Curves and Superelevation

Mr Manivel M

Department of Civil Engineering

Pandit Deendayal Petroleum University

Introduction• Horizontal curves are provided when a change

in the direction of the track is required andvertical curves are provided at points wheretwo gradients meet or where a gradient meetslevel ground.

• To provide a comfortable ride on a horizontalcurve, the level of the outer rail is raised abovethe level of the inner rail. This is known assuperelevation.

Circular Curves

Maximum permissible degree of curves

Setting Out a Circular Curve

Superelevation

• Superelevation or cant Superelevation or cant(Ca) is the difference in height between the outerand the inner rail on a curve.

• It is provided by gradually lifting the outer railabove the level of the inner rail.

• The inner rail is taken as the reference rail and isnormally maintained at its original level.

• The inner rail is also known as the gradient rail.

• Actual cant for BG track limited to 165mm

The main functions of superelevation

(a) To ensure a better distribution of load onboth rails

(b) To reduce the wear and tear of the rails androlling stock

(c) To neutralize the effect of lateral forces

(d) To provide comfort to passengers

Equilibrium speed• When the speed of a vehicle negotiating a curved

track is such that the resultant force of the weightof the vehicle and of radial acceleration isperpendicular to the plane of the rails, the vehicleis not subjected to any unbalanced radialacceleration and is said to be in equilibrium.

• This particular speed is called the equilibriumspeed. The equilibrium speed, as such, is thespeed at which the effect of the centrifugal force iscompletely balanced by the cant provided.

Maximum permissible speed

• This is the highest speed permitted to a trainon a curve taking into consideration the radiusof curvature, actual cant, cant deficiency, cantexcess, and the length of transition.

• On cruves where the maximum permissiblespeed is less than the maximum sectionalspeed of the section of the line, permanentspeed restriction becomes necessary.

Cant deficiency

• Cant deficiency (Cd) occurs when a traintravels around a curve at a speed higher thanthe equilibrium speed. It is the differencebetween the theoretical cant required for suchhigh speeds and the actual cant provided.

Cant excess

• Cant excess (Ce) occurs when a train travelsaround a curve at a speed lower than theequilibrium speed. It is the differencebetween the actual cant provided and thetheoretical cant required for such a low speed.

• The limiting values of cant excess have alsobeen prescribed. Cant excess should not bemore than 75 mm on broad gauge and 65 mmon metre gauge for all types of rolling stock.

Cant gradient and cant deficiency gradient

• These indicate the increase or decrease in thecant or the deficiency of cant in a given lengthof transition.

• A gradient of 1 in 1000 means that a cant or adeficiency of cant of 1 mm is attained or lostin every 1000 mm of transition length.

Rate of change of cant or cant deficiency

• This is the rate at which cant deficiencyincreases while passing over the transitioncurve, e.g., a rate of 35 mm per second meansthat a vehicle will experience a change in cantor a cant deficiency of 35 mm in each secondof travel over the transition when travelling atthe maximum permissible speed.

Centrifugal Force on a Curved Track

• A vehicle has a tendency to travel in a straightdirection, which is tangential to the curve,even when it moves on a circular curve. As aresult, the vehicle is subjected to a constantradial acceleration:

Radial acceleration = g = V2/R

• where V is the velocity (metres per second)and R is the radius of curve (metres).

• This radial acceleration produces a centrifugalforce which acts in a radial direction away fromthe centre. The value of the centrifugal force isgiven by the formula

• Force = mass × acceleration

F = m × (V2/R)

= (W/g) × (V2/R)

• where F is the centrifugal force (tonnes), W is theweight of the vehicle (tonnes), V is the speed(metre/sec), g is the acceleration due to gravity(metre/sec2), and R is the radius of the curve(metres).

• To counteract the effect of the centrifugal force,the outer rail of the curve is elevated withrespect to the inner rail by an amount equal tothe superelevation

• A state of equilibrium is reached when both thewheels exert equal pressure on the rails and thesuperelevation is enough to bring the resultantof the centrifugal force and the force exerted bythe weight of the vehicle at right angles to theplane of the top surface of the rails. In this stateof equilibrium, the difference in the heights ofthe outer and inner rails of the curve known asequilibrium superelevation.

