POLITECNICO DI MILANO
School of Industrial and Information Engineering
Master of Science in Mechanical Engineering
Modelling and Testing of an Embedded Rail
System for the Analysis of Train-Track
Dynamic Interaction
Supervisor: Roberto Corradi
Co-supervisor: Qianqian Li
Author:
Matteo Colombo 883167
Academic Year: 2018/2019
Table of Contents
i
Table of Contents
Table of Contents ................................................................................................ i
List of Figures ................................................................................................... iii
List of Tables ..................................................................................................... ix
Abstract.............................................................................................................. x
Sommario ........................................................................................................ xii
List of Abbreviations ....................................................................................... xiii
Chapter 1 Introduction ................................................................................... 1
Chapter 2 State of the art ................................................................................ 9
2.1 Track modelling and train-track interaction ............................................................. 9
2.2 Literature review of track resilient elements modelling .......................................... 16
2.2.1 Introduction to rheology of viscoelastic materials ................................................ 17
2.2.2 Linear models ....................................................................................................... 22
2.2.3 Non-linear models ................................................................................................ 32
Chapter 3 Laboratory tests ........................................................................... 50
3.1 The test bench .......................................................................................................... 51
3.2 Static characterization tests ..................................................................................... 53
3.3 Dynamic characterization tests ................................................................................ 57
3.3.1 Dynamic tests with monoharmonic force input................................................... 57
3.3.2 Dynamic tests with displacement input simulating a bogie/train passage ......... 61
Chapter 4 Non-linear rheological model of the Embedded Rail System ........ 67
4.1 Model identification ................................................................................................. 68
4.2 Model validation....................................................................................................... 78
Chapter 5 2D Track models and train-track dynamic interaction simulation 85
5.1 Track models and frequency domain analysis ......................................................... 86
5.1.1 Analytical models of a single beam on elastic foundation ................................... 87
Table of Contents
ii
5.1.2 Finite element models of a single beam on elastic foundation ............................ 95
5.2 Moving load simulation.......................................................................................... 107
5.2.1 Simulation conditions ........................................................................................ 108
5.2.2 Simulation results ............................................................................................... 110
5.3 Train-track dynamic interaction simulation ........................................................... 112
5.3.1 Proposed models ................................................................................................. 113
5.3.2 Influence of track model parameters ................................................................. 122
5.3.3 Simulation conditions ........................................................................................ 126
5.3.4 Simulation results ...............................................................................................127
Conclusions and future developments ........................................................... 138
Ringraziamenti ............................................................................................... 141
Appendix A ......................................................................................................142
Bibliography .................................................................................................. 148
List of Figures
iii
List of Figures
Figure 1-1 Typical track layout. (left) Superstructure components. (right) lateral view with rail, sleeper and substructure component, courtesy of C. Esveld [2] .................................... 2
Figure 1-2 Cross-section of a Ballasted track structure, courtesy of C. Esveld [2]................ 4
Figure 1-3 Cross-section of a superstructure slab track with an asphalt concrete, courtesy of C. Esveld [2]....................................................................................................................... 4
Figure 1-4 Cross-section of an embedded rail superstructure, courtesy of C. Esveld [2] ..... 5
Figure 1-5 Static load-deflection curve (left) and derived tangent stiffness (right), courtesy of D.J. Thompson [4]............................................................................................................. 6
Figure 1-6 Stiffness (top) and loss factor (bottom) of a railpad for different preload values, courtesy of A. Fenander [3] ................................................................................................... 7
Figure 2-1 Components of vehicle/track system model, courtesy of K. Knothe [18] ......... 10
Figure 2-2 Illustration of an Hertzian spring ....................................................................... 11
Figure 2-3 Sleeper support models, courtesy of K. Knothe [18] ........................................ 13
Figure 2-4 Different models for describing track dynamic properties, courtesy of C. Esveld [2] ........................................................................................................................................ 14
Figure 2-5 Typical train-track interaction excitation modelling approaches, courtesy of K. Knothe [18] .......................................................................................................................... 16
Figure 2-6 Illustration of Hookean spring (left) and Newtonian dashpot (right) .............. 19
Figure 2-7 Typical viscoelastic solid response. (a) Stress and strain histories in the stress relaxation test. (b) Stress and strain histories in the creep test, courtesy of H. Banks [29] ............................................................................................................................................. 20
Figure 2-8 Stress and strain curves during cyclic loading-unloading. (left): Hookean elastic solid; (right): linear viscoelastic solid depicted by the solid line, courtesy of H. Banks [29] ............................................................................................................................................. 21
Figure 2-9 Schematic representation of the Maxwell element ........................................... 22
Figure 2-10 Maxwell element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29] .................................................................................................... 24
Figure 2-11 Schematic representation of the Kelvin-Voigt element .................................... 25
List of Figures
iv
Figure 2-12 Kelvin-Voigt element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29]..................................................................................... 26
Figure 2-13 Schematic representation of the Standard Linear Solid element .................... 27
Figure 2-14 Standard Linear Solid element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29] .......................................................................... 28
Figure 2-15 Schematic representation of the Standard Linear Solid conjugate element .... 29
Figure 2-16 Schematic representation of the Generalized Kelvin-Voigt rheological model29
Figure 2-17 Modelling the viscoelastic component. (a) 4-parameter Zener model. (b) Generalized 4-parameter Zener model ............................................................................... 30
Figure 2-18 Comparison of the optimized Zener model with target values of equivalent stiffness and damping, courtesy of S. Bruni and A. Collina [30] ........................................ 31
Figure 2-19 Comparison between the measured and simulated amplitude frequency response of the track, courtesy of S. Bruni and A. Collina [30] .......................................... 32
Figure 2-20 Schematic representation of the loads on the pad and ballast, courtesy of T.X Wu and D.J. Thompson [31] ............................................................................................... 34
Figure 2-21 Foundation deflection and reaction force under 75 kN wheel load. —, stiff ballast; – · –, medium ballast; - - - , soft ballast, courtesy of T.X Wu and D.J. Thompson [34] ...................................................................................................................................... 35
Figure 2-22 Force on sleeper versus sleeper displacement when rail is loaded with sinusoidal force, measured by Banverket [6]. The measurement results are reported for three different sleepers along the track, courtesy of T. Dahlberg [6].................................. 37
Figure 2-23 Rail displacement when bogie of high-speed train passes on track, courtesy of T. Dahlberg [6] .................................................................................................................... 39
Figure 2-24 Dynamic characteristics of railpad at preloads 500, 750 and 1000 N: — measurement; - - - - adapted P–T material model, courtesy of J. Maes [5] ....................... 42
Figure 2-25 Schematic representation of the Modified Poynthing-Thomson model according to Koroma ........................................................................................................... 43
Figure 2-26 Rail discretely supported on a non-linear modified Poynthing-Thomson elastic foundation and subjected to a moving load [24] ..................................................... 44
Figure 2-27 Schematic representation of Sjoberg's resilient element ................................ 45
Figure 2-28 (left) Amplitude dependencies of measured and simulated dynamic stiffness and damping, frequency is 0.05 Hz. (right) Frequency dependence of measured and simulated quantities, amplitude is constant at 0.01 mm, courtesy of M.M. Sjoberg [35] .. 48
Figure 3-1 Edilon ERS cross-section and main features ..................................................... 51
Figure 3-2 Laboratory experimental layout, frontal view scheme (not in scale) ................ 52
Figure 3-3 Measurement system, detail on instruments measuring the rail deformation . 53
