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Track Modernisation Chapter V Curves and Superelevation

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Track Modernisation Chapter V Curves and Superelevation TRACK MODERNISATION CHAPTER V CURVES AND SUPERELEVATION 5.1 NECESSITY OF A CURVE A curve is provided to by-pass obstacles, to have longer and easier gradients or to route the line through obligatory or desirable locations. 5.2 RADIUS OR DEGREE OF A CURVE A curve is defined either by radius or by its degree. The degree of a curve (D) is the angle subtended as its center by a chord of 30.5 meters or 100 feet. The value of the degree of the curve can be found out as indicated below: (i) Circumference of a circle = 2ᴫR (ii) Angle subtended at the center by a circle having the above circumference = 3600 (iii) Angle subtended at the center By a 30.5 m chord or degree of curve =360/2ᴫR x 30.5 = 1750/R approx. (R is in meters) (iv) Angle subtended at the center by a = 360/2ᴫR1x 100 100 feet arc or degree of curve = 5730/R1 (R1 is in feet) In case when Radius is very large, an 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 formula: D = 1750/R When R is in meters D = 5730/R1 Where R1 is in feet. A 20 curve has, therefore, a radius of 1750/2 = 875meters. Or 5730/2 = 2865 feet. 5.3 Relationship between radius and Versine of a curve. The relationship between radius and versine can be established as indicated below. (Fig.5.1) Let R be the Radius of the curve. Let C be the length of the chord. Let V be the versine of a chord of length C. As AC and DE are two chords meeting perpendicular at common point B, it can be proved from simple geometry that: AB x BC = DB x BE Or V (2R-V) = C/2 x C/2 Or 2RV-V2 = C2/4 V being very small, V2 can be neglected, 2 RV = C2/4 or V = C2/8R…………(i) In the above equation V, C & R are in the same unit say meter or cms. This general equation can be used to find out the versines, once the chord and radius of a curve are known. 5.4 Determination of Degree of the Curve in the field. For determining the degree of the curve in the field, a chord length of either 11.8 meters or 62 ft. is adopted. These are the chord lengths, where the relationship between the degree and versine of a curve is very simple as indicated below:- (a) Versine in cm on a chord of 11.8 meters = Degree of the curve. (b) Versine in inches on a chord of 62 ft. = Degree of the curve. This important relationship is utilized in finding out the degree of the curve at any point by measuring versines either in cms on a chord of 11.8 meters length or in inches on a chord of 62 feet length. The curve is of as many degree as there are cms or inches of the versine of the above chord lengths. 5.5 Elements of a circular curve (Fig. 5.2) In the figure 13.2 AO and BO are two tangents of the circular curve which meet or intersect at a point O, called point of intersection of apex. T1 and T2 are the points where the curve touches the tangents and are called tangent points (T.P.) OT1 and OT2 are the tangent length of the curve and are equal in case of a simple curve. T1 T2 is the long chord and EF is the versine on the same. Angle AOB formed between the tangent line AO an OB is called the angle of intersection (I) and the angle BOO1 is the angle of defection. The following are some of the important relations between various elements (Fig. 5.2). (i) <I+<f = 1800 (ii) Tangent length = OT1 = OT2 = R tanf/2 (iii) T1 T2 = length of the long chord = 2 R sinf/2 (iv) Length of the curve = 2ᴫR/360 x f = ᴫRf/180 5.6 SETTING OUT A CIRCULAR CURVE A circular curve is generally set out by any of the following methods. 5.6.1 Tangential Offset Method (Fig.5.3) This is the method employed for setting out a short curve of about 100 meter (300 ft.) length. It is generally used for laying turn-out curves. In the Fig.5.3, let PT be the straight alignment and let T be the tangent point for a curve of known radius. Let AA’, BB’, CC’, etc. be the perpendicular offsets from the tangent. It can be proved by simple geometry that: 2 Value offset O1 C1 /2R Where C1 is the length of the chord along the tangent. 