RAIL STRESSES

- Nilmani, Prof Track

Stresses in Rail

• Residual Stress

• Thermal stress

• Flexural stress

• Contact stress

Assessment Methodology Adopted

• Determination of Ultimate Tensile Strength of rail steel

• Determination of Yield Strength of rail steel• Calculation of induced stresses due to rolling

stock, thermal variation, manufacturing of rail, and other unforeseen factors.

• Sum of induced stresses should not exceed yield strength

RDSO’s Rational Approach based on Yield criteria

• All unfavorable factors do not occur together• Identify critical locations based on studies,

trials and experiments.• Use the criteria to broadly ascertain stress

magnitude.• Analyze failures.• Observe effects and apply corrections, if

required.

Typical Values Adopted on IR

Ultimate Tensile Strength (UTS)

• Regularly measured at Steel Plants – Mean UTS of rail steel for 880 grade rails

is 931Mpa with standard deviation of 15.5MPa.

• Average UTS obtained during investigations by M&C Directorate of RDSO for used rails – 90 UTS (880 grade) - 893MPa – 72UTS - 812Mpa

• Values considered – 770MPa for 72UTS – 900 MPa for 90UTS rails

Yield Strength

• By convention, yield strength of 72 UTS rails has been taken as 40kg/mm2 , taking 52% of the average value of UTS obtained for rail procured i.e. 72 to 82 kg/mm2

• As per provisions of relevant IS Code for steel having chemical composition similar to rail, yield strength should be taken as 55% of the minimum UTS which works out to 0.55X72= 39.6kg/mm2

Yield Strength

• By a similar logic, yield strength of 90UTS rails works out to 0.55x90 = 49.5kg/mm2.

Residual Stresses

• Result of rail manufacturing process- due to differences in time of phase transformations in railhead, web and foot during rail cooling

• Tensile in railhead and base and compressive in web

• Value of maximum residual stress in IRS T 12 Specification - 14% of rail UTS ( 12.6 kg/mm2 for 90 UTS)

• Assumed value at the bottom flange are 6 kg/mm2 - this is an ad-hoc value

Unforeseen Reasons

• A factor of safety of 10% of yield strength ( 4 kg/mm2 in 72 UTS rails and 4.8 kg/mm2 in 90 UTS rails) has been included – This is to cater for unforeseen factors, namely

flexed laying of rails in curve, one sided sun radiation etc

• The stresses on account of impact of wheel irregularities, wheel flats, rail/weld irregularities are not taken into account, considering them to be of instantaneous nature

Thermal Stresses

Earlier calculations– Maximum annual range of variation in rail

temperature in India is about 76 degree C– Distressing temperature range - tm to

tm+5 – Maximum temperature difference

between the distressing temperature and minimum rail temperature works out to 43 degree C

– Thermal stress of 10.75 kg/mm2

Flexural Stresses

These stresses are introduced in the rail on account of action of rolling stock– Pure bending due to vertical load– Bending action due to eccentricity of

vertical load– Twisting due to flange force– Lateral bending due to flange force

Flexural Stresses• The Eisenmann method used on Indian Railway for

calculation of rail stress due to rolling stock is based on theory of beam on elastic foundation by Timoshenko. The method uses double track modulus.

• The solution of the differential equation was first proposed by Zimmermann

d4w dx4EI = -uw

u

Qx

w(x)

