Railway simple curve design and maintenance

Railway Curves
Anil Kumar Khare
Sr.Professor/Track-3/IRICEN
By

At the end of the session, we may be
able to -
• Define a Curve
• State Why Curves are necessary
• Calculate Curve Design Parameters
Objectives of the Course

 a line, which is not straight; and
 changes direction without angles
(No sharp Edges); or
 line, which gradually deviates from being
straight
What is a Curve ?

Why Curves ?
For Change in direction
e.g. for
 Perpendicular crossing for river – reduce bridge length
 Pass through stable reaches of a river

Why Curves ?
 Rising a hilly terrain – Max. permissible gradient & load

Why Curves ?
Follow the natural ground – reduces cost of
formation/bridge etc

Why Curves ?
Presence of any obstruction/built-up area in
straight path

Why Curves ?
• To pass through defined points like important cities/towns
etc.

Necessary
• Due to physical & geographical features
• Techno –economical considerations
• Sources of traffic
→ Approx. 16% of BG tracks on IR is in curve
Necessary evil !
• Forces exerted on vehicle moving on curve are more than
straight track & resultant effect on track & passengers
• Maintenance effort in curved tracks is approx. 25% more
than straight tracks
Curves are problematic feature for higher speeds ?
Thus understanding the vehicle movement and proper laying,
maintenance, realignment of curves etc is important
Why Curves ?

Type of Curve on IR
• Curve adopted on IR is a CIRCULAR CURVE
• Circular Curve – Uniform curvature thus Easy to lay
and manage

Understanding Circular Motion
Centrifugal force
(Reqd for object
to rotate in a
curve)

Effects of curve: Centrifugal force
•Vehicle Running at Speed V on a Curve of
Radius R experiences-
Centrifugal Force = MV2
/R
C.F.
V
Movement

Movement of a Car on Curve
Frictional Force is the
Centripetal Force
Centrifugal
Force

Movement of a Train on Curve
Centrifugal Force
Centrepetal Force
by Rail
Position of Wheel on
Outer Rail of Curved
Track
Outer Rail

•Undesirable Effects
Possible passenger discomfort (in Passenger trains)
Possible displacement of loads (in Goods Train)
Higher Lateral forces on track structure from
wheel flange (rubbing against outer rail)
Maintenance problems (geometrical/structural
disturbances)
Wear of rail & wheel flange
Risk of derailment by wheel mounting on outer
rail
Risk of vehicle overturning about outer rail
Effects of curve: Centrifugal force

Effects of curve: Curve Resistance
Direction
of Velocity
Direction of
movement of
train
P
Loco to apply extra force for changing direction & overcoming
frictional force between rail & wheel → Curve resistance

•Straight Track
•Sinusoidal motion due to coning of wheels
Guidance of wheel on track
1 in 20

•Curved Track
 Centrifugal force shifts the
wheel set to outer rail →
resulting in Large Diameter
of wheel on outer rail &
Smaller on inner rail →
Outer wheel will travel more
distance than inner one
 Requirement – travelling of
larger distance by wheel on
outer rail (with larger length)
of curve than inner one
Length of Outer Rail >
Length of inner rail

•Curved Track
Coning of wheels thus helps the wheel negotiate the
curve (for a certain degree) for which Difference in
outer & inner rail is equal to difference between
circumference.
For other degrees either slipping or skidding of
outer/inner wheel may take place ( as difference
between length of outer rail & inner rail will be less or
more respectively) – Damage rail top
Note:
a) Slipping – Rotational movement > Translational movement
b) Skidding – Translational movement > Rotational movement

•Curved Track (Curve Resistance)-
•Wheel movement is guided by wheel flange, with
rail head counteracting the centrifugal force,
causing force (in turn causing frictional force =
μ*R) on wheel flange → Curve resistance
Where μ = coefficient of friction
•To reduce curve resistance & wear, CANT is
provided
-Cant reduces such wheel flange forces by adding
weight component opposite to centrifugal force

