1

CURVES

2

CURVES• WHY??

•WHAT??

•PARAMETERS??

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•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

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ANGULARITY OF AXLE WHILE NEGOTIATING A CURVE

•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

Angle of attackα

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CURVE DESIGNATION

• CURVES ARE DESIGNATED BY THEIR RADII, EXCEPT ON IR & US RAIL ROADS

• ON IR DEGREE OF CURVE FOR DESIGNATION– RADII FOR CALCULATION

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DEGREE OF CURVE IS THE ANGLE SUBTENDED BY 30.5m

CHORD AT THE CENTRE

D = 1750/R

100 feet=30.5m

D RR

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2R

ON IR, THE CURVES ARE MEASURED BY VERSINE- WHICH IS MID CHORD OFFSET ON

20m CHORD

V*(2R-V) = C/2*C/22RV=C2/4V = C2/8R

C

V

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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

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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

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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

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COMPENSATION FOR CURVATURE ON GRADIENT

• COMPENSATION ALLOWED ON GRADIENTS DUE TO CURVATURE–70/R–0.04% PER DEGREE

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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

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VEHICLE ON A CANTED TRACK

W

GSE

Centrifugal Force

Centripetal Force Wsinθ

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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

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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

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Equilibrium Speed

• Russian Formula:

– ni: no of trains, wi: weight of such train, Vi: speed of such train, m: types of trains

ii

m

i

iii

m

ieq

Wn

VWnV

2

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ACTUAL CANT Ca

• MAINTENANCE CRITERIA– 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

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MAXIMUM VALUE OF CANT

COUNTRY GAUGE (mm)CANT Cmax (mm) Cmax/G

INDIA -B.G. -M.G.

1676 1000

165 90

0.098 0.09

DB 1435 150 0.104SNCF(TGV) 1435 180 0.125BR 1435 150 0.104JNR 1435 180 0.125JNR(MG) 1067 105 0.098

THE MAXM VALUE OF CANT PROVIDED ON WORLD RAILWAYS

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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

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Cant Deficiency

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CANT DEFICIENCY Cd

• SAFETY – 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

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MAXIMUM VALUE OF CANT DEFICIENCY

CANT DEFICIENCY '(mm)

THEORETICAL VALUE OF UNBALANCED LATERAL

ACCELERATION m/sec2

WDM4 ICF ALL COILED COACH

75 0.42 0.51 0.49100 0.56 0.68 0.67125 0.70 0.80 0.81150 0.84 0.98 0.91

ACTUAL VALUE OF UNBALANCED LATERAL

ACCELERATION IN m/sec2

UIC RECOMMENDATION 0.4 TO 0.7 m/sec2

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Cant Excess

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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.

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SNCFIMPORTANCE 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 IMPORTANCE TONNAGE PER DAY CANT EXCESS

VERY IMP. >60,000 50

IMPORTANT 30,000 TO 60,000 70

MEDIUM 10,000 TO 30,000 90

UNIMPORTANT <10,000 110

Cant Excess values

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IRPWM PROVISIONS

• CANT DEFICIENCY(406(2))– 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

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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.

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TRANSITION CURVES• CURVATURE VARIES UNIFORMLY

WITH DISTANCE

• VERSINES VARY UNIFORMLY

• CANT VARIES UNIFORMLY

VERSINE AND CANT DIAGRAM OF A CURVE

CaV

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SHIFT ON TRANSITION CURVE

CIRCULARCURVE WITHOUT

TRANSITION

TRANSITION CURVE

C D B

E

F GA L/2 L/2

S/2

S

EXTENDED CIRCULAR

CURVE

CIRCULARCURVE WITH TRANSITION

H TANGENT

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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

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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

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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

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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

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V

VIRTUAL TRANSITION

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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

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COMPOUND AND REVERSE CURVES

For Reverse Curve: Length of transition will be MAX. of

L =0.008*(Ca1+Ca2)*VmL =0.008 (Cd1+Cd2)*VmL =0.72 (Ca1+Ca2)

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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

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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

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Vertical Curves

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VERTICAL CURVES• 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

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Extra Clearances on curves

45

Ca

H

L

CLEARANCE ON CURVES- LEAN

L = H*Ca/G

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ADDITIONAL ALLOWANCE DUE TO SWAY

• INSIDE–¼ OF LEAN DUE TO

SUPERELEVATION

• OUTSIDE–NIL

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Extra Clearances on Curves

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ADDITIONAL CLEARANCES ON CURVES

OVER THROW Vo

END THROW VE

CENTRE LINE

OF TRACK

BOGIE CENTRE

BOGIE CENTRE

14.785

21.340

CENTRE LINE

OF COACH

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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.

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TURNOUTS TAKING OFF FROM CURVES

• SIMILAR FLEXURE– EQUIVALENT RADIUS Re=Rm*Rs/(Rm+Rs)Rm: MAIN LINE RADIUSRs: SWITCH RADIUSRe: EQUIVALENT

RADIUS

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• CONTRARY FLEXURE

– EQUIVALENT RADIUS

Re=Rm*Rs/(Rm-Rs)

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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

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CLEAR-ANCE

CHECK RAIL

CHECK RAILS ON CURVES

• PARA 426 IRPWM• CL. 4, CHAPTER 1, BG SOD

• ON CURVES OF 8 DEGREES & ABOVE

• MIN. CLEARANCE = 44 mm

• CLEARANCE = 44 + (G-1676)/2

TOO MUCH WEAR ON OUTER RAIL IN SHARP CURVES

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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)

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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

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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

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IRPWM PROVISIONSAFTER 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

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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

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IRPWM PROVISIONSIN 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

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IRPWM PROVISIONSIN 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

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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)

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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 )

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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

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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))

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LIMITING RADII

• BG : 175 M

• MG: 109 M

• NG: 44 M– Item 2, Chapter I, Schedule I of SOD

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IRPWM PROVISIONS• CURVE BOARDS(409(1))

–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

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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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