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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 the bridge Mid-point vertical displacement (mm) -55 0 10 20 30 40 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 tillstnd av Kungl Tekniska Hgskolan i Stockholm framlgges till offentlig granskning fr avlggande av teknologie doktorsexamen fredagen den 12 februari 1999 kl 10.00 i Kollegiesalen, Valhallavgen 79, Stockholm. Avhandlingen frsvaras p svenska. Fakultetsopponent: Docent Sven Ohlsson Huvudhandledare: Professor Hkan 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 tillstnd av Kungliga Tekniska Hgskolan i Stockholm framlgges till offentlig granskning fr avlggande 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

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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 Hkan 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 analysis............................................................................................. 141 10 Numerical Examples 143 10.1 Simply supported bridge ............................................................................ 144 10.2 The Great Belt suspension bridge .............................................................. 149 10.2.1 Static response during erection and natural frequency analysis ... 151 10.2.2 Dynamic response due to moving vehicles................................... 154 10.3 Medium span cable-stayed bridge.............................................................. 158 10.3.1 Static response and natural frequency analysis............................. 159 10.3.2 Dynamic response due to moving vehicles parametric study.... 162 10.3.2.1 Response due to a single moving vehicle .............................. 163 10.3.2.2 Response due to a train of moving vehicles, effect of bridge- vehicle interaction and cable modeling.................................. 165 10.3.2.3 Speed and bridge damping effect........................................... 166 10.3.2.4 Effect of surface irregularities at the bridge entrance ............ 167 10.3.2.5 Effect of tuned vibration absorbers ........................................ 168 vii 11 Conclusions and Suggestions for Further Research 181 11.1 Conclusions of Part B................................................................................. 181 11.1.1 Nonlinear finite element modeling technique............................... 181 11.1.2 Response due to moving vehicles ................................................. 182 11.2 Suggestions for further research................................................................. 184 A Maple Procedures 187 A.1 Cable element............................................................................................. 187 A.2 Beam element............................................................................................. 188 Bibliography of Part B 189 viii ______________________________________________________________________ General Introduction and Summary ______________________________________________________________________ Due to their aesthetic appearance, efficient utilization of structural materials and other notable advantages, cable supported bridges, i.e. cable-stayed and suspension bridges, have gained much popularity in recent decades. Among bridge engineers the popularity of cable-stayed bridges has increased tremendously. Bridges of this type are now entering a new era with main span lengths reaching 1000 m. This fact is due, on one hand to the relatively small size of the substructures required and on the other hand to the development of efficient construction techniques and to the rapid progress in the analysis and design of this type of bridges. Ever since the dramatic collapse of the first Tacoma Narrows Bridge in 1940, much attention has been given to the dynamic behavior of cable supported bridges. During the last fifty-eight years, great deal of theoretical and experimental research was conducted in order to gain more knowledge about the different aspects that affect the behavior of this type of structures to wind and earthquake loading. The recent developments in design technology, material qualities, and efficient construction techniques in bridge engineering enable the construction of lighter, longer, and more slender bridges. Thus nowadays, very long span cable supported bridges are being built, and the ambition is to further increase the span length and use shallower and more slender girders for future bridges. To achieve this, accurate procedures need to be developed that can lead to a thorough understanding and a realistic prediction of the structural response due to not only