Vibration mitigation of high-speed railway bridges1090042/... · 2017. 4. 21. · vibrations of high-speed railway bridges”. International Journal of Rail Transportation, 5 (1):

SARAH TELL Vibration m

itigation of high-speed railway bridges

KTH

2017

LICENTIATE THESIS IN STRUCTURAL ENGINEERING AND BRIDGESSTOCKHOLM, SWEDEN 2017

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENTwww.kth.se

TRITA-BKN. BULLETIN 147, 2017ISBN 978-91-7729-383-5ISSN 1103-4270ISRN KTH/BKN/B–147–SE

Vibration mitigation of high-speed railway bridgesApplication of � uid viscous dampers

SARAH TELL

Vibration mitigation of high-speed railway bridges,Application of fluid viscous dampers

SARAH TELL

Licentiate ThesisStockholm, Sweden 2017

TRITA-BKN. Bulletin 147, 2017ISSN 1103-4270ISRN KTH/BKN/B–147–SE

KTH School of ABESE-100 44 Stockholm

Sweden

Akademisk avhandling som med tillstånd av Kungl. Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie licentiatexamen i brobyggnadonsdagen den 17 maj 2017 klockan 13:00 i sal M108, Kungl. Tekniska högskolan,Brinellvägen 23, Stockholm.

© Sarah Tell, May 2017

Tryck: Universitetsservice US-AB

Abstract

At the moment of writing, an expansion of the Swedish railway network has started,by constructions of new lines for high-speed trains. The aim is to create a high-speedconnection between the most populous cities in Sweden - Stockholm, Göteborgand Malmö, and the rest of Europe. Thereby, the likelihood of faster, longer andheavier foreign trains crossing the Swedish lines is increased. However, this couldbe problematic since the dynamic response in railway bridges and, consequently,the risk of resonance increases with increasing train speeds.

Bridges are usually designed based on contemporary conditions and future require-ments are rarely considered, due to e.g. cost issues. Prospectively, the dynamicperformance of existing bridges may become insufficient. Hence, the current expan-sion of the high-speed railway network results in an increased demand of innovativedesign solutions for new bridges and cost-efficient upgrading methods for existinglines.

The aim of the present thesis is to propose a vibration mitigation strategy suitablefor new and existing high-speed railway bridges. The main focus is a retrofit methodwith fluid viscous dampers installed between the bridge superstructure and thesupports, which is intended to reduce the vertical bridge deck acceleration belowthe European design code limits. Furthermore, the intention is to investigate theefficiency of such a system, as well as to identify and analyse the parameters anduncertainties which could influence its functionality.

In order to examine the applicability of the proposed retrofit, case studies, statis-tical screenings and sensitivity analyses are performed and analysed. Two differentmodels, a single-degree-of-freedom system and a finite element model, are devel-oped and compared. From the different models, it is possible to study the influencefrom the damper parameters, the variability of the material properties and differentmodelling aspects on the bridge response. After the installation of the fluid viscousdampers, it is found that the acceleration level of the bridge deck is significantlyreduced, even below the design code requirements.

Keywords: dynamics, vibration mitigation, railway bridge, high-speed trains,bridge deck acceleration, passive damping, fluid viscous dampers, retrofit

i

Sammanfattning

I skrivande stund har en utbyggnad av det svenska järnvägsnätet initierats. Måletär att skapa en höghastighetsanslutning mellan de folkrikaste städerna i Sverige- Stockholm, Göteborg och Malmö, och vidare ut i Europa. Därmed ökar san-nolikheten att snabbare, längre och tyngre utländska tåg korsar de svenska järn-vägslinjerna. Dock kan detta bli problematiskt i och med att järnvägsbroars dy-namiska respons och, följaktligen, risken för resonans ökar med ökad tåghastighet.

Broar dimensioneras ofta utifrån nuvarande förutsättningar och hänsyn tas säl-lan till framtida hållbarhetskrav, exempelvis p.g.a. kostnadsbesparingar. Ur ettframtidsperspektiv kan därför det dynamiska beteendet hos befintliga broar kommaatt bli otillräckligt. Utbyggnaden av höghastighetsnätverket ökar därmed behovetav innovativa konstruktionslösningar för nya broar och kostnadseffektiva uppgrader-ingsmetoder för befintliga sträckor.

Syftet med föreliggande avhandling är att föreslå en metod för att minska de vi-brationsnivåer som kan uppstå i både nybyggda och befintliga järnvägsbroar förhöghastighetståg. Huvudfokus är en eftermonteringsmetod med viskösa dämpare,som har installerats mellan brons överbyggnad och landfästen, för att minskabrobanans vertikala acceleration under gällande europeiska dimensioneringskrav.Vidare avses att undersöka effektiveteten av ett sådant system, samt att identifieraoch analysera de parametrar och osäkerheter som kan påverka dess funktionalitet.

Fall- och parameterstudier, samt statistiska metoder används och utvärderas för attundersöka tillämpbarheten av den föreslagna vibrationsdämpningsmetoden. Tvåolika modeller, ett enfrihetsgradssystem och en finit elementmodell, har skapatsoch jämförts. Utifrån dessa modeller kan påverkan av dämparens parametrar, vari-abiliteten hos materialegenskaperna och behandlingen av olika modelleringsaspek-ter studeras. Från resultaten är det tydligt att brobanans accelerationsnivå avsevärtreduceras efter monteringen av viskösa dämpare, till och med under dimensioner-ingskraven.

Nyckelord: dynamik, vibrationsdämpning, järnvägsbro, höghastighetståg,brobaneacceleration, passiv dämpning, viskösa dämpare, eftermontering

iii

Preface

The work presented in this thesis was financed by the Swedish research councilFormas. All underlying research was conducted at the Department of Civil and Ar-chitectural Engineering, KTH Royal Institute of Technology, under the supervisionof Professor Raid Karoumi, Dr. Mahir Ülker-Kaustell and Dr. Andreas Anders-son. I am highly grateful for all their support, dedication and professional guidancewhich has made me grow as a researcher.

A special thanks to my colleagues and friends at the Division of Structural Engi-neering and Bridges for valuable advice and discussions. Truly, it is a team of kindpeople that contribute to an enjoyable atmosphere at work and beyond. Especially,I would like to direct my profound gratitude to Professor Jean-Marc Battini forspending time reviewing the present thesis.

My sincerest appreciation is expressed to my beloved parents with their significantothers, siblings, grandparents and parents-in-law for always being encouraging andhelpful. Finally, I wish to thank my husband, Viktor, for his love and supportthroughout my studies.

Stockholm, May 2017Sarah Tell

v

Publications

The current thesis is based on one conference proceeding and two journal papers,labelled Paper I-III.

Appended conference proceedings:

Paper I Rådeström, S., Andersson, A., Ülker-Kaustell, M., Tell, V., andKaroumi, R., “Structural control of high-speed railway bridges bymeans of fluid viscous dampers”. In 19th IABSE Congress on Chal-lenges in Design and Construction of an Innovative and SustainableBuilt Environment, IABSE 2016 (pp. 2535–2542).

Appended journal papers:

Paper II Rådeström, S., Ülker-Kaustell, M., Andersson, A., Tell, V., andKaroumi, R., “Application of fluid viscous dampers to mitigatevibrations of high-speed railway bridges”. International Journal ofRail Transportation, 5 (1): 47–62.

Paper III Tell, S., Leander, J., Andersson, A., and Ülker-Kaustell, M., “Sen-sitivity analysis of a high-speed railway bridge with supplementalfluid viscous dampers”. To be submitted to Engineering Structures.

Other relevant publications:

1) Rådeström, S., Tell, V., Ülker-Kaustell, M., and Karoumi, R.,“Parametric evaluation of viscous damper retrofit for high-speedrailway bridges”. In Conference of 5th ECCOMAS Thematic Con-ference on Computational Methods in Structural Dynamics andEarthquake Engineering, COMPDYN 2015 (pp. 1672–1681). Op-tical Society of America.

vii

Contents

Preface v

Publications vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Scientific contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Damping devices 72.1 Passive, active and semi-active control systems . . . . . . . . . . . . 72.2 Fluid viscous dampers . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Magnetorheological dampers . . . . . . . . . . . . . . . . . . . . . . . 112.4 Dynamic vibration absorbers . . . . . . . . . . . . . . . . . . . . . . 16

3 Modelling aspects of the bridge 193.1 Introduction to the proposed models . . . . . . . . . . . . . . . . . . 193.2 The single-degree-of-freedom system . . . . . . . . . . . . . . . . . . 203.3 The finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Comparison of the models . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Modelling details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Reliability assessment 414.1 The concept of reliability assessments . . . . . . . . . . . . . . . . . 414.2 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Probability of exceedance . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Concluding remarks 455.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

References 51

Paper I 59

Paper II 69

Paper III 87

ix

Chapter 1

Introduction

1.1 Background

HelsingborgHelsingør

Malmö

Linköping

Jönköping

Stockholm

København

Hamburg

Göteborg

Luleå

Umeå

Järna

Nyland

Ostlänken

Botniabanan

{{

Figure 1.1: An illustration of the extentof the European Corridor, which is re-designed from Europakorridoren (2016).

Currently, the existing main railwaylines in Sweden are carrying mixedtrain traffic (Trafikverket, 2015a).Thus, the utilisation of passenger trainspeeds are limited by the slow run-ning freight trains. This is a causefor accessibility difficulties and delays,which may reduce the willingness totravel by rail. As a result, mea-sures are taken to increase the capac-ity of the Swedish railway network byconstructing new lines for high-speedtrains (Trafikverket, 2015b).

So far, this has resulted in the north-easterly situated Bothnia Line (Bot-niabanan), which is the first subsec-tion that is prepared to operate atspeeds up to 250 km/h (Lindblom,2010). At the moment of writing, theplanning of the East Link (Ostlänken)is initiated. The East Link will bea double-track, high-speed passengerrailway line with a maximum allowedspeed of 320 km/h (Banverket, 2010;Trafikverket, 2010). This line will bepart of the European Corridor (Eu-ropakorridoren), with a future rout-ing from Stockholm to Göteborg andMalmö, and will, thereafter, connectto the rest of Europe. An illustrationof the European Corridor is shown inFigure 1.1.

1

CHAPTER 1. INTRODUCTION

Generally, the European trains are longer, heavier and faster than their Swedishcounterparts. Hence, a connection may result in European trains crossing thebridges along the Swedish lines, which are not designed for these loads and speeds.Accordingly, this could lead to problems in terms of serviceability, such as passengerdiscomfort, or safety, e.g. train derailment (ERRI D214, 1999; Frýba, 2008).

Laboratory tests conducted by the European Rail Research Institute (ERRI) showthat ballast starts to destabilize at an acceleration level close to 0.7g-0.8g (ERRID214, 1999). Similarly, for unballasted slab-tracks, the structural integrity dueto train wheel unloading decreases at approximately 1.0g. Employing a safetyfactor of 2 results in acceleration levels of 3.5 m/s2 and 5.0 m/s2 for ballastedand unballasted tracks, respectively. These values are used in the Eurocode (CEN,2003) as limits for the vertical bridge deck acceleration. Furthermore, limits oftraffic safety also exist for deck twist, deflections of the deck and displacements atthe deck ends. When dynamic loads are involved, the acceleration limit is usuallythe first one to be exceeded (Zacher and Baeßler, 2009). Hence, the vertical bridgedeck acceleration is normally the decisive parameter in dynamic analyses.

The vertical bridge deck acceleration reaches its maximum value at resonance, whichis a state of dynamic response originating from the repetitive nature of the traininduced loading. Resonance occurs when the forcing frequency of the train ap-proaches or coincides with the natural frequency of the structure. At resonance,the response of the bridge deck may reach unacceptable levels and the risk increaseswith increasing train speeds.

Typically, the first bending mode is the one mainly contributing to the response ofrailway bridges, but several modes of vibration could also have an impact (Muserosand Alarcón, 2005). Due to this, it is stated in the Eurocode that either the threefirst modes of vibration or all modes up to 30 Hz should be included in dynamicanalyses (CEN, 2003). In fact, the critical train speed vcr at which resonance occursfor a certain bridge is a function of the natural frequency fn of the nth mode ofvibration and a characteristic length λ corresponding to multiples or submultiplesof the axle or bogie distance d

vcr ≈ fnλ, λ = d/i, i = 1, 2, 3... (1.1)

On this basis, it is clear that changes in the natural frequency of the bridge or thetrain geometry may result in a critical speed occurring within the design interval ofallowed train speeds. In conclusion, the demands for increased passenger capacityand higher speeds of trains may result in a greater probability of exceeding thelimits for vertical bridge deck acceleration. Thus, it is desirable to prepare existingbridges for higher speeds and, at the same time, upgrade the structural resistanceto meet with future demands, which is rarely accounted for at the design stagedue to e.g. increased project costs. Consequently, new, cost-efficient procedures toreduce the acceleration levels in railway bridges are required.

2

1.2. AIMS AND SCOPE

1.2 Aims and scope

The overall aim of this project is to propose a vibration mitigation strategy that issuitable as a retrofit method for existing bridges, but also as a cost-efficient alterna-tive in the design of new bridges. In order to achieve this aim, the objectives are tostudy the effect of fluid viscous dampers (FVDs), which are installed between thebridge deck and the abutments, on the vertical bridge deck acceleration, as well asinvestigating the influence of changes and uncertainties in different parameters onthe damper efficiency. To analyse these effects, a single degree-of-freedom (SDOF)model of the bridge-damper system is derived and evaluated. Moreover, a corre-sponding finite element (FE) model is also produced and the results from the twodifferent models are compared.

The research work in this thesis is subject to some assumptions and limitations.Firstly, the bridge is idealised as a simply supported Euler-Bernoulli beam andbehaves according to the linear theory of elasticity. Secondly, a maximum of twodimensions are studied, which means that lateral and torsional modes are not con-sidered. However, this is a reasonable assumption for single-track, non-skewedbridges. Thirdly, the influence from surrounding structures and ground conditionsare neglected, which means that the abutments are considered as fixed and soil-structure interaction (SSI) is not accounted for. Nevertheless, the influence fromthe stiffness in the abutments is included as a minor supplemental study in thisextended summary. Finally, verifications of the numerical models by field measure-ments are out of the scope of this licentiate project, but will be included in thecontinuation of the research work.

