Bridge Interaction on Railway Lines

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Train‐Bridge Interaction on Freight Railway Lines

DAVIDE MARTINO

Master of Science Thesis

Stockholm, Sweden 2011

Train-Bridge Interaction on FreightRailway Lines

DAVIDE MARTINO

Master’s Thesis at ABESupervisor: Prof. Raid Karoumi

TRITA 339 2011-10

TRITA-BKN. Master Thesis 339, 2011 ISSN 1103-4297 ISRN KTH/BKN/EX-339-SE

Preface

This thesis is the result of half a year study carried out at the Division of StructuralEngineering and Bridges at KTH, Royal Institute of Technology, Sweden. It has been anextremely good experience to work here with all the people I met of this department. Myspecial gratitude goes to Prof. Raid Karoumi for founding my interest in bridge dynamicsthrough his courses at KTH and supervising this project, and to John Leander for hisgreat help and constant availability. I also would like to thank Andreas Andersson andElias Kassa for the interesting technical conversation we had.

Stockholm, October 2011.

Davide Martino

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Abstract

This study investigates the dynamic response of a railway bridge under train passage.Three load models designed around the Swedish Steel Arrow freight train are tested andcompared. A series of Concentrated Forces, a succession of single degree of freedomSprung-Masses, and a sequence of complex multi-degree of freedom Train Wagons. Theincrease in accuracy of the representation corresponds to taking into account the inertialproperties of the wagons. The track-bed layer is substitute by a sequence of regularlyspaced couple of springs and dampers at the sleeper distance. Under the assumption ofthis work, a portion of the ballast vibrates with the sleeper during train passage. Bothbridge and rail are modelled under Bernoulli-Euler beam theory. The dynamic behaviorof the bridge is investigated in presence or absence of vertical track irregularities. Themain conclusions of the report can be summarized as:

• the dynamic amplification attains its maximum value, for every train model, at thecritical train speeds of 120 km/h. Proper resonance has also been detected at thespeed of 60 km/h in all the simulations;

• the Concentrated Forces model provided an upper boundary of the accelerationresponse of the bridge while the Sprung-Mass systems a lower boundary. The re-sponse of the two models is in very good agreement at non resonance speeds. Thesimulation with Train Wagons loading does not fit completely this trend, it addstwo peaks on the diagram; Besides that, the bridge response lies between the twolimits;

• the presence of track irregularities determines a variation of the bridge dynamicsonly if combined with Train Wagon load model. The Concentrated Force patterncouldn’t detect the modification of the profile while the Sprung-Masses case provideda diagram of maximum acceleration similar to the one over flat rail simply shiftedupwards;

• the position of the track irregularities along the bridge influence its dynamics.

Keywords: train-bridge interaction, dynamics, acceleration, freight train models, trackmodels.

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Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Steel-concrete composite bridge . . . . . . . . . . . . . . . . . . . . 31.2.2 Railway bridge dynamics . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Track-bed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Aim of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Description of the Case Study 92.1 The Bothinia Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Power Cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Coaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 The Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Cross-Section Properties . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 The Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Realization of the Model 213.1 The Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Concentrated Forces Model . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Train Wagons Model . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Sprung-Masses Model . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 The Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Cross-Section Properties . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 The Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Numerical Analysis 354.1 Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Direct Integration Technique . . . . . . . . . . . . . . . . . . . . . . 374.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Bridge Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2 Sleepers Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.3 Track-Bridge Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 41

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5 Model Checking 435.1 Acceleration and Displacement at Midspan . . . . . . . . . . . . . . . . . . 435.2 Vertical Reaction Force at the Bearings . . . . . . . . . . . . . . . . . . . . 465.3 Moving Wheel-Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 DynStArr, a MATLAB® Toolbox 516.1 Examples of Function Calls . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Interrelation between Output commands . . . . . . . . . . . . . . . . . . . 546.3 Automatized Approach to the Dynamic Problems . . . . . . . . . . . . . . 55

6.3.1 The MainRail_SteelArrow.m file . . . . . . . . . . . . . . . . . . . 55

7 Assignation of Damping and Convergence Analysis 597.1 Damping Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Convergence Analysis of the Models . . . . . . . . . . . . . . . . . . . . . . 65

8 Results 698.1 Rail without Vertical Irregularities . . . . . . . . . . . . . . . . . . . . . . 698.2 The Rail with Vertical Irregularities . . . . . . . . . . . . . . . . . . . . . . 79

9 Conclusion 879.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.2 Suggestion for Future Research . . . . . . . . . . . . . . . . . . . . . . . . 88

A Example of ABAQUS® Input File 93

B MATLAB® Toolbox DynStArr 97

Chapter 1

Introduction

1.1 Background

In this high time of debating about how to fight the climate change and re-launch theeconomy in the European Union, the transport sector should be a top priority to be im-proved. Recent studies conducted at the European Environment Agency (EEA), statethat this is the only branch in which the green house gases (GHGs) emission have con-tinued to rise, despite their decrease in all other industry compartments. At the end of2008, the EU-15 member countries produced 20% of GHGs from transport more thanwhat monitored at the end of 1990. In this statistic the road transport is, by far, thelargest contributor to these emissions with the 93% in 2008 and, accompanied by the airtransport, the fastest growing contributor [EEA11]. Oil and fossil energy are still thecauses of pollution in a sector that struggle to evolve. Those figures can be reduced. Thewhole transport sector offers significant room for improvements in efficiency, as each modeis still far from reaching their full emissions reduction potential.

As cross border freight transport continues to grow, rail freight has a key role to play increating a mental shift in favor of a more sustainable transport system in Europe. As sug-gested by the European Rail Infrastructure Managers (EIM) Secretary-General, MichaelRobson "the relationship between rail and road transport should become increasinglycomplementary in the future, with rail using its obvious strengths over long distances androad freight playing its critical role for regional feeders and distribution" [tb]. The Euro-pean Commission has supported this view by granting EU funds to shift freight off theEuropean roads via the Marco Polo Programme. This instrument is aimed at improvingthe efficiency and the environmental performance of the freight transport system acrossEurope by reducing the road congestion and increasing the reliability of the service. Ithas also been shown that the railway is by far the safest mode of transport. Having lessheavy trucks from the European roads, would then mean saving lives. Another relevantcommercial advantage of rail freight transport compared to road haulage is the less directexposure to fluctuations in fuel prices. The further electrification of rail lines will makethis advantage even more apparent.

An example of this approach comes from Sweden. The Port of Göteborg is, nowadays,the most important Swedish intermodal connection for goods tracks. To match the newenvironment demand with a reliable distribution service, the completely electrified Rail-

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2 CHAPTER 1. INTRODUCTION

Port connects its Terminals with 25 train shuttles to the most important logistics centersaround Scandinavia. Rail and road are then efficiently combined. Despite the recession, inMay 2009, 31280 Twenty-foot Equivalent Units (TEUs) passed through the rail terminal,an increase of 10% over the previous year and an all-time best performance [EIM].

(a) The Lines (b) The Port

Figure 1.1: The Role of the Port of Göteborg in Scandinavia.

In order to extend the co-modality principle in Europe, the European Commission launchedrecently the Green Corridors project. Such a concept denote long-distance freight trans-port corridors where advanced technology and co-modality are used to achieve energyefficiency and reduce environmental impact. The characteristics of a green corridor in-clude [rege]:

• sustainable logistics solutions with documented reductions of environmental andclimate impact, high safety, high quality and strong efficiency;

• integrated logistics concepts with optimal utilization of all transport modes, so calledco-modality;

• harmonized regulations with openness for all actors;

• a concentration of national and international freight traffic on relatively long trans-port routes;

1.2. LITERATURE REVIEW 3

• efficient and strategically placed trans-shipment points, as well as an adapted, sup-portive infrastructure;

• a platform for development and demonstration of innovative logistics solutions, in-cluding information systems, collaborative models and technology.

As always, turning good intentions into reality is the tricky phase of any ambitious project.

1.2 Literature Review

The dynamic behavior of railway bridges has been a topic of numerous researches in thefield of civil engineering since the beginning of the past century. At that time the firstaccidents occurred in metal bridges showed the narrowness of the old level of knowledgeand stimulated the work of the scientific community. An exhaustive report about thehistory and the improvement done on this subject is due to Frýba and can be found inReference [Fr96]. By the end of 1996 the physical model most frequently used was theso-called "moving load model". The assumption of this simulation was to replace thetrain actions at the contact with the rail by a series of concentrated, constant-valuedforces. The regularly set of loads would then be moved at the constant speed v over abridge sketched as a beam-like structure. For such a model the closed-form solution of theequation of motion can be derived but with the obvious limit of disregarding the inertiaeffects of the moving vehicles. The results computed at non-resonance speeds, though, arein good agreement with the experimental data while at resonance speeds displacementsand accelerations can be significantly magnified leading to dangerous scenarios, [Fr01] and[MRPA]. The natural step towards the more complex bridge-train interaction problems,was introducing moving masses in place of the constant forces of the equivalent weight ofthe axle load. In most of the studies dealing with this issue, the railway track is usuallymodeled either as a beam on Winkler elastic foundation or a beam supported on a seriesof discrete springs and dampers. Nowadays, the accuracy of the simulations reached levelreally close to the reality. Examples of such achievements can be found in the works ofGalvín, Romero, and Domíniques [GRD], Xia and Zhang [XZ]. This recent increase incomplexity has been enhanced by the construction of new high-speed railway line and theupgrade of the existing ones. The demand in railway transportation has indeed risen forpeople and goods all over the world. To match those requirements the Swedish authoritieshave developed new high speed passenger lines and planned to increase the axle load ofthe freight wagons. This thesis considers the idea of speeding up the freight train fasterthan the usual limit of 120 km/h.

Many articles contributes to idealize and test different track types, bridges and trains. Inthe following a short introduction is given for those that have been used in this project.

1.2.1 Steel-concrete composite bridge

Collin, Johansson and Sundquist [CJS] illustrate in a textbook for graduate coursestheory and technique to design steel-concrete composite bridges. Construction anderection methods, Structural analysis, Different design solutions and Practical designdescription are some of the main chapter of the book.


ScandiaConsult AB was responsible of the design of the Banafjäl Bridge. In referencecan be found the reports with the detailed static calculations and the final drawingsof the superstructure [Lun1] and [Lun2].

1.2.2 Railway bridge dynamics

Frýba is responsible of many contributes in the field of bridge dynamics. A detailed com-pendium about railway bridges, which himself improved with his original works,can be found in [Fr96]. Special attention is given on bridges and railway vehi-cles modelling. Sensitivity analysis of the main parameters that are involved inthe phenomena are presented. Several sections discuss horizontal, longitudinal andtransverse effects due to track irregularities. The study interests also fatigue in steelrailway bridges. The same Author evaluated in [Fr01] the maximum values of ver-tical acceleration, displacement, and bending moment for a simply supported beamsubjected to equidistant moving loads. The calculations satisfy the Euler-Bernoullibeam theory. These maximum values are given for an infinitely long train and canbe regarded as a first rough estimation of the bridge dynamics.

Galvín, Romero and Domíniquez [GRD] formulated a general three dimensional multi-body-finite element-boundary element model, to predict the vibration in time do-main due to train passage. The vehicle is modeled as a multi-body system, whilethe track as a sequence of layers meshed by finite elements. The soil is considered asa homogeneous half-space by the boundary element method. This methodology cantake into account local soil discontinuities, underground construction and couplingwith nearby structures that break the uniformity of the geometry along the trackline. The numerical method is validate by comparison with experimental recordsfrom two high-speed train line.

Xia and Zhang [XZ] investigated the dynamic interaction between high-speed train andbridges comparing theoretical analysis with in-situ measurements. The numericalmodel introduced considers each vehicle as a 27 degrees of freedom object. Thebridge dynamics is described by modal superposition technique. The measured trackirregularities are considered as the most important self-excitation to the vibration ofthe train–bridge system, second only to the moving gravity loading of the vehicles.The dynamic responses presented are: lateral and vertical accelerations, dynamicdeflections, lateral amplitudes, lateral pier amplitudes, and vehicle responses suchas derail factors, offload factors, wheel/rail forces and car-body accelerations. Themodel gives good results if compared to experimental data.

Majka and Hartnett [MH] investigated the effects of various parameters influencingthe dynamic response of railway bridges, such as random track irregularities andbridge skewness. A versatile and computationally efficient numerical model wasdeveloped for this purpose. The model incorporates three-dimensional multi-bodytrain and finite element bridge subsystems. The corresponding equations of motionare integrated numerically by applying the Newmark’s method combined with amodified Newton-Raphson iterative procedure. The model was verified by compar-isons with analytical and numerical solutions available in the literature and goodagreement was found. A parametric analysis was carried out to establish the keyvariables influencing the dynamic response of railway bridges. The speed of the

1.2. LITERATURE REVIEW 5

train, train-to-bridge frequency, mass and span ratios, as well as bridge dampingwere identified as significant variables. Vehicle damping was found to have negligi-ble influence on bridge response. Random track irregularities, as well, were foundto have minor effects on the dynamic amplification factors and bridge accelerations.However, lateral responses are considerably affected by irregularities. Particularlystrong dynamic amplification was found for train with shortly and regularly spacedaxles travelling at the critical speeds.

ERRI gives guidelines for the design of railway bridges for speeds over 200 km/h and howto consider track irregularities in [ERRI99] and [ERRI00]. Those include: designingrequirements, dynamic behavior of case studies, actions to consider, train propertiesand influence of some bridge parameters .

UIC lists requirements that a railway bridge should satisfy. The limits on accelerationand displacements are introduced for safety reason, passenger comfort, and lifetimeof the structures [UIC03].

1.2.3 Track-bed models

Riguero, Rebelo and Simoes da Silva [RRS] investigated the numerical, dynamic re-sponse of medium-span railway viaducts by taking into account the influence of theballasted track and the load modeling methodologies. Three models for the trackand two loading procedures (the Moving Forces procedure and the Train-structureInteraction procedure) were used. The comparison was made using three case stud-ies. Model calibration using available response-acceleration measurements showedthat time variation of the first natural frequency must be assumed to reproducethe measured time histories. The influence of track models was detectable in thetime domain for only the maximum accelerations. When response is analyzed inthe frequency domain, results showed that track models act as a filter for the high-frequency components.

Sun and Dhanasekar [SD] developed a dynamic model to examine the vertical interac-tion of the rail track and the wagon system. Wagon with four wheelsets representingtwo bogies is modeled as a 10 degree of freedom subsystem, the track is modeled asa four-layer subsystem and the two subsystems are coupled together via the non-linear Hertz contact mechanism. The current model is validated using several fieldtest data and other numerical models reported in the literature.

Cheng, Au and Cheung [CAC] studied the role of track structures on the vibration ofrailway bridges. The moving train is modeled as a series of two-degree of freedommass-spring-damper system at the axle locations. The bridge-track element consists,instead,in one upper beam element to model the rails and a lower beam elementto model the bridge deck. The two beam element are interconnected by a series ofsprings and dampers to model the rail bed. The investigation shows that the effectof track structure on the dynamic response of the bridge is insignificant. However,the effect of the bridge structure on the dynamic response of the track structure isconsiderable.


1.3 Aim of the Work

The aim of this thesis is to set up a numerical 2D model that plugs a lack of informationabout the dynamic behavior of an existing bridge usually traveled by freight trains. Themotion of a train over a railway track can be described with different levels of accuracy.Models with higher level of complexity should be able to catch the actual behavior of thesystem and reproduce the physical observation. Such a scheme is always a compromise interms of cost (CPU time and PC memory) and target of the analysis. Within this thesisthree different train models, designed around the Swedish Steel Arrow freight train, aregoing to be tested over two railway track-bridge system, differing only on the presence ofvertical irregularities. Two of the models suggested consider the interaction bridge-train.Comparing the dynamic response of the superstructure will circumscribe the validity ofeach train model. The goal of the work is to determine, whether and when, the usual,efficient but rough, habit of considering the train action over the bridge as concentratedand constant forces can describe the real behavior of a medium span bridge traveled byheavy train. The commercial finite element package ABAQUS®, worldly known for beingpowerful and robust, is customized and controlled by mean of a series of MATLAB® func-tions built on purpose. More in details the MATLAB® environment provides the propersequence of commands in input to construct and solve the FE model in ABAQUS®, whichreturns the results of the calculation. Back to MATLAB®, those data are convenientlyfiltered and organized in plots.

1.4 Limitations

The cross section parameters of the bridge are assumed to be constant along its lengthand are designed over the more representative sections of the superstructure. Concreteand steel are thought to be un-cracked and to have elastic behavior under service. Thedynamic analysis is performed on a 2D model that considers the track-bed as a seriesof coupled spring-dampers with lumped masses free to vibrate at the level of the rail.The planar nature of the model cannot capture lateral or torsional dynamic vibration,fairly unusual on an very slightly curved bridge. The vertical track irregularities areintroduced by modifying the rail profile. The dynamic response focuses on accelerationsand displacements at midspan nodes of the bridge. The train wagons are supposed tocross the bridge at constant speed. Three schemes of moving loads are suggested toreproduce the train passage. The first, and simplest, one regards the train as a series ofConcentrated Forces, the second sketches each train-axle as a single degree of freedomsystem composed by two masses connected by mean of spring and damper (the SprungMasses), while the last provides a complex sequence of multi degree of freedom bodiesmade with rigid elements, springs and dashpots, after recalled as Train Wagons model.The increase in accuracy correspond with the take in consideration of the inertial massand, at last, the rotational inertia. Unfortunately fields measurements are not availableyet to calibrate the results of the simulations.

1.5. STRUCTURE OF THE THESIS 7

1.5 Structure of the thesis

Chapter 1 introduces the general background that motivated the dynamic study. Theaims and scope of the thesis are described, as well as the limitations that the analysisconsidered.

Chapter 2 presents the case study. The success of the modeling phase that will followhighly depend on the amount and quality of information here collected. A propermodel should then be able to catch the essential features to describe the actualdynamic behavior. Three are the main objects involved: the bridge, the track andthe train.

Chapter 3 describes how a passage of a train over a simply supported bridge can besimulated using the FEM Software ABAQUS®. All the key commands are listedand described following the order required by the compiler. A special attention isgiven to the train models. All the three levels of accuracy mentioned are treated indifferent sections.

Chapter 4 aims at describing how the FE model built in the previous Chapter wassolved. Firstly, is given an insight of the integration method that solves the equationsof the motion with an overall view of the stability criteria to satisfy. Then, a briefdiscussion of the modal analysis of the track-bridge system is presented in orderto justify the choice of Rayleigh functions that damp out the vibration of rail andbridge.

Chapter 5 discusses the quality tests that validate the model. Acceleration and dis-placement are plotted for the midspan span of the bridge, while at its boundariesthe variation in vertical reaction force was checked. To ensure contact betweenwheels and rail, the time variation of contact force has been asked. The series ofchecks have been performed also on the Sprung-Mass model but to avoid repetitionthe graph presented are only those relative to the Train Wagon.

Chapter 6 describes the MATLAB® toolbox developed with this thesis. Examples offunction calls are commented to show the potentiality of the code.

Chapter 7 deals with the assignation of structural damping and presents the convergenceanalysis. Each of the moving train models required a separate iterative study whichconcluded with the selection of the proper couple of values (fsampling,lelem) for thefinal simulation.

Chapter 8 comments the dynamic behavior of the railway bridge by looking at theresults of the numerical study.

Chapter 9 summarizes the results and evokes possible further research.

Additional material and input data are provided in Appendices. Appendix A containsthe text file with the list of commands to solve the eigenvalue problem for the power carof the Train Wagon model. A similar sheet has been written to investigate the coachesdynamics. In Appendix B is attached the MATLAB® code of the toolbox developed. Allthe functions were written by the Author, with the exception of the two, "AmplitudeCF"and "Loads", that describe the motion of a constant point load which are due to the PhDStudent John Leander.


Chapter 2

Description of the Case Study

2.1 The Bothinia Line

The Bothinia Line (in Swedish Botniabanan) is the new high speed line that serves theeast coast of Sweden. This is approximately 190 km long and, as Figure 2.1 illustrates, isdelimited between the river Ångermanälven just north of Kramfors and the city of Umeå.The line connects a series of significant industrial locations all situated along the Swedishcoastline in order to improve the efficiency in transport of freight and halve the travelingtime for passengers. This line is being built as a single track (although prepared for asecond track) with maximum axle weight of 25 tons at 120 km/h (freight trains) and 250km/h permitted for passenger trains. On the entire line can be counted 147 bridges and25 km of tunnels [bb].

Figure 2.1: The Bothnia Line.

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10 CHAPTER 2. DESCRIPTION OF THE CASE STUDY

2.2 The Train

The Steel Arrow train is the Swedish more common model of a iron ore freight train. Itis usually composed of 2 power cars and 26 coaches with the first axle distant 388 m formthe last. Both power cars and coaches have a length of 10.4 m and are constituted by onecar body and 2 bogies separated respectively by 7.7 m and 8.6 m. Within the same bogiethe wheels are located at 2.7 m from each other in the power car while in the coach thisdistance is 1.8 m. Each axle of the power car is loaded by 19.5 tons and the coaches havea total mass per axle of 25 tons.

2.2.1 Power Cars

The Power cars of the Steel Arrows are, usually, the Sweden’s most famous Rc4. Alto-gether there have been 8 versions of the Rc-locomotive. The fourth edition was deliveredto Statens Järnvägar (SJ, the Swedish State Railways) during the 1975th. A total of 130pieces were manufactured and 128 of them are still in service [wiki]. All the Rc mod-els are used both in freight and passenger trains. The success of the locomotive abroadstarted already since 1976 when the Rc4 1166 was given on loan to Amtrak, the Amer-ican National Railroad Passenger Corporation. The positive 10 years of driving testsled to a consistent order in locomotives, which engine would become the basis for theAEM-7 power car. The Rc and Rc-based locomotives have been exported to many othercountries. The Austrian Federal Railways bought 10 Rc2 with extra brakes for alpineconditions. A slightly altered Rc4 has been sold to Norwegian State Railways (NSB),nowadays known as El 16. Another exemplar of Rc4 reached even Iran, when in the earlyeighties, a robust locomotive had to serve the electrified stretches near the then-Sovietborder. The Rc4-locomotives weigh 78,0 tons, slightly more than the two previous ver-sions. The maximum speed is the same as the Rc1 and Rc2-locos, and reaches 135 km/h.The Rc4 1322, though, has been powered up to 160 km/h to pull mail train and newerversions have been test driven in near 200 km/h [jarn]. The Rc4-s were used, from thebeginning, in all sorts of trains, but now, are mostly involved in pulling freight trains.The orange paint with white stripes has been used to recognize the Rc4-s. The last onepainted with this scheme is the 1290 model, that can be seen in Figure 2.2. The Swedishcourt itself wanted the orange to commemorate the Swedish railway’s 150th anniversary.Nowadays the company Green Cargo bought all the locomotives destined to freight trainand required a green painting, Figure 2.3. Meanwhile the outfit of Rc4 was changing, agreat technological development in electronics equipment lead to the early versions of Rc4.This differs considerably from the old manufactured, mainly regarding the placement ofappliances and electrical equipment in the engine room [gc].

2.2.2 Coaches

The Steel Arrow train can be equipped by very different coach types. Within this projecteach wagon is thought to be geared by a generic Eaos807 car body and two Y25 freightbogies, see Figures 2.4 and 2.5. This specification is essential since to different couple carbody-bogie corresponds different values of stiffness and inertial properties. The Eaos carbody is intended for the transport of bulk goods, such as scrap, coal, crushed glass, or

2.2. THE TRAIN 11

Figure 2.2: Green Cargo Rc4 1290 i Malmö 2006. Foto Frederik Tellerup.

Figure 2.3: Green Cargo Rc4 1191 i Malmö 2009. Foto Frederik Tellerup.

long goods, which are in fact the format mostly moved in the Scandinavia. The trolleyhas solid sides and end walls of the plate while the floor is fitted and reinforced with ametal coating. Each long side wall is equipped with two doors and It has attached ringsfor nets or tarpaulin. The Y25 bogie is the most diffuse type in Europe that equips freighttrains. It has been a UIC standard freight bogie since 1967 and it is well designed forratios of loaded to tare weight of 5:1 [MHK]. It originated from the Y21A designed bythe French railway during the 60s.


Figure 2.4: The Eaos car body type.

Figure 2.5: The Y25 freight bogie type.

2.3 The Bridge

The superstructure under dynamic investigation is a railway bridge located at the km22+392 of the Bothnia line. It carries the name of the stream Banafjäl that crosses, inthe municipality of Örnsköldsvik. The Banafjäl bridge, in Figure 2.6, is a single-trackcomposite bridge with span length of 42 m and width of 7.7 m. A reinforced concreteslab transfers the load from the track to two main steel beams. The steel beams aresimply supported at their ends with respect to the vertical bending moment and free totwist under torsional moments. The bridge has a small curvature of its neutral axis of

2.3. THE BRIDGE 13

4000 m radius. The original design was made by "ScandiaConsult" and built by "Skanska"between August 2001 and June 2002. Figure 2.7 and 2.8 collect the drawings of prospectiveand plan view of the bridge as it was built.

Figure 2.6: The Banafjäl bridge.

2.3.1 Cross-Section Properties

Section Area and Inertia

While the concrete slab has constant cross-section area along the whole span the dimensionof the two steel I-beams changes with the position. As can be seen in Figure 2.10 thethickness of the concrete slab is 350 mm along the mid axle and 250 mm at the extremeedges. The slab is supported by two steel I-shaped beams of 2.5 m height. The grade ofthe flanges is S460M while for the web panel was chosen the S420M type [EN04].

Figure 2.9 presents the variation in geometry of the girders. The elevation line of the innerbeam, sketched on the top, distinguishes the three type of steel beams (balk in Swedish)from left to right. Table 2.1 gathers the data of the actual beams.

Beam Part Length tu bu tw bw tl bl(mm) (mm) (mm) (mm) (mm) (mm) (mm)

Exterior1 13857 45 900 2415 21 40 9502 14300 55 900 2395 17 50 9503 13857 45 900 2415 21 40 950

Interior1 13843 45 920 2415 21 40 9702 14300 45 920 2395 17 50 9703 13843 45 920 2415 21 40 970

Table 2.1: Cross-section dimensions of the Steel Beams.

The schematic plan view of the bridge, at the bottom of Figure 2.9 instead, indicateshow and where the two steel beams are connected to each other via bracings. In the S


Figure

2.7:The

elevationview

ofthebridge

fromScandiaC

onsultdraw

ings.

2.3. THE BRIDGE 15

Figure2.8:

The

plan

view

ofthebridge

from

Scan

diaC

onsultdraw

ings.


Figure

2.9:Sketches

inelevation

andplan

viewofthe

bridgefrom

ScandiaConsult

drawings.

2.3. THE BRIDGE 17

Figure 2.10: The Bridge Section chosen for modeling.

type (typ in Swedish), located at the bearings, the bracings have Z-shape as Figure 2.11asuggests. The rest of the bracings, along the span, have instead a triangular shape Figure2.11b.

(a) Section A 26, Type S (b) Section B 26, Type F

Figure 2.11: The two shapes of rod connection of the girders, end and midspan of thebridge.

Density

The density of reinforced concrete has a weight density usually set, for design purposes,to 2500 kg/m3 while the density of plain mild steel is thought to be 7850 kg/m3. A moreaccurate data should depend on the grade of steel adopted. A density of 7880 kg/m3 isoften quoted for mild steel as well [Haw97]. If you add alloying elements such as tungsten,chrome or manganese to improve the steel, the density will change. So 7850 kg/m3 is anaverage value picked from the interval 7750 - 8050 kg/m3.


Young’s modulus

At Serviceability Limit State it is assumed, for both Steel and Concrete, a linear elasticbehavior. The secant modulus of elasticity of concrete at 28 days Ecm, in the absence ofdirect experiments, increases conventionally with the class of material according to thefunction [EC2]:

Ecm = 22000

(fck + 8

10

)0.3

N/mm2

where fck is the characteristic cylinder compressive strength of concrete at 28 days. Theclass of the concrete used for the slab of the Banafjäl Bridge is C32/40 that means:

fck = 32 N/mm2

Ecm ' 33300 N/mm2 (2.1)

It should be noted that the given value is a mean value that in reality can considerablybe influenced by type of binder, water-binder ratio, type of aggregate and amount of air(voids in concrete).

