Rail Stress PHII-1011
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Transcript of Rail Stress PHII-1011
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