1.3 Scientific contribution

This research project, including the present licentiate thesis with appended papers,scientifically contributes to the field by the proposal of a new methodology to reduceexcessive vibrations of the bridge deck. A more detailed overview of the scientificcontribution is summarized in the following list1. First and foremost, a conceptual proof of the applicability of FVDs installedbetween the bridge superstructure and the abutments is presented.

2. To obtain this proof, a novel analytical model is derived, including the dis-tance from the neutral axis of the bridge to the supports and damper connec-tions, which is entirely based on an arbitrary mode shape of the bridge.

3. A statistical screening of the bridge-damper system is conducted and thesensitivity of different parameters on the response is evaluated. The resultsfrom this study facilitate the design of a functioning bridge-damper system.

4. The analyses highlight the importance of correct modelling of the bearingmechanisms. In fact, it is found that the FVDs may become inoperable if thefriction of the moveable bearings are not overcome.

3

CHAPTER 1. INTRODUCTION

1.4 Outline of the thesis

This thesis is based on the three appended papers and provides an extended sum-mary of the associated research. Initially, a brief introduction to the subject is givenin Chapter 1. Chapter 2 includes information about different damping devices, aswell as previous research within structural control of railway bridges. In Chapter 3,the theoretical frameworks, with important key concepts and limitations, of differ-ent bridge modelling techniques are stated and explained. Further, the performedanalyses and important results are also presented, along with interim findings fromessential modelling aspects connected to the bridge and the damper. Chapter 4forms the basis for the applied statistical methodology adopted in Paper III. Fi-nally, discussions and conclusions emerging from the research work are compiled inChapter 5, which is followed by the appended papers. The connection between thisthesis and the different papers is outlined in Table 1.1. A brief description of eachpaper is given as follows

Paper I includes a study of a 2D FE Euler-Bernoulli beam equipped with FVDs.A parametric study is conducted, where the inclination and position of the FVDsare varied, in order to find the influence on the required damping coefficient toreduce the vertical bridge deck acceleration below the Eurocode limit. From theresults, it is concluded that the acceleration level in the bridge is substantiallyreduced using the proposed retrofit. An optimal combination of the investigatedparameters exists for a specific bridge, allowing for the development of design curvesfor each individual case.

Paper II presents a SDOF model of a simply supported bridge with inclined FVDsinstalled between the superstructure and the abutments. The novelty of this paperis the formulation of a SDOF system to include the rotation of the cross section ofthe bridge due to the distance from the neutral axis to the connection points of theFVDs, henceforth referred to as the eccentricity. It is found that the eccentricityhas a major effect on the efficiency of the FVDs at resonance. Conclusively, thepresented model allows for finding the impact of the dampers on the maximumvertical bridge deck acceleration and damper force within the train speed range.

Paper III provides statistical screenings and sensitivity analyses of an extension ofthe SDOF model presented in Paper II. The model is further expanded to includethe eccentricity arising from the distance between the neutral axis of the bridgedeck to the supports as well. Further, the input mode shapes of the bridge aregeneralised which enables verifications of the FVD behaviour when the friction ofthe moveable bearings is not overcome. The mass, bending stiffness and structuraldamping of the bridge, as well as the bearing friction and damping of the FVDs, arevaried during the analyses to find the influence of these parameters on the bridgedynamic response. The results show that the FVDs required in a deterministicanalysis may be supernumerary in a stochastic model and that a distribution forthe acceleration limit and model uncertainty must be properly assessed.

4

1.4. OUTLINE OF THE THESIS

Table 1.1: The connections between the different papers and the present thesis.

Paper I Paper II Paper III Thesis

Models

FE • •

SDOF • • •

Modal analysis • • •

Dampers

FVDs • • • •

Studies

Damper parameters • • • •

- Damper inclination • •

- Position of the damper • •

- Damping coefficient • • • •

Uncertainties • (•)

- Material properties •

- Bearing friction • (•)

Modelling details • •

- Stiffness in damper •

- Flexibility of abutment •

- Bearing friction • •

5

Chapter 2

Damping devices

2.1 Passive, active and semi-active control systems

Damping devices for structural control can be divided into three main groups:passive, active and semi-active control systems, see Figure 2.1 for conceptual de-scription. The views about the general definitions separating these devices havediffered among researchers throughout the years. However, there are two essentialdifferences between a passive and an active control device: 1) a passive system isincapable of inserting mechanical forces into the primary structure, in contrast toan active device which experiences real-time adjustments and controls the structureby adjustable input energy and 2) the active damper requires power supply for itsoperation, unlike the passive device (Symans and Constantinou, 1999). Accordingto recently accepted definitions, the semi-active device is considered as a compro-mise between the passive and active systems. The reason for this is that it has thechangeability of an active device, but does not impose any input energy into thecontrolled structure and has very low power requirements (Spencer Jr and Nagara-jaiah, 2003). It is widely known that active control systems improve the structuralperformance at a higher level than the equivalent passive counterpart. On the otherhand, passive damping devices are superior in terms of reliability, robustness, main-tenance and cost-efficiency. Although the scope of this thesis only covers passiveFVDs, concepts and backgrounds of two other types of common damping devices,the passive/semi-active dynamic vibration absorber/tuned mass damper (TMD)and the semi-active magnetorheological (MR) damper, are also further describedin this chapter.

Excitation Structure Response

a) Structure with passive system

Passive system

b) Structure with active/semi-active control

Sensors Computer

Actuators

Sensors

Passive system

StructureExcitation Response

Figure 2.1: Conceptual description of structures with a) passive systems and b) ac-tive (no passive system included) and semi-active control systems (include a passivesystem), redesigned from Soong and Spencer (2002).

7

CHAPTER 2. DAMPING DEVICES

2.2 Fluid viscous dampers

A FVD generally consists of a cylinder housing which encloses a compressive siliconefluid, as visualised in Figure 2.2. By means of a piston with small orifices, thefluid may be transferred between the two chambers within the cylinder that arelocated on different sides of the piston head (Constantinou and Symans, 1993). Asa consequence of the viscosity of the damper fluid, the motion of the FVD willdissipate energy from the controlled structure and generate output damper forcesthat are out-of-phase with the input displacement (Housner et al., 1997). In thecase of a linear, purely viscous damper fluid with damping coefficient cd, the outputforce Fd will be proportional to the piston velocity ud

Fd = cdud (2.1)

Initially, the FVD was commonly used within military and aerospace applications(Soong and Spencer, 2002). However, starting from the 1990s, the use of FVDsfor mitigating excessive atructural vibrations reached the field of civil engineering.However, the main scope of application was fairly limited to structures subjected toseismic activities (Makris and Constantinou, 1990, 1991; Constantinou and Symans,1993; Symans and Constantinou, 1998). For this application, Shinozuka et al.(1992) and Seleemah and Constantinou (1997) studied both passive and active, aswell as linear and nonlinear, FVDs on the dynamic response of structures inducedby earthquakes. Moreover, FVDs have been studied for other types of structuralsystems and load cases. For example, the London Millennium Footbridge wasequipped with a hybrid FVD-TMD system to reduce violent oscillations caused bypedestrian interlocking (Dallard et al., 2001). The response of beams and bars withviscous dampers induced by general loads have been extensively studied as well. Inmany of these cases, the question of how to suitably account for damping in themodels has been discussed.

Piston rod CylinderCompressivesilicone fluid

Accumulatorhousing

Seal retainerSeal

Chamber 1 Rod make-upaccumulatorChamber 2Piston head

with orifices

Control valve

Figure 2.2: A sketch of a type of fluid viscous damper, reproduced from Taylor andConstantinou (1998).

8

2.2. FLUID VISCOUS DAMPERS

The traditional approach to account for damping in bridges is to assume that theenergy is equally dissipated along the beam (classical or proportional damping).This simplification is justified if the involved eigenmodes are orthogonal and un-coupled. If this applies, it is fair to assume that the damping is proportional to themass and stiffness of the system and, hence, that the eigenvalue problem could besolved by a linear combination of these entities. However, the procedure becomesmore complex if a system is non-proportionally damped.

Based on the original work by Foss (1956), the orthogonality conditions for non-proportionally damped systems and the decoupling of the equation of motion ofan unloaded, damped system with lumped parameters were derived and proved.In contrast to traditional methods, the damped eigenfrequencies and mode shapescould be calculated using the presented procedure. Later on, Hull (1994) managedto derive an analytical solution for a point load acting on a vibrating bar with aviscous damper as boundary condition.

Oliveto et al. (1997) solved the complex eigenvalue problem and decoupled theequation of motion for a vibrating, simply supported beam with rotational viscousdampers as end conditions. The rotational dampers at the beam ends clearly re-duced the dynamic response of this structural system. The results also showedthat carefully selected damper constants resulted in an optimised design. Mean-while, Greco and Santini (2002) extended this complex mode superposition methodwith rotational viscous dampers at the beam ends to deal with excitations from aconstant moving load as well. From this, it was found that the damper efficiencydepended on the loading speed and that a cautiously chosen value of the damp-ing coefficient resulted in a substantial reduction of the dynamic response. Com-pared to traditional modal analysis, the response obtained by considering com-plex modes was more reliable and should, thus, be employed when evaluatingnon-proportionally damped systems. The current model was further generalisedby Krenk (2004) to also include continuous beams with discrete viscous dampers.However, the method was only illustrated for impulse loading, in addition to concen-trated and distributed modal loads. The main conclusion proved that the dampedeigenmodes could be utilized to identify both the maximum attainable dampingand the optimal tuning of the dampers for a specific system.

Svedholm et al. (2016) continued to develop the complex mode superposition methodto be applicable for both simply supported and continuous beams with general endconditions. The usability of the model was demonstrated with some examples,out of which one corresponded to the case with supplemental damping devices. Anumber of non-classical (overdamped) modes arose as a consequence of the non-proportionality of the discrete dampers. It was clearly highlighted that if the nat-ural frequency of a non-classical mode coincided with a bending mode, maximumobtainable damping of the system would occur.

9


Recently, several authors have conducted studies with response reduction of high-speed railway bridges equipped with supplemental FVDs as main focus. Muserosand Martínez-Rodrigo (2007) investigated the influence of linear FVDs connectingthe bridge deck to a longitudinally spanned, simply supported auxiliary beam, asvisualised in Figure 2.3. The study was conducted in two steps. Firstly, a harmonicload was added to the main beam, in order to obtain closed-form expressions of thedamper properties that minimised the dynamic response at resonance. Secondly,the obtained expressions were, now, applied to the structure which was subjectedto realistic moving train loads. It was found that the bridge deck response wassignificantly reduced through the proposed procedure. This method was furtherrevised in Luu et al. (2014a), in which the optimal damper constants were obtainedby basis of the H∞ norm covering the relevant range of frequencies.

The promising results motivated further studies within the subject. Martínez-Rodrigo et al. (2010) created 3D orthotropic plate FE models of short simply sup-ported bridges retrofitted according to the presented procedure. A difference was,though, that several auxiliary beams and FVDs were distributed along the trans-verse direction of the deck. As a continuation, Martínez-Rodrigo and Museros(2011) developed an analytical, dimensionless solution to the orthotropic platemodel to find the optimal FVD parameters. The outcomes from this study wereconsistent with former findings, i.e. that reduction of the deck acceleration wasattainable and that neither the local deformation of the slab, nor the force capacityof the dampers, would be compromised using the introduced approach. Due to an-other conclusion, that the natural frequency of the auxiliary beam must exceed thatof the bridge deck, Lavado et al. (2014) studied the influence of adopting auxiliarybeams that were clamped to the abutment. The results implied that a clampedauxiliary beam is preferred over a simply supported one.

Bridge girder

Auxiliary beamFVD placed ininternal section

Gap allowing FVDdifferential displacement

Abutment

Figure 2.3: The retrofitted bridge with the auxiliary beam system, reproduced fromMuseros and Martínez-Rodrigo (2007).

10

2.3. MAGNETORHEOLOGICAL DAMPERS

2.3 Magnetorheological dampers

According to classical definitions, viscoelasticity is the capability of a material todisplay both viscous and elastic characteristics when subjected to applied forces(Weiss et al., 1994). As a consequence, a viscoelastic material can, at constant de-formation, absorb a portion of the input energy and release it as heat. Nevertheless,the material will try to restore parts of its original shape if the imposed forces areremoved (Ferry, 1980). If the rheological properties of a viscoelastic (or any other)material can be altered by exposure to external impact, it can be classified as asmart material (Carlson and Jolly, 2000). Further, a certain material is referred toas MR if the properties change due to the presence of a magnetic field, in contrastto electrorheological (ER) materials which react to electric activities.

Owing to the multiphase behaviour of MR fluids, which arises from their ability torapidly and reversibly change between states of a viscous liquid and a viscoplas-tic semi-solid (Soong and Spencer, 2002), they are classified as viscoelastic smartmaterials (Weiss et al., 1994). MR fluids generally consist of micron-sized, mag-netizable particles dispersed in a non-magnetic carrier liquid (Jolly et al., 1996).The behaviour of the fluid depends on the particle size, composition and saturationmagnetization. A high saturation magnetization increases the MR effect of thefluid, while the size of the particles decides its stability and the reversibility of par-ticle agglomeration. Consequently, larger particles of either pure iron or iron/cobaltalloys are to be preferred (Carlson and Jolly, 2000). Regarding the carrier fluid,the rheologial, tribological and temperature-related characteristics determine whichsolution is more suitable. Usually, polyesters, polyethers, water or different oils ofeither silicone, mineral or synthetic hydrocarbon types are used.

