# Parametric study of bridge response to high speed trains SHAHBAZ RASHID

i

## Parametric study of bridge response to high speed trains

Ballasted track on concrete bridges

## SHAHBAZ RASHID

## Master of Science Thesis

## Stockholm, Sweden 2011

## Parametric study of bridge response to high speed trains

### Ballasted track on concrete bridges

## Shahbaz Rashid

### October 2011

### TRITA-BKN. Master Thesis 341, 2011

### ISSN 1103-4297

### ISRN KTH/BKN/EX-341-SE

© Shahbaz Rashid, 2011

KTH Royal Institute of Technology

Department of Civil and Architectural Engineering

Division of Structural Engineering and Bridges

Stockholm, Sweden, 2011

## Preface

This master thesis is based upon the studies conducted during February 2011 to

October 2011 at the Division of Structural Engineering and Bridges, KTH Royal

Institute of Technology, Stockholm.

MATLAB model used in this thesis was developed in collaboration with Yashar

Daroudi. I would like to express my sincere gratitude to my leading supervisor Raid

Karoumi. Without his advice and unique support this thesis would never had become a reality. Further I would like to thank John Leander for his great co-operation and help in ABAQUS modeling.

Finally, I wish to express my greatest thanks to my family, friends and colleagues, who have supported me at all stage of my studies.

Stockholm, October 2011

*Shahbaz Rashid* i

ii

## Abstract

When a train enters a bridge, passenger sitting inside will feel a sudden bump in the track, which not only affect the riding comfort of the passengers but also put a dynamic impact on the bridge structure. Due to this impact force, we have very serious maintenance problems in the track close to the bridge structure. This sudden bump is produced when train travelling on the track suddenly hit by a very stiff medium like bridge structure. In order to reduce this effect, transition zones are introduced before the bridge so that the change in stiffness will occur gradually without producing any bump.

This master thesis examine the effect of track stiffness on the bridge dynamic response under different train speeds from 150 to 350 km/h with interval 5 km/h and also estimate the minimum length of transition zones require to reduce the effect of change in stiffness on the bridge. Study also gives us some guidelines about the choice of loading model of the train, location of maximum vertical acceleration, effect of ballast model on the results and minimum length of transition zone needs to include in the bridge-track FE-model, for dynamic analysis of the concrete bridges. To carry out this research MATLAB is used to produce an input file for the ABAQUS FEM program.

ABAQUS will first read this file, model the bridge and then analysis the bridge.

MATLAB will again read the result file, process the result data and plot the necessary graphs.

The Swedish X2000 train is used for this study, which has been modeled with two different methods: moving load model and sprung mass model, in order to see the difference in results. For verification of the MATLAB-ABAQUS model, a 42m long bridge is analysed and results are compared with known results. In this study, concrete simply supported bridges with spans of 5, 10, 15, 20, 25 m have been analysed.

**Keywords**:

Ballast stiffness, transition zones, Railway bridges, the Swedish X2000 train, vertical deck acceleration, MATLAB-ABAQUS model, Finite element analysis. iii

iv

## Nomenclature

DAF Dynamic amplification factor

FE-model Finite element model

*A* Cross section area (m

2

)

*ω*

1

*ω*

2

First natural frequency of vibration (Hz)

Second natural frequency of vibration (Hz)

ERRI

HSLM

*I*

European Rail Research Institute

High-Speed Load Model

Second moment of inertia (m

4

)

*L*

*M*

*α*

Span length (m)

Cross section mass (kg/m)

Rayleigh damping coefficient

β Rayleigh damping coefficient

*v *

*ζ *

damping ratio (%)

F Concentrated Load (N)

Lt Length of transition zone (m)

TCRP Transit cooperative research program v

vi

## Contents

Preface ......................................................................................................................... i

Abstract..................................................................................................................... iii

Nomenclature...............................................................................................................v

1

Introduction .........................................................................................................1

1.1

General background ..................................................................................... 1

1.2

Aim and Scope ............................................................................................. 3

1.3

Assumptions ................................................................................................ 4

2

Literature Review.................................................................................................5

3

Methodology.......................................................................................................11

3.1

Factors influencing the bridge dynamic behaviour......................................11

3.1.1

Damping of the bridge.....................................................................11

3.1.2

Stiffness of track..............................................................................13

3.1.3

Transition zones..............................................................................13

3.1.4

Type of element ..............................................................................13

3.1.5

Filtering the data ............................................................................14

3.1.6

Section properties............................................................................14

3.2

Loading model.............................................................................................14

3.2.1

Moving load model..........................................................................14

3.2.2

Sprung mass model .........................................................................14

3.3

Bridge-track model .....................................................................................15

3.4

Convergence study......................................................................................16

3.4.1

Time Step........................................................................................16

3.5

ABAQUS Modeling.....................................................................................17

vii

3.5.1

Description of the MATLAB-ABAQUS program for Moving load model 17

3.5.2

Description of the MATLAB-ABAQUS program for Sprung mass model 19

3.6

MATLAB-ABAQUS model verification .....................................................19

3.6.1

General............................................................................................19

3.6.2

Banafjäl bridge, Single axle moving load model ..............................20

3.6.3

Banafjäl bridge, Moving load model of HSLM-A1 train..................22

3.6.4

Banafjäl bridge, X2000 train sprung mass model and moving load model 23

4

Results and discussions.......................................................................................25

4.1

Influence of the change in track stiffness on the bridge response.................25

4.1.1

Short span bridges...........................................................................25

4.1.2

Long span bridges ...........................................................................28

4.2

Influence of the transition zone ...................................................................30

4.2.1

Results for Span L=5 m ..................................................................30

4.2.2

Results for Span L=10 m ................................................................33

4.2.3

Results for Span L=15 m ................................................................35

4.2.4

Results for Span L=20 m ................................................................37

4.2.5

Results for Span L=25 m ................................................................39

4.2.6

Summary of the results ...................................................................41

4.3

Comparison of Moving load model and sprung mass model ........................44

4.3.1

Short span bridges...........................................................................44

4.3.2

Long span bridges ...........................................................................46

4.4

Comparison of bridge model with track and without track.........................47

4.4.1

Short span bridge ............................................................................48

4.4.2

Long span bridge.............................................................................49

4.5

Acceleration along the Rail .........................................................................50

5

Conclusions and further research ........................................................................53

5.1

Conclusions .................................................................................................53

5.2

Further research..........................................................................................54

Bibliography ..............................................................................................................55

Appendix A Modes of vibration included in the results .............................................58

5m span bridge .....................................................................................................58

viii

10m span bridge ...................................................................................................57

15m span bridge ...................................................................................................58

20m span bridge ...................................................................................................59

25m span bridge ...................................................................................................60

Appendix B................................................................................................................62

MATLAB codes for Moving load model .....................................................................62

MATLAB codes for Sprung mass model.....................................................................74

ix

x

## 1** ** Introduction

## 1.1** **General background

Now a days improvement of railway infrastructure is a main concern due to constantly increasing demand for the high-speed railway lines in different parts of the world especially in Sweden. There are no such clear standards to define high-speed line.

Several concepts exist for the high-speed line. UIC defines the high-speed line as a line, which allows the train to operate above 250 km/h throughout the journey or a significant part of the journey [1].

In July 2011, according to [2], there are 16954 km of high-speed lines in operation in the world, 8040 km under construction and 17643 km planned. This gives a total of

42637 km, expected by the UIC by 2025. In the coming years High-speed railway lines are planned to be the standard of the railways. Some Maps of the existing high-speed railway system and planned projects around the world are shown in figure 1.1.

Figure 1.1 High speed railway systems around the world-2009 [3]

1

CHAPTER 1. INTRODUCTION

Figure 1.2 High speed railway systems forecast in 2025 [3]

Figure 1.3 High speed railway systems for Sweden in 2010 [3]

2

1.1 General background

Bridges are built e.g. when railway line crosses a river or an existing road. Special attention should be given while selecting the design loads for the bridge because now railway track is not in contact with the ground. All the loads have to carry by the bridge structure.

Bridges constructed for high speed lines should take care of the dynamic loads and resonance effects. Special standards and codes are available for the high speed train loads. Dynamic effects are usually considered in terms of dynamic amplification factor

(DAF). DAF is a measure of dynamic response with respect to static response for a moving load [4]. However, DAF does not include the effect from the resonance, which may occur due to e.g. repeatedly moving axle loads. To include the resonance effect in our calculations detailed dynamic analysis of the bridge is required.

The Eurocode [4] specifies the conditions under which a dynamic analysis is required.

In the dynamic analysis of the bridge, Train is modeled as a series of moving axle loads travelling over the bridge at different speeds. From this analysis maximum vertical acceleration is calculated against resonance speed. Maximum peak deck acceleration due to train load should fulfill the safety criteria according to EN 1990 [5].

Ballastless tracks perform better than traditional ballasted tracks on high-speed lines.

Reason to use ballasted track is that they are much cheaper to build. But they require frequent maintenance, which can cost a lot in the long time run.

One of the main difficulties in modeling ballasted-track railway bridges is that the influence of track superstructure which is composed of rail, sleeper and ballast is not very much known. For example there are no clear recommendations in the design code weather to include ballast in to account for dynamic analysis or not. Many studies have been done before in modeling bridge-track system [6, 7, 8, 9], where track and bridge has been modeled by two beams and the effect of ballast has been introduced with a more advanced system of visco-elastic springs/dampers and mass between two beams. In [10] a special finite element is developed which also include the ballast layer and accounts for the slip between the ballast and the bridge deck.

## 1.2** **Aim and Scope

Stiffness of the track on soil foundation is different from the stiffness over the bridge.

The purpose of this study is to see the effect on dynamic response of the bridge due to change in track stiffness without including the transition zones and to see the effect by introducing transition zones. Also, the aim is to estimate the minimum length of transition zones require to reduce the effect of change in stiffness on the bridge. Study will also give us some guidelines about the choice of loading model of the train, location of maximum vertical acceleration, effect of ballast model on the results and minimum length of transition zone needs to include in the bridge-track FE-model.

Scope of this study is limited to only concrete bridges and the Swedish X2000 train, which is running at a speed of 150 to 350 km/h. Train has been model with two different methods: moving load model and sprung mass model in order to see the

3

CHAPTER 1. INTRODUCTION difference in results. In this study, concrete simply supported bridges with spans of 5,

10, 15, 20, 25 m have been analysed.

## 1.3** **Assumptions

Main assumptions considered during this study and for modeling the bridge-track model are listed below.

The dynamic analysis is performed on a 2D-model. Since a plane model is adopted in the present analysis, the two rails are treated as one and whole bridge is treated as one beam. The influence from the rail irregularities are neglected in this study. Bridge structure is model with Timoshenko beam elements resting on simple supports.

Damping ratio for each span length is selected according to the guidelines provided by the Eurocode. All the sleepers are placed at equidistance. Track is symmetric in the longitudinal direction and lateral motion of the train is neglected. The Swedish X2000 train is uses for analysis, which moves with constant speed from (150-350 km/h) with interval 5 km/h. All wheels are assume to contact rigidly and continuously with the track as they roll over. Element size for B21 element is taken as 0.6m and results are assumed to be converging at this element size. Some assumptions are also made for sprung mass model of the train. Coupling provided by bogies and vehicle box is neglected. Rocking motion of the vehicle box is neglected.

4

## 2** ** Literature Review

In this chapter, a short description of important literature is presented. The purpose of this section is to provide an overall view of the existing literature related to bridge track model, loading model and methods used for the dynamic analysis of the bridge by different researchers. Unfortunately, there is not much literature related to our subject, but somehow they can help at different stages of the research.

B. Biondi, G. Muscolino, A. Sofi [6] has presented a numerical procedure for dynamic analysis of train-track-bridge system by a substructure approach. In order to carry out this investigation they modeled rail and bridge as Bernoulli-Euler beams, train as a sequence of identical vehicles moving at constant speed, while ballast as viscoelastic foundation. Basically the idea is to treat rail, bridge deck and train as three separate substructures. The problem with vehicle-bridge dynamic interaction is solved by applying a particular variant of the traditional component-mode synthesis method.

The purpose for applying these variant is to enable condensation of the axle degree of freedom into those of rail in contact taking into account the interaction effect from all three substructures, Which helps in reducing number of variables involve in dynamic analysis of railway bridge. Accuracy of this method has been check by a case study done by finite element method and result proves to be extremely well. This numerical procedure allows us to deal with vehicle models of various complexity and different boundary conditions.

P. Museros, M.L. Romero, A. Poy, E. Alarcon [11] have worked on the problem using moving load model for short span bridges. Because moving load model is considered not a good option for the study of short span bridges (L ≤ 20-25 m) since the results obtained (acceleration and displacement) from this model is much conservative than obtained by experiment. In their research they have studied two factors, which are believed to have influence on the dynamic behaviour of the short span bridges. These factors are the distribution of load through sleepers and ballast layer, and the interaction between bridge and train model. These factors are usually ignored in the moving load model. After running several numerical simulations, they have found that the distribution of load through sleepers and ballast does not have any influence on the results, While the interaction between bridge and train cause a considerable reduction in acceleration and displacement of the short span bridges.

To support their finding they study 25 numbers of simple supported bridges with 10m span length. From their investigation they have found that the reductions obtained in bridges with different natural frequencies and moment of inertia are almost proportional to each other. Coefficients of proportionality computed for acceleration and displacement are called intensities of reduction. These intensities of reduction can

5

CHAPTER 2. LITERATURE REVIEW be approximated accurately by numerical expression. Comparison is made between impact coefficient and maximum acceleration values obtained from interaction model, which gives very satisfactory results.