Figure 3-4 Track static reaction force per unit length due to the presence of a train bogie along the x coordinate. (Left) ETR500, wheelbase length 3 m and static load 170 kN. (Right) Freight train, wheelbase length 1.8 m and static load 250 kN ............................... 54
Figure 3-5 Track cumulative reaction force. (Left) ETR500. (Right) Freght train ............. 55
Figure 3-6 Input and output time histories of static test. (Left) Input force measured by load cell. (Right) Output rail displacement measured by laser transducers ....................... 56
Figure 3-7 Static Load-deflection characteristic curve of the specimen, obtained considering only the third cycle of the static test. (top left) static case with maximum load
List of Figures
v
equal to 64 kN. (top right) static case with maximum load equal to 55 kN. (bottom) static case with maximum load equal to 37 kN. ............................................................................ 56
Figure 3-8 Stiffness-deflection curve derived from the 64 kN static characterization test 57
Figure 3-9 Dynamic test time histories for the 55 kN preload and 1 Hz frequency case. (left) Averaged signal derived from the measurement of the four transducers. (right) Load cell signal ............................................................................................................................. 59
Figure 3-10 Specimen dynamic behavior during monoharmonic excitation of 55 kN and 1 Hz. (Left) Complete spectral content of the output displacement and highlight of the amplitude at the exciting frequency. (Right) Complete hysteresis cycle and rectified monoharmonic hysteretic cycle ........................................................................................... 60
Figure 3-11 Dynamic characterization test results. (Right) Equivalent stiffness. (Left) Equivalent viscous damping ................................................................................................ 61
Figure 3-12 Example of a simulated displacement time history of a rail section suspended on elastic foundation due to a moving bogie travelling at 200 km/h. (Blue line) Response computed from a dynamic simulation using a finite element with moving loads model. (Red line) Repsonse generated using the Winkler foundation model ................................ 63
Figure 3-13 Displacement time history of repeated bogie passage tests considering an ETR500 bogie train model moving at 200 km/h. The repetition of the bogie passage simulation is performed to obtain a steady-state response ................................................ 64
Figure 3-14 Comparison between the time histories of the input command reference and the actual rail displacement ................................................................................................ 65
Figure 3-15 Output force time history of a bogie passage test considering an ETR500 bogie train model moving at 200 km/h ........................................................................................ 66
Figure 3-16 Output force time history of a train passage test considering an ETR500 train model comprised of eleven wagons moving at 200 km/h................................................... 66
Figure 4-1 Schematic representation of the rheological element models considered. (a) Three-parameters Standard Linear Solid model. (b) Four-parameters Zener model ........ 69
Figure 4-2 𝐾𝑡𝑜𝑡 and 𝐶𝑡𝑜𝑡 fitting performance, preload value equal to 55 kN. (top) Three-parameters Standard Linear Solid model. (bottom) Four-parameters Zener model .......... 71
Figure 4-3 Schematic representation of the Standard Linear Solid preload dependent models. (a) Model with a single preload dependent variable. (b) Model with two preload dependent variables. (c) Model with three preload dependent variables ........................... 72
Figure 4-4 𝐾𝑡𝑜𝑡 and 𝐶𝑡𝑜𝑡 fitting performace of modified De Man model (a). Preload equal to 55 kN................................................................................................................................ 74
Figure 4-5 4 𝐾𝑡𝑜𝑡 and 𝐶𝑡𝑜𝑡 fitting performace of modified De Man model (b). Preload equal to 55 kN ...................................................................................................................... 74
Figure 4-6 𝐾𝑡𝑜𝑡 and 𝐶𝑡𝑜𝑡 fitting performance of modified De Man model (c). Preload equal to 55 kN................................................................................................................................ 75
Figure 4-7 K_tot and C_tot fitting performance of modified De Man model (a) for different values of preload. .................................................................................................. 76
Figure 4-8 Schematic representation of the linear Kelvin-Voigt rheological model .......... 77
Figure 4-9 𝐾𝑡𝑜𝑡 and 𝐶𝑡𝑜𝑡 fitting performance of linear Kelvin-Voigt model . Preload equal to 55 kN................................................................................................................................ 78
Figure 4-10 Schematic representation of the modified De Man model coupled with a lumped mass ........................................................................................................................80
List of Figures
vi
Figure 4-11 Schematic representation of the linear Kelvin-Voigt model coupled with a lumped mass ........................................................................................................................ 81
Figure 4-12 Experimental load cell signal. Input simulating an ETR500 bogie passage at a speed of 200 km/h .............................................................................................................. 82
Figure 4-13 𝐹𝑒𝑥𝑡 generated by the non-linear modified De Man sectional model. Input simulating an ETR500 bogie passage at a speed of 200 km/h ........................................... 82
Figure 4-14 𝐹𝑒𝑥𝑡 generated by the linear Kelvin-Voigt sectional model. Input simulating an ETR500 bogie passage at a speed of 200 km/h ............................................................. 83
Figure 4-15 Comparison of the numerical and experimental force output with displacement input simulating a bogie/train passage: linear Kelvin-Voigt model of tested ERS sample (750 mm), non-linear modified De Man model of tested ERS sample (750 mm), laboratory test with ERS sample (750 mm). Input simulating an ETR500 bogie passage at a speed of 200 km/h .......................................................................................... 83
Figure 4-16 Comparison of the numerical and experimental force output with displacement input simulating a train passage: linear Kelvin-Voigt model of tested ERS sample (750 mm), non-linear modified De Man model of tested ERS sample (750 mm), laboratory test with ERS sample (750 mm). Input simulating an eleven-wagons ETR500 train passage at a speed of 200 km/h ................................................................................. 84
Figure 5-1 Analytical track model: an infinite beam suspended on a linear Kelvin-Voigt foundation ........................................................................................................................... 87
Figure 5-2 Accelerance of the force application point of an infinite beam suspended on Kelvin-Voigt foundation ...................................................................................................... 89
Figure 5-3 Analytical track model: a finite beam of length L suspended on a linear Kelvin-Voigt foundation .................................................................................................................. 90
Figure 5-4 Accelerance of the force application point of a 2-meter-long finite beam suspended on Kelvin-Voigt foundation ............................................................................... 91
Figure 5-5 Accelerance of the force application point of a finite beam suspended on Kelvin-Voigt foundation. Analysis with different lengths ................................................... 92
Figure 5-6 Accelerance of the force application point of a 10-meter-long finite beam suspended on Kelvin-Voigt foundation. Analysis with different 𝐸𝐽/𝑘𝐾𝑉 ratios ................. 93
Figure 5-7 Analytical track model: a finite beam of length L suspended on a linear Kelvin-Voigt foundation and constrained at one end by concentrated impedances ...................... 94
Figure 5-8 Accelerance of the force application point of a 2-meter-long finite beam suspended on Kelvin-Voigt foundation and constrained by properly tuned anechoic impedances .......................................................................................................................... 95
Figure 5-9 FE track model: a rail continuously supported on a linear Kelvin-Voigt elastic foundation and subjected to fixed monoharmonic load .................................................... 98
Figure 5-10 Accelerance of the force application point of a 2-meter-long finite element model of element beams suspended on Kelvin-Voigt foundation ...................................... 99
Figure 5-11 FE track model: a rail continuously supported on a linear Kelvin-Voigt elastic foundation with anechoic constraints and subjected to fixed monoharmonic load ........ 100
Figure 5-12 Accelerance of the force application point of a 2-meter-long finite element model of element beams suspended on Kelvin-Voigt foundation and constrained by properly tuned anechoic impedances ................................................................................. 101
Figure 5-13 Accelerance of the force application point of a finite element model of element beams suspended on Kelvin-Voigt foundation. Analysis with different structural damping parameters ......................................................................................................................... 102
List of Figures
vii
Figure 5-14 FE track model: a rail continuously supported on linear Kelvin-Voigt rheological elements and subjected to an applied load .................................................... 103
Figure 5-15 Accelerance of the force application point of a finite element model of Euler-Bernoulli element beams suspended on Kelvin-Voigt foundation and with clamped edges ........................................................................................................................................... 104
Figure 5-16 FE track model: a rail discretely supported on modified De Man rheological elements and subjected to an applied load ....................................................................... 105
Figure 5-17 FE track model: a rail continuously supported on a linear Kelvin-Voigt elastic foundation with moving loads simulating a bogie passage ............................................... 109
Figure 5-18 FE track model: a rail discretely supported on modified De Man rheological elements with moving loads simulating a bogie passage .................................................. 109