The change in phase from liquid to solid state occurs due to the induced magneticinteraction of the constituent particles. This causes a formation of columnar, chain-like structures aligned parallel to the magnetic field (Jolly et al., 1996), restrictingthe flow of the fluid. This reformation process is visualised in Figure 2.4. Themechanical energy required to yield these chain-like alignments of particles increaseswith the intensity of the applied magnetic field (Carlson and Jolly, 2000).

a) b) c)Figure 2.4: The magnetorheological fluid a) without magnetic field, b) under ref-ormation and c) with reformed particles, revised from Truong and Ahn (2012).

11


Electrochemistry does not tend to affect the magnetic mechanism and interactionbetween the particles (Spencer Jr et al., 1997). Hence, MR fluids are not highlysusceptible to impurities resulting from manufacturing and usage, which meansthat different additives can be employed to improve their stability and life-span.Further, this could also be employed to reduce the risk of particle sedimentation,despite the considerable density mismatch between the components within the fluid(Spencer et al., 1997).

The application of MR fluids extends to devices in which the response may be con-trolled in real-time due to the rapid and reversible rheological behaviour (Carlsonand Jolly, 2000). Such an example is the MR damper (MRD). As shown in Fig-ure 2.5, a typical MRD consists of a pair of wires connected to a cylinder housing,which encircles the MR fluid, a magnetic coil, a piston and an accumulator (Truongand Ahn, 2012). During operation, an electric current is supplied from the wiresto the piston-embedded coil that generates the magnetic field. Thus, the MRDproduces controllable damping forces which to a great extent are influenced bothby the magnetic field and the velocity of the piston. The output damper force isonly slightly varied within a temperature interval from -40◦C to 150◦C (Spencer Jret al., 1997). In the absence of a magnetic field (due to e.g. control hardware mal-function or power failure), the MRD will continue to operate in a reliable, fail-safe,passive manner similar to that of a linear viscous damper (Symans and Constanti-nou, 1999; Soong and Spencer, 2002). Another advantage with the MRD is thelow power and voltage requirements, enabling for the use of batteries for powersupply. However, difficulties related to proper modelling of the nonlinear and hys-teretic characteristics of the MR fluid reduces the applicability of MRDs (Truongand Ahn, 2012). Hence, the search for accurate models and control algorithms hasbeen widely undertaken by several authors.

a) Hardware structure

b) Working principle

Accumulator Diaphragm Coil MR Fluid Wires to Coil

Magnetorheological Fluid Damper

Generated Force

Displacement (Velocity)

Current(Magnetic Field)

Figure 2.5: Conceptual sketch of a) the hardware structure of a magnetorheologicaldamper and b) its working principle, reproduced from Truong and Ahn (2012).

12


On behalf of the National Bureau of Standards, the MR fluids and devices wereinitially developed by Rabinow (1948). Further, Carlson (1996) invented a MRDwhich was capable of producing damping motion along several longitudinal or lat-eral axes, for a total of six degrees-of-freedom. To this point, lateral and torsionalmovements had not been a design consideration, resulting in devices limited to sin-gle axial motions. It was also proposed to replace the light fluids with a gel, inorder to avoid settling of particles.

In order to obtain desired control performance, it is of highest importance to apply aforce model that accurately captures the hysteretic behaviour of MRDs (Dominguezet al., 2004). Different parametric and non-parametric models have been proposedto simulate this phenomenon (Yang et al., 2002). Although there exist many differ-ent models within each of these main categories, the focus herein is directed towardsthe most commonly used parametric models.

From experimental studies performed by Spasojević et al. (1974), it was found thatferromagnetic fluids have a rheological behaviour that can be modelled as Binghamplastics with variable yield strengths. The flow of the fluid is governed by Bingham’sequation at stress levels τ exceeding the field dependent yield stress τy (Carlson andJolly, 2000). Otherwise, the behaviour of the material is viscoelastic. This relationcan be expressed as

τ ={τy + ηy, τ > τy

Gy, τ < τy(2.2)

where G is the complex material modulus. Stanway et al. (1987) proposed theBingham idealized mechanical model to describe the behaviour of an ER damper.Generally, this model includes a Coulomb friction element in parallel with a viscousdamper and is governed by

F = FC sign(u) + c0u+ F0 (2.3)

where F , F0 and FC are the force generated by the device, the offset force andthe frictional force, respectively. Moreover, Gamota and Filisko (1991) investigatedthe damping behaviour of an ER material. It was found that this material did notonly correspond to a standard solid (Shames, 1997), but was also dominated byboth Coulomb and viscous damping. Hence, the actual behaviour of the systemcorresponded to a Bingham model and a standard solid in series, which could bedescribed as

F−F0 ={k1(u2− u1)+ c1(u2− u1)= c0u1+ FC sign(u1)= k2(u3− u2), |F |> FC

k1(u2− u1)+ c1u2= k2(u3− u2), |F |≤ FC(2.4)

in which kj and uj are the spring stiffness and the resulting displacement of memberj, respectively. The concepts of the original and modified Bingham models arevisualised and clarified in Figure 2.6.

13


c0

F − F0

u1c1

k1

u2 u3

k2

Original Bingham model

Figure 2.6: Modified Bingham model developed by Gamota and Filisko (1991). Theoriginal Bingham model is presented in the dashed box, but with F −F0 at u = u1.

However, the idealized Bingham element has shown to differ from the actual be-haviour of a MRD at low velocities (Symans and Constantinou, 1999). The un-derlying reason for this was, according to Jolly et al. (1999), that MR fluids haverelatively linear relationships for low currents and begin to saturate beyond thelinear state. Thus, an improved model was developed which consists of an arrange-ment of linear springs and dashpots, together with a Bouc-Wen hysteresis element(Wen, 1976), given by

F = c0u+ k0(u− u0) + αz, z = −γ|u|z|z|n−1 − βu|z|n +Au (2.5)

where γ, β and A are adjustable parameters and z is the evolutionary variable.However, both the Bingham and Bouc-Wen models still lacked the ability to cor-rectly capture the nonlinear behaviour of the force roll-off behaviour at close to zerovelocities (Spencer Jr et al., 1997). Consequently, Spencer Jr et al. (1997) proposeda phenomenological model, given by Equation (2.6), for MRDs with a dashpot in se-ries with the Bouc-Wen model. A spring was introduced in parallel with this seriessystem, in order to account for the stiffness of the accumulator. Figure 2.7 showsthe mechanical systems of the original Bouc-Wen and phenomenological models.

F = c0(u2 − u1) + k0(u2 − u1) + k1(u2 − u0) + αz = c1u1 + k1(u2 − u0) (2.6)

The proposed model more accurately predicted the response obtained from exper-imental data of a MRD with an applied load, compared to the original Bouc-Wenand the Bingham models. Furthermore, this method was further applied in Spenceret al. (1997), where it was compared to the response obtained from a developed200-kN large-scale MRD capable of supplying adequate damping for structural ap-plications. The preliminary results from the experimental characterisation showedthat the design of this MRD was promising for the intended area of use. Tu et al.(2011) performed experiments and simulations of a developed 500-kN large-scaledamper. Inside this MRD, a new type of MR fluid was included, which exhib-ited excellent sedimentation proof stability, maximum obtainable yield stress andresponse time requirements for full-size applications.

14


k0

c0

c1

F

u1

k1

u2

Original Bouc-Wen model

Figure 2.7: Phenomenological model of a magnetorheological damper (Spencer Jret al., 1997). The original Bouc-Wen hysteresis model (Wen, 1976) is described inthe dashed box, with the displacement u = u2 and a clamped end at u0 = u1.

An investigation of the same MRD composition as proposed by Spencer et al. (1997)was conducted by Yang et al. (2002). This study strengthened the claim that thephenomenological model outmatched the Bingham model in terms of compliancewith experimental results. Additionally, a design proposal with a parallel connec-tion of several coils was suggested, which proved to provide a faster response time.Wang and Liao (2011) reviewed different parametric, dynamical, modelling ap-proaches for MRDs. The conclusions were that the force-displacement behaviour ofMRDs could be well described by most of the examined dynamical models. How-ever, no simple parametric model for MRDs gave highly accurate results, whichemphasized the need for further studies within the subject.

Regarding the use of MRDs to mitigate vibrations caused by traffic loads, Jiangand Christenson (2010) performed a real-time hybrid test of a simulated highwaybridge with inclined supplemental dampers. The bridge was modelled as a simplysupported Euler-Bernoulli beam and the vibration control forces was obtained fromtwo large-scale MRDs connected to one actuator and load cell each. The commandcurrent to each damper was controlled in real-time as a heavy truck crossed thebridge. In the simulated response, the dynamic free vibrations were virtually elimi-nated. Linking to the reduction of dynamic response in high-speed railway bridges,Luu et al. (2014b) proposed a combination of the auxiliary beam system (in Figure2.3) and MRDs. An H∞ control algorithm was applied to obtain suitable MRDparameters. The proposed system further decreased the dynamic response as com-pared to FVDs, since optimal control responses could be generated for both theauxiliary and main beams.

15


2.4 Dynamic vibration absorbers

cdkd

md

mb

cbkb

Figure 2.8: Illustration of a tunedmass damper with massmd, dampingcd and stiffness kd installed to a mainstructure with mass mb, damping cband stiffness kb.

Generally, dynamic vibration absorbers arecategorised into three groups: tuned massdampers (TMDs), tuned liquid dampers(TLDs) and tuned liquid column dampers(TLCDs) (Constantinou et al., 1998). Theconventional TMD is the most commonwithin civil engineering applications. It canbe described as a mechanical mass-spring-damper system that is designed to resonatewith and, hence, counteract the motion ofthe main structure by generating equal andopposing forces (Marioni, 2007). Figure 2.8illustrates the concept of a TMD installedto a SDOF system corresponding to themain structure. Traditionally, TMDs havebeen installed to several major buildings(Soto and Adeli, 2013), in order to reducevibrations mainly caused by wind loadingand seismic events.

The concept of mass-damping devices dates back to 1909, when Frahm inventeda dynamic vibration absorber that reduced the vibrations of bodies subjected toperiodic forces (Frahm, 1911). However, this device did not include a dampingelement, which resulted in deficiencies of the vibrating regime if the excitationfrequency deviated from the frequency of the absorber. In order to reduce thisproblem, Ormondroyd and Den Hartog (1928) placed a damper in parallel with themass-supporting spring subjected to harmonic loading. The results showed a de-crease in amplitude for a wider frequency range compared to the original undampedabsorber. Further, Brock (1946) and Den Hartog (1947) presented procedures todetermine the optimal damping ξd and tuning frequency fd of the TMD, respec-tively, given by

fd =1

1 + μ, ξ2

d =3μ

8(1 + μ) (2.7)

Ayorinde and Warburton (Ayorinde and Warburton, 1980; Warburton, 1981, 1982)derived optimal values of the tuning frequency and damping for a given mass ratio μof the TMD subjected to random excitation. By minimizing the response variance,it was confirmed that the optimal TMD parameters varied with the type of exter-nal loading. While structural damping was neglected in all of the aforementionedstudies, Bishop and Welbourn (1952) modified the method to consider dampingrelated to the main mass as well. Moreover, Snowdon (1959) replaced the viscousdamping element and original spring with a rubber material, which resulted in animproved performance of the system.

16

2.4. DYNAMIC VIBRATION ABSORBERS

In previous studies, only vibration suppression of harmonic and random excitationshad been investigated. Kwon et al. (1998) studied moving load-induced vibrationcontrol of bridges equipped with TMDs. From the results, it was found that thevertical bridge deck displacement was significantly reduced. However, the vehicleacceleration was practically unchanged, since the passage time was too short toincrease the TMD motion. Furthermore, Wang et al. (2003) investigated the effectof a passive TMD (PTMD) on the bridge acceleration, displacement and end rota-tions, as well as train body acceleration. The interaction between the bridge andtrain resulted in a detuning effect, due to the difference between the narrow-bandedexcitation frequency and the bridge fundamental frequency. For this case, however,the train-bridge interaction (TBI) effects were not significant for the PTMD per-formance. Detuning may also occur due to changes in environmental conditions,uncertainties in the structural properties or during the manufacturing process ofthe devices etc.

Marano et al. (2010) compared deterministic and stochastic optimization approachesfor TMDs, in which uncertain properties of the main structure were accounted for.The results implied that the uncertainties had an influence on the optimal damperproperties and must, thus, be accounted for to obtain a robust design solution forTMDs. Another measure that has been proposed for reducing the detuning effect isto install a multiple TMD (MTMD) system, as e.g. the one visualised in Figure 2.9,that covers a wider frequency band than a single TMD. Das and Dey (1992) studieda MTMD system on a bridge. The general conclusion was that a single TMD wasefficient when the eigenfrequencies were widely separated and that a MTMD systemtuned to different frequencies was preferred for closely spaced eigenmodes. Giventhe assumption of closely spaced frequencies, Igusa and Xu (1994) investigated theinfluence of MTMDs. It was confirmed that the MTMD system is more effectivecompared to a single TMD of equal mass and has more robust properties duringdetuned conditions.

The influence of several MTMD subsystems on the multiple resonant response of acontinuous railway truss bridge was studied by Yau and Yang (2004). One set withineach subsystem was designed to cover a certain frequency bandwidth, in which themean was tuned to cover the first natural frequencies of the bridge. The proposedmethod efficiently reduced the multi-mode resonance of the continuous structure.Nevertheless, the risk of detuning was not eliminated, but could be decreased bywidening the frequency bandwidth within each MTMD subsystem. Luu et al. (2012)proposed a new method for the optimization of MTMD parameters based on theH2 norm, which was applied to statically indeterminate high-speed railway bridgesfor reducing the multi-resonant peaks. This approach proved to further mitigatethe structural response compared to the procedure suggested by Den Hartog.

17


Additionally, the structural system could be controlled by real-time adjustments ofthe stiffness and damping in the absorbers. Samani and Pellicano (2009) comparedthe efficacy of linear and nonlinear dynamic vibration absorbers by minimisingthe maximum deck deflection and maximising the energy absorbed by the TMDs.Nonlinear dampers had a greater influence on the amplitude, while linear dampersmore efficiently absorbed the energy of the system.

