in uence of the vertical support sti ness on the dynamic behavior of

Inuence of the Vertical Support

Stiness on the Dynamic Behavior

of High-Speed Railway Bridges

Rui Afonso de Sousa Tavares Carneiro Santos

Dissertation for the degree of Master of Science in

Civil Engineering

Supervisor: Prof. Raid Karoumi

Jury

President: Prof. José Manuel Matos Noronha da Câmara

Assessor: Prof. Luís Manuel Coelho Guerreiro

Co-supervisor: Prof. Jorge Miguel Silveira Filipe Mascarenhas Proença

September 2007

c©Rui Afonso Tavares 2007

À minha Avó Xanoca,

Abstract

When designing a bridge, the modelling of the soil and the soil-structure interactionis commonly far from reality. Furthermore, the studies undertaken by the EuropeanRail Research Institute (ERRI) point out that it is essential to model the supportconditions, if realistic predictions of the dynamic behavior are to be made. Whena bridge is subjected to the loads of a high-speed train, the dynamic response ofthe structure is, obviously, inuenced by the soil beneath the foundations and sur-rounding the bridge. The magnitude of that inuence was studied, namely fromthe analysis of the variations obtained for the eigenfrequencies, displacements andaccelerations. Greater attention was given to how changes in the vertical supportstiness inuence the dynamic behavior of bridges subjected to loads travelling ata speed that induces the resonant response.

As a rst approach to the problem, a theoretical analysis of two common railwaybridges from the Bothnia Line (the new Swedish high-speed line) was made. After-wards, as a case study, another railway bridge was analyzed using the Finite ElementMethod. The results obtained with this method were compared with data measuredin situ, to validate the model. Theoretical estimation of the support stiness andmodel updating were also performed.

The results obtained with the updated Finite Element model were found to bevery satisfactory. The ndings suggest that models with sti supports can greatlyunderestimate the maximum responses of high-speed railway bridges and may notbe reliable. Furthermore, it was concluded that simple 2D beam models are ableto simulate reasonably well the behavior of real bridges. With the help of eldmeasurements and model updating, these simulations can be increasingly accurate,and particulary meaningful for structures under constant monitoring.

Key words: bridge dynamics, mechanical vibration, resonance, vertical supportstiness, high-speed railway bridges

v

Resumo

Ao projectar uma ponte, a modelação do solo e a interacção solo-estrutura sãogeralmente menosprezados. No entanto, estudos conduzidos pelo European RailResearch Institute (ERRI) indicam que é essencial modelar as condições de apoio, deforma a avaliar de modo realista o comportamento dinâmico da estrutura. Quandouma ponte é sujeita às cargas dos eixos de um comboio de alta velocidade, a respostadinâmica da estrutura é, obviamente, inuenciada pelo solo sob as fundações. Aimportância dessa inuência foi estudada através de análises de sensibilidade dasfrequências, deslocamentos e acelerações. Maior atenção foi dada à inuência quealterações na rigidez vertical das fundações podem ter no comportamento dinâmicode estruturas sujeitas à acção de cargas móveis com velocidade que origine efeitosde ressonância.

Numa primeira abordagem ao problema, procedeu-se a uma análise numérica eteórica de duas pontes ferroviárias correntes da Bothnia Line (a nova linha fer-roviária de alta velocidade da Suécia). Em seguida, como caso de estudo, outraponte ferroviária, mais complexa, foi analisada usando o Método dos ElementosFinitos. Os resultados obtidos com este método foram comparados com mediçõesobtidas in situ, para validar o modelo. Foram também efectuadas estimativas teóri-cas da rigidez vertical das fundações, utilizadas na actualização do modelo.

Os resultados obtidos com o modelo actualizado de Elementos Finitos foram muitosatisfatórios. As conclusões sugerem que modelos numéricos com apoios rígidos po-dem seriamente menosprezar as respostas máximas de pontes ferroviárias de altavelocidade e não devem ser utilizados, sob prejuízo de avaliar erradamente o com-portamento estrutural. Concluiu-se ainda que modelos simples bi-dimensionais po-dem simular relativamente bem o comportamento de estruturas reais. Com a ajudade medições in situ e actualização do modelo, estas simulações podem ser pro-gressivamente mais exactas, e particularmente úteis para pontes sob monitorizaçãoconstante.

Palavras chave: dinâmica estrutural, vibração mecânica, ressonância, rigidez ver-tical dos apoios, pontes ferroviárias de alta velocidade

vii

Preface

The research presented in this dissertation was carried out at the Department ofStructural Engineering, Structural Design and Bridges, at the Royal Institute ofTechnology (KTH) in Stockholm, Sweden and the report nished in the InstitutoSuperior Técnico (IST), in Lisboa, Portugal. The dissertation was performed underthe scope of the co-operation protocol between the Rede Ferroviária de Alta Ve-locidade (RAVE) and IST, and supervised by Associate Professor Raid Karoumi,PhD Student Mahir Ülker and Professor Jorge Proença.

I would like to express my sincere gratitude to:

- Associate Professor Raid Karoumi for being always available to help mending the right path to follow, for his words of encouragement when theywere more needed, and for his scientic guidance and valuable advice;

- Professor Jorge Proença for suggesting the subject and for the orientation inmy home university;

- Mahir Ülker, because this dissertation would not be done without him. Forall his support and all the days working until exhaustion, the everyday discus-sions on structural dynamics or the messages on Sunday morning, solving theproblem that had taken my sleep all weekend;

- Associate Professor Anders Bodare for his knowledge on soil dynamics andhis help solving all the questions that I found in the way;

- João Henriques for the terric and more than helpful discussions on bridgedynamics, even though he was 3000 km away;

- Diogo, for sharing the interest on poker odds, the lunches when everyone elsewas thinking about dinner and, most of all, the passion for Sporting CP;

- My parents and my sisters, for supporting my decision of becoming a CivilEngineer, when everyone else in the family wanted a doctor;

- But, most of all, my Avó Xanoca, just for being who she is.

Lisboa, September 2007

Rui Afonso Tavares

ix

Contents

Abstract v

Resumo vii

Preface ix

Contents xi

List of Figures xv

List of Tables xxiii

List of Acronyms xxv

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 General structure of the dissertation . . . . . . . . . . . . . . . . . . 3

2 State-of-the-art Review 5

2.1 General studies on the subject . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Soil-structure interaction and soil modeling . . . . . . . . . . . . . . . 13

2.3 The UIC reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Important Theoretical Concepts 33

3.1 Dynamic behavior of HSR bridges . . . . . . . . . . . . . . . . . . . . 33

3.2 Theoretical estimation of the support stiness . . . . . . . . . . . . . 35

xi

3.3 Load models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 2D beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Euler-Bernoulli beam . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.2 Timoshenko beam . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Mode Superposition Method . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 Modeling in Abaqus/Brigade . . . . . . . . . . . . . . . . . . . . . 50

3.6.1 Basic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6.2 Analysis type . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6.3 Visualization of the results . . . . . . . . . . . . . . . . . . . . 52

3.6.4 Brigade add-ons . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Theoretical Behavior of Simple Bridges 55

4.1 Single span bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 The bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.2 FE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.3 Frequency analysis and convergence study . . . . . . . . . . . 57

4.1.4 Comparison of the methods . . . . . . . . . . . . . . . . . . . 65

4.1.5 Resonance eects . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.6 Sensitivity to vertical support stiness . . . . . . . . . . . . . 67

4.1.7 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 79

4.2 Double span bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 The bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.2 FE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.4 Sensitivity to vertical support stiness . . . . . . . . . . . . . 86

4.2.5 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 94

5 Case Study: the Sagån Bridge 97

5.1 The bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 FE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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5.3 In situ measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.1 The Gröna Tåget . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.2 Instrumentation of the bridge . . . . . . . . . . . . . . . . . . 103

5.3.3 Measurement data . . . . . . . . . . . . . . . . . . . . . . . . 104

5.4 Inuence of the vertical support stiness . . . . . . . . . . . . . . . . 106

5.4.1 Eigenfrequencies sensitivity . . . . . . . . . . . . . . . . . . . 107

5.4.2 Model with theoretical support stiness . . . . . . . . . . . . . 108

5.4.3 Model updating . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Conclusions and Suggestions for Further Research 117

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Suggestions for further research . . . . . . . . . . . . . . . . . . . . . 119

References 121

A Mode Shapes of the Banafjäl Bridge 127

B Mode Shapes of the Lögdeälv Bridge 131

C Mode Shapes of the Sagån Bridge 135

D Drawings of the Sagån Bridge 139

xiii

List of Figures

1.1 European high-speed network (from www.uic.asso.fr). . . . . . . . . . 1

1.2 The Japanese HST: Shinkansen. . . . . . . . . . . . . . . . . . . . . . 2

2.1 Dynamic model of articulated vehicles (from Xia et al. [66]). . . . . . 7

2.2 High speed Thalys train composition (from Xia and Zhang [64]). . . . 7

2.3 Dynamic interaction of vehicle and bridge (from Xia and Zhang [64]). 8

2.4 Non-stochastic parameters in the track model (from Oscarsson [48]). . 11

2.5 Coupling of nite elements with a boundary element domain. Vectornj is the unit normal of element j representing the subsurface Sj ofthe BE domain. Vk is the volume represented by nite element k(from Andersen et al. [3]). . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Cross section of a ballasted track model (from Lombaert et al. [42]). . 15

2.7 Alternative models of a ballasted track (from Lombaert et al. [42]). . 16

2.8 Rayleigh wave propagation from a vertical harmonic rectangular load(the encircled area) moving along the surface of an elastic half-spacein the direction indicated by the vector and at the speeds: (a) v =0 m/s, (b) v = 100 m/s and (c) v = 200 m/s. Dark and lightshades of grey indicate negative and positive vertical displacements,respectively (from Andersen et al. [3]). . . . . . . . . . . . . . . . . . 17

2.9 Finite element mesh of the bridge and foundations (from Ju [36]). . . 18

2.10 Surface displacements of a nite element analysis under the trainspeed of 240 km/h (from Ju [36]). . . . . . . . . . . . . . . . . . . . . 19

2.11 Eurocode envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.12 Comparison of the Eurocode envelope with real trains. . . . . . . . . 21

2.13 Comparison between the Eurocode envelope and the HSLM's signature. 21

2.14 Flow chart to determine whether a dynamic analysis is necessary(from [22]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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2.15 Maximum mid-span acceleration on a single-track bridge with dier-ent mass (from [19]). . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.16 Maximum mid-span displacement on a single-track bridge with dif-ferent mass (from [19]). . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.17 Maximum mid-span acceleration on a single-track bridge with dier-ent bending stiness (from [19]). . . . . . . . . . . . . . . . . . . . . . 28

2.18 Maximum mid-span displacement on a single-track bridge with dif-ferent bending stiness (from [19]). . . . . . . . . . . . . . . . . . . . 28

2.19 Transfer function between bridge deck acceleration (Bm) and ballastacceleration (Ba) (from [21]). . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Dynamic amplication factor as a function of the frequency ratio β(from Clough and Penzien [12]). . . . . . . . . . . . . . . . . . . . . . 34

3.2 Coecients βx, βz and βψ for rectangular footing (from Whitman andRichart [58]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Dynamic stiness and dashpot coecients for arbitrary shaped foun-dations on homogeneous half-space surface (from Gazetas et al. [30]). 37

3.4 Graphs accompanying table 3.3 (from Gazetas et al. [30]). . . . . . . 38

3.5 Dynamic stiness and dashpot coecients for surface foundations onhomogeneous stratum over bedrock (from Gazetas et al. [30]). . . . . 39

3.6 Graphs accompanying table 3.5 (from Gazetas et al. [30]). . . . . . . 40

3.7 Conventional Train. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8 Articulated train with single axle bogie. . . . . . . . . . . . . . . . . . 41

3.9 Articulated train with Jakobs-type bogie. . . . . . . . . . . . . . . . . 41

3.10 SW/0 and SW/2 load models. . . . . . . . . . . . . . . . . . . . . . . 42

3.11 LM71 load model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.12 High speed load model A axle conguration (from [10]). . . . . . . . . 43

3.13 High speed load model B axle conguration (from [10]). . . . . . . . . 43

3.14 Parameters N and d as functions of L, for HSLM-B (from [10]). . . . 43

3.15 Beam element with internal forces and applied loading in positivedirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 FE model of the single span bridge. . . . . . . . . . . . . . . . . . . . 56

4.2 Maximum responses for the rst 100 modes (left) and zoomed in therst 25 modes (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xvi

4.3 Maximum displacement of the whole beam (left) and zoomed in mid-span (right), with increasing mode number as parameter. . . . . . . . 59

4.4 Maximum bending moment of the whole beam (left) and zoomed inmid-span (right), with increasing mode number as parameter. . . . . 59

4.5 Maximum acceleration of the whole beam (left) and zoomed in mid-span (right), with increasing mode number as parameter. . . . . . . . 59

4.6 Calculated response with a time step too large. . . . . . . . . . . . . 60

4.7 Maximum displacement (left) and bending moment (right) of thebeam, with time step as parameter. . . . . . . . . . . . . . . . . . . . 61

4.8 Maximum acceleration of the beam, with time step as parameter. . . 61

4.9 Maximum displacement (left) and bending moment (right) on thebeam, with speed step as parameter. . . . . . . . . . . . . . . . . . . 63

4.10 Maximum acceleration on the beam, with speed step as parameter. . 63

4.11 Maximum responses normalized, for speed steps of 5 km/h (left) and2.5 km/h (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.12 First 10 bending frequencies (left) and zoomed in frequencies 7 to 10(right) of the Banafjäl bridge, with mesh type as parameter. . . . . . 64

4.13 Vertical displacement at mid-span, for HSLM-A1 running at crawling(left) and resonant (right) speed. . . . . . . . . . . . . . . . . . . . . 66

4.14 Acceleration at mid-span, for HSLM-A1 running at crawling (left)and resonant (right) speed. . . . . . . . . . . . . . . . . . . . . . . . . 66

4.15 Maximum displacement on the beam, for dierent train speeds. . . . 66

4.16 Vertical displacement, over time and space, for HSLM-A1 running atcrawling speed (5 km/h). . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.17 Vertical isplacement, over time and space, for HSLM-A1 running atresonant speed (155 km/h). . . . . . . . . . . . . . . . . . . . . . . . 68

4.18 Bending moment, over time and space, for HSLM-A1 running atcrawling speed (5 km/h). . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.19 Bending moment, over time and space, for HSLM-A1 running at res-onant speed (155 km/h). . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.20 Vertical acceleration, over time and space, for HSLM-A1 running atcrawling speed (5 km/h). . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.21 Vertical acceleration, over time and space, for HSLM-A1 running atresonant speed (155 km/h). . . . . . . . . . . . . . . . . . . . . . . . 70

xvii

4.22 Eect of the end supports vertical stiness in the bending frequenciesof the rst ten bending modes of vibration of the Banafjäl bridge. . . 71

4.23 Maximum downwards (left) and upwards (right) displacement of thebeam, as function of the train speed. . . . . . . . . . . . . . . . . . . 73



4.26 Displacement at mid-span, for HSLM-A1 running at 5 km/h. Wholetime span (left) and zoomed in the high frequency vibrations (right). 75



4.29 Vertical displacement at mid-span, for HSLM-A1 running at resonantspeed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.30 Vertical displacement at mid-span, for HSLM-A1 running at resonantspeed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.31 Maximum downwards (left) and upwards (right) acceleration, forHSLM-A1 running at resonant speed. . . . . . . . . . . . . . . . . . . 78



4.34 Sketch of the Lögdeälv bridge. . . . . . . . . . . . . . . . . . . . . . . 81

4.35 FE model of the Lögdeälv bridge. . . . . . . . . . . . . . . . . . . . . 81

4.36 Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with element length as parameter. . . . . . . . . . 82

4.37 Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with element length as parameter. . . . . . . . . . 83

4.38 Eigenfrequencies, with element length as parameter. . . . . . . . . . . 83

4.39 Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with time step and number of modes as parameter. 84

4.40 Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with time step and number of modes as parameter. 85

xviii

4.41 Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with time step and number of modes as parameter. 85

4.42 Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with time step and number of modes as parameter. 86

4.43 Eect of the vertical support stiness on the frequencies of the rstten modes of vibration of the Lögdeälv bridge. . . . . . . . . . . . . . 87

4.44 Maximum upwards (left) and downwards (right) displacement of thebeam (cases 1 to 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 89



4.47 Maximum upwards (left) and downwards (right) displacement of thebeam (cases 0 to 6 with speed step of 1 km/h). . . . . . . . . . . . . 90

4.48 Maximum upwards (left) and downwards (right) displacement of thebeam (cases 7 and 8 with speed step of 1 km/h). . . . . . . . . . . . . 90

4.49 Maximum upwards (left) and downwards (right) displacement of thebeam (case 9 with speed step of 1 km/h). . . . . . . . . . . . . . . . . 90

4.50 Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 1 to 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 92



4.53 Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 0 to 2, 4 and 5 with speed step of 1 km/h). . . . . . . . . 93

4.54 Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 3 and 6 with speed step of 1 km/h). . . . . . . . . . . . . 93

4.55 Maximum upwards (left) and downwards (right) acceleration of thebeam with dierent number of modes (cases 4 to 6 with speed stepof 5 km/h). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.56 Maximum displacement (cases 1 to 9) and acceleration (cases 0 to 3)of the beam (with speed step of 1 km/h). . . . . . . . . . . . . . . . . 96

5.1 The Sagån bridge, view from northwest (a column belonging to anolder bridge behind the studied bridge can be seen in the middle). . . 97

5.2 The Sagån bridge, view from east. . . . . . . . . . . . . . . . . . . . . 98

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5.3 The short columns (left) and the embankments (right) of the Sagånbridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4 The cross-section of the Sagån bridge. . . . . . . . . . . . . . . . . . . 99

5.5 Join connector used to connect the beam to the columns. . . . . . . . 100

5.6 FE model of the Sagån bridge. . . . . . . . . . . . . . . . . . . . . . . 101

5.7 3D render (left) and picture of the interior (right) of the Gröna Tågettest train. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.8 3D render of one accelerometer (left) and one LVDT (right). . . . . . 104

5.9 Position of the accelerometers and LVDTs. . . . . . . . . . . . . . . . 104

5.10 PSD estimated from accelerations, when the Gröna Tåget crosses theSagån bridge (from Ülker [40]). . . . . . . . . . . . . . . . . . . . . . 105

5.11 Vertical displacements measured with the LDVTs. . . . . . . . . . . . 105

5.12 Rotation over east bearing, calculated using the signal of the twoLVDT sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.13 Modied FE model of the Sagån bridge (with spheres representingthe springs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.14 Eect of the end supports stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge. . . . . 107

5.15 Eect of the short columns stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge. . . . . 108

5.16 Eect of the main columns stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge. . . . . 108

5.17 Frequencies of the rst ten (left) and three (right) modes of vibrationof the Sagån bridge (model 1 and 2). . . . . . . . . . . . . . . . . . . 110

5.18 Frequencies of the rst ten (left) and three (right) modes of vibrationof the Sagån bridge (with the updated model). . . . . . . . . . . . . . 112

5.19 Vertical displacement at mid-span of span 1 (see gure 5.9). . . . . . 112

5.20 Vertical displacement over east bearing (east). . . . . . . . . . . . . . 113

5.21 Vertical displacement over east bearing (west). . . . . . . . . . . . . . 113

5.22 Rotation over bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.1 Modes shapes of the Banafjäl bridge (common to all models). . . . . 127



xx

A.4 Modes shapes of the Banafjäl bridge (only present in the stier models).128

A.5 Modes shapes of the Banafjäl bridge (only present in the stier models).128

A.6 Modes shapes of the Banafjäl bridge (only present in the less stimodels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.7 Modes shapes of the Banafjäl bridge (only present in the less stimodels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

B.1 Modes shapes of the Lögdeälv bridge (common to all models). . . . . 131



B.4 Modes shapes of the Lögdeälv bridge (only present in the stier models).132

B.5 Modes shapes of the Lögdeälv bridge (only present in the stiest model).132

B.6 Modes shapes of the Lögdeälv bridge (only present in the less stimodels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.7 Modes shapes of the Lögdeälv bridge (only present in the less stimodel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.8 Modes shapes of the Lögdeälv bridge (only present in the least stimodel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

C.1 Modes shapes of the Sagån bridge (common to all models). . . . . . . 135




C.5 Modes shapes of the Sagån bridge (only present in model 1). . . . . . 136

C.6 Modes shapes of the Sagån bridge (only present in model 1). . . . . . 137

C.7 Modes shapes of the Sagån bridge (only present in model 2 and 3). . 137

C.8 Modes shapes of the Sagån bridge (only present in model 2 and 3). . 137

xxi

List of Tables

2.1 Non-stochastic track parameters used in the numerical simulations(from Oscarsson [48]). . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 The mean values (Mean) and the standard deviations (S.D.) of thestochastic variables evaluated at the two dierent test sites, Gåsakullaand Grundbro (from Oscarsson [48]). . . . . . . . . . . . . . . . . . . 11

2.3 Values of critical damping to be assumed for design purpose (from[16]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Values to adopt for SW/0 and SW/2 load models. . . . . . . . . . . . 42

3.2 High speed load model A (from [10]). . . . . . . . . . . . . . . . . . . 43

4.1 Maximum responses, with time step as parameter. . . . . . . . . . . . 62

4.2 First 10 bending frequencies [Hz] of the Banafjäl bridge, with elementtype and size as parameters. . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Eect of the end support stiness in the frequencies [Hz] of the rstten bending modes of vibration of the Banafjäl bridge. . . . . . . . . 71

4.4 Resonant speed, for the dierent support stiness, obtained fromequation 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Resonant speed, for dierent vertical support stiness. . . . . . . . . 74

4.6 Maximum displacement at mid-span, for dierent support stiness. . 76

4.7 Maximum acceleration, for dierent vertical support stiness. . . . . 79

4.8 The weight of the Lögdeälv bridge. . . . . . . . . . . . . . . . . . . . 80

4.9 The material and sectional properties for the Lögdeälv bridge. . . . . 80

4.10 Maximum responses, with element length as parameter. . . . . . . . . 83

4.11 Maximum responses, with time step and number of modes as parameter. 84

4.12 Maximum responses, with speed step as parameter. . . . . . . . . . . 86

xxiii

4.13 Denition of the cases for the study. . . . . . . . . . . . . . . . . . . . 87

4.14 Eect of the vertical support stiness in the frequencies [Hz] of therst ten bending modes of vibration of the Lögdeälv bridge. . . . . . 87

4.15 Eect of the vertical support stiness on the bending frequencies ofthe rst 2 bending modes of vibration of the Lögdeälv bridge and onthe resonant speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.16 Maximum displacements with dierent speed steps (resonant speed[km/h] is indicated between parenthesis). . . . . . . . . . . . . . . . . 91

4.17 Maximum acceleration [m/s2] (using the speed step of 5 km/h). . . . 93

4.18 Maximum responses for the cases where resonance was found (reso-nant speed [km/h] is indicated between parenthesis). . . . . . . . . . 95

4.19 Resonant speeds [km/h] obtained from equation 4.2 and with the FEM. 96

5.1 Gröna Tåget axle loads and distances. . . . . . . . . . . . . . . . . . 103

5.2 Estimation of the eigenfrequencies, when Gröna Tåget crosses theSagån bridge (from Ülker [40]). . . . . . . . . . . . . . . . . . . . . . 105

5.3 Theoretical estimation of vertical stiness of the supports. . . . . . . 109

5.4 Frequencies [Hz] of the rst ten modes of vibration with the percent-ages, comparing with the measured values, between parenthesis (withthe updated model). . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Maximum responses for the 3 models (between parenthesis are thepercentages, comparing with the measured values). . . . . . . . . . . 114

xxiv

List of Acronyms

BE Boundary Element

BEM Boundary Element Method

DAF Dynamic Amplication Factor

DOF Degree of Freedom

ERRI European Rail Research Institute

FE Finite Element

FEM Finite Element Method

HSLM High Speed Load Model

HSR High Speed Railway

HST High Speed Train

IOS Initial Operating Segment

IST Instituto Superior Técnico

KTH Kungliga Tekniska Högskolan (Royal Institute of Technology)

LVDT Linear Variable Dierential Transformers

MEMS Micro Electro Mechanical Systems

MSc Magister Scientiae (Master of Science)

ORE Oce of Research and Experiments

PhD Philosophiae Doctor (Doctor of Philosophy)

RAVE Rede Ferroviária de Alta Velocidade

SASW Spectral Analysis of Surface Waves

SDOF Single Degree of Freedom

UIC International Union of Railways

WIB Wave Impeding Barrier

xxv

Chapter 1

Introduction

1.1 Background and motivation

Nowadays, in modern societies, the need to move people and goods is growing fast,both in number of transactions and in traveled distance. Therefore, transport in-frastructures are being built all over the world, to cope with the market claim.

The railways have been used for many decades to assure the conveyance of people andgoods. With the launch of the high-speed trains (HST), this way of transportationbecame even more useful. From the early 1980's, when the Paris-Lyon railway wasbuilt, with a total distance of 410 km, the high-speed railway (HSR) have grownand spread to all the world. The current HSR infrastructures, and the new linesplanned can be seen in gure 1.1, and totalize more than 10000 km.

Figure 1.1: European high-speed network (from www.uic.asso.fr).

1

CHAPTER 1. INTRODUCTION

With the appearance of the TransRapid05, in 1979, a new type of HST was born, theMagnetic Levitation Train or Maglev. The rst commercial Maglev was opened in1984 in Birmingham, covering 600 meters between its airport and railhub, but waseventually closed in 1995 due to technical problems. At the time of this dissertation,the only operating high-speed maglev line of note is the Initial Operating Segment(IOS) demonstration line of Shanghai, that transports people through 30 km tothe airport in just 7 minutes 20 seconds, achieving a top velocity of 431 km/h andaveraging 250 km/h.

The oriental civilizations have always been in the front edge of the HSRs. In Japan,the Maglev experimental trains have achieved, in 2003, 581 km/h, but the high costof its tracks makes it unprotable for conventional passenger lines. While engineersare trying to lower the expenses involved with Maglev trains, the Shinkansen (gure1.2) spreads its tracks around Japan. Since the initial Shinkansen opened in 1964running at 210 km/h, the network (2,459 km) has expanded to link most majorcities with running speeds of up to 300 km/h, in an earthquake and typhoon proneenvironment.

Figure 1.2: The Japanese HST: Shinkansen.

Despite not being in the front edge, countries like Sweden and Portugal are nowstarting to implement HSR networks. The Bothnia Line, the new Swedish railwayfrom Nyland, north of Sundsvall, to Umeå, will provide a direct rail link for the rsttime between Sundsvall, Örnsköldsvik and Umeå, serving about 350,000 people. Itwill also double the rail capacity between central and northern Sweden. This HSRconsists of 190 km of railway with 150 bridges and 30 km of tunnels and it is designedfor operation by 120 km/h freight trains and 250 km/h passenger trains, makingthis Sweden's rst line capable of this speed. The line will be single track with 22two or three-track, 1 km long, passing loops.

