1 Geometric Design of

Railway Track

Chapter 1
1.1 INTRODUCTION
y The concept of railway is the movement of train wagons or passenger
bogies fitted with steel wheels running over two parallel steel rails of
the railway track.
y The main advantage of the railway is that it can transport many
passengers or large quantities of goods at a time.
y Rail transportation is economical.
y Indian Railways was introduced in 1853. The Network of Indian railway is
one of the largest in the world.

1.2 GAUGE IN RAILWAY TRACK

Definitions

Gauge: The clear distance between inner or running faces of two-track

rails is called the gauge in railway track. Gauge is also defined as the
minimum distance between the inner running of gauge faces of the
two rails.

S.No. Type of Gauge Gauge Width

1) Standard gauge 1.67 m

or
Broad gauge (BG)

2) Metre gauge (MG) 1.0 m

3) Narrow gauge (NG) 0.762 m

4) Feeder track gauge (LG) 0.610 m

or
Light gauge
Geometric Design of Railway Track
Table 1.1 Different Types of Gauge

Factor which governs the choice of the gauge are:

1) Cost of construction
2) Volume and nature of traffic
3) Development of the area
4) Physical features of the country
5) Speed of movement

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Chapter 1

1.2.1 Uniformity of Gauges

y Uniform gauges should be used throughout the country to avoid
difficulties experienced in a non-uniform system.
y The advantages of uniformity of gauges are:
1) Transfer of passengers and goods from the vehicles of one gauge
to another results in the delay. Additional costs and efforts can be
avoided if uniform gauges are adopted.
2) Negligible breakage of goods, as there is no transshipping.
3) Frequent unloading and loading can be prevented, which reduces
the labour expense.
4) During shifting from one train to another, there might be events of
robbery and malposition, which can be avoided in uniform gauges.
5) Provisions of large sheds are not needed to store goods.
6) Surplus wagons can be used anywhere if the gauge is uniform.
7) Uniform gauges help in the effective utilisation of locomotives on all
the tracks.
8) Additional expense because of duplication of equipment like
platforms, sanitary arrangements, clocks, etc., can be evaded.
9) During military emergencies, rapid movement of personnel and
equipment can be achieved.
10) New rolling stock, fresh construction and widening of bridges and
tunnels are required as a conversion of one gauge into another at a
later stage is uneconomical.
11) There is a waste of time for passengers because of delays in the
arrival of trains at the junction, where a change of gauge is involved.

1.3 RAILS

Definitions
Geometric Design of Railway Track

Rails: Steel girder which provides a smooth and hard surface for
movement of locomotive and wagons are called rails.

1.3.1 Functions of rails

y Rails deliver an invariable, smooth and hard surface which results in
minimum friction between the steel rails and steel wheels during heavy
loads movement.
y Rails bear the various stresses, i.e., stresses arising from the lateral
force due to braking, vertical force due to heavy loads and thermal
stresses.

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 Chapter 1
y The material used in rails should be such that it offers minimum wear
so that frequent replacement and wearing failure of rails is avoided.
y Rails transmit the loads to sleepers and consequently reduce pressure
on the ballast and formation below.

Types of rail sections

Double headed rails Bull headed rails Flat footed rails

Head
F ishing
Head
angle

Bull head Height

Web Web Web

Foot

Double headed rail Bull headed rail Flat footed rail

Fig. 1.1 Types of Rail Section

C
Compression

E
1:3
side

1:3

D
Neutral Horizontal axis Geometric Design of Railway Track
A
Vertical axis
Height

Tension
side

1 :6 1:6
F

Fig. 1.2 Standard Flat Foot Section

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Chapter 1

y Selection of rails
Since weight per unit length is the vital parameter in the design of rail.
The following factors govern the suitability of weight of rail:
a) Speed of the train
b) The gauge of the track
c) The axle load and nature of traffic
d) Type of rails
e) Spacing of sleepers (sleeper density)
f) Maximum permissible wear on top of rails (5% of the weight of rail
is allowed)
y Kinks in rails
When the end of adjoining rails moves slightly out of position, ‘shoulder‘
or ‘kinks’ are formed.
Kinks formation

Loose packing Defects in gauge Defect in cross Uneven wear

of joints and alignment level at joints of rail heads

y Length of Rails
The rails of larger length are preferred because they are economical and
provide more strength. The joint between two rails is the weakest point
of a track. For fewer joints, the less number of fish plates are required,
which means minimal maintenance cost along with the smoother and
comfortable ride.
On Indian Railways, the standard lengths are:
12.80 m (42 ft) for B.G. (say 13 m)
11.89 m (39 ft) for M.G. (say 12 m)
Geometric Design of Railway Track

y Hogged rails

Definitions

Hogged rails: The rails which get deflected due to the battering action
of wheel over the ends of the rails are called hogged rails.

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 Chapter 1
y The hogging at the end of the rail is due to loose packing under the
joints or a loose fish plate.

