1. curvature of track

1. CURVATURE
OF
TRACK

2. DEFINITION
 A railway track on a straight is an ideal condition.
 However, this ideal condition may not be continued
in a track.
 Therefore, curvatures are provided inevitably on a
railway track
 to bypass obstacles,
 to provide longer and easily traversed gradients, and
 to pass a railway line through obligatory or desirable
locations.

3. DEFINITION
 Horizontal curves are provided when a change in
the direction of the track is required and
 Vertical curves are provided at points where two
gradients meet or where a gradient meets level
ground.

4. DISADVANTAGES OF PROVIDING CURVATURE
 Restriction in speed, limiting the length of trains
and prevent the use of heavy type of locomotives.
 Maintenance cost of track increases due to increase
in the wear and tear of parts of tracks.
 Danger of collision, derailment or other form of
accident is increased.
 Running of train is not smooth.

5. RESTRICTION OF PROVIDING CURVATURE
 Bridge and tunnels
 Approaches to bridges
 Steep gradients
 Stations and yards
 Level crossing

6. DEGREE / RADIUS OF CURVATURE
 A simple curve is designated either by its degree or
by its radius.
 The degree of a curve (ϴ) is the angle subtended at
its centre by a chord of 30 m length.

7. DEGREE OF CURVATURE
 If AB = 30 m and ∠𝐴𝐴𝐴 = 1° the degree of curvature
of this curve is 1 degree.
 The greater the degree of curvature, the curve will
be sharper and consequently, the smaller will be its
radius.

8. RELATIONSHIP BETWEEN RADIUS AND DEGREE
OF CURVATURE
 Circumference of a circle = 2𝜋R
 Angle subtended at the centre by a circle with this
circumference = 360°
 Angle subtended at the centre by a 30 m chord
D =
360
2 𝜋R
× 30
𝐷 ≈
1719
𝑅
(𝑅 𝑖𝑖 𝑚𝑚𝑚𝑚𝑚)

9. SUPERELEVATION OR CANT
 When a train is moving on a curved path, it has a
constant radial acceleration which produces
centrifugal force.
 In order to counteract this force, the outer rail of the
track is raised slightly higher than the inner rail.
This is known as the Super-elevation or Cant.

10. PURPOSES OF PROVIDING SUPER-ELEVATION
 To ensure safe and smooth movements of
passengers and goods on the track.
 It counteract the effect of the centrifugal force by
producing centripetal force on the train.
 It prevents derailment and reduces the creep and
as well as side wear of rails.
 It provides equal distribution of wheel loads on
two rails.
 It results in the decrease of maintenance cost of
the track.

11. EQUILIBRIUM SUPERELEVATION
v = velocity in m/s
W = weight of the moving train
F = centrifugal force acting on the vehicle
g = acceleration due to gravity in m/s
R = radius of curvature in m
G = gauge of track
e = super-elevation in m
 = angle of inclination
S = length of inclined surface

12. EQUILIBRIUM SUPERELEVATION

13. EQUILIBRIUM SUPERELEVATION
F=
𝑊𝑣2
gR

14. EQUILIBRIUM SUPERELEVATION
𝑒 =
𝑉 × 100,000
60 × 60
2
×
𝐺 × 100
981 × R × 100
If V = velocity in km/hr, then e would be
𝑒 =
𝑉2
𝐺
1.27 R

15. CALCULATION OF SUPER-ELEVATION
 Calculate the superelevation for a 2° BG transitioned
curve on a high-speed route. The speed for calculating
the equilibrium superelevation as decided by the chief
engineer is 80 km/h.

16. CANT DEFICIENCY
 Under a certain conditions, it is not possible to
provide the equilibrium cant.
 In figure ??, a branch line diverges from a main line.
AP and BQ are the inner and outer rails respectively
of main line.
 BD and AC are the inner and outer rails respectively
of the branch line.
 Let S1 and S2 be the amounts of the super-elevation
required form main and branch lines respectively.

17. CANT DEFICIENCY
 Therefore, following condition should be satisfied:
 Considering main line, the point B should be higher than
point A by amount S1.
 Considering branch line, the point A should be higher
than point B by amount S2.

18. CANT EXCESS
 ???

19. CANT DEFICIENCY / CANT EXCESS
 It is obvious that it is impossible to comply with both
the conditions simultaneously.
 Therefore, under such condition a small amount of
deficiency in super-elevation is permitted without
reducing speed.
 This is known as “cant deficiency” or “deficiency in
super-elevation”.

