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Creep of Rails
By
Dr. T. Vijaya Gowri
Associate Professor,
B.V. Raju Institute of Technology, Narsapur
Definition
 Creep is defined as the longitudinal movement
of rails with respect to sleepers in a track.
 Creep is common to all railway tracks, but
varies in magnitude considerably, the rail, in
some places, moves by several centimetres in a
month while in other locations the movement of
rails may be negligible.
 It is observed that the rails have tendency to
move gradually in the direction of dominant
traffic.
2
Indications of Creep
 Occurrence of creep can be noticed from the
following observations :
1. 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.
2. Marks on flanges and webs of rails made by
spike heads, by scraping or scratching as the rails
slide.
3
Theories of Creep
 Various theories are propounded to explain the different causes of creep but
none of them gives the true picture for real cause (or root cause) of creep in
rails.
The various theories propounded for explaining the probable causes of creep in
rails are described in following paragraphs:
1. Wave Action or Wave Theory
2. Percussion Theory
3. Drag or Dragging Theory
4. Starting, Accelerating, Slowing down or Stopping of a Train
5. Expansion or Contraction of Rails due to Temperature
6. Unbalanced Traffic
4
1. Wave Action or Wave Theory
 Wave motion is set up by moving loads of wheels.
The vertical reverse curve ABC (as shown in Fig.) is
formed in the rails ahead of the wheels, resulting
from the rail deflection under the load, is the chief
cause of creep.
 The wheels push the wave with a tendency to force
the rail in the direction of traffic.
 On a particular rail, the joint action by several
wheels causes creep.
 As the wheels move, the lift in front of the moving
load is thus carried forward by the wheels and
causes creep, whereas the lift at the rear of the
wheel gets back to its normal position.
5
1. Wave Action or Wave Theory
 The pitch and depth of wave depend upon
the following :
1. Track modulus
2. Stiffness of track
3. Stability of formation.
 The wave action can be reduced (i.e. creep is
reduced) by adopting following measures :
1. Angular and heavy ballast-which develops
good interlock
2. Increased stiffness of track
3. Lesser sleeper spacing
4. Bigger, section of the rail
6
2. Percussion Theory
 This theory states that the creep is due to impact
of wheels at the rail end ahead at joints.
 The horizontal component ‘P’ of ‘R’ tends to
cause creep while the vertical component tends
to bend down the rail end vertically, i.e., to make
a battered rail end.
 Hence as and when the wheels leave the trailing
rail and strike the facing rail end at each joint, it
pushes the rail forward resulting in creep.
 Though the creep is very small in single impact
but cumulative effect of number of wheels in
quick succession results in sufficient creep.
7
2. Percussion Theory
 The creep by this theory will increase due to :
1. weak and loose fish bolts
2. worn out fish plates
3. loose packing at joints
4. wide expansion gap
5. heavy axle loads moving at high speed.
8
3. Drag or Dragging Theory
 It states that backward thrust on driving wheels
of the locomotive of train has got a tendency to
push the rail off the track backward while the
other wheels of the locomotive and the vehicles
(i.e. wheels of coaches and wagons) push the
rail in the direction of travel as explained in
Wave Action Theory and they have greater
effect (as compared to drag effect).
 This results in creep of rails in the direction of
movement of trains.
9
4. Starting, Accelerating, Slowing down or
Stopping of a Train
 When a train is starting or accelerating,
the backward thrust of the engine
driving wheels tends to push the rails
backward.
 When it is slowing down (i.e.
decelerating) or coming to a stop, the
braking effect tends to push the rails
forward.
10
5. Expansion or Contraction of Rails due to
Temperature
 Creep also occurs due to variation in temperature.
 The creep in this case is influenced by the range in temperature
variation, location of track, whether exposed or shady
surroundings, etc.
11
6. Unbalanced Traffic
 (a) In a single line system if heavy equal traffic (both in number and load)
runs in both directions, the creep is almost balanced. Otherwise, heavy
traffic in one direction will cause creep, which is partly balanced by light
traffic in opposite direction.
 (b) In the double line system, trains on a particular line being unidirectional,
creep occurs in both the lines. The above mentioned causes are main but
by no means the only causes of creep.