The behavior of the steel can be better approximated with the Hooke’s law. It has beenproved that the steel grade does not vary sensibly the Young’s modulus which is set equalto Esk = 210000 N/mm2 for all the steel with carbon percentage lower than 5%.

2.3.2 Damping

The structure damping has an important role in dynamic analysis. It acts reducing theamplitude of oscillations of an exited system. Sources of damping for bridges are: theinternal friction of the building materials, the presence and width of cracks, the friction atsupports and bearings, the aerodynamic resistance of the structure, and the soil propertiesat piers and abutment. The magnitude of damping depends also on the amplitude of thevibrations [Fr96]. Due to its high complexity, the characteristic structural damping wasdetermined statistically form measurements on existing bridges. The UIC suggested threecategories of bridges collected in Table 2.2 [UIC03].

Type of bridge Lower limit of the percentage of critical damping (%)

Span length L < 20 m Span length L ≥ 20

Metal and mixed 0.5 + 0.125(20− L) 0.5Encased steel girders andreinforced concrete 1.5 + 0.07(20− L) 1.5

Pre-stressed concrete 1.0 + 0.07(20− L) 1.0

Table 2.2: Critical damping coefficients according to Eurocode.

Field measurements on this bridge, confirmed that the critical damping ξ of the first 5eigenmodes can be roughly estimated equal to 0.5% as the UIC formula suggested.

2.4. THE TRACK 19

2.4 The Track

The line is composed by a ballasted track-bed and a couple UIC 60 profile rails. Thoseare supported by concrete sleepers regularly located at the nominal distance of 0.65 m, asresolved by the Swedish National Rail Administration in 2008. The upcoming numericalsimulations will consider, at first, the rails perfectly flat and, afterwards, irregular (i.e.shaped like a sine function graph). The depth of the ballast layer is approximately 0.60 mand the rock particles of which is composed have a diameter of 5 cm circa. The maximumwidth of the ballast layer is 6.2 m. Figure 2.12 shows a typical trapezoidal transversesection. Over the bridge the trackbed does not carry a sub-ballast layer of bigger rockparticles.

Figure 2.12: Typical section of a ballasted rail track.


Chapter 3

Realization of the Model

As briefly described in the introductory Chapter, the dynamic problem was modelled andsolved using the commercial package ABAQUS®. The recurrent geometry of train andrail track, that can be both seen as a succession of units (wagons and sleepers), madethe implicit definition of the problem the most convenient way to proceed. The problemwas therefore translated in a proper series of instructions in ABAQUS® syntax, writtenin a text file, and sent to the compiler. The length of train and track, then, made theautomatized approach via MATLAB® the only way to control the process. This Chapterpresents the relevant instructions implemented.

The global geometry of the model is characterized by three parts:

1. a train model;

2. a bridge;

3. a railway;

and it was created node by node using the command *Node. This generates a singlenode per line by defining its position in the x, y(, z) space domain. Each commandstarts with a star "*" while two stars "**" characterize a comment, everything writtenafter it will therefore be disregarded by the compiler. The *Nset command instead wasused to assign a label to critical nodes, later on recalled to print out their acceleration,displacement, stress, strain and/or reaction forces in order to solve dynamic problemor check the reliability of the model. The nodes were then coupled through a finiteelement, appropriately chosen with the command *Element. The bridge and the railwere represented by a beam element type B23. This element type applies the Euler-Bernoulli beam theory formulation with cubic interpolation order. Within the hypothesisof this theory the cross-section of the deformed element remain normal to the undeformedbeam axis, this does not lead to a contribution to the strain due to the transverse shear.This elements is generally considered useful for thin beams, whose shear flexibility maybe disregarded. The section properties of the rail and the bridge slab were assignedusing the command *Beam General Section, which allows complex section definition.The line that follows the command contains, in order: beam area A, moment of inertiafor bending about the 1-axis Ixx, moment of inertia for cross bending Ixy, moment ofinertia for bending about the 2-axis Iyy, torsional rigidity J , sectorial moment, warpingconstant (the last two only needed in Abaqus/Standard when the section is associated

21

22 CHAPTER 3. REALIZATION OF THE MODEL

with open-section beam elements).

3.1 The Train

The main goal of this work is to compare the effects on the dynamic behavior of the bridgeof different moving load models. The object that has to be simulate is a train composedof 2 locomotives and 26 coaches. Three different alternatives are going to be illustratedand compared:

1. a set of concentrated forces;

2. a set of independent, multi degree of freedom systems, train wagons;

3. a set of independent, single degree of freedom systems, sprung-masses.

3.1.1 Concentrated Forces Model

This first case study follows the indications of UIC [UIC03] concerning the way of sketch-ing the train in a design of a railway bridge. This is indeed the simplest technique toevaluate the dynamic effects of a running train over a superstructure since it disregardsany interaction phenomena bridge-wagon. The complexity of the train has been reducedto a moving set of constant value forces located at the contact point of the wheel on therail. The magnitude of each vector Fi is equal to the nominal axle load, that is a quarterof the self weight.

Fpower car =(mcar + 2mbogie + 4mwheel set)

4g = 19.5 tons× 9.81

ms2

= 191295 N

Fcoach =(mcar + 2mbogie + 4mwheel set +mgoods)

4g = 25 tons× 9.81

ms2

= 245250 N

Figure 3.1: Nodal force time history definition for an axle load F moving at velocity v.

3.1. THE TRAIN 23

A simple procedure to define the motion of the forces along the rail, applies load historiesto each node on the way. For a time step ti and a constant axle load F , a nodal loadFj is assigned to the node j if the axle is above an element with one of its nodes j.The magnitude of Fj depends linearly on the distance from the axle to the node. Thisprocedure is outlined in Figure 3.1 for a single load.

3.1.2 Train Wagons Model

A second case study considers the train as a sequence of unconnected wagons representedby complex objects made by the union of three rigid bodies and seven lumped masses, fourof which represent the wheel sets. The rigid body elements are in place of the car bodyand the two bogies, and its masses are lumped at the center of gravity of the element. Thiselement type can be assigned in ABAQUS® Standard by typing type=RB2D2 after, andon the same line, the command *Element. The structure of the command then requiresto indicate in the following lines name of the rigid element and the nodes connected.After those lines the option *Rigid Body is then used to bind the set of elements into arigid body and assign a reference node to it. By constructing each unit over three nodesand assigning mass and rotational inertia in form of concentrated properties to its centerof gravity, it was possible to assign the inertial properties of each wagon part accordingto the construction specifics. The element types Mass and Rotaryi, and the commands*Mass and *Rotaryi Inertia have been used for this purpose.

Figure 3.2: The Rc4 power car with its 2.7 m long bogie.

The car and the bogies are bonded by a spring and a viscous damper at their externalnodes. This is the secondary suspension set of the wagon. Those particular elementdefinitions are available among the ABAQUS® tools with the commands type=spring2and dashpot2 after the text *Element. The wagon model completes when the four pointmasses (of the equivalent weight of two wheels) are connected to the bogies through theprimary suspension set of springs and dashpots. According with [MHK] the stiffness ofthe springs of the Y25 is load-dependent and so is the (friction) damping. A criteria wasneeded to complete a realistic model. The characteristic constants of springs for primaryand secondary suspension sets were, therefore, selected fitting the six eigenfrequencies ofthe train model so depicted in the interval [1, 10] Hz with the Power Cars generally stifferbut lighter than the Coaches. Regarding the dampers, instead, this model adopts thewidely used habit of substituting the friction dampers with viscous ones. The character-istic values were selected according the available literature [UIC03], [GRD], and [XZ].


Figure 3.3: The freight wagon with Eaos802 car body.

The Steel Arrow transports iron ore between mines and steel mills in the northern partsof Sweden. It is usually equipped with specialized wagons Fanoo040 type which, in thisstudy, were substituted by the general purpose Eaos802. This choice came to compensatethe lack of information on the actual coach type. It is important to underline that thetwo cars have similar axles disposition. Scientific articles provided geometry, mass, andinertia data for the couple: Rc4 - 2.7 m bogie and the empty Eaos802 - Y25 freightbogie, in Figure 3.2 and 3.3 respectively, [SC], [MHK], and [SD]. The unknown inertialproperties of the Power Car and its bogies, though, were defined assuming that the 2.7 mbogie described in [KAN] were equivalent to those installed on the Rc4. The Rotationalinertial of the Rc4 car body itself was calculated assuming it to be a parallelepiped of13.15× 2.20 m2 with evenly distributed mass, m = 61560 kg. These geometrical data areestimated over the only sketch available of the Rc4 locomotive presented in Figure 3.2.

Iload =m

12(l2 + h2) = 911922 kg m2

This value is congruent with those available in literature [GRD], and [XZ]. The data inTable 3.1 was therefore the starting point to construct the Swedish Steel Arrow.

Power Car Empty Coachm (kg) I (kg m2) m (kg) I (kg m2)

Wheelset 1510 - 1380 -Bogie frame 5200 5900 1990 1484Car body 61560 911922 11400 219667

Table 3.1: Assumed mass and moment of inertia for the empty Swedish Steel Arrow.

The data given for the Power Cars can immediately be used for the analysis, the inertiaof one Coach instead has to be corrected imagining to have it fully loaded. Since the

3.1. THE TRAIN 25

specifications allow a maximum axle load of 25 tons, such a freight wagon has a maximumload capacity of 79.1 tons. Assuming it transporting Anthracite Coal, broken, with thedensity of 1110 kg/m3 the volume occupied by the full load bulk material is around 71.3m3, which is exactly the limit of the specifications. Imagining this load as a rigid body (norelative motion of the bulk material) of the shape of a parallelepiped of 12.79× 2.02 m2,the inertia to add at the one of the empty car body is:

Iload =m

12(l2 + h2) = 1105188 kg m2

Under those simplification the loaded wagon will have the inertia:

Ifull = Itare + Iload = 1324855 kg m2

Those parameter were fitted with the geometrical length of Power Cars and Coaches inorder to obtain a reasonable set of vibration modes and correspondent frequencies. Ateach eigenvalues extraction a small variation in stiffness (primary and secondary) followeduntil the result in Tables 3.2 and 3.3 were considered to be reasonably accurate if comparedwith publications [SC].

Figure 3.4: The Train Wagon model.

Power Car Coachk(N

m

)c(N s

m

)k(N

m

)c(N s

m

)Primary Suspension 4.9× 106 108× 103 3.2× 106 30× 103

Secondary Suspension 1.9× 106 152× 103 2.7× 106 40× 103

Table 3.2: Assumed stiffness and damping data for the Swedish Steel Arrow.

The list of instructions to solve this problem for the Power Cars is presented in the inputfile Power_Car.inp attached in Appendix A. An analogous has been written for the caseof Coaches. This procedure fulfill the lack of information about the suspension systems ofthe Steel Arrow train. Table 3.3 presents the eigenfrequencies of the Steel Arrow wagonsdesigned with the ultimate stiffness properties. The six images collected in Figure 3.5present the modes of vibrating associated to the six eigenfrequencies of each wagon. Toavoid redundancy, only the case of the Power Car is sketched.

The movement of the Train Wagon model (and the Sprung-Mass systems) over the railhas been implemented through a so-called contact analysis procedure. The first call of thecommand *Surface define the Master, element based, line where the moving masses will


Power Car Coach

f1 1.79 1.10f2 2.65 2.31f3 4.49 2.92f4 4.53 2.92f5 4.53 3.79f6 4.91 4.71

Table 3.3: Eigenvalues of the Swedish Steel Arrow wagons. Figures in cycle/s.

(a) 1st Eigenmode, f1 (b) 2nd Eigenmode, f2

(c) 3rd Eigenmode, f3 (d) 4th Eigenmode, f4

(e) 5th Eigenmode, f5 (f) 6th Eigenmode, f6

Figure 3.5: The vibrating mode of the Power Car.

3.1. THE TRAIN 27

be driven. The calls that follow specify, instead, the Slave node of the moving systemsdirectly connected to the Master. With *Surface Interaction, *Friction, *SurfaceBehavior and *Contact Pair some fundamental contact properties are set.

The movement of the train was controlled through an appropriate amplitude function.Distance on the rail and time to reach it were combined by the parameter velocity varyingfrom 50 until 200 km/h. Besides, by knowing the distance of the first axle of the i-thwagon from the first axle of the locomotive, it has been possible to design each wagonover the same platform (named "Station1") and avoid the onerous construction of thetrain in its natural extension. This distance in fact was converted in delay that the i-thwagon had to wait still after the first locomotive had left the Station1. The ABAQUS®

command to describe, by coordinate, the variation in distance against time of the firstaxle of each wagon from the starting point of the analysis (the first node of the rail) is*Amplitude. Each wagon is literally pushed by its nodes by resetting their horizontalboundary conditions to varying according to the due amplitude function. The train startsto move only after the application of the gravity acceleration and the end of the staticanalysis.

3.1.3 Sprung-Masses Model

The third case study, not in order of complexity but constructed over the property of theprevious, considers the train as a series of independent Single Degree of Freedom Systems.The train complexity was therefore reduced to a couple of lumped masses, mquart wag andmwheel set, aligned with the train axles connected by a spring and a viscous damper. Assuggested by the subscript, the upper mass weight one quarter of a wagon and the lowermass exactly two wheels.

Figure 3.6: The equivalent moving Axle, a Sprung Mass System.

keq(N

m

)ceq(N s

m

)Power Car 1.96× 106 152× 103

Coaches 1.24× 106 40× 103

Table 3.4: Constant Values of the Equivalent Suspension Set.

In order to make this model equivalent to the previous the stiffness of the Sprung MassSystem was computed as:

keq = mquart wag(2πf1)2


In other words the two models have in common the first resonance frequency, f1 accordingto the notation previously introduced. The damping coefficient, instead, was simplyestimated as the maximum value between the primary and secondary suspension systems,cp and cs:

ceq = max(cs, cp)

Table 3.4 presents the constants of damping and stiffness used for the 8 Axles of the 2Power Cars and the 104 Axles of the 26 Coaches.

The property of the contact analysis procedure as well as the technique to move eachSprung Mass System have been written using the same approach described for the TrainWagons model.

3.2 The Bridge

The single spanned bridge was modeled as a simply supported beam with infinitely stiffbearings along the vertical direction. The choice of a 2D model is considered to be accurateenough since the actual bridge has an extremely low curvature of its neutral axes. TheEuler-Bernoulli beam element was applied to mesh the geometry of the bridge.

3.2.1 Cross-Section Properties

The cross section properties are assumed to be constant along the bridge. The sectionchosen to be representative for the entire bridge is the more frequent along the bridgeitself.

Section Area and Inertia

In order to simplify the modeling process, an equivalent steel section of the concrete slabis calculated and summed to the original existent in steel. The new composite sectionthat is created is then entirely in steel and will have the following properties:

Acomp =Ac

α+ As = 0.57 m2

Icomp = Is + Asd2s +

Icα

+Acd

2c

α= 0.62 m4

where, A indicates a cross-section area, I a cross-section moment of inertia and α theratio of the steel Young’s modulus to the effective concrete Young’s modulus, accordingto the formula:

α =Esk

Ecm

(1 + φeff, dyn)

where Esk = 210.0 GPa, Ecm = 33.3 GPa, and φeff, dyn = 0 is the creep factor for adynamic load configuration imposed equal to the correspondent value for a static case.This assumption applies in a more conservative way the prescription of BBK04 to assume

3.2. THE BRIDGE 29

the characteristic concrete Young’s modulus in presence of dynamic load Ecm, dyn, 20%greater than the value suggested for static analysis Ecm, sta [BBK04].

Ec, eff, dyn = 1.2Ec, eff, sta = 1.2Ecm

(1 + φeff, sta)

Density

Assuming that half of the layer of ballast firmly vibrates with the bridge, the density ofthe simply supported beam has been calculated adding at the mass of the bridge the massof half of the layer of ballast. The remaining half layer has been lumped at the nodes inposition of the sleepers. From the project drawings the values of interest are:

LBridge = 42 m

mBridge = 10.7× 103 kgm

ρBallast = 2.0× 103 kgm3

hBallast = 0.6 m

wBallast = 6.2 m

Where mBridge is the mass per unit of length of the bridge (sum of the amount of steeland concrete) without ballast, ρBallast is the density of the ballast, hBallast the thicknessand wBallast the with of the ballast layer. The increased density of the bridge has then tobe evaluated as:

ρBridge =mBridge + ρBallast

hBallast

2wBallast

Acomp

= 25.3× 103 kgm3

Young’s modulus

Since the cross section of the composite bridge has been converted in an equivalent onemade in steel only, the Young’s modulus to apply has to be the one of the steel. Thereare several qualities (grades) of steel, with associated different properties. The value ofthe Young’s modulus assumed it is not considered to vary in function of the load type:

Es, eff, dyn ' Esk

3.2.2 Damping

The direct integration command *Dynamic allows application of damping properties onlyin form of Rayleigh function. The command to recall, among the material properties ofthe beam elements, is *Damping followed by the specification of the two Rayleigh dampingfactors: α for mass proportional damping and β for stiffness proportional damping. Thisway of damping out the vibrations does not corresponds to the design recommendationform UIC, which assess for such a structure a damping ratio of 0.5%, independently tothe frequency of vibration. This restriction leaded to research a range of interest withinthe frequency spectrum, where most probably were the frequencies corresponding to the


essential modes of vibration. A modal analysis showed that with the 5 first modes ofvibration the Cumulative Participation Factor reaches the 91%. High importance has thefirst one with almost the 80% of mass participation over the mass of the entire model.For a given mode i the fraction of critical damping ξi can be expressed in terms of thedamping factors α and β as:

ξi =α

2ωi

+βωi

2(3.1)

where ωi is the natural frequency at this mode. This equation implies that, generallyspeaking, the mass proportional Rayleigh damping, α, damps the lower frequencies andthe stiffness proportional Rayleigh damping, β, damps the higher frequencies. Figure3.7 visualizes the relative importance of those two terms. This way of applying materialdamping does not give the same results as prescribing a constant damping ratio to anyrandom vibration of the structure, as suggested by UIC [UIC03], an equivalent dampinghas, therefore, to be sought. The Rayleigh function adopted for the bridge gives a differentlogarithmic decrement for different train models since each of them determine a differentvibration response of the bridge axial. A better description of the damping assignationis postponed to Chapter 7 since a preliminary eigenmodes extraction of the model isrequired.

Frequency (rad/s)

Dam

ping

(−

)

+

+

(ωj,ξ

j)

(ωk,ξ

k)

← Stiffness Proportional, α = 0

← Mass Proportional, β = 0

Combined

Figure 3.7: The Rayleigh Damping Function of a Structural Object.

3.3 The Track

The high complexity of the real track-bridge system, couldn’t avoid the construction of anintermediate object to connect the rail with the superstructure. In order to consider thecontribution of the ballast to the vibration of the rail, only half of the mass of the layerwas summed to the bridge by increasing its density. The remaining half ballast layer was

3.3. THE TRACK 31

divided in parallelepiped with length equal to the sleeper distance. Its mass, then, waslumped and assigned at the nodes located along the rail where the sleepers are positioned.The mass of the sleepers itself was summed and applied to those nodes. The passive forcethat this amount of soil develops when pushed by an external load was substituted by thereaction force of a linear spring, located under the "sleeper node". Such a scheme assumesthat beneath the sleeper half of the ballast rigidly moves downwards with it under trainpassage. At the interface with the second half a reaction force coming from the ballast incontact with the bridge acts to attenuate this motion. The dissipative phenomena usuallymonitored in soils are considered by mean of a linear damper. Figure 3.8 visualizes thetrack model just described. The idea of concentrating the ballast mass at the sleepernode was suggested by noticing that similar track models in literature connected rail andsleeper (with augmented mass) with a spring more than two times stiffer than the springused to connect sleepers and bridge deck [RRS].

Figure 3.8: The Track Model Assumed in Comparison with the Reality.

The values of damper constant and spring stiffness are:

krail = 150× 106Nm

crail = 100× 103Nm.

Those were chosen according to both previous analysis along similar lines and the avail-able literature [And11] and [RRS]. The track model suggested comes as a compromisebetween level of complexity and amount of information available. A higher level in com-plexity of the model requires also a better knowledge of the property and the parameterintroduced with additional concentrated systems and the relative effects of the variationsof those property to be tested. Such a model considers the contribution of the ballastand leaves 3 constants to choose: stiffness rail/bridge, damping rail/bridge and amount ofvibrating ballast (lumped mass). Introducing an additional layer between rail and bridgewould have introduced three more constant to calibrate: stiffness ballast/bridge, dampingballast/bridge, and amount of vibrating ballast as independent vibrating mass. As didfor the two train models that consider the inertial properties, the commands *Spring,*Dashpot, *Mass were used to define all those property.

In previous analysis without such a ballast model, the moving system (train) was con-


structed over the first hinged support and then "pushed" over the bridge span. In sucha model the midspan vibrated vigorously since the first centimeter of train motion. Thatproblem could have been due to the fact that those bridge boundaries were fixed, or ratherinfinitely stiff along the vertical direction. Further analysis showed that the best way tomodel the whole interaction was to add an extra railway path to the bridge in order toestablish contact between slave and master, stabilize it and assure a more realistic initialacceleration to the train elements at the entrance of the bridge. This entryway to thebridge (called "WayIn" in the MATLAB code) was thought as a strait rail laying oversprings and dampers (equal in all the features to the one laying over the bridge) andconnected to a series of fixed nodes, representing the motionless soil. The train model isconstructed before this entryway rail, over a specific platform named, in the MATLABroutine, "Station1". Such a station has the length of a wagon and is the starting pointof all the cars. The nodes on rail at the position of the wheel set are clamped to facilitatethe adhesion of the wheels to the rails. An identical station has been designed ad the endof the model to collect and park the running train.

The second group of simulations considers the presence of vertical track irregularities.Even a newly built line can present a non flat rail profile. The main source of irregularity is,anyway, due to wagon braking. While slowing down the heavy car body bounces over thesmall natural peaks increasing their magnitude. Slowly the rail assumes a periodic shapewhich usually can be considered a sine function. The model introduces the irregularitiesexactly by shaping the rail as a graph of a sine function. In Figure 3.9 are compared thetwo models tested.

(a)

(b)

Figure 3.9: The Complete Track-Bridge Model with an without Vertical Irregularities.

3.4 Boundary Conditions

After the command *Boundary are defined the conditions at the boundary of each object.There is an hinged support at the beginning of the bridge, while its last node has beenfixed in the vertical direction by a roller support. Each mass of the train model is at this

3.4. BOUNDARY CONDITIONS 33

point of the analysis (before moving) fixed over the station platform. With the beginningof the dynamic step the nodes will move horizontally according to the amplitude functionrelative to the car. The command *Equation, defined after the definition of the rigidelement, constrain wheel nodes and middle node of bogies to follow the horizontal motionof the node located at the center of gravity of the car body. This explains why only thenumber of those reference nodes are listed below the *Boundary, Op=NEW command.


Chapter 4

Numerical Analysis

4.1 Dynamic Analysis

As the previous Chapter described how to define the geometry of the model and assign thematerial properties to nodes and elements, this one focuses on the blocks of instructionsthat close the ABAQUS® input file with the purpose of solving the contact analysis.

The command *Step starts a single task. In Step_1_Gravity the gravity load is applied toeach mass object of the model by typing *Dload. The gravity acceleration raises followinga ramp function, where zero is the starting value (at step time zero) and −9.81m/s2 isthe end value at step time 1 s. An extract from the original input file is presentedbelow. It shows the order with which the instructions should be invoked. *Output,field and *Node Output request the Reaction Forces (RF) and the component of thevector Displacements (U) for each node of the model constrained or free to move underself weight.

*Step, name=Step_1_Gravity_SteelArrow*Static*Dload, GRAV, 9.81, 0, -1

*Output, field*Node OutputRF, U*End Step

The second step definition called Step_2_EigFrequency starts a modal analysis. A setof instructions similar to those presented below was described in the previous chapterto calculate the natural frequencies of vibration of the train model. Here it is appliedonly to the structural part of the model that need material damping, the track systemand the bridge. This step is required before dynamic analysis and it is performed onlyonce. Section 4.2 investigates and describes the reason and the results of such analysis.Including Perturbation on the first line of the step definition means that load, boundary,and temperature changes should be given and that the results will be change relativelyto the previous step. *Frequency performs the eigenvalue extraction up to 100 Hz asspecified in the data line that follows. The Lanczos Eigensolver algorithm requested isan iterative algorithm invented by Cornelius Lanczos that adapts the power methods to

35

36 CHAPTER 4. NUMERICAL ANALYSIS

find eigenvalues and eigenvectors of a square matrix or the singular value decompositionof a rectangular matrix. The power iteration algorithm starts with a vector x0, whichmay be an approximation to the dominant eigenvector or a random vector. The methodis described by the iteration

xn+1 =Axn

||Axn||where at each step the vector xn is multiplied by the matrix A and normalized. Underthe two assumptions: the matrix A has an eigenvalue that is strictly greater in magnitudethan its other eigenvalues, and the starting vector x0 has a nonzero component in thedirection of an eigenvector associated with the dominant eigenvalue, then a subsequenceof (xn) converges to an eigenvector associated with the dominant eigenvalue φdom.

limn→∞

Axn

||Axn||= φdom

After the extraction of the first (dominant) couple eigenvector/value, the algorithm canbe applied to the restricted space of the lower eigenvalue to get the others couple eigen-vector/values. The option Normalization=Displacement normalize the eigenvectors sothat the largest displacement, rotation entry in each vector is unity.

*Step, name=Step_2_EigFrequency, Perturbation*Frequency, Eigensolver=Lanczos, Normalization=Displacement, , 100 , , ,

*Output, field, variable=PRESELECT*Output, history, variable=PRESELECT*End Step

The last step block is named Step_3_MoveCF (or Step_3_MoveSM or Step_3_MoveT de-pending on the train model moved) and contains the procedure to solve the dynamicstress/displacement response by direct integration. The parameter Inc set the upperbound number of increments in a step. This depends on the train speed, model lengthand time step integration dt according to the inequality:

Inc ≥ tpassagedt

.

Only like this the complete train passage is ensured before ending the analysis. Thecommand *Dynamic select a direct time step integration to solve the problem. Section4.1.1 presents an insight on the numerical method adopted. The word Direct that followsstands for direct user control of the incrementation through the step. If this parameteris used, constant increments of the size defined on the beneath data line are used. Nohafsuppress the calculation of the half-step residual and saves some of the solution cost. Withspecifying Initial=No the calculation of initial accelerations at the beginning of the stepis bypassed. It will be assumed, instead, that the initial accelerations for the current stepare zero. The Alpha value indicates the numerical (artificial) damping control parameter.It has been set equal to −0.05 according to the prescription in ABAQUS® manual "Sucha value is used by default because it introduces just enough artificial damping in thesystem to allow the automatic time stepping procedure to work smoothly".

*Step, Name=Step_3_MoveCF_SteelArrow, Inc=9677*Dynamic, Direct, Nohaf, Initial=No, Alpha=-0.050.001, 9.6767, 0.001, 0.001

4.1. DYNAMIC ANALYSIS 37

The Alpha parameter introduces damping that grows with the ratio of the time incrementto the period of vibration of a mode. Only negative values of α provide damping. Thisparameter can take any value in the interval

[−1

3, 0]as it will be explained in Section

4.1.1. To simulate the movement of the force/axle/wagon, amplitudes functions were usedto push the point force or the boundary conditions of wheel and wagon nodes. *Cloadcontrols the motion of a single concentrated force while *Boundary, Op=New set a movingboundary condition of the Sprung Mass system or Wagon. The *Output, History com-mand followed by *Node Output saves the vertical acceleration and displacement historyof the node indicated in Nset in a .dat file at job solved.

The data requested to the FEM software are:

• vertical acceleration,

• vertical displacement,

• natural frequencies,

• mode shapes.