Moreover, semi-active vibration absorbers, with controllable properties, have beenstudied by several authors. Abe and Igusa (1996) derived analytical control algo-rithms for the optimization of semi-active vibration absorbers with time-varyingdamping. These devices showed to be superior compared to the conventional pas-sive absorbers. Variable damping absorbers were further studied by Pinkaew andFujino (2001), in the form of semi-active TMDs (STMDs) under harmonic loading.By means of the STMD, the improvement of the structural response correspondedto a four time increase of the PTMD mass. Aldemir (2003) replaced the viscousdamper in the TMD with a MRD, in order to reduce the response of a structure ex-cited by different seismic loading conditions. The results were consistent with theprevious studies regarding the potential of STMDs within structural engineeringapplications.

In contrast to prior studies, Lin et al. (2015) recently developed a STMD withresettable, controllable stiffness which did not include any inherent damping. Still,the damping of the system could be adjusted by altering the properties of theresettable element. Using this procedure, it was possible to avoid the detuning effectand, thus, ensure optimal control even during changes in the structural propertiesof the main system.

L/2

L/4 L/4

cdkd

md

cdkd

md

cdkd

md

F (t)

Figure 2.9: The concept of the multiple tuned mass damper system proposed byDas and Dey (1992). The mass, damping and stiffness of the dampers are given bymd, cd and kd, respectively. L is the span length and F (t) is the external force.

18

Chapter 3

Modelling aspects of the bridge

3.1 Introduction to the proposed models

In the present context, the neutral axis of the bridge is located close to the topsurface, resulting in an eccentricity from the supports which is important to considerin the models. During deflection of the bridge, this eccentricity may cause thebeam edges to rotate when the friction of the moveable bearing is overcome. As aconsequence, all points located along the bridge in the vicinity of the supports willreceive extra motional contribution in the horizontal direction. This means that theFVDs can be placed closer to the supports where the movement otherwise would betoo small for the dampers to operate. The proposed vibration mitigation strategyincludes a set of FVDs installed between the bridge deck and the abutments, asshown in Figure 3.1.

Fr(t)Fr±1(t)v

cdα

xd

yd

e

Figure 3.1: The proposed FVD system with the damper inclination α, the dampingcoefficient cd, the eccentricity e, the load from the rth train axle Fr(t) and horizontaland vertical coordinates describing the position of the FVD xd and yd, respectively.

To provide a more comprehensive view of the method, two different models aredeveloped, evaluated and described in this chapter. Firstly, a simplified SDOFmodel of the system is derived and explained in Section 3.2. Secondly, in Section3.3, a corresponding FE model is created. Thirdly, the two models are comparedand assessed in Section 3.4. Finally, relevant modelling details which may have animpact on the dynamic behaviour of the system are highlighted and interpreted inSection 3.5. Some parts in this chapter add nothing new to the field of research.Nevertheless, these are included to clarify the comprised procedures, while at thesame time keeping the extended summary self-contained from the appended papers.

19

CHAPTER 3. MODELLING ASPECTS OF THE BRIDGE

3.2 The single-degree-of-freedom system

3.2.1 Theoretical background

The majority of all structures can be idealised into an arbitrary number of individuallumped masses connected to stiff, damped and massless supporting systems. Thenumber of degrees-of-freedom (DOFs) of a structure is defined as all the independenttranslational and rotational movements that are required to describe the displacedpositions of all masses in relation to their origin (Chopra, 2012). Therefore, aSDOF model is an idealisation of a structure into a single lumped mass which isrestricted to movement in one direction only. As mentioned, the vertical bridge deckacceleration is generally the limiting factor in dynamic analyses. Thus, a SDOF inthe vertical direction is usually sufficient to describe the dynamical behaviour of asimply supported bridge, while rotational, horizontal and lateral displacements areassumed negligible. Figure 3.2 shows the idealisation of a simply supported bridgeinto a SDOF system.

The governing equation of motion of an Euler-Bernoulli beam could be described bya linearly elastic, viscously damped oscillator in terms of the displacement u, as wellas its time derivatives, velocity u and acceleration u. By means of modal analysis,the dynamic response can be expressed as a weighted sum of different naturalvibration modes that, in combination, are able to describe any possible deformedshape of the structure. This is obtained by separation of variables according to theexpansion theorem. The displacement and its time derivatives are, consequently,given in terms of the generalised coordinates qn, qn and qn of the nth mode ofvibration φn(x)

u(x, t) =∞∑

n=1φn(x)qn(t)

u(x, t) =∞∑

n=1φn(x)qn(t)

u(x, t) =∞∑

n=1φn(x)qn(t)

(3.1)

By multiplication with φn(x), the following expression for the equation of motionin modal coordinates is obtained

mb,nqn + cb,nqn + kb,nqn = Fn(t) (3.2)

20

3.2. THE SINGLE-DEGREE-OF-FREEDOM SYSTEM

mb,n

cb,nkb,n

Fn(t)

m,E, I, L=⇒

F (t)v

u

Figure 3.2: Idealisation of a simply supported bridge into a SDOF system. Thevariablesm, E, I and L are the mass per unit length, modulus of elasticity, momentof inertia and span length of the bridge. Further, kb,n, cb,n, mb,n and Fn(t) are themodal stiffness, damping, mass and force in the SDOF system.

where the modal mass mb,n, damping cb,n and stiffness kb,n are given as

mb,n =∫ L

0m(x)φn(x)2 dx

cb,n =∫ L

0c(x)φn(x)2 dx

kb,n =∫ L

0EI(x)φ′′

n(x)2 dx

(3.3)

In Equation (3.3), m is the mass per unit length, c is the structural damping, Eis the modulus of elasticity, I is the second area moment of inertia, L is the spanlength and x is a spatial coordinate along the bridge. The expression for the externalmodal force Fn(t) may vary with the source of excitation that the system is exposedto. For the present application, i.e. railway traffic, one approach is to model thetrain loads as a group of concentrated forces with a certain set of characteristicdistances in between. These distances are typically equidistant and appear withrepetitive intervals. On that account, the modal force could be expressed as thesum of R amplitude functions

Fn(t) =R∑

r=1Frεr(t)φn(vt− dr) (3.4)

where v is the train speed, Fr is the load from the rth axle and dr is the distancefrom the rth axle to the first point of the beam. Further, εr is defined as

εr(t) = H[t− dr/v]−H[t− (dr + L)/v] (3.5)

where H(t) is the Heaviside step function governed by

H(t) ={0, t < 01, t ≥ 0

(3.6)

21


3.2.2 Extension of the modelAlthough the applied SDOF model in this thesis is based on analogous approaches,some additions must be made to account for different phenomena. Firstly, theadditional damping from the FVDs must be properly included. Hence, by assumingthat the mode shapes are unaffected by the dampers (which is reasonable for lightlydamped systems such as the present), the damping coefficient from the FVDs cd,n

could be included in the modal equation of motion

mb,nqn + (cb,n + cd,n)qn + kb,nqn = Fn(t) (3.7)

The second phenomenon that must be included in the model is the eccentricity,which can be incorporated as a modal coordinate along the line of action of theFVDs. The derivation of this coordinate is based on the vertical component of thebridge displacement

uv,n = φn(x)qn (3.8)

and due to the eccentricity, the resulting horizontal component is described as

uh,n = eφ′n(x)qn (3.9)

The total length l of the FVDs at any time instant is the sum of the initial lengthl0 and the change in length Δl due to contraction or elongation of the dampers. Inthis model, it is assumed that the abutment is entirely fixed. Hence, with referenceto Figure 3.3, l is given by

l = l0 +Δl =√[xd + uh,n(xd) + uh,s,n]2 + [yd + uv,n(xd)]2 (3.10)

where uh,s,n is the horizontal displacement at the moveable bearing. According tosimple geometry, the following condition applies

l20 = x2d + y2

d (3.11)

and by omitting the remaining second order terms due to their negligible influence,Equation (3.10) results in

Δl = [xduh,n(xd) + xduh,s,n + yduv,n(xd)]l0

(3.12)

By once again applying the rules of geometry, the following variables are reformu-lated

l0 =xdcosα,

ydxd

= tanα = sinαcosα (3.13)

Consequently, the change in length of the FVDs may be expressed as

Δl = [uh,n(xd) cosα+ uh,s,n cosα+ uv,n(xd) sinα] (3.14)

22

3.2. THE SINGLE-DEGREE-OF-FREEDOM SYSTEM

cd

l0

α

xd

yd

e

cd

l0

α

xd

yd

e

cd

uh(xd) + uh,s

uv(xd)l0

Δl

Figure 3.3: The change in length Δl of the fluid viscous damper due to the deflectionof the bridge (with vertical uv(xd) and horizontal uh(xd) components) and theresulting displacement due to the rotation at the bearing uh,s. l0 is the initallength of the dampers, α is the inclination and xd and yd are initial coordinates ofthe damper connection point.

and, since Δl is equivalent to a displacement along the line of action of the dampers,it may be denoted ud,n. Inserting Equations (3.8) and (3.9) into (3.14) results in

ud,n = [eφ′n(xd) cosα+ eφ′

s,n cosα+ φn(xd) sinα]qn (3.15)

Differentiation with respect to time gives the velocity along the line of action ofthe dampers, which is required to obtain an estimate of the supplemental dampingfrom the FVDs

ud,n = [eφ′n(xd) cosα+ eφ′

s,n cosα+ φn(xd) sinα]qn (3.16)

which can be rewritten to emphasise that, due to the eccentricity, a new coordinateis derived which is entirely based on the original mode of vibration

ud,n = ψd,nqn (3.17)

and, for a number of J FVDs, the total supplemental damping coefficient for thenth mode of vibration is obtained as

cd,n =J∑

j=1cd,jψd,n(αj , xd,j)2 (3.18)

This is inserted into Equation (3.7), which means that the two essential phenomenaare included in the model. The resulting response in terms of u, u and u may beobtained from different solution methods (e.g. numerical time-stepping methods),followed by mode superposition according to Equation (3.1).

23


3.3 The finite element model

3.3.1 Introduction to finite elementsFE analysis is the numerical methodology used to determine the spatial distributionof a system by dividing it into an arbitrary, but finite, number of elements (Cooket al., 2002). The points connecting the elements are called nodes, which all containthe DOFs required to define the movement of that individual node. The formationof the elements are called the mesh and the smaller the mesh size, the betteris the resemblance between the responses of the simulated and actual structures.Thus, the size of the mesh is often a trade-off between the computational time thatcan be tolerated and the required resolution of the structural response. Figure 3.4shows the discretisation of the beam shown in Figure 3.1, without any supplementaldampers. The beam in Figure 3.4 may be described by the following equation ofmotion in matrix form

Mbu +Cbu +Kbu = F(t) (3.19)which includes the mass Mb damping Cb and stiffness Kb matrices of the beam,respectively. For the present application, a consistent local mass matrix mb anda local stiffness matrix kb considering both translational and bending forces areused. After assembling mb and kb into global matrices, the global damping matrixcan be obtained as a linear combination of the mass and stiffness using the methoddefined by Rayleigh

Cb = αMb + βKb (3.20)where α and β are the Rayleigh coefficients. For a train crossing the beam, the forcevector F(t) may be created through interpolation between the two nodes adjacentto the rth axle load as described by

Fj(t) =R∑

r=1Frεr(t)

[vt− dr − xj

xj+1 − xj

]

Fj+1(t) =R∑

r=1Frεr(t)

[1− vt− dr − xj

xj+1 − xj

] (3.21)

where xj is the distance to the jth vertical DOF, which is the one that the forcemost recently passed.

...

Fr(t)Fr±1(t)v

Figure 3.4: An example of a discretised beam, in which Fr(t) is the force from therth train axle and v is the train speed.

24

3.3. THE FINITE ELEMENT MODEL

3.3.2 Application to fluid viscous dampersWith the most basic knowledge of FE analysis described, the same phenomena asmentioned in Section 3.2.2 must be included in the FE model as well. As a first step,the FVDs are implemented into Equation (3.19) by means of a local supplementaldamping matrix cd given by

cd =[cd −cd−cd cd

](3.22)

which is transformed and assembled into the global matrix Cd. Hence, the equationof motion in matrix form for the entire bridge-FVD system may be expressed as

Mbu + (Cb +Cd)u +Kbu = F(t) (3.23)

Secondly, a correct representation of the behaviour arising due to the eccentricityis obtained with rigid links connecting the neutral axis of the bridge to the supportpoints and to the connection points of the dampers. A similar approach is foundin Ülker-Kaustell and Karoumi (2013), which was applied to obtain corrects con-straints at the beam ends. Instead of creating elements with high stiffness, a rigidlink could be modelled using transformation methods for constraints that connectsthe relations between the relevant DOFs. A detailed description of the underlyingtheory behind this procedure is found in Cook et al. (2002). To describe the theoryin brief, the two DOFs that are subject to the constraint equations are assignedas either a master or a slave. The slave DOF is restricted to follow the motionof the master according to a predetermined pattern. For this particular case, theadditional horizontal motion of the slave due to the rotational movement of themaster is relevant, as shown in Figure 3.5.

cd

e θi

θi+1

uh,j

uh,j+1

Figure 3.5: The relationship between the master and slave degrees-of-freedom, withthe rotation θ and horizontal displacement uh at master i and slave j.

With reference to Figure 3.5, the relationship between the horizontal and rotationaldisplacements are given as

uh,j = uh,i − e sin θi (3.24)

25


Assuming that the vertical displacement due to the rotation of the rigid link issmall and applying the small-angle approximation, the following is valid

uh,j ≈ uh,i − eθi (3.25)

Hence, a local transformation matrix T may be arranged, which contains the rela-tions between the master and slave DOFs

T =

⎡⎣1 0 −e0 1 00 0 1

⎤⎦ (3.26)

After assembling, the columns containing the slave DOFs are condensed from theglobal transformation matrix T. In order to obtain the new element matrices whichcontain the influence from the eccentricities, the original matrices are transformedaccording to

Mb = TTMbTKb = TTKbTCtot = TTCdT+ αMb + βKb

(3.27)

which are inserted into Equation (3.23) to obtain the final expression for the equa-tion of motion with rigid links and supplemental FVDs

Mbu +Ctotu +Kbu = F(t) (3.28)

The solution to this equation is found using numerical time stepping procedures,such as Newmark’s average acceleration method.

