Stress in Track

University of Pretoria etd - Naude, F P (2005)
DEVELOPMENT OF A METHODOLOGY FOR CALCULATING

STRESSES IN TRACK COMPONENTS
FRANCOIS PAULUS NAUDÉ
Submitted in fulfilment of part of the requirements, for the degree of
MASTER OF ENGINEERING (MECHANICAL)
in the
FACULTY OF ENGINEERING
UNIVERSITY OF PRETORIA
October 2004
Development of a methodology for calculating stresses in track components
DEVELOPMENT OF A METHODOLOGY FOR CALCULATING STRESSES IN TRACK

COMPONENTS
FRANCOIS PAULUS NAUDÉ
SUMMARY
Supervisor: Professor NJ Theron

Co-Supervisor: Doctor RD Fröhling
Department: Mechanical and Aeronautical Engineering
University: University of Pretoria
Degree: Master of Engineering (Mech)
An existing analytical model, in use by Spoornet for the past two decades for calculating rail stresses
on railway track, was revisited and improved. The model provided engineers with an easy-to-use
program for evaluating track capacity and authorizing heavier loads on track. The model was
modified to calculate rail and track component stresses more accurately. These modifications include
the incorporation of current best practices and presentation of guidelines for the engineer on how to
determine some input parameters which are normally difficult to obtain.
Firstly it was determined which input parameters the model was the most sensitive to. Thereafter it
was determined whether or not the correct information would generally be readily available for those
sensitive parameters. The most sensitive parameters were further investigated and test results, as well
as best practice analytical methods, were used to establish nominal input values and guidelines for
determining such values.
This research was necessary to establish whether or not the currently used analytical model still
provided railway engineers with a useful tool and whether or not more modern and popular tools
could validate or replace it.
After some modifications to the analytical model, it was proved that it provides engineers with a
suitably accurate tool for calculating rail and track component stresses, without the need to build
time-consuming models of the track under investigation. It showed that the model, after some
modifications, is current with calculational methods in recent publications and provides an immediate
answer to "what-if" questions without the need to run lengthy analyses.
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ACKNOWLEDGEMENTS
I would like to thank and acknowledge contributions by the following persons:

x All my former Spoornet colleagues for their invaluable assistance and support during this
research.
x Mr Hannes Maree for supporting this project and for providing valuable information and
guidelines.
x Mr Mike Tomas for sharing his expertise and unlimited supply of test data.
x Professor Nico Theron for his support, enthusiasm and guidance.
x Dr Robert Fröhling for his consistent vision, technical contributions and for his driving force
without which this project may not have been completed!
x Spoornet and Land Mobility Technologies (Pty) Ltd for making this research possible.
x I wish to thank my wife for all the hours I spent in seclusion and for her everlasting
understanding, support and love, and my daughter, Andrea, whose birthdate marked a D-date
for the delivery of this dissertation and forced many a late night's work!
Lastly, I want to echo my wife and thank my Lord and Saviour, in heaven, here and everywhere, for
the opportunity to have an occupation that keeps me busy (Ecclesiastes 3:10).
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TABLE OF CONTENTS
LIST OF FIGURES ........................................................................................................................ VI
LIST OF TABLES .......................................................................................................................... IX
NOMENCLATURE.......................................................................................................................... X
1 INTRODUCTION..................................................................................................................1
1.1. Background ..............................................................................................................................1
1.2. Problem statement ....................................................................................................................2
1.3. Objectives and scope of research .............................................................................................3
1.4. Conclusion................................................................................................................................4
2 LITERATURE STUDY.........................................................................................................5
2.1. History of the railways .............................................................................................................5
2.2. Track design philosophy ..........................................................................................................6
2.3. Engineering Modelling of track ...............................................................................................6
2.4. Available approaches relevant to this project ..........................................................................8
2.4.1. Finite element modelling.....................................................................................................9
2.4.2. Dynamic modelling ...........................................................................................................10
2.4.3. Analytical methods............................................................................................................11
2.4.4. Combination of methods ...................................................................................................11
2.5. Conclusion..............................................................................................................................12
3 THE ANALYTICAL METHOD ........................................................................................13

3.1. Input parameters.....................................................................................................................13
3.1.1. Dynamic factor ..................................................................................................................14
3.1.2. Support condition ..............................................................................................................16
3.1.3. Rail stresses .......................................................................................................................18
3.1.3.1. Comparison of rail stress calculations from the analytical method to those of
Finite Element Analysis results 24
3.1.3.2. Rail stress results from field measurements 32
3.1.4. Sleeper reaction force:.......................................................................................................34
3.2. Limitations of the analytical method...................................................................................... 35
3.3. Sensitivity analyses ................................................................................................................37
3.3.1. Quantitative sensitivity analysis ........................................................................................37
3.3.2. Qualitative sensitivity analysis ..........................................................................................39
3.4. Conclusion..............................................................................................................................43
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4 IMPROVEMENTS TO THE ANALYTICAL METHOD...............................................44

4.1. Investigations into the improvements of the analytical method.............................................44
4.1.1. Dynamic Factor .................................................................................................................44
4.1.1.1. Empirical Formulas 49
4.1.1.2. Empirical Tests 51
4.1.1.3. Calculating the dynamic factor 63
4.1.1.4. Worked out example on Sweden test data (dynamic factor) 65
4.1.1.5. Dynamic factor overview 68
4.1.2. Lateral Force......................................................................................................................70
4.1.2.1. Empirical test results 71
4.1.2.2. Lateral force analysis 85
4.1.3. Eccentricity of vertical wheel load ....................................................................................85
4.1.3.1. Eccentricity analysis 86
4.1.3.2. Summary of the methodology to determine eccentricity vs. lateral load values 90
4.1.4. Rail stresses calculation improvement ..............................................................................90
4.2. Sensitivity study on modified analytical method ...................................................................94
4.3. Conclusion..............................................................................................................................95
5 METHODOLOGY FOR CALCULATING TRACK COMPONENT STRESSES.......97

5.1. Methodology ..........................................................................................................................97
5.1.1. Dynamic factor ..................................................................................................................97
5.1.2. Support condition ............................................................................................................100
5.1.3. Material............................................................................................................................101
5.1.4. Rail section properties .....................................................................................................102
5.1.5. Vertical load eccentricity.................................................................................................102
5.1.6. Bogie setup ......................................................................................................................104
5.1.7. Rail seat stress .................................................................................................................105
5.1.8. Railhead and flange stresses............................................................................................106
5.2. Conclusion............................................................................................................................107
6 CONCLUSION...................................................................................................................108
6.1. Summary of contributions....................................................................................................108
6.2. Recommendations for future work.......................................................................................109
6.3. Conclusion............................................................................................................................110
REFERENCES...............................................................................................................................111
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APPENDIX A 113
Stochastic rail stress distributions (Different iterations with the same model)
APPENDIX B 115
Histograms of Leandra test data
APPENDIX C 117
Lateral force graphs of Iron Ore Export line data
APPENDIX D 123
Summary of dynamic factors
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LIST OF FIGURES
Figure 3-1: Typical train speed distribution 16
Figure 3-2: Load decomposition for calculation of rail stresses 20
Figure 3-3: Position of shear centre / rotational axis 22
Figure 3-4: Finite element model – combination of 3D 8-node elements and 1D beam elements 25
Figure 3-5: Deflection curve of FEA models and analytical methods 26
Figure 3-6: Centred and off-centred load cases 28
Figure 3-7: Axial stress results from analytical and FEM model with centred vertical load and
different support stiffnesses 28
Figure 3-8: Axial stress results from FEM model with 100kN off-centre load and 50kN lateral
load on different support stiffnesses of 20, 50 and 100MPa/m (right side of rail) 29
Figure 3-9: Axial stress results from FEM model with 100kN off-centre load and 50kN lateral
load on different support stiffnesses of 20, 50 and 100MPa/m (left side of rail) 29
Figure 3-10: Graphical representation of typical stress results from a FEM model with an
off-centre load with a simultaneous lateral load (left) and a centre load only (right) 30
Figure 3-11: Strain gauge positions as pasted onto rail for field tests (23) 33
Figure 3-12: Rail stresses under different vertical and lateral loads (eccentricities unknown) (23) 34
Figure 3-13: Sensitivity analysis of parameters in component stress calculation spreadsheet 39
Figure 3-14: Stochastic distribution of rail stresses from the analytical method 40
Figure 3-15: Stochastic distribution of rail stresses with a number of parameters held constant 41
Figure 3-16: Stochastic distribution of rail stresses with a number of parameters held constant 42
Figure 4-1: Skid mark on rail running surface 46
Figure 4-2: Skid mark on wheel tread (“flat wheel”) 46
Figure 4-3: The presence of water and excessive track deformation pump mud to the surface 47
Figure 4-4: Measuring the depth of a dip joint (soft flash butt weld) 47
Figure 4-5: Dip joint identified by two wider patches on the contact band 48
Figure 4-6: Track geometry vertical alignment vs. dynamic loads 48
Figure 4-7: Comparison of dynamic factors from different empirical formulas 50
Figure 4-8: A complete unmanned weigh-in-motion weighbridge. 52
Figure 4-9: Calibration curves for weighbridge on No1. line at Saaiwater 53
Figure 4-12: Wheel impact load presentation 54
Figure 4-13: Dynamic impact loading distribution (number of wheels) – Line 1 55
Figure 4-16: Spoornet mobile laboratory for measuring strains, deflections and accelerations. 57
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Figure 4-17: Instrumented wheelset installed in 30t per axle iron ore wagon 58
Figure 4-18: Leandra test site track layout 59
Figure 4-19: Leandra test site track geometry (curve radius) 60
Figure 4-20: Instrumented wheelset with “red eye” 60
Figure 4-21: Vertical forces measured at 20km/h with instrumented wheelset and 15t axle load 62
Figure 4-22: Vertical forces measured at 60km/h with instrumented wheelset and 15t axle load 62
Figure 4-23: Sweden test track geometry (curvature) 66
Figure 4-24: Sweden wheel-rail vertical forces @ 40km/h. 67
Figure 4-25: Dynamic factors from tests at Leandra and Sweden with different methods 67
Figure 4-26: Summary of all dynamic factors from this study 69
Figure 4-27: Track geometry lateral alignment vs. dynamic loads 71
Figure 4-28: Lateral left wheelset force at different speeds [km/h] 72
Figure 4-29: Lateral right wheelset force at different speeds [km/h] 73
st
Figure 4-30: Lateral left wheelset force of 1 100m at different speeds [km/h] 73
Figure 4-31: Lateral right wheelset force of 1st 100m at different speeds [km/h] 74
Figure 4-32: Statistical summary of lateral left forces at Leandra test site 75
Figure 4-33: Statistical summary of lateral right forces at Leandra test site 76
Figure 4-34: Leandra test data: Vertical vs. Lateral Forces 77
Figure 4-35: Test train setup with three instrumented wheelsets at Iron Ore Export line tests 79
Figure 4-36: Statistical summary of lateral left forces at different speeds and axle loads 80
Figure 4-37: Statistical summary of lateral right forces at different speeds and axle loads 80
Figure 4-38: 30km/h test results of vertical and lateral loads (Iron Ore Export line tests) 81
Figure 4-39: 60km/h test results of vertical and lateral loads (Iron Ore Export line tests) 81
Figure 4-40: Iron Ore Export line vertical rail-wheel forces vs. lateral rail-wheel forces for
15t/axle tests 82
Figure 4-41: Iron Ore Export line vertical rail-wheel forces vs. lateral rail-wheel forces for
30t/axle tests 83
Figure 4-42: Statistical summary of lateral left forces at different speeds 84
Figure 4-43: Track layout for Medyna analysis 87
Figure 4-44: Eccentricity and lateral force sign convention of Medyna results 88
Figure 4-45: Vertical force eccentricity vs. lateral load 88
Figure 4-46: Profile properties of a 57kg/m rail 91
Figure 4-47: Rail stress comparison with railhead depth chosen at 47.5mm (100kN off-centred
vertical load and 50kN lateral load) 92
Figure 4-48: Rail stress comparison with railhead depth chosen at 43.17mm vs. 47.5mm (100kN
off-centred vertical load and 50kN lateral load) 93
Figure 4-49: Rail stress comparison with railhead depth chosen at 43.17mm (full profile scale) 94
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Figure 4-50: Sensitivity analysis of parameters in component stress calculation spreadsheet 96

Figure 5-1: Dynamic factor parameters 98
Figure 5-2: Support condition parameters 101
Figure 5-3: Material parameters 101
Figure 5-4: Rail section properties and load position input 103
Figure 5-5: Rail section property list 104
Figure 5-6: Bogie setup 104
Figure 5-7: Rail deflection due to wheel spacing 105
Figure 5-8: Track setup 105
Figure 5-9: Rail axial stresses (without temperature stresses) 106
Figure 5-10: Rail axial stresses (with temperature stresses) 107
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LIST OF TABLES
Table 3-1: Input parameters of the analytical method 14
Table 3-2: Conversion of foundation modulus values to support stiffness 27
Table 3-3: Stress comparison between FEM and analytical model 31
Table 3-4: Input parameter sensitivity analysis 38
Table 3-5: Order of sensitive parameters 39
Table 3-6: List of parameters modified in stochastic simulation results of Figure 3-15 and
Figure 3-16 42
Table 3-7: Most sensitive input parameters of the analytical method 43
Table 4-1: List of empirical formulas for the calculation of a dynamic factor 49
Table 4-2: Comparison of dynamic factors from different empirical formulas 50
Table 4-3: Track setup at Saaiwater weighbridges 53
Table 4-4: Test section lengths for statistical analysis 65
Table 4-5: Dynamic factors as calculated with different methods 68
Table 4-6: Statistical results from lateral left forces [kN] of Leandra test data 74
Table 4-7: Statistical results from lateral right forces [kN] of Leandra test data 75
Table 4-8: Correlation between vertical and lateral force data 77
Table 4-9: Statistical results from lateral left forces [kN] of Iron Ore Export line test data 79
Table 4-10: Statistical results from lateral right forces [kN] of Iron Ore Export line test data 79
Table 4-11: Correlation between vertical and lateral force data 83
Table 4-12: Statistical results from lateral left forces of Sweden test data 84
Table 4-13: Possible worst-case scenarios for eccentricity and lateral load pairs (rail stress model
coordinate system: positive eccentricity opposing positive lateral force) 89
Table 4-14: New proposed railhead depths for different rail sections 94
Table 4-15: Input parameter sensitivity analysis 95
Table 4-16: Order of sensitive parameters 96
Table 5-1: Test section lengths for statistical analysis 99
Table 5-2: Foundation modulus values 100
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NOMENCLATURE
A Section area
Ae Effective area
Ars Baseplate area
a Sleeper spacing
a0 Coefficient of locomotive maintenance
a1 Railhead depth
a2 Top of rail to top of flange
D Coefficient of thermal expansion
b1 Width of railhead
b2 Width of rail flange
E Inverse of the characteristic length
b0 Coefficient of track maintenance
C Foundation modulus
C1 Torsion stiffness
c Actual super elevation
D Wheel diameter [m]
d Section height
G Running top condition
E Elasticity modulus
e Eccentricity of vertical forces
F0 Pretension force of fastening system
f Flange thickness
fL Factor of static load
g Gravitational acceleration
h Head NA to flange NA
h Track cant
h1 Distance from head NA to section rotation point
h2 Distance from flange NA to section rotation point
K Influence factor of adjoining wheels
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NOMENCLATURE
I Second moment of area

Iyy Second moment of area yy
Izz Second moment of area zz
Iyy Head Second moment of area yy head
I1 or Izh Second moment of area zz head
I2 or Izf Second moment of area zz flange
I Dynamic factor
k Equivalent spring stiffness (MN/m)
N Characteristic value
O Characteristic length
L Lateral load
NA Neutral Axis
n Position of lateral force below rail top
P Static wheel load
pc Vehicle centre of gravity height
Q Dynamic wheel load
q’ Reaction force per unit area
R Curve radius
R’ Sleeper reaction force
s Track width (gauge)
V Stress
Vrs Mean contact pressure between rail and sleeper
T Torsion
¨T Temperature change
t Web thickness
t’ Statistical confidence level
'T Max temp change
UTS Ultimate Tensile Strength
U Track modulus
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NOMENCLATURE
X Poisson
V Speed (km/h)
v Type of sleeper
x Distance from wheel
x1 Distance from wheel in front
x2 Distance from wheel behind
x3 Distance from third wheel
Y Lateral load
yk Top to section NA
z Deflection of beam
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1 INTRODUCTION
Railway track deteriorates over time due to the dynamic loading of passing rolling stock. To ensure
the safe passing of trains, the track has to be properly designed and maintained. This study will focus
on the development of a methodology on how to calculate stresses in track components. Emphasis
will be placed on how to determine representative input parameter values, with the focus on
identifying the parameters of highest importance. An existing analytical method will be utilised and
improved with which track component stresses may be calculated in a fast and reliable manner.
Before continuing with the study on the development of a methodology on how to calculate track
component stresses, it is important to understand the railway track structure and how it behaves as a
system. Some background will also be given on the methods used in the past to calculate track
component stresses.
1.1. Background
As railway track deteriorates over time under dynamic loading (1, p. 399), it is important to know
which components are critical to maintain. Sometimes, however, it is not the components that
deteriorate but the operating conditions could change. The need could arise to increase the traffic
volumes, loads or speed on existing lines. It would then be necessary to determine and evaluate if any
components should be replaced.
Since the rail is the single most expensive item on track, it is necessary to be able to calculate the
stresses in the rail under operating conditions to evaluate its capability to fulfil its function.
Maintenance personnel should thus, throughout the life of the track, evaluate its condition and
determine whether it is still capable of performing to its design standards. In order to do this, a
practical tool is needed to calculate stresses in all the track components, which could also help to
identify limits for a number of track parameters.
Under the influence of dynamic track loads, the response and deterioration of the track and the
mechanism governing this phenomenon are rather complex. Lombard (2) compiled a method on how
to calculate the stresses of a rail under loading conditions using analytical calculations. This method
used a number of parameter values describing the track and its condition to calculate the rail and rail
seat stresses under a quasi-static load. A computer program was written in GW BASIC, which
prompted the user for input and calculated the relevant stresses. The program was named
“MOMENT” and is commonly known as “Lombard’s model”.
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The Lombard model will be referred to as the ‘analytical method’ throughout this study since it is a
compilation of generic analytical methods to calculate track component stresses and deflections. It is
also called the ‘analytical model’, since it represents the track structure and loads as a system of
different components.
The model originally only made provision for standard rail types and was used by SPOORNET
Engineering for many years to evaluate rail and rail seat stresses under dynamic conditions. The
dynamic loads are taken into account by factoring the static load with a dynamic factor; therefore it
can be regarded as a quasi-static analysis.
Recent requests, however, were made to be able to evaluate worn rails under various support and
track conditions and it became necessary to revise the program. In order to revise the program to have
the options of defining worn rails, it was necessary to re-investigate the theory and assumptions made
in the model. In this process, it was found that a MS EXCEL spreadsheet could be written to replace
the BASIC program, which made the model fast and easy to use, especially when investigating the
effect of changing input values.
The derivation of the analytical formulas with which the rail and component stresses (and deflections)
are calculated, are presented in detail in report BBB2540ver1 (3). When all the original sources of
information on the theory as presented by Lombard (2) were consolidated into a useable format to
create the spreadsheet, it became apparent that the model had some shortcomings. The model did not
include some important parameters and was sensitive to some parameters for which many users
would have difficulty in obtaining representative values. Assumptions in the theory regarding the
calculation of rail stresses were also scrutinised.
This dissertation will investigate shortcomings of the analytical model, identify sensitive parameters
and propose a methodology on choosing the best values for these critical parameters. The analytical
model is not the only or most accurate way to determine rail stresses under dynamic conditions but it
will become apparent through this dissertation why this method is favoured and focused on in this
study.
1.2. Problem statement
To calculate track component stresses under dynamic loading a practical tool is needed to determine
the actual load-carrying capacity of the track and its components. The model currently in use by
Spoornet excludes important parameters, such as curve radius, and is sensitive towards certain
parameters, such as the dynamic factor. The user may be unaware of these sensitive parameters and
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neglect to obtain proper values for them. This could lead to incorrect results regarding the capacity of
the track to fulfil its function. Since it is important to use appropriate values for the most sensitive
parameters of the model, a methodology on how to calculate stresses in track components is required
with specific reference to the method of determining reasonable values for the most sensitive input
parameters.
1.3. Objectives and scope of research
The primary objective of the study is:

x to present a methodology to calculate track component stresses accurately and reliably under
different operating conditions.
The secondary objectives are:

