Railway Curves Suggested Further Readings IRPWM Chapter 4

Railway Curves

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

Railway Curves What is a Curve ? Why are Curves necessary ? What are Curve Design Parameters ?

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

Why Curves ? Necessary due to physical & geographical features

Why Curves ?

Why Curves ?

Why Curves ?

Why Curves ? Necessary due to physical & geographical features (Necessary evil !) Curves are impediments for higher speeds ?

Curvilinear Motion

Effects of curve: Centrifugal force • Vehicle Running at Speed V on a Curve of Radius R experiences- Centrifugal Force = MV 2/R

Effects of curve: Centrifugal force • Undesirable Effects – Risk of vehicle overturning – Possible passenger discomfort – Possible displacement of loads – Risk of derailment – Higher Lateral forces on track structure – Higher lateral forces resulting in maintenance problems – Wear of rail & wheel flange

Effects of curve: Curve Resistance

Guidance of wheel on track • Straight Track – Sinusoidal motion • Curved Track – Shifting of center of gravity of wheel set – Actual guidance of the wheel flange by the outer rail in Curves • Slipping/skidding of wheels on sharp curves

Wheel set on Curve

Wheel-sets on Curve

Angularity of axle while negotiating a curve α Angle of attack

Negotiating a curve • Guidance – Preferably be from the track and not flange contact • Play helps the wheel negotiate curve • 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) Optimum Option II (Max. Radius) Choosing Appropriate Curve

Design Parameters of Curves • Radius, R and Degree of Curve, D • Actual Cant (Super-elevation), Ca • • • Equilibrium Cant, Ce / Equilibrium Speed, Ve Cant Deficiency, Cd Cant Excess, Cex Rate of Change of Actual Cant, RCa Rate of Change of Cant Deficiency, RCd Cant Gradient, i • Length of Transition, L

Curve Designation • Curves are Designated by their Radii – Except On IR & US rail roads • On US degree of curve – Designation & Calculation • On IR degree of curve for designation – Radii For Calculation

Degree of Curve The Angle Subtended 30. 5 m (100 feet ) by a 30. 5 m Chord at the Centre of Curve D = 1750/R R D R

Degree of Curve - Exercise Find Radius. If Degree of Curve is • • 0. 50° 2° 4° 5°

Degree of Curve - Exercise Find Radius. If Degree of Curve is • • 0. 50° 2° 4° 5° - 3500 m 875 m 437. 5 m 350 m

Curve Measurement Versine (Mid Chord Offset On 20 m Chord) By Property Of Circle, V*(2 R-V) = C/2*C/2 2 2 RV=C 2/4 i. e. [Neglecting V , being very small] 2 i. e. Versine, V = C /8 R V C 2 R 2 R-V C V R

Versine - Exercise Find Versine on 20 m Chord If Degree of Curve is • • 0. 50° 1° 4° 5°

Versine - Exercise Find Versine on 20 m Chord – If Degree of Curve is • • 0. 50° 1° 4° 5° - 14. 28 mm 28. 57 mm 114. 28 mm 142. 80 mm

Versine - Exercise Find Versine on 11. 832 m Chord If Degree of Curve is • • 0. 50° 1° 4° 5°

Versine - Exercise Find Versine on 11. 832 m Chord – If Degree of Curve is • • 0. 50° 1° 4° 5° ≈ ≈ 0. 5 cm 1. 0 cm 4. 0 cm 5. 0 cm

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

Super-elevation/Cant

Super-elevation/Cant

Vehicle on a Canted Track Centrifugal Force θ W Sinθ θ SE G θ W Co W sθ

Super-elevation/Cant • A force is generated, by raising of the outer rail, by the mass of the body counters the Centrifugal Force • Raising of the outer rail (w. r. t. Inner Rail) to counter the effect of Centrifugal Force (elimination/reduction) is known as Super-elevation/ Cant

Equilibrium Cant/Speed • When on circular motion • If the resultant of Weight & Centrifugal Force is perpendicular to the plane of rail & passes through the centre of track The corresponding speed is known as Equilibrium Speed; and cant is known as Equilibrium Cant

Equilibrium Cant Weight Component = W*sinθ Centrifugal Force Comp. = M*(V 2/R)*cosθ i. e. W*sinθ = M*(V 2/R)*cosθ i. e. W*tanθ = M*(V 2/R) i. e. M*g*SE/G (for small θ, sinθ ≈ tanθ) = M*V 2/R i. e. Equilibrium Cant, SE=G*V 2/(g*R) SE=G*V 2/(127*R) Para 406(a) of IRPWM

Equilibrium Cant - Exercise Find Cant for – BG Speed 100 Kmph Degree of Curve = 2° Dynamic Gauge = 1750 mm SE = GV 2/g. R SE = 157. 31 mm (c/c of Rail heads)

