Geometric design of tracks aims to provide smooth and safe running of trains at maximum speed while carrying heavy loads. This involves proper design of gradients, curvature, and super elevation (cant). There are different types of gradients - ruling gradient which is the maximum gradient permitted, momentum gradient which is steeper and uses train momentum, and pusher gradient requiring extra locomotives. Gradients are designed considering train performance and load. Curvature introduces greater resistance requiring grade compensation of ruling gradients. Super elevation (cant) involves raising the outer rail on curves to counteract centrifugal forces. Equilibrium cant provides equal wheel load distribution. Higher speeds result in cant deficiency which must be limited for passenger safety. Contrary flexures like

1. GEOMETRIC DESIGN
OF THE TRACK

3. Necessity of Geometric Design:
 Smooth & safe running of trains.
 Maximum speed.
 Carrying heavy axle loads.
 Avoid accidents & derailments.
 Less maintenance efforts.
 Good aesthetic value.
There, if all the above elements are properly
designed, the possibility of derailments due to defects
in the track can be avoided.

4.  Gradients
The amount of slope in longitudinal direction of railway track
is called gradient or grade.
 Gradients are provided to negotiate the rise or fall in the level
of the railway track.
 Rising gradient rises the track in the direction of movement,
whereas, falling gradient cause the track to go down in the
direction of movement.
 A gradient is represented by the distance traveled for a rise or
fall of one unit.
It is written as; 1 in ‘X’ or 1 in ‘n’ or as percent

6.  Gradients are provided on the track due to the
following reasons.
 To provide a uniform rate of rise or fall as far as
possible.
 To reach the various stations located at different
elevation.
 To reduce the cost of earth work.

7.  Gradients – Types
 Ruling gradient
 Momentum gradient
 Pusher or Helper gradient
 Gradients in Station Yards

8. A) Ruling gradient: This is the design gradient basically,
because it is determined on the basis of the performance
of the locomotive and at the same time, it tries to look at
the total amount of load which that locomotive can take
up along with it, while negotiating any gradient without
any loss or major loss in the speed of the movement.

9.  Ruling gradient;
 It is a maximum gradient (steepest gradient), which may
be permitted on the section of the track.
 It is determined by maximum load that a locomotive can
haul with maximum permissible speed.
 Extra pull required by locomotive on gradient with ‘ ’
inclination. P = W Sin = W * gradient
For Ex:-
Weight of train (W)=500 tonnes
Gradient = 1 in 100
Extra power (P) = 5 tonnes

10.  Ruling gradient with one locomotive.
• In Plane area = 1 in 150 to 1 in 250
• In hilly area = 1 in 100 to 1in 150
 Once a ruling gradient is specified for a section,
then all other gradients provided in that section
should be flatter than the ruling gradients (after
making due to compensation for curvature)

11.  Gradients – Types ….
b) Momentum gradient:
 Momentum gradient steeper than ruling gradient that is
overcome by momentum gathered while having a run in
plane or on falling gradient in valleys.
 Use additional kinetic energy received during run on a
section.
 No obstruction like signals are provided on section with
these gradients.(means the train should not be stopped
at that territory)

13. For example, in valleys, a falling gradient is usually
followed by rising gradient acquires sufficient
momentum.
This momentum gives additional kinetic energy to
the moving train which would enable the train to
overcome a steeper rising gradient than the ruling
gradient for a certain length of the track.
This rising gradient is called momentum gradient
and this gradient is steeper than ruling gradient.

15. C) Pusher or Helper gradient:
 Gradient steeper than ruling gradient requiring extra
locomotive.
• It reduces the length of a railway section.
• It also reduces the overall cost.
Examples : In Darjeeling Railways 1 in 37 Pusher gradient is
used on Western Ghats, B.G tracks & N.G tracks 1 in 25 is
provided.