x to identify parameters not included in the current analytical method that may have a significant
effect on track component stresses
x to identify the parameters to which the analytical method is the most sensitive
x to provide nominal values to be used for the sensitive parameters where uncertainty exists
regarding appropriate values to use when calculating track component stresses
x to validate the analytical method with other proven methods and empirical results.
In order to achieve the objectives, a brief reference will be made to different methods that can be
utilised when determining the load-carrying capacity of the track structure. The focus will, however,
remain on the analytical method currently used by Spoornet. Different methods will be used to test
the analytical method's sensitivity towards each of the input parameters and, furthermore, the results
from empirically developed formulas will be compared with field measurements to determine
nominal values for each of the most sensitive parameters. As part of the scope of this project, results
from the instrumented wheelset tests and wayside measurement tests, executed by Spoornet
Engineering, will be analysed to determine actual input loads to the track structure. The results of
these tests will primarily be used to validate or enhance certain theoretical and popular empirical
formulas used in the analytical method and to determine values for the most sensitive model
parameters.
It should be emphasised that in this study structural dynamics will not be considered since the
operating conditions on Spoornet track only allow for low speed and high axle loads in which case the
track dynamics do not play a significant part. The "dynamic effect" of moving loads will, however, be
taken into account by means of a factored value (dynamic factor) in a quasi-static analysis.
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1.4. Conclusion
This study will focus on the development of a methodology on how to calculate stresses in track
components. Emphasis will be placed on how to obtain representative input parameter values with the
focus on identifying the parameters of highest importance. An existing analytical method will be
utilised and improved with which track component stresses may be calculated in a fast and reliable
manner.
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2 LITERATURE STUDY
To understand modern railways and current engineering practices, it is important to look into the
history of the railways and how they were developed from earliest concepts to modern-day track.
2.1. History of the railways
Like so much in present day society, the railroad, as we know it, was a product of the industrial
revolution although the idea of a special 'track' for hauling goods dates back about 2,000 years (4).
The ancient Greeks built roads paved with stone blocks that had grooves cut in them. Wagons that had
wheels the widths of the grooves were pulled over the roads by horses. These grooves kept the
wagons on the road, and the stone paving was much smoother than dirt roads, so heavier loads could
be transported much faster.
Roads of rails, called ‘wagon ways’, were being used as early as 1550 in Germany. These primitive
railed roads consisted of wooden rails over which horse-drawn wagons or carts moved with greater
ease than over dirt roads (5). By 1776, iron had replaced the wooden wheels and rails. Wagon ways
evolved into tramways and spread throughout Europe. Horses still provided all the pulling power and,
in 1789, William Jessup designed the first wagons with flanged wheels (5). The invention of the
steam engine was then critical to the invention of the modern railroad and trains. In 1803, a man
named Samuel Homfray decided to fund the development of a steam-powered vehicle to replace the
horse-drawn carts on the tramways (5) and a new era dawned.
Railways had a much greater influence on modern day life than we normally realise. Railways in a
sense have 'created' time as is evidenced from this extract from 'This day in American History' by E.
Gross (6): "On November 18, 1883, four standard time zones for the continental U.S.A. were
introduced at the instigation of the railroads. At noon on this day, the U.S. Naval Observatory
changed its telegraphic signals to correspond to the change. Until the invention of the railway, it took
such a long time to get from one place to another that local 'sun time' could be used. When travelling
to the east or to the west, a person would have to change his or her watch by one minute every twelve
miles. When people began travelling by train, sometimes hundreds of miles in a day, the calculation
of time became a serious problem. Operators of the new railroad lines realised that a new time plan
was needed in order to offer a uniform train schedule for departures and arrivals.".
In this way, the railway industry helped to revolutionise our world by presenting us with a fast and
efficient transport method to haul heavy bulk goods in a most efficient way. To continue this
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life-enhancing curve, railway engineers should continue to develop ways to reduce maintenance costs
and to prolong the life of the track structure, thereby reducing the costs and increasing the benefits.
2.2. Track design philosophy
The railway track structure, or so-called "permanent way", is all but permanent and is a maintenance
intensive item, which adds to the overall transporting costs. The sole purpose of the track is to
transport rail vehicles safely from one point to another but it has a double function, namely, to support
and guide the vehicles simultaneously (4). The track is designed according to the expected traffic
loads and the load bearing approach (1, p. 14), (7) is utilised to ensure that the concentrated loads of
the wheels are transferred to the formation while ensuring that the strength of the components are not
exceeded (8), (9). The most important strength parameters are: rail stresses; sleeper stresses and
ballast subgrade pressures. The Guidelines to Best Practices for Heavy Haul Railway Operations (10,
p. 1-2) state that: "The materials must be strong enough to resist the vertical forces introduced by very
heavy loads and the dynamic response at the wheel-rail interface introduced by vertical accelerations
of the car induced by track and wheel irregularities”. Both the wheel and rail are responsible for
transmitting the static and dynamic loads from the car body through to the track structure. At the
contact patch between the wheel and rail, the vertical load of the wheel must be supported, and the
steering, braking, and traction forces transmitted (10, p. 1-2). The following conditions (1, p. 14)
should be met by the design objectives:
x Component dimensions should be sufficient for successful fulfilment of their function
under the traffic load.
x Correct geometry of the track must be maintained, whether the track is loaded or not.
x Cost over service life should be as low as possible.
It is important that railway engineers design and evaluate cost effective track components that will not
fail under prevailing traffic conditions (11). It is therefore necessary for them to utilise a method that
would calculate component stresses under prevailing operating conditions, to enable them to judge
whether or not the components will survive.
2.3. Engineering Modelling of track
As demands for improvement in technology have increased, research has developed to the point
where the traditional theoretical tools for the classroom have now moved into the construction and
maintenance departments of railway companies. These tools assist in the improvement of safety and
the reduction of construction and maintenance costs. Advances in computer technology have made it
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possible to create elaborate models of the railway track with which its behaviour could be predicted.
However, modelling railway track is not an easy task since different models serve different purposes.
Modelling railway track is necessary to optimise¥ the design of the components and to assist
engineers in understanding and maintaining the track. Due to fatigue, wear, settlement and
degradation of components of the structure, it is also necessary to quantify the condition of the track
and its components in order to determine if corrective action should be taken (1, p. 409). 200 years
after the steam engine revolutionised the railway industry, in an age where paying passengers can
enter earth's orbit, it is ironic that railway track is still a complex structure for engineers to describe
mathematically.
Traditionally, in the railway industry, mechanical engineers were more involved with vehicle
dynamics and civil engineers were more involved with the track structure. For a civil engineer, a
structure, such as a road, building or bridge, is stationary. Railway track consisting of the formation,
ballast, sleepers and rails is also deemed a stationary structure. Due to the impact of the dynamic
loads, however, it cannot be analysed or modelled in the same way. This causes technical problems
that should be solved in both static and dynamic terms (13, p. 1). When analysing the track structure,
its complexity is evident. A major problem the engineer is faced with is the elasticity of the track and
defining its boundary conditions. To mathematically describe fixed or freely supported beams is quite
straightforward, but defining the continuous beams (the rails) whose ends are neither fixed nor free
and which are supported by discrete elastic supports (pads-sleeper-ballast combination), is quite a
different matter. Although the track structure consists of easily identifiable components with clear
functions, the interaction between them under dynamic loading is not always fully understood. To
simplify complex structures, engineers often use symmetry in structures to represent
three-dimensional structures in two dimensions. The rails and supporting structure cannot be
adequately modelled by a two dimensional system, since the rails do not only bend in-plane but can
also bend or move out-of-plane, with the elastic fastening system as a restraint. This phenomenon has
a major influence on the rail stresses and thus cannot be ignored.
Modelling is called an engineering art and a general rule is: "Models should be as simple as possible
and as accurate as necessary regarding the task they serve for" (14). The demand for model
simplicity is especially important with respect to computational efficiency. Although computational
effort becomes less and less time-consuming as higher computational power becomes available, the
need for models that are more detailed will always catch up with the speed of computer processors
and more efficient models will always be in demand. The second-mentioned requirement for a
sufficient model is accuracy. Every type of model introduces its own limitations and subsequent
¥
Apologies to Marczyk (12) who strongly disagrees with the optimising philosophy, which is so deeply rooted in the field
of engineering. Marczyk promotes ‘robust design’ as superior to ‘optimisation’.
Page 7
errors. In general, there is not a single adequate model of a system but many different ones, depending
on the task they are created for (14). Modelling the track structure is a good example of the diversity
of models that can be used to represent a real structure. These models could range from the classical
continuous models, like the Bernouli-Euler beam, to disturbance and damage models where the
modelling involves more than the mere feeding of information to mathematical equations. Some
models are based on static principles and others on dynamic behaviour of structures. Continuous
models, like the ones that date back to Timeshenko, represent the track as an infinite continuous
model on an elastic Winkler foundation (1, p. 71).
Apart from the continuous models, there are the discrete models, where the sleepers, for instance, are
modelled as separate supports as in the real structure. These models, where elements are of finite
length, are mostly based on the finite element method. Finite element modelling is yet another art and
can be applied in an unlimited number of different ways. As the range of models and their
applications become endless, first and foremost, the question to answer is what we want out of a
model in order to determine which model would suit the task best. J. Box said, as quoted by Marczyk
(12, p. iii):"All models are wrong. Some are useful".
This study will focus on a model (or practical tool) that will be useful to the engineer:
x to ensure the safe functioning of the track structure,
x to assist in the design thereof by calculating stresses in the components,
x to help identify limits for a number of track parameters.
2.4. Available approaches relevant to this project
Several tools are currently available to describe railway track mathematically, each with its own
advantages and disadvantages. Since the rail is the single most expensive item on track, the fatigue
life of rails is an important issue to consider and the engineer should throughout the rail’s life evaluate
its capability to fulfil its function. It is therefore necessary to be able to calculate the stresses in the rail
and other track components under all significant loading conditions.
The tools currently available to describe the track mathematically are: finite element modelling,
dynamic modelling, quasi-static analytical methods and any combination of these.
Page 8
2.4.1. Finite element modelling
Finite element modelling (FEM) is a method that can be utilised to model discrete track components
and determine the interaction between them, as well as stresses, under both static and dynamic
conditions. FEM is not limited to static analyses. However, dynamic analyses are computationally
expensive. Another disadvantage of finite element analyses (FEA) is that one model can represent
only one chosen track layout and cannot easily be adapted to model a track with different geometry.
This means that a new model should be created for each new layout.
Finite element analyses can be useful to calculate stresses of components in complicated structures.
The challenge when building finite element models is to determine the amount of detail needed to
represent the reality accurately. In order to do this the engineer needs to know the behaviour of the
structure to some extent to ensure the correct elements, boundary conditions and loads in order to
represent the structure adequately.
There is currently no set of rules on how to model railway track structures with finite element
methods and different researchers have used different methods to obtain results. According to Popp,
Kruse and Kaiser (14), it is the frequency range of interest that dictates which track model is best
suited to the investigation of a certain task. For investigating vehicle behaviour in curves, their
stability or passenger comfort, only frequencies below 20Hz have to be considered and therefore
simple track models are sufficient. Ballast and subsoil properties have a strong effect on track
dynamics in the range up to 250Hz and, for higher frequencies up to 700Hz, the geometry of the rails
has to be taken into account. It is concluded that, for the mid frequency range of 50Hz to ±500Hz,
detailed models are needed for pads, ballast and the subgrade. According to the Guidelines to best
practices for heavy haul railway operations (10, p. 2-11), the typical exciting mechanisms are:
x Vehicle body dynamics in the frequency range between 1 and 30 Hz
x Out-of round wheels (10 to 20 Hz)
x Wheel flats (10 to 20 Hz)
x Rail irregularities, such as rail joints and skid marks*
The European Rail Research Institute (ERRI) (15) built a 3D model for wheel-rail contact studies
where the principle of the rail, being a simple beam on a continuously supported elastic foundation,
was examined. In their study of rail rolling contact fatigue cracks, they classified the stresses in the
rails as either “bulk” or “contact” stresses. In order to study the contact stresses, the bulk stresses as a
result of wheel loads, thermal forces and residual stresses had to be examined. Of particular interest
for their study were the rail and sleeper deflections as well as the sleeper-ballast pressures under static
*
No frequency range was published in the source document, supposedly because irregularities could be highly discreet
(which result in infinitely high frequencies) or continuous and small (low frequency vibration).
Page 9
loads. The ERRI used a 3D solid FEA model and included the rail, Polyethylene pad, base plate,
rubber pad, timber sleeper and ballast. The model consisted of 3309 solid elements representing a
track model symmetric about the longitudinal axis and 9.7m in length. The loads were spaced
according to an EU 07 Locomotive wheelbase and the stresses calculated for a static load case.
Finite element analyses are therefore useful when studying detailed phenomena and when
computational efficiency does not come at a high premium.
2.4.2. Dynamic modelling
Many different dynamic models have been developed to simulate the actual field conditions and to
determine the dynamic loads on track (16, p. 2.1). Some models are more complex than others;
depending on the specific purpose they fulfil (14). When a vehicle is moving over the track, it
experiences vibrations of varying frequencies, which excite the various modes of the vehicle
structure, body and the payload (10, p. 2-11). The dynamic modes are generally in bounce, roll, pitch,
nosing, and sway. Some of the exciting mechanisms are long wavelength track alignment
irregularities in the vertical profile, track twist and rail discontinuities. These irregularities typically
result in vehicle input frequencies between 0.5 and 30 Hz. High frequency impacts at rail
discontinuities often excite the vehicle body vertical modes to induce the so-called lower frequency
reaction forces. A multi-body model representing a typical vehicle and track by means of springs and
dampers could be used to simulate the movement of a vehicle over an irregularity (17).
Another method that can be utilised is kinematic modelling such as used by software packages like
Medyna, DADS or MSC.ADAMS. The advantage of these packages is in the possibility to accurately
model a specific railway vehicle and to have it run on a specific track with pre-defined geometry,
even including irregularities. From these analyses the forces acting on the track while the vehicle is
travelling over it can be determined. These packages do not, however, calculate stresses in the rail or
track components as yet, but the acting forces can be exported into finite element packages with
which the stresses can be determined (see paragraph 2.4.4).
As with finite element modelling, these dynamic models cannot be changed quickly to represent
different track layouts or different loading conditions and can only present results for the specific
vehicle and track under consideration.
Page 10
2.4.3. Analytical methods
Analytical methods normally assume universal boundary conditions but use all geometrical input in
the form of input parameters. Analytical models do not solve derivative equations (such as the
equations of motion), which involve iterative calculations, and modelling is done statically. This
makes analytical methods more economical to use than finite element methods when the same results
are required for different track layouts (provided that the applicable parameters are supported in the
model).
Since these models represent static loading, some models incorporate a dynamic factor, which
artificially represents the magnified effect of dynamic forces and deflections due to moving loads.
These dynamic factors are always a point of discussion since the amplified effect of forces and
deflections due to dynamic forces is neither constant nor linear and depend on the characteristics of
the whole system.
When designing track in general, or evaluating a railway line under changing operating conditions,
analytical model are the most economical to use. Analytical models make it possible to quickly
evaluate different layouts and loading conditions without the necessity to build detailed models for
each scenario.
2.4.4. Combination of methods
Bezin, Iwnicki and Stow (18) approached the problem of defining input loads for a detailed track
model using the following methodology: The wheel-rail forces of a particular vehicle were predicted
on a range of curves with MSC.ADAMS/Rail. The results were then validated with test
measurements and the predicted forces were then transferred to two different finite element models.
The first model was called the “Global model” and represented the track in a simplified manner to
predict bending stresses in the rails. The second model was called the “Contact stress model” and
represented a short piece of railhead and a portion of a wheel rim to simulate rail/wheel contact. With
the MSC.ADAMS/Rail model, the vehicle dynamics and resulting track loads were determined for
specific operating conditions and track geometry. The supporting track conditions were then
accounted for in the ‘Global model’ and the rail bending stresses were determined.
By using these methods in combination, each method complemented the other by being used for the
purpose it was developed for. Through this interaction, global dynamic forces were translated into
detailed wheel-rail contact forces and subsequently detailed wheel and rail stresses.
Page 11
2.5. Conclusion
Following the history of the railways and how they helped to shape the world as we know it today
makes us realise that there is an ever-increasing drive to improve on existing technologies. In an era
where detailed designs have to precede construction and maintaining infrastructure becomes a
financial challenge, modelling railway track - in the various phases of its life - has become a
necessity. It is thus imperative to improve on existing methods of modelling railway track to find cost
effective ways to improve the life of track components and the track as a whole.
Page 12
3 THE ANALYTICAL METHOD
Lombard (2) compiled an analytical method with which to calculate the stresses of a rail with a
number of input parameters that described the loading conditions. This included the static load,
factored by a dynamic factor, the track geometry condition and support conditions. A computer
program was written in GWBASIC, which prompted the user for input and subsequently calculated
the rail and rail seat stresses. The model of Lombard made provision only for standard rail types and
recent requests were made to do the evaluations on worn rails to determine at what stage these rails
had to be replaced. In order to revise the software program to have the option of defining worn rails,
the theory was re-visited to understand and modify the program. In this process, it was found that a
spreadsheet could be written to replace the GWBASIC program, which made the process more
understandable and the model easy to use. When all the different sources of information on the
theory, as presented by Lombard, were consolidated into a useable format for creating the
spreadsheet, it became apparent that the model was sensitive to some parameters. For some of these
parameters many users would not have direct access to correct or realistic values and may mistakenly
assume that a thumb-suck value would suffice. When using any method to model railway track, it is
not only important to know the model’s limitations, but also to know which parameters the results are
the most sensitive to, to ensure that proper values for those parameters are sought.
3.1. Input parameters
Table 3-1 lists all the input parameters for the analytical method with an indication of whether or not
the input value is currently uniquely identifiable. It is important to know which parameter values are
not currently uniquely identifiable in order to later identify for which of these parameters research
will be needed to determine nominal values or to put proper guidelines in place for obtaining such
values. It is most often encountered that engineers tend to use "default" values for those
difficult-to-obtain values, unaware of whether or not the results from the method are sensitive to these
values.
In order to understand the analytical method under investigation and identify shortcomings and
possible areas for improvement, the underlying theory will be described in sufficient detail. For
convenience it will be divided into the following sections: “Dynamic factor”, “Support condition”,
“Rail stresses” and “Sleeper reaction force”.
Page 13
Table 3-1: Input parameters of the analytical method

Uniquely
No. Symbol Parameter
Identifiable?
1 G Running top condition 8
2 t' Confidence level parameter 8
3 V Speed ( V ) 9
4 P Static wheel load 9
5 C Foundation modulus 8
6 a Sleeper spacing 9
7 Q Type of sleeper 9
8 Ae Effective area of sleeper 9
9 E E modulus rail 9
10 X Poisson’s ratio 9
11 UTS Ultimate tensile strength 9
12 e Eccentricity of vertical force Q 8
13 n Position of lateral force below top 8
14 A Section area 9
15 b1 Width of head 9
16 b2 Width of flange 9
17 t Web thickness 9
18 d Section height 9
19 a1 Head depth 9
20 f Flange thickness 9
21 yk Top to section NA 9
22 h Head NA to flange NA 9
23 a2 Top to top of flange depth 9
24 Ixx Second moment of area xx 9
25 Izz Second moment of area zz 9
26 Ixx Head Second moment of area xx head 9
27 I1 or Izh Second moment of area zz head 9
28 I2 or Izf Second moment of area zz flange 9
29 x1 Distance from wheel in front 9
30 x2 Distance from wheel behind 9
31 fL Factor of static load 8
32 'T Max temperature change 9
3.1.1. Dynamic factor
The first calculation of the analytical method is that of the dynamic (amplification) factor (1, p. 83).
The dynamic factor converts a static wheel load to that of an equivalent dynamic wheel load. The
empirical formula of Eisenmann is used in the analytical method and is a function of not only the
static wheel load but of the speed of the vehicle, the running top (geometric condition) of the track
Page 14
and the statistical confidence. This empirical calculation has been accepted by European railway
companies (1, p. 83) and is based on the following observations and assumptions:
x Measurements have shown that the stresses in the rail foot, from a statistical point of view,
have a normal distribution about the mean value.
x The mean value is independent of running speed and can be determined with sufficient
accuracy using Zimmermann's (19) longitudinal beam theory.
x The standard deviation is dependent on the running speed and the state of the track.
The dynamic load (Q) is calculated by (2, p. 65):

Q P 1 G t ' if V d 60 km/h (3.1)
§ V 60 ·
Q P ¨1 G (1 ) t '¸ if 60 d V d 200 km/h (3.2)
© 140 ¹
with P the static load§, G the running top condition (0.1 = Excellent track, 0.3 = Bad track), V the
vehicle speed in km/h and t’ being a multiplication factor of standard deviation which depends on the
confidence interval. In equations 3.1 and 3.2 the bracketed expressions constitute the dynamic factor
M. For a statistical confidence level of 68.3%, t’=1, for 95.4%, t’=2 and for 99.7% t’ = 3. It is
recommended by Esveld (1, p. 84) to use t'=3 when investigating rail, pad and sleeper stresses, t'=2
when only looking at ballast bed stresses and t'=1 for the subgrade stresses. With t'=3 there is only a
0.15% probability that the maximum calculated rail stresses may be exceeded.
This relationship may prove to be irrelevant or inaccurate for use with Spoornet operating conditions
since it was developed in a European high-speed environment while local conditions are more in the
low speed and high axle load environment. Figure 3-1 indicates a typical train speed distribution from
which it can be derived that approximately 70% of the trains travel at 60km/h or slower on the
particular section of track. According to the Eisenmann formula (3.2), the speed of the vehicle affects
the dynamic factor only at speeds above 60km/h.
It will thus be investigated in this study whether or not a new relationship between static and dynamic
forces should be developed and if other parameters should be used, like, for instance, track design life
or rolling stock type. Furthermore, the formula used here to determine the dynamic factor includes
two parameters, namely, the running top condition and the statistical confidence level for which
values are not uniquely identifiable. Firstly, the value for running top condition is subject to the
interpretation and experience of the engineer doing the analysis. It is therefore advisable that a
scientific method is developed to quantify the condition of the running top by incorporating one of the
§
Esveld (1, p. 58) proposes the use of the static wheel load multiplied by an additional factor (paragraph 4.1.1.2, page 63)
to account for the increase in wheel loads in curves due to cant deficiency or excess.
Page 15
existing quality measurement parameters into the dynamic factor's formula. It is intended in this study
to identify typical dynamic factors measured in service and to compare these to the results from the
Eisenmann formula.
300 100%
90%
250
80%
70%
200
Number of trains
60%
# of trains
150 50%
Cumulative %
40%
100
30%
20%
50
10%
0 0%
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
Train Speed (km/h)
Figure 3-1: Typical train speed distribution
Based on this comparison it is foreseeable that a clearer guideline would be established, regarding
specific values that would be appropriate for various circumstances, or that a methodology would be
defined with which to determine dynamic factors.
The next section of the analytical method defines the support condition and how it affects the way the
input load is transferred to the substructure.
3.1.2. Support condition
The localised input load from the wheel is transferred directly to the rail and then via the pads,
sleepers and ballast to the substructure (8). In the analytical model the rail is defined as an infinite
beam on an elastic support (20, p. B3.4) without differentiating between the different rail support
components. The components which contribute largely to the elasticity of the rail support are the
ballast and substructure. Due to the moving loads, which cause ballast break down, and drainage
Page 16
conditions, which cause ballast fouling, the elasticity of the track (or support condition) changes with
time.
The support condition is quantified in the analytical method and described by two parameters: the
foundation modulus (C) and the track modulus (U) (2, p. 36).
N C (3.3)
U
a
Ae Q (3.4)
N
2
with Ae the effective area of the sleeper, Q the type of sleeper (Q=1: Concrete sleeper, Q=0.91: Wood
sleepers with C> 60MPa/m), a the sleeper spacing and N being the characteristic value.
According to Lombard (2, p 36-37), values for C range from 20 MPa/m, for a very bad † clay
formation and bad ballast condition, to 100+ MPa/m, for a very good formation and ballast condition.
According to Esveld (1, p. 79), the value for C is also 20MPa/m for a poor foundation but 200MPa/m
for a good foundation.
EI
According to Lombard (2, p. 10), Hetényi stated that if U % 1.522 then the effect of discrete
a4
supports is negligible and the theory of a continuous beam on a continuous elastic foundation is valid
(I is the second moment of area of the rail about its horizontal axis). Lombard states that if
EI
U % 2.23 the error of the stresses in the rail will be less than 10% under the assumptions that
a4
the sleeper spacing is in the order of 700mm and the track modulus (U) is in the order of between 6
and 30MPa.
The analytical methods assume that the discrete supports may be represented by continuous supports;
therefore results from finite element models with discrete support conditions will be compared to the
results from analytical calculations in order to test this assumption's validity.
In the above calculations the value for the foundation modulus is a subjective value, based on testing
experience, and may be interpreted differently by different analysts. The effect of using a high or low
value can incidentally have opposite effects in the results investigated, based on whether the
deflection or rail stresses are investigated. Thus, an operator of the model may think a conservative
answer is being obtained whereas the opposite may be true. Many tests have been executed on
†
Unfortunately, these terms: “very bad”, “very good”, etc., are not quantified by the authors and this is identified as one of
the shortcomings of this method.
Page 17
Spoornet track with the BSSM (Track Stiffness Measuring System) and MDD's (Multi-Depth
Deflection metres) but all these tests are expensive, time-consuming and only give results for the
immediate area under investigation. Other methods are being developed that include the usage of the
track geometry measuring car to relate the deflection of the track under the car to the track stiffness.
One such possibility is to vary the weight distribution of the vehicle while measuring the
displacement of the track. However, this option is undesirable since it would require the vehicle to run
multiple passes over a given piece of track while the first objective of the geometry car is to occupy
the track for as short a time as possible.
The rails are the most important component of the track structure and the theory regarding the
calculation of rail stresses will be presented in the following section.
3.1.3. Rail stresses
The rail stresses are calculated using the classic Zimmerman-theory, based on the principle of a beam
on an elastic Winkler foundation (20, p. B3.4).
q' C z (3.5)
with q’ = reaction force per unit area that the rail will experience from its support under a deflection of
z. C is thus the Foundation modulus and expressed in MPa/m, which is the amount of stress under the
rail that will result from a unit deflection of the rail. In order to calculate the deflection of a long
continuous beam on an elastic foundation, the foundation modulus C is converted to the track
modulus U using equation 3.3. U has the units of MPa, or rather MN/m/m, which is the distributed
force reaction of the continuous rail support due to a unit deflection of the rail.
The solution of the deflection of a long continuous beam on an elastic foundation and a single point
load (Q) is (21); (2, p. 6):
Q E E x
z e cos( E x ) sin( E x ) ( 3.6 )
2 U
where
U
E 4 ( 3.7 )
4 E I
The so-called characteristic length is defined by
O 1 E ( 3.8 )
4 E I
O 4 ( 3.9 )
U
and x is the distance from the point load Q.
Page 18
Following from the deflection, the bending moment (M)
d 2z Q
cos( E x ) sin( E x )
E x
M E I 2 e ( 3.10 )
dx 4E
In equations 3.6 and 3.10, x was substituted with x due to symmetry conditions and since the
equations are only mathematically valid for x t 0 .
The stresses in the rail are then calculated from the bending moment M. The load case on the rail is
seen as the sum of different load cases as illustrated in Figure 3-2. When the stresses are then
calculated for each of these separate load cases, it can be summed up again with careful consideration
of the sign at each location along the rail profile.
Under a centre vertical load the stresses V in the rail, a distance y from the neutral axis will be
M ( 3.11 )
V
Z
I
Z ( 3.12 )
y
with Z being the section modulus and I the second moment of area of the whole section about the
horizontal axis. Due to the Timosenko effect, the web of the rail is regarded as an elastic foundation to
the head of the rail, which influences the stress distribution throughout the rail section as indicated in
Figure 3-2. Figure 3-2 indicates how the rail stresses under an off-centre vertical load and
simultaneous lateral load, could be calculated by decoupling the off-centre loads into a centred
vertical and lateral load and moments about the neutral axis. The stresses at various points on the rail
profile are then calculated as follows:
Va V 1 V 3 V 4 V 6 V 8 ( 3.13 )
Vb Vc V3 ( 3.14 )
Va V3 Vd
Vc Va V3 a1 ( 3.15 )
d
Vd V 2 V5 V7 V9 ( 3.16 )
Ve V 1 V 3 V 4 V 6 V 8 ( 3.17 )
Vf V g V3 ( 3.18 )
Ve V3 Vh
Vg Ve V3 a1 ( 3.19 )
d
Vh V2 V5 V7 V9 ( 3.20 )
Page 19
V1+V3
V1 V3
V3 V3

