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Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis Methods and Practical Modeling Techniques

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Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis Methods and Practical Modeling Techniques v = 110 km/h 146 m 335 m 146 m - 25 15 5 -5 -15 -25 -35 with tuned mass damper (TMD) -45 without tuned mass damper (TMD) the truck leaves bridge Mid-point vertical displacement (mm) -55 0 10203040 Time (s) Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Royal Institute of Technology Department of Structural Engineering TRITA-BKN. Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B--44--SE Doctoral Thesis Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden Akademisk avhandling Som med tillstånd av Kungl Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari 1999 kl 10.00 i Kollegiesalen, Valhallavägen 79, Stockholm. Avhandlingen försvaras på svenska. Fakultetsopponent: Docent Sven Ohlsson Huvudhandledare: Professor Håkan Sundquist TRITA-BKN. Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B--44--SE Stockholm 1999 Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden _____________________________________________________________________ TRITA-BKN. Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B--44--SE Doctoral Thesis To my wife, Lena, to my daughter and son, Maria and Marcus, and to my parents, Faiza and Sabah. Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari 1999. Raid Karoumi 1999 KTH, TS- Tryck & Kopiering, Stockholm 1999 ______________________________________________________________________ Abstract ______________________________________________________________________ This thesis presents a state-of-the-art-review and two different approaches for solving the moving load problem of cable-stayed and suspension bridges. The first approach uses a simplified analysis method to study the dynamic response of simple cable-stayed bridge models. The bridge is idealized as a Bernoulli-Euler beam on elastic supports with varying support stiffness. To solve the equation of motion of the bridge, the finite difference method and the mode superposition technique are used. The second approach is based on the nonlinear finite element method and is used to study the response of more realistic cable-stayed and suspension bridge models considering exact cable behavior and nonlinear geometric effects. The cables are modeled using a two-node catenary cable element derived using “exact” analytical expressions for the elastic catenary. Two methods for evaluating the dynamic response are presented. The first for evaluating the linear traffic load response using the mode superposition technique and the deformed dead load tangent stiffness matrix, and the second for the nonlinear traffic load response using the Newton-Newmark algorithm. The implemented programs have been verified by comparing analysis results with those found in the literature and with results obtained using a commercial finite element code. Several numerical examples are presented including one for the Great Belt suspension bridge in Denmark. Parametric studies have been conducted to investigate the effect of, among others, bridge damping, bridge-vehicle interaction, cables vibration, road surface roughness, vehicle speed, and tuned mass dampers. From the numerical study, it was concluded that road surface roughness has great influence on the dynamic response and should always be considered. It was also found that utilizing the dead load tangent stiffness matrix, linear dynamic traffic load analysis give sufficiently accurate results from the engineering point of view. Key words: cable-stayed bridge, suspension bridge, Great Belt suspension bridge, bridge, moving loads, traffic-induced vibrations, bridge-vehicle interaction, dynamic analysis, cable element, finite element analysis, finite difference method, tuned mass damper. – i – – ii – ______________________________________________________________________ Preface ______________________________________________________________________ The research presented in this thesis was carried out at the Department of Structural Engineering, Structural Design and Bridges group, at the Royal Institute of Technology (KTH) in Stockholm. The project has been financed by KTH and the Axel and Margaret Ax:son Johnson Foundation. The work was conducted under the supervision of Professor Håkan Sundquist to whom I want to express my sincere appreciation and gratitude for his encouragement, valuable advice and for always having time for discussions. I also wish to thank Dr. Costin Pacoste for reviewing the manuscript of this report and providing valuable comments for improvement. Finally, I would like to thank my wife Lena Karoumi, my daughter and son, and my parents for their love, understanding, support and encouragement. Stockholm, January 1999 Raid Karoumi – iii – – iv – ______________________________________________________________________ Contents ______________________________________________________________________ Abstract i Preface iii General Introduction and Summary 1 Part A State-of-the-art Review and a Simplified Analysis Method for Cable- 7 Stayed Bridges 1 Introduction 9 1.1 General............................................................ 9 1.2 Review of previous research........................................ 15 1.2.1 Research on cable-stayed bridges............................ 15 1.2.2 Research on other bridge types............................... 22 1.3 General aims of the present study.................................... 27 2 Vehicle and Structure Modeling 29 2.1 Vehicle models.................................................... 29 2.2 Bridge structure................................................... 31 2.2.1 Major assumptions......................................... 32 2.2.2 Differential equation of motion.............................. 33 2.2.3 Spring stiffness............................................ 34 2.3 Bridge deck surface roughness...................................... 38 3 Response Analysis 43 3.1 Dynamic analysis.................................................. 43 3.1.1 Eigenmode extraction....................................... 43 – v – 3.1.2 Response of the bridge...................................... 45 3.2 Static analysis..................................................... 49 4 Numerical Examples and Model Verifications 51 4.1 General........................................................... 51 4.2 Simply supported bridge, moving force model........................ 52 4.3 Multi-span continuous bridge with rough road surface ................. 57 4.4 Simple cable-stayed bridge......................................... 63 4.5 Three-span cable-stayed bridge...................................... 72 4.6 Discussion of the numerical results.................................. 80 5 Conclusions and Suggestions for Further Research 83 5.1 Conclusions of Part A.............................................. 83 5.2 Suggestions for further research..................................... 85 Bibliography of Part A 87 Part B Refined Analysis Utilizing the Nonlinear Finite Element Method 97 6 Introduction 99 6.1 General ......................................................................................................... 99 6.2 Cable structures and cable modeling techniques ....................................... 101 6.3 General aims of the present study .............................................................. 103 7 Nonlinear Finite Elements 105 7.1 General ....................................................................................................... 105 7.2 Modeling of cables..................................................................................... 106 7.2.1 Cable element formulation............................................................ 107 7.2.2 Analytical verification................................................................... 111 7.3 Modeling of bridge deck and pylons.......................................................... 113 – vi – 8 Vehicle and Structure Modeling 117 8.1 Vehicle models........................................................................................... 117 8.2 Vehicle load modeling and the moving load algorithm............................. 121 8.3 Bridge structure.......................................................................................... 123 8.3.1 Modeling of damping in cable supported bridges......................... 123 8.3.2 Bridge deck surface roughness...................................................... 126 8.4 Tuned vibration absorbers.......................................................................... 127 9 Response Analysis 133 9.1 Dynamic Analysis...................................................................................... 133 9.1.1 Linear dynamic analysis................................................................ 134 9.1.1.1 Eigenmode extraction and normalization of eigenvectors..... 135 9.1.1.2 Mode superposition technique ............................................... 136 9.1.2 Nonlinear dynamic analysis.......................................................... 138 9.2 Static
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