Computational Fluid-Structure Interaction Applied to ...

Computational Fluid-Structure Interaction Applied to Long-Span Bridge Design

J ANNETTE B EHRNDTZ F RANDSEN

Dissertation submitted to the University of Cambridge in partial fulfillment of the requirements for the degree of D OCTOR OF PHILOSOPHY

P ETERHOUSE October 1999

To T HE MEMORY

OF MY LOVING GRANDMOTHER ,

Marie Elise Jørgensen, MY MOTHER , Lissi Frandsen,

MY SISTER ,

Helle Frandsen, MY NIECE , Pernille Frandsen, AND MY NEPHEW,

Søren Frandsen.

Preface “At vove er at miste fodfæste for en tid. ...... Ikke at vove er at miste sig selv.”

- Søren Kirkegaard (1813-1855)

This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Except from otherwise stated, this dissertation describes work undertaken solely by the author. No part of this dissertation has been submitted to any other university. This dissertation describes fluid-structure interaction behaviour related to long-span bridges and involves both applied numerical work and experimental work based on full-scale measurements carried out between October 1996 and October 1999. During this research, large oscillations were measured on the full-scale Great Belt East suspension bridge (Denmark) with a main span of 1624 metres, and this leaves no doubt of the need for greater understanding of wind-induced bridge motion. Computational fluid dynamics appears to offer additional insight into the complex fluid-structure interaction behaviour, which is observed, when a long-span bridge moves due to windaction. It is hoped that numerical methods can shed further light on the scale effect problems which are encountered with conventional wind tunnel tests of scale models of long-span bridges. There are several computational methods available to model the fluid-structure interaction mechanism. Here the finite element method is applied to the incompressible Navi´er-Stokes equations. The primary goal of this dissertation is to validate (or otherwise) a Finite Element code to be used in relation to aeroelastic long-span bridge design. With the aim of achieving practical use of the finite element method, the work is carried out on moderate computational resources, which would be available in most of today’s bridge companies. Except for the full-scale measurements, all work was carried out at Cambridge University Engineering Department under the guidance of Allan McRobie, to whom I am grateful for supervision and advice during the preparation of this dissertation. Danish-English translation: “To dare is to lose footing for a time. ...... Not to dare is to lose yourself.”

iii I would also like to acknowledge the kind permission of the bridge owner Great Belt A=S to use the Great Belt Bridge as a case study. A very special thanks to Dr. Guy L. Larose, The Danish Maritime Institute, both for numerous fruitful discusions and for making wind tunnel test results available as validation material for the numerical experiments. During this research a funding application was made to conduct further wind tunnel experiments. In preparing the grant-documents, advice and background information were given by various experts. A telephone conversation with Dr. Svend Ole Hansen on flow visualisation methods used in wind tunnels turned into a very special offer and I quote: “Would you not prefer to do full-scale measurement instead?”. I am truly grateful for having got the oppurtunity to carry out full-scale measurements on the currently longest spanning box-girder bridge in the world. The measurements on-site were a major turning point for the numerical experiments and this is the motivation behind presenting the full-scale measurements before the computational flow simulations. I would also like to thank Lars T. Thorbæk, Svend Ole Hansen ApS (Denmark), for advice on the use of the experimental equipment. It proved a far from easy task to learn the commercially available MultiPhysics finite element code Spectrum, distributed by ANSYS Inc. (USA), formerly Centric Engineering Systems Inc. (USA), for the aeroelastic flow simulations past bridge decks. A very special thanks to Arthur Muller, currently Director of Finite Element Technology at Visual Kinematics (formerly at Centric), who helped with code queries on countless occasions. At a later stage in the doctoral research, Dolf van der Heide, Ansys Inc. (USA), continued to help on code queries and moreover I was whisked into the world of finite elements at the speed of sound. This extremely valuable help is hard to describe. Discussions and suggestions around the theoretical finite element chapter is gratefully acknowledged. Thanks are also due to Professor Thomas J. R. Hughes, Standford University (USA), Department of Mechanical Engineering, for supplying me with his books and lecture-notes. Countless discusions on fluid-structure interaction, as related to the Spectrum code, with Dr. Knut Morten Okstad, SINTEF Applied Mathematics (Norway), have been a very great help. Views from proof-reading the theoretical and numerical case studies have also proved extremely valuable. Besides applying the finite element approach, the grid-free Discrete Vortex Method was tested on the aeroelastic bridge case study for comparison purposes. The visit by Dr. Allan Larsen, COWIconsult (Denmark), was very much appreciated, during which he introduced the PC-based Discrete Vortex Method code, DVMflow.

iv Dr. John Huber, Cambridge University Engineering Department, whom I had many fruitful discussions with played an important role, especially in the last phase of the Ph.D.-period. The feed back on the full-scale measurement chapter did also prove very valuable. Judith Harvey, Cambridge University Engineering Department, leaves me many happy memories of invaluable discussions. Although no background in fluid-structure interaction, she nevertheless managed to picturise a ribbon glued to the moving bridge deck surface as the fluid-structure interaction no-slip boundary conditions at the interfaces. The timely completion of this thesis is in part due to her quick proof-reading and subsequent comments and suggestions. Never have I met a greater librarian than Margaret Cosnett, Cambridge University Engineering Department, who certainly looked after me; she claimed and I quote: “You are our best customer!”. Meeting Diana Galletly and Patrick Gosling, Cambridge University Engineering Department, has been an experience I would not have liked to be without; certainly they managed to take me into the world of computers. I’m extremely grateful for their support and continuous help in a range of areas throughout the Ph.D.-period. During my studies at Imperial College, London (UK), I met my friend Nick Fuchs, Halcrow (UK). Although we studied steel structures, i.e. not fluid mechanics, I constantly heard him saying: “Von K´arm´an had an equation as complex as it may be it killed all the Romans and now its killing me”

I should later realise that my friend spoke the truth. My final thoughts go to Professor Rodney Eatock Taylor, Oxford University, Engineering Science Department. The outcome of our meeting cannot be expressed in words, but certainly it did put the final positive dot on the research here presented and at the same time made me realise that research never ends. This study was supported by the Engineering and Physical Sciences Research Council (UK). Cambridge University Engineering Department England, October 1999 Jannette Behrndtz Frandsen

Abstract This thesis describes an investigation into the use of numerical methods to predict the aeroelastic behaviour of long-span bridges, and provides an account of measurements of the dynamic behaviour of a suspension bridge obtained on-site in Denmark. That there still is a problem with the conventional experimental methods in predicting the full-scale behaviour of long-span bridges was confirmed when, as a part of this research, large oscillations were measured on the Great Belt East suspension bridge. The bridge was instrumented with accelerometers and wind-pressure sensors. Lock-in samples are presented and evaluated. For the first time pressures at deck surface and accelerations of the bridge deck are measured simultaneously. Correlations at and off lock-in are estimated. The experimental data are compared with the results of the numerical analyses. The main numerical investigations are based around the finite element method (FEM) using the commercial code Spectrum which employs an arbitrary LagrangianEulerian formulation to simulate the fluid-structure interaction, using both stationary and moving two-dimensional grids. The FE-analyses are restricted to those that can be solved on a fast work-station. In comparative studies, the Discrete Vortex Method (DVM) is applied using the COWIconsult code DVMflow which has been developed for bridge applications. The aim is to explore whether the FE approach can provide a supplementary tool for designers and can aid the understanding of the physics of flow processes, potentially reducing the large number of expensive physical model tests that are currently required. The numerical models show the ability to self-excite into vortexinduced oscillations and flutter, and the detailed flow visualisations obtained could contribute to further understanding of the fluid-structure interaction behaviour. These flow details would be difficult to obtain in a wind tunnel. In the flow regime associated with vortex-induced vibrations, the numerical model exhibited mesh-dependency and difficulties were found in simulations of the shedding frequency. Modelling of flutter in the higher wind speed flow regime showed more promising results, even though they were obtained on relatively coarse meshes. The intended end-use of the FE-method by bridge aerodynamicists forms the motivation behind these studies. However, this research suggests that successful modelling would require the use of parallel processing, and therefore the discrete vortex method appears to be the more promising approach for application to bridge design. Keywords: Long-Span Bridges, Full-Scale measurements, Lock-in and Pressure-Acceleration Correlations, 2D Discrete Vortex Method and Finite Element Simulations, Laminar Incompressible Navi´er-Stokes Flow, Implicit Galerkin/Least-Squares approximation, Arbitrary Lagrangian-Eulerian Method, VortexInduced Oscillations, Flutter.

Contents

Preface

ii

Abstract

v

Contents

vi

List of Figures

xi

List of Tables

xvi

Nomenclature

xviii

Introduction

1

1 Review of Bridge Aerodynamics

6

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2 Aeroelastic Flow Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3 Wind Tunnels Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4 Full-scale Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.5 Semi-Empirical Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Computational Bridge Aerodynamics . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

vii 2 Full-Scale Measurements

20

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Wind and Structural Monitoring System . . . . . . . . . . . . . . . . . . . . 23 2.3 Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Data Acquisition and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 Main Findings and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.1

Wind Climate at the Site . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2

Structural Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.3

Measured Vortex-Induced Oscillations . . . . . . . . . . . . . . . . . 33

2.5.4

Strouhal number estimates for cross-wind oscillations . . . . . . . . 37

2.5.5

Structural and Aerodynamic Damping estimates . . . . . . . . . . . 40

2.5.6

Resonant Measurement Sample . . . . . . . . . . . . . . . . . . . . . 43

2.5.7

Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.8

Pressure and Acceleration Correlations

. . . . . . . . . . . . . . . . 52

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Finite Element Formulation

57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Fluid Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.1

Strong form of the fluid flow equations . . . . . . . . . . . . . . . . . 59

3.2.2

Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.3

Weak Form of the Fluid Equations . . . . . . . . . . . . . . . . . . . 65

3.3 Structural Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Method to solve the Fluid-Structure Interaction Problem . . . . . . . . . . 70 3.5 Mesh Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.6 The Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

viii 3.7 Numerical Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.7.1

Fixed Bridge Decks: Segregated Solver System . . . . . . . . . . . . 84

3.7.2

Moving Bridge Decks: Coupled Solver System . . . . . . . . . . . . . 86

3.7.3

Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.8 Spectrum Code Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.9 Spectrum Code Validation by Others . . . . . . . . . . . . . . . . . . . . . . 91 3.10 Classic Test Case: Flow Past a Fixed Circular Cylinder . . . . . . . . . . . . . 93 3.10.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.10.2 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 95 3.10.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.10.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4 Finite Element Experiments

102

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2 Model Assumptions and Limitations . . . . . . . . . . . . . . . . . . . . . . 104 4.2.1

Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.2

Two-Dimensional Simulations . . . . . . . . . . . . . . . . . . . . . 104

4.2.3

Laminar Flow Assumption . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3 Modelling Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.1

Element Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3.2

Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 Bridge Aerodynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Validation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 The Great Belt East Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 Suspension Bridge (semi-streamlined box-girder) . . . . . . . . . . . . . . 115 4.8 Fixed Suspension Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . . . 116

ix 4.8.1

Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 116

4.8.2

Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.8.3

Fluid models without boundary layer modelling . . . . . . . . . . . 117

4.8.4

Fluid models with boundary layer modelling . . . . . . . . . . . . . 125

4.8.5

Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.9 Moving Suspension Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.9.1

Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 135

4.9.2

Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.9.3

Models without guide vanes . . . . . . . . . . . . . . . . . . . . . . . 135

4.9.4

Models with guide vanes . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.9.4.1

Fluid model (with vanes) . . . . . . . . . . . . . . . . . . . 140

4.9.4.2

Fluid-Structure Interaction (with vanes) . . . . . . . . . . . 145

4.9.5

Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.9.6

Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.10 Approach Bridges (bluff box-girder) . . . . . . . . . . . . . . . . . . . . . . . 153 4.10.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.10.2 Fixed Approach Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . 154 4.10.3 Moving Approach Bridge Deck . . . . . . . . . . . . . . . . . . . . . 159 4.10.4 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5 Comparative Studies

163

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2 The Discrete Vortex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.3 Fixed suspension Bridge Decks . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.4 Moving Suspension Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.4.1

Vortex-Induced Oscillations . . . . . . . . . . . . . . . . . . . . . . . 168

x 5.4.2

Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.5 Fixed Approach Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.6 Moving Approach Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.7 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Conclusions and Suggestions for Future Work

179

Bibliography

184

List of Figures 1

The Great Belt East suspension bridge oscillating in a single vertical mode

2

2

Overview of thesis contents and contributions to current knowledge . . . .