V2 V2
Q
e
V4 V8
V6
Y
n
T T
Y
V5 V9
V7
Y Va Ve
Vb Vf
Vc Vg
Vd Vh
Left Right
Figure 3-2: Load decomposition for calculation of rail stresses
Page 20
M
V1 ( 3.21 )
Zk
M
V2 ( 3.22 )
Zf
I xx
Zk ( 3.23 )
yk
I xx
Zf ( 3.24 )
yf
and with yk and yf being the distance from the neutral axis to the top of the railhead and bottom of the
flange respectively. Ixx is the second moment of area of the rail profile about the horizontal axis. Then,
due to the Timoshenko effect (2, p. 14),
P O'
V3 ( 3.25 )
4 Zk
with
1 4 E Ik
O' '
4 ( 3.26 )
E Uw
and Ik being the second moment of area of the railhead about its neutral axis and
tE
Uw
§a · ( 3.27 )
2.3 a1 log¨¨ 2 ¸¸
© a1 ¹
where a1 is the railhead depth and a2 is the distance from the top of the rail to the top of the rail flange.
Due to torsion of the rail, the stresses can be calculated with the definition of the rotational centre as in
Figure 3-3: The position of the shear centre or rotational axis is given by:
h I2
h1 ( 3.28 )
I1 I 2
h I1
h2 ( 3.29 )
I1 I 2
with I1 = Izh = Second moment of area of the head about the vertical axis
and
with I2 = Izf = Second moment of area of the flange about the vertical axis.
Page 21
Head
centre
h1
Shear
centre
h2
Flange
centre
b2
Figure 3-3: Position of shear centre / rotational axis
Then the stresses in the railhead are:
E I 1 h1 T
Vk J ( 3.30 )
Z1 C1
and in the flange

E I 2 h2 T
Vf J ( 3.31 )
Z2 C1
with
A4 G
C1 ( 3.32 )
4 S 2 J x
E
G ( 3.33 )
2 1 X
I1
Z1
§ b1 · ( 3.34 )
¨ ¸
©2¹
Page 22
I2
Z2
§ b2 · ( 3.35 )
¨ ¸
©2¹
Y n Pe
T ( 3.36 )
2
and
C1
J ( 3.37 )
D h2
with
I1 I 2
D E ( 3.38 )
I1 I 2
Axial stresses due to torsion (constrained warping) of the lateral force Y alone:
Y n
T ( 3.39 )
2
E I h h1 Y n
V4 J ( 3.40 )
Zh 2 C1
E I f hf Y n
V5 J ( 3.41 )
Zf 2 C1
Axial stresses due to torsion (constrained warping) of the vertical force Q alone:
Qe
T ( 3.42 )
2
E I h hh Q e
V6 J ( 3.43 )
Zh 2 C1
E I 2 h2 Q e
V7 J ( 3.44 )
z2 2 C1
Due to sideways bending, the stresses V8 and V9 are given as:

Y a
V8 ( 3.45 )
6 Z8
Y a
V9 ( 3.46 )
6 Z9
Page 23
with
I yy
Z8
§ b1 · ( 3.47 )
¨ ¸
©2¹
and
I yy
Z9 ( 3.48 )
§ b2 ·
¨ ¸
©2¹
After the stresses in the rail due to the load are calculated, the additional stresses in the rail due to
temperature are calculated with
V D 'T E ( 3.49 )
with D being the coefficient of thermal expansion of the material and 'T being the change in
temperature.
The final stresses in the rail at the various positions are then simply calculated by adding the relevant
stresses from V1 to V 9, (while taking the sign of each into account) to additional axial stress (V) from
equation 3.49.
Due to the assumption that the above theory for a continuously supported and infinitely long beam
will be valid for a discretely supported beam of finite length (under certain conditions), it was deemed
necessary to compare the stress results from the analytical method with those from finite element
analyses where a beam of finite length is discretely supported.
The following sections will compare the results from the analytical method to results from finite
element methods and to actual rail stress measurements for validation purposes.
3.1.3.1. Comparison of rail stress calculations from the analytical method to those of Finite
Element Analysis results
A finite element model as depicted in Figure 3-4 was used to test the assumption that the derived
theory for calculating stresses in continuously supported and infinitely long beams will be valid for
discretely supported beams of finite length (under certain conditions). According to the Zimmerman
theory of a continuous beam on an elastic support (24, p. 111), the deflection and bending moment
formulas will also be valid for a finite beam of length greater than 2SO, with O as defined in (3.9). For
Page 24
a 57kg/m rail, the characteristic length (O ranges from 0.696m (U=200MPa) to 1.238m (U=20MPa)
which results in a beam longer than 4.4m (U=200MPa) or 7.8m (U=20MPa).
The choice of length for the finite element model should therefore be longer than the maximum length
of these restrictions. For convenience, and to ensure that the constraints on the beam ends do not
influence the bending in the middle of the model, an 11.05m section (a multiple of 0.650m - the
chosen sleeper spacing) of a 57kg/m rail was modelled. Since the model and load cases were
symmetrical with respect to the length of the rail, a symmetry plane was used as indicated in Figure
3-4 to reduce the length of the model to 5.525m. The symmetry plane was chosen to be halfway
between sleeper supports and a 100kN (50kN for symmetry) wheel load was applied in this area. This
position gave higher stresses than when the wheel load was applied directly on top of a sleeper. The
model consisted of a combination of approximately 25000 3D 8-node elements and 60 1D beam
elements.
Fixed point
1D Beam
Elements
1D Spring
Elements
650mm
Symmetry
plane
8 node 3D
elements
Figure 3-4: Finite element model – combination of 3D 8-node elements and 1D beam elements
Page 25
At the symmetry plane the complete rail profile was modelled with 3D Hex (8 node) elements to
enable reporting on detailed stresses along the varying profile. The 3D elements were used for a half
sleeper spacing distance of 325mm, from the end where the load was applied, to the first spring
element, which represented a sleeper support. Spring elements were used at 650mm intervals to
represent the discrete support of the rail pads and sleepers. The rest of rail was modelled with simple
beam elements, which had the same nominal section properties as the 57kg/m rail to reduce
computational cost. In order to evaluate the effect of the rigid connection between the solid elements
and beam elements, a vertical displacement plot was produced for the nodes on the neutral axis of the
rail (see Figure 3-5). From the deflection results of the FEA models, it can be seen that there is a
smooth transition in the deflection from the solid elements to the beam elements. These
displacements have been compared with analytically determined deflections, which are also indicated
in Figure 3-5.
0.001
0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-0.001
Deflection (m)
-0.002
-0.003
-0.004
-0.005
Distance (m)
Analytical 20 MPa/m Analytical 50 MPa/m Analytical 100 MPa/m
FEA 6.01 MN/m FEA 15.03 MN/m FEA 30.06 MN/m
Figure 3-5: Deflection curve of FEA models and analytical methods
For the analytical results, a Foundation Modulus (C) of 20, 50 and 100MPa/m were used where the
units (MPa/m) indicate the stress experienced by the foundation under a unit deflection. For the FEA
models, the Foundation Modulus had to be converted to a spring stiffness with units in MN/m.
According to Esveld (22, p. 59), the recommended conversion method is to multiply the Foundation
Modulus (C) with half of the effective sleeper support area, thus
k C. Ae ( 3.50 )
2
Page 26
For the case where C = 20 MPa/m, and a PY or FY concrete sleeper is used:
k 20. 0.6012 ( 3.51 )

2
k 6.012 MN / m ( 3.52 )
Table 3-2 indicates a number of support stiffness values, which include those used by Spoornet in the
analytical calculations, and those used in the FEA models. The deflections of a 57kg/m rail were then
calculated with both the analytical and finite element methods under different support conditions and
the results are tabulated in Table 3-2. With these three different support stiffnesses, two different load
cases were applied to give a total of 6 different result sets. The first load case represented a 100kN
centred vertical load (spread over an area of 20mm x 20mm) and the second represented an
off-centred vertical load of 100kN with a simultaneously applied lateral load of 50kN over the same
20mm x 20mm area (see Figure 3-6). The actual loads and application area were halved since a
symmetry plane was used in the model.
Table 3-2: Conversion of foundation modulus values to support stiffness
A B C
Track support
stiffness units Stiffness Maximum Stiffness Maximum Stiffness Maximum
deflection deflection deflection
Foundation
modulus 20 50 100
Analytical
[MPa/m]
y=4.4mm y=2.2mm y=1.3mm
Track modulus 9.249 23.123 46.246
[MPa]
Support
FEM
stiffness 1 6.012 y=4.5mm 15.03 y=2.4mm 30.06 y=1.4mm

[MN/m]
From the deflection results it is clear that there is a strong correlation between the two methods.
The FEA stress results for the different support stiffnesses under a 100kN centred vertical load are
indicated in Figure 3-7, overlaid on the results of the analytical model for the same load condition.
The stress results for the different support stiffnesses under a 100kN off-centred vertical load and
simultaneous 50kN lateral load are indicated in Figure 3-8 and Figure 3-9, overlaid on the results of
the analytical model for the same load condition.
Page 27
Figure 3-6: Centred and off-centred load cases
0.175
0.155
0.135
Height of Rail (m)
0.115
0.095
0.075
0.055
0.035
0.015
-0.005
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15
Compression Stress (GPa) Tension
Analytical 20 MPa/m Analytical 50 MPa/m Analytical 100 MPa/m

FEM 6.265 kN/mm FEM 16.209 kN/mm FEM 33.944 kN/mm
Approximate 57 kg/m Rail Profile
Figure 3-7: Axial stress results from analytical and FEM model with centred vertical load and different
support stiffnesses
Page 28
0.175
0.155
Left Right
0.135
Height of Rail (m)
0.115
0.095
0.075
0.055
0.035
0.015
-0.005
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10
Compression Tension
Stress (GPa)
Analytical 20 MPa/m Right Analytical 50 MPa/m Right Analytical 100 MPa/m Right
FEM 6.265 kN/mm Right FEM 16.209 kN/mm Right FEM 33.944 kN/mm Right
Figure 3-8: Axial stress results from FEM model with 100kN off-centre load and 50kN lateral load on
different support stiffnesses of 20, 50 and 100MPa/m (right side of rail)
0.175
0.155
Left 0.135 Right

Height of Rail (m)
0.115
0.095
0.075
0.055
0.035
0.015
-0.005
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Compression Stress (GPa) Tension
Analytical 20 MPa/m Left Analytical 50 MPa/m Left Analytical 100 MPa/m Left
FEM 6.265 kN/mm Left FEM 16.209 kN/mm Left FEM 33.944 kN/mm Left
Figure 3-9: Axial stress results from FEM model with 100kN off-centre load and 50kN lateral load on
different support stiffnesses of 20, 50 and 100MPa/m (left side of rail)
Page 29
116 MPa 46 MPa
0 MPa
0 MPa
-370 MPa -264 MPa
Figure 3-10: Graphical representation of typical stress results from a FEM model with an off-centre
load with a simultaneous lateral load (left) and a centre load only (right)
The rail stresses of the finite element model and the analytical model were compared and are
presented in Table 3-3.
The difference between the analytically determined stresses and those determined from the FEA,
ranged from 2MPa to 106MPa for the different support stiffnesses under the centred load. For the
off-centred load case, the difference ranged from 8MPa to 386MPa for the different support
stiffnesses. The biggest difference is in the railhead area where the FEA model reported high stresses
because of the close vicinity of the stress concentration under the applied load. It is clear from Figure
3-7 to Figure 3-9 that the stresses correlate well in the railhead area (below the top corner) and the
so-called Timoshenko effect is also well represented.
The rail web area shows less correlation than the flange area since the analytical calculations do not
account for the discontinuity between the web and flange and thus approximate a uniform stress
distribution from below the railhead to the bottom of the flange.
It should also be noted that the overall stresses of the FEA and analytical model have the best
correlation in the centred load case, whereas the off-centred load case presents higher differences in
stresses. The overall correlation is, however, quite acceptable if the fundamental differences between
the two methods are taken into account.
Page 30
Table 3-3: Stress comparison between FEM and analytical model
Difference
Maximum Rail Stress, type and FEM Analytical
location [GPa] [MPa] [MPa] [MPa] [%]
20 MPa/m
Railhead compressive stress -138 -159 -20 13%
Railhead tensile stress -9 7 16 229%
Centre load 100kN
Rail flange tensile stress 90 92 2 2%

50 MPa/m
Railhead tensile stress 2 17 16 91%
100 MPa/m
20 MPa/m Right
Railhead compressive stress -386 -256 131 -51%
Railhead tensile stress -58 -76 -18 24%
50 MPa/m Right
100 MPa/m Right
Off-centre load 100kN

20 MPa/m Left
Rail flange tensile stress 137 135 -1 -1%
50 MPa/m Left
Rail flange tensile stress 117 116 -1 0%
100 MPa/m Left
It is important to re-emphasise the fundamental differences between the analytical method and the
FEA method:
x The analytical method sees the rail as a infinite beam on a continuous elastic foundation
whereas the FEA model has a finite length and is discretely supported.
Page 31
x The analytical method simplifies the complex rail profile to a profile built up by 3 rectangular
profiles (railhead, rail web and rail flange) whilst the FEA model uses the precisely defined
profile in the area of highest stresses (solid elements) and simple beam elements in the rest of
the model. In both cases the simplified profiles have the same areas and second moment of
areas as the actual rail profile.
x In the analytical method the Timoshenko effect is taken into account by decomposing the
model into firstly a beam with the same height as the rail profile and continuously supported
by an elastic foundation, and secondly as a model where the railhead is continuously
supported by the elastic rail web. The stress results from these two models are then again
superimposed to produce the final stresses. In the FEA the profile geometry and material
properties define the varying elasticity throughout the rail profile, with the resultant varying
axial stresses throughout the profile.
x The deflection of the rail in both methods is a function of the rail properties and the support
stiffness, although the calculations are derived from totally different avenues.
Apart from these fundamental differences between the analytical model and the finite element
analysis, the results in Figure 3-7 to Figure 3-9 and Table 3-2 and Table 3-3 indicate that the two
methods are in accord.
Another method to validate the results from the analytical method was a comparison with rail stress
results from field measurements.
3.1.3.2. Rail stress results from field measurements
In preparation for field tests (23), several strain gauges were pasted onto a rail in track along the rail
profile (Refer Figure 3-11). The rail stresses were then measured under passing trains. Unfortunately,
the exact locations (eccentricity) of the loads are unknown, which makes it difficult to compare the
results with those of the previous analyses. However, the vertical loads were measured, as well as the
lateral loads. A summary of maximum rail stresses measured during the test is presented in Figure
3-12 for the two different load cases with vertical loads of 118kN and 108kN and lateral loads of 25kN
and 19kN. From the results it became clear that the measured stresses are in the same order of
magnitude as those of the previous analyses and that the stress patterns show similarities to those of
the analytical results.
Page 32
GAUGE SIDE FIELD SIDE

158.3
148.3
138.3
128.3
118.3
108.3 A
98.3 C B
88.3
NEUTRAL AXIS 78.3
68.3
58.3
48.3
38.3
28.3
18.3
12.6
3.0
0
Figure 3-11: Strain gauge positions as pasted onto rail for field tests (23)
Although a number of assumptions were made in the derivation of the theory for calculating rail
stresses, the results are in accord with finite element analyses and field measurements. Adjustments to
the input will, however, be investigated to improve on the correlation.
The last section of the analytical method, the sleeper reaction force, is described in the next section.
Page 33
Rail Stresses (Field measurements)
0.175
0.155
Approximate 57 kg/m
0.135
Rail Profile
0.115
Height of Rail (m)
0.095
118kN Vertical; 25kN
Lateral; unknown
0.075
eccentricity
0.055 108kN Vertical; 19kN

Lateral; unknown
0.035 eccentricity
0.015
-0.15 -0.10 -0.05 -0.0050.00 0.05 0.10 0.15 0.20
Stress (GPa)
Figure 3-12: Rail stresses under different vertical and lateral loads (eccentricities unknown) (23)
3.1.4. Sleeper reaction force:
The sleeper reaction forces (20, p. B3.9) are the forces from the rail acting on the sleeper as the
vertical wheel load is distributed from the rail to a number of sleepers. The sleeper reaction force in
the analytical method is calculated by distributing the vertical wheel load from the rail to a number of
sleepers. The sleeper reaction force is dependent on the stiffness of the rail, the sleeper spacing and
the foundation modulus.
If the deflection of the sleeper is presumed to be proportional to the deflection of the rail, the force
carried by the sleeper can be calculated by (20, p. B3.13)‡:
a Q O
K R' ( 3.53 )
2
with R’ the sleeper reaction force [kN], a the sleeper spacing [m], Q the vertical dynamic wheel load
[kN] and K the influence factor of the adjoining wheels.
The formula for K is:
O x § §x · § x ··
K e ¨¨ cos¨ ¸ sin ¨ ¸ ¸¸ ( 3.54 )
© © O ¹ © O ¹¹
‡
Due to (20, p. B3.7) inverting the definition of O, by using E instead of O to define the characteristic length and defining
O as the inverse of E, equations 3.53 and 3.54 were modified to have O in the nominator part of the fractions in order to
keep the definition of O constant throughout this document.
Page 34
where x is the distance from the wheel. In the event of more than one wheel being calculated, K is
replaced by K', with
K ' K ( x1 ) K ( x 2 ) ..... K ( x n ) ( 3.55 )

and the distance to each wheel being x1, x2 to xn respectively.
The mean contact pressure between rail and sleeper on the most heavily loaded sleeper can be
approximated by:
F0 R'
V rs ( 3.56 )
Ars
with Fo being the pretension force of the fastening system and Ars being the base plate area in contact
with the rail or sleeper, whichever is the smallest.
According to (1, p. 91), the maximum permissible contact pressures between rail and sleeper are:
x Softwood sleepers: V rs d 1.0 1.5MPa
x Hardwood sleepers: V rs d 1.5 2.5MPa
x Concrete sleepers: V rs d 4MPa
The vertical stresses between the sleeper and ballast bed should be kept below 0.5MPa, with the
formula for the sleeper ballast stress (Vsb):
R'
V rs ( 3.57 )
Asb
with Asb being half the contact area between sleeper and ballast.
By investigating the theory behind the analytical model, its limitations have been identified and will
be discussed in the next section.
3.2. Limitations of the analytical method
Preliminary investigations showed that important parameters such as curve radius and super elevation
were not included in the analytical method and that some calculated values did not correlate well with
measurements performed on Spoornet track (23). Tomas (23) thoroughly explained the manner in
which the measurements did not correlate with the theoretical results. The most important unknown
parameters were the vertical load positions (eccentricity) and track stiffness for which values had to
Page 35
be chosen based on experience and good engineering judgment. Tomas further questions the
assumption that the rail has only one rotation centre at the bottom of the rail web and proposes that the
rail may actually have an additional rotation centre at the top of the rail web. Proving this theory,
however, falls beyond the scope of this study.
The following areas for investigation into the analytical method have been identified:
x The calculation of the dynamic factor is performed with an empirical formula that was not
developed or validated locally, although widely accepted.
x The formula for the dynamic factor does not include the effect of factors such as track cant or
curve radius, nor does it include any suspension parameters of the rolling stock.
x The parameter indicating the running top condition is a subjective value and not uniquely
specifiable.
x There is no indication which value for the running top condition will result in a “safer” result
if the user is uncertain of the correct value to enter.
x There is no guideline on how the statistical confidence level should be chosen and how its
choice will affect the results.
x The value for the track support condition is a subjective value and not easily determinable.
x There is no indication which value for the track support condition will result in a “safer” result
if the user is uncertain of the correct value to use.
x The eccentricity of the vertical force is used as an input from the user and not as a function of
the rail type, rail profile, wheel profiles, track geometry or steering and hunting characteristics
of the vehicle.
x There is no specific guideline on typical eccentricity values of the vertical load.
x There is no indication which value for the eccentricity will result in a “safer” result if the user
is uncertain of the correct value.
x The lateral load in the analytical method is calculated as a fixed percentage of the vertical
force, which could not be the case since the lateral force is a function of a number of other
parameters, such as track geometry, rolling stock steering ability and speed of the vehicle.
x The analytical method acknowledges the fact that an error is introduced with the assumption
that the rail is continuously supported.
In order to prioritise and focus the investigation on the shortcomings of the analytical method, it was
decided to perform a sensitivity study on the model to determine which parameters have the greatest
effect on the results under the current relationships.
Page 36
3.3. Sensitivity analyses
Sensitivity analyses help to focus the attention on the input parameters that have the greatest effect on
the output by recording the range of results according to a predetermined range of input.
It is well understood that a sensitivity study will not highlight or indicate any incorrect relationships
between parameters and thus any areas of concern in this regard will have to be dealt with in separate
investigations by comparing results with empirical results or other established methods. It is thus also
imperative that another sensitivity study should be performed on the model, after changes to the
model have been introduced, to ensure that the final list of important parameters is well documented
and guidelines for obtaining proper input are given.
Two independent sensitivity studies were performed on the analytical method namely a quantitative
study and a qualitative study.
3.3.1. Quantitative sensitivity analysis
In this study, the rail stresses were monitored while each parameter in the model was set to a median
(middle) value and each in turn was changed from its minimum to its maximum value. The results did
not include the effect of combinations of high or low parameter values, but included the effect of a
single parameter value on the rail stresses, with the other parameter values equal to their respective
median values. The range of stresses of the rail was then noted for each parameter as indicated in
Table 3-4.
Figure 3-13 indicates the information from Table 3-4 graphically where it is clear that the ‘Static
Wheel load’, ‘Lateral force factor’, ‘Max Temperature change’, ‘Eccentricity of Vertical force Q’,
‘Dynamic Factor’, the ‘Wheel distances’ and the ‘Rail section’ are the most important parameters to
investigate since the results are the most sensitive to changes in these parameters. Table 3-5 indicates
the parameters from Table 3-4 in order of importance as a function of the sum of the railhead and rail
flange stress ranges. The stress ranges for each parameter are also indicated as a percentage of the
maximum value and used here as an indication of relative importance.
According to the result from Table 3-5 it can be concluded that the current model is most sensitive to
the static wheel load and least sensitive to the type of sleeper used. One must keep in mind that, once
the model is further developed, results from a sensitivity analysis on the updated model may differ
from the results shown.
Page 37
Table 3-4: Input parameter sensitivity analysis