17. d) Gradient at Station Yards:
As per as possible the track along the stations & yards
should be level or gradients should be sufficiently low.
• To prevent standing vehicle from rolling & moving
away from the yard due to combined effect of gravity &
strong winds.
• To reduce additional resistive forces required to start a
locomotive to the extent possible.
• Minimum gradient from drainage consideration.
• On Indian railways, maximum gradient permitted is 1 in
400 in station yards & minimum gradient permitted is 1 in
1000

19.  Grade Compensation ( On Curves):-
 If a curve is provided on a track with ruling
gradient, the resistance of the track will be increased on this
curve. In order to avoid resistance beyond the allowable
limits, the gradients are reduced on curves & this reduction in
gradient is known as grade compensation for curves.
In India, Compensation for curvature is given by.
o BG track: 0.04% per degree of curve
o MG track: 0.03 % per degree of curve
o NG track: 0.02 % per degree of curve

20.  Grade Compensation ( On Curves);.
Example :If the ruling gradient is 1 in 250 on a particular section
of B.G & at the same time a curve of 4 degree is situated on this
ruling gradient, what should be the allowable ruling gradient?
Solution:
As per Indian railway recommendation, the grade compensation
for of B.G track is 0.04% per degree of curve.
Therefore, Grade compensation for 4 degree curve
= 0.04 * 4 = 0.16%
Ruling gradients is 1 in 250
= 1/250 *100 = 0.4%
Therefore, Required ruling gradient or Actual gradient
= Ruling gradient – grade compensation
= 0.4- 0.16
= 0.24% or 1 in 417

21.  Radius & Degree of a curve:-
 The main curved portion of a railway track is kept circular
i.e, the radius at every point of the curve is same.
 The radius of a railway curve is sometimes represented by the
degree of the curve.
Degree of a railway curve:-
The angle subtended at the center of the curve by an arc of 30.0m
length is defined as a defined as a degree of the curve.
Relationship between the radius & degree of a curve.
Let, R=radius of the curve in meters, D = degree of the curve
Now, total circumference 2πR makes
360o at the center.
Therefore, for 30.0 m arc makes an angle.
D/30 = 360/2πR
D = 360*30/2πR
D = 1718.87/R
D = 1720/R

22.  Safe speed on curves...
 Safe speed for all practical purposes means a speed which
is safe from the danger of overturning & derailment with a
certain margin of safety. This speed, to negotiate curves
safely, depends upon the following factors.
• The gauge of track.
• The radius of the curve.
• Amount of super elevation provided.
In India, using safe speed V in kmph, formula.
B.G & M.G : V = 4.35 R- 67
N.G: V = 3.65 R- 6

23.  Superelevation or Cant….
When a train moves round a curve, it is subjected to a centrifugal
force acting horizontally at the centre of gravity of each vehicle
radially away from the centre of the curve. This increases the
weight on the outer rails.
To counteract the effect of centrifugal forces, the level of
the outer rail is raised above the inner rail by a certain amount to
introduce the centripetal force. This raised elevation of outer rail
above the inner rail at a horizontal curve is called “Cant”.

24.  Superelevation or Cant….
 It is the difference in elevation (or height) between the outer
rail and inner rail at a horizontal curve is called “Cant”.
 Inner rail is taken as a reference rail & is maintained at its
original level.
 Inner rail is also known
as ‘gradient rail’.
 It is denoted by “e”.

25.  Superelevation or Cant….
 Objectives of Superelevation.
 To neutralize the effect of centrifugal force.
 Equal distribution of wheel loads.
 Providing smooth track, improving passenger comfort.
 To reduce wear & tear of the rails & rolling stock.

28.  Equilibrium Cant….
 The cant or super elevation as given by equation e=(GV2/1.27R)
cm, the load carried by both the wheels will be the same, the
springs will be equally compressed & the passengers will not
tend to lean in either direction, such cant is known as the
“Equilibrium cant”.

29.  Equilibrium Cant….
 The cant is provided on the basis of average speed of the
trains.
 The majority of Indian Railways provide super elevation for
equilibrium speed or average speed under condition of level
track.
Average speed or Weighted average speed
= n1V1+ n2V2+ n3V3 / n1+ n2+ n3
Where, n1, n2,n3 = Number of trains
V1, V2,V3 = Speed of trains in kmph

30.  Equilibrium Cant….
 The cant is provided on the basis of average speed of the
trains.
 The majority of Indian Railways provide super elevation for
equilibrium speed or average speed under condition of level
track.
Average speed or Weighted average speed
= n1V1+ n2V2+ n3V3 / n1+ n2+ n3
Where, n1, n2,n3 = Number of trains
V1, V2,V3 = Speed of trains in kmph

