLINEAR METHODS of setting out CurvesThe following are the methods of setting out simple circular curves by the use of chain and tape :- By offsets from the tangents. By successive bisection of arcs. By offsets from chords produced.45CURVESLINEAR METHODS of setting out Curves1. By offsets from the tangents. When the deflection angle and the radius of the curve both are small, the curves are set out by offsets from the tangents. Offsets are set out either (i) radially or (ii) perpendicular to the tangents according as the centre of the curve is accessible or inaccessible 46CURVESBBT1T2ORCA Fig. 12 By Radial OffsetsLINEAR METHODS of setting out CurvesOxxPP190047CURVESB By Radial OffsetsLINEAR METHODS of setting out CurvesOffsets is given by :

Ox = R2 +x2 R .. (Exact relation.)When the radius is large ,the offsets may be calculated by the approximate formula which is as underOx = x2 (Approximate ) 2R

48CURVESBO(ii) By offsets perpendicular to the Tangents LINEAR METHODS of setting out CurvesOxxPP1P2BABT2T1Fig. 13. 49CURVESLINEAR METHODS of setting out Curves1. (ii) By offsets perpendicular to the TangentsOx= R R2 x2 (Exact)

Ox = x2 (Approximate ) 2R

50CURVESLINEAR METHODS of setting out CurvesBy offsets from the tangents: Procedure

Locate the tangent points T1 and T2.

Measure equal distances , say 15 or 30 m along the tangent fro T1. (iii) Set out the offsets calculated by any of the above methods at each distance ,thus obtaining the required points on the curve.

51CURVESLINEAR METHODS of setting out CurvesBy offsets from the tangents: Procedure.

Continue the process until the apex of the curve is reached.(v) Set out the other half of the curve from second tangent.(vi) This method is suitable for setting out sharp curves where the ground outside the curve is favourable for chaining.

52CURVESExample. Calculate the offsets at 20m intervals along the tangents to locate a curve having a radius of 400m ,the deflection angle being 600 .

Solution . Given:Radius of the curve ,R = 400mDeflection angle, = 600Therefore tangent length = R. tan /2 = 400 x tan 600 = 230.96 mRadial offsets. (Exact method)

Ox= R2 + x2 - R (Exact)53CURVESRadial offsets. (Exact method)

Ox= R2 + x2 - R (Exact)

O20 = 4002+202 - 400 = 400.50 - 400 = 0.50 m

O40 = 4002+402 - 400 = 402.00 - 400 = 2.00 m

O60 = 4002+602 - 400 = 404.47 - 400 = 4.47 m

O80 = 4002+802 - 400 = 407.92 - 400 = 7.92 m

O100 = 4002+1002- 400 = 412.31 - 400 = 12.31 m

And so on.

54CURVESB) Perpendicular offsets (Exact method)

Ox = R R2 x2 (Exact)

O20 = 400 - 4002 - 202 = 400 -399.50 = 0.50 m

O40 = 400 - 4002 - 402 = 400 -398.00 = 2.00 m

O60 = 400 - 4002 - 602 = 400 -395.47 = 4.53 m

O80 = 400 - 4002 - 802 = 400 -391.92 =8.08 m

O100 = 400 - 4002 -1002 = 400 -387.30 =12.70 m

And so on..

55CURVESBy the approximate Formula (Both radial and perpendicular offsets)

Ox = 2RTherefore O20 = 202 = 0.50 m 2x400

x2O40 = 402 = 2.00 m 2x400O60 = 602 = 4.50 m 2x400

O80 = 802 = 8.00 m 2x 400O100 = 1002 = 12.50 m 2 x 400 and so on.5657THANKS