Equilibrium Superelevation

• where e is the superelevation in millimetres, Vis the speed in km/h, R is the radius of thecurve in metres, and G is the dynamic gaugein millimetres,

• which is equal to the sum of the gauge andthe (C/C) width of the rail head in millimetres.

• This is equal to 1750 mm for BG tracks and1058 mm for MG tracks.

Negative Superelevation

• When the main line lies on a curve and has aturnout of contrary flexure leading to a branchline, the superelevation necessary for theaverage speed of trains running over the mainline curve cannot be provided.

• AB, which is the outer rail of the main linecurve, must be higher than CD.

• For the branch line, however,CF should behigher than AE or point C should be higherthan point A. These two contradictoryconditions cannot be met within one layout.

• In such cases, the branch line curve has anegative superelevation and, therefore,speeds on both tracks must be restricted,particularly on the branch line.

Safe Speed on CurvesMartin’s formula: Earlier

New Formula for Determining Maximum PermissibleSpeed on Transitioned Curves

New Formula for Determining Maximum PermissibleSpeed on Transitioned Curves

Maximum Permissible Speed on a Curve

• The maximum permissible speed on a curve is theminimum value of the speed that is calculatedafter determining the four different speed limitsmentioned here.

• The first three speed limits are taken into accountfor the calculation of maximum permissiblespeed, particularly if the length of the transitioncurve can be increased.

• For high-speed routes, however, the fourth speedlimit is also very important, as cases may arisewhen the length of the transition curve cannot bealtered easily.

(i) Maximum sanctioned speed of the section

• This is the maximum permissible speedauthorized by the commissioner of railwaysafety. This is determined after an analysis ofthe condition of the track, the standard ofinterlocking, the type of locomotive androlling stock used, and other such factors.

(ii) Maximum speed of the section taking into consideration cant deficiency

• Equilibrium speed is decided after takingvarious factors into consideration and theequilibrium superelevation (Ca) calculated.The cant deficiency (Cd) is then added to theequilibrium superelevation and the maximumspeed is calculated as per this increasedsuperelevalion (Ca + Cd).

(iii) Maximum speed taking into consideration speed of goods train and cant

excess

• Cant (Ca) is calculated based on the speed of slowmoving traffic, i.e., goods train. This speed isdecided for each section after taking variousfactors into account, but generally its value is 65km/h for BG and 50 km/h for MG.

• The maximum value of cant excess (Ce) is addedto this cant and it should be ensured that the cantfor the maximum speed does not exceed thevalue of the sum of the actual cant + and the cantexcess (Ca + Ce).

(iv) Speed corresponding to the length of the transition curves

• This is the least value of speed calculated after taking into consideration the various lenths of transition curves given by the formulae listed in Table 13.6.

The following points may be noted when calculating the maximum permissible speed on a curve.

a) Criterion (iv) is to be used only in cases where the length of thetransition curve cannot be increased due to site restrictions.The rate of change of cant or cant deficiency has beenpermitted at a rate of 55 mm/sec purely as an interim measurefor the existing curves on BG tracks.

(b) For high-speed BG routes, when the speed is restricted as aresult of the rate of change of cant deficiency exceeding 55mm/sec, it is necessary to limit the cant deficiency to a valuelower than 100 mm in such a way that optimum results areobtained. In this situation, the maximum permissible speed isdetermined for a cant deficiency less than 100 mm, but gives ahigher value of the maximum permissible speed.

• Example 13.1 Calculate the superelevation andthe maximum permissible speed for a 2° BGtransitioned curve on a high-speed route with amaximum sanctioned speed of 110 km/h. Thespeed for calculating the equilibriumsuperelevation as decided by the chiefengineer is 80 km/h and the booked speed ofgoods trains is 50 km/h.

• Example 13.2 Calculate the superelevation,maximum permissible speed, and transitionlength for a 3° curve on a high-speed BGsection with a maximum sanctioned speed of110 km/h. Assume the equilibrium speed to be80 km/h and the booked speed of the goodstrain to be 50 km/h.

Reference

• Railway Engineering book written by

SATISH CHANDRA,Professor, Department of Civil Engineering,Indian Institute of Technology Roorkee.