Circular Curves
This section describes the defining parameters, elements, and methods of setting out circular curves.
Radius or degree of a curve
A curve is defined either by its radius or by its degree. The degree of a curve (D) is the angle subtended at its centre by a 30.5m or 100ft chord.
The value of the degree of the curve can be determined as indicated below.
Circumference of a circle = 2pR
Angle subtended at the centre by a circle with this circumference = 360° Angle subtended at the centre by a 30.5m chord, or degree of curve
2pR
= 1750/R (approx., R is in metres)
In cases where the radius is very large, the arc of a circle is almost equal to the chord connecting the two ends of the arc. The degree of the curve is thus given by the following formulae
D = 1750/R (when R is in metres)
D = 5730/R (when R is in feet)
A 2° curve, therefore, has a radius of 1750/2 = 875 m.
Relationship between radius and versine of a curve
The versine is the perpendicular distance of the midpoint of a chord from the arc of a circle. The relationship between the radius and versine of a curve can be established as shown in Fig. 13.1. Let R be the radius of the curve, C be the length of the chord, and V be the versine of a chord of length C.
Fig. 13.1 Relation between radius and versine of a curve
In Eqn (13.1), V, C, and R are in the same unit, say, metres or centimetres. This general equation can be used to determined versines if the chord and the radius of a curve are known.
Case II: Values in fps units When R_{x} is the radius in feet, C_{1} is the chord length in feet, and V_{1} is the versine in inches, Formula (13.1) can be written as
Using formulae (13.2) and (13.3), the radius of the curve can be calculated once the versine and chord length are known.
Determination of degree of a curve in field
For determining the degree of the curve in the field, a chord length of either 11.8 m or 62 ft is adopted. The relationship between the degree and versine of a curve is very simple for these chord lengths as indicated below.
Versine on a 11.8m chord
From the two equations given above, the degree of the curve for a 11.8m chord can be determined as follows. Substituting the value of R = 12.5C ^{2}/V,
The degree of the curve for a 62ft chord can be determined as follows. Substituting the value of
This important relationship is helpful in determining the degree of the curve at any point by measuring the versine either in centimetres on a 11.8m chord or in inches on a 62ft chord. The curve can be of as many degrees as there are centimetres or inches of the versine for the chord lengths given above.
Maximum Degree of a Curve The maximum permissible degree of a curve on a track depends on various factors such as gauge, wheel base of the vehicle, maximum permissible superelevation, and other such allied factors. The maximum degree or the minimum radius of the curve permitted on Indian Railways for various gauges is given in Table 13.1.
Table 13.1 Maximum permissible degree of curves
Gauge 
On plain track 
On turnouts 

Max. degree 
Min. radius (m) 
Max. degree 
Min. radius (m) 

BG 
10 
175 
8 
218 
MG 
16 
109 
15 
116 
NG 
40 
44 
17 
103 
Elements of a circular curve
In Fig. 13.2, AO and BO are two tangents of a circular curve which meet or intersect at a point O, called the point of intersection or apex. T_{1} and T_{2} are the points where the curve touches the tangents, called tangent points (TP). OT_{1} and OT_{2} are the tangent lengths of the curve and are equal in the case of a simple curve. T_{1}T_{2} is the chord and EF is the versine of the same. The angle AOB formed between the tangents AO and OB is called the angle of intersection (Zl) and the angle BOO_{1 }is the angle of deflection (Zf). The following are some of the important relations between these elements:
Setting Out a Circular Curve
A circular curve is generally set out by any one of the following methods.
Tangential offset method
The tangential offset method is employed for setting out a short curve of a length of about 100 m (300 ft). It is generally used for laying turnout curves.
In Fig. 13.3, let PQ be the straight alignment and T be the tangent point for a curve of a known radius. Let AA', BB', CC', etc. be perpendicular offsets from the tangent. It can be proved that
Fig. 13.3 Tangential offset method
The various steps involved in the laying out of a curve using this method are as follows.
(a) Extend the straight alignment PT to TQ with the help of a ranging rod. TQ is now the tangential direction.
(b) Measure lengths C_{1}, C_{2}, C_{3}, etc. along the tangential direction and calculate the offsets 0_{1}, O_{2},0_{3}, etc. for these lengths as per the formulae given above. For simplicity, the values of C_{1}, C_{2}, C_{3}, etc. may be taken in multiples of three or so.
(c) Measure the perpendicular offsets 0_{1}, 0_{2}, 0_{3}, etc. from the points A, B, C, etc. and locate the points A', B', C', etc. on the curve.
In practice, sometimes it becomes difficult to extend the tangent length beyond a certain point due to the presence of some obstruction or because the offsets become too large to measure accurately as the length of the curve increases. In such cases, the curve is laid up to any convenient point and another tangent is drawn out at this point. For laying the curve further, offsets are measured at fixed distances from the newly drawn tangent.
Long chord offset method
The long chord offset method is employed for laying curves of short lengths. In such cases, it is necessary that both tangent points be located in such a way that the distance between them can be measured, and the offsets taken from the long chord.
In Fig. 13.4, let T_{1}T_{2} be the long chord of a curve of radius R.