The method to solve the FE problem is direct integration in order to catch the nonlinearity of the deflection. All the three train model have been run at constant speedvarying in the range 50 - 150 km/h, with the speed step of 5 km/h.

4.1.1 Direct Integration Technique

In the ABAQUS® syntax the command *Dynamic stands for Hilbert-Huges-Taylor (HHT)direct integration method. The HHT method, also known as the α-method, is widely usedin the structural dynamics community for the remarkable improvement to the Newmarkβ-methods. In 1959 Newmark proposed a family of integration formulas that depend ontwo parameters β and γ so defined [New59]:

qn+1 = qn + hq̇n +h2

2[(1− 2β)q̈n + 2βq̈n+1] (4.1a)

q̇n+1 = q̇n + h[(1− γ)q̈n + γq̈n+1] (4.1b)

This method is implicit and A-stable (stable in the whole left-hand plane) provided:

γ ≥ 1

2β ≥

(γ + 1

2

)24

Table 4.1 presents some direct time integration methods based on the Newmark formula.

Method Type β γ

Trapezoidal rule Implicit 1/4 1/2Linear acceleration Implicit 1/6 1/2Central difference Explicit 0 1/2

Table 4.1: Different Newmark methods.


The only combination of β and γ that leads to a second-order integration formula is γ = 12

and β = 14. This choice of parameters produces the trapezoidal method, which is both

A-stable and second order. The drawback of the trapezoidal formula is that it does notinduce any numerical damping in the solution, which makes it impractical for problemsthat have high-frequency oscillations. The HHT method came as an improvement becauseit preserved the A-stability and the numerical damping properties, while achieving secondorder accuracy. The idea proposed in [HHT] actually does not pertain the expression ofthe Newmark integration formulas, but rather the form of the discretized equation ofmotion. As indicated in [Hug87], though, the HHT method will possess the advertisedstability and order properties provided

γ =1− 2α

2β =

(1− α)2

4

andα ∈

[−1

3, 0

]The smaller the value of α, the more damping is induced in the numerical solution. Notethat the choice of α = 0 corresponds to an energy preserving analysis and it is exactly thetrapezoidal rule. α = −1

3provides instead the maximum artificial damping available from

this operator. The ABAQUS® reference manual states, for the method implemented, adamping ratio of about 6 % when the time increment is 40 % of the period of oscillationof the mode being studied and smaller if the oscillation period increases. Therefore theartificial damping set to −0.05 is never very substantial for realistic time increments.

4.2 Modal Analysis

Controlling the damping ratio in a direct time step integration method can be a diffi-cult task. In these cases the damping property of the model are a combination of theconcentrated dampers and the material property of the elastic objects. In ABAQUS®

this feature is provided in form of Rayleigh function that damps out each frequency ofvibration with a different factor. Since the motion of any object can be seen as a linearcombination of its modes of vibration (each of them stimulated by a different naturalfrequency), the damping ratio applied to the structure varies with the response of thesystem to the external forces. Each traveling train model, then, affects the assignationof the damping ratio by producing a different oscillation of the system. Not only, a big-ger inconvenient arises when the prescription to follow sets for a general vibration of thesystem a constant value of damping ratio [UIC03].

To overcome this problem the dynamic behavior of the model has to be forecast. A modalanalysis of the track-bridge system, made by rail, sleepers, and bridge, was carried outin order to extract its eigenfrequencies and relative eigenmodes. By means of the MassParticipation Factor (MPF) it was possible to predict which ones of the infinite eigenmodesof a structure will, presumably, have a higher importance in describing a generic vibrationunder whatever train model. The MPF indicate how much of the total mass (expressedas a fraction) is active in a specific direction for a given eigenmode. Hence, modes with alarge MPF have a greater influence on describing the structure response due to dynamicload. Usually, a cumulative MPF greater than 90 % indicates that enough eigenmodes

4.2. MODAL ANALYSIS 39

have been considered in the dynamic description of the case. It is important to noticethat a symmetric eigenmode will not affect this value. The number of modes and therelative frequencies of interest was noted down. Moreover a Fast Fourier Transformationof the acceleration history response of the midspan of the bridge, will also detect which(odd) eigenmodes are mainly involved in describing the vibration of the system. Thoseeigenfrequencies will form an interval of interest that the two Rayleigh functions willhave to fit considering the actual level of damping and the prescription from normative.The assignation of a representative damping, at a certain speed, comes as a result of aniteration. For each Train model a contact analysis was run and the global damping ratioevaluated by logarithmic decrement in midspan of the Bridge in free vibration. Whilethe Rail had a fixed Rayleigh function, the Rayleigh parameters αBridge and βBridge (theobject with higher mass) changed value in order to seek the damping ratio of 0.5 %.

A modal analysis of the each part of the Track-Bridge system was considered solely topoint out the interaction phenomenas.

4.2.1 Bridge Dynamics

In Table 4.2 are collected the eigenfrequencies and the absolute MPF of the relativeeigenmode. By its formulation the MPF doesn’t express an importance of the symmetriceigenmodes of the bridge. The bridge is considered on its own hinged at the first nodeand supported by a bearing at the opposite end. The total mass of the model is 605634kg, which does not correspond to the real case since half of the layer of ballast has beenlumped at the position of a sleeper. For the same reason the eigenvalues extracted are notequal to the measurements on field. A series of frames of each eigenmodes are presentedin Figure 4.1. The vertical displacement of the nodes is denoted by a red vector statingfrom the deformed mesh in green. The cumulative MPF in the vertical direction up tothe sixth frequency is approximately equal to 93 %.

Frequency (Hz) MPF (%)

f1 2.68 81f2 10.70 -f3 24.08 9f4 42.81 -f5 66.89 3f6 96.33 -

Table 4.2: Eigenvalues of the Bridge solely.

4.2.2 Sleepers Dynamics

The natural resonance frequency of the sleeper fn seen as a Single Degree of Freedomsystem can be calculate analytically by the formula:

fn =1

2π

√k

m= 40.54 Hz





Figure 4.1: The vibrating mode of the Bridge.

4.2. MODAL ANALYSIS 41

This assumes the spring free to extend only in the vertical direction, as already demandedin FE model, and connected to the ground. The stiffness k is equal to 150 × 106 N/m.The lumped mass weight m = 2312 kg and it includes the mass of one sleeper, and thatof a parallelepiped of ballast of length 0.7 m, depth 0.3 m, and width 6.2 m.

Figure 4.2: One Sleeper as Single Degree of Freedom System.

4.2.3 Track-Bridge Dynamics

In Table 4.3 are collected the results from modal analysis of the whole track-bridge system.If compared with the MPFs of the bridge independent, those percentage are really low justbecause the complete length of the track is almost 200 m. There is a high probability thatthose six eigenmodes will have a great interest in describing any motion of the bridge sinceup to the 5th eigenmodes activates the 92 % of the mass of the model comprised betweenthe two supports. The remaining 47 % of the model is not of great interest to describethe vibration in midspan of the bridge, but exists to assign a realistic acceleration to thetrain masses before entering the bridge, and avoid reaction forces at the hinged end nodesof the rail. Therefore to control the percentage of damping of the bridge will be enoughfitting the Rayleigh function as close as possible to the requested value of damping in theinterval [2.370, 37.371]. Figure 4.3 collects the screen shots of those modes.

By comparing the eigenfrequency of the bridge independent it can be seen that the inter-action bridge-sleepers lowered the natural frequencies of the track and concentrated moremodes around the natural frequency of the sleeper fn. This effect can be detected directlyby looking at the Figure 4.3d and the two that follows. The relative displacement of thenodes on the rail is greater than the one of the nodes lying on the axle of the bridge.This is the sign that the ballast is vibrating. The Eurocode imposes that the safety checkon the maximum peak values of acceleration has to be run considering all the structuralmembers supporting the track and with frequencies up to the greater of: 30 Hz, 1.5 timesthe frequency of the fundamental mode of vibration of the member being considered, orthe frequency of the third mode of vibration of the member. This correspond to theapplication of a filter on the results for frequencies beyond 30.00 Hz. In this case, then,the contribution of the fourth mode of vibration, with f4 = 32.106 Hz, will be smoothedin post-processing. A filter with low order is suggested to save part of this signal. It isimportant to underline that the vibrating mass of the system between the two bearingsdue to the 1st and the 3rd mode is still the 90 %.


Frequency (Hz) MPF (%)

f1 2.370 47f2 9.419 -f3 20.543 5f4 32.106 -f5 37.371 1f6 38.883 -

Table 4.3: Eigenvalues of the whole Track-Bridge.




Figure 4.3: The vibrating mode of the Track-Bridge system.

Chapter 5

Model Checking

5.1 Acceleration and Displacement at Midspan

The variation in acceleration and displacement of a point over the bridge deck can revealif mistakes occurred during the analysis. The first series of graphs presents the variationin acceleration of the bridge midspan due to the movement of four Steel-Arrow wagons.In Figure 5.1 are plotted the results relative to the passage of the Train Wagons model.

The initial time equal to zero corresponds to the train standing still at 66.0 + 4.8 m tothe bridge, in contact to the rail (a succession of Bernoulli beam elements) laying over asequence of springs and dampers at the constant distance of 0.6 m, the distance of twoconsequent sleepers. The dynamic analysis starts when a constant speed is assigned tothe axles and the train moves towards the bridge. From the top graph on Figure 5.1 itis possible to recognize that the midspan node starts to accelerate already at t1 ' 2.25 s.Since the train moves at the constant speed the position of its axles on the model can bedetermined through the plot by the formula:

s(t) = vt.

Notice that the model remains still for 1 s after the application of the gravity load to letthe acceleration become null. therefore the relation space-time is valid for all the timeson abscissa reduced by one. During the time t1 the first axle of the first wagon on thebridge is entering the bridge. The oscillation at the node 21 m further away is due to theseries of springs that, connecting the rail to the bridge, propagates the vibration generatedon the rail by the wheels in contact. The first peak in acceleration, with negative sign,occurs at t2 ' 2.50 s. By this time the first wagon is on the bridge, since its last axlehas just run s2 ' 83 m. Looking at the same graph it is possible to recognize that thedisturbance on the signal (filtered or unfiltered) smooths out after t3 ' 4.10 s. At thatmoment the fourth axle of the fourth (and last, for this simulation) wagon leaves thebridge. The superstructure is free to vibrate until the end of the analysis, which occurs att4 ' 5.30 s when the last axle moved of 232 m approximately. The signal is composed byfrequency within two distinctive intervals. The first [0,10] Hz and the second [7.5,10] Hz.The filter applied at 30 Hz as requested by Eurocode is, in this case, useless. The cuttingoff frequency that produced the filtered data in Figure 5.1 has been lowered to 15 Hz toshow the importance of the second group of frequencies that generates the accelerationsignal. The filter is a low band passing Butterworth type of order 6.

43

44 CHAPTER 5. MODEL CHECKING

The second check on this node regards the vertical displacement. In Figure 5.2 are plottedthe results of the direct integration. The global shape reflects what expected by staticanalysis. A hand check can be carried out to verify the static deflection in static conditions.Assuming the model as a simply supported beam loaded by the dead weight of concrete,ballast and sleepers, the theoretical deflection in midspan is calculated by the formula:

δTmax =5

384

qL4

EI= 5.616 cm

where:

q = g(Aballastρballast + Abridgeρbridge + Arailρrail +msleepers/l) = 180444 N/m

L = 42 m

E = 2.1× 1011 N/m2

I = 0.62 m4

which is slightly higher but in accordance with the calculated one by ABAQUS:

δFEMmax = 5.606 cm.

5.1. ACCELERATION AND DISPLACEMENT AT MIDSPAN 45

0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Train Model TW, Speed = 200 km/h

0 5 10 15 20 25 30 35 40 45 500

100

200

300

400

f (Hz) − The Highest Frequency is the Nyquist Frequency

|FF

T(f

)|

Unfiltered DataFiltered Data

Figure 5.1: Acceleration at Midspan due to 4 Train Wagons.

0 1 2 3 4 5 6−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

Train Model TW, Speed = 200 (km/h)

Time (s)

Dis

plac

emen

t at M

idS

pan

(m)

Figure 5.2: Displacement at Midspan due to 4 Train Wagons.


5.2 Vertical Reaction Force at the Bearings

An effective way to check the contact between moving wheel and bridge can be analyzingthe variation of the reaction forces at the bridge support. Figure 5.3 presents these data.The graph contains both the results of the gradual application of the gravity accelerationto the model objects (Static step) and the results from the movement of the Trains Wagons(Dynamic step). The diagram shows a series of peaks. Each of them corresponds to theeffect of an axle moving over the bridge. The magnitude of the reaction forces during thestatic step can be confirmed by hand calculation:

RF T =qL

2= 3.789 MN

with q and L as previously described. The value from FE model is instead:

RF FEM = 3.847 MN.

greater than the theoretical one because in such a case the supports are also loaded by apart of the track just outside the bridge.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

Time (s)

Rea

ctio

n F

orce

(M

N)

Vertical Reaction Forces at Bridge Boundaries

HingeRoller

Figure 5.3: Vertical Reaction Forces at the Supports, Static and Dynamic Steps.

5.3. MOVING WHEEL-NODES 47

5.3 Moving Wheel-Nodes

Contact is assured when the force at the nodes involved increase. Interaction exists if thisproperties are not constant. The evolution of the contact force at the moving wheel-nodeof each axle can be visualized selecting the output request CPRESS. Figures 5.4, 5.5, 5.6,5.7 present in order the contact pressure under the four running wheels of the first wagon.Each of the mentioned diagrams show a sudden increase in oscillation of the contactpressure around 3.30− 3.50 s. This time corresponds with the entrance of the train axleon the bridge. The vibration stops instead at around 5.20 − 5.40 s since, by this time,each axle stops at the end of the model. The easy prove comes again from the formulas(t) = vt.

The average values of the axle force is easily confirmed by simple calculation:

F Tpower =

mpower

4g = 1.91× 105 N

F Tcoach =

mcoach

4g = 2.45× 105 N

In Figure 5.8 are collected all the diagrams previously showed in separated figures, whileFigure 5.9 presents the contact pressure under the first wheel set of the first four wagons.In Figure 5.10 is plotted the distance, function of time, that each axle travels from itsstarting point at 70.8 m distance from the bridge.

All the data have been sampled at the frequency of 50 Hz.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the 1st Wheel Set of the 1st Wagon

Figure 5.4: Variation of Force at the Wheel-Node of the 1st Axle.


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the 2nd Wheel Set of the 1st Wagon

Figure 5.5: Variation of Force at the Wheel-Node of the 2nd Axle.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the 3rd Wheel Set of the 1st Wagon

Figure 5.6: Variation of Force at the Wheel-Node of the 3rd Axle.

5.3. MOVING WHEEL-NODES 49

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the 4th Wheel Set of the 1st Wagon

Figure 5.7: Variation of Force at the Wheel-Node of the 4th Axle.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the Wheels of the 1st Wagon

1st Wheel Set2nd Wheel Set3rd Wheel Set4th Wheel Set

Figure 5.8: Variation of Force at the Wheel-Nodes, cumulative plot.


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

−3.4

−3.2

−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2x 10

5

Time (s)

F(N

)

Contact Force at the First Wheel Set

1st Wagon2nd Wagon3rd Wagon4th Wagon

Figure 5.9: Variation of Force at the Wheel-Nodes, cumulative plot.

1 2 3 4 5 6 7−200

−150

−100

−50

0

50

Time (s)

Dis

tanc

e (m

)

Contact Slip at the First Wheel Set

1st Wagon2nd Wagon3rd Wagon4th Wagon

Figure 5.10: Horizontal Movement of the Wheel-Nodes, cumulative plot.

Chapter 6

DynStArr, a MATLAB® Toolbox

The following Chapter presents the MATLAB® toolbox written to build the input fileand solve the dynamic problem in ABAQUS® environment. Examples of function callsare given to show its purposes as well as an introduction to its main subroutines.

In its general formulation the toolbox DynStArr is defined by:

DynStArr(Output1,LBridge,m_Bridge,A_Bridge,I33_Bridge,E_Bridge,LWay,......,lambda,A,phi,Output2,Output3,Output4)

different tasks, tough, require different way of recalling the function. In several cases,later described, some of those variable are omitted and the required data are asked tobe introduced from keyboard by the user. DynStArr calculates the dynamic response ofa simply supported bridge under the passage of the freight train called Steel Arrow atconstant speed. Since the bridge is defined by its general properties the toolbox can han-dle concrete, steel or composite type sections. The toolbox builds an input file and asksABAQUS® to compile it. At analysis completed the results stored in a file by the FiniteElement Software are collected and are organized in MATLAB® environment. Section6.3 here below lists and describes all the possible entries of the toolbox. The next sectionwill describe, instead, how to use the function for specific tasks.

Output1 = %s String that specifies the type of analysis to perform. Five possibilitiesare allowed.

TrainImpact collects in a plot the sequence of maximum accelerations and dis-placements on the bridge caused by the train traveling at different constantspeed. The interval of interest varies from vMin = 50 km/h to vMax = 150km/h with increment step of 5 km/h. Use in combination with Output2 set toReadAccDisp_MidSpan.

SingleAnal performes a single analysis. The user will be asked the additional datarequired to carry on the analysis. Those are:

1. Train speed;2. Sampling frequency;3. Element length.

51

52 CHAPTER 6. DYNSTARR, A MATLAB® TOOLBOX

When used in combination with ReadAccDisp_MidSpan it prints on video themaximum acceleration of the bridge midspan recorded during analysis. Addingthe facultative parameter Output4 to the call will instead calculate the globaldamping ratio of the bridge.

ConvAnal performs the convergence analysis of the model. The train speed isfixed to 200 km/h while the sampling frequency varies between 100 and 300Hz (with step of 50 Hz), and the element length changes to: 0.60, 0.30, 0.15,0.075 m. Use in combination with Output2 set to ReadAccDisp_MidSpan. Theadditional parameter Output4 will perform instead the convergence analysis ofdamping ratio.

PlotHistory plots the variation in acceleration and displacement during train pas-sage of a single node. Such a call produces charts from data previously savedin a .txt file without solving any dynamic problem. On the other hand theresults must have been saved in a standard format. For example, to plot thetime history acceleration (A) of the Train Wagon model traveling at 195 km/hwith sampling frequency 100 Hz and element length of 0.3 m the name of thefile has to be: "Results_TW_A_MidSpan-v_195-fMax_100-dL_0.3". Theadditional information entered by user are:

1. Train type (CF,SM,TW);2. Train speed;3. Sampling frequency;4. Element length.

Model builds only the track-bridge systems only and performs eigenfrequency andeigenmodes extraction of such a model. The maximum frequency is 100 Hzand each beam element is 0.3 m long. The results are stored in a .odb file.

LBridge = %f Length of the bridge, number expressed in m. To have equally spacedsleepers over the track the code requires this length to be a multiple of the sleeperdistance (0.6 m).

m_Bridge = %f Mass per unit of length of the bridge without ballast, only steel andconcrete have to be considered (kg/m).

A_Bridge = %f Cross-sectional area normalized to concrete or steel, (m2).

I33_Bridge = %f Rotational inertia normalized to concrete or steel, (m4).

E_Bridge = %f Young’s modulus of the material used to normalize the section, (N/m2).

LWay = %f Length of the track before and after the bridge, (m). To have equally spacedsleepers over the track the code requires this length to be a multiple of the sleeperdistance (0.6 m).

lambda = %f Length of the wave that reproduces the vertical track irregularities, (m).

A = %f Amplitude of the wave, (m).

phi = %f Integer number used to shift the wave and reproduce different phase angles.This is achieved by extending the entrance way to the bridge of phi×0.6 m (sleeperdistance).

6.1. EXAMPLES OF FUNCTION CALLS 53

Output2 = %s String to select the train model:

CF stands for Concentrated Forces. The train model comes as a path of movingconstant value forces.

SM stands for Sprung-Masses. This model sketches the train as a series of movingaxles. Two lumped masses are connected by mean of a couple of spring anddamper. The vertical inertial effect is considered.

TW stands for Train Wagons. This is the more accurate model that considers alsothe rotational inertia of bogies and car body.

Output3 = %s String to select variables and nodes to investigate:

Visualize asks for:

1. Contact analysis results (force and displacement of the wheel nodes);2. Reaction forces at the bridge supports;3. Accelerations and Displacements for each node belonging to the bridge.

The results are stored in a .odb file.

ReadAccDisp_MidSpan asks for Acceleration and Displacement in bridge midspan.The history data on acceleration and displacement are stored in separated .txtfiles.

Output4 = %s can assume only one parameter: Set_Damping that ask for thecalculation of the global damping ratio.

6.1 Examples of Function Calls

As introduced the toolbox performs very different analysis, each of them distinguished bya different call. Let’s illustrate through some examples the alternatives.

DynStArr(’Model’,..Bridge Properties..)performs the modal analysis of the track-bridge system. This step is fundamentalto set the proper Rayleigh damping function to the structural parts of the model.It has to be done before any dynamic investigation.

DynStArr(’TrainImpact’,..Bridge Properties..,’CF’,’ReadAccDisp_MidSpan’)draws the dynamic effects on bridge under CF train model passage at differentconstant speeds. From the minimum velocity of 50 km/h up to 150 km/h, themaximum acceleration/displacement at midspan of the bridge will be saved every 5km/h of increment.

DynStArr(’ConvAnal’,..Bridge Properties..,’CF’,’ReadAccDisp_MidSpan’)carries out the convergence analysis of the model. To test the accuracy of the results,the sampling frequency is increased from 100 until 300 Hz, with 50 Hz of step, whilethe element length lowered with the rule 0.6, 0.3, 0.15, and 0.075 m. This callmodels the train as a set of Concentrated Forces.

DynStArr(’SingleAnal’,..Bridge Properties..,’SM’,’Visualize’)performs the calculation of acceleration and displacement of every nodes belonging


to the bridge due to Sprung Mass systems passage. The contact parameters (CPRESSand CSLIP are also checked for all the wheel nodes. The results are stored in a .odb.

DynStArr(’SingleAnal’,..Bridge Properties..,’SM’,’ReadAccDisp_MidSpan’)calculates acceleration and displacement in center of the bridge. The user will beasked to insert the missing parameters to run the analysis: train speed, samplingfrequency and element length. As a result of the analysis it plots on video thefiltered and the unfiltered maximum acceleration and displacement detected duringthe passage of the SM system.

DynStArr(’SingleAnal’,..Bridge Properties..,’SM’,’ReadAccDisp_MidSpan’,...,Set_Damping’) calculates the damping ratio over free vibration of the bridgeat its midspan due to the train modeled as a series of Sprung-Masses. The user willbe asked to insert the missing parameters to run the analysis: train speed, samplingfrequency and beam element length. As a result of the analysis it plots on video theglobal damping ratio of the bridge under the load previously defined.

DynStArr(’PlotHistory’)plots the variation of acceleration and displacement of the bridge midspan previouslysaved in a text file. The user will be asked to insert the information to fetch theproper file.

The whole MATLAB code is attached in Appendix B.

6.2 Interrelation between Output commands

Even tough the Output2 can assume any value with the Output1 and the Output3 anyset, there are some limitation in the use of the other two outputs. The Output1 equalto TrainImpact, for example is subordinate to the the use of Output2 set to ReadAc-cDisp_MidSpan. This because the variable requested is only one while the commandVisualize doesn’t save any value in a text file. The grade of interdependency of thosevariables are explained in the following chart.

Output1 Output3 Output4

TrainImpact

##

SingleAnal

''

// V isualize

ConvAnal

++PlotHistory ReadAccDisp_MidSpan // Set_Damping

Model

6.3. AUTOMATIZED APPROACH TO THE DYNAMIC PROBLEMS 55

Only selecting a single analysis allows two different choices of Output3. PlotHistory andModel don’t need any further output definition since don’t perform dynamic calculationof acceleration and displacement in any bridge section.

6.3 Automatized Approach to the Dynamic Problems

The routine introduced and described in the previous section treats the dynamic problemin three different parts. The first one takes place in MATLAB® environment with theobjective of writing, in a text file, the sequence of commands to build the model, move thetrain, and solve the dynamic problem with ABAQUS®. The second one, in ABAQUS®

environment, solves the equation of the motion derived with FEM approach and saves theresult in a proper text file with .dat extension. Finally, back to MATLAB®, those resultare filtered and presented in plots. The method does not solve the dynamic equationsin MATLAB® environment but uses the ABAQUS® solver to do that. The strength ofthis approach is to avoid the manual construction of elaborated finite element models bywriting the list of (iterative) implicit instruction with MATLAB®. The train velocityis given as a parameter that the user can vary. The method is then customized andautomatized. Figure 6.1 presents a diagram of the instructions.

Figure 6.1: The Method to Solve the Dynamic Problem.

The core function of DynStArr is called Main. This function has all the information todefine nodes through its coordinate Nodes, construct the finite element Elements andassign tasks Step. Once the list of instruction is completed, the ABAQUS® compileris activated, straight in MATLAB® environment by the command dos(Abaqus Job =Model interactive). The solution of the equation of the motion are stored in Model.data text file conveniently scanned, back to the MATLAB® environment, by the routineRead. Acceleration and displacements are then easily plotted and saved.

6.3.1 The MainRail_SteelArrow.m file

The function MainRail_SteelArrow, previously shortly called Main, is responsible of allthe results that can be obtained by DynStArr. A further description of the contents ofthis file is required to facilitate future improvements of the toolbox.


[a_Bridge,b_Bridge] = Rayleigh_coeff(w(1),w(n),z_1,z_n) This portion of codecalculates the material damping coefficients α (= a_Bridge) and β (= b_Bridge)by forcing the Rayleigh equation to pass through the point (w1, ξ1) and (xn, ξn) of thefrequency-damping plane. According to the notation in the code those two pointsare, respectively, (w(1),z_1) and (w(n),z_n). The eigenfrequencies of the Track-Bridge model are stored in the vector w and are the result of the modal analysispreviously performed. The same function has been recalled to design the structuraldamping of the railway.

[LastN_Station1,Axles1] = NodesStation(MainID,FirstN_Station1_Name,......,FirstN_Station1_Pos,’Station1’) This routine writes on the text file openedwith identifier MainID the coordinates of the nodes belonging to the starting sta-tion, Station1. The function requires name and position of the first node. All theothers coordinates are derived by this definition. In this way it is possible to controlthe names of nodes not to interfere with other parts of the model. The position ofthe nodes in this part of the model is designed to fit the axle distance. Laying thewheels over clamped nodes will facilitate their contact to the master elements (therailway). ’Station1’ is just the label to identify the last node of the station.

LastElementSet_Station1 = ElementsStation(MainID,FirstSpringName,......,FirstDashpotName,Axles1,k_Station,c_Station,’Station1’) The functiondefines springs and dashpots for the first station. It requires the names of the firstcouple since the name of all the others are derived from those. The name of theelements are chosen not to interfere with other object built in the same model.

LastN_EstaCont = NodesRail(MainID,FirstN_EstaCont_Name,......,FirstN_EstaCont_Pos,LEstaCont,dLRail,0,lambda,’EstaCont’) This is theequivalent of NodeStation that builds nodes of the rail bed. As the previous one,the nodes are defined in couple: a lower node on the ground and the upper one usedto define the rail beam element. What changes from the station model will be theboundary condition assigned. All the nodes laying on the ground will be clampedwhile those belonging to the rail will be free to move. This function is recalled lateron to build all the other rail paths: ’WayIn’, ’Bridge’, and ’WayOut’. The lowernodes over the bridge will not be constrained. In this case FirstN_EstaCont_Nameand FirstN_EstaCont_Pos stand for name and position of the first node of a shortway after the first station. In analysis with corrugated rail it was not possible to es-tablish contact (from which the name) without a preliminary path free from verticalirregularities. 0 and lambda are indeed amplitude and length of the wave. Withinthis 4.8 m the wheel can be constrained to run over a straight line.