5

2.1 Great Belt East suspension bridge (Denmark): General full-scale measurement layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Equipment used for the full-scale measurements. . . . . . . . . . . . . . . 24 2.3 Free Stream velocity estimate based on numerical flow solution . . . . . . 25 2.4 Full-scale pressure measurement set-up. . . . . . . . . . . . . . . . . . . . . 27 2.5 Three different types of full-scale pressure measurements. . . . . . . . . . 28 2.6 The pressure measurement series consisted of five sequences. The tube connections are shown, illustrating how each sequence was measured. . . 29 2.7 Typical pressure-signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Wind rose data from the measurement period at Sprogø (70m above sealevel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.9 Comparison of r.m.s. vertical amplitudes between full-scale and modelscale response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.10 Full-scale cross-flow response for vortex-induced oscillations. . . . . . . . 38 2.11 Lock-in sample (cross-flow). . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.12 Maximum full-scale displacements . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories before lock-in. . . . . . . . . . . . . . . . . 46 2.14 Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories at lock-in. . . . . . . . . . . . . . . . . . . . 47

xii 2.15 Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories after lock-in. . . . . . . . . . . . . . . . . . 48 2.16 Pressure distribution on the 31m wide deck surface. . . . . . . . . . . . . . 50 2.17 Spanwise pressure correlations. . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.18 Simultaneous pressure correlations and pressure-acceleration correlations. 53 3.1 Domains and fluid-structure interaction boundary conditions . . . . . . . 64 3.2 Finite element schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Illustration of the ALE-scheme applied to a wind-induced oscillating bridge deck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4 Illustration of particle movement in space . . . . . . . . . . . . . . . . . . . 73 3.5 Illustration of mesh movement in space . . . . . . . . . . . . . . . . . . . . 74 3.6 Illustration of particle and mesh point movement in space (the ALE method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.7 Mixed mesh techniques for ALE . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.8 The fluid-structure interaction illustration. . . . . . . . . . . . . . . . . . . 81 3.9 Stagger ordering for weak and strongly incompressible coupled related to bridge deck problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.10 The stagger loop for fixed bridge deck. . . . . . . . . . . . . . . . . . . . . . 85 3.11 The stagger loop for moving bridge deck. . . . . . . . . . . . . . . . . . . . . 87 3.12 Alternative FSI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.13 Submerged tunnel project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.14 Discretization of the fluid region. Mesh with 4334 elements and 2 4454 nodes, after Okstad and Mathisen (1998a). (a) Whole mesh. (b) Close-up. . 94 3.15 The boundary conditions for flow past a fixed circular cylinder . . . . . . . 96 3.16 Initial flow features for flow past a fixed circular cylinder . . . . . . . . . . . 98 3.17 Force coefficents for the 4334-element mesh for Re= 200 and Re= 1000. . . 99 3.18 Vorticity contours for flow past fixed cylinder . . . . . . . . . . . . . . . . . 100

xiii 4.1 Flow zones around a wide bridge deck. . . . . . . . . . . . . . . . . . . . . . 107 4.2 Relevant element library. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Bridge deck notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 Great Belt East Bridge (Denmark) during construction . . . . . . . . . . . . 113 4.5 Great Belt East Suspension Bridge . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6 The boundary condition for flow past a fixed bridge deck . . . . . . . . . . 117 4.7 Discretization of the fluid region (mesh 1) around the suspension bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.8 Fluid-only model solution (mesh 1) . . . . . . . . . . . . . . . . . . . . . . . 119 4.9 Discretization of the fluid region (mesh 2) around the suspension bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.10 Fluid-only model solution (mesh 2) . . . . . . . . . . . . . . . . . . . . . . . 121 4.11 Discretisation of the fluid region (mesh 3) around the suspension bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.12 Vorticity contours (mesh 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.13 Comparison of flow from smoke visualization and Spectrum pressure contour plot based on LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.14 Discretisation of the fluid region (mesh 4) around the suspension bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.15 Strouhal number estimate for mesh 4 . . . . . . . . . . . . . . . . . . . . . . 126 4.16 Pressure distribution for the fixed suspension deck ( = 0) . . . . . . . . . 129 4.17 Fluid only solution: Flow details at Re(B)=1.65

107 .

. . . . . . . . . . . . . 130

4.18 Discretization of the fluid region (mesh 5) around the suspension bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.19 Strouhal number estimate for mesh 5 . . . . . . . . . . . . . . . . . . . . . . 132 4.20 Wake details mesh 4 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.21 5th heave mode spring-mass deck model . . . . . . . . . . . . . . . . . . . 136 4.22 5th mode (heave) lock-in simulation . . . . . . . . . . . . . . . . . . . . . . 138

xiv 4.23 Fluid-structure interaction solution with no downstream vortex street . . . 139 4.24 Installation of guide vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.25 FE-model with guide vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.26 Pressure contours for the fluid-only models, with and without vanes . . . . 143 4.27 Numerical and wind tunnel flow visualsation comparison for the bridge deck with and without guide vanes . . . . . . . . . . . . . . . . . . . . . . . 144 4.28 [Pressure contours for the fluid-structure interaction models, with and without vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.29 Flutter deck model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.30 The flutter motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.31 Flutter displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.32 Pressure history near trailing edge, mesh deformation and pressure contours overlaying the deformed mesh (enlarged by a factor 5) for an inflow velocity of 70m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.33 Great Belt East Approach Bridge . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.34 Tuned Mass Dampers in the Great Belt East Approach Bridges . . . . . . . 154 4.35 Discretization of the fluid region around the approach bridge of the Great Belt East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.36 The approach bridge cross-section . . . . . . . . . . . . . . . . . . . . . . . 155 4.37 FEM fluid-only solutions at different Reynold numbers for the approach span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.38 The approach bridge: location of springs and dashpots. . . . . . . . . . . . 159 4.39 Displacement histories at and off lock-in for the approach bridge . . . . . 160 4.40 Hysteresis in the approach bridge simulations

. . . . . . . . . . . . . . . . 161

4.41 The approach bridge of the Great Belt East: Comparison of numerical and wind tunnel test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.1 Fluid-only solution comparison between the FEM and DVM for the suspension bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

xv 5.2 DVM 5th mode (heave) simulation . . . . . . . . . . . . . . . . . . . . . . . 169 5.3 The flutter motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.4 Flutter solution obtained from DVMflow . . . . . . . . . . . . . . . . . . . . 171 5.5 Fluid-only solutions at different Reynolds numbers for the approach span 174 5.6 Strouhal number and drag coefficient as a function of Reynolds number for the approach bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.7 DVM 1st mode (heave) simulation for the approach bridge . . . . . . . . . 177

List of Tables 2.1 Full-scale natural vertical frequencies . . . . . . . . . . . . . . . . . . . . . 34 2.2 Structural damping ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1 Boundary layer modelling details for the mesh shown in Figure 3.14 . . . . 95 3.2 Force data for the flow past a stationary circular cylinder at Reynolds number 200 and 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1 Classification of flow regimes, after Ansys Inc. (1993). Note this is a general guideline for flow past arbitrary bodies. . . . . . . . . . . . . . . . . . . 106 4.2 Time-averaged force coefficients for the fixed Suspension bridge. . . . . . 128 4.3 Mesh statistics with resulting Strouhal Number for the stationary Suspension Bridge. Mesh 1, 4 and 5 contains hexahedron elements whereas mesh 2-3 contains triangular prism elements. The bridge deck in mesh 14 is embedded in a fluid domain of 3.2B 6.5B. Mesh 5 has a fluid domain size of 4B 9B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.4 Properties of Suspension Bridge used in 5th mode heave simulations. . . . 136 4.5 Properties (in air values) of the Suspension Bridge, after DMI and SINTEF (1993d) used in the 1st symmetric coupled flutter mode simulations. . . . 147 4.6 Flutter limit for the Great Belt East Suspension bridge. . . . . . . . . . . . . 150 4.7 Spectrum and wind tunnel predictions for the approach and suspension bridge. Note: time-average static force coefficients given. . . . . . . . . . . 157 4.8 Properties of the Approach Bridge, after Larsen (1997) and Larsen et al. (1995), used in 1st vertical mode simulations related to vortex-induced oscillations. Note that the actual properties differ slightly from the ones reported in (DMI and SINTEF 1993a). . . . . . . . . . . . . . . . . . . . . . . 160

xvii 5.1 DVMflow input parameters specified in all flow simulations. Note that the NSE’s are non-dimensionalised relative to the bridge width (B=1) and the free stream velocity (ux =1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2 Properties of Suspension Bridge used in the DVMflow 5th heave mode simulations, see Table 4.4 for other properties. . . . . . . . . . . . . . . . . 168 5.3 Properties of Suspension Bridge used in the DVMflow flutter simulations corresponding to an inflow velocity of 65m/s, see Table 4.5 for other properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.4 Comparison of drag and Strouhal number for different Reynolds number. 173 5.5 Properties of the Approach Bridge used in the DVMflow 1st vertical mode simulations related to vortex-induced oscillations at 24m/s. See Table 4.8 for other properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Nomenclature Roman

A B Cp CD CL CM Cs D D

F g Im Iu k ky k kp m ne ns Na p po Re St u v

Area Bridge-deck width Pressure coefficient Drag coefficient. Lift coefficient. Moment coefficient. Structural damping coefficient (2Ms !). Bridge-deck height. Domain or region. Wind load. Acceleration of gravity. Mass moment of inertia. Turbulence intensity ( uu ). Structural stiffness. Structural stiffness in vertical bending. Structural stiffness in torsion. -u ) Peak factor ( (Umax u ). Mass. Natural frequency of structure. Frequency of vortex shedding. Finite element shape function. Pressure. Far upstream pressure. Reynolds number. Strouhal number. Particle fluid velocity in horizontal direction in the fluid domain. Particle fluid velocity in vertical direction in the fluid domain.

Nomenclature

w

um uALE conv

u U U10 Ur t

xix Particle fluid velocity in out-of-plane direction in the fluid domain. Mesh velocity vector. Velocity of particle relative to mesh motion (u - um ). Fluctuating wind velocity in the wind direction. Free stream velocity. 10-minute mean wind velocity. Reduced wind velocity. time.

Greek

 t x y



()  !

Angle of incidence. Time increment. Separation length across the span. Separation length along the span. Logarithmic decrement (2). Kinematic viscosity (  ). Structural damping relative to critical. Fluid density. Correlation coefficients. Standard deviation. Angular frequency.

Subscripts

a crit f i g m p pa s

acceleration. critical vortex shedding. Fluid. i’th pressure point or i’th mode of vibration. generalised. mesh. pressure. pressure-acceleration. structure.

Nomenclature

sta

xx static.

Superscripts

T ()˙ ()¨ 

Transpose of. Time derivative. Twice time derivative. Non-dimensional value.

Abbreviations

acc ALE GLS FO FSI CFD DVM BEM FEM FVM FDM HHT HP IINSE 0s LES NS NSE 0s pres r:m:s: SUPG

Acceleration. Arbitrary Lagrangian-Eulerian. Galerkin Least-Squares. Fluid only. Fluid Structure Interaction. Computational Fluid Dynamics. Discrete Vortex Method. Boundary Element Method. Finite Element Method. Finite Volume Method. Finite Different Method. Hilber-Hughes-Taylor. Hewlett Packard. Incompressible isothermal Navi´er-Stokes equations. Large Eddy Simulation. Navi´er-Stokes. Navi´er-Stokes equations. Pressure. Root-Mean square. Streamline-Upwind Petrov-Galerkin.

Introduction Long-span bridge design is dominated by aeroelastic stability considerations involving a complex interaction between bluff-body unsteady fluid dynamics and structural response. The complexity of this interaction was evident with the collapse of the 854m main span of the original Tacoma Narrows bridge (USA) in 1940 (a plate-girder type of structure). Research on bridge aerodynamics then began and the collapse initiated wind tunnel test studies on bridge decks. Today, this is still the conventional research method and forms the basis for long-span bridge design. One major problem with wind tunnels is the prediction of full-scale flow properties. The high Reynolds number present in the full-scale cannot be reproduced at model-scale level. Another sensitive parameter is the prediction of structural damping. Although this parameter has a small magnitude, it has a major influence on the prediction of vortex-induced oscillations. That there still is a problem with the traditional experimental methods in predicting the full-scale fluid-structure interaction behaviour was confirmed (see Figure 1) when large oscillations were measured as a part of this research (1998) on the 1624m main spanning box-girder bridge of the Great Belt East suspension bridge (Denmark). These oscillations were associated with a single vertical mode of vibration following large but non-catastrophic displacements. Harmonic oscillations were identified with vortexinduced oscillations which occurred at low wind speeds of 8m/s, low turbulence intensities, and wind directions predominantly perpendicular to the bridge line. Recently, other long-span bridges have been identified with troublesome vortex-induced oscillations (Dragland 1999). The increasing need for longer spans is still a challenge for the professional engineer. Today, at the end of the 20th century, the suspension bridge has reached a free span of almost 2000m main span, the record being the 1991m truss-girder Akashi-Kaikyo bridge (Japan). This is more than four times the maximum span of this bridge type at the beginning of the century. With the aim of designing still longer spans, for example the proposed Messina and Gibraltar crossings, there is a clear need to try to resolve the problems by modelling fluid-structure interaction behaviour. The design of any long-span bridge will always require physical model tests, but supple-

Introduction

2

stationary car visible

Main span

Main span

Figure 1: The Great Belt East suspension bridge oscillating (May 4th 1998) in a single vertical mode. Full-scale measurement records are reported in Chapter 2. Photo: courtesy Michael Koch, Aalborg Stiftstidende.

mentary methods to support wind tunnel test results are highly desirable. Experimental testing typically reveals a great sensitivity of the bridge behaviour to minor details in the cross-section geometry. Such sensitivities cannot be predicted by semi-empirical analytical models. Moreover, it is not unusual that more than twenty variations of section models are tested in order to establish the influence of geometrical modifications of section depth and edge configurations for optimising the aerodynamic stability. In the initial design phase, this becomes time-consuming and expensive. With the price/performance ratio of computer chips halving every 18 months, numerical solutions are becoming increasingly attractive, and appear to offer increased insight into the complex processes involved in fluid-structure interaction. This generates hopes that the combination of quantitative predictions and improved understanding