Figure 5-19 Deflection time history of the rail midspan simulated by the FE track model with linear Kelvin-Voigt model with moving loads simulating a bogie passage (ETR 500 at 200 km/h) .......................................................................................................................... 110
Figure 5-20 Deflection time history of the rail midspan simulated by the FE track model with proposed non-linear model with moving loads simulating a bogie passage (ETR 500 at 200 km/h) ...................................................................................................................... 110
Figure 5-21 Comparison of the simulated deflection time history of the rail midspan between the FE linear track model and the non-linear FE track model with moving loads simulating a bogie passage (ETR 500 at 200 km/h).......................................................... 111
Figure 5-22 Spatial distribution of variable 𝛾1 when the moving bogie locates at midspan of the finite element domain (ETR 500 at 200 km/h) ....................................................... 112
Figure 5-23 Train-track dynamic interaction model: a finite element rail continuously supported on a linear Kelvin-Voigt elastic foundation with rail irregularity input and a spring-mass system ............................................................................................................ 113
Figure 5-24 Train-track dynamic interaction model: a finite element rail discretely supported on a modified De Man foundation, with a moving loaded spring-mass system and rail irregularity input ................................................................................................... 116
Figure 5-25 Train-track dynamic interaction model: a finite element rail continuously supported on a linear Kelvin-Voigt foundation, with a moving loaded spring-mass system and rail irregularity input .................................................................................................. 120
Figure 5-26 Spectra of the acceleration time history of the midspan of the rail due to an impulse excitation on the mass obtained with different domain lengths ......................... 123
Figure 5-27 Displacement time history of the midspan of the rail due to an impulse excitation on the mass obtained with the domain length equal to 20 m. ......................... 123
Figure 5-28 Spectra of the acceleration time history of the midspan of the rail due to an impulse excitation on the mass obtained with the domain length equal to 40 m. Analysis with different values of structural damping ...................................................................... 124
Figure 5-29 First mode shape of model 1, relative resonant frequency equal to 59 Hz. Red circle: Oscillator node. Blue circles: Track finite element nodes. Dashed green line: Undisturbed configuration ................................................................................................ 125
Figure 5-30 Second mode shape of model 1, relative resonant frequency equal to 183 Hz. Red circle: Oscillator node. Blue circles: Track finite element nodes. Dashed green line: Undisturbed configuration ................................................................................................ 125
Figure 5-31 A typical example of the response of the midspan of the rail, in terms of displacement time history, due to a bogie passage and highlight of two different time intervals. Red line: During passage. Purple line: After passage .........................................127
List of Figures
viii
Figure 5-32 Track response of the midspan of the rail for simulation condition case a (ETR 500 at 200 km/h). Top left: Displacement time history. Top right: Rail acceleration time history. Bottom: Rail acceleration levels expressed in third octave-bands ...................... 128
Figure 5-33 Contact force of simulation condition case a. Left: Contact force time history. Right: Contact force spectral content ................................................................................ 129
Figure 5-34 Track response of the midspan of the rail for simulation condition case b: Displacement time history response ................................................................................. 129
Figure 5-35 Track response of the midspan of the rail for simulation condition case b. Top left: Rail acceleration time history during the bogie passage. Top right: Rail acceleration levels expressed in third octave-bands during the bogie passage. Bottom left: Rail acceleration time history after the bogie passage. Bottom right: Rail acceleration levels expressed in third octave-bands after the bogie passage. ................................................. 130
Figure 5-36 Contact force of simulation condition case b. Left: Contact force time history. Right: Contact force spectral content ................................................................................. 131
Figure 5-37 Track response of the midspan of the rail for simulation condition case c: Displacement time history response ................................................................................. 132
Figure 5-38 Track response of the midspan of the rail for simulation condition case c. Top left: Rail acceleration time history during the bogie passage. Top right: Rail acceleration levels expressed in third octave-bands during the bogie passage. Bottom left: Rail acceleration time history after the bogie passage. Bottom right: Rail acceleration levels expressed in third octave-bands after the bogie passage. ................................................. 133
Figure 5-39 Contact force of simulation condition case c. Left: Contact force time history. Right: Contact force spectral content ................................................................................ 133
Figure 5-40 Track response of the midspan of the rail for simulation condition case d: Displacement time history response ................................................................................. 134
Figure 5-41 Track response of the midspan of the rail for simulation condition case d. Top left: Rail acceleration time history during the bogie passage. Top right: Rail acceleration levels expressed in third octave-bands during the bogie passage. Bottom left: Rail acceleration time history after the bogie passage. Bottom right: Rail acceleration levels expressed in third octave-bands after the bogie passage. ................................................. 135
Figure 5-42 Contact force of simulation condition case d. Left: Contact force time history. Right: Contact force spectral content ................................................................................ 136
List of Tables
ix
List of Tables
Table 4-1 Single preload dependent variable modified De Man model identified parameters ........................................................................................................................... 76
Table 4-2 Linear Kelvin-Voigt model identified parameters .............................................. 78
Table 5-1 Train-track interaction models considered ........................................................ 114
Table 5-2 Simulation condition cases considered for the train-track interaction simulation ........................................................................................................................................... 126
Abstract
x
Abstract
Train-track dynamic interaction is a complex phenomenon which involves the
vehicle, the railway structure and the wheel/rail contact force. Numerical models and
simulations are useful tools to comprehend this complex phenomenon and the related
issues. Currently, most of the track models are linear. However, it is important and
necessary to properly account for the non-linear dynamic characteristics of the
constituting components, such as the resilient elements, for the track modelling. This
thesis proposes a new methodology to include the non-linear dynamic behavior of the
track resilient elements in the train-track dynamic interaction simulation by referring
to an embedded rail system (ERS). A systematic review of track resilient materials
modelling is performed. Laboratory tests are performed on an ERS specimen and the
obtained data is then employed to identify and validate a non-linear rheological model
which reproduces well the dynamic behavior of the ERS specimen. The non-linear
model is then integrated into a 2D finite element track model with which a
comprehensive analysis about the influence of the track model parameters is
performed. Simulations with both moving loads and train-track dynamic interaction
are performed with linear/non-linear track models. The simulation results
demonstrate that the non-linear track model leads to significantly different track
response with respect to the linear model.
Keywords: non-linear, track, resilient element, rheological model, train-track
interaction
Abstract
xi
Sommario
xii
Sommario
L’interazione ruota-rotaia è un fenomeno complesso che riguarda il veicolo,
l’armamento ferroviario e la forza di contatto ruota/rotaia. Modelli numerici e
simulazioni sono strumenti utili per comprendere questo complesso fenomeno e anche
problemi legati ad esso. Attualmente, la maggior parte dei modelli di armamento sono
modelli lineari. Tuttavia, è di particolare importanza considerare correttamente le
caratteristiche dinamiche non-lineari delle componenti costituenti, come gli elementi
elastici, per modellare accuratamente il comportamento dell’armamento. In questa
tesi viene proposta una nuova metodologia in grado di includere il comportamento
dinamico non-lineare degli elementi elastici all’interno di simulazioni di interazione
ruota-rotaia facendo riferimento ad un Embedded Rail System (ERS). Viene
presentata una revisione sistematica della modellazione dei materiali elastici degli
armamenti ferroviari. Test di laboratorio sono svolti su un provino ERS e i dati
sperimentali ottenuti sono successivamente utilizzati per identificare e validare un
modello reologico non-lineare in grado di riprodurre adeguatamente il
comportamento del provino. Il modello non-lineare è integrato in un modello 2D di
armamento ferroviario con il quale viene proposta un’analisi dell’influenza dei
parametri del modello armamento. In aggiunta, sono eseguite simulazioni con carichi
viaggianti e interazione ruota-rotaia considerando modelli lineari e non-lineari di
armamento. I risultati delle simulazioni dimostrano come il modello di armamento
non-lineare comporta una risposta della rotaia che differisce in maniera significativa
da quella ottenuta utilizzando il modello lineare classico.
Keywords: non-lineare, armamento ferroviario, elemento elastico, modello
reologico, interazione ruota-rotaia
List of Abbreviations
xiii
List of Abbreviations
ERS Embedded Rail System
FE Finite Element
LVDT Linear Variable Differential Transformer
List of Abbreviations
xiv
Introduction
1
Chapter 1 Introduction
The daily operation of a rail transportation system is affected by problems of
different nature. Issues related to the railway system are typically: comfort of the
passengers, ground borne vibration and airborne noise, runnability of bridges and
viaducts, design of switches and crossings and other track sections with complex
geometry.