LastEl_Bridge = ElementsBeam(MainID,FirstN_Bridge_Name-2,......,LastN_Bridge(1)-1,FirstEl_Bridge,ElementType_Bridge,......,Density_Bridge,A_Bridge,I33_Bridge,E_Bridge,G_Bridge,......,a_Bridge,b_Bridge,’Bridge’) As said the lower nodes of the bridge have tobe treated differently. This function defines the beam element of the bridge underthe Euler-Bernoulli hypothesis. Following the order of definition the second andthe third variables are the first and the last node name of the bridge, it follows thename of the first element of the bridge, its type, and the sectional properties suchas: density, cross-sectional area, rotational inertia, Young’s modulus and Torsionalelastic modulus (not required in planar model but introduced for further 3D de-

6.3. AUTOMATIZED APPROACH TO THE DYNAMIC PROBLEMS 57

velopments). a_Bridge and b_Bridge are the Rayleigh coefficient of the materialdamping of the bridge previously calculated.

SleepersPos = ElementsRail(MainID,LastN_Station1(1)-1,......,LastN_WayOut(1)-1,LastElementSet_Station2(1),......,LastElementSet_Station2(2),dLSleeper,dLRail,k_Rail,c_Rail)Like ElementsStation does for the stations, ElementsRail defines geometry andproperty of the track bed. At the constant distance of dLSleeper a couple of springand damper elements connects the lower node with the upper one. k_Rail andc_Rail specify their characteristic coefficient. dLRail is instead the length of theelement chosen according convergence analysis.

VibratingMass(MainID,SleepersPos,mVibratingMass) This function assigns to the up-per nodes in place of the sleepers the mass mVibratingMass. Lumped with thesleeper weight there is also the equivalent mass of the vibrating ballast.

AmplitudesCF(MainID,Crds,Pa,v) To move the set of constant forces a variation in timeof its position has to be defined. Crds specifies the coordinate of the nodes thatform the railway path. Pa is instead the nominal axial load and v the train speed.

NodesSM(MainID,AxPos,AxNum,FirstN_Station1_Pos) Each object as to be defined bynodes, and so is for the single degree of freedom moving masses. Each axle issketched by a couple of nodes over the ’Station1’ starting form its first nodeFirstN_Station1_Pos. AxPos and AxNum help to locate the SMs over the station.An equivalent function has been written for the Train Wagon model.

ElementsSM(MainID,AxNum,k_P,c_P,k_C,c_C,mWheelP,mWheelC,mAxle_P,mAxle_C)This function designs the sprung-mass models. k_P, c_P, k_C, and c_C definesrespectively constant stiffness and damping factor of springs and dashpots of powercars and coaches. An equivalent function has been written for the Train Wagonmodel.

Surface(MainID,AxNum) Any contact analysis requires a definition of a master surfaceto guide the slave objects. This function supplies this definition for both SprungMass and Train Wagon models.

AmplitudesSM(MainID,v,tTrans,tFree,LastN_Station2(2),AxNum,AxPos) This func-tion controls the motion of the wagons modeled by SMs. Among the parametersrequired tTrans and tFree indicate the waiting time needed to stabilize the ac-celeration of the masses after the application of the gravity load and the time offree vibration during the evaluation of the critical damping. An equivalent functionexists to move the TWs.

Boundary_Model(MainID,Clamp,Hinge,Bearing,Axles1,SleepersPos,......,LastN_Wayin(1),LastN_Bridge(1),Axles2) It sets the boundary conditionsof the nodes of the model. The first bridge node is hinged while the last one issimply supported.

BoundarySM(MainID,AxNum) At the moment of assignation of the gravity load each sprungmass node is fixed in its design position. This allows also to establish contact withthe rail. An equivalent function exists to fix the TW nodes before the start of theanalysis.


Step_1_Gravity(MainID,g,Output2) The function assigns the gravity load to all thenodes with mass property.

Step_2_EigFrequency(MainID,fMax) When required from user this function calculatesthe eigenvalues of the model. This part of the code belongs exclusively to Output1= Model option.

Step_3_MoveCF(MainID,fMax,AxPos,v,tTrans,tFree,LastN_Station2(2),......,Crds,Output2) When the model has been built the interaction analysis canstart. This function literally pushes the set of concentrated forces to travel over thebridge. All the parameters of interest in defining the motion of the train model havealready been defined. The constant fMax states the sampling ratio applied duringthe analysis.

Step_3_MoveSM(MainID,fMax,AxPos,v,tTrans,tFree,LastN_Station2(2),......,Clamp,Hinge,Bearing,Axles1,SleepersPos,LastN_WayIn(1),......,LastN_Bridge(1),Axles2,Output2) This is the function that writes on theABAQUS® input file how to perform the dynamic task. The motion of the SMs iscontrolled through AxPos, v, tTrans, and tFree. The strategy to move the coupleof nodes is to modify their boundary conditions. An analogous function has beenwritten for the TW model type.

Results = ReadAccDisp_MidSpan(NameFile,fMax,fFilter,ButtOrder,v,......,tFree,dLRail,NumWag,Output1,Output3) To scan the results stored in the.dat file, this function has been written. It fetches the proper file by using its name(NameFile) and filters the signal with a low band pass filter with the filtering fre-quency fFilter and the order ButtOrder. Acceleration and displacement data arethen saved in separated text files. When Output3 is set to ’ReadAccDisp_MidSpan’,the variable in output (Results) contains the maximum acceleration and displace-ment of the filtered and unfiltered signal. When ’Set_Damping’ is also inserted,Results holds the damping ratio of the track-bridge system.

Chapter 7

Assignation of Damping andConvergence Analysis

7.1 Damping Investigation

The reference normative, [EC2], sets the global damping ratio of this superstructureto 0.5%. In direct step integration analyses, however, only dissipative objects (such asdashpots) have fixed damping coefficient. The remain of the model requires a dampingdefinition in form of Rayleigh function, which happens to be a material property varyingwith the frequency of vibration of the object. Matching the code requirement becomesdifficult: the structural damping is now depending on the actions. An iterative test couldhelp to calibrate the assignation of damping. At each trial Rayleigh function appliedto the bridge, the effective damping ratio can be calculated by mean of the logarithmicdecrement over free vibration of the midspan and suggest how to modify the next entry.This value of damping is just an estimation, it depends in fact only on the contribute tothe vibrations of the symmetric eigenmodes.

Many variable can influence the Rayleigh damping:

I) train speed, vtrain;

II) number of wagons, #wag;

III) ABAQUS® numerical damping, αartificial;

IV) time of free vibration, tfree;

V) frequency of sampling, fsampling;

VI) length of the beam elements, lelem;

VII) presence of track irregularities.

It can be said that the first four of those have, mainly, a relevance in defining the accu-racy of the result, while the last two influence its precision. A different number of wagonspushed at different train speeds, in fact, excite the bridge deck each time in a different way.The set of eigenmodes involved in the vibration, and their relative importance, change,modifying also the damping assigned to the bridge. For the same reason each way of mod-eling the train, concentrated forces, sprung-mass systems, and train wagons, requires a

59

60 CHAPTER 7. ASSIGNATION OF DAMPING AND CONVERGENCE ANALYSIS

different setting of Rayleigh coefficients. Even the time of free vibration modifies the cal-culation of the logarithmic decrement. A minimum number of peaks is, therefore, neededto capture the real damping of the structure. As previously mentioned, the accelerationresponse at midspan of the deck gives the feed back information to control the actualbehavior of the whole model. For such a purpose the oscillation in displacement couldhave been considered as well. In Figure 7.1 are plotted the fluctuation in accelerationand displacement during free vibration. It can be seen that the two graphs differs onlyin magnitude. Finally, it is assumed that the (known) track irregularities do not have agreat effect on the application of damping. Therefore the further analysis are only runover a model with flat rail.

12 12.5 13 13.5 14 14.5 15 15.5 16 16.5−0.1

−0.05

0

0.05

0.1

Acc

eler

atio

n (m

/s2 )

Time (s)

Free Vibration at Bridge MidSpan

12 12.5 13 13.5 14 14.5 15 15.5 16 16.5−0.0564

−0.0562

−0.056

−0.0558

−0.0556

Dis

plac

emen

t (m

)

Figure 7.1: Acceleration and Displacement at Bridge Midspan during free oscillation.

In order to reduce the computational time, a preliminary series of tests determinatethe value to assign to each of the important parameters mentioned above. The results,common to all the three train type models, were:

1. At the train speed of 200 km/h the number of moving wagons can be reduced. Themeasured damping ratio due to the passage of 6 wagons only (2 power cars and4 coaches) is extremely close to the damping value obtained by running all the 28wagons. The maximum estimated error is around 10%.

2. At the train speed of 200 km/h the damping ratio calculated after the 6 wagon-train passage is influenced by the αartificial parameter. Keeping the suggested valueαartificial = −0.05 or setting it equal to zero brings to results different approximatelyby the same number.

3. The damping ratio can be evaluated over a time of free vibration of 5 s, with thecare of removing the first peaks still disturbed by the moving load and averagingthe damping over three different values.

Based on those observations, the model tested iteratively had these common characteris-tics:

7.1. DAMPING INVESTIGATION 61

◦ vtrain = 200 km/h;◦ #wag = 6;◦ αartificial = −0.05;◦ tfree = 5 s;◦ perfectly flat rail;◦ Rayleigh damping function of the rail as in Figure 7.2.

50 100 150 2003

4

5

6

7

8

9

10x 10

−3

ω (rad/s)

ξ (%

)

Rayleigh Function

Damping Function of the RailValues at Model Eigenfrequencies

Figure 7.2: The Rayleigh function chosen for the Rail Line.

Sampling frequency and element length instead didn’t have same value for the differenttrain models. Tables 7.1a, 7.1b, and 7.1c present the convergence analysis for dampingratio run with each different train type. The time needed by each simulation is referredto the CPU: Intel(R) Core(TM)2 CPU 6600 2.40GHz, with 32-bit Operating System.The data refers to the ultimate Rayleigh function selected for each train type. The sameinformation are presented in Figure 7.3 in form of a graph.

For the train modeled by Concentrated Forces it is possible to conclude that the dampingratio reaches convergence already with (100 Hz, 30 cm). The result from the last iterationξ(3) = 0.50633% is, in fact, only 0.38% greater than the first one ξ(1) = 0.50443%. ForSprung-Masses and Train Wagons, instead, the situation is different. To reach conver-gence in the result a sampling frequency of 400 Hz and an element length of 3.75 cm arerequired. The values on the diagonals and the few over it are enough to conclude that theimplementation of damping in form of Rayleigh function depends mainly on the samplingfrequency. Once the proper Rayleigh function at the speed of 200 km/h has been selectedit is time to check the variability of the damping ratio at different train speed. Accordingto the tests run with the Concentrated Forces model, the damping ratio barely decreasesat lower speeds. Those results are presented is Table 7.2. The mean value of damping canbe said to be well representative. Such analysis has not been run for the other two load


lelem (cm)30.0 15.0 7.5

fsampling (Hz)

100 0.50443 % - -32’

200 - 0.50584 % -1 h 48’

300 - - 0.50633 %4 h 18’

(a) Train as Concentrated Forces.

lelem (cm)30.0 15.0 7.5 3.75

fsampling (Hz)

100 0.70957 % - 0.70122 % -43’ 1 h 47’

200 - 0.55747 % - -2 h 48’

300 0.50153 % - 0.50564 % -1 h 59’ 6 h 42’

400 - - - 0.50210 %12 h 6’

(b) Train as Sprung-Masses.

lelem (cm)30.0 15.0 7.5 3.75

fsampling (Hz)

100 0.77543 % - 0.77527 % -55’ 1 h 58’

200 - 0.61026 % - -3 h 23’

300 0.55516 % - 0.55503 % -2 h 21’ 5 h 31’

400 - - - 0.51695 %18 h 45’

(c) Train as Train Wagons.

Table 7.1: The Convergence Analysis for Damping Ratio and Simulation Times.

7.1. DAMPING INVESTIGATION 63

1 1.5 2 2.5 3 3.5 40.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Frequency and Element Length (Hz,cm)

ξ (%

)

Convergence Damping Ratio

CFSMTEC limitation

Figure 7.3: The Convergence Analysis for Damping Ratio.

cases with the belief that once convergence in damping has been reached at the speedof 200 km/h its value doesn’t change considerably at different train speeds, just as inthe case of the Concentrated Forces. Figure 7.4 presents the ultimate Rayleigh functionsapplied to the Bridge when crossed by one of the three train types at the time.

Concentrated Forcesξ (%) CPU Time

vtrain (km/h)

50 0.50380 % 1 h 11’100 0.50412 % 45’150 0.50425 % 37’200 0.50443 % 32’

Average Damping, ξ̄CF 0.50415 %Range 6.3× 10−4

Standard Deviation, σCF 2.3× 10−4

Coefficient of Variation, CV 0.04564

Table 7.2: Damping Ratio and Simulation Times at Different Train Speeds.

The analysis of the component of the free vibration signal showed that only the firsteigenmode is excited. This explain the invariance at different train speed of the moni-tored damping. The assignation of damping summarized in Table 7.1, then, succeededin calibrating the damping ratio of the first eigenmode. In Figure 7.5 are plotted thefree vibration signal and the exponential curve reproduced with the damping ratio ξ esti-mated by logarithmic decrement method. The load case running is the Train Wagon withsampling frequency of 300 Hz and element length 7.5 cm.


50 100 150 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

ω (rad/s)

ξ (%

)

Rayleigh Damping Functions

CFValues at Model EigenfrequenciesSMValues at Model EigenfrequenciesTValues at Model EigenfrequenciesEC limitation

Figure 7.4: The Ultimate Rayleigh Damping Functions of the Bridge.

5 6 7 8 9 10 11−0.4

−0.2

0

0.2

0.4

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Free Vibration − Train Model SM, Speed = 200 km/h

0 50 100 1500

500

1000

1500

2000

2500

fsampling

= 300 Hz, lelem

= 7.5 cm


|FF

T(f

)|

Numerical Simulation

e(ξωnt) with ξ = 0.5 %

Figure 7.5: An Example of Free Vibration Signal and Calculated Damping Ratio.

7.2. CONVERGENCE ANALYSIS OF THE MODELS 65

7.2 Convergence Analysis of the Models

Once the proper Rayleigh damping function has been applied to each of the models aconvergence analysis will say which couple of values (fsampling,lelem) has to be consideredto obtain accurate results. The maximum accelerations and displacements in center spanwere therefore compared. The model tested had this common properties:

◦ vtrain = 200 km/h;

◦ #wag = 14;

◦ αartificial = −0.05;

◦ low band passing filtering of the signals with Butterworth function at 30 Hz andorder 6.

The choice of the couple of values to be implemented in the next analysis has necessaryto be a compromise in simulation time and reliability of the results. With the purpose ofcomparing the different alternatives, in Table 7.3 are collected the results of the conver-gence analysis on acceleration and the relative simulation times at each different coupleof values (fsampling,lelem). Since the convergence analysis has been run with 14 wagonsonly, the simulation times of the whole model have to be approximately doubled. Withthis case the train speed is set to be the maximum among all.

The values chosen to run the next analysis are the same for all the model types:

fsampling = 100 Hz;

lelem = 0.3 m.

This choice introduces an error in the calculation of the accelerations that varies with thetrain model. Moreover, comparing the effective damping ratio of the first eigenmode inTable 7.1 with the maximum acceleration in Table 7.3, the increase in maximum acceler-ation at Sprung-Masses and Train Wagons passage is due to the variation in assignationof damping. The values of displacement, instead, are almost non influential by those vari-ables. Assuming acorrect = a(300, 0.075) the relative errors have been calculated accordingto the following formula:

ε =a(300, 0.075)− a(100, 0.3)

a(100, 0.3)

εCF = +0.08 %;

εSM = +2.76 %;

εTW = +8.87 %;

The positive sign indicates that the approximated results are underestimated.


00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

3000.35

0.4

0.45

0.5

0.55

0.6

0.65

dl (m)f (Hz)

Accele

ration (

m/s

2)

(a)

00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

300−0.0806

−0.0805

−0.0804

−0.0803

−0.0802

−0.0801

dl (m)f (Hz)

Dis

pla

ce

me

nt

(m

)

(b)

Figure 7.6: The Convergence Analysis for the CF.

00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

3000.35

0.4

0.45

0.5

0.55

0.6

0.65

dl (m)f (Hz)

Accele

ration (

m/s

2)

(a)

00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

300−0.0806

−0.0805

−0.0804

−0.0803

−0.0802

−0.0801

dl (m)f (Hz)

Dis

pla

ce

me

nt

(m

)

(b)

Figure 7.7: The Convergence Analysis for the SM.

7.2. CONVERGENCE ANALYSIS OF THE MODELS 67

00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

3000.35

0.4

0.45

0.5

0.55

0.6

0.65

dl (m)f (Hz)

Accele

ration (

m/s

2)

(a)

00.1

0.20.3

0.40.5

0.60.7

100

150

200

250

300−0.0806

−0.0805

−0.0804

−0.0803

−0.0802

−0.0801

dl (m)f (Hz)

Dis

pla

ce

me

nt

(m

)

(b)

Figure 7.8: The Convergence Analysis for the TW.


lelem (cm)60.0 30.0 15.0 7.5

fsampling (Hz)

100 0.55249 0.54091 0.54441 0.5447613’ 18’ 28’ 49’

200 0.55330 0.54147 0.54158 0.5420526’ 36’ 57’ 1 h 39’

300 0.55398 0.54084 0.54103 0.5413538’ 54’ 1 h 26’ 2 h 34’

(a) Train as Concentrated Forces.

lelem (cm)60.0 30.0 15.0 7.5

fsampling (Hz)

100 0.56045 0.43101 0.39802 0.3996725’ 34’ 54’ 1 h 31’

200 0.58388 0.45988 0.42621 0.4305851’ 1 h 10’ 1 h 48’ 3 h 4’

300 0.59650 0.48186 0.44294 0.442921 h 19’ 1 h 44’ 2 h 42’ 4 h 34’

(b) Train as Sprung-Masses.

lelem (cm)60.0 30.0 15.0 7.5

fsampling (Hz)

100 0.54443 0.44632 0.44798 0.4484828’ 38’ 57’ 1 h 41’

200 0.58500 0.47173 0.47339 0.4740156’ 1 h 13’ 1 h 55’ 3 h 12’

300 0.60199 0.51106 0.48515 0.48591 h 25’ 1 h 54’ 2 h 50’ 4 h 47’

(c) Train as Train Wagons.

Table 7.3: The Convergence Analysis on Acceleration (in m/s2) and Simulation Times.

Chapter 8

Results

8.1 Rail without Vertical Irregularities

In Figure 8.1 are collected the maximum accelerations at bridge midspan due to trainpassage at different constant speeds. The velocity of the train varies from 50 up to 200km/h and increases each 5 km/h. The plot compares the three train types moving over aflat rail. The maximum acceleration is caused by CF model traveling at the speed of 120km/h and it reaches:

aCF, 120 = aCF, 120, model(1 + εCF ) = 4.18 m/s2.

With the notation introduced in the previous chapter εCF = 0.08 % is the error in accuracythat has been introduced when calculating the time step integration over the frequencyof 100 Hz and requiring the element length of 0.3 m. The remaining train models havetheir own, different, error factors denoted by εSM = 2.76 % and εTW = 8.87 % thatshift upwards the diagram of maximum acceleration. The CF model stimulates anotherresonance peak at 60 km/h with corrected amplitude:

aCF, 60 = aCF, 60, model(1 + εCF ) = 1.76 m/s2.

Looking at the general shape of the diagrams, the SM model shows many similarities withthe one due to CF model. The two resonance peaks, highlighted by the simpler model,are still detected even if the first one, at the speed of 60 km/h, is barely recognizable.The TW model does not contradict the conclusion of the previous two, but adds twoother peaks at the critical speed of 95 and 190 km/h. Assuming the results of the TWmodel, for its level of details, the most accurate among the three, then the CF modeloverestimate the resonance peaks by the two factors:

SF60 =aCF, 60

aTW, 60

=aCF, 60, model(1 + εCF )

aTW, 60, model(1 + εTW )= 2.02

SF120 =aCF, 120

aTW, 120

=aCF, 120, model(1 + εCF )

aTW, 120, model(1 + εTW )= 1.84

Unfortunately the absence of field-measurements forbade the calibration of the track-model and the results have only importance within this numerical simulation.

69

70 CHAPTER 8. RESULTS

50 60 95 120 190 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison of Different Train Models on Flat Rail

CFSMTW

Figure 8.1: Maximum Acceleration at Bridge Midspan.

50 60 95 120 190 2000.95

1

1.05

1.1

1.15

1.2

1.25

Train Speed (km/h)

DA

F a

t Brid

ge M

id S

pan

(−)

Comparison of Different Train Models on Flat Rail

CFSMTW

Figure 8.2: The DAF over Displacements at Midspan.

8.1. RAIL WITHOUT VERTICAL IRREGULARITIES 71

To have a closer look at the nature of the peaks, the variation in acceleration againsttime have been asked for the three train models at the critical speeds. Figures 8.4, 8.5,8.6, 8.7, 8.8, and 8.8 present the history variation of acceleration and displacement atbridge mid span respectively due to CF, SM, and TW models traveling at 120 km/h. Theshape of the diagrams reflect the one that can be seen at 60 km/h, not printed to avoidredundancy.

According to the formula:vcritical = fnlcritical

each critical speed vcritical can be related to a natural frequency fn of the infrastructureand to a critical distance lcritical among the set of actions imposed by the moving train. AFast Fourier Transformed (FFT) of the acceleration signal suggested which frequencies arestimulated and the characteristic length could be calculated. Since the measurements aretaken at bridge midspan, the description of the bridge dynamics disregards the contribu-tion of the even eigenmodes. The FFT plot is not able to detect any "even" eigenfrequencyof the bridge, responsible of a antisymmetric eigenmode. At the train speed of 60 and 120km/h the bridge goes in resonance at its first eigenfrequency and the two critical lengthinvolved are:

l60 =60/3.6

f1' 7 m

l120 =120/3.6

f1' 14 m

The calculation of the critical lengths is affected by an error related to the uncertainty ofindividuating the peak. One data every 5 km/h of speed step could not be enough. Inthis case the characteristic lengths are confirmed by overlapping the path of forces withsine functions with wave lengths λ = lcritical. Figure 8.3 helps to visualize this recurrentgeometry.

(a) λ = 7.0 m

(b) λ = 14.0 m

Figure 8.3: Critical lengths.


0 2 4 6 8 10 12 14 16 18−5

0

5

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Train Model CF, Speed = 120 km/h

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2x 10

4


|FF

T(f

)|


Figure 8.4: History Acceleration at Bridge Midspan.

0 2 4 6 8 10 12 14 16 18−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

Train Model CF, Speed = 120 (km/h)

Time (s)

Dis

plac

emen

t at M

idS

pan

(m)

Figure 8.5: History Displacement at Bridge Midspan.


0 2 4 6 8 10 12 14 16 18 20−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Train Model SM, Speed = 120 km/h

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000


|FF

T(f

)|



0 2 4 6 8 10 12 14 16 18 20−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

Train Model SM, Speed = 120 (km/h)

Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

Time (s)

Acc

eler

atio

n at

Mid

span

(m


0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000


|FF

T(f

)|



0 2 4 6 8 10 12 14 16 18 20−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04


Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



Something different happens at 95 and 190 km/h. While SM and CF models do not detectany considerable variation in maximum acceleration, the TW diagram rises suddenly. InFigures 8.10, 8.12, and 8.14 are presented the variation in acceleration against time duringtrain model passing at 95 km/h. The three graphs show little similarities. One resides inthe first part of each diagram. The three models oscillates in the same way for the firstthree seconds of train motion over the bridge. At the speed of 95 km/h seven wagonshave the time to enter and pass it. The second similarity can be seen in the FFT of thesignal. The main frequencies that compose the signal are the same three:

fs1, 95 = 1.87 Hz;

fs2, 95 = 2.37 Hz;

fs3, 95 = 3.73 Hz.

From the modal analysis of the track-bridge system it is known that fs2, 95 is the firstresonance frequency of the model, the other two are instead related to the sequence oftrain axles:

l1, 95 =95/3.6

1.87' 14 m

l2, 95 =95/3.6

3.73' 7 m

the same critical lengths involved in the resonance phenomenas. At these speeds the trainmodels are not in resonance with any symmetric eigenmodes of the bridge. It could bepossible though that the second eigenfrequency of the model had been stimulated andonly the TW model, which fully considers the inertia of the wagons, shifted in midspan ofthe deck some vibrations. This view could be supported by the fact that the two peaks arelocated speeds that are multiple to each other, 95 and 190 km/h just like the resonancepeaks for the first eigenmode. If a geometric feature in the loads can stimulate the system,even its double (that considers half of the action of the previous) will have a part in that.An analysis at quarter span of the bridge will confirm this hypothesis.


0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Train Model CF, Speed = 95 km/h

0 5 10 15 20 25 30 35 40 45 500

200

400

600

800

1000

1200

1400


|FF

T(f

)|



0 5 10 15 20 25−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

Train Model CF, Speed = 95 (km/h)

Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



0 5 10 15 20 25−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (s)

Acc

eler

atio

n at

Mid

span

(m

/s2 ) Train Model SM, Speed = 95 km/h

0 5 10 15 20 25 30 35 40 45 500

200

400

600

800

1000


|FF

T(f

)|



0 5 10 15 20 25−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

Train Model SM, Speed = 95 (km/h)

Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



0 5 10 15 20 25−2

−1

0

1

2

Time (s)

Acc

eler

atio

n at

Mid

span

(m


0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

3500


|FF

T(f

)|



0 5 10 15 20 25−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04


Time (s)

Dis

plac

emen

t at M

idS

pan

(m)


8.2. THE RAIL WITH VERTICAL IRREGULARITIES 79

8.2 The Rail with Vertical Irregularities

Figure 8.16 presents the effects of an irregular rail track on the bridge dynamics underthe three moving models. The rail profile has been shaped according to the sine function"Irr1" in Figure 8.18. Amplitude and wave length are taken from the requirements of anewly built railway, and are respectively 2 mm and 6 m. In a second group of simulationsdifferent phase angles have been tested to determine the importance of this unknownparameter. In Figure 8.18 are sketched the rail profiles studied.

To easily compare the response with flat rail, the sequence of maximum acceleration atmid span of CF, SM, and TW model were plotted separately in Figures 8.19, 8.20, and8.21. Only the TW model can detect the modifications on the rail profile. Its resonancepeak at 60 km/h on flat rail, has been split in two at the speeds of 55 and 65 km/h, whilethe one at 120 Km/h remain unchanged even in magnitude.

The variation in acceleration at bridge mid span, with CF and SM models pushed at thecritical speed, have the same distribution of those obtained before the application of thetrack irregularities. No additional plots are presented for those train types. The TWmodel, instead, offers and interesting different behavior. The FFT of the accelerationat 55 km/h presents a rather broad peak around 2.40 Hz, while at 65 km/h the signalis mainly compose by the frequency 2.58 Hz. Trying to find a critical length for thosefrequency gives:

l55 = 6.4 m

l65 = 7.0 m.

Since 7 m is the critical length responsible of resonance at 60 km/h, with a flat rail, and6 m is the wave length of the rail irregularity, it could be that the twisting path, byactivating the rotation around the center of gravity of the car bodies (pitching mode),and, that particular shape of the rail, modified the distribution of forces on the bridgedeck preventing resonance. Further analysis have to be conducted with a smaller velocityincrement step to obtain a more accurate distribution of the peaks on the graph.

Other analysis were conducted shifting the rail corrugation along the bridge. The trainwas, of course, modeled by the TW approach. In Figure 8.26 are presented the maximumacceleration at midspan over different corrugated rail. The position of the peaks of thesine function over the rail, even if with small amplitude, changes reasonably the dynamicsof the system. Such importance cannot avoid an accurate survey of the condition of therail before any numerical simulations.


556065 85 120 185 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison of Different Train Models on Irregular Rail

CFSMTW


556065 85 120 185 2000.95

1

1.05

1.1

1.15

1.2

1.25

Train Speed (km/h)

DA

F a

t Brid

ge M

id S

pan

(−)

Comparison of Different Train Models on Irregular Rail

CFSMTW

Figure 8.17: The Dynamic Amplification Factor over Displacements at Midspan.


Figure 8.18: Positions of the Vertical Track Irregularities along the Bridge.