Introduction

3

could lead to more efficient use of physical testing facilities, saving expense and time during the design phase by reducing the number of physical model tests required. The background outlined, together with potentially attractive numerical solutions, are the motivation for choosing a numerical method for investigations of the physical problem of oscillating long-span bridges. In the fifty years since the beginning of Computational Fluid Dynamics (CFD), great progress has been made. During this time numerous codes for the solution of the unsteady Navi´er-Stokes equations have been developed. Research in simulating incompressible flows is comparatively much less advanced than that within the compressible flow regimes. In the initial stage of this research, following the review on semi-empirical mathematical models, there was a choice between developing our own code or using a commercial code. It appeared that most researchers who took the route of developing their own codes could, within a three-year period, reach the stage of two-dimensional laminar flow solvers for flow past fixed bodies. However, the long-span aerodynamic bridge design problem required a level of complexity which could deal fully with fluidstructure interaction. This is the reason why this dissertation involves presentation of flow solutions using a commercial solver, the multi-physics finite element code Spectrum (Ansys Inc. 1993). It should be mentioned that, in the initial stages of this research (1996), there were to the author’s knowledge no other available commercial codes which possessed fluid-structure interaction capabilities. The research undertaken consists of a critical investigation into the numerical modelling of vortex-induced oscillations and coupled flutter of bridge cross-sections using finite elements on stationary and moving two-dimensional grids. Investigations into the finite element approach are made by applying the Spectrumy code to case studies based around the Great Belt East bridges. In particular, both the main suspension bridge and the approach bridges are considered. Prior to the fluid-structure interaction simulations, fluid-only simulations are undertaken to observe the unsteady flow patterns around stationary bridge decks. The finite element method used in this research could prove to be a future tool for guiding the development of bridge decks prior to wind tunnel tests and thus reduce the number of physical model tests currently required. Eventually, the primary objective of this type of work is to compare full-scale results with model-scale tests with the long y The computational

code, Spectrum is distributed by ANSYS Inc. (USA). The flow simulations are run on moderate computational resources (as opposed to a supercomputer), comprising the fast work station: HP C200 having a single 200MHz processor with a peak performance of 800 Mflops and 768 Mb RAM.

Introduction

4

term goal of validating numerical models. As demonstrated in this thesis, linking fullscale measurement with wind tunnel test results and a numerical model, which possesses fluid-structure interaction capabilities, is crucial to the development of bridge aerodynamics. To summarise, this dissertation contains fluid-only and fluid-structure interaction simulations applied to long-span bridge decks using a commercially-available program, invoking a finite element approach. The numerical experiments are validated through comparison with the results of full-scale measurements carried out by the author (containing lock-in samples) and with results of wind tunnel tests carried out by others. The contribution of this study to current knowledge is outlined in the flow-chart in Figure 2. The layout of this dissertation is as follows: Chapter 1: Review of Bridge Aerodynamics; Chapter 2: Full-Scale Measurements; Chapter 3: Finite Element Formulation; Chapter 4: Finite Element Experiments; Chapter 5: Comparative Studies;

Introduction

5 Simulation Involvement Vortex-induced Oscillations Coupled Flutter BEM DVM

2D Model

Numerical Methods

FDM

FEM

3D Model

FVM

1

Full-scale Measurements

Numerical Model Experiments

Validation Conventional Method Wind-Tunnel Tests (Model-Scale)

2

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INTERACTION

Existing Link

00000000 11111111 11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

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STRUCTURE

WIND

Semi-Empirical Analytical Models Lift Oscillator

Wake Oscillator

Fixed Navier-Stokes Eqn’s Compressible Flow

Simple Geometries

Moving

Fixed

Bridge Decks

Moving

Incompressible Flow

Flow Regime Abbreviations: FEM : BEM : DVM : FDM : FVM :

Finite Element Method Boundary Element Method Discrete Vortex Method Finite Difference Method Finite Volume Method

Re. no. : Reynolds Number : Two-Dimensional 2D

Low Re.no.

High Re.no.

ASSUMPTION Laminar Flow Model

ASSUMPTION Turbulence Flow Model

Areas of contribution to current knowledge Areas treated in detail in thesis Areas mentioned in review or discussions only Crude Approximation in Numerical Simulations Long-term Aim

Figure 2: Overview of thesis contents and contributions to current knowledge viewed in the light of the bridge aerodynamics field. The research contributes to current understanding as follows: 1: Full-scale measurement data on a long-span bridge, which includes lock-in samples. The measurement focuses on linking the fluid-structure interaction behaviour through simultaneous pressures and accelerations of the bridge deck in addition to records of the wind speed and wind direction measured simultaneously. 2: Numerical experiments of vortex-induced oscillations and coupled flutter of bridge sections using finite elements on stationary and moving two-dimensional grids. The arbitrary Lagrangian-Eulerian method is applied in the fluid-structure interaction simulations.

Besides the full-scale measurement, validation of the numerical model is done through comparison with results of wind tunnel tests carried out by others.

Review of Bridge Aerodynamics

1.1

1

I NTRODUCTION

Wind-induced phenomena in flexible structures are numerous. They happen at different wind speed ranges, some of them can occur concurrently and also the different modes of oscillation of the structure may be susceptible to excitation by different mechanisms. Comprehensive backgrounds to the effects of wind on structures are given by Simiu and Scanlan (1996) and Dyrbye and Hansen (1997). This dissertation is concerned with aeroelastic design considerations related to longspan bridges. A general state-of-the-art background to bridge engineering is covered in the proceedings of Larsen (1992) and Larsen and Esdahl (1998). Governing design criteria for long-span bridges involve the aeroelastic phenomena of vortex-induced oscillations, buffeting and flutter (Scanlan 1981; Dowell et al. 1995). To address the problem of the fluid-structure interaction behaviour in bridge aerodynamics a review of the current methods (wind tunnels) for long span bridges are given. Semi-empirical analytical models whose formulation depend on aerodynamic forces derived from the wind tunnel will also be addressed. The combination of wind tunnel testing and analytical models have been the tool for solving fluid-structure interaction problems for many decades. Much research in bridge aerodynamics is to a large extent based on investigations into the flow past arbitrary bodies such as airfoils, flat plates and cylinders with both round and sharp edges. Comprehensive investigations have been done to establish analytical models. However, these models cannot predict the change in fluid forces due to minor changes in leading edge details. This is of major importance in bridge aerodynamic design. As an alternative, investigations into the use of numerical methods have received increasing interest both in research and industry. These numerical methods will briefly be reviewed here, concentrating on the applica-

1.2. Aeroelastic Flow Phenomena

7

tion to bridge aerodynamics.

1.2

A EROELASTIC F LOW PHENOMENA

Wind on long-span bridges causes vortex-induced oscillations occurring at low wind speed and flutter at high wind speeds. Vortex-induced oscillations do not cause sudden collapse, but must be designed against as they can cause discomfort to users of the bridge and result in long-term fatigue damage. In contrast, when the flutter limit is reached the amplitude of vibration can increase without limit until the bridge collapses. These design conditions are equivalent to designing a structure against a serviceability (vortex-induced oscillations) and an ultimate (flutter) limit state design criterion. On a historical note the original focus of aeroelastic studies on bridge models was the problem of flutter, as this caused the collapse of the original Tacoma Narrows. Extensive investigations into the collapse of the bridge were carried out by Farquharson (1952) supported by the pioneering work of Von K´arm´an and Sears (1938) and Bleich (1949) who explored the application of airfoil theory to bridge structures. In the literature, it is evident that most work in bridge aerodynamics is based around flutter design evaluation. The design case of vortex-induced oscillations is reported less although this is still an unresolved aerodynamic design issue, and many modern semistreamlined long-span box-grider decks suffer from these oscillations. Because this flow phenomenon occurs at the more frequent low wind speeds and does not result in sudden collapse, it has the potential for experimental study. This was the case recently when large vortex-induced oscillations were measured full-scale on the Great Belt East suspension bridge as reported in Chapter 2. In this research, besides the problem of flutter at high wind speeds, specific attention is given to the flow regime of vortex-induced oscillations occurring at low wind speed. It should be noted that forces induced by vortex-shedding occur whether or not the structure is moving. Flutter occurs at a wind velocity that has a motion-induced windload at which the vertical and torsional vibration modes couple. The risk of flutterinduced vibrations is significant when the torsional natural frequency is only slightly larger than the vertical natural frequency, which is often the case on slender long-span bridge decks. An acceptable flutter limit is one of the principal design criteria for longspan bridges and Scanlan’s theory of flutter derivatives is widely used to estimate this.

1.3. Wind Tunnels Tests

8

For many decks, initial estimates of the derivatives may be obtained using Theodorsen’s inviscid flat plate theory and the very useful approximate formula of Selberg (1961). A background to flutter analysis is also given by Fung (1993).

1.3

WIND TUNNELS TESTS

Currently wind tunnel tests form the basis of the design of long-span bridges. The tests are carried out either on section models, taut-strip models or full aeroelastic models. The modelling techniques for each of the model types are outlined by Tanaka (1992). In the following each of these models are briefly described. In most cases, the test method of choice is the section model. This method is simple and comparatively cheap. The section model contains a three-dimensional part of a bridge deck and can usually, depending on the size of the wind tunnel, allow reasonable modelling of geometric details avoiding questionable Reynolds number problems of smaller-scale models. A detailed discussion on advantages and limitations of section models is presented by Hjort-Hansen (1992). In the late 1960’s Davenport developed the idea of carrying out wind tunnel tests on a taut strip model of the entire main span of the bridge. A taut strip model enables modelling of the three dimensional dynamic response of a full bridge based on a simplified structure with sinusoidal modes of vibration. Advantages and limitations of taut strip models are discussed by Davenport et al. (1992). A review on full-aeroelastic models is given by Irwin (1992), who points out that the best outcome from such tests is only obtained when section model tests and analytical studies are conducted in parallel. Aeroelastic experimental investigations may be grouped in two categories: those in which the bridge is free to oscillate and those where it may be subjected to prescribed motion with the frequency and amplitude of vibration controlled independently. Independent control over the frequency and amplitude of the deck oscillations makes it easier to observe the effect of either parameter, and it is for this reason that prescribed motion tests have often been preferred to free vibration tests, particularly for flutter studies. Flow visualisation and measurement of velocities in the flow field are important for validation of numerical models related to bridges. Comparisons between fluid flow patterns is an important issue for the development of bridge aerodynamics. Much data from model-scale tests refers to structural motions rather than fluid flow patterns,

1.4. Full-scale Measurements

9

which makes it difficult when validating analytical and numerical models; these models tend to describe the fluid behaviour in great detail. There are a variety of ways of measuring and visualising the flow patterns around structures, which are thoroughly described by Yang (1989), Merzkirch (1987) and Goldstein (1996). As a supplement to the velocity measurements, a visualisation method to study various parts of the flow provides additional crucial validation material for the very detailed fluid-structure interaction analyses. A literature survey revealed that flow visualisation experiments around bridge decks are only carried out on rare occasions. Both air and water can be used as the fluid medium for wind engineering experiments, as shown in the fluid album of Van Dyke (1982) covering arbitrary bodies. For the bridge engineering application see for example the hydrogen bubble techniques used in water-channels (Su and Li 1993) or smoke in wind tunnels (Larose 1998b). Li (1995) discusses the advantages of using water as the fluid medium instead of air addressing the difficulties of high Reynolds number flow. Su and Li (1993) used the hydrogen bubble techniques in a water-channel to derive flutter derivatives on section models in prescribed motion tests.

1.4

F ULL - SCALE MEASUREMENTS

Comparisons between full-scale measurements and wind tunnel testing are crucial to the development of bridge aerodynamics. Full-scale measurements on long-span bridges are rare. If they are carried out, the measurements usually consist of direct measurements of acceleration, wind speed and wind direction. Examples of such valuable measurements have now been carried out on long-span bridges such as the West Gate Bridge (Melbourne 1979), the Humber Bridge (Brownjohn et al. 1994), the Normandie Bridge (Delaunay et al. 1999) and the H¨oga Kusten bridge (Larose et al. 1998). Reports also exist on full-scale measurements carried out on British bridges such as Cleddau, Erskine and Severn bridges (Hay 1992). Beside full-scale measurement carried out on these long-span bridge box-girder decks it is also necessary to test the pylons in the free-standing situation during construction. Such full-scale tests were carried out on one of the pylons of the Great Belt East bridge (Larose et al. 1998). Full-scale bridge behaviour is still somewhat uncertain despite these full-scale data and

1.4. Full-scale Measurements

10

the extensive experimental model-scale data available. As a part of this research fullscale measurements were carried out on the Great Belt East suspension bridge. The unique feature of these measurements is that wind-pressures are measured simultaneously with structural accelerations aiming at gaining further insight into the fluidstructure interaction behaviour. Details are given in Chapter 2.