In Portugal, the Rede Ferroviária de Alta Velocidade (RAVE) was constituted to

2

1.2. AIMS AND SCOPE

build and manage the Portuguese HSR, in 2000. After the Iberian Meetings withSpain, in 2003 and 2005, it was decided to make 4 international HSR connections:Lisboa-Madrid, opening in 2013; Porto-Vigo, to be started after 2009; Aveiro-Salamanca, after 2015; and Évora-Faro-Huelva, in the 3rd decade of the century.The connection Lisboa-Porto, very important for the Portuguese transport networkwill allow people to commute between the 2 cities in 1 hour and 35 minutes.

Both the networks will need a large number of bridges and viaducts. In the caseof trains running at high-speed, the risk of resonance in the structures is largerthan classical trains, and assessment of vibration problems in the high-speed rail-way bridge is required during its design, to guarantee the safety of the crossingtrain, which is subordinated to strict crossing conditions. Therefore, performing dy-namic analysis that investigates resonance of the bridge induced by the bridgetraininteraction constitutes an essential element of the design.

1.2 Aims and scope

When designing a bridge, the modelling of the soil and the soil-structure interactionis commonly far from reality. Furthermore, the studies undertaken by ERRI andCommittee D214 (see section 2.3) point out that it is essential to model the supportconditions, if realistic predictions of the dynamic behavior are to be made. Hereby,the importance of the accurate estimation of the vertical support stiness, on thedynamic behavior of the bridge, is the subject of this MSc dissertation.

When a bridge is subjected to the loads of a high-speed train, the dynamic responseof the structure is, obviously, inuenced by the soil beneath the foundations andsurrounding the supports. The magnitude of that inuence is going to be stud-ied, namely from the analysis of the variations obtained for the eigenfrequencies,displacements and accelerations. Greater attention will be given to the eect thatchanges in the vertical support stiness have on the dynamic behavior of structuressubjected to loads travelling at a speed that induces the resonant response.

1.3 General structure of the dissertation

The dissertation will be divided in chapters, sections and subsections, properly enu-merated.

In chapter 2, a review of the state-of-the-art investigation on the subject of high-speed railway (HSR) will be presented. Most of the information discussed in thatchapter will not be used in this dissertation, but the goal is to give general andhistorical approach to the whole problem.

Important results achieved in the past, but essentially related to the studies under-taken in this dissertation, will be presented in chapter 3.

3

CHAPTER 1. INTRODUCTION

In the 4th chapter, a theoretical analysis of two common railway bridges will be done.One single span and one double span composite bridge from the Bothnia Line (thenew Swedish HSR) will be used for the study. No comparison with experimentalmeasurements will be made at this stage.

To adjust the theoretical model to reality, a case study with the Sagån bridge willbe focused on chapter 5. Theoretical predictions will be compared with the resultsfrom the experimental measurements in situ.

Chapter 6 states the general conclusions inferred from the studies undertaken andrecommends some directions for further investigation on the subject.

4

Chapter 2

State-of-the-art Review

In this chapter, a state-of-the-art on the subject of high-speed railway (HSR) willbe presented. Most of the information discussed here will not be used in this dis-sertation, but the goal is to give general and historical background to the wholeproblem.

The chapter will be partitioned in three sections. The rst will refer to the mainresults on high-speed structural behavior, obtained in the last years. The secondsection will be reserved to investigation involving the soil-structure interaction andthe modelling of the soil. The last part will be dedicated exclusively to the work de-veloped by the UIC, that greatly extended the knowledge on the high-speed subject,all over the world.

Important results achieved in the past but essentially related to the studies under-taken in this dissertation will be presented in chapter 3.

2.1 General studies on the subject

Since the 19th century, when the rst accidents with metal railway bridges occurred,many scientist and engineers tried to explain the phenomenon, both through exper-imental and analytical studies. The rst publications about the dynamic behaviorof bridges came from the mid-nineteenth century, following the works of Willis [60]in investigating the collapse of the Chester rail bridge, over the river Dee, in Eng-land, 1847, the rst case of collapse of a railway bridge in history. In this pioneerwork, the inertial eect of the beam was ignored, and the vehicle was modeled as aconcentrated moving mass travelling at constant speed. Although for this particularcase, an exact solution could be obtained, its applicability remains rather limited,due to the omission of the inertial eect of the beam. Nevertheless, the contributionof Willis is considered historical, since he is among the rst to bring the problem ofvehicle impacts to the design desks of bridge engineers.

In the beginning of the 20th century, the theoretical view of Timoshenko [55] gavea complete solution for the problem of the dynamic behavior of a prismatic bar

5

CHAPTER 2. STATE-OF-THE-ART REVIEW

acted upon by a moving harmonic pulsating force moving with a constant velocity.The publication of Inglis [35], also provided extremely important basis for the studyof the dynamic behavior of high-speed railway bridges. Later in the 20th century,the investigations in that eld by Ladislav Frýba (see Frýba [26] and Frýba [27])have greatly developed the knowledge on the subject. Thus, and with the growingnumber of high-speed railway lines, all across the world, the dynamical problemsassociated with moving of the loads across the structure have caught the attentionof both investigators and designers.

Ladislav Frýba, in the past decade, published some very interesting analytical so-lutions (Frýba [28]) for a simply supported beam subjected to any set of loads,travelling to a certain speed. The complete derivation of that solution and the de-nition of all the variables used will be deeply discussed in section 3.4.1. In the samearticle, there are also some very interesting propositions for the calculation of theinteroperability, dened as the capability of a bridge to carry a particular train orvehicle running at certain speed or, in other way, the technical conditions whichascertain that the train could move on a given railway line, including bridges, atthe designed speed. Using both denitions, Frýba proposed ways to calculate theinteroperability constant of the bridge (B1) and the vehicle (V1 and V2), sepa-rately. For bridges with ballast, the quantication of these constants is such thatthe maximum vertical bridge deck acceleration is less than the limits conducting tothe destabilization of ballast. The maximally accepted values for the acceleration ofthe bridge deck, ault, are specied by [21]:

ault = 3.5 m/s2 or ault = 5 m/s2 (2.1)

for bridges with ballast or without ballast, respectively.

The formulas suggested for the amplitudes of the deection, bending moment andacceleration include the most important parameters: speed, span, natural frequency,damping, length of vehicles axle load and permanent load of the bridge. As theydepend on the square of the speed it is, therefore, explained why the resonancevibration appears at high speeds only. Furthermore, the results obtained by Frýbapoint that the amplitudes of resonance vibration depend on the square of speedand on the span of the bridge and inversely on damping, vehicle length and bridgerigidity. Moreover, the acceleration depends also on the ratio of the axle load to thepermanent load of the bridge.

Furthermore, two reasons for the resonance vibration of railway bridges on high-speed lines were discovered: repeated action of axle loads and loss of stability undermoving forces. While the rst reason appears actually on high-speed lines at today'sspeeds, the second one is not yet actual.

Apart from Ladislav Frýba, there is another investigator that has contributed forthe scientic community with a large number of articles about the dynamic behaviorof railway bridges. He Xia, from the Beijing Jiaotong University, has investigated,both analytically and experimentally, the dynamic phenomena occurring in bridgescrossed by HSTs, namely in Xia et al. [61, 62, 63, 65, 66] and and Xia and Zhang

6

2.1. GENERAL STUDIES ON THE SUBJECT

[64]. His partnership with Guido de Roeck and Nan Zhang has been particularyproductive.

In Xia et al. [61, 66], the Thalys articulated train (gure 2.2) passing along theAntoing bridge on the ParisBrussels high-speed railway line was analyzed. Thetrain was modeled as 17 rigid bodies and 85 degrees-of-freedom (DOFs) in total(gure 2.1). With the 30 DOFs of the two locomotives, the total number of theDOFs of the whole train model is 115. The bridge was modeled by nite elements.

Figure 2.1: Dynamic model of articulated vehicles (from Xia et al. [66]).

The Newmark-β Method was used in the step-by-step integration of the combinedvehicle and bridge system.

After the experimental and analytical results comparison, it was concluded that:

- The dynamic analytical model of the bridge-articulated- train system and thecomputer simulation method proposed could well reect the main vibrationcharacteristics of the bridge and the articulated train vehicle;

- The calculated results were well in accordance, both in response curves, in am-plitudes and in distribution tendencies, with the in situ measured data, whichveried the eectiveness of the analytical model and the computer simulationmethod;

- The articulated train vehicles have a rather smooth running at high-speed,which also helps to reduce the impact on the bridge structures.

Figure 2.2: High speed Thalys train composition (from Xia and Zhang [64]).

In Xia et al. [62], Xia and Zhang [64] the same type of analysis was used, and someassumptions are referred. Regarding the vehicle, it was assumed that:

7


- The eects of the the elastic deformations of the car bodies and the wheel setswas negligible;

- All vehicles could be simplied into the suspension system and the single-set-springs and dashpots and its coecients divided over the wheels;

- The conguration of each vehicle body could be specied by 5 DOFs: lateralmovement, rolling, yawing, oating and nodding; and each wheel had 3 DOFs:lateral moving, rolling and oating. Hence, a six-axle locomotive represented23 DOFs and each regular coach 17 DOFs.

For the bridge model, it was assumed that:

- There was no displacement between the track and the bridge deck, and therail pad eect could be neglected;

- The vibration modes of the bridge girders were the same as the modes of thebridge deck;

- The inuence of the masses of the vehicles was much lower that the bridgeweight and, therefore, negligible;

- The cross-section deformation of the girder was negligible, thus its movementcould be expressed with 3 DOFs: lateral and vertical displacement and rota-tion.

Figure 2.3: Dynamic interaction of vehicle and bridge (from Xia and Zhang [64]).

These assumptions can be very important for the modelling and dynamic analysis ofbridges in countries where the high-speed railway is starting, as happens in Portugaland Sweden. From the dynamic studies, it was concluded that the models, althoughwith the simplications, could rather well reect the main vibration characteristics of

8


the train-bridge interaction system. The calculated results were well in accordance,regarding the response curves, amplitudes and distribution tendencies, with theexperimental data, which veried the eectiveness of the analytical model and thecomputer simulation method.

In Xia et al. [63], a three-dimensional nite element model was used to representa long suspension bridge and each 4-axle vehicle in a train was modeled by a 27DOFs dynamic system. The measured track irregularities and the wheel huntingdescribed by a sinusoid function were used to represent the two most important self-excitations in the coupled train bridge system. The degrees-of-freedom for all wheelswere eliminated from the basic coupled equations of motion to reduce computationeorts and the Mode Superposition technique was then applied, only to the bridge.By using the Mode Superposition technique it was assumed that the bridge wasoperating in a linear range.

Similar studies on the dynamic behavior of bridges subjected to high-speed loadshave been made in Korea. Kwark et al. [39] presents a study of the amplied dynamicresponses of bridges crossed by the Korean high-speed train (KHST).

The bridge used was representative of the bridges adopted for the Korean HSR: asimple box continuous concrete bridge of 80 m length constituted by two-span of 40m length. Three dimensional space frame elements constituted by two-nodes, eachnode with 6 degrees-of-freedom, were established to numerically model the bridge.First order Lagrange interpolation function and third order Hermite Interpolationfunction as displacement shape functions were used for axial directional degrees-of-freedoms and exural degrees-of-freedoms, respectively. Applying classical niteelement method, the solution of the equation of motion of the bridge was obtainedeasily.

The equation of motion of the high-speed train could be expressed in matrix formsuch as equation 2.2.

Mu(t) + Cu(t) + Ku(t) = Pt (2.2)

where u represents the displacement, u stands for the velocity and u represents theacceleration. M, C, K and Pt represent the mass, damping and stiness matrices,and the external force vector which includes the interaction vectors of vehicles,respectively. In order to solve equation of motions for the bridgevehicle system, themagnitude of interacting forces at the end of each time interval should be determined.For the vehicle model used, the interacting force between a given axle and bridgesurface was a function of the stiness of its suspension spring assembly and thedeformation that was applied to the spring assembly. The displacement, velocityand acceleration of each axle in previous time step was used as the initial conditionfor the next time step.

Using the proposed method, the dynamic behavior of the bridge according to themodelling method of the train system was investigated. Eects of the idealizationmethod of the train were studied by comparing bridge responses resulting from

9


the simplest idealization method, i.e., using constant moving forces, and the three-dimensional model, which takes into account the bouncing, pitching and rolling ofeach component of the train. To obtain the solution of the bridgetrain interactionproblem, a direct integration method: the Newmark-β method, adopting a predictor-corrector iteration scheme, was applied.

The conclusions achieved from the analysis were the following:

- The dynamic response of the bridge crossed by trains running at high-speed issignicantly amplied at the vicinity of the critical speed, itself closely relatedto the fundamental natural frequency of the bridge and the eective beatinginterval produced by the train. Consequently, safety verication related tothe dynamic behavior at speeds close to the critical speed shall be necessaryperformed for bridges intended to be crossed by trains running at high-speed;

- During the analysis of dynamic behavior of structures, damping is essential.Especially, when resonance occurs, responses show very sensitive variations.During the verication of the dynamic behavior of the bridge expected tooperate at resonance under the crossing of the high-speed train, damping shallnecessarily be selected carefully and rationally;

- Compared with the actual eld test results, the proposed numerical analy-sis method led to reasonable results. As the use of three-dimensional modelswithout vehicle-bridge interactions may produce conservative results, it seemsadvisable to use a model that takes into account interactions during the veri-cation of the dynamic behavior of the bridge;

- Bridgetrain interaction eects appeared signicantly at every speed of theKHST;

- The maximum deection of the bridge being generally produced by locomotivesheavier than coaches, the passengers loading of the coaches does not aectthe maximum deection. When resonance occurs due to train running at thecritical speed, the resonance induced by the coaches may amplify the maximumdeection produced by the locomotives. Such amplication depends on thenumber of coaches, that is, the duration of resonance, and the intensity of theloading.

Oscarsson [48] took the next step by studying properties such as rail pad stiness,ballast stiness, dynamic ballast-subgrade mass and sleeper spacing, and their in-uence on the dynamic behavior of the vehicle-bridge interaction. A perturbationtechnique was used to investigate the inuence of the selected track properties. Thetrain-track interaction problem was numerically solved by use of an extended state-space vector approach in conjunction with a complex Modal Superposition for thewhole track structure. All numerical simulations were carried out in the time-domainwith a moving mass model.

10


Figure 2.4: Non-stochastic parameters in the track model (from Oscarsson [48]).

The non-stochastic parameters (gure 2.4) were studied in situ and in laboratory,and the values obtained are expressed in table 2.1. A stochastic model was used tond the remaining parameters, shown in table 2.2.

Table 2.1: Non-stochastic track parameters used in the numerical simulations (fromOscarsson [48]).

Track Element Parameter Parameter ValueRail Pad Damping [kNs/m] cp 20Ballast Damping [kNs/m] cb 500Subgrade Stiness [MN/m] ks 600Subgrade Damping [kNs/m] cs 650Subgrade Shear Stiness [MN/m] kss 700Subgrade Shear Damping [kNs/m] css 150

Table 2.2: The mean values (Mean) and the standard deviations (S.D.) of thestochastic variables evaluated at the two dierent test sites, Gåsakullaand Grundbro (from Oscarsson [48]).

Gåsakulla GrundbroTrack Element Mean S.D. Mean S.D.Sleeper Spacing [m] 0.652 0.017 0.650 0.020Ballast Stiness [MN/m] 255 16 186 22Ballast-Subgrade Mass [kg] - - 11600 5520

In the Iberian Peninsula, the research about high-speed railway eects have beenparticulary intensive in the Faculdade de Engenharia da Universidade do Porto, withthe work of professors Rui Calçada and Raimundo Delgado, and in the UniversidadPolitécnica de Madrid, through professors José Ma Goicolea and Felipe GabaldónCastillo.

After the rst approach of dos Santos [25] in 1994, the work of Delgado and dosSantos [24] started the discussion in Portugal about the HST eects on structures,

11


even though the rail trac in the country could not achieve high-speeds. Thedierential equation for the models with and without train-structure interactionwere obtained and solved with the Newmark Method. Track irregularities werestudied as well and various parameters were identied, such as: the stiness andlength of the bridge, the existence of ballast or the structural damping. The resultssuggested that the stiness of the structure and track irregularities were responsiblefor greater changes in the response. Roughness in the track was found to increasethe amplications as well, especially if there was a coincidence in the periods of thebridge or train. In the beginning of the 21th century, the work of Ribeiro [50] andPinto [49] carried on with the research on the subject.

In Spain, Goicolea et al. [31] suggested new dynamic analysis methods for railwaybridges to cover the possibility of resonance in the structure. The suggestions of thenew Eurocode [10] and the Spanish norms were studied and compared with otherEuropean norms. The results obtained suggested that:

1. The design of high-speed railroad bridges, because of the real possibility ofresonance, require consideration of the dynamic vibration under moving loads;

2. It is essential to apply dynamic analysis methods in order to improve theknowledge about the dynamic response of the bridges from the designer pointof view, as well as to be able to develop engineering design methods and codeswhich are suciently practical, secure and simple to use;

3. The nal draft of Eurocode 1 of actions in bridges [10] cover adequately thenecessity of dynamic analysis for the high-speed lines.

Museros et al. [45] focused only in the analysis of short span railway bridges. Themoving loads model is not well suited for the study of short bridges (L ≤ 20-25 m)since the results it produces (displacements and accelerations) are much greater thanthose obtained from more sophisticated ones (see [17] for more information). Theinvestigation was undertaken in order to decide on the inuence of the distribution ofthe loads due to the presence of the sleepers and ballast layer, and the trainbridgeinteraction. The results shown that the maximum accelerations of the deck are notsignicantly aected by the load distribution through the sleepers and ballast layer,except for the shorter values of the wavelength. However, the trainbridge interac-tion causes reductions of considerable importance in the maximum displacementsand accelerations of short bridges, since it decreases the concentration of the loads.It has been found that the reductions obtained in bridges with dierent natural fre-quency and moment of inertia are nearly proportional to each other. Nevertheless,further study is required in order to prove the validity of the procedure for dierenttrains and span lengths.

12

2.2. SOIL-STRUCTURE INTERACTION AND SOIL MODELING

2.2 Soil-structure interaction and soil modeling

Soil-structure interaction has been described as the group of phenomena inuencingthe dynamic behavior of structures while interacting with the soil, induced by theapplication of loads in the system. In the past years, some work on this topic hasbeen made, specially concerning railway induced vibrations.

According to Bayoglu [9], one of the key points in soil structure interaction is toevaluate the behavior of the soil under external loads. For designing purposes,the stresses and strains in the soil around the structures to be designed are moreimportant than the behavior of the whole medium. Hence, the tendency is to modelthe soil by surface deection due to external forces.

The simplest one of these linear elastic models is the Winkler soil model, wherethe soil is idealized with springs. The spring stiness k is known as the modulus ofsubgrade reaction and the pressure at soil surface, p, is:

p = k · z

where z is the deformation at the foundation surface.

However, non-linear elastic soil models are considered to be more realistic. To usethis approach, the material parameters from the theory of linear elasticity shouldbe modied through increments, to simulate the non-linear behavior. The materialparameters are often taken to be function of the current stress state.

Nowadays, numerical methods make it easier to apply the rules of mechanics tosoil-structure interaction of a very complex medium. Most of that analysis involvesthe following methods:

- Finite Element Method (FEM): The soil is divided into discrete elements, withspecied material properties and deformation behavior. Individual elementstiness are derived and assembled to yield a global system of equations, fromwhich displacements, strains and stresses are obtained. The FEM is veryexible and can be applied to more generalized soil models. However, whenthe material is linear elastic, the boundary conditions are known and thestresses/displacements elds in the interior can be found easily, making itunnecessary to use FE;

- Boundary Element Method (BEM): The governing dierential equation aretransformed to integral equations, which are solved numerically on the bound-ary regions. The number of physical dimensions to be considered is reduced byone, resulting in a smaller system of equation and a more ecient solution. Thenumerical technique is very similar to the one used with FEM. However, theboundary functions are approximated and solved and the stress/displacementvectors on the boundary are primary unknowns hence, of equal accuracy. BEM

13


is best suited for homogeneous bodies, and can simulate the behavior of inniteand semi-innite domains;

- Combined FEM and BEM: Coupling of FE and BE is a very used techniquewhen considering large or innite domains with mostly linear behavior, exceptin a small portion. The modelation is easier, the computation faster and theresults more accurate (see gure 2.5).

Figure 2.5: Coupling of nite elements with a boundary element domain. Vector nj

is the unit normal of element j representing the subsurface Sj of the BEdomain. Vk is the volume represented by nite element k (from Andersenet al. [3]).

Lombaert et al. [42] and Takemiya [54] both present very interesting models topredict the train-track and nearby ground-borne vibrations, and compare the resultswith experimental data. Lombaert et al. [42] used measurements from the new HSTtrack on the line L2 between Brussels and Köln to measure the soil transfer functions,the tracksoil transfer functions and the track and free eld vibrations during thepassage of a Thalys high-speed train. Ground-borne railway induced vibrations aregenerated by a large number of excitation mechanisms. For HST tracks on soft soils,the train speed can be close to or even larger than the critical phase velocity of thecoupled tracksoil system. In this case, the quasi static contribution of the load isimportant for both the track and the free eld response. High vibration levels andtrack displacements are obtained, aecting track stability and safety.

For the model, the rails were assumed to behave as EulerBernoulli beams and thesleepers were supposed to be rigid in the plane of the track cross-section, so thatthe vertical sleeper displacements along the track were determined by the verticaldisplacement usl(y, t) at the center of gravity of the sleeper and the rotation βsl(y, t)about this center. The sleepers were presumed not to contribute to the longitudinalstiness of the track, so that they can be modelled as a uniformly distributed massalong the track. The tracksoil interface was assumed to be rigid in the plane of thetrack cross-section. The model used for the numerical calculations can be seen ingure 2.6 and alternative models are shown in gure 2.7.

A boundary element method is used to calculate the soil tractions at the tracksoilinterface, assuming that the track is located at the soil's surface. The bound-

14


Figure 2.6: Cross section of a ballasted track model (from Lombaert et al. [42]).

ary element formulation is based on the boundary integral equations in the fre-quencywavenumber domain, using the Green's functions of a horizontally layeredsoil. Each layer in the half-space model is characterized by its thickness d, thedynamic soil characteristics E and ν or the longitudinal and transversal wave veloc-ities Cp and Cs, the material density ρ and a material damping ratio βp and βs involumetric and deviatoric deformation, respectively.

An elaborate measurement campaign was performed to validate the numerical model.The measurement campaign consisted of experiments that were performed to iden-tify model parameters and experiments that were used to validate the numericalmodel. A Spectral Analysis of Surface Waves (SASW) test and a track receptancetest were used to determine the dynamic soil and track parameters. The transferfunctions between a steel foundation and the free eld and between the track andthe free eld were subsequently used to validate the numerical model.

With the analysis, it was shown that the experimental and numerical track-free eldtransfer functions show a relatively good agreement, although at small distances anoverestimation of the experimental response is observed. The sleeper response andthe free eld vibrations due to the passage of the Thalys HST was also predicted andvalidated for two train speeds. The results emphasize the crucial role of the dynamicsoil properties. Given the large number of modelling uncertainties, the numericalresults of the free eld vibrations shown good agreement with the experimentalresults.

Takemiya [54] studied the train-track and nearby ground-borne vibrations producedby the Swedish high-speed train X-2000 at Ledsgard. As the result of the soft soildeposits at Ledsgard, the high train speed is almost in the trans-Rayleigh wave stateso that a large amplied track response appeared due to the resonance between thetrack behavior and the Rayleigh wave propagation in the ground. Field measurementat the low train speed of 70 km/h shown that the track and ground deformations weresmall, while at the high train speed of 200 km/h, they were very large, indicatingthat appreciable non-linear soil behavior had occurred at the soft clay layers.

15


Figure 2.7: Alternative models of a ballasted track (from Lombaert et al. [42]).

Kaynia et al. [38] investigated the track behavior by increasing the rigidity of thetrack as a potential remedy. An alternative measure is to improve the soft subsoilbeneath the track embankment. Takemiya et al. [53] proposed the wave imped-ing barrier (WIB) procedure for the mitigation of track vibration and proved itseectiveness.

According to Takemiya [54], a series of impact forces act on the rails at the momentwhen the train wheels roll over them, with specied time delays occurring accordingto the train geometry, the sleeper spacing, and the speed of motion. These axle loadsare transmitted through the rails to the ballast bed and then into the underlyingground. Depending on the train speed, an inertia force can be generated in the wholetrack including the ballast bed and embankment, as a result of the dynamics of thetrackground system. The consequence is the wave elds in the whole system, i.e.,rail, track and ground. These train-induced vibrations at the track dier signicantlydepending on train geometry and speed. At low speeds the response is quasi staticso that the track response due to train axle loads appears mostly downward at thepoint of their action. On the other hand, at high-speeds the train-induced responsebecomes dynamic due to the inertia generated in the trackground system, so thatthe track vibrations appear evenly in both upward and downward directions.

Through computer simulations the train-induced responses were interpreted in aquasi static sense for the low speed (70 km/h), while in a dynamic sense for thehigh-speed (200 km/h). At the low speed, the response appeared to the axle loadsat the instantaneous position so that the moving eect on track was given by thetime shifting of the static response conguration accordingly. At the high-speed, onthe other hand, wave motions occur due to the inertia generated in the track andground system. The time histories of ground responses are reasonably dierent atlow and high train speeds. For low speed trains, a clear-cut individual axle loadeect appears. However, as the distance from the track to the observation locationincreases, those corresponding displacements disappear, so that the response turns

16


out to be a smooth long-span displacement due to the total train weight. Thevelocity response follows such a change of displacement with distance. A similartendency also holds true for the acceleration response. For high-speed trains, onthe other hand, due to a true wave eld generation in the trackground system, thevibration appears in all response quantities at both the track and the nearby groundlocations.

A crude model was investigated in which a soft layer was replaced by an equiva-lent sti layer with the WIB installation. The results suggested that a signicantresponse reduction can be expected in the case where the WIB is installed over asubstantial area across the transverse section. The improved soil model has led toa dramatic response reduction. Surprisingly, the response features were brought tosimilar values to those experienced in the quasi static state for low train speedsso that the wave propagation disappeared as the distance from the track increased.This suggested that installing the WIB may be a very promising method for vibra-tion mitigation.