50 60 95 120 190 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison on CF

Flat RailIrregular Rail



50 60 95 120 190 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison on SM



50556065 95 120 185190 2000

0.5

1

1.5

2

2.5

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison on TW




0 5 10 15 20 25 30 35 40−2

−1

0

1

2

Time (s)

Acc

eler

atio

n at

Mid

span

(m


0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000


|FF

T(f

)|



0 5 10 15 20 25 30 35 40−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04


Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



0 5 10 15 20 25 30 35−1

−0.5

0

0.5

1

Time (s)

Acc

eler

atio

n at

Mid

span

(m


0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000


|FF

T(f

)|



0 5 10 15 20 25 30 35−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04


Time (s)

Dis

plac

emen

t at M

idS

pan

(m)



50 100 150 2000

0.5

1

1.5

2

2.5

Train Speed (km/h)

Max

imum

Acc

eler

atio

n at

Brid

ge M

id S

pan

(m/s

2 )

Comparison of Different Rail Irregularities with TW Model

Irr1Irr2Irr3


50 100 150 2000.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Train Speed (km/h)

DA

F a

t Brid

ge M

id S

pan

(−)

Comparison of Different Rail Irregularities with TW Model

Irr1Irr2Irr3

Figure 8.27: The Dynamic Amplification Factor over Displacements at Midspan.


Chapter 9

Conclusion

This study investigated the dynamic response of a railway bridge under train passage.Three load models designed around the Swedish Steel Arrow freight train were testedand compared. A series of Concentrated Forces, a succession of single degree of freedomSprung-Masses, and a sequence of complex multi-degree of freedom Train Wagons. Theincrease in accuracy corresponded with the take in consideration of the inertial properties.The slenderness of the infrastructure, suggested to use Bernoulli-Euler beam element forits mesh. Finally, a regularly spaced sequence of springs and dampers interconnected therail with the bridge and simulated the behavior of the ballast layer. The dynamic responseof the bridge is investigated in presence or absence of vertical track irregularities.

9.1 Concluding Remarks

The following conclusions can be drawn from this thesis:

1. The approach of binding MATLAB® and ABAQUS® was successful and encourag-ing. A correct calibration of the model, that is left, can produce a very convenienttool applicable to a wide range of simply supported bridges;

2. The dynamic amplification attains its maximum value, for every train model, at thecritical train speeds of 120 km/h. Proper resonance has also been detected at thespeed of 60 km/h in all the simulations. At those velocities the set of moving axlescan be overlapped by a sine function of wave length 7 and 14 m respectively;

3. The acceleration response at midspan caused by the Concentrated Forces modelrunning over flat rail can be regarded as an upper boundary of the values obtainedwith the other models while the Sprung-Mass systems as lower boundary of theresults. The response of the two models is in very good agreement at non resonancespeeds and the peaks are located at the same train speeds. The simulation withTrain Wagons does not fit completely this trend, it adds two peaks on the diagram.Besides that, the bridge response lies between the curves due to CF and SM models;

4. The Train Wagon model running over flat rail at the speed of 95 and 190 km/h doesnot excite any symmetric eigenmode of the bridge. A sudden increase of accelerationat midspan on the acceleration/train speed diagram is, anyway, recorded.

87

88 CHAPTER 9. CONCLUSION

5. The presence of track irregularities determines a variation of the bridge dynamicsonly if combined with train wagon load model. The CF model cannot detect themodification of the profile while the SM load case provides a diagram of maximumacceleration similar to the one over flat rail simply shifted upwards.

6. The position of the irregularities along the bridge has an high impact on the dynamicresponse of the structure. It was not possible to relate position of wave with somefeature in the output acceleration diagrams but the position of the peaks changeswith the position of the track irregularities only for train speeds is lower than 120km/h

9.2 Suggestion for Future Research

1. A calibration of the model should be carried out in order to validate the track-bed structure implemented in this model. The wide range of bridge types that thetoolbox can study should allow multiple confronts. Comparing the results with amore sophisticated track-bed model could also be relevant.

2. Since this thesis studied only the mid span of the bridge, it could be really inter-esting, and complementary, investigate the vibration under moving load at quarterspan of the bridge. In this point the second eigenmode has its maximum contribu-tion. The toolbox can be easily modified and adapted to this task.

Bibliography

[jarn] www.jarnvag.net

[wiki] www.wikipedia.org

[gc] www.greencargo.com

[rege] www.regeringen.se/infrastruktur Green Corridors. Green Corridors.

[bb] www.botniabanan.se

[tb] www.transportbusiness.net Goods get on train. Transport Business International.

[EEA11] European Environment Agency Transport emissions of greenhouse gases - As-sessment published Jan 2011. EEA, Copenhagen, 2011.

[EIM] European Rail Infrastructure Managers On track for a sustainable future - EIMresponse to the public consultation on the Communication on a Sustainable Future forTransport. EIM, Bruxelles, 2011.

[ERRI99] ERRI D-214 Committee Numerical investigation of the effect of track irregular-ities at bridge resonance: draft Report 5. European Rail Research Institute (ERRI),Utrecht, 1999.

[ERRI00] ERRI D-214 Committee Rail bridges for speed > 200 km/h. Final Report. Eu-ropean Rail Research Institute (ERRI), Utrecht, 2000.

[EC2] European Committee for Standardization (CEN). Eurocode 2 - Actions on Struc-tures: Traffic loads on bridges. prEN 1991-2. Comité Européen de Normalisation(CEN), Brussels, 2002.

[EN04] EN 10025 - 4 : 2004. Technical delivery conditions for thermomechanical rolledweldable fine grain structural steels.

[UIC03] Union Internationale des Chemins de fer (UIC) Design requirements for railbridges based on interaction phenomena between train, track, bridge, and in partic-ular, speed. UIC Code 776-2, Paris, 2003.

[Ban] Banverket BVS 583.10 Broregler för nybyggnad – BV Bro, utgåva 9. 2008. (inSwedish)

[BBK04] Boverket. Boverkets handbok om betongkonstruktioner, BBK 04. Stockholm,2004.

[Fr96] Frýba L. Dynamics of Railway Bridges. Thomas Telford, London, 1996.

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90 BIBLIOGRAPHY

[Fr01] Frýba L. A rough assessment of railway bridges. Engineering Structures, 23(5):548-56, 2001.

[MRPA] Museros P., Romero M.L., Poy A., Alarcón E. Advances in the analysis of shortspan railway bridges for high-speed lines. Computer and Structures, July 2002.

[BMS] Biondi B., Moscolino G., Sofi A. A substructure approach for the dynamic analysisof train-track-bridge system. Computer and Structures, August 2005.

[CAC] Cheng Y.S., Au F.T.K., Cheung Y.K. Vibration of railway bridges under a movingtrain by using bridge-track-vehicle element. Engineering Structures 23 (2001) 1597-1607, May 2001.

[GRD] Galvín P. Romero A., Domíniquez J. Fully three-dimensional analysis of high-speedtrain-track-soil-structure dynamic interaction. Journal of Sound and Vibration, June2010.

[XZ] Xia H, Zhang N. Dynamic analysis of railway bridge under high-speed trains. Com-puters and Structures 83 (2005) 1891–1901, April 2005.

[RRS] Riguero C., Rebelo C., Simoes da Silva L. Influence of ballast models in the dynamicresponse of railway viaducts. Journal of Sound and Vibration, February 2010.

[SD] Sun Y.Q., Dhanasekar M. A dynamic model for the vertical interaction of the railtrack and wagon system. International Journal of Solids and Structures 39 (2002)1337–1359, December 2000.

[MH] Majka M., Hartnett M. Effects of speed, load and damping on the dynamic responseof railway bridges and vehicles. Computers and Structures, February 2008.

[CJS] Collin P., Johansson B., Sundquist, H. Steel Concrete Composite Bridges - Report121. Division of Structural Design and Bridges KTH, Stockholm, 2008.

[Bjö05] Björklund L. Dynamic Analysis of a Railway Bridge - Master of Science Thesis218. Structural Design and Bridges KTH, Stockholm, 2005.

[KW] Karoumi R., Wiberg J. Kontroll av dynamiska effecter av passerande tåg på Bot-niabananas broar - Sammanfattning . TRITA-BKN Rapport 97, Brobyggnad KTH,Stockholm, 2006.

[Lun1] Lundmark T. Bro over Banafjälsån. Konstruktionsberäkninbgar - Stål. Scandia-Consult Sverige AB, Luleåkontoret, 2001. (In Swedish)

[Lun2] Lundmark T. Bro over Banafjälsån. Statiska Beräkningar, Underbyggnad. Scan-diaConsult Sverige AB, Luleåkontoret, 2001. (in Swedish)

[Haw97] Hawkins S. What is the density of steel?. MadSci Network, 11 September 1997.

[MHK] Molatefi H., Hecht M., and Kadivar M. H. Critical speed and limit cycles in theempty Y25-freight wagon. Journal of Rail and Rapid Transit 220:347, 25 April 2006.

[KAN] Kassa E., Andersson C., and Nielsen J. Simulation of dynamic interaction betweentrain and railway turnout. Vehicle System Dynamics, March 2006.

[And11] Andersson A., Inledande analyser av dynamiskt verkningssätt för ballastfriaspaårkonstrutioner. Teknisk Rapport, Trafikverket, March 2011.

BIBLIOGRAPHY 91

[SC] Sun Y.Q., Cole C. E. Vertical dynamics behavior of three-piece bogie suspensionswith two types of friction wedge. Multibody System Dynamic, 4 August 2007.

[Neg07] Negrut D., Rampalli R., Ottarsson G., Sajdak A. On an Implementation of theHilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equa-tions of Multibody Dynamics. Journal of Comput. Nonlinear Dynam., Volume 2, Issue1, 73 (13 pages), January 2007.

[HHT] Hilber H. M., Hughes T. J. R., Taylor R. L. Improved numerical dissipation fortime integration algorithms in structural dynamics. Earthquake Eng. and Struct. Dy-namics, 5:283 - 292, 1977.

[Hug87] Hughes J. R. Finite Element Method - Linear Static and Dynamic Finite ElementAnalysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1987.

[New59] Newmark N. M. A method of computation for structural dynamics. Journal ofthe Engineering Mechanics Division, ASCE, pages 67 - 94, 1959.

92 BIBLIOGRAPHY

Appendix A

Example of ABAQUS® Input File

Listing A.1: PowerCar.inp1 ∗Heading

Modal Ana lys i s3 ∗∗ Generated by : Abaqus/Standard 6 .9−EF1

∗Prepr int , echo=no , h i s t o r y=yes , model=yes5 ∗∗

∗∗7 ∗∗ Power Car

∗∗9 ∗Node

5001 , 10 .4 , 111 5002 , 10 .4 , 2

5003 , 7 .7 , 113 5004 , 7 .7 , 2

6001 , 9 .05 , 215 6002 , 9 .05 , 3

5005 , 2 .7 , 117 5006 , 2 .7 , 2

5007 , 0 , 119 5008 , 0 , 2

6003 , 1 .35 , 221 6004 , 1 .35 , 3

7001 , 5 .2 , 323 ∗∗

∗Nset , Nset=FrontBogie125 5002

500427 ∗∗

∗Nset , Nset=RearBogie129 5006

500831 ∗∗

∗Nset , Nset=Car133 6002

600435 ∗∗

∗Element , type=spr ing2 , E l s e t=SpringPrimary_P37 4001 , 5001 , 5002

4002 , 5003 , 5004

93

94 APPENDIX A. EXAMPLE OF ABAQUS® INPUT FILE

39 4003 , 5005 , 50064004 , 5007 , 5008

41 ∗Element , type=spr ing2 , E l s e t=SpringSecondary_P4501 , 6001 , 6002

43 4502 , 6003 , 6004∗∗

45 ∗Element , type=dashpot2 , E l s e t=DashpotPrimary_P5001 , 5001 , 5002

47 5002 , 5003 , 50045003 , 5005 , 5006

49 5004 , 5007 , 5008∗Element , type=dashpot2 , E l s e t=DashpotSecondary_P

51 5501 , 6001 , 60025502 , 6003 , 6004

53 ∗∗∗Element , Type=Mass , E l s e t=mWheel_P

55 6001 , 50016002 , 5003

57 6003 , 50056004 , 5007

59 ∗Element , Type=Mass , E l s e t=mBogie_P7001 , 6001

61 7002 , 6003∗Element , Type=Mass , E l s e t=mCar_P

63 7501 , 7001∗∗

65 ∗Element , Type=Rotaryi , E l s e t=IBogie_P8001 , 6001

67 8002 , 6003∗Element , Type=Rotaryi , E l s e t=ICar_P

69 8501 , 7001∗∗

71 ∗Spring , E l s e t=SpringPrimary_P2 , 2

73 4900000∗Spring , E l s e t=SpringSecondary_P

75 2 , 21900000

77 ∗∗∗Dashpot , E l s e t=DashpotPrimary_P

79 2 , 2108000

81 ∗Dashpot , E l s e t=DashpotSecondary_P2 , 2

83 152000∗∗

85 ∗Mass , E l s e t=mWheel_P1510

87 ∗Mass , E l s e t=mBogie_P5200

89 ∗Mass , E l s e t=mCar_P61560

91 ∗∗∗Rotary In e r t i a , E l s e t=IBogie_P

93 0 , 0 , 5900 , 0 , 0 , 0∗Rotary In e r t i a , E l s e t=ICar_P

95 0 , 0 , 911922 , 0 , 0 , 0∗∗

95

97 ∗∗∗∗ Wagon Body 1

99 ∗∗∗Element , type=RB2D2, E l s e t=FB1

101 9001 , 5002 , 60019002 , 6001 , 5004

103 ∗Rigid Body , E l s e t=FB1 , Ref Node=6001 , Pos i t i on=Input , PIN Nset=FrontBogie1∗∗

105 ∗Element , type=RB2D2, E l s e t=RB19003 , 5006 , 6003

107 9004 , 6003 , 5008∗Rigid Body , E l s e t=RB1, Ref Node=6003 , Pos i t i on=Input , PIN Nset=RearBogie1

109 ∗∗∗Element , type=RB2D2, E l s e t=C1

111 9005 , 6002 , 70019006 , 7001 , 6004

113 ∗Rigid Body , E l s e t=C1 , Ref Node=7001 , Pos i t i on=Input , PIN Nset=Car1∗∗

115 ∗∗∗Boundary

117 ∗∗∗∗ Train

119 ∗∗5001 , 1 , 2

121 5003 , 1 , 25005 , 1 , 2

123 5007 , 1 , 2∗∗

125 ∗∗∗Step , name=Step_1_Gravity_SteelArrow

127 ∗ S t a t i c∗Dload

129 , GRAV, 9 .81 , 0 , −1∗Output , f i e l d

131 ∗Node OutputRF, U

133 ∗End Step∗∗

135 ∗∗∗Step , name=Step_2_EigFrequency , pe r tu rbat i on

137 ∗Frequency , e i g e n s o l v e r=Lanczos , a c ou s t i c coup l ing=on , norma l i za t i on=disp lacement

, , 30 , , ,139 ∗Restart , write , f requency=0

∗Output , f i e l d , v a r i ab l e=PRESELECT141 ∗Output , h i s to ry , v a r i ab l e=PRESELECT

∗End Step143 ∗∗

∗∗

96 APPENDIX A. EXAMPLE OF ABAQUS® INPUT FILE

Appendix B

MATLAB® Toolbox DynStArr

Listing B.1: DynStArr.mf unc t i on DynStArr (Output1 , LBridge , m_Bridge , A_Bridge , I33_Bridge , E_Bridge ,

LWay, lambda ,A, phi , Output2 , Output3 , Output4 )2

% This i s a Matlab too lbox to study the Dynamic behaviour o f a4 % Bridge t r a v e l l e d by the Swedish f r e i g h t t r a i n " S t e e l Arrow" .

% Each coach has 25 t o f ax l e load (1 metr ic ton = 1000 kg ) and i t6 % i s cons ide r ed to be loaded by bulk mat e r i a l s up to i t s maximum

% al lowed volume.8 %

% INPUT PARAMETERS10 %

% Bridge − Example f o r Banaf ja ld Bridge12 %

% LBridge = 42 ; (m) Length o f the Bridge14 % m_Bridge = 10700; ( kg/m) Mass per l enght o f the Bridge without

% ba l l a s t , only s t e e l and c on c r e t e .16 % A_Bridge = 0 .57 ; (m^2) Sec t i on Area Normalized

% I33_Bridge = 0 .62 ; (m^4) Rotat iona l I n e r t i a Normalized18 % E_Bridge = 2 .1e11 ; (N/m^2) Young ' s Modulus o f the Mater ia l used

% to normal ize the s e c t i o n .20 %

% Entrance and Exit Track to the Bridge22 %

% LWay = 66 (m) Length o f the WayIn and WayOut24 %

% Rai l I r r e g u l a r i t i e s26 %

% lambda = 6 ; (m) Length o f the wave28 % A = 2e−3; (m) Amplitude o f the wave

% phi = 0 ; (−) I n t eg e r number used to s h i f t the wave30 % and reproduce d i f f e r e n t phase a n g l e s .

% This i s ach ieved by extending the32 % entrance way to the br idge o f phi ∗0 . 6 m

%34 % To avoid f i l e s with long names the in fomat ions about the presence

% and the nature o f the t rack i r r e g u l a r i t y are not s p e c i f i e d .36 % To avoid c a n c e l l a t i o n o f r e s u l t s change dyrectory and name i t

% p rop e r l y .

97

98 APPENDIX B. MATLAB® TOOLBOX DYNSTARR

38 %% Analys i s

40 %% Output1 = ' . . . ' St r ing to s e l e c t the type o f a n a l y s i s .

42 %% ' TrainImpact ' c o l l e c t s in a p l o t the sequence

44 % of maximum a c c e l e r a t i o n s and d i sp lacements on% the br idge caused by the t r a i n t r a v e l l i n g at

46 % the speed vary ing from vMin = 50 km/h to% vMax = 150 km/ h. One data every 5 km/ h.

48 % Use in combination with Output2 s e t to% 'ReadAccDisp_MidSpan ' .

50 %% ' SingleAnal ' per formes a s i n g l e a n a l y s i s .

52 % The user w i l l be asked the add i t i ona l data% r equ i r e to car ry on the a n a l y s i s . Those are :

54 %% 1) Train speed ;

56 % 2) Sampling f requency ;% 3) Element l e n g t h .

58 %% 'ConvAnal ' performs the convergence ana l y s i s

60 % of the model . The t r a i n speed i s f i x ed to 200% km/h whi le the sampling f requency va r i e s

62 % between 100 and 300 Hz ( with step o f 50 Hz) ,% and the element l ength changes to : 0 .60 , 0 .30 ,

64 % 0 .15 , 0 .075 m. Use in combination with Output2% se t to 'ReadAccDisp_MidSpan ' .

66 %% ' PlotHistory ' p l o t s the va r i a t i o n in

68 % a c c e l e r a t i o n and disp lacement o f a s i n g l e node.% The data have to be saved in a . t x t f i l e with

70 % the name format :%

72 % "Results_TW_A_MidSpan−v_195−fMax_100−dL_0.3"%

74 % To f e t ch t h i s f i l e , p r ev i ou s l y saved , the user% has to in t roduce :

76 %% 1) Train type (CF,SM,TW) ;

78 % 2) Train speed ;% 3) Sampling f requency ;

80 % 4) Element l e n g t h .%

82 % 'Model ' bu i l d s only the Track−Bridge systems% and performs the e i g en f r equency and the

84 % eigenmodes ex t r a c t i on o f such a model . The task% i s c a r r i e d out with 100 Hz o f sampling

86 % frequency and 0 . 3 m of element l e n g t h .% The r e s u l t s are s to r ed in a .odb f i l e .

88 %% Output2 = ' . . . ' St r ing to s e l e c t the t r a i n model :

90 %% 'CF' stands f o r Concentrated Forces

92 % 'SM' stands f o r Sprung−Masses% 'TW' stands f o r Train Wagons

94 %% Output3 = ' . . . ' St r ing to s e l e c t v a r i a b l e s and nodes to

99

96 % i n v e s t i g a t e :%

98 % ' Visua l i z e ' asks f o r :%

100 % 1) Contact an a l y s i s r e s u l t s such as% contact f o r c e s and motion o f the

102 % wheel nodes ;% 2) Reaction f o r c e s at the br idge

104 % supports ;% 3) Acc e l e r a t i on s and Displacements f o r

106 % each node be long ing to the b r i d g e .%

108 % The r e s u l t s are s to r ed in a .odb f i l e .%

110 % 'ReadAccDisp_MidSpan ' asks f o r :%

112 % 1) Acce l e r a t i on and Displacement in% br idge midspan ;

114 %% The h i s t o r y data on a c c e l e r a t i o n and

116 % disp lacement are s to r ed in separated% . t x t f i l e .

118 %% Output4 = ' . . . ' St r ing that asks f o r the eva lua t i on o f the

120 % loga r i thmi c decrement over f r e e v i b r a t i on at% br idge midspan. I t i s an op t i ona l parameter to

122 % use in combination with Output3 s e t to% 'ReadAccDisp_MidSpan ' . The only entry

124 % recogn i z ed i s 'Set_Damping ' .

126 f F i l t e r = 30 ; % (Hz)ButtOrder = 6 ;

128 i f ~ e x i s t ( ' Output4 ' , ' var ' )Output4 = ' ' ;

130 end

132 %% Plot Acce l e r a t i on and Displacement accord ing to Output r eques t

134 switch Output1

136 case 'Model 'f = 100 ;

138 l = 0 . 3 ;MainRail_SteelArrow ( LBridge , m_Bridge , A_Bridge , I33_Bridge ,

E_Bridge ,LWay, lambda ,A, phi , f , l , Output1 ) ;140

case 'ConvAnal '142 v = 200 ; % (km/h) Train speed

fMin = 100 ; % (Hz) Frequency to c a l c u l a t e the s tep time144 fMax = 300 ; % during ana l y s i s

df = 100 ;146 dLRail = 0 . 6 ; % (m)

dl = [ dLRail , dLRail /2 , dLRail /4 , dLRail / 8 ] ;148

A_all = [ ] ;150 U_all = [ ] ;

Af_all = [ ] ;152 Uf_all = [ ] ;


154 f o r f = fMin : df : fMaxA = [ ] ;

156 U = [ ] ;Af = [ ] ;

158 Uf = [ ] ;f o r l = dl

160 Resu l t s = MainRail_SteelArrow ( LBridge , m_Bridge , A_Bridge, I33_Bridge , E_Bridge ,LWay, lambda ,A, phi , f , l , Output1 ,Output2 , Output3 , v , f F i l t e r , ButtOrder , Output4 ) ;

A = [A, Resu l t s (1 ) ] ;162 U = [U, Resu l t s (2 ) ] ;

Af = [ Af , Resu l t s (3 ) ] ;164 Uf = [ Uf , Resu l t s (4 ) ] ;

end166 A_all = [ A_all ; A ] ;

U_all = [ U_all ; U ] ;168 Af_all = [ Af_all ; Af ] ;

Uf_all = [ Uf_all ; Uf ] ;170 end

172 dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_A−v_ ' num2str ( v ) '. t x t ' ] , A_all ) ;

dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_U−v_ ' num2str ( v ) '. t x t ' ] , U_all ) ;

174 dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_Af−v_ ' num2str ( v ) '. t x t ' ] , Af_all ) ;

dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_Uf−v_ ' num2str ( v ) '. t x t ' ] , Uf_all ) ;

176f i g u r e ;

178 s u r f ( dl , fMin : df : fMax , Af_all )x l ab e l ( ' Element Length (m) ' )

180 y l ab e l ( ' Sampling Frequency (Hz) ' )z l a b e l ( ' Acce l e r a t i on (m/ s ^2) ' )

182 saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Af−v_ ' num2str ( v )' . e p s ' ] , ' psc2 ' ) ;

saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Af−v_ ' num2str ( v )' . f i g ' ] ) ;

184f i g u r e ;

186 s u r f ( dl , fMin : df : fMax , Uf_all )x l ab e l ( ' Element Length (m) ' )

188 y l ab e l ( ' Sampling Frequency (Hz) ' )z l a b e l ( ' Displacement (m) ' )

190 saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Uf−v_ ' num2str ( v )' . e p s ' ] , ' psc2 ' ) ;

saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Uf−v_ ' num2str ( v )' . f i g ' ] ) ;

192case ' TrainImpact '

194 vMin = 150 ;vMax = 150 ;

196 dv = 5 ;f = 100 ;

198 l = 0 . 3 ;

200 A = [ ] ;

101

U = [ ] ;202 Af = [ ] ;

Uf = [ ] ;204

f o r v = vMin : dv : vMax206 Resu l t s = MainRail_SteelArrow ( LBridge , m_Bridge , A_Bridge ,

I33_Bridge , E_Bridge ,LWay, lambda ,A, phi , f , l , Output1 ,Output2 , Output3 , v , f F i l t e r , ButtOrder , Output4 ) ;

A = [A, Resu l t s (1 ) ] ;208 U = [U, Resu l t s (2 ) ] ;

Af = [ Af , Resu l t s (3 ) ] ;210 Uf = [ Uf , Resu l t s (4 ) ] ;

end212

dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_A−vMin_ ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . t x t ' ] ,A) ;

214 dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_U−vMin_ ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . t x t ' ] ,U) ;

dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_Af−vMin_ ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . t x t ' ] , Af ) ;

216 dlmwrite ( [ ' Results_ ' Output1 '_ ' Output2 '_Uf−vMin_ ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . t x t ' ] , Uf ) ;

218 f i g u r e ;p l o t (vMin : dv : vMax , Af ) ;

220 g r id ont i t l e ( [ ' Train Impact − ' Output1 ' Model ' ] )

222 x l ab e l ( ' Train Speed (km/h) ' )y l ab e l ( 'Maximum Acce l e r a t i on at Bridge Mid Span (m/ s ^2) ' )

224 saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Af−vMin_ ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . e p s ' ] , ' psc2 ' ) ;

saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Af−vMin− ' num2str (vMin) '_vMax_ ' num2str (vMax) ' . f i g ' ] ) ;

226f i g u r e ;

228 p l o t (vMin : dv : vMax , Uf ) ;g r i d on

230 t i t l e ( [ ' Train Impact − ' Output1 ' Model ' ] )x l ab e l ( ' Train Speed (km/h) ' )

232 y l ab e l ( 'Maximum Displacement at Bridge Mid Span (m/ s ^2) ' )saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Uf−vMin_ ' num2str (

vMin) '_vMax_ ' num2str (vMax) ' . e p s ' ] , ' psc2 ' ) ;234 saveas ( gcf , [ ' Results_ ' Output1 '_ ' Output2 '_Uf−vMin_ ' num2str (

vMin) '_vMax_ ' num2str (vMax) ' . f i g ' ] ) ;

236 case ' PlotHi s tory 'Train = input ( ' \nTrain Type (CF,SM,TW) = ' , ' s ' ) ;

238 v = input ( ' Train Speed (km/h) = ' , ' s ' ) ;fMax = st r2doub l e ( input ( ' Sampling Frequency (Hz) = ' , ' s ' ) ) ;

240 dL = input ( ' Element Length (m) = ' , ' s ' ) ;f p r i n t f ( ' \n ' ) ;

242A = dlmread ( [ ' Results_ ' Train '_A_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . t x t ' ] ) ;244 U = dlmread ( [ ' Results_ ' Train '_U_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . t x t ' ] ) ;

246 [K,P] = butte r ( ButtOrder , f F i l t e r /(0 . 5 ∗10∗ fMax) , ' low ' ) ;


Af = f i l t e r (K,P,A) ;248

NumSteps = length (A) ;250 TimeStep = 1/(10∗ fMax) ;

FreqStep = ( fMax∗10) /NumSteps ;252 TimePassage = TimeStep∗NumSteps ;

time = 0 : TimeStep : TimePassage−TimeStep ;254 f r e q = 0 : FreqStep : fMax−FreqStep ;

Ny = c e i l ( l ength ( f r e q ) /2) ;256

LW = 1 .3 ;258 FS = 12 ;

260 f i g u r e ;subplot ( 2 , 1 , 1 ) , p l o t ( time ,A, '−b ' , time , Af , '−r ' , ' LineWidth ' ,LW)

262 x l ab e l ( 'Time ( s ) ' , ' FontSize ' ,FS)y l ab e l ( ' Acce l e r a t i on at Midspan (m/ s ^2) ' , ' FontSize ' ,FS)

264 t i t l e ( [ ' Train Model ' Train ' , Speed = ' v ' km/h ' ] , ' FontSize ' ,FS)

g r id on266

FFT_A = abs ( f f t (A) ) ;268 FFT_Af = abs ( f f t (Af ) ) ;

270 subplot ( 2 , 1 , 2 ) , p l o t ( f r e q ( 1 :Ny) ,FFT_A(1 :Ny) , '−b ' , f r e q ( 1 :Ny) ,FFT_Af( 1 :Ny) , '−r ' , ' LineWidth ' ,LW)

x l ab e l ( ' f (Hz) − The Highest Frequency i s the Nyquist Frequency' , ' FontSize ' ,FS)

272 y l ab e l ( ' |FFT( f ) | ' , ' FontSize ' ,FS)g r id on

274 h_legend = legend ( ' Un f i l t e r e d Data ' , ' F i l t e r e d Data ' , ' Locat ion ' ,' NorthEast ' ) ;

s e t ( h_legend , ' FontSize ' ,FS) ;276 saveas ( gcf , [ ' Results_ ' Train '_AccHist_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . e p s ' ] , ' psc2 ' ) ;saveas ( gcf , [ ' Results_ ' Train '_AccHist_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . f i g ' ] ) ;278

f i g u r e ;280 p l o t ( time ,U, ' LineWidth ' ,LW)

gr id on282 t i t l e ( [ ' Train Model ' Train ' , Speed = ' v ' (km/h) ' ] , '

FontSize ' ,FS)x l ab e l ( 'Time ( s ) ' , ' FontSize ' ,FS)

284 y l ab e l ( ' Displacement at MidSpan (m) ' , ' FontSize ' ,FS)saveas ( gcf , [ ' Results_ ' Train '_DispHist_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . e p s ' ] , ' psc2 ' ) ;286 saveas ( gcf , [ ' Results_ ' Train '_DispHist_MidSpan−v_ ' v '−fMax_ '

num2str ( fMax) '−dL_ ' dL ' . f i g ' ] ) ;

288 case ' Sing leAnal 'v = st r2doub l e ( input ( ' \nTrain Speed (km/h) = ' , ' s ' ) ) ;

290 f = st r2doub l e ( input ( ' Sampling Frequency (Hz) = ' , ' s ' ) ) ;l = s t r2doub l e ( input ( ' Element Length (m) = ' , ' s ' ) ) ;

292 f p r i n t f ( ' \n ' ) ;MainRail_SteelArrow ( LBridge , m_Bridge , A_Bridge , I33_Bridge ,

E_Bridge ,LWay, lambda ,A, phi , f , l , Output1 , Output2 , Output3 , v ,f F i l t e r , ButtOrder , Output4 ) ;

103

294end

296end

Listing B.2: MainRail_SteelArrow.m1 func t i on [ Resu l t s ] = MainRail_SteelArrow ( LBridge , m_Bridge , A_Bridge ,

I33_Bridge , E_Bridge ,LWay, lambda ,A, phi , fMax , dLRail , Output1 , Output2 ,Output3 , v , f F i l t e r , ButtOrder , Output4 )

3 % Code that c r e a t e s an Abaqus input f i l e f o r s imu la t ing the passage% of the Swedish S t e e l Arrow Fre ight Train over a simply supported

5 % br idge with gene ra l s e c t i o n prope r ty . The name o f the Output f i l e% i s s to r ed in the va r i ab l e c a l l e d "NameFile" .