12
Factors those govern the magnitude and
direction of creep
(i) Alignment of Track
Creep is observed greater on curves
than on tangent railway track
(ii) Grade of Track
Creep is more with steep gradient
particularly if the trains move
downward with heavy loads.
Though in the opposite direction, it is
not impossible for creep to develop.
(iii) Type of Rails
The old rails have more creep than
new rails.
(iv) Direction of Heaviest Traffic
If the loaded trains run in one direction (such as
from production centres to the markets) and
empty trains in opposite direction, creep will
usually be found in the direction of loaded
trains.
(v) Poor Maintenance of Track Components
Poor Maintenance of Track Components and
ill-design of super-elevation, curves, joints, etc.
will also increase the creep.
It is observed that creep is not constant over
any given time nor does it vary at uniform rate,
nor does it continue in one direction, nor do
both the rails creep by an equal amount.
In fact, the direction and extent of creep
cannot be predicted. Such variations may
occur daily or may be seasonal.
13
Effects of Creep
Following are the main effects of the
development of creep:
• Development of disturbance in gauge and
alignment.
• Widening of gaps.
• Disturbance at points and crossing
• The interlocking mechanism of signals is
disturbed and thrown out of gear due to
creep.
• If a crept rail is removed from the track, it
becomes difficult to refix it. either it is found
too short or too long due to creep.
• Bucking of the track takes place in extreme
cases.
• Rail end gets battered and suspended
joints start becoming supported joints.
14
Measurement of Creep
 “Creep Indicator” is used to measure the
creep of rails.
 A chisel mark is made at the side of bottom
flange of rail and two rail posts are fixed to the
formation with their top levels flushing with top
of sleepers.
 A fishing string is stretched below the rails on
the marks on the top of posts and creep is thus
measured.
 The distance between chisel mark to the string
is the amount of creep.
15
Measurement of Creep
• As per Indian Railways practice, the creep
should be measured frequently at an interval
of about 3-months.
• A creep in excess of 150 mm (15 cm) should
not be permitted on standard track and at
any location more than 6 consecutive rails
should not be found jammed in a single
railway track.
• Moreover, no creep should be permitted on
approaches of points and crossings.
16
Track Portions Susceptible to Creep
 The following portions of track are found to be more susceptible to
should be frequently attended :
1. Steel Sleeper Track (CST – 9) joining/melting wooden sleeper track.
2. Dips in stretches with long gradients.
3. Approaches to major structures.
4. Approaches to level crossings and points crossings.
17
Remedies / Prevention of Creep
 Prevention is always better than cure. If creep is not prevented in
time, it will result in derailment.
Following are the common methods adopted to prevent creep.
 1. Pulling Back the Rails
 2. Provision of Anchors or Anti-creepers
 3. Use of Steel Sleepers
18
1. Pulling Back the Rails
 If creep is distinctly visible, the remedy is to pull back the rails to their
original position.
 For doing this, first inspect the track, note the extent of pulling back
distance and determine the point from which to begin.
 Now start pulling the rails back to their original positions by means of
crow bars and hooks provided through the fish bolt holes of rail.
 In pulling back, the positions of joints relative to sleepers must be
maintained, and both the rail joints must be in their relative positions.
 Pulling back the rails is a very slow and tedious process and is only
possible when a small length is to be dealt.
 It has, moreover, been noticed that rails start creeping immediately
after pulling back.
19
2. Provision of Anchors or Anticreepers
 The creep of the track can be prevented by use of Anchors
and sufficient crib ballast. For creep of 7.5 cm to 15 cm, in a
month 4-anchors per rail and for creep of 22.5 cm to 25 cm.
6-anchors per rail are used in the Indian practice.
 Anchors are fastened to the foot of rail and kept in perfect
contact with the side of the sleeper being the side opposite
to the direction of creep. If creep occurs in both directions,
anticreepers are provided on both the sides of sleepers,
starting from the centre of the rail and should never be
fixed near the joints.
 Anchors are fixed to rails either (i) by wedging action, (ii) by
clamping (iii) by a spring grip. There are several varieties of
anchors based on these three types.
20
The following points should be considered for
efficient maintenance of Anchors:
 The creep anchors should be strong enough to resist the movement of rails due
to creep.
 The anchors should butt against the sleepers otherwise they do not function
effectively.
 The anchors should be uniformly distributed over the entire rail length.