Parameter Value Max Rail Head Tension (MPa) Sum Head
Band Width
Parameter Min Mid Max Max Rail Flange Tension (MPa) & Flange
Max - Min
A B C A B C Band width
233.23 248.16 263.1 29.87
Running Top Condition 0.1 0.2 0.3 64.21
174.66 191.83 209 34.34
238.2 248.16 258.12 19.92
Confidence Level 1 2 3 42.81
180.38 191.83 203.27 22.89
242.89 248.16 253.43 10.54
Speed ( V ) 60 90 120 22.66
185.77 191.83 197.89 12.12
196.7 248.16 239 51.46
Dynamic Factor 1.1 1.49 1.73 102.86
149.9 191.83 201.3 51.4
198.57 248.16 324.46 125.89
Static Wheel load 67.5 100 150 205.3
160.54 191.83 239.95 79.41
239.48 248.16 250.17 10.69
Foundation Modulus 20 50 100 30.5
207.92 191.83 188.11 19.81
241.98 248.16 254.27 12.29
Sleeper Spacing 500 600 700 36.79
179.65 191.83 204.15 24.5
247.85 248.16 247.96 0.31
Type of Sleeper Wood Concrete Steel 0.88
192.4 191.83 192.2 0.57
247.14 248.16 249.05 1.91
Effective Area of Sleeper 0.4 0.48 0.6 5.45
193.71 191.83 190.17 3.54
245.95 248.16 250.38 4.43
E modulus rail 200 205 210 9.53
189.28 191.83 194.38 5.1
247.99 248.16 248.62 0.63
Poisson 0.289 0.3 0.33 1.42
192.04 191.83 191.25 0.79
281.14 248.16 215.18 65.96
Eccentricity of Vertical force Q -0.022 0 0.022 107
203.29 191.83 232.87 41.04
250.59 248.16 245.39 5.2
Position of Lateral force below top 0 0.007 0.015 11.67
188.81 191.83 195.28 6.47
271.44 248.16 258.97 23.28
Rail section 48 57 S60 65.11
218.95 191.83 177.12 41.83
250.4 248.16 220.79 29.61
Wheel distances 1.56 1.91 10.06 84.5
187.68 191.83 242.57 54.89
220.86 248.16 277.74 56.88
Max Temperature change 30 42 55 113.77
164.52 191.83 221.41 56.89
166.45 248.16 329.88 163.43
Lateral force factor 0 0.4 0.8 194.16
177.04 191.83 207.77 30.73
This exercise should therefore be seen as an initial step to highlight important parameters in order to
focus the attention on areas where the greatest impact can be made in the improvement of the current
model. The quantitative sensitivity analysis can be regarded as a simplistic method for determining
parameter sensitivity since it does not take the effect of different combinations of high and low
parameter values into account. It was therefore decided to validate the findings of the quantitative
sensitivity analysis with a qualitative sensitivity analysis where random combinations of parameter
values are used in identifying sensitive parameters.
Page 38
Sensitivity Analysis
350
300 Head
Max tensile Stess (MPa)
250
200
150 Flange
Sensitivity
100 A
Mid
50
C
Max Temperature change

Type of Sleeper
Foundation Modulus
Eccentricity of Vertical force Q

Dynamic Factor
Position of Lateral force below top

Effective Area of Sleeper
Confidence Level
Speed ( V )
Lateral force factor

Running Top Condition
Sleeper Spacing
E modulus rail
Poisson
Rail section
Static Wheel load
Wheel distances
Parameter
Figure 3-13: Sensitivity analysis of parameters in component stress calculation spreadsheet
Table 3-5: Order of sensitive parameters

S um of He a d &
O rd e r o f P e rc e n ta g e o f
P a ra m e te r F l a n g e S tre ss
I m p o rta n c e h i g h e st v a l u e
B a n d w i d th s
1 S t a t ic W h e e l lo a d 205 100%
2 L a t e ra l fo rc e fa c t o r 194 95%
3 M a x Te m p e ra t u re c h a n g e 114 55%
4 E c c e n t ric it y o f V e rt ic a l fo rc e Q 107 52%
5 D y n a m ic F a c t o r 103 50%
6 W h e e l d is t a n c e s 85 41%
7 R a il s e c t io n 65 32%
8 R u n n in g To p C o n d it io n 64 31%
9 C o n fid e n c e L e ve l 43 21%
10 S le e p e r S p a c in g 37 18%
11 F o u n d a t io n M o d u lu s 31 15%
12 S peed ( V ) 23 11%
13 P o s it io n o f L a t e ra l fo rc e b e lo w t o p 12 6%
14 E m o d u lu s ra il 10 5%
15 E ffe c t ive A re a o f S le e p e r 5 3%
16 P o is s o n 1 1%
17 Ty p e o f S le e p e r 1 0%
3.3.2. Qualitative sensitivity analysis
Marczyk’s book, “Beyond Optimization in CAE (Computer Aided Engineering)” (12), inspired the
use of his philosophy of stochastic simulation as an alternative method to evaluate the sensitivity of
Page 39
the analytical method towards its input parameters. In this case, the input parameters of the analytical
method were randomly changed between their lowest and highest values and the stress results of the
rails evaluated accordingly for each random combination of parameters. Figure 3-14 indicates the
railhead stresses vs. the rail flange stresses for a number of random combinations of input parameters.
The tensile stresses at the railhead and flange were plotted against each other in Figure 3-14 to Figure
3-16, to investigate not a correlation between these two output parameters, but the spread of their
values due to changes in the input parameters. Using such 2D plots creates a visual aid for
investigating the effect of the changing input parameters on the output parameters chosen. Since each
data point is generated from randomly changed input parameters, numerous graphs like this can be
generated, with each graph displaying a unique data scatter. A point to note, however, is that the data
form “point clouds” which are a phenomenon found in each generated graph. Additional graphs
(iterations from the same model) are presented in Appendix A, which indicate not only the
similarities between such randomly generated data points graphs, but also the possible changes in
outliers.
600
500
Railhead stress (tension) [MPa]
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Rail flange stress (tension ) [MPa]
Figure 3-14: Stochastic distribution of rail stresses from the analytical method
The main objective of stochastic simulation is to get an impression of possible ranges of answers to a
problem instead of focusing on only one answer. The data scatter produced in Figure 3-14 will
collapse into one data point if each input parameters is changed to one specific value. Studying the
patterns of the data scatter in these results therefore provides the engineer with a more holistic
impression of the problem than studying only a few isolated data points or fixed combinations of
input parameters.
Page 40
The next step was to eliminate parameters by holding them constant in turn and evaluating the effect
on the data scatter. In using this method, the sensitivity of the model towards the different parameters
was thus qualitatively evaluated, instead of quantitatively as in the previous method.
Figure 3-15 indicates the data scatter when a number of input parameters are held constant as
indicated in Table 3-6. Figure 3-16 indicates the resulting data scatter when the parameters, which
were held constant in Figure 3-15, are changed randomly while the remaining parameters are held
constant.
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Figure 3-15: Stochastic distribution of rail stresses with a number of parameters held constant
The number of unique data points was subsequently reduced but the spread of data points (or rail
stress ranges) was still similar to those of Figure 3-14 and Figure 3-15. Figure 3-16 indicates a
dramatic reduction in data scatter due to the parameters as listed in Table 3-6 being held constant.
From Figure 3-15 and Figure 3-16 it is clear that the parameters that affect the rail stresses the most
are indeed those that were determined earlier and indicated here in Table 3-7.
Page 41
Table 3-6: List of parameters modified in stochastic simulation results of Figure 3-15 and Figure 3-16
Held Constant?
Parameter
Figure 3-15 Figure 3-16
1 Static Wheel load No Yes
2 Lateral force factor No Yes
3 Max Temperature change No Yes
4 Eccentricity of Vertical force Q No Yes
5 Dynamic Factor No Yes
6 Wheel distances No Yes
7 Rail section No Yes
8 Running Top Condition Yes No
9 Confidence Level Yes No
10 Sleeper Spacing Yes No
11 Foundation Modulus Yes No
12 Speed ( V ) Yes No
13 Position of Lateral force below top Yes No
14 E modulus rail Yes No
15 Effective Area of Sleeper Yes No
16 Poisson Yes No
17 Rail steel type Yes No
18 Type of Sleeper Yes No
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Figure 3-16: Stochastic distribution of rail stresses with a number of parameters held constant
Page 42
Table 3-7: Most sensitive input parameters of the analytical method
Most Sensitive Input Parameters Uniquely Identifiable?
1 Static Wheel load Yes

2 Lateral force factor No
3 Max Temperature change Yes
4 Eccentricity of Vertical force No
5 Dynamic Factor No
6 Wheel distances Yes
7 Rail section Yes
Table 3-7 also indicates whether or not the input parameter values are uniquely identifiable (or
uniquely obtainable), which is of importance for the purpose of this study.
Because it is vitally important that the parameters, to which the model is most sensitive and for which
confidence is lacking in the values used, are correctly entered, the following three parameters will be
researched: dynamic factor, lateral force factor and the eccentricity of the vertical force.
3.4. Conclusion
The analytical method is a method utilised to calculate rail stresses under dynamic loading by
factoring the static load with an appropriate amount and calculating track component stresses by
means of a static analysis. This method employs input parameters, which describe the loading
conditions and track design and then calculate track component stresses directly. The analytical
method is fast and easy-to-use in evaluating different loading conditions or different track designs
without having to modify the model and proved capable of providing results, which correlate well
with other proven methods and empirical tests. For this reason, the shortcomings and limitations of
the method were investigated in order to improve on them and further enhance the analytical method
for calculating track component stresses.
Page 43
4 IMPROVEMENTS TO THE ANALYTICAL METHOD
The analytical method has proven to be valuable in calculating track component deflections and
stresses under vehicle loads with reasonable accuracy as validated with empirical and other
well-established methods. It was, however, discovered that the current analytical method had its
limitations. These limitations were mostly related to input parameters for which the user would have
difficulty in supplying reasonable (or correct) values and to which the results were the most sensitive.
It is the intention of this chapter to identify the shortcomings of the analytical method, to improve on
it where possible and to provide the user with guidelines and some nominal values for input
parameters that may be used in future stress calculations.
4.1. Investigations into the improvements of the analytical method
From the sensitivity analyses it was discovered that the dynamic factor, lateral force factor and the
eccentricity of the vertical force were the three parameters that the analytical method was the most
sensitive to and for which the user may have the greatest difficulty in obtaining reasonable (or
correct) values. A number of empirical tests were conducted in separate studies which were utilised to
determine the best practice approaches in obtaining these input parameter values. It will further
provide the user with some nominal values and/or ranges of values to be used in the event that
empirical testing is not possible.
The following section will focus on the dynamic factor which is an important parameter in the
analytical method since it is the parameter to which the model is the most sensitive.
4.1.1. Dynamic Factor
The influence of moving loads on the track structure cannot be ignored since loads under dynamic
conditions are higher than the same loads under static conditions (24, p. 107), (25, p. 1). This increase
in load is referred to as the dynamic factor or dynamic amplification (26, p. 5), (10, p. 1-4) and is
determined, either empirically or numerically, with appropriate models by comparing the static wheel
loads on the rail or structure to the loads when the same wheels are moving over at speed.
Since the analytical method is the most sensitive towards the static load, which directly affects the
dynamic factor, the dynamic factor will be investigated here under the assumption that the static load
is known. The loads affecting the track can be divided into horizontal and vertical forces. The vertical
forces are a function of the static wheel load and wheel load shifting due to the slanting and swaying
Page 44
of the vehicle (25, p. 1). The slanting of the vehicle is affected by the radius of the curve (27), the
amount of super elevation of the track, the travelling speed, and the height of the centre of gravity of
the vehicle. Timoshenko (21, p. 412) introduced a further possible cause to the magnifying effect of
moving loads, namely, the vibration of the rails under moving loads. Wilson (24, p. 125), however,
regards this contribution as negligible under low speeds. Dynamic loads are not only higher than
static loads because of the above-mentioned factors, but also because of impact loads. According to
Wang (17), there are two kinds of factors that cause high impacts between wheel and rail, namely,
those resulting from the vehicle, and those resulting from the track. Impacts resulting from the vehicle
include: wheel abrasion, wheel ovality, flat wheels and wheel eccentricity. The impacts due to the
track include: track geometry irregularities, rail joints and track defects underneath the rail such as
slacks. The support condition therefore also causes moving loads to be higher than static loads.
Figure 4-1 indicates a skid mark on a rail, which causes impact forces under moving loads. Skid
marks are caused by locomotive traction wheelsets that slip and “burn” the rail metal away. Figure
4-2 indicates a flat wheel (skid mark on a wheel thread) that causes impact forces when the wheel
rolls over the rail. Wheel flats are the result of excessive braking or brakes that fail to release the
wheel, when the vehicle is pulling away. Surface defects on the rail have a detrimental effect on the
track structure, and, with the presence of water, may cause "mud holes" such as the one indicated in
Figure 4-3. In an investigation on the Sishen Saldanha line, Spoornet found that there is a linear
relationship between the vertical track profile and the depth of dip joints (flash butt welds that
developed local soft zones) (28). This indicates that the impact due to the surface defects has a
detrimental effect on the track geometry, which in turn affects the dynamic loading of the track.
Figure 4-4 and Figure 4-5 indicate typical dip joints.
Figure 4-6 indicates an attempt to relate track vertical alignment to the resulting dynamic vertical
forces. The results show that there exists no definite relationship for this specific data set, which were
measured in Sweden by the Spoornet Engineering team (29).
The results in Figure 4-6 underline the statement in the UIC Code 518 OR (30, p. D-1) which states
that"…it is difficult to draw a simple relation between track-geometry and vehicle response. As a
result, it is not possible either to relate a geometrical value to a track geometric quality or a given
track geometrical error to a vehicle-response."
One of the options of calculating a dynamic factor using a quasi-static calculation is to use an
empirically determined formula. Many such formulas have been developed in the past and will be
discussed in the next section.
Page 45
Figure 4-1: Skid mark on rail running surface
Figure 4-2: Skid mark on wheel tread (“flat wheel”)
Page 46
Figure 4-3: The presence of water and excessive track deformation pump mud to the surface
Figure 4-4: Measuring the depth of a dip joint (soft flash butt weld)
Page 47
Figure 4-5: Dip joint identified by two wider patches on the contact band
18
Vertical Profile (90th Percentiles of 100m sections)
16
14
12
10
40 km/h
[mm]
50km/h
60km/h
8
70km/h
0
148 150 152 154 156 158 160 162 164
Vertical Dynamic Load (90th Percentile of 100m sections) [kN]
Figure 4-6: Track geometry vertical alignment vs. dynamic loads
Page 48
4.1.1.1. Empirical Formulas
In order to investigate the magnitude of typical dynamic factors, current widely accepted empirical
formulas were investigated and compared to measurements. According to Esveld (1, p. 83), it is
common practice to carry out strength or fatigue calculations for a static system in which the dynamic
effects are taken into account by a speed coefficient or a dynamic amplification factor. Researchers
over the world have investigated the magnifying effect of dynamic loads and produced various
empirically determined formulas. According to Lombard (31), various researchers demonstrated that
the average results obtained from extensive track measurements are in accord with theoretical
predictions using static wheel loads as input. A number of empirical formulas were investigated,
namely, that of Eisenmann (19), Schramm (25), Klöckner (25), Clintock (25), ORE and AREA as
summarised by Wang (17) and listed in Table 4-1.
Table 4-1: List of empirical formulas for the calculation of a dynamic factor
§ V 60 · '
1. Eisenmann I 1 G ¨1 ¸t
© 140 ¹
2. Schramm M 1 45 10 6 V 2 15 10 8 V 3
Klöckner M 1 20 10 4 V 5 10 6 V 2
3.
(Light traffic)
Klöckner M 1 25 10 4 V 7 10 6 V 2
4.
(Heavy traffic)
5. AREA M 1 5.21 V D
3 3
§ V · V 2 h c 2 c h §¨
2
§ V · ·¸
6. ORE M 1 0.04 ¨ ¸ 0.1 0.017¨ ¸ a 0 b0
© 100 ¹ 127 R g2 ¨ © 100 ¹ ¸¹
©
3
ORE § V ·
7. M 1.29 0.04 ¨ ¸
(V<140km/h) © 100 ¹
Figure 4-7 and Table 4-1 indicate a comparison by Wang (17) of empirical formulas which calculate
dynamic factors. A number of results from the Eisenmann formula are indicated in Figure 4-7 and
Table 4-2. The results are distinguished by two numbers in the legend, which relate to the track
geometry condition and statistical confidence. These factors are not included in the other empirical
formulas. In Figure 4-7 and Table 4-2, "Eisen 1 0.1" indicates the results from the Eisenmann formula
for a good track geometry {0.1}, and a low statistical confidence {1}, whereas "Eisen 3 0.3" indicates
the results from the Eisenmann formula with a bad track geometry {0.3} and high statistical
confidence {3}.
Page 49
Comparison of dynamic factors

AREA
3.00 Eisen 1 0.1
Eisen 1 0.2
Eisen 1 0.3
2.50
Eisen 2 0.1
Eisen 2 0.2
2.00
Eisen 2 0.3
Dynamic factor
Eisen 3 0.1
1.50
Eisen 3 0.2
Eisen 3 0.3
1.00 ORE
Dynamic: Loco
0.50 Dynamic: Wag
Schram
Klöckner Light
0.00
30 50 70 90 110 130 150 Klöckner Heavy
Speed (km/h) Clintock
Figure 4-7: Comparison of dynamic factors from different empirical formulas
Table 4-2: Comparison of dynamic factors from different empirical formulas

Speed (km/h) 40 50 60 70 80 90 100 110 120 130 140
AREA 1.17 1.21 1.25 1.29 1.33 1.38 1.42 1.46 1.50 1.54 1.58
Eisen 1 į=0.1 1.09 1.09 1.10 1.11 1.11 1.12 1.13 1.14 1.14 1.15 1.16
Eisen 1 į=0.2 1.17 1.19 1.20 1.21 1.23 1.24 1.26 1.27 1.29 1.30 1.31
Eisen 1 į=0.3 1.26 1.28 1.30 1.32 1.34 1.36 1.39 1.41 1.43 1.45 1.47
Eisen 2 į=0.1 1.17 1.19 1.20 1.21 1.23 1.24 1.26 1.27 1.29 1.30 1.31
Eisen 2 į=0.2 1.34 1.37 1.40 1.43 1.46 1.49 1.51 1.54 1.57 1.60 1.63
Eisen 2 į=0.3 1.51 1.56 1.60 1.64 1.69 1.73 1.77 1.81 1.86 1.90 1.94
Eisen 3 į=0.1 1.26 1.28 1.30 1.32 1.34 1.36 1.39 1.41 1.43 1.45 1.47
Eisen 3 į=0.2 1.51 1.56 1.60 1.64 1.69 1.73 1.77 1.81 1.86 1.90 1.94
Eisen 3 į=0.3 1.77 1.84 1.90 1.96 2.03 2.09 2.16 2.22 2.29 2.35 2.41
ORE 1.29 1.30 1.30 1.30 1.31 1.32 1.33 1.34 1.36 1.38 1.40
LOCO *2 1.04 1.07 1.08 1.09 1.16 1.24 1.33 1.42 1.52 1.62 1.72
WAGON *3 1.02 1.04 1.05 1.06 1.08 1.10 1.12 1.13 1.16 1.19 1.22
Schram 1.06 1.09 1.13 1.17 1.21 1.26 1.30 1.34 1.39 1.43 1.47
Klöckner Light 1.07 1.09 1.10 1.12 1.13 1.14 1.15 Only valid for V<=100km/h
Klöckner Heavy 1.09 1.11 1.12 1.14 1.16 1.17 1.18
Clintock 1.08 1.09 1.11 1.13 1.15 1.17 1.19 1.21 1.23 1.25 1.27
The average value of the dynamic factors listed in Table 4-2 is equal to 1.38, with the minimum being
1.086 as calculated at 40 km/h with the Eisenmann formula (with an excellent track geometry and a
68.3% statistical confidence) and a maximum of 2.414 as calculated at 140 km/h with the Eisenmann
formula (with a bad track geometry and 99.7% statistical confidence).
Page 50
The majority of empirical formulas is in agreement with values between 1 and 1.5, the exception
being the Eisenmann formula (values of 1.7 to 2.4) when a statistical confidence of 99.7% is chosen.
Esveld (1, p. 84) suggests the use of a statistical confidence of 99.7% when investigating rail stresses
and a lower statistical confidence when investigating ballast and sub-formation stresses.
Because of uncertainty regarding the validity of these empirical formulas for local track and rolling
stock conditions, data from field tests were investigated to determine typical dynamic factors
4.1.1.2. Empirical Tests
Data from empirical tests are useful when determining dynamic factors since they represent local
track and operating conditions. Dynamic factors can be determined from field measurements where
static loads from vehicles are known and are then compared to their measured load under dynamic
conditions. These tests can be performed with wayside (weigh-in-motion) measurements or with
instrumented wheelset measurements.
(i) Weigh-in-motion weighbridges
The weighbridges on Spoornet track measure the load of each passing truck wheel at operational
speed. These weighbridges are specially calibrated to determine the static loads of each passing wheel
by utilising a dynamic factor to convert the vertical force of the wheel to a static load value. Thus,
extracting the dynamic factors used in the weighbridges provides representative dynamic factors
applicable for local track and rolling stock conditions. The weighbridges are firstly statically
calibrated at installation with a load cell rig, which is used to load the track to a known value and
calibrate the system accordingly. After this is performed, the software analyses the signals as a train
passes and reports the measured values. These values are not the static loads of the vehicles but the
dynamic loads as measured by the weigh-in-motion system. The system is calibrated with the use of a
test train, with wagons which were measured at an SABS certified static weighbridge. The calibration
is done at various speeds and in both directions. The dynamic factors are then calculated by
comparing the measured values with the known static wheel loads. If the weighbridge is installed in
track where operating conditions permit trains to pass without tractive or braking effort from the
locomotives, the measured weight of the locomotives can be compared to design loads as an
additional method for calibration. Unfortunately, when the locomotives apply braking or traction
while travelling over the weigh bridges, weight transfer takes place on the locomotive bogies and a
dynamic or calibration factor cannot be calculated.
Page 51
Figure 4-8: A complete unmanned weigh-in-motion weighbridge.
At the Saaiwater weigh-in-motion weighbridges, there are three railway lines, each with its own
independent weighbridge. Each weighbridge has a number of calibration curves for each direction of
travel and for different weight groups of the vehicles. It was found that vehicles from different weight
groups (static loads) need different calibration curves since light vehicles tend to have higher
dynamic loads in relation to their static loads than heavier vehicles. This is due to a higher undamped
mass to damped mass ratio in unloaded vehicles. Because the coal export line is a dedicated line with
a closed fleet and only a small amount of general freight traffic, vehicles of similar mass are normally
of the same type, which would also contribute to the fact that the different groups have different
dynamic factor calibration curves. The following graphs in Figure 4-9 to Figure 4-11 indicate the
dynamic calibration factor curves for the three weighbridges at Saaiwater.
The results from the calibration curves indicate dynamic factors of less than 1.2 for the whole range of
vehicles and speeds, which is less than the dynamic factors calculated with the empirical formulas.
This could, however, be due to the high standard of track maintenance in these areas.
Page 52
Line 1 Down Direction Line 1 Up Direction

1.2 1.2
Wagon mass Wagon mass
categories categories
Dynamic factor
Dynamic factor
1.15 1.15
0-20t 0-20t
20-70t 20-70t
1.1 1.1
70-90t 70-90t
1.05
90t+ 1.05
90t+
1 1
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Speed (km/h) Speed (km/h)
Figure 4-9: Calibration curves for weighbridge on No1. line at Saaiwater

1.2 1.2
Dynamic factor
1.15
0-20t Dynamic factor 1.15
0-20t
20-70t 20-70t
1.1 1.1
70-90t 70-90t
1.05
90t+ 1.05
90t+
1 1
0 20 40 60 80 100 120 0 20 40 60 80 100 120

1.2 1.2
Dynamic factor
Dynamic factor
1.15 1.15
0-20t 0-20t
20-70t 20-70t
1.1 1.1
70-90t 70-90t
1.05
90t+ 1.05
90t+
1 1
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Table 4-3: Track setup at Saaiwater weighbridges
Track setup Line 1 Line 2 Line 3
Rail type 57kg/m 57kg/m S60kg/m
Sleeper spacing 700mm 700mm 650mm
Page 53
As mentioned before, wheel irregularities may also cause high impact loads on the track, resulting in
loads higher than the static wheel loads.
Figure 4-12 indicates a typical wheel load signal as captured by a weigh-in-motion weigh bridge. As
the wheel rolls along the track, the wheel load is recorded and should, under perfect circumstances,
follow the ‘pink’ line as portrayed by the “perfectly round wheel” data series. If the wheel however
has a flat spot or any other imperfection, the captured data may look like the ‘blue’ data series
portrayed by the “Wheel with a flat spot” data series. The impact load is clearly distinguishable from
the nominal wheel load since it occurs within a short distance compared to the wheel circumference.
Wheel load signal