1.5. Semi-Empirical Analytical Models

1.5

11

SEMI -E MPIRICAL A NALYTICAL MODELS

Physics of the vortex-shedding phenomena for flows past arbitrary bodies has been investigated through experiments on scale models since the beginning of the century when von K´arm´an did his pioneering work on vortex methods. Many investigators have since given comprehensive reviews on vortex induced oscillations (Parkinson 1989; Bearman 1984; Sarpkaya 1979; Blevins 1990). It is important to realize that vortex induced oscillations for a moving body are not a simple resonance problem: the response is known to exhibit a nonlinear limit cycle. The complexity occurs when the structure interacts with the vortex-shedding of the fluid flow. Experiments have shown that elastic structures near resonance can develop flow-induced vibration by extracting energy from the flow around them. Bishop and Hassan (1964) were amongst the first to show that the oscillations of a structure modify the flow and give rise to non-linear interaction. The theory of non-linear oscillations admits the possibility of oscillation hysteresis, which involves jumps in the lift forces, the amplitudes and the phase between lift force and displacement of the structure. These intriguing effects were first observed in experiments carried out by Feng (1968) in the vortex-induced vibration of circular and D-section cylinders. Since then a considerable effort has been made in establishing mathematical models in order to explain these effects (Parkinson 1989). The search for mechanical models representing vortex-induced vibrations was very active in the late sixties and in the seventies. The outcome of this work was a series of models describing the fluid forces acting on a fixed or moving body resulting in equations of motion for a fluid and structural oscillator which can be solved analytically. These mathematical models are of an empirical nature as the non-dimensional parameters such as the lift-coefficients are derived from results of wind-tunnel tests. These equations of motion can take account of the negative aerodynamic damping induced by vortex-shedding. Mathematically, this is included in the equation of motion by some non-linear factor in the damping term. Amongst these mathematical models are the liftand wake-oscillator models for vortex-shedding. Hartlen and Currie (1970) proposed the first lift-oscillator model. In this model the lift-coefficient is not determined from the flow-field theory of the Navi´er-Stokes equations, but is instead regarded as a nonlinear oscillator representing the wake vortex-effect and governed by a second order differential equation. In contrast to lift-oscillator models, the wake oscillator models employ a semi-empirical flow-field theory by representing the effects of the vortex wake from flow past a body by a finite cavity attached to the structure in torsional oscillation.

1.5. Semi-Empirical Analytical Models

12

For further details on this type of model see Tamura and Matsui (1979), whose model is based on the model by Birkhoff (1953). Most of the above-mentioned research covers extensive studies on incompressible flow past arbitrary bodies in connection with simple geometries such as circular, square and rectangular cylinders and airfoils (usually related to the compressible flow regime). This knowledge has been applied in the field of bridge aerodynamics. Traditional analytical methods for long-span bridge design have been based upon structural models excited by external forces, prior to wind-tunnel testing. These models include liftand wake-oscillator models for vortex-shedding and the flutter stability criteria based largely on the classical inviscid theory of Theodorsen. In relation to vortex-induced oscillations on long-span bridges a comprehensive review of lift-and wake oscillator models is given by Ehsan (1988) and Billah (1989). Recently, Srinivasan (1997) developed a vortex-induced oscillation model for long-span bridge application using an expanded two-dimensional wake oscillator model of Tamura and Matsui (1979). Also in connection with vortex-induced oscillation models for long-span bridges, Simiu and Scanlan (1996) established a suitable forcing function in their model, recognising the non-linear characteristic of vortex-induced oscillations. The suggested analytical model includes a non-linear damping term following the Van der Pol oscillator concept (Van der Pol 1920) with a cubic term, ensuring prediction of vortex-induced oscillatory response. Such models are not based on an understanding of the underlying physics of the flow processes, but are merely equations known to exhibit self-excited limit cycles which can be fitted to suit the experimental data. Furthermore, Larsen (1993b) expands the forcing function of Simiu and Scanlan (1996) and applies it in connection with design and optimization of tuned mass dampers installed in the Great Belt East Approach bridges (see Section 4.10).

1.6. Numerical Methods

13

1.6

N UMERICAL METHODS

Design of long-span bridges is dominated by aeroelastic stability considerations involving a complex interaction between bluff-body unsteady fluid dynamics and structural response. In comparison to the rich complexity of the fluid processes, the traditional mathematical descriptions of the various aeroelastic phenomena involve gross simplifications. By contrast, numerical solutions of the full field equations appear to offer the possibility of greater realism, and the ability to graphically visualise the results appears to offer the prospect of obtaining more insight into the geometry-sensitive subtleties of fluid-structure interaction. If such models could be demonstrated to successfully replicate the correct physics of these problems, their usefulness as an aid to design is evident, particularly in helping to reduce the number of physical tests required. The mathematical theory underlying the problem of bridge design and analysis is based on partial differential equations. Techniques that have successfully been developed over the last decades include:

   

Finite Difference Method (FDM); Finite Element Method (FEM); Finite Volume Method (FVM); Boundary Element Method (BEM);

It is outside the scope of this dissertation to give a detailed, quantitative description of each of these methods. Detailed descriptions can be found in the literature (Wendt 1996; Gresho and Sani 1998; Johnson 1998). However, all methods can be used to approximate the initial boundary value problems that arise in bridge design. The FEM, FVM and FDM are similar in that the entire domain must be discretised, while in the BEM only the bounding surfaces is discretised. The FEM and FVM are superior to BEM for representing nonlinearity, while the BEM are superior to the FEM/FVM for problems where the boundary solution is of interest or where solutions are desired at a set of points spaced in a highly-irregular manner throughout the domain. Because the computational mesh is simpler for the BEM than for the grid-based methods, the BEM requires less CPU and in this respect becomes more attractive from a practical point of view.

1.6. Numerical Methods

14

The finite element method is a technique for finding approximate solutions of partial differential equations (PDEs). The domain is divided into elements which typically have either triangular or a quadrilateral form. The grid need not be structured. Due to this unstructured form, very complex geometries like bridge decks can be handled with ease. This is the most important advantage of the method over the finite difference method (FDM) which needs a structured grid. The FEM and the FVM (Zienkiewicz and Onate 1991; Gresho and Sani 1998) have similar advantages and disadvantages, though the Finite Element method appears to be the more robust. Boundary element methods such as the discrete vortex methods appear as appropriate methods to apply to bridge aerodynamics problems such as oscillating long-span bridges. Using such a method the time-consuming task of mesh generation can be avoided and more importantly there are potential savings in CPU requirements. Inspired by the ideas of Hughes and Jansen (1993), the FEM for both the fluid and the structure is used for the main numerical investigations (Chapter 4) of wind-induced bridge motion presented in this thesis. The grid-free Discrete Vortex Method (DVM) is used in comparative studies (Chapter 5). Note that there are two classes of Finite Element Methods for solving the timedependent problem: semi-discrete methods and fully-discrete space-time methods. The idea behind the semi-discrete methods is to have the time (t) continuous and develop a spatial variational formulation for the time derivative terms in the strong formulation. This results in a coupled system of ordinary differential equations (ODE). ODE integrators are then used to solve the problem. The semi-discrete method is applied in the research presented. For theoretical details see Section 3.2.2 and for application see the numerical case studies in Chapter 4. In the fully-discrete space-time methods the space-time domain is discretized into space-time “slabs”, i.e. t intervals in (x,t)-space. The discontinuous Galerkin method is used to weakly enforce the continuity of the solution across the time slabs. The fully-discrete space-time methods are described in detail by Johnson (1987), Thompson (1994), Masud and Hughes (1997) and in the state-of-theart review by Harari et al. (1996). To the author’s knowledge, no commercial FEM code has yet implemented a fully-discrete space-time method.

1.7. Computational Bridge Aerodynamics

1.7

15

C OMPUTATIONAL B RIDGE A ERODYNAMICS

The flow around any structure depends on its external shape, the Reynolds number and the turbulence in the approaching flow and to some extent on the amplitude, mode and frequency of structural oscillations. Before reviewing numerical studies of flow past bridge decks it should be emphasized that numerious numerical studies have been carried out on flow past arbitrary bodies, see the review by Matsumoto (1999). Although these objects have simple geometries, the flow patterns past these objects are complex, the flow tending to become more complex for increasing Reynolds numbers. In addition, flow complexity is added when the bodies begin to oscillate. Most research carried out is related to low Reynolds number flow, with the flow regime Re=100-200 receiving much attention. When dealing with higher Reynolds numbers the influence of turbulence becomes evident. Some form of turbulence model (Murakami 1990; Laurence and Mattei 1993) is usually required to capture the high frequency components. Note that this has a knock-on effect as to mesh requirements and ideally requires three dimensional flow models (Kalro and Tezduyar 1997). Current research has reached this level but uncertainties around turbulence modelling are present. However, little work is published (Farhat et al. 1999) on the high Reynolds number flow regime for the combination of moving bodies and inclusion of a turbulence model (even on two-dimensional idealised models). Flow past circular cylinders and high angles-of-attack airfoils appear to be the test cases most used in the field of bluff-body aerodynamics (Tezduyar et al. 1994). A review of CFD techniques to model flow around bridge decks revealed that mainly fluid-only analyses were undertaken. Results from these analyses are typically steadystate coefficients (drag, lift and moment) and Strouhal number. Comprehensive studies also involve analyses with prescribed deck motion allowing linearised aerodynamic motional force coefficients (flutter derivatives) to be determined. It should be noted that validation of a numerical model involves more than comparison of aerodynamic derivatives. Forces and pressure distribution on the deck together with flow field data and flow visualization are required. A literature survey revealed that there is no numerical model which have undertaken such validation. However, the prediction of flutter wind speed for a bridge deck is the primary aim for adopting a numerical method for bridge arodynamics. This is because the current aim is to have such a method which can reduce the number physical model tests required when selecting an optimized aerodynamic cross-section (which is based on the flutter limit).

1.7. Computational Bridge Aerodynamics

16

Current advances in computer technology (CPU speed and memory) make it possible to investigate the fluid-structure interaction process numerically. Within the last couple of years it appears that most researchers are choosing either the fluid-structure interaction direction with the assumption of laminar flow or flow simulation with sensitivity studies of turbulence models. Currently, parallel computing is required when including both a moving structure and a turbulence model in grid-based methods. Comparable studies to this research applying alternative numerical schemes to longspan bridge problems include the Jenssen and Kvamsdal (1999) work with a finite volume method on moving unstructured regular grids with Large Eddy Simulation (LES), and the work of Enevoldsen et al. (1999) using a finite volume method on moving grids with multiblock mesh generation. Enevoldsen et al. (1999) and Jenssen and Kvamsdal (1999) show results based on parallel computing techniques from flutter analyses in good agreement with wind tunnel tests. Note that in contrast to the fully-coupled fluid-structure interaction solutions presented in this research, Enevoldsen et al. (1999) and Jenssen and Kvamsdal (1999) have a coupling module included in their numerical approach (see Figure 3.12) between the finite volume method (fluid) and the finite element method (structure). Jenssen and Kvamsdal (1999) have done studies on threedimensional models including LES on stationary grids expanding a 2D model of 75000 cells to 4 75000 cells and conclude that the steady-state lift and pitching moment are in closer agreement with the results of the wind tunnel. However, it does not appear evident that 3D simulations are required. Bruno et al. (1999) apply the finite volume method adopting the (k-)-model when solving the flow past the more complicated leading edge detail including railings of the Normandy cable-stayed bridge. Using parallel computing (Bruno et al.) present 2D flow results for a stationary bridge deck. Their optimised fluid model consist of an unstructured irregular grid with domain size of 50B 40B. The mesh has 29000 nodes with a cell thickness at the wall of 2 10-3 B, where B21.2m. Bruno et al. (1999) reported good agreement with the wind tunnel when comparing force coefficients for their fluid-only solutions. Recently De Foy (1998) applied his unsteady incompressible finite volume code to the Great Belt East bridge deck section. Fluid-only simulations were performed for a Reynolds number (Re(B)) of 1:38 108 assuming laminar flow. De Foy (1998) reported a Strouhal number, St(D) of 0:17 which compares reasonable well with the section model of the wind tunnel (0.11-0.15). Previous numerical work using the Finite Element Method has been carried out by Lee et al. (1995), Mendes and Branco (1998) and Selvam and Bosch (1999). Lee et al. (1995) use the FEM on moving structured regular grids adopting an arbitrary Lagrangian Eu-

1.7. Computational Bridge Aerodynamics

17

lerian formulation. A (k-)-model is used and a Streamlined Upwind Petrov Galerkin (SUPG) formulation is assumed. Lee et al. (1995) investigated the FE-approach on several bridges. The static force coefficents agreed with the results obtained in the wind tunnel. However, some discrepancies were found with regard to the onset velocity of vortex-induced oscillations. Mendes and Branco (1998) carried out flow investigations on the Vasco da Gama cable-stayed bridge. They assumed laminar flow. However in recognition of the high prototype Reynolds number, an incorrect low Reynolds of 3 103 was used in the demonstration of the flow solutions. Their studies demonstrated that a cross-section with baffles aid the suppression of torsional instability. Furthermore, Selvam (1998) applied the FEM to the approach bridges of the Great Belt East and in recent analysis to the suspension bridge (Selvam and Bosch 1999). Selvam and Bosch present both 2D and 3D flow solutions assuming fixed bridge decks and report good agreement with the results of the wind tunnel in terms of static coefficients and Strouhal number. Research related to the Finite Difference Method for bridge applications has been explored by Fujiwara et al. (1993) both on stationary and moving two-dimensional grids. The flow solutions presented were for low Reynolds numbers in the range of 2100 to 4000 following wind tunnel experiments. Onset wind speed prediction agrees with the wind tunnel experiments, but discrepancies were found in the amplitudes, these being overestimated by the numerical model. They reported that a possible explanation could be the loss of 3D effects, the 2D solution predicting larger fluctuation of lift. Furthermore, Onyemelukwe (1993) developed a finite difference code and comprehensive studies were undertaken in application to bridge design. The 2D Discrete Vortex method has been applied to bridge decks by Walther (1994) who developed his own code. This code has been extensively validated, see for example Walther and Larsen (1997a). The code is applied in this research in connection with comparisons with the FEM (see Chapter 5 for details). Taylor and Vezza (1999) have also developed their own 2D Discrete Vortex method code and present results on stationary and oscillating bridge decks. In addition, Taylor and Vezza (1999) show results of the flutter motion being suppressed by inclusion of active control vanes. Furthermore, the Discrete Vortex method in 2D was applied by Larsen et al. (1998) to long-span bridges with extreme span, i.e. the proposed Gibraltar and Messina crossings. Such crosssections contain multi-deck sections to achieve the necessary torsional resistance. The flow solutions using the DVM method were reported to predict the flutter wind speed in good agreement with the wind tunnel. Finally various investigators have used combined numerical methods, see for example the studies carried out by Brar (1997) who proposed a coupled finite difference and

1.8. Summary

18

vortex method scheme.