 Defective anchors must be renewed to prevent accumulation of creep.
 Anchors should be fixed to good sound sleepers only.
 Anchors, which depend on spring effect for their grip, must not be driven along a
rail as this will destroy the spring.
 As far as possible, creep anchors should not be provided on the railway bridges.
 It is a good practice to use number of anchors at approaches to the yards, in
yards, at level crossings or at places where heavy brake applications are made,
since the creep adversely affects the alignment, gauge and position of points
and crossings at these locations.
21
3. Use of Steel Sleepers
 Sleepers should be of such a type and
with such fittings that they effectively
prevent the rail from creeping on them.
 Secondly, the sleepers must have a
good grip with the ballast to resist the
movement of the sleepers in the ballast.
 Steel trough sleepers are the best for
this purpose.
 Increase in the number of sleepers will,
therefore, also help in the prevention of
creep.
22
Coning of wheels
 The rim or flanges of the wheels are never made flat but
they are in the shape of a cone with a slope of about 1 in
20.
 This is known as coning of wheels.
 The coning of wheels is manly done to maintain the
vehicle in the central position with respect to the track.
 When the vehicle is moving on leveled track then the
flanges of wheels have equal circumference.
23
Theory of conning
 If the tread diameter on both the rails is same, the amount of slip will be
given by
 Slip = Ѳ (R2 – R1)
where,
 Outer radius, R2 = R + (G/2)
 Inner radius, R1 = R – (G/2)
 G = Gauge
 Ѳ = Angle at centre in radians
Therefore, Slip = Ѳ x G
 For B.G. track, G = 1.676 m
Thus, Slip = (2πѲ° / 360) x 1.676
where,
 Ѳ° = angle at centre in degrees, say 1°
 Slip = 0.029 (approx. for 1° of central angle)
Therefore, the slip is about 0.029 m per degree of the central angle.
24
Theory of conning
• Chamfering of wheels on bends is not useful as the principal axle if owing to
centrifugal force proceeds towards the outward rail the back axle will precede
towards the inside rail and the complete benefit of coning wheels cannot be
availed.
• Put differently, there will be no free sidelong movement of wheels.
• This results in the disadvantages of the outer rail will have more pressure while
the inner rail will have lesser pressure.
• Owing to the central strength, the parallel components incline to turn the rail
out and the gauge has broadening tendency.
• Due to this condition if the voids sleepers have no base plate beneath the
edge of the rail they will be damages. In order to get rid of or minimize the
supra mentioned faults “angling of rails” is done.
• When the rails are tilted then the base plate or sleeper is not placed
horizontally. It is laid at a slope of 1 in 20 towards the inner side.
25
26
Coning of Wheels
Advantages
1. Smooth riding- help vehicle to
negotiate curves smoothly
2. Reduces wear and tear of wheel
flanges. Damage is caused
because of the friction action of
rims with inner faces of the rail
top.
3. It gives an option of lateral drift of
the hinge with its wheels
4. It prevents, to some extent, the
slipping of the wheels
Disadvantages
1. The pressure on the horizontal
component of force near the
inner edge of outer rail has a
tendency to wear the rail
quickly
2. The horizontal component has
to turn the rail outwards and
hence the gauges may be
widened.
3. If no base plates are provided,
sleepers under the outer edge
of the rail may be damaged.
27
Sleeper Density
 Sleeper density is the number of sleepers per rail length.
 It is specified as M + x or N + x, where M or N is the length of the rail in
metres and x is a number that varies according to factors such as
(a) axle load and speed,
(b) type and section of rails,
(c) type and strength of the sleepers,
(d) type of ballast and ballast cushion, and
(e) nature of formation.
28
Sleeper Density
 If the sleeper density is M + 7 on a broad gauge route and
the length of the rail is 13 m,
 it means that 13 + 7 = 20 sleepers will be used per rail on
that route.
 The number of sleepers in a track can also be specified by
indicating the number of sleepers per kilometre of the
track.
 For example, 1540 sleepers/km.
 This specification becomes more relevant particularly in
cases where rails are welded and the length of the rail
does not have much bearing on the number of sleepers
required.
 This system of specifying the number of sleepers per
kilometre exists in many foreign countries and is now being
adopted by Indian Railways as well.