A dierent investigation was made by Andersen et al. [3] and Ju [36], where nu-merical solutions were provided, using nite elements. Andersen et al. [3] used anite-element time-domain analysis in convected coordinates with a simple upwindscheme, including a special set of boundary conditions permitting the passage ofoutgoing waves in the convected coordinate system. The modication of frequency-dependent damping to convected coordinates is described, and the convected for-mulation of boundary elements is presented and used for illustrating the eect ofhigh-speed motion.

Figure 2.8: Rayleigh wave propagation from a vertical harmonic rectangular load(the encircled area) moving along the surface of an elastic half-space inthe direction indicated by the vector and at the speeds: (a) v = 0 m/s,(b) v = 100 m/s and (c) v = 200 m/s. Dark and light shades of greyindicate negative and positive vertical displacements, respectively (fromAndersen et al. [3]).

Figure 2.8 shows the results for a load moving along the surface of a homogeneoushalf-space. The half-space has a mass density of 1550kg/m3, and the P- and S-wave speeds are 539 and 308 m/s, respectively. The load is applied vertically at thefrequency 40 [Hz] and is distributed uniformly over a 3 · 3 m2 rectangular area witha total intensity of 1 N.

As an alternative to the implementation of local transmitting boundary conditionsin a nite-element scheme for an innite domain, the FE scheme may be coupledwith a boundary-element scheme, as said before. In this way, a computation model is

17


achieved that combines the adaptability of the FEM with the radiation capabilitiesof the BEM. However, the coupling is not straightforward, since exterior loads areapplied as nodal forces in the FEM and as surface traction in the BEM.

Ju [36] used nite element analysis to investigate the behavior of the building vi-bration induced by high-speed trains moving on bridges. The model included thebridge, nearby building, soil and train.

Figure 2.9: Finite element mesh of the bridge and foundations (from Ju [36]).

A 3D semi-innite soil prole (−∞ < x < ∞; −∞ < y < ∞; −∞ < z < 0)supported a continuous railroad bridge along the x-axis with pile foundations. They-axis is perpendicular to the railroad bridge, and the negative z-axis is the soildepth direction. The top surface of piles is connected by a reinforced concrete capwith, which is buried underground. The simple bridge beam with the cross-sectionis supported on the rectangular pier using four bearing plates (gure 2.9). Otherthan the beam mass itself, the bridge beam supports an extra-mass for the railway,parapet and devices. The high-speed train investigated in the study was the modiedJapan SKS-700.

The results (partially shown in gure 2.10) demonstrated that trainload frequenciesare more important than the natural frequencies of bridges and trains for buildingvibrations. If the building natural frequencies approach the trainload frequencies,which equal an integer times the train speed over the compartment length, theresonance occurs and the building vibration will be large. Moreover, the vibrationshape is similar to the mode shape of the resonance building frequency.

Three common types of the foundation were used to isolate the building vibrationinduced by moving trains. Retaining walls cannot reduce signicantly either the hor-izontal or the vertical vibration. Pile foundations were found to reduce the verticalvibration somewhat but could not reduce the horizontal vibration successfully. Soilimprovement around the building reduced the building vibration eectively both inhorizontal and vertical directions. Furthermore, the last method can be constructedafter the building is built.

18

2.3. THE UIC REPORTS

Figure 2.10: Surface displacements of a nite element analysis under the train speedof 240 km/h (from Ju [36]).

2.3 The UIC reports

With the objective of dening the standards for the dynamic amplication factorson the high-speed railway bridges, in the early 1970's, the International Union ofRailways (UIC), through its Oce of Research and Experiments (ORE), studied thedynamic behavior of bridges, subjected to high-speed loads. During that decade,more than 350 measurements on 37 bridges, using dierent trains, were made, to-gether with studies on bridge models loaded with scaled prototypes. The resultswere ltered and numerical simulations were performed to extrapolate scenarios notpredicted in the measurements. From these studies, an expression for the dynamicamplication factor (DAF) was obtained:

1 + ϕ = 1 + ϕ′ + λϕ′′ (2.3)

where 1+ϕ′ is the DAF in an ideal track and K is a variable used for its calculation:

ϕ′ =K

1−K +K4, K =

v

2LΦn0

(2.4)

ϕ′′ includes irregularities in the track, and is the following, for speeds greater than22 m/s:

ϕ′′ =1

100

[56e

−(

LΦ10

)2

+ 50

(LΦn0

80− 1

)e−(

LΦ20

)2]

(2.5)

λ is a function related to the type of maintenance of the track, v is the speed of thetrain, n0 the rst natural frequency of the structure and LΦ the determinant length.For n0 between certain limits, as a function of LΦ, these coecients were establishedin the UIC leaet 776-1R [57]. Whenever these limits were not veried, a completedynamic analysis would have to be performed for the structure. From the eort of

19


trying to simplify the design calculation, the train model LM71 (see section 3.3 fordetails) was developed. Using the DAF and the static responses obtained with theLM71, the dynamic displacements were reasonable for the train speed achieved inthat decade.

With increasing train speeds, the dynamic amplications started to become muchhigher than predicted. In 1992, due to resonance in the structure, a bridge with 44 mbetween Hannover andWürzburg registered dynamic displacements 35% higher thanthe static ones. The reasons founded to explain this behavior were the following:

- Amongst the trains used by UIC to study the DAF, only one could achievespeeds up to 300 km/h. This train, called the Turbo Train, was 38.4 m longwith loads of 170 kN per axle. Nowadays, almost all the HST can achievethat speed and are much longer, causing the axles to excite the structure ina frequency close to resonance. The resulting amplication depends on thenumber of coaches, i.e., the duration of excitation, and the intensity of theloading;

- The damping on modern structures is lower than what was common on oldbridges.

Similar problems were experienced in short span bridges, where high-speed tracaected the ballast stability. Subsequent testing and analysis demonstrated thatthe ballast was subjected to accelerations of 0.7− 0.8g (7− 8 m/s2).

Acknowledging that the safety of the structures and the passengers was not assuredfor the moderns trains, the UIC decided to study the dynamic behavior of struc-tures subjected to high-speed trains in detail. Therefore, in 1999, the SpecialistSub-Committee D214 was formed by the, then known as, European Rail ResearchInstitute (ERRI). Its aim was to give guidance on the additional criteria to be sat-ised to ensure that the performance of structures on or about high-speed lines metthe necessary safety criteria to ensure satisfactory dynamic performance in serviceat speeds up to 300 km/h. From the studies conducted by the Committee, 9 reportswere produced. The subject and ndings of these reports will be discussed in thefollowing.

One of the studies undertaken by the D214 Committee was the denition of uniedmodels that could cover the dynamic eects caused by modern trains. The rst trywas made with new Univ-A load model, obtained from the Eurostar with dierentdistances between axles. However, the dynamic signature, S0, obtained with theUniv-A was not high enough to cover the responses obtained with the Talgo andVirgin trains. Hence, a new envelope was developed and adopted for the Eurocode[10]. This new envelope (gure 2.11) contemplated the possibility of new Virgintrains beign built with dierent spacing between axles, adding a constant value of5700 kN for wave lengths between 24 and 27 m. The comparison between the newenvelope and the dynamic signatures of common HSTs is shown in gure 2.12.

Subsequent studies were started with the objective of dening load models thatwould produce a signature as close as possible to that envelope. The result was the

20


Figure 2.11: Eurocode envelope.

Figure 2.12: Comparison of the Eurocode envelope with real trains.

Figure 2.13: Comparison between the Eurocode envelope and the HSLM's signature.

21


HSLM-A1 to A10 that will be properly dened in section 3.3. The signatures of theten trains are compared with the Eurocode envelope in gure 2.13

The HSLM-A was developed for almost every bridge, except short simply supportedbridges, for which was developed yet another load model, the HSLM-B. Once more,this model will be suitably dened in section 3.3.

The studies undertaken covered a large number of aspects concerning the dynamicbehavior of railway bridges. The study of the dynamic behavior can be made con-sidering a single moving force, a single moving mass, moving forces, or modellingthe complex vehicle-structure interaction. From the work of the Committee, it wasconcluded that greater accuracy in the prediction of resonance and dynamic eectsof HST travelling over typical bridges is obtained if the nature of the spacing ofthe axles is taken into account. Where the engineer wishes to investigate mass in-teraction, structure nite element updating techniques are more suitable for realtrains.

The rst report published [14] is a literature survey and report 2 gives recommen-dations for calculating bridge deck stiness [15]. In the third report [16], the impor-tance of damping was studied. The measured damping results show that:

- Reducing the damping to a single value is an over-simplication;

- Increasing the number of bridge categories does not produce satisfactory re-sults;

- A rule that takes into account the span of the bridge allow the estimates tobe rened.

The values of critical damping to be used for design are shown in table 2.3. Thelower limits should be used for design.

Table 2.3: Values of critical damping to be assumed for design purpose (from [16]).

Span Lower Limit of ξ(%)Steel and L<20 m ξ = 0.5 + 0.125(20− L)Composite L>20 m ξ = 0.5Prestressed L<20 m ξ = 1.0 + 0.07(20− L)Concrete L>20 m ξ = 1.0

Reinforced Concrete L<20 m ξ = 1.5 + 0.07(20− L)and Filler Beams L>20 m ξ = 1.5

The 4th report [17] focused on structure-vehicle interaction. According to this report,the eects of this phenomena are reduced by the presence of the vehicle suspension.Typical primary and secondary suspensions limit approximately 90% of the vehiclemass to accelerations around 0.1g. The full complex behavior of the vehicle primaryand secondary suspension eect and associated masses was modeled, and its dynamicbehavior studied. The results obtained shown that:

22


- Away from resonance, including or excluding vehicle mass interaction phenom-ena in the analysis made negligible dierence;

- At resonance, the analytical techniques based on the interaction predict re-duced dynamic responses in the structure, when compared to moving forcemodels. Both peak accelerations and deection are reduced;

- The reduction in peak eect displayed by interactive models, when comparedwith moving force models, at resonance, for continuous spans and for a se-quence of simply supported spans, is less than the reduction for a simplysupported span;

- The interaction models predict that resonance eects will occur at slightlylower speeds than travelling force models;

- The eects of interaction diminish with spans over 30 m.

Furthermore, the studies indicated that the use of an increase in structural damping,to simulate the interaction, was feasible. It is also mentioned that the single masscalculation techniques fail to represent the eects of a train of axles with varyingspacing, and therefore are not recommended. The more complex mass interactionmodels show only a small advantage over travelling force models. Concerning theapproximation provided by equation 2.3, it was concluded that, at train speeds awayfrom resonance, the over estimation of ϕ′ compensates the underestimation of thedynamic eects due to track irregularities ϕ′′, validating the formula for the vastmajority of cases.

The structure-rail interaction and the load distribution due to the track structurewere briey studied. Because of the composite action between track, ballast andbridge, the overall eective stiness of the structure is increased above that dueto the structure alone. As a result, the natural frequencies increase. This eectis more pronounced for short spans. The same is believed to happen with theload distribution phenomena, which is more signicant for spans of less than 20 m(according to Museros et al. [45]).

The studies undertaken show that it is essential to have an accurate estimate ofthe natural frequencis and mode shapes of the structure if realistic prediction ofthe dynamic behavior are to be made. Modeling support conditions, skew andbending stiness accurately is, therefore, mandatory.

The design criteria, requires that:

- The verication of maximum peak deck acceleration shall be regarded as aserviceability limit state for trac safety for the prevention of track and ballastinstability;

- The static eects of loads should be enhanced by dynamic factor to allow fordynamic increment due to impact, resonance and track irregularities. Nor-mally, the DAF (from equation 2.3) should be used;

23


- A check shall be carried out to ensure that the dynamic loading eects includ-ing resonance are covered by the DAF. If not, the greater eects shall be usedin all calculations;

- A check should be carried out to ensure that the fatigue loading at resonanceis covered by consideration of the stresses due to the load eects, consideringthe DAF.

A ow chart that summarises simple conservative checks is given in gure 2.14.

As mentioned previously, structures operating at resonance show extremely sensitivedynamic responses, depending on their damping level. Damping in structures occurbecause of energy losses during cycles of oscillation. As a result, the free vibrationsof structures diminish with time. All structures exhibit damping, and it is mainlydue to:

- Energy dissipation through bending of materials;

- Friction at supports and along structural boundaries;

- Energy dissipation from soil-structure interaction at the ends of bridges;

- Energy dissipation in ballast;

- Opening and closing of cracks in the material (especially concrete).

The 5th report [18] concentrated on the eect of track irregularities. These irregu-larities aect the dynamic behavior of railway bridges and can increase the dynamicload eects. The increase in the dynamic loading due to track irregularities increaseswith speed and decreases for longer bridge spans, and is mainly due to the loadingeects developed in the unsprung axle masses of the vehicles traversing the track.In order to avoid complex bridge and track prole specic dynamic calculations,railway bridge engineers increase the live load static eects by a factor of ϕ′′ asshown in equation 2.3. The prime purpose of the study was to investigate if theformula proposed by ORE Committees D23 and D128 was valid for the dynamiceects resulting from HST, due to resonance phenomena, on modern structures.

During the course of the D128 studies, the importance of damping was realizedand a number of calculations were carried out with zero damping as a lower boundvalue. However, some of the results obtained from such calculations were consideredto be unrepresentative, as the measured damping values of existing structures weresignicantly higher. Recent test results indicate that lower damping values shouldbe taken into account than those assumed in the derivation of UIC 776-1R [57]criteria for the design of bridges.

The validity of equation 2.5 was studied using numerical methods with and withouttrack defects at resonance. The values provided by equation 2.5 were compared

24


Figure 2.14: Flow chart to determine whether a dynamic analysis is necessary (from[22]).

25


with the calculated increase in deection and acceleration due to a track defect,respectively:

ϕ′′δ =δmax,dipδmax

− 1; ϕ′′acc =accmax,dipaccmax

− 1 (2.6)

where δmaxdip and accmaxdip represent the maximum responses of the beam with atrack dip, while δmax and accmax represent the same responses without the dip.

From the comparison of the values, it was concluded that the displacements fromthe trials were always below or comparable to the UIC values. For the accelerations,the calculated values where also below or comparable to the values obtained withequation 2.5, in the vicinity of train speed corresponding to resonance. Away fromresonance, the values appear to increase with decreasing speed. However, at lowspeeds, the deck acceleration caused by the impact of the heaviest axles is usually lessthan at higher speeds. Thus, any errors obtained from the usage of the UIC formulawill be less signicant, providing the standard of track maintenance is sucient toprevent wheel lift o.

However, from the studies undertaken and presented in this report, it may be con-cluded that the existing criteria for ϕ′ severely underestimates the dynamic factorfor an individual train when resonance phenomena occurs. Therefore, it is recom-mended that the limit of validity of ϕ′ should be taken as 200 km/h. Specic checksshall be made for trains travelling at higher speeds.

The eect of track irregularities were found to be very signicant and should betaken into account in the designing of railway bridges for speeds up to 350 km/h.From the calculations undertaken, it was shown that:

- Wheel lift o is likely to occur when large track defects are present, for speedsabove 260 km/h, which causes high frequency bridge deck accelerations toincrease considerably due to the impact from the unloaded wheel regainingcontact with the rail;

- Calculated values for ϕ′′ are always below or comparable to the values providedby equation 2.5, providing the standard of track maintenance is sucient toprevent wheel lift o.

Moreover, and in accordance with UIC 776-1R [57], when the track maintenanceis suciently exhaustive, the usage of ϕ′′/2 (λ = 0.5 in equation 2.3) is permitted,both for the dynamic deection and for the bridge deck acceleration.Dynamic Am-plication Factors such as Φ2 (for very well maintained tracks) and Φ3 (for commontracks) have also been suggested by past UIC studies, and are dened as:

Φ2 =1.44√LΦ − 0.2

+ 0.82 (2.7)

Φ3 =2.16√LΦ − 0.2

+ 0.73 (2.8)

26


where the equivalent span (or determinant length) LΦ coincides with the real one fora simply supported isostatic element, and an equivalence table is provided for otherstructural types. These factors have been developed in conjunction with LM71 andmust only be used with their associated load model. Φn represents the increase inLM71 necessary to cover the dynamic eects of all trains on all spans. Thus, it mustnot be applied to an individual load train, as it would most likely underestimate thedynamic eects of that particular train. On the other hand, dynamic factors suchas ϕ′ and ϕ′′ have been specically developed to allow the engineer to estimate thelikely maximum dynamic eects of a series of forces representing the axle loads of aparticular train.

It is suggested that the dynamic analysis of a structure should consider at least therst 3 or 4 modes of vibration. Filtering at a cut-o frequency of 20 Hz may notidentify critical behaviors.

Report 6 [19] studies the inuence of the mass and the stiness of the bridge, onthe magnitude of the displacement and acceleration. The results indicate that themaximum response of the structure at resonance is inversely proportional to its dis-tributed mass. The resonance condition produced by a series of forces at equidistantdistances d, is calculated from the time necessary for crossing the distance d at speedc which is equal to the k-multiple of the period of natural vibration 1/fj:

d

c=

k

fj, j = 1, 2, 3.., k = 1, 2, 3... (2.9)

Equation 2.9 provides the critical speeds:

ccr =dfjk, j = 1, 2, 3.., k = 1, 2, 3... (2.10)

Figures 2.15 and 2.16 show the inuence of the mass on the responses of a single-track bridge, for the k-multiple of the period of natural vibration of the structure.Therefore, it is recommended for designers to estimate the lower bound of the mass

Figure 2.15: Maximum mid-span acceleration on a single-track bridge with dierentmass (from [19]).

27


Figure 2.16: Maximum mid-span displacement on a single-track bridge with dierentmass (from [19]).

Figure 2.17: Maximum mid-span acceleration on a single-track bridge with dierentbending stiness (from [19]).

Figure 2.18: Maximum mid-span displacement on a single-track bridge with dierentbending stiness (from [19]).

28


considering the maximum deck accelerations permitted, and the higher bound con-sidering the lowest speeds at which resonant eects are likely to occur.

Concerning the stiness, the relation is proportional, i.e., increasing the stiness ofthe structure is benecial because it raises the critical speeds at which resonanceeects occur. Figures 2.17 and 2.18 show the inuence of the stiness on the a single-track bridge response, L being the span and f the deection at mid-span. However,increasing the stiness has a major impact upon cost. Hence, for economic design, itis necessary to make an accurate prediction of the correct stiness for the structure.Central to achieving a suciently accurate understanding of the behavior of anystructure is the employment of realistic values in the equations for bending andtorsional stiness. Yet, accuracy of calculation of reinforced concrete structures iscomplicated when signicant amounts of cracking occur. Where possible, in thedesign of new structures, the strategy should be to ensure that tensile stresses arecontrolled to the extent that cracked sections need not be taken into account.

The 7th report [20] undertakes studies related with the calculation of very complexstructures and suggests programs to solve the problem. For structures being de-signed that show unsatisfactory dynamic behavior, the designer should consider thefollowing options:

- Using a more rened analysis technique, that could overcome some simplica-tions adopted in the previous analysis;

- Using added damping to simulate vehicle-bridge interaction;

- Using a program which allows for vehicle-bridge interaction;

- Using a better estimate for damping, based on tests on identical structures;

- Increasing the mass of the structure;

- Allowing for the distribution of point axle forces by the track by using movingdistributed loads (particulary important in short spans);

- Increasing the stiness of the structure, through a reduction on the extend ofthe cracking in concrete, increasing the quality of the concrete and its modulusof elasticity or increasing the cross-section;

- Changing the structure;

- Restraining trains speeds.

In the assessment of existing lines for high-speed, the Train Signature Method en-ables the dynamic eects of dierent trains at resonance and away from resonanceto be compared without reference to bridge or track characteristics. The train sig-nature represents the dynamic excitation characteristics of a particular trains and isindependent of the structure. If the magnitude of the train signature for a new trainis less than that of the trains using the structure, then it will be satisfactory for thenew train. Furthermore, the requirements to ensure trac safety when assessing anexisting structure are similar to the requirements for designing a new structure:

29


1. A check should be carried out to ensure that the DAF including resonanceand excessive vibration eects is covered by the load eects from Φ·LM71 or(1+ϕ)·HSLM. When the load eects of particular trains exceed the load eectspreviously referred, the greater eects shall be used in all of the assessment;

2. The verication of maximum peak deck acceleration shall be regarded as aserviceability limit state for trac safety, to prevent track instability;

3. A check should be carried out to ensure that the fatigue loading at resonanceis covered by the stresses due to the static load eect, increased by the DAF.

Report 8 [21] focuses on the maximum permitted bridge deck accelerations. Fromthe results and the observed behavior of track and ballast subjected to resonantloading, it can be seen that critical behavior starts for bridge deck accelerations of0.7−0.8g. Applying a factor of safety of 2, the maximum permitted deck accelerationwas xed at 3.5 m/s2 for ballasted tracks. For direct fastening decks with track andstructural decks designed for high-speed trac, that limit is 5 m/s2.

Once again, it was concluded that the 20 Hz frequency limit for cut o is not validfor a large number of structures. It is recommended that bridge deck accelerationshould be checked for all elements supporting the track at frequencies up to 30 Hzor 2 times the frequency of the rst mode of vibration.

Figure 2.19: Transfer function between bridge deck acceleration (Bm) and ballastacceleration (Ba) (from [21]).

An amplication factor was dened as the ratio between the maximum accelerationwithin the ballast layer and the applied acceleration at the bottom of the ballastlayer and is shown in gure 2.19. This transfer function shows the non-linearityof the ballast bed for bridge deck accelerations higher that 0.6g. The existence ofballast mats increases the amplifying eects: the maximum amplication factor was15% without mats and 60% with ballast mats.

30


The minimum ballast depth recommended is 250 mm, to ensure the maintainabil-ity of the track, a satisfactory transition between formation stiness on and o thebridge and other items relevant for the satisfactory performance of the track struc-ture. However, it may be noted that on a number of high-speed lines, a ballastdepth of 350 mm beneath the underside of sleepers has been adopted with satisfac-tory results.

The last report [22], synthesizes the results obtained by the D214 Research Com-mittee and proposes further research in the area.

With the studies and experiments developed in report D214, the UIC, through thatCommittee, found the answer for numerous problems related to HSTs and placedthe foundations for the norms used nowadays in the vast majority of the Europeancountries, namely, the Eurocodes.

31

Chapter 3

Important Theoretical Concepts

The aim of this chapter is to introduce the theoretical concepts that will be used inthe investigations undertaken in chapters 4 and 5.

The two rst sections bring the most important concepts for understanding the workundertaken in this dissertation. The dierences between the static and the dynamicbehavior of a structure will be referred in the rst section, so that the reader canfully understand the inuence of the dynamic loads and the occurrence of resonance.Section 3.2 will explain how to estimate the stiness of bridge supports, used for thestiness of the springs in the FE analysis of chapters 4 and 5.

Afterwards, the most common high-speed load models will be dened in section 3.3.Section 3.4 will shortly present the theoretical approach to the 2D beam element,that gave the basis to the 2D nite beam element. The analytical solution pro-posed by Frýba and used for the theoretical studies undertaken in chapter 4 will bepresented in subsection 3.4.1.

The last three sections will be focused on the methods used (Mode Superpositionand FEM) for the numerical calculations obtained with Abaqus/Brigade.

3.1 Dynamic behavior of HSR bridges

From the late 1950's, the train speed record has been increasing continuously and theexperimental maximum speed achieved by the TGV train, in France, is already 515km/h. However, while the speed increases, the safety and the comfort of passengertrains have to be kept at the same high level. To do so, the interaction between thehigh-speed train and the structure has to be deeply studied. This interaction canbe seen in both ways: the dynamic impact of the running axles aects the servicelife and the working state of the structure and, at the same time, the vibration ofthe structure has an eect on the stability and the safety of the train. The way bywhich the two subsystems interact with each other is determined primarily by theinherent frequencies of the two subsystems and the driving frequency of the movingvehicles.

33

CHAPTER 3. IMPORTANT THEORETICAL CONCEPTS

Occurrence of resonance phenomenon in bridges can be observed for trains runningat speeds close to the critical speed. As the passengers loading is not a deterministicquantity and it is believed to by low, the bridges response obtained by means ofanalytical method ignores such loading and considers only the load induced by thetrain itself. Generally, the maximum response induced by the crossing train isproduced by the axles located between the heaviest locomotive and motorized carand the increase of response brought by the additional loading due to passengersdoes not aect the maximum response of the bridge. However, in the case of trainscrossing at critical speed, resonance, occurring due to the repeated loading generatedby wheel loads of the coaches crossing continuously at regular interval, ampliessignicantly the maximum response of the bridge produced by the locomotive andmotorized car.

Consequently, when resonance due to train running at the critical speed occurs, themaximum response generated by the locomotive is amplied to a scale dependingon the duration of the resonance produced by the coaches. Furthermore, structuresoperating at resonance show extremely sensitive dynamic responses according totheir damping level.

Figure 3.1: Dynamic amplication factor as a function of the frequency ratio β (fromClough and Penzien [12]).

When a SDOF system is subjected to periodic excitation with a frequency similarto the natural one of the system (β = w/p ≈ 1), resonance will occur and thestatic response will be amplied up to ust/2ξ

√1− ξ2 (gure 3.1), depending on the

damping, where ust is the static response, as can be seen in equation 3.2. FromChopra [11] we have that:

DAF =udynustat

= [(1− β2)2 + (2ξβ)2]−12 (3.1)

where β represent the ratio between the applied load frequency and the natural freevibration frequency of the bridge. Hence:

∂

∂βDAF = 0⇔ β =

√1− 2ξ2

34

3.2. THEORETICAL ESTIMATION OF THE SUPPORT STIFFNESS

Thus:

DAF (β =√

1− 2ξ2) =1

2ξ√

1− ξ2(3.2)

The dynamic response will reach the limit value, steady-state amplitude, for a certaintime, depending on the damping. The more the damping increase, the less time isneeded to reach the steady-state amplitude. Therefore, the more numbers of coachesare arranged successively with equally spaced axles, the more the response will beamplied up to ust/2ξ (considering low damping). Therefore, during the design ofhigh-speed railway bridges, studying the dynamic behavior of bridge arises as anessential issue to guarantee safety of passengers and riding conditions.

3.2 Theoretical estimation of the support stiness

The estimation of the support stiness as a function of the dynamic proprieties ofthe soil is a very complex problem that was deeply studied by Richart and Lysmerin the decade of 1960 (Lysmer and Richart [43] and Richard et al. [51]), giving veryimportant contribution on the subject for the scientic community. In 1962, Barkan[7] had already suggested one expression to calculate the vertical spring stiness forrigid rectangular footings resting on elastic half-space:

Kz =G

1− νβz√Ab (3.3)

where Ab represents the area of footing, G the soil shear modulus, ν the soil Poison'sratio and βz is given in gure 3.2.

Figure 3.2: Coecients βx, βz and βψ for rectangular footing (from Whitman andRichart [58]).

