CURVES

CURVES WHY?? WHAT?? PARAMETERS??

NECESSITY ARISES DUE TO PHYSICAL & GEOGRAPHICAL FEATURES NECESSARY EVIL CURVE IS A LINE OF WHICH NO PART IS STRAIGHT AND WHICH CHANGES DIRECTION WITHOUT ANGLES POSITIVE IMPEDIMENTS FOR HIGHER SPEEDS

ANGULARITY OF AXLE WHILE NEGOTIATING A CURVE Angle of attack α PLAY HELPS THE WHEEL NEGOTIATE CURVE A BOGIE WHICH CAN TAKE RADIAL POSITION ON THE CURVES IS BETTER GUIDANCE SHALL BE FROM THE TRACK GUIDANCE AND NOT FLANGE CONTACT

CURVE DESIGNATION CURVES ARE DESIGNATED BY THEIR RADII, EXCEPT ON IR & US RAIL ROADS ON IR DEGREE OF CURVE FOR DESIGNATION RADII FOR CALCULATION

DEGREE OF CURVE IS THE ANGLE SUBTENDED BY 30.5m CHORD AT THE CENTRE 100 feet=30.5m D = 1750/R D R

V*(2R-V) = C/2*C/2 2RV=C2/4 V = C2/8R ON IR, THE CURVES ARE MEASURED BY VERSINE- WHICH IS MID CHORD OFFSET ON 20m CHORD V*(2R-V) = C/2*C/2 2RV=C2/4 V = C2/8R 2R V C

PARAMETERS OF CURVES RADIUS, R AND DEGREE OF CURVE, D ACTUAL CANT Ca CANT DEFICIENCY Cd CANT EXCESS Cex EQUILIBRIUM CANT Ce / EQUILIBRIUM SPEED Ve RATE OF CHANGE OF ACTUAL CANT rca RATE OF CHANGE OF CANT DEFICIENCY rcd CANT GRADIENT, i LENGTH OF TRANSITION, L

EFFECTS OF CURVES VEHICLE RUNNING AT A SPEED OF V IN A CURVE OF RADIUS R EXPERIENCES A CENTRIFUGAL FORCE = MV2/R UNDESIRABLE EFFECTS POSSIBLE PASSENGER DISCOMFORT POSSIBLE DISPLACEMENT OF LOADS RISK OF VEHICLE OVERTURNING

EFFECTS OF CURVES RISK OF DERAILMENT HIGH LATERAL FORCE RESULTING IN MAINTENANCE PROBLEMS CURVE RESISTANCE WEAR OF RAIL & WHEEL FLANGE LATERAL FORCE ON TRACK STRUCTURE

COMPENSATION FOR CURVATURE ON GRADIENT COMPENSATION ALLOWED ON GRADIENTS DUE TO CURVATURE 70/R 0.04% PER DEGREE

SUPERELEVATION / CANT THE EFFECT OF CENTRIFUGAL FORCE IS ELIMINATED/REDUCED BY RAISING THE OUTER RAIL BY A SPECIFIED AMOUNT. THIS RAISING OF OUTER RAIL OVER INNER RAIL IS CALLED SUPERELEVATION/CANT THE FORCE DUE TO THE RAISING OF THE OUTER RAIL IS EXERTED INWARDS AND IS CALLED CENTRIPETAL FORCE

VEHICLE ON A CANTED TRACK G SE Centrifugal Force   W Centripetal Force Wsinθ 

EQUILIBRIUM CANT 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 CALLED EQUILIBRIUM SPEED & THE CANT IS CALLED EQUILIBRIUM CANT

EQUILIBRIUM CANT IRPWM STIPULATION EQUILIBRIUM SPEED IS TO BE DECIDED BY CE CONSIDERING MAX. SPEEDS OF FAST & SLOW MOVING TRAINS PERMANENT SPEED RESTRICTION JUNCTIONS STOPPING PLACES GRADIENT AFFECTING SPEED OF GOODS TRAIN

