Survey 2 curves1
- 2. • INTRODUCTION • DEFINITION & EXPLANATIONS OF DIFFERENT TERMS • TYPES OF HORIZONTAL CURVES • NOTATION USED WITH CIRCULR CURVES • PROPERTIES OF SIMPLE CIRCULAR CURVES
- 3. 10.1 INTRODUCTION During the survey of the alignment of a project involving roads or railways, the direction of the line may change due to some unavoidable circumstances. The angle of the change in direction is known as the deflection angle. For it to be possible for a vehicle to run easily along the road or railway track, the two straight lines (the original line and the deflected line) are connected by an arc (Fig. 10.1) which is known as the curve öf the road or track.
- 4. • When the curve is provided in the horizontal plane, it is known as a horizontal curves. • Again, along the alignment of any project the nature of the ground may not be uniform and may consist of different gradients (for instance, rising gradient may be followed by falling gradient and vice versa). In such a case, a parabolic curved path is provided in the vertical plane in order to connect the gradients for easy movement of the vehicles. This curve is known as a vertical curve.
- 6. 10.2 DEFINITIONS AND EXPLANATIONS OF DIFFERENT TERMS 1.Degree of curve The angle a unit chord of length subtends at the centre of the circle formed by the curve is known as the degree of the curve. It is designated as D 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°, is called a one-degree curve, and so on.
- 7. •It may be calculated that the radius of a one-degree curve is 1,719 m.
- 8. 2. Relation between radius and degree of curve Let AB be the unit chord of 30 m O the centre, R the radius and D the degree of the curve (Fig. 10.3) Here OA=R AB=30 m AC = 15 m <AOC = D/2 From triangle OAC, sin D/2=AC/OA=15/R R=15/(Sin D/2)
- 9. When D is very small, sin D/2 may be taken as D/2 radians. R=15/(D/2)*(Π/180) =15 * 360/ Π D = 1,718.9/D R=1,719/D (approx.)
- 10. SUPERELEVATION when a particle moves in a circular path, then a force (known as centrifugaL force) acts upon it, and tends to push it away from the centre 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. This is because there is no component force to counterbalance this centrifugal force.To counterbalance 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 counterbalance the overturning force The height through which the outer edge of the road or rail is raised is known is superelevation or cant. n Fig. 10.4, P is the centrifugal force, W sin 0 is the component of the weight of the vehicle, and h is the superelevation given to the road or rail. For equilibrium
- 12. centrifugal ratio The ratio between the centrifugal force and the weight of the vehicle is known as centrifugal ratio. Allowable value for CR in roads = 1/4 Allowable value for CR in railways = 1 /8
- 13. 10.2 TYPES OF HORIZONTAL CURVES The following are the different types of horizontal curves: 1.Simple circular curve When a curve consists of a single arc with a constant radius connecting the two tangents, it is said to be a circular curve.
- 14. 2. Compound curve When a curve consists of two or more arcs with different radii, it is called a compound curve. Such a curve lies on the on same a side of a common tangent and the centres of the different arcs lie on the same side of their respective tangents.
- 15. •3. Reverse curve A reverse curve consists of two arc bending in opposite directions.Their centres lie on opposite sides of the curve. Their radii may be either equal or different, and they have
- 16. 4. Transition curve A curve of variable radius is known as a transition curve. It is also called a spiral curve or easement curve. In railways, such a curve is provided of both sides of a circular curve to minimize superelevation . Excessive SE may cause wear and tear of the rail section and discomfort to passengers .
- 17. • 5. Lemniscate curve A lemniscate curve is similar to a transition curve, and is generally adopted in city roads where the deflection angle is large. OPD shows the shape of such a curve. The curve is designed by taking a major axis OD, minor axis PP', with origin O, and axes OA and OB. OP(ρ) polar ray, α polar angle.