1.8

SUMMARY

The review revealed that the traditional analytical models developed have limitations in terms of detecting changes in the flow caused by small variations of leading edge details. The following Chapter contains full-scale measurement on a long-span suspension bridge (the Great Belt East suspension bridge) carried out as a part of this research. It illustrates that predicting full-scale behaviour of long-span bridges is surrounded by great uncertainty. Wind tunnels form the basic approach to long-span bridge design but high Reynolds numbers can not be reproduced and therefore a source of uncertainty is present. In addition, a minimum of approximately twenty different sections are usually required to be tested in the wind tunnel before an optimal cross-section can be selected. This is costly and time-consuming. With the aim of reducing the number of physical tests required and to aid further insight into the complex flow processes, this research undertakes numerical investigations. This may provide a supplementary tool which can guide the development of bridge cross-section and thus reduce the physical model tests. Computational bridge aerodynamics is divided into grid based and grid-free methods. Most numerical work related to the grid-based methods are carried out using parallel processing. This make the grid-free methods more attractive from a practical perspective, as personal computers or work stations have proved sufficient for such methods. Note that this is for two-dimensional flow simulations on both fixed and moving bridge decks. The literature study revealed that work related to the grid-based methods are mainly carried out for flow past stationary bridge decks. Fluid-structure interaction simulations and inclusion of a turbulence model due the high prototype Reynolds number add to the computational requirements. Currently CPU speed puts constraints onto the progress on numerical grid-based methods. However over the last couple of years major progress has been achieved. Currently research is undertaken in both 2D and 3D with inclusion of a turbulence models. Oscillating bridge decks are mainly studied in 2D. This research involves 2D FEM investigations using moderate computer resources aiming to achieve practical use by bridge aerodynamicists. Subsequently the fluid model

1.8. Summary

19

are then constrained to laminar flow assumptions with two-dimensional stationary and moving grids. Chapter 4 presents case studies carried out on both the main suspension bridge and the approach bridges of the Great Belt East. The discrete vortex method flow solver (developed by Walther (1994)) is applied for comparison with the FEM studies (Chapter 5). This numerical method is proving to be the first practical computational method for use in bridge design, the Personal Computer based nature of the code being particularly attractive. Finally, it should be noted that both conventional and numerical methods display uncertainties due to the presence of high Reynolds number flow.

Full-Scale Measurements

2.1

2

I NTRODUCTION

In this chapter, results from measurements of simultaneous pressures and accelerations induced by wind action on the Great Belt East suspension bridge are presented. At the time of measurement the bridge was in finished condition but not officially opened for traffic. The full-scale measurements took place between 24/4/1998 to 7/6/1998. The bridge had no wind screens installed, the only flow obstruction being due to the outer and centre crash barriers. The general measurement layout is outlined in Figure 2.1. The actual full-scale accelerations were measured 222.5m from the centre. The simultaneous pressure measurements took place in various cross-sections offset (by 15-62m) from the cross-section of the accelerometers. Simultaneous measurements of wind speed, wind direction, pressures on the deck and accelerations of the deck were recorded. From these measurements selected data are presented, including a sample of resonant suspension bridge behaviour which occurred due to vortex-induced oscillations. Most pressure measurements took place in the emergency stopping lane and the traffic lane near the trailing edge on the main span of the suspension bridge. Wind-induced phenomena in flexible structures are numerous. They happen at different wind speed ranges, some of them can occur concurrently and also the different modes of oscillation of the structure may be susceptible to different excitation. The full-scale data was recorded for wind speeds up to 17 m/s and thus the phenomena of vortex-induced oscillations and buffeting are the key aeroelastic phenomena recorded. In the context of providing validation material for the two-dimensional numerical simulations (chapter 4), the results presented focus on the vortex-induced oscillations. The full-scale results are compared with wind-tunnel test results obtained from section models, a taut strip model and full aeroelastic model tests carried out by others at the

2.1. Introduction

21

Danish Maritime Institute (DMI). The primary objective of this work is to compare full-scale results with model-scale tests with the long term goal of validating numerical models. Among the key parameters for comparison are the Strouhal number, the critical wind speed at which vortexinduced oscillations occur, the pressure distributions around the bridge deck, spanand chord-wise pressure correlations, turbulence intensities, structural accelerations, natural frequencies, mode shapes, structural and aerodynamic damping, and pressureacceleration correlations. The results presented are aiming to investigate and report whether there is a connection between each of these parameters, and whether that changes during a lock-in situation. As lock-in is the important design case, particular attention to these results will be given. If strong correlation between pressure and acceleration along the bridge is found during lock-in, it would imply one justification for carrying out two-dimensional simulations of vortex-induced resonance. This is rather important as all computational flow simulations in this thesis are two-dimensional, see case studies in Chapter 4. Also addressed will be the pressure-correlations along (and across) the span, which may not have the same correlation patterns as the pressureacceleration correlations such that care should be taken before drawing conclusions from pressure-correlations only. The full-scale measurements presented are selected with an emphasis on the simultaneous measurement of fluid pressures and structural accelerations since the major issue in this dissertation is modelling the interaction between fluid and structure. Six days of useful data were recorded during the measurement period consisting of simultaneous pressures at and near the deck surface, accelerations of deck, wind speeds and wind directions . Only these data are extracted from the measurement period and presented here, although a continuous record of the accelerations, wind speeds and wind directions also exists.

2.1. Introduction

22 CL

222.5m

accelerometers

1624m

535m

535m

Elevation

C.G. S.C.

2.35m 3.06m

accelerometers

C.G.: Centre of Gravity S.C.: Shear-Centre

3.0m 1.0m

27.0m 23.6m

19.0m 31.0m Cross Section

12.0m

6.0m

6.0m

24.0m

10.0m

CL

0110 1010 1010 1010 1010

01 1010 1010 10 accelerometers 1010 1010 10 1010 centre of 1010 main span 10 1010 1010 10 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 Pressure measurement 2 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 area on deck: 13.5x50m 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 hangers 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 10 man hole 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 00 11 000000000000000000 111111111111111111 00 11 000000000000000000 111111111111111111 001010 11 000000000000000000 111111111111111111 10 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 wind vane 1010 000000000000000000 111111111111111111 cup anemometer 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 (268.5m from main span centre) 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 111111111111111111 000000000000000000 111111111111111111 1010 000000000000000000 10111111111111111111 000000000000000000 10111111111111111111 1010 000000000000000000 10111111111111111111 000000000000000000 10111111111111111111 crash barrier 1010 000000000000000000 10111111111111111111 000000000000000000 10111111111111111111 1010 000000000000000000 111111111111111111 1010111111111111111111 000000000000000000 10 000000000000000000 10111111111111111111 10 10

01 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010

N

0110 1010 10

0110 1010 10

Plan

Figure 2.1: Great Belt East suspension bridge (Denmark): General full-scale measurement layout. The full-scale accelerations were measured at a distance of 222.5m from the centre of the main span. The simultaneous pressure measurements took place in various cross-sections offset (by 15-62m) from the cross-section of the accelerometers. Wind speeds and wind directions were recorded by a cup anemometer and wind vane situated 268.5m from the central node at main span, located 2m aboved deck surface.

2.2. Wind and Structural Monitoring System

2.2

WIND

AND

23

STRUCTURAL MONITORING SYSTEM

A cup anemometer and a wind vane were mounted on the outer crash barrier to continuously measure the wind speed and direction at 2m above deck level, see Figure 2.2. The presence of the bridge deck gave rise to a risk that the wind data did not correspond to the true free stream velocities. To ensure that these wind data related to the free stream a calibration with available meteorological data from Risø National Laboratory (Denmark) was carried out. Risø staff had a reference mast placed in the middle of Storebælt on the Sprogø island (approximately 3km west of the central node of the suspension bridge). The available data from Risø consisted of 10 minute average values of wind speeds, 3s gusts, wind directions and temperature gradients at 10m and 70m (bridge girder level at mid span) above ground. A comparison of the author’s own wind data with that of Risø showed that the wind data measured 2m above deck level did correspond to free stream velocities. Therefore the wind data presented do not contain any Risø data, but are the wind measurements directly connected to the structural acquisition unit. Note that a theoretical estimate yielded a minimum distance of approximately 2.3m above the deck on the upwind side to the free stream velocity region based on the finite element method described in Chapter 4, see Figure 2.3. This analysis also indicates that for flow over the downwind side of the deck, this is less critical. Only two accelerometers (Bruel and Kjær accelerometers) were available for the measurement program and these were mounted on the bottom flange located inside the bridge girder, see Figure 2.2. One accelerometer measured horizontal accelerations and one measured vertical accelerations. Note that only the vertical accelerations are important for the findings reported in this thesis. Throughout the measurement period continuous and simultaneous recordings of the acceleration signals from the sensors were made at a sampling frequency of 5Hz. The sampling frequency was chosen in order to detect all frequencies below 2Hz. All sampled time histories were stored in files containing 30-minute time series on a data acquisition laptop located inside the bridge girder, see Figure 2.2.

2.2. Wind and Structural Monitoring System

24

cup anemometer

254m

wind vane

(2m above deck level)

Flow

Pitot Tube Reference Pressure (3m above deck level)

72m The coast of Zealand (Denmark)

Flow 30cm

Actual pressures

(Type 2, see Figure 2.5)

Static Pipe 8-channel Pressure Transducer Inside the Bridge Girder

Electronic Assembly-Box (8 channels) Actual Pressures Ref. Pressure Data Storage

Vertical acc. Accelerometer-Box (2 no.) Horizontal acc. Figure 2.2: Equipment used for the full-scale measurements. Simultaneous pressures and accelerations measured for CFD/FSI and model-scale validation.

2.2. Wind and Structural Monitoring System Free Stream velocity

25

Free stream velocity minimum distance: 2.3m Pitot tube (3m above deck level) Cup anemometer and wind vane (2m above deck level)

High suction

High pressure 1m

Detail 1 (leading edge) velocity vectors superimposed onto pressure contours

Detail 1

(a)

(b)

Figure 2.3: Free Stream velocity estimate based on finite element fluid-only solution (mesh 5) with a free stream velocity of 8m/s, see Chapter 4 for details. (a) Pressure contours with streamlines superimposed. (b) Vorticity contours with streamlines superimposed.

2.3. Pressure Measurements

2.3

26

P RESSURE MEASUREMENTS

A pitot tube was mounted on the crash barrier to measure the reference pressure in terms of static and dynamic pressure, see Figure 2.2. The pitot tube was located 3m above deck and pointed into the wind. It was turned manually when the wind direction changed, such that it always pointed into the wind. The pressures on deck were measured with an 8-channel pressure transducer (Simens transducers), see Figure 2.2, which recorded the pressure differences between the static reference pressure of the crash barrier pitot tube and the actual pressure on or near the deck surface. As well as always measuring the static reference pressure, the instrument had four available channels of which one channel was constantly measuring the total pressure (static plus dynamic) connected to the reference pitot tube. The remaining channels made it possible to measure pressures on the deck at 3 points simultaneously. The majority of the pressure measurements on the bridge deck took place near the trailing edge, in the traffic lanes and emergency stopping lane. The simultaneous pressure measurements at 3 locations took place on the deck surface itself, 0.30m or 1.37m above. The flow inlet locations of the pipes near the deck surface were fixed distances due to equipment constraints. The pressure measurements took place either across or along the deck; some in both directions at the same time, see Figure 2.4, such that the pressure at points along and across the deck were recorded simultaneously. The different types of pressure measurements are outlined in Figure 2.5. The purposes of the pressure measurements on the bridge deck were: 1. To investigate the importance of the simultaneous response of acceleration and pressures in the flow which could shed light on the fluid-structure interaction process and thus prove valuable for the validation of the computational fluidstructure interaction simulations; 2. To produce a time-average pressure distribution on the deck surface; 3. To evaluate the pressure correlation along and across the bridge deck.