EI

Flexural Stresses

• Maximum Bending Moment

= 0.318*P*x1

• X1 = (¶/4)(4*E*I/U)**0.25

Track Modulus

• Track modulus is defined as load per unit

length of rail to cause unit depression of track

• Load deflection graph consists of 2 straight

lines – one for initial loading and other for

loading in elastic range

• RDSO undertook trials on PSC sleeper track,

density 1540/1660 per km, in June-Aug 2005

y = 0.1719x + 0.4129

y = 0.2421x - 0.00360

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10 12 14 16 18Load

De

fle

cti

on

Typical Observations on Load-Deflection

5.8 t

Pi = 5.8 t

Ui = 214.85 kg/cm/cm

Ue = 339.18 kg/cm/cm

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14 16 18Load

Stre

ss

Observed

Calculated

Pi = 5.8 t

Ui = 214.85 kg/cm/cm

Ue = 339.18 kg/cm/cm

Correlation between observed and calculated stresses in rail

Recommended Track Modulus Values

Sleeper Density

Track Modulus in kg/cm/cm (BG)

Initial Elastic

Old Values 75 300

New Values at 1540/km

125 425

New Values at 1660/km

135 540

Rail section properties

 • For working out rail stresses, the

properties like moment of inertia and section modulus of rails are assumed 10% lesser than the properties for new rails.

• The wear is assumed to be 5%.

Speed/impact factor

• For BG vehicles, the graphs plotted in RDSO’s report C-100 are being followed for different vehicles.

• In case of M.G. vehicles, similar graphs plotted in Report C-92 are being used.

• Where the measured values of dynamic augment are not available , the dynamic augment values for similar vehicles are to be adopted.

Management Strategies

• Rail Grinding• Rail Lubrication• High Yield/UTS rails • In Motion Weigh Bridges• Wheel Impact Load Detection System

(WILD)

AT 75KMPH ,

ON 52KG, M+7(1540nos/km),

250MM TRACK

(WITH DOUBLE TRACK MODULUS METHOD)

Calculation of 4 types of stresses

a) Rail stresses due to vertical bending

b) Stresses due to eccentricity of vertical load

c) Stress due to twisting by flange force

d) Stresses due to lateral deflection under flange force

• For the first 4 tonnes live wheel load,

Initial track modulus (i) = 125Kg/cm/cm

 • For rest of live wheel load

Elastic track modulus (e) = 425 Kg/cm/cm

• Assume

I & Z of 5 % worn rail are reduced by 10 %

Moment of inertia about xx-axis,

Ixx (worn) = 1942. 2cm4

Compression ,

Zc (worn) = 241.65 cm 3

Tension ,

Zt (worn) = 256.95cm 3

X1 = The distance from the load to the point of contra flexure of the rail in cm.

= 42.33 * ( Ixx(worn) / µ)1/4

For initial load of 4t,

X1i = 42.33 *(1942.2 / 125)1/4

= 84.04 cm

For rest of the load,

X1e = 42.33 * (1942.2 / 425)1/4

= 61.89 cm

Impact factor at 75Km/h = 53 %

(taken from graph plotted in RDSO report no. C-100.)

Axle Load = 21.82 tonnes

 

Wheel Load = 10.91 tonnes

 

Dynamic wheel load = 10.91 * 1.53

= 16.69t

Tlv considering initial load of 4 tonnes

1 2 3 4

200.0 452.4 200.0

Effect of Wheel no.1

4 -0.76 -- --

Effect of Wheel no.2

-0.76 4 -- --

Effect of Wheel no.3

-- -- 4 -0.76

Effect of Wheel no.4

-- -- -0.76 4

Virtual wheel load 3.24 3.24 3.24 3.24

B.M Co-efficient from master diagram

Effects of adjacent wheels

With Xli = 84.04cm

With X1e = 61.89cm

For 200 cm (-) 0.19 (-) 0.10

For 452.4 cm (-) 0 0

For 652.4 cm 0 0

For initial load of 4 t For elastic loading(balance of dynamic wheel load)

4 * (-).19 = - .76t 12.69 * (-) .10 = -1.269t

If the distance between adjacent axle is more than 6 *Xli , Effect of leading wheel is to be taken in consideration,

In the present case, 6 * Xli = 6 * 84.04 = 504.24 cm

No wheel except wheel nos. 1 and 4 is having effect of leading axle

Adding 10% for the effect of leading wheel 

Tl v = 3.24 + 0.324 = 3.564 t.