35


Before that, Timoshenko [56] had derived the vertical spring stiness if the rigidfooting was circular, also resting on elastic half-space:

Kz =Gr

1− ν(3.4)

However, if one denes r as the equivalent radius:

r =

√Abπ

it is possible to estimate the stiness of rectangular footing using Timoshenko'sexpression.

In the beginning of the 21th century, George Gazetas [29, 30] studied the dynamicinteraction of spread footings and pile groups, suggesting expressions, not only forthe static stiness, but also for the frequency dependent stiness. For the surfacefoundation on homogeneous half-space, Gazetas suggested that the static stinessshould be obtained from:

Kz =2GL

1− ν

(0.73 + 1.54

(B

L

)0.75)

(3.5)

where B and L represent the semi-width and semi-length of the circumscribed rect-angle, L>B.

The presence of bedrock at shallow depth, rather than having natural soil depositsextending to practically innite depth as the homogeneous half-space implies, modi-es the static stiness. Particularly sensitive to variations in the depth of the bedrockis the vertical stiness. Thus, for footings lying on an homogeneous stratum withheight H, over a bedrock, the expression is:

Kz =2GL

1− ν

(0.73 + 1.54

(B

L

)0.75)(

1 +B/H

0.5 +B/L

)(3.6)

To obtain a dynamic stiness, Gazetas suggested that the values for the staticstiness, Kz, should be multiplied by a factor depending on the frequency.

The expression for the calculation of the dynamic stiness and dashpot for arbitraryshaped foundations on homogeneous half-space is shown in table 3.3, together withthe graphs in gure 3.4. In table 3.4 and gure 3.5, the solutions for the dynamicstiness and dashpot coecients for surface foundations on homogeneous stratumover bedrock are presented. For the usage of the tables: G, ν, Ab, B, L and H are asalready dened and Iaa is the area moments of inertia about the a-axis of the actualsoil-foundation contact surface. The solution for the partially and fully embeddedfoundations and for the foundation on soil stratum over half-space were also found,but will not be discussed in this study.

36


Figure 3.3: Dynamic stiness and dashpot coecients for arbitrary shaped founda-tions on homogeneous half-space surface (from Gazetas et al. [30]).

37


Figure 3.4: Graphs accompanying table 3.3 (from Gazetas et al. [30]).

38


Figure 3.5: Dynamic stiness and dashpot coecients for surface foundations onhomogeneous stratum over bedrock (from Gazetas et al. [30]).

39


Figure 3.6: Graphs accompanying table 3.5 (from Gazetas et al. [30]).

40

3.3. LOAD MODELS

3.3 Load models

Whenever the designer knows in advance which trains will load the track or bridgestructure, he or she should use the axle load induced by that train for design purpose.

Nowadays, the vast majority of HSTs are grouped in three train architectures:

1. Conventional, where each carbody is supported by two independent bogies.Two high-speed trains of this type are the VIRGIN and the ICE2;

Figure 3.7: Conventional Train.

2. Articulated with single-axle bogie, where each bogie, placed between two sec-tions, has one single axle. The new Talgo is an example;

Figure 3.8: Articulated train with single axle bogie.

3. Articulated with Jakobs-type bogie, named after Wilhelm Jakobs (1858-1942),where each bogie is placed between two carbody sections. Thalys, Eurostarand TGV share this architecture.

Figure 3.9: Articulated train with Jakobs-type bogie.

Some HSTs are equipped with tilting mechanisms that enables increased speed onregular railroad tracks. As any vehicle rounds a curve at speed, independent objectsinside it exert centrifugal force since their inherent momentum forward no longerlies along the line of the vehicle's course. Tilting trains are designed to counteractthis discomfort. This type of trains may be constructed such that inertial forcesthemselves cause the tilting, commonly referred to as passive tilt, or it may beactively induced by a computer-controlled mechanism, referred to as active tilt.This improvement is shared by the Talgo, the ICE-T, the Pendolino and the SwedishX2000, amongst others.

41


Although, some times, the train loading the structures is known, whenever the trackis designed for dierent types of trains, load models, that produce an envelope largeenough to contemplate the dynamic eects of all trains, have to be found. The his-torical background about the development of most commonly used load models wasreferred in 2.3. To perceive the static eects of normal trac in continuous bridgeand heavy train trac, load models SW/0 and SW/2 were developed, respectively.The loads for the two load models are dened in table 3.1 and should be multipliedby a factor α, depending on the type of trains the line will serve. For lines subjectedto heavy train loads, α can be as high as 1.46. For light loaded lines, the value candecrease to 0.75.

Figure 3.10: SW/0 and SW/2 load models.

Table 3.1: Values to adopt for SW/0 and SW/2 load models.

Load Model qvk [kN/m] a [m] c [m]SW/0 133 15.0 5.3SW/2 150 25.0 7.0

Besides de two SW models, based on the eects observed in structures due to 6 realtrains, the LM71 load model was developed (see 2.3 for more information). Theloads, likewise happens with SW/0 and SW/2, should be multiplied by α.

Figure 3.11: LM71 load model.

From the studies undertaken for the Eurocode [10], yet another load model wasdeveloped. The HSLM-A was dened with the architecture shown in gure 3.12,but dierent axle characteristics. The 10 models are dened in table 3.2. TheHSLM-A should be used for almost every bridge, except short simply supportedbridges, for which the HSLM-B should be used instead.

42

3.3. LOAD MODELS

Figure 3.12: High speed load model A axle conguration (from [10]).

Table 3.2: High speed load model A (from [10]).

HSLM N D [m] d [m] P [kN]A1 18 18 2.0 170A2 17 19 3.5 200A3 16 20 2.0 180A4 15 21 3.0 190A5 14 22 2.0 170A6 13 23 2.0 180A7 13 24 2.0 190A8 12 25 2.5 190A9 11 26 2.0 210A10 11 27 2.0 210

Figure 3.13: High speed load model B axle conguration (from [10]).

Figure 3.14: Parameters N and d as functions of L, for HSLM-B (from [10]).

43


3.4 2D beam theory

The structural analysis of bridges is usually based on the linear elasticity theory.The classic elasticity theory assumes a linear dependence between strain and stress,disregarding the non-linear behavior of some materials.

A beam in a plane frame structural model can be aected by transversal and longi-tudinal loadings. Standard equilibrium equations are thereby formulated comparingthe internal forces N , V and M with the applied external loads px and py.

Figure 3.15: Beam element with internal forces and applied loading in positive di-rection.

The equation of equilibrium can be obtained from gure 3.15 and are the following:

∂N

∂x+ px = 0 (3.7)

∂V

∂x+ pz = 0 (3.8)

∂M

∂x− V = 0 (3.9)

For the 2D analysis of a beam, two theories are commonly used: Euler-Bernoulliand Timoshenko beam theory. A brief description of the two theories will be pre-sented in sections 3.4.1 and 3.4.2. Further information on the subject is available inTimoshenko [56].

3.4.1 Euler-Bernoulli beam

According to Hibbitt et al. [32], Euler-Bernoulli beams do not allow for transverseshear deformation, plane sections initially normal to the beam's axis remain plane (if

44

3.4. 2D BEAM THEORY

there is no warping) and normal to the beam axis. They should be used only to modelslender beams, i.e., the beam's cross-sectional dimensions should be small comparedto typical distances along its axis (such as the distance between support points orthe wavelength of the highest mode that participates in a dynamic response).

In Abaqus/Brigade, the Euler-Bernoulli beam elements use a consistent massformulation and cubic interpolation functions, which makes them reasonably accu-rate for cases involving distributed loading along the beam. Therefore, they are wellsuited for dynamic vibration studies, where the d'Alembert (inertia) forces providesuch distributed loading. The cubic beam elements are written for small-strain,large-rotation analysis. They may not be appropriate for torsional stability prob-lems due to the approximations in the underlying formulation and cannot be usedin analysis involving very large rotations, quadratic or linear beam elements shouldbe used instead.

Analytical solution for the Euler-Bernoulli beam

The analytical response of the Euler-Bernoulli beam was studied by Frýba [28], fora simply supported beam subjected to any set of loads, travelling to a certain speed.The derivation of that solution is included in the following.

If a beam of span l is subjected to a row of N axle forces Fn, moving at constantspeed c, its behavior is described by the Euler-Bernoulli partial dierential equation:

EI∂4v(x, t)

∂x4+ µ

∂2v(x, t)

∂t2+ 2µωd

∂v(x, t)

∂t=

N∑n=1

εn(t)∂(x− xn)Fn (3.10)

where EI is the bending stiness, µ the mass per meter, v(t, x) represents the verticaldeection of the beam, ωd is the damped circular frequency, Fn is the nth axle forceand:

εn(t) = h(t− tn)− h(t− Tn) (3.11)

with h(t) being the Heaviside unit function:

h(t) =

0 for t < 01 for t ≥ 0

(3.12)

tn and Tn being, respectively, the instants Fn enters and leaves the beam:

tn =dnc

(3.13)

Tn =l + dnc

(3.14)

and ∂(x) is the Dirac delta function, where:

xn = ct− dn (3.15)

45


with dn being the distance between the nth axle and the rst axle of the train.

Taking in account the boundary conditions of the simply supported beam, and theinitial values of the problem being zero, the natural frequencies, ωj, fj and shapesof natural vibration vj(x) are:

ω2j =

j4π4

l4EI

µ(3.16)

fj =1

2πwj (3.17)

vj(x) = sin

(jπx

l

)(3.18)

Using the mutual relations of the Fourier integral transformation and the Laplace-Carlson integral transformation, the solution for the problem comes in the followingform:

v(x, t) =∞∑j=1

N∑n=1

v0FnFjww2

i [f(t− tn)h(t− tn)−

−(−1)jf(t− Tn)h(t− Tn)]sin

(jπx

l

)(3.19)

where (note that there is an error in the expression shown in Frýba [28]):

f(t) =1

ω′jD

[ω′jjωsin(jωt+ λ)− eωdtsin(ω′jt+ ϕ)

](3.20)

with v0 representing the deection of a simple beam at its center due to the forceF = Fn place at the same point:

v0 ≈Fl3

48EI(3.21)

The rst term between the parenthesis on equation 3.20 stands for forced vibrationdue to the moving loads and the second term represent the free damped vibration.

Knowing that the bending moment on the beam can be expressed by:

M(x, t) = −EI ∂2v(x, t)

∂x2(3.22)

the result can be calculated in a similar way, expressed as:

M(x, t) =∞∑j=1

N∑n=1

M0FnFj3ww2


−(−1)jf(t− Tn)h(t− Tn)]sin

(jπx

l

)(3.23)

46

3.4. 2D BEAM THEORY

where M0 is the bending moment at the center of a simple beam due to the forceF = Fn applied to the same point:

M0 ≈Fl

4(3.24)

In a similar way, and knowing that the acceleration can be expressed as:

a(x, t) =∂2v(t, x)

∂t2(3.25)

the acceleration comes as:

a(x, t) =∞∑j=1

N∑n=1

v0FnFjww2


−(−1)j f(t− Tn)h(t− Tn)]sin

(jπx

l

)(3.26)

where:

f(t) = −ω′2j − w2

d

ω′jd

[jωω′j

ω′2j − w2d

sin(jωt+ λ)− eωdtsin(ω′jt+ ϕ)

](3.27)

In the response equations, the following relations were used:

ω =πc

l(3.28)

ω′j = ω2j − ω2

d − 2iωdωj (3.29)

Ω2j = ω′2j + ω2

d ≈ ω2j (3.30)

D2 = (Ω2j − j2ω2)2 + 4j2ω2ω2

d ≈ (ω2j − j2ω2)2 (3.31)

λ = arctan

(−2jωωd

Ω2j − j2ω2

)(3.32)

ϕ = arctan

(2ω′jωd

ω2d − ω′2j + j2ω2

)(3.33)

ϕ′ = ϕ+ arctan

(2ω′jωd

ω′2j − ω2d

)(3.34)

Note that the approximations used with Ωj and D are valid only for low dampingsystems.

With equation 3.19, 3.23 and 3.26, Frýba was able to describe the response of asimply supported beam to a train passing at constant speed c.

47


3.4.2 Timoshenko beam

Timoshenko beams allow for transverse shear deformation. They can be used forthick as well as slender beams. For beams made from uniform material, shearexible beam theory can provide useful results for cross-sectional dimensions up to1/8 of typical axial distances or the wavelength of the highest natural mode thatcontributes signicantly to the response. The Timoshenko beams can be subjectedto large axial strains but the axial strains due to torsion are assumed to be small.In combined axial-torsion loading, torsional shear strains are calculated accuratelyonly when the axial strain is not large.

The FE beam elements used for most part of the research undertaken in this disser-tation were based on the Timoshenko theory. For further research in the subject,one should study Cook et al. [13] or amd W. Wunderlich [2]. Abaqus assumes thatthe transverse shear behavior of Timoshenko beams is linear elastic with a xedmodulus and, thus, independent of the response of the beam section to axial stretchand bending. For most beam sections Abaqus will calculate the transverse shearstiness values required in the element formulation.

3.5 Mode Superposition Method

The Mode Superposition Method provide higher accuracy, and requires less com-putational eort than the traditional step-by-step integration solution technique.Besides, the user can choose the number of modes to use in the response. Otheradvantage is that modal damping can be used in addition to (or instead of) Rayleighdamping. When experimental data are available, modal damping gives a more ac-curate representation of the damping in the system.

However, Modal Superposition is not suitable for problems such as shock loads orimpacts, where the higher frequency modes are excited. In these cases, many modesare required, and the cost to calculate these modes can signicantly oset the sav-ing in time. The method is best suited to structures where the lower frequenciesdominate the response. Typically, 10 modes will provide good accuracy for theseproblems, although prior frequency analysis is required. Yet, the most serious dis-advantage of this approach is that it is not capable of handling any non-linearity inthe solution.

For its simplicity, this method was chosen as the basic method for solving the dy-namic equations of the bridges studied, since it is a very powerful method used toreduce the number of equations to be solved in a dynamic response analysis. Alltypes of loading can be accurately approximated by linear functions within a smalltime increment. An exact solution exists for this type of loading and this solutioncan be computed with a trivial amount of computer time for equal time increments.Using the Mode Superposition Method, responses of the individual modes of thestructure are calculated separately and then combined to produce the total responseof the structure. The method will be briey presented here, for more information

48

3.5. MODE SUPERPOSITION METHOD

one should see Bathe [8].

To solve the linear dynamic response of structures subjected to periodic loading itis only necessary to add a corrective solution to the transient solution for a typicaltime period of loading. Hence, only one numerical algorithm is required to solvea large number of dierent dynamic response problems in structural engineering.Participating mass factors can be used to estimate the number of vectors requiredin an elastic analysis. The use of mass participation factors to estimate the accuracyof a non-linear analysis can introduce signicant errors. This is mainly due to theexistence of internal non-linear forces, in opposite directions, that do not producea base shear. A dynamic load participation ratio is dened and can be used toestimate the number of vectors required for other types of loading.

The dynamic force equilibrium equation can be written in the following form as aset of N second order dierential equations:

Mu(t) + Cu(t) + Ku(t) = F(t) =J∑j=1

fj · gj(t) (3.35)

All possible types of time-dependent loading, including wind, wave and seismic, canbe represented by a sum of J space vectors fj, which are not a function of time, andJ time functions gj(t), where J cannot be greater than the number of displacementsN . M, C, K and F represent the mass, damping and stiness matrices, and theexternal force vector; u represents the displacement, u stands for the velocity and urepresents the acceleration.

The number of dynamic degrees-of-freedom is equal to the number of lumped massesin the system. Many publications advocate the elimination of all massless displace-ments by static condensation prior to the solution of equation 3.35. The static con-densation method reduces the number of dynamic equilibrium equations to solve;however, it can signicantly increase the density and the bandwidth of the condensedstiness matrix.

The fundamental mathematical method that is used to solve equation 3.35 is theseparation of variables. This approach assumes the solution can be expressed in thefollowing form:

u(t) = ΦY(t) (3.36)

Where Φ is an N by L matrix containing L spatial vectors which are not a functionof time, and Y(t) is a vector containing L functions of time. From equation 3.36 itfollows that:

u(t) = ΦY(t) (3.37)

u(t) = ΦY(t) (3.38)

49


Prior to solution, we require that the space functions satisfy the following mass andstiness orthogonality conditions:

ΦTMΦ = I (3.39)

ΦTKΦ = Ω2 (3.40)

Where I is a diagonal unit matrix and Ω2 is a diagonal matrix which may or maynot contain all the natural frequencies. It should be noted that the fundamentalsof mathematics place no restrictions on these vectors, other than the orthogonalityproperties.

After substitution of equations 3.36, 3.37 and 3.38 into equation 3.35 and the pre-multiplication by ΦT, the following L equations is produced:

IY(t) + DY(t) + Ω2 =J∑j=1

pj · gj(t) (3.41)

Where pj = ΦTfj and are dened as the modal participation factors for time func-tion j. The term pnj is associated with the n− th mode. For all real structures theL by L matrix D is not diagonal; however, in order to uncouple the modal equationsit is necessary to assume that there is no coupling between the modes. Therefore,it is assumed to be diagonal with the modal damping terms dened by:

Dnn = 2ξnωn (3.42)

Where ξn is dened as the damping ratio in the n− th mode. For more informationon the construction of matrix D see Karoumi [37].

A typical uncoupled modal equation, for linear structural systems, has the followingform:

yn(t) + 2ξnωnyn(t) + ω2nyn(t) =

J∑j=1

pj · gj(t) (3.43)

3.6 Modeling in Abaqus/Brigade

The models of the three structures studied in chapters 4 and 5 were made in thegraphical interface of Abaqus. Abaqus/Cae is a user-friendly template dividedby dierent modules. Each module is a step in the modelling process, presentedin logical sequence. The rst modules, described in section 3.6.1, give shape andmaterial properties to the structure. After the model is dened, the analysis starts,described in section 3.6.2. The 3rd section describes how the results are visualized

50

3.6. MODELING IN ABAQUS/BRIGADE

and the last section explain the advantages of the Brigade pre-processor. Fordetailed information on the subject, one should see Hibbitt et al. [32] and [1].

3.6.1 Basic modules

The rst module of Abaqus is the Part module, where the dierent parts of thestructure are drawn (possibly with the help of the Sketch module), without con-nections between each other. In the Property module, the sections and proles aredened, as well as the properties and material behavior (elastic, plastic, thermal,acoustic or electric, between others). The Assembly module assembles the dierentparts of the structure, with sti connections or with connectors dened in the In-teraction module. In this module almost all the real structural connectors can besimulated from a vast interaction library.

The next level is the denition of the applied loads and the boundary conditions,both done with the Load module. All kind of mechanical, electric or acoustic bound-ary conditions and applied loads are available in the pre-dened library. The lastbasic denition is done in the Mesh module. Dierent mesh types can be applied inall the structure or in partitioned parts of it.

3.6.2 Analysis type

The analysis type is dened in the Step module. There are two basic procedures todetermine the response of the structure: general analysis steps, which can be usedfor both linear or non-linear analysis, and linear perturbation steps, which can beused only to analyze linear problems. Loading conditions are dened dierently forthe two cases, and the time steps are dierent as well. Therefore, the results shouldbe interpreted dierently.

General static analysis

A general analysis step is one in which the eects of any non-linearities present inthe model can be included. The starting condition for each general step is the endingcondition from the last general step, with the state of the model evolving throughoutthe history of general analysis steps as it responds to the history of loading. Thenon-linearity of the real structure can arise from: material non-linearity, geometricnon-linearity and boundary non-linearity.

The general analysis procedures in ABAQUS oer two approaches for controllingincrementation: automatic control or direct user control. In non-linear problems,the challenge is always to obtain a convergent solution in the least possible compu-tational time. In these cases automatic control of the time increment is usually moreecient because Abaqus can react to non-linear response that you cannot predictahead of time.

51


Linear eigenvalue analysis

The frequency extraction procedure is a particular linear perturbation procedurethat performs eigenvalue extraction to calculate the natural frequencies and thecorresponding mode shapes of a system. The linear perturbation response has noeect as the general analysis is continued. The step time of linear perturbation steps,which is taken arbitrarily to be a very small number, is never accumulated into thetotal time. The frequency extraction procedure uses eigenvalue techniques to extractthe frequencies of the current system, and two dierent algorithms are available: theLanczos method and the subspace iteration. The rst is generally faster when a largenumber of eigenmodes is required for a system with many degrees-of-freedom. Thesubspace iteration method may be faster when only a few (less than 20) eigenmodesare needed.

3.6.3 Visualization of the results

The Visualization module provides graphical display of nite element models andresults. It obtains model and result information from the output database andprovides control over the information that is placed in the output database, bymodifying the output requests in the Step module. Amongst others, it is possibleto plot:

- Undeformed and deformed shape;

- Contours: the values of an analysis variable such as stress or strain at a spec-ied step and frame of your analysis;

- Symbols: the magnitude and direction of a particular vector or tensor variableat a specied step and frame of your analysis;

- XY data;

- Time history, scale factor and harmonic animation;

- Paths through the model, with the respective values.

3.6.4 Brigade add-ons

Abaqus supplement Brigade1 adds 3 new modules to the graphical interface. Twoof them give the possibility of adding static live loads and load combinations, butwere not used in this dissertation. The other one, though, was very useful for thestudies undertaken. It added a pre-processor, that automatically calculated theamplitude functions for each nodes of the mesh, for a train passing any given speedor group of speeds. It is called Dynamic Live Load module and has 3 main options:

1Supplement developed by Scanscot Technology, see http://www.scanscot.se/

52

3.6. MODELING IN ABAQUS/BRIGADE

Track Manager, Vehicle Manager and Dynamic Live Loads Manager. The trackdenition, with the entrance point of the train in the bridge, the possibility of using2 or only 1 rail (for 2D analysis), the delay of the rail loading (to simulate curves)and the load direction, are set in the Track Manager. The Vehicle Manager has themost common load models already dened but allows user dened load models. Theuser can dene any new train by adding each axle load and its distance to the rstaxle. Finally, the Dynamic Live Loads Manager puts together the track and theload model, for a speed or a span of speeds dened by the user. One can also choosethe analysis procedure: modal or direct time integration and the time period of thetrial. If Mode Superposition is chosen, the time increment, the damping (direct orcomposite) and the number of eigenmodes to account for during the analysis canbe changed. On the other hand, if it was chosen to analyze the model with directtime integration, the characteristics of the increment should the dened, and thepossibility of non-linear geometry is added.

53

Chapter 4

Theoretical Behavior of Simple

Bridges

In this chapter, a theoretical analysis of two common railway bridges will be pre-sented. One single span and one double span composite bridges from the BothniaLine (the new Swedish HSR) will be used for the study. No comparison with exper-imental measurements will be made at this stage.

The study of the two bridges will not follow the same pattern. The dynamic behaviorof a bridge is such a large subject, that it was chosen to present the results indierent ways. Both studies will be mainly based on the FEM but, for the singlespan bridge, a comparison with the analytical solution for the Euler-Bernoulli beamwill be performed.

The FE program used is Abaqus with a pre-processing unit for the dynamic liveloads: Brigade. The pre-processor will be used for the denition of the amplitudefunctions of the dynamic loads. To solve the numerical equations of the analyticalsolutions and nd the maximum responses of the FEM matrices, the commercialsoftware MatLab is used (see MathWorks [44] and Ortigueira [47]).

4.1 Single span bridge

4.1.1 The bridge

The Banafjäl bridge is a 42 m long, simply supported bridge, which carries oneballasted track. The bridge is a composite structure, with an ordinary reinforcedconcrete deck supported by two steel beams, and has the following physical proper-ties:

- Mass of the composite section, m = 10700 kg/m;

- Density of ballast, ρballast = 2000 kg/m3;

55

CHAPTER 4. THEORETICAL BEHAVIOR OF SIMPLE BRIDGES

- Thickness of ballast, hballast = 0.6 m;

- Width of ballast, bballast = 6.2 m.

The composite cross-section was homogenized, so that the material properties ofsteel could be used for the calculations. After the homogenization, the followingcharacteristic values were used on the model:

- Modulus of elasticity, E = 210 · 109 Pa;

- Moment of inertia, I = 0.62 m4;

- Area, A = 0.57 m2;

- Linear mass, µ = 1814 kg/m;

- Density, ρ = 31824.6 kg/m3.

4.1.2 FE model

The FE model of the bridge was developed using Abaqus FE graphical interface,with the pre-processor Brigade. Information about the general usage of the pro-grams is described in section 3.6.

The cross-section of the single span beam model was a generalized prole with theproperties of the Banafjäl bridge. The supports were, initially, considered sti tostudy its dynamic behavior and so that the results could be compared with theanalytical solution. The rst model is shown in gure 4.1, represented with stisupports.

Z

Y

X

K

K

1

2

3

Figure 4.1: FE model of the single span bridge.

56

4.1. SINGLE SPAN BRIDGE

As can be seen in gure 4.1, only the permanent parts of the structure were mod-eled. International System units were used in the model denition and analysis.The damping was considered as direct damping (see Hibbitt et al. [32] for moreinformation on the subject) and equal for the whole beam. The value chosen wasξ = 0, 5%, according to [16] and table 2.3.

The embankments were considered sti enough to lock the longitudinal DOFs at theends of the beam. Nevertheless, a study was made without restraining the longi-tudinal displacement, and the discrepancies in the results were negligible. Becausethe bridge is simply supported, the rotations at the ends were dened as free.

For the sensitivity study, the supports were replaced by springs connected to the soil,representing the foundations. The springs permitted free rotation and horizontaldisplacement of the nodes, and the stiness was only noted for vertical displacement.

The dynamic loads were added in the Dynamic Live Load module of Brigade,where the HSLM-A1 (see section 3.12 and table 3.2) was already dened. It wasdecided to run only the HSLM-A1, because the objective of the study was a com-parison of the responses for dierent support condition, and not a complete studyof the structural behavior.

4.1.3 Frequency analysis and convergence study

The eigenfrequencies extraction was made using both the FE model and the an-alytical solution proposed by Frýba (described in 3.4.1). To solve the numericalequations, the commercial software MatLab was used.

The exact circular bending frequencies, for mode j, of a simply supported Euler-Bernoulli beam are given by the equation:

wj =

√(jπ

l

)4EI

µ(4.1)

For the Banafjäl bridge, the exact eigenfrequencies [Hz] for the rst ten bendingmodes, using equation 4.1, were calculated as:

f exact =

2, 399, 5421, 4738, 1759, 6485, 88116, 90152, 68193, 24238, 57

57


To nd convergence in the responses, dierent parameter studies were made, usingboth FEM calculations and the Frýba analytical solution. The rst study contem-plates the number of modes needed to get convergence in the responses, for a singlepoint load moving on the bridge. For the subsequent studies, the bridge was loadedwith HSLM-A1 model proposed by the Eurocode [10].

Number of modes

Mode Superposition Method gives the possibility of using only a dened number ofmodes in the analysis. Yet, the accuracy of the response depends on the numberof modes used. Therefore, a brief study on the number of modes needed to getconvergent results was made.