Wheel on Straight Outer Wheel on Curve

Negotiating a curve
•Guidance while negotiating a curve -
By wheel coning
By Play between Rail & Wheel flange
helps the wheel negotiate curve
Preferably be from the track and not flange contact
→ part of load providing the centripetal force (by
way of Cant)
•If bogie can take radial position on the curves,
it will be better
• (present design of rigid bogies cannot)

Apex Distance
Option I
(Min. Radius)
Option II
(Max. Radius)
Choosing Appropriate Curve
Optimum Option

Choosing curve radius
• Sharp or Flat ?
Sharper Curve → less space requirement, but high
lateral forces on vehicle and rail → high rate of wear of
wheel and rail, more disturbance to track due to higher
forces, less safe, discomfort etc
Flatter curve takes more space but has less
maintenance issue.
Optimise curve radius – based on Available land,
Sectional speed etc

Design Parameters of Curves
• Curve has two parts- Circular & Transition
•Circular portion – Radius and Cant
Radius, R or Degree of Curve, D
 Cant
 Equilibrium Cant, Ce / Equilibrium Speed, Ve
Actual Cant (Super-elevation), Ca
Cant Deficiency, Cd
Cant Excess, Cex
•Transition Portion – Length, keeping in view -
Rate of Change of Actual Cant, RCa
Rate of Change of Cant Deficiency, RCd
Cant Gradient, i

A. Curve radius
• Minimum curve radius is decided based on -
 Speed potential desired
 Permissible Max cant value
 Permissible Cant deficiency
 Permissible Cant excess
 Negotiation capability of vehicle
 Land Availability

Curve Designation
Curves are Designated by their Radii –
Generally Except On IR & US rail roads
On IR, Degree of curve for designation. but Radii
For Calculation

Relationship between Degree & Radius of
Curve
D = 1750/R
30.5m (100 feet )
D
R
R
• Angle Subtended by a 30.5m
Arc/Chord (on OUTER RAIL) at the
Centre of Curve
• Angle subtended by complete
circle of length 2πR = 3600
• ⸫ Angle subtended by an arc (arc
≈ chord for large radius) of length
30.5m, D = 360* 30.5/ 2πR
• i.e. D= 1746.8/R
Or

Find Radius-
If Degree of Curve is
• 0.50°
• 1°
• 2°
• 4°
Degree of Curve - Exercise
R=1750/D
Degree Radius (m)
0.5 3500
1 1750
2 875
4 437.5

Limiting radii on IR
•BG* : 175 M
•MG: 109 M
•NG: 44 M
*Item 2, Chapter I, Schedule I of SOD

2R
Curve Measurement
 Neither by Degree nor Radius as both are impracticable.
 Curve measurement is done by Versine (Mid Chord Offset)
By Property Of Circle, v*(2R-v) = C/2*C/2
i.e. 2Rv=C2
/4 [Neglecting v as compared to 2Rbeing very small]
i.e. Versine, v(m) = C
2
/8R or v (mm) = 125C
2
/R
C
V
2R-v
V
R
C
A B
D
D=2
R
C/2 C/2

Measurement of Versines of curves (Para 401
of IRPWM)
• On plain tracks, versines are measured on 20m
overlapping chords with stations at 10m interval
v= 125*20*20/R = 50000/R
where R=1750/D
⸫ v = 28.5D
• For turnout & turn-in curves, versines are measured
on 6m overlapping chords with stations at 3m interval
• v = 125* 6*6/R = 4500/R
v = 2.57 D
0
10
20 30

Tools for measurement of Versine
Rail touching
location Thread
location
25mm
Zero of scale starts at
25mm
Versine
Holders
String coil

Find Versine on 20m Chord -
If Degree of Curve is
• 0.50°
• 1°
• 4°
• 5°
Versine - Exercise
V= 28.5*D
Degree Versine (mm)
0.5 14
1 28.5
4 114
5 142.5