Measure to rectify hogging

Cropping Replacing Welding De-hogging

1.3.2 Buckling of rails

y The phenomenon in which a track from its original position goes out is
known as buckling of rails, also in some cases where the expansion of
rails in hot weather is prevented, then alignment changes and leads to
the buckling of rails.
y The causes of buckling are as follows:
i) Insufficient expansion gap, or
ii) The fish plates being bolted so tight that the rials are not allowed
to slip or expand.
iii) Excessive expansion resulting from the longer welded rails on weak tracks.

y Various precautions or measures to be taken to prevent buckling are:

i) The ballast section, sleeper density and the rail section must be
designed and checked for safety under various stresses.
ii) Number of welded rails should not be very large.
iii) Provision of steel sleepers or anchoring of welded rails should be done.
iv) Proper lubrication of contact surfaces of fish plates and rails should
be done at regular intervals (once in a year or two).
v) Expansion gap should be provided by considering the expansion of
rails due to the rise in temperature in that region.
vi) Slight expansion or contraction of rails is allowed; to achieve this
fish bolts are tightened loosely.
Geometric Design of Railway Track
1.3.3 Creep of rails

Definitions

Creep of rails: It is the longitudinal movement of rails with respect to

sleepers in a track.

y Generally, in the direction of dominant traffic, there is a slight movement

of the rail.

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Chapter 1

y Creep is identified from the following observations:

i) Closing of successive expansion spaces at rail joints in the direction
of creep and opening out of joints at the point from where the
creep starts.
ii) Marks on flanges and webs of rails made by spike heads, by scarping
or scratching as the rails slide.
y Theory of Creep:
1) Wave action or wave theory:
y The movement of wheel load causes wave motion. This wave is pushed
by the wheel to force the rail in the direction of traffic.
y As the wheel waves, the lift in front of the moving load is thus carried
forward by the wheels and causes creep, whereas generates the lift at
the rear of the wheel.
2) Percussion theory:
y The creep, according to this theory, is due to the impact of wheels at
the rail and ahead at joints.
y When the wheel leaves the trailing rail and strikes the facing rail end at
each joint, it pushes the rails forward, resulting in creep.
3) Drag theory:
y This theory states that backward thrust on the driving the wheels of the
locomotive of the train has got a tendency to push the rail off the track
backwards and the other wheels of the locomotive and the wheels of
wagons push the rail in the direction of travel.
y The outcome of this theory is the creep of rails in the direction of the
movement of the train.

Creep Formation

Starting, stopping, Expansion or Unbalanced traffic

accelerating and contraction
Geometric Design of Railway Track

deceleration of of rails due to

locomotive temperature
1.3.4 Welding of rails:
1) Purpose of welding:
y Welding of rails serves the following purposes:
i) To increase the length of the rail by joining two or more rails
and thus to reduce the number of joints and requirement of fish
plates, which lead to economy and strength.
ii) To repair the worn out or damaged rails and thus increase their life.

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 Chapter 1
iii) To build up worn-out points and rails on the sharp curves.
iv) To build up the burnt portion of the railhead, which is caused due to
slippage of wheels over the rails or other defects or spots in rail steel.
2) Advantages of welding rails:
y Welding satisfies the condition of the perfect joint and hence
increases the life of the rail.
y There is a reduction in the maintenance cost of the track by about
20 to 40 percent.
y It reduces the creep due to an increase in the length of the rail and,
in turn, friction as well.
y Expansion effect due to temperature is reduced, which in turn also
reduces the creep.
y Due to discontinuity of joints, a source of track weakness is reduced.
The defects, such as hammering at rail joints, displacement of joints,
disturbance in alignment and running surface, which result in bad
riding quality, are eliminated.
y Long rail lengths being heavier, dampen the intensity of high
frequency vibration due to moving loads.
y Welding increased the life of rails due to a decrease in the wear of
rails at joints.
y Welding facilitates track circuiting on electrified tracks.
y Welded rails provided on large bridges for the span length are helpful
as they result in better performance.
y Welded rails provision on curves is under investigation. However,
maximum curve length may be welded depending upon the resistance
and lateral displacement of the track.
3) Length of welded rails:
y Joints in the rail track are the weakest part.
y The lesser the number of joints lesser is the cost of maintenance
required.
y So, the length of rails should be as long as possible, but the
limitations are:
i) Facilities available for rail manufacturing at a reasonable cost. Geometric Design of Railway Track
ii) Length of rail that can be transported using the longest wagon
available.
iii) Limitations arising due to the availability of machinery for
handing and shifting of long rails.
4) Welded rails:
y Increase in length of rail due to expansion
dl = l × a × Dt
l = length of rail
a = Coefficient of expansion in per °C
Dt = Rise in temperature above the temperature at the time of
construction
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Chapter 1

y The rail fittings have a tendency to hold down the rails to the sleeper,
hence, restricting their movement and transferring the longitudinal
stress in the transverse and vertical direction.
y These stresses way pull the alignment of the track due to buckling
and may result in derailment of trains.
Consider:
E = Modulus of elasticity of steel (kg/cm2)
A = Cross-sectional area of rail (cm2)
F = Force in kg, required to prevent expansion due to change in
temperature.
F.l = dl.A.E
Fl = l × a × Dt.A.E
F = α∆tAE
y This is referred ‘Locking up of longitudinal thermal stresses’ and has
proved that longitudinal movement of rails takes place only at the
ends (known as breathing length) due to temperature variation and
the absence of the resisting force of track while the central portion
of the rails remain fixed (known as fixed length) due to resistance
offered by rails by means of sleeper, rail fastening and ballast against
any expansion due to change in temperature.