Equilibrium Speed Russian Formula: ni: no of trains, wi: weight of such train, Vi: speed of such train, m: types of trains

ACTUAL CANT Ca MAINTENANCE CRITERIA OVERTURNING AT INNER RAIL HIGH CANT WILL CAUSE ROLLING OF BALLAST & FLATTENING OF INNER RAIL OVERTURNING AT INNER RAIL NOT VERY SENSITIVE TO WIND FORCE SAFETY AGAINST DERAILMENT EMPTY WAGON STOPPED & STARTED COMFORT CRITERIA NO APPRECIABLE DISCOMFORT UPTO 180 mm LIMITED TO 1/8 TO 1/10 OF DYNAMIC GAUGE

MAXIMUM VALUE OF CANT

IRPWM PROVISIONS EQUILIBRIUM CANT (406(1)(a)) C=GV2/(127 R) MAXIMUM CANT (406(1)(d)(i)) 165 MM FOR GROUP A, B AND C ROUTE 185 MM FOR STRUCTURES 140 MM FOR GROUP D AND E ROUTE

Cant Deficiency

CANT DEFICIENCY Cd SAFETY COMFORT CRITERIA UPTO 175 mm SAFE WITH CRITICAL WIND VELOCITY COMFORT CRITERIA DISCOMFORT IF UNBALANCED LATERAL ACCELERATION IS GREATER THAN 0.1g CANT DEFICIENCY SHOULD BE LESS THAN 0.1G OBSERVED VALUE OF ULA IS MORE THAN THE THEORETICAL VALUE

MAXIMUM VALUE OF CANT DEFICIENCY

Cant Excess

CANT EXCESS Cex NO COMFORT CONSIDERATION MAINTENANCE CONSIDERATION EXCESS WEAR ON INNER RAIL MORE THE VOLUME OF GOODS TRAFFIC LESSER WOULD BE CANT EXCESS WORKED OUT FOR BOOKED SPEED OF GOODS TRAINS. NORMALLY 65 KMPH ON BG.

Cant Excess values SNCF DB IMPORTANCE TONNAGE PER DAY CANT EXCESS VERY IMP. >45000 70 IMPORTANT 25,000 TO 45,000 80 MEDIUM 10,000 TO 25,000 90 UNIMPORTANT <10,000 100 DB  >60,000 50 30,000 TO 60,000 10,000 TO 30,000 110

IRPWM PROVISIONS CANT DEFICIENCY(406(2)) CANT EXCESS (406(3)) FOR GROUP A AND B ROUTES FOR SPEEDS IN EXCESS OF 100 KMPH FOR NOMINATED STOCK RUNS WITH PERMISSION OF CE : 100 MM FOR OTHERS : 75 MM CANT EXCESS (406(3)) 75 MM

MAXIMUM VALUE OF Cex

TRANSITION CURVE TRANSITION CURVE IS AN EASEMENT CURVE INTRODUCED BETWEEN STRAIGHT & CURVED TRACK TO FACILITATE GRADUAL CHANGE OF VERSINES & SUPERELEVATION ON INDIAN RAILWAYS, IT IS CUBICAL PARABOLA WITH THE EQUATION: Y = KX3 THE SPIRAL, WHICH CHANGES THE DIRECTION ANGLE UNIFORMLY, IS THE IDEAL TRANSITION THERE IS NOT MUCH DIFFERENCE IN THE LAYOUT OF THE SPIRAL AND CUBIC PARABOLA UNTIL THE DEFLECTION FROM STRAIGHT IS APPROX 4 M.