•To counteract the effect of centrifugal force,
outer rail is raised w.r.t. to inner rail.
•Raising of the outer rail (w.r.t. Inner Rail) to counter
(eliminate/reduce) the effect of Centrifugal
Force is known as
Super-elevation/ Cant (+/-)
A force is generated, by raising of the outer rail, by
the mass of the body countering the Centrifugal
Force
Super-elevation/Cant

G
SE or Cant
Centrifugal Force =
MV2
/R
θ
θ
Vehicle on a Canted Track
V
W sinθ
W cos θ
+
θ
(MV2
/R) cosθ
MV2
/R sinθ
θ
W
Counteracts
the
Centrifugal
force

Purpose of Cant ?
•Neutralization of lateral forces leading to
better comfort
•Better distribution of load on both Rails
•Reduction of wear of Rails

Equilibrium Cant/Speed
•When on circular motion
• If the component of Weight & Centrifugal Force along the
plane of the rail are equal and opposite in direction i.e.
W sinө = MV2
/R cos ө
then the corresponding
speed is known as Equilibrium Speed; and
cant is known as Equilibrium Cant
(MV2
/R) cosθ

Weight Component along the plane of track
= W*sinθ
CF=M*(V2
/R)
Centrifugal Force Comp (CF) along plane= M*(V2
/R) cosθ
i.e. W*sinθ = M*(V2
/R)*cosθ
i.e. W*sinθ = M*(V2
/R) (for small θ, cosθ ≈ 1)
i.e. M*g*SE/G = M*V2
/R
i.e. Equilibrium Cant, SE=G*V2
/(g*R),
if G is in mm, V in Kmph, R in m, then SE=G*V2
/(127*R) mm
(Substituting g=9.8m/s2
, multiplying ‘V’ by 5/18 for converting km/h to m/s)
SE = 13.76* V2
/R (For BG) (where G= 1750, dynamic gauge)
Equilibrium Cant

Find Equilibrium Cant for –
BG
Speed 100 Kmph
Degree of Curve = 2°,
R= 1750/2 = 875 m
Equilibrium Cant - Exercise
SE = 157.26 mm
Equilibrium Cant, SE = 13.76*V2
/R
SE = 13.76* 100* 100/875

Equilibrium Cant/Speed
• For a particular Speed of train, there will be one Equilibrium
Cant or for a particular Cant there will be one Equilibrium
Speed for a Curve of Radius R .
• In such equilibrium situation :
Load will be equal on both rails
Wear on both rails will be same
Maintenance of track Geometry will be easier.
Fittings and Fixtures are subjected to less stress.

Considerations of Mixed Traffic
• For what speed should the Cant be provided ?
• Maximum speed ?
• Minimum Speed ?
• Average Speed?

• Schramm’s Formula:
• Li : Load of ith
train,
• Vi : Speed of ith
train,
• n : Number of trains
Equilibrium Speed

• Russian Formula:
• ni : No of trains of type i,
• Wi : Weight of such train,
• Vi : Speed of such train,
• m : Total types of such train group
Equilibrium Speed

Equilibrium Speed
•IRPWM Stipulation (Para 404(b))
Equilibrium speed is to be decided by PCE
considering
Max. Speeds of fast & slow moving trains
(actually attainable)
Proximity to Permanent speed restriction
Junctions
Stopping places
Gradient affecting speed of goods train
- without appreciably affecting the speed of fast trains and
their relative importance

Equilibrium Speed (Para 404) contd…
• Entire section may be divided into a certain number of sub
sections with a nominated equilibrium speed.
• Equilibrium speed fixed to be such that there is no need for
imposing any speed restrictions for limiting the cant excess
for slow trains and cant deficiency for fast trains.
• On sections where all trains run at about the same
maximum permissible speeds like suburban section, it will
be preferable to provide cant for that speed.
• Cant to be provided on the basis of this Equilibrium
speed.