Using the beam response to a single point load of 170 kN, moving along a simplysupported Euler-Bernoulli beam, proposed by Frýba [28], and already described insection 3.4.1, the number of bending modes needed to get convergent results, wasstudied. To make this, the analytical responses were calculated, using an increasingnumber of modes. The study was undertaken from the moment the rst axle of theload model enters the bridge, to the moment the last axle is at a distance of twospans from the structure.

The results for a single point load, moving on the beam at 250 km/h, are shownin gures 4.3, 4.4 and 4.5. The x-axis represent the longitudinal position along thebeam, and the y-axis show the maximum magnitude of the response, in time. Thenormalized results against each maximum are shown in gure 4.2.

It is clear that the displacement converges to the correct values for a small numberof modes and the bending moment converges, in a well behaved way, some modesafter. The accelerations were only roughly approximated if less than 100 modeswere considered. Its convergence has many uctuations and higher modes have animportant role in the nal values, unlike the displacement and the bending moment.

20 40 60 80 100

0.5

0.6

0.7

0.8

0.9

1

Number of modes

Nor

mal

ized

mag

nitu

de

DisplacementAccelerationBending moment

5 10 15 20 25

0.5

0.6

0.7

0.8

0.9

1

Number of modes

Nor

mal

ized

mag

nitu

de


Figure 4.2: Maximum responses for the rst 100 modes (left) and zoomed in the rst25 modes (right).

58


0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5x 10

−3

x [m]

Dis

plac

emen

t [m

]

1 mode5 modes25 modes50 modes100 modes

14 16 18 20 22 24 26 28

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1x 10

−3

x [m]

Dis

plac

emen

t [m

]


Figure 4.3: Maximum displacement of the whole beam (left) and zoomed in mid-span(right), with increasing mode number as parameter.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5x 10

6

x [m]

Ben

ding

mom

ent [

kNm

]


14 16 18 20 22 24 26 282.2

2.4

2.6

2.8

3

3.2x 10

6

x [m]

Ben

ding

mom

ent [

kNm

]


Figure 4.4: Maximum bending moment of the whole beam (left) and zoomed in mid-span (right), with increasing mode number as parameter.

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x [m]

Acc

eler

atio

n [m

/s2 ]


14 16 18 20 22 24 26 280.1

0.15

0.2

0.25

0.3

0.35

x [m]

Acc

eler

atio

n [m

/s2 ]


Figure 4.5: Maximum acceleration of the whole beam (left) and zoomed in mid-span(right), with increasing mode number as parameter.

59


Analyzing the convergence of the values, the number of modes needed to get a goodapproximation of the response, was chosen, as follows:

- Displacement: 5 modes;

- Bending moment: 25 modes;

- Acceleration: 32 modes.

Time step

The choice of the correct time step size is important to ensure that the completeresponse of the structure is captured by the solution and that the solution is stableand free from divergence. Usually, the smaller the time step the more accurate thesolution. However, there are practical limits on how small the step can be, as moresolution steps will be required for a smaller time step and the required computationaltime will increase accordingly. Therefore, a limit may be set by the time requiredfor the solution. The size of the time step is also limited by the number of sets ofresults that can be physically stored.

If the time step is too large then much of the higher frequency response of thestructure will be missed and the solution may not adequately represent the realbehavior of the structure as shown in gure 4.6.

Figure 4.6: Calculated response with a time step too large.

The analytical responses for a set of point loads, using dierent time steps, wasfound, and the convergence of the solutions was studied. The point loads corre-sponded to the HSLM-A1 axle loads proposed by the Eurocode [10] and describedin section 3.3. Three time steps were considered: 0.0002 s, 0.002 s and 0.02 s. Thenumber of modes used was chosen depending on the response being studied andaccording to the earlier performed mode convergence study.

60


100 150 200 250 3000.005

0.01

0.015

0.02

0.025

0.03

Train speed [km/h]

Dis

plac

emen

t [m

]

Time step of 0.02 sTime step of 0.002 sTime step of 0.0002 s

100 150 200 250 3000.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

7

Train speed [km/h]

Ben

ding

mom

ent [

kNm

]


Figure 4.7: Maximum displacement (left) and bending moment (right) of the beam,with time step as parameter.

100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.8: Maximum acceleration of the beam, with time step as parameter.

Figures 4.7 and 4.8 show very high responses for the same train speed. That speedcorresponds to the resonant speed of the bridge for the HSLM-A1, and is the samefor the displacement, acceleration and bending moment. The resonance responseis usually dominated by the rst frequency. Hence, an expression to determine theresonant speed for the bridge, is to notice that the frequency induced by the axlesis obtained by the following relation:

f =v

λ(4.2)

where λ is the distance between axles.

Thus:v = f · λ = 2, 39 · 18 = 43.02 m/s ≈ 155 km/h

it is interesting to notice that the discrepancies in the plots for the displacementand bending moment were almost negligible. That is due to the small inuence of

61


higher modes of vibration. The lower modes were responsible for the large majorityof the displacement and the bending moment. However, for accelerations, the dis-similarities between the 3 time steps are evident. This is explained by the inuenceof higher modes of vibration on the accelerations of the beam. These modes havehigher frequency rates that were not fully captured by the larger time steps.

The responses, for the dierent time steps are shown in table 4.1. The train speed[km/h], at which those responses were achieved, is given in parenthesis.

Table 4.1: Maximum responses, with time step as parameter.

Time Step [s] Displacement [cm] Acceleration [m/s2] Moment [MNm]0.02 2.28 (155) 4.03 (155) 20.3 (155)0.002 2.28 (155) 4.32 (155) 20.5 (155)0.0002 2.28 (155) 4.38 (155) 20.5 (155)

After the analysis of the results, it was decided to proceed the studies using a timestep of 0.02 s for the displacement and the bending moment. For the accelerations,a time step of 0.002 s was considered to be accurate enough, without increasing thecomputational time of the trials too much.

Speed step

To see how sensitive the dynamic responses peaks were to the speed step used, theanalytical calculations of the Frýba solution were made with increments of 2.5 km/hand compared with the results using a speed step of 5 km/h. The span of the speedswas from 100 km/h to 300 km/h in both cases. Results can be seen in gures 4.9and 4.10. Normalized amplitudes are compared in gure 4.11.

Examining the gures, the similarity between the plots of the displacement and thebending moment is obvious. Furthermore, one can notice that, for the displacementand bending moment, the inuence of the smaller speed step is not important.However, for the accelerations, the dierences between the two speed steps wereevident. That can be explained, once more, by the inuence of higher modes ofvibration on the accelerations of the beam. As a result, higher frequencies areexcited when the train runs at dierent speeds, and the inuence of higher modesgenerate higher accelerations. Nevertheless, these uctuations occur mainly for thehigher speeds, and the main resonant speed of 155 km/h was not aected.

The maximum responses, for the dierent speed steps, were the same. Thus, onecan conclude that the resonant speed is closer to a multiple of 5 km/h than 2.5 km/hand, therefore, it was decided to proceed the study with the speed step of 5 km/h.

Mesh type

The inuence of the mesh type on eigenfrequencies was studied as well. For theFE model, four meshes were tested: Timoshenko beam elements of 1 and 2 m and

62


100 150 200 250 3000.005

0.01

0.015

0.02

0.025

0.03

Train speed [km/h]

Dis

plac

emen

t [m

]

Speed step of 5 km/hSpeed step of 2.5 km/h

100 150 200 250 3000.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

7

Train speed [km/h]

Ben

ding

mom

ent [

kNm

]


Figure 4.9: Maximum displacement (left) and bending moment (right) on the beam,with speed step as parameter.

100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.10: Maximum acceleration on the beam, with speed step as parameter.

100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Train speed [km/h]

Nor

mal

ized

mag

nitu

de


100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Train speed [km/h]

Nor

mal

ized

mag

nitu

de


Figure 4.11: Maximum responses normalized, for speed steps of 5 km/h (left) and2.5 km/h (right).

63


Euler-Bernoulli beam elements of 1 and 2 m. The values obtained were comparedwith the exact frequencies provided by the exact solution of an Euler-Bernoulli beam(see Frýba [28]).

Figure 4.12 shows the values [Hz] of the eigenfrequencies for the rst ten bendingmodes, obtained using dierent elements and mesh sizes. The same frequencies areshown in table 4.1.

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

Mode

Freq

uenc

y [H

z]

ExactTimoshenko of 1 mTimoshenko of 2 mEuler−Bernoulli of 1 mEuler−Bernoulli of 2 m

7 8 9 10

120

140

160

180

200

220

240

Mode

Freq

uenc

y [H

z]

ExactTimoshenko of 1 mTimoshenko of 2 mEuler−Bernoulli of 1 mEuler−Bernoulli of 2 m

Figure 4.12: First 10 bending frequencies (left) and zoomed in frequencies 7 to 10(right) of the Banafjäl bridge, with mesh type as parameter.

Table 4.2: First 10 bending frequencies [Hz] of the Banafjäl bridge, with elementtype and size as parameters.

Timoshenko Beam Euler-Bernoulli BeamMode Exact 1 [m] 2 [m] 1 [m] 2 [m]

1st 2.39 2.36 2.36 2.39 2.392nd 9.54 9.17 9.18 9.54 9.543rd 21.47 19.73 19.75 21.47 21.474th 38.17 33.20 33.23 38.17 38.175th 59.64 48.80 48.79 59.64 59.656th 85.88 65.89 65.74 85.89 85.927th 116.90 84.00 83.51 116.90 116.988th 152.68 102.77 101.67 152.70 152.869th 193.24 121.93 120.67 193.27 193.5910th 238.57 142.58 137.77 238.62 239.23

After analyzing the frequencies, one can notice that the model with the Timoshenkoelements provides solutions lower than the Euler-Bernoulli solution. As expected,the Euler-Bernoulli beam elements provide a solution very close to the exact analyt-ical frequencies (also based on the Euler-Bernoulli theory), approaching them fromhigher values. For the model with the 1 m long Euler-Bernoulli beam elements, theprecision of the results is greater than 0.1Hz, for the 10th mode of vibration. Themodels meshed with Timoshenko beam elements show a relation almost linear with

64


the mode number. On the other hand, the Euler-Bernoulli solution has a quadraticrelation with the mode number, as can be seen from equation 4.1:

wj =

√(jπ

l

)4EI

µ= j2 · constant (4.3)

Therefore, the dierence between the Timoshenko models and the Euler-Bernoullianalytical solution increases very rapidly with the number of modes and is around100 Hz for the 10th bending frequency. Comparing the mesh size for the same beamelements, the dierences are only visible for higher modes, and can be considerednegligible.

The degree of the interpolation functions has a great inuence on the computationaltime needed to compute one trial. To run trials with the Euler-Bernoulli theoryand the cubic interpolation functions, very fast processors are needed to get resultsin reasonable time. Hence, and because this study focus on the inuence of thevertical support stiness, it was decided to proceed the study meshing the modelwith Timoshenko beam elements of 1 m. It permitted to simulate the transverseshear deformation of the girders and lower the computational time, through the useof linear interpolations functions.

4.1.4 Comparison of the methods

To validate the FEM solution, the responses obtained from the FE model werecompared with the analytical solution based on Frýba's equations.

Figures 4.13 and 4.14 show the responses using the analytical and the FEM numericalsolutions, with innite support stiness. The responses were calculated using thenumber of modes and the time step dened during the parametric study (section4.1.3), both for the analytical solution and the FEM solution. The analytical solutionwas numerically solved inMatLab using the same beam properties of the FE model.

The similarities are clear, even though the FE model uses Timoshenko beam theory,and the analytical solution uses the Euler-Bernoulli principles. The plots on theleft show this comparison for the case with crawling speed, and the values are veryclose, both in magnitude and time. For the train running at resonant speed, theagreement is, once more, very good, both in magnitude and time. However, onemay notice that the analytical solution shows more damping, and the responseseems to lag behind, for the free vibration, when the train has left the beam. Thatinstant can be acknowledged in the displacement plot, when its magnitude becomesa reection in the x-axis. The very good agreement between the FEM and theanalytical method can also be concluded form the observation of gure 4.15, wherethe maximum displacement for the dierent train speeds is shown.

65


0 100 200 300 400 500−10

−8

−6

−4

−2

0

2x 10

−3

Time [s]

Dis

plac

emen

t [m

]

AnalyticalFEM

0 5 10 15

−0.02

−0.01

0

0.01

0.02

Time [s]

Dis

plac

emen

t [m

]

AnalyticalFEM

Figure 4.13: Vertical displacement at mid-span, for HSLM-A1 running at crawling(left) and resonant (right) speed.

0 100 200 300 400 500−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time [s]

Acc

eler

atio

n [m

/s2 ]

AnalyticalFEM

0 5 10 15−4

−2

0

2

4

Time [s]

Acc

eler

atio

n [m

/s2 ]

AnalyticalFEM

Figure 4.14: Acceleration at mid-span, for HSLM-A1 running at crawling (left) andresonant (right) speed.

100 150 200 250 3000.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

Train speed [km/h]

Dis

plac

emen

t [m

]

AnalyticalFEM

Figure 4.15: Maximum displacement on the beam, for dierent train speeds.

66


4.1.5 Resonance eects

The inuence of the dynamic interaction is clear when the resonant responses (trainmoving at 155 km/h) are compared with the responses with the train running atcrawling speed (5 km/h). The trials were conducted until the train was one spanaway from the bridge, to allow the identication of the free vibration.

Plots of the displacement, bending moment and acceleration on the bridge, both intime and space, are shown in gures 4.16 to 4.21.

Analyzing gure 4.16, one can notice that the maximum displacement increases from1 cm to, approximately, 2.5 cm, when the resonance is reached. Furthermore, it isinteresting to note from gure 4.17 that, although for the rst 5 s the displacementswere all negative, this behavior changed for the next seconds, and values up to 1 cmwere obtained for the upwards displacement, when the train has already passed thebridge.

The plots for the bending moment show almost the same relation of the displace-ment, as expected. The maximum bending moment occurs always at mid-span,and the maximum negative bending moment is obtained in the nal seconds, whenthe train has already left the bridge. The values for the positive bending moment(corresponding to the downwards displacement) rise up to 20 · 109 Nm. For thequasi static case, that value is around 8 · 109 Nm.

The accelerations are, on the other hand, far more complex. Yet, the resonance isclear, with the magnitudes rising very rapidly, until the train leaves the bridge. Atthe resonant speed, the acceleration of the bridge deck rises up to 4 m/s2, and itwould be greater for longer trains. This peak acceleration, around 0.4g, can causedestabilization of the ballast, as it is higher than the valued recommended in theERRI D214 reports [21]: ault=3.5 m/s2 for bridges with ballast.

One can also notice that the maximum acceleration is not always registered at mid-span, as might be expected. In subsequent studies, namely when the accelerationsensitivity to the vertical displacement is studied, this phenomenon will be furtherdiscussed.

4.1.6 Sensitivity to vertical support stiness

In the following, the eect of the support stiness on the dynamic behavior ofthe simply supported bridge is studied. The vertical support stiness was variedaccording to the values found to provide changes in the behavior of the structure:from 1 · 108 N/m, for soft clays, to 1 · 1011 N/m, for foundation on solid bedrock orpile groups. To achieve that eect on the FE model, the sti boundary conditionsat the end of the beam were replaced by non-homogeneous boundary condition,representing the soil-foundation interaction. The stiness of the springs was thenvaried within the chosen interval (1 · 108 to 1 · 1011 N/m).

To see how the vertical stiness of the supports aected the dynamic behavior of

67


Figure 4.16: Vertical displacement, over time and space, for HSLM-A1 running atcrawling speed (5 km/h).

Figure 4.17: Vertical isplacement, over time and space, for HSLM-A1 running atresonant speed (155 km/h).

68


Figure 4.18: Bending moment, over time and space, for HSLM-A1 running at crawl-ing speed (5 km/h).

Figure 4.19: Bending moment, over time and space, for HSLM-A1 running at reso-nant speed (155 km/h).

69


Figure 4.20: Vertical acceleration, over time and space, for HSLM-A1 running atcrawling speed (5 km/h).

Figure 4.21: Vertical acceleration, over time and space, for HSLM-A1 running atresonant speed (155 km/h).

70


a bridge crossed by a HST, numerous trials were made and are presented in thefollowing.

Eigenfrequencies

With the decrease of the vertical stiness at the end supports, the bending frequen-cies started to decrease. Table 4.3 shows the eect of the supports stiness on theeigenfrequencies. The last column shows the ratio between the frequencies with theless sti model and with the stiest model. The same relations are also shown ingure 4.22, with the normalized frequencies in the left plot.

Table 4.3: Eect of the end support stiness in the frequencies [Hz] of the rst tenbending modes of vibration of the Banafjäl bridge.

Supportstiness [N/m] 1 · 1011 1 · 1010 5 · 109 1 · 109 5 · 108 1 · 108 Ratio1st frequency 2.36 2.35 2.35 2.28 2.21 1.79 76%2nd frequency 9.16 9.05 8.94 8.07 7.16 4.14 45%3rd frequency 19.68 19.22 18.71 15.00 12.28 7.44 38%4th frequency 33.07 31.83 30.32 22.01 18.48 14.85 45%5th frequency 48.54 45.85 42.57 30.98 28.38 26.43 54%6th frequency 65.45 60.48 54.85 43.38 41.83 40.73 62%7th frequency 83.33 75.12 67.68 58.53 57.58 56.88 68%8th frequency 101.83 89.76 81.93 75.38 74.75 74.28 73%9th frequency 120.69 104.65 97.82 93.26 92.82 92.48 77%10th frequency 141.15 139.70 120.28 114.97 111.74 111.42 79%

Infinite 1E11 1E10 5E9 1E9 5E8 1E80

0.25

0.5

0.75

1

Support stiffness [N/m]

Nor

mal

ized

fre

quen

cy

1st mode2nd mode3rd mode4th mode5th mode6th mode7th mode8th mode9th mode10th mode

2 4 6 8 10

25

50

75

100

125

150

Mode

Freq

uenc

y [H

z]

Infinite1E11 N/m1E10 N/m5E9 N/m1E9 N/m5E8 N/m1E8 N/m

Figure 4.22: Eect of the end supports vertical stiness in the bending frequenciesof the rst ten bending modes of vibration of the Banafjäl bridge.

The plot on the left of gure 4.22 shows the eigenfrequencies plotted against theend supports stiness. The values are normalized against the frequency of the stimodel. The plot to the right shows the dierent modes on the x-axis, and the variouscolors represent each vertical stiness.

71


From the analysis of the plots, one can notice that there is hardly any change in therst three eigenfrequencies, for stiness higher than 5 · 109 N/m. However, the 3rd

and 4th eigenfrequencies change very much between support stiness of 5 · 109 N/mand 1·108 N/m. The same type of behavior is observed for frequencies above the 4th:barely altered for the higher stiness, and with greater variations for stiness lowerthan 5 ·109 N/m. On the other hand, the rst two modes shows a dierent behavior,with large changes happening only when the stiness decreases from 5 · 108.

Resonant speed

As seen before, the resonant speed is a function mainly of the 1st eigenfrequencyof the structure. Therefore, with the highest variations of the 1st eigenfrequencystarting for vertical stiness lower than 1 · 109 N/m, the highest variations on theresonant speed were expected for lower stiness. Table 4.4 shows the dierent reso-nant speeds, obtained not from the FE model but from equation 4.2.

Table 4.4: Resonant speed, for the dierent support stiness, obtained from equation4.2.

Support stiness [N/m] 1st Frequency [Hz] Resonant speed [km/h]∞ 2.36 153

1 · 1011 2.36 1531 · 1010 2.35 1525 · 109 2.34 1521 · 109 2.28 1485 · 108 2.21 1431 · 108 1.78 116

The results of table 4.4 can be compared with the plots of the responses, for dierenttrain speeds. Figures 4.23 to 4.25 show the maximum displacement of the beam,at mid-span. The trials were run in Abaqus/Brigade, the number of modes usedwas 5 and the time step 0.02 s, in accordance to the study on mode convergence(section 4.1.3). The frequency for the 5th mode of vibration is, approximately, 50Hz, and its period around 0.02 Hz. Hence, the time step is small enough to capturefrequencies lower than that. The speed step is 5 km/h, since it gives acceptableresults, without increasing the computational time too much.

The trials were divided in groups of two, to make the presentation of the resultsclear. The response for the model with innite support stiness is shown in allgures, for the purpose of comparison. The resonant speeds obtained from the trialsare shown in table 4.5, and compared the ones obtained with expression 4.2.

The rst similarity one can see is between the resonant speed obtained from equation4.2 and from the FEM solution. The match is very good for all the cases, within thereach of the speed step. From the analysis of gure 4.23, it can also be seen that thedisplacement is hardly aected by the decrease of the vertical support stiness, from∞ to 1 · 1010. Only a slight decrease in the maximum displacement is registered,

72


100 150 200 250 300−0.025

−0.02

−0.015

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]

Infinite1E11 N/m1E10 N/m

100 150 200 250 3000

0.005

0.01

0.015

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]


Figure 4.23: Maximum downwards (left) and upwards (right) displacement of thebeam, as function of the train speed.

100 150 200 250 300−0.025

−0.02

−0.015

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]


100 150 200 250 3000

0.005

0.01

0.015

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]



100 150 200 250 300−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]


100 150 200 250 3000

0.01

0.02

0.03

0.04

0.05

Train speed [km/h]

Dis

plac

emen

t [m

]



73


Table 4.5: Resonant speed, for dierent vertical support stiness.

Support Resonant speed [km/h]stiness [N/m] from the FEM from equation 4.2

∞ 155 1531 · 1011 155 1531 · 1010 155 1525 · 109 155 1521 · 109 150 1485 · 108 145 1431 · 108 115 116

which will be analyzed in the next section.

For lower values of stiness, major changes start to appear, not only in the resonantspeed, but also in the magnitude of the response. Changes in the resonant speedwere expected. However, the decrease of the maximum displacement on resonance,for lower vertical stiness, was an interesting nding. Moreover, the resonance isnot as clear as it appears for higher stiness. Instead, for a stiness of 5 · 108 N/m,the relation between the displacement in resonance and the displacement for speedsup to 200 km/h decreases signicantly. In fact, almost no resonance eects appearon displacements.

Displacement

To compare the displacements, with dierent support stiness, a point at mid-spanwas studied. The bridge was loaded with the HSLM-A1 train running at resonantspeed. The resonant speed of each trial depended on the vertical stiness of thesupports and is shown in table 4.5. The time step used was 0.02 s and 5 bendingmodes were included, in accordance to the convergence study (section 4.1.3). Sincethere was one longitudinal mode between the rst 5 bending modes, the trials wererun with a total of 6 modes.

The inuence of the vertical stiness of the supports on the displacements due tothe HSLM-A1, running at crawling speed (5 km/h), is shown in gures 4.26, 4.27and 4.28. The results for resonant speed are shown in gures 4.29 and 4.30.

From the analysis of gure 4.26, it is clear that, as suggested from gure 4.23, themaximum displacement at mid-span, with the train running at crawling speed, isonly slightly aected with the decrease of vertical support stiness.

Figure 4.27 shows dierent results. For the displacement with the train running atcrawling speed, although no major changes take place for lower stiness, one can seethat, for a stiness of 1·109 N/m, the vibrations were not only due to bending modes,but also to rigid body movements. These rigid body movements were produced bythe vibration of the whole bridge over the support springs. However, for the resonantresponse, the displacement decreases with the stiness, as it was already discussed

74


0 100 200 300 400−10

−8

−6

−4

−2

0

2x 10

−3

Time [s]

Dis

plac

emen

t [m

]


88 90 92 94 96 98−6.3

−6.2

−6.1

−6

−5.9

−5.8

−5.7x 10

−3

Time [s]

Dis

plac

emen

t [m

]


Figure 4.26: Displacement at mid-span, for HSLM-A1 running at 5 km/h. Wholetime span (left) and zoomed in the high frequency vibrations (right).

0 100 200 300 400−12

−10

−8

−6

−4

−2

0

2x 10

−3

Time [s]

Dis

plac

emen

t [m

]


88 90 92 94 96 98−6.8

−6.6

−6.4

−6.2

−6

−5.8

−5.6x 10

−3

Time [s]

Dis

plac

emen

t [m

]



0 100 200 300 400−20

−15

−10

−5

0

5x 10

−3

Time [s]

Dis

plac

emen

t [m

]


88 90 92 94 96 98−13

−12

−11

−10

−9

−8

−7

−6

−5x 10

−3

Time [s]

Dis

plac

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t [m

]



75


when analyzing the speed trials (previous section). The plot on the right, from gure4.29, shows some decrease on the displacement for the 5 · 109 N/m stiness and alarge reduction for 1 · 109 N/m. For this last case, one can also notice the lowerfrequency of the response, due to the lower speed of the load model (148 km/h, seetable 4.5).

When the stiness decreases below 1 · 109 N/m, the displacement at crawling speedincreases very much, with a greater contribution of modes with higher frequencies,as can be seen in 4.28. The maximum static displacement is almost 2 times greaterfor a vertical support stiness of 1 · 108 N/m than for innite sti supports.

What happens at resonant speed is contradictory (gures 4.29 and 4.30). On onehand, when the vertical support stiness decreases to 1 · 108 N/m, the displacementrises more than double, when compared to the sti support model. On the otherhand, the displacement, for a stiness of 5 · 108 N/m, is less than the displacementwith sti supports. In fact, this is in accordance with gure 4.25, where the resonantis almost impossible to identify, for a stiness of 5 · 108 N/m. Table 4.6 summarizesthe results.

Table 4.6: Maximum displacement at mid-span, for dierent support stiness.

Support Maximum displacement [cm]stiness [N/m] Crawling speed Resonant speed

∞ 0.99 2.291 · 1011 0.99 2.281 · 1010 0.99 2.255 · 109 1.00 2.181 · 109 1.08 1.795 · 108 1.15 1.281 · 108 1.83 5.64

0 5 10 15−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time [s]

Dis

plac

emen

t [m

]


0 5 10 15−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time [s]

Dis

plac

emen

t [m

]


Figure 4.29: Vertical displacement at mid-span, for HSLM-A1 running at resonantspeed.

76


0 5 10 15−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

Dis

plac

emen

t [m

]


Figure 4.30: Vertical displacement at mid-span, for HSLM-A1 running at resonantspeed.

Acceleration

As it was mentioned previously and can be seen in gure 4.21, the maximum ac-celeration is not always found at mid-span. Hence, the accelerations for dierentvalues of support stiness were studied at each node along the bridge model. Theconvergence study (section 4.1.3) shown that 32 bending modes and a time step of0.002 s were need to get reliable results. However, there were 30 longitudinal modesbetween the rst 32 bending modes and, therefore, to ensure the convergence of theaccelerations, a total of 62 modes were included for the FE trials.