- 18. Considering the properties of polar coordinates, the polar equation of the curve is given by where p = polar ray of any point r= radius of curvature at that point α = polar deflection angle At the origin, the radius of curvature is infinity. It then gradually decreases and becomes minimum at the apex D . Length of curve OPD = 1.3115 K where
- 19. 10.4 NOTATION USED WITH CIRCULAR CURVES ‘’’’’’’ ϕ
- 20. 10.4 NOTATION USED WITH CIRCULAR CURVES 1. AB and BC are known as the tangents to the curve 2. B is known as the point of intersection or vertex. 3. The angle is known as the angle of deflection. 4. The angle I is called the angle of intersection. 5. Points T1 and T2 are known as tangent points. 6. Distances BT1 and BT2 are known as tangent lengths. 7. When the curve deflects to the right, it is called a right-hand curve, when it deflects to the left, it is said to be a left-hand curve. 8. AB is called the rear tangent and BC the forward tangent. 9. The straight line T1DT2 is known as the long chord. 10. The curved line T1ET2 is said to be the length of the curve.
- 21. 11. The mid-point E of the curve T1ET2 is known as the apex or summit of the curve. 12. The distance BE is known as the apex distance or external distance. 13. The distance DE is called the versed sine of the curve. 14. R is the radius of the curve. 15. < TI OT2 is equal to the deflection angle . 16. The point TI is known as the beginning of the curve or the point of curve. 17 .The end of the curve (T2) is known as the point of tangency.
- 22. 10.5 PROPERTIES OF SIMPLE CIRCULAR CURVE Consider Fig. 10.10. I .If the angle of intersection is given, then ϕ=180°- l (l = Angle of intersection) 2. If radius is not given, then R = 1,719/D (D = degree of curve) 3. Tangent length BT1 or BT2 = R tan ϕ /2 4 Length of curve = length of arc T ET2 = R x ϕ radians =(ΠR ϕ°/180°)m Again, length of curve = 30ϕ/ D (if degree of curve D is given)
- 23. 5. Length of long chord = 2T1D =20T1 sin ϕ /2 =2R sin ϕ /2 m 6. Apex distance BE = OB -OE = R sec ϕ /2 - R = R (sec ϕ /2 - 1) m 7. Versed sine of curve = DE = OF - OD = R - R cos ϕ /2 = R (1 - cos ϕ /2)
- 24. 8. Ful chord (peg interval) Pegs are fixed at regular intervals along the curve. Each interval is said to equal the length of a full chord or unit chord. The curve is represented by a series of chords, instead of arcs Thus, the length of the chord is practically equal to the length of the arc. In usual practice, the length of the unit chord should not be more than 1/20th of the radius of the curve. In railway curves, the unit chords (peg intervals) are generally taken between 20 and 30 m In road curves, the unit chord should be 10 m or less It should be remembered that the curve will be more accurate if short unit chords are taken 8. Ful chord (peginterval) Pegs are fixed atregularintervalsalongthe curve. Each intervalis said to equal thelength of a fullchord or unit chord. Thecurveis representedby a series of chords, insteadof arcs Thus, the length of thechord is practicallyequalto thelength of the arc. In usualpractice, the lengthof the unitchord should notbe more than 1/20th of the radiusof the curve. In railway curves, theunitchords(pegintervals) are generallytaken between 20 and 30 m In roadcurves, the unit chord shouldbe 10 m or less It shouldbe remembered that thecurve will be more accurate if short unit chords are taken
- 25. 9. Initial subchord: Sometimes the chainage of the first tangent point works out to be a very odd number. To make it a round number, a short chord is introduced at the beginning. This short chord is known as the initial subchord. 10. Final subchord: Sometimes it is found that after introducing a number of full chords, some distance still remains to be covered in order to reach the second tangent point. The short chord introduced for covering this distance is known as the final subchord 11 Chainage of first tangent point = chainage of intersection point – tangent length 12. Chainage of second tangent point =chainage of first tangent point + curve length
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