7g = 9 .81 ; % (m/ s ^2) Gravity Acce l e r a t i on

9%% The Train

11 i f strcmp (Output1 , 'Model ' ) == 1 % Modal Ana lys i sNameFile = [ 'Main_ ' Output1 '_−fMax_ ' num2str ( fMax) '−dL_ ' num2str (

dLRail ) ] ;13 a_Bridge = 0 ;

b_Bridge = 0 ;15 a_Rail = 0 ;

b_Rail = 0 ;17 e l s e

i f ~ e x i s t ( ' Results_Modal_Analys is .x ls ' , ' f i l e ' )19 f p r i n t f ( [ ' \n ERROR:\ n\n Modal Ana lys i s i s Required in ' . . .

' Advance. Save the Natural \n Frequenc ie s ' . . .21 ' ( in c y c l e s / time ) o f the Bridge−Track Model in \n ' . . .

' an Excel F i l e 1997−2003 Named ' . . .23 ' "Results_Modal_Analysis " . \n\n ' ] )

r e turn25 e l s e

f = x l s r ead ( ' Results_Modal_Analysis ' ) ;27 w = 2∗ pi ∗ f ;

end29

AxPos=[ 0 ; 2 . 7 ; 7 . 7 ; 10 . 4 ;31 15 .52 ; 18 .22 ; 23 .22 ; 25 .92 ;

30 .23 ; 32 .03 ; 38 .83 ; 40 .63 ;33 44 .13 ; 45 .93 ; 52 .73 ; 54 .53 ;

58 .03 ; 59 .83 ; 66 .63 ; 68 .43 ;35 71 .93 ; 73 .73 ; 80 .53 ; 82 .33 ;

85 .83 ; 87 .63 ; 94 .43 ; 96 .23 ;37 99 .73 ; 101 .53 ; 108 .33 ; 110 .13 ;

113 .63 ; 115 .43 ; 122 .23 ; 124 .03 ;39 127 .53 ; 129 .33 ; 136 .13 ; 137 .93 ;

141 .43 ; 143 .23 ; 150 .03 ; 151 .83 ;41 155 .33 ; 157 .13 ; 163 .93 ; 165 .73 ;

169 .23 ; 171 .03 ; 177 .83 ; 179 .63 ;43 183 .13 ; 184 .93 ; 191 .73 ; 193 .53 ;

197 .03 ; 198 .83 ; 205 .63 ; 207 .43 ;45 210 .93 ; 212 .73 ; 219 .53 ; 221 .33 ;

224 .83 ; 226 .63 ; 233 .43 ; 235 .23 ;47 238 .73 ; 240 .53 ; 247 .33 ; 249 .13 ;


252 .63 ; 254 .43 ; 261 .23 ; 263 .03 ;49 266 .53 ; 268 .33 ; 275 .13 ; 276 .93 ;

280 .43 ; 282 .23 ; 289 .03 ; 290 .83 ;51 294 .33 ; 296 .13 ; 302 .93 ; 304 .73 ;

308 .23 ; 310 .03 ; 316 .83 ; 318 .63 ;53 322 .13 ; 323 .93 ; 330 .73 ; 332 .53 ;

336 .03 ; 337 .83 ; 344 .63 ; 346 .43 ;55 349 .93 ; 351 .73 ; 358 .53 ; 360 .33 ;

363 .83 ; 365 .63 ; 372 .43 ; 374 .23 ;57 377 .73 ; 379 .53 ; 386 .33 ; 388 .13 ] ;

59 AxNum = length (AxPos ) ;NameFile = [ 'Main_ ' Output2 '−v_ ' num2str ( v ) '−fMax_ ' num2str ( fMax)

'−dL_ ' num2str ( dLRail ) ] ;61

% Data Power Car63 mWheelP = 1510 ; % ( kg ) Mass o f one Wheelset

mBogieP = 5200 ; % ( kg ) Mass o f one Boogie Frame65 mCarP = 61560 ; % ( kg ) Mass o f one Car Body

67 % Data CoachmWheelC = 1380 ; % ( kg ) Mass o f one Wheelset

69 mBogieC = 1990 ; % ( kg ) Mass o f one Boogie FramemCarC = 90500; % ( kg ) Mass o f one Car Body

71switch Output2

73case 'CF '

75 mAxle_P = (2∗mBogieP+mCarP+mWheelP) /4 ;mAxle_C = (2∗mBogieC+mCarC+mWheelC) /4 ;

77i f strcmp (Output4 , 'Set_Damping ' ) == 1

79 tTrans = 0 ; % ( s ) Waiting time to s t a b i l i z e the ForcestFree = 5 ; % ( s ) Waiting time a f t e r t r a i n passage to

81 % eva luate Damping Ratio over f r e e v i b r a t i o n se l s e

83 tTrans = 0 ;tFree = 0 ;

85 end

87 z_1_Bridge = 0 .62 /100 ; % Damping Ratio f o r the 1 s t eigenmodez_3_Bridge = 0 .60 /100 ; % Damping Ratio f o r the 3 rd eigenmode

89 [ a_Bridge , b_Bridge ] = Rayle igh_coe f f (w(1 ) ,w(3 ) , z_1_Bridge ,z_3_Bridge ) ;

91 case 'SM 'f_P = 1 .142 ; % 1 s t Natural Frequency o f the Power Car

93 % ( Ve r t i c a l Mode o f Vibrat ion )mAxle_P = (2∗mBogieP+mCarP) /4 ;

95 k_P = mAxle_P∗(2∗ pi ∗f_P) ^2; % (N/m) S t i f f n e s s o f the Springc_P = 152 .00e3 ; % (Ns/m) Damping Co e f f i c i e n t

97 f_C = 1 .029 ; % 1 s t Natural Frequency o f the Coach% ( Ve r t i c a l Mode o f Vibrat ion )

99 mAxle_C = (2∗mBogieC+mCarC) /4 ;k_C = mAxle_C∗(2∗ pi ∗f_C) ^2; % (N/m) S t i f f n e s s o f the Spring

101 c_C = 40 .00e3 ; % (Ns/m) Damping Co e f f i c i e n t

103 i f strcmp (Output4 , 'Set_Damping ' ) == 1

105

tTrans = 1 ; % ( s ) Waiting time to s t a b i l i z e the masses105 tFree = 5 ; % ( s ) Waiting time a f t e r t r a i n passage to

% eva luate damping r a t i o over f r e e v i b r a t i o n s107 e l s e

tTrans = 1 ;109 tFree = 0 . 1 ;

end111

z_1_Bridge = 0 .50 /100 ;113 z_3_Bridge = 0 .50 /100 ;

[ a_Bridge , b_Bridge ] = Rayle igh_coe f f (w(1) ,w(3) , z_1_Bridge ,z_3_Bridge ) ;

115case 'TW'

117 % Data Power CarIBogieP = 5900 ; % ( kg m^2) Rotat iona l I n e r t i a , Boogie System

119 ICarP = 909716; % ( kg m^2) Rotat iona l I n e r t i a , Car BodykP_P = 4 .90e6 ; % (N/m) S t i f f n e s s , Primary Spring

121 cP_P = 108 .00e3 ;% (Ns/m) Damping Coe f f i c i e n t , Primary DashpotkS_P = 1 .90e6 ; % (N/m) S t i f f n e s s , Secondary Spring

123 cS_P = 152 .00e3 ;% (Ns/m) Damping Coe f f i c i e n t , Secondary Dashpot

125 % Data CoachIBogieC = 1484 ; % ( kg m^2) Rotat iona l I n e r t i a , Boogie System

127 ICarC = 1324855;% ( kg m^2) Rotat iona l I n e r t i a , Car BodykP_C = 3 .20e6 ; % (N/m) S t i f f n e s s , Primary Spring

129 cP_C = 30 .00e3 ; % (Ns/m) Damping Coe f f i c i e n t , Primary DashpotkS_C = 2 .70e6 ; % (N/m) S t i f f n e s s , Secondary Spring

131 cS_C = 40 .00e3 ; % (Ns/m) Damping Coe f f i c i e n t , Secondary Dashpot

133 i f strcmp (Output4 , 'Set_Damping ' ) == 1tTrans = 1 ; % ( s ) Waiting time to s t a b i l i z e the masses

135 tFree = 5 ; % ( s ) Waiting time a f t e r t r a i n passage to% eva luate Damping Ratio over f r e e v i b r a t i o n s

137 e l s etTrans = 1 ;

139 tFree = 0 . 1 ;end

141z_1_Bridge = 0 .55 /100 ;

143 z_3_Bridge = 0 .55 /100 ;[ a_Bridge , b_Bridge ] = Rayle igh_coe f f (w(1) ,w(3) , z_1_Bridge ,

z_3_Bridge ) ;145 end

147 z_1_Rail = 0 .75 /100 ; % Damping Ratio f o r the 1 s t eigenmodez_5_Rail = 0 .75 /100 ; % Damping Ratio f o r the 5 th eigenmode

149 [ a_Rail , b_Rail ] = Rayle igh_coe f f (w(1) ,w(5 ) , z_1_Rail , z_5_Rail ) ;end

151

153 %% The Rai lElementType_Rail = 'B23 ' ; % Euler−Bernou l l i beam element , that uses

155 % cubic i n t e r p o l a t i o ndLSleeper = 0 . 6 ; % (m) Distance between S l e ep e r s

157 h_Ballast = 0 . 6 ; % (m) Ba l l a s t he ightw_Ballast = 6 . 2 ; % (m) Ba l l a s t width

159 Dens ity_Bal last = 2000 ; % ( kg/m^3) Density o f the Ba l l a s t


v ibr_Bal la s t = 0 . 5 ; % (−) Ratio o f v i b r a t i ng Ba l l a s t161 m_vibr_Ballast = vibr_Bal la s t ∗h_Ballast ∗w_Ballast∗dLSleeper ∗

Density_Bal last ;% ( kg ) Mass o f Ba l l a s t that v i b r a t e s

163 % with the Rai lmSleeper = 300 ; % ( kg ) Mass o f one S l e epe r in Concrete

165 mVibratingMass = m_vibr_Ballast+mSleeper ;k_Rail = 150 e6 ; % (N/m) S t i f f n e s s o f the Rai l Spr ing

167 c_Rail = 100 e3 ; % (Ns/m) Damping Co e f f i c i e n t o f the% Rai l Dashpot

169 Density_Rail = 7850 ; % ( kg/m^3) Density o f the Rai lA_Rail = 2∗7 .687e −3; % (m^2) Sec t i on Area

171 % − Data f o r 2 r a i l s UIC 60 −I33_Rail = 2∗3 .060e −5; % (m^4) Rotat iona l I n e r t i a

173 E_Rail = 2 .1e11 ; % (N/m^2) Young ' s ModulusPoisson = 0 . 3 ; % (−) Poisson ' s r a t i o

175 G_Rail = E_Rail /(2∗(1+ Poisson ) ) ; % (N/m^2) Tors iona l Shear Modulus

177%% The Bridge

179 ElementType_Bridge = 'B23 ' ; % Euler−Bernou l l i beam element , that uses% cubic i n t e r p o l a t i o n

181 m_non_vibr_Ballast = (1−v ibr_Bal la s t ) ∗h_Ballast ∗w_Ballast∗Density_Bal last ;

% ( kg/m) Mass over l ength o f the Ba l l a s t183 % that has to be inc luded in the

% dens i ty o f the Bridge185 Density_Bridge = (m_Bridge + m_non_vibr_Ballast ) /A_Bridge ;

% ( kg/m^3) Density o f the Composite187 % Sect i on in c l ud ing the mass o f

% the Vibrat ing Ba l l a s t Layer189 Hinge = ' Pinned ' ; % BC on the 1 s t node o f the Bridge

Bearing = ' 2 , 2 ' ; % BC on the l a s t node o f the Bridge191 G_Bridge = E_Bridge/(2∗(1+ Poisson ) ) ;% (N/m^2) Tors iona l Shear Modulus

193%% The Sta t i on s

195 Clamp = ' Encastre ' ; % BC on the 1 s t node o f the Model% BC on the l a s t node o f the Model and on

197 % a l l the Track nodes l a i d on the groundk_Station = ( k_Rail/ dLSleeper ) ∗0 .45 ;

199 c_Station = c_Rail ;

201%% Length o f the Tracks

203 LEstaCont = 4 . 8 ;LWay = LWay + phi ∗dLSleeper ;

205

207 %% Open NameFile . inpMainID = fopen ( [ NameFile ' . i n p ' ] , 'w+ ' ) ;

209

211 %% HeadingHeading (MainID)

213

215 %% Stat ion 1

107

FirstN_Station1_Name = 101 ; % (−) Name o f the 1 s t node o f the model217 FirstN_Station1_Pos = 0 ; % (m) Coordinate o f the 1 s t node o f model

[ LastN_Station1 , Axles1 ] = NodesStat ion (MainID , FirstN_Station1_Name ,FirstN_Station1_Pos , ' Stat ion1 ' ) ;

219FirstSpringName = 10001;

221 FirstDashpotName = 15001;LastElementSet_Station1 = ElementsStat ion (MainID , FirstSpringName ,

FirstDashpotName , Axles1 , k_Station , c_Station , ' Stat ion1 ' ) ;223

225 %% Estab l i sh ContactFirstN_EstaCont_Name = LastN_Station1 (1 ) +1;

227 FirstN_EstaCont_Pos = LastN_Station1 (2 )+dLRail ;LastN_EstaCont = NodesRail (MainID , FirstN_EstaCont_Name ,

FirstN_EstaCont_Pos , LEstaCont , dLRail , 0 , lambda , ' EstaCont ' ) ;229

231 %% WayInFirstN_WayIn_Name = LastN_EstaCont (1 ) +1;

233 FirstN_WayIn_Pos = LastN_EstaCont (2 )+dLRail ;LastN_WayIn = NodesRail (MainID , FirstN_WayIn_Name , FirstN_WayIn_Pos ,

LWay, dLRail , A, lambda , 'WayIn ' ) ;235

237 %% BridgeFirstN_Bridge_Name = LastN_WayIn (1 ) +1;

239 FirstN_Bridge_Pos = LastN_WayIn (2 )+dLRail ;LastN_Bridge = NodesRail (MainID , FirstN_Bridge_Name , FirstN_Bridge_Pos ,

LBridge , dLRail , A, lambda , ' Bridge ' ) ;241

FirstEl_Bridge = 1 ;243 LastEl_Bridge = ElementsBeam (MainID , FirstN_Bridge_Name−2, LastN_Bridge

(1 )−1, FirstEl_Bridge , ElementType_Bridge , Density_Bridge , A_Bridge ,I33_Bridge , E_Bridge , G_Bridge , a_Bridge , b_Bridge , ' Bridge ' ) ;

245%% WayOut

247 FirstN_WayOut_Name = LastN_Bridge (1 ) +1;FirstN_WayOut_Pos = LastN_Bridge (2 )+dLRail ;

249 LastN_WayOut = NodesRail (MainID , FirstN_WayOut_Name , FirstN_WayOut_Pos ,LWay, dLRail , A, lambda , 'WayOut ' ) ;

251%% Stat ion 2

253 FirstN_Station2_Name = LastN_WayOut(1 ) +1;FirstN_Station2_Pos = LastN_WayOut(2 ) ;

255 [ LastN_Station2 , Axles2 ] = NodesStation (MainID , FirstN_Station2_Name ,FirstN_Station2_Pos , ' Stat ion2 ' ) ;

LastElementSet_Station2 = ElementsStat ion (MainID ,LastElementSet_Station1 (1 ) , LastElementSet_Station1 (2 ) , Axles2 ,k_Station , c_Station , ' Stat ion2 ' ) ;

257

259 %% Rai lF i r s tEl_Rai l = LastEl_Bridge ;

261 ElementsBeam (MainID , FirstN_Station1_Name+1, LastN_Station2 (1 ) ,FirstEl_Rai l , ElementType_Rail , Density_Rail , A_Rail , I33_Rail ,


E_Rail , G_Rail , a_Rail , b_Rail , ' Rai l ' ) ;S l eeper sPos = ElementsRai l (MainID , LastN_Station1 (1 )−1, LastN_WayOut(1 )−1, LastElementSet_Station2 (1 ) , LastElementSet_Station2 (2 ) ,dLSleeper , dLRail , k_Rail , c_Rail ) ;

263 VibratingMass (MainID , S leepersPos , mVibratingMass ) ;

265%% Train

267 i f strcmp (Output2 , 'CF ' ) == 1Crds = Coordinates ( LastN_Station1 (1 ) , FirstN_Station2_Name+1,

LastN_Station1 (2 ) , dLRail ) ;269 Pa = AxleForces (AxPos ,mAxle_P ,mAxle_C, g ) ;

AmplitudesCF (MainID , Crds , Pa , v ) ;271 e l s e i f strcmp (Output2 , 'SM ' ) == 1

NodesSM(MainID , AxPos , AxNum, FirstN_Station1_Pos )273 ElementsSM(MainID , AxNum, k_P, c_P, k_C, c_C, mWheelP , mWheelC ,

mAxle_P , mAxle_C)Sur face (MainID , AxNum)

275 AmplitudesSM(MainID , v , tTrans , tFree , LastN_Station2 (2 ) , AxNum,AxPos)

e l s e i f strcmp (Output2 , 'TW' ) == 1277 NodesT(MainID , AxPos , AxNum, FirstN_Station1_Pos )

ElementsT (MainID , AxNum, kP_P, cP_P, kS_P, cS_P , kP_C, cP_C, kS_C,cS_C, mWheelP , mWheelC , mBogieP , IBogieP , mCarP, ICarP , mBogieC ,

IBogieC , mCarC, ICarC )279 Sur face (MainID , AxNum)

AmplitudesT (MainID , v , tTrans , tFree , LastN_Station2 (2 ) , AxNum,AxPos)

281 end

283%% Boundary Condit ions

285 Boundary_Model (MainID , Clamp , Hinge , Bearing , Axles1 , S leepersPos ,LastN_WayIn (1 ) , LastN_Bridge (1 ) , Axles2 )

287 i f strcmp (Output2 , 'SM ' ) == 1BoundarySM(MainID ,AxNum)

289 e l s e i f strcmp (Output2 , 'TW' ) == 1BoundaryT(MainID ,AxNum)

291 end

293%% Step_1 Gravity Forces

295 Step_1_Gravity (MainID , g , Output3 )

297%% Step_2 EigFrequency

299 i f strcmp (Output1 , 'Model ' ) == 1Step_2_EigFrequency (MainID , fMax)

301 end

303%% Step_3 MoveLoad

305 i f strcmp (Output2 , 'CF ' ) == 1Step_3_MoveCF(MainID , fMax , AxPos , v , tTrans , tFree , LastN_Station2 (2 ) ,

Crds , Output3 )307 e l s e i f strcmp (Output2 , 'SM ' ) == 1

109

Step_3_MoveSM(MainID , fMax , AxPos , AxNum, v , tTrans , tFree ,LastN_Station2 (2 ) , Clamp , Hinge , Bearing , Axles1 , S leepersPos ,LastN_WayIn (1 ) , LastN_Bridge (1 ) , Axles2 , Output3 )

309 e l s e i f strcmp (Output2 , 'TW' ) == 1Step_3_MoveT(MainID , fMax , AxPos , AxNum, v , tTrans , tFree ,

LastN_Station2 (2 ) , Clamp , Hinge , Bearing , Axles1 , S leepersPos ,LastN_WayIn (1 ) , LastN_Bridge (1 ) , Axles2 , Output3 )

311 end

313%% Close Main.inp

315 f c l o s e (MainID) ;

317%% Run on ABAQUS

319 dos ( [ ' abaqus job= ' NameFile ' i n t e r a c t i v e ' ] )

321%% Read Acce l e r a t i on and Displacement

323 i f strcmp (Output3 , 'ReadAccDisp_MidSpan ' ) == 1Resu l t s = ReadAccDisp_MidSpan ( NameFile , fMax , f F i l t e r , ButtOrder , v ,

tFree , dLRail , Output2 , Output4 ) ;325 end

327 end

Listing B.3: Rayleigh_coeff.mf unc t i on [ alpha , beta ] = Rayle igh_coe f f (w_1, w_n, z_1 , z_n)

2z = [ z_1 , z_n ] ' ;

4 A = [ 1/(2∗w_1) , w_1/21/(2∗w_n) , w_n/ 2 ] ;

6x = A\z ;

8alpha = x (1) ;

10 beta = x (2) ;

12 end

Listing B.4: log_decrement.mf unc t i on [ ze ta ] = log_decrement (A, fMax , tFree , Train , v , dL)

2LW = 1 .3 ;

4 FS = 12 ;

6 NumSteps = length (A) ;TimeStep = 1/(10∗ fMax) ;

8 TimePassage = TimeStep∗NumSteps ;time = 0 : TimeStep : TimePassage−TimeStep ;

10A = A( end−tFree /TimeStep:end) ;

12 time = time ( end−tFree /TimeStep:end) ;


14 NumSteps = length (A) ;FreqStep = ( fMax∗10) /NumSteps ;

16 f r e q = 0 : FreqStep : fMax−FreqStep ;Ny = c e i l ( l ength ( f r e q ) /2) ;

18f i g u r e (1 ) ;

20 subplot ( 2 , 1 , 1 ) , p l o t ( time ,A, '−b ' , ' LineWidth ' ,LW)x l ab e l ( 'Time ( s ) ' , ' FontSize ' ,FS)

22 y l ab e l ( ' Acce l e r a t i on at Midspan (m/ s ^2) ' , ' FontSize ' ,FS)t i t l e ( [ ' Free Vibrat ion − Train Model ' Train ' , Speed = ' num2str ( v ) '

km/h ' ] , ' FontSize ' ,FS)24 g r id on

hold on26

FFT_A = abs ( f f t (A) ) ;28

subplot ( 2 , 1 , 2 ) , p l o t ( f r e q ( 1 :Ny) ,FFT_A(1 :Ny) , '−b ' , ' LineWidth ' ,LW)30 t i t l e ( [ ' f_{ sampling } = ' num2str ( fMax) ' Hz , l_{elem} = ' num2str (dL

∗100) ' cm ' ] , ' FontSize ' ,FS)x l ab e l ( ' f (Hz) − The Highest Frequency i s the Nyquist Frequency ' , '

FontSize ' ,FS)32 y l ab e l ( ' |FFT( f ) | ' , ' FontSize ' ,FS)

g r id on34

e l = [ ] ;36 ind = [ ] ;

[ el_new , ind_new ] = max(A) ;38 e l = [ e l , el_new ] ;

A = A( ind_new:end) ;40 ind = [ ind , ind_new−1] ;

42 whi l e l ength (A) ~= 1i f el_new > 0

44 [ el_new , ind_min ] = min (A) ;A = A( ind_min:end) ;

46 e l s e[ el_new , ind_new ] = max(A) ;

48 i f el_new > 0e l = [ e l , el_new ] ; % the l a s t max could not be a max !

50 ind = [ ind , ind_new+ind_min−2] ;A = A( ind_new:end) ;

52 e l s eA = A( ind_new:end) ;

54 endend

56 end

58 ind_sum (1) = ind (1) ;f o r i = 2 : l ength ( ind )

60 ind_sum( i ) = ind ( i )+ind_sum( i −1) ;end

62time_max = ind_sum∗TimeStep+TimePassage−tFree ;

64c = length ( e l ) ;

66delta_1 = (1/( c−5) ) ∗ l og ( e l (4 ) / e l ( c−1) ) ; % exc lude the f i r s t three peaks

68 delta_2 = (1/( c−6) ) ∗ l og ( e l (4 ) / e l ( c−2) ) ;

111

delta_3 = (1/( c−7) ) ∗ l og ( e l (4 ) / e l ( c−3) ) ;70 de l t a = ( delta_1+delta_2+delta_3 ) /3 ;

72 zeta = 1/ sq r t (1+(2∗ pi / de l t a ) ^2) ;

74 f_n = 1/(( ind (5 ) ) ∗TimeStep ) ;w_n = 2∗ pi /f_n ;

76y = [ e l (4 )

78 e l ( c−3)e l ( c−2)

80 e l ( c−1) ] ;

82 M = [ exp(−ze ta ∗w_n∗time_max (4) ) 1exp(−ze ta ∗w_n∗time_max( c−3) ) 1

84 exp(−ze ta ∗w_n∗time_max( c−2) ) 1exp(−ze ta ∗w_n∗time_max( c−1) ) 1 ] ;

86x = M\y ;

88f i g u r e (1 ) ;

90 subplot ( 2 , 1 , 1 ) , p l o t ( time , x (2 )+x (1) ∗exp(−ze ta ∗w_n∗ time ) , ' r ' , ' LineWidth ' ,LW)

h_legend = legend ( ' Numerical S imulat ion ' , [ ' e^{(\ x i \omega_nt ) } with \ x i= ' num2str ( round ( zeta ∗10000) /100) ' % ' ] , ' Locat ion ' , ' SouthEast ' ) ;

92 s e t ( h_legend , ' FontSize ' ,FS) ;saveas ( gcf , [ ' Results_ ' Train '_FreeVibr_MidSpan−v_ ' num2str ( v ) '−fMax_ '

num2str ( fMax) '−dL_ ' num2str (1000∗dL) ' . e p s ' ] , ' psc2 ' ) ;94 saveas ( gcf , [ ' Results_ ' Train '_FreeVibr_MidSpan−v_ ' num2str ( v ) '−fMax_ '

num2str ( fMax) '−dL_ ' num2str (1000∗dL) ' . f i g ' ] ) ;

96 end

Listing B.5: Heading.mf unc t i on Heading ( Fi le ID )

2Text = [ ' ∗Heading \nDynamic Ana lys i s \n ' . . .

