RAILWAY CURVES Master Copy for IRTMTC by IRICEN

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 (Necessary evil !) Curves are positive impediment for higher speeds ?

Why Curves ?

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 • Possible passenger discomfort • Possible displacement of loads • Higher Lateral forces on track structure • Maintenance problems (geometrical/structural disturbances) • Wear of rail & wheel flange • Risk of derailment by wheel mounting on outer rail • Risk of vehicle overturning

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

CHOOSING CURVE RADIUS Sharper Curve has an advantage of less space requirement, however it results in high lateral forces on vehicle and rail resulting in high rate of wear of wheel and rail. Also maintaining sharper curve is more difficult. Flatter curve takes more space but has less maintenance issue. Optimise curve radius?

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

CHOOSING CURVE RADIUS The minimum curve radius is decided on 1. Speed potential desired. 2. Max cant value. 3. Cant deficiency. 4. Cant excess permissible. 5. Negotiation capability of vehicle.

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

LIMITING RADII ON IR • BG* : 175 M • MG: 109 M • NG: 44 M *Item 2, Chapter I, Schedule I of SOD

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

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

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

SUPERELEVATION/CANT

VEHICLE ON A CANTED TRACK Centrifugal Force θ G SE W Sinθ θ =G θ W Co sθ W

Super-elevation/Cant • A force is generated, by raising of the outer rail, by the mass of the body countering 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

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 i. e. Equilibrium Cant, SE=G*V 2/(g*R), if G in mm, V in Kmph, R in m, then 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°

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)

EQUILIBRIUM CANT/SPEED • For a particular Speed of train, there will be one Equilibrium Cant or for a particular Equilibrium Cant there will be one Equilibrium Speed. • 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.

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 potential to increase speed in future. (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 • Increased Lateral Forces on Track • More wear on outer rail gauge face

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 EXCESS: SLOW TRAINS

EFFECTS OF VEHICLE WITH CANT EXCESS Speed less than equilibrium speed Centrifugal Force Component < Weight Component • Creq < Ca • Cd = Ca - Creq θ • R o < R i SE G θ • More wear on inner rail gauge face

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.

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

Transition Curves An easement curve, introduced between straight & curved track to facilitate gradual change of Curvature & Superelevation from Straight Track to Curved Track

Transition Curves • On Indian Railways for Transition Curves, it is cubical parabola with the equation: Y = KX 3 (Y= X 3/6 R*L)

Desirable Versine and Cant Diagram Cubic Parabola – Curvature changes linearly v Transition Cant variation – Linear Transition (Linear ramp) Cubic Parabola – Ease in setting/laying/maintaining

Shift Transition Curves Inserting Transition Curves

SHIFT DUE TO TRANSITION CURVE Circular Curve With Transition Extended Circular Curve C D Circular Curve Without Transition B E Transition Curve Tangent H S S/2 A L/2 F Shift = S = L 2/24 R L/2 G BG=L 2/6 R DE=L 2/8 R

Length of Transition Curve • Length of transition will be maximum of L 1 = C a * V m / R Ca or L 2 = C d * V m / R Cd or L 3 = Ca / i

Length of Transition Curve • Length of transition will be maximum of L 1 = 0. 008 Ca* Vm (m, mm, kmph, RCa=35 mm/s) or L 2 = 0. 008 Cd*Vm (m, mm, kmph, RCd=35 mm/s) or L 3 = 0. 72 Ca (m, mm, i = 1 in 720)

Length of Transition Curve • In exceptional circumstances, minimum length of transition will be maximum of 2/3 rd of L 1 or 2/3 rd of L 2 or ½ of L 3

REVERSE AND COMPOUND CURVES

VERTICAL CURVES

TYPES OF VERTICAL CURVES

Realignment Of Curve (ROC) The alignment of curve needs to be corrected when- • Unsatisfactory running (FP/BV/IC/TRC inspections) • Based on results of curve inspection • Service limits are laid down in IRPWM

CRITERIA FOR REALIGNMENT OF CURVE Speed range Limits of station to station variation (mm) 10 mm (15 mm for speed of 110 km/h) Below 140 km/h; and or 20% of average Versine on circular upto 110 km/h portion, Whichever is more Below 110 km/h; and 20 mm or 20% of average Versine on upto 50 km/h circular portion, Whichever is more Below 50 km/h 40 mm or 20% of average Versine on circular portion, Whichever is more

CRITERIA FOR REALIGNMENT OF CURVE • Local adjustment to be done if the variation is at isolated few locations. • If more than 20% station are having versine variations above the limits prescribed, complete realignment of curve should be planned within a month. So it is the Station to Station variation of versine which is more important and not the absolute value of versine.

Thank You