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

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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

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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

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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

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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

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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.

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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

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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:

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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)

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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

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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:

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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

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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)

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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

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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

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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 ).

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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

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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],

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[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).

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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 ).

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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/