VERSINE AND CANT DIAGRAM OF A CURVE TRANSITION CURVES CURVATURE VARIES UNIFORMLY WITH DISTANCE VERSINES VARY UNIFORMLY CANT VARIES UNIFORMLY V Ca VERSINE AND CANT DIAGRAM OF A CURVE

SHIFT ON TRANSITION CURVE CIRCULAR CURVE WITHOUT TRANSITION CIRCULAR CURVE WITH TRANSITION EXTENDED CIRCULAR CURVE d2y/dx2=k x. dy/dx=kx2/2. y = kx3/6. at L, 1/R=kL. k=1/RL. Y=x3/6RL. At L/2, S/2= L2/48R. S=L2/24R C D B E F G A S TRANSITION CURVE TANGENT H S/2 L/2

LENGTH OF TRANSITION COMFORT CRITERIA RATE OF CHANGE OF ULA LESS THAN 0.03g RATE OF CHANGE OF Cd <0.03G FIXED AS 35 mm /sec RATE OF CHANGE OF Ca IS JUST NOTICEABLE AT 65 TO 75mm/sec BUT FIXED AS 35 mm /sec UNDER EXCEPTIONAL CIRCUMSTANCES IT CAN BE INCREASED TO 55 mm /sec

LENGTH OF TRANSITION SAFETY CRITERIA CANT GRADIENT CAUSES TWIST IN TRACK LIMITED TO 1.4 mm/m CANT GRADIENT – 1 IN 720 IN EXCEPTIONAL CASES IT CAN BE 1 IN 360 FUTURE LAYOUTS WITH 1 IN 1200

LENGTH OF TRANSITION LENGTH OF TRANSITION WILL BE MAX. OF L1 =0.008 Ca*Vm L2 =0.008 Cd*Vm L3 =0.72 Ca MINIMUM LENGTH OF TRANSITION WILL BE MAX OF 2/3RD OF L1 2/3RD OF L2 ½ OF L3

SUITABILITY OF CUBIC PARABOLA AS TRANSITION CUBIC PARABOLA IS SIMPLE TO LAY AND MAINTAIN AS VERSINES VARY LINEARLY. THERE IS SHOCK AT ENTRY AND EXIT OF THE CUBIC PARABOLA DUE TO SUDDEN CHANGE IN THE VERSINES. END CORRECTION IS TO BE APPLIED AT THE 0 STATION AND SECOND LAST STATION.

SUITABILITY OF CUBIC PARABOLA AS TRANSITION FOR SPEEDS ABOVE 130 KMPH, CUBIC PARABOLA IS NOT SUITABLE AS THE EFFECT OF END CONDITIONS GETS PRONOUNCED. ALTERNATIVE SHAPES ARE FOURTH ORDER PARABOLA HALF SINE CURVE FULL SINE CURVE

V VIRTUAL TRANSITION

VIRTUAL TRANSITION IF THERE IS NO SPACE FOR TRANSITION, THE BOGIE LENGTH BECOMES THE TRANSITION VIRTUALLY For BG- 14.6 M For MG-13.7 M

COMPOUND AND REVERSE CURVES For Reverse Curve: Length of transition will be MAX. of L =0.008*(Ca1+Ca2)*Vm L =0.008 (Cd1+Cd2)*Vm L =0.72 (Ca1+Ca2)

COMPOUND AND REVERSE CURVES For Compound Curves: Length of transition will be MAX. of L =0.008 (Ca1-Ca2)*Vm L =0.008 (Cd1-Cd2)*Vm L =0.72 (Ca1-Ca2) If length is coming less than virtual transition then common transition is deleted and the cant is run out on the length of virtual transition

REVERSE CURVES For high speeds in Group A and B routes a straight of 50m length shall be kept Otherwise, increase the transition length to eliminate the straight If neither of the above two are possible than speed restriction of 130 KMPH on BG

Vertical Curves

VERTICAL CURVES ROUTE MIN.RADIUS (m) A 4000 B 3000 C,D,E &MG 2500 VERTICAL CURVES ARE PROVIDED AT THOSE LOCATIONS WHERE ALGEBRAIC DIFFERENCE BETWEEN THE GRADES IS EQUAL TO OR MORE THAN 0.4% (4mm/m) ROUTE MIN.RADIUS (m) A 4000 B 3000 C,D,E &MG 2500