25
20
Impact
wheel load
Wheel load [kN]
15
Wheel with flat spot

Perfectly round wheel
10
Dynamic
5 wheel load
0
0 0.5 1 1.5 2 2.5 3
Distance along track [m]
Figure 4-12: Wheel impact load presentation
Figure 4-13 to Figure 4-15 indicate the distribution and magnitude of significant impact loads (higher
than 5 tons) as identified at the Saaiwater weigh-in-motion stations. However, these impact loads,
although substantially higher than the static wheel loads (up to 44 tons in one instance), only occurred
on 0.3% of 1.9 million wheels (Line 1), 0.03% of 1.8 million wheels (Line 2) and 0.06% of 2.6
million wheels (Line 3) respectively. It is important to take note of these high impact loads when
designing railway track, but, due to their rare occurrences, it would be uneconomical to design the
track to be able to carry an infinite number of these high loads.
The cumulative percentages of wheel impact loads indicated in Figure 4-13 to Figure 4-15 (secondary
vertical axes) indicate the contribution of significant wheel impact loads (higher than 5 tons) to the
Page 54
total number of wheel impact loads. The distributions of impact loads are only indicated for impact
loads up to 31 tons.
800 100%
90%
700
80%
600
70%
Number of wheels
500
60%
# of wheels
400 50%
Cumulative %
40%
300
30%
200
20%
100
10%
0 0%
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Wheel impact (ton)
Figure 4-13: Dynamic impact loading distribution (number of wheels) – Line 1
The impact loads from the above figures represent the whole fleet travelling over the Saaiwater weigh
bridges (coal export line) which could be much lower than impact loads on other less maintained
fleets. Such information should therefore be obtained from weighbridges from the area of interest, if
an area specific design is sought, and from weighbridges across the country, if a design is necessary
for the worst-case scenario.
Page 55
700 100%
90%
600
80%
500
70%
Number of wheels
60%
400
# of wheels
50%
Cumulative %
300
40%
200 30%
20%
100
10%
0 0%
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Wheel impact (ton)
600 100%
90%
500
80%
70%
400
Number of wheels
60%
# of wheels
300 50%
Cumulative %
40%
200
30%
20%
100
10%
0 0%
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Wheel impact (ton)
Page 56
(ii) Way-side measurement tests
The way-side measurement tests use the technology developed for the weigh-in-motion bridges to
measure vertical and lateral loads of passing trains, as well as deflections or accelerations of track
components. The only difference in the way-side measurements is that the mobile laboratory is used
to capture and analyse the data and that the equipment is removed after testing. Figure 4-16 indicates
the mobile laboratory with which temporary tests sites can be set up to measure strains, deflections or
accelerations of track components.
Figure 4-16: Spoornet mobile laboratory for measuring strains, deflections and accelerations.
(iii) Instrumented wheelset
Another method that can be utilised to determine dynamic loads is the instrumented wheelset. With
the instrumented wheelset installed in a rolling stock vehicle, dynamic wheel loads can be measured
for both vertical and lateral loads. Figure 4-17 indicates the instrumented wheelset as installed on a
30ton per axle iron ore wagon. Figure 4-35 indicates the train setup. The wheelsets as well as the
Page 57
vehicle body and bogie are instrumented to measure strains, deflections and accelerations. The wheel
strains are analysed to produce vertical and lateral loads at any desired speed, which can then be
compared to the static wheel loads in order to calculate dynamic factors under different conditions.
Instrumented
wheelset
Figure 4-17: Instrumented wheelset installed in 30t per axle iron ore wagon
Results from tests (28) on the Iron Ore Export line indicated that the dynamic factor at 70km/h could
typically be 1.13 for a 30ton axle load and 1.43 for a 26-ton axle load. Possible reasons for the lower
dynamic factor associated with the higher axle load include the following: the suspensions of the
vehicles behaved differently at that particular speed, or the unsprung mass (wheels, axles and bogies§)
for a heavier loaded vehicle is lower in relation to its sprung mass (bogies§, wagon body and load)
than for a lighter loaded vehicle. In each case the sprung mass acts as a shock absorber to
accelerations of the unsprung mass.
Another test was executed at Leandra on the Springs-Bethal line. Figure 4-18 indicates the track
layout at the test site and Figure 4-19 indicates the measured track alignment in terms of the curve
radius for the 800m length of track at Leandra on the Springs-Bethal line. It is clear that the test
§
Depending on the bogie design, parts of it can be unsprung (i.e. unsupported by the suspension), and the other parts can
form part of the sprung mass (i.e. supported by the suspension).
Page 58
section consists of two adjacent curves. The first curve has a radius of ±950m for 200m and the
second a curve of ±800m radius. In between the curves exists a piece of track which is almost straight
(large radius) and is immediately followed by a sharp curve with a radius of 600m. The sharp
transition from a straight track to a 600m-radius track is evidence that the track is moving in the
direction of the ruling traffic over time.
28.934
28.933 N
28.932
R±950m
28.931
28.93
28.929
28.928 R±800m Direction of

28.927 ruling traffic
28.926
28.925
28.924
-26.374 -26.372 -26.37 -26.368 -26.366 -26.364 -26.362
Figure 4-18: Leandra test site track layout
Eight runs of data were obtained over an 800m length of track. Measurements were taken at 20, 30, 40
and 60 km/h in one direction only and two tests were conducted at each speed.
A reflective plate triggered a "red eye", which is an optical instrument, as the train passed over to
indicate the start and end of the test section for comparison purposes. The position of the red eye is as
indicated in Figure 4-20 on the wagon body and reflects a beam from the stationary reflective plates at
the start and end of the test section. The impulse from the red eye was recorded in the measured data
as a peak or “spike” and was used to superimpose data from different runs.
Page 59
1600
1400
1200
1000
Radius (m)
800
600
400
200
0
0 100 200 300 400 500 600 700 800 900
Distance (m)
Figure 4-19: Leandra test site track geometry (curve radius)
Wagon Bogie
Instrumented wheelset
“Red eye”
Stationary reflective
plate placed on ground
Figure 4-20: Instrumented wheelset with “red eye”
Page 60
It is important to note that the only difference between the tests was the operating speed. Figure 4-21
and Figure 4-22 indicate the vertical forces between wheel and rail as measured with the instrumented
wheelset under a 15 ton /axle load. The results from Figure 4-21 were measured whilst travelling at
20km/h. Due to the super elevation of the track (the test section was in a curve), there is a difference
between the average left and right wheel loads. The data from the left leg are the outside leg of the
curve, also referred to as the high leg. It is clear from the results that the low leg (right rail) carries the
highest vertical load due to the excessive super elevation of the track which shifts the centre of the
train mass at this low speed to the inside leg. The results from Figure 4-22 indicate the same test
section and measuring wheelset at 60km/h. The results indicate the increase in loads on the high leg
(left rail) and decrease of loads on the low leg (right leg). The average load on the right leg is still,
however, higher than the load on the high leg due to the excessive super elevation of the track and the
relatively low speed of the vehicle. From the data, the difference in the dynamic vertical forces from
the static 75kN wheel load can clearly be seen. Local geometry errors and vehicle dynamics cause the
measured loads to vary about the mean values of 60kN for the left wheel and 90kN for the right wheel.
The shift of the average from 75 to 60 and 90kN in Figure 4-21 can be attributed to the super elevation
of the track, possible skew loading of the vehicle or suspension defects. In Figure 4-22 the value of
90kN decreases to 80kN on the right rail (low leg) due to the increase in speed that shifts some of the
weight to the outside leg (left rail).
Turnouts also present areas of high dynamic impact, as is seen from the data at approximately 240m
and at 800m into the test section. This effect is greatly enhanced by the increase in speed (Compare
Figure 4-21 and Figure 4-22).
One of the shortcomings of the analytical model is the fact that the macrotrack geometry is not
accounted for in the determination of the dynamic factor. These results clearly indicate the effect of
the track geometry on the vertical track forces. It is therefore suggested that the super elevation of the
track should be considered, together with the speed and centre of mass of the vehicle, when
calculating the dynamic factor. Esveld (1, p. 58) suggested the use of an additional factor to include
the effect of excess cant or cant deficiency.
Page 61
0
0 100 200 300 400 500 600 700 800
-20
Vertical Left
-40
Vertical Right
Force (kN)
-60 100 per. Mov. Avg.

(Vertical Left)
(Vertical Right)
-100
-120
-140
Distance [m]
Figure 4-21: Vertical forces measured at 20km/h with instrumented wheelset and 15t axle load
0
0 100 200 300 400 500 600 700 800
-20
Vertical Left
-40
Vertical Right
Force (kN)

(Vertical Left)
(Vertical Right)
-100
-120
-140
Distance [m]
Figure 4-22: Vertical forces measured at 60km/h with instrumented wheelset and 15t axle load
Page 62
The reason for using only one factor to account for both excess cant and cant deficiency is:
x In the first instance, the low leg is loaded with a higher load than the static wheel load, due to
the centre of gravity being closer to the low leg, and the centrifugal forces are being
overcompensated for
x In the second instance, the centrifugal forces are not being compensated for and thus the high
leg will experience a higher load than the static wheel load.
The factor Qc to account for excess cant and cant deficiency may be calculated by
§ s v2 ·
p c ¨¨ h' ¸¸
Qc 1 © gR ¹
s 2 ( 4.1 )
where pc is the vehicle centre of gravity distance (vertical) from the track, s is the track gauge (width),
v is the speed, g is gravitational acceleration, R is the curve radius and h’ is the track cant. According
to Esveld, Qc are normally in the order of 1.1 to 1.25 which correlates well with the empirical data that
showed an increase of 1.18 on the low leg at 20km/h.
The results of the lateral forces on these tests are presented in Section 4.1.2.1 and are shown in Figure
4-28 to Figure 4-31.
4.1.1.3. Calculating the dynamic factor
When executing a track stress analysis, one normally has one of two objectives in mind: to design a
new track that does not currently exist or to evaluate an existing track (e.g. for increasing wheel load
or speed capacity).
These two mainstream design objectives would result in different methods of obtaining the necessary
information to calculate the track stresses. When designing a new track, typical track parameters such
as sleeper spacing and rail size would be design parameter variables and dynamic factors would be
obtained from formulas such as the empirically determined Eisenmann formula. The other design
objective, to evaluate an existing track, would provide the opportunity to measure the track quality as
well as to determine the dynamic factors for the relevant conditions and rolling stock.
With the availability of the instrumented wheelset, it is possible to empirically determine a dynamic
factor from the vertical loads. A methodology is henceforth provided on how to use the instrumented
wheelset to determine a vertical dynamic factor.
Page 63
In order to determine the dynamic factor from the instrumented wheelset results, there are two
possible approaches to follow:
x The first approach ignores the existence of a moving average (i.e. there is no distinction made
between straight and curved track) and calculates the dynamic factor from the 99.85th
percentile (3 times the standard deviation) of the whole track, including irregularities such as
turnouts. This method is preferred when the railway line is designed as a whole, without
distinguishing between straight and curved track (and even turnouts).
x The second approach is based on the procedure specified in UIC Code 518 (30, p. 28) for
testing and accepting railway vehicles based on their dynamic behaviour. The dynamic factor
is calculated for short sections of the whole track individually. This method is preferred when
evaluating shorter sections of track, where the macrogeometry is taken into account by other
means like additional force factors etc.
When determining a dynamic factor that includes macrogeometric deviations (first approach), the
dynamic factor should be calculated as follows:
1. A piece of representative track should be measured with the instrumented wheelset at different
speeds (up to and including the maximum line speed). It is, however, seldom possible to find a
short section of track which is sufficiently representative of the track as a whole and it is thus
preferred to measure the whole track under consideration.
2. Calculate the dynamic factor with either of the following equations:
3 Standard Deviation ( Vertical Forces ) mean( Vertical Forces )
M ( 4.2 )
mean ( Vertical Forces )
or
99.85 th Percentile ( Vertical Forces )
M ( 4.3 )
With currently available calculation tools, both formulas could easily be utilised and the results
evaluated to determine which value would be more appropriate to use. Using 3 times the Standard
Deviation of the vertical forces could present a less realistic result, when a short piece of track is
measured, if the maximum values are a result of localised geometry defects (which would result in a
high deviation from the mean). If a sufficiently long piece of track is measured and a Gaussian
distribution is present, the results of equations 4.2 and 4.3 should be the same.
When determining a dynamic factor that does not include the effect of macrogeometric deviations
(second approach), the dynamic factor should be calculated as follows:
1. The test section should be divided into three zones (30, p. 28), namely:
x tangent track and large radius curves (R>1200m),
x large radius curves (600m<R<1200m),
x small radius curves (R<400m).
Page 64
The procedure in (30, p. 30) specifies that each zone should have a total of at least 10km, since
it is seldom possible to find a short section of track which is sufficiently representative of the
track as a whole.
2. Each of these zones should then be divided into sections with a maximum length as indicated
in Table 4-4 (for this purpose, the transient curves may be included in the curve sections).
Table 4-4: Test section lengths for statistical analysis
Test Zones Section Lengths

Tangent track and large radius curves (R>1200m) 250m
Large radius curves (600<R<1200m) 100m
Small radius (400<R<600m) 100m

curves (R<400m) 70m
3. For each of these sections, the following should be calculated: Mean value, Standard
Deviation and 99.85th percentile of the measured vertical loads.
4. A dynamic factor is then calculated for each section with the use of equations 4.2 and 4.3 and
then calculated for the line as a whole by taking the average of all the values calculated (for
either the standard deviation method or the percentile method).
4.1.1.4. Worked out example on Sweden test data (dynamic factor)
As an example, test results from Sweden, measured by Spoornet Engineering (29), were analysed
with this method to determine the dynamic factor of this line. The dynamic factors from the Iron Ore
Export line data as well as for the Leandra data were also calculated and are presented here. The
available data from the Sweden line were for a 5km section of track, consisting of straight and curved
track as indicated in Figure 4-23. The curvature can be related to curve radius R by the following
relationship**:
10000
R ( 4.4 )
0.875 Curvature
where curve radius R is in metres and Curvature is measured by the IM2000 track geometry car as the
lateral deviation of the track to the vehicle wheelbase in millimetres.
Figure 4-24 indicates the vertical wheel rail forces measured at 40km/h. An indication is also given of
the 99.85th percentile and 3*Standard Deviation values for each section, as well as the average and the
**
This formula is specific to the IM2000 track geometry vehicle from Plasserrail which measures curvature as a function
of the IM2000’s wheelbase. Modifications to the vehicle wheelbase or measuring system will result in a different formula.
Page 65
99.85th percentile and 3*Standard Deviation values for the measured track as whole. From the results
it is clear the 3*Standard Deviation value is somewhat higher than the 99.85th percentile for both
approaches. The 99.85th percentile and 3*Standard Deviation for the whole line are much larger than
the averages of the sections' values. This clearly indicates how the one approach can be used to
include the effect of the moving average or vice versa.
25
R=583
20 Curve
R=780 Radius [m]
15
R=1505 R=1515
Curvature [mm] (see eq. 4.4)
10
0
0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 3500.00 4000.00 4500.00 5000.00
-5
R=2833
-10
-15
R=762
-20
R=680
-25
Distance [m]
Figure 4-23: Sweden test track geometry (curvature)
Table 4-5 and Figure 4-25 indicate the calculated dynamic factors for the Sweden, Leandra and Iron
Ore Export line data as calculated by the two different approaches. From the results in Table 4-5 it is
clear that the standard deviation method versus the percentile method provided similar answers on the
long tracks (Sweden and Iron Ore Export line) but not on the short track (Leandra data) due to local
force peaks in the Leandra data that influenced the results.
The dynamic factors, as calculated for the tests conducted at higher speeds, sometimes show lower
values than the trend set by the lower speeds. It is therefore not safe to extrapolate dynamic factor
values at speeds for which values were not measured from these results since the test vehicle's body
motions could influence the data at particular speeds.
Page 66
180
Vertical force
170 Sections: 99.85th percentile
Sections: 3 std dev + mean

160
Whole track: 99.85th
Vertical force (kN)
percentile
Whole track: 3 std dev
150
Avg. sections: 99.85th
percentile
Avg. sections: 3 std dev +
140
mean
Sections
130 Sections Section Mean
120
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 4000.0 4500.0 5000.0
Distance (m)
Figure 4-24: Sweden wheel-rail vertical forces @ 40km/h.
1.6
1.5
Sweden: Whole line : f(Std
Dev)
Sweden: Whole line : f(99.85th
P)
1.4 Sweden: Sections : f(Std Dev)
Dynamic Factor
Sweden: Sections : f(99.85th

P)
1.3
Leandra: Whole line : f(Std
Dev)
Leandra: Whole line : f(99.85th
P)
1.2
Leandra: Sections : f(Std Dev)
Leandra: Sections : f(99.85th

1.1 P)
Sweden: Optimised track and

suspension
1
15 25 35 45 55 65 75
Test Speed (km/h)
Figure 4-25: Dynamic factors from tests at Leandra and Sweden with different methods
Page 67
Table 4-5: Dynamic factors as calculated with different methods
1st Approach (whole line) 2nd Approach (average of sections)

Tests
M=f(3*Std Dev) M=f(99.85th M=f(3*Std Dev) M=f(99.85th
Percentile) Percentile)
40km/h 1.167 1.150 1.053 1.074

50km/h 1.118 1.111 1.080 1.067
Sweden
60km/h 1.103 1.106 1.093 1.073
70km/h 1.132 1.142 1.117 1.092
20km/h 1.377 1.365 1.200 1.126
30km/h 1.362 1.393 1.246 1.147
Leandra
40km/h 1.378 1.413 1.277 1.158
60km/h 1.450 1.547 1.324 1.181
30km/h 1.167 1.146 1.163 1.132
Iron Ore
Export line 40km/h 1.188 1.161 1.186 1.152
15t
50km/h 1.196 1.174 1.192 1.160
30km/h 1.131 1.115 1.123 1.104
40km/h 1.155 1.137 1.149 1.125
Iron Ore
Export line 50km/h 1.179 1.157 1.170 1.143
26t
60km/h 1.183 1.162 1.177 1.151
70km/h 1.196 1.178 1.192 1.163
30km/h 1.166 1.149 1.161 1.132
40km/h 1.187 1.164 1.181 1.148
Iron Ore
Export line 50km/h 1.195 1.171 1.188 1.153
30t
60km/h 1.223 1.195 1.216 1.174
70km/h 1.252 1.223 1.246 1.196
4.1.1.5. Dynamic factor overview
A dynamic factor cannot always be empirically determined, for example, when a new track and or
rolling stock is designed and simulation resources are limited. It is therefore worth evaluating the
range of possible values in order to be able to make an educated guess or use a suitable formula.
For this purpose, all the calculated dynamic factors, be it from empirical formulas or measured
wheel–rail forces, are indicated in Figure 4-26. Some of the dynamic factors were grouped together
and only the envelopes were presented (e.g. the Eisenmann formulas which are dependent on a
number of factors beside speed). The ungrouped dynamic factors are presented in Appendix D.
Page 68
From the processed data, although limited in scope, it is evident that the Eisenmann formula might be
too conservative for local conditions and should not be used without the insight gained from this
study.
2.4
2.2
AREA
ORE
2 Dynamic: Loco
Dymnamic Factor
Dynamic: Wagon
Schram
1.8 Klöckner Light
Klöckner Heavy
Clintock
1.6
Eisenmann
Sweden
1.4 Leandra
Orex
Saaiwater
1.2
1
10 30 50 70 90 110 130
Speed (km/h)
Figure 4-26: Summary of all dynamic factors from this study
It is important to realise that the empirically determined dynamic factors (Saaiwater, Leandra, Iron
Ore Export line and Sweden), as displayed in Figure 4-26, are relevant only to the conditions under
which they were determined.
The dynamic factors from the wayside measurements (Saaiwater) represent the entire COALlink fleet
but the condition of the track at the measured site is of a very high standard, which might not be
representative of the whole track. The Leandra, Iron Ore Export line and Sweden data, on the other
hand, represent different conditions of track quality but again may only be characteristic of the type of
vehicle utilised for the measurements.
The next section will investigate lateral wheel-rail forces.
Page 69
4.1.2. Lateral Force
The lateral load parameter from the analytical method is the parameter to which the rail stresses are
the second most sensitive, as indicated in Table 3-5. Ironically, the lateral load is also the parameter to
which the least attention is given in the analytical method and was taken only as a fixed percentage of
the vertical force.
Figure 4-27 indicates an attempt to relate lateral track geometry to lateral loads and shows that no
definite relationship exists for this dataset measured in Sweden (29) as was the case with the vertical
loads – see paragraph 4.1.1.
The lateral force is not only a function of the vertical load but also of a number of parameters such as
the curve radius, the speed of the vehicle, the steering ability of the vehicle and the rigidity of the
fastening system (31). Lombard (2, p. 57) specified the use of a lateral load of 40kN, with no
indication of whether or not this may be used with any vertical load. In the program "Moment", the
lateral force was calculated as 40% of the vertical force. It is unknown whether this was done
intentionally or in error, since Lombard used a 100kN vertical force in his worked-out example, which
could possibly have led to this relationship, conveniently taken to be linear for higher vertical loads.
In this study, the lateral forces will be determined statistically from field measurements and compared
to the simultaneous vertical forces at various speeds.
Page 70
10
9
(90th Percentiles of 100m sections) [mm]
7
Lateral Allignment
5 40 km/h
50km/h
4 60km/h
70km/h
3
0
4 5 6 7 8 9 10 11 12 13
Lateral Dynamic Load (90th Percentile of 100m sections) [kN]
Figure 4-27: Track geometry lateral alignment vs. dynamic loads
4.1.2.1. Empirical test results
Lateral wheel loads will be investigated empirically in order to determine typical values under local
conditions.
Lateral force measurements were obtained from the instrumented wheelset from three different test
sites, namely, Leandra (General Freight Line), Loop 12 (Iron Ore Line) and from measurements
performed in Sweden. The instrumented wheelset data from Loop 12 were available on three different
vehicles, each with a different axle load.
(i) Leandra Test site
For the tests at Leandra, eight runs of data were available. Figure 4-18 indicates the track layout at the
test site and Figure 4-19 indicates the measured track alignment in terms of the curve radius for the
800m length of track. Wheel load measurements were taken at 20, 30, 40 and 60 km/h in one direction
Page 71
only. Two tests were conducted at each speed. Figure 4-28 and Figure 4-29 indicate the lateral forces
on the instrumented wheelset at the Leandra test site at different speeds. Positive lateral loads are
from the inside of the rail towards the outside. When comparing the test results, the change in lateral
force can mostly be attributed to the change in speed, although the lateral position of the wheelset
changes and plays a part in the magnitude of the lateral forces (see paragraph 4.1.3 on page 85). The
negative lateral force peaks in the presented graphs indicate impulses due to the discontinuity of the
rails at turnouts, which introduces large forces onto the wheelsets.
In order to study the effect of the vehicle speed on the magnitude of the lateral forces, statistical
indexes have been determined for each test.
30
20
10 20 km/h a
20 km/h b
30 km/h a
Force (kN)
0 30 km/h b
0 100 200 300 400 500 600 700 800 900 40 km/h a
40 km/h b
-10 60 km/h a
60 km/h b
-20
-30
First Turnout Second Turnout