2.3. Pressure Measurements

27

P1

11 00 11 00 11 00

Δy Δz P3 P2 Δx P4

U

11 00 11 00

00 11 11 00 00 11

00 11 11 00 00 11

z x y

Figure 2.4: Full-scale pressure measurement set-up. Pressure correlations were recorded in x-, y- and z directions. Pressure and acceleration correlations are evaluated in section 2.5.8

2.3. Pressure Measurements

28

North wind measurements on the south part of the bridge deck

Flow The coast of Zealand (Denmark) Pylon

72m Big pitot tube

Flow

free stream pitot tube Flow big pitot tube static pipe 30cm

1.37m

2.91m

Static pipe

accelerometers

72m (above mean water level) North side of the deck

Type 1

30cm

2.91m

free stream pitot tube

Flow 3 static pipes

accelerometers

North side of the deck

Type 2 (illustrated in Figure 2.7)

2.91m

free stream pitot tube

Flow

3 lamina plates accelerometers

North side of the deck

Type 3

z x y

Figure 2.5: Three different types of full-scale pressure measurements were carried out. Type 1: A big pitot tube together with one static pipe. Pressure measurements away from the deck surface were measured. xz type of pressure correlation coefficients are evaluated, see Section 2.5.8. Type 2: Three static pipes used. Pressure measurements away from the deck surface were measured. Type 3: Three lamina plates used. Pressures at the deck surface were measured. x, y and xy type of pressure correlation coefficients are evaluated for the type 2 and 3 measurements, see Section 2.5.8.

2.3. Pressure Measurements

29 Flow

Pitot Tube (3m above deck)

closed circuit

manifold

manifold

Static Tube Pressure Instrument

Static Tube Pressure Instrument Electrical Signal Output (Volts)

Electrical Signal Output (Volts) (a) Offset Measurement

equivalent to

1ρ 2 U 2

(b) Dynamic Pressure Measurement

Pitot Tube

Flow 3 lamina plates

Plate Detail

(3m above deck)

Total Pressure Flow

Static Pressure, Psta

0.7 2

#1 #2 #3 #4

10x70x2 plate Deck surface (mm)

Pressure Instrument

Electrical Signal Output (Volts)

P1 - Psta

1 ρ U2 2

P2 - Psta P3 - Psta

(c) Set-up for 10min. type 3 Pressure Measurement on the bridge deck surface

Figure 2.6: The pressure measurement series consisted of five sequences. The tube connections are shown, illustrating how each sequence was measured. (a) Offset measurement. (b) Dynamic pressure measurement via the pitot tube in the free airstream. (c) The actual pressure measurement, here measured on the bridge deck surface itself using thin lamina plates (Type 3 pressure measurement).

In general, each pressure measurement series consisted of five sequences of measurements, as follows: 1. Approximately 1 minute (1-5 minutes in practice) offset measured and recorded on the four channels before measuring actual pressures, which was carried out in a closed circuit as shown in Figure 2.6 (a); 2. Approximately 1 minute total reference pressure (static + dynamic pressure) was measured and recorded for all four channels before measuring actual pressures, see Figure 2.6 (b);

2.3. Pressure Measurements

30

3. Approximately 10 minutes pressure measurements on deck at three chosen locations, i.e. at the deck surface itself, 0.30m or 1.37m above. During these 10 minutes, one channel recorded continuously the total reference pressure, see Figure 2.6 (c); 4. Repetition of 1 minute total pressure after the actual pressure measurements for verifying the starting total pressure measured, see setup in Figure 2.6 (b); 5. Repetition of 1 minute offset on the channels after the actual pressure measurements for verifying the starting total pressure measured, see Figure 2.6 (a). To illustrate a typical output from the four channels, two measurement series from type 2 measurements (see Figure 2.5) each having five sequences of measurement are shown in Figure 2.7. These pressures were measured near the trailing edge in the emergency stopping lane. The first measurement series in Figure 2.7 recorded on channel 1, shows the starting offset with a duration of 227s, then followed by measurement of the total reference pressure (between 275 to 335s). Then there is approximately 15 min. (from 475 to 1348s) pressure measurements at 0.3m from deck surface. The total reference pressure measurement (from 1560 to 1621s) is a repetition to verify the first total reference pressure recordings for each channel. The series is finished by repeating the offset measurement (here from 1679 to 1754s) again as a check of the offset measured when the series began. Similar recording took place for channels 2 and 3. In these series of records the inlets for the static pipes (connected to channels 2 and 3) were located at distances of 1m and 2m respectively from the static pipe of channel 1 as shown in Figure 2.7. Varying the location of the static pipes relative to the leading edge in a straight line across the deck, several pressure measurements of this type were obtained. The last pressure measurement channel (channel 4), see Figure 2.7, was continuously recording the reference pressure using a pitot tube located 2.91m from the deck surface. The pressure coefficients at various locations on the deck could then be calculated by knowing the offset, dynamic and actual pressures measured from the corresponding channel, see Section 2.5.7 for results. All types of pressure measurement followed this procedure of five measurement sequences in a series. The pressure measurements on deck were always carried out simultaneously with measurements of the accelerations, wind speed, wind direction at a sampling frequency of 10Hz.

2.3. Pressure Measurements

31

North wind measurements on the south part of the bridge deck 72m (above mean water level)

Railings

Zealand

Flow

Flow

Flow 1m

1m actual pressure channel 3

actual pressure channel 1

actual pressure channel 2 Type 2 pressure measurements

total ref. pressure

0

actual pressures

offset

−10

channel #2: Actual pressure

Pressures (Volts)

Pressures (Volts)

channel #1: Actual pressure 10

offset

1 series

0

1 series

1800 Time (sec.)

3600

10

0 actual pressures

−10

Pressures (Volts)

Pressures (Volts)

−10

1 series

0

offset

1 series

1800 Time (sec.)

0

1 series

1800 Time (sec.)

3600

Pitot tube in free stream

0 actual pressures

1 series

channel #4: Reference pressure

total ref. pressure

offset

offset

offset

channel #3: Actual pressure 10

total ref. pressure

3600

10 actual ref. pressures

0 offset

−10

total ref. pressure

1 series

0

offset

1 series

1800 Time (sec.)

3600

Figure 2.7: Typical pressure-signals simultaneously recorded through the total of 4 channels showing two pressure measurement series, each consisting of five sequences. The specific records are from the type 2 pressure measurements, in which the actual pressures are measured using 3 static pipes with an inlet located 30cm above the deck surface. Pressure coefficients calculated from these measurements contributed to the time averaged pressure distribution shown in Figure 2.16.

2.4. Data Acquisition and Analysis

2.4

32

D ATA ACQUISITION

AND

A NALYSIS

The data presented here should be seen in the context of the computational fluidstructure interaction analyses carried out in chapter 4, which contain simulations involving vortex-induced oscillation around the Great Belt East suspension bridge deck (two-dimensional flow simulation). 2048-point Fast Fourier Transforms (FFT) spectral analyses of the 10-minute samples of raw data were carried out to determine the character of the response, i.e. vortex-shedding or buffeting. Spectra related to wind speeds greater than 12m/s were generally broad-banded and so identified with buffeting, whereas spectra related to wind speeds of 4-12m/s were usually narrow-banded and so associated with vortex-shedding. Moreover the lower wind speeds in wind directions almost perpendicular to the bridge axis revealed large harmonic oscillations. Six days of data of simultaneous wind speeds, wind directions, pressures and accelerations turned out to contain valuable information on vortex-shedding, including occurrences of lockin, and are thus presented here. The pressure spectra were also used for identifying shedding frequencies from which the Strouhal number was evaluated. Eigenfrequencies were determined from acceleration spectra, and these were then combined with available information about the mode shapes (Larsen 1999).

2.5

MAIN F INDINGS

AND

D ISCUSSIONS

2.5.1 Wind Climate at the Site The wind climate for the period covered by the field measurement campaign was characterised by several days with wind perpendicular to the longitudinal bridge axis, conditions which resulted in vortex-induced oscillations. At lock-in the wind speed was typically 8m/s coming from the North. Figure 2.8 shows the wind-rose based on 10 minutes of mean wind speed data measured at Sprogø (70m above sea-level) during the period of the measurement campaign. The most frequent winds came from the NorthWest, in the speed range 6-9m/s, and South-East with wind speeds lower than 6m/s. The strongest winds came from South-West, with a maximum value of 16.92m/s at 70m. Low turbulence intensities, with an average of 6%, were recorded during the measurement campaign (also at lock-in) revealing smooth stable atmospheric conditions.

2.5. Main Findings and Discussions

33

Troublesome north wind causing lock-in (resonant measurement sample reported)

The wind directions referred to in this work are all relative to the bridge axis.

North (0o) No. of counts per 15 degree sector 500 400 300 200

Bridge axis (77−257 deg)

o

East (90o)

West (270 )

12−17 m/s 9−12 m/s 6−9 m/s 0−6 m/s

South (180o)

Figure 2.8: Wind rose data from the measurement period at Sprogø (70m above sealevel).

2.5.2 Structural Behaviour With only two accelerometers available it was decided to measure vertical and horizontal motions simultaneously. From site observations there were no torsional motions evident during the measurement campaign. The horizontal motion records typically showed insignificant accelerations and are not reported here. The vertical motions were severe and on some occasions the structure went into a resonance mode of vibration. The natural frequencies of the bridge deck are evaluated from structural acceleration spectra, see Table 2.1. The frequencies obtained from the measured accelerations are in close agreement with the results obtained by Larsen et al. (1999) from a 3D FEM structural analysis. Notice the closely spaced modal frequencies.

2.5.3 Measured Vortex-Induced Oscillations Wind speeds in the range of 4-12m/s were identified with vortex-induced oscillations corresponding to a flow regime of Re(B) = 8:3x106 to 2:5x107 . Vortex-induced resonance occurred on several occasions for North and South wind directions, i.e. for wind directions perpendicular (or nearly perpendicular) to the longitudinal bridge axis, illustrated by the increased acceleration around zero degrees, see Figure 2.10 (f ). The oscilla-

2.5. Main Findings and Discussions No. of waves in main span 3 4 5 5

Freq. Predicted (Hz) (Larsen et al. 1999) 0.128 0.177 0.205 0.235

34 Freq. Measured (Hz) (Author) 0.13 0.17 0.205 0.23

Table 2.1: Full-scale natural vertical frequencies obtained from acceleration spectra and related mode shapes recorded by Larsen (1999).

tions were typically associated with a single vertical mode and large harmonic displacements. The maximum r.m.s. acceleration levels were of the order of 0:03g (maximum 0:04g) and were experienced on-site as extremely uncomfortable, and arguably unacceptable. Note that an r.m.s. acceleration level of 0:15g was established as the maximum acceptable level for the Great Belt East bridges for frequencies below 1Hz (Larsen 1993a). Based on the site experience, one may suggest that this comfort design criterion should be lowered for future long-span bridge designs. In Figure 2.9, vertical amplitudes are plotted as a function of the reduced velocity from the six selected days of measurements.

2.5. Main Findings and Discussions

35

0.08 Full-Scale Full−Scale

δ = 1%

BritishDesign DesignRules Rules British

Section Model (scale 1:60)

RMS amplitude/height (-)

DMIWind WindTunnel TunnelTests Tests DMI 0.06

Section Model (scale 1:80) 0.04

δ = 3% Full Aeroelastic Model (scale 1:200) 0.02

0

0.2

0.6

1.0

1.4

1.8

U10 (-) Reduced Velocity, ne B [Hz]

no. of waves in main span

Reduced U10 Velocity ne B [-]

5

9.2 8.9 4.9 8.0-8.6 11.0 12.2 10.9-11.5 6.9-7.3 5.9 6.1-6.6

0.17 0.205 0.13 0.205 0.23 0.23 0.205 0.205 0.205 0.23

4 5 3 5 5 5 5 5 5 5

1.7 1.4 1.3 1.2-1.4 1.5 1.7 1.7-1.8 1.1 0.94 0.6

6

4.5

0.13

3

1.1

Event

1 2 3 4

U10

ne

[m/s]

Mode

Turbulence Peak Factor Wind dir. offset Intensity, IU kp from bridge axis [%] [-] o 5.2 2.4 7.3o 2.6 6.4 6.0 o 17.1 2.4 3.2 o o

5.0-5.9 3.1 2.4 2.1-3.8 6.9-9.0 8.7 7.7-8.4 22.2

2.6-3.0 6.8 - 8.4 o 3.0 10.1 o 28.8 3.5 o o 3.3-3.4 17.7 - 23.7 o o 2.4-3.8 7.2 - 14.4 o 2.6 4.0 o o 2.2-2.7 20.0 - 27.1 2.4

48.1o

Max. (RMS) Amplitude [m]

0.037 0.036 0.31 0.13-0.25 0.23 0.11 0.08 0.17-0.23 0.08 0.01 0.23

Observation No vortex-shedding excitation Vortex-induced Resonance (observed visually) Vortex-induced Resonance (observed visually) Vortex-induced Resonance (observed visually) No vortex-shedding excitation Vortex-shedding excitation (no visual evidence)

Figure 2.9: Comparison of r.m.s. vertical amplitudes between full-scale and modelscale response (all wind tunnel test results here are associated with low structural damping levels of the order of 1% logarithmic decrement). The full-scale lock-in amplitudes correspond mainly to the 5th vertical mode of vibrations (0.205Hz) although an amplitude of 0.31m occurred due to a 3rd vertical mode (0.13Hz). The section model results compare well with the full-scale measured amplitudes. The full-aeroelastic model test result reported little or no vortex-shedding exicitation for the 5th mode. The 8th vertical mode (0.39Hz at Ucrit approximately 12-14m/s) was referred to as an upper bound for the vortex shedding responses of the prototype (DMI and SINTEF 1992a). The British Design Rules (Department of Transport 1993) give a good prediction of the vortex-shedding response for low structural damping.