2 2 2 Similarly O2 = C2 /2R; O3 = C3 / 2 R; On = Cn /2R The various steps involved in laying out the curve are as follows: (1) Extend the straight alignment PT to TO with the help of a ranging rod. TO is now the tangential direction. (2) Measure out lengths equal to C1, C2, C3, etc. along the tangential direction and calculate the lengths O1, O2, O3, etc. for these lengths as per the formula given above. For simplicity purposes the values of C1, C2, C3, etc. may be taken in multiples of 3 m or so. (3) Measure the perpendicular offsets O1, O2, O3, etc. from the points A, B, C, etc. and thus locate the points A’.B’.C’. etc. on the curve. Sometimes in the actual practice, it becomes difficult to extend the tangent length beyond a certain point due to some obstruction or as the length of the curve increases, the offsets become too large to be measured accurately. 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, the offsets are measured at fixed distances from the newly drawn out tangent. 5.6.2 Long Chord Offset Method (Fig.5.4) The method is employed for laying out curves of short lengths. In this case it is necessary that both the tangent points are allocated in such a way that the distance between them can be measured and the offsets taken from the long chord. In the field long chord is first marked on the ground and its length measured Points A, B, C, etc. are then fixed by dividing this long chord in 8 equal parts. The values of perpendicular offsets from the long chord are then measured and points located on the curve. The method is used rarely in the field on the railways 5.6.3. Chord deflection method (Fig.5.5) The chord deflection method is one of the most popular methods of laying the curves on the Indian Railways. The method is particularly suited to confined situations as most of the work is done in the immediate proximity of the curve. In the Fig.5.5, T2 be the tangent point and A,B,C and D etc. to be successive points on the curve. Let this length of chords T1A, AB, BC and CD be X1,X2, X3, and X4. In actual practice, all chords are taken of equal lengths and let their value be c. the last chord may be of different length and its value be “c1. It can be proved by simple geometry that. 2 1st offset A’A = X21/2R =C /2R 2 2nd offset B’B =X1X2/2R + X22/2R = C /2R 2 3rd offset C’C =X2X3/2R + X23/2R = C /2R 2 Last offset N’N =Xn-1Xn/2R + X2n/2R = CC1/2R + C1 /2R =C1(C+C1)/2R Note: Setting of various points on the curve has to be done with great precision because if any point is inaccurately fixed, its error is carried forward through all subsequent points. 5.6.4 Theodolite Method (Fig.5.6) The theodolite method is a most popular method for setting out a curve on the Indian Railways, particularly when accuracy is required. This method is also known as Rankine’s Methods of tangential angles. In this method the curve is set out by tangential angles with the help of a theodolite and a chain or tape. In the figure 4.6 let A, B,C & D etc. be the successive points on the curve having lengths T1A=x1, AB = x2, BC = x3, CD=x4 etc. Let δ1, δ2, δ3, δ4 be the tangential angles OT1A, AT1, B, BT1C AND CT1D which the successive chords made amongst themselves. Let Δ1, Δ2, Δ3, Δ4 be the deflection angles (OT1A, OT2B, OT3C and OT3D) of chord from the deflection line Tangential angle for x ft chord = D/2 x 1/100 x degree = 5730/2 x R x 1/100 x 60 x minutes = 1719 x/R Where deflection angle in minutes, X = chord in feet, R = radius in feet. It would be seen that, Δ1 = δ1 Δ2 = δ1 + δ2 = Δ1+ δ2 Δ3 = δ1 + δ2 + δ3 = Δ2 + δ3 etc. The procedure followed for setting of the curve is as indicated below (1) Set the theodolite on the tangent point T1 and sight it in the direction to T2O. (2) Rotate the theodolite by an angle which is already calculated and set the line T1A1. (3) Measure the distance X1, on this line T1 A1 to locate the point A. (4) Now rotate the theodolite by a deflection angle to sight it in the direction T1B1 and locate the point B by measuring AB as the chord length X2.
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