Unfortunately, because such a small time step increases the computational timevery much, it was not possible to present plots with the maximum acceleration fordierent train speeds. Hence, and based on gure 4.11 and the studies undertakenin the model with sti supports, it was assumed that the resonant speed for theacceleration was the same as for the displacement. Hereby, the trials were madewith the train running at a speed that excited the rst frequency of the structure.That speed is shown in table 4.5.

It was seen from gure 4.21 that the maximum acceleration is not always fround atmid-span. Hereby, it was chosen to plot the maximum accelerations, in time, foreach one of the nodes along the bridge model. Figures 4.31, 4.32 and 4.33 showthe variation of the maximum acceleration. Table 4.7 summarizes the maximumacceleration obtained for dierent stiness.

From the analysis of gure 4.31, it can be seen that, for vertical support stinesslower than 1 · 1011 N/m, the higher accelerations were found at the supports. This

77


5 10 15 20 25 30 35 40−7

−6

−5

−4

−3

−2

−1

0

x [m]

Acc

eler

atio

n [m

/s2 ]


5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

x [m]

Acc

eler

atio

n [m

/s2 ]


Figure 4.31: Maximum downwards (left) and upwards (right) acceleration, forHSLM-A1 running at resonant speed.

5 10 15 20 25 30 35 40−10

−8

−6

−4

−2

0

x [m]

Acc

eler

atio

n [m

/s2 ]


5 10 15 20 25 30 35 400

2

4

6

8

10

12

x [m]

Acc

eler

atio

n [m

/s2 ]



5 10 15 20 25 30 35 40−12

−10

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x [m]

Acc

eler

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/s2 ]


5 10 15 20 25 30 35 400

2

4

6

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x [m]

Acc

eler

atio

n [m

/s2 ]



78


is due to the oscillation of the support springs of the FE model. The accelerationregistered for the model with innite stiness and for the model with stiness of1 · 1011 N/m, were similar, growing from the supports to mid-span, but usually lessthan the limit of 3.5 m/s2 proposed in the ERRI reports [21]. For stiness of 1 ·1010

N/m and 5 · 109 N/m, the maximum accelerations were greater, exceeding the limitproposed for bridges, even without ballast (5 m/s2). For lower values of stiness,as shown in gure 4.32 and 4.33, the acceleration in the supports were greater thangravity, which may denitely cause destabilization of the ballast and the rails, ifproved that these magnitude of accelerations are possible to appear in reality.

Table 4.7: Maximum acceleration, for dierent vertical support stiness.

Support Maximumstiness [N/m] acceleration [m/s2]

∞ 3.221 · 1011 3.581 · 1010 6.605 · 109 7.751 · 109 10.25 · 108 10.91 · 108 10.4

4.1.7 Discussion of the results

The study on the single span bridge started with a convergence study on importantparameters. The results show, on one hand, that accurate values for the displace-ment could be found with only 5 bending modes and a time step of 0.02 s. On theother hand, 32 bending modes and a time step of 0.002 were essential to get conver-gence for accelerations. The reason for this is that the displacement is dominatedby the lower modes, while the maximum accelerations have greater contributionsfrom higher modes.

The study on the inuence of the element type and size provided interesting con-clusions. Results using Timoshenko beam theory were compared with the solutionsprovided by the Euler-Bernoulli theory. Timoshenko beam elements allow for trans-verse shear deformations, can be subjected to large axial strains and use linearinterpolations functions. Euler-Bernoulli beams do not allow for transverse sheardeformation, use a consistent mass formulation and cubic interpolation functions,which makes them reasonably accurate for cases involving distributed loading alongthe beam. However, the cubic interpolation functions greatly increase the compu-tational time. From the study, it was seen that Timoshenko beam elements of 1m provided accurate results, for the bridge in question, reducing the computationaltime. The accuracy of the FEM solutions was found to be very good, when com-pared to the analytical solutions for the Euler-Bernoulli beam, even though the FEmodel was meshed with Timoshenko beam elements.

79


The study on the inuence of the vertical support stiness provided the most im-portant result: it is mandatory to model the stiness of the supports if realisticpredictions of the dynamic behavior are to be made. Large changes are obtained inthe eigenfrequencies and mode shapes, resonant speeds of the trains and magnitudeof displacements and accelerations.

For a vertical stiness of 5 · 108 N/m in the supports, resonance eects are notobserved. In fact, the maximum displacement at resonant speed, for stiness of5 · 108 and 1 · 109 N/m, is lower than the displacement with sti supports. This factwas surprising and its reasons should be deeply investigated in future studies.

In what concerns the maximum accelerations, they were always found at the sup-ports, for models with stiness lower than 1 ·1011 N/m. This is due to the oscillationof the support springs in the FE model, but should be validated with further re-search. It is uncertain if this behavior happens in real structures.

4.2 Double span bridge

4.2.1 The bridge

The bridge over Lögdeälv is a two span composite bridge along the Bothnia Line.The concrete deck is supported by two steel girders, connected to each other bytransversal steel beams, weighting 0.2 N/m. Each of the girders is composed of 4dierent beam cross-sections, with variations in cross-sectional area and momentof inertia. A sketch of the bridge is shown in gure 4.34 and the properties ofthe girders are presented in tables 4.8 and 4.9. The weight of the ballast can beconsidered roughly constant with the value of 30 N/m and the weight of the concreteslab is 30.8 N/m.

Table 4.8: The weight of the Lögdeälv bridge.

Beam type gbeam [N/m] gtransv [N/m] gconc [N/m] gbal [N/m] gtot [N/m]1 8.2 0.2 30.8 30.0 69.22 10.0 0.2 30.8 30.0 71.03 9.7 0.2 30.8 30.0 70.74 16.0 0.2 30.8 30.0 77.0

Table 4.9: The material and sectional properties for the Lögdeälv bridge.

Beam type ρ [ kg/m3] E [Pa] A [m2] I [m4] 2·A [m2] 2·I [m2]1 34343 2.10·1011 0.205 0.177 0.411 0.3552 31223 2.10·1011 0.232 0.220 0.464 0.4403 32059 2.10·1011 0.225 0.209 0.450 0.4194 25509 2.10·1011 0.308 0.312 0.615 0.624

80

4.2. DOUBLE SPAN BRIDGE

Figure 4.34: Sketch of the Lögdeälv bridge.

4.2.2 FE model

The bridge was modeled as a double span beam, supported at the ends and atthe middle node. The bending stiness of the beam was assumed to be twice thebending stiness of one girder, the same happening with the area. Four dierentbeam cross-sections were dened, with changes in the moment of inertia and area,in accordance with table 4.9 (see also gure 4.35).

The model of the bridge is shown in gure 4.35, with dierent colors representingdierent type of girders. The model shows the supports, later replaced by springs.

Z

Y

XK

K

K

1

2

3

Figure 4.35: FE model of the Lögdeälv bridge.

The stiness of the supports was made variable with the substitution of the stiboundary conditions for spring connected to the soil, representing the foundations ofthe structure. The springs, located at the ends of the beam and mid-span, permittedfree rotation and horizontal displacement of the nodes, and the stiness was onlynoted for vertical displacement.

The eective axial stiness of the main column was neglected (it was believed to be

81


much greater than the values in question) and only the support stiness, i.e., thestiness of the foundation and soil beneath, was modeled. The deformation of thebearing was neglected as well. The bending stiness of the column was consideredvery small and, therefore, the middle node was free to move longitudinally. Therotation was also permitted at the center of the beam, because the Lögdeälv bridgehad a bearing over the main column.

To simulate the transverse shear deformation of the girders, the Timoshenko beamelements were used, with linear interpolations functions (more information can befound in section 3.4.2).

The damping considered was, according to [16] and table 2.3, ξ = 0, 5%. This valueswas the same for every part of the structure, and was dened as direct damping (seeHibbitt et al. [32] for more information on the subject).

4.2.3 Convergence study

To achieve convergent results, dierent parametric studies were made using the FEmodel developed in Abaqus/Brigade. All the trials were run using the HSLM-A1and the responses registered until the load model was, at least, one span away fromthe bridge, to ensure the high accelerations produced immediately after the trainhas left the bridge were included.

Mesh elements

The size of the mesh elements needed to get convergence for displacement andacceleration was studied, using ten vibrating modes. Since the eigenfrequency ofthe 10th mode was, approximately, 33 Hz, a time step of 0.02 s was considered smallenough to get accurate results and was, therefore, used. Figures 4.36 and 4.37 showthe responses as functions of train speed, with element length as parameter.

100 150 200 250 3000.004

0.006

0.008

0.01

0.012

0.014

0.016

Train speed [km/h]

Dis

plac

emen

t m

Mesh of 1 mMesh of 2 mMesh of 4 m

100 150 200 250 300−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

Train speed [km/h]

Dis

plac

emen

t m


Figure 4.36: Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with element length as parameter.

82


100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


100 150 200 250 300−5

−4

−3

−2

−1

0

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.37: Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with element length as parameter.

Table 4.10: Maximum responses, with element length as parameter.

Mesh Maximum displacement [cm] Maximumsize [m] Downwards Upwards acceleration [m/s2]

1 2.05 1.53 4.232 2.04 1.53 4.274 2.05 1.55 4.19

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

Mode

Freq

uenc

y [H

z]

FEM 1 mFEM 2 mFEM 4 m

Figure 4.38: Eigenfrequencies, with element length as parameter.

83


Table 4.10 summarized the results from gures 4.36 and 4.37, showing the maximumdisplacement and acceleration for the dierent trials. The inuence of the elementsize in the eigenfrequencies of the rst ten modes was studied and is shown in gure4.38. The dierences are negligible.

From the analysis of the results, one can see that the displacement almost does notshow any sensitivity to the mesh size, and the changes in the acceleration were lessthan to 0.1 m/s2. The inuence on the eigenfrequencies is negligible as well. Thus,it was decided to proceed the studies with an element length of 2 m.

Time step and number of modes

In contrast to what happened in the single span bridge study (section 4.1), thenumber of modes and the time step convergence were studied together. The FEsolution uses Timoshenko beam elements of 2 m and a speed step of 5 km/h wasused. Three trials were run: 0.02 s and 10 modes, 0.01 s and 25 modes, 0.005 s and40 modes. The time step was chosen so that it would be less than the period of thehighest mode included.

The results are shown in gure 4.39 to 4.40 and table 4.11.

100 150 200 250 3000.004

0.006

0.008

0.01

0.012

0.014

0.016

Train speed [km/h]

Dis

plac

emen

t [m

]

Trial 1Trial 2Trial 3

100 150 200 250 300−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]


Figure 4.39: Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with time step and number of modes as parameter.

Table 4.11: Maximum responses, with time step and number of modes as parameter.

Trial Maximum displacement [cm] Maximumnumber Downwards Upwards acceleration [m/s2]

1 2.05 1.55 4.192 2.06 1.56 4.383 2.06 1.56 4.50

One can notice that the time step and the number of modes does not aect themaximum displacement, assuring the convergence for the rst trial. Therefore, for

84


100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


100 150 200 250 300−5

−4

−3

−2

−1

0

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.40: Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with time step and number of modes as parameter.

studying the maximum displacement of the bridge, 10 modes and a time step of 0.02s were used.

For the acceleration, the dierences between trial 2 and 3 were not large, so it wasconsidered that the response for trial 3 were close enough to the exact. Hence, forthe subsequent studies concerning acceleration, a time step of 0.005 s was used and40 modes were considered.

Speed step

From the time step study, one can notice that the peaks of the responses were notobtained, for a speed step of 5 km/h. Hence, trials with a speed step of only 1 km/hwere made, only for the speeds around the resonant peaks. The same three trialswere run: 0.02 s and 10 modes, 0.01 s and 25 modes, 0.005 s and 40 modes. Thestructure responses are shown in gures 4.41 and 4.42 and table 4.12.

135 140 1450.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]


135 140 145−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

Train speed [km/h]

Dis

plac

emen

t [m

]


Figure 4.41: Maximum upwards (left) and downwards (right) displacement as func-tion of train speed, with time step and number of modes as parameter.

85


220 222 224 226 228 2302.5

3

3.5

4

4.5

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


220 222 224 226 228 230−5

−4.5

−4

−3.5

−3

−2.5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.42: Maximum upwards (left) and downwards (right) acceleration as func-tion of train speed, with time step and number of modes as parameter.

Table 4.12: Maximum responses, with speed step as parameter.

Speed Trial Maximum displacement [cm] Maximumstep number Downwards Upwards acceleration [m/s2]

1 2.05 1.55 4.195 km/h 2 2.06 1.56 4.38

3 2.06 1.56 4.501 2.34 1.84 4.34

1 km/h 2 2.35 1.84 4.493 2.35 1.84 4.64

It can be seen that both the displacement and the acceleration are aected by thespeed step. Hence, it was decided, for subsequent studies, to run trials with 5 km/hfor the whole range of speeds, and then analyze the resonant areas with a smallerspeed step of 1 km/h.

4.2.4 Sensitivity to vertical support stiness

After the analysis of the results provided by the study of the single span bridge,it was decided to perform the sensitivity study on the double span bridge only forstiness varying between 5·1010 and 5·108 N/m, since this interval of values providedvery interesting results in the rst study. Therefore, it was settled to run 9 trials onthe bridge. Each trial is identied by the case number, and the cases are dened intable 4.13.

Table 4.13 is organized as a matrix. For instance, case 4 has a vertical stiness of5 · 109 N/m for the main column and 5 · 1010 N/m for the end supports. The modelwith innite stiness both in the end supports and in the center column is referredto as case 0.

86


Table 4.13: Denition of the cases for the study.

Center column End supports stiness [N/m]stiness [N/m] 5 · 1010 5 · 109 5 · 108

5 · 1010 Case 1 Case 2 Case 35 · 109 Case 4 Case 5 Case 65 · 108 Case 7 Case 8 Case 9

Eigenfrequencies

With the element size of 2 m, the inuence of the support stiness on the eigen-frequencies was studied. Table 4.14 shows the frequencies for the FE trials withdierent supports conditions (cases 0 to 9). The same information is shown in gure4.43.

Table 4.14: Eect of the vertical support stiness in the frequencies [Hz] of the rstten bending modes of vibration of the Lögdeälv bridge.

EigenfrequencyCase 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

0 2.11 3.43 8.16 10.45 15.51 15.51 17.72 20.59 30.16 33.331 2.12 3.49 8.41 10.93 15.51 18.93 22.27 29.84 33.74 37.522 2.12 3.44 8.41 10.38 15.51 18.93 19.63 29.84 30.52 33.743 2.12 2.96 6.86 8.41 14.14 15.51 18.93 26.36 29.84 33.744 2.12 3.48 8.34 10.81 15.51 18.58 21.8 29.84 32.59 36.135 2.12 3.43 8.34 10.29 15.51 18.58 19.34 29.79 29.84 32.596 2.12 2.96 6.83 8.34 13.96 15.51 18.58 25.69 29.84 32.597 2.08 3.37 7.63 9.59 15.05 15.51 17.32 25.02 28.09 29.848 2.08 3.32 7.63 9.25 15.05 15.51 16.14 24.31 25.02 29.849 2.08 2.9 6.52 7.63 12.06 15.05 15.51 20.09 25.02 29.84

0 1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

Case

Freq

uenc

y [H

z]

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Mode

Freq

uenc

y [H

z]

Case 0Case 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9

Figure 4.43: Eect of the vertical support stiness on the frequencies of the rst tenmodes of vibration of the Lögdeälv bridge.

87


From the analysis of the results, one can conclude that the greatest changes occurwhen the stiness of the end supports decreases to 5 · 108 N/m (cases 3, 6 and 9).For the stier support conditions (cases 0, 1 and 2) the lower frequencies show verysmall changes. However, for higher frequencies, namely the 7th and the 8th, thedierences between cases 0 and 1 are clear. The reason why, for higher modes (fromthe 6th), the eigenfrequencies are higher in the models with spring than in the modelwith sti supports is that there are two modes shapes with the same frequency (5th

and 6th, both longitudinal modes, but in opposite directions: 15.51 Hz) in the stiermodel. For the models with springs, that frequency is only due to the 6th modeshape, thus disarranging the correspondence between models.

One can also obtain the speed of excitation for the rst frequencies. Table 4.15shows the dierent resonant speeds, obtained from equation 4.2.

Table 4.15: Eect of the vertical support stiness on the bending frequencies of therst 2 bending modes of vibration of the Lögdeälv bridge and on theresonant speeds.

1st mode 2nd modeCase Frequency [Hz] Speed [km/h] Frequency [Hz] Speed [km/h]0 2.11 137 3.42 2221 2.12 137 3.49 2262 2.12 137 3.44 2233 2.12 137 2.96 1924 2.12 137 3.48 2265 2.12 137 3.43 2226 2.12 137 2.96 1927 2.08 135 3.37 2188 2.08 135 3.32 2159 1.08 135 1.90 188

Displacement

To analyze the inuence of the vertical stiness on the maximum displacement ofthe bridge deck, trials were made in Abaqus/Brigade, for train speeds from 100 to300 km/h. The model was meshed using Timoshenko beam elements with 2 m andthe speed step was 5 km/h. 10 modes were used, with a time step of 0.02 s, becausethe parameter study (section 4.2.3) shown it was enough to get convergence for thedisplacements. Figures 4.44 to 4.46 show the maximum displacement as function ofthe train speed, for the 10 cases.

Analyzing the plots in gures 4.44 to 4.46, one can see that the peaks were notalways found, due to an excessively coarse speed step (5 km/h). Hence, the trialswere rerun for the speed interval were the resonance occurs. The speed step usedwas 1 km/h and the results are presented in gures 4.47 (cases 0 to 6, speed range:135 to 145 km/h), 4.48 (cases 7 and 8, speed range: 210 to 220 km/h) and 4.49(case 9, speed range: 185 to 195 km/h).

88


100 150 200 250 3000

0.005

0.01

0.015

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]

Case 0Case 1Case 2Case 3

100 150 200 250 300−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]


Figure 4.44: Maximum upwards (left) and downwards (right) displacement of thebeam (cases 1 to 3).

100 150 200 250 3000

0.005

0.01

0.015

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]


100 150 200 250 300−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]



100 150 200 250 3000.004

0.006

0.008

0.01

0.012

0.014

0.016

Train speed [km/h]

Dis

plac

emen

t [m

]


100 150 200 250 300−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

Train speed [km/h]

Dis

plac

emen

t [m

]



89


135 140 1450.005

0.01

0.015

0.02

Train speed [km/h]

Dis

plac

emen

t [m

]

Case 0Case 1Case 2Case 3Case 4Case 5Case 6

135 140 145−0.026

−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

Train speed [km/h]

Dis

plac

emen

t [m

]

Case 0Case 1Case 2Case 3Case 4Case 5Case 6

Figure 4.47: Maximum upwards (left) and downwards (right) displacement of thebeam (cases 0 to 6 with speed step of 1 km/h).

210 212 214 216 218 2200.006

0.008

0.01

0.012

0.014

0.016

Train speed [km/h]

Dis

plac

emen

t [m

]

Case 7Case 8

210 212 214 216 218 220−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

Train speed [km/h]

Dis

plac

emen

t [m

]

Case 7Case 8

Figure 4.48: Maximum upwards (left) and downwards (right) displacement of thebeam (cases 7 and 8 with speed step of 1 km/h).

185 190 1950.004

0.006

0.008

0.01

0.012

0.014

0.016

Train speed [km/h]

Dis

plac

emen

t [m

]

185 190 195−0.024

−0.022

−0.02

−0.018

−0.016

−0.014

−0.012

Train speed [km/h]

Dis

plac

emen

t [m

]

Figure 4.49: Maximum upwards (left) and downwards (right) displacement of thebeam (case 9 with speed step of 1 km/h).

90


Table 4.16 show the values of displacements and between parenthesis the speeds[km/h] at which these maximum were obtained. With the speed increment of 1km/h, more accurate results for the displacement were obtained from the trials,with an increase of about 5%.

Table 4.16: Maximum displacements with dierent speed steps (resonant speed[km/h] is indicated between parenthesis).

Downwards [cm] Upwards [cm]Case 5 km/h 1 km/h 5 km/h 1 km/h0 2.04 (140) 2.34 (139) 1.53 (140) 1.84 (139)1 2.24 (140) 2.34 (139) 1.73 (140) 1.82 (139)2 2.24 (140) 2.35 (139) 1.74 (140) 1.82 (139)3 2.37 (140) 2.47 (139) 1.60 (140) 1.69 (139)4 2.18 (140) 2.30 (139) 1.67 (140) 1.76 (138)5 2.17 (140) 2.31 (138) 1.67 (140) 1.77 (139)6 2.31 (140) 2.42 (139) 1.53 (140) 1.64 (138)7 2.23 (220) 2.33 (218) 1.50 (220) 1.58 (218)8 2.30 (215) 2.30 (215) 1.56 (215) 1.57 (214)9 2.08 (190) 2.24 (188) 1.45 (185) 1.56 (187)

Acceleration

To analyze the inuence of the vertical stiness on the maximum acceleration of thebridge deck, trials were made in Abaqus/Brigade, for train speeds from 100 to300 km/h. In the rst approach, the speed step was 5 km/h, with only 10 modesbeing used, and a time step of 0.02 s. This rough approach will later be rened, forthe critical train speeds. The gures 4.50 to 4.52 show the rst results.

Figures 4.50 and 4.51 are very similar. They show a decrease in the maximumacceleration for case 3 and 6, when the stiness of the central column is decreased.Furthermore, the resonant peaks were obtained for the same trains speeds. Figure4.52 shows a very dierent behavior. The resonant speeds were many more and theaccelerations change from about 3 to 8 m/s2. For these cases (with end supportsstiness of 5 · 108 N/m), the maximum acceleration is obtained at the ends of thebeam, where the springs, with low stiness, were located. This behavior, present inthe model, needs to be validated with experimental results from bridges with verylow support stiness (if such bridges can be found). Therefore, cases 7 to 9 will notbe followed in the subsequent studies.

The maximum acceleration is summarized in table 4.17, with the speed at which thecorresponding value was achieved between parenthesis. To get more reliable results,the number of modes considered should be at least 40, according to the parameterstudy made before. Therefore, the trials were rerun (from cases 0 to 6) with a speedstep of 1 km/h, time step of 0.005 s and considering 40 modes. The results arepresented in gures 4.53 (cases 0 to 2, 4 and 5, speed range: 215 to 230 km/h) and4.54 (cases 3 and 6, speed range: 185 to 200 km/h).

91


100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


100 150 200 250 300−5

−4

−3

−2

−1

0

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


Figure 4.50: Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 1 to 3).

100 150 200 250 3000

1

2

3

4

5

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


100 150 200 250 300−5

−4

−3

−2

−1

0

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]



100 150 200 250 3000

2

4

6

8

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]


100 150 200 250 300−8

−6

−4

−2

0

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]



92


Table 4.17: Maximum acceleration [m/s2] (using the speed step of 5 km/h).

Case Downwards Upwards0 4.13 (220) 4.26 (220)1 4.41 (225) 4.43 (225)2 3.92 (220) 3.95 (220)3 3.14 (190) 3.02 (190)4 4.52 (225) 4.54 (225)5 4.24 (220) 4.20 (220)6 3.35 (190) 3.22 (190)7 7.83 (160) 7.89 (140)8 7.28 (175) 7.16 (150)9 7.81 (130) 7.93 (130)

215 220 225 230

0

2

4

6

8

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 0Case 1Case 2Case 4Case 5

215 220 225 230

−10

−8

−6

−4

−2

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 0Case 1Case 2Case 4Case 5

Figure 4.53: Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 0 to 2, 4 and 5 with speed step of 1 km/h).

185 190 195 2001

2

3

4

5

6

7

8

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 3Case 6

185 190 195 200−7

−6

−5

−4

−3

−2

−1

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 3Case 6

Figure 4.54: Maximum upwards (left) and downwards (right) acceleration of thebeam (cases 3 and 6 with speed step of 1 km/h).

93


Cases 1 to 3 shown a good behavior, with the maximum being obtained for thepredicted resonant speed. For cases 4 to 6, however, the results were not the ex-pected. Instead of resonant peaks similar to what were obtained using 10 modes,the acceleration show diuse peaks at dierent speeds, analogous to the behaviorobtained for cases 7 to 9. To get the bigger picture of the problem, trials with 40modes and a time step of 0.005 s were made for speeds between 150 and 250 km/h.The results are shown in gure 4.55. Analyzing the gure, it becomes clear that aphenomenon similar to what happened for cases 7 to 9 is noted here. The peaksbecome diuse and one cannot dene a clear resonant speed. That is due to theinuence of higher modes. The importance of higher modes is demonstrated by thedierent maximum accelerations obtained, when 10 and 40 modes were considered.

150 200 250

−2

0

2

4

6

8

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 4 (40 modes)Case 5 (40 modes)Case 6 (40 modes)Case 4 (10 modes)Case 5 (10 modes)Case 6 (10 modes)

150 200 250

−8

−6

−4

−2

0

2

Train speed [km/h]

Acc

eler

atio

n [m

/s2 ]

Case 4 (40 modes)Case 5 (40 modes)Case 6 (40 modes)Case 4 (10 modes)Case 5 (10 modes)Case 6 (10 modes)

Figure 4.55: Maximum upwards (left) and downwards (right) acceleration of thebeam with dierent number of modes (cases 4 to 6 with speed stepof 5 km/h).

4.2.5 Discussion of the results

From the convergence study undertaken in the beginning of section 4.2, it was pos-sible to conclude that mesh elements of 2 m were short enough to get convergentresults for displacements and accelerations. The inuence of the element size on theeigenfrequencies was found to be negligible. The same was inferred for the inuenceof the time step and number of modes on the displacements. 10 modes and a timestep of 0.02 s were found to be good enough. However, to ensure the convergence ofthe accelerations, a time step of 0.005 s was used and 40 modes were considered.

After the study on the inuence of the support stiness one can infer, once more,that it is essential to consider the vertical stiness of the supports, when designing

94


or studying HSR bridges. The vertical stiness of the supports, alone, is responsi-ble for major alterations in the dynamic behavior of the structure. Regarding theeigenfrequencies, it was seen that large dierences occur when the stiness of theend supports decrease to 5 ·108 N/m. The inuence of the central column stiness isnot not so clear, when analyzing gure 4.43. Even though, the 10th eigenfrequency,for instance, decreased around 4% between stiness 5 ·1010 and 5 ·109 N/m and 10%between stiness 5 · 109 and 5 · 108 N/m.

Changes in maximum displacement and maximum acceleration were obvious. Table4.18 shows the maximum values obtained for displacement and acceleration, withthe HSLM-A1 running at resonance, whenever resonant speed was found.

Table 4.18: Maximum responses for the cases where resonance was found (resonantspeed [km/h] is indicated between parenthesis).