-40
Distance [m]
Figure 4-28: Lateral left wheelset force at different speeds [km/h]
Page 72
40
35
30
25
20 km/h a
Force (kN)
20 20 km/h b
30 km/h a
15 30 km/h b
40 km/h a
40 km/h b
10
60 km/h a
60 km/h b
5
0
0 100 200 300 400 500 600 700 800
-5
Distance [m]
Figure 4-29: Lateral right wheelset force at different speeds [km/h]
30
25
20
20 km/h a
Force (kN)
20 km/h b
30 km/h a
15
30 km/h b
40 km/h a
40 km/h b
10 60 km/h a
60 km/h b
0
0 10 20 30 40 50 60 70 80 90 100
Distance [m]
Figure 4-30: Lateral left wheelset force of 1st 100m at different speeds [km/h]
Figure 4-30 and Figure 4-31 present the data from Figure 4-28 and Figure 4-29 for the first 100m. At
this scale, patterns in the results are easier to identify. At first glance there seems to be some
correspondence between peaks and valleys of the different runs.
Page 73
26
24
22
20 km/h a
Force (kN)
20 km/h b
30 km/h a
20
30 km/h b
40 km/h a
40 km/h b
18 60 km/h a
60 km/h b
16
14
0 10 20 30 40 50 60 70 80 90 100
Distance [m]
Figure 4-31: Lateral right wheelset force of 1st 100m at different speeds [km/h]
At closer inspection, there is a phase shift noticeable in the lateral force patterns of the different
speeds. It seems that the lateral dynamics of the vehicle systems are a contributing factor in this case.
In order to investigate other trends in the data that could be related to the speed of the vehicle, the data
were analysed statistically. Table 4-6 indicates the statistical results of the lateral left forces and Table
4-7 indicates the results of the lateral right forces.
Table 4-6: Statistical results from lateral left forces [kN] of Leandra test data
20km/h 20 km/h 30 km/h 30 km/h 40 km/h 40 km/h 60 km/h 60 km/h

Mean 13.61 14.65 14.91 15.71 15.80 15.73 13.19 12.80
Median (50th percentile) 13.96 15.04 15.23 16.02 15.92 15.72 13.38 13.18
Range 40.14 42.68 56.93 57.23 56.64 62.99 60.25 74.22
Minimum -19.43 -20.51 -32.71 -31.15 -27.15 -31.25 -29.98 -43.55
Maximum 20.70 22.17 24.22 26.07 29.49 31.74 30.27 30.66
Standard deviation 2.59 2.93 3.68 4.18 4.67 5.00 5.12 5.10
Sample variance 6.73 8.61 13.55 17.50 21.81 25.05 26.17 25.98
99.85th percentile 18.76 20.80 23.14 25.25 27.39 29.82 28.84 29.15
3*standard deviation + mean 26.30 23.46 25.95 28.25 29.81 30.74 28.54 28.09
Page 74
Table 4-7: Statistical results from lateral right forces [kN] of Leandra test data
20km/h 20 km/h 30 km/h 30 km/h 40 km/h 40 km/h 60 km/h 60 km/h

Mean 19.19 19.37 19.28 19.16 19.04 18.89 18.45 18.01
Median (50th percentile) 19.24 19.53 19.43 19.34 19.24 19.04 18.65 18.36
Range 48.73 49.61 47.46 46.88 44.63 46.09 41.41 41.02
Minimum -9.96 -10.64 -9.47 -9.08 -7.52 -9.47 -6.45 -5.96
Maximum 38.77 38.96 37.99 37.79 37.11 36.62 34.96 35.06
Standard deviation 3.07 3.17 3.10 3.17 3.15 3.21 3.21 3.61
Sample variance 9.44 10.06 9.62 10.07 9.94 10.31 10.30 13.00
99.85th percentile 35.78 34.65 34.02 34.20 33.37 33.50 31.89 30.21
3*standard deviation + mean 32.91 28.88 28.58 28.68 28.50 28.53 28.08 28.83
The same data are also indicated in Figure 4-32 and Figure 4-33, which indicate the difference
between the patterns of the left and right wheel forces.
40.00
35.00
30.00
Lateral load [kN]
Mean
Maximum
25.00
99.85th Percentile
3*Std Dev + mean
20.00
15.00
10.00
15 20 25 30 35 40 45 50 55 60 65
Speed [km/h]
Figure 4-32: Statistical summary of lateral left forces at Leandra test site
Figure 4-32 clearly indicates an increase in lateral force with an increase in speed while Figure 4-33
indicates a reduction in forces with an increase in speed. As was the case with the vertical force
(paragraph 4.1.1.2(iii)), the excess super elevation of the track caused an excess weight shift to the
right leg (low leg) which increased the lateral towards the outside of the rail. As the speed increased,
the centrifugal forces of the vehicle increased the lateral forces to the outside of the left rail (high leg)
while reducing the right lateral forces. At the maximum speed of 60km/h, however, the results from
the left and right leg converged and did not follow the trend set by the results from the lower speeds.
Page 75
40.00
35.00
30.00
Lateral load [kN]
Mean
Maximum
25.00
99.85th Percentile
3*Std Dev + mean
20.00
15.00
10.00
15 20 25 30 35 40 45 50 55 60 65
Speed [km/h]
Figure 4-33: Statistical summary of lateral right forces at Leandra test site
Force distribution graphs have been drawn up for each test and are indicated in Appendix B to
indicate the spread of lateral forces.
In order to establish whether or not a relationship exists between the lateral and vertical forces, the
data from the different runs are presented in Figure 4-34. For each of the scatters, a linear fit was
performed to quantify a possible relationship between the vertical and lateral forces.
It is evident that the data points about the linear fits are widely scattered and therefore the statistical
correlation between the vertical and lateral forces had to be determined to provide an indication of
confidence in a linear relationship. The correlation values are presented in Table 4-8.
From the correlation data in Table 4-8 it is clear that there is a 60% probability of a linear relationship
between the vertical and lateral forces on the left but only about an 8% probability of a linear fit on the
right. The negative correlation of the left data indicates a decreasing lateral load for an increasing
vertical load.
From the results in Table 4-8 it is evident that no definite relationship exists between the vertical and
lateral wheel-rail forces. It may be concluded that the suggestion by Lombard (2, p. 57) to use a
constant lateral force of 40kN is a better option than trying to predict the lateral force as a function of
the vertical force or the speed of the vehicle.
Page 76
140
120
20km/h Left
20km/h Right
30km/h Left
30km/h Right
Vertical Forces (kN)
100 40km/h Left

40km/h Right
60km/h Left
60km/h Right
80
Linear (20km/h Left)
Linear (20km/h Right)
60 Linear (30km/h Right)
40
20
-40 -30 -20 -10 0 10 20 30 40
Lateral Forces (kN)
Figure 4-34: Leandra test data: Vertical vs. Lateral Forces
Table 4-8: Correlation between vertical and lateral force data
Correlation
Speed [km/h]
Left Right
20 -0.678 0.080
30 -0.665 0.087
40 -0.626 0.097
60 -0.654 0.061
Lombard’s proposal of 40kN is too high in this case for the left leg forces but is acceptable for the
right leg when including the maximum lateral force at 40km/h. As will become clearer from
following results, this measured piece of track is too short to make any definite decisions since the
measured portion does not represent the whole line.
According to the limit prescribed by the International Union of Railways (UIC) (30, p. 23) when
evaluating railway vehicles for acceptance on track, the sum of the guiding (lateral) forces over 2m of
track should not exceed the following:
¦Y
D 10 P 3 ( 4.5 )
Page 77
where Y is the lateral forces and P is the vertical axle load, both in kN and D = 1 for traction units and
D = 0.85 for wagons. This results in a total lateral force limit of 57kN for a locomotive and 48kN for a
wagon under a 140kN/axle load. It is, however, also mentioned that these values may be exceeded by
agreement between the infrastructure manager and railway operator. It is further stated under the
heading of track fatigue that the quasi-static lateral force limit for small radius curves (Radius <
600m) is 60kN.
It should be made clear that these limits are not per rail, but the total lateral forces (both left and right
rails) over a distance of 2m, which is in contrast to the limit set by Lombard (2, p. 56) where each rail
is evaluated separately.
The reason why the UIC states a lateral load limit per 2m track and not per rail is that its values are
primarily based on empirically determined values, governed by track maintenance (alignment)
strategies, and not individual rail stresses. When these limits are thus exceeded, track maintenance
should be adjusted accordingly.
It can therefore be concluded that in the case of the limits set by the UIC, the maintenance of the track
alignment is determined by the force limits and not the rail stresses.
(ii) Iron Ore Export line Test site
The instrumented wheelset tests conducted on the Sishen – Saldanha line (iron ore export line)
consisted of three instrumented wheelsets on three different wagons. Figure 4-35 indicates the test
train setup with the three instrumented wheelsets under 15, 26 and 30ton axle loadings. The tests were
conducted on a 3km section of the line at speeds of 30, 40, 50, 60 and 70km/h and the vertical and
lateral forces for each wheelset were investigated. Figure 4-38 and Figure 4-39 indicate the wheel rail
forces for the tests at 30 and 70km/h. The force results from each test were analysed statistically and
the results are tabulated in Table 4-9 and Table 4-10.
The data from Table 4-9 and Table 4-10 are also indicated in Figure 4-36 and Figure 4-37, which
indicate the difference between the patterns of the left and right wheel forces. The vertical scale of
Figure 4-36 was purposefully set to match the vertical scale of Figure 4-37 in order to highlight the
difference between the left and right lateral forces.
From the results there is a clear discrepancy between the lateral loads of the 26ton/axle results and the
rest of the results. In order to investigate this phenomenon, the lateral loads are displayed graphically
in Figure 4-38 (30km/h tests) and Figure 4-39 (60km/h tests).
Page 78
26t/axle wheelset
15t/axle wheelset
30t/axle wheelset
Locomotive Test coach
Figure 4-35: Test train setup with three instrumented wheelsets at Iron Ore Export line tests
Table 4-9: Statistical results from lateral left forces [kN] of Iron Ore Export line test data
15t/axle 26t/axle 30t/axle

Speed [km/h] 30 50 60 70 30 40 50 60 70 30 40 50 60 70
Mean -0.77 -0.52 -0.56 -0.58 -18.24 -17.83 -19.70 -19.36 -19.33 0.21 0.34 0.62 0.45 0.58
Median -0.78 -0.58 -0.58 -0.58 -17.96 -17.48 -19.53 -19.14 -18.94 0.00 0.19 0.58 0.39 0.58
Range 5.96 6.24 6.43 6.54 22.41 25.63 26.84 26.01 25.74 41.56 44.46 49.02 34.97 20.79
Minimum -3.81 -3.61 -3.80 -3.71 -32.19 -33.73 -36.01 -34.79 -34.72 -30.69 -33.55 -37.99 -24.52 -10.05
Maximum 2.15 2.63 2.63 2.83 -9.79 -8.10 -9.17 -8.77 -8.98 10.87 10.91 11.03 10.45 10.74
Sample Variance 0.81 1.11 1.23 1.29 13.15 16.17 18.81 18.48 21.70 15.88 17.50 20.14 18.31 17.95
Standard Deviation 0.90 1.05 1.11 1.14 3.63 4.02 4.34 4.30 4.66 3.98 4.18 4.49 4.28 4.24
th
99.85 Percentile 1.76 2.34 2.34 2.44 -10.54 -9.08 -9.86 -9.66 -9.57 9.57 10.25 10.35 9.96 10.24
3*Std Dev +mean 1.94 2.64 2.76 2.83 -7.36 -5.77 -6.69 -6.46 -5.35 12.16 12.89 14.08 13.29 13.29
Table 4-10: Statistical results from lateral right forces [kN] of Iron Ore Export line test data
15t/axle 26t/axle 30t/axle

Speed [km/h] 30 50 60 70 30 40 50 60 70 30 40 50 60 70
Mean -0.34 -0.16 -0.18 -0.47 -8.81 -10.19 -6.38 -8.05 -8.86 5.72 5.96 6.80 5.69 5.73
Median -0.39 -0.19 -0.19 -0.48 -10.64 -12.10 -7.61 -10.15 -10.83 5.07 5.27 6.05 5.27 5.46
Range 6.41 6.62 6.83 7.40 31.14 31.78 39.06 37.47 43.07 24.70 31.51 37.06 25.68 23.13
Minimum -3.68 -3.51 -3.61 -4.10 -18.65 -18.56 -18.35 -18.45 -18.84 -5.35 -6.33 -5.10 -5.57 -4.75
Maximum 2.73 3.11 3.22 3.30 12.48 13.23 20.71 19.02 24.23 19.36 25.18 31.96 20.12 18.38
Sample Variance 0.93 1.28 1.50 1.53 38.48 32.01 57.12 51.94 43.84 18.71 21.83 30.58 21.01 20.35
Standard Deviation 0.97 1.13 1.22 1.24 6.20 5.66 7.56 7.21 6.62 4.33 4.67 5.53 4.58 4.51
th
99.85 Percentile 2.34 2.73 2.83 2.83 9.37 8.78 13.08 13.08 13.96 17.67 19.23 27.73 18.85 17.57
3*Std Dev +mean 2.56 3.23 3.50 3.25 9.80 6.78 16.29 13.57 11.00 18.70 19.97 23.39 19.44 19.26
Page 79
40
30
20
Lateral load [kN]
10 Mean
Maximum
99.85th Percentile
0
3*Std Dev + mean
30 50 60 70 30 40 50 60 70 30 40 50 60 70
-10
-20
15 ton/axle 26 ton/axle 30 ton/axle
-30
Speed [km/h]
Figure 4-36: Statistical summary of lateral left forces at different speeds and axle loads
40
30
20
Lateral load [kN]
10 Mean
Maximum
99.85th Percentile
0
3*Std Dev + mean
30 50 60 70 30 40 50 60 70 30 40 50 60 70
-10
-20
15 ton/axle 26 ton/axle 30 ton/axle
-30
Speed [km/h]
Figure 4-37: Statistical summary of lateral right forces at different speeds and axle loads
Page 80
Figure 4-38: 30km/h test results of vertical and lateral loads (Iron Ore Export line tests)
Figure 4-39: 60km/h test results of vertical and lateral loads (Iron Ore Export line tests)
Page 81
From the above lateral load results, it is evident that the 26on/axle wagon results do not follow the
trend set by the 15ton/axle and 30ton/axle results. It is also evident from Figure 4-38 and Figure 4-39
that the axle loads have greater effects on the lateral loads than a change in speed. In comparison with
the Leandra lateral load results (paragraph 4.1.2.1 (i)), which were measured in an 800m curve, the
26ton/axle results are of the same order of magnitude, although being measured on straight track and
large radius curves (R>1200m).
It was concluded that this particular wheelset or wagon bogie was faulty and can be described as a
"skew" bogie which means that the wagon steers as if it is in a curve, even though it is travelling on a
straight track. As described earlier in the document, this could be due to a wheel diameter difference
between the left and right wheels of either axle.
Figure 4-40 and Figure 4-41 indicate the vertical vs. lateral forces for the 15ton/axle and 30ton/axle
loads respectively as measured on the Iron Ore Export line line. Linear curve fits through the data
scatter are also indicated. The correlations for each of the data sets have been calculated and are
presented in Table 4-11.
Figure 4-40: Iron Ore Export line vertical rail-wheel forces vs. lateral rail-wheel forces for 15t/axle tests
Page 82
Figure 4-41: Iron Ore Export line vertical rail-wheel forces vs. lateral rail-wheel forces for 30t/axle tests
Table 4-11: Correlation between vertical and lateral force data

30km/h 40 km/h 50 km/h 60 km/h 70 km/h
Left Right Left Right Left Right Left Right Left Right
15t
-0.101 0.156 -0.209 0.299 -0.233 0.344 -0.189
30t 0.420 -0.475 0.466 -0.496 0.466 -0.479 0.497 -0.538 0.459 -0.461
According to the correlation results there is little confidence in the apparent linear relationship
between the vertical and lateral wheel-rail forces. The right leg forces at 60km/h of the 30ton/axle
loads have the highest correlation of 54%.
It therefore seems to be the best approach to account for the lateral forces in rail stress calculations by
choosing a constant value. Based on the results, the suggested value of 40kN seems to be too high.
This may be due to the fact that the piece of measured track contained only straight and large radius
curves. If this piece of track were representable of the track as a whole, a lateral load of 30kN could be
used in rail stress calculations. In light of the UIC limits as presented in paragraph 4.1.2.1(i), the total
lateral forces in small curves (Radius < R600) may be accepted as high 60kN. This is, however, the
total lateral force and not the force on each rail. In the above test results, lateral forces with the same
sign, on each rail, are actually in opposing directions on the track (refer Figure 4-44).
Page 83
(iii) Sweden test results
Similar tests to those conducted at Leandra and on the Iron Ore Export line were conducted by
Spoornet in Sweden. The Sweden lateral force results were also evaluated statistically and are
presented in Table 4-12 and in Figure 4-42.
Table 4-12: Statistical results from lateral left forces of Sweden test data
14.6t
Axle Load
Speed [km/h] 40 50 60 70
Mean 15.72093 15.47391 14.39602 14.50645
Median 16.30859 15.91797 14.55078 14.45313
Range 237.9883 236.3281 231.0547 236.8164
Minimum -0.87891 0.390625 0.585938 2.832031
Maximum 37.98828 36.32813 31.05469 36.81641
Sample Variance 34.1233 29.96871 29.09316 25.71806
Standard Deviation 21.56245 20.94828 19.78983 19.57774
99.85th Percentile 33.44727 32.24609 28.90625 29.00391
3*Std Dev +mean 33.24548 31.89702 30.57745 29.72034
40
35
30
Lateral load [kN]
Mean
Maximum
25
99.85th Percentile
3*Std Dev +mean
20
15
10
35 40 45 50 55 60 65 70 75
Speed [km/h]
Figure 4-42: Statistical summary of lateral left forces at different speeds
When referring back to the Sweden curve geometry (see Figure 4-23 on page 66) which included a
number of representative curves, the lateral load results indicate all values to be below 40kN. For this
section of line, 40kN would therefore be an acceptable value to use for the lateral forces in rail stress
calculations.
Page 84
4.1.2.2. Lateral force analysis
With detailed vehicle and track models, typical lateral forces can also be determined with software
packages such as Medyna and MSC.ADAMS. An example of such an analysis will be presented in
section 4.1.3.1 under the heading of “Eccentricity analysis”. A first order analysis was executed with
Medyna. A vehicle was modelled travelling at a speed of 60km/h with a new wheel profile through a
500m radius curve to the right with a super elevation of 45mm on a new rail profile. The results
presented are the eccentricity of the vertical wheel forces as well as the lateral track forces.
4.1.3. Eccentricity of vertical wheel load
The eccentricity of the vertical wheel force on the rail is the third factor that the analytical method of
determining rail stresses is the most sensitive to and for which no guideline is currently available
regarding minimum and maximum values. In the sensitivity study, the eccentricity was varied
between –22mm and +22mm, in conjunction with a lateral force. From Figure 3-2 it can be seen that
the position of the vertical force could either contribute to the moment about the rail shear centre due
to the lateral force, or oppose it. In the current analytical model, a positive eccentricity opposes the
lateral force and vice versa in determining the torsional moment.
Determining nominal values for eccentricity is not an easy task since wheel rail interaction is an
engineering science in its own right and a focused study in the field of vehicle design, especially
bogie design. Furthermore, it is not only a function of the bogie design and the wheel and rail profile
design, but also of the maintenance plans of both the rolling stock and infrastructure departments. The
vertical wheel force is a function of especially the wheel and rail profile designs and each railway
could have its own design and maintenance philosophy. Maintenance plans are important since they
affect the amount of wheel and rail wear that will be allowed before corrective action is taken.
Maintaining the rail profile means the deployment of on-track grinding machines that profile the
railhead running surface to a predetermined profile on a predetermined cycle. Maintaining wheel
profiles means the removal of vehicles from service, removal of complete wheel-and-axle sets and
machining them back to a predetermined profile on a predetermined cycle.
Typical worn wheel profiles become "hollow" and two point contact may occur between wheel and
rail. This means that the wheel no longer has only one point of contact with the rail and determining
the resultant point of contact of the vertical wheel rail force becomes quite complex. A worn rail
profile also changes the behaviour of wheel rail interface, complicating matters further. A method
that is used more often these days is to use computer software that is explicitly designed for this
Page 85
purpose, to determine not only the wheel rail contact points in a centre static position, but also during
the dynamic negotiating of a track.
Medyna and ADAMS/Rail are two such packages that can be used to investigate the effect of rail and
wheel profiles on the lateral and vertical forces, as well as vertical load eccentricity. The following
section will show how typical eccentricity values were determined from an analysis with Medyna.
4.1.3.1. Eccentricity analysis
In order to be able to supply eccentricity values for a user of the analytical rail stress model, a first
order analysis was executed with Medyna. A vehicle was modelled travelling at a speed of 60km/h
with a new wheel profile through a 500m radius curve to the right with a super elevation of 45mm on
a new rail profile. The software analyses the behaviour of the vehicle due to the rail wheel interaction
and reports the wheel forces as discrete points with time. These results are presented in Figure 4-45
and the sign conventions are as indicated in Figure 4-44. Positive eccentricity values indicate vertical
forces on the outside of the rails and positive lateral forces are also towards the outside of the rails.
This means that a positive eccentricity value of the vertical load would contribute to a positive lateral
load in increasing the moment about the rail centre, which is contrary to the current analytical
model’s sign definition where a positive eccentricity value of the vertical load opposes a positive
lateral load.
It would be recommended that a more detailed study into representative wheel and rail profiles should
be performed and that dynamic wheel rail interaction studies should then be performed on such
profiles or a variety of profiles to determine nominal eccentricity values. This should then be
performed on a variety of track conditions and curve radii to ensure realistic results.
Another reason why extensive research should be performed on this subject is that it is not necessarily
true that the extreme eccentricity values would occur simultaneously with the maximum lateral loads.
This could be assumed by an inexperienced user of the current analytical model, since the user is
simply prompted for these two values and one may intuitively assume that two maximum values
might represent the worst-case scenario, which is not the case.
Page 86
Travelling
Direction
R500m
High leg
Low leg
(left rail)
(Right rail)
Figure 4-43: Track layout for Medyna analysis
It must be emphasised that this is an analysis on only one specific case. It is not the intention to study
all possible cases but rather study trends of eccentricity vs. lateral force magnitude for this nominal
case study.
This first order analysis of a new wheel on a new rail profile is not only a good starting point but also
represents conditions on a railway line that has a good maintenance schedule with regular re-profiling
of wheels and rails back to their design profiles.
The results from Figure 4-45 indicate the complicated pattern of possible combinations of the
position of the vertical load and the magnitude of the lateral load (even for a new rail and wheel
profile!). It would therefore be incorrect to specify a range of eccentricity values without including a
relationship to the magnitude of the lateral load. The high eccentricity values are only occurring with
opposing lateral loads while the lower eccentricity values may occur with contributing high lateral
loads. Depending on the magnitude of the vertical load, either combination could result in the highest
rail stresses.
From the above results it became clear that for the high leg rail a positive eccentricity of 9mm could
occur with a lateral load of 1.65kN in an opposing direction (point A in Figure 4-45) or a 13kN load in
a contributing direction (B). A maximum eccentricity of 33mm can occur together with a 31kN
opposing lateral load (D) or at 31mm with a 58kN opposing lateral load (C). It is also important to note
that an eccentricity of 32mm could occur with a very low opposing lateral load of 3.7kN (E), which
could also result in the highest rail stresses under a high vertical load.
Page 87
Q Q
e e
+ L L +
Field side Gauge Side Field side
Right Rail
Left Rail
Figure 4-44: Eccentricity and lateral force sign convention of Medyna results
Vertical wheel force eccentricity vs. lateral force magnitudes