Extra Clearances on curves

CLEARANCE ON CURVES- LEAN H Ca L = H*Ca/G

ADDITIONAL ALLOWANCE DUE TO SWAY INSIDE ¼ OF LEAN DUE TO SUPERELEVATION OUTSIDE NIL

Extra Clearances on Curves

ADDITIONAL CLEARANCES ON CURVES OVER THROW Vo CENTRE LINE OF COACH END THROW VE BOGIE CENTRE 14.785 21.340 CENTRE LINE OF TRACK

EXTRA CLEARANCE DUE TO CURVATURE PLATFORMS/ STRUCTURES INSIDE OF CURVE (VO+L+S-51) OUTSIDE OF CURVE (VE-25)mm BETWEEN ADJACENT TRACKS VO+ VE+2*(L/4) In new works, if c/c track is 5300 mm, extra clearance is to be provided for curves beyond 5 degrees only.

TURNOUTS TAKING OFF FROM CURVES SIMILAR FLEXURE EQUIVALENT RADIUS Re=Rm*Rs/(Rm+Rs) Rm: MAIN LINE RADIUS Rs: SWITCH RADIUS Re: EQUIVALENT RADIUS

CONTRARY FLEXURE EQUIVALENT RADIUS Re=Rm*Rs/(Rm-Rs)

TURNOUTS OF CONTRARY FLEXURE CURVES OF CONTRARY FLEXURE EQUILIBRIUM SUPER ELEVATION FOR TURNOUT SIDE C= G * V2 / 127 * R; V: SPEED KPMH FOR TURNOUT, R: Metres, C: MM, G: MM NEGATIVE SUPERELEVATION FOR TURNOUT IS 75-C

CHECK RAILS ON CURVES PARA 426 IRPWM CL. 4, CHAPTER 1, BG SOD TOO MUCH WEAR ON OUTER RAIL IN SHARP CURVES CHECK RAIL CLEAR-ANCE PARA 426 IRPWM CL. 4, CHAPTER 1, BG SOD ON CURVES OF 8 DEGREES & ABOVE MIN. CLEARANCE = 44 mm CLEARANCE = 44 + (G-1676)/2

IRPWM PROVISIONS INSPECTIONS AEN TO INSPECT ONE CURVE IN EACH PWI JURISDICTION EVERY QUARTER (107(4)) PWI INCHARGE AND HIS ASSISTANT SHALL INSPECT CURVES ONCE IN SIX MONTHS ALTERNATELY EXCEPT AT GROUP A AND B ROUTES WHERE INSPECTION IS TO BE DONE ONCE IN FOUR MONTHS (124(4), 139(4)) FOR CONCRETE SLEEPER TRACK, ONCE IN SIX MONTHS BY ROTATION (124A, 139A)

IRPWM PROVISIONS LATERAL WEAR ON RAILS IN CURVES (301(b)(iv)) GROUP A AND B : 8 MM GROUP C AND D : 10 MM MINIMISING WEAR ON OUTER RAIL (427(2)) RAIL FLANGE LUBRICATORS SHALL BE PROVIDED IN CURVES LESS THAN 600 M RADIUS. FIRST ONE SHALL BE AHEAD OF THE CURVE

IRPWM PROVISIONS CUT IN RAILS IN SWR TRACK (424 ) TO MAKE THE JOINTS SQUARE WHEN THE GAIN OF INNER RAIL BECOMES EQUAL TO PITCH OF THE FIRST BOLT HOLE GAIN OF INNER RAIL GIVEN BY: d= LG/R, L IS LENGTH OF CURVE. ON CURVES LESS THAN 400 M RADIUS, THE JOINTS SHALL BE LAID STAGGERED

IRPWM PROVISIONS AFTER RELAYING VERSINE VARIATION OVER THEORETICAL (316) 600 M RADIUS AND MORE: 5 MM LESS THAN 600 M RADIUS: 10 MM GAUGE IN TRACK ON CURVES (403(1)) MORE THAN 350 M RADIUS: -5 MM TO + 3 MM LESS THAN 350 M RADIUS: +10 MM