29
References
 Satish Chandra and Agarwal, M.M. (2007) "Railway Engineering" Oxford
Higher Education, University Press New Delhi.
30
31

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Creep of Rails.pdf

  • 1. Creep of Rails By Dr. T. Vijaya Gowri Associate Professor, B.V. Raju Institute of Technology, Narsapur
  • 2. Definition  Creep is defined as the longitudinal movement of rails with respect to sleepers in a track.  Creep is common to all railway tracks, but varies in magnitude considerably, the rail, in some places, moves by several centimetres in a month while in other locations the movement of rails may be negligible.  It is observed that the rails have tendency to move gradually in the direction of dominant traffic. 2
  • 3. Indications of Creep  Occurrence of creep can be noticed from the following observations : 1. 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. 2. Marks on flanges and webs of rails made by spike heads, by scraping or scratching as the rails slide. 3
  • 4. Theories of Creep  Various theories are propounded to explain the different causes of creep but none of them gives the true picture for real cause (or root cause) of creep in rails. The various theories propounded for explaining the probable causes of creep in rails are described in following paragraphs: 1. Wave Action or Wave Theory 2. Percussion Theory 3. Drag or Dragging Theory 4. Starting, Accelerating, Slowing down or Stopping of a Train 5. Expansion or Contraction of Rails due to Temperature 6. Unbalanced Traffic 4
  • 5. 1. Wave Action or Wave Theory  Wave motion is set up by moving loads of wheels. The vertical reverse curve ABC (as shown in Fig.) is formed in the rails ahead of the wheels, resulting from the rail deflection under the load, is the chief cause of creep.  The wheels push the wave with a tendency to force the rail in the direction of traffic.  On a particular rail, the joint action by several wheels causes creep.  As the wheels move, the lift in front of the moving load is thus carried forward by the wheels and causes creep, whereas the lift at the rear of the wheel gets back to its normal position. 5
  • 6. 1. Wave Action or Wave Theory  The pitch and depth of wave depend upon the following : 1. Track modulus 2. Stiffness of track 3. Stability of formation.  The wave action can be reduced (i.e. creep is reduced) by adopting following measures : 1. Angular and heavy ballast-which develops good interlock 2. Increased stiffness of track 3. Lesser sleeper spacing 4. Bigger, section of the rail 6
  • 7. 2. Percussion Theory  This theory states that the creep is due to impact of wheels at the rail end ahead at joints.  The horizontal component ‘P’ of ‘R’ tends to cause creep while the vertical component tends to bend down the rail end vertically, i.e., to make a battered rail end.  Hence as and when the wheels leave the trailing rail and strike the facing rail end at each joint, it pushes the rail forward resulting in creep.  Though the creep is very small in single impact but cumulative effect of number of wheels in quick succession results in sufficient creep. 7
  • 8. 2. Percussion Theory  The creep by this theory will increase due to : 1. weak and loose fish bolts 2. worn out fish plates 3. loose packing at joints 4. wide expansion gap 5. heavy axle loads moving at high speed. 8
  • 9. 3. Drag or Dragging Theory  It states that backward thrust on driving wheels of the locomotive of train has got a tendency to push the rail off the track backward while the other wheels of the locomotive and the vehicles (i.e. wheels of coaches and wagons) push the rail in the direction of travel as explained in Wave Action Theory and they have greater effect (as compared to drag effect).  This results in creep of rails in the direction of movement of trains. 9
  • 10. 4. Starting, Accelerating, Slowing down or Stopping of a Train  When a train is starting or accelerating, the backward thrust of the engine driving wheels tends to push the rails backward.  When it is slowing down (i.e. decelerating) or coming to a stop, the braking effect tends to push the rails forward. 10
  • 11. 5. Expansion or Contraction of Rails due to Temperature  Creep also occurs due to variation in temperature.  The creep in this case is influenced by the range in temperature variation, location of track, whether exposed or shady surroundings, etc. 11
  • 12. 6. Unbalanced Traffic  (a) In a single line system if heavy equal traffic (both in number and load) runs in both directions, the creep is almost balanced. Otherwise, heavy traffic in one direction will cause creep, which is partly balanced by light traffic in opposite direction.  (b) In the double line system, trains on a particular line being unidirectional, creep occurs in both the lines. The above mentioned causes are main but by no means the only causes of creep. 12