Example 1.1: If the temperature rise is 29°C, then the increase in the length
of rail of 12.8 m will be _____. [Given: a= 1.18 × 10–5 per °C]
a) 4.38 mm b) 3.97 mm
c) 4.12 mm d) 5.83 mm

Sol:
Increase in length (dl) = l × a × Dt
= 12.8 × 1.18 × 10–5 × 29
= 4.38 × 10–3 m = 4.38 mm
Correct answer is a).
Example 1.2: Find the length of track
Geometric Design of Railway Track

a) to overcome temperature stress

b) to prevent creep for equilibrium using the following data
A = 600 cm2
a = 1.16 × 10–5/ °C
E = 21.45 × 105 kg/cm2
Change in temperature = 30 °C
Consider 700 kg/m resistance to tack movement?

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 Chapter 1
Sol:
Force required to prevent the expansion due to change in temperature
F = atAE
F = 1.16 × 10–5 × 30 × 60 × 21.45 × 105
F = 44787.6 kg
a) Length of track to overcome temperature stress
F
(Lt) =
resistance of track
44787.6
Lt =
700
Lt = 63.98 m
b) Length of welded track to prevent creep for equilibrium
= 2Lt
= 2 × 63.98
= 127.96 m

1.4 SLEEPERS
1) Function of sleepers:
y Hold the rail to the correct gauge.
y To hold the rail at the proper level, i.e., at turnouts and cross-overs.
y To act as an elastic medium in between the ballast and rails to
absorb the blows and vibration of moving loads.
y To distribute the load from the rails to the underlying ballast or to
the girders in case of bridges.

Definitions

Sleepers: Sleepers are members generally laid transverse to the rail

on which the rails are supported and fixed, to transfer the load from Geometric Design of Railway Track
rails to the ballast and subgrade below.

y To provide longitudinal and lateral stability to the permanent track.

y Sleepers help to rectify track geometry during service life.

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Chapter 1

2) Types of sleepers:

S.No. Point of Wooden Sleeper Cast Iron Steel Sleeper Concrete

Comparison Sleeper Sleeper
1) Cost per Low Medium High Depends
sleeper upon design
2) Life 10–15 years 35–50 years 35–50 years 40–60 years
for untreated
sleepers.
20–25 years
for treated
sleepers.
3) Weight per Low Heavy Medium Depends
sleeper for upon design
B.G. track but heavier
than other
4) Maintenance Higher than Minimum Moderate Moderate
cost other sleepers
Geometric Design of Railway Track

5) Overall Cheaper in the Costlier in Same as for Under trail

economy initial cost but first cost but C.I.
expensive in cheaper in
long run. long run.
6) Handling Not liable to Liable Not liable to With
break under to break break, if clip improved
rough handling under rough and bolts are design, not
handling used liable to
break

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 Chapter 1
S.No. Point of Wooden Sleeper Cast Iron Steel Sleeper Concrete
Comparison Sleeper Sleeper
7) Track fittings Requires less Required Requires less Requires less
fittings more fittings fittings fittings
8) Elasticity Good Not so good Not so good Not good
9) Laying and Easiest Difficult Easy due to Difficult
Relaying due to large light weight. by manual
number of labour. Easy
fittings if mechanical
devices are
used
10) Rigidity of Poor both Better than Better than Best
track laterally and timber timber because of
longitudinally sleepers sleepers heavy dead
weight
11) Suitability of Generally Suitable only Suitable only Suitable for
track suitable in for stone for stone any location
all locations ballast. ballast. on railway
except areas Unsuitable in Unsuitable track.
of vermins station yards for station
and white yards and
ants. Specially coastal areas
suitable for
points, bridges,
station yards
and level
crossings.
12) Track Best Restricted, Restricted Moderate
circuiting insulating insulating
Geometric Design of Railway Track
pads are pads are
necessary necessary
13) Scrap value Very little Highest Next to C.I. Nil
14) Gauge Does not Slight shift Maintains Depends
maintain proper in gauge due proper gauge upon design,
gauge to play in improved
the bar and design
socket. maintains
proper
gauge.

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Chapter 1

S.No. Point of Wooden Sleeper Cast Iron Steel Sleeper Concrete

Comparison Sleeper Sleeper

15) Renewal Easy Difficult Difficult Difficult

16) Gauge Difficult Easy Easy Easy

adjustment

17) Creep of rials Heavy, anchors Fittings No difficulty No anchor is

are necessary in function in clip and necessary
large number as creep bolt type
anchors.
Requires
constant
attention
in C.S.T-9
sleepers

Table 1.2 Comparision of Different Types of Sleepers

3) Spacing of sleepers and sleeper density:
y The space between two adjacent sleepers decides the effective
span of the rail over the sleepers
y Dependency of sleeper density is as shown:
D

r
y The number of sleepers per rail varies in India from M+4 to M+7 for
the main track, here M is length of rail in meters.
y Length of rail for B.G track is 12.8 m approx. 13 m.

Example 1.3: Which of the following is the expression for sleeper density for
Geometric Design of Railway Track

a B.G. track if 18 sleepers are used under a rail length?

a) M + 4   b) M + 5   c) M + 6        
d) M + 7
Sol: b)
Length of rail in BG track = 13 m
Sleeper density = M + x
M → length of rail
18 = M + x
18 = 13 + x
x = 5
So, sleeper density is M + 5 Correct answer is b).