A
F 10 B
G
5
Contributing quadrant
0
-10 0 10 20 30 40 50 60
H
-5
Eccentricity (mm)
-10
Left rail
Right rail
-15
Opposing quadrant
-20
-25
-30
C
-35 E D
Lateral force (kN)
Figure 4-45: Vertical force eccentricity vs. lateral load
Page 88
For the low leg rail, a maximum eccentricity of 9.2mm occurred together with an opposing lateral
load of 7.5kN (F) and, at an eccentricity of 6.8mm, the highest contributing lateral load of 32kN (G)
was reached.
Table 4-13 indicates the above combinations of eccentricity and lateral load combinations that could
result in worst-case scenarios for rail stresses. Each corner of the result envelopes (refer Figure 4-45)
was taken as a possible worst-case scenario and is indicated with the letters A to H. The coordinate
system from the above results was translated to the rail stress model input coordinate system where
the lateral load is always positive with the eccentricity being either positive (opposing the lateral load)
or negative (contributing to the lateral load).
Table 4-13: Possible worst-case scenarios for eccentricity and lateral load pairs (rail stress model
coordinate system: positive eccentricity opposing positive lateral force)
A B C D E
High leg e (mm) +9 -6.5 +31 +33 +32
L (kN) 1.7 14.3 58 31 3.7
F G
Low leg e (mm) 9.2 -6.8
L (kN) 7.5 32
The highest stresses will occur on either of the extreme points in Figure 4-45 (F,G, C, D or E) since
the rail stresses are directly related to the magnitude of the lateral forces and the eccentricity of the
vertical forces. Since it was shown that no relationship exists between the lateral and vertical wheel
loads, it is important to realise that the magnitude of the vertical forces will determine which
combination of lateral loads and eccentricity for a specific scenario would result in the highest rail
stresses. For this reason, all the possible worst-case scenarios should be identified and analysed to
determine the highest rail stresses.
The above results of this specific case study gave rise to the notion that at least four worst-case
scenarios could typically occur namely:
x Maximum lateral force with maximum opposing eccentricity (point C and/or D)
x Zero lateral force with maximum eccentricity (~ point E)
x 5% of the maximum lateral force with 30% of the maximum opposing eccentricity (point F)
x 50 % of the maximum lateral force with 20% of the maximum contributing eccentricity (point
G)
Page 89
4.1.3.2. Summary of the methodology to determine eccentricity vs. lateral load values
Based on these results, it is evident that a rail stress analysis should not only be determined for one
maximum eccentricity along with one maximum lateral load. The maximum lateral load does not
coincide with the maximum contributing eccentricity of the vertical load and a high eccentricity with
a low opposing lateral force could prove to be the worst-case scenario. The reason why the case of a
high eccentricity with a high opposing lateral force is still regarded as a possible worst-case scenario
(Table 4-13), includes the following:
x the vertical force is always significantly larger than the lateral force
x the high eccentricity, together with the high vertical force, could still result in a moment
about the centre of the rail that is higher than in the case where the lateral load contributes to
the moment of the vertical force with a lower eccentricity.
The methodology to follow includes the execution of an eccentricity analysis on either a range of
wheel and rail profiles, or on a statistically representative wheel and rail profile pair. From such an
analysis the envelope of eccentricity values vs. lateral load values should be determined. A rail stress
analysis should then be performed for each possible worst-case scenario as was done in the example
of Table 4-13.
4.1.4. Rail stresses calculation improvement
When comparing the rail stresses from the finite element model to the analytical model (3.1.3.1), it
was found that there was a slight offset in the rail stresses in the bottom corner of the railhead. After
investigation it was found that the definition of the railhead depth (height), as used in the analytical
calculations, was causing this phenomenon.
It is generally acceptable to define the railhead as the shaded area in Figure 4-46 where the bottom
boundary of the railhead is defined by extending the inclined lines at the bottom of the railhead, until
they intersect (Point A in Figure 4-46). The railhead depth was then subsequently defined as the
maximum distance between this intersection and the top of the rail. For a 57kg/m rail, this distance is
47.5mm.
Page 90
Railhead
Area
Figure 4-46: Profile properties of a 57kg/m rail
In Figure 4-47, the stress results from both the FEA and analytical method are compared, with the
railhead depth chosen as 47.5mm. The inconsistency of the rail stresses between the two methods can
clearly be seen.
Page 91
0.135
0.13
Height of Rail (m)
Analytical (railhead 47.5mm)

0.125 FEM 6.265 kN/mm Left
0.12
0.115
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Stress (GPa)
Figure 4-47: Rail stress comparison with railhead depth chosen at 47.5mm (100kN off-centred vertical
load and 50kN lateral load)
The stress results in Figure 4-47 indicate that the maximum local tensile stresses of the analytical
method with the railhead depth taken as 47.5mm are higher than the FEA results (±30MPa) and also
at a lower position (±10mm). The fact that the FEA results indicate the maximum tensile stress to be
at a higher location suggests that the definition of the railhead depth, as used in the analytical method,
is wrong and should be re-defined. The reason for this is that, although this distance represents the
extremities of the railhead, it does not represent the height of a rectangular section of equal area and
second moment of area (as is assumed in the calculations).
It could be noted that the analytical calculation gives a rapid change in stress at the bottom corner of
the railhead due to the simplification of the rail profile as separate rectangular beams supporting one
another as described in 3.1.3. The FEA model on the other hand uses a more accurate profile, thus the
smooth transition in stresses at the bottom of the railhead.
Since the railhead area is defined as in Figure 4-46 and is easily calculated with any commercially
available CAD package together with the second moment of area, it is proposed to derive the railhead
depth and width from the relationships:
b A/ h ( 4.6 )
and
12 I
h ( 4.7 )
A
Page 92
with A being the railhead area and I being the second moment of area of the railhead about its
horizontal axis through its centroid.
This results in dimensions equivalent to that of a rectangular beam section with the same area and
second moment of area, which therefore makes a better approximation of the rail stresses.
The railhead depth for a 57kg/m rail therefore changes to 43.17mm and gives the resultant stress
comparison with the FEA results as in Figure 4-48 which differ by approximately 4MPa (13%
improvement) in magnitude and ±4mm in location (50% improvement).
0.135
0.13
Height of Rail (m)

FEM 6.265 kN/mm Left
0.125
0.12
0.115
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Stress (GPa)
Figure 4-48: Rail stress comparison with railhead depth chosen at 43.17mm vs. 47.5mm (100kN
off-centred vertical load and 50kN lateral load)
The proposed definition of railhead depth will thus result in new railhead depths for all the different
rail profiles as indicated in Table 4-14.
This modified definition of the railhead depth clearly improves the correlation between the results of
the FEM and the analytical method.
Page 93
Table 4-14: New proposed railhead depths for different rail sections
Rail Profile Current Railhead depth (mm) Proposed Railhead depth (mm)
48 kg/m 43.0 38.3
57 kg/m 47.5 43.2
S60 kg/m 47.9 41.1
UIC 60kg/m 51.0 43.8
0.18
0.16
0.14
Height of Rail (m)
0.12
0.1 Analytical 20 MPa/m Left

FEM 6.265 kN/mm Left
0.08
0.06
0.04
0.02
0
-0.10 -0.05 0.00 0.05 0.10 0.15
Stress (GPa)
Figure 4-49: Rail stress comparison with railhead depth chosen at 43.17mm (full profile scale)
4.2. Sensitivity study on modified analytical method
A sensitivity study was performed on the new model with all the applied modifications in order to
identify the parameters to which the modified model was the most sensitive to to enable future users
and researchers to focus their efforts on the acquisition of those parameters. Table 4-15 indicates the
input parameters and their minimum and maximum values used in the analysis with their individual
effect on the rail stresses. Each parameter was held constant at its median (middle) value while
varying each one at a time while reporting the effect on the rail stresses. The parameters to which
some changes were made in the modified model are indicated in blue: Foundation Modulus (Changed
range of values), Eccentricity (Changed range of values and model now automatically use in different
combinations with Lateral load), Rail section (change of railhead depth definition) and the Lateral
load (Changed factor of static value to force value).
Page 94
Table 4-15: Input parameter sensitivity analysis

Parameter Value Max Rail Head Tension (MPa) Sum Head
Band Width
Parameter Min Mid Max Max Rail Flange Tension (MPa) & Flange
Max - Min
A B C A B C Band width
163.12 185.6 199.75 36.63
Dynamic Factor 1.1 1.49 1.73 86.37
193.58 224.1 243.32 49.74
157.48 185.6 228.89 71.41
Static Wheel load 67.5 100 150 168.41
185.89 224.1 282.89 97
174.21 185.6 186.72 12.51
Foundation Modulus 20 100 200 35.1
244.68 224.1 222.09 22.59
185.9 185.6 185.3 0.6
Sleeper Spacing 500 600 700 12.23
218.29 224.1 229.92 11.63
185.43 185.6 185.47 0.17
Type of Sleeper Wood Concrete Steel 0.49
224.42 224.1 224.34 0.32
185.24 185.6 185.96 0.72
Effective Area of Sleeper 0.4 0.48 0.6 2.02
224.76 224.1 223.46 1.3
183.23 185.6 187.97 4.74
E modulus rail 200 205 210 9.73
221.61 224.1 226.6 4.99
185.49 185.6 185.91 0.42
Poisson 0.289 0.3 0.33 0.65
224.04 224.1 224.27 0.23
195.13 185.6 199.19 13.59
Eccentricity of Vertical force Q 0 0.02 0.03 71.35
185.6 224.1 243.36 57.76
185.6 185.6 185.6 0
Position of Lateral force below top 0 0.007 0.015 3.24
222.59 224.1 225.83 3.24
202.64 185.6 195.15 17.04
Rail section 48 57 S60 73.92
261.24 224.1 204.36 56.88
191.34 185.6 166.56 24.78
Wheel distances 1.56 1.91 10.06 69.54
213.74 224.1 258.5 44.76
157.31 185.6 216.25 58.94
Max Temperature change 30 42 55 117.88
195.81 224.1 254.75 58.94
185.6 185.6 204.65 19.05
Lateral Load 0 20 40 34.06
216.6 224.1 231.61 15.01
The resultant stress ranges, as a result of changing each parameter, are also graphically displayed in
Figure 4-50, with the order of importance indicated by the resultant range in stresses as listed in Table
4-16. The parameters from the Eisenman formula (eq. 3.2) namely the running top condition,
confidence level and speed are omitted from the parameter list in Table 4-16 since the Dynamic factor
may now be specified without the use of the Eisenmann formula. The rail steel type is also omitted
since it influences only the fatigue limit and not the rail stresses.
4.3. Conclusion
By investigating the input parameters to which the analytical model is the most sensitive, the user is
now provided with nominal values which he may compare with new input values. The user is also
made aware of possible methods that could determine these important parameters and possible
problems to avoid. This contribution has improved the available analytical method significantly and
enhanced its suitability as a method to calculate track component stresses.
Page 95
300
Max tensile Stess (MPA)
250 Head
200
Sensitivity
150
A
Flange
100
Mid
C
50
Max Temperature change

Dynamic Factor
Foundation Modulus
Type of Sleeper
Poisson
Sleeper Spacing
E modulus rail
Eccentricity of Vertical force Q
Rail section
Position of Lateral force below top
Effective Area of Sleeper
Static Wheel load
Wheel distances
Parameter
Figure 4-50: Sensitivity analysis of parameters in component stress calculation spreadsheet
Table 4-16: Order of sensitive parameters
Sum of Head &

Order of Parameter Flange Band Percentage of
Importance widths highest value
1 Static Wheel load 168.41 100%
2 Max Temperature change 117.88 70%
3 Dynamic Factor 86.37 51%
4 Rail section 73.92 44%
5 Eccentricity of Vertical force Q 71.35 42%
6 Wheel distances 69.54 41%
7 Foundation Modulus 35.1 21%
8 Lateral Load 34.06 20%
9 Sleeper Spacing 12.23 7%
10 E modulus rail 9.73 6%
11 Position of Lateral force below top 3.24 2%
12 Effective Area of Sleeper 2.02 1%
13 Poisson 0.65 0%
14 Type of Sleeper 0.49 0%
Page 96
5 METHODOLOGY FOR CALCULATING TRACK COMPONENT STRESSES
Different investigations were launched regarding the input parameters to which the analytical method
was the most sensitive and best practice procedures were identified that could determine input
parameter values to ensure acceptable results.
This chapter will summarise the methodology to calculate stresses in track components with
reference to the developed spreadsheet that incorporates all the latest modifications to the analytical
method.
5.1. Methodology
When the user assesses the process of determining proper input values for the model, it should be
emphasised that the available resources should be spent on obtaining proper input for those
parameters that the model is the most sensitive to (paragraph 4.2. page 94). From those parameters,
the following is the most important (in order of sensitivity): static wheel load, maximum temperature
change, dynamic factor, rail section, eccentricity of the vertical wheel load and wheel distances.
Portions from the developed spreadsheet, with which a quasi-static analysis could be performed to
calculate track component stresses, are indicated in Figure 5-1 to Figure 5-6. The spreadsheet
incorporates the latest modifications to the analytical method. Drop-down boxes indicate where the
user has the option to choose input from a list of values and yellow cells indicate where the user may
enter a value.
The developed spreadsheet comprises of a multitude of input cells and formulas. These and additional
information are grouped under seven headings namely: “Dynamic factor”, “Support condition”,
“Material”, “Rail section properties”, “Bogie setup”, “Rail seat stress” and “Railhead and flange
stresses”. The methodology to calculate track component stresses will be presented in this order.
5.1.1. Dynamic factor
The dynamic factor is one of the parameters that the analytical model is the most sensitive to
(paragraph 4.2. page 94) and which has to be chosen (or calculated) carefully in order to obtain
acceptable stress results.
Figure 5-1 indicates the input parameters for calculating a dynamic factor with the Eisenmann
formula. This research has shown that the Eisenmann formula is a conservative method to use and
Page 97
may result in a high dynamic factor (paragraph 4.1.1.5, page 68) depending on the chosen input. The
user is encouraged to compare the result of the Eisenmann formula with the results from actual tests
(Figure 4-26, page 69).
Running Top Condition

G Excellent 1
0.1
0.1
Confidence Level
t' 95.4% 3
2
2
Dynamic Factor
Speed ( V ) 80
K s= 1.142857143
GK Coefficient of variation 0.114285714
Q Dynamic Wheel load 110.6 kN
Cant deficiency/excess factor (additional) 1.2

P Static Wheel load 75 kN
M Dynamic factor (Eisenmann) 1.23
Figure 5-1: Dynamic factor parameters
In order to determine the dynamic factor from the instrumented wheelset results, there are two
possible approaches to follow:
x The first approach ignores the existence of a moving average (i.e. there is no distinction made
between straight and curved track) and calculates the dynamic factor from the 99.85th
percentile (3 times the standard deviation) of the whole track, including irregularities such as
turnouts. This method is preferred when the railway line is designed as a whole, without
distinguishing between straight and curved track (and even turnouts).
x The second approach is based on the procedure specified in UIC Code 518 (30, p. 28) for
testing and accepting railway vehicles based on their dynamic behaviour. The dynamic factor
is calculated for short sections of the whole track individually. This method is preferred when
evaluating shorter sections of track, where the macrogeometry is taken into account by other
means like additional force factors etc.
When determining a dynamic factor that includes macrogeometric deviations (first approach), the
dynamic factor should be calculated as follows:
1. A piece of representative track should be measured with the instrumented wheelset at different
speeds (up to and including the maximum line speed). It is, however, seldom possible to find a
short section of track which is sufficiently representative of the track as a whole and it is thus
preferred to measure the whole track under consideration.
Page 98
2. Calculate the dynamic factor with either of the following equations:

3 Standard Deviation ( Vertical Forces ) mean( Vertical Forces )
M ( 5.1 )
or
99.85 th Percentile ( Vertical Forces )
M ( 5.2 )
With currently available calculation tools, both formulas could easily be utilised and the results
evaluated to determine which value would be more appropriate to use. Using 3 times the Standard
Deviation of the vertical forces could present a less realistic result, when a short piece of track is
measured, if the maximum values are a result of localised geometry defects (which would result in a
high deviation from the mean). If a sufficiently long piece of track is measured and a Gaussian
distribution is present, the results of equations 5.1 and 5.2 should be the same.
When determining a dynamic factor that does not include the effect of macrogeometric deviations
(second approach), the dynamic factor should be calculated as follows:
1. The test section should be divided into three zones (30, p. 28), namely:
x tangent track and large radius curves (R>1200m),
x large radius curves (600m<R<1200m),
x small radius curves (R<400m).
The procedure in (30, p. 30) specifies that each zone should have a total of at least 10km, since
it is rarely possible to find a short section of track which is sufficiently representative of the
track as a whole.
2. Each of these zones should then be divided into sections with a maximum length as indicated
in Table 5-1 (for this purpose, the transient curves may be included in the curve sections).
Table 5-1: Test section lengths for statistical analysis
Test Zones Section Lengths

Tangent track and large radius curves (R>1200m) 250m
Large radius curves (600<R<1200m) 100m
Small radius (400<R<600m) 100m

curves (R<400m) 70m
3. For each of these sections, the following should be calculated: Mean value, Standard
Deviation and 99.85th percentile of the measured vertical loads.
Page 99
4. A dynamic factor is then calculated for each section with the use of equations 5.1 and 5.2 and
then calculated for the line as a whole by taking the average of all the values calculated (for
either the standard deviation method or the percentile method).
When stress calculations for a new track (non-existing track) are carried out, the user may choose to
use the available Eisenmann formula and values for the track condition that match the intended
maintenance plans. The alternative is to utilise dynamic factors measured on existing lines which
carry the same rolling stock under the same operating conditions as the newly designed track would.
5.1.2. Support condition
Although the support condition does not appear in the “top six” list of parameters to which the
analytical model is the most sensitive (paragraph 4.2. page 94) and therefore does not affect the rail
stress result as much as the other mentioned input parameters, it does affect the rail deflection
significantly (refer Figure 3-5, page 26).
It is important to realise that a stiff support increases the rail stresses while decreasing the rail
deflection and vice versa (24, p. 122). However, excessive rail deflection accelerates ballast
breakdown, which enhances ballast fouling and eventually stiffens the track again. Unfortunately,
this process does not stabilise or lead to an optimally stiff track. Due to this uncontrolled process,
portions of the track may foul faster, resulting in irregular support stiffnesses. These irregularities,
fuelled by excessive movement of the track, are eventually detrimental to the track geometry.
Choosing the foundation modulus for a piece of track can be done by following the guidelines in
Table 5-2 or by extensive testing with the BSSM. The user may then simply choose a value from the
drop-down list in the spreadsheet (refer Figure 5-2).
Table 5-2: Foundation modulus values

Foundation Modulus (C) [MPa/m] Condition description (1, p. 79) (2, p 36-37)
20 Very bad clay formation and bad ballast condition††
50 Bad to average formation and ballast condition
100 Good formation and ballast condition
200 Other: (200=Very good formation and ballast condition)
The next parameter to specify is the sleeper spacing (refer Figure 5-2). The user may simply enter a
value for the sleeper centre-to-centre distance. A warning will be displayed if the value does not
satisfy the equation:
††
As mentioned earlier, these unquantified terms do make using these criteria vulnerable to poor user interpretation.
Page 100
S O
a% ( 5.3 )
4
with O being the characteristic length as defined by (3.9). Equation 5.3 should be true to ensure that
the theory of a continuously supported beam holds for a discretely supported rail.
The type of sleeper may then be chosen from the available list (drop-down box) which then
automatically fills in the appropriate effective area for the sleeper and calculates the track modulus.
C Foundation Modulus 1 MPa/m

20
20
a Sleeper Spacing 500 mm
Support Condition
Q Type of Sleeper 1
Concrete
1
2
PY, FY
2
Ae Effective Area of whole Sleeper 0.6012 m
N Characteristic Value 0.3006
k track stiffness 27.88460038 MN/m
U Track Modulus 12.024 MPa
Figure 5-2: Support condition parameters
5.1.3. Material
The material properties are specified as indicated in Figure 5-3. The user may select the type of
commercial rail steel from the list in the drop-down box which sets the ultimate tensile strength of the
material for calculating the fatigue stress limit. The rail fatigue stress limit is calculated as 30% of the
UTS.
E E modulus rail 205 GPa

Material
G Stiffness modulus 79.51900698 GPa

X Poisson 0.289
UTS Ultimate Tensile Strength 5
HH (UICA)
1200
Figure 5-3: Material parameters
Page 101
5.1.4. Rail section properties
An extensive list of available rail profiles is available in the spreadsheet, which includes standard
profiles as well as worn rail profiles. Due to the limitations of the software, only 8 profiles are
available from the drop-down box (refer Figure 5-4) at a time but the user may replace any of these
with any of the 45 available profiles and their properties.
When the user selects a rail profile, its properties (refer Figure 5-5) are automatically updated in the
spreadsheet (refer Figure 5-4).
After selecting the rail profile, the user needs to enter the maximum eccentricity of the vertical force
as well as the maximum lateral force.
5.1.5. Vertical load eccentricity
The proposed method to follow when determining eccentricity values is to simulate the wheel-rail
interaction (refer 4.1.3, page 85) for the applicable vehicle and track conditions. These simulations
can be carried out by using suitable software, such as Medyna or MSC.ADAMS. The maximum
eccentricity and maximum lateral forces, as well as the possible worst-case combinations thereof, can
then be determined.
Based on these results, it is evident that a rail stress analysis should not be determined only for one
maximum eccentricity along with one maximum lateral load. The maximum lateral load does not
coincide with the maximum contributing eccentricity of the vertical load and a high eccentricity with
a low opposing lateral force could prove to be the worst-case scenario. The reason why the case of a
high eccentricity with a high opposing lateral force is still regarded as a possible worst-case scenario
(Table 4-13), includes the following:
x the vertical force is always significantly larger than the lateral force
x the high eccentricity, together with the high vertical force, could still result in a moment
about the centre of the rail that is higher than in the case where the lateral load contributes to
the moment of the vertical force with a lower eccentricity.
The methodology to follow includes the execution of an eccentricity analysis on either a range of
wheel and rail profiles, or on a statistically representative wheel and rail profile pair. From such an
analysis the envelope of eccentricity values vs. lateral load values should be determined.
From this point on in the spreadsheet, all calculations are done in fourfold to calculate rail stresses for
the four possible worst-case scenarios simultaneously. Although the spreadsheet contains default
Page 102
values for the lateral loads and vertical load eccentricity, based on this research, the user has the
option to change these.
The user needs to specify the maximum eccentricity and the maximum lateral load. The four different
scenarios (worst-case scenarios) are then made up with percentages of these maximum values. A
positive percentage of eccentricity of the vertical force is in an opposing direction to the lateral force
with regards to the resulting moment about the railhead centre.
2
57
42% 42% 57% 55%

OK, safety envelope: OK, safety envelope: OK, safety envelope: OK, safety envelope:
e Max Eccentricity of Vertical force Q 0.033 0.033 -0.0099 -0.0066

L or Y Max Lateral Force (Default = 40kN) 40.0 0 2 20
n' Position of Lateral force below top 0.005 0.005 0.005 0.005
Worst-case 1 Worst-case 2 Worst-case 3 Worst-case 4
% of eccentricity 100% 100% -30% -20%
% of lateral force 100% 0% 5% 50%
Eccentricity Case 1 Eccentricity Case 2 Eccentricity Case 3 Eccentricity Case 4
A Section Area 0.007324 0.007324 0.007324 0.007324
Ah Railhead Area 0.002942 0.002942 0.002942 0.002942
b1 Width of Head 0.068142063 0.068142063 0.068142063 0.068142063
b2 Width of Flange 0.14 0.14 0.14 0.14
t Web Thickness 0.016 0.016 0.016 0.016
Rail Section Properties
d Section height 0.165 0.165 0.165 0.165

a1 Head Depth 0.043174507 0.043174507 0.043174507 0.043174507
f Flange thickness 0.0174 0.0174 0.0174 0.0174
yk Top to Section NA 0.0864 0.0864 0.0864 0.0864

h Head NA to flange NA 0.1336 0.1336 0.1336 0.1336
a2 Top to top of flange depth 0.1392 0.1392 0.1392 0.1392
Ixx Second Moment of Area xx 0.000026508 0.000026508 0.000026508 0.000026508

Izz Second Moment of Area zz 0.00000442 0.00000442 0.00000442 0.00000442
Ixx Head Second Moment of Area xx Head 0.000000457 0.000000457 0.000000457 0.000000457
I1 or Izh Second Moment of Area zz head 0.00000119 0.00000119 0.00000119 0.00000119
I2 or Izf Second Moment of Area zz flange 0.00000319 0.00000319 0.00000319 0.00000319
J Polar Inertia moment 0.000030928 0.000030928 0.000030928 0.000030928