IRPWM PROVISIONS IN SERVICE PARAMETERS GAUGE (224(2)(e)(v)) ON STRAIGHT : (-) 6 MM TO (+) 6 MM ON CURVES RADIUS 350 M OR MORE: (-) 6 MM TO (+) 15 MM ON CURVES RADIUS LESS THAN 350 M: UPTO (+) 20 MM

IRPWM PROVISIONS IN SERVICE TOLERANCES (SPEED >100 KMPH, <140 KMPH) TWIST (607(2)(iii)) ON STRAIGHT/CURVES NORMAL 2 mm/M ISOLATED 3.5 mm/M ON TRANSITIONS NORMAL 1mm/M ISOLATED 2.1 mm/M

IRPWM PROVISIONS ALIGNMENT (607(2)(i)) 7.5 M CHORD IN SERVICE TOLERANCES (SPEED >100 KMPH, <140 KMPH) ALIGNMENT (607(2)(i)) 7.5 M CHORD ON STRAIGHT NORMAL 5 mm ISOLATED 10 mm ON CURVES NORMAL ±5 mm OVER AVERAGE VERSINE ISOLATED ±7 mm OVER AVERAGE VERSINE CHORD TO CHORD > 10 mm

IRPWM PROVISIONS RADIUS OF CURVE (401(1)) R = 125C2/V FOR MEASURING VERSINES (401(3)) NORMAL: 20 M CHORD OVERLAPPING WITH STATIONS AT 10 M P & Cs: 6 M CHORD OVERLAPPING WITH STATIONS AT 1.5 M INNER RAIL IS TAKEN AS REFERENCE AND OUTER RAIL IS RAISED BY AMOUNT OF SUPERELEVATION(402)

IRPWM PROVISIONS SAFE SPEED ON A CURVE (405(1)(a)) V = 0.27 √R (Ca + Cd) NON TRANSITIONED CURVE (405(2)(a)) LENGTH OF TRANSITION : 14.6 M ON OUTSIDE OF CURVES, EXTRA WIDTH OF BALLAST TO BE PROVIDED 150 MM: FOR LWR TRACK (ANN 2/11 AND 2/12) 95 MM: ON FISH PLATED TRACK, RADIUS LESS THAN 400 M (ANN 2/13 )

LEAD CURVE FOLLOWING TURNOUT MINIMUM RADIUS OF LEAD AND TURN IN CURVE (410(2)) : 350 M MIN RADIUS IN LEAD 220 M, IN EXCEPTIONAL CIRCUMSTANCES (WITH PSC/ST SLEEPERS AND FULL ML BALLAST PROFILE) THERE SHALL BE NO CHANGE IN SUPERELEVATION 20 M ON EITHER SIDE OF THE TOE OF SWITCH AND NOSE OF CROSSING TURNOUT FOLLOWED BY REVERSE CURVE, CHANGE IN CANT BEHIND CROSSING IS PERMITTED. CANT LIMITED TO 65 MM, RUN OUT AT 2.8 MM/M

LEAD CURVE FOLLOWING TURNOUT The variation in versines on two successive stations in lead curve and turn in curve portions should not be more than 4mm. And versine at each station should also not be beyond ± 3mm. from its designed value (237(4)(d))

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

IRPWM PROVISIONS CURVE BOARDS(409(1)) CURVE POSTS (409(2)) CURVE BOARD SHALL BE PROVIDED AT TANGENT POINT ON OUTSIDE OF THE CURVE CURVE POSTS (409(2)) RAIL POSTS INDICATING BEGINNING AND END OF TRANSITION, PAINTED RED AND WHITE SHALL BE PROVIDED

IRPWM PROVISIONS SUPERELEVATION MARKING ON RAILS (409(3)) VALUE OF SUPERELEVATION SHALL BE MARKED ON THE WEB OF INSIDE RAIL AT VERSINE STATIONS IN TRANSITION PORTION VALUE AS ABOVE SHALL BE MARKED AT THE BEGINNING AND END OF THE CIRCULAR CURVE FOR LONGER CURVES, THE VALUE OF SUPERELEVATION SHALL BE REPEATED AS ABOVE AT INTERVALS NOT EXCEEDING 250 M

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