20. PROCEDURES OF PROVIDING SUPER-ELEVATION
 The equilibrium cant on branch line is calculated by usual
formula by assuming suitable speed on branch line.
 The permissible cant deficiency is deducted from the
equilibrium cant.
 The result thus obtained will represent the negative super-
elevation to be given on the branch line.
 Evidently, the negative cant on branch line will be equal to
the maximum cant permitted on the main line.
 The permissible cant deficiency is added to the maximum
cant permitted on the main line and correspondingly, the
restricted speed on the main line is worked out.

21. PERMISSIBLE CANT DEFICIENCY

22. WORKOUT PROBLEM
 A 6 degrees curve branches off from a 3 degrees main
curve in an opposite direction in a layout of a BG line.
If the speed on the branch line is restricted to 35 kmph,
determine the speed restriction on the main line.
Permissible cant deficiency is 75mm.

23. WORKOUT PROBLEM
 SE for branch line, 𝑒 =
𝑉2 𝐺
1.27 R
𝑒 =
352 × 1.676
1.27 ×
1719
6
𝑒 = 5.622 𝑐𝑐
Negative super-elevation = (5.622 – 7.5) = - 1.878 cm
Maximum super-elevation can be provided on the main line =
= 1.878+7.5 = 9.378 cm
Therefore, speed for main line, 𝑒 =
𝑉2 𝐺
1.27 R
9.378 =
𝑉2 1.676
1.27×
1719
3
V = 63.93 kmph

24. WORKOUT PROBLEM
Negative super-elevation = (5.622 – 7.5) = - 1.878 cm
Maximum super-elevation can be provided on the main line =
= 1.878+7.5 = 9.378 cm
Therefore, speed for main line, 𝑒 =
𝑉2 𝐺
1.27 R
9.378 =
𝑉2 1.676
1.27×
1719
3
V = 63.93 kmph

25. FACTORS AFFECTING SUPER-ELEVATION
 Frictional resistance
 Coning of wheel
 Weight of the vehicle
 Speed of the train

26. BENDING OF RAILS ON CURVE
 If the curvature is less than
3˚ then the curve is
considered as the flat curve.
 In that case, the rails are
placed in the curved position
by sleepers
 If the curvature is greater 3˚
then the rails are bend.

27. VERSINE OF A CURVE
 The versine of a curve is
the perpendicular distance
of the midpoint of a chord
from the arc of a circle.

28. RELATIONSHIP BETWEEN RADIUS
AND VERSINE OF A CURVE
 The relationship between the
radius and versine of a curve
can be established from this
figure.
 Let R be the radius of the
curve, C be the length of the
chord, and
 V be the versine of a chord of
length C.

29. RELATIONSHIP BETWEEN RADIUS
AND VERSINE OF A CURVE
 AC and DE being two chords
meeting perpendicularly at a
common point B, simple
geometry can prove that –
AB × BC = DB × BE

30. RELATIONSHIP BETWEEN RADIUS
AND VERSINE OF A CURVE
Or, V(2R – V) = (C/2) × (C/2)
Or, 2RV – V2 = C2/4
V being very small, V2 can be
neglected. Therefore,

31. RELATIONSHIP BETWEEN RADIUS
AND VERSINE OF A CURVE
In this Eq. V, C, and R are in
the same unit, say, metres or
centimetres.
This general equation can be
used to determined versines if
the chord and the radius of a
curve are known.

32. CASE I: VALUES IN METRIC UNITS
The versine formula can also be
written as-
where R is the radius of the curve, C is the chord length
in metres, and V is the versine in centimetres, or

33. CASE II: VALUES IN FPS UNITS
When R1 is the radius in feet, C1
is the chord length in feet, and
V1 is the versine in inches, the
above formula can be written as

34. SAFE SPEED ON CURVES
 For all practical purposes safe speed means a
speed which protects a carriage from the danger
of overturning and derailment and provides a
certain margin of safety.
 Earlier it was calculated empirically by applying
Martin’s formula

35. SAFE SPEED ON CURVES
 For BG and MG
Transitioned curves –
where V is the speed in km/h and R is the radius in
metres.
 For NG
Transitioned curves –

36. TRANSITION CURVE
 Definition
 Requirements
 Formula