In order to properly understand and predict these issues, it is categorical to
identify and comprehend the characteristics of the participating elements . The vehicle
system, the track system and the contact force coupling the train wheel to the railhead
are the main components affecting the train-track interaction phenomenon and are
thus responsible for the problems related to the operation of the rail transportation
system. The railway structure is a complex apparatus comprised of multiple elements,
each one dedicated to one or more specific purposes necessary for the safe and correct
passage of the rail vehicle. The behavior of the track structure heavily depends on the
type of structure, the applied components and materials, the accuracy of installation
and the level of maintenance.
The track components can be generally classified into two main categories:
superstructure and substructure. The parts of the track comprised of rails, rail pads,
sleepers, and fastening systems are considered as the superstructure while the
Introduction
2
substructure consists in a geotechnical system consisting of ballast, sub-ballast and
subgrade formation ( Figure 1-1 ). Both superstructure and substructure are mutually
important in ensuring the safety and comfort of passengers and quality of the ride [1].
Considering the typical railway layout configuration ( [1], [2], Figure 1-1):
• rails are longitudinal steel members installed to guide the train vehicle. Their
strength and stiffness must be sufficient to resist various forces exerted by
travelling rolling stock;
• sleepers are transverse beams resting on ballast. The sleeper main purposes are
to uniformly transfer and distribute loads from the rail to the underlying ballast
bed;
• fasteners are the clipping components of the rails. They keep the rails in place,
withstanding the forces and moments in different directions;
• railpads are placed on the rail seat (the bottom plane of the rail) to filter and
transfer the dynamic forces from rails and fasteners to the sleepers;
• the ballast is a layer of free draining coarse aggregate used as a tensionless
elastic support for resting sleepers. It not only provides support but also
transfers the load from the track to the sub-ballast;
• sub-ballast is a layer of granular material between the ballast and underlying
subgrade. The main functions of the sub-ballast are to reduce stress at the
bottom of the ballast and to prevent interpenetration between the different
interfacing layers;
• the subgrade includes the existing soil, rock and other structures or materials
within. This deep layer must have sufficient bearing capacity and yield a
tolerably smooth settlement in order to prolong track serviceability;
Figure 1-1 Typical track layout. (left) Superstructure components. (right) lateral view with rail, sleeper and substructure component, courtesy of C. Esveld [2]
Introduction
3
In reality, multiple track structure types exist for different applications.
Nonetheless, rail, fastening element, resilient element, sleeper and base supporting
layer can always be identified in each type. In some particular rail systems, specific
physical elements can incorporate functionalities of different rail superstructure and
substructure components. For example, in the Embedded Rail System, an elastomeric
material embeds continuously the rail beam inside a steel channel and the function of
this resilient element is equivalent to the fastening and railpad components.
The dynamic behavior of the track system to excitations typical of the train-track
interaction phenomenon is non-linear. It is known from literature that these non-
linear characteristics are associated mainly to the fastening system ( [3], [4], [5] ),
ballast layer ( [6], [2], [7]) and soil layer ( [8], [9] ). It is thus important to study in
detail the non-linear behavior derived from each of these components in order to
correctly understand and predict the behavior of the track system.
A specific type of components which introduces the nonlinear dynamic behavior
of the track is the resilient materials.
The resilient elements are necessary for the safe and comfortable operation of
the rail vehicle, since they provide both filtering and energy absorption capabilities.
Additionally, their viscoelastic effects are paramount in the track system as they
include the majority of the elastic and damping properties of the railway, especially in
the higher frequency domain [2]. They also provide protection from wear, fatigue and
impact loads due to the train passage [3].
Being generally made of studded rubber or polyurethane, the resilient materials
are affected by typical non-linearities of polymers. Such non-linearities are identified
as dependencies with respect to preload, temperature, frequency, strain amplitude [4],
and strain persistency [10]. It is important to notice that the investigating and
modelling of the main non-linearities associated with vehicle and track interaction
implies that temperature dependency, static and very low dynamic characteristics are
of secondary importance. In fact, these are generally neglected when considering the
real-time dynamic of the train-track interaction.
Resilient materials are widely used in different types of track system.
Considering a ballasted track (Figure 1-2) the resilient element is present in the
form of rail pads inserted between the rail foot and the sleeper. In this figure, it is
shown that rail and sleeper are connected by fastenings; the sleeper is supported by
the ballast layer; the ballast bed rests on the sub-ballast layer which forms the
transition layer. Typical material for these railpads are rubber bonded cork, EVA
(lupolen V 3510 K), studded elastomer etc.
Introduction
4
Considering a ballastless slab track (Figure 1-3) the resilient element is present
as pads in a configuration equivalent to the ballasted track. In addition, a second type
of pad is placed between the sleeper and the baseplate. Given the absence of other
added resilient components inside the railway system, such as the ballast, the railpad
is the only filtering component above the very low frequency range [2].
Considering an embedded railway track (Figure 1-4) the resilient element is
present in the form of a continuous volume of one or multiple kinds of hyper-
elastomeric materials poured inside a steel channel, embedding the rail system. The
resilient materials are generally cork and polyurethane [2].
Figure 1-2 Cross-section of a Ballasted track structure, courtesy of C. Esveld [2]
Figure 1-3 Cross-section of a superstructure slab track with an asphalt concrete, courtesy of C. Esveld [2]
Introduction
5
Figure 1-4 Cross-section of an embedded rail superstructure, courtesy of C. Esveld [2]
In literature are present several researches related to the comprehension of the
non-linear behavior of railway resilient elements. The general consensus is that track
structures are affected by non-linearities typical of polymeric material when resilient
elements are present.
Different authors, such as Grassie et al. [11], Grassie and Cox [12], Dalenbring
[13] have pointed out the influence of the railway behavior when railpads are present.
While Fermér and Nielsen quantified on a full scale experimental setup the influence
of soft and stiff pads on wheel-rail contact force, on the sleeper end acceleration and
on the rail head acceleration [14].
The measurement of railpads dynamic stiffness as function of frequency and
preload has been carried out by authors Thompson et al. [4], Zand [15],Knothe et al.
[16] and Maes et al. [5].
The procedure utilized by the four papers is described by the ISO norm [17],
which defines three different methods for extracting vibro-acoustic transfer properties
of resilient elements. This allows for the quantification of damping and stiffness
properties of visco-elastic materials, such as railpads, at different preloads and
frequencies. In Appendix A a more detailed review of railpad tester laboratory setups
and test procedures is presented.
Thompson also measured the rail static deflection using a dial gauge to an applied
load generated by a hydraulic actuator and the results are presented in Figure 1-5 .
From the static characterization it is demonstrated that the tangent stiffness of the
material changes as a function of the applied load: the static stiffness is constant up to
a certain preload value, after which it increases sharply.
Introduction
6
Figure 1-5 Static load-deflection curve (left) and derived tangent stiffness (right), courtesy of D.J. Thompson [4]
The dynamic characterization results of Fenander are presented in Figure 1-6.
The dynamic stiffness is shown to increase slightly with frequency, while the influence
of the preload is more pronounced. At low frequencies the loss factor is about the same
for all preloads. It increases slightly with frequency for high preloads, while for low
preloads the loss factor increases more at high frequencies.
Introduction
7
Figure 1-6 Stiffness (top) and loss factor (bottom) of a railpad for different preload values, courtesy
of A. Fenander [3]
The non-linear behavior of the studded rubber resilient elements to frequency,
preloads and static load amplitude is shown in the presented figures above.
The dependences of the dynamic stiffness and loss factor on preloads and
frequency are the main features affecting the train-track interaction while the effects
of strain persistency and temperature are not strictly related to the coupled train-
railway dynamic interaction.
Numerical models are useful and mature tools to study the train-track interaction
and the related problems. The coupled track-vehicle dynamic is caused by the mutual
interaction of moving vehicle and rail track by means of the contact force present
between wheel and railhead. Dynamic modelling of the railway track and interaction
with vehicle requires the investigation of the different agents affecting the coupled
system. In literature several researches are focused on the interaction phenomenon
problem ( [18], [18], [19], [23] ).
Of particular interest is the modelling of the track structure. Different approaches
( [2] ) found in literature are capable of accounting for the dynamic behavior of the
Introduction
8
railway both in terms of its superstructure and substructure components. However,
most of the available track models are linear and do not account for the railway
nonlinearities such as the ones of the resilient materials.