4 ' ∗∗ Generated by : Abaqus/Standard 6 .9−EF1\n ' . . .' ∗Prepr int , echo=no , h i s t o r y=no , model=no\n∗∗\n∗∗\n ' ] ;

6 f p r i n t f ( Fi leID , Text ) ;

8 end

Listing B.6: NodesStation.mf unc t i on [ CoordLastNode , FixedNodes ] = NodesStat ion ( FileID , FirstNodeName ,

FirstNodePos , Label )2

Text = [ ' ∗∗\n∗∗\ t ' Label ' \n ' . . .4 ' ∗∗\n∗∗\n ' . . .

' ∗Node\n ' ] ;6 f p r i n t f ( Fi leID , Text ) ;

8 i f strcmp ( Label , ' Stat ion1 ' ) == 1


Text = [ num2str ( FirstNodeName ) ' , ' num2str ( FirstNodePos ) ' , 0 \n '. . .

10 num2str ( FirstNodeName+1) ' , ' num2str ( FirstNodePos ) ' , 1 \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

12 FixedNodes = FirstNodeName+1;e l s e

14 FixedNodes = FirstNodeName−1;end

16Axle = FirstNodePos + [ 0 , 1 .8 , 2 .7 , 7 .7 , 8 .6 , 10 . 4 ] ;

18f o r k = 1 : l ength ( Axle )−1

20 FixedNodes = [ FixedNodes , TrainAxle ( FileID , FirstNodePos , Axle ( k ) ,Axle ( k+1) , FixedNodes(end)+1) ] ;

end22

CoordLastNode = [ FixedNodes(end) , Axle(end) ] ;24 Text = ' ∗∗\n∗∗\n ' ;

f p r i n t f ( Fi leID , Text ) ;26

end

Listing B.7: ElementsStation.m1 func t i on [ LastElementSet ] = ElementsStat ion ( FileID , FirstSpringName ,

FirstDashpotName , FixedNodes , k_Rail , c_Rail , Label )

3 Text1 = [ ' ∗Element , type=spr ing2 , E l s e t= ' Label ' _Spring \n ' ] ;Text2 = [ ' ∗Element , type=dashpot2 , E l s e t= ' Label '_Dashpot \n ' ] ;

5 T i t l e = {Text1 ; Text2 } ;j = 1 ;

7 LastElementSet = [ ] ;

9 f o r k = [ FirstSpringName , FirstDashpotName ]f p r i n t f ( Fi leID , char ( T i t l e ( j ) ) ) ;

11 j = j +1;f o r z = 1 : l ength ( FixedNodes )−1

13 f o r i = FixedNodes ( z ) +1:2 : FixedNodes ( z+1)−2Text = [ num2str ( k ) ' , ' num2str ( i ) ' , ' num2str ( i +1) ' \n '

] ;15 f p r i n t f ( Fi leID , Text ) ;

k = k+1;17 end

Text = ' ∗∗\n ' ;19 f p r i n t f ( Fi leID , Text ) ;

end21 LastElementSet = [ LastElementSet , k ] ;


end25

Text = [ ' ∗Spring , E l s e t= ' Label ' _Spring \n ' . . .27 ' 2 , 2\n ' num2str ( k_Rail ) ' \n∗∗\n∗∗\n ' . . . % 2 , 2 =

Degree o f freedom of the end nodes which the spr ing i s connected' ∗Dashpot , E l s e t= ' Label '_Dashpot \n ' . . .

29 ' 2 , 2\n ' num2str ( c_Rail ) ' \n∗∗\n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

113

31end

Listing B.8: NodesRail.mf unc t i on CoordLastNode = NodesRail ( Fi leID , FirstNodeName , FirstNodePos , L ,

dL , A, lambda , Label )2

Text = [ ' ∗∗\n∗∗\ t ' Label ' \n∗∗\n∗∗\n∗Node\n ' ] ;4 f p r i n t f ( Fi leID , Text ) ;

6 LastNodePos = L+FirstNodePos−dL ;i = FirstNodeName ;

8f o r k = FirstNodePos : dL : LastNodePos

10 y = A∗ s i n ( (2∗ pi /lambda ) ∗(k−10.4−4. 8 ) )+1;Text = [ num2str ( i ) ' , ' num2str ( k ) ' , 0 \n ' . . .

12 num2str ( i +1) ' , ' num2str ( k ) ' , ' num2str ( y ) ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

14 i = i +2;end

16Text = ' ∗∗\n∗∗\n ' ;


20 i f strcmp ( Label , ' Bridge ' ) == 1CoordLastNode = [ i −1,k ] ;

22 MidNodeName = (FirstNodeName−2+CoordLastNode (1 )−1) /2 ;Text = [ ' ∗Nset , Nset=Bridge_Nodes_Concrete , Generate \n ' . . .

24 num2str ( FirstNodeName ) ' , ' num2str ( CoordLastNode (1 )−1) ' , 2 \n∗∗\n∗∗\n ' . . .

' ∗Nset , Nset=Bridge_MidNode_Concrete \n ' . . .26 num2str (MidNodeName) ' \n∗∗\n∗∗\n ' . . .

' ∗Nset , Nset=Bridge_EndNode_Soil \n ' . . .28 num2str ( CoordLastNode (1 )−1) ' \n∗∗\n∗∗\n ' ] ;

f p r i n t f ( Fi leID , Text ) ;30 e l s e

CoordLastNode = [ i −1,k ] ;32 Text = [ ' ∗Nset , nset= ' Label '_EndNode_Soil \n ' . . .

num2str ( CoordLastNode (1 )−1) ' \n∗∗\n∗∗\n ' ] ;34 f p r i n t f ( Fi leID , Text ) ;

end36

end

Listing B.9: ElementsBeam.m1 func t i on [ LastElementBeam ] = ElementsBeam ( FileID , FirstNodeName ,

LastNodeName , FirstElemName , ElementType , Density , A, I , E, G, alpha ,beta , Label )

3 Elements = (LastNodeName−FirstNodeName ) /2 ;Text = [ ' ∗∗\n∗∗\ t ' Label ' _Elements\n∗∗\n∗∗\n ' . . .

5 ' ∗Element , Type= ' ElementType ' \n ' . . .num2str ( FirstElemName ) ' , ' num2str ( FirstNodeName ) ' , ' num2str (

FirstNodeName+2) ' \n ' . . .


7 ' ∗Elgen , E l s e t= ' Label ' _Elements\n ' . . .num2str ( FirstElemName ) ' , ' num2str ( Elements ) ' , 2 \n∗∗\n∗∗\n ' . . .

9 ' ∗Beam General Sect ion , E l s e t= ' Label '_Elements , Density= ' num2str( Density ) ' , S ec t i on=General \n ' . . .

num2str (A) ' , ' num2str ( I ) ' , 0 , 0 , 0\n ' . . .11 ' 0 , 0 , −1\n ' . . .

num2str (E) ' , ' num2str (G) ' \n∗∗\n∗∗\n ' . . .13 ' ∗Damping , Alpha= ' num2str ( alpha ) ' , Beta= ' num2str ( beta ) ' \n∗∗\n

∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

15 LastElementBeam = FirstElemName+Elements ;

17 end

Listing B.10: ElementsRail.m1 func t i on [ S l eeper sPos ] = ElementsRai l ( Fi leID , FirstNodeName , LastNodeName ,

FirstSpringName , FirstDashpotName , dLs leeper , dLnode , k_Rail , c_Rail )

3 Text1 = ' ∗Element , type=spr ing2 , E l s e t=Rail_Spring \n ' ;Text2 = ' ∗Element , type=dashpot2 , E l s e t=Rail_Dashpot \n ' ;

5 T i t l e = {Text1 ; Text2 } ;j = 1 ;

7 S l eeper sPos = FirstNodeName+(dLs l eeper /dLnode ) ∗2 : ( dLs l eeper /dLnode ) ∗2 :LastNodeName−(dLs l eeper /dLnode ) ∗2 ;

9 f o r k = [ FirstSpringName , FirstDashpotName ]f p r i n t f ( Fi leID , char ( T i t l e ( j ) ) ) ;

11 j = j +1;f o r i = S leeper sPos

13 Text = [ num2str ( k ) ' , ' num2str ( i ) ' , ' num2str ( i +1) ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

15 k = k+1;end

17 Text = ' ∗∗\n∗∗\n ' ;f p r i n t f ( Fi leID , Text ) ;

19 end

21 Text = [ ' ∗Spring , E l s e t=Rail_Spring \n ' . . .' 2 , 2\n ' num2str ( k_Rail ) ' \n∗∗\n∗∗\n ' . . . % 2 , 2 =

Degree o f freedom of the end nodes which the spr ing i s connected23 ' ∗Dashpot , E l s e t=Rail_Dashpot \n ' . . .

' 2 , 2\n ' num2str ( c_Rail ) ' \n∗∗\n∗∗\n ' ] ;25 f p r i n t f ( Fi leID , Text ) ;

27 end

Listing B.11: Coordinates.m1 func t i on [ Crds ] = Coordinates ( FirstNodeName , LastNodeName , FirstNodePos , dL)

3 Crds = [ ] ;k = FirstNodePos ;

5 f o r i = FirstNodeName : 2 : LastNodeNameCrds_New = [ i , k , 1 , 0 ] ;

7 Crds = [ Crds ; Crds_New ] ;

115

k = k+dL ;9 end

11 end

Listing B.12: AxleForces.m1 func t i on [ Pa ] = AxleForces (AxPos ,mAxle_P ,mAxle_C, g )

3 Force = [ ] ;f o r i = 1 :8

5 Force = [ Force ;mAxle_P∗g /1000 ] ; % Forces in kNend

7whi l e i < length (AxPos )

9 Force = [ Force ;mAxle_C∗g /1000 ] ;i = i +1;

11 end

13 Pa = [ AxPos , Force ] ;

15 end

Listing B.13: AmplitudesCF.m1 func t i on AmplitudesCF ( FileID , crds , Pa , v )

3 v = v/3 . 6 ; % Train Speed in m/ snn = length ( crds ( : , 1) ) ;

5 f p r i n t f ( Fi leID , ' ∗∗\n∗∗\ tAmplitudes \n∗∗\n ' ) ;s = ze ro s (nn , 1) ;

7f o r n = 2 : nn

9 [ th , s1 , z1 ] = ca r t 2po l ( c rds (n , 2)−crds (n−1, 2) , . . .c rds (n , 3)−crds (n−1, 3) , c rds (n , 4)−crds (n−1, 4) ) ;

11 s (n) = s (n−1) + s1 ;end

13 f o r n = 1 : nnf p r i n t f ( Fi leID , [ ' ∗Amplitude , name=AmpN ' num2str ( c rds (n , 1) ) ' \n '

] ) ;15 i f n > 1 ; s1 = s (n) − s (n−1) ; e l s e s1 = s (n+1) − s (n) ; end

i f n < nn ; s2 = s (n+1) − s (n) ; e l s e s2 = s (n) − s (n−1) ; end17 amp = ze ro s ( l ength (Pa ( : , 1) ) ∗ 3 , 2) ;

f o r m = 1 : l ength (Pa ( : , 1) )19 amp(3∗m−2, 1) = ( s (n) + Pa(m, 1) − s1 ) / v ;

amp(3∗m−2, 2) = 0 ;21 amp(3∗m−1, 1) = ( s (n) + Pa(m, 1) ) / v ;

amp(3∗m−1, 2) = Pa(m, 2) ;23 amp(3∗m, 1) = ( s (n) + Pa(m, 1) + s2 ) / v ;

amp(3∗m, 2) = 0 ;25 end

AmpS = ' ' ;27 f o r m = 1 : l ength (amp ( : , 1) )

i f ~isempty (AmpS)29 AmpS = [AmpS ' , ' num2str (amp(m, 1) ) ' , ' num2str (amp(m, 2)

) ] ;


e l s e31 AmpS = [ num2str (amp(m, 1) ) ' , ' num2str (amp(m, 2) ) ] ;

end33 i f mod(m, 4) == 0

f p r i n t f ( Fi leID , [AmpS ' \n ' ] ) ;35 AmpS = ' ' ;

end37 end

i f ~isempty (AmpS) ; f p r i n t f ( Fi leID , [AmpS ' \n ' ] ) ; end39

end41

Text = ' ∗∗\n∗∗\n ' ;43 f p r i n t f ( Fi leID , Text ) ;

45 end

Listing B.14: NodesSM.m1 func t i on NodesSM( FileID , AxPos , AxNum, FirstNodePos )

3 Text = [ ' ∗∗\n∗∗\ tNodeT_SteelArrow\n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

5p = 5001 ;

7 j = 1 ;z = 1 ;

9 f o r k = 4 : 4 : 8Text = [ ' ∗∗\n∗∗\ tPower Car ' num2str ( z ) ' \n∗∗\n∗Node\n ' . . .

11 num2str (p) ' , ' num2str (AxPos (k ) − AxPos( j ) + FirstNodePos ) ' ,1 \n ' . . .

num2str (p+1) ' , ' num2str (AxPos (k ) − AxPos( j ) + FirstNodePos ) ', 2 \n ' . . .

13 num2str (p+2) ' , ' num2str (AxPos (k ) − AxPos( j +1) + FirstNodePos )' , 1 \n ' . . .

num2str (p+3) ' , ' num2str (AxPos (k ) − AxPos( j +1) + FirstNodePos )' , 2 \n ' . . .




num2str (p+7) ' , ' num2str (AxPos (k ) − AxPos( j +3) + FirstNodePos )' , 2 \n ' ] ;

19 f p r i n t f ( Fi leID , Text ) ;p = p+8;

21 j = j +4;z = z+1;

23 end

25 f o r k = 12 : 4 :AxNumText = [ ' ∗∗\n∗∗\ tCoach ' num2str ( z−2) ' \n∗∗\n∗Node\n ' . . .



117






num2str (p+7) ' , ' num2str (AxPos (k ) − AxPos( j +3) + FirstNodePos )' , 2 \n ' ] ;


37 j = j +4;z = z+1;

39 end


43end

Listing B.15: ElementsSM.mf unc t i on ElementsSM( FileID , AxNum, k_P, c_P, k_C, c_C, mWheel_P, mWheel_C,

mAxle_P , mAxle_C)2

Text = ' ∗Element , type=spr ing2 , E l s e t=Spring_P \n ' ;4 f p r i n t f ( Fi leID , Text ) ;

i = 4001 ;6 j = 5001 ;

whi l e i <= 4000+88 Text = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;

f p r i n t f ( Fi leID , Text ) ;10 i = i +1;

j = j +2;12 end

14 Text = ' ∗Element , type=spr ing2 , E l s e t=Spring_C \n ' ;f p r i n t f ( Fi leID , Text ) ;

16 whi l e i <= 4000+AxNumText = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;

18 f p r i n t f ( Fi leID , Text ) ;i = i +1;

20 j = j +2;end

22Text = ' ∗∗\n∗∗\n∗Element , type=dashpot2 , E l s e t=Dashpot_P \n ' ;

24 f p r i n t f ( Fi leID , Text ) ;i = 5001 ;

26 j = 5001 ;whi l e i <= 5000+8

28 Text = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

30 i = i +1;j = j +2;


32 end

34 Text = ' ∗Element , type=dashpot2 , E l s e t=Dashpot_C \n ' ;f p r i n t f ( Fi leID , Text ) ;

36 whi l e i <= 5000+AxNumText = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;


40 j = j +2;end

42Text = ' ∗∗\n∗∗\n∗Element , type=mass , E l s e t=mWheel_P \n ' ;


46 j = 5001 ;whi l e i <= 6000+8

48 Text = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

50 i = i +1;j = j +2;

52 end

54 Text = ' ∗∗\n∗∗\n∗Element , type=mass , E l s e t=mWheel_C \n ' ;f p r i n t f ( Fi leID , Text ) ;

56 whi l e i <= 6000+AxNumText = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;


60 j = j +2;end

62Text = ' ∗Element , type=mass , E l s e t=mAxle_P \n ' ;


66 j = 5002 ;whi l e i <= 7000+8


70 i = i +1;j = j +2;

72 end

74 Text = ' ∗Element , type=mass , E l s e t=mAxle_C \n ' ;f p r i n t f ( Fi leID , Text ) ;

76 whi l e i <= 7000+AxNumText = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;


80 j = j +2;end

82Text = [ ' ∗∗\n∗∗\n∗Spring , E l s e t=Spring_P\n ' . . .

84 ' 2 , 2\n ' num2str (k_P) ' \n ' . . . % 2 , 2 = Degree o ffreedom of the end nodes which the spr ing i s connected

' ∗Spring , E l s e t=Spring_C\n ' . . .86 ' 2 , 2\n ' num2str (k_C) ' \n ' . . . % 2 , 2 = Degree o f

freedom of the end nodes which the spr ing i s connected' ∗∗\n∗∗\n∗Dashpot , E l s e t=Dashpot_P\n ' . . .

119

88 ' 2 , 2\n ' num2str (c_P) ' \n ' . . .' ∗Dashpot , E l s e t=Dashpot_C\n ' . . .

90 ' 2 , 2\n ' num2str (c_C) ' \n ' . . .' ∗∗\n∗∗\n∗Mass , E l s e t=mWheel_P\n ' . . .

92 num2str (mWheel_P) ' \n ' . . .' ∗Mass , E l s e t=mWheel_C\n ' . . .

94 num2str (mWheel_C) ' \n ' . . .' ∗Mass , E l s e t=mAxle_P\n ' . . .

96 num2str (mAxle_P) ' \n ' . . .' ∗Mass , E l s e t=mAxle_C\n ' . . .

98 num2str (mAxle_C) ' \n∗∗\n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

100 j = 5001 ;whi l e j <= 5000+AxNum∗2

102 Text = [ ' ∗Equation \n ' . . .' 2 \n ' num2str ( j ) ' , 1 , 1 , ' num2str ( j +1) ' , 1 , −1 \n∗∗\n ' ] ;

104 f p r i n t f ( Fi leID , Text ) ;j = j +2;

106 end

108 end

Listing B.16: Loads.mf unc t i on Loads ( Fi leID , crds )

2f p r i n t f ( Fi leID , ' ∗∗\n∗∗\ tLoads\n∗∗\n ' ) ;

4nn = length ( crds ( : , 1) ) ;

6 f o r n = 1 : nnf p r i n t f ( Fi leID , [ ' ∗Cload , amplitude=AmpN ' num2str ( c rds (n , 1) ) ' \n ' ] )

;8 f p r i n t f ( Fi leID , [ num2str ( c rds (n , 1) ) ' , 2 , −1000 \n ' ] ) ;

end10


14 end

Listing B.17: Surface.mf unc t i on Sur face ( Fi leID , AxNum)

2Text = [ ' ∗Surface , Type=ELEMENT, Name=Master \n ' . . .

4 ' Rail_Elements , SNEG \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

6i = 5001 ;

8 j = 1 ;whi l e i <= 5000+AxNum∗2

10 Text = [ ' ∗Surface , Type=NODE, Name=Slave ' num2str ( j ) ' \n ' . . .num2str ( i ) ' , 1 \n ' ] ;


14 j = j +1;


end16


20 Text = [ ' ∗ Sur face In t e r a c t i on , Name=IntProp \n ' . . .' 1 \n ' . . .

22 ' ∗ Fr i c t i on \n ' . . .' 0 \n ' . . .

24 ' ∗ Sur face Behavior , No Separat ion \n∗∗\n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

26j = 1 ;

28 whi l e j <= AxNumText = [ ' ∗Contact Pair , I n t e r a c t i o n=IntProp \n ' . . .

30 ' Slave ' num2str ( j ) ' , Master \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

32 j = j +1;end

34Text = ' ∗∗\n∗∗\n ' ;


38 end

Listing B.18: AmplitudesSM.mf unc t i on AmplitudesSM( FileID , v , tTrans , tFree , L , AxNum, AxPos)

2v = v/3 . 6 ; % To convert the Train Speed in m/ s

4 sWagon = AxPos (4 )−AxPos (1 ) ;sPassage = L−sWagon ;

6 tPassage = sPassage /v ;sEnd = AxPos(end)+L−2∗sWagon ;

8 tEnd = sEnd/v+tFree ;

10 Text = [ ' ∗Amplitude , Name=MoveWagon1 \n ' . . .' 0 , 0 , ' num2str ( tTrans ) ' , 0 , ' num2str ( tTrans+0. 1 ) ' , ' num2str ( v

∗0 . 1 ) ' , ' num2str ( tTrans+tPassage−0. 1 ) ' , ' num2str ( sPassage−v∗0 . 1 ) ' \n ' . . .

12 num2str ( tTrans+tPassage ) ' , ' num2str ( sPassage ) ' , ' num2str ( tTrans+tEnd ) ' , ' num2str ( sPassage ) ' \n ' ] ;


sDelay = 0 ;16 k = 2 ;

18 f o r j = 4 : 4 :AxNum−4sDelay = AxPos( j +4)−AxPos( j )+sDelay ;

20 tDelay = sDelay/v ;sWagon = AxPos ( j +4)−AxPos( j +1) ;

22 sPassage = L−sWagon ;tPassage = ( sPassage /v ) ;

24Text = [ ' ∗Amplitude , Name=MoveWagon ' num2str ( k ) ' \n ' . . .

26 ' 0 , 0 , ' num2str ( tTrans+tDelay ) ' , 0 , ' num2str ( tTrans+tDelay+0. 1 ) ' , ' num2str ( v∗0 . 1 ) ' , ' num2str ( tTrans+tDelay+tPassage

121

−0. 1 ) ' , ' num2str ( sPassage−v∗0 . 1 ) ' \n ' . . .num2str ( tTrans+tDelay+tPassage ) ' , ' num2str ( sPassage ) ' , '

num2str ( tTrans+tEnd ) ' , ' num2str ( sPassage ) ' \n ' ] ;28 f p r i n t f ( Fi leID , Text ) ;

30 k = k+1;end

32Text = ' ∗∗\n∗∗\n ' ;


36 end

Listing B.19: NodesT.mf unc t i on NodesT( FileID , AxPos , AxNum, FirstNodePos )

2Text = ' ∗∗\n∗∗\ tNodeT_SteelArrow\n∗∗\n ' ;


6 p = 5001 ;s = 6001 ;

8 m = 7001 ;j = 1 ;

10 z = 1 ;

12 f o r k = 4 : 4 : 8Text = [ ' ∗∗\n∗∗\ tPower Car ' num2str ( z ) ' \n∗∗\n∗Node\n ' . . .





18 num2str ( s ) ' , ' num2str ( (AxPos (k )−AxPos( j )+AxPos(k )−AxPos( j +1) )/2 + FirstNodePos ) ' , 2 \n ' . . .

num2str ( s+1) ' , ' num2str ( (AxPos (k )−AxPos( j )+AxPos(k )−AxPos( j+1) ) /2 + FirstNodePos ) ' , 3 \n ' . . .





24 num2str ( s+2) ' , ' num2str ( (AxPos (k )−AxPos( j +2)+AxPos(k )−AxPos( j+3) ) /2 + FirstNodePos ) ' , 2 \n ' . . .

num2str ( s+3) ' , ' num2str ( (AxPos (k )−AxPos( j +2)+AxPos(k )−AxPos( j+3) ) /2 + FirstNodePos ) ' , 3 \n ' . . .

26 num2str (m) ' , ' num2str ( (AxPos (k ) − AxPos( j ) ) /2 + FirstNodePos )' , 3 \n∗∗\n ' . . .

' ∗Nset , Nset=FrontBogie ' num2str ( z ) ' \n ' . . .28 num2str (p+1) ' \n ' . . .

num2str (p+3) ' \n∗∗\n ' . . .


30 ' ∗Nset , Nset=RearBogie ' num2str ( z ) ' \n ' . . .num2str (p+5) ' \n ' . . .

32 num2str (p+7) ' \n∗∗\n ' . . .' ∗Nset , Nset=Car ' num2str ( z ) ' \n ' . . .

34 num2str ( s+1) ' \n ' . . .num2str ( s+3) ' \n∗∗\n ' ] ;


38 p = p+8;s = s+4;

40 m = m+1;j = j +4;

42 z = z+1;end

44f o r k = 12 : 4 :AxNum

46 Text = [ ' ∗∗\n∗∗\ tCoach ' num2str ( z−2) ' \n∗∗\n∗Node\n ' . . .num2str (p) ' , ' num2str (AxPos (k ) − AxPos( j ) + FirstNodePos ) ' ,

1 \n ' . . .48 num2str (p+1) ' , ' num2str (AxPos (k ) − AxPos( j ) + FirstNodePos ) '

, 2 \n ' . . .num2str (p+2) ' , ' num2str (AxPos (k ) − AxPos( j +1) + FirstNodePos )

' , 1 \n ' . . .50 num2str (p+3) ' , ' num2str (AxPos (k ) − AxPos( j +1) + FirstNodePos )

' , 2 \n ' . . .num2str ( s ) ' , ' num2str ( (AxPos (k )−AxPos( j )+AxPos(k )−AxPos( j +1) )

/2 + FirstNodePos ) ' , 2 \n ' . . .52 num2str ( s+1) ' , ' num2str ( (AxPos (k )−AxPos( j )+AxPos(k )−AxPos( j

+1) ) /2 + FirstNodePos ) ' , 3 \n ' . . .num2str (p+4) ' , ' num2str (AxPos (k ) − AxPos( j +2) + FirstNodePos )


' , 2 \n ' . . .num2str (p+6) ' , ' num2str (AxPos (k ) − AxPos( j +3) + FirstNodePos )


' , 2 \n ' . . .num2str ( s+2) ' , ' num2str ( (AxPos (k )−AxPos( j +2)+AxPos(k )−AxPos( j

+3) ) /2 + FirstNodePos ) ' , 2 \n ' . . .58 num2str ( s+3) ' , ' num2str ( (AxPos (k )−AxPos( j +2)+AxPos(k )−AxPos( j

+3) ) /2 + FirstNodePos ) ' , 3 \n ' . . .num2str (m) ' , ' num2str ( (AxPos (k ) − AxPos( j ) ) /2 + FirstNodePos )

' , 3 \n∗∗\n ' . . .60 ' ∗Nset , Nset=FrontBogie ' num2str ( z ) ' \n ' . . .

num2str (p+1) ' \n ' . . .62 num2str (p+3) ' \n∗∗\n ' . . .