wind and earthquake loading but also traffic loading. It is well known that large deflections and vibrations caused by dynamic tire forces of heavy vehicles can lead to bridge deterioration and eventually increasing maintenance costs and decreasing service life of the bridge structure. The recent developments in bridge engineering have also affected damping capacity of bridge structures. Major sources of damping in conventional bridgework have been largely eliminated in modern bridge designs reducing the damping to undesirably low levels. As an example, welded joints are extensively used nowadays in modern bridge designs. This has greatly reduced the hysteresis that was provided in riveted or bolted 1 joints in earlier bridges. For cable supported bridges and in particular long span cable-stayed bridges, energy dissipation is very low and is often not enough on its own to suppress vibrations. To increase the overall damping capacity of the bridge structure, one possible option is to incorporate external dampers (discrete damping devices such as viscous dampers and tuned mass dampers) into the system. Such devices are frequently used today for cable supported bridges. However, it is not believed that this is always the most effective and the most economic solution. Therefore, a great deal of research is needed to investigate the damping capacity of modern cable supported bridges and to find new alternatives to increase the overall damping of the bridge structure. To consider dynamic effects due to moving traffic on bridges, structural engineers worldwide rely on dynamic amplification factors specified in bridge design codes. These factors are usually a function of the bridge fundamental natural frequency or span length and states how many times the static effects must be magnified in order to cover the additional dynamic loads. This is the traditional method used today for design purpose and can yield a conservative and expensive design for some bridges but might underestimate the dynamic effects for others. In addition, design codes disagree on how this factor should be evaluated and today, when comparing different national codes, a wide range of variation is found for the dynamic amplification factor. Thus, improved analytical techniques that consider all the important parameters that influence the dynamic response, such as bridge-vehicle interaction and road surface roughness, are required in order to check the true capacity of existing bridges to heavier traffic and for proper design of new bridges. Various studies, of the dynamic response due to moving vehicles, have been conducted on ordinary bridges. However, they cannot be directly applied to cable supported bridges, as cable supported bridges are more complex structures consisting of various structural components with different properties. Consequently, more research is required on cable supported bridges to take account of the complex structural response and to realistically predict their response due to moving vehicles. Not only the dynamic behavior of new bridges need to be studied and understood but also the response of existing bridges, as governments and the industry are seeking improvements in transport efficiency and our aging and deteriorating bridge infrastructure is being asked to carry ever increasing loads. 2 The aim of this work is to study the moving load problem of cable supported bridges using different analysis methods and modeling techniques. The applicability of the implemented solution procedures is examined and guidelines for future analysis are proposed. Moreover, the influence of different parameters on the response of cable supported bridges is investigated. However, it should be noted that the aim is not to completely solve the moving load problem and develop new formulas for the dynamic amplification factors. It is to the authors opinion that one must conduct more comprehensive parametric studies than what is done here and perform extensive testing on existing bridges before introducing new formulas for design. This thesis contains two separate parts, Part A (Chapter 1-5) and Part B (Chapter 6-11), where each has its own introduction, conclusions, and reference list. These two parts present two different approaches for solving the moving load problem of ordinary and cable supported bridges. Part A, which is a slightly modified version of the licentiate thesis presented by the author in November 96, presents a state-of-the-art review and proposes a simplified analysis method for evaluating the dynamic response of cable-stayed bridges. 