The main purpose of this thesis is to propose a new methodology to account for
the typical non-linearities of the resilient elements employed in the track system for
the train-track interaction numerical simulation by making reference to an ERS. To
this end, a literature review of the existing techniques and modelling approaches of
track system, train-track interaction and track resilient materials is presented .
Inspired by the literature, a non-linear rheological model of unit length of
resilient material is proposed which accounts for the non-linear dynamic properties
function of both preload and frequency.
In order to identify the parameters of the non-linear rheological model and to
verify whether it can correctly reproduce the non-linear track dynamic behavior of the
resilient material, a laboratory experimental campaign including different types of
tests is performed on an ERS sample. A frequency domain identification procedure is
implemented, while a validation of the model is carried out in time domain.
The validated non-linear rheological model is then integrated into a 2D track
model by substituting the original linear foundation. To ensure that the track model is
adequate for train-track dynamic simulation, an investigation of the effect of track
model parameters is performed in frequency domain.
The 2D linear and non-linear track models are employed in moving loads
simulations in order to study the effect of the bogie passage on the track structure. In
particular, the bogie is modelled as two separate forces acting on the rail. Each of these
forces accounts for the load transmitted to a single wheelset.
Finally, the 2D linear and non-linear track models are implemented in 2D train-
track interaction simulations. In particular, the vehicle is modelled by means of
multiple independent moving spring-mass systems (also called oscillators), each one
accounting for the effect of the train unsprung mass, load on wheelset, and a Hertzian
spring modelling the wheel/rail contact. In addition, the implementation of rail
irregularities is performed in order to obtain realistic excitations for the train-track
dynamic interaction.
.
State of the art
9
Chapter 2 State of the art
There have been numerous studies on the topic of track modelling. Depending on
the specific study objectives, track models of different level of complexity have been
proposed and applied to the train-track dynamic interaction simulation. The
differences between models mainly lie in the modelling choices of each constituting
elements. In particular, resilient elements, strongly influencing the dynamic properties
of the whole track system, have been modelled in different manners which could lead
to distinct track response. In addition, the track response is not influenced only by the
resilient element model but also by all the other simulation parameters. Consequently,
appropriate simulation parameters are essential to interpret the simulation results and
comprehend the difference led by the choice of resilient element model.
2.1 Track modelling and train-track interaction
In this paragraph an overview of the models historically employed for the
simulation of the train-track interaction is presented.
Authors Knothe and Grassie ‘s renowned paper Modelling of Railway Track and
Vehicle/Track Interaction at High Frequencies [18] presents a clear image of the
classical problems associated with railway structure and rail vehicle interaction.
In Figure 2-1 a scheme of the typical train-track interaction components is
presented. In particular a train-track interaction model is generally divided into three
main parts: the vehicle, the contact force and the railway track structure.
State of the art
10
Figure 2-1 Components of vehicle/track system model, courtesy of K. Knothe [18]
The vehicle model employed in the train-track interaction modelling may assume
different forms depending on the required simulation accuracy. The vehicle can be
represented by simple forces acting on the railhead when only the response of the track
to a dynamic or static loading is necessary [18]. Another approach is to consider the
vehicle as a group independent unsprung masses and discrete forces acting on them,
where the single unsprung mass represents the wheelset and each discrete force
represents the force transmitted from the bogie to the wheelset. This approach is useful
for modelling a simplified train-track interaction [18]. Full vehicle models can also be
employed in the coupled train-track system [18] and this methodology is the most
accurate but also more complex and computationally demanding.
The contact force couples the train and track. It is a complex phenomenon
derived by the wheel-rail contact (which is often regarded as a Hertzian contact
ar od
o ie
heelset
leeper
allast
u rade
ontact
ail
ad astenin
State of the art
11
problem) [2]. Several approaches are available for modelling this force. The simplest
and most diffused approach for modelling the contact interaction is the use of a
Hertzian mechanical spring (Figure 2-2). In this way, the contact force is modelled as
a vertical force proportional to the displacements of the wheel and railhead
Figure 2-2 Illustration of an Hertzian spring
Other more complex approaches for modelling the contact phenomenon may be
employed [19] which also accounts for longitudinal and lateral contact forces. For
example, FASTSIM [20] is a numerical algorithm which employs Kalker's simplified
theory by approximating the contact by means of a multitude of independent springs.
When it comes to the track model, in the most diffused railway structure
configurations, the components considered for the rail dynamic modelling are
generally: rail beam, fastening system (fastener and railpad), sleeper, ballast and
subgrade ( Figure 2-1 ).
For the rail beam there are different classic modelling approaches available, each
guaranteeing a different performance in terms of accuracy and complexity. Euler-
Bernoulli beam is the historical and simplest beam element used in literature [21], [22].
It is best suited for static and stability analysis. In the frequency domain it is adequate
up to 500 Hz [23]. However, such a model is no longer satisfactory for the response to
vertical forces at higher frequencies, as shear deformation of the rail becomes
increasingly important. If only vertical and longitudinal vibrations are of interest, the
rail can be modelled as a Timoshenko-Rayleigh beam up to 2.5 kHz and for
ailh ead
heel
State of the art
12
wavelengths greater than 0.4 m. For lateral and torsional modes, however, railhead
and foot have to be modelled at least as independent Timoshenko beams
interconnected by continuous rotational springs [18]. More complex three-
dimensional finite elements may also be employed for an even more accurate analysis
[16].
The fastening clip component is generally considered as a single spring connected
in parallel with the railpad, or regarded simply as a constant static load impinging on
the rail head together with the rail beam weight [4], [24].
The railpads and resilient elements, which generally behave in a non-linear
manner, are historically modelled linearly by means of springs and dampers about the
s stem’s equili rium position [18].
The sleeper is modelled as a rigid body with mass and moment of inertia when
considering the 2D transvers problem. The main problem to account for is in the third
dimension, where the sleeper variable thickness in the railway lateral direction
becomes non-negligible.
The ballast bed consists of a layer of loose, coarse grained material [2], while the
subgrade represents the general terrain and foundation composition found below the
railway structure itself. They deflect in a highly non-linear manner under load. In
particular, there may be voids between sleeper and the structure below. Energy
dissipation occurs due to dry friction and from wave radiation through the substrate.
Despite this, in most researches the of use a simple two-parameters rheological model
is the preferred approach (Figure 2-3 Type A).
More complex linear models presented in literature can be employed for more
accurate representations, for example: the ballast and substrate are considered
together as an elastic or viscoelastic half-space, especially useful when modelling
ground borne vibrations [25]( Figure 2-3 Type B ); the ballast and subgrade are
modelled as a single viscoelastic Pasternak element, which is a viscoelastic rheological
model accounting for shear loads [26], ( Figure 2-3 Type C ); the layer of ballast is
placed on a three-dimensional subgrade half-space [27] ( Figure 2-3 Type D ). These
four configurations are represented in the scheme below.
State of the art
13
Figure 2-3 Sleeper support models, courtesy of K. Knothe [18]
In general, the 2D implementation of different track components inside a track
structure model may be performed in different ways (Figure 2-4). For example, for a
rail suspended on a single foundation the supporting aspect of the structure is
completely accounted by a single bed component. The increase in complexity for what
concern discerning the different track component generally results in more accurate
simulation results. For example, the track structure can be regarded as a system
consisting of rails which are elastically supported by means of rail pads on sleepers
spaced at a fixed distance, where the sleepers are supported by a viscoelastic
substructure consisting of ballast plus subgrade [2] .
pe A
iscrete sleeper support
allast sprin and damper
pe
iscrete sleeper support
alf space modellin of allast
and su rade
pe
iscrete sleeper support
allast sprin and damper
Interconnected allast masses
u rade sprin and damper
pe
iscrete sleeper support
ontinuous allast la er
halfspace su rade model
State of the art
14
In literature are also present more complex 3D track models capable of modelling
the system behavior in a more accurate manner ( [28] ).
Another main distinction in modelling approaches for the track system is
between models with a completely continuous rail support and those with a discrete
support, shown in Figure 2-4. The employment of one or the other kind is dependent
on the type of system to be modelled and the necessary complexity required to obtain
the correct results.