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 Chapter 1
Example 1.4: Using sleeper density of M + 6 find out the number of railway
sleepers required for a track of 648 meters in length. (BG Track).
Sol:
Length of each rail on a B.G track = 12.8 m ~ 13 m
Total number of rails required will be
648
=
13
= 49.846 ~ 50
Sleeper density = M + 6
Number of sleeper under each rail = 13 + 6 = 19
Total number of sleeper = 50 × 19 = 950 sleepers

1.5 TRACK FITTING USED IN PERMANENT WAY

Geometric Design of Railway Track

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Chapter 1

1.6 BALLAST

Definitions
Ballast: Granular material is placed and packed below and around the
sleepers. This granular material transmits wheel load from sleepers to
the formation. Ballast also helps in drainage in the track.

1) Functions of ballast are:

i) Transfer load to subgrade and distribute in a large area
ii) To hold the sleepers in position
iii) Impart elasticity to the track
iv) Provide easy means of maintenance to correct level of two line-track.
v) Provide a good drainage foundation.
� Ballast should be such, which fulfils the characteristics of
strength, cleanability, durability, drain ability, economy and
stability and is workable with a specific size with no harm to
rails and sleepers.
� The size of the ballast used varies from 1.9 cm to 5.1 cm gauge,
stone of large size is not desirable, and the maximum size of 5.1 cm
is preferable as interlocking of stones of this size is better than the
stone of larger sizes.
� The best gradation of stone is the one which has stone of size
from 1.9 cm to 5.1 cm.
2) Minimum depth of ballast section:
y The lines of equal pressure in ballast through wheel loads are in the
shape of a ‘bulb’.
y To simplify the calculations, the 45° load dispersion is considered.
The depth of ballast is provided such that the dispersion lines do
not overlap and uniform distribution of load is achieved.
W 2Db W
Geometric Design of Railway Track

P P
Sleeper spacing (S)

45° 45°
Db Db

Fig. 1.3 Minimum Depth of Ballast

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 Chapter 1
Sleeper Spacing (S) = Width of sleeper (W) + 2 × depth of ballast (Db)
or S = W + 2 × Db
S–W
∴ Db = = Minimum Depth of ballast
2
y For example, if wooden sleepers are used in track laying with the sleeper
density as (M + 7), the sleeper spacing is 65 cm, and the width of the
sleeper is 25 cm.
y Then, the minimum depth of ballast from above works out to be 20 cm
which is the minimum depth of ballast generally prescribed on Indian
Railways.
Example 1.5: Which of the following statements are correct with respect to
the use of ballast ?
a) Ballast facilitates easy drainage.
b) Ballast provides a firm, resilient and level bed for the sleepers.
c) Ballast carries the load and distributes uniformly over the formation.
d) Ballast provides lateral and longitudinal stability to track.
Sol: (a, b, c, d)
Ballast is used because of following the reasons:
i) Protect formation against rains and winds.
ii) Does not allow free vegetation growth.
iii) Protect the sleeper from capillary moisture of formation.
So, all the options are correct.

1.7 PERMANENT WAY

Definitions

Permanent way: The combination of rails, fitted on sleepers and resting

on ballast and subgrade is called the railway track or permanent way.

y The permanent way consists of a series of rails joined by fishplates and Geometric Design of Railway Track
bolts.
y Rails are then fixed to sleepers by different types of fastenings.
y The rails act as girders to transmit the wheel load to the sleeper.
y The rails are fitted in a specific position considering the tilt, gauge and
level.
y Sleepers transmit the load from rails to the ballast.
y The ballast holds the sleeper in position and distributes the load over
the formation.
y The sleepers are properly placed, packed and boxed with ballast.

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Chapter 1

Ballast shoulder
Gauge
Ballast Cess
Rails
Sleeper

Trolley reuse

1 .5
:1

2.5m
Ballast cushion
1 .5

:1
Sub-ballast
:1

2
2

:1
G.L. of murum G.L.

Ballast base
Formation width

Fig. 1.4 Cross-Section of a Permanent Way on Embankment

y Following are some of the basic requirements of a permanent way:

i) The gauge should be correct and uniform.
ii) The rails should be at a proper level. In the case of curves, the
proper superelevation should be provided along with the transition
at the junction.
iii) The alignment should be correct.
iv) The gradients should be uniform and as gentle as possible. To
provide smooth riding quality, any change of gradient should be
followed by a smooth vertical curve.
v) The track should absorb the shock and vibration of a running train,
so it has to be resilient and elastic.
vi) There should be sufficient lateral strength in the track in order for
the alignment to be maintained even due to the effects of:
a) side thrust on tangent lengths and centrifugal force on curves.
b) lateral forces due to the expansion of rails, particularly in case
of the welded rails.
Geometric Design of Railway Track

vii) The radii and superelevation on curves should be properly designed

and maintained.
viii) The good drainage system should be available, which will improve
the safety and durability of the track.
ix) Weaker links like joints, points and crossings need to be properly
designed and maintained.
x) 
The preventive measures should be taken to avoid any damage due
to creep.