Limitations on Maximum value of Actual Cant Ca
A. Maintenance criteria
Higher cant causes rolling of ballast
(loss of lateral ballast resistance and alignment
disturbances)
In vehicle standing position, excess force on inner
rail. (flattening of inner rail head)
• Based on above maintenance criteria - Cant is
generally limited to G/10 = 167 mm on BG
ACTUAL CANT

Limitations on Maximum value of Actual
Cant Ca contd…
B. Overturning about inner rail
Vehicle at rest/low speed on canted track –
 No/very less centrifugal force to counter wt.
component
wind blowing from outside
vibration or other disturbance.
• Max. Cant with Factor of safety of 3 comes
to 304 mm for BG, 200mm after taking into
account wind force.

Limitations on Maximum value of
Actual Cant Ca
C. Safety against derailment by Flange
climbing*
Empty wagon stopped on Canted
Track & just starts moving →
Reaction on outer rail is less than
inner rail while Frictional forces &
angularity favours the wheel
climbing
Adverse L/V (>1) ratio
with positive angle of attack
especially for empty vehicles on
sharp curves
(added adversity by twisted track
condition) → High permissible values
of Ca

Limitations on Maximum value of
Actual Cant Ca
D. Comfort criteria
Maximum discomfort/load displacement when
stopped at Canted Track
No appreciable discomfort upto 180 mm
UIC (International Union of Railways) Recommends
160 mm (for SG of 1435mm) ≈ 185 mm (BG)
• Actual Cant is generally Limited to 1/8 (0.125) to
1/10 (0.1) of Gauge

IRPWM Provisions
•Maximum Cant
(Para-404 of IRPWM)
•185 mm - Group A & B routes
• On existing track, approval of Chief Track Engineer
to be taken based on clearance study & with
corresponding increase in length of transition
•165 mm – Other Routes

Maximum Value Of Cant
The Maximum Value of Cant provided on the World Railways
185 0.11

Cant Deficiency: Fast Trains
CANT DEFICIENCY- Fast Moving Train

Effects On Vehicle With Cant Deficiency
G
SE
θ
When Speed is more than equilibrium speed
Centrifugal Force Component > Weight Component
θ
• Creq > Ca
• Cant Deficiency,
Cd = Creq – Ca
• Ro > Ri
• Increased Lateral &
vertical Forces on
outer rail of Track
• More wear on outer
Ro
Ri

Criteria for Limitation on Cant Deficiency Cd
A. Track Stresses & Lateral stability
Cd affects vertical and lateral stresses in Outer Rail (Rail
damage is likely to occur)
Lateral forces should not exceed track resistance to
lateral deformation -
A function of type of ballast material; and
Degree of ballast consolidation (post tamping ???)
Apart from Self Weight
 Vertical forces are less for high speed trains, which are
generally passenger trains
Large Cd Values can be permitted, thus not a
limiting criteria for Cant Deficiency

Criteria for for Limitation on Cant Deficiency Cd
B. Safety (overturning about the outer Rail)
Larger values can be permitted
 up to 175 mm, safe with wind velocity of 35 m/s (126
km/h) on 2500 m curve; and 560 mm for zero wind speed)
C. Comfort criteria*
•Discomfort, if Unbalanced Lateral Acceleration is
greater than 0.1g i.e. 1.0 m/ s2
• UIC recommends 0.4 m/s2
to 0.7 m/s2
Observed value of ULA is more than the theoretical value. Why?