Let the length of the long chord be C and let it be divided into eight equal parts Tj A, AB, BC, CD, etc., where each part has a length x = C/8. Let PW be a line parallel to the long chord and let O_{1}, O_{2}, and O_{3} be the offsets taken from points R, Q, and P
Versine V from the long chord C is calculated by the formula
Offset 0_{1} from the line PW is calculated by the formula
(13.4)
(13.5)
Using formulae (13.4) and (13.5), the values of the perpendicular offsets V_{1}, V_{2}, V_{3}, etc. can be calculated as follows:
During fieldwork, first the long chord is marked on the ground and its length measured. Then points A, B, C, etc. are marked by dividing this long chord into eight equal parts. The values of the perpendicular offsets V_{x}, V_{2}, V_{3}, etc. are then calculated and the points A', B', C', etc. identified on the curve.
Quartering of versine method
The quartering of versine method (Fig. 13.5) is also used for laying curves of short lengths, of about 100 m (300 ft). In this method, first the location of the two tangent points (T_{x} and T_{2}) is determined and then the distance between them is measured. The versine (V) is then calculated using the formula
V is measured in the perpendicular direction at the central point O of the long chord. The tangent points T_{x} and T_{2} are joined and the distance AT_{X} measured. As
AT_{1} is almost half the length of chord T_{1}T_{2} and as versines are proportional to the square of the chord, the versine of chord AT_{1} is V/4.
For laying the curve in the field, the versine V/4 is measured at the central point B on chord AT_{1} and the position of point B is thus fixed. Similarly, a point is also fixed on the second half of the curve. AB is further taken as a subchord and the versine on this subchord is measured as V/16. In this way the points D and F are also fixed. The curve can thus be laid by marking halfchords and quartering the versines on these halfchords.
Chord deflection method
The chord deflection method of laying curves is one of the most popular methods with Indian Railways. The method is particularly suited to confined locations, as most of the work is done in the immediate proximity of the curves. In Fig. 13.6, let T_{1} be the tangent point and A, B, C, D, etc. be successive points on the curve. Let X_{1}, X_{2}, X_{3} and X_{4} be the length of chords T_{1}A, AB, BC, and CD. In practice, all the chords are of equal length. Let the value of these chords be c. The last chord may be of a different length. Let its value be c_{1}. It can be proved that
Fig. 13.6 Chord deflection method
1. The position of the tangent point Tj is located by measuring a distance equal to the tangent length R tanf/2 from the apex point O. In this case, f is the deflection angle.
2. A length equal to the first chord (c) is measured along the tangent line T_{1}O and point A' is marked.
3. The zero end of the tape is placed at the tangent point T_{1}. It is then swung and the arc A_{1}A marked. Then the first offset on the arc is measured. The value of the offset is c^{2}/2R. The position of point A is thus fixed.
4. The chord T_{1}A is extended to point B and AB' is marked as the second chord length equal to c.
5. The position of point B is then fixed on the curve since the value of the second offset is known and is equal to c^{2}/R.
6. Similarly, the positions of other points C, D, etc. are also located.
7. The last point on the curve is located by taking the value of the offset as c_{j} (c + c_{1})/R , where c_{1} is the length of the last chord.
The various points on the curve should be set with great precision because if any point is fixed inaccurately, its error is carried forward to all subsequent points.
Theodolite method
The theodolite method for setting out curves is also a very popular method with Indian Railways, particularly when accuracy is required. This method is also known as Rankine's method of tangential angles. In this method, the curve is set out using tangential angles with the help of a theodolite and a chain or a tape.
In Fig. 13.7, let A, B, C, D, etc. be successive points on the curve with lengths T_{1}A = x_{1}, AB = x_{2}, BC = x_{3}, CD = x_{4}, etc. Let d_{j}, d_{2}, d_{3}, d_{4} be the tangential angles OT_{1}A, AT_{1}B, BT_{1}C, and CT_{1}D made by the successive chords amongst themselves.
Let D_{j}, A_{2}, D_{3}, and A_{4} be the deflection angles of the chord from the deflection line.
Angle subtended at centre by a 100ft chord = D°
Tangential angle for a 100ft chord = D/2 Tangential angle for an xft chord, S = (D/2) x (1/100)x degree = (5730/2R) x (1/100) x 60x minutes = 1719(x/R)
where 8 is the deflection angle in minutes, x is the chord length in feet, and R is the radius in feet. It is seen that
^{A}i = ^{8}i
^{A} 2 = ^{8}1 + ^{8}2 = ^{A}1 + ^{8}2 A3 = 81 +82 +83 = A 2 + 83
The procedure followed for setting the curve is as follows.
(a) The theodolite is set on the tangent point T_{1} in the direction of T_{2}O.
(b) The theodolite is rotated by an angle 8_{1}, which is already calculated, and the line T_{1}A_{1} is set.
(c) The distance x_{1} is measured on the line T_{1}A_{1} in order to locate the point A.
(d) Now the theodolite is rotated by a deflection angle 8_{2} to set it in the direction of T_{1}B_{1} and point B is located by measuring AB as the chord length x_{2}.
(e) Similarly, the other points C, D, E, etc. are located on the curve by rotating the theodolite to the required deflection angles till the last point on the curve is reached.
(f) If higher precision is required, the curve can also be set by using two theodolites.
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