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 Chapter 1
xi) The components of the track, i.e., the rails, fittings, sleepers, ballast
and formation should fulfil their intended purpose.
xii) There should be adequate provision for easy renewals and
replacement.
xiii) The track structure should be strong with low initial and
maintenance cost.
3.4m

2.7m
1.676 m

12.7cm
20.3cm
1.8 1.8
m m
2
:1
:1

d 30.5cm
2

Turfed 4.4m
5.5m 2d 6.1 m 2d 5.5m
Temporary
land for
Permanent land narrow
(17.1 + 4 × Embankment depth in meters) pits

Fig. 1.5 The Cross-Section of a B.G. Track in Embankment (on Straight Track)

Permanent land
(21.6 + 3 × Depth of cutting + spoil bank)

12.5m + Depth of cutting 5.5m

12.5m
Spoil 10.2m
bank 3.6m
4.3m
3.9m 3.0m 1.8m
0.6m Geometric Design of Railway Track
1.3m 3.4m
2.7m
1.7m
0.9m

Turfed
Fig. 1.6 The Cross-Section of a B.G. Track in Cutting
for Double Line (on Straight Track)

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Chapter 1

1.8 CONING OF WHEELS

y A gap is provided between the wheel flange and running edges of the
rails by keeping the distance between the inside edges of wheel flanges
less than the gauge of the track.
y This gap is generally 1 cm on either side.
y The wheel is coned so as to keep tread in the centre of the head of the
rails position automatically.
y The coning of wheels is done at a slope of 1 in 20.
y The advantage of coning of wheels are:
i) Reduces the wear and tear of the wheel flanges and rails, which is
caused due to rubbing action of flanges with inside faces of the rail
head.
ii) To provide a possibility of lateral movement of the axle with its
wheels.
iii) To some extent coning of wheels prevents slipping of wheels.

Flanges of
Slope 1 in 20 wheels Slope 1 in 20
Wheel Axle Wheel
diameter = D diameter = D

Slope 1 in 20 G Slope 1 in 20
B

Rails
Slope 1 in 20 Slope 1 in 20
Sleeper
:1

1.5
1.5

Ballast
:1
Geometric Design of Railway Track

Fig. 1.7 Coning of Wheels on Level-track

1.9 GEOMETRIC DESIGNING OF TRACK

y A Railway track is designed for load and speed of the train considering
the safety and economy.

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 Chapter 1
Cause of Derailment

On Straight track On Curved track

i) Defective cross-levels i) Improper superelevation

ii) Defective gauge length ii) Improper speed

iii) Defective alignment iii) Improper radius of the curve

iv) Loosening of joints iv) Unequal distribution

of load on two rails

y The elements involved in the geometric designing of the track are:

i) Gradients and grade compensation
ii) Speed of train
iii) Radius or degree of curves
iv) Superelevation (cant)
v) Widening of Gauge on curves
vi) Curves

1.9.1 Gradient:
y Any departure of the track from the level surface is known as grade or
gradient.
y Reasons for providing gradient are:
i) To give a uniform rate of rise and fall
ii) To reach various stations located at different locations.
iii) To reduce the cost of construction

g
u t
Geometric Design of Railway Track

g g g s y
a) Ruling gradient:
� The gradient which helps in determining the maximum load that the
engine can haul on the rail section is known as the ruling gradient.
� While determining the ruling gradient of a section, it will not only
be the severity of the gradient that will come into play but also the
length of the gradient and its position.

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Chapter 1

y Following gradient are generally adopted:

in plain terrain 1 in 150 to 1 in 200
In hilly regions 1 in 100 to 1 in 150

b) Momentum gradient:
� The gradient, which is steeper than the ruling gradient, does not
determine the maximum load of the train but is used to find a
suitable position on the track. The train, before reaching these
gradients, it obtains sufficient momentum to negotiate them, which
are referred to as momentum gradient.
� The rising gradient is called the momentum gradient, and in some
cases, a steeper gradient more than the ruling grade can be adopted.

c) Pusher or helper gradient:

� The important effect of a ruling gradient is its limit on a locomotive
capacity.
� But if the grade is concentrated in a specific section such as a
mountainous region, instead of limiting the trainload, it may be
operationally easy or even be economical to run the train on the
basin of load that the engine can carry on the remaining portion of
the track and arrange for an assisting engine or pusher engine for
the portion where the gradient is severe. Such gradient are known
as ‘Pusher’ or ‘helper’ gradient.
d) Gradient at station yard:
y The gradient at the station yard must be sufficiently low because:
i) To prevent the movement of a standing vehicle on track due to
the effect of gravity.
ii) To prevent additional resistance due to grade on the starting vehicle.
iii) Grade resistance of vehicle in motion is half as compared to the
resistance at the start.
y In Indian railways, for all the gauges, the maximum gradient permitted
in station yards is 1 in 400, while a minimum gradient of 1 in 1000 is
Geometric Design of Railway Track

recommended from the drainage point of view.

1.9.2 Grade compression on curves:
y The ruling gradient is the maximum gradient on a particular section,
but if a curve lies on a ruling gradient, the resistance due to gradient
is increased by that due to curvature, and this further increases the
resistance beyond the ruling gradient.
y To avoid resistances beyond the allowable limits, the gradients are
reduced on curves, and this reduction in gradients is known as grade
compensation for curves.

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 Chapter 1
y In India, compensation for curvature is given at 0.04% per degree of
curve for B.G., 0.03% for M.G. and 0.02% for N.G. In terms of radius of
curves in meters 4 is 70/R for B.G., 52..5/R for M.G. and 35/R for N.G.