G
SE
θ
θ
Reaction on
Outer Rail > Inner Rail
Deflection of
Outer Spring > Inner Spring
Vehicles with Cant Deficiency Cd
If Speed > Equilibrium Speed
Centrifugal force Component > Weight Component
Ro
Ri

G
SE
θ
Centrifugal force Component > Weight Component
θ
Vehicles with Cant Deficiency Cd
Actual Cant deficiency
experienced by vehicle is
more than calculated
value.
Hence ULA is kept near
lower limit

IRPWM Provisions
• Max. Cant Deficiency
(Para 404(2))
On all routes
 For Nominated Rolling Stock: 100/150 mm (ULA =
0.56/0.84 m/s2
)# , 115mm on track with T/O with crossing on
outer rail & on track with SEJ (from impact load consideration) for
nominated rolling stocks based on oscillation trials & specified in
speed certificate issued by RDSO
For Other cases : 75 mm (ULA = 0.42 m/s2
)
#As ∆p = Cd * g/G
= 100*9.81/1750 = 0.56 m/s2

Speed for curve with Cd
Equilibrium, SE=G*V2
/(127*R)
For curve with Cd
Cd = Ceq - Ca
i.e.
Cd=G*V2
/(127*R)- Ca =13.76 V2
/R - Ca
or V2
=R* (Ca + Cd)/13.76
Vmax=0.27*√ {R*(Ca+Cd)}

Find Maximum Permissible Speed for –
BG,
Rajdhani Route and
Degree of Curve = 2°
Cant Deficiency - Exercise
Max. Speed = Vmax=0.27*SQRT{R*(Ca+Cd)}
= 130 Kmph
= 146 kmph for Ca = 185mm, Cd = 150mm
Ca = 165 mm
Cd = 100mm
R=1750/2 = 875m

Effects Of Vehicle With Cant Excess
If Speed < Equilibrium speed
Centrifugal force Component < Weight Component
• Creq < Ca
• Cant Excess, Cex = Ca -Creq
• Ri > Ro
• More wear on
inner rail top table
G
SE
θ
θ
Ro
Ri

Effects Of Vehicle With Cant Excess
Actual Cant excess
experienced by vehicle is
more than calculated
value
Speed less than equilibrium speed
Centrifugal force Component < Weight Component
G
SE
θ
θ

Criteria for Cant Excess (Cex)
•Comfort Criteria
For min speed = 0, Cex = Ca
already taken in max. Actual Cant considerations, thus
Max Cex = Max. Ca = 185mm
- Comfort Consideration not a governing
criteria : ULA inwards & for goods train
•Safety Criteria – Same as for Ca, Cex = 304mm
•Maintenance consideration* (Crushing & Metal flow)
Inner rail stresses (Freight Trains with higher axle load)
Excess wear on inner rail

• Max. Cant Excess - 75 mm
(Para 404)
•Sections carrying predominantly goods
traffic shall have less cant excess to reduce
wear on inner rail
•Worked out for booked speed of goods
trains.
IRPWM Provisions

Find Maximum Speed for goods train with full Cant Excess
–
Degree of Curve = 2°
SE = 140 mm
Cant Excess - Exercise
Equilibrium Cant = 140 -75 = 65mm
From, C = GV2
/127R
Speed = 64.39 Kmph

Remove
SE
For
Balancing
Reduce
SE
Balanced
Condition
Increase
SE
Increasing Speed
Lateral
Accn
< 0 < 0 = 0 > 0
Cant Cex Cex Ceq Cd
Speed = 0

RECAP
• Curves – Necessity
• Curve Design parameters like
Degree of curve – angle subtended by arc of 30.5m
Radius of curve, R=1750/D
Versine- for measurement of curve, v= 28.5D
Equilibrium speed (Veq) & Eq. Cant (Ceq) – Equal stresses
on both rails
Actual Cant, Ca – Limit (185mm) governed by comfort criteria
& not safety criteria
Cant Deficiency (Cd) – Limit (75 to 150mm) governed by
comfort criteria
Cant Excess (Cex) – Limit (75mm) governed by maintenance
criteria

Suggested Further Readings
• IRPWM (Chapter 4 &
relevant)
• IR Schedule of Dimensions
• IRICEN publication
“RAILWAY CURVES”
• Etc.

Railway simple curve design and maintenance

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