Zk Section modulus Ixx/yk 0.000306806 0.000306806 0.000306806 0.000306806
Zf Section modulus Ixx/yf 0.000337252 0.000337252 0.000337252 0.000337252
Zxk Railhead modulus xx 2.11699E-05 2.11699E-05 2.11699E-05 2.11699E-05
Z1 / Zzk Railhead modulus zz 3.4927E-05 3.4927E-05 3.4927E-05 3.4927E-05
Z2 / Zzf Rail flange modulus zz 4.55714E-05 4.55714E-05 4.55714E-05 4.55714E-05
h1 Head mmp from rotation point 0.097302283 0.097302283 0.097302283 0.097302283

h2 Flange mmp from rotation point 0.036297717 0.036297717 0.036297717 0.036297717
Figure 5-4: Rail section properties and load position input
Page 103
Different Rail Sections
48 57 S60 UIC60 s60cw_7.5sw_15 S60 s60cw 10sw_10 60 60

Section Area A 6.1250E-03 7.3240E-03 7.6760E-03 7.6710E-03 6.8130E-03 7.6760E-03 6.7850E-03 7.6130E-03 7.6130E-03
Railhead Area Ah 2.5140E-03 2.9420E-03 2.8960E-03 3.0850E-03
Width of Head b1 6.5566E-02 6.8142E-02 7.0433E-02 7.0448E-02 6.0700E-02 7.28E-02 6.4600E-02 7.00E-02 7.00E-02
Width of Flange b2 1.2700E-01 1.4000E-01 1.5000E-01 1.5000E-01 1.5000E-01 1.50E-01 1.5000E-01 1.40E-01 1.40E-01
Web Thickness t 1.4100E-02 1.6000E-02 1.6000E-02 1.6500E-02 1.6000E-02 1.600E-02 1.6000E-02 1.900E-02 1.900E-02
Section height d 1.5000E-01 1.6500E-01 1.7200E-01 1.7200E-01 1.6450E-01 1.720E-01 1.6210E-01 1.710E-01 1.710E-01
Head Depth a1 3.8343E-02 4.3175E-02 4.1117E-02 4.3791E-02 4.0000E-02 4.790E-02 3.7600E-02 4.750E-02 4.750E-02
Flange thickness f 1.7300E-02 1.7400E-02 1.8000E-02 1.7900E-02 1.8000E-02 1.800E-02 1.8000E-02 1.940E-02 1.940E-02
Top to Section NA yk 7.8900E-02 8.6400E-02 9.3500E-02 9.1500E-02 9.6700E-02 9.350E-02 9.4500E-02 7.900E-02 7.900E-02
Head NA to flange NA h 1.2100E-01 1.3360E-01 1.4010E-01 1.3840E-01 1.3560E-01 0.1401 1.3490E-01 1.333E-01 1.333E-01
Top to top of flange depth a2 1.2480E-01 1.3920E-01 1.4320E-01 1.4050E-01 1.3580E-01 1.432E-01 1.3330E-01 1.288E-01 1.288E-01
Second Moment of Area xx Ixx 1.8506E-05 2.6508E-05 3.0815E-05 3.0350E-05 2.4290E-05 3.0815E-05 2.3802E-05 2.703E-05 2.703E-05
Second Moment of Area zz Izz 3.2160E-06 4.4200E-06 5.4160E-06 5.1130E-06 4.8910E-06 5.4160E-06 4.9380E-06 4.454E-06 4.454E-06
Second Moment of Area xx Head Ixx Head 3.0800E-07 4.5700E-07 4.0800E-07 4.9300E-07 1.8700E-07 4.0800E-07 1.6600E-07 4.570E-07 4.570E-07
Second Moment of Area zz head I1 or Izh 8.6800E-07 1.1900E-06 1.1080E-06 1.1820E-06 5.8900E-07 1.1080E-06 6.2600E-07 1.190E-06 1.190E-06
Second Moment of Area zz flange I2 or Izf 2.3140E-06 3.1900E-06 4.2410E-06 3.8800E-06 4.2410E-06 4.2410E-06 4.2410E-06 3.190E-06 3.190E-06
Figure 5-5: Rail section property list
5.1.6. Bogie setup
Regarding the bogie setup, the user has to enter the distances between four wheels (or axles). Figure
5-7 indicates how four wheels would each cause the rail to deflect. Since the deflections caused by the
four wheels overlap, these are added to present the total deflection as indicated in Figure 5-7. From
this example it is clear that, although the fourth wheel is relatively far away from the second, it still
influences the total displacement of the rail below the second wheel. It should be noted that the total
deflection caused by two or more wheels is higher than by one wheel alone but that the bending
moment is lower than under a single wheel. Therefore, multiple wheels reduce rail stresses, while
increasing the maximum deflection (24, p. 123).
x1 Distance from wheel in front 1.83 1.83 1.83 1.83 m

Bogie set-up
x2 Distance from wheel behind 3.66 3.66 3.66 3.66 m

x3 Distance from 4th wheel 5.49 5.49 5.49 5.49 m
K Influence factor (wheel in front) -0.207868198 -0.207868198 -0.207868198 -0.207868198
K Influence factor (wheel behind) -0.041941405 -0.041941405 -0.041941405 -0.041941405
K Influence factor (4th wheel) 0.008978909 0.008978909 0.008978909 0.008978909
Figure 5-6: Bogie setup
Page 104
0.001
0
-10 -8 -6 -4 -2 0 2 4 6 8 10
-0.001
-0.002
Deflection [m]
-0.003
-0.004
-0.005
-0.006
-0.007
Distance [m]
2nd Wheel (at Max total deflection) 1st Wheel 3rd Wheel 4th Wheel Total Deflection
Figure 5-7: Rail deflection due to wheel spacing
5.1.7. Rail seat stress
The user needs to enter the pre-tension force of the rail clips as well as the baseplate area (contact area
between rail and sleeper). From these values and the calculated rail seat force, the rail seat stress (pad
stress) is calculated. This stress is then compared to guidelines from Esveld (1, p. 91) depending on
the type of sleeper chosen. The sleeper-ballast stress is also calculated and compared to acceptable
values as specified by (1, p. 92).
M Rail Bending Moment 24.334 24.33363739 24.33363739 24.33363739 kNm
R Rail Seat Force 18.098 18.09823023 18.09823023 18.09823023 kN

Rail Seat Stress
Ro Design reaction force 36.19646047 36.19646047 36.19646047 36.19646047 kN

Mmax Sleeper bending moment 108.81 108.8065602 108.8065602 108.8065602 kNm
Fo Pretension Force (due to clips) 10 10 10 10 kN
2
Baseplate area 0.036 0.036 0.036 0.036 m
Rail Seat Stress (Pad stress) 0.78 0.780506395 0.780506395 0.780506395 MPa
OK for Concrete OK for Concrete OK for Concrete OK for Concrete
Sleeper Ballast stresses 0.015 0.015 0.015 0.015 MPa
Ok Ok Ok Ok
Figure 5-8: Track setup
Page 105
5.1.8. Railhead and flange stresses
From the available theory (see 3.1.3, page 18), the rail stresses are calculated along the rail profile for
the four different loading scenarios (see 4.1.3.1, page 89). Figure 5-9 indicates the stress results as
calculated for the four different loading scenarios (E1 to E4) and presents the axial stresses of the rail
on both the left and right side of the profile.
Rail Stresses (Without Temp.)
0.175
0.155
0.135
Approximate Rail Profile
Fatigue Tensile Limit
0.115
Height of Rail (mm)
Left E1
Right E1
0.095
Left E2
Right E2
0.075
Left E3
Right E3
0.055
Left E4
Right E4
0.035
0.015
-300 -200 -100 -0.005 0 100 200 300 400
Stress (MPa)
Figure 5-9: Rail axial stresses (without temperature stresses)
The last input parameter for the user to specify is the maximum temperature change the rail may
experience. This value is dependent on the geographic position of the rail and the rail stress relief
strategy followed by the maintenance personnel. After the static wheel load, this parameter has the
single most influence on the resultant rail stresses and should be chosen carefully.
Rail temperature ranges were not investigated as part of this study but are easily obtainable from the
various maintenance depots throughout the country which are managing rail stress relief
programmes. Typical temperature ranges are also obtainable from the rich archive of local research
on the subject.
Figure 5-10 indicates the resultant rail stresses with the added tensile stress due to a maximum
decrease in rail temperature.
Page 106
It should be noted that the stress-free temperature as maintained by the maintenance depot, minus the
minimum rail temperature due to ambient temperature drop, determines the value to be used in these
calculations. Using the maximum temperature range is representative of a rail which is stress-free at
the maximum rail temperature (due to ambient conditions). This case is, however, unlikely to occur
due to rail de-stressing strategies which do not allow rails to be de-stressed at maximum ambient
temperatures.
Rail Stresses (Incl. Temp.)
0.175
0.155
0.135
Approximate Rail Profile
Fatigue Tensile Limit
Height of Rail (mm)
0.115
Left E1
Right E1
0.095
Left E2
Right E2
0.075
Left E3
Right E3
0.055
Left E4
Right E4
0.035
0.015
-0.005
-200 -100 0 100 200 300 400
Stress (MPa)
Figure 5-10: Rail axial stresses (with temperature stresses)
The stress results in Figure 5-10 can be compared to those of Figure 5-9 to indicate the effect of
temperature on rail stresses.
5.2. Conclusion
This chapter summarised the methodology for calculating stresses in track components in reference to
the developed spreadsheet. From the different investigations into the input parameters to which the
analytical method is the most sensitive, best practice procedures were identified that could determine
input parameter values to ensure acceptable results.
Page 107
6 CONCLUSION
Railway track deteriorates over time due to the dynamic loading of passing rolling stock over its
elastic structure. To ensure the safe passing of trains, the track has to be properly designed and
maintained. This study focused on the development of a methodology for calculating stresses in track
components and emphasis has been placed on ways to determine representative input parameter
values, with the focus on identifying the parameters of highest importance. An existing analytical
method has been utilised and improved that could calculate track component stresses in a fast and
reliable manner.
The primary objective of the study was:

x To present a methodology for reliably calculating track component stresses under current and
future operating conditions.
The secondary objectives were:

x to identify parameters not included in the current analytical method that may have a significant
effect on track component stresses.
x to identify the parameters to which the analytical method is the most sensitive
x to provide nominal values to be used for the sensitive parameters where uncertainty exists
regarding appropriate values to use when calculating track component stresses
x to validate the analytical method with results from finite element models and field test results.
6.1. Summary of contributions
The following contributions have been made through this study:
x The analytical method proved to be a valuable tool in the calculation of track component
stresses.
x The assumption of a continuously supported beam of infinite length holds for a discretely
supported beam of finite length under certain conditions.
x The user’s attention has been focused on the input parameters that the analytical method is the
most sensitive to. This will help to eliminate unnecessary allocation of resources to obtain
input to unimportant parameters and places the focus on the parameters which proved to be
the most important.
Page 108
A number of enhancements have been made to the analytical method itself and the final model was
validated against empirical results and other well-established methods. The following enhancements
have been made:
x The upper limit of C (Foundation modulus) will be changed from 100+ for a very good
foundation and track condition, to 200MPa/m in accordance with (1, p. 79).
x The definition of the railhead depth (height) was changed to improve the analytical
approximation of stresses.
x A methodology was developed to determine dynamic factors from measured wheel-rail
forces.
x Running top condition is only an important parameter when using Eisenmann’s dynamic
factor formula
x A guideline has been presented to indicate when to use the relevant statistical confidence
levels together with the Eisenmann formula
x For low speed operations, the dynamic factor does not exceed 1.2.
x Eccentricity should not be regarded only as a maximum together with an opposing lateral
load, but should be used in different combinations to cover all possible worst-case scenarios.
x A methodology was developed on how to study wheel rail eccentricity and to determine
possible worst-case scenarios in conjunction with a maximum lateral load
x The spreadsheet version of the analytical model was enhanced by including 4 worst-case
scenarios of eccentricity vs. lateral load.
x The lateral load is not a function of the vertical load and should be used as a design maximum
for track component calculations.
x An additional vertical load factor has been introduced to account for higher loads in curves.
x It was shown that a fixed lateral load limit is sufficient to account for loads in all curves since
a proper track design should limit the lateral forces.
x Rail – sleeper stress limits have been introduced into the spreadsheet version of the analytical
model
x The fastening system's pretension force has been taken into account in the rail seat stress
calculations.
The analytical method is thus a very useful method that could calculate track component stresses.
6.2. Recommendations for future work
Valuable contributions to work done in this study could be made if future field test results
(instrumented wheelset, weighbridge or wayside tests) were to be analysed according to the presented
methodology in order to broaden the empirical base of input data.
Page 109
Finite element methods are used extensively throughout the railway industry and additional
comparisons of track model results with the analytical model would surely enrich both methods and
strengthen the applicability of the analytical method or indicate areas for improvement.
The possibility of integrating available calculation tools for managing rail stress relief programmes
could be investigated as an enhancement to the presented spreadsheet to be able to have zone specific
temperature change data available for calculating track component stresses.
6.3. Conclusion
It may be concluded that the results of this study proved the usefulness of the analytical method as a
trustworthy method with which to design and maintain railway track to ensure the safe passing of
trains.
Page 110
REFERENCES
1. Esveld C., Modern Railway Track, 2nd ed., MRT Productions, Zaltbommel, ISBN:
90-800324-3-3, 2001.
2. Lombard P. C., Spannings in die Spoorbaan struktuur, Spoornet (handwritten internal
document), 1972.
3. Naudé F.P., Track & Rail Stress Calculations - A Step By Step Guide, Spoornet, submitted
to Spoornet Engineering, report no. BBB2540ver1, 2001.
4. RAILROAD HISTORY, National Railroad Museum,
http://inventors.about.com/gi/dynamic/offsite.htm?site=http://www.nationalrrmuseum.org/
EdPacket/html/Tguide1.htm, 1999, Date downloaded 30/4/2002.
5. Bellis M., The History of Railroad Inventions,
http://inventors.about.com/library/inventors/blrailroad.htm, 1996, downloaded 30/4/2002.
6. Gross E., This Day in American History, Neal-Schuman Publishers, Inc., New York, 1990.
7. International Union of Railways , Loading Diagram to be taken into consideration for the
calculation of rail carrying structures on lines used by international services, UIC CODE,
UIC, 2nd, 702-O, 1974.
8. International Union of Railways, Factors affecting track maintenance costs and their relative
importance, UIC CODE, UIC, 3rd, 715-R, 1992.
9. Ebersohn W, Selig E.T., Track design and maintenance, Railway Geotechnology course
notes, University of Pretoria, 1992.
10. International Heavy Haul Association, Guidelines to best practices for heavy haul railway
operations: Wheel and rail interface issues, 1st ed., IHHA, Virginia Beach, 2001
11. International Union of Railwails, Loading Diagram to be taken into consideration for the
calculation of rail carrying structures on lines used by international services, UIC CODE,
UIC, 2nd ed., 702-O, 1974.
12. Marczyk J., Beyond Optimization in CAE, International Center for Numerical Methods in
Engineering (CIMNE), Barcelona, ISBN: 84-95999-11-0, 2002.
13. Fastenrath F., Railroad track, 1st ed., Frederick Ungar Publishing Co., New York, ISBN:
0-8044-4231-2, 1977.
14. Popp K., Kruse H., Kaiser I., Vehicle-Track Dynamics in the Mid-Frequency Range,
Vehicle System Dynamics - International Journal of Vehicle Mechanics and Mobility, vol.
31, no. 5-6, p. 423, 1999.
15. European Rail Research Institute, Rail rolling contact fatigue - Stress analysis of rail rolling
contact fatigue cracks – Interim Report, ERRI D 173/RP11, 1995
16. Maree J. S., Dynamic behaviour of track with resilient rail pads, thesis for Masters in
Engineering, lead by Prof. Vernon Marshall, Civil Engineering, University of Pretoria,
Pretoria, 1993.
Page 111
17. Wang J., The influence of track and vehicle properties on dynamic impact forces, Spoornet,
submitted to Spoornet Engineering, 2002.
18. Bezin Y, Iwnicki S, Stow J, Development of a method to predict stresses in rails using
ADAMS/Rail and ABAQUS, Manchester Metropolitan University, Manchester, 1998
19. Eisenmann, J., Germans gain a better understanding of track structure, Railway Gazette
International, 1972.
20. Ebersohn W., Selig E. T., Introduction to Multi Disciplinary Concepts in Railway
Engineering, Chair in Railway Engineering, University of Pretoria, 1994.
21. Timishenko, S., Method of analysis of statical and dynamical stresses in rail, Proceedings of
the Second International Congress of Applied Mechanics, Zurich, 1927.
22. Esveld C, Modern Railway Track, 1st ed., MRT-Productions, Duisburg, ISBN:
90-800324-1-7, 1965.
23. Tomas M. D., Ogies - Saaiwater project. Stresses in 48, 57, and 60 kg/m rails from traffic-
1st interim report, submitted to Spoornet Engineering, Spoornet, Report no. BBB2501,
2001.
24. Wilson B.W., On strength of railway track, Proceedings of the South African Society of
Civil Engineers, 1941
25. Brinkman, Stresses in the rail as a result of dynamic forces, Klöckner-Werke AG Werk
Osnabrück, Osnabrück, 1972
26. Igwemezie, J.O., Morgan, C.A., Stress analysis of transposed 102.4, 115, 136 LB/YD and
jointed 100 LB/YD rails, Applied Rail Research Technologies Inc., submitted to Canadian
Pacific Rail Systems, ARRT Report No. 1093, 1993.
27. Labuschagne, T.J., The calculation of bending stresses in rails on curved track, The Civil
Engineer in South Africa, 1960.
28. Frohling R. D., Maartens S. J., Pretorius G. J., Dip Joint Analysis On The Orex Line,
Spoornet, submitted to Spoornet Engineering, 2002.
29. Fröhling, R.D., Summer commissioning and verification tests on the MTAB iron ore
prototype wagon, submitted to Spoornet Engineering, Report E/MT/TMO/040, 1999.
30. International Union of Railways, Test and acceptance of railway vehicles form the point of
view of dynamic behaviour, safety, track fatigue and quality of ride, UIC CODE 518 OR,
2nd edition, International Union of railways, 1999.
31. Lombard, P.J., Track structure – optimization of design, Heavy Haul Railway Conference,
Institute of Engineers - Australia, Perth, 1978.
32. Eisenmann J.,The rail as travelling way, Technishe Universität München, München , 1970
Page 112
APPENDIX A
Stochastic rail stress distributions (Different iterations with the same model)
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Figure A-1: Stochastic rail stress distributions
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Page 113
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
600
500
400
300
200
100
0
50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490
Page 114
APPENDIX B
Histograms of Leandra test data
Figure B-1: Histograms of lateral forces at 20km/h
Page 115
Page 116
APPENDIX C
Lateral force graphs of Iron Ore Export line data
Figure C-1: Lateral left and right forces at 30 km/h, for 15 ton axle
Page 117
Page 118
Page 119
Figure C-7: Vertical left and right forces at 30 km/h, for 15 ton axle
Page 120
Page 121
Page 122
APPENDIX D
Summary of dynamic factors
AREA Eisen 1 0.1
Eisen 1 0.2 Eisen 1 0.3
2.4 Eisen 2 0.3 Eisen 3 0.1
ORE Dynamic: Loco
Dynamic: Wagon Schram
Klöckner Light Klöckner Heavy
2.2 Clintock Sweden 1st Approach; (3*Std Dev)

Sweden 1st Approach, (99.85th P) Sweden 2nd Approach, (3*Std Dev)
Sweden 2nd Approach, (99.85th P) Leandra 1st Approach; (3*Std Dev)
Leandra 1st Approach, (99.85th P) Leandra 2nd Approach, (3*Std Dev)
2
Dymnamic Factor
Leandra 2nd Approach, (99.85th P) Orex, 15t, 1st Approach; (3*Std Dev)
Orex, 15t, 1st Approach, (99.85th P) Orex, 15t, 2nd Approach, (3*Std Dev)
Orex, 15t, 2nd Approach, (99.85th P) Orex, 26t, 1st Approach; (3*Std Dev)
Orex, 26t, 1st Approach, (99.85th P) Orex, 26t, 2nd Approach, (3*Std Dev)
Orex, 26t, 2nd Approach, (99.85th P) Orex, 30t, 1st Approach; (3*Std Dev)
1.8 Orex, 30t, 1st Approach, (99.85th P) Orex, 30t, 2nd Approach, (3*Std Dev)
Orex, 30t, 2nd Approach, (99.85th P) LINE 1 DOWN 0-20
LINE 1 DOWN 20-70 LINE 1 DOWN 70-90
LINE 1 DOWN 90+ LINE 1 DOWN 125
LINE 1 DOWN 200 LINE 1 UP 0-20
1.6 LINE 1 UP 20-70 LINE 1 UP 70-90
LINE 1 UP 90+ LINE 1 UP 125
LINE 1 UP 200 LINE 2 DOWN 0-20
LINE 2 DOWN 20-70 LINE 2 DOWN 70-90
1.4 LINE 2 DOWN 90+ LINE 2 DOWN 125
LINE 2 UP 20-70 LINE 2 UP 70-90
LINE 2 UP 200 LINE 3 DOWN 0-20
1.2 LINE 3 DOWN 20-70 LINE 3 DOWN 70-90
LINE 3 DOWN 90+ LINE 3 DOWN 125
LINE 3 UP 20-70 LINE 3 UP 70-90
1 LINE 3 UP 200 Orex 15t
0 50 100 Orex Orex 30t
Speed (km/h) Saaiwater

Saaiwater Line 3
Saaiwater Line 2
Figure D-1: Summary of all dynamic factors
Page 123

Stress in Track

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What is the research about?

What is the research about?

What modifications were made to the existing analytical model?

What modifications were made to the existing analytical model?

University of Pretoria etd - Naude, F P (2005)

DEVELOPMENT OF A METHODOLOGY FOR CALCULATING

FRANCOIS PAULUS NAUDÉ

Submitted in fulfilment of part of the requirements, for the degree of

MASTER OF ENGINEERING (MECHANICAL)

DEVELOPMENT OF A METHODOLOGY FOR CALCULATING STRESSES IN TRACK

FRANCOIS PAULUS NAUDÉ

Supervisor: Professor NJ Theron

I would like to thank and acknowledge contributions by the following persons:

LIST OF FIGURES ........................................................................................................................ VI

LIST OF TABLES .......................................................................................................................... IX

3 THE ANALYTICAL METHOD ........................................................................................13

4 IMPROVEMENTS TO THE ANALYTICAL METHOD...............................................44

5 METHODOLOGY FOR CALCULATING TRACK COMPONENT STRESSES.......97

Figure 4-50: Sensitivity analysis of parameters in component stress calculation spreadsheet 96

I Second moment of area

1.2. Problem statement

1.3. Objectives and scope of research

The primary objective of the study is:

The secondary objectives are:

2.1. History of the railways

2.2. Track design philosophy

2.3. Engineering Modelling of track

2.4. Available approaches relevant to this project

2.4.1. Finite element modelling

2.4.2. Dynamic modelling

2.4.3. Analytical methods

2.4.4. Combination of methods

3 THE ANALYTICAL METHOD

3.1. Input parameters

Table 3-1: Input parameters of the analytical method

3.1.1. Dynamic factor

The dynamic load (Q) is calculated by (2, p. 65):

Train Speed (km/h)

Figure 3-1: Typical train speed distribution

3.1.2. Support condition

3.1.3. Rail stresses

Following from the deflection, the bending moment (M)

Figure 3-2: Load decomposition for calculation of rail stresses

Figure 3-3: Position of shear centre / rotational axis

Then the stresses in the railhead are:

and in the flange

Due to sideways bending, the stresses V8 and V9 are given as:

Figure 3-5: Deflection curve of FEA models and analytical methods

For the case where C = 20 MPa/m, and a PY or FY concrete sleeper is used:

k 20. 0.6012 ( 3.51 )

Table 3-2: Conversion of foundation modulus values to support stiffness

stiffness 1 6.012 y=4.5mm 15.03 y=2.4mm 30.06 y=1.4mm

Figure 3-6: Centred and off-centred load cases

Analytical 20 MPa/m Analytical 50 MPa/m Analytical 100 MPa/m

Left 0.135 Right

116 MPa 46 MPa

-370 MPa -264 MPa

Table 3-3: Stress comparison between FEM and analytical model

Rail flange tensile stress 90 92 2 2%

K ' K ( x1 ) K ( x 2 ) ..... K ( x n ) ( 3.55 )