3.4 Comparison of the models

As a verification to ensure the accuracy of the results from the proposed SDOFsystem, a comparison between the two aforementioned models is carried out. Thesingle-track, ballasted Banafjäl Bridge shown in Figure 3.6, which is also the casestudy in Papers I-III, is used as a reference. Table 3.1 contains the relevant proper-ties of this steel-concrete composite girder bridge, which are obtained from guide-lines in CEN (2003) and construction drawings.

Table 3.1: Properties of the Banafjäl bridge.

E I L e m ξ

(GPa) (m4) (m) (m) (kg/m) (%)

200 0.61 42 2.20 18400 0.50

26

3.4. COMPARISON OF THE MODELS

Figure 3.6: The Banafjäl Bridge.

Four different models are tested in the comparative study - the FE model and threedifferent cases of the SDOF model. The content of the different tests are

- FE: The complete model with eccentricities at the connections to the supportsand dampers.

- SDOF Case I: The bridge only has eccentricities at the points connecting theFVDs to the bridge, as in Paper II.

- SDOF Case II: The bridge eccentricity is present both at the connection pointsand the supports, as in the model presented here and in Paper III.

- SDOF Case III: The conditions for the eccentricity are the same as in CaseII. However, the modal stiffness is updated based on the natural frequencyobtained from the FE model.

For Case III of the SDOF model, the natural frequency is retrieved from the FEmodel and used to alter the modal stiffness. The reason for this is that thereis a small difference in frequency between the models. Instead of calculating thestiffness using Equation (3.3), it is obtained as

kb,n = ω2nmb,n (3.29)

The change in frequency will also result in a slight alteration of the modal damping.

This supplemental study consists of comparisons of the maximum vertical bridgedeck acceleration and resulting damper force for crossings with all HSLM trains(CEN, 2003). All succeeding figures show the envelopes as clarified in Paper I,Figure 3. Further, the damping coefficient of the FVDs is fixed to 1.77 MNs/m,which is required to decrease the maximum acceleration level in the FE modelbelow the Eurocode requirement of 3.5 m/s2 (CEN, 2003) for all train passageswhen the FVDs are placed at xd = 1 m with inclination α = π/4 radians. Table3.2 summarises the results that are shown in Figures 3.7-3.9. The results are alsocompared to the FE model without any supplemental dampers to find the influenceof the FVDs on the bridge response.

27


Table 3.2: The maximum bridge deck acceleration and maximum required damperforce obtained from the different cases.

Model Acceleration Peak damper force,moveable bearing

Peak damper force,pinned bearing

(m/s2) (kN) (kN)

FE, no damper 6.35 - -

FE 3.49 131.93 24.23

SDOF Case I 4.32 92.91 93.72

SDOF Case II 3.64 132.35 25.08

SDOF Case III 3.57 134.31 26.74

Based on the results, it is clear that a correct implementation of the eccentricitiesis essential. Not only the eccentricities at the damper connections, but also theeccentricities at the supports, have a major impact on the vertical bridge deckacceleration (see Figure 3.7). The difference between the FE model and SDOFCase III occurs due to the treatment of the structural damping. In the SDOFmodel, the damping is modal and includes only the considered modes of vibration.However, Rayleigh damping may include higher-order modes as well, which is thecase for the FE model. Regarding the resulting damper force, the results for SDOFCase I differ significantly as compared to the other models. The reason behind thisis that the eccentricities at the supports are not included, which results in equallylarge movements at both ends of the bridge. It is more likely that the moveablebearing will entail a higher motion and, hence, an increased damper force in theFVDs placed closer to this point. This phenomenon is captured in the other models,resulting in a damper force in each individual FVD of equal magnitude.

Figure 3.7: The envelope of the maximum vertical bridge deck acceleration for alltrain speeds.

28

3.4. COMPARISON OF THE MODELS

Further, the peaks for the maximum required (resulting) damper force obtainedfrom the FE model and SDOF Case III in each of Figures 3.8 (at moveable bearing)and 3.9 (at pinned bearing) are of similar amplitudes. Yet, the difference seemsto increase with the train speed. The implementation of the structural dampingand the influence of higher-order modes may be the cause for this, since higherfrequencies results in an increased critical train speed. Higher-order modes couldbe removed from the FE model by employing a low-pass filter. In conclusion, whenmoving from Case I to Case III, the resemblance with the acceleration in the FEmodel increases for each step. Even though there is a slight difference betweenthe FE model and Case II or III, the simplified SDOF model overestimates theacceleration and damper forces. Therefore, it is guaranteed (for this particularbridge) that the SDOF model is conservative and applicable for the intended FVDapplication.

Figure 3.8: Maximum resulting damper force at the moveable bearing during thetrain passages over the bridge.

Figure 3.9: The resulting damper force at the pinned bearing for the train passages.

29


3.5 Modelling details

Since the models are verified and applicable to the intended bridge-FVD system,some modelling details that may have an influence on the dynamic response areinvestigated. Firstly, it must be assured that the stiffness in the dampers and theabutments do not have a major impact on the functionality of the FVDs. Secondly,as it is shown in Section 3.4 that the damper force is higher for the FVD placedclose to the moveable bearing, it is important to study the behaviour of the bearingfriction. In all of the following studies, the analyses are based on passages with allHSLM trains on the Banafjäl Bridge with supplemental FVDs having a dampingcoefficient of 1.77 MNs/m (xd = 1 m and α = π/4 radians), for the same reasonsas described in Section 3.4.

3.5.1 Stiffness in the damperIn contrast to the common assumption, the FVD is not exclusively composed of adashpot element. A FVD also has stiffness components, caused by the incompress-ibility of the damper liquid during expansion and compression, as well as due to therigidity of the mounts. The FVD may, hence, be included in the model by meansof a Maxwell element. Basically, the Maxwell element is a dashpot and spring inseries without any interconnecting inertia, as shown in Figure 3.10.

cdks

Figure 3.10: The concept of the Maxwell model, with spring stiffness ks and damp-ing coefficient cd.

Hence, the equation of motion of a SDOF system with a Maxwell element is givenby

mbu+ cbu+ kbu+ P (t) = F (t) (3.30)where P (t) is the restoring force of the Maxwell model, which can be derived ac-cording to the following procedure. The total stress within the element σ(t) isassumed to be of the same magnitude as the individual parts, i.e. the damper andthe spring

σ(t) = σs(t) = σd(t) (3.31)Further, the total strain of the element ε(t) is the sum of the individual strains ofthe spring and the damper

ε(t) = εs(t) + εd(t) (3.32)Applying Hooke’s law, the strain of the spring is expressed as a function of thespring stiffness ks

εs(t) = σ(t)/E, {ks = EA/L} =⇒ εs(t) =σ(t)AksL

(3.33)

30

3.5. MODELLING DETAILS

and since the restoring force in the damper Pd(t) is equal to the damper force, whichis proportional to the damping coefficient cd, the rate of change of the damper strainis calculated as

εd(t) = u/L, {Pd(t) = cdu, σ(t) = Pd(t)/A} =⇒ εd(t) =σ(t)AcdL

(3.34)

From Equations (3.33) and (3.34), the total strain rate and resulting velocity of theMaxwell element are

ε(t) = σ(t)AksL

+ σ(t)AcdL

, {ε(t) = u/L} =⇒ u = P (t)ks

+ P (t)cd

(3.35)

and by reformulating Equation (3.35), the final expression of the restoring force isobtained

P (t) + kscdP (t) = ksu (3.36)

The procedure to account for the Maxwell element and its restoring force is de-scribed by Hatada et al. (2000). According to their descriptions, by differentiatingEquation (3.36) and inserting the result into Equation (3.30), the equation of mo-tion of the SDOF system will be a third-order differential equation. All solutions tothis system will require extra DOFs, which will deviate from the simplified SDOFmodel presented in this thesis. Consequently, the FE model is applied in this sectionto account for the stiffness associated with the FVDs. To begin with, by assumingthat the change of the restoring force P (t) is linear within each time increment Δt,the following is valid

P (t+ 1) = P (t) + Δt2 [P (t+ 1) + P (t)] (3.37)

which is inserted into Equation (3.36)

P (t+ 1) + P (t) + kscd[P (t+ 1) + P (t)] = ks [u(t+ 1) + u(t)] (3.38)

From Equation (3.37), P (t+1) is extracted and the resulting expression is insertedinto Equation (3.38)

2Δt [P (t+ 1)− P (t)] + ks

cd[P (t+ 1) + P (t)] = ks [u(t+ 1) + u(t)] (3.39)

which can be rewritten as2cd + ksΔt

cdΔtP (t+ 1) + ksΔt− 2cd

cdΔtP (t) = ks [u(t+ 1) + u(t)] (3.40)

Solving for P (t+ 1) gives

P (t+ 1) = cdksΔt2cd + ksΔt

[u(t+ 1) + u(t)]− ksΔt− 2cd2cd + ksΔt

P (t) (3.41)

31


and by assigning new variables,

α1 =cdksΔt

2cd + ksΔt, α2 =

2cd − ksΔt2cd + ksΔt

(3.42)

the restoring force of the Maxwell element is

P (t+ 1) = α1 [u(t+ 1) + u(t)] + α2P (t) (3.43)

Since the Maxwell element is unable to sustain forces in any direction other thanthe axial, the local element matrices resemble that of a bar

α1 =[α1 −α1−α1 α1

], α2 =

[α2 −α2−α2 α2

](3.44)

The equation of motion in matrix form including the Maxwell element is denotedas

Mbu +Cbu +Kbu + P(t) = F(t) (3.45)

and from Equations (3.42) to (3.44), the restoring force vector of the Maxwellelement at time t+ 1 could be obtained

P(t+ 1) = A1 [u(t+ 1) + u(t)] +A2P(t) (3.46)

where A1 and A2 are the global matrices assembled from Equation (3.44). Thissystem of equations can be solved for by means of numerical time stepping methods.The described procedure is adopted to study the influence of the stiffness in theFVD on the damper efficiency. Accordingly, the spring stiffness in the Maxwellelement is increased, as seen in Table 3.3, and compared with the results obtainedwith a dashpot element.

Table 3.3: The maximum bridge deck acceleration and maximum required damperforce obtained with different spring stiffness in the Maxwell element.

Spring stiffness Acceleration Peak damper force,moveable bearing


(MN/m) (m/s2) (kN) (kN)

1 6.27 236.75 43.75

10 4.61 171.79 31.84

100 3.52 131.80 24.27

1000 3.49 131.50 24.15

Dashpot only 3.49 131.93 24.23

32


Figures 3.11-3.13 shows the maximum vertical bridge deck acceleration and requireddamper force, respectively, for an increasing spring stiffness. According to theresults, a spring stiffness exceeding 100 MN/m will scarcely add any reductivecontribution to the response. Nevertheless, if the design of the FVD is purelybased on a model not accounting for the stiffness in the spring, there is a riskof under-dimensioning the damper configuration. From Table 3.3, it is clear thata FVD with 100 MN/m exceeds the Eurocode limit when the design is optimisedwith only a dashpot element. A FVD type 80 (Dellner Dampers, 2016) has a springstiffness of 552 MN/m, depending on the input stroke length and cylinder diameter.This would result in a maximum acceleration of 3.49 m/s2. Hence, the spring in thisFVD is sufficiently rigid to reduce the response of the bridge below the regulatoryguidelines.

Figure 3.11: The envelope of the obtained acceleration response of the bridge tra-versed by the trains, with varying stiffness within the dampers.

Figure 3.12: The maximum required damper force at the moveable bearing whenthe trains cross the bridge and the stiffness of the damper is varied.

33


Figure 3.13: The resulting damper force at the pinned bearing for all train passagesover the bridge. The stiffness in the fluid viscous damper is varied.

Simultaneously, the resulting maximum damper forces are 131.48 kN and 24.15 kNfor the moveable and pinned bearings, respectively, as shown in Figures 3.12 and3.13. This means, in other words, that a spring in series with the dashpot elementis a slightly favourable factor since the FVD has to endure lower forces.

3.5.2 Flexibility of abutments

Another important aspect to consider is the flexibility of the abutments, whichcould have a reductive influence on the relative displacement between the ends ofthe FVD. Hence, the elongation/contraction of the FVD may not be achieved. Onthis basis, a 3D solid model of the abutment with surrounding soil is created usingthe commercial FE software Abaqus 6.13-3 (Dassault Systémes, 2013). By applyinghorizontally and vertically aligned static unit loads to the point where one damperconnects to the abutment, it is possible to extract the displacements along thesame directions. The stiffness of the abutment could, hence, be calculated, since itis commonly known that

k = F

u(3.47)

However, the material properties must be properly assigned to the different parts ofthe structure. By assuming a flexible concrete structure to describe the abutmentitself, the modulus of elasticity is set to 30 GPa. For the ulterior soil and backfill,Möller et al. (2000) provides estimated limits of the shear modulus based on thespeed of the shear waves. The lowest shear wave speed gives an approximation ofthe shear modulus of 40 MPa, which is a low value for cohesionless soil. Applyingthe relation between the shear modulus and the modulus of elasticity

E = 2G(1 + ν) (3.48)

34


and using a Poisson’s ratio of 0.15, the modulus of elasticity equals 92 MPa. Fromthe output displacements and Equation (3.47), the stiffness in the horizontal andvertical directions are in the order of magnitude of 1000 MN/m. These values areused as input to the 2D numerical FE model, in terms of vertical and horizontalsprings attached to each of the FVDs. Regarding the inertia separating the endsof the FVDs and the springs representing the abutments, a value that will notsignificantly influence the lower natural frequencies of the bridge is chosen.

To find an estimate of the critical spring stiffness, a quick parameter test is per-formed. The same FE model is used, but the horizontal and vertical stiffness arevaried as given by Table 3.4. As shown in the table, a spring stiffness close to1000 MN/m will not significantly affect the final results, proving that a flexibleabutment will still withstand the additional forces from the FVDs and at the sametime increase the structural damping in the system.