31.  Cant Deficiency (Cd)….
 The equilibrium cant is provided on the basis of equilibrium speed
(or Average speed) of different trains. But this equilibrium cant or
super elevation falls short of that required for the high speed trains.
This shortage of cant is called “ Cant Deficiency”.
 In other words, it is the difference between the equilibrium cant
necessary for the maximum permissible speed on a curve and the
actual cant provided. Higher cant deficiency causes more
unbalanced centrifugal force and discomfort to the passengers.
 Maximum value of cant deficiency prescribed for Indian Railways.
B.G = 7.6cm, M.G = 5.1cm, N.G = 3.8cm

32.  Negative Superelevation….
 When a branch line diverges from a main line on a curve of
contrary flexure, the super elevation necessary for the average
speed of trains running over the main line, cannot be
provided.
 The speed of the trains over the diverging track and main line
track has to be reduced considerably.
 The reason for the reduction of speed is that, on the branch
line track, the inner rail remains at higher level than the outer
rail.

33.  Negative Superelevation….
D
Outer rail
Inner rail
Inner rail

34.  Negative Superelevation….
 Ref fig: AD which is the outer rail of the main line curve must be
higher than inner rail BC or in other words, the point A should be
higher than point B.
 For the branch line, however, BE should be higher than AF or the
point B should be higher than point A.
 These two contradictory conditions cannot be meet at the same
time within one layout . So, outer rail BE on branch line is kept
lower than the inner rail AF. In such case branch line curve has a
negative super elevation & therefore speeds on both tracks must be
restricted, particularly on branch line.

35.  Negative Superelevation….
 Calculation of restricted speed on the main line & branch line.
 Super elevation for branch line can be calculated
 Calculate equilibrium super elevation for branch line (eb).
 Find super elevation, X = eb – Cd for branch line.
This is to be provided on main line.
 To calculate the maximum permissible speed for main line,
Cant should be, em = X + Cd
 Super elevation for main line can be calculated
 Calculate equilibrium super elevation for main line (em).
 Find super elevation, X = em – Cd for main line.
 Provided cant on branch line as ( -X ).
 To calculate the maximum permissible speed for branch line,
Cant should be, eb = (-X ) + Cd

36.  Transitional Curves….
 Transition curve is defined as a curve of parabolic nature
which is introduced between a straight and a circular curve or
between two branches of a compound curve.
 It is necessary to provide an easy change from a tangent to the
radius selected for a particular curve.
 It is essential that the curvature and superelevation in the
outer rail and the curvature in the inner rail are attained
gradually, by the use of easement curve or transition curve.

37.  Transitional Curves….
 Objects:
 Primary objects:
 To decrease the radius of curve gradually from infinite at the
straight to that of circular curve of selected radius.
 To attain gradual rise for the desired superelevation. This is
applicable for outer rail.
 Secondary objects:
 The gradual increase or decrease of the centrifugal force on
the vehicle by use of this curve provides smooth running of
vehicle & comfort to the passengers.
 No sudden application, so the chances of derailment are gently
reduced.

38.  Transitional Curves….
 Requirements of a Transition Curve;
 It should be perfectly tangential to the straight.
 The length of the transition curve should be such that
curvature may increase at the same rate as the superelevation .
 This curve should join the circular arc tangentially i.e.,
curvature of transition curve should conform with that of
circular curve.

39.  Superelevation for B.G, M.G & N.G….
 Relationship of superelevation (e), with gauge(G),
speed (V) and radius of the curve ( R ).
W= Weight of moving vehicle in kg.
v = Speed of vehicle in m/sec
V = Speed of vehicle in km.ph
R = Radius of curve in meters
G = Gauge of track in meters
g = Acceleration due to gravity in m/sec2
α = Angle of inclination
S = Length of inclined surface in meters
Centrifugal force is given by the following expression.
F = Wv2/gR -------------(1)
Now resolving the forces along the inclined surface we get
F cosα = W sinα -------- (2)

40.  Superelevation for B.G, M.G & N.G….
F = Wv2/gR , cosα = G/S & sinα = e/S
Therefore equation (2) becomes Wv2/gR * G/S = W*e/S
Therefore, e = v2/gR * G meters
Where, v in m/sec.
= G ( 0,278V)2/9.81R
= GV2/127 R meter
e = GV2/1.27 R centimeter
For B.G., e = GV2/1.27 R = 1.676 V2/1.27 R centimeter
For M.G., e = GV2/1.27 R = 1.0 V2/1.27 R centimeter
For N.G., e = GV2/1.27 R = 0.762 V2/1.27 R centimeter