Jose N. Varandas, Paul Hölscher, Manuel A.G. Silva [12] have studied the dynamic behavior of railway track on transition zones. In their study they have presented a numerical solution for dynamic loads on the ballast by train passing over the transition zone. Numerical model has been checked with the field measurement data collected from two transition zones in the Netherlands. Results from both methods are quite similar with each other.This means that the numerical model describes the dynamic behavior of the track on transition zone by train passage very well. It also takes in to account the long term track deformation, the non-constant stiffness of the support and the possibility of voids under the sleepers.

Constanca Rigueiro, Carlos Rebelo, Luıs Simoes da Silva [7] have done an investigation about the influence of ballast model in the dynamic response of railway viaducts. They carry out their investigation using three models for track and two loading models, the moving load and train-structure interaction model. Three real structures whose modal parameters and acceleration response under real traffic was available have been used for comparison with the response from these models. The computed acceleration response has been compared in time domain. While track models were analyzed in frequency domain and results were compared with model having no track model to see the difference. The results show that track model does not affect the frequency content when frequency is between 10-15 Hz. But for higher frequencies track model act like a filter.

K. Liu, G. De Roeck, G. Lombaert [13] have investigated in their research which conditions train-bridge interaction model should be considered for the dynamic analysis of a bridge by passing train. Also they have studied the effect of several other parameters related to bridge and train model. Like the ratio of the mass of the vehicle and the bridge, the ratio of the natural frequency of the vehicle and the bridge, the train speed and the damping ratio of the bridge are considered to be most important factors that determine the effect of train-bridge interaction on the dynamic response of bridge. From their results it has been seen that at critical speed or at resonance speed the train-bridge interaction model gives less values for acceleration as compared to moving load model. This reduction is large for acceleration at mid span as compared to corresponding displacement results.

Dynamic response of the bridge can be accurately estimated by moving load model when ratio of the natural frequency of the vehicle and the bridge is much smaller than one. With the increase in this ratio the dynamic analysis by interaction model becomes more and more important. Also interaction model becomes more important for dynamic analysis when the ratio of mass of the vehicle to the mass of the bridge is relatively high. For low values of mass ratio moving load model is enough for dynamic analysis. An increase in damping ratio of the bridge results in a decrease in dynamic response.

6

CHAPTER 2. Literature review

ERRI D 214/RP 9 Part A [14] main author of this part is I. Bucknall. Part A of the report mainly present the methods for calculating dynamic effects (Acceleration, displacement, etc.), criteria which needs to be verified, the dynamic signature of a train and recommend some values for key parameters used in calculations and measurements. Flow chart has been presented in the report which decides whether a dynamic analysis is required or not. In case where dynamic analysis is required methods have been presented ranging from simplest to more complex ones. Methods presented in the report have been tested on different span of bridges. Calculations are compared with the actual measurements collected from field tests, to ensure that the dynamic calculations are sufficiently representative of the actual results.

7

CHAPTER 3. METHODOLOGY

10

3.1 Factor influencing the bridge dynamic behaviour

## 3** ** Methodology

## 3.1** **Factors influencing the bridge dynamic behaviour

### 3.1.1** **Damping of the bridge

Fryba (1996) [15] Damping is describe as a property of building material and structure, which in most cases reduce the dynamic response and helps the bridge to reach to its state of equilibrium after the train passage. In bridge structures, damping comes from many sources which has been divided into two main categorise internal source and external source. Internal source of damping comes from the internal friction, cracks and non-homogeneous properties of building material etc. External source of damping in the bridges comes from friction between supports and bearing, friction in the ballast, friction in the joints of the structure, viscoelastic properties of soil, foundation and abutments and so on. As we can see that damping of the structure depends upon many factors so it is almost impossible to make any engineering calculations for damping. Code recommend some values for initial assessment of the bridge damping but real damping values should be determined from measurements.

Bridge Type

Steel and Composite

Pre-stressed Concrete

ζ Lower limit of percentage of critical damping [%]

Span L < 20m Span L ≥ 20m

ζ =0.5+0.125(20-L)

ζ =1.0+0.07(20-L)

ζ =0.5

ζ =1.0

Filled beam and reinforced

Concrete

ζ =1.5+0.07(20-L) ζ =1.5

Table 3.1 Code recommendation for new bridges from ERRI D214/RP-9 [14]

11

CHAPTER 3. METHODOLOGY

In ABAQUS critical damping values recommended by the code as shown above are used in modal dynamic method. However, for time integration or dynamic method equivalent Rayleigh Damping is calculated defined as Rayleigh mass proportional damping for that purpose damping coefficients are calculated.

Rayleigh damping is a classical method of idealising damping ratios into damping coefficients, which is use in the finite element model and it is sufficient for linear analysis [16].

Figure 3.1 variation of modal damping ratios with natural frequency [16]

2ζω

1

ω

2

/ω

1

+ω

2

(3.1)

2ζ/ω

1

+ω

2

(3.2)

= α and =β

ω

1

= First natural frequency of vibration and ω

2

= Second natural frequency of vibration

Following damping values are used as an input in the current models for dynamic analysis

Span Length

[m]

5

10

15

20

25

Critical damping

Ratio %

0,0255

0,022

0,0185

0,015

0,015

First frequency

[Hz]

32,5

13,34

8,4

5,3

4,06

Second frequency

[Hz]

110,19

46,306

28,88

18,6

14,156

Rayleigh damping

Alpha

Beta

1,2799770

0,0003574

0,4556847

0,0007377

0,2407700

0,0009925

0,1237406

0,0012552

0,0946531

0,0016469

Table 3.2 Damping in the concrete beams according to code recommendation

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3.1 Factor influencing the bridge dynamic behaviour

### 3.1.2** **Stiffness of track

The FE models of the track over the bridge include elements for the ballast, sleepers and the connections between the rails and sleepers. Each element is a combination of all these specific functions. These elements behave as a complete track when subjected to train passage.

A study is performed in section 4.1 to see the influence from the track stiffness on the bridge dynamic response.

After reading different literature about the track stiffness and our own study, the following values are selected as an input for this study. A stiffness value of 400 MN/m is used for springs over the bridge; 100 MN/m spring stiffness is used for track before and after the bridge, while for transition zones an average spring stiffness value of 250

MN/m is used.

### 3.1.3** **Transition zones

Transitions zone is defined as interface points between ballasted track and bridge structure or locations of sudden changes in track stiffness. Locations where track stiffness changes abruptly got serious problem of vertical alignment and the passengers sitting inside the train can feel a sudden bump in the track due to change in vertical acceleration [17]. In order to smoothen out this effect transition zones are introduce between the track and the bridge. Going from soft to stiff track is worse than going from stiff to soft track. In North America, an effect was made to compensate for the stiffness difference by using a reinforced concrete slab (also called approach track) just before the bridge. These transition slabs are 6 meter long and embedded in the ballast at 300 millimetres from the bottom of the sleeper [17].

In this study, we will try to find out the minimum length of transition zone require to be included in the track to reduce the effect of change in stiffness. For that purpose different lengths of transition zones (0L, 0.25L, 0.5L, 0.75L, L; L=bridge span) will be used in the model to see which one is more appropriate. An average stiffness value of bridge and track is used for the stiffness of the transition zones.

### 3.1.4** **Type of element

In ABAQUS there are two types of 2D beam elements available: The Euler-Bernoulli beam element called B23 element and Timoshenko beam element called B21 element.

Main difference between these two elements is that the Timoshenko beam element consider the shear deformation in the calculation, while in Euler-Bernoulli beam element the shear deformations is ignored. In this study we will use B21 element in our model.

13

CHAPTER 3. METHODOLOGY

### 3.1.5** **Filtering the data

Higher frequency accelerations or displacements do not have any significant effect on the ballast and need to be filtered out. [4] Recommend us that all the modes with frequencies higher than 30 Hz or 1.5 times the first frequency should be excluded from the results. Different filtering techniques can be used depending upon the method of analysis. When using numerical methods time step is used as a cutoff frequency for getting sufficient accuracy. For example 0.002 sec time is sufficient for modal frequencies up to 50 Hz [14]. In the modal dynamic method there is a provision for max frequency of interest and number of modes to be included in the analysis. For sprung mass model butterworth filter is uses to remove the higher frequencies.

### 3.1.6** **Section properties

Section properties of the bridges used in this study are calculated by the graphs presented below. These graphs are taken from Christoffer’s work [19]. Regression line is used to calculate the section properties of the bridges. These graphs include the mass of the ballast.

Figure 3.2: Frequency for different spans of Reinforced and pre-stressed concrete bridges [19].

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3.1 Factor influencing the bridge dynamic behaviour

Figure 3.3: Mass for different spans of Reinforced and pre-stressed concrete bridges

[19].

Following table is produced from the above graphs.

Span Length

[m]

5

10

25

15

20

Poision

Ratio

(vi)

0,2

0,2

0,2

0,2

0,2

Conc. Density

[Kg/m3]

2400

2400

2400

2400

2400

Modulus of elasticity E

[Gpa]

25

25

25

25

25

First frequency from graph

[Hz]

33

14,2

8,9

5,6

4,3

M from graph

[Kg/m]

9000

12222

15000

17777

20555

A

[m2]

I

[m4]

1

0,10

1

0,40

1

0,99

1

1,46

1

2,43

Table 3.3 Section properties of the bridges used in the current study

15

CHAPTER 3. METHODOLOGY

## 3.2** **Loading model

### 3.2.1** **Moving load model

The simplest method of calculating dynamic response in the railway bridge is develop by Frỳba and Naprstek [15]. In this method, train is modeled with a series of axle forces moving at a constant speed over the bridge. This method considers the components of forced and free vibration. The only short come of this method is that it does not take in to account the inertia effect of the train mass and dynamic interaction between the train and the track [14].

### 3.2.2** **Sprung mass model

The simplified interaction model is easier to construct and less time consuming as compared to detailed interaction model. ERRI D214 / RP 9 (sec 13.9) [14] recommend us that for span less than 30 meter simplified interaction model and detailed interaction model produce almost same results. So for that reason simplified interaction model is selected for this thesis work.

Each axle of simplified interaction model consists of two masses connected by a spring and a damper. Upper mass represent the suspended mass of the bogie, lower mass represent the unsprung mass of the wheel set, while spring and damper represent the primary suspension system as shown in figure below.

Suspended mass m1 k

Primary suspension system m2

Unsprung mass of wheel set

Figure 3.4: Sprung mass axle

Complete X2000 train is model with these axles for the current analysis.

14

3.3 Track model

## 3.3** **Bridge-track model

Track structure is added before and after the bridge in order to include the dynamic effect from the track structure over the bridge. Transition zones are modeled between bridge and track structure with average stiffness value of bridge and track model as shown in the figure below. v y Rail

K

3

K

2

K

1 x

L

X

L t

Bridge

L b

L t

L

X

Figure 3.5: Bridge-track model with sprung mass model of the train v y

K

3

K

2

K

1

Rail x

L

X

L t

L b

L t

L

X

Bridge

Figure 3.6: Bridge-track model with moving load model of the train

Lx shows the length of track on which train is standing, Lt represent transition zone length and L bridge. b

represent the length of bridge structure, similarly on the other side of the

K

1 stiffness of track (Lx)

K

3 stiffness of the bridge (L b

)

K

2

stiffness of the transition part (Lt)

Bridge structure is modeled with Timoshenko beam elements resting on simple supports. The track system lying on the bridge is modeled with an infinite length of rail supported by a continuous and homogeneous viscoelastic foundation of springs and dampers. As we are using 2D model for this study, the two rails are replaced with one and whole bridge is replaced with one beam. Damping of the ballast is kept constant for the whole model.

15

CHAPTER 3. METHODOLOGY

## 3.4** **Convergence study

Accuracy of the result and total analysis time are very much dependent on the time step and should be selected carefully. Convergence study on time step has been carried out for each span length and loading model. As an example the analysis of a 10 m long concrete bridge has been presented below. The convergence study have been performed with both loading models of the train: moving load model and sprung mass model at a certain speed.

### 3.4.1** **Time Step

Analysis is made with different time steps by using moving load model of the train at train speed 170 km/h. As an example, results are plotted for 10m span bridge. As shown below.

Velocity 170 km/h

Figure 3.7 Absolute maximum vertical acceleration vs Time step for moving load model of the train

We can see from the graph that there is not that much difference in the results even for larger time step. To be more precise in the results a time step of 0.001 sec is selected for the analysis. All the other bridges in the thesis also converge at this time step.

16

3.4 Convergence study

Convergence is also made for same 10 m span bridge with sprung mass model of the train at a speed of 160 km/h as shown below.

Velocity 160 km/h

Figure 3.8 Maximum absolute vertical acceleration vs Time step for sprung mass loading model of the train

From the above results it can be seen that a good convergence is not achieved. In order to achieve a good convergence a smaller time step is required. A smaller time step means a longer analysis time, which is not possible for this master’s thesis. So a reasonable time step 0.0005 sec is selected for analysis.

