**Horizontal Curves**

**11,1 Introduction**

Horizontal curves are mainly used in the setting out of curved road centerlines. The principles involved, however, could be used for setting out any type of curved

structure. This section will be restricted to the setting out of road centreline curves using deflection angles.

Figure 4.29 – Horizontal curve terminology.

**11.2 Terminology**– Refer to Figure 4.29

Notes to Figure 4.29

IP - Intersection point. The point where two

Straights intersect.

TC - “Tangent changing to cuve”. The tangent

point at the start of the curve.

CT - “Curve changing to tangent”. The tangent

Point at the end of the curve.

Tangent length - The distance from IP to both tangent points

Length of curve - The arc length from one tangent point to the

Other

Radium - The radius of the circular arc.

I - Intersection Angle. The angle between the

1. first straight produced and the second

straight. It represents the total change in

direction.

Central angle - Sometimes called ‘Radius Angle’. This is

The angle between the intersecting radii

From the tangent points.

Deflection angle - An angle at the TC subtended by an arc.

Chord length - The chord length for a given arc.

Initial sub arc - An arc, less than standard length, between

TC and the first point on the curve.

Final sub arc - An arc, less than standard length, between the

Last point on the curve and CT.

Chainage - Accumulative distance to a point along a road

Center line.

Initial sub chord - The chord between TC and the first point on the

Curve.

Final sub chord - The chord between the last point on the curve

And the CT.

Figure 4.30 – Horizontal curve terminology

**11.3 Calculation of a horizontal curve**

(a) To calculate a horizontal curve by the deflection angle method, the following data is required.

(i) Intersection angle (1)

(ii) Radius ®

(iii) Chainage of a point

(iv) Curve hand.

(b) Calculate the following:

Tangent length = Rxtan (1/2)

Long chord = 2 x Rxsin (1/2)

1

Curve length = ------------ x 2 x iixR.

360

© Determine chainage of tangent points

(i) Subtracting tangent length from IP chainage gives chainage of TC.

(ii) Add curve length to TC chainage gives chainage to CT

Figure 4.31 – Calculation of TC chainage

Figure 4.32 – Calculation of CT chainage.

(c) Determine sub-arc lengths.

(i) Decide on tndard arc length – usually 10 or 5 m.

(ii) Subtract TC chainage from chainage of first point on curve, using even chainage, gives initial sub-arc.

(iii) Subtract last even chainage, gives initial sub-arc.

(b) Calculte deflection angles

Deflection angle =

__a x 90__

R ii

Where a = arc length

R = radius of curve.

(d) Calculate chord lengths.

(e) When the arc length is greater than one twentieth of the radius, a chord length should be calculated from:

Chord length = 2 x radius x sine of deflection angle.

(f) For ease of calculation and for providing a permanent record, use a purpose made calculation sheet. An example is given in Figure 4.34 and a worked

example will illustrate its use.

**11.4**

**Example calculation**

Given data

Rectangular Co-ordinates A(100) 263.400 mE 653.200 mN

B(101) 294 100 mE 708.200 mN

C(102) 382.000 mE 690.000 mN

Radius = 40 m

Point 100 has zero chainage

Figure 4.33 – Road cenreline layout

(a) Compute WCB & distance 100 to 101

(b) Compute WCB & distance 101 to 102

(c) Compute Intersection angle from whole circle bearings

(d) Distance 100 to 101 is added to the chainage of 100 i.e. 0.000 to give the chainage to IP.

(e) Complete the calculation sheet in numbered order.

**11.5**

**Alignment sheet**

An alignment sheet of accumulative deflection angles is produced for setting out purposes and an example is given in Figure 4.354

HORIZONTAL CURVE CALCULATION SHEET Rect Co-ords : Sin A (100) 263.400 E 653.200

Figure 4.35 – Alignment sheet.

NB: The plate readings for a right hand curve increase from 0, whilst those for a left hand curve decrease from 360.

11.4

11.4

**Setting out procedure**

(a) The straights may be defined by co-ordinates and are set out as described in paragraph 6.3.

(b) Measure the intersection angle an c9mpare with designed value.

(c) Calculate and set out BOTH tangent points by measuring the tangent length along each straight.

(d) Check the position of the tangent points by measuring the length of the long chord and the angle between the tangent and the long chord at TC, and

comparing them with the computed values.

(e) Now set out the curve from the TC using accumulative deflection angle and chord lengths as follows:

(f) A theodolite is set up over the TC and sighted on to the IP with a zero plate reading.

(g) The first plate reading is tuned off. The theodolite is now pointing in the direction of the first peg on the curve.

(h) A chord length equivalent to the initial sub arc is now measured along this line and a peg established.

(i) The next plate reading is turned off. The theodolite is now pointing in the direction of the second point on the curve.

(j) Measure a chord length equivalent to the standard arc length from the previously established peg and locate another peg at the intersection of the chord

length with the theodolite line of sight.

Figure 4.36 – Setting out first point on curve.

Figure 4.37 – Setting out second point on curve.

Figure 4.38 – Setting out a curve by deflection angles

(vi) This procedure continues until the end of the curve is reached.

(vii) The final check on the setting out is done by measuring the final sub chord and comparing it with its computed value.