Maximum displacement [cm] Maximum acceleration [m/s2]Case Downwards Upwards Downwards Upwards0 2.34 (139) 1.84 (139) 4.64 (225) 4.64 (225)1 2.34 (139) 1.82 (139) 4.73 (225) 4.53 (225)2 2.35 (139) 1.82 (139) 4.88 (222) 4.76 (222)3 2.47 (139) 1.69 (139) 3.49 (191) 3.51 (191)4 2.30 (139) 1.76 (138) - -5 2.31 (138) 1.77 (139) - -6 2.42 (139) 1.64 (138) - -7 2.33 (218) 1.58 (218) - -8 2.30 (215) 1.57 (214) - -9 2.24 (188) 1.56 (187) - -

The reason why table 4.18 does not show the maximum accelerations for cases 4 to9 is that no simple resonance peak was clear as for the other cases. In cases 4 to 6,although these peaks were found using 10 modes, they were not clear with 40 modes,as can be seen in gure 4.55. That is due to the importance of higher modes on themaximum accelerations. The values from table 4.18 are plotted in gure 4.56. Thedownwards responses are shown reected over the abscissa (absolute values).

One can see the inuence the vertical stiness of the supports has on the responsesof the bridge, especially on the maximum accelerations. When the stiness of theend supports decreases, the acceleration drops from almost 4.9 m/s2 to 3.5 m/s2

(case 3). The inuence of the main column stiness is not that evident. For cases 1to 2, the decrease of the maximum acceleration is only around 3%. The resemblancebetween the downwards and upwards accelerations is also clear. The inuence tothe support stiness is very similar on both.

Large changes in the displacement, due to the supports stiness, can also be seen,especially from cases 2 to 3 and 5 to 6. When the stiness of the end supports de-creases, the downwards and the upwards displacement show opposite behavior. Theless sti the supports are, the more settlement will occur and, therefore, the absolutevalue of the downwards displacement will rise while the upwards displacement will

95


0 2 4 6 81.5

2

2.5Maximum Displacement of the Beam

Case

Dis

plac

emen

t [cm

]

Downwards (absolute value)Upwards

0 1 2 33

3.5

4

4.5

5Maximum Acceleration of the Beam

Case

Acc

eler

atio

n [m

/s2 ]

Downwards (absolute value)Upwards

Figure 4.56: Maximum displacement (cases 1 to 9) and acceleration (cases 0 to 3) ofthe beam (with speed step of 1 km/h).

decrease. Another fact to be noted is the inuence of the rst and second modes tothe response of the structure. Table 4.19 shows the resonant speed obtained fromequation 4.2, using the two rst eigenfrequencies, and the resonant speeds obtainedfrom the trials run in Abaqus.

Table 4.19: Resonant speeds [km/h] obtained from equation 4.2 and with the FEM.

from equation 4.2 from the FEMCase 1st mode 2nd mode for displacements for accelerations0 137 222 139 2251 137 226 139 2252 137 223 139 2223 137 192 139 1914 137 226 139 -5 137 222 138 -6 137 192 139 -7 135 218 218 -8 135 215 215 -9 135 188 188 -

As can be seen in table 4.19, for case 0 to 6, the maximum displacement is clearlydue to the rst mode. However, for lower stiness at the central column (cases 7to 9), the second mode becomes the most important. This can be explained fromthe analysis of the mode shapes (appendix B), where one can see that the verticalstiness of the main column has a great inuence on the 2nd mode shape. Thecomparison of the resonance can also be seen in gures 4.44 to 4.46, were the peaksoccur for the resonant speeds corresponding to the modes referred.

In the rst 3 cases, where resonance speed for acceleration could be established,the second mode dominated the response. For cases 4 to 9, higher modes (bothbending modes and rigid body modes) have a great inuence on the response andthe resonance peaks were not clear for the studied speed interval.

96

Chapter 5

Case Study: the Sagån Bridge

In this chapter, one railway bridge will be analyzed using the FE program Abaquswith a pre-processing unit for the dynamic live loads: Brigade, with special em-pathisis in the inuence of the vertical support stiness. The results obtained fromthe FEM will be compared with data measured in situ, to validate the model. Theo-retical estimation of the support stiness and model updating will also be peformed.

5.1 The bridge

The bridge over Sagån is located in Sweden, on the railway connecting Enköping toVästerås and serving the airport.

Figure 5.1: The Sagån bridge, view from northwest (a column belonging to an olderbridge behind the studied bridge can be seen in the middle).

The structure is a three-span post-tensioned concrete girder bridge, carrying one

97

CHAPTER 5. CASE STUDY: THE SAGÅN BRIDGE

Figure 5.2: The Sagån bridge, view from east.

ballasted track. The main span is 26 m long and the two side spans are 18.5 m long.Between the short columns and the embankments, on both sides, there is a span of2.2 m. Therefore, the total length of the bridge is 67.4 m.

The end supports are not structural and the facing walls only support the abutments.The loads are distributed between the two main columns and the two short columnsnear the embankments (gure 5.3). This fact will be further discussed in the modelsection.

Figure 5.3: The short columns (left) and the embankments (right) of the Sagånbridge.

98

5.2. FE MODEL

Figure 5.4: The cross-section of the Sagån bridge.

5.2 FE model

For the modelling and the results extraction, using the Finite Element Method, theFE program Abaqus/Brigade was used. The general usage of the program isdescribed in section 3.6.

The model consisted only of the permanent parts of the bridge: two main columnssupporting the continuous beam. The beam was placed along the x-axis (or axis1), which was directed from west to east. The main columns had the direction ofthe y-axis (or axis 2). The z-axis (or axis 3) from the global cartesian coordinatesystem, points in the transversal direction.

For the main beam, it was used a generalized prole with the characteristics of theSagån bridge cross-section (see gure 5.4). The material of the beam was concreteand the physical properties were the following:

- Specic weight, γ=25 kN/m;

- Modulus of elasticity, E = 32 · 109 Pa;

- Moment of inertia, Ixx = 1.74 m4;

- Area, A = 6.3 m2;

- Poison's ratio, ν = 0.25.

The concrete slab was covered with 3.6 m2/m of ballast, with a weight, γ, of 20kN/m3. It was decided to increase the density of the concrete, to include the addi-tional weight of the ballast. Thus, the density of the cross-section was consideredas 3600 kg/m3. The section of the columns (see appendix D) was modeled as arectangular shape of 4.8·1.2 m2, with a density of 2500 kg/m3. The left column wasmodeled as 7.95 m high and the right column with 6.7 m of height.

99


To simulate the behavior of the bearings, joins were used to constrain the interactionbetween the main columns and the beam. The short columns were modeled assupports and, therefore, no connectors were necessary. The joins (or connectors) aredened in the Interaction Module of Abaqus, and the interaction diagram of a joinconnector can be seen in gure 5.5.

Figure 5.5: Join connector used to connect the beam to the columns.

The interaction of the structure with the soil was, initially, considered sti. Thebehavior of the bridge-embankment interaction was discussed and it was decided torestrain only the vertical displacement on the end of the beam. The rotation andlongitudinal displacement were permitted because it was believed that the facingwalls of the embankment were not thick enough to constrain the correspondingDOFs. Nevertheless, a study was made on the subject, and it was concluded thatthose constrains had negligible inuence on the bridge behavior, for sti supports.

The bending stiness of the short columns was considered innite, since the lon-gitudinal forces were not believed to be strong enough to change the behavior ofthe structure. Thus, the longitudinal displacement at the short columns supportswas locked. The foundation of the main columns was believed to be sti enough torestrain the longitudinal displacement, and that DOF was also locked. The rotationof those foundations probably allow some rotation. However, it was decided not toinclude rotational springs in this study. Therefore, the rotation at the lower end ofthe columns was restrained.

The model was meshed with Timoshenko beam elements of 0.3 m on the mainbeam and 1 m on the columns. The interpolations functions used were linear. Thedamping was considered as direct damping and equal for the whole model. Thevalue chosen was, according to [16] and table 2.3, ξ = 1%.

The nal model, with sti supports, can be seen in gure 5.6.

The loads of the Gröna Tåget presented in section 5.3.1 were used to dene a newtrain in the Dynamic Live Load model of Brigade. The load model was then runover the structure, at dierent speeds.

100

5.3. IN SITU MEASUREMENTS

Z

Y

X

Join

Join

K

K

K

K

K

K

1

2

3

Figure 5.6: FE model of the Sagån bridge.

5.3 In situ measurements

5.3.1 The Gröna Tåget

The train used for the measurements was a testing train, called the Gröna Tåget (orGreen Train), property of Banverket, the authority responsible for administratingall railway trac in Sweden. It is a modied Regina train, capable of achievingspeeds up to 300 km/h and designed specically for the Nordic climate and Nordictracks.

Figure 5.7: 3D render (left) and picture of the interior (right) of the Gröna Tågettest train.

101


The special equipment of the test train includes:

- Traction motors on all 8 axles;

- Motor reduction gears for higher speeds;

- Speedometers and pacesetters for speeds up to 300 km/h;

- Pantograph tops for higher speeds on existing catenary;

- Track-friendly bogies for improved running stability as well as low track forcesand wheel-rail wear;

- Brake pads for higher temperatures and speed;

- Bogie fairings for reduced external noise and improved aerodynamics.

The high-speed tests were run on the railway lines between Skövde-Töreboda andEnköpingVästerås. In this last line, the train crossed the Sagån bridge, and theeects on the structure were measured. During the test periods, the behavior of thetrain and the track is constantly monitored, namely:

- Running stability;

- Lateral and vertical track forces;

- Forces and motions on pantograph;

- External noise: inuence of bogie skirts and low noise barriers;

- Internal noise and vibration;

- Passenger comfort;

- Air ow under the train;

- Behavior on curves;

- High speed behavior;

- Bogie fairings for reduced external noise and improved aerodynamics;

- Aerodynamics and pressure variations in tunnels.

The loads induced by the axles of the Gröna Tåget, and the distances between axlesare shown in table 5.1.

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Table 5.1: Gröna Tåget axle loads and distances.

Axle number Axle load [N] Axel distance [m]1 179 000 0.02 177 000 2.73 176 000 19.04 175 000 21.75 162 000 26.66 161 000 29.37 177 000 45.68 178 000 48.3

5.3.2 Instrumentation of the bridge

The instrumentation of the bridge consisted of (see gure 5.9):

- 4 accelerometers: micro electro mechanical systems (MEMS) based upon abulk micro-machined silicon element. The core consists of a 3-terminal vari-able capacitance sensor in a custom ceramic package sealed under vacuum (seegure 5.8, left). Applied acceleration or tilt to the sensitive axis, changes theinertia, causing the mass to move between the upper and lower electrodes,which results in a change to the values of the capacitors. As dierential vari-ation of the sensing capacitors is measured, a restoring electrostatic force isapplied to maintain the proof mass in a central (neutral) position. The out-put signal of the sensor is derived directly from the correction signal used tokeep the center-mass in the neutral position. This correction signal is linearlyproportional to the acceleration applied (by the ground motion) to the sensor.This type of closed loop design generally provides a better degree of outputlinearity than open-loop sensors;

- 4 linear variable dierential transformers (LVDTs): electric transformers usedfor measuring linear displacements. Each transformer contains three solenoidalcoils placed end-to-end around a tube. A cylindrical ferromagnetic core, at-tached to the object whose position is to be measured, slides along the axisof the tube (see gure 5.8, right). The magnitude of the output voltage isproportional to the distance moved by the core, which is why the device isdescribed as linear. The phase of the voltage indicates the direction of thedisplacement. Because the sliding core does not touch the inside of the tube,it can move without friction, making the LVDT a highly reliable device. Theabsence of any sliding or rotating contacts allows the LVDT to be completelysealed against the environment.

The placement of the instrumentation can be seen on gure 5.9. The four accelerom-eters were placed on the edges of the beam and were later used to determine theeigenfrequencies. The LVDTs were used to measure the vertical displacement of therst span, the longitudinal displacement over one of the bearings at the rst support

103


Figure 5.8: 3D render of one accelerometer (left) and one LVDT (right).

Figure 5.9: Position of the accelerometers and LVDTs.

and the rotation at the rst support. In addition to the mentioned sensors, straintransducers and geophones were also mounted on the bridge, but not used in thisstudy.

5.3.3 Measurement data

The eigenfrequencies were identied using the Peak Picking Method from gure5.10 (information about this method and the identication of the mode shapes canbe found in Ülker [40]). The mode types can be seen in table 5.2 and have beennamed in the following way: vertical bending modes (VB), transversal bendingmodes (TB) and torsional modes (T), followed by the number of the mode. Closelyspaced modes with similar shape have been labeled with +. The mean value andthe standard deviation of the 6 passages of the Gröna Tåget are displayed in thebottom of the table.

104


Figure 5.10: PSD estimated from accelerations, when the Gröna Tåget crosses theSagån bridge (from Ülker [40]).

Table 5.2: Estimation of the eigenfrequencies, when Gröna Tåget crosses the Sagånbridge (from Ülker [40]).

VB 1 TB 1 TB 1+ TB 2 VB 2 VB 2+ VB 3 T 1 T 2time f [Hz] f [Hz] f [Hz] f [Hz] f [Hz] f [Hz] f [Hz] f [Hz] f [Hz]1.51 5.76 6.6 9.17 9.76 10.3 12.9 14.52.08 5.79 6.61 7.35 7.79 9.15 9.72 10.5 12.7 14.32.26 5.76 6.6 7.15 7.73 9.24 9.73 10.3 12.9 14.53.08 5.76 6.6 7.16 7.71 9.76 10.3 13 14.53.25 5.84 6.61 7.33 7.79 9.2 9.74 10.5 12.6 14.34.05 6.62 7.34 7.79 9.15 10.5 12.7 14.3m 5.78 6.61 7.27 7.76 9.18 9.74 10.4 12.8 14.4σ 0.03 0.01 0.1 0.04 0.04 0.02 0.1 0.2 0.1

The eigenfrequencies chosen and later compared with the FEM results were the onesidentied with VB 1, VB 2+ and VB 3, respectively, 5.78, 9.74 and 10.40 Hz. Thedisplacements were obtained from the LVDTs and are shown in gure 5.11, for theGröna Tåget passing, approximately, at 230 km/h.

0 0.5 1 1.5 2 2.5−6

−4

−2

0

2x 10

−4

Time [s]

Dis

plac

emen

t [m

]

mid−spanover bearing (east)over bearing (west)

Figure 5.11: Vertical displacements measured with the LDVTs.

The rotation of the bridge deck over the east bearing was calculated, based on thedisplacement registered at east and west of the bearing and the distance between the

105


LVDTs. Figure 5.12 shows the rotation over the east bearing, for the same GrönaTåget run.

0 0.5 1 1.5 2 2.5−8

−6

−4

−2

0

2x 10

−5

Time [s]

Rot

atio

n [r

ad]

Figure 5.12: Rotation over east bearing, calculated using the signal of the two LVDTsensors.

5.4 Inuence of the vertical support stiness

To study the behavior of the modeled structure and the inuence of the verticalsupport stiness, and afterwards compare the numerical response with the measuredvalues, the FE model was modied. All vertical supports were replaced by springs,and their stiness varied. The formulas 3.3, 3.4, 3.5 and 3.6 suggested by Richartand Lysmer in 1966 [43, 51] and Gazetas [29, 30] are fully explained in chapter 3.2and were used in the following. The new model is shown in gure 5.13, where thespheres represent the springs.

Z

Y

X

Join

Join

K

K

K

K

K

K

1

2

3

Figure 5.13: Modied FE model of the Sagån bridge (with spheres representing thesprings).

106

5.4. INFLUENCE OF THE VERTICAL SUPPORT STIFFNESS

5.4.1 Eigenfrequencies sensitivity

The sensitivity of the eigenfrequencies to the stiness of the supports was studied.The supports conditions were grouped in 3 sets: end supports, short columns andmain columns. From the results obtained in chapter 4, it was decided to study aspan of stiness between 5·108 and 5·1010 N/m. The stiness of each set of supportswas changed, while the others were maintained sti.

Figure 5.14 shows the variation of the frequencies of the rst ten modes of vibration,for a support stiness changing within the dened interval.

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Mode

Freq

uenc

y [H

z]

MeasuredStiff supports5E10 N/m5E9 N/m5E8 N/m

1 2 32.5

5

7.5

10

12.5

Mode

Freq

uenc

y [H

z]


Figure 5.14: Eect of the end supports stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge.

For end support stiness of 5 · 1010 N/m, the eigenfrequencies are very similar tothe model with sti supports. For lower stiness, the main changes occur for the2nd and 3rd modes and for the 6th and 7th. This can be explained by the analysis ofthe modes shapes shown in appendix C. The 6th and 7th mode shapes imply higherdisplacements for the nodes at the beam ends. Since lower vertical stiness on thesupports unlocks that degree-of-freedom, the beam can vibrate at low frequencies.

A similar behavior happens for variations in the stiness of the short columns, thatcan be seen in gure 5.15. Once again, when the stiness is reduced to 5 · 1010

N/m, the frequencies are very close to the model with sti supports. Moreover, thesame modes are more aected by the reducing of the stiness. This behavior canalso be explained because modes 2, 3, 6 and 7 are responsible for the most part ofthe displacement at the beam ends. To conclude, it was seen that decreasing thestiness of the short columns has similar inuence to the reduction of the stinesson the end supports. However, the dierences in the value of the frequencies, ismuch greater for the 2nd study.

Figure 5.16 shows the inuence, on the same frequencies, of the vertical stiness ofthe supports/soil below the main columns. The dierences between the model withsti supports and the new model with vertical stiness under the main columns of5 · 1010 N/m are only noted for the 7th and 8th modes. For the second model, witha stiness of 5 · 109 N/m, all the frequencies decrease, expect the 2nd mode. This

107


1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Mode

Freq

uenc

y [H

z]


1 2 32.5

5

7.5

10

12.5

Mode

Freq

uenc

y [H

z]


Figure 5.15: Eect of the short columns stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge.

happens because that mode shape (see appendix C) has nodes very close to the joinsof the beam with the main columns. Therefore, because the displacement of thatpoints is very small, the inuence of the stiness of the main columns supports is notrelevant for that frequency. All the other frequencies decrease, specially from mode6 to 8. In the try with less sti supports, all the frequencies show great variations,and even the mode shapes change because the supports are too soft to restrain thedisplacements at the lower end of the columns.

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Mode

Freq

uenc

y [H

z]


1 2 32.5

5

7.5

10

12.5

Mode

Freq

uenc

y [H

z]


Figure 5.16: Eect of the main columns stiness in the frequencies of the rst ten(left) and three (right) modes of vibration of the Sagån bridge.

5.4.2 Model with theoretical support stiness

After the study on the eect that support stiness had on mode shapes and frequen-cies of the structure, the eective stiness of the bridge supports was estimated.

Some of the proprieties of the soil were not obtained from the in situ measurementsand were assumed between the usual span of values. That span and the values as-

108


sumed were discussed with Associate Professor Anders Bodare, from the Division ofSoil and Rock Mechanics of KTH, who commented and gave valuable suggestionsfrom his professional experience. Since this dissertation only tries to show the in-uence of the support stiness on the behavior of the structure, these assumptionsare not very important. However, when designing a bridge, preliminary studies toevaluate the dynamic response of the soil should be performed, for greater accuracyof the model.

The soil supporting the foundations is a rock subtract partially covered by sand. Toevaluate the theoretical stiness of the supports, we can assume that the density ofthe water, ρw, is 1000 kg/m3, the compact density of sand, ρs, 2650 kg/m3 and thevoid ratio, e, 0.5. The eective density [kg/m3] was obtained using formulae 5.1 fordry material and 5.2 for saturated material:

ρ =ρs

1 + e= 1767 (5.1)

ρ =e · ρw + ρs

1 + e= 2100 (5.2)

The shear modulus, G, is:

G = ρ · c2s (5.3)

The shear wave velocity in the soil, cs, was assumed as 250 m/s and the height ofthe soil layer above the rock subtract 0.8 m. The foundation dimensions can beobtained from the bridge drawings in appendix D. Hence, using formulae 3.3, 3.4,3.5 and 3.6, the values of the shear modulus, G [Pa], and vertical stiness of thesupports, Kz [N/m], can be evaluated (see table 5.3).

Table 5.3: Theoretical estimation of vertical stiness of the supports.

Richart and Lysmer GazetasCondition Circular Rectangular Half Stratumof the soil G foundation foundation space over bedrockdry 1.10 · 108 2.3 · 109 2.2 · 109 2.3 · 109 9.4 · 109

saturated 1.31 · 108 2.7 · 109 2.7 · 109 2.8 · 109 1.1 · 1010

The formulas referred by Richard and Lysmer and the estimation by Gazetas forthe vertical stiness of a surface foundation on homogeneous half-space, do notcontemplate the existence of a rock subtract underneath the soil layer. Therefore,the theoretical vertical stiness of the main columns and short columns supportswas roughly estimated from the last column of table 5.3, as 1 · 1010 N/m. For theend supports, to simulate the inuence of the bridge-embankment interaction, astiness of 1 · 109 N/m was assumed. Using these stiness, a new FE model wasmade (designated by model 2) and its eigenfrequencies were compared with theeigenfrequencies obtained from the model with sti supports (designated from this

109


point forward by model 1) and with the measured frequencies. Figure 5.17 showsthat relation.

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Mode

Freq

uenc

y [H

z]

MeasuredModel 1Model 2

1 2 32.5

5

7.5

10

12.5

Mode

Freq

uenc

y [H

z]

MeasuredModel 1Model 2

Figure 5.17: Frequencies of the rst ten (left) and three (right) modes of vibrationof the Sagån bridge (model 1 and 2).

As expected, comparing model 1 with model 2, all the frequencies decrease a lot,except the 5th. This is because the 5th mode is a longitudinal mode of vibration.The relation between the rst 3 frequencies is very similar to the relation betweenthe measured frequencies, although all the values are lower. This can be explainedby a low estimation of the modulus of elasticity or an excessively high value assumedfor the mass of the bridge, but this will be further discussed in the next section.

5.4.3 Model updating

In the last section, the resemblance between the experimental values and the nu-merical values of the model with springs (model 2), on what concerns the rstfrequencies, was discussed. In fact, the values are lower, but that is probably due toan unprecise estimation of important properties such as the mass of the ballast orthe modulus of elasticity of the material. Hence, it was decided to increase the E to42 GPa increasing, therefore, the bending stiness of the beam. With this increaseon the stiness of the model, the frequencies for the rst modes became very close tothe measured values (gure 5.18). However, due to the very low stiness of the endsupports, the nodes at the beam ends show large displacements, whenever a newtrain axle enters the bridge (see gure 5.20), demonstrating that the model shouldbe updated. For model 1, with sti supports, no changes were made to the modulusof elasticity, since it would stien the structure even more, thus reducing the, al-ready too low, displacements. These displacements will be compared in gures 5.19to 5.22, illustrating why the bending stiness was not changed for model 1.

FE model updating is a very useful tool for designers and investigators. In 1995,Imregun et al. studied the problem of model updating and the accuracy of experi-mental data required [33, 34]. In the beginning of this century, Sinha et al. [52] andZapicoa et al. [67], followed the studies undertaken by M. Imregun and, in 2007, the

110


investigation by Asma and Bouazzouni [4] and Bakir et al. [5] provided new resultsabout the subject. In the same year, Zivanovic et al. [68] studied the complete pro-cess of modelling and updating a footbridge and Daniell and Macdonald [23] focusedin the problem of systematic manual tuning.

In the past years, the software company SDTools developed a Structural DynamicsToolbox to work together with FEMLink and expand the possibilities of MatLabinto the FEM analysis. More information on the subject can be found in Balmèset al. [6]. Although powerful algorithms have already been implemented for com-puter aided model update, the usage of that algorithms is not a subject of this studyand, therefore, the update was made by trying a reduced number of pre-establishedcombinations of stiness, and studying the variations. The study of the eigenfre-quencies sensitivity to changes in the dierent supports stiness (section 5.4.1) wasparticulary useful for the update, since it gave a clear idea on how the bridge wouldrespond and what stiness should be used. After 6 trials, a FE model was foundthat gave suciently accurate results for displacement. That model (designated bymodel 3) had the following properties:

- Modulus of elasticity: 42 GPa;

- End supports stiness: 3 · 109 N/m;

- Short columns stiness: 2 · 1010 N/m;

- Main columns stiness: 1 · 1010 N/m.

The vertical support stiness values were reasonable, considering the results fromtable 5.3. Moreover, recent experimental studies undertaken on the Sagån bridgecolumns by PhD Student Mahir Ülker, from the Division of Structural Design andBridges of KTH, for estimating the support stiness, suggested a stiness around9 ·109 N/m, very close to the values adopted for model 2 and 3. However, the exper-imental method used for that calculation needs further investigation and validation.The report explaining the method and showing the results will be available throughKTH, as soon as complete (Ülker [41]).

The inuence of the updating on the frequencies of the rst ten modes of vibrationis shown in gure 5.18. Table 5.4 shows the eigenfrequencies of the dierent modelsand between parenthesis the relations between them and the measured frequencies.The displacements of the FE models were compared with the measured values andare shown in gures 5.19 to 5.21.

One can see that the eigenfrequencies for model 2, with E=42 GPa, are very closeto the measured frequencies, conrming the idea that the modulus of elasticity ofthe concrete was initially assumed too low. For model 3, the 2nd and 3rd frequenciesare higher than model 2, and not so close to the measured values. However, as canbe seen in gures 5.19 to 5.22, the displacements and the rotations show very goodaccuracy.

111


1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Mode

Freq

uenc

y [H

z]

MeasuredModel 1 (E=32 GPa)Model 2 (E=42 GPa)Model 3 (E=42 GPa)

1 2 32.5

5

7.5

10

12.5

Mode

Freq

uenc

y [H

z]


Figure 5.18: Frequencies of the rst ten (left) and three (right) modes of vibrationof the Sagån bridge (with the updated model).

Table 5.4: Frequencies [Hz] of the rst ten modes of vibration with the percentages,comparing with the measured values, between parenthesis (with the up-dated model).

Measured Model 1 Model 2 Model 3(E=32 GPa) (E=42 GPa) (E=42 GPa)

5.78 (100%) 5.34 (92%) 5.66 (98%) 5.78 (100%)9.74 (100%) 11.10 (114%) 9.82 (101%) 10.53 (108%)10.40 (100%) 12.45 (120%) 10.76 (103%) 11.40 (110%)

- 17.90 17.70 17.72- 23.23 26.44 26.61- 31.21 26.61 26.90- 33.24 27.63 28.46- 35.28 32.42 33.08- 42.49 41.78 42.76- 48.21 48.51 48.68

0 0.5 1 1.5 2 2.5−6

−4

−2

0

2

4x 10

−4

Time [s]

Dis

plac

emen

t [m

]


Figure 5.19: Vertical displacement at mid-span of span 1 (see gure 5.9).