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 utilization of the beam on elastic bed analogy makes the presented approach also suitable for analysis of the dynamic response of railway tracks subjected to moving trains. In Part B, a more general approach, based on the nonlinear finite element method, is adopted to study more realistic cable-stayed and suspension bridge models considering, e.g., exact cable behavior and nonlinear geometric effects. A beam element is used for modeling the girder and the pylons, and a catenary cable element, derived using exact analytical expressions for the elastic catenary, is used for modeling the cables. This cable element has the distinct advantage over the traditionally used elements in being able to approximate the curved catenary of the real cable with high accuracy using only one element. 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. Damping characteristics and damping ratios of cable supported bridges are discussed and a practical technique for deriving the damping 3 matrix from modal damping ratios, is presented. Among other things, the effectiveness of using a tuned mass damper to suppress traffic-induced vibrations and the effect of including cables motion and modes of vibration on the dynamic response are investigated. To study the dynamic response of the bridge-vehicle system in Part A and B, two sets of equations of motion are written one for the vehicle and one for the bridge. The two sets of equations are coupled through the interaction forces existing at the contact points of the two subsystems. To solve these two sets of equations, an iterative procedure is adopted. The implemented codes fully consider the bridge-vehicle dynamic interaction and have been verified by comparing analysis results with those found in the literature and with results obtained using a commercial finite element code. The following basic assumptions and restrictions are made: elastic structural material two-dimensional bridge models. Consequently, the torsional behavior caused by eccentric loading of the bridge deck is disregarded as the damage to bridges is done mostly by heavy moving trucks rather than passenger cars, only vehicle models of heavy trucks are used simple one dimensional vehicle models are used consisting of masses, springs, and viscous dampers. Consequently, only vertical modes of vibration of the vehicles are considered it is assumed that the vehicles never loses contact with the bridge, the springs and the viscous dampers of the vehicles have linear characteristics, the bridge-vehicle interaction forces act in the vertical direction, and the contact between the bridge and each moving vehicle is assumed to be a point contact. Moreover, longitudinal forces generated by the moving vehicles are neglected. Based on the study conducted in Part A and B, the following guidelines for future analysis and practical recommendations can be made: for preliminary studies using very simple cable-stayed bridge models to determine the feasibility of different design alternatives, the approach presented in Part A can 4 be adopted as it is found to be simple and accurate enough for the analysis of the dynamic response. However, for analysis of more realistic bridge models where e.g. exact cable behavior, nonlinear geometric effects, or non-uniform cross- sections are to be considered, this approach becomes difficult and cumbersome. For such problems, the finite element approach presented in Part B is found to be more suitable as it can easily handle such analysis difficulties for cable supported bridges, nonlinear static analysis is essential to determine the dead load deformed condition. However, starting from this position and utilizing the dead load tangent stiffness matrix, linear static and linear dynamic traffic load analysis give sufficiently accurate results from the engineering point of view it is recommended to use the mode superposition technique for such analysis especially if large bridge models with many degrees of freedom are to be analyzed. For most cases, sufficiently accurate results are obtained including only the first 25 to 30 modes of vibration correct and accurate representation of the true dynamic response is obtained only if road surface roughness, bridge-vehicle interaction, bridge damping, and cables vibration are considered. For the analysis, realistic bridge damping values, e.g. based on results from tests on similar bridges, must be used care should be taken when the dynamic amplification factors given in the different design codes and specifications are used for cable supported bridges, as it is not believed that these can be used for such bridges. For some cases it is found that design codes underestimate the additional dynamic loads due to