NOTES ON CURVES FOR RAILWAYS BY V B SOOD PROFESSOR BRIDGES INDIAN RAILWAYS INSTITUTE OF CIVIL ENGINEERING PUNE- 411001 Notes on —Curves“ Dated 040809 1 COMMONLY USED TERMS IN THE BOOK BG Broad Gauge track, 1676 mm gauge MG Meter Gauge track, 1000 mm gauge NG Narrow Gauge track, 762 mm or 610 mm gauge G Dynamic Gauge or center to center of the running rails, 1750 mm for BG and 1080 mm for MG g Acceleration due to gravity, 9.81 m/sec2 KMPH Speed in Kilometers Per Hour m/sec Speed in metres per second m/sec2 Acceleration in metre per second square m Length or distance in metres cm Length or distance in centimetres mm Length or distance in millimetres D Degree of curve R Radius of curve Ca Actual Cant or superelevation provided Cd Cant Deficiency Cex Cant Excess Camax Maximum actual Cant or superelevation permissible Cdmax Maximum Cant Deficiency permissible Cexmax Maximum Cant Excess permissible Veq Equilibrium Speed Vg Booked speed of goods trains Vmax Maximum speed permissible on the curve BG SOD Indian Railways Schedule of Dimensions 1676 mm Gauge, Revised 2004 IR Indian Railways IRPWM Indian Railways Permanent Way Manual second reprint 2004 IRTMM Indian railways Track Machines Manual , March 2000 LWR Manual Manual of Instructions on Long Welded Rails, 1996 Notes on —Curves“ Dated 040809 2 PWI Permanent Way Inspector, Refers to Senior Section Engineer, Section Engineer or Junior Engineer looking after the Permanent Way or Track on Indian railways. The term may also include the Permanent Way Supervisor/ Gang Mate etc who might look after the maintenance work in the track. BC Beginning of Curve EC End of Curve CC Center of Curve L Length of transition Equivalent A conceptual circular curve without considering any circular curve transitions. The transitions are added later on at either end to get the actual proposed curve. Slew Shifting the alignment of curve. It can be outwards or inwards and is generally expressed with a sign depending on the sign convention. Notes on —Curves“ Dated 040809 3 UNDERSTANDING CURVES 1.1 Introduction: Curves form an indispensable component of railway alignment. It is desirable, but not always possible, to follow the straight alignment. The alignment has to be changed by introduction of curve(s) due to various techno-economic reasons such as: 1.1.1 The sources of traffic i.e. ports, industries, mines etc have to be connected 1.1.2 The important towns/ cities near the alignment have to be connected, even by changing the track alignment. 1.1.3 If some area in straight alignment is already built up, or there are obstructions such as water bodies, hills, utility services etc, the track has to be shifted by introducing curves. 1.1.4 Certain geological formations and faults etc are inherently unstable and if the track is laid on these, there will be problems in maintenance. Such formations are avoided by changing the alignment and track is laid on better geological formations. 1.1.5 From technical and economic considerations, rivers shall be crossed at the locations where the river alignment is stable, and length of the bridge to be provided is minimum. The track alignment has to be changed to cross the river at the best possible location. 1.1.6 While climbing steep hills, it is possible that the straight alignment has excessive gradients. In such cases, track length has to be increased by introducing curves so as to keep the gradients within desirable limits. Due to the above and other similar reasons, approximately 16% of track is in curve on BG/MG on Indian Railways, and substantially higher percentage on NG is located on curves. Nature of forces experienced by vehicle is different on curved track as compared to straight track. The forces and their effect on track, vehicle and passengers are required to be systematically studied so that curves are properly designed and easily maintained. Due to the interplay of the forces on a curved track, it has been estimated that the maintenance effort on curves is about 25% extra over that of a straight track. Therefore, understanding the vehicle movement on curves, laying, maintenance, realignment of curves etc is important. With better understanding of curves, track engineers will be in better position to properly manage the curves on the railway system. The curves have, therefore, been described in literature as necessary evil. The railway curves have the property that their radii are generally quite large and this property is used for making many approximations, which are approximately valid especially when seen in comparison with the methods of measurement employed in field. 