Figure 2-4 Different models for describing track dynamic properties, courtesy of C. Esveld [2]
For example, considering a ballasted track, discrete support appears more
representative of reality for the majority of tracks, in which rails are laid on discrete
sleepers. The corresponding continuous support is obtained by "smearing out" the
discrete support along the track [18].
State of the art
15
The continuous model principal deficiency is its inability to show the track's
behavior around the frequency of the so-called "pinned-pinned" resonance which is
present in the spectral domain where the wavelength of the flexural waves travelling in
the rail beam is equal to the sleeper spacing. To be able to perform such prediction it
is mandatory to account for the discrete support nature of the sleepers.
If instead an Embedded Rail system is considered, which is generally installed on
metro lines or train lines passing on bridges, then single or double beam configurations
are preferred and the continuous support is employed correctly.
The railway track system can also be modelled considering a finite or an infinite
length. The type of structure is closely connected to the solution technique. Infinite
length configuration is preferred in the frequency domain analysis, whereas finite track
structures are preferred when time domain solutions are required. While the second
solution allows for the modelling of an extremely diversified range of problems, its
main issue is related to the undesired boundary effects intrinsically associated with its
space limited nature.
The modelling of the train-track excitation may be tackled in multiple ways [18]
( Figure 2-5 ). Each of the different cases approaches the excitation problem by
modelling in a more or less complex manner the vehicle’s equivalent mass-spring
system interacting with the railroad. The most realistic model of vertical excitation
arising in wheel-rail contact is that of a wheel rolling over irregularities on the track,
( case d. in the figure below) . However, this model is also the most difficult, thus
simpler models have many attractions in particular circumstances. Case a. is
appropriate for comparing the calculated and measured response of the track excited
by a stationary periodic or transient force. Case b. is regarded to be more useful when
a theoretical investigation of moving-load excitation is required. Case c. can be
regarded physically as a model in which the wheelset remains in a fixed position on the
rail, and a strip containing the irregularities on the railhead and wheel tread is
effectively pulled at a steady speed between wheel and rail. This last model is mainly
employed when studying the response of wheelset on discretely supported track and
wheel/rail interaction.
State of the art
16
Figure 2-5 Typical train-track interaction excitation modelling approaches, courtesy of K. Knothe [18]
2.2 Literature review of track resilient elements modelling
The presented literature review was derived from a literature search performed
on the platform Scopus. The search was performed using the string: TITLE-ABS-KEY
( nonlinear OR non-linear OR "nonlinearity of" OR "non-linearity of" ) AND TITLE-
ABS-KEY ( ( railway W/5 track ) OR ( railroad W/5 track ) ). The review was also
focused on the time period between the years 1980 and 2019.
The resilient element material models introduced in the literature review in terms
of their mechanical behavior can be categorized as linear and non-linear model.
Before describing the models in detail, an introduction to the rheology of
viscoelastic materials is proposed due to the importance of rheological modelling
inside the literature review.
State of the art
17
2.2.1 Introduction to rheology of viscoelastic materials
Constitutive equations are equations linking two physical quantities, the stress
and the strain, through one or more parameters or functions which represent the
characteristic response of the material per unit volume regardless of size or shape [29].
Such equations are commonly used to describe the behavior of resilient materials.
The equation 𝜎 = 𝑓(𝜀) portrays the response of resilient materials in terms of the
stress variable 𝜎 to the external input of strain 𝜀. To this end, the subject studying the
general relationship 𝑓(∙) between stress and strain is the rheology.
In rheological modelling the modelled physical system is replaced by idealized
counterparts of its actual constitutive elements. The components affecting the
mechanical s stem’s ehavior are sprin s and damper with a parallel or serial
relationship with respect to each other. It is, in this way, the mechanical analog of
electric circuit theory.
It must be emphasized, however, that the representation of constitutive equations
by means of springs and dashpots doesn’t imply that these elements in any manner
reflect molecular mechanisms in the material whose behavior they model. In fact, it is
possible to demonstrate that the observed behavior of a material can generally be
represented by a multiplicity of rheological elements.
This introduction describes concepts which have been validated and established
in literature. A multitude of books and academical lectures can be found describing in
detail the concepts that will briefly be presented in the following paragraphs. The main
sources used for this review are Banks et al, [29], and Tschoegl’s ook [10].
a. Perfect elastic materials and pure viscous materials
Elasticity is the physical property of a material that when it deforms under
external load, it returns to its original shape when the stress is removed. For an elastic
material, the stress-strain curve is the same for the loading and unloading process, and
the stress only depends on the current strain, not on its history.
In case of linear elastic solids the stress and strain are related in a proportional
manner followin ook’s law. considering a monoaxial configuration:
𝜎(𝑡) = 𝑘 𝜀(𝑡) (2.2-1)
Where 𝜎 is the stress generated, 𝜀 is the strain and 𝑘 is the elastic modulus of the
material.
State of the art
18
In case of a non-linear elastic material in a general way, the relationship between
stress and strain is non-linear. Different categories of non-linear elastic materials
exists. For example, hyperelastic (or Green elastic) material, is an ideally elastic
material for which the strain energy density function (a measure of the energy stored
in the material as a result of deformation) can be explicitly derived. The behavior of
unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal.
A purely viscous material instead is characterized by a constitutive stress
behavior affected by the strain rate. For a linear viscous material this effect is
proportional. In fact, for a monoaxial load deforming in time, with 𝑟 being the viscous
constant of the material:
𝜎(𝑡) = 𝑟 𝑑(𝜀(𝑡))
𝑑𝑡
(2.2-2)
In a similar fashion to non-linear elastic material, for non-linear purely viscous
material the stress-strain rate relationship is non-linear. For example, a non-
Newtonian fluid is a fluid affected by a non-linear relationship between stress and
viscosity.
Considering an input sinusoidal strain law:
𝜀(𝑡) = 𝜀0 sin (𝜔𝑡) (2.2-3)
Where 𝜀0 is the harmonic amplitude and 𝜔 is the angular frequency of excitation.
In case of a purely elastic solid the stress response is in-phase with the strain with
an input-dependent amplitude:
𝜎(𝑡) = 𝑘 𝜀0 sin (𝜔𝑡) (2.2-4)
In case of a purely viscous material instead the response is 90° out-of-phase with
respect to the strain and its amplitude is both input and frequency-dependent:
State of the art
19
𝜎(𝑡) = 𝑟𝜀0𝜔 sin (𝜔𝑡 +𝜋
2) (2.2-5)
Figure 2-6 Illustration of Hookean spring (left) and Newtonian dashpot (right)
b. Viscoelastic materials
The distinction between nonlinear elastic and viscoelastic materials is not always
easily discerned and definitions may vary. However, it is generally agreed that
viscoelasticity is the property of materials that exhibit both viscous and elastic
characteristics when undergoing deformation.
Viscoelastic materials are those for which the relationship between stress and
strain depends on time, and they possess the following three important properties:
stress relaxation (a step constant strain results in decreasing stress), creep (a step
constant stress results in increasing strain), and hysteresis (a stress-strain phase lag).
Considering a stress relaxation test, in which the input strain law 𝜀(𝑡) is imposed
from time 𝑡0 , one can obtain an output time-dependent stress response 𝜎(𝑡) .
Performing such test one can define the stress function 𝐺(𝑡):
𝐺(𝑡) =𝜎(𝑡)
𝜀(𝑡) (2.2-6)
If the input strain law is a unit step strain, then 𝐺(𝑡) is referred to as relaxation
modulus.
In a similar fashion, but using the stress as input, one can perform a creep test.
Defining the strain function 𝐽(𝑡) as:
State of the art
20
𝐽(𝑡) =𝜀(𝑡)
𝜎(𝑡) (2.2-7)
If the input strain law is a unit step stress, then 𝐽(𝑡) is referred to as creep
compliance.
In a stress relaxation test, viscoelastic solids gradually relax and reach an
equilibrium stress greater than zero. In a creep test, the resulting strain for viscoelastic
solids increases until it reaches a nonzero equilibrium value ( Figure 2-7 ). In other
words:
lim𝑡→+∞
{ 𝐺(𝑡) = 𝐺∞ > 0
𝐽(𝑡) = 𝐽∞ > 0
(2.2-8)
Hysteresis can be show from a stress-strain relationships stand point. It reveals
that for a viscoelastic material the loading process is different than in the unloading
process. From a phenomenological stand point it is visible in case of cyclic excitation
as an enclosed area defined by the loading and unloading phases of excitation in the
stress-strain plain ( Figure 2-8 ).