1.9.3 Speed of the train:

y The power of the locomotive and strength of the track governs the
speed of the train.
y Safe speed is a speed at which there is no risk of over-turning and
derailment.
y Dependency of train speed is represented in the flow chart:

w
v

y In India, Martin’s formula is used to work the safe speed empirically.

a) When a transition curve exists, i.e., on the transition curve for BG or MG
track
V = 4.35 R – 67 Geometric Design of Railway Track
R → Radius of the curve (m)
b) When the transition curve is absent for BG and MG track
4
V= of speed specified above
5
c) For high-speed trains
V = 4.58 R
here,
V = speed in kmph
R = Radius of curve (m)

21
Chapter 1

Grey Matter Alert!

The empirical formula given by Martin’s for the calculation of safe speed
on curves are no longer followed by Indian Railways.
The maximum speed for the transition curve is now determined in
Indian Railway as per the revised formula given as:
For BG track

V=
(Ca + Cd ) R
13.76
V = Maximum speed in kmph
Ca = Actual cant (superelevation) mm
Cd = Cant deficiency permitted in mm
R = Radius in meters

Note:
The above equation for maximum speed for BG tracks is derived from the
GV2
formula of equilibrium superelevation i.e. e = on assumption that G
127R
for BG is 1750 mm (Centre of Centre distance between rails)
The maximum cant deficiency value for different gauges for Indian Railway
are:

S.No. Track Cant deficiency

1) BG 75 mm

2) MG 50 mm
Geometric Design of Railway Track

3) NG 40 mm
Table 1.3 Cant Deficiency

1.9.4 Degree and radius of curve:

y Curves on railway tracks are mostly circular; that is, each curve
should have the same radius at every point of the curve.

22
 Chapter 1
30 m

Circular Curve

R
D°

y The radius of a curve is sometimes represented by the degree of the

curve of curvature. The degree of the curvature is defined as the degree
of curvature (D) value in terms of radius of the curve will be:
D 360°
= (where length of arc = 30 m)
30 2πR

1720
D= (R is the radius in metre)
R
y The smallest radius and the largest degree for the curvature are
restricted on the basis of:
i) Wheel base of the vehicle:
There is a chance of derailment when the degree of curve is large
for the length of the wheel base, resisting the free manoeuvre of
the vehicle along the curve.

Grey Matter Alert!

In India, curves on through tracks are limited to the following maximum

radii: Geometric Design of Railway Track
i) Maximum degree of curvature for BG = 10° (min. R = 175 m)
ii) Maximum degree of curvature for MG = 16° (min R = 109 m)

ii) Sharpness of curve: In case of a sharp curve, a greater effort is

required by engines in hauling vehicles than on straights.

23
Chapter 1

1.9.5 Superelevation or Cant:

Objective of providing superelevation:

Geometric Design of Railway Track

Fig. 1.8 Superelevation on Curves

24
 Chapter 1
i) To introduce the centripetal force for counteracting the effect of
centrifugal force, which enables the faster movement of trains on
curves. This will also prevent derailment and reduce the side wear
and creep of rails.
ii) To provide equal distribution of wheel loads on two rails so that
there is no tendency for the track to move out of position due
to more load on the outer rail. This reduces the wear of rails,
equipment and results in saving in maintenance costs.

Fig. 1.9 Relationship Between Superelevation, Gauge and Curve

iii) For the safe movement of goods and comfortable ride to passengers
by providing a smooth and even track.
Relationship of superelevation (e), with gauge (G), speed (V) and
radius of the curve (R). Using the following notations,
W = Weight of moving vehicle in kg.
n = Speed of vehicle in m/sec.
V = Speed of vehicle in kmph
R = Radius of curve in meters.
G = Gauge of track in meters.
g = Acceleration due to gravity in m/sec2 Geometric Design of Railway Track
a = Angle oof inclination.
S = Length of inclined surface in metres.
Centrifugal force is given by
Wv 2
F = …(1)
gR
Resolving the forces along the inclined surface, we get
F cos a = W sin a …(2)
Wv 2 G
where F = , cos α =
gR S

25
Chapter 1

e
and sin a =
S
Therefore, equation 2) becomes
Wv 2 G e
× = W×
gR S S
v2
Therefore, e = × G metres …(3)
gR
Where v is in m/sec.

( )
2
G 0.278V
= m. where V is in kmph
9.81R
GV2
= m. …(4)
127R
GV2
= cm. …(5)
1.27 R
Where G is in metres.
V is in kmph.
R is in metres.
In India G for B.G. = 1.676 m
M.G. = 1.0 m
and N.G. = 0.762 m
1.676V2 V2
so, for B.G., e = = 1.315 cm
1.27R R
y The cant or superelevation obtained from equations 4) and 5) is known
as equilibrium cant.
y The cant is said to be in equilibrium when the lateral force and wheel
loads are almost equal. This equilibrium cant is provided on the basis
of the average speed of the train.
y A superelevation is to be provided such that faster trains pass safely
Geometric Design of Railway Track

without overturning or discomfort to the passenger, and slower

trains should pass safely without fear of derailment due to excessive
superelevation.
y On some railways, “Weighted average” is calculated for finding out the
equilibrium speed of the trains.
n1V1 + n2 V2 + n3 V3 + ....
Equilibrium speed =
Σ(n1 + n2 + n3 ....)
Here → n1, n2, n3 denotes the no. of trains running on the track at the
speed of V1, V2, V3 etc., respectively

26
 Chapter 1
1.9.6 Limit of superelevation and cant-deficiency
y Superelevation should be such that the track is suitable for various
trains running at different speeds.
y There are limits to the amount of superelevation which may be provided
safely.
y Normally, the maximum superelevation value permissible to railway
board is 1/10th of gauge.
y The maximum permissible value in India for different gauge are:

S.No. Gauge Maximum S.E.