2.5. Main Findings and Discussions

36

The amplitudes are obtained by double integration of the acceleration signals obtained at 222.5m from the centre node of the main span. From available mode shapes (Larsen 1999) the maximum amplitudes were estimated for the mode excited. The full-scale amplitudes compare well with the section model results, especially the 1:80 scale model. Most of the vortex-induced resonance amplitudes relate to a “5th mode” of vibration with a natural frequency of 0.205Hz (“5th mode” meaning five half-waves in the main span) resulting in a maximum amplitude (r.m.s.) of 0:25m at a reduced velocity of 1:35. Excitation of the 3rd mode shape was also recorded (observed to occur after the 5th mode event) and resulted in a r.m.s. amplitude of 0:31m at a reduced velocity of 1:25, which compares well with the section model (scale 1:80) test result, i.e. 0:30m at a reduced velocity of 1:27 as reported in DMI and SINTEF (1993d). Apparently, the fullaeroelastic model test results found the 8th mode with a natural frequency of 0.39Hz (eight number of half-waves in the main span) to be the upper bound giving a maximum amplitude of 0:13m (DMI and SINTEF 1992a). Perhaps this is an indication of the uncertainty in wind tunnel tests predicting full-scale behaviour for vortex-induced oscillations. In the British Design Rules (BDR) of the Department of Transport (1993), a formula is suggested for predicting the critical wind speed for vortex-shedding excitation (assuming the width B=31m and depth D=4.4m):

Ucrit = (1:1 DB + 1)neD (for 5  DB < 10) 31m + 1)  0:205Hz  4:4m = 7:9m/s (5th mode) = (1:1 4:4 m From this, the reduced velocity can be estimated to be formula for predicting the maximum amplitude:

1:24.

(2.1)

The BDR also suggest a

1 5 2 2 ymax = air8BMD  1 5 kg/m3  (31m) 2  (4:4m) 2 1:23 = 8    23687kg/m  0:0016 = 0:29m

(2.2)

The BDR prediction of a critical wind speed of 7.9m/s (Ur = 1:24) and a maximum amplitude of 0.29m (assuming a logarithmic decrement, , of 1%) compares well with the full-scale measured data. Peak events from the six days of measurements are listed in the table attached to Figure 2.9 from which the lock-in flow regime causing large harmonic oscillations can be observed to occur within a reduced velocity, 1.09 < nUe10B < 1.48, where U10 is the average wind speed over 10 minutes, ne is the frequency of the mode excited and B is the deck width. The lock-in range of 1.0 < nUe10B < 1.4 found in the section model, scale 1:80 (DMI

2.5. Main Findings and Discussions

37

and SINTEF 1993d), and 1.1 < nUe10B < 1.4 from the taut strip model (Larose 1992) compares well with the full-scale recorded lock-in flow regime. Notice the low wind speed of 4.5-8.6m/s and low turbulence intensities of 2-9% (smooth incoming wind flow) which gave rise to the large vortex-shedding amplitudes. Vortex-induced oscillations were also observed visually at wind speed as high as 12m/s (event 3) but the resulting amplitudes were not maximal. Full-scale acceleration-only measurements were also carried out by COWI and Rambøll (1998), who reported 10 occasions of vortex shedding excitations occurring between 14/1/1998 and 16/5/1998. Lock-in occurred within 1.0 < nUe10B < 1.5 with maximum amplitudes ranging from 0.14-0.32m related to the modes listed in Table 2.1. These results, also reported by Larsen et al. (1999), compare well with the full-scale measurements carried out by the author. Although the observations between full-scale and model-scale are in fairly good agreement the amplitudes were found to be larger than anticipated. Possible reasons for this are that the structural damping was lower than expected and that there are uncertainties in predicting full-scale flow properties related to Reynolds number effects. Not only do these large oscillations lead to visual and physical discomfort, but it could give an increased risk of fatigue in connections on the bridge structure. Storebælt A/S decided to install guide vanes to suppress the vortex shedding. From the results of the model-scale tests, it was reported (Larose 1998b) that complete suppresion of the vortex-induced resonance oscillation could be achieved by the addition of vanes.

2.5.4 Strouhal number estimates for cross-wind oscillations Large vibrations occurred when the natural frequency, ne , was the same as the frequency of the vortex shedding, ns , causing the structure to vibrate in a mode in the cross-wind direction. The critical wind velocity, defined when ne = ns , occurred over a reduced velocity range of 0.94 < nUe10B < 1.81, see Figure 2.10 (b), but large displacements (0.2-0.3m rms amplitudes) only occurred over the narrow range of 1.1 < nUe10B < 1.5, see Figure 2.10 (a).

2.5. Main Findings and Discussions

38 2

(a) Frequency ratio, ns (-) e

U 1.1 < n 10 < 1.5 eB

0.1

Correlation Coefficient, ρpa (-)

0

1

0

1.8 U Reduced Velocity, n 10 (-) eB

0.5

0

1.8 U Reduced Velocity, n 10 (-) B e

0

0

1.8

U Reduced Velocity, 10 (-) ne B

(d) ST = 0.11

0.12 0.08

0.2

1

U10 1.8 ne B (-)

1

2

3

77 Reynolds no., Re(B) (-) xx10 10

(e)

5

1

0 0

1

10

0

0.2

RMS Accelerations (m/s2)

Turbulence Intensity, IU (%)

0

U10 ns ne = ST(B) ne B

Reduced Velocity,

Pressures and Accelerations

Lock-in Plateau

1

0

1

(c)

Lock-in Plateau U 0.94 < n 10 < 1.81 eB

(b)

n

Lock-in regime related to large limited amplitudes

Strouhal no., ST(D) (-)

RMS Amplitude (m)

0.3

(f)

0.1

0

−50

0

50

Wind Direction (deg.) (offset from bridge axis)

Figure 2.10: Full-scale cross-flow response for vortex-induced oscillations. (a) Vertical amplitudes. (b) Classical frequency characteristics of vortex-shedding. (c) Pressure and acceleration correlations. (d) Strouhal number as a function of the Reynolds number. (e) Turbulence intensity. (f) Acceleration at measurement location (222.5m from centre of main span) as a function of wind direction relative to the longitudinal bridge axis (zero wind direction corresponds to perpendicular wind direction, i.e. the critical condition for vortex-induced oscillations). For further event details see attached tables in Figure 2.9 and Figure 2.18.

2.5. Main Findings and Discussions

39

In the lock-in flow regime, the fluid-structure interaction mechanism becomes more important as the pressures and accelerations tend to become more correlated than outside lock-in as illustrated in Figure 2.10 (c). The correlation between pressures (p) and accelerations (¨y) is expressed here through the correlation coefficient ( pa ) :

pa = COVp(t)y¨ (t)] p a

(2.3)

where COV[ ] is the covariance between pressures and accelerations:

COV p(t)y¨(t)] = E(p(t) - p)(y¨ (t) - a)] = Ep(t)y¨ (t)] - pa

(2.4)

and p and a are the standard deviations of the measured pressures and accelerations and p and a are the corresponding mean values. Theoretically, full correlation takes place when pa = 1. Furthermore, notice that lock-in generally occurred at low turbulence intensity, Iu = uu , as shown in Figure 2.10 (e). The turbulence intensities are based on 10-minute time series and during the measurement period were found to average approximately 6%, indicative of smooth stable atmospheric conditions. A Strouhal number St(D) was calculated based on the frequency of the peak of the pressure spectra. These ranged from 0.08 to 0.11 for the events connected with large vortexinduced resonance oscillations, see Figure 2.10 (d), where the Strouhal number is shown as a function of the Reynolds number. Outside the lock-in flow regime, the pressure spectra were typically of a broad-banded nature with most of the energy concentrated below 0.05Hz and thus it was difficult to estimate shedding frequencies. Therefore some scatter does exist in Figure 2.10 (b) and thus the graph as a whole should be interpreted as an approximation to the classical frequency characteristic of vortex-shedding. Although some scatter is present outside lock-in, it appears that the Strouhal number is constant within the Reynolds number flow regime of Re(B) = 8:3x106 to Re(B) = 2:5x107 recorded in the measurement period. In conclusion, from Figure 2.10 (b,d) a Strouhal number of 0.11 is estimated. A constant Strouhal number was also reported in the model-scale measurements carried out by DMI and SINTEF (1993d), and in the numerical experiments by Frandsen and McRobie (1999b). This follows the common assumption in wind engineering that the flow around bluff and sharp edged bodies is independent of the Reynolds number. The Strouhal number range 0.11-0.15 obtained from the section model (1:80) in the wind tunnel (DMI and SINTEF 1993d), and the value St = 0.11 from the Taut Strip Model

2.5. Main Findings and Discussions

40

(1:300) by Larose (1992) compare well with the estimated Strouhal number from the full-scale measurements.

2.5.5 Structural and Aerodynamic Damping estimates The dynamic response caused by vortex-induced oscillations is a function of damping which is amplitude dependent. Estimation of damping is a key parameter when evaluating cross-wind vortex-shedding response. For both full-scale and model-scale measurements the prediction of damping in long-span bridges is associated with high uncertainty and there is a danger that a bridge model with too high a level of structural damping in comparison with what is experienced at full-scale can fail to predict vortex-shedding excitation. Identifying structural damping from full-scale acceleration measurements involves estimation of the aerodynamic damping; i.e. the damping associated with the bridge deck’s own movement in air. It is the total damping (i total ) which is extracted from the raw acceleration data, consisting of the following two contributions:

itotal = istruct + iair

(2.5)

Observations in the wind tunnels suggest that the relationship between aerodynamic damping and wind speed is not necessarily linear. At wind velocities close to the critical wind velocity the aerodynamic damping can become negative, tending to reduce the total damping, and leading to significant vortex-induced vibrations (see for example the results from the wind tunnel experiments carried out by Bogunovi´c Jakobsen and Hjort-Hansen (1998)). When estimating structural damping from acceleration spectra one must thus ensure that the acceleration spectra do not contain dominant frequencies which could be associated with vortex-shedding at lock-in. For a first estimate one could assume small vibration amplitudes and a linear dependence between aerodynamic damping and wind speed following the formula of Davenport (1962) which is based on quasi-steady aerodynamic considerations:

iair

2 U 1 dC

B L = 8 d m n B ei

(2.6)

For larger amplitudes, the non-linear nature of damping becomes important, i.e. when the amplitudes are greater than approximately 10 to 20% of structural width, as suggested by Dyrbye and Hansen (1997). As reported by Parkinson (1989), many investigators have suggested analytical models with a non-linear damping term following the Van der Pol oscillator concept, see Van der Pol (1920), with a cubic term, ensuring prediction

2.5. Main Findings and Discussions

41

of vortex-induced oscillatory response. Such models are not based on an understanding of the underlying physics of the flow processes, but are merely equations known to exhibit self-excited limit cycles which can be fitted to suit the experimental data. Dealing with a non-linear aeroelastic phenomenon, one must therefore be extremely careful as to which technique is used for extracting damping values. Several modal analysis identification methods are available for estimation of damping (and natural frequencies) from ambient vibration measurements, see Maia et al. (1997), page 188. For example there are the AutoRegressive Moving Average (ARMA) time series models (Jensen 1990) and the Random Decrement Technique (RDT) which is also applied in the time domain (Asmussen 1997). Comparing the RDT and ARMA techniques, the RDT technique has the advantage of providing damping estimates as a function of the amplitude. As structural damping usually increases with vibration amplitude, the RDT-technique is preferable for evaluation of structural damping of bridges undergoing vortex-induced oscillations. Analysis of the raw acceleration data showed that the measurement accuracy of 0:005m/s2 was insufficient for extracting consistent total damping values. For this reason evaluation of aerodynamic damping was not possible. For the sake of completeness of the full-scale results from the Great Belt East suspension bridge, the reader is referred to the structural damping value estimates of Jensen et al. (1999), who used the ARMA model to extract structural damping estimates from wind-induced vibration measured in the last construction phase of the finished erected bridge, see Table 2.2 . Jensen et al. (1999) also found scatter in damping values possibly due to the uncertainty related to the close spaced modes, but with a longer period of measurement, the data did show patterns from which estimation of aerodynamic damping values were possible. Initial measurement consisting of one day (Brincker, Frandsen, and Andersen 2000) were carried out to test and compare various techniques for damping extraction on more accurate samples. Note that these data are based on traffic induced vibrations. The wind speeds were lower than 4m/s and the wind direction was not perpendicular to the bridge line, therefore the measured data were mainly traffic induced as opposed to wind induced vibrations. Eight accelerometers were mounted on the bridge deck for mode shape extraction. The natural frequencies evaluated compare well with existing full-scale extracted natural frequencies, however a longer period of measurement is required to obtain the comprehensive samples needed for the aerodynamic damping evaluation.

2.5. Main Findings and Discussions

No. waves 3 4 5 5

Freq. Measured (Hz) (Author) 0.13 0.17 0.205 0.23

Max. Amplitude (m) (Author) 0.31 0.037 0.25 0.23

42

Mean  (%) Measured (Jensen et al. 1999) 0.26 0.11 0.25 0.11

log.dec.  (%) 1.63 0.69 1.57 0.69

Table 2.2: Structural damping ratios,  (mean values) extracted by using ARMA model based on measured full-scale accelerations, after Jensen et al. (1999), related to the natural frequencies and (r.m.s.) vertical amplitudes measured by the author.

2.5. Main Findings and Discussions

(m/s)

10

decrease in wind speed

Wind Direction (2m above deck surface)

180

North Wind: (0o)

lock-in (deg.)