Figure 2-7 Typical viscoelastic solid response. (a) Stress and strain histories in the stress relaxation test. (b) Stress and strain histories in the creep test, courtesy of H. Banks [29]
a
State of the art
21
Figure 2-8 Stress and strain curves during cyclic loading-unloading. (left): Hookean elastic solid; (right): linear viscoelastic solid depicted by the solid line, courtesy of H. Banks [29]
Performing a stress relaxation test with equation (2.2-3) as input, the steady-state
stress response of a viscoelastic material is defined as:
𝜎(𝑡) = 𝜎0sin (𝜔𝑡 + 𝛿) (2.2-9)
Where the phase shift 𝛿 is between zero and π/2, and the stress amplitude σ0
depends on the frequency ω and the input amplitude. This is because the viscoelastic
material is both affected by the viscosity and elasticity phenomenon.
Equation (2.2-9) is composed by the superposition of an in-phase response and
an out-of-phase response. The former component, in a steady-state configuration,
produces no net work when integrated over a cycle, while the latter results in a
dissipative action.
Considering the same behavior but in frequency domain. Taking into account
Equations (2.2-6),(2.2-3) ,(2.2-9) and expressing as complex variable one can obtain
the complex dynamic modulus 𝐺∗:
𝐺∗ =𝜎0
𝜀0 𝑒𝑖𝛿 =
𝜎0
𝜀0cos(𝛿) + 𝑖
𝜎0
𝜀0sin(𝛿) = 𝐺′ + 𝑖𝐺′′ (2.2-10)
Where the in-phase element 𝐺′ is the storage modulus, measuring the energy
stored and recovered each cycle, and the out-of-phase element 𝐺′′ is the loss modulus,
a characterization of the energy dissipated in the material by internal damping.
State of the art
22
It is important to notice that the typical material used in track systems
demonstrate a viscoelastic behavior.
2.2.2 Linear models
The three basic models typically used for describing linear viscoelastic material
behaviors are the Maxwell model, the Kelvin-Voigt model and the Standard Linear
Solid model. Each of these is defined by a specific number and arrangement of
Hookean springs and Newtonian dashpot elements.
a. Maxwell model
The Maxwell model is represented by a dashpot and an elastic spring connected
in series. Due to this configuration the model is also called an iso-stress model ( Figure
2-9 ).
Figure 2-9 Schematic representation of the Maxwell element
The total strain is the sum of the elastic and viscous strain contributions:
1
𝑟𝜎 +
1
𝑘
𝑑𝜎
𝑑𝑡=𝑑𝜀
𝑑𝑡 (2.2-11)
To describe the stress relaxation function of the element, it is important to define
the necessary input step function and initial condition:
State of the art
23
𝜀(𝑡) = 𝜀0 𝐻(𝑡 − 𝑡0) ; 𝜎(0) = 0 (2.2-12)
Where 𝐻(∙) is the Heaviside step function, different than zero at 𝑡 ≥ 𝑡0 .
Introducing (2.2-12) in (2.2-11) , with 𝛿(∙) as the Dirac delta function:
1
𝑟𝜎 +
1
𝑘
𝑑𝜎
𝑑𝑡= 𝜀0 𝛿 (𝑡 − 𝑡0)
(2.2-13)
Given that the system is linear and the initial condition is zero, it is possible to
easily obtain the complete time response of equation (2.2-13) by means of the Laplace
transform and then inverse transform. The complete solution will then be:
𝜎(𝑡) = 𝑘 𝜀0 𝐻(𝑡 − 𝑡0) 𝑒−𝑘𝑟(𝑡−𝑡0)
(2.2-14)
If instead the creep function of the element is required, the input step function
and initial condition will be:
𝜎(𝑡) = 𝜎0 𝐻(𝑡 − 𝑡0) ; 𝜀(0) = 0 (2.2-15)
Similar to how (2.2-14) was obtain, the creep response to a stress step input of
the Maxwell element is:
𝜀(𝑡) = [ 1
𝑘+1
𝑟(𝑡 − 𝑡0)] 𝜎0 𝐻(𝑡 − 𝑡0)
(2.2-16)
In the complex domain the Maxwell model defines the following storage and loss
modulus:
𝐺′ =𝑘 (𝑟𝜔)2
𝑘2 + (𝑟𝜔)2 ; 𝐺′′ =
𝑘2 𝑟𝜔
𝑘2 + (𝑟𝜔)2 (2.2-17)
State of the art
24
The two step response functions are reported in the time domain in Figure 2-10.
Figure 2-10 Maxwell element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29]
From the above considerations, it is observable how the Maxwell model predicts
that the stress decays exponentially with time, which is accurate for many materials,
especially polymers. However, a serious limitation of this model is its inability to
correctly represent the creep response of solid material which does not increase
without bound. Indeed, polymers frequently exhibit decreasing strain rate with
increasing time.
b. Kelvin-Voigt model
The Kelvin-Voigt model, found in literature as simply the Kelvin model, consist
of a spring and a dashpot connected in parallel. Due to its configuration it is also known
as an iso-strain model (Figure 2-11).
a
State of the art
25
Figure 2-11 Schematic representation of the Kelvin-Voigt element
The total stress is the sum of the stress in the spring and the stress in the dashpot,
so that:
𝜎(𝑡) = 𝑘𝜀 + 𝑟𝑑𝜀
𝑑𝑡 (2.2-18)
Considering a procedure equal to the one described in the previous section the
relaxation in function of time can be obtained:
𝜎(𝑡) = 𝑘𝜀0𝐻(𝑡 − 𝑡0) + 𝑟𝜀0𝛿(𝑡 − 𝑡0) (2.2-19)
Also, the creep function is computed as:
𝜀(𝑡) =1
𝑘[1 − 𝑒−
𝑘𝑟(𝑡−𝑡0)] 𝜎0𝐻(𝑡 − 𝑡0)
(2.2-20)
In the complex domain, the Maxwell model defines the following storage and loss
modulus:
State of the art
26
𝐺′ = 𝑘 ; 𝐺′′ = 𝑟𝜔 (2.2-21)
The two step response functions for the Kelvin-Voigt element in time domain are
visible in Figure 2-12.
Figure 2-12 Kelvin-Voigt element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29]
The Kelvin-Voigt model is accurate in modelling creep phenomenon in many
materials. However, the model has limitations in its ability to describe the commonly
observed relaxation of stress in numerous strained viscoelastic materials.
It is important to notice that the Kelvin-Voigt and Maxwell models are not
equivalent, in fact the two elements describe different complex moduli.
c. Standard Linear Solid model
The Standard Linear Solid model, also known as the 3-parameter Zener model,
is defined as the combination of a Maxwell model and a Hookean spring connected in
parallel ( Figure 2-13 ).
a
State of the art
27
Figure 2-13 Schematic representation of the Standard Linear Solid element
The stress-strain relationship is computed as:
𝜎 + 𝜏𝜀𝑑𝜎
𝑑𝑡= 𝑘1 (𝜀 + 𝜏𝜎
𝑑𝜀
𝑑𝑡) (2.2-22)
where 𝜏𝜀 = 𝑟2/𝑘2 and 𝜏𝜎 = 𝑟2(𝑘1 + 𝑘2)/𝑘1𝑘2 . The stress relaxation function and
the creep function for the Standard linear model are computed in a similar fashion to
the Maxwell and Kelvin-Voigt models.
The stress relaxation function of the Standard Linear Solid is:
𝜎(𝑡) = [𝑘1 + 𝑘2 𝑒− 𝑡−𝑡0𝜏𝜀 ] 𝜀0 𝐻(𝑡 − 𝑡0)
(2.2-23)
while the creep function of the Standard Linear Solid instead is:
𝜀(𝑡) =1
𝑘1[1 + (
𝜏𝜀𝜏𝜎− 1) 𝑒
− 𝑡−𝑡0𝜏𝜎 ] 𝜎0𝐻(𝑡 − 𝑡0)
(2.2-24)
State of the art
28
In Figure 2-14 it is shown that model is accurate in predicating both creep and
relaxation responses for many materials of interest.