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

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

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

Equilibrium Speed • IRPWM Stipulation (Para 406(b)) Equilibrium speed is to be decided by CE considering • Max. Speeds of fast & slow moving trains (actually attainable) • Proximity to Permanent speed restriction • Junctions • Stopping places • Gradient affecting speed of goods train

Limitations on Maximum value of Actual Cant Ca • Maintenance criteria* – High cant will cause rolling of ballast (loss of lateral ballast resistance and alignment disturbances) – Counters 1 in 20 cross slope (flattening of inner rail head) • Overturning at inner rail – wind blowing from outside – Vehicle at rest on canted track – Maximum Cant - Not very sensitive to wind forces

Limitations on Maximum value of Actual Cant Ca • Safety against derailment* – Empty wagon stopped on Canted Track & just starts moving – Absence of centrifugal forces – Adverse L/V (>1) ratio + angle of attack

Limitations on Maximum value of Actual Cant Ca • Comfort criteria – Maximum discomfort when stopped at Canted Track – No appreciable discomfort upto 180 mm • Limited to 1/8 to 1/10 of Track Gauge

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

IRPWM Provisions • Maximum Cant (Para 406(1)(d)(i)) – 165 mm for group A, B and C routes • 185 MM for locating permanent structures on group A routes with speed increase potential (new works and doubling) incl. TL – 140 mm for group D and E routes

Cant Deficiency: Fast Trains

Effects Of Vehicle With Cant Deficiency Speed more than equilibrium speed Centrifugal Force Component > Weight Component • Creq > Ca • Cd = Creq - Ca θ • R o > R i SE G θ • More wear on outer rail gauge face

Criteria for Cant Deficiency Cd • Safety (overturning about the outer Rail) – Larger values can be permitted – not governing criteria • Comfort criteria* – Discomfort, if Unbalanced Lateral Acceleration is greater than 0. 1 g • UIC recommends 0. 4 m/s 2 to 0. 7 m/s 2 – (1. 0 m/s 2 to 1. 5 m/s 2 for fast passenger trains UIC 703 -R) • SNCF/TGV adopted 0. 56 m/s 2 (trials 0. 7 m/s 2 ) Observed value of ULA is more than theoretical value (Why ? )

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

Vehicles with Cant Deficiency Cd Centrifugal force Component > Weight Component Actual Cant deficiency experienced by vehicle is more than calculated value θ Roll flexibility Coefficient SE G 0. 4 for most modern vehicles (full load) θ

Excessive Cant Deficiency Cd

Excessive Cant Deficiency Cd

Maximum Value of Cant Deficiency

IRPWM Provisions • Max. Cant Deficiency (Para 406(2)) • On routes – with track maintained to C&M, Vol-I standard; – For Nominated Rolling Stock; – With Permission of PCE : 100 mm – For Other cases : 75 mm

Cant Deficiency - Exercise Find Maximum Permissible Speed for – BG, Rajdhani Route (Group “A”), and Degree of Curve = 2° Max. Speed = 129. 79 Kmph

Cant Deficiency - Exercise Find Minimum Permissible Radius on BG High Speed Route Speed 200 Kmph Cd = 75 mm Cd = 100 mm Radius = 2294 m Radius = 2077 m

Cant Excess: Slow Trains

Effects Of Vehicle With Cant Excess Speed less than equilibrium speed Centrifugal force Component < Weight Component • Creq < Ca • Cex = Ca - Creq θ • R i > R o SE G θ • More wear on inner rail top table

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

Criteria for Cant Excess (Cex) • Comfort Consideration not a governing criteria • ULA inwards • For min speed = 0 – already taken in max. Actual Cant considerations • Maintenance consideration* – Excess wear on inner rail

IRPWM Provisions • Max. Cant Excess - 75 mm (Para 406(3)) – Sections carrying predominantly goods traffic shall have less cant excess to reduce wear on inner rail – Worked out for booked speed of goods trains.

Cant Excess - Exercise Find Maximum Speed for goods train with full Cant Excess – Degree of Curve = 2° SE = 140 mm Speed = 64. 39 Kmph

Resultant SE

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

How to introduce Curvature and Super-elevation (Cant) ?

Transition Curves

Rate of Change of Actual Cant (RCa) mm/s, Cant Deficiency (RCd) mm/s, Cant Gradient (i) mm/m; and Length of Transition (L)

Thank You

The UIC (French: Union internationale des chemins de fer), or International Union of Railways, is an international rail transport industry body TGV (French: Train à Grande Vitesse), or high speed train. SNCF (French: Société Nationale des Chemins de fer Français), or French National Railways