The results from this study are also visualised in Figure 3.14 for the vertical bridgedeck acceleration, as well as in Figures 3.15 and 3.16 for the forces in the FVDsat the moveable and pinned bearings, respectively. Noteworthy is the fact thatalthough a low spring stiffness is applied to represent the flexibility of the abutment,the damper forces increases as compared to a more rigid structure. Furthermore, theacceleration level is at the same time not appreciably reduced. Consequently, thisdemonstrates that the FVDs are activated, but inefficient, for flexible structures.

Table 3.4: The maximum bridge deck acceleration and maximum required damperforce with varying vertical and horizontal spring stiffness for the idealised abutment.

Spring stiffness Acceleration Peak damper force,moveable bearing


(MN/m) (m/s2) (kN) (kN)

1 6.33 239.51 44.31

10 5.81 217.12 40.17

100 3.64 135.42 24.97

1000 3.50 131.88 24.24

Dashpot only 3.49 131.93 24.23

35


Figure 3.14: The maximum vertical bridge deck acceleration during the train pas-sages when the stiffness of the abutment is altered.

Figure 3.15: The resulting damper force at the moveable bearing during the trainpassages with varying stiffness of the abutment.

Figure 3.16: The required damping force at the pinned bearing, when the bridge issubjected to the trains and the stiffness of the abutment is altered.

36


3.5.3 Friction of bearingsAs clearly shown in Figure 3.3, the change in length of the FVDs are strongly de-pendent on the displacements arising from the rotation at the supports. Hence, theinfluence of the friction that may constrain the moveable bearing on the efficiencyof the dampers must be properly assessed. A different background to this theoryis visualised in Figure 3.17, which shows the change in fundamental frequency ofthe bridge when an element with stiffness along its axial direction is added to themoveable bearing. Accordingly, if the friction in the moveable bearing behaveslike a spring, the fundamental frequency will move between two extreme states asthe stiffness in the bearings increase. The first state occurs when the moveablebearing is free to move, i.e. when the friction is overcome. This resembles the be-haviour of a simply supported bridge (with pinned-free supports). The second statearises when the bearing friction is not overcome, corresponding to a pinned-pinnedsupport condition. Because of the lack of previous research within this field, thebehaviour of the bearing in between these two extremes are not explicitly known.A plausible scenario is that the bearings are fixed in the translational directionsuntil the friction is overcome and when this occurs, they will be able to move fric-tionlessly. Hence, the present study is bound to investigations of the pinned-freeand pinned-pinned cases. The following results are obtained from the FE model,in which all translational DOFs at the supports are adjusted to be fixed for thepinned-pinned case.

Figure 3.17: The change in the fundamental frequency of the bridge due to arestraint of the moveable bearing.

37


With reference to Figure 3.17, it appears as the frequency due to an increase inspring stiffness (restraints of bearings) is initially unaffected, but increases rapidlyafter 108 N/m until it reaches a full pinned-pinned state. However, as previouslymentioned, the mechanisms behind the bearing friction are not generally known.Therefore, the results in the figure should only be interpreted as a visualisation ofthe concept for the transition from pinned-free to pinned-pinned support conditions.

Moreover, the maximum vertical bridge deck accelerations for the pinned-free andpinned-pinned cases are shown in Figure 3.18. Clearly, the increased fundamentalfrequency of the restrained beam shifts the acceleration peak outside the relevantrange of train speeds. Hence, supplemental FVDs may not be useful if the bearingfriction is not overcome. Although pinned bearings result in a decreased amplitudeat lower speeds, the frequency may change due to overcoming of the bearing fric-tion. This increases the risk of resonance within the considered train speed range.Consequently, the FVDs could be applied as a safety measure in case of a frequencychange due to the bearing mechanisms.

Another demonstration of this may be inferred from the damper forces at themoveable and pinned bearings shown in Figures 3.19 and 3.20. For the pinned-pinned case, the maximum required damper forces are close to zero within therelevant train speed interval. Hence, this suggests that the dampers are disabledfor the present inclinations and positions of the FVDs. As a consequence, the FVDsshould be installed closer to the mid-span. Alternatively, it must be verified that thebearing friction is overcome, which requires knowledge of the bearing mechanisms.Table 3.5 shows the maximum acceleration and peak damper forces for the pinned-free and pinned-pinned end conditions of the bridge, with and without supplementalFVDs.

Table 3.5: The maximum acceleration and required damper force for the bridgewith pinned-free and pinned-pinned end conditions.

Conditions Acceleration Peak damper force,moveable bearing


(-) (m/s2) (kN) (kN)

Pinned-free,no damper

6.35 - -

Pinned-free 3.49 131.93 24.23

Pinned-pinned,no damper

2.13 - -

Pinned-pinned 2.02 6.98 7.14

38


Figure 3.18: The acceleration of the pinned-free and pinned-pinned cases of thebridge, which is traversed by the different train models.

Figure 3.19: The maximum required damper force for the FVDs at the moveablebearing during the train passages over the pinned-free and pinned-pinned bridges.

Figure 3.20: The required damper force for the FVDs installed at the pinned bearingwhen the trains cross the pinned-free and pinned-pinned versions of the bridge.

39

Chapter 4

Reliability assessment

4.1 The concept of reliability assessments

During the modelling steps of different structures, it is commonly adopted to assignfixed values to the material properties, dimensions and loads etc. Consequently, theuncertainties which may arise due to e.g. temperature related changes or degrada-tion of materials, inaccurate assumptions or modelling errors, are rarely accountedfor. Nevertheless, these are partly included in the safety factors and limits given byCEN (2003). By conducting reliability analyses, these uncertainties are includedin the models and the probability that the structure fails to fulfil the required de-mands can be estimated. This means that it is possible to ensure the structuralintegrity to a higher degree, as well as revising demands that are exaggerated orunderestimated. In this chapter, the theoretical background to the statistical ap-proach in Paper III is provided. Further, the important key concepts are describedmore thoroughly. The procedures forming the Monte Carlo method is explained inSection 4.2, followed by the concept of probability of exceedance in Section 4.3.

4.2 Monte Carlo simulation

The Monte Carlo method is a simulation technique, which is used to obtain nu-merical results without having to perform any physical testing (Nowak and Collins,2012). By known information about the problem at hand, the important param-eters are assigned to different probability distributions. From the distributions, itis possible to randomly select variables to generate samples of the numerical datathat are implemented as input to the simulations. One sample corresponds to a setof random parameters and each sample is evaluated one at a time. Hence, the vari-ables are varied during the numerical experiments (so called stochastic), in contrastto deterministic parameters which are assigned a certain constant value.

In JCSS (2001), the probabilistic approach is divided into the load models (e.g.self weight and external loads) and the resistance models (e.g. capacities related tomaterial properties and dimensions). These values are often obtained from repeatedexperimental tests from which it is possible to determine the probability distributionfor each structural or loading property. For the current application, the loads areconsidered as deterministic, while the stochastic resistance parameters are limited tothe normal, lognormal and uniform distributions described in the following sections.

41

CHAPTER 4. RELIABILITY ASSESSMENT

4.2.1 Normal distributionA random variable X belongs to the normal (or Gaussian) distribution if it islikely to adopt the mean value μX with a certain standard deviation σX , which ismathematically described as X ∈ N(μX ,σX). The cumulative distribution function(CDF) for a normal random variable is given by (Rychlik and Rydén, 2006)

FX(x) =∫ x

−∞fX(x) dx (4.1)

where fX(x) is the probability density function (PDF) for a normal random variableexpressed as (Nowak and Collins, 2012)

fX(x) =1

σX

√2πe− 1

2

( x−μXσX

)2

(4.2)

For the special case with zero mean and a standard deviation equal to 1, Z ∈ N(0, 1),the random variable Z is standard normal. As a simplifying tool, the CDF ofthe standard normal variables are tabulated and can be employed to any normaldistribution. On account of this, the PDF of the standard normal variable z iscommonly denoted φ(z) and defined as

φ(z) = 1√2πe− z2

2 (4.3)

The most important property of a normal random variable is the symmetrical be-haviour of the PDF about the mean. Hence, the sum of two values, equidistantlylocated on opposite sides of the mean, will equal 1. This relation is commonly usedfor the CDFs given in standard normal tables, see e.g. Nowak and Collins (2012);Rychlik and Rydén (2006). To describe the concepts visually, with reference toFigure 4.1, the CDF shows the probability that the variable X will adopt any valuex up to a specific limit. The PDF is simply the first derivative of the CDF. Thenormal distribution is the most widely used within structural reliability (Nowakand Collins, 2012) and could also preferably be applied if the information regardingthe distribution of a random variable is limited.

4.2.2 Lognormal distributionThe random variable X is lognormally distributed if lnX ∈ N(μX ,σX) (Rychlikand Rydén, 2006), which, equivalently, means that (Nowak and Collins, 2012)

σln X =

√ln

(σ2

X

μ2X

+ 1)

μln X = lnμX − σ2ln X

2

(4.4)

42

4.2. MONTE CARLO SIMULATION

The CDF and PDF of the lognormal distributed random variables are shown inFigure 4.1. Lognormally distributed random variables are commonly used withinreliability analyses of structures. According to JCSS (2001), some material proper-ties of several structural members in a bridge could be approximated as lognormallydistributed. For example, the moduli of elasticity for concrete and structural steelfollow the lognormal distribution. Furthermore, the structural damping may alsobe assumed as a lognormal random variable (Gonzalez and Karoumi, 2014; Rochaet al., 2014, 2016).

4.2.3 Uniform distributionFor a uniform distribution, it is equally likely that the random variable may takeany value from 0 to 1 within a range from a to b. Hence, the PDF is given by

fX(x) =

⎧⎨⎩1

b− aa ≤ x ≤ b

0 otherwise(4.5)

The PDF and CDF are visualised in Figure 4.1. From this, it is clear that the meanand standard deviation are given as

μX = a+ b

2

σX = b− a√12

(4.6)

In accordance with previous studies by Rocha et al. (2014, 2016), the mass of theoverlaying ballast on the track may be assumed to follow a uniform distribution.

Figure 4.1: The cumulative distribution functions FX(x) (dashed lines) and theprobability density functions fX(x) (solid lines) for the a) normal, b) lognormaland c) uniform random variables.

43

CHAPTER 4. RELIABILITY ASSESSMENT

4.3 Probability of exceedance

The aim of structural reliability assessments are typically to determine the prob-ability that the structure will fail, given a number of samples that includes therandom variables (uncertainties). Generally, the measure of this is given by theprobability of failure Pf , which is the sum of all events when the impact exceedsthe capacity compared to the total number of samples. However, the probabilityof failure is the term used for the ultimate limit state (ULS), i.e. associated withstructural collapse or similar failure (CEN, 2002). For the serviceability limit state(SLS), the correct term is probability of exceedance Pe. The acceleration limit forballasted tracks is a serviceability limit, since the passenger comfort is the designthreshold.

One difficulty in performing statistical experiments is to determine the requirednumber of samples to obtain a correct representation of the given distribution func-tion. According to Au and Wang (2014), a reasonable estimate of the number ofsamples Ns to achieve a certain target value for the probability of exceedance isdetermined as

Ns >10Pe

(4.7)

If this condition is not fulfilled, the results will be uncertain. The limit for thepermitted probability of failure is associated with the established confidence levels(e.g. 99%, 98% or 95%). 95% is commonly adopted for characteristic materialproperties and hypothesis testing, which corresponds to a limit for Pe of 5%. Forthis reason, it is considered as an appropriate limit for the analyses presentedin Paper III. In this research, the simulations are statistically dependent in thesense that the random variables are not regenerated for each HSLM train passagei. Hence, the estimated probability of exceedance is given by the probability ofthe union of all events where the limit state function g(x, v) is exceeded. This isdescribed as

Pe =⋃

i

Pe,i, Pe,i =1Ns

Ns∑j=1

(I[g(xj, v)] ∈ [0, 1]) (4.8)

where the indicator function I[g(xj, v)] is 1 if the limit state function defined by

g(x, v) = alim − a(x, v)[1 + C] (4.9)

is negative and 0 otherwise. In the limit state function, alim is the accelerationlimit, a(x, v) is the acceleration of sample x occurring at the train speed v and Cis the model uncertainty accounting for inaccurate assumptions in the models etc.Once the probability of exceedance is estimated, it is possible to verify if the FVDconfiguration is sufficient for the proposed application or if the damper parametersmust be adjusted to meet with the regulations. Further, the design speed, for upto which the acceleration limit is not exceeded, can also be determined.

44

Chapter 5

Concluding remarks

5.1 Discussion

The studies in the previous chapters have provided additional insights as comparedto the appended papers. Starting from the papers, the main novelty is the derivedSDOF model in Paper II and III containing the influence of the eccentricity on theresponse of the bridge-damper system. The key findings from the derived modelare 1) that the FVDs could be placed closer to the supports, due to the movementof the bridge bearing and rotation of the bridge cross-section during deflection and2) that the vertical bridge deck acceleration could be substantially reduced usingthe proposed retrofit method. Moreover, an optimal damper position, in termsof inclination and distance between the fluid viscous damper and the support, ispossible to obtain for a specific bridge in accordance with Papers I and II. Movingon to the supplemental information in the present thesis, the various studies arefurther discussed below.

5.1.1 Comparison of the models

Comparing the results from the different cases in Section 3.4, the applicability of theproposed retrofit method is justified and the accuracy of the two applied modelsis verified. As the results deviate slightly, it is important to bear in mind thatthe input variables to the models are introduced according to different theories(single-degree-of-freedom systems and finite element modelling). This will mainlyaffect the stiffness and damping of the bridge, which is an effect arising from 1)the introduction of the eccentricity and 2) higher-order modes that are not equallyincluded in the models. The difference in stiffness will affect the critical trainspeed, which means that a margin of safety must be included if the accelerationlevel exceeds given requirements. However, this is an established approach that isalready included in regulatory documents.

