Curves

 In the geometric design of motorways , railways , pipelines , etc , the design and setting out of curves is an important aspect of the engineer’s work 

The initial design is usually based on a series of straight sections whose positions are defined largely by the topography of the area 

The intersection of pairs of straights are then connected by horizontal curves 

In the vertical design , intersecting gradients are connected by curves in the vertical plane 

Curves can be listed under three main heading , as follows : 

(1) Circular curves of constant radius 

(2) Transition curves of varying radius (spirals ) 

(3) Vertical curves of parabolic from 

Circular curves 

Two straights D1 T1 and D2 T2 are connected by a circular curve of radius R :

(1) The straights when projected forward , meet at I : the intersection point 

(2) The angle at I is called the angle of intersection or the deflection angle , and equals the angle T1 0T2 subtended at the centre of curve 0 

(3) The angle at I is called the apex angle , but is little used in curve computations 

(4) The curve commences from T1 and ends at T2 ; these points are called the tangent points 

(5) Distances T1 I and T2 I are the tangent lengths and are equal to R tan /2 

(6) The length of curve T1 AT2 is obtained from : 

(7) Distance T1 T2 is called the main chord (C) and from figure 10 .1 : 

(8) La is called the apex distance and equals

These equations should be deduced using a curve diagram ( Figure 10.1 ) 

Curve designation 

Curves are designated either by their radius ( R) or their degree of curvature ( D) 

The degree of curvature is defined as the angle subtended at the centre of a circle by an arc of 100 m( Figure 10.2 ) 

Through chainage 

Through chainage is the horizontal distance from the start of a scheme for route construction 

Consider Figure 10 . 3 

If the distance from the start of the route ( Chn 0.00 m ) to the tangent point T1 is 2115. 50 m then it is said that the chainage of T1 is 2115.50 m written as (Chn 2115 . 50 m )

If the route centre - line is being staked out at 20 - m chord intervals , then the peg immediately prior to T1 must have a chainage of 2100 m ( an integer number of 20 m intervals ) 

The next peg on the centre - line must therefore have a chainage of 2120 m 

If follows that the length of the first sub - chord on the curve from T1 must be ( 2120 _ 2115. 50 ) 4.50 m 

Similarly , if the chord interval had been 30 m , the peg chainage prior to T1 must be 2100 m and the next peg ( on the curve ) 2130 m , thus the first sub chord will be ( 2130 - 2115.50 ) = 14.50 m 

A further point to note in regard to chainage is that if the chainage at I1 is known , then the chainage at T1 = Chn I1 distance I1 T1 , the tangent length 

However the chainage at T2 = Chn T1 + curve length ,as chainage is measured along the route under construction 

Setting out curves 

This is the process of establishing the centre line of the curve on the ground by means of pegs at 10 m to 30 m intervals 

In order to do this the tangent and intersection points must first be fixed in the ground in their correct positions 

Consider Figure 10 . 3 

The straights OI1 , I1 I2 , I2 I3 , etc will have been designed on the plan in the first instance 

Using railway curves , appropriate curves will now be designed to connect the straights 

The tangent points of these curves will then be fixed , making sure that the tangent lengths are equal i . e T1 I1 = T2 I1 and T3 I2 = T4 I2 

The coordinates of the origin point O and all the intersection points anly will now be carefully scaled from the plan 

Using these coordinates the bearings of the straights are computed and using the tangent lengths on these bearings the coordinates of the tangent points are also computed 

The difference of the bearings of the straights provides the deflection angles of the curves which combined with the tangent length enables computation of the curve radius through chainage and all setting out data 

Now the tangent and intersection points are set out from existing control survey stations and the curves ranged between them using the methods detailed below