1) BG 0.165 m or 16.5 cm
2) MG 0.1 m or 10 cm
3) NG 0.076 or 7.6 cm
Table 1.4 Maximum Superelevation

y The superelevation should be provided smoothly and uniformly by the

use of transition curves in between the straight track and circular curve.
Superelevation varies from zero at the beginning of a transition curve to
the fall amount at the junction of the transition curve and circular curve
and the correct amount is obtained at every point of the transition
curve by applying cant gradient over transition length.
1.9.7 Cant deficiency
y The equilibrium cant is provided on the basis of average speed of
different trains. But there is a possibility that this cant value can be
less than required for the high-speed trains. This lack of cant is called
‘Cant Deficiency’.
y Cant deficiency is the difference between the equilibrium cant necessary
for the maximum permissible speed on the curve and the actual cant
provided.
y Maximum value of cant deficiency prescribed by the Indian Railway are: Geometric Design of Railway Track

S.No. Gauge Cant deficiency for Cant deficiency of speed

speed upto 100 kmph higher than 100 kmph
1) BG 7.6 cm 10 cm
2) MG 5.1 cm -
3) NG 3.8 cm -
Table 1.5 Cant Deficiency

27
Chapter 1

Maximum permissible speed on a curve is taken as the minimum value of

following speeds:
i) Maximum sanctioned speed of the section:
y This is the maximum speed authorised by the additional commissioner
of railways.
y The speed is based on track condition, type of traction, standards of
signalling and interlocking etc.
ii) Safe speed over the curve:
y This speed is calculated by the formula of equilibrium superelevation,
considering superelevation as theoretical superelevation (i.e. full amount
of cant deficiency + the actual superelevation.)
iii) Speed based on the consideration of S.E:
y This speed is calculated by the formula of equilibrium superelevation,
considering superelevation as theoretical superelevation (i.e. full amount
of cant deficiency + the actual superelevation.)
y When there is a possibility of an increase in the length of the transition
curve, the above three methods are used to calculate maximum speed.
But when the length of the transition curve is fixed following method is
taken in to consideration.
iv) Speed from the length of the transition curve. This is the lesser value of
the speed given by the following two formulas:
a) For normal speed upto 100 kmph:
134 × L
Vmax =
e

134 × L
Vmax =
D
where L = length of transition curve based on the rate of change of cant as
38 mm/sec for normal speed and 55 mm/sec for high speeds.
b) For high speeds above 100 kmph:
Geometric Design of Railway Track

198 × L
Vmax =
e

198 × L
Vmax =
D
1.9.8 Negative superelevation:
y When the curved mainline has a turnout on the opposite side leading
to a branch line, the required superelevation for the average speeds of
trains running over the mainline cannot be provided.

28
 Chapter 1
Outer rail
Outer
S.E. rail O S.E.
S

Inner R P
Inner

Y
rail X rail
Bra

Y
nc

k
ac
ht

tr
ra c

in
Ma
k

Crossing

M N Points

Fig. 1.10 Negative Superelevation

y From the above figure, MO, which is the outer rail of the mainline curve
must be higher than inner rail NP or point M should be higher than point N.
y For the branch line, however, NS should be higher than MR or the point
N should be higher than point M. These two contradictory conditions
cannot be met at the same time within are layout.
y So, instead of the outer rail NS on the branch line being higher, it is kept
lower than the inner rail MR.
y In such cases, the branch line curve has a negative superelevation and
therefore speed on both tracks must be restricted, particularly on a
branch line.
y The methodology of working out the speed on main line, branch line,
and negative superelevation on branch line are: Geometric Design of Railway Track
a) The equilibrium superelevation or cant on branch line is calculated
GV2
by formula, e = , after assuming a speed on branch line.
127R
b) The permissible cant deficiency is deducted from the equilibrium
cant as obtained in step a).
c) The difference obtained (Equilibrium cant-permissible cant
deficiency) will give the negative superelevation to be used on the
branch line.

29
Chapter 1

d) This negative superelevation is also equal to the maximum superelevation

permitted on the main curved track.
e) The restricted speed on the curved track is obtained by adding
permissible deficiency in maximum cant on the main track and applying
GV2
the equation e = .
127R

Example 1.6: Calculate the equilibrium cant on a MG curve of 6° for an

average speed of 50 km/hr. Also, find out the maximum permissible speed
after allowing maximum cant deficiency?