' ∗Nset , Nset=RearBogie ' num2str ( z ) ' \n ' . . .64 num2str (p+5) ' \n ' . . .

num2str (p+7) ' \n∗∗\n ' . . .66 ' ∗Nset , Nset=Car ' num2str ( z ) ' \n ' . . .

num2str ( s+1) ' \n ' . . .68 num2str ( s+3) ' \n∗∗\n ' ] ;


p = p+8;72 s = s+4;

m = m+1;74 j = j +4;

123

z = z+1;76 end


80end

Listing B.20: ElementsT.m1 func t i on ElementsT ( FileID , AxNum, kP_P, cP_P, kS_P, cS_P , kP_C, cP_C, kS_C,

cS_C, mWheel_P, mWheel_C, mBogie_P , IBogie_P , mCar_P, ICar_P , mBogie_C ,IBogie_C , mCar_C, ICar_C)

3 Text = ' ∗∗\n∗Element , Type=Spring2 , E l s e t=SpringPrimary_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

5 i = 4001 ;j = 5001 ;

7 whi l e i <= 4000+8Text = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;


11 j = j +2;end

13Text = ' ∗Element , Type=Spring2 , E l s e t=SpringPrimary_C \n ' ;

15 f p r i n t f ( Fi leID , Text ) ;whi l e i <= 4000+AxNum


19 i = i +1;j = j +2;

21 end

23 Text = ' ∗∗\n∗Element , Type=Spring2 , E l s e t=SpringSecondary_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

25 i = 4501 ;j = 6001 ;



31 j = j +2;end

33Text = ' ∗Element , Type=Spring2 , E l s e t=SpringSecondary_C \n ' ;

35 f p r i n t f ( Fi leID , Text ) ;whi l e i <= 4500+AxNum/2


39 i = i +1;j = j +2;

41 end

43 Text = ' ∗∗\n∗Element , Type=Dashpot2 , E l s e t=DashpotPrimary_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

45 i = 5001 ;


j = 5001 ;47 whi l e i <= 5000+8

Text = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;49 f p r i n t f ( Fi leID , Text ) ;

i = i +1;51 j = j +2;

end53

Text = ' ∗Element , Type=Dashpot2 , E l s e t=DashpotPrimary_C \n ' ;55 f p r i n t f ( Fi leID , Text ) ;

whi l e i <= 5000+AxNum57 Text = [ num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( j +1) ' \n ' ] ;


j = j +2;61 end

63 Text = ' ∗∗\n∗Element , Type=Dashpot2 , E l s e t=DashpotSecondary_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

65 i = 5501 ;j = 6001 ;



71 j = j +2;end

73Text = ' ∗Element , Type=Dashpot2 , E l s e t=DashpotSecondary_C \n ' ;



79 i = i +1;j = j +2;

81 end

83 Text = ' ∗∗\n∗Element , Type=Mass , E l s e t=mWheel_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

85 i = 6001 ;j = 5001 ;

87 whi l e i <= 6000+8Text = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;


91 j = j +2;end

93Text = ' ∗Element , Type=Mass , E l s e t=mWheel_C \n ' ;

95 f p r i n t f ( Fi leID , Text ) ;whi l e i <= 6000+AxNum


99 i = i +1;j = j +2;

101 end

103 Text = ' ∗∗\n∗Element , Type=Mass , E l s e t=mBogie_P \n ' ;

125

f p r i n t f ( Fi leID , Text ) ;105 i = 7001 ;

j = 6001 ;107 whi l e i <= 7000+4

Text = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;109 f p r i n t f ( Fi leID , Text ) ;

i = i +1;111 j = j +2;

end113

Text = ' ∗Element , Type=Mass , E l s e t=mBogie_C \n ' ;115 f p r i n t f ( Fi leID , Text ) ;

whi l e i <= 7000+AxNum/2117 Text = [ num2str ( i ) ' , ' num2str ( j ) ' \n ' ] ;


j = j +2;121 end

123 Text = ' ∗∗\n∗Element , Type=Rotaryi , E l s e t=IBogie_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

125 i = 8001 ;j = 6001 ;



131 j = j +2;end

133Text = ' ∗Element , Type=Rotaryi , E l s e t=IBogie_C \n ' ;



139 i = i +1;j = j +2;

141 end

143 Text = ' ∗∗\n∗Element , Type=Mass , E l s e t=mCar_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

145 i = 7501 ;j = 7001 ;



151 j = j +1;end

153Text = ' ∗Element , Type=Mass , E l s e t=mCar_C \n ' ;



159 i = i +1;j = j +1;

161 end


163 Text = ' ∗∗\n∗Element , Type=Rotaryi , E l s e t=ICar_P \n ' ;f p r i n t f ( Fi leID , Text ) ;

165 i = 8501 ;j = 7001 ;



171 j = j +1;end

173Text = ' ∗Element , Type=Rotaryi , E l s e t=ICar_C \n ' ;



179 i = i +1;j = j +1;

181 end

183 Text = [ ' ∗∗\n∗Spring , E l s e t=SpringPrimary_P\n ' . . .' 2 , 2\n ' num2str (kP_P) ' \n ' . . . % 2 , 2 = Degree o f

freedom of the end nodes which the spr ing i s connected185 ' ∗Spring , E l s e t=SpringSecondary_P\n ' . . .

' 2 , 2\n ' num2str (kS_P) ' \n ' . . .187 ' ∗Spring , E l s e t=SpringPrimary_C\n ' . . .

' 2 , 2\n ' num2str (kP_C) ' \n ' . . . % 2 , 2 = Degree o ffreedom of the end nodes which the spr ing i s connected

189 ' ∗Spring , E l s e t=SpringSecondary_C\n ' . . .' 2 , 2\n ' num2str (kS_C) ' \n∗∗\n ' . . .

191 ' ∗Dashpot , E l s e t=DashpotPrimary_P\n ' . . .' 2 , 2\n ' num2str (cP_P) ' \n ' . . .

193 ' ∗Dashpot , E l s e t=DashpotSecondary_P\n ' . . .' 2 , 2\n ' num2str (cS_P) ' \n ' . . .

195 ' ∗Dashpot , E l s e t=DashpotPrimary_C\n ' . . .' 2 , 2\n ' num2str (cP_C) ' \n ' . . .

197 ' ∗Dashpot , E l s e t=DashpotSecondary_C\n ' . . .' 2 , 2\n ' num2str (cS_C) ' \n∗∗\n ' . . .

199 ' ∗Mass , E l s e t=mWheel_P\n ' . . .num2str (mWheel_P) ' \n ' . . .

201 ' ∗Mass , E l s e t=mWheel_C\n ' . . .num2str (mWheel_C) ' \n ' . . .

203 ' ∗Mass , E l s e t=mBogie_P\n ' . . .num2str (mBogie_P) ' \n ' . . .

205 ' ∗Mass , E l s e t=mBogie_C\n ' . . .num2str (mBogie_C) ' \n ' . . .

207 ' ∗Mass , E l s e t=mCar_P\n ' . . .num2str (mCar_P) ' \n ' . . .

209 ' ∗Mass , E l s e t=mCar_C\n ' . . .num2str (mCar_C) ' \n∗∗\n ' . . .

211 ' ∗Rotary In e r t i a , E l s e t=IBogie_P\n ' . . .' 0 , 0 , ' num2str ( IBogie_P ) ' , 0 , 0 , 0 \n ' . . .

213 ' ∗Rotary In e r t i a , E l s e t=IBogie_C\n ' . . .' 0 , 0 , ' num2str ( IBogie_C ) ' , 0 , 0 , 0 \n ' . . .

215 ' ∗Rotary In e r t i a , E l s e t=ICar_P\n ' . . .' 0 , 0 , ' num2str ( ICar_P) ' , 0 , 0 , 0 \n ' . . .

217 ' ∗Rotary In e r t i a , E l s e t=ICar_C\n ' . . .

127

' 0 , 0 , ' num2str ( ICar_C) ' , 0 , 0 , 0 \n∗∗\n ' ] ;219 f p r i n t f ( Fi leID , Text ) ;

221 i = 9001 ;j = 5002 ;

223 k = 6001 ;m = 7001 ;

225 f o r z = 1 :AxNum/4Text = [ ' ∗∗\n∗∗\tWagon Body ' num2str ( z ) ' \n∗∗\n ' . . .

227 ' ∗Element , type=RB2D2, E l s e t=FB ' num2str ( z ) ' \n ' . . .num2str ( i ) ' , ' num2str ( j ) ' , ' num2str ( k ) ' \n ' . . .

229 num2str ( i +1) ' , ' num2str ( k ) ' , ' num2str ( j +2) ' \n ' . . .' ∗Rigid Body , E l s e t=FB ' num2str ( z ) ' , Ref Node= ' num2str ( k ) ' ,

Po s i t i on=Input , PIN Nset=FrontBogie ' num2str ( z ) ' \n∗∗\n '. . .

231 ' ∗Element , type=RB2D2, E l s e t=RB ' num2str ( z ) ' \n ' . . .num2str ( i +2) ' , ' num2str ( j +4) ' , ' num2str ( k+2) ' \n ' . . .

233 num2str ( i +3) ' , ' num2str ( k+2) ' , ' num2str ( j +6) ' \n ' . . .' ∗Rigid Body , E l s e t=RB ' num2str ( z ) ' , Ref Node= ' num2str ( k+2) '

, Po s i t i on=Input , PIN Nset=RearBogie ' num2str ( z ) ' \n∗∗\n '. . .

235 ' ∗Element , type=RB2D2, E l s e t=C ' num2str ( z ) ' \n ' . . .num2str ( i +4) ' , ' num2str ( k+1) ' , ' num2str (m) ' \n ' . . .

237 num2str ( i +5) ' , ' num2str (m) ' , ' num2str ( k+3) ' \n ' . . .' ∗Rigid Body , E l s e t=C ' num2str ( z ) ' , Ref Node= ' num2str (m) ' ,

Po s i t i on=Input , PIN Nset=Car ' num2str ( z ) ' \n∗∗\n ' . . .239 ' ∗Equation \n ' . . .

' 2 \n ' num2str ( j−1) ' , 1 , 1 , ' num2str (m) ' , 1 , −1 \n ' . . .241 ' ∗Equation \n ' . . .

' 2 \n ' num2str ( j +1) ' , 1 , 1 , ' num2str (m) ' , 1 , −1 \n ' . . .243 ' ∗Equation \n ' . . .



' 2 \n ' num2str ( k ) ' , 1 , 1 , ' num2str (m) ' , 1 , −1 \n ' . . .249 ' ∗Equation \n ' . . .

' 2 \n ' num2str ( k+2) ' , 1 , 1 , ' num2str (m) ' , 1 , −1 \n∗∗\n ' ] ;251 f p r i n t f ( Fi leID , Text ) ;

i = i +6;253 j = j +8;

k = k+4;255 m = m+1;

end257


261 end

Listing B.21: AmplitudesT.m1 func t i on AmplitudesT ( FileID , v , tTrans , tFree , L , AxNum, AxPos)

3 v = v/3 . 6 ; % To convert the Train Speed in m/ ssWagon = AxPos (4 )−AxPos (1 ) ;

5 sPassage = L−sWagon ;


tPassage = sPassage /v ;7 sEnd = AxPos(end)+L−2∗sWagon ;

tEnd = sEnd/v+tFree ;9

Text = [ ' ∗Amplitude , Name=MoveWagon1 \n ' . . .11 ' 0 , 0 , ' num2str ( tTrans ) ' , 0 , ' num2str ( tTrans+0. 1 ) ' , ' num2str ( v

∗0 . 1 ) ' , ' num2str ( tTrans+tPassage−0. 1 ) ' , ' num2str ( sPassage−v∗0 . 1 ) ' \n ' . . .

num2str ( tTrans+tPassage ) ' , ' num2str ( sPassage ) ' , ' num2str ( tTrans+tEnd ) ' , ' num2str ( sPassage ) ' \n ' ] ;


15 sDelay = 0 ;k = 2 ;

17f o r j = 4 : 4 :AxNum−4

19 sDelay = AxPos( j +4)−AxPos( j )+sDelay ;tDelay = sDelay /v ;

21 sWagon = AxPos( j +4)−AxPos( j +1) ;sPassage = L−sWagon ;

23 tPassage = ( sPassage /v ) ;

25 Text = [ ' ∗Amplitude , Name=MoveWagon ' num2str ( k ) ' \n ' . . .' 0 , 0 , ' num2str ( tTrans+tDelay ) ' , 0 , ' num2str ( tTrans+tDelay+0

. 1 ) ' , ' num2str ( v∗0 . 1 ) ' , ' num2str ( tTrans+tDelay+tPassage−0. 1 ) ' , ' num2str ( sPassage−v∗0 . 1 ) ' \n ' . . .

27 num2str ( tTrans+tDelay+tPassage ) ' , ' num2str ( sPassage ) ' , 'num2str ( tTrans+tEnd ) ' , ' num2str ( sPassage ) ' \n ' ] ;


k = k+1;31 end


35end

Listing B.22: Boundary_Model.mf unc t i on Boundary_Model ( Fi leID , Clamp , Hinge , Bearing , Axles1 , S leepersPos ,

Bridge_First , Bridge_Last , Axles2 )2

Text = [ ' ∗Boundary\n ' . . .4 ' ∗∗\n∗∗\ tModel\n∗∗\n ' ] ;


f o r i = 1 : l ength ( Axles1 )8 Text = [ num2str ( Axles1 ( i ) ) ' , ' Clamp ' \n ' ] ;

f p r i n t f ( Fi leID , Text ) ;10 end

12 Text = ' ∗∗\n ' ;f p r i n t f ( Fi leID , Text ) ;

14k = 1 ;

16 f o r i = Axles1 (1 ) −1:2: Axles1(end)−1

129

i f i == Axles1 (k )−118 Text = ' ∗∗\n ' ;

f p r i n t f ( Fi leID , Text ) ;20 k = k+1;

e l s e22 Text = [ num2str ( i ) ' , ' Clamp ' \n ' ] ;


end26


30 k = 1 ;f o r i = Axles1(end)+1:2: Bridge_First−3

32 i f i == Sleeper sPos (k )Text = [ num2str ( i ) ' , ' Clamp ' \n ' ] ;

34 f p r i n t f ( Fi leID , Text ) ;k = k+1;

36 e l s eText = ' ∗∗\n ' ;

38 f p r i n t f ( Fi leID , Text ) ;end

40 end

42 Text = [ ' ∗∗\n ' num2str ( Bridge_First −1) ' , ' Hinge ' \n ' . . .num2str ( Bridge_Last−1) ' , ' Bearing ' \n∗∗\n ' ] ;


46 f o r i = Bridge_First −1:2 : Bridge_Last−1i f i == Sleeper sPos (k )

48 k = k+1;end

50 end

52 f o r i = Bridge_Last +1:2 : S l eeper sPos(end)i f i == Sleeper sPos (k )

54 Text = [ num2str ( i ) ' , ' Clamp ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

56 k = k+1;e l s e


60 endend

62k = 1 ;

64 f o r i = Axles2 (1 ) −1:2: Axles2(end)−1i f i == Axles2 (k )−1


68 k = k+1;e l s e


72 endend

74



78 f o r i = 1 : l ength ( Axles2 )Text = [ num2str ( Axles2 ( i ) ) ' , ' Clamp ' \n ' ] ;


82Text = ' ∗∗\n ' ;


86 end

Listing B.23: BoundarySM.mf unc t i on BoundarySM( FileID , AxNum)

2Text = ' ∗∗\ tTrain \n∗∗\n ' ;


6 p = 5001 ;f o r z = 1 : 1 :AxNum/4

8 Text = [ num2str (p+1) ' , 1 , 1 \n ' . . .num2str (p+3) ' , 1 , 1 \n ' . . .

10 num2str (p+5) ' , 1 , 1 \n ' . . .num2str (p+7) ' , 1 , 1 \n ' ] ;


14 end


18end

Listing B.24: BoundaryT.m1 func t i on BoundaryT( FileID , AxNum)

3 Text = ' ∗∗\ tTrain \n∗∗\n ' ;f p r i n t f ( Fi leID , Text ) ;

5m = 7001 ;

7 f o r z = 1 : 1 :AxNum/4Text = [ num2str (m) ' , 1 , 1 \n∗∗\n ' ] ;

9 f p r i n t f ( Fi leID , Text ) ;m = m+1;

11 end


15end

131

Listing B.25: Step_1_Gravity.mf unc t i on Step_1_Gravity ( Fi leID , g , Output )

2Text = [ ' ∗Step , name=Step_1_Gravity_SteelArrow\n ' . . .

4 ' ∗ S t a t i c \n ' . . .' ∗Dload\n ' . . .

6 ' , GRAV, ' num2str ( g ) ' , 0 , −1 \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

8%% Output

10 i f strcmp (Output , ' Vi s u a l i s e ' ) == 1Text = [ ' ∗Output , f i e l d \n ' . . .

12 ' ∗Node Output \n ' . . .'RF, U \n ' . . .

14 ' ∗End Step\n∗∗\n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;

16 e l s eText = ' ∗End Step\n∗∗\n∗∗\n ' ;


20end

Listing B.26: Step_2_EigFrequency.m1 func t i on Step_2_EigFrequency ( FileID , FreqMax)

3 Text = [ ' ∗Step , name=Step_2_EigFrequency , pe r tu rbat i on \n ' . . .' ∗Frequency , e i g e n s o l v e r=Lanczos , a c ou s t i c coup l ing=on ,

norma l i za t i on=disp lacement \n ' . . .5 ' , , ' num2str (FreqMax) ' , , , \n ' . . .

' ∗Restart , write , f requency=0 \n ' . . .7 ' ∗Output , f i e l d , v a r i ab l e=PRESELECT\n ' . . .

' ∗Output , h i s to ry , v a r i ab l e=PRESELECT\n ' . . .9 ' ∗End Step\n∗∗\n∗∗\n ' ] ;


end

Listing B.27: Step_3_MoveCF.mf unc t i on Step_3_MoveCF( FileID , FreqMax , AxPos , v , tTrans , tFree , Lmodel , Crds ,

Output )2

TimeStep = 1/(10∗FreqMax) ; %Increment o f time

4 TimePassage = (AxPos(end)+Lmodel−2∗AxPos (4 ) ) /(v/3 . 6 ) + tTrans + tFree ;% Time needed by the t r a i n to pass the br idge p lus tTrans = 1 s

and tS ta t i on2 = 0 . 1 s extraTimeIncMIN = TimeStep ;

6 TimeIncMAX = TimeStep ;NumInc = c e i l ( TimePassage/TimeIncMIN) ;

8Text = [ ' ∗Step , Name=Step_3_MoveCF_SteelArrow , Inc= ' num2str (NumInc) ' \

n ' . . .


10 ' ∗Dynamic , Direct , Nohaf , I n i t i a l=No , Alpha=−0.05 \n ' . . .num2str ( TimeStep ) ' , ' num2str ( TimePassage ) ' , ' num2str (TimeIncMIN

) ' , ' num2str (TimeIncMAX) ' \n ' ] ;12 f p r i n t f ( Fi leID , Text ) ;

Loads ( Fi leID , Crds ) % Function to wr i t e the Loads in Main.inp14

%% Output16 i f strcmp (Output , 'ReadAccDisp_MidSpan ' ) == 1

Text = [ ' ∗Output , History , Frequency=1 \n ' . . .18 ' ∗Node Print , Nset=Bridge_MidNode_Concrete \n ' . . .

'A2 , U2 \n ' . . .20 ' ∗End Step\n∗∗\n∗∗\n ' ] ;

f p r i n t f ( Fi leID , Text ) ;22 e l s e i f strcmp (Output , ' Vi s u a l i s e ' ) == 1

Text = [ ' ∗Output , Fie ld , Frequency=1 \n ' . . .24 ' ∗Node Output , Nset=Bridge_Nodes_Concrete \n ' . . .

'A, U \n ' . . .26 ' ∗Output , Fie ld , Frequency=1 \n ' . . .

' ∗Node Output , Nset=WayIn_EndNode_Soil \n ' . . .28 'RF \n ' . . .

' ∗Node Output , Nset=Bridge_EndNode_Soil \n ' . . .30 'RF \n ' . . .

' ∗Contact Output \n ' . . .32 'CSTRESS, CDISP \n ' . . .

' ∗End Step\n∗∗\n∗∗\n ' ] ;34 f p r i n t f ( Fi leID , Text ) ;

end36

end

Listing B.28: Step_3_MoveSM.m1 func t i on Step_3_MoveSM( FileID , FreqMax , AxPos , AxNum, v , tTrans , tFree ,

Lmodel , Clamp , Hinge , Bearing , Axles1 , S leepersPos , Bridge_First ,Bridge_Last , Axles2 , Output )

3 TimeStep = 1/(10∗FreqMax) ; %Increment o f time

TimePassage = (AxPos(end)+Lmodel−2∗AxPos (4 ) ) /(v/3 . 6 ) + tTrans + tFree ;% Time needed by the t r a i n to pass the br idge p lus tTrans = 1 s

and tS ta t i on2 = 0 . 1 s extra5 TimeIncMIN = TimeStep ;

TimeIncMAX = TimeStep ;7 NumInc = c e i l ( TimePassage/TimeIncMIN) ;

9 Text = [ ' ∗Step , Name=Step_3_MoveSM_SteelArrow , Inc= ' num2str (NumInc) ' \n ' . . .' ∗Dynamic , Direct , Nohaf , I n i t i a l=No , Alpha=−0.05 \n ' . . .

11 num2str ( TimeStep ) ' , ' num2str ( TimePassage ) ' , ' num2str (TimeIncMIN) ' , ' num2str (TimeIncMAX) ' \n ' . . .

' ∗Boundary , Op=NEW \n ' . . .13 ' ∗∗\n∗∗\ tModel\n∗∗\n ' ] ;


f o r i = 1 : l ength ( Axles1 )17 Text = [ num2str ( Axles1 ( i ) ) ' , ' Clamp ' \n ' ] ;

f p r i n t f ( Fi leID , Text ) ;

133

19 end


23k = 1 ;



29 k = k+1;e l s e


33 endend

35Text = ' ∗∗\n ' ;


39 k = 1 ;f o r i = Axles1(end)+1:2: Bridge_First−3



45 e l s eText = ' ∗∗\n ' ;


49 end

51 Text = [ ' ∗∗\n ' num2str ( Bridge_First −1) ' , ' Hinge ' \n ' . . .num2str ( Bridge_Last−1) ' , ' Bearing ' \n∗∗\n ' ] ;


55 f o r i = Bridge_First −1:2 : Bridge_Last−1i f i == Sleeper sPos (k )

57 k = k+1;end

59 end

61 f o r i = Bridge_Last +1:2 : S l eeper sPos(end)i f i == Sleeper sPos (k )


65 k = k+1;e l s e


69 endend

71k = 1 ;




77 k = k+1;e l s e


81 endend

83Text = ' ∗∗\n ' ;




91Text = ' ∗∗\n ' ;


95 p = 5001 ;f o r z = 1 : 1 :AxNum/4

97 Text = [ ' ∗Boundary , Op=NEW, Amplitude=MoveWagon ' num2str ( z ) ' \n '. . .num2str (p+1) ' , 1 , 1 , 1 \n ' . . .

99 num2str (p+3) ' , 1 , 1 , 1 \n ' . . .num2str (p+5) ' , 1 , 1 , 1 \n ' . . .

101 num2str (p+7) ' , 1 , 1 , 1 \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

103 p = p+8;end

105%% Output

107 i f strcmp (Output , 'ReadAccDisp_MidSpan ' ) == 1Text = [ ' ∗Output , History , Frequency=1 \n ' . . .

109 ' ∗Node Print , Nset=Bridge_MidNode_Concrete \n ' . . .'A2 , U2 \n ' . . .


113 e l s e i f strcmp (Output , ' Vi s u a l i s e ' ) == 1Text = [ ' ∗Output , Fie ld , Frequency=1 \n ' . . .

115 ' ∗Node Output , Nset=Bridge_Nodes_Concrete \n ' . . .'A, U \n ' . . .

117 ' ∗Node Output , Nset=WayIn_EndNode_Soil \n ' . . .'RF \n ' . . .

119 ' ∗Node Output , Nset=Bridge_EndNode_Soil \n ' . . .'RF \n ' . . .

121 ' ∗Contact Output \n ' . . .'CSTRESS, CDISP \n ' . . .


125 end

127 end

Listing B.29: Step_3_MoveT.mf unc t i on Step_3_MoveT( FileID , FreqMax , AxPos , AxNum, v , tTrans , tFree ,

Lmodel , Clamp , Hinge , Bearing , Axles1 , S leepersPos , Bridge_First ,

135

Bridge_Last , Axles2 , Output )2

TimeStep = 1/(10∗FreqMax) ; %Increment o f time

4 TimePassage = (AxPos(end)+Lmodel−2∗AxPos (4 ) ) /(v/3 . 6 ) + tTrans + tFree ;% Time needed by the t r a i n to pass the br idge p lus tTrans = 1 s

and tS ta t i on2 = 0 . 1 s extraTimeIncMIN = TimeStep ;

6 TimeIncMAX = TimeStep ;NumInc = c e i l ( TimePassage/TimeIncMIN) ;

8Text = [ ' ∗Step , Name=Step_3_MoveT_SteelArrow , Inc= ' num2str (NumInc) ' \n

' . . .10 ' ∗Dynamic , Direct , Nohaf , I n i t i a l=No , Alpha=−0.05 \n ' . . .

num2str ( TimeStep ) ' , ' num2str ( TimePassage ) ' , ' num2str (TimeIncMIN) ' , ' num2str (TimeIncMAX) ' \n ' . . .

12 ' ∗Boundary , Op=NEW \n ' . . .' ∗∗\n∗∗\ tModel\n∗∗\n ' ] ;




20Text = ' ∗∗\n ' ;


24 k = 1 ;f o r i = Axles1 (1 ) −1:2: Axles1(end)−1

26 i f i == Axles1 (k )−1Text = ' ∗∗\n ' ;


30 e l s eText = [ num2str ( i ) ' , ' Clamp ' \n ' ] ;


34 end


38k = 1 ;

40 f o r i = Axles1(end)+1:2: Bridge_First−3i f i == Sleeper sPos (k )


44 k = k+1;e l s e


48 endend

50Text = [ ' ∗∗\n ' num2str ( Bridge_First −1) ' , ' Hinge ' \n ' . . .

52 num2str ( Bridge_Last−1) ' , ' Bearing ' \n∗∗\n ' ] ;f p r i n t f ( Fi leID , Text ) ;


54f o r i = Bridge_First −1:2 : Bridge_Last−1

56 i f i == Sleeper sPos (k )k = k+1;

58 endend

60f o r i = Bridge_Last +1:2 : S l eeper sPos(end)



66 e l s eText = ' ∗∗\n ' ;


70 end

72 k = 1 ;f o r i = Axles2 (1 ) −1:2: Axles2(end)−1

74 i f i == Axles2 (k )−1Text = ' ∗∗\n ' ;


78 e l s eText = [ num2str ( i ) ' , ' Clamp ' \n ' ] ;


82 end


86f o r i = 1 : l ength ( Axles2 )

88 Text = [ num2str ( Axles2 ( i ) ) ' , ' Clamp ' \n ' ] ;f p r i n t f ( Fi leID , Text ) ;

90 end


94m = 7001 ;

96 f o r z = 1 : 1 :AxNum/4Text = [ ' ∗Boundary , Op=NEW, Amplitude=MoveWagon ' num2str ( z ) ' \n '

. . .98 num2str (m) ' , 1 , 1 , 1 \n∗∗\n ' ] ;

f p r i n t f ( Fi leID , Text ) ;100 m = m+1;

end102

%% Output104 i f strcmp (Output , 'ReadAccDisp_MidSpan ' ) == 1

Text = [ ' ∗Output , History , Frequency=1 \n ' . . .106 ' ∗Node Print , Nset=Bridge_MidNode_Concrete \n ' . . .

'A2 , U2 \n ' . . .108 ' ∗End Step\n∗∗\n∗∗\n ' ] ;

f p r i n t f ( Fi leID , Text ) ;110 e l s e i f strcmp (Output , ' Vi s u a l i s e ' ) == 1

137

Text = [ ' ∗Output , Fie ld , Frequency=1 \n ' . . .112 ' ∗Node Output , Nset=Bridge_Nodes_Concrete \n ' . . .

'A, U \n ' . . .114 ' ∗Node Output , Nset=WayIn_EndNode_Soil \n ' . . .

'RF \n ' . . .116 ' ∗Node Output , Nset=Bridge_EndNode_Soil \n ' . . .

'RF \n ' . . .118 ' ∗Contact Output \n ' . . .

'CSTRESS, CDISP \n ' . . .120 ' ∗End Step\n∗∗\n∗∗\n ' ] ;


124 end

Documents

Bridge Interaction on Railway Lines