What is Curve, how many types of curves in Surveying Engineering

CIRCULAR CURVES

* The circular curves used in roads and railways are of three types, namely, simple, compound and

reverse curves.

* Definitions:

1. Right hand curve: It is the curve which deflects to the right of the direction of progress of

route. Similarly, left hand curve may be defined.

2. Back tangent: The tangent before the commencement of curve.

3. Forward tangent: The tangent after the end of curve.

4. Point of intersection: The intersection of back and forward tangents.

5. Intersection angle: Angle of deflection at point of intersection from back to forward tangent.

6. Point of curve (PC): The point where curve begins.

7. Point of tangency: The point where curve ends.

8. Apex or summit of curve: The mid-point of curve.

9. Mid-ordinate: Distance between the mid-point of the long chord and apex.

10. External distance: The distance between apex and point of intersection.

Elements of Simple Curve

If R is radius of a simple curve and D is angle of deflection

1. Length of curve: l = RD, if D is in radians

= RD , if D is in degrees.

2. Tangent length: T = R tan

3. Length of long chord: L = 2R sin

4. Mid-ordinate: M = R (1 – cos )

5. External distance: E = R (sec – 1)

Setting out Simple Circular Curves

1. Linear methods: (a) Offsets from long chords, (b) successive bisection method, (c) Offsets from

the tangent-perpendicular or radial and (d) offsets from the chord produced.

2. Angular methods: (a) Rankine’s method of tangential angles, (b) two-theodolite method, and (c)

tacheometric method.

Compound curves A compound curve consists of two or more simple curves of different radii. In

practice it normally consists of two curves. This type of curve is used to avoid the obstruction.

Reverse curves In a reverse curve two circular arcs having radius of curvatures in opposite

directions meet at a point tangentially. The common point is called the point of reverse curvature or

contrary flexure. As far as possible such curves should be avoided.

Transition curves

* It is a curve provided to bring about a transition between a straight and a circular curve or between

two branches of compound or reverse curves. It is also known as a spiral or easements curve. In

these curves gradual change of super elevation is introduced.

* Length of transition curve is decided on any one of the following basis:

1. Uniform rate of 1 in n super elevation L = ne where e is super elevation required in circular

curve.

2. Time rate: Super elevation e is applied at a time rate of m units/second. If v is the designed

speed of vehicle, then

L = 

3. Rate of change of radial acceleration: For comfortable journey of passengers the rate of change

of radial acceleration should be 0.3 m/sec

2

From this consideration

L = metres V is kmph.

In practice, instead of radial acceleration, the centrifugal ratio (centrifugal force divided by

weight of vehicle) is kept constant.

For roads = \ L = 12.80

For railway = \ L = 4.525

* Clothoid spiral is known as an ideal curve. In such a curve at any point in transition curve

l = .

The coordinates of such curves are

x =

y =

where f =

L being total length of transition curve.

If only the first term is taken in the above expression, y = . Such a curve is called cubic parabola.

If only first term of x and y are taken in ideal curve,

x = l; y = = .

Such a curve is known as cubic parabola.

* Radius of curvature r in cubic parabola goes on reducing till f 24° 5¢ 41″ and then starts increasing

with f and hence do not serve purpose of a transition curve.

* Bernaulli’s lemniscate curve is a transition curve used in roads. It is well adopted when the

deflection angle is large. This curve is preferred to the spiral for the following reasons:

1. In this the radius of curvature decrease more gradually.

2. Rate of increase of curvature diminishes towards the end of the transition curve.

3. Its shape correspondence to the actual path traced by a vehicle tuning freely on the curve.

* For small angles, the length of Bernaulli’s lemniscate may be taken as

l = 6r µ

where µ is in radians

Vertical curves

* The change of grade is made smooth by introducing curves in vertical planes

* To make change of grade uniform,

y = ax

2 + bx

* The grade is considered +ve, if it is upward

* Types of vertical curves are:

1. Summit or crest curve

2. Sag or valley curve

Length of vertical curve, L =

where r = rate of change of grade. The recommended rate of change of grade is 0.06% for 20 m

station at summit and 0.03% for 20 m station at sags.

* The methods used for setting vertical curves are:

1. Tangent correction method.

2. Chord gradients method.

* Length of summit curve is based on the minimum sight distance requirement. It is given by

L =

where S = minimum sight distance

(g1 – g2

) total change in gradient sign of g1

is +ve for upward gradient and –ve for downward

gradient.

h1 = Height of driver’s eye above the road.

h2 = Height of obstacle above the road.

* In sag curve, visibility of road is not obstructed but they are to be designed so that headlight of the

vehicle illuminates a minimum of stopping sight distance. For this headlight is assumed at a height

of 0.75 m above the road and its beam tilled upward by an angle 1° to the horizontal.

From these considerations if S < L,

L =

if S > L, then

L = 2S –