1.2 Identification of a curve: The change of alignment through curves is best attained through a circular curve. A circular curve has the advantage of uniform curvature i.e. uniform change of direction which makes the task of Notes on —Curves“ Dated 040809 4 management of forces during change of alignment easier. A circular curve used on railway system is identified with the following parameters: • Radius, R or Degree, D • Direction of Curve (LH or RH) The radius R is the radius of the circle at the center line of the track, part of which is the curved alignment for the railway track. Degree of curve D is the angle subtended by an arc of 30.5m length at the centre of the same circle (Fig.1.1). The circumference of the curve, 2* π *R, for which the angle at the center of the curve is 3600. Therefore, the angle subtended by 30.5 m chord (which is taken approximately equal to the 30.5 m arc as the radius of the curve is quite large) at the center of curve can be worked out as follows: 2πR → 360 0 , 2πR 2πRD ∴ → 10 and → D 0 360 360 Since arc length of 30.5m correspeon ds to D 0 , 2πRD 2πRD i.e. = 30.5. → D 0 solving the equation we get 360 360 1750 D = .......... .......... .....(1.1) R The direction of curve is determined by the change in direction as seen in the direction of movement of trains. Left hand (LH) curve is there if the change in direction of the curve is in counterclockwise direction when seen in the direction of travel in multiple lines or in the direction of increasing kilometers in case of single line. Similarly, Right hand (RH) curve is there if the change in direction of the curve is in clockwise direction when seen in the Notes on —Curves“ Dated 040809 5 direction of travel in multiple lines or in the direction of increasing kilometers in case of single line. 1.3 Versine of a Curve: The curves are identified by the degree or radius. But both of these are difficult to measure in the field due to the very large radii of the railway curves. The versine is a very easy measurement which can be used in the field for measurement and rectification of the geometry. The versine of a curve stands for the ordinate from the mid point of a chord on the curve1 . Figure 1.2 illustrates the concept of versine in a curve. If the chord AB of length, C, on a circle having radius R is considered, then R the ordinate EF will be the versine, E v. A C B Now, circle has a property that if V two chords intersect, the product of the two parts is equal for both the F chords. Using this property for the Fig 1.2 chords DF and AB, DE * EF = AE * EB i.e. (2R œ v) * v = (C/2) * (C/2) i.e. 2Rv-v2 = C2/4. Since value of versines is normally in Fig 1.1 millimetres or centimeters, whereas value of R is in metres, v2 is very small compared with 2Rv, and can be neglected. i.e. 2Rv=C2/4 C 2 i.e. v = …………….(1.2) 8R On Indian Railways, measurement of curve is done at nominated points on the curve, which are usually paint marked on the track. These points are called stations. 1.4 Movement of Vehicle on Curved Track: When a vehicle moves over the curved track, following are to be achieved: 1 The versine mentioned here shall not be confused with versed sine which is also sometimes called versine Notes on —Curves“ Dated 040809 6 1.4.1 Continuous change in direction: The change in alignment of a vehicle on a curved track is affected by the rails. The rail closer to the center of track is called inner rail and the rail farther away from the center of curve is called outer rail. The vehicle moving over the curve is pushed in proper alignment by the rails which are changing direction continuously. The leading wheel of a bogie (or trolley) in case of a bogied vehicle and the leading wheel of a vehicle in case of a four- wheeler vehicle moves with positive angularity, attacking the outer rail of the curve. The outer rail causes the wheel to change direction (see figure 1.3). Due to this, there are large lateral forces on the track as well as vehicle. Fig 1.3 1.4.2 Movement without slip: On a curved track, the outer wheel travels with a larger radius as compared to the inner wheel. Therefore, a mechanism is required to ensure that the outer wheel does not slip during movement on a curve. The actual vehicle movement on curve is quite complicated but the vehicle traverses the curved path without appreciable slip due to coning in the wheels, which causes larger diameter to travel on the outer rail and smaller diameter on the inner rail by slight shifting of the center of gravity of the vehicle towards the outer rail.