FACULTY OF SCIENCE AND TECHNOLOGY MASTER’S THESIS Study program/ Specialization: Spring semester, 2012 Masters Degree in Mechanical and Structural Engineering – Civil Engineering Open Writer: Jan Tveiten (Writer’s signature) Faculty supervisor: Jasna Bogunovic Jakobsen External supervisor(s): Title of thesis: Dynamic analysis of a suspension bridge. Credits (ECTS): 30 Key words: Pages: 90 Suspension bridge Steel bridge + enclosure: 80 Eigen value analysis Dynamic analysis Stavanger, 11/6-2012 Date/year PREFACE This master thesis is the final part of my Master degree in structural and material science at the University of Stavanger. The thesis consists of collecting and studying data for the Lysefjord Bridge in light of the deterioration of the main cables and to carry out some static and dynamic analysis of the bridge to see if any load case could indicate reasons for the wire breakage. Data is given from the Norwegian Public Roads Administration (Statens Vegvesen) The existing finite element model of the bridge is modeled in ABAQUS. The model is analyzed with emphasis on the dynamic aspects of the bridge. Studies of the eigenfrequencies and eigenmodes have been done, both from the finite element model results and by simpler theoretical calculations. Wind loading and vortex induced vibrations of the bridge girder has been looked into. Cables forces from the loads have been checked. The wire fractures observed in the main cables have been summarized and analyzed with respect to weather conditions in the area. Data on the weather in the area has been provided by the Lysefjord Weather Station, downloaded through www.wunderground.com. The thesis consists of; A presentation of the static and dynamic characteristics of a suspension bridge A summary of the wire fractures in the main cables on Lysefjord Bridge A weather analysis at the time of the wire fractures Verification of the existing finite element model in ABAQUS An eigenvalue, wind, temperature and traffic analysis of the bridge model A verification of the finite element model results by theoretical calculations Contents Faculty of Science and Technology ................................................................................... …………...i Preface..................................................................................................................................... …………...i Table of Figures .................................................................................................................... …………..ii Table of Tables ...................................................................................................................... ………….iv Acknowledgments .............................................................................................................. ... .………......v 1 Introduction ........................................................................................................... ... .…. ………1 2 Suspension bridges ........................................................................................ …… …………..3 2.1. Stiffening girder .............................................................................................................. 4 2.2. Support of stiffening girder .......................................................................................... 5 2.3. Main Cables ..................................................................................................................... 6 2.4. Hanger cables .................................................................................................................. 7 2.5. Towers .............................................................................................................................. 7 2.6. Anchor piers .................................................................................................................... 9 3 Fracturing of main cables .................................................................................. … …………10 3.1. The main cables ............................................................................................................ 10 3.2. Possible causes for wire fractures .............................................................................. 11 3.3. SoundPrint® acoustic monitoring system ............................................................... 12 3.4. Weather around Lysefjord Bridge ............................................................................. 15 4 Finite element model ........................................................................................ …. …………22 4.1. ABAQUS software ...................................................................................................... 22 4.2. ABAQUS model .......................................................................................................... 23 4.3. Basic model loads ......................................................................................................... 33 5 Results of ABAQUS analysis ................................................................................ …………42 5.1. Deformed model .......................................................................................................... 42 5.2. Temperature loads ....................................................................................................... 43 5.3. Traffic load model ........................................................................................................ 44 5.4. Motion induced wind loads ........................................................................................ 44 5.5. Eigenfrequencies and eigenmodes ............................................................................ 46 5.6. Vortex Induced Vibrations ......................................................................................... 59 6 Validation of eigenfrequencies and eigenmodes ........................................... … …………72 6.1. Basic definitions ............................................................................................................ 72 6.2. Basic assumptions ........................................................................................................ 73 6.3. Vertical asymmetric modes ........................................................................................ 73 6.4. Vertical symmetric modes .......................................................................................... 74 6.5. Torsional eigenfrequencies and modes .................................................................... 76 7 Conclusion ............................................................................................................... …………79 8 Future recommendations ....................................................................................... …………81 Bibliography ............................................................................................................................ …………82 APPENDIX A: Drawings ................................................................................................... …………84 APPENDIX B: Girder cross section calculations........................................................... …………93 APPENDIX C: Calculation of shear center ..................................................................... …………95 APPENDIX D: Calculation of Steinman eigen modes ................................................. …………98 APPENDIX E: ABAQUS files .......................................................................................... ………..100 APPENDIX F: FORTRAN files ....................................................................................... ………..158 i TABLE OF FIGURES Figure 1-1: Plan and section of Lysefjord Bridge[1] ............................................................................. 1 Figure 2-1: Bakke bridge ............................................................................................................................ 3 Figure 2-2: Main components of a suspension bridge [30] ................................................................. 4 Figure 2-3: Cross section of bridge girder at Lysefjord Bridge [7] .................................................... 5 Figure 2-4: Pendulum support north side [30] ...................................................................................... 5 Figure 2-5: Fixed support north side [30] .............................................................................................. 6 Figure 2-6: Pendulum support south side [30] ...................................................................................... 6 Figure 2-7: Detail of hanger attachment to main cables Lysefjord Bridge [30] .............................. 7 Figure 2-8: North tower Lysefjord Bridge [30] ..................................................................................... 8 Figure 3-1: Main cable cross section Lysefjord Bridge [7] ................................................................ 10 Figure 3-2: Cumulative wire breaks since bridge opening ................................................................ 11 Figure 3-3: Surface defects on examined z wire from Lysefjord Bridge[27] ................................. 12 Figure 3-4: Acoustic sensor on main span ........................................................................................... 13 Figure 3-5: Acoustic sensor on backstay cable .................................................................................... 13 Figure 3-6: Wire breaks east cable [3] ..................................................................................................