Figure 2-14 Standard Linear Solid element response. (a) Stress relaxation function. (b) Creep function, courtesy of H. Banks [29]
In the complex domain the Standard Linear Solid model defines the following
storage and loss modulus:
𝐺′ =𝑘1 𝑘2
2 + Ω 𝑟2 2(𝑘1 + 𝑘2)
𝑘2 2 + Ω2𝑟2
2 ; 𝐺′′ =
Ω ∙ 𝑟2 ∙ 𝑘2 2
𝑘2 2 + Ω2 ∙ 𝑟2
2 (2.2-25)
d. Structure of a generalized rheologic model
The Standard Linear Solid model shows that for the correct stress-strain behavior
modelling of a solid viscoelastic material a minimum of three elements is required.
Even when considering such a low number of parameters, different rheologic models
can be obtained by different combinations of springs and dashpots. Furthermore, not
all configurations are capable of describing solid material behavior [10].
In literature it is possible to find several rules adequate to generate models
capable of describing linear solid materials. Another important characteristic of the
rheological approach is that if the system contains more than two parameters, then for
that specific configuration it exists a conjugate model of equal complexity that is
a
State of the art
29
capable of describing the same relaxation and creep functions as the considered
system. As a reference, the scheme of the conjugate to the Standard Linear Solid model
is presented below ( Figure 2-15 ).
Figure 2-15 Schematic representation of the Standard Linear Solid conjugate element
A generalized rheological model is defined by connecting in a parallel and serial
manner N rheological elements, in order to obtain indiscriminately a more complex
behavior of the material. As an example, the generalized Kelvin-Voigt model is
presented in Figure 2-16. This can be an attractive approach for obtaining better fitting
results and more realistic performance. However, it is undeniable that the physical
poignance of the model parameters is lost in the process and leads to a drastic increase
in complexity.
Figure 2-16 Schematic representation of the Generalized Kelvin-Voigt rheological model
State of the art
30
e. Bruni, Collina model
runi and ollina’s work [30] is one of the first in literature to focus on the
implementation of generalized rheological elements inside the railway for the
modelling of railpads. Their research is primarily focused on the low-mid frequency
range, up to 500 Hz, in which the effects of the railpad on the train-track interaction is
preponderant.
Collina and Bruni employed the particular rheological family of ‘Generalized
Zener models’ for their own railpads. This implies the combination of a certain number
of Zener elements placed in parallel ( Figure 2-17 ). It is important to notice that their
implementation of the generalized Zener model considered also a non-linear Coulomb
friction element placed in parallel with the rest of the elements. Though, its effects are
secondary in the behavior of material and not quantifiable in the frequency domain.
Figure 2-17 Modelling the viscoelastic component. (a) 4-parameter Zener model. (b) Generalized 4-parameter Zener model
The identification of the model’s parameters was performed usin the results
obtained with frequency domain characterization tests. The procedure was performed
for different complexities of the generalized model, and the results were accurate even
for models with very few elements. The resulting fitting for a model composed by a
spring, a dashpot, two coulomb friction elements and two Zener elements is reported
below ( Figure 2-18 ).
State of the art
31
Figure 2-18 Comparison of the optimized Zener model with target values of equivalent stiffness and damping, courtesy of S. Bruni and A. Collina [30]
The mathematical model of the train-track interaction proposed by Bruni and
Collina is based on a finite element description of a ballastless slab-track railway and a
multi-body model of the rail vehicle. The rails are modelled as Euler-Bernoulli beam
elements while the steel plates are modelled as lumped masses. The pads under the
baseplates are represented by the optimized rheological models.
The numerical results derived from the simulation are then compared to the
impact hammer response of a 3.5 m long track segment set up. These two results are
also compared to a finite element system with pads represented by simple Kelvin-Voigt
elements. In this second case the parameters are tuned on experimental track
resonance, which was performed for two different corner frequencies of 130 Hz and
400 Hz. These are the lowest and highest resonance frequencies in the spectral interval
of interest ( Figure 2-19 ).
A comparison between the measured and simulated amplitude frequency
response of the track is shown in Figure 2-19. The track inertance provided by the
Zener rheologic model fits very well with the first and third resonances. In the second
resonance range at about 160 Hz a high sensitivity to the boundary conditions of the
track model was found. For the Kelvin-Voi t models, la elled ‘Viscous model’ in the
figure below, the accuracy is good about the resonance used for the tuning procedure
and in the non-amplified intervals, while it is over or under estimated for the other
resonant ranges.
odel
xperimental v alues
State of the art
32
Figure 2-19 Comparison between the measured and simulated amplitude frequency response of the track, courtesy of S. Bruni and A. Collina [30]
2.2.3 Non-linear models
Even though it is of crucial importance to properly define the behavior of resilient
elements inside the railway system, the vast majority of literature investigation
revolving around resilient element non-linearities has been carried out by few scholars.
Even less effort has been spent by the scientific community to try and improve the
models developed by these researchers.
The presented review is comprised of different solution ideas, each one may be
preferred due to its applicability for specific problems of interest or because of the
author’s school of thoughts. They can be divided into two main categories:
• the first one, developed by authors Wu and Thompson ( [31], [32], [33], [34] )
and T. Dahlberg ( [6] ), is primarily focused on the definition of constitutive
models of railpads capable of accounting for the dynamic preload-stiffness
relationships of resilient materials.
• the second category is instead interested in the use of rheological element
modelling to approximate the complex dynamic mechanical behavior of
resilient materials. Main contributors in the field of railway applications are
Sjoberg ( [35], [36] ) and Zhu ( [37] ) for employing a fractional derivative plus
frictional force model, De Man ( [38] ), Maes ( [5] ) and Koroma et al.( [39],
State of the art
33
[24] ) for defining a modified Poynting–Thomson rheological model capable of
accounting for both preload and frequency non-linearities typical of resilient
materials.
a. Wu, Thompson model
Wu and Thompson ( [31], [32], [33], [34] ) derived a model for the prediction of
railpad’s ehavior after an experimental survey on the static and dynamic effects of
preload on resilient materials. Their study shows that when the preloaded pad is
considered, the results are quite different from the model in which only the uniform
values of the pad is employed.
Wu and Thompson performed static and dynamic characterization tests in [4].
Following the experimental static load-deflection curve and its derived local tangent,
the static stiffness can be computed (Figure 1-5). For the dynamic stiffness the values
are extracted from experimental measurements at specific frequencies of harmonic
excitation using the procedure presented in [4]. Appendix A presents a more detailed
explanation of how the laboratory data was retrieved.
The model derived by the authors is applied to a track system consisting of UIC
60 rails on monobloc concrete sleepers. The railpads are Pandrol studded 10 mm pads.
The author introduces also the effect of preload on the ballasted section of the
railway [40]. In particular both pad and ballast will be affected by their own load
amplitude non-linear behavior (Figure 2-20).
State of the art
34
Figure 2-20 Schematic representation of the loads on the pad and ballast, courtesy of T.X Wu and D.J. Thompson [31]
In order to determine the deflection of the track foundation due to wheel passage,
the wheel load can be represented by a concentrated load while the railway track is
simplified as a finite uniform beam supported by a continuous non-linear elastic
foundation. The differential equation for the rail deflection has the form of:
𝐸𝐽𝑑4𝑢
𝑑𝑥4= −
𝑓(𝑢)
𝑑
(2.2-26)
where 𝑓(𝑢) is the reaction force of the non-linear foundation as a function of
deflection 𝑢. In addition, 𝑑 is the span length, 𝐸𝐽 is the bending stiffness of the beam
and 𝑥 is the distance along the track. In order to compute a solution of equation
(2.2-26), boundary constraints are required.
Equation (2.2-26) describes a non-linear boundary value problem which may be
solved numerically ( results presented in Figure 2-21 ).
State of the art
35
Figure 2-21 Foundation deflection and reaction force under 75 kN wheel load. —, stiff ballast; – · –, medium ballast; - - - , soft ballast, courtesy of T.X Wu and D.J. Thompson [34]
To account for the dynamic effects of the railpad and ballast, the authors perform
an assumption: knowing the behavior of the static stiffness-deflection curve and the
dynamic/static stiffness ratios at different frequencies, it is assumed that the
dynamic/