The eccentricity arising due to the fact that the supports do not connect directly tothe neutral axis of the bridge has a major impact on the efficiency of FVDs installedin the proposed manner. This is especially true for bridges with a large eccentricityor for composite bridges for which the centre of gravity is shifted due to notabledifferences between the properties of the constituent materials. The deflection ofthe bridge deck in combination with the movement of the bearings result in anadditional motion along the line of action of the FVDs.

45

CHAPTER 5. CONCLUDING REMARKS

From findings in Papers I and II, the rotation of the bridge cross-section seemsto increase when approaching the supports. As a consequence, the FVDs could beplaced closer to the supports with a smaller inclination towards the horizontal planeand still perform with very good results. It is, though, important to implement theeccentricities properly in the models, both at the supports and the connectionpoints between the bridge deck and the FVDs, as highlighted in Figure 3.7. Ifthe eccentricities at the supports are neglected, the maximum vertical bridge deckacceleration is overestimated. However, this is a conservative approach for the deckacceleration that will not violate the safety of the system. On the other hand, therequired damper force at the moveable bearing will be underestimated, leading toan under-dimensioned FVD which may be a safety issue. This will also result in anoverestimation of the required damping at the pinned bearing.

5.1.2 Stiffness in the damper

For a FE model, the stiffness resulting from the incompressibility of the damperliquid and the rigidity of the mounts could be implemented in terms of a spring inseries with a dashpot element. A Maxwell model is an example of an element whichcontains these two parts, without any interconnecting mass. From the analysis ofthe bridge model equipped with Maxwell elements, as described in Section 3.5.1, aspring with stiffness of 100 MN/m will have an impact on the results with less than1%. It is found that the stiffness provided from a FVD of type 80 (see Paper II forfurther details) will not affect the vertical bridge deck acceleration, as compared toa purely viscous dashpot. Considering the damper forces, the spring will cause adecrease in the velocity along the line of action of the FVD and, thus, reduce thedamper force slightly. Although it is demonstrated that the effect from the stiffnessin the FVD will not be an issue, it is still important to consider in the models ifthe results are close to the permitted level or if another FVD type is preferred.

5.1.3 Flexibility of abutments

Assuming that the abutment could be represented by springs in the horizontal andvertical planes, the flexibility affecting the relative displacement between the endsof the FVDs could be investigated. According to the numerical simulations, a springstiffness of 1000 MN/m is sufficient to maintain the efficiency of the FVDs. Thisvalue is also obtained from a check of the spring stiffness in a 3D solid FE modelcreated in Abaqus. Hence, it is confirmed that the flexibility of the abutmentswill not influence the damper capacity. However, for more flexible structures, thedamper forces increase while the acceleration level is not significantly affected. Forsuch cases in which the restraining structure is not sufficiently rigid, the proposedvibration mitigation method with FVDs is not recommended. This is due to thefact that the damper is degraded but not activated.

46

5.2. CONCLUSIONS

5.1.4 Friction of bearingsDue to the fact that the behaviour of the bearings during a train passage is notwell established, the two extreme states of the beam are studied. These are thesimply supported case, in which the moveable bearing is free to rotate and moveparallel to the surface, and the pinned-pinned case for which all bearings are fixed inthe translational directions. For the same bridge (Banafjäl) the change in the firstbending frequency between these two states is close to 50%. This clearly affects theresults and the applicability of the proposed vibration mitigation strategy. To beginwith, the FVDs must be placed further from the supports, in order to be efficientwhen installed to a pinned-pinned beam. Since the angular change at the supportsis smaller, the deflection of the beam in the vicinity of these points will be negligible.Hence, the necessary motion in the FVDs will not be attained. Furthermore, thedrastic change in frequency due to the frictional locking/unlocking of the bearingsis significant and shifts the critical train speed outside the design range of speeds.Due to this, there is no need for vibration mitigation and supplemental FVDs are,thus, not required. Consequently, this behaviour is found to be the most criticalaspect to consider for this application of FVDs.

5.2 Conclusions

The main conclusions from the appended papers, as well as from the supplemen-tary studies included in the present thesis, are summarised in this section. Drawninferences are based on the reasoning in Section 5.1 and each of Papers I-III.

5.2.1 Inferences based on appended papers- As a proof of concept, a vibration mitigation strategy is proposed which canbe applied to high-speed railway bridges.

- By installing fluid viscous dampers between the abutments and the bridgedeck, the acceleration level of bridges may be significantly reduced.

- In terms of scientific contributions, analytical and numerical tools are createdfor analyses of the efficiency of the fluid viscous dampers.

- The influence of the eccentricities on the efficiency of the fluid viscous dampersis important to consider, especially for bridges with a significant distance fromthe supports to the deck neutral axis.

- Regarding the positioning of the fluid viscous dampers, an increasing inclina-tion and decreasing distance from the supports result in an increased requireddamping coefficient to reduce the deck acceleration below the requirements.

- For a specific bridge, it is possible to obtain design curves that enable thedesign of the fluid viscous damper retrofit.

- The changes in bending stiffness, structural damping, model uncertainty andacceleration limit affects the probability of exceedance and should, hence, betreated as stochastic variables.

47

CHAPTER 5. CONCLUDING REMARKS

- From a stochastic point of view, supplemental fluid viscous dampers maybe supernumerary in order to reduce the vertical bridge deck acceleration toan acceptable level. Hence, the variations in the obtained probabilities of ex-ceedance emphasise the need for a probabilistic assessment of the accelerationlimit and model uncertainties.

5.2.2 Implications from supplementary findings- It is possible to simplify the system into a single-degree-of-freedom modelproviding the same degree of accuracy as a full-scale finite element model.

- The implementation of the eccentricities in the model must be properly as-sessed, in order to avoid safety problems such as under-dimensioning of thecapacity of the fluid viscous dampers.

- Above a certain level, the stiffness within the fluid viscous dampers has nomajor impact on the results. If using a Type 80 fluid viscous damper (pro-duced by Dellner Dampers (2016)), the resulting maximum vertical bridgedeck acceleration will be virtually the same as for a purely viscous damper.

- According to the performed simulations, the flexibility of the abutment willnot significantly influence the damper efficiency. However, if the stiffness isunreasonably low, the dampers will be activated, but inefficient.

- If the friction of the bearings is not overcome, the bridge will exhibit a re-strained behaviour. This will result in inactivation of fluid viscous dampersplaced close to the supports, since the angular change is counteracted. How-ever, the beam with pinned-pinned support conditions will have a funda-mental frequency greater than that of the pinned-free case. Consequently,the critical train speed will be shifted towards higher velocities, maybe evenoutside the relevant speed range.

5.3 Further research

During the progress of the project, several different research directions within thesubject have been identified. To this point, only idealised models have been used toinvestigate the effect of the FVDs on the dynamic response of the bridge. Hence, itis necessary to validate the results from previous studies with field measurementsbefore this application could be realised. The aim of the author is, thus, to

1. conduct a hybrid simulation test to ensure that the bridge displacement atthe connection between the bridge and damper is large enough to activate thestroke motion of the FVDs. The main advantage with this test is the real-timeinterchange between the bridge displacement and the resulting force in thedamper. In other words, the response of the bridge is numerically calculatedand the resulting displacements are then applied to a test specimen (Carrionand Spencer, 2007), which for this case is an actual damper.

48

5.3. FURTHER RESEARCH

2. perform a full-scale test with a vibration exciter acting on an existing bridgewith post-installed FVDs and compare the results with a 3D model of thesystem. Meanwhile, this enables an additional, connected study that includesmodel updating to investigate the influence of certain modelling details onthe damper efficiency, e.g. the behaviour of the moveable bearings, stiffnessof the abutments and soil-structure interaction (SSI).

3. develop optimization and design procedures for the proposed retrofit method.Also, a generalisation of the models to include a wider range of bridges maybe necessary to demonstrate the applicability of FVDs in this context.

4. carry out a comparative study between different types of dampers, to highlightthe benefits and disadvantages with the proposed procedure compared to e.g.MRDs, TMDs and nonlinear FVDs.

49

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87

List of Bulletins from the Department of Structural Engineering,The Royal Institute of Technology, Stockholm

TRITA-BKN. Bulletin

1. Pacoste, C., On the Application of Catastrophe Theory to Stability Analysesof Elastic Structures. Doctoral Thesis, 1993.

2. Stenmark, A-K., Dämpning av 13 m lång stålbalk – ”Ullevibalken”. Utprov-ning av dämpmassor och fastsättning av motbalk samt experimentell bestäm-ning av modformer och förlustfaktorer. Vibration tests of full-scale steel girderto determine optimum passive control. Licentiate thesis, 1993.

3. Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifier-ade avjämningsmassor. 1993.

4. Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors andGeometry – Tests and Theory. Doctoral Thesis, 1993.

5. Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licen-tiate Thesis, 1993.

6. Krus, J., Hållfasthet hos frostnedbruten betong. Licentiate thesis, 1993.

7. Wiberg, U., Material Characterization and Defect Detection by QuantitativeUltrasonics. Doctoral Thesis, 1993.

8. Lidström, T., Finite Element Modelling Supported by Object Oriented Meth-ods. Licentiate Thesis, 1993.

9. Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Con-crete Beams without Stirrups. Licentiate Thesis, 1994.

10. Krus, J., Betongbalkars lastkapacitet efter miljöbelastning. 1994.

11. Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Struc-tures. Licentiate Thesis, 1994.

12. Sanne, L., Information Transfer Analysis and Modelling of the StructuralSteel Construction Process. Licentiate Thesis, 1994.

13. Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin-Walled Columns.Doctoral Thesis, 1994.

14. Kjörling, M., Dynamic response of railway track components. Measurementsduring train passage and dynamic laboratory loading. Licentiate Thesis, 1995.

15. Yang, L., On Analysis Methods for Reinforced Concrete Structures. DoctoralThesis, 1995.

16. Petersson, Ö., Svensk metod för dimensionering av betongvägar. Licentiatethesis, 1996.

17. Lidström, T., Computational Methods for Finite Element Instability Analy-ses. Doctoral Thesis, 1996. Bulletin 17.

18. Krus, J., Environment- and Function-induced Degradation of Concrete Struc-tures. Doctoral Thesis, 1996.

19. Silfwerbrand, J., (editor), Structural Loadings in the 21st Century. SvenSahlin Workshop, June 1996. Proceedings.

20. Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of FrameType Structures. Licentiate Thesis, 1996.

21. Troive, S., Optimering av åtgärder för ökad livslängd hos infrastruktur-konstruktioner. Licentiate thesis, 1996.

22. Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Mov-ing Vehicles. Licentiate Thesis, 1996.

23. Hallgren, M., Punching Shear Capacity of Reinforced High Strength ConcreteSlabs. Doctoral Thesis, 1996.

24. Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Alu-minium Extrusions. Licentiate Thesis, 1996.

25. Yagi, T., Wind-induced Instabilities of Structures. Doctoral Thesis, 1997.

26. Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Ex-treme Events – Proceedings from a Symposium, May 1997.

27. Paulsson, J., Effects of Repairs on the Remaining Life of Concrete BridgeDecks. Licentiate Thesis, 1997.

28. Olsson, A., Object-oriented finite element algorithms. Licentiate Thesis, 1997.

29. Yunhua, L., On Shear Locking in Finite Elements. Licentiate Thesis, 1997.

30. Ekman, M., Sprickor i betongkonstruktioner och dess inverkan på beständig-heten. Licentiate Thesis, 1997.

31. Karawajczyk, E., Finite Element Approach to the Mechanics of Track-DeckSystems. Licentiate Thesis, 1997.

32. Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams.Licentiate Thesis, 1997.

33. Edlund, S., Arbitrary Thin-Walled Cross Sections. Theory and ComputerImplementation. Licentiate Thesis, 1997.

34. Forsell, K., Dynamic analyses of static instability phenomena. LicentiateThesis, 1997.

35. Ikäheimonen, J., Construction Loads on Shores and Stability of HorizontalFormworks. Doctoral Thesis, 1997.

36. Racutanu, G., Konstbyggnaders reella livslängd. Licentiate thesis, 1997.

37. Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stöd för byggprocessen. Licentiate thesis, 1997.

38. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete BeamsSubjected to Combined Mechanical and Thermal Loading. Licentiate Thesis,1998.

39. Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: NordicSeminar on Computational Mechanics, October 1998.

40. Luo, Y., On some Finite Element Formulations in Structural Mechanics. Doc-toral Thesis, 1998.

41. Troive, S., Structural LCC Design of Concrete Bridges. Doctoral Thesis,1998.

42. Tärno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hypar-form Suspended Roofs. Licentiate Thesis, 1998.

43. Hassanzadeh, G., Betongplattor på pelare. Förstärkningsmetoder och dimen-sioneringsmetoder för plattor med icke vidhäftande spännarmering. Licenti-ate thesis, 1998.

44. Karoumi, R., Response of Cable-Stayed and Suspension Bridges to MovingVehicles. Analysis methods and practical modeling techniques. DoctoralThesis, 1998.

45. Johnson, R., Progression of the Dynamic Properties of Large SuspensionBridges during Construction – A Case Study of the Höga Kusten Bridge.Licentiate Thesis, 1999.

46. Tibert, G., Numerical Analyses of Cable Roof Structures. Licentiate Thesis,1999.

47. Ahlenius, E., Explosionslaster och infrastrukturkonstruktioner – Risker,värderingar och kostnader. Licentiate thesis, 1999.

48. Battini, J.-M., Plastic instability of plane frames using a co-rotational ap-proach. Licentiate Thesis, 1999.

49. Ay, L., Using Steel Fiber Reinforced High Performance Concrete in the In-dustrialization of Bridge Structures. Licentiate Thesis, 1999.

50. Paulsson-Tralla, J., Service Life of Repaired Concrete Bridge Decks. DoctoralThesis, 1999.

51. Billberg, P., Some rheology aspects on fine mortar part of concrete. LicentiateThesis, 1999.

52. Ansell, A., Dynamically Loaded Rock Reinforcement. Doctoral Thesis, 1999.

53. Forsell, K., Instability analyses of structures under dynamic loads. DoctoralThesis, 2000.

54. Edlund, S., Buckling of T-Section Beam-Columns in Aluminium with or with-out Transverse Welds. Doctoral Thesis, 2000.