Sol:
Degree of curve = 6°

GV2
eactual =
127R
G = 1 m for MG
V = 50 km/hr
1720
R=
6
1 × 502
eacutal =
1720
127 ×
6
eactual = 0.06867 m or 6.87 cm
for speed <100 kmph, for MG limit of cant deficiency is equal to 5.1 cm
eth = ea + ed

= 6.87 + 5.1
eth = 11.97 cm
2
GVmax
so, eth =
127R
Geometric Design of Railway Track

2
1 × Vmax
0.1197 =
1720
127 ×
6
Vmax = 66 kmph
Checking by Martin’s formula
Vmax = 4.35 R –67

30
 Chapter 1
1720
= 4.35 –67
6
Vmax = 64.47 kmph
Therefore, maximum permissible speed Vmax = 64.47 kmph

Example 1.7: A 8° curve diverges from a 4° main curve in the reverse

direction. What will be the maximum allowable speed on the main line,
considering the restricted speed on the branch line is 25 kmph?
Sol:
i) Superelevation on diverging track (or branch line) is given as
equilibrium superelevation
GV2
e =
1.27R
where G = 1.676 m
V = 25 kmph
1720
R =
8
1.676 × 25 × 25 8
e = ×
1.27 1720
ii) So, the negative cant =Equilibrium cant – cant deficiency
= (3.84 – 7.60) cm
= – 3.76
Because can deficiency for B.G. permitted is equal to 7.6 cm.
iii) Negative cant = Max. permissible superelevation on the main line
= 3.76 cm
iv) Theoretical superelevation which can be provided on main line
= (3.76 + 7.6) cm = 11.36 cm
v) Hence speed on the main track can be calculated from the formula of
equilibrium
GV2 Geometric Design of Railway Track
Superelevation =
1.27R
1.676 × V2 × 4
11.36 =
1.27 × 1720
10.49 × 1.27 × 1720
∴ V2 = = 4550 kmph
1.676 × 3
∴ V = 60.84 km.p.h
So, the restricted speed on main track = 60.44 kmph
Say, 60 kmph

31
Chapter 1

1.10 TRANSITION CURVE

Definitions

Transition Curve: A curve of parabolic nature which is introduced

between a straight and a circular curve or between two branches of a
compound curve is called a transition curve.

y The radius of the transition curve gradually decreases from infinity to

the minimum value, so that it should achieve full superelevation and
curvature.
1.10.1 Requirement of transition curve
i) It should be perfectly tangential to the straight
ii) Along the transition curve, curvature and superelevation should vary at
the same rate.
iii) The transition curve should join the circular arc tangentially.

1.10.2 Types of transition curves

t c

c p

Bernoulli’s
is
ax

Leminiscate
or
aj
M

Spiral
Geometric Design of Railway Track

45° Cubic
45° parabola
φ = 3α
0
Fig. 1.11 Types of Transition Curves

32
 Chapter 1
1.10.3 Length of transition curve
y The length of the transition curve is the length along the centre line of
the track from the intersection of the straight to the circular curve.
y Half of the transition curve is provided in the straight and the other half
in the circular curve.
y Indian Railway specifies that length of transition curve should be
maximum of the following:
i) L = 7.2e
Where e = actual super-elevation in centimeters
ii) L = 0.073D × Vmax
(Based on rate of change of cant deficiency)
iii) L = 0.073e × Vmax
(Based on rate of change of superelevation)
Here,
L = Length of transition curve
E = Actual cant or superelevation in cm
D = Cant deficiency for maximum speed in cm
V = Maximum speed in kmph

Shifted
u rve
circular c Original
L y ∆ circular curve
X= L/2

L/2 C .S.
S.C.
P.C. P.T. S.T
P.S . Shift S(Shift)
or .
or T T1
R–S
R

Straight Transition (I–2∆) Straight

curve ∆ ∆
I
Geometric Design of Railway Track

Fig. 1.12 Layout of Transition Curve

Example 1.8: What is the length of transition curve for a B.G. curved track having
5° curvature and a cant of 11 cm ? The maximum permissible speed on
curve is 90 kmph.

33
Chapter 1

Sol:
Length of curve is maximum of following
i) L = 7.2e
= 7.2 × 11
= 79.2 m ….(i)
ii) L = 0.073D × Vmax
= 0.073 × 7.6 × 90
= 49.93 m ….(ii)
here, D → cant deficiency for BG track is 7.6 cm

iii) L = 0.073e × Vmax

= 0.073 × 11 × 90 = 72.27 m ….iii)
Length of transition curve is maximum of i), ii), iii)
L = 79.2 m

Previous Years’ Question

For a broad gauge railway track on a horizontal curve of radius R (in

m), the equilibrium cant e required for a train moving at a speed of V
(in km per hour) is

V2 V2
a) e = 1.676 b) e = 1.315
R R

V2 V2
c) e = 0.80 d) e = 0.60
R R
Sol: b) [2017, SET-II]

Previous Years’ Question

Geometric Design of Railway Track

A broad gauge railway line passes through a horizontal curved section

(radius = 875 m) of length 200 m. The allowable speed on this portion
is 100 km/h. For calculating the cant, consider the gauge as the centre-
to-centre distance between the rail heads, equal to 1750 mm. The
maximum permissible cant (in mm, round off to 1 decimal place) with
respect to the centre-to-centre distance between the rail heads is
Sol: 157.5 (157.2-157.6) [GATE-2019, Set-II]

34
 Chapter
Previous Years’ Question

For a 2° curve on a high speed Broad Gauge (BG) rail section, the
maximum sanctioned speed is 100 km/h, and the equilibrium speed
is 80 km/h. Consider the dynamic gauge of BG rail as 1750 mm. The
degree of curve is defined as the angle subtended at its centre by
a 30.5 m are. The cant deficiency for the curve (in mm, round off to
integer) is_____________.
Sol: 57 [2021-Set-II]

Keywords

 Gauge in railway track

 Rails
 Sleepers
 Track fitting used in a permanent way
 Ballast
 Permanent way
 Coning of wheels
 Geometric designing of track
 Transition curve

35