Tlv considering remaining load.

1 2 3 4

200.0 452.4 200.0

Effect of Wheel no.1

12.69 -1.269 -- --

Effect of Wheel no.1

-1.269 12.69 -- --

Effect of Wheel no.1

-- -- 12.69 -1.269

Effect of Wheel no.1

-- -- -1.269 12.69

Virtual wheel load

11.421 11.421 11.421 11.421

6 * X1e = 6 * 61.89 = 371.34 cm

Wheel 2 & 3 will also have leading wheel effect 

Tlv for wheel nos. 1 & 4

= 11.421+1.1421 = 12.5631 t

Tlv for wheel nos. 2 & 3

= 11.421+1.1421 = 12.5631 t

For max. Value of Tlv, the Tlv due to initial and elastic loading will be added.

Max. Tlv = 16.1271 t

1 2 3 4

Tlv due to initial loading

3.564 3.24 3.24 3.564

Tlv due to elastic loading

12.5631 12.5631 12.5631 12.5631

Total Tlv 16.1271 15.8031 15.8031 16.1271

B.M. in rail = 0.318*P*x1

=(0.318 * 84.04 * 3.564)+

(0.318 * 61.89 * 12.5631)

  =95.246 + 247.254 = 342.50 t.cm

Stress in head (comp)

= 342.5 / 241.65 = 1.417 t /cm2

= 14.17 Kg/mm2 --------- (a)

Stress in foot (tensile)

= 342.5 / 256.95 t /cm2 = 1.332 t /cm2

= 13.32 Kg/mm2 -------- (a)

Eccentricity (e) = 1.5cm (assumed)

Torque = 2Mt = TLv * e

Mt = 16.1271 * 1.5 / 2

= 12.095325 t.cm

Note : (E*I1*h1*r) / C*Z1 = 0.0351

(E*I *h1*r) / C*Z2 = 0.0247

(i) Stress in head

= (Mt*E*I1*h1*R) / C*Z1

= 12.095325 *0.0351

= 0.427 t/cm2

= 4.27 Kg/mm2 ------- (b)

ii) Stress in foot

= (Mt*E*I1*h1*R) / C*Z2

= 12.095325 * 0.2470

= 0.298 t/cm2

= 2.98 Kg/mm2 -------- (b) 

Prud Homme’s Formula

HY ≤ 0.85 (1+ P/3)

HY - Lateral force per axle

P – Axle load

Max. Flange force/axle

= 0.85(1 + 21.82 / 3)

= 7.032t

Flange force / wheel = 3.516 t

2 Mt = 3.516 * hf

Mt = 3.516 * 10.958 / 2

= 19.264164 t.cm

(i) Stress in head = (Mt*E*I1*h1*R) / C*Z1

=0.676 t/cm2 = 6.76kg/mm2 --------------- (c)

ii) Stress in foot = (Mt*E*I1*h1*R) / C*Z2

= 0.485 t/cm2

= 4.85 kg/mm2 --------------- (c) 

Flange force / wheel = 3.516 t

  Sleeper spacing = 68 cm

BM =Lateral force*spacing/4

= (3.516 * 68) / 4

= 59.772 t-cm

Lateral Modulus (head) Z3 - 108.253cm3

Stress (head) = 59.772 / 108.253

= 5.52 Kg/mm2  --- (d)

Lateral Modulus(foot) Z4 - 53.324 cm3

Stress (foot) = 59.772 / 53.324

= 11.20 Kg/mm2 ---(d)

For head

(a + c + d - b) = 14.17+ 6.76+ 5.52 - 4.27 = 22.18Kg/mm2

For foot

(a + b + d - c) =13.33+ 2.98+11.20 – 4.85 = 22.66 Kg/mm2

 

Contact Stresses

• The contact of rail-wheel results in small contact patch and high contact pressure.