112


0 0.5 1 1.5 2 2.5−6

−4

−2

0

2x 10

−5

Time [s]

Dis

plac

emen

t [m

]


Figure 5.20: Vertical displacement over east bearing (east).

0 0.5 1 1.5 2 2.5−15

−10

−5

0

5x 10

−5

Time [s]

Dis

plac

emen

t [m

]


Figure 5.21: Vertical displacement over east bearing (west).

0 0.5 1 1.5 2 2.5−10

−5

0

5x 10

−5

Time [s]

Rot

atio

n [r

ad]


Figure 5.22: Rotation over bearing.

113


From the analysis of the displacement at mid-span 1 (gure 5.19), it is possible tosee that all the models show reasonable agreement with the measured values. Formodel 1, with sti supports, the displacements are lower, as expected. The oppositehappens for model 3, with the less sti supports.

The displacements over the bearing (gures 5.20 and 5.21) are 10 times lower thanthe observed displacement at mid-span, and show greater sensitivity whenever atrain axle enters the bridge. Model 1 shows almost no displacement, implying thata model with sti supports does not correctly simulate the behavior of the structure.For model 2, the computed displacements are higher than the experimental and showvery high variations for the east LDVT. The reason for this is believed to be that thevery low stiness of the end supports leave the ends of the beam too loose, contraryto what happens in reality.

The rotation over the bearing (gure 5.22) is, once again, very low for model 1and model 2 shows values greater than the measured. However, model 3 showsperfect agreement, the same happening for the displacements. Table 5.5 showsthe maximum responses of the structure, for dierent support conditions. Betweenparenthesis are the percentages, comparing with the measured values.

Table 5.5: Maximum responses for the 3 models (between parenthesis are the per-centages, comparing with the measured values).

Rotation Displacement Displacement Displacementover bearing at mid-span 1 over bearing over bearing[10−3 rad] [mm] (east) [mm] (west) [mm]

Measured 0.06(100%) 0.46(100%) 0.02(100%) 0.08(100%)Model 1 0.02(34%) 0.39(85%) 0.01(25%) 0.02(26%)Model 2 0.09(141%) 0.53(117%) 0.05(216%) 0.11(148%)Model 3 0.06(96%) 0.46(101%) 0.02(103%) 0.07(95%)

5.5 Discussion of the results

The objective of chapter 5 was to show the signicance of the supports stiness onthe dynamic behavior of HSR bridges. From the formulas suggested by Lysmer andRichart [43], Richard et al. [51] and Gazetas et al. [29, 30], values for the verticalsupports stiness of the foundations of the Sagån bridge were estimated. The modelwith theoretical support stiness was compared with the experimental displacementsand frequencies obtained in situ with the Gröna Tåget at 230 km/h, and with amodel with sti supports. From the comparison, the rst conclusion was that amodel with sti supports may not show the actual behavior of the real structure.The values of the frequencies obtained from the model with sti supports and,even more important, the relation between them, were far from the measured. Forinstance, the 3rd mode of vibration had a frequency 2 Hz (20%) greater than themeasured one. Moreover, the maximum vertical displacement for the model with

114

5.5. DISCUSSION OF THE RESULTS

sti supports (model 1) was approximately between 70% and 15% lower than themeasured values.

The relation between the rst 3 frequencies, for the model with theoretical stiness,was very similar to the relation between the measured frequencies, although thevalues were somewhat lower. Lower eigenfrequencies are believed to be due to anunderestimation of the bending stiness of the beam. It is common that the concretesupplied by the contractor has a modulus of elasticity greater than that stated ondrawings. Hereby, the stiness of the beam was increased, considering a modulusof elasticity for concrete of 42 GPa. After this change, the eigenfrequencies becamevery close to the 3 measured frequencies, with an error of around 2%. However,when the displacements were compared, the results were not as good. In fact, forthe sensor on the left of the bearing, the FE model 2 shown very high oscillations,whenever a train axle entered the bridge. That was found to be due to the verylow vertical stiness assumed for the end supports (1 · 109 N/m). Using knowledgeof model updating and the know-how provided by previous studies on the bridge,a 3rd model was proposed, with a support stiness of 3 · 109 N/m for the bridge-embankment interaction.

The 3rd model had eigenfrequencies 8 to 10% higher than the measured, but showna very good agreement for the displacement and the rotation over bearing. Themaximum displacement at mid-span 1 was only 1% higher than the displacementregistered with the LVDT and the rotation at east bearing was 4% lower.

With this case study is was shown that, on one hand, a model with sti supportsmay not be reliable if realistic predictions of the dynamic behavior are to be made.On the other hand, that with a simple 2D beam model it is possible to simulate verywell the behavior of the structure. Furthermore, with the help of eld measurementsand model updating, these simulations can be increasingly accurate.

However, one should be conscious of the simplications made to the structure. Thebridge-embankment interaction clearly represents an elastic rotational spring at theend of the bridge deck, that was not accounted for in the model. Rotational springsare also more accurate to simulate the interaction at the lower end of the columns,instead of locking the rotational DOFs. More investigation about this subject shouldbe done. Moreover, a 2D beam analysis does not take into account transversalor torsional modes of vibration. This fact is not so harmful because the bridgeis transversally symmetrical, with a single track, but a 3D analysis of the bridgeshould be made. Furthermore, the train-structure interaction and the distributionof the axle loads through the ballast were neglected, the same happening with theexibility/stiness of the bearings. The track-ballast bed may also give some extrastiness to the structure, that was not accounted for. Studies on these questionsseem very interesting and should be undertaken in the future.

Finally, there are uncertainties inherent to any experimental study. Uncertainties,on the measured train speeds or LVDTs measurements, are always expected and hardto avoid. Nevertheless, the results of the study were found to be very satisfactoryand hopefully helpful for future investigations on the subject.

115

Chapter 6

Conclusions and Suggestions for

Further Research

6.1 Conclusions

The work undertaken has presented the inuence of vertical support stiness onthe behavior of high-speed railway bridges. Theoretical and numerical studies ontwo common bridges: one single span simply supported bridge and one double spancontinuous composite bridge on the Bothnia Line, were conducted. For the simplysupported bridge, the exact responses proposed by Frýba [28] for an Euler-Bernoullibeam were used, and compared with the results obtained form the FEM. For thedouble span bridge, no analytical results were extracted, and only FEM was used inthe study. From the two studies undertaken, the following conclusions can be made:

• Filtering at a cut-o frequency of 20 Hz may not identify critical behaviors.Even the value of 30 Hz or 2 times the frequency of the rst mode of vibra-tion, suggested in the D214 reports; [21], seems too low to include relevantcontributions of higher modes of vibration for the acceleration response;

• The inuence of the vertical stiness of the supports on the eigenfrequenciesis very large, and most of all, the relation between the frequencies changesignicantly for dierent support stiness;

• Increasing the vertical stiness of the supports moves the resonance peaks tohigher speeds, possibly outside the speed range of the train, decreasing theprobability of resonant behavior of the structure;

• Flexible supports bring the resonance peaks to lower speeds, but increase themass participation. Furthermore, mode shapes change and new rigid bodymodes can occur;

• Large settlements may occur at the supports with low vertical support stiness,when loaded systematically with train axles, resulting in opening of cracks inthe concrete and fatigue related problems;

117

CHAPTER 6. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

• Accelerations may be greatly underestimated by a model with sti supports,which may mislead the designer to think that the bridge-track is stable andfollows the ERRI recommendations on bridge deck accelerations [21]. This can,afterwards, instigate ballast and track destabilization, when the structure isat service and loaded with HSTs;

• Train resonant speeds are highly inuenced by the stiness of the supports.Using a simple formula, designers are able to check the train speed at whichresonance eects may occur. Moreover, increasing the stiness of the structureand of the supports, the resonant speeds can be altered;

• Resonance speeds may become more dicult to distinguish, if the stiness ofthe supports is too low, due to the greater inuence that higher modes (bothbending modes and rigid body modes) have on the structural behavior;

• Most of all, it should be mandatory to model the stiness of the supports whendesigning or studying HSR bridges.

In the 2nd part of the study, an investigation was conducted to investigate theagreement of the FE models with experimental records. The Sagån bridge, a three-span post-tensioned concrete girder bridge, carrying one ballasted track was used asa case study. Studies on how sensitive were the eigenfrequencies, displacements andaccelerations, to changes in the supports stiness were performed. From the results,one can conclude:

• Models with sti supports can greatly underestimate the maximum responsesof HSR bridges and may not be reliable if realistic predictions of the dynamicbehavior are to be made;

• The formulas suggested by Lysmer and Richart [43], Richard et al. [51] andGazetas et al. [29, 30] give reasonably accurate estimations for the verticalstiness of the supports;

• Model updating is a very useful tool for designers and researchers. Powerful al-gorithms have already been implemented for computer aided model updating,with reasonable results, speeding the updating process;

• Very simple 2D beam models are able to simulate reasonably well the behav-ior of real structures. Furthermore, with the help of eld measurements andmodel updating, these simulations can be increasingly accurate, and particu-lary meaningful for structures under constant monitoring.

The results obtained with the updated FE model were found to be very satisfactory.Nevertheless, one should be aware of the simplications in the model. Moreover,a 2D analysis may excessively underestimate the complexity of the structure, andprovide misleading results, especially for bridges carrying more than one track.

118

6.2. SUGGESTIONS FOR FURTHER RESEARCH

6.2 Suggestions for further research

Some problems occurred when performing this study, mainly related to computa-tional processing of the information. All the analysis done with FEM used the ModeSuperposition Method to reduce the number of unknowns and uncouple the modalequations. However, for complex structures and if a large number of modes is to betaken into account, computational time increases. Moreover, the physical hard diskand RAM space needed to compute so much information, is sometimes unbearable.Problems with trials, that did not seem to end, happened some times and were thereason why some interesting plots are not shown, namely concerning accelerations.Hence, the rst suggestion for further research concerns the inuence of the supportsstiness on the resonant speeds for accelerations. Plots of the maximum accelera-tions, for speeds from 100 to 300 km/h (or more), using at least 40/50 modes anda time step ensuring the convergence of the values, would be very interesting toanalyze.

Interesting results found in this investigation can also be helpful for further research.The fact that no resonance eects happened for vertical support stiness of 5 ·108 N/m, in the Banafjäl bridge, even though very high resonance was found forstiness of 1 · 108 N/m, was very surprising and more investigation on the subjectshould be undertaken. Moreover, the importance of higher modes of vibration inthe magnitude of accelerations, as was obtained for trial cases 4 to 6 of the Lögdeälvbridge, may be a very interesting eld of study.

The eect of the large amount of simplications used for this study could be interest-ing to research, as well. Even though the Sagån bridge is transversally symmetrical,with one single track, a 3D analysis, accounting for transversal and torsional modesof vibration, could be used to compare with the results obtained in this study. Fur-thermore, as it was mentioned before, the bridge-embankment interaction clearlyrepresents an elastic rotational spring at the end of the bridge deck. Investigationsabout the stiness of that spring and the eect that rotational springs may haveon the behavior of HSR bridges should bring up enough questions for a new MScdissertation. Springs at the lower ends of the columns should also be accounted for.Moreover, accelerations, both on the bridge deck and on the ballast bed, and strainsin the concrete were also registered in situ for the Sagån bridge. Studies on theaccuracy of FE models to predict these eld results can also be undertaken.

Other studies, such as the inuence of vehicle-bridge interaction, distribution of axleloads through the ballast bed or the increase in structural stiness due to the track-sleepers-ballast interaction, are ideas to try in any of the three bridges studied here,or even in others.

Nevertheless, the author hopes that the study undertaken here and the results ob-tained can be helpful to everyone interested in bridge dynamic response due tohigh-speed trains.

119

References

[1] Scanscot Technology AB. Brigade/Plus User's Manual. Lund, 2006.

[2] W.D. Pilkey amd W. Wunderlich. Mechanics of Structures Variational andComputational Methods. CRC Press, Florida, 1993.

[3] L. Andersen, S.R.K. Nielsen, and S. Krenk. Numerical methods for analysis ofstructure and ground vibration from moving loads. Computers & Structures,Volume 85, Issues 1-2:4358, 2007.

[4] Farid Asma and Amar Bouazzouni. Finite element model updating using vari-able separation. European Journal of Mechanics A/Solids, Volume 26:728735,2007.

[5] Pelin Gundes Bakir, Edwin Reynders, and Guido De Roeck. Sensitivity-basednite element model updating using constrained optimization with a trust re-gion algorithm. Journal of Sound and Vibration, Volume 305:211225, 2007.

[6] Etienne Balmès, Jean-Philippe Bianchi, and Jean-Michel Leclère. StructuralDynamics Toolbox FEMLink (for use with MatLab). Paris, 2007.

[7] D.D. Barkan. Dynamics of bases and foundations. McGraw Hill Co., New York,1962.

[8] K.J. Bathe. Finite Elements Procedure in Engineering Analysis. Prentice Hall,Englewood Clis, 1982.

[9] E. Bayoglu. Soil-Structure Interaction for Integral Bridges and Culverts. Ph.D.Thesis, Royal Institute of Technology, Stockholm, Sweden, 2004.

[10] CEN. EC1 - Actions on Structures. Part 2: Trac Load on Bridges (prEN1991-2). Brussels, 2003.

[11] Anil K. Chopra. Dynamics of Structures: Theory and Applications to Earth-quake Engineering. Prentice Hall, New Jersey, 1981.

[12] Ray W. Clough and Joseph Penzien. Dynamics of Structures. McGraw-HillBook Company Inc., New York, 1975.

[13] R.D. Cook, D.S. Malkus, and M.E. Plesha. Concepts and Applications of FiniteElement Analysis. John Willey & Sons, New York, 1989.

121

REFERENCES

[14] UIC-ERRI Committee D214. Report 1. Literature Summary on Dynamic be-haviour of railway bridges, 1999.

[15] UIC-ERRI Committee D214. Report 2. Recommendations for calculation ofbridge deck stiness, 1999.

[16] UIC-ERRI Committee D214. Report 3. Recommendations for calculatingdamping in rail bridge decks, 1999.

[17] UIC-ERRI Committee D214. Report 4. Train-bridge interaction, 1999.

[18] UIC-ERRI Committee D214. Report 5. Numerical investigation of the eect oftrack irregularities at bridge resonance, 1999.

[19] UIC-ERRI Committee D214. Report 6. Calculation for bridges with simply-supported beams during the passage of a train, 1999.

[20] UIC-ERRI Committee D214. Report 7. Calculation of bridges with a complexstructure for the passage of trac - Computer programs for dynamic calcula-tions, 1999.

[21] UIC-ERRI Committee D214. Report 8. Conrmation of values against experi-mental data. Part A: Rig tests to investigate ballast behaviour on bridges due tohigh acceleration levels - Conrmation of the acceleration limit for the ballast.Part B: Comparison of calculations and measurements using simplied modelsof rail bridges - Conrmation of the validity of the calculated values, 1999.

[22] UIC-ERRI Committee D214. Report 9. Part A: Synthesis of the results of D214 research Part B: Proposed UIC Leaet, 1999.

[23] Wendy E. Daniell and John H.G. Macdonald. Improved nite element mod-elling of a cable-stayed bridge through systematic manual tuning. EngineeringStructures, Volume 29:358371, 2007.

[24] R. Moreno Delgado and S.M. dos Santos. Modelling of railway bridge-vehicleinteraction on high-speed tracks. Computers & Structures, Volume 63, Issue 3:511523, 1997.

[25] Sara Maria dos Santos. Comportamento Dinâmico de Pontes Ferroviárias emVias de Alta Velocidade (in Portuguese). M.Sc. Thesis, Universidade do Porto,Porto, Portugal, 1994.

[26] Ladislav Frýba. Vibration of solids and structures under moving loads. Noord-ho International Publishing, Groningen, 1972.

[27] Ladislav Frýba. Dynamics of Railway Bridges. Thomas Telford, Prague, 1996.

[28] Ladislav Frýba. A rough assessment of railway bridges for high-speed trains.Engineering Structures, Volume 23, Issue 5:548556, 2001.

122

REFERENCES

[29] George Gazetas, Ke Fan, Amir Kaynia, and Eduardo Kausel. Dynamic inter-action factors for oating pile groups. Journal of Geotechnical Engineering,Volume 117, Number 10:15311548, 1991.

[30] George Gazetas, George Mylonakis, Sissy Nikolaou, and Andre Chauncey. De-velopment of Analysis and Design Procedures for Spread Footings. Universityof Bualo, New York, 2002.

[31] J.M. Goicolea, J. Domínguez, J.A. Navarro, and F. Gabaldón. New DynamicAnalysis Methods For Railway Bridges In Codes IAPF And Eurocode 1. InCongreso de Puentes de Ferrocarril, Grupo Español de IABSE, Madrid, June2002.

[32] Hibbitt, Karlsson, and Sorensen. Abaqus User's Manual. Pawtucket, 2006.

[33] M. Imregun, K.Y. Sanliturk, and D.J. Ewins. Finite element model updatingusing frequency response function data: case study on a medium-size niteelement model. Mechanical Systems and Signal Processing, Volume 9, Number2:203213, 1995.

[34] M. Imregun and S. Ziaei-Rad. On the accuracy required of experimental datafor nite element model updating. Journal of Sound and Vibration, Volume196, Number 3:323336, 1996.

[35] C.E. Inglis. A mathematical teatrise on vibrations in railway bridges. CambridgeUniversity Press, Cambridge, 1934.

[36] S.H. Ju. Finite element analysis of structure-borne vibration from high-speedtrain. Soil Dynamics and Earthquake Engineering, Volume 27, Issue 3:259273,2007.

[37] R. Karoumi. Response of Cable-Stayed and Suspension Bridges to Moving Vehi-cles, Analysis Methods and Pratical Modeling Techinques. Ph.D. Thesis, RoyalInstitute of Technology, Stockholm, Sweden, 1998.

[38] A.M. Kaynia, C. Madshus, L. Harvik, and P. Zackrisson. Ground vibration fromhigh-speed trains: prediction and countermeasure. Journal of Geotechnical andGeoenvironmental Engineering, Volume 126, Issue 6:531537, 2000.

[39] J.W. Kwark, E.S. Choi, Y.J. Kim, B.S. Kim, and S.I. Kim. Dynamic behaviorof two-span continuous concrete bridges under moving high-speed train. Com-puters & Structures, Volume 82, Issues 4-5:463474, 2004.

[40] Mahir Ülker. The Dynamic Response of two Concrete Railway Bridges duringthe testing of Gröna Tåget. Royal Institute of Technology, Stockholm, Sweden,2007.

[41] Mahir Ülker. Measuring the Dynamic Stiness of Bridge Column Foundations.Royal Institute of Technology, Stockholm, Sweden, 2007.

123

REFERENCES

[42] G. Lombaert, G. Degrande, J. Kogut, and S. François. The experimental val-idation of a numerical model for the prediction of railway induced vibrations.Journal of Sound and Vibration, Volume 297, Issues 3-5:512535, 2006.

[43] J. Lysmer and F. E. Richart. Dynamic response of footings to vertical loading.Journal of the Soil Mechanics and Foundations Division, Volume 92, Number1:6591, 1966.

[44] MathWorks. MatLab Notebook User's Guide. Natick, 1998.

[45] P. Museros, M.L. Romero, A. Poy, and E. Alarcón. Advances in the analysisof short span railway bridges for high-speed lines. Computers & Structures,Volume 80, Issues 27-30:21212132, 2002.

[46] Tobias Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl. The Not SoShort Introduction to LATEX2ε, 2006.

[47] Manuel Duarte Ortigueira. Manual de Introdução ao MatLab. Lisboa, 2000.

[48] Johan Oscarsson. Dynamic train-track interaction: variability attributable toscatter in the track properties. Vehicle System Dynamics, Volume 37, Issue 1:5979, 2007.

[49] José Rui Cunha Matos Lopes Pinto. Dinâmica de Pontes em Viga Caixão emLinhas Ferroviárias de Alta Velocidade (in Portuguese). M.Sc. Thesis, Univer-sidade do Porto, Porto, Portugal, 2007.

[50] Diogo Rodrigo Ferreira Ribeiro. Comportamento Dinâmico de Pontes sob Acçãode Tráfego Ferroviário a Alta Velocidade (in Portuguese). M.Sc. Thesis, Uni-versidade do Porto, Porto, Portugal, 2004.

[51] F.E. Richard, R.D. Woods, and J.R. Hall. Vibration of soil and foundations.Prentice Hall, New Jersey, 1970.

[52] Jyoti K. Sinha, A. Rama Rao, and R.K. Sinha. Advantage of the updated modelof structure: a case study. Nuclear Engineering and Design, Volume 232:16,2004.

[53] H. Takemiya, Y. Shiotsu, and S. Yuasa. Features of ground vibration inducedby high-speed train and the counter vibration measure x-wib. Proceedings ofthe Japan Society of Civil Engineers, Volume 661:3342, 2000.

[54] Hirokazu Takemiya. Simulation of track-ground vibrations due to a high-speedtrain: the case of X-2000 at Ledsgard. Journal of Sound and Vibration, Volume261, Issue 3:503526, 2003.

[55] S.P. Timoshenko. Vibration Problems in Engineering. D. Van Nostrand Com-pany, New York, 1928.

[56] S.P. Timoshenko. Theory of Elasticity. McGraw-Hill Book Co., New York,1934.

124

REFERENCES

[57] UIC. Code 776-1 R: Loads to be considered in railway bridge design, 1979.

[58] R. V. Whitman and F. E. Richart. Design procedures for dynamically loadedfoundations. Journal of the Soil Mechanics and Foundations Division, Volume93, Number 6:169193, 1967.

[59] David R. Wilkins. Getting Started with LATEX, 1995.

[60] R. Willis. Appendix to the report of the commissioners appointed to inquire intothe application of iron to railway structures. Stationery oce, London, 1849.

[61] H. Xia, G. De Roeck, and J. Maeck N. Zhang. Experimental analysis of a high-speed railway bridge under Thalys trains. Journal of Sound and Vibration,Volume 268:103113, 2003.

[62] H. Xia, G. De Roeck, H.R. Zhang, and N. Zhang. Dynamic analysis of train-bridge system and its application in steel girder reinforcement. Computers &Structures, Volume 79, Issues 20-21:18511860, 2001.

[63] H. Xia, Y.L. Xu, and T.H.T. Chan. Dynamic interaction of long suspensionbridges with running trains. Journal of Sound and Vibration, Volume 237:263280, 2000.

[64] H. Xia and N. Zhang. Dynamic analysis of railway bridge under high-speedtrains. Computers & Structures, Volume 83, Issues 23-24:18911901, 2005.

[65] H. Xia, N. Zhang, and R. Gao. Experimental analysis of railway bridge underhigh-speed trains. Journal of Sound and Vibration, Volume 282, Issues 1-2:517528, 2005.

[66] H. Xia, N. Zhang, and G. De Roeck. Dynamic analysis of high-speed railwaybridge under articulated trains. Computers & Structures, Volume 81, Issues26-27:24672478, 2003.

[67] J.L. Zapicoa, M.P. González, M.I. Friswell, C.A. Taylor, and A.J. Crewe. Finiteelement model updating of a small scale bridge. Journal of Sound and Vibration,Volume 268:9931012, 2003.

[68] Stana Zivanovic, Aleksandar Pavic, and Paul Reynolds. Finite element mod-elling and updating of a lively footbridge: The complete process. Journal ofSound and Vibration, Volume 301:126145, 2007.

125

Appendix A

Mode Shapes of the Banafjäl Bridge

Stiff model (Mode 1:2.36 Hz)1E10 N/m (Mode 1:2.35 Hz)1E9 N/m (Mode 1:2.28 Hz)1E8 N/m (Mode 1:1.79 Hz)


Figure A.1: Modes shapes of the Banafjäl bridge (common to all models).

Stiff model (Mode 3:19.73 Hz)1E10 N/m (Mode 3:19.22 Hz)1E9 N/m (Mode 3:15 Hz)1E8 N/m (Mode 3:7.44 Hz)



127



Stiff model (Mode 5:33.2 Hz)1E10 N/m (Mode 5:31.82 Hz)


Figure A.4: Modes shapes of the Banafjäl bridge (only present in the stier models).


Stiff model (Mode 9:84 Hz)1E10 N/m (Mode 9:75.16 Hz)

Figure A.5: Modes shapes of the Banafjäl bridge (only present in the stier models).

1E9 N/m (Mode 4:22.01 Hz)1E8 N/m (Mode 4:14.85 Hz)


Figure A.6: Modes shapes of the Banafjäl bridge (only present in the less sti mod-els).



Figure A.7: Modes shapes of the Banafjäl bridge (only present in the less sti mod-els).

Appendix B

Mode Shapes of the Lögdeälv Bridge

Stiff model (Mode 1:2.13 Hz)Case 1 (Mode 1:2.12 Hz)Case 5 (Mode 1:2.12 Hz)Case 9 (Mode 1:2.08 Hz)


Figure B.1: Modes shapes of the Lögdeälv bridge (common to all models).




131



Stiff model (Mode 4:11 Hz)Case 1 (Mode 4:10.93 Hz)Case 5 (Mode 4:10.29 Hz)

Stiff model (Mode 9:33.86 Hz)Case 1 (Mode 9:33.74 Hz)Case 5 (Mode 10:32.59 Hz)

Figure B.4: Modes shapes of the Lögdeälv bridge (only present in the stier models).

Stiff model (Mode 6:15.51 Hz)

Figure B.5: Modes shapes of the Lögdeälv bridge (only present in the stiest model).

Case 1 (Mode 8:29.84 Hz)Case 5 (Mode 9:29.84 Hz)Case 9 (Mode 10:29.84 Hz)

Figure B.6: Modes shapes of the Lögdeälv bridge (only present in the less sti mod-els).

Case 9 (Mode 3:6.52 Hz)


Figure B.7: Modes shapes of the Lögdeälv bridge (only present in the less stimodel).



Figure B.8: Modes shapes of the Lögdeälv bridge (only present in the least stimodel).

Appendix C

Mode Shapes of the Sagån Bridge

Model 1 (Mode 1:5.34 Hz)Model 2 (Mode 1:5.66 Hz)Model 3 (Mode 1:5.78 Hz)


Figure C.1: Modes shapes of the Sagån bridge (common to all models).




135






Model 1 (Mode 7:33.24 Hz)


Figure C.5: Modes shapes of the Sagån bridge (only present in model 1).



Figure C.6: Modes shapes of the Sagån bridge (only present in model 1).

Model 2 (Mode 5:26.44 Hz)Model 3 (Mode 6:26.9 Hz)


Figure C.7: Modes shapes of the Sagån bridge (only present in model 2 and 3).



Figure C.8: Modes shapes of the Sagån bridge (only present in model 2 and 3).

Appendix D

Drawings of the Sagån Bridge

139

in uence of the vertical support sti ness on the dynamic behavior of

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