55. Löfsjögård, M., Functional Properties of Concrete Roads – General Interre-lationships and Studies on Pavement Brightness and Sawcutting Times forJoints. Licentiate Thesis, 2000.

56. Nilsson, U., Load bearing capacity of steel fibre reinforced shotcrete linings.Licentiate Thesis, 2000.

57. Silfwerbrand, J. and Hassanzadeh, G., (editors), International Workshop onPunching Shear Capacity of RC Slabs – Proceedings. Dedicated to ProfessorSven Kinnunen. Stockholm June 7-9, 2000.

58. Wiberg, A., Strengthening and repair of structural concrete with advanced,cementitious composites. Licentiate Thesis, 2000.

59. Racutanu, G., The Real Service Life of Swedish Road Bridges – A case study.Doctoral Thesis, 2000.

60. Alavizadeh-Farhang, A., Concrete Structures Subjected to Combined Me-chanical and Thermal Loading. Doctoral Thesis, 2000.

61. Wäppling, M., Behaviour of Concrete Block Pavements – Field Tests andSurveys. Licentiate Thesis, 2000.

62. Getachew, A., Trafiklaster på broar. Analys av insamlade och Monte Carlogenererade fordonsdata. Licentiate thesis, 2000.

63. James, G., Raising Allowable Axle Loads on Railway Bridges using Simulationand Field Data. Licentiate Thesis, 2001.

64. Karawajczyk, E., Finite Elements Simulations of Integral Bridge Behaviour.Doctoral Thesis, 2001.

65. Thöyrä, T., Strength of Slotted Steel Studs. Licentiate Thesis, 2001.

66. Tranvik, P., Dynamic Behaviour under Wind Loading of a 90m Steel Chim-ney. Licentiate Thesis, 2001.

67. Ullman, R., Buckling of Aluminium Girders with Corrugated Webs. Licenti-ate Thesis, 2002.

68. Getachew, A., Traffic Load Effects on Bridges. Statistical Analysis of Col-lected and Monte Carlo Simulated Vehicle Data. Doctoral Thesis, 2003.

69. Quilligan, M., Bridge Weigh-in-Motion. Development of a 2-D Multi-VehicleAlgorithm. Licentiate Thesis, 2003.

70. James, G., Analysis of Traffic Load Effects on Railway Bridges. DoctoralThesis, 2003.

71. Nilsson, U., Structural behaviour of fibre reinforced sprayed concrete anchoredin rock. Doctoral Thesis, 2003.

72. Wiberg, A., Strengthening of Concrete Beams Using Cementitious CarbonFibre Composites. Doctoral Thesis, 2003.

73. Löfsjögård, M., Functional Properties of Concrete Roads – Development ofan Optimisation Model and Studies on Road Lighting Design and Joint Per-formance. Doctoral Thesis, 2003.

74. Bayoglu-Flener, E., Soil-Structure Interaction for Integral Bridges and Cul-verts. Licentiate Thesis, 2004.

75. Lutfi, A., Steel Fibrous Cement Based Composites. Part one: Material andmechanical properties. Part two: Behaviour in the anchorage zones of pre-stressed bridges. Doctoral Thesis, 2004.

76. Johansson, U., Fatigue Tests and Analysis of Reinforced Concrete BridgeDeck Models. Licentiate Thesis, 2004.

77. Roth, T., Langzeitverhalten von Spannstählen in Betonkonstruktionen. Li-centitate Thesis, 2004.

78. Hedebratt, J., Integrerad projektering och produktion av industrigolv – Metoderför att förbättra kvaliteten. Licentiate thesis, 2004.

79. Österberg, E., Revealing of age-related deterioration of prestressed reinforcedconcrete containments in nuclear power plants – Requirements and NDTmethods. Licentiate Thesis, 2004.

80. Broms, C.E., Concrete flat slabs and footings. New design method for punch-ing and detailing for ductility. Doctoral Thesis, 2005.

81. Wiberg, J., Bridge Monitoring to Allow for Reliable Dynamic FE Modelling– A Case Study of the New Årsta Railway Bridge. Licentiate Thesis, 2006.

82. Mattsson, H.-Å., Funktionsentreprenad Brounderhåll – En pilotstudie i Up-psala län. Licentiate Thesis, 2006.

83. Masanja, D. P, Foam concrete as a structural material. Doctoral Thesis, 2006.

84. Johansson, A., Impregnation of Concrete Structures – Transportation andFixation of Moisture in Water Repellent Treated Concrete. Licentiate Thesis,2006.

85. Billberg, P., Form Pressure Generated by Self-Compacting Concrete – In-fluence of Thixotropy and Structural Behaviour at Rest. Doctoral Thesis,2006.

86. Enckell, M., Structural Health Monitoring using Modern Sensor Technology– Long-term Monitoring of the New Årsta Railway Bridge. Licentiate Thesis,2006.

87. Söderqvist, J., Design of Concrete Pavements – Design Criteria for Plain andLean Concrete. Licentiate Thesis, 2006.

88. Malm, R., Shear cracks in concrete structures subjected to in-plane stresses.Licentiate Thesis, 2006.

89. Skoglund, P., Chloride Transport and Reinforcement Corrosion in the Vicinityof the Transition Zone between Substrate and Repair Concrete. LicentiateThesis, 2006.

90. Liljencrantz, A., Monitoring railway traffic loads using Bridge Weight-in-Motion. Licentiate Thesis, 2007.

91. Stenbeck, T., Promoting Innovation in Transportation Infrastructure Mainte-nance – Incentives, Contracting and Performance-Based Specifications. Doc-toral Thesis, 2007.

92. Magnusson, J., Structural Concrete Elements Subjected to Air Blast Loading.Licentiate Thesis, 2007.

93. Pettersson, L., G., Full Scale Tests and Structural Evaluation of Soil SteelFlexible Culverts with low Height of Cover. Doctoral Thesis, 2007.

94. Westerberg, B., Time-dependent effects in the analysis and design of slenderconcrete compression members. Doctoral Thesis, 2008.

95. Mattsson, H.-Å., Integrated Bridge Maintenance. Evaluation of a pilot projectand future perspectives. Doctoral Thesis, 2008.

96. Andersson, A., Utmattningsanalys av järnvägsbroar. En fallstudie av stål-broarna mellan Stockholm Central och Söder Mälarstrand, baserat på teo-retiska analyser och töjningsmätningar. Licentiate thesis, 2009.

97. Malm, R., Predicting shear type crack initiation and growth in concrete withnon-linear finite element method. Doctoral Thesis, 2009.

98. Bayoglu Flener, E., Static and dynamic behaviour of soil-steel compositebridges obtained by field testing. Doctoral Thesis, 2009.

99. Gram, A., Numerical Modelling of Self-Compacting Concrete Flow: Discreteand Continuous Approach. Licentiate Thesis, 2009.

100. Wiberg, J., Railway bridge response to passing trains. Measurements and FEmodel updating. Doctoral Thesis, 2009.

101. Ngoma, A., Characterization and consolidation of historical lime mortars incultural heritage buildings and associated structures in east Africa. DoctoralThesis, 2009.

102. Ülker-Kaustell, M., Some aspects of the dynamic soil–structure interaction ofa portal frame railway bridge. Licentiate Thesis, 2009.

103. Vogt, C., Ultrafine particles in concrete: Influence of ultrafine particles onconcrete properties and application to concrete mix design. Doctoral Thesis,2010.

104. Selander, A., Hydrophobic Impregnation of Concrete Structures: Effects onConcrete Properties. Doctoral Thesis, 2010.

105. Ilina, E., Understanding the application of knowledge management to safetycritical facilities. Doctoral Thesis, 2010.

106. Leander, J., Improving a bridge fatigue life prediction by monitoring. Licen-tiate Thesis, 2010.

107. Andersson, A., Capacity assessment of arch bridges with backfill - Case of theold Årsta railway bridge. Doctoral Thesis, 2011.

108. Enckell, M., Lessons Learned in Structural Health Monitoring of Bridges Us-ing Advanced Sensor Technology. Doctoral Thesis, 2011.

109. Hansson, H., Warhead penetration in concrete protective structures. Licenti-ate Thesis, 2011.

110. Gonzalez Silva, I., Study and Application of Modern Bridge Monitoring Tech-niques. Licentiate Thesis, 2011.

111. Safi, M., LCC Applications for Bridges and Integration with BMS. LicentiateThesis, 2012.

112. Du, G., Towards sustainable construction, life cycle assessment of railwaybridges. Licentiate Thesis, 2012.

113. Hedebratt, J., Industrial Fibre Concrete Slabs - Experiences and Tests onPile-Supported Slab. Doctoral Thesis, 2012.

114. Ahmed, L., Models for analysis of shotcrete on rock exposed to blasting.Licentiate Thesis, 2012.

115. Le, T., N., Corotational formulation for nonlinear dynamic analysis of flexiblebeam structures. Licentiate Thesis, 2012.

116. Johansson, C., Simplified dynamic analysis of railway bridges under high-speed trains. Licentiate Thesis, 2013.

117. Jansson, R., Fire Spalling of Concrete - Theoretical and Experimental Studies.Doctoral Thesis, 2013.

118. Leander, J., Refining the fatigue assessment procedure of existing steel bridges.Doctoral Thesis, 2013.

119. Le, T., N., Nonlinear Dynamics of Flexible Structures using CorotationalBeam Elements. Doctoral Thesis, 2013.

120. Ülker-Kaustell, M., Essential Modelling Details in Dynamic FE-analyses ofRailway Bridges. Doctoral Thesis, 2013.

121. Safi, M., Life-Cycle Costing, Applications and Implementations in BridgeInvestment and Management. Doctoral Thesis, 2013.

122. Arvidsson, T., Train-Bridge Interaction - Literature Review and ParameterScreening. Licentiate Thesis, 2014.

123. Rydell, C., Seismic high-frequency content loads on structures and compo-nents within nuclear facilities. Licentiate Thesis, 2014.

124. Bryne, L., E., Time dependent material properties of shotcrete for hard rocktunnelling. Doctoral Thesis, 2014.

125. Wennström, J., Life Cycle Costing in Road Planning and Management - ACase Study on Collision-free Roads. Licentiate Thesis, 2014.

126. Gonzalez Silva, I., Application of monitoring to dynamic characterization anddamage detection in bridges. Doctoral Thesis, 2014.

127. Sangiorgio, F., Safety Format for Non-linear Analysis of RC Structures Sub-jected to Multiple Failure Modes. Doctoral Thesis, 2015.

128. Gram, A., Modelling of Bingham Suspensional Flow - Influence of Viscos-ity and Particle Properties Applicable to Cementitious Materials. DoctoralThesis, 2015.

129. Du, G., Life cycle assessment of bridges, model development and case studies.Doctoral Thesis, 2015.

130. Zhu, J., Towards a Viscoelastic Model for Phase Separation in Polymer Mod-ified Bitumen. Licentiate Thesis, 2015.

131. Chen, F., The Future of Smart Road Infrastructure - A Case Study for theeRoad. Licentiate Thesis, 2015.

132. Ahmed, L., Models for analysis of young cast and sprayed concrete subjectedto impact-type loads. Doctoral Thesis, 2015.

133. Albrektsson, J., Durability of fire exposed concrete - Experimental StudiesFocusing on Stiffness & Transport Properties. Licentiate Thesis, 2015.

134. Wallin, J., Systematic planning and execution of finite element model updat-ing. Licentiate Thesis, 2015.

135. Wadi, A., Flexible culverts in sloping terrain - Research advances and appli-cation. Licentiate Thesis, 2015.

136. McCarthy, R., Self-compacting concrete for improved construction technol-ogy. Licentiate Thesis, 2015.

137. Veganzones Muñoz, J., J., Bridge Edge Beams - LCCA and Structural Anal-ysis for the Evaluation of New Concepts. Licentiate Thesis, 2016.

138. Abbasiverki, R., Analysis of underground concrete pipelines subjected to seis-mic high-frequency loads. Licentiate Thesis, 2016.

139. Gasch, T., Concrete as a multi-physical material with applications to hydropower facilities. Licentiate Thesis, 2016.

140. Mohammadi Mohaghegh, A., Use of Macro Basalt Fibre Concrete for MarineApplications. Licentiate Thesis, 2016.

141. Döse, M., Ionizing Radiation in Concrete and Concrete Buildings – EmpiricalAssessment. Licentiate Thesis, 2016.

142. Khan, A., Fundamental investigation to improve the quality of cold mix as-phalt. Licentiate Thesis, 2016.

143. Zhu, J., Storage Stability and Phase Separation Behaviour of Polymer-ModifiedBitumen. Doctoral Thesis, 2016.

144. Chen, F., Sustainable Implementation of Electrified Roads. Doctoral Thesis,2016.

145. Svedholm, C., Efficient modelling techniques for vibration analyses of railwaybridges. Doctoral Thesis, 2017.

146. Solat Yavari, M., Slab Frame Bridges, Structural Optimization ConsideringInvestment Cost and Environmental Impacts. Licentiate Thesis, 2017.

The bulletins enumerated above, with the exception for those which are out of print,may be purchased from the Department of Civil and Architectural Engineering,Royal Institute of Technology, SE-100 44 Stockholm, Sweden.

The department also publishes other series. For full information see our home pagehttp://www.byv.kth.se

SARAH TELL Vibration m

itigation of high-speed railway bridges

KTH

2017

LICENTIATE THESIS IN STRUCTURAL ENGINEERING AND BRIDGESSTOCKHOLM, SWEDEN 2017

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENTwww.kth.se

TRITA-BKN. BULLETIN 147, 2017ISBN 978-91-7729-383-5ISSN 1103-4270ISRN KTH/BKN/B–147–SE

Vibration mitigation of high-speed railway bridgesApplication of � uid viscous dampers

SARAH TELL

Documents

Vibration mitigation of high-speed railway bridges1090042/... · 2017. 4. 21. · vibrations of high-speed railway bridges”. International Journal of Rail Transportation, 5 (1):