  • 13. Factors those govern the magnitude and direction of creep (i) Alignment of Track Creep is observed greater on curves than on tangent railway track (ii) Grade of Track Creep is more with steep gradient particularly if the trains move downward with heavy loads. Though in the opposite direction, it is not impossible for creep to develop. (iii) Type of Rails The old rails have more creep than new rails. (iv) Direction of Heaviest Traffic If the loaded trains run in one direction (such as from production centres to the markets) and empty trains in opposite direction, creep will usually be found in the direction of loaded trains. (v) Poor Maintenance of Track Components Poor Maintenance of Track Components and ill-design of super-elevation, curves, joints, etc. will also increase the creep. It is observed that creep is not constant over any given time nor does it vary at uniform rate, nor does it continue in one direction, nor do both the rails creep by an equal amount. In fact, the direction and extent of creep cannot be predicted. Such variations may occur daily or may be seasonal. 13
  • 14. Effects of Creep Following are the main effects of the development of creep: • Development of disturbance in gauge and alignment. • Widening of gaps. • Disturbance at points and crossing • The interlocking mechanism of signals is disturbed and thrown out of gear due to creep. • If a crept rail is removed from the track, it becomes difficult to refix it. either it is found too short or too long due to creep. • Bucking of the track takes place in extreme cases. • Rail end gets battered and suspended joints start becoming supported joints. 14
  • 15. Measurement of Creep  “Creep Indicator” is used to measure the creep of rails.  A chisel mark is made at the side of bottom flange of rail and two rail posts are fixed to the formation with their top levels flushing with top of sleepers.  A fishing string is stretched below the rails on the marks on the top of posts and creep is thus measured.  The distance between chisel mark to the string is the amount of creep. 15
  • 16. Measurement of Creep • As per Indian Railways practice, the creep should be measured frequently at an interval of about 3-months. • A creep in excess of 150 mm (15 cm) should not be permitted on standard track and at any location more than 6 consecutive rails should not be found jammed in a single railway track. • Moreover, no creep should be permitted on approaches of points and crossings. 16
  • 17. Track Portions Susceptible to Creep  The following portions of track are found to be more susceptible to should be frequently attended : 1. Steel Sleeper Track (CST – 9) joining/melting wooden sleeper track. 2. Dips in stretches with long gradients. 3. Approaches to major structures. 4. Approaches to level crossings and points crossings. 17
  • 18. Remedies / Prevention of Creep  Prevention is always better than cure. If creep is not prevented in time, it will result in derailment. Following are the common methods adopted to prevent creep.  1. Pulling Back the Rails  2. Provision of Anchors or Anti-creepers  3. Use of Steel Sleepers 18
  • 19. 1. Pulling Back the Rails  If creep is distinctly visible, the remedy is to pull back the rails to their original position.  For doing this, first inspect the track, note the extent of pulling back distance and determine the point from which to begin.  Now start pulling the rails back to their original positions by means of crow bars and hooks provided through the fish bolt holes of rail.  In pulling back, the positions of joints relative to sleepers must be maintained, and both the rail joints must be in their relative positions.  Pulling back the rails is a very slow and tedious process and is only possible when a small length is to be dealt.  It has, moreover, been noticed that rails start creeping immediately after pulling back. 19
  • 20. 2. Provision of Anchors or Anticreepers  The creep of the track can be prevented by use of Anchors and sufficient crib ballast. For creep of 7.5 cm to 15 cm, in a month 4-anchors per rail and for creep of 22.5 cm to 25 cm. 6-anchors per rail are used in the Indian practice.  Anchors are fastened to the foot of rail and kept in perfect contact with the side of the sleeper being the side opposite to the direction of creep. If creep occurs in both directions, anticreepers are provided on both the sides of sleepers, starting from the centre of the rail and should never be fixed near the joints.  Anchors are fixed to rails either (i) by wedging action, (ii) by clamping (iii) by a spring grip. There are several varieties of anchors based on these three types. 20