moving vehicles. Consequently, each bridge of this type, particularly those with long spans, should be analyzed as made in Part B of this thesis. For the final design, such analysis should be performed more accurately using a 3D bridge and vehicle models and with more realistic traffic conditions to reduce damage to bridges not only maintenance of the bridge deck surface is important but also the elimination of irregularities (unevenness) in the approach pavements and over bearings. It is also suggested that the formulas for dynamic amplification factors specified in bridge design codes should not only be a function of the fundamental natural frequency or span length (as in many present design codes) but also should consider the road surface condition. 5 It is believed that Part A presents the first study of the moving load problem of cable-stayed bridges where this simple modeling and analysis technique is utilized. For Part B of this thesis, it is believed that this is the first study of the moving load problem of cable-stayed and suspension bridges where results from linear and nonlinear dynamic traffic load analysis are compared. In addition, such analyses have not been performed earlier taking into account exact cable behavior and fully considering the bridge-vehicle dynamic interaction. Most certainly this study has not provided a complete answer to the moving load problem of cable supported bridges. However, the author hopes that the results of this study will be a help to bridge designers and researchers, and provide a basis for future work. 6 Part A

State-of-the-art Review and a Simplified Analysis Method for Cable-Stayed Bridges 7 8 Chapter

______________________________________________________________________ Introduction

______________________________________________________________________ 1.1 General Studies of the dynamic effects on bridges subjected to moving loads have been carried out ever since the first railway bridges were built in the early 19th century. Since that time vehicle speed and vehicle mass to the bridge mass ratio have been increased, resulting in much greater dynamic effects. In recent years, the interest in traffic induced vibrations has been increasing due to the introduction of high-speed vehicles, like the TGV train in France and the Shinkansen train in Japan with speeds exceeding 300 km/h. The increasing dynamic effects are not only imposing severe conditions upon bridge design but also upon vehicle design, in order to give an acceptable level of comfort for the passengers. Modern cable-stayed bridges with their long spans are relatively new and have been introduced widely only since the 1950, see Table 1.1 and Figure 1.2. The first modern cable-stayed bridge was the Strmsund Bridge in Sweden opened to traffic in 1956. For the study of the concept, design and construction of cable-stayed bridges, see the excellent book by Gimsing [27] and also [28, 68, 75, 76, 79]. Cable supported bridges are special because they are of the geometric-hardening type, as shown in Figure 1.3 on page 16, which means that the overall stiffness of the bridge increases with the increase in the displacements as well as the forces. This is mainly due to the decrease of the cable sag and increase of the cable stiffness as the cable tension increases. Compared to other types of bridges, the dynamic response of cable-stayed bridges subjected to moving loads is given less attention in theoretical studies. Static analysis and dynamic response analysis of cable-stayed bridges due to earthquake and wind loading, received, and have been receiving most of the attention, while only few 9 studies, see section 1.2.1, have been carried out to investigate the dynamic effects of moving loads on cable-stayed bridges. However, with increasing span length and increasing slenderness of the stiffening girder, great attention must be paid not only to the behavior of such bridges under earthquake and wind loading but also under dynamic traffic loading as well. The dynamic response of bridges subjected to moving vehicles is complicated. This is because the dynamic effects induced by moving vehicles on the bridge are greatly influenced by the interaction between vehicles and the bridge structure. The important parameters that influence the dynamic response are (according to previous research conducted in this field, see section 1.2): vehicle speed road (or rail) surface roughness characteristics of the vehicle, such as the number of axles, axle spacing, axle load, natural frequencies, and damping and stiffness of the vehicle suspension system the number of vehicles and their travel paths characteristics of the bridge