## 3.5** **ABAQUS Modeling

### 3.5.1** **Description of the MATLAB-ABAQUS program for

### Moving load model

A small description about the developed MATLAB-ABAQUS program is presented in figure 3.9

17

CHAPTER 3. METHODOLOGY

New velocity

Input variables

Create bridge structure

Create Loads and Amplitudes

Write ABAQUS Input file

FE analysis

Read Acceleration and displacement

Figure 3.9 Schematic diagram for moving load MATLAB-ABAQUS model

*Input variables*: input variables are provided in the MATLAB, which remain constant for one type of the bridge. These input variables include damping ratio, sleeper spacing, stiffness of ballast, stiffness of bridge and rail structure, span length and element size etc.

*Create bridge structure*: Bridge structure is created from these inputs during the analysis

*Create Loads and Amplitudes*: Moving load function (by john Leander) is used to create loads and amplitudes. These loads and amplitudes create moving load model of the train.

*Write ABAQUS input file*: Loads and Amplitudes are then copied in the main input file, which is then used for analysis in the ABAQUS.

*FE analysis*: Model dynamic method of analysis is used for this model. Main input file is given as an input in the ABAQUS. ABAQUS will first read the file, create the model, run the analysis and produce result files.

*Read Acceleration and displacement*: After the completion of the analysis, MATLAB will read the DAT file and extract the required accelerations, displacements and store them in a vector.

*Loop for velocity*: ‘’For loop’’ is introduced in the program in order to get results for different velocities from 150 km/h to 350 km/h with an interval of 5 km/h. Therefore, after storing the data the program overwrites the main input file for next velocity and goes on up to the last velocity.

18

3.5 ABAQUS modelling

### 3.5.2** **Description of the MATLAB-ABAQUS program for

### Sprung mass model

New velocity

Input variables

Create bridge structure

Create sprung mass system

Write ABAQUS Input file

FE analysis

Filter the DATA

Read Acceleration and displacement

Figure 3.10 Schematic diagram for Sprung mass MATLAB-ABAQUS model

Most of the steps in this diagram have already been described in previous section. The steps which are different from moving load model are presented below.

*Create Sprung mass system*: Each axle is created with two masses connected by a spring and a damper. Complete train is modeled with these axles and move over the rail called master surface.

*FE analysis*: Direct integration method is used for this analysis.

*Filter the Data*: Butterworth filter is used to exclude the higher frequencies from the results.

## 3.6** **MATLAB-ABAQUS model verification

### 3.6.1** **General

A railway bridge on the Bothnia Line is selected for verification of this model. All the bridges on this line are design for train speed of 300 km/h. The bridge selected for verification of the MATLAB-ABAQUS program is a 42 m long simply supported composite bridge with single ballasted track called Banafjäl Bridge.

19

CHAPTER 3. METHODOLOGY

Main reason for selecting this bridge for verification of the model is that this bridge was studied in structural dynamics course (Exercise B, AF 2011) also a lot of research has been done before on this bridge. This means, we can rely on the results comfortably in order to check our program.

### 3.6.2** **Banafjäl bridge, Single axle moving load model

As an initial step, single axle load is run over the bridge with a constant speed of 250 km/h; the results are compared with exercise results and with the numerical method named as exact method presented in [18] at mid span of the bridge. As shown in the figure below.

A simply supported bridge subjected to a constant force F, moving at constant speed v, is one of the few moving force problem which can be solve analytically.

Figure 3.11 simply supported Banafjäl bridge subjected to constant moving force,

Equivalent steel section inputs taken from exercise B2 (AF 2011 March 2010)

Using notations of figure 3.11, the analytical solution (presented in Karoumi’s PhD thesis 1998 as exact solution [18]) for displacement is given below.

(3.3)

Where і is the mode number, the circular frequency for ith mode of vibration and α non-dimensional speed parameter and α are defined as

(3.4)

(3.5)

20

3.6 MATLAB-ABAQUS model verification

Simply supported bridge model presented in figure 3.11 was solved using analytical solution (exact method) and with the MATLAB-ABAQUS program.

Results have been collected for first three natural frequencies and displacement at mid span and compared with exercise results.

Mode number Analytical method MATLAB-ABAQUS program

Exercise B2 Task 1

1 2.3838 2.3856 2.3857

2 9.5352 9.5426 9.5426

3 21.4542 21.4710 21.4711

Table 3.4 Comparison of first three natural frequencies (Hz) of the Banafjäl bridge

Figure 3.12 Vertical displacement versus time at mid span of the bridge deck

The above figure shows a good resemblance in the results for mid span vertical displacement. In order to be more sure about the model, bridge deck vertical acceleration is also plotted at mid span of the bridge and compare with exercise results.

As shown in figure 3.13.

21

CHAPTER 3. METHODOLOGY

Figure 3.13 Bridge deck vertical mid span acceleration in time domain

From the figure, we can say that the program is working well and we can move to the next step for modeling complete trains.

### 3.6.3** **Banafjäl bridge, Moving load model of HSLM-A1 train

After getting the satisfactory results from single axle force model, complete moving force model of HSLM-A1 train is constructed and run over the bridge with different speeds (120-250 km/h at 5 km/h interval). Results are compared with the exercise results, which are shown below.

Figure 3.14 Abs. max bridge deck vertical displacement against train speed

22

3.6 MATLAB-ABAQUS model verification

Figure 3.15 Abs. max bridge deck vertical acceleration against train speed

From the above two figures we can say that the program is good enough to be used for this study.

### 3.6.4** **Banafjäl bridge, X2000 train sprung mass model and moving load model

In this section we will compute the same results as in section 3.6.3 but with X2000 train and for moving force model and sprung mass model. Comparison of the results is presented in figure 3.16 and 3.17.

23

CHAPTER 3. METHODOLOGY

Figure 3.16 Absolute maximum bridge deck vertical acceleration against train speed

Figure 3.17 Absolute maximum bridge deck vertical displacement against train speed

The above figures show that moving load model provide more conservative results than sprung mass model at resonance speed.

24

## 4** ** Results and discussions

After verification of the model, results are produced for five simply supported concrete bridges of span lengths [5 10 15 20 and 25m]. Two type of loading models: moving load model and sprung mass model of the Swedish X2000 train are used in this study to see the difference in results. Five different lengths of transition zone [0L, 0.25L, 0.5L,

0.75L and L; L=bridge span] are studied for each beam to find out minimum length of transition zone required to include its effect in the dynamic analysis of the bridge.

Some other parameters, which can affect the dynamic analysis results are also investigated like change in stiffness of the track, type of loading model, effect of track structure, location of maximum acceleration peak etc.

## 4.1** **Influence of the change in track stiffness on the bridge response

There is no specific single value for the track stiffness in the literature. Every author uses a different number for the track stiffness depending upon their model. But from the literature we can extract a range of this value (100-400 MN/m). To study more about this subject bridge-track model as describe in section 3.3 is uses to plot maximum vertical acceleration and displacement of the bridge against different track stiffness values. As shown in figures 4.1, 4.2, 4.5 and 4.6.

Results are produced for 5m and 25m span with both type of loading models.

### 4.1.1** **Short span bridges

To study the influence from track stiffness in short span bridges. A bridge of 5 m span length described in table 3.3 is used for this study. Following graphs are plotted at a train speed of 275 km/h.

25

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.1 Absolute maximum vertical acceleration versus vertical track stiffness

Figure 4.2 Absolute maximum vertical displacement versus vertical track stiffness

It can be seen from figures 4.1 and 4.2 that the absolute maximum vertical acceleration and displacement of the bridge increases with the increase in stiffness difference between the track and the bridge up to a certain number and then become constant, in both type of loading models.

26

4.1 Influence of the change in track stiffness

To get a more clear picture about the influence of the change in track stiffness on the bridge dynamic response, vertical accelerations and displacements are plotted against train speed for three different stiffness values. As shown below

Figure 4.3 Vertical acceleration as a function of train speed, for 5m span bridge, using moving load model of the train with three different vertical bridge track stiffness values.

Figure 4.4 Vertical displacement as a function of train speed, for 5m span bridge, using moving load model of the train with three different vertical bridge track stiffness values

27

CHAPTER 4. RESULTS AND DISCUSSIONS

From the above results, we can say that the change in track stiffness value has a considerable effect on the results and should be selected carefully for short span bridges.

### 4.1.2** **Long span bridges

Same results are plotted for 25m span bridge, in order to see the effect of track stiffness on longer spans.

Figure 4.5 vertical acceleration against vertical track stiffness for 25m span bridge with train speed of 180 km/h

Figure 4.6 vertical displacement against vertical track stiffness for 25m span bridge with train speed of 180 km/h

28

4.1 Influence of the change in track stiffness

Figure 4.7 Absolute maximum vertical acceleration vs train speed for 25 m span bridge, using moving load model of the train for three different bridge track stiffness values

K=200 MN/m

K=400 MN/m

K=600 MN/m

Figure 4.8 Absolute maximum vertical displacement vs train speed for 25m span bridge, using moving load model of the train for three different bridge track stiffness values

29

CHAPTER 4. RESULTS AND DISCUSSIONS

Above results for 25 m span bridge can be summarise as that, for longer span bridges the change in stiffness of the track has a negligible effect on the acceleration and displacement. From this study, we can conclude that the change in track stiffness value should be carefully studied, when analysing the short span bridges. While in long span bridges the change in track stiffness values does not affect the results. After reading literature and the above results, a track stiffness value of 400 MN/m is selected on the bridge for this thesis.

## 4.2** **Influence of the transition zone

### 4.2.1** **Results for Span L=5 m

Results for 5m span bridge are presented below for both type of loading models: moving load model and sprung mass model.

Moving load model

Figure 4.9 Abs. max. vertical acceleration of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

By looking at the figure 4.9, we can say that 25 % length of the span is enough to include the effect from the transition zone in the dynamic analysis of the bridge.

However, in order to be sure about the conclusion vertical acceleration of the bridge is also plotted in time domain at 275 km/h for three different transition lengths 0L, 0.25L and L, as shown in figure below.

30

4.2 Influence of the transition zone

Figure 4.10 vertical mid span acceleration of the bridge at 275 km/h train speed, using moving load model of the train for different lengths of transition zone (LT).

Figure 4.10 satisfy the same conclusion, as it has been describe for figure 4.9 that, only

0.25L length of transition zone is required to include the effect from transition zone in vertical acceleration of the bridge.

Figure 4.11 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

31

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.12 vertical mid span displacement of the bridge at 275 km/h train speed, using moving load model of the train, for different lengths of transition zone (LT).

Figure 4.11 and 4.12 shows the results for vertical displacement of the bridge, when running moving load model of the train. Results show the same behaviour as discussed for vertical acceleration of the bridge.

Same bridge span and input parameter are used to plot the results for sprung mass model of the X2000 train.

Sprung mass model

Figure 4.13 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

32

4.2 Influence of the transition zone

Figure 4.14 Abs. max. vertical displacement of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

Almost the same behaviour can be seen in the sprung mass model of the train. Only peak of the acceleration figure 4.9 is shifted to lower speed in figure 4.13.

### 4.2.2** **Results for Span L=10 m

Moving load model

Figure 4.15 Abs. max. vertical acceleration of the bridge against train speed using moving load model of the train, for different lengths of transition zone (LT).

33

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.16 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train for different lengths of transition zone (LT).

Sprung mass model

Figure 4.17 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

34

4.2 Influence of the transition zone

Figure 4.18 Abs. max. vertical displacement of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

### 4.2.3** **Results for Span L=15 m

Moving load model

Figure 4.19 Abs. max. vertical acceleration of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

35

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.20 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

Sprung mass model

Figure 4.21 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

36

4.2 Influence of the transition zone

Figure 4.22 Abs. max. vertical displacement of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

### 4.2.4** **Results for Span L=20 m

Moving load model

Figure 4.23 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

37

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.24 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

Sprung mass model

LT=L

LT=0.5L

LT=0

Figure 4.25 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

38

LT=L

LT=0.5L

LT=0

4.2 Influence of the transition zone

Figure 4.26 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

### 4.2.5** **Results for Span L=25 m

Moving load model

Figure 4.27 Abs. max. vertical acceleration of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

39

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.28 Abs. max. vertical displacement of the bridge against train speed, using moving load model of the train, for different lengths of transition zone (LT).

Sprung mass system

LT=L

LT=0.5L

LT=0

Figure 4.29 Abs. max. vertical acceleration of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

40

4.2 Influence of the transition zone

LT=L

LT=0.5L

LT=0

Figure 4.30 Abs. max. vertical displacement of the bridge against train speed, using sprung mass model of the train, for different lengths of transition zone (LT).

### 4.2.6** **Summary of the results

Results for bridge spans of 5, 10, 15, 20 and 25 m are plotted and shown in section 4.2.1 to 4.2.5 for both type of loading models: moving load model and sprung mass model of the X2000 train running at constant speed 150 km/h to 350 km/h with interval 5 km/h.

Transition zones are modeled at start and end of the bridge with different lengths [0L,

0.25L, 0.5L, 0.75L and L; L=span length]. Results are collected for each transition zone length.

From the results, we can say that the dynamic response (acceleration and displacement) have a little influence due to introduction of transition zones in the model only in short span bridges [5 m to 15 m span length]. This effect decrease rapidly, from 5 m span to 15 m span and become almost zero in 20 and 25m span bridges in both type of loading models.

In addition, results describe that only 0.25L length of transition zone is enough to include the effect from transition zone in the bridge dynamic parameters.

It has been notice from the graphs, that there is a large difference in the graph shapes between both loading models. However, this difference decreases from 5m span to 20 m span bridge and become almost zero in 25 m span bridge.

Max. vertical acceleration graph of 25 m span bridge for moving load model and for sprung mass model is almost of the same shape. Which means moving load model is

41

CHAPTER 4. RESULTS AND DISCUSSIONS good enough for dynamic analysis of the long span bridges and for short spans, simple interaction model is more suitable.

Moving load model

Transition Zone Length(Lt)=0.25L

Transition Zone Length(Lt)=0

5

5

4

4

3

3

2

2

1

1

0

350

300 25

0

350

300 25

250

20

250

20

15

15

200 200

10 10

Train speed (Km/h)

150

5

Span length (m)

Train speed (Km/h)

150

5

Span length (m)

Figure 4.31 Absolute maximum vertical acceleration as a function of train speed and span length, for transition zone length (LT) =0 & LT=0.25L

Transition Zone Length(Lt)=0.25L

Transition Zone Length(Lt)=0

10

10

8

8

6

6

4

4

2

2

0

350

0

350

300 25

300 25

20

20

250 250

15

15

200 200

10

10

Train speed (Km/h)

150

5

Span length (m)

Train speed (Km/h)

150

5

Span length (m)

Figure 4.32 Absolute maximum vertical displacement as a function of train speed and span length, for transition zone length (LT) =0 & LT=0.25L

Above two figures, 4.31 & 4.32 describe the summary of section 4.2 for moving load model of the train.

42

4.2 Influence of the transition zone

Sprung mass model

Transition Zone Length(Lt)=0.25L

Transition Zone Length(Lt)=0

0

350

4

3

2

1

6

5

300 25

2

1

4

3

0

350

6

5

300 25

250

Train speed (Km/h)

200

150

5

10

15

20

Span length (m)

250

Train speed (Km/h)

200

150

5

10

15

20

Span length (m)

Figure 4.33 Absolute maximum vertical acceleration as a function of train speed and span length, for transition zone length (LT) =0 & LT=0.25L

Transition Zone Length(Lt)=0.25L

Transition Zone Length(Lt)=0

8

8

6

6

4

4

2

2

0

350

300 25

0

350

300 25

20

250

20

250

15

15

200

200

10 10

Train speed (Km/h)

150

5

Span length (m)

Train speed (Km/h)

150

5

Span length (m)

Figure 4.34 Absolute maximum vertical displacement as a function of train speed and span length, for transition zone length (LT) =0 & LT=0.25L

Above two figures, 4.33 & 4.34 describe the summary of section 4.2 for sprung mass model of the train.

43

CHAPTER 4. RESULTS AND DISCUSSIONS

## 4.3** **Comparison of Moving load model and sprung mass model

As it has been noticed in section 4.2 that moving load model and sprung mass model gives almost same results for 25m span length. This study is carried out to investigate more about this conclusion. Results are plotted for moving load model and sprung mass model on the same graph. Study is performed on 5m span and 25 m span bridges at resonance speed.

### 4.3.1** **Short span bridges

For 5m span bridge first mid span acceleration and displacement is compared at a train speed of 275 km/h and then absolute maximum moment is plotted against train speed for both type of loading models. See also figure 4.9 & 4.13

Figure 4.35 Mid span vertical acceleration of the 5m span bridge at a resonance train speed of 275 km/h.

44

4.3 Comparison of loading models

Figure 4.36 Mid span vertical displacement of the 5m span bridge at a resonance train speed of 275 km/h.

In figure, 4.35 and 4.36 mid span vertical acceleration and vertical displacement are plotted for moving load model and for sprung mass model on the same diagram so that the difference in results can be analysed in detail. The results clearly show that the moving load model of the train provide more conservative results than sprung mass model.

In order to get an idea how the absolute maximum moment in the bridge changes with the train speed. Moment diagrams are plotted for moving load and sprung mass model against train speeds.

Figure 4.37 Absolute maximum moment in the bridge against train speed for 5m span bridge.

45

CHAPTER 4. RESULTS AND DISCUSSIONS

The above figure shows that sprung mass model will calculate slightly lower moments at resonance speeds than moving load model.

### 4.3.2** **Long span bridges

Comparison is also made for 25 m span bridge at train speed of 180 km/h. Results are plotted for accelerations, displacements and moments. As shown below.

Figure 4.38 Mid span vertical acceleration of the 25m span bridge, at a train speed of

180 km/h.

Figure 4.39 Mid span vertical displacement of the 25 m span bridge, at a train speed of

180 km/h.

46

4.3 Comparison of loading models

In figure, 4.38 and 4.39 the mid span vertical acceleration and vertical displacement is plotted for moving load model and sprung mass model, for 25 m span bridge with a resonance train speed of 180 km/h.

As we can see in the graphs that the difference in results of two models is very small as compared to 5m span bridge results. That means we can use moving load model of the train for long span bridges.

Figure 4.40 Absolute maximum moment in the bridge against train speed for 25m span bridge.

Results are also plotted for absolute maximum moments in the bridge structure for several speeds as shown in figure 4.40. Results show that for higher speed sprung mass model produce less moments as compared to moving load model.

## 4.4** **Comparison of bridge model with track and without track

To study the effect of track structure in the FE modeling, two different models with track and without track structure have been prepared for 5m and 25m span bridges.

Track bridge model is shown in figure 3.6 and bridge without track model is shown in figure 4.41 below. y v

Bridge

Figure 4.41 concrete bridge without track for moving load model of the train

47

CHAPTER 4. RESULTS AND DISCUSSIONS

### 4.4.1** **Short span bridge

In order to check the effect of track model on the bridge dynamic response for short span bridges, 5m span bridge is selected for this study and results are collected.

Figure 4.42 Absolute maximum vertical acceleration of the bridge against train speed for moving load model.

Figure 4.43 Absolute maximum vertical displacement of the bridge against train speed for moving load model

48

4.4 Comparison of bridge model with and without track

In the above figures 4.42 and 4.43, there is almost 50% decrease in the peak for acceleration and 27% in displacement due to the introduction of track structure in the model. We can say that the track structure will distribute the load.

### 4.4.2** **Long span bridge

In order to check the effect of track model in the bridge dynamic response for long span bridges, 25m span bridge is uses for the analysis.

Figure 4.44 Absolute maximum vertical acceleration of the bridge against train speed for moving load model

Figure 4.45 Absolute maximum vertical displacement of the bridge against train speed for moving load model

49

CHAPTER 4. RESULTS AND DISCUSSIONS

In the above figures 4.44 and 4.45, there is no significant difference in the values.

Which means that for long span bridges, the presence of track model will not affect the results much.

## 4.5** **Acceleration along the Rail

In order to understand better about the dynamic behaviour of the bridge track model.

Absolute maximum acceleration at each point on the rail is plotted against distance for moving load model and sprung mass model of the train. y

Rail

140+L b

Bridge

L b

140+L b

Figure 4.46 Bridge track model

In the following graph, the concrete bridge is between distances 145 to 150 m. x

Figure 4.47 Absolute maximum vertical acceleration of the bridge at each node along the rail, for 5 m span bridge, at a train speed of 275 km/h.

50

4.5 Acceleration along the rail

Figure 4.47 shows that high acceleration peaks occur outside and just before the bridge and in the transition zones, which can cause track maintenance problems.

Figure 4.48 Absolute maximum vertical acceleration of the bridge at each point along the rail, for 25m span bridge, at a train speed of 180 km/h.

In the above figure 4.48 the concrete bridge is between 165 to 190 m distance.

The above two graph shows that moving load model cannot capture the bump in the track when it enters the bridge structure but sprung mass model gives us an idea how vertical acceleration changes along the length. We can see a disturbance in the transition zone in figures 4.47 & 4.48, which may cause track maintenance problem.

Maximum peak in the vertical acceleration can be seen at start and end of the bridge structure.

51

CHAPTER 5. CONCLUSIONS AND FURTHER RESEARCH

52

CHAPTER 5. CONCLUSIONS AND FURTHER RESEARCH

## 5** ** Conclusions and further research

## 5.1** **Conclusions

In this study, five different bridge span lengths 5, 10, 15, 20 and 25m were studied.

Two loading models: moving load model and sprung mass model have been used to idealise the Swedish X2000 train, which is running at constant speeds from 150-350 km/h with interval 5 km/h.

In order to estimate the minimum length of transition zone required to be included in the dynamic analysis, four different lengths of transition zones 0L, 0.25L, 0.5L and L have been tried. From this research, following conclusions can be drawn.

1. Short span bridges (L≤ 15m) behave differently under dynamic loading from long span bridges (L> 15m).

2. Stiffness of track over the bridge has a large effect on the dynamic response of short span bridges than long span bridges. That means in longer spans the dynamic response is not sensitive to track stiffness value.

3. From the results of section 4.2, we can say that a length of 0.25L transition zone should be included in the model when analysing short span bridges. However, for long span bridges we do not need any transition length to be included in the model for dynamic analysis.

4. A study had also been performed to compare moving load model of the train with sprung mass model in order to see which model is most efficient and reliable. Results shows that, for short span bridges interaction model or sprung mass model give more realistic results than, from moving load model. However, for long span bridges, results are quite similar from both models and it is better to use moving load model of the train because it is easier to construct and require less analysis time.

5. In order to see the effect of modelling track, as an elastic layer of springs and dampers, two different models with track and without track were prepared.

Dynamic response (acceleration and displacement) were plotted against train speed. Results show that introduction of elastic layer will decrease the maximum peak in the short span bridges but in long span bridges its effect is negligible.

6. At resonance the values of dynamic response (accelerations and displacements) predicted from sprung mass model are less as compared to moving force model.

53

CHAPTER 5. CONCLUSIONS AND FURTHER RESEARCH

7. The results shows that in sprung mass model the resonance will occur at lower speeds than moving force model.

8. From this research, it is recommended to use moving load model of the train if designing a new bridge because it will be on safe side. However, when calculating the capacity of an existing bridge it is better to use sprung mass model of the train because it will produce results close to the real behaviour, especially this is important for short span bridges.

## 5.2** **Further research

1. The study presented in this thesis should be extended to 30m span on ward and for other types of bridges like continuous bridges, frame bridges, composite bridges, steel bridges, etc. to see the difference in results between sprung mass model and moving load model.

2. Field tests can be made to verify the results obtain from this thesis.

3. Results from this thesis could be compare with the results from 3D model.

4. There is a need to study more about the behaviour of interaction between wheel and rail structure and its effect on the results.

5. Some results have been produced in section 4.5 to find out the peak acceleration along the rail in the whole bridge-track model. More bridges should be analysed to make a generalized conclusion and identify the area of interest for this problem.

6. Refine the same model with at least three beam elements between each spring pair, track model with ballast mass in the middle and make convergence analysis for the size of the element. Use this model to study other parameters.

7. Sprung mass model require larger analysis time than moving load model. A study can be made to reduce this analysis time.

8. During this research, it has been found that the length of track in the bridgetrack model has a large influence on the results. Research can be made to investigate the length of track required to be included in the bridge model.

9. There are no clear guidelines for the track stiffness and damping values in the literature. An investigation could be made to calculate the track stiffness and damping values to be used for FE analysis.

10. A comparison between simplified sprung mass model and more complicated detailed sprung mass model of the train could be studied to see the difference in the results.

54

BIBLIOGRAPHY

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[16] Anil K. chopra, Dynamics of structures. Theory and Applications to Earthquake

Engineering, University of California at Berkeley.

[17] TCRP rpt 57-b, 2000. Track design handbook for light rail transit, Transit cooperative research program (TCRP) Report 57b/section 4.3.3

[18] Karoumi, R., 1998. Response of Cable-Stayed and Suspension Bridges to Moving

Vehicle. Analysis methods and practical modeling techniques. Doctoral Thesis, Royal

Institute of Technology(KTH).

[19] Höghastighetsprojekt Bro, delrapport 1, KTH Kungliga Tekniska Högskolan,

TRITA-BKN Rapport 139, 2010.

[20] CarineMellier, 2010. Optimal design of bridges for high-speed trains. Single and double-span bridges. Masters thesis, Royal Institute of Technology(KTH).

56

## Appendix A

## Modes of vibration included in the results

All the mode shapes, which are included in the above results for both type of loading models, are describe in this appendix. Mode shapes are presented according to span length and frequency range.

## 5m span bridge

Mode 1, f n

=8.7664 Hz

Mode 2, f n

=17.533 Hz

Mode 3, f n

=26.299 Hz

Mode 4, f n

=31.269 Hz

Mode 5, f n

=35.065 Hz

58

APPENDIX A

Mode 6, f n

=43.830 Hz

Figure A.1 Modes 1-6 of the two-dimensional model with frequency range 0-50 Hz for 5 m span bridge.

## 10m span bridge

Mode 1, f n

=8.3422 Hz

Mode 2, f n

=13.355 Hz

Mode 3, f n

=16.684 Hz

Mode 4, f n

=25.026 Hz

Figure A.2 Modes 1-4 of the two-dimensional model with frequency range 0-30 Hz for 10 m span bridge

57

APPENDIX A

## 15m span bridge

Mode 1, f n

=7.9572 Hz

Mode 2, f n

=8.4104 Hz

Mode 3, f n

=15.914 Hz

Mode 4, f n

=21.513 Hz

Mode 5, f n

=23.871 Hz

Mode 6, f n

=28.888 Hz

Figure A.3 Modes 1-6 of the two-dimensional model with frequency range 0-30 Hz for 15 m span bridge

58

APPENDIX A

## 20m span bridge

Mode 1, f n

=5.3114 Hz

Mode 2, f n

=7.6062 Hz

Mode 3, f n

=14.822 Hz

Mode 4, f n

=15.212 Hz

Mode 5, f n

=18.612 Hz

Mode 6, f n

=22.818 Hz

Figure A.4 Modes 1-6 of the two-dimensional model with frequency range 0-30 Hz for 20 m span bridge

59

APPENDIX A

## 25m span bridge

Mode 1, f n

=4.0641 Hz

Mode 2, f n

=7.2848 Hz

Mode 3, f n

=11.028 Hz

Mode 4, f n

=14.158 Hz

Mode 5, f n

=14.569 Hz

Mode 6, f n

=21.854 Hz

60

APPENDIX A

Mode 7, f n

=27.197 Hz

Mode 8, f n

=29.139 Hz

Figure A.5 Modes 1-8 of the two-dimensional model with frequency range 0-30 Hz for 25 m span bridge

61

APPENDIX B

## Appendix B

## MATLAB codes for Moving load model

**Input variables**

clear clc

L=25; %Span Length (m)

Xi=0.015; %Damping Ratio

Frqmax=30; %Maximum Frequency of Request(HZ)

T_Inc=0.001; %Dynamic Time Increment(Sec)

E=25e9; %Uncracked Concrete Young's Modulus(N/m2) vi=0.2; %Poisson's Ratio (concrete)

Con_D=20555; %Concrete Density including ballast(Kg/m3)

Es=210e9; %Steel Young's Modulus(N/m2) vis=0.3; %Poisson's Ratio (steel)

Con_S=7850; %Steel Density(Kg/m3)

Ball_T=0.6; %Ballast Thickness(m)

Ball_w=6.2; %Ballast width(m)

Ball_D=2000; %Ballast Density(Kg/m3)

K_t=400e6; %Vertical Spring Stiffness of approach track(N/m)

K_b=400e6; %Vertical Spring Stiffness of ballast(N/m)

C_b=100e3; %Vertical Damping for ballast and approach track(Ns/m)

%% ABAQUS INPUT FILE

Sleeper_s=0.625; %Sleeper spacing or element size

Lt=0.5*L; %transition zone length

ExTrack_L=140+(L-Lt); %Track length

GroundNodes_Left=101+(ExTrack_L/Sleeper_s);

%Ground nodes before transition zone on left side

AppGroundNodes_Left=GroundNodes_Left+(Lt/Sleeper_s);

%Ground nodes for transition zone on left side

BridgeNodes=AppGroundNodes_Left+(L/Sleeper_s);

%Bridge Nodes

AppGroundNodes_Right=BridgeNodes+(Lt/Sleeper_s);

%Ground nodes for transition zone on Right side

GroundNodes_Right=AppGroundNodes_Right+(ExTrack_L/Sleeper_s);

%Ground nodes after transition zone on Right side

ExRailNodes_Left=GroundNodes_Right+1+(ExTrack_L/Sleeper_s);

%Rail Nodes on track Left side

AppRailNodes_Left=ExRailNodes_Left+(Lt/Sleeper_s);

%Rail Nodes on Transition zone Left side

62

APPENDIX B

RailNodes=AppRailNodes_Left+(L/Sleeper_s);

%Bridge Rail Nodes

AppRailNodes_Right=RailNodes+(Lt/Sleeper_s);

%Rail Nodes on Transition zone Right side

ExRailNodes_Right=AppRailNodes_Right+(ExTrack_L/Sleeper_s);

%Rail Nodes on track Right side

**ELEMENTS**

%**************************************************************************

BridgeEl=BridgeNodes-AppGroundNodes_Left; %Bridge Elements

ExRailEl_Left=ExRailNodes_Left-(GroundNodes_Right+1);

%Track Rail Elements left side

AppRailEl_Left=AppRailNodes_Left-ExRailNodes_Left;

%Transition zone Rail Elements left side

RailEl=RailNodes-AppRailNodes_Left; %Bridge Rail Elements

AppRailEl_Right=AppRailNodes_Right-RailNodes;

%Transition zone Rail Elements Right side

ExRailEl_Right=ExRailNodes_Right-AppRailNodes_Right;

%Track Rail Elements Right side

GroundSpringEl_Left=(ExTrack_L/Sleeper_s);

AppGroundSpringEl_Left=(Lt/Sleeper_s);

BridgeSpringEl=(L/Sleeper_s)+1;

AppGroundSpringEl_Right=(Lt/Sleeper_s);

GroundSpringEl_Right=(ExTrack_L/Sleeper_s);

GroundDashpotEl_Left=(ExTrack_L/Sleeper_s);

AppGroundDashpotEl_Left=(Lt/Sleeper_s);

BridgeDashpotEl=(L/Sleeper_s)+1;

AppGroundDashpotEl_Right=(Lt/Sleeper_s);

GroundDashpotEl_Right=(ExTrack_L/Sleeper_s);

%% Define coordinates

S=[];j=1; for i1=-(140+L):0.625:L+L+140

S(j,:)=[GroundNodes_Right+j,i1,0];

j=j+1; end crds=[S zeros(length(S),1)]; %coordinates of the nodes

%% Define Loads (Train:X2000)

Pa=[0,182.5;2.9,182.5;9.5,182.5;12.4,182.5;17.13,117.5;20.03,117.5;

34.83,117.5;37.73,117.5;42.08,117.5;44.98,117.5;59.78,117.5;

62.68,117.5;67.03,117.5;69.93,117.5;84.73,117.5;87.63,117.5;

91.98,117.5;94.88,117.5;109.68,117.5;112.58,117.5;117.27,137.5;

120.17,137.5;131.77,137.5;134.67,137.5];

**GENERAL SECTIONS**

%**************************************************************************

G=E/(2*(1+vi)); %Shear modulus for Bridge

I11=2.43;I12=0;I22=0; %General Section definition for Bridge

A=1;

Gr=Es/(2*(1+vis)); %Shear modulus for Rail

Ir11=0.00006120;Ir12=0;Ir22=0; %stiffness for two Rails

Ar=0.015374; %cross section area for two Rails

63

APPENDIX B

%% Generate MovingLoads (Amp&CLoad) (Train:X2000)

A2=[];

U2=[]; for v=150:5:350 %Velocity in Km/h v_msec=v/3.6;

L_totload=Lt+(2*L)+(2*ExTrack_L);

T_tot=(L_totload/(v_msec));

MovingLoad( 'input.inp' ,crds,Pa,v/3.6)

%% separate Amplitude and CLoads to copy in ABAQUS input file

fid1=fopen([ 'input' , '.inp' ] , 'r' );

f=fscanf(fid1, '%c' );

success=strfind(f, '** LOADS **' ); fclose( 'all' ); fid=fopen([ 'ML25mLt=0,5L' , '.inp' ] , 'w' );

%% HEADING

%......................................................................

Text=[ '*HEADING\n' ...

'** Job name: ML25mLt=0,5L Model name: ML25mLt=0,5L\n' ...

'**\n' ]; fprintf(fid,Text);

%ground NODES Left side

%......................................................................

Text=[ '*Node\n' ...

'101,-' num2str(Lt+ExTrack_L) ',-1\n' ...

% node nb, 1st coordinate, 2nd coordinate

num2str(GroundNodes_Left-1) ',-' num2str(Lt+Sleeper_s) ',-1\n' ...

'*NGEN, Nset=GroundNodes_Left\n' ...

% generates nodes incrementally, name of the nset

'101,' num2str(GroundNodes_Left-1) ',1\n' ...

% nb of the 1st end node, 2nd end node, increment (the default is 1)

]; fprintf(fid,Text);

%APPROACH NODES LEFT SIDE

%......................................................................

Text=[ '*Node\n' ...

num2str(GroundNodes_Left) ',-' num2str(Lt) ',-1\n' ...

num2str(AppGroundNodes_Left-1) ',-' num2str(Sleeper_s) ',-1\n' ...

'*NGEN, Nset=AppGroundNodes_Left\n' ...

num2str(GroundNodes_Left) ',' num2str(AppGroundNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

%BRIDGE NODES

%......................................................................

Text=[ '*Node\n' ...

num2str(AppGroundNodes_Left) ',0,-1\n' ...

num2str(BridgeNodes) ',' num2str(L) ',-1\n' ...

'*NGEN, Nset=BridgeNodes\n' ...

num2str(AppGroundNodes_Left) ',' num2str(BridgeNodes) ',1\n' ...

]; fprintf(fid,Text);

64

APPENDIX B

%BRIDGE ELEMENTS

%.....................................................................

Text=[ '*Element, type=B21\n' ...

'1,' num2str(AppGroundNodes_Left) ',' num2str(AppGroundNodes_Left+1)

'\n' ...

% nb of the element, 1st node forming the element, 2nde node

'*ELGEN, ELSET=Bridgebeam\n' ...

% generates elements incrementally, ELSET= give name to the beam

'1,' num2str(BridgeEl) '\n' ...

% master element nb, nb of elements to be defined

]; fprintf(fid,Text);

% Bridge beam SECTION

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_D) ',ELSET

=Bridgebeam\n' ...

% (ELSET:defines to which elements the bloc is assigned) num2str(A) ',' num2str(I11) ',' num2str(I12) ',' num2str(I22) ', 0, 0,

0\n' ...

% A, I11, I12, I22, J, (gamma)0,(gamma)w

'0, 0, -1\n' ...

% beamsectionorientation. for planar beams: 0,0,-1 num2str(E) ',' num2str(G) '\n' ...

% Young´s modulus E, shear modulus G

'**\n' ...

]; fprintf(fid,Text);

%APPROACH NODES Right SIDE

%......................................................................

Text=[ '*Node\n' ...

num2str(BridgeNodes+1) ',' num2str(L+Sleeper_s) ',-1\n' ...

num2str(AppGroundNodes_Right) ',' num2str(L+Lt) ',-1\n' ...

'*NGEN, Nset=AppGroundNodes_Right\n' ...

num2str(BridgeNodes+1) ',' num2str(AppGroundNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

%ground NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(AppGroundNodes_Right+1) ',' num2str(L+Lt+Sleeper_s) ',-1\n' ...

num2str(GroundNodes_Right) ',' num2str(L+Lt+ExTrack_L) ',-1\n' ...

'*NGEN, Nset=GroundNodes_Right\n' ...

num2str(AppGroundNodes_Right+1) ',' num2str(GroundNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

%Extra Rail NODES left side

%......................................................................

Text=[ '*Node\n' ...

num2str(GroundNodes_Right+1) ',-' num2str(Lt+ExTrack_L) ',0\n' ...

num2str(ExRailNodes_Left-1) ',-' num2str(Lt+Sleeper_s) ',0\n' ...

'*NGEN, Nset=ExRailNodes_Left\n' ...

num2str(GroundNodes_Right+1) ',' num2str(ExRailNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

65

APPENDIX B

% Extra Rail ELEMENTS left side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+1) ',' num2str(GroundNodes_Right+1)

',' num2str(GroundNodes_Right+2) '\n' ...

'*ELGEN, ELSET=ExRail_Left\n' ...

num2str(BridgeEl+1) ',' num2str(ExRailEl_Left) '\n' ...

]; fprintf(fid,Text);

% Extra Rail SECTION Left side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=ExRail_Left\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) ', 0, 0,

0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Approach Rail NODES Left side

%......................................................................

Text=[ '*Node\n' ...

num2str(ExRailNodes_Left) ',-' num2str(Lt) ',0\n' ...

num2str(AppRailNodes_Left-1) ',-' num2str(Sleeper_s) ',0\n' ...

'*NGEN, Nset=AppRailNodes_Left\n' ...

num2str(ExRailNodes_Left) ',' num2str(AppRailNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

% Approach Rail ELEMENTS Left side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(ExRailEl_Left+BridgeEl+1) ',' num2str(ExRailNodes_Left)

',' num2str(ExRailNodes_Left+1) '\n' ...

'*ELGEN, ELSET=AppRail_Left\n' ...

num2str(ExRailEl_Left+BridgeEl+1) ',' num2str(AppRailEl_Left) '\n' ...

]; fprintf(fid,Text);

% Approach Rail SECTION Left side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=AppRail_Left\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) ', 0, 0,

0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

66

APPENDIX B

%Rail NODES

%.....................................................................

Text=[ '*Node\n' ...

num2str(AppRailNodes_Left) ',0,0\n' ...

num2str(RailNodes) ',' num2str(L) ',0\n' ...

'*NGEN, Nset=RailNodes\n' ...

num2str(AppRailNodes_Left) ',' num2str(RailNodes) ',1\n' ...

]; fprintf(fid,Text);

% Rail ELEMENTS

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+1)

',' num2str(AppRailNodes_Left) ',' num2str(AppRailNodes_Left+1) '\n' ...

'*ELGEN, ELSET=Rail\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+1) ',' num2str(RailEl)

'\n' ...

]; fprintf(fid,Text);

% Rail SECTION

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=Rail\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) '

, 0, 0, 0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Approach Rail NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(RailNodes+1) ',' num2str(L+Sleeper_s) ',0\n' ...

num2str(AppRailNodes_Right) ',' num2str(L+Lt) ',0\n' ...

'*NGEN, Nset=AppRailNodes_Right\n' ...

num2str(RailNodes+1) ',' num2str(AppRailNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

% Approach Rail ELEMENTS Right side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+1) ',

' num2str(RailNodes) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppRail_Right\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+1) ',

' num2str(AppRailEl_Right) '\n' ...

]; fprintf(fid,Text);

67

APPENDIX B

% Approach Rail SECTION Right side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',

ELSET=AppRail_Right\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22)

', 0, 0, 0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Extra Rail NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(AppRailNodes_Right+1) ',' num2str(L+Lt+Sleeper_s) ',0\n' ...

num2str(ExRailNodes_Right) ',' num2str(ExTrack_L+L+Lt) ',0\n' ...

'*NGEN, Nset=ExRailNodes_Right\n' ...

num2str(AppRailNodes_Right+1) ',' num2str(ExRailNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

% Extra Rail ELEMENTS Right side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+1) ',

' num2str(AppRailNodes_Right) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=ExRail_Right\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+1) ',

' num2str(ExRailEl_Right) '\n' ...

]; fprintf(fid,Text);

% Extra Rail SECTION Right side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=ExRail_Right\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) '

, 0, 0, 0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground left side

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+1) ',101,' num2str(GroundNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundSpring_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+1) ',' num2str(GroundSpringEl_Left) '\n' ...

68

APPENDIX B

'*Spring, elset=GroundSpring_Left\n' ...

'2,2\n' ...

'100e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground left side

%.....................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+1)

'\n' ...

',101,' num2str(GroundNodes_Right+1)

'*ELGEN, ELSET=GroundDashpot_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+1) ',' num2str(GroundDashpotEl_Left) '\n' ...

'*Dashpot, elset=GroundDashpot_Left\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground left side approach track

%.....................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai

',' lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+1) num2str(GroundNodes_Left) ',' num2str(ExRailNodes_Left) '\n' ...

'*ELGEN, ELSET=AppGroundSpring_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+1) num2str(AppGroundSpringEl_Left) '\n' ...

'*Spring, elset=AppGroundSpring_Left\n' ...

'2,2\n' ...

'250e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground left side approach track

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

','

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+1

) ',' num2str(GroundNodes_Left) ',' num2str(ExRailNodes_Left) '\n' ...

% nb of the element, 1st node forming the element, 2nde node

'*ELGEN, ELSET=AppGroundDashpot_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+1

) ',' num2str(AppGroundDashpotEl_Left) '\n' ...

'*Dashpot, elset=AppGroundDashpot_Left\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

69

APPENDIX B

% Spring ELEMENTS for bridge beam

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+1) ',' num2str(AppGroundNodes_Left) ',' num2str(AppRailNodes_Left) '\n' ...

'*ELGEN, ELSET=BridgeSprings\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+1) ',' num2str(BridgeSpringEl) '\n' ...

'*Spring, elset=BridgeSprings\n' ...

'2,2\n' ...

'400e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for bridge beam

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+1) ',' num2str(AppGroundNodes_Left)

',' num2str(AppRailNodes_Left) '\n' ...

'*ELGEN, ELSET=BridgeDashpots\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+1)

'\n' ...

'*Dashpot, elset=BridgeDashpots\n' ...

'2,2\n' ...

',' num2str(BridgeDashpotEl)

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground right side approach track

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+1) num2str(BridgeNodes+1) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppGroundSpring_Right\n' ...

','

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+

AppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+1) num2str(AppGroundSpringEl_Right) '\n' ...

'*Spring, elset=AppGroundSpring_Right\n' ...

'2,2\n' ...

'250e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground right side approach track

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

','

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A

70

APPENDIX B ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+1) ',' num2str(BridgeNodes+1) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppGroundDashpot_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+1) ',' num2str(AppGroundDashpotEl_Right) '\n' ...

'*Dashpot, elset=AppGroundDashpot_Right\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground Right side

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+1) ',' num2str(AppGroundNodes_Right+1) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundSpring_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+1) ',' num2str(GroundSpringEl_Right) '\n' ...

'*Spring, elset=GroundSpring_Right\n' ...

'2,2\n' ...

'100e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground Right side

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+GroundSpringEl_Right+1) ',' num2str(AppGroundNodes_Right+1) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundDashpot_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig

',' ht+AppGroundDashpotEl_Right+GroundSpringEl_Right+1) num2str(GroundDashpotEl_Right) '\n' ...

'*Dashpot, elset=GroundDashpot_Right\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

71

APPENDIX B

% Load amplitudes

%.....................................................................

Text=[num2str(f(1:success-2)), '\n' ...

'**\n' ...

]; fprintf(fid,Text);

BOUNDARY CONDITIONS

%......................................................................

Text=[ '*BOUNDARY\n' ...

'GroundNodes_Left,Encastre\n' ...

%node nb, type of boundary

'GroundNodes_Right,Encastre\n' ...

'AppGroundNodes_Left,Encastre\n' ...

'AppGroundNodes_Right,Encastre\n' ...

num2str(AppGroundNodes_Left) ',2\n' ...

num2str(BridgeNodes) ',PINNED\n' ...

num2str(GroundNodes_Right+1) ',1\n' ...

num2str(ExRailNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

% STEPS

%......................................................................

Text=[ '** STEP: frequency\n' ...

'**\n' ...

'*Step, name=frequency, perturbation\n' ...

'*Frequency, eigensolver=Ltnczos, acoustic coupling=on, normalization=displacement\n' ...

', ,' ,num2str(Frqmax), ', , ,\n' ...

'** \n' ...

'** OUTPUT REQUESTS\n' ...

'**\n' ...

'*Restart, write, frequency=0\n' ...

'** \n' ...

'** FIELD OUTPUT: F-Output-2\n' ...

'** \n' ...

'*Output, field, variable=PRESELECT\n' ...

'*End Step\n' ...

'** -----------------------------------------------------------\n' ...

'** \n' ...

'** STEP: moving load\n' ...

'** \n' ...

'*Step, name="moving load", perturbation\n' ...

'*Modal Dynamic, continue=NO\n' ...

,num2str(T_Inc), ',' num2str(T_tot) '\n' ...

'*Modal Damping\n' ...

'1,8,' ,num2str(Xi), '\n' ...

,num2str(f(success:end)), '**\n' ...

% Copy moving Loads

'** OUTPUT REQUESTS\n' ...

'**\n' ...

'**\n' ...

'** FIELD OUTPUT: F-Output-2\n' ...

'**\n' ...

'*Output, field, variable=PRESELECT\n' ...

'**\n' ...

'** HISTORY OUTPUT: H-Output-1\n' ...

72

APPENDIX B

'**\n' ...

'*Output, history, variable=PRESELECT\n' ...

'*Node output, nset=BridgeNodes\n' ...

'A2,U2\n' ...

'*NODE PRINT, nset=BridgeNodes\n' ...

'A2,U2\n' ...

'*END STEP\n' ]; fprintf(fid,Text);

dos( 'abaqus job=ML25mLt=0,5L.inp interactive' ); fclose( 'all' ); fid=fopen([ 'ML25mLt=0,5L' , '.dat' ], 'r' );

UMAXValues=[];

AMAXValues=[];

LineText=fgetl(fid); while strcmp(LineText, ' THE ANALYSIS HAS BEEN COMPLETED' )~=1

LineText=fgetl(fid); if (length(LineText))>8 if strcmp(LineText(1:8), ' MAXIMUM' )==1

%disp('Saving acceleration maximum');

AMAXValues=[AMAXValues; sscanf(LineText, '%*s %f %*f' )];

UMAXValues=[UMAXValues; sscanf(LineText, '%*s %*f %f %*f' )]; elseif strcmp(LineText(1:8), ' MINIMUM' )==1

%disp('Saving acceleration maximum');

AMAXValues=[AMAXValues; sscanf(LineText, '%*s %f %*f' )];

UMAXValues=[UMAXValues; sscanf(LineText, '%*s %*f %f %*f' )]; end end end

AMAXabs=abs(AMAXValues); %Take the absolute value of the values

A2=[A2;max(AMAXabs)]; %Take the maximum value of the absolute values

UMAXabs=abs(UMAXValues); %Take the absolute value of the values

U2=[U2;max(UMAXabs)]; %Take the maximum value of the absolute values fclose( 'all' ); end

**Plot results**

save( 'Acceleration.txt' , 'A2' , '-ASCII' ) save( 'Displacement.txt' , 'U2' , '-ASCII' ) v=150:5:350; plot(v,A2, 'r' );title( 'TRAIN SPEED VS. VERTICAL ACCELERATION' );xlabel( 'TRAIN

SPEED (Km/h)' );ylabel( 'VERTICAL ACCELERATION (m/s2)' );grid; saveas(gcf, 'Acceleration.fig' ) figure(2); plot(v,U2, 'b' );title( 'TRAIN SPEED VS. VERTICAL DISPLACEMENT' );xlabel( 'TRAIN

SPEED (Km/h)' );ylabel( 'VERTICAL DISPLACEMENT (m)' );grid; saveas(gcf, 'Displacement.fig' )

73

APPENDIX B

## MATLAB codes for Sprung mass model

**Input variables**

clear clc

L=5; %Span Length (m)

Xi=0.0255; %Damping Ratio

Frqmax=50; %Maximum Frequency of Request(HZ)

T_Inc=0.0005; %Dynamic Time Increment(Sec)

E=25e9; %Uncracked Concrete Young's Modulus(N/m2) vi=0.2; %Poisson's Ratio (concrete)

Con_D=9000; %Concrete Density including ballast(Kg/m3)

Es=210e9; %Steel Young's Modulus(N/m2) vis=0.3; %Poisson's Ratio (steel)

Con_S=7850; %Steel Density(Kg/m3)

Ball_T=0.6; %Ballast Thickness(m)

Ball_w=6.2; %Ballast width(m)

Ball_D=2000; %Ballast Density(Kg/m3)

K_t=400e6; %Vertical Spring Stiffness of approach track(N/m)

K_b=400e6; %Vertical Spring Stiffness of ballast(N/m)

C_b=100e3; %Vertical Damping for ballast and approach track(Ns/m)

%% ABAQUS INPUT FILE

Sleeper_s=0.625; %Sleeper spacing or element size

Lt=0.5*L; %transition zone length

ExTrack_L=140+(L-Lt); %Track length

GroundNodes_Left=101+(ExTrack_L/Sleeper_s);

%Ground nodes before transition zone on left side

AppGroundNodes_Left=GroundNodes_Left+(Lt/Sleeper_s);

%Ground nodes for transition zone on left side

BridgeNodes=AppGroundNodes_Left+(L/Sleeper_s);

%Bridge Nodes

AppGroundNodes_Right=BridgeNodes+(Lt/Sleeper_s);

%Ground nodes for transition zone on Right side

GroundNodes_Right=AppGroundNodes_Right+(ExTrack_L/Sleeper_s);

%Ground nodes after transition zone on Right side

ExRailNodes_Left=GroundNodes_Right+1+(ExTrack_L/Sleeper_s);

%Rail Nodes on track Left side

AppRailNodes_Left=ExRailNodes_Left+(Lt/Sleeper_s);

%Rail Nodes on Transition zone Left side

RailNodes=AppRailNodes_Left+(L/Sleeper_s);

%Bridge Rail Nodes

AppRailNodes_Right=RailNodes+(Lt/Sleeper_s);

%Rail Nodes on Transition zone Right side

ExRailNodes_Right=AppRailNodes_Right+(ExTrack_L/Sleeper_s);

%Rail Nodes on track Right side

%%ELEMENTS%

%**************************************************************************

BridgeEl=BridgeNodes-AppGroundNodes_Left; %Bridge Elements

ExRailEl_Left=ExRailNodes_Left-(GroundNodes_Right+1);

%Track Rail Elements left side

AppRailEl_Left=AppRailNodes_Left-ExRailNodes_Left;

%Transition zone Rail Elements left side

RailEl=RailNodes-AppRailNodes_Left; %Bridge Rail Elements

74

APPENDIX B

AppRailEl_Right=AppRailNodes_Right-RailNodes;

%Transition zone Rail Elements Right side

ExRailEl_Right=ExRailNodes_Right-AppRailNodes_Right;

%Track Rail Elements Right side

GroundSpringEl_Left=(ExTrack_L/Sleeper_s);

AppGroundSpringEl_Left=(Lt/Sleeper_s);

BridgeSpringEl=(L/Sleeper_s)+1;

AppGroundSpringEl_Right=(Lt/Sleeper_s);

GroundSpringEl_Right=(ExTrack_L/Sleeper_s);

GroundDashpotEl_Left=(ExTrack_L/Sleeper_s);

AppGroundDashpotEl_Left=(Lt/Sleeper_s);

BridgeDashpotEl=(L/Sleeper_s)+1;

AppGroundDashpotEl_Right=(Lt/Sleeper_s);

GroundDashpotEl_Right=(ExTrack_L/Sleeper_s);

**GENERAL SECTIONS**

%**************************************************************************

G=E/(2*(1+vi)); %Shear modulus for Bridge

I11=0.1;I12=0;I22=0; %General Section definition for Bridge

A=1;

Gr=Es/(2*(1+vis)); %Shear modulus for Rail

Ir11=0.00006120;Ir12=0;Ir22=0; %stiffness for two Rails

Ar=0.015374; %cross section area for two Rails

%% ABAQUS input file

A2=[];

U2=[]; for v=150:5:350 %Velocity in Km/h v_msec=v/3.6;

L_totload=La+(2*L)+ExTrack_L;

T_tot=(L_totload/(v_msec)); t_inc_dyn=0.0005; inc=ceil(T_tot/t_inc_dyn)+500; fid=fopen([ 'SM5mLa=0,5L' , '.inp' ] , 'w' );

%% HEADING

%......................................................................

Text=[ '*HEADING\n' ...

'** Job name: SM5mLa=0,5L Model name: SM5mLa=0,5L\n' ...

'**\n'

]; fprintf(fid,Text);

%ground NODES Left side

%......................................................................

Text=[ '*Node\n' ...

'101,-' num2str(Lt+ExTrack_L) ',-1\n' ...

% node nb, 1st coordinate, 2nd coordinate

num2str(GroundNodes_Left-1) ',-' num2str(Lt+Sleeper_s) ',-1\n' ...

'*NGEN, Nset=GroundNodes_Left\n' ...

% generates nodes incrementally, name of the nset

'101,' num2str(GroundNodes_Left-1) ',1\n' ...

% nb of the 1st end node, 2nd end node, increment (the default is 1)

]; fprintf(fid,Text);

%APPROACH NODES LEFT SIDE

%......................................................................

Text=[ '*Node\n' ...

num2str(GroundNodes_Left) ',-' num2str(Lt) ',-1\n' ...

75

APPENDIX B

num2str(AppGroundNodes_Left-1) ',-' num2str(Sleeper_s) ',-1\n' ...

'*NGEN, Nset=AppGroundNodes_Left\n' ...

num2str(GroundNodes_Left) ',' num2str(AppGroundNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

%BRIDGE NODES

%......................................................................

Text=[ '*Node\n' ...

num2str(AppGroundNodes_Left) ',0,-1\n' ...

num2str(BridgeNodes) ',' num2str(L) ',-1\n' ...

'*NGEN, Nset=BridgeNodes\n' ...

num2str(AppGroundNodes_Left) ',' num2str(BridgeNodes) ',1\n' ...

]; fprintf(fid,Text);

%BRIDGE ELEMENTS

%.....................................................................

Text=[ '*Element, type=B21\n' ...

'1,' num2str(AppGroundNodes_Left) ',' num2str(AppGroundNodes_Left+1)

'\n' ...

% nb of the element, 1st node forming the element, 2nde node

'*ELGEN, ELSET=Bridgebeam\n' ...

% generates elements incrementally, ELSET= give name to the beam

'1,' num2str(BridgeEl) '\n' ...

% master element nb, nb of elements to be defined

]; fprintf(fid,Text);

% Bridge beam SECTION

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_D) ',ELSET

=Bridgebeam\n' ...

% (ELSET:defines to which elements the bloc is assigned) num2str(A) ',' num2str(I11) ',' num2str(I12) ',' num2str(I22) ', 0, 0,

0\n' ...

% A, I11, I12, I22, J, (gamma)0,(gamma)w

'0, 0, -1\n' ...

% beamsectionorientation. for planar beams: 0,0,-1 num2str(E) ',' num2str(G) '\n' ...

% Young´s modulus E, shear modulus G

'**\n' ...

'*DAMPING,ALPHA=1.28,BETA=0.000357\n' ...

'**\n' ...

]; fprintf(fid,Text);

%APPROACH NODES Right SIDE

%......................................................................

Text=[ '*Node\n' ...

num2str(BridgeNodes+1) ',' num2str(L+Sleeper_s) ',-1\n' ...

num2str(AppGroundNodes_Right) ',' num2str(L+Lt) ',-1\n' ...

'*NGEN, Nset=AppGroundNodes_Right\n' ...

num2str(BridgeNodes+1) ',' num2str(AppGroundNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

%ground NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(AppGroundNodes_Right+1) ',' num2str(L+Lt+Sleeper_s) ',-1\n' ...

76

APPENDIX B

num2str(GroundNodes_Right) ',' num2str(L+Lt+ExTrack_L) ',-1\n' ...

'*NGEN, Nset=GroundNodes_Right\n' ...

num2str(AppGroundNodes_Right+1) ',' num2str(GroundNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

%Extra Rail NODES left side

%......................................................................

Text=[ '*Node\n' ...

num2str(GroundNodes_Right+1) ',-' num2str(Lt+ExTrack_L) ',0\n' ...

num2str(ExRailNodes_Left-1) ',-' num2str(Lt+Sleeper_s) ',0\n' ...

'*NGEN, Nset=ExRailNodes_Left\n' ...

num2str(GroundNodes_Right+1) ',' num2str(ExRailNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

% Extra Rail ELEMENTS left side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+1) ',' num2str(GroundNodes_Right+1)

',' num2str(GroundNodes_Right+2) '\n' ...

'*ELGEN, ELSET=ExRail_Left\n' ...

num2str(BridgeEl+1) ',' num2str(ExRailEl_Left) '\n' ...

]; fprintf(fid,Text);

% Extra Rail SECTION Left side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=ExRail_Left\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) ', 0, 0,

0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Approach Rail NODES Left side

%......................................................................

Text=[ '*Node\n' ...

num2str(ExRailNodes_Left) ',-' num2str(Lt) ',0\n' ...

num2str(AppRailNodes_Left-1) ',-' num2str(Sleeper_s) ',0\n' ...

'*NGEN, Nset=AppRailNodes_Left\n' ...

num2str(ExRailNodes_Left) ',' num2str(AppRailNodes_Left-1) ',1\n' ...

]; fprintf(fid,Text);

% Approach Rail ELEMENTS Left side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(ExRailEl_Left+BridgeEl+1) ',' num2str(ExRailNodes_Left)

',' num2str(ExRailNodes_Left+1) '\n' ...

'*ELGEN, ELSET=AppRail_Left\n' ...

77

APPENDIX B

num2str(ExRailEl_Left+BridgeEl+1) ',' num2str(AppRailEl_Left) '\n' ...

]; fprintf(fid,Text);

% Approach Rail SECTION Left side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=AppRail_Left\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) ', 0, 0,

0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Rail NODES

%.....................................................................

Text=[ '*Node\n' ...

num2str(AppRailNodes_Left) ',0,0\n' ...

num2str(RailNodes) ',' num2str(L) ',0\n' ...

'*NGEN, Nset=RailNodes\n' ...

num2str(AppRailNodes_Left) ',' num2str(RailNodes) ',1\n' ...

]; fprintf(fid,Text);

% Rail ELEMENTS

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+1)

',' num2str(AppRailNodes_Left) ',' num2str(AppRailNodes_Left+1) '\n' ...

'*ELGEN, ELSET=Rail\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+1) ',' num2str(RailEl)

'\n' ...

]; fprintf(fid,Text);

% Rail SECTION

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=Rail\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) ', 0, 0,

0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Approach Rail NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(RailNodes+1) ',' num2str(L+Sleeper_s) ',0\n' ...

num2str(AppRailNodes_Right) ',' num2str(L+Lt) ',0\n' ...

'*NGEN, Nset=AppRailNodes_Right\n' ...

num2str(RailNodes+1) ',' num2str(AppRailNodes_Right) ',1\n' ...

78

APPENDIX B

]; fprintf(fid,Text);

% Approach Rail ELEMENTS Right side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+1) ',

' num2str(RailNodes) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppRail_Right\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+1) ',

' num2str(AppRailEl_Right) '\n' ...

]; fprintf(fid,Text);

% Approach Rail SECTION Right side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',

ELSET=AppRail_Right\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22)

', 0, 0, 0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

%Extra Rail NODES Right side

%......................................................................

Text=[ '*Node\n' ...

num2str(AppRailNodes_Right+1) ',' num2str(L+Lt+Sleeper_s) ',0\n' ...

num2str(ExRailNodes_Right) ',' num2str(ExTrack_L+L+Lt) ',0\n' ...

'*NGEN, Nset=ExRailNodes_Right\n' ...

num2str(AppRailNodes_Right+1) ',' num2str(ExRailNodes_Right) ',1\n' ...

]; fprintf(fid,Text);

% Extra Rail ELEMENTS Right side

%.....................................................................

Text=[ '*Element, type=B21\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+1) ',' num2str(AppRailNodes_Right) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=ExRail_Right\n' ...

num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+1) ',' num2str(ExRailEl_Right) '\n' ...

]; fprintf(fid,Text);

% Extra Rail SECTION Right side

%.....................................................................

Text=[

'*Beam General Section, Density=' num2str(Con_S) ',ELSET=ExRail_Right\n' ...

num2str(Ar) ',' num2str(Ir11) ',' num2str(Ir12) ',' num2str(Ir22) '

79

APPENDIX B

, 0, 0, 0\n' ...

'0, 0, -1\n' ...

num2str(Es) ',' num2str(Gr) '\n' ...

'**\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground left side

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+1) ',101,' num2str(GroundNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundSpring_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+1) ',' num2str(GroundSpringEl_Left) '\n' ...

'*Spring, elset=GroundSpring_Left\n' ...

'2,2\n' ...

'100e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground left side

%.....................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+1)

'\n' ...

',101,' num2str(GroundNodes_Right+1)

'*ELGEN, ELSET=GroundDashpot_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+1) ',' num2str(GroundDashpotEl_Left) '\n' ...

'*Dashpot, elset=GroundDashpot_Left\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground left side approach track

%.....................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai

',' lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+1) num2str(GroundNodes_Left) ',' num2str(ExRailNodes_Left) '\n' ...

'*ELGEN, ELSET=AppGroundSpring_Left\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+1) num2str(AppGroundSpringEl_Left) '\n' ...

'*Spring, elset=AppGroundSpring_Left\n' ...

'2,2\n' ...

'250e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground left side approach track

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

','

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+1

80

APPENDIX B

) ',' num2str(GroundNodes_Left) ',' num2str(ExRailNodes_Left) '\n' ...

% nb of the element, 1st node forming the element, 2nde node

'*ELGEN, ELSET=AppGroundDashpot_Left\n' ...

) ',' num2str(AppGroundDashpotEl_Left) '\n' ...

'*Dashpot, elset=AppGroundDashpot_Left\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for bridge beam

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+1) ',' num2str(AppGroundNodes_Left) ',' num2str(AppRailNodes_Left) '\n' ...

'*ELGEN, ELSET=BridgeSprings\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+1) ',' num2str(BridgeSpringEl) '\n' ...

'*Spring, elset=BridgeSprings\n' ...

'2,2\n' ...

'400e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for bridge beam

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+1) ',' num2str(AppGroundNodes_Left)

',' num2str(AppRailNodes_Left) '\n' ...

'*ELGEN, ELSET=BridgeDashpots\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A

',' num2str(BridgeDashpotEl) ppGroundDashpotEl_Left+BridgeSpringEl+1)

'\n' ...

'*Dashpot, elset=BridgeDashpots\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground right side approach track

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+1) ',' num2str(BridgeNodes+1) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppGroundSpring_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+

81

APPENDIX B

AppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+1) num2str(AppGroundSpringEl_Right) '\n' ...

'*Spring, elset=AppGroundSpring_Right\n' ...

'2,2\n' ...

'250e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground right side approach track

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

','

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+1) ',' num2str(BridgeNodes+1) ',' num2str(RailNodes+1) '\n' ...

'*ELGEN, ELSET=AppGroundDashpot_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+1) ',' num2str(AppGroundDashpotEl_Right) '\n' ...

'*Dashpot, elset=AppGroundDashpot_Right\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

% Spring ELEMENTS for connection to ground Right side

%......................................................................

Text=[ '*Element, type=SpringA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+1) ',' num2str(AppGroundNodes_Right+1) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundSpring_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+1) ',' num2str(GroundSpringEl_Right) '\n' ...

'*Spring, elset=GroundSpring_Right\n' ...

'2,2\n' ...

'100e+06\n' ...

]; fprintf(fid,Text);

% Dashpot ELEMENTS for connection to ground Right side

%......................................................................

Text=[ '*Element, type=DashpotA\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+GroundSpringEl_Right+1) ',' num2str(AppGroundNodes_Right+1) ',' num2str(AppRailNodes_Right+1) '\n' ...

'*ELGEN, ELSET=GroundDashpot_Right\n' ...

,num2str(BridgeEl+ExRailEl_Left+AppRailEl_Left+RailEl+AppRailEl_Right+ExRai lEl_Right+GroundSpringEl_Left+GroundDashpotEl_Left+AppGroundSpringEl_Left+A

82

APPENDIX B ppGroundDashpotEl_Left+BridgeSpringEl+BridgeDashpotEl+AppGroundSpringEl_Rig ht+AppGroundDashpotEl_Right+GroundSpringEl_Right+1) num2str(GroundDashpotEl_Right) '\n' ...

'*Dashpot, elset=GroundDashpot_Right\n' ...

'2,2\n' ...

'100000\n' ...

]; fprintf(fid,Text);

','

**Sprung mass model of X2000 train**

%......................................................................

Text=[ '*Node, Nset=wsn\n' ...

'8001,-' num2str(L+2) ',0\n' ...

'8002,-' num2str(L+4.9) ',0\n' ...

'8003,-' num2str(L+11.5) ',0\n' ...

'8004,-' num2str(L+14.4) ',0\n' ...

'8005,-' num2str(L+19.13) ',0\n' ...

'8006,-' num2str(L+23.03) ',0\n' ...

'8007,-' num2str(L+36.83) ',0\n' ...

'8008,-' num2str(L+39.73) ',0\n' ...

'8009,-' num2str(L+44.08) ',0\n' ...

'8010,-' num2str(L+46.98) ',0\n' ...

'8011,-' num2str(L+61.78) ',0\n' ...

'8012,-' num2str(L+64.68) ',0\n' ...

'8013,-' num2str(L+69.03) ',0\n' ...

'8014,-' num2str(L+71.93) ',0\n' ...