  • 21. The following points should be considered for efficient maintenance of Anchors:  The creep anchors should be strong enough to resist the movement of rails due to creep.  The anchors should butt against the sleepers otherwise they do not function effectively.  The anchors should be uniformly distributed over the entire rail length.  Defective anchors must be renewed to prevent accumulation of creep.  Anchors should be fixed to good sound sleepers only.  Anchors, which depend on spring effect for their grip, must not be driven along a rail as this will destroy the spring.  As far as possible, creep anchors should not be provided on the railway bridges.  It is a good practice to use number of anchors at approaches to the yards, in yards, at level crossings or at places where heavy brake applications are made, since the creep adversely affects the alignment, gauge and position of points and crossings at these locations. 21
  • 22. 3. Use of Steel Sleepers  Sleepers should be of such a type and with such fittings that they effectively prevent the rail from creeping on them.  Secondly, the sleepers must have a good grip with the ballast to resist the movement of the sleepers in the ballast.  Steel trough sleepers are the best for this purpose.  Increase in the number of sleepers will, therefore, also help in the prevention of creep. 22
  • 23. Coning of wheels  The rim or flanges of the wheels are never made flat but they are in the shape of a cone with a slope of about 1 in 20.  This is known as coning of wheels.  The coning of wheels is manly done to maintain the vehicle in the central position with respect to the track.  When the vehicle is moving on leveled track then the flanges of wheels have equal circumference. 23
  • 24. Theory of conning  If the tread diameter on both the rails is same, the amount of slip will be given by  Slip = Ѳ (R2 – R1) where,  Outer radius, R2 = R + (G/2)  Inner radius, R1 = R – (G/2)  G = Gauge  Ѳ = Angle at centre in radians Therefore, Slip = Ѳ x G  For B.G. track, G = 1.676 m Thus, Slip = (2πѲ° / 360) x 1.676 where,  Ѳ° = angle at centre in degrees, say 1°  Slip = 0.029 (approx. for 1° of central angle) Therefore, the slip is about 0.029 m per degree of the central angle. 24
  • 25. Theory of conning • Chamfering of wheels on bends is not useful as the principal axle if owing to centrifugal force proceeds towards the outward rail the back axle will precede towards the inside rail and the complete benefit of coning wheels cannot be availed. • Put differently, there will be no free sidelong movement of wheels. • This results in the disadvantages of the outer rail will have more pressure while the inner rail will have lesser pressure. • Owing to the central strength, the parallel components incline to turn the rail out and the gauge has broadening tendency. • Due to this condition if the voids sleepers have no base plate beneath the edge of the rail they will be damages. In order to get rid of or minimize the supra mentioned faults “angling of rails” is done. • When the rails are tilted then the base plate or sleeper is not placed horizontally. It is laid at a slope of 1 in 20 towards the inner side. 25
  • 26. 26
  • 27. Coning of Wheels Advantages 1. Smooth riding- help vehicle to negotiate curves smoothly 2. Reduces wear and tear of wheel flanges. Damage is caused because of the friction action of rims with inner faces of the rail top. 3. It gives an option of lateral drift of the hinge with its wheels 4. It prevents, to some extent, the slipping of the wheels Disadvantages 1. The pressure on the horizontal component of force near the inner edge of outer rail has a tendency to wear the rail quickly 2. The horizontal component has to turn the rail outwards and hence the gauges may be widened. 3. If no base plates are provided, sleepers under the outer edge of the rail may be damaged. 27
  • 28. Sleeper Density  Sleeper density is the number of sleepers per rail length.  It is specified as M + x or N + x, where M or N is the length of the rail in metres and x is a number that varies according to factors such as (a) axle load and speed, (b) type and section of rails, (c) type and strength of the sleepers, (d) type of ballast and ballast cushion, and (e) nature of formation. 28
  • 29. Sleeper Density  If the sleeper density is M + 7 on a broad gauge route and the length of the rail is 13 m,  it means that 13 + 7 = 20 sleepers will be used per rail on that route.  The number of sleepers in a track can also be specified by indicating the number of sleepers per kilometre of the track.  For example, 1540 sleepers/km.  This specification becomes more relevant particularly in cases where rails are welded and the length of the rail does not have much bearing on the number of sleepers required.  This system of specifying the number of sleepers per kilometre exists in many foreign countries and is now being adopted by Indian Railways as well. 29
  • 30. References  Satish Chandra and Agarwal, M.M. (2007) "Railway Engineering" Oxford Higher Education, University Press New Delhi. 30
  • 31. 31