• Typically, contact is made over a quasi-elliptical contact patch of the size of a coin of 13mm diameter.

• Immediately below the contact patch, the steel is under tremendous pressure from all directions as the contact pressure is supported by reaction pressure from surrounding material.

Contact Stress• The contact shear stress developed at a depth of

5-7mm below top table in the head of the rail •   Tmax = 4.13 *(Q / R)1/2

– Tmax = Contact shear stress in Kg/mm2

– Q = wheel pressure in kg

(static wheel load + 1 tonne to account for constant on-loading on the curves)

– R = wheel radius in mm ( Worn wheel radius)

• The permissible value for this stress is taken as 30% of UTS of rail ,i.e. 21.6 kg/mm2 for 72UTS rails

Formation PressurePmax = (2P*S / π*D*L)* (U / (64EI))1/4

where,

Pmax = Maximum formation pressure in kg/cm2

P = Dynamic wheel load in kgs.

S = Sleeper spacing in cm

D = Depth of ballast in cm.

L = Effective length of sleeper per rail seat in cm, 76/63 cm for BG/MG

contd.

Formation PressureU= Track Modulus kg/cm/cm(using elastic

track modulus value only)

E= Modulus of elasticity of rail steel in kg/cm2.

I = M.I. of rail in cm4.

Permissible limits of formation pressure

  Locomotives Other stock like,

Wagons, coaches only

3.5 kg/cm2 3.0 kg/cm2

Joint Loading

Due to the effect of unsuspended masses and speeds the joints are subjected to severe overloads given by the formula:

  F = Fo + 0.1188 V (W)1/2

  Where,

F = Dynamic overload at joint in tonnes

Fo = Static wheel load including sprung and unsprung masses in tones

V = Velocity in km/h

W = Unsprung mass per wheel in tonnes.

Joint Loading

Permissible values being adopted

Locomotive EMU stock Wagons & coaches

B.G. 27 tonnes 23 tonnes 19 tonnes

M.G 17 tonnes 14 tonnes 11 tonnes

WHEEL IMPACT LOAD DETECTOR

Defective Wheel

Instrumented Track Pictures

Signal from Instrumented track

Calculation

Calculation

L Rail Load T R Rail Load T

L1 10.3 R1 10.2

L2 11.1 R2 10.9

L3 10.5 R3 10.5

L4 10.1 R4 10.7

L5 22.3 R5 28.3

L6 11.5 R6 10.4

L7 10.4 R7 10.8

L8 10.7 R8 10.1

L9 10.8 R9 10.3

L10 10 R10 10.1

L11 42.3 R11 48.3

L12 10.6 R12 10.1

L Rail Load T R Rail Load T

L1 10.3 R1 10.2

L2 11.1 R2 10.9

L3 10.5 R3 10.5

L4 10.1 R4 10.7

L5 22.3 R5 28.3

L6 11.5 R6 10.4

L7 10.4 R7 10.8

L8 10.7 R8 10.1

L9 10.8 R9 10.3

L10 10 R10 10.1

L11 42.3 R11 48.3

L12 10.6 R12 10.1

Management Strategies –Proposed WILD Alarm Levels

• Maintenance Alarm: ILF 2.0-4.5 or Impact Load 20-35t – Attention in next Schedule maintenance

• Critical Alarm : ILF > 4.5 or Impact Load > 35t – Attention at next TXR point orr detachment within next 50 km

List of Currently installed WILD Systems on Indian Railways

(www.irwild.net)

• SER-Mahalimarup-18/07/2007• SWR-Hospet-22/07/2007• SECR-Bhilai-28/11/2007• SR-Arakkonam-21/12/2007• ECR-Mugalsarai-10/04/2008• ECR-Mugalsarai-10/04/2008• SCR-Guntakal-25/02/2008• ER-Asansol-24/04/2008• ECoR-Vishakapatnam-30/05/2008