'8015,-' num2str(L+86.73) ',0\n' ...

'8016,-' num2str(L+89.63) ',0\n' ...

'8017,-' num2str(L+93.98) ',0\n' ...

'8018,-' num2str(L+96.88) ',0\n' ...

'8019,-' num2str(L+111.68) ',0\n' ...

'8020,-' num2str(L+114.58) ',0\n' ...

'8021,-' num2str(L+119.27) ',0\n' ...

'8022,-' num2str(L+122.17) ',0\n' ...

'8023,-' num2str(L+133.77) ',0\n' ...

'8024,-' num2str(L+136.67) ',0\n' ...

'*Node, Nset=csn\n' ...

'9001,-' num2str(L+2) ',1\n' ...

'9002,-' num2str(L+4.9) ',1\n' ...

'9003,-' num2str(L+11.5) ',1\n' ...

'9004,-' num2str(L+14.4) ',1\n' ...

'9005,-' num2str(L+19.13) ',1\n' ...

'9006,-' num2str(L+23.03) ',1\n' ...

'9007,-' num2str(L+36.83) ',1\n' ...

'9008,-' num2str(L+39.73) ',1\n' ...

'9009,-' num2str(L+44.08) ',1\n' ...

'9010,-' num2str(L+46.98) ',1\n' ...

'9011,-' num2str(L+61.78) ',1\n' ...

'9012,-' num2str(L+64.68) ',1\n' ...

'9013,-' num2str(L+69.03) ',1\n' ...

'9014,-' num2str(L+71.93) ',1\n' ...

'9015,-' num2str(L+86.73) ',1\n' ...

'9016,-' num2str(L+89.63) ',1\n' ...

'9017,-' num2str(L+93.98) ',1\n' ...

'9018,-' num2str(L+96.88) ',1\n' ...

'9019,-' num2str(L+111.68) ',1\n' ...

'9020,-' num2str(L+114.58) ',1\n' ...

'9021,-' num2str(L+119.27) ',1\n' ...

'9022,-' num2str(L+122.17) ',1\n' ...

83

APPENDIX B

'9023,-' num2str(L+133.77) ',1\n' ...

'9024,-' num2str(L+136.67) ',1\n' ...

]; fprintf(fid,Text);

% ELEMENTS

.......................................................................

Text=[

'*Surface, type=ELEMENT, name=master\n' ...

'ExRail_Left,SPOS\n' ...

'AppRail_Left,SPOS\n' ...

'Rail,SPOS\n' ...

'AppRail_Right,SPOS\n' ...

'ExRail_Right,SPOS\n' ...

'*Surface, type=NODE, name=slave\n' ...

'wsn, 1.\n' ...

'** INTERACTION PROPERTIES\n' ...

'**\n' ...

'*Surface Interaction, name=ContactProp\n' ...

'1.\n' ...

'*Friction\n' ...

'0.1\n' ...

'*Surface Behavior, pressure-overclosure=HARD\n' ...

'**\n' ...

]; fprintf(fid,Text);

%.......................................................................

Text=[ '*Element, type=MASS, elset=m1_wheel\n' ...

'5001,8001\n' ...

'5002,8002\n' ...

'5003,8003\n' ...

'5004,8004\n' ...

'*Element, type=MASS, elset=m2_wheel\n' ...

'5005,8005\n' ...

'5006,8006\n' ...

'5007,8007\n' ...

'5008,8008\n' ...

'5009,8009\n' ...

'5010,8010\n' ...

'5011,8011\n' ...

'5012,8012\n' ...

'5013,8013\n' ...

'5014,8014\n' ...

'5015,8015\n' ...

'5016,8016\n' ...

'5017,8017\n' ...

'5018,8018\n' ...

'5019,8019\n' ...

'5020,8020\n' ...

'*Element, type=MASS, elset=m3_wheel\n' ...

'5021,8021\n' ...

'5022,8022\n' ...

'*Element, type=MASS, elset=m4_wheel\n' ...

'5023,8023\n' ...

'5024,8024\n' ...

'*Mass, elset=m1_wheel\n' ...

'2050., \n' ...

'*Mass, elset=m2_wheel\n' ...

'1650., \n' ...

'*Mass, elset=m3_wheel\n' ...

'1650., \n' ...

84

'*Mass, elset=m4_wheel\n' ...

'1800., \n' ...

'*Element, type=MASS, elset=m1_body\n' ...

'6001,9001\n' ...

'6002,9002\n' ...

'6003,9003\n' ...

'6004,9004\n' ...

'*Element, type=MASS, elset=m2_body\n' ...

'6005,9005\n' ...

'6006,9006\n' ...

'6007,9007\n' ...

'6008,9008\n' ...

'6009,9009\n' ...

'6010,9010\n' ...

'6011,9011\n' ...

'6012,9012\n' ...

'6013,9013\n' ...

'6014,9014\n' ...

'6015,9015\n' ...

'6016,9016\n' ...

'6017,9017\n' ...

'6018,9018\n' ...

'6019,9019\n' ...

'6020,9020\n' ...

'*Element, type=MASS, elset=m3_body\n' ...

'6021,9021\n' ...

'6022,9022\n' ...

'*Element, type=MASS, elset=m4_body\n' ...

'6023,9023\n' ...

'6024,9024\n' ...

'*Mass, elset=m1_body\n' ...

'16200., \n' ...

'*Mass, elset=m2_body\n' ...

'10100., \n' ...

'*Mass, elset=m3_body\n' ...

'9850., \n' ...

'*Mass, elset=m4_body\n' ...

'14200., \n' ...

'*Element, type=dashpotA, Elset=dashpot\n' ...

'7001,8001,9001\n' ...

'7002,8002,9002\n' ...

'7003,8003,9003\n' ...

'7004,8004,9004\n' ...

'7005,8005,9005\n' ...

'7006,8006,9006\n' ...

'7007,8007,9007\n' ...

'7008,8008,9008\n' ...

'7009,8009,9009\n' ...

'7010,8010,9010\n' ...

'7011,8011,9011\n' ...

'7012,8012,9012\n' ...

'7013,8013,9013\n' ...

'7014,8014,9014\n' ...

'7015,8015,9015\n' ...

'7016,8016,9016\n' ...

'7017,8017,9017\n' ...

'7018,8018,9018\n' ...

'7019,8019,9019\n' ...

'7020,8020,9020\n' ...

'7021,8021,9021\n' ...

85

APPENDIX B

APPENDIX B

'7022,8022,9022\n' ...

'7023,8023,9023\n' ...

'7024,8024,9024\n' ...

'*Dashpot, elset=dashpot\n' ...

'2 ,2\n' ...

'30000.\n' ...

'*Element, type=springA, ELSET=spring-1\n' ...

'4001,8001,9001\n' ...

'4002,8002,9002\n' ...

'4003,8003,9003\n' ...

'4004,8004,9004\n' ...

'*Element, type=springA, ELSET=spring-2\n' ...

'4005,8005,9005\n' ...

'4006,8006,9006\n' ...

'4007,8007,9007\n' ...

'4008,8008,9008\n' ...

'4009,8009,9009\n' ...

'4010,8010,9010\n' ...

'4011,8011,9011\n' ...

'4012,8012,9012\n' ...

'4013,8013,9013\n' ...

'4014,8014,9014\n' ...

'4015,8015,9015\n' ...

'4016,8016,9016\n' ...

'4017,8017,9017\n' ...

'4018,8018,9018\n' ...

'4019,8019,9019\n' ...

'4020,8020,9020\n' ...

'*Element, type=springA, ELSET=spring-3\n' ...

'4021,8021,9021\n' ...

'4022,8022,9022\n' ...

'*Element, type=springA, ELSET=spring-4\n' ...

'4023,8023,9023\n' ...

'4024,8024,9024\n' ...

'*Spring, elset=spring-1\n' ...

'2 ,2\n' ...

'1.45e+06\n' ...

'*Spring, elset=spring-2\n' ...

'2 ,2\n' ...

'1.05e+06\n' ...

'*Spring, elset=spring-3\n' ...

'2 ,2\n' ...

'1.05e+06\n' ...

'*Spring, elset=spring-4\n' ...

'2 ,2\n' ...

'1.45e+06\n' ...

]; fprintf(fid,Text);

**BOUNDARY CONDITIONS**

%......................................................................

Text=[ '*BOUNDARY\n' ...

'GroundNodes_Left,Encastre\n' ...

%node nb, type of boundary

'GroundNodes_Right,Encastre\n' ...

'AppGroundNodes_Left,Encastre\n' ...

'AppGroundNodes_Right,Encastre\n' ...

num2str(AppGroundNodes_Left) ',2\n' ...

num2str(BridgeNodes) ',PINNED\n' ...

num2str(GroundNodes_Right+1) ',1\n' ...

num2str(ExRailNodes_Right) ',1\n' ...

86

APPENDIX B

'wsn,1,1\n' ...

'csn,1,1\n' ...

'** Interaction: ContactCondition\n' ...

'*Contact Pair, interaction=ContactProp\n' ...

'slave,master\n' ...

]; fprintf(fid,Text);

Text=[ '**STEP:gravity\n' ...

'**\n' ...

'*Step, name=gravity\n' ...

'*Static\n' ...

'**\n' ...

'** LOADS\n' ...

'**\n' ...

'** Name: GRAVITY-1 Type: Gravity\n' ...

'*Dload\n' ...

'm1_wheel, GRAV, 10., 0., -1.\n' ...

'm2_wheel, GRAV, 10., 0., -1.\n' ...

'm3_wheel, GRAV, 10., 0., -1.\n' ...

'm4_wheel, GRAV, 10., 0., -1.\n' ...

'm1_body, GRAV, 10., 0., -1.\n' ...

'm2_body, GRAV, 10., 0., -1.\n' ...

'm3_body, GRAV, 10., 0., -1.\n' ...

'm4_body, GRAV, 10., 0., -1.\n' ...

'**\n' ...

'** OUTPUT REQUESTS\n' ...

'**\n' ...

'** FIELD OUTPUT: F-Output-1\n' ...

'**\n' ...

'*Output, field, variable=PRESELECT\n' ...

'**\n' ...

'** HISTORY OUTPUT: H-Output-1\n' ...

'**\n' ...

'*Output, history, variable=PRESELECT\n' ...

'*End Step\n' ...

'** -----------------------------------------------------------\n' ...

'** \n' ...

'** STEP: Step-2-dynamic\n' ...

'**\n' ...

'*Step, name=Step-2-dynamic, inc=' num2str(inc) '\n' ...

'*Dynamic,alpha=-0.05,direct\n' ...

num2str(t_inc_dyn) ',' num2str(T_tot) ',\n' ...

'** \n' ...

'** BOUNDARY CONDITIONS\n' ...

'*Boundary\n' ...

'wsn, 1, 1,' num2str(L_totload) '\n' ...

'csn, 1, 1,' num2str(L_totload) '\n' ...

'**\n' ...

'** OUTPUT REQUESTS\n' ...

'**\n' ...

'**\n' ...

'** FIELD OUTPUT: F-Output-2\n' ...

'**\n' ...

'*Output, field, variable=PRESELECT\n' ...

'**\n' ...

'** HISTORY OUTPUT: H-Output-1\n' ...

'**\n' ...

'*Output, history, variable=PRESELECT\n' ...

'*Node output, nset=BridgeNodes\n' ...

'A2,U2\n' ...

87

APPENDIX B

'*NODE PRINT, nset=BridgeNodes\n' ...

'A2,U2\n' ...

'*END STEP\n' ]; fprintf(fid,Text);

dos( 'abaqus job=SM5mLa=0,5L.inp interactive' ); fclose( 'all' );

Amax=[];

Umax=[]; for n=334:340

AValues=[];

UValues=[]; fid=fopen([ 'SM5mLa=0,5L' , '.dat' ], 'r' );

LineText=fgetl(fid); while strcmp(LineText, ' THE ANALYSIS HAS BEEN COMPLETED' )~=1

LineText=fgetl(fid); if (length(LineText))>11 if strcmp(LineText(1:11),[ ' ' num2str(n)])==1

%disp('Saving all acceleration values for each node');

AValues=[AValues; sscanf(LineText, '%*s %f %*f' )];

UValues=[UValues; sscanf(LineText, '%*s %*f %f %*f' )]; end end end

rate=1000;

Wn=50;

N=6;

Wnr=(Wn*2)/rate; % relative cut-off frequency

[b,a]=butter(N,Wnr); av=filter(b,a,AValues); uv=filter(b,a,UValues);

Amax=[Amax;max(abs(av))];

Umax=[Umax;max(abs(uv))]; end

A2=[A2;max(Amax)];

U2=[U2;max(Umax)]; fclose( 'all' ); end

**Plot result**

save( 'Acceleration.txt' , 'A2' , '-ASCII' ) save( 'Displacement.txt' , 'U2' , '-ASCII' ) v=150:5:350; plot(v,A2, 'r' );title( 'TRAIN SPEED VS. VERTICAL ACCELERATION' );xlabel( 'TRAIN

SPEED (Km/h)' );ylabel( 'VERTICAL ACCELERATION (m/s2)' );grid; saveas(gcf, 'Acceleration.fig' ) figure(2); plot(v,U2, 'b' );title( 'TRAIN SPEED VS. VERTICAL DISPLACEMENT' );xlabel( 'TRAIN

SPEED (Km/h)' );ylabel( 'VERTICAL DISPLACEMENT (m)' );grid; saveas(gcf, 'Displacement.fig' )

88

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project