12

Wind Speed (2m above deck surface)

43

8

0

6 4

0

(a)

10845

18000

Time (s)

(c)

(volts)

0

4730

10845

0

18000

Time (s)

4730

10845

18000

Time (s) Channel #1: Actual Pressures (7 series)

5

0

−0.4

lock-in

(b)

Vertical Accelerations lock-in

0.6 0.4 (m/s 2 )

4730

−180

0

−5 (d)

lock-in

0

4730

10845

18000

Time (s)

Figure 2.11: Lock-in sample (cross-flow), May 4 1998 (14:28 to 18:58pm). (a) Lock-in occurred due to a decrease in wind speed. During lock-in (1 hour 42 minutes) the wind speed was around 8m/s and almost constant. (b) The wind direction was perpendicular to the bridge axis (critical wind direction for lock-in). Notice: the big fluctuations occurred around 0 deg. due to equipment sensitivity, but should be interpreted as North Wind (0 deg.). (c) Vertical accelerations of the bridge deck measured 222.5m from central node of main span. (d) Actual pressures from one specific channel (other channels shows similar pressures in this case). The pressures are measured using a static pipe (Type 2, see Figure 2.5) and in total, 7 pressure measurement series are here shown.

2.5.6 Resonant Measurement Sample A measurement sample of resonant suspension bridge behaviour is illustrated in Figure 2.11. A sample of five hours of simultaneous measurements of the wind speed, wind direction, acceleration and pressure time histories is shown. During this period of measurements the structure experienced resonant oscillations due to vortex-shedding, which occurred for a Northerly wind, i.e. a perpendicular wind direction, at 8m/s over a period of about 1 hour 42 minutes, see Figure 2.11 (a,b). The mode shape observed was characterised by five half-waves along the main span between the pylons with a corresponding natural frequency of 0.205Hz. The acceleration

2.5. Main Findings and Discussions

44

time history in Figure 2.11 (c) shows the increase in acceleration due to the vortexinduced resonance oscillations. The largest acceleration measured at 222.5m from central node of the main span is approximately three percent of gravity (0:03g). This corresponds to 0:04g maximum acceleration (r.m.s. of 0:03g) for this particular mode, i.e occurring at 323.2m from centre of main span. The pressure time history in Figure 2.11 (d) shows seven pressure measurement series of the actual pressures 30cm above deck. The gap between the pressure measurement series is due to measurements of offset and reference pressures, which are not shown. During lock-in, the amplitude remains constant and is large but limited, which is characteristic for vortex-induced oscillations (i.e. not a catastrophic aeroelastic phenomena), see Figure 2.12 (a). At the lock-in regime the displacements becomes harmonic as shown in the 1-minute time history in Figure 2.12 (c), with an average amplitude of approximately 0:35m, whereas outside the lock-in flow regime the oscillations are of a more non-periodic nature, see Figure 2.12 (b). Figures 2.13 to 2.15 relates to raw data obtained at the measurement location. These Figures show extracted simultaneous pressure and acceleration time histories before, at and after lock-in respectively, together with the corresponding evaluated pressure and acceleration spectra. Before lock-in, see Figure 2.13, the acceleration is not noticeable. The corresponding pressures are irregular and a range of shedding frequencies can be detected, i.e. no distinct Strouhal shedding frequency can be evaluated. At lock-in, see Figure 2.14, the accelerations are constant and have grown to the order of 0:3m/s2 . The shedding frequency has become regular and a distinct shedding frequency of 0.205Hz can be observed in the pressure spectrum. The simultaneous acceleration spectrum also shows a natural frequency of 0.205Hz and therefore (also as visually observed) the structure experienced large motion-induced displacements, see Figure 1 in the Introduction Chapter. Note that this lock-in sample was associated with a wind speed decrease from approximately 9.3 m/s to 8 m/s average mean free stream velocity. After the period of lock-in the vortex shedding frequencies have again become irregular and the accelerations are no longer noticeable, see Figure 2.15. The pressure spectrum is again broad-banded.

Maximum displacements (m)

2.5. Main Findings and Discussions

45

Lock-in 0.3 0.3

0 0

Symmetric Vertical 5th Mode 0.205Hz

−0.3 -0.3

00

8m/s Smooth Flow 4730 10845 18000 4730 10845 18000 Time (s)

At Lock-in Maximum displacements (m)

Maximum displacements (m)

Off Lock-in

0.2 0.2

00 −0.2 -0.2

00

(b)

30 30 Time (s)

(a)

60 60

0.4 0.4

0.2 0.2

00 −0.2 -0.2

−0.4 -0.4 6100 6100

6130 6130 Time (s)

6160 6160

(c)

Figure 2.12: Maximum estimated full-scale displacements (323.2m from centre of main span) scaled from the accelerations at measurement location (222.5m from centre of main span), see Figure 2.11c. (a) 5 hours record from May 4 1998 (14:28 to 18:58pm). (b) 1-minute time trace off lock-in: Displacements are small and irregular. (c) 1-minute time trace at lock-in: Displacements are large harmonic oscillations.

2.5. Main Findings and Discussions

channel #1: Actual pressure

50

Pressure Spectrum

(volts

x

(volts)

s)

5

46

0

−5 500

0.3

800 Time (s)

25

0

1100

Vertical Accelerations

0.30

0.1 0.4 Frequency (Hz) - log scale Acceleration Spectrum

(m/s 2 x s)

(m/s 2)

0.205Hz

0

−0.3 500

800 Time (s)

1100

0.15

0

0.1 0.2 Frequency (Hz) - log scale

0.4

Figure 2.13: Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories before lock-in at measurement location. Small amplitudes and irregular shedding with a corresponding broad-banded pressure spectrum characterise the response before lock-in.

2.5. Main Findings and Discussions

5

47

channel #1: Actual pressure

400

Pressure Spectrum

(volts x s)

(volts)

0.205Hz

0

−5 8400

8800 Time (s)

200

0 0.1

9200

Vertical Accelerations

50

0.3

0.2 0.3 Frequency (Hz) - log scale Acceleration Spectrum

x

0

−0.3 8400

(m/s 2

(m/s 2)

s)

0.205Hz

8800 Time (s)

9200

25

0 0.1

0.2 Frequency (Hz) - log scale

0.3

Figure 2.14: Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories at lock-in at measurement location. Large harmonic amplitudes and regular shedding with a pressure spectrum showing a distinct shedding frequency coinciding with the natural frequency of the bridge deck characterise the response at lock-in.

2.5. Main Findings and Discussions

5

48

channel #1: Actual pressure

5

Pressure Spectrum

x

0

−5 14500

0.3

(volts

(volts)

s)

0.205Hz

14800 Time (s)

0

15100

Vertical Accelerations

0.25

0.1 0.2 Frequency (Hz) - log scale

1

Acceleration Spectrum

(m/s 2 x s)

(m/s 2)

0.205Hz

0

−0.3 14500

14800 Time (s)

15100

0.1 0.12Hz 0 0.1

0.2 Frequency (Hz) - log scale

1

Figure 2.15: Cross-flow oscillations showing simultaneous wind-pressure and structural acceleration time histories after lock-in at measurement location. Small amplitudes and irregular shedding with a corresponding broad-banded pressure spectrum characterises the response after lock-in.

2.5. Main Findings and Discussions

49

2.5.7 Pressure Distribution Evaluating an average pressure distribution on as much of the bridge deck as possible was one purpose of carrying out the pressure measurements on the bridge deck. The pressure, p, refers here to the actual pressure measured either at the deck surface, 0.30m or 1.37m above the deck surface. It is usual to refer all measured pressures to the reference pressure, or rather the dynamic pressure ( 12 U2 ) and the pressure coefficient is defined as the ratio of the pressure or suction on the surface of a structure (or at a distance above the surface in this case) and the dynamic pressure of the undisturbed approaching air flow:

Cp = p1 -Up2o 10 2

(2.7)

where U is the mean free stream wind speed, here averaged over 10 minutes and p - po refer to the pressure difference between local and far upstream pressure po . For details on how the pressures were measured (including the reference pressure) see the description in Section 2.3 together with the illustration in Figures 2.5 and 2.6 (note: po is the same as psta ). The pressure coefficients from the measurements are displayed in Figure 2.16. The time averaged pressure coefficients were found to be near zero on the top flange trailing edge part. As mentioned previously the turbulence intensity was in general recorded to be low and thus there was an almost laminar flow measurement condition. Pressure measurements were carried out on a taut strip model (Larose 1992) both for smooth and turbulent flow. In Figure 2.16, the average pressure coefficients for zero angle-of-attack, obtained at model-scale, are compared with the full-scale measurements. The time average pressure coefficients compare well between full-scale measurements and model-scale predictions. The model-scale prediction shows slight suction in smooth flow on the top flange trailing edge side of the deck, which gets further reduced in turbulence flow. The pressure distribution from the full-scale measurements shown in Figure 2.16 was obtained during lock-in and measured in a straight line across the deck, see Figure 2.7. On other occasions the measurement took place along the deck with the purpose of finding out how the pressures between two points along the span were correlated at and outside the lock-in flow regime. Figure 2.17 shows that, for large amplitudes (at lock-in), the pressure correlations increase for all separation lengths measured (maximum separation length of 47m). It can also be observed that the pressures tend to become more correlated for increasing structural motions. At lock-in, when large vibration amplitudes are present (in this case when the accelerations are approximately between 0:3 - 0:4m/s2 ), the pressure correlation increases for all the measured separation distances ( y=43-47m, y=24m and y=17-19m). This does

2.5. Main Findings and Discussions

1

50

Turbulent flow: Taut Strip Model Smooth flow: Taut Strip Model Smooth flow: Full-scale

Cp [−]

0.5 0 −0.5 −1

Leading edge

−13.5

Trailing edge

0 x [m]

13.5

Figure 2.16: Pressure distribution on the 31m wide deck surface. A comparison is made between the full-scale time-averaged pressure coefficients and the model scale results from the taut strip model (1:300) carried out by Larose (1992). The full-scale pressure coefficients are from the lock-in event of May 4 1998 (14:28 to 18:58pm). During this occasion pressure measurements were carried out on the trailing edge part of the deck surface. As expected the pressures are low at this location.

indicate that the flow is well correlated within a length of approximately 50m when the structure exhibits resonance oscillations. This is one justification for carrying out twodimensional fluid-structure interaction flow simulations around a bridge deck when simulating vortex-induced resonance. The pressure correlation is evaluated using an equation similar to eqn. (2.3):

p = COVp1(t)p2(t)] p1 p2

(2.8)

where COV[ ] is the covariance between pressures measured at two locations:

COV p1(t)p2 (t)] = E(p1 (t) - p1)(p2 (t) - p2)] = Ep1(t)p2 (t)] - p1p2

(2.9)

and p1 and p2 are the standard deviations of the pressures at locations 1 and 2 respectively and p1 , p2 are the corresponding means.

2.5. Main Findings and Discussions

51

P1 11 00 Δy

U 11 00 P2

Leading Edge

Trailing Edge

0.12m < Δ y < 47.3m

Spanwise Pressure Pressure Correlation Spanwise Correlation

Correlation Coefficient [−]

Correlation Coefficient, ρp p (-) 1 2

1.0

0.8

Δy Δy Δy Δy

= 43-47m = 24m = 17-19m = 0.12m

Lock-in (Event 3)

0.4

+ static pipes o lamina plates 0

0.1

0.3 0.4 2 2 Max. acceleration [m/s ] Maximum acceleration (m/s )

Figure 2.17: Spanwise pressure correlation data points, each obtained during 10minute measurement periods. The pressure correlation coefficients, p (= p1 p2 ), related to pressures at two points along the span are plotted as a function of the accelerations of the deck. Note that there is no particular difference in pressure correlations along the span when comparing the use of static pipes (type 2) with lamina plates (type 3), see Figure 2.5 for illustration of types of pressure measurement. Further results on pressure correlations can be found in Figure 2.18.

2.5. Main Findings and Discussions

52

2.5.8 Pressure and Acceleration Correlations Pressure correlation measurements were carried out both across and along the bridge deck, see the full-scale pressure measurement set-up in Figure 2.4. Pressures at three locations were measured simultaneously with acceleration measurements of the structure. One purpose of carrying out these measurements was to investigate whether the pressures were closely linked to structural motions and thus to gain some further insight into the fluid-structure interaction mechanism span- and chord-wise. One measure is the correlation between pressures and accelerations . The definition of the simultaneous correlation coefficients ( pa ) is the covariance between pressure at point 1 and the acceleration at approximately the same location (offset varied by 15-62m) divided by the product of the standard deviation for the pressure at point 1 and the simultaneous acceleration (resulting in p1a ), see eqn. (2.3). The pressures at points 1 and 2 (separated by either y, x or xy , see Figure 2.18) were recorded simultaneously. Correlating the pressure at point 2 with the simultaneous acceleration (assumed to be the same as at point 1) results in p2a . Figure 2.18 (b) illustrates the coupling of p1a and p2a separated by y for the particular mode excited, and the corresponding average wind speed. For each 10-minute measurement of this kind p1a  p2a . For the separation length encountered during the measurement period, the exponential decay mainly occurred due to the variation in structural motions as the pressure and acceleration became less correlated outside lock-in. Similar pressure-acceleration correlation conclusions can be drawn from Figure 2.18 (d,f ). The simultaneous pressure correlations ( p ) and pressure-acceleration correlations ( pa ) shown in Figure 2.18 suggest well-correlated pressures (0.5< y p