structure, such as the bridge geometry, support conditions, bridge mass and stiffness, and natural frequencies. For design purpose, structural engineers worldwide rely on dynamic amplification factors (DAF), which are usually related to the first vibration frequency of the bridge or to its span length. The DAF states how many times the static effects must be magnified in order to cover additional dynamic loads resulting from the moving traffic (DAF is usually defined as the ratio of the absolute maximum dynamic response to the absolute maximum static response). Because of the simplicity of the DAF expressions specified in current bridge design codes, these expressions cannot characterize the effect of all the above listed parameters. Moreover, as these expressions are originally developed for ordinary bridges, it is believed that for long span bridges like cable-stayed bridges the additional dynamic loads must be determined in more accurate way in order to guarantee the planned lifetime and economical dimensioning. Figure 1.1 shows the variation of the DAF with respect to the fundamental frequency of the bridge, recommended by different standards [66]. For cases where the DAF was related to the span length, the fundamental frequency was approximated from the span length. It is apparent from Figure 1.1 that the national design codes disagree on the 10 evaluation of the dynamic amplification factors, and although the specified traffic loads vary in these codes, this does not explain such a wide range of variation for the DAF. In the Swedish design code for new bridges, the Swedish National Road Administration (Vgverket) includes the additional dynamic loads, due to moving vehicles, in the traffic loads specified for the different types of vehicles. This gives a constant DAF that is totally independent on the characteristics of the bridge. For bridges like cable-stayed bridges that are more complex and behave differently compared to ordinary bridges, this approach can lead to incorrect traffic loads to be used for designing the bridge. This part of the thesis presents a state-of-the-art review and a simplified analysis method for evaluating the dynamic response of cable-stayed bridges. 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 utilization of the beam on elastic bed analogy makes the presented approach also suitable for analysis of the dynamic response of railway tracks subjected to moving trains. 2.0 Canada CSA-S6-88m OHBDC Swiss SIA-88, single vehicle ) Swiss SIA-88, lane load 1.8 AASHTO-1989 (DAF India, IRC or Germany, DIN1075 act U.K. - BS5400 (1978) 1.6 France LCPC D/L=0.5 ion f France LCPC D/L=5 plificat D/L = Dead load / Live load 1.4 c am ami Dyn 1.2 1.0 0 1 2 3 4 5 6 7 8 9 10 Bridge fundamental frequency (Hz) Figure 1.1 Dynamic amplification factors used in different national codes [66] 11 Bridge name Country Center span Year of Girder (m) completion material Tatara Japan 890 1999 Steel Pont de Normandie France 856 1995 Steel Qingzhou Minjiang China (Fuzhou) 605 1996 Composite Yangpu China (Shanghai) 602 1993 Composite Xupu China (Shanghai) 590 1996 Composite Meiko-Chuo Japan 590 1997 Steel Skarnsund Norway 530 1991 Concrete Tsurumi Tsubasa Japan 510 1994 Steel resund Sweden/Denmark 490 2000 Steel Ikuchi Japan 490 1991 Steel Higashi-Kobe Japan 485 1994 Steel Ting Kau Hong Kong 475 1997 Steel Seohae South Korea 470 1998 unknown Annacis Island Canada 465 1986 Composite Yokohama Bay Japan 460 1989 Steel Second Hooghly India (Calcutta) 457 1992 Composite Second Severn England 456 1996 Composite Queen Elizabeth II England 450 1991 Composite Rama IX Thailand (Bangk.) 450 1987 Steel Chongqing Second China (Sichuan) 444 1996 Concrete Barrios de Luna Spain 440 1983 Concrete Tongling China (Anhui) 432 1995 Concrete Kap Shui Mun Hong Kong 430 1997 Composite Helgeland Norway 425 1991 Concrete Nanpu China (Shanghai) 423 1991 Composite Vasco da Gama Portugal 420 1998 unknown Hitsushijima Japan 420 1988 Steel Iwagurujima Japan 420 1988 Steel Yuanyang Hanjiang China (Hubei) 414 1993 Concrete Uddevalla Sweden 414 2000 Composite Meiko-Nishi Ohashi Japan 405 1986 Steel S:t Nazarine France 404 1975 Steel Elorn France 400 1994 Concrete Vigo-Rande Spain 400 1978 Steel Table 1.1 Major cable-stayed bridges in the world 12 Dame Point USA (Florida) 396 1989 Concrete Houston Ship Channel USA (Texas) 381 1995 Composite Luling, Mississippi USA 372 1982 Steel Duesseldorf-Flehe Germany 368 1979 Steel Tjrn (new) Sweden 366 1981 Steel Sunshine Skyway USA (Florida) 366 1987 Concrete Yamatogawa Japan 355 1982 Steel Neuenkamp Germany 350 1970 Steel Ajigawa (Tempozan) Japan 350 1990 Steel Glebe Island Australia 345 1990 Concrete ALRT Fraser Canada 340 1985 Concrete West Gate Australia 336 1974 Steel Talmadge Memorial USA (Georgia) 335 1990 Concrete Rio Parana (2 bridges) Argentina 330 1978 Steel Karnali Nepal 325 1993 Composite Khlbrand Germany 325 1974 Steel Guadiana Portugal/Spain 324 1991 Concrete Kniebruecke Germany 320 1969 Steel Brotonne France 320 1977 Concrete Mezcala Mexico 311 1993 Composite Erskine Scotland 305 1971 Steel Bratislava Slovakia 305 1972 Steel Severin Germany 302 1959 Steel Moscovsky Ukraine (Kiev) 300 1976 Steel Faro Denmark 290 1985 Steel Dongying China (Shandong) 288 1987 Steel Mannheim Germany 287 1971 Steel Wadi Kuf Libya 282 1972 Concrete Leverkusen Germany 280 1965 Steel Bonn Nord Germany 280 1967 Steel Speyer Germany 275 1974 Steel East Huntington USA 274 1985 Concrete Bayview USA 274 1990 Composite River Waal Holland 267 1974 Concrete Theodor Heuss Germany 260 1958 Steel Yonghe China (Tianjin) 260 1987 Concrete Table 1.1 (continued) 13 Oberkassel Germany 258 1975 Steel Rees-Kalkar Germany 255 1967 Steel Weirton-Steubenville USA 250 1986 Steel Chaco/Corrientes Argentina 245 1973 Concrete Papineau-Leblanc Canada 241 1971 Steel Krkistensalmi Finland 240 1996 Composite Maracaibo Venezuela 235 1962 Concrete Pasco Kennewick USA 229 1978 Concrete Jinan Yellow River China (Shandong) 220 1983 Concrete Toyosato-Ohashi Japan 216 1970 Steel Onomichi-Ohashi Japan 215 1968 Steel Strmsund Sweden 183 1956 Steel Table 1.1 (continued) 1000 900 Steel girder 800 Composite girder Concrete girder 700 600 500 400 Length of center span (m) 300 200 100 1950 1960 1970 1980 1990 2000 Year of completion Figure 1.2 Span length increase of cable-stayed bridges in the last fifty years 14 1.2 Review of previous research 1.2.1 Research on cable-stayed bridges In recent years the dynamic behavior of cable-stayed bridges has been a source of interesting research. This includes free vibration and forced vibration due to wind and earthquakes, see for example [2, 9, 47]. However, literature dealing with the dynamics of these bridges due to moving vehicles is relatively scarce. For a cable-stayed footbridge, theoretical and experimental study on the effectiveness of tuned mass dampers, TMDs, was carried out in [6]. In this study, tests with one and two persons jumping or running were performed, and acceleration responses with the TMD locked and unlocked were compared. In [59, 60], modal testing of the Tjrn bridge, a cable-stayed bridge in Sweden with a 366 m main span, is described. And in [11], dynamic load testing on the Riddes-Leytron bridge, a cable-stayed bridge in Switzerland with a 60 m main span, is presented. Previous investigations on the dynamic response of cable-stayed bridges subjected to moving loads are summarised in the following: Fleming and Egeseli (1980) [21, 22] compared linear and nonlinear dynamic analysis results for a cable-stayed bridge subjected to seismic and wind loads. The nonlinear dynamic response due to a single moving constant force was also studied. A two-dimensional (2-D) harp system cable-stayed bridge model with a main span of 260 m was adopted, and the bridge was discretized using the finite element method. The nonlinear behavior of the cables due to sag effect and the nonlinear behavior of the bending members due to the interaction of axial and bending deformations, were considered. Fleming et al. showed that although there is significant nonlinear behavior during the static application of the dead load, the structure can be assumed to behave as a linear system starting from the dead load deformed state for both static and dynamic loads, as illustrated in Figure 1.3. This means that influence lines and superposition technique can be used in the design process. Considering only seismic loading a similar comparison was conducted in [2] and the same conclusion was made. 15 Generalized force ic dynamic namic load eigenvalue problem ear dy lin linear dynamic nonlinear dynam cable structures dead load non-cable structures linear static nonlinear static Generalized displacement Figure 1.3 Schematic diagram showing the difference between the behavior of cable structures and non-cable structures and also the accuracy in the results from different analysis procedures Wilson and Barbas (1980) [89] performed theoretical and experimental works on cable-stayed bridge models to determine the dynamic effects due to a moving vehicle. For the theoretical bridge model, a 2-D undamped continuous Bernoulli-Euler beam resting on discrete evenly spaced elastic supports, was adopted. The vehicle was modeled using one or two constant forces travelling at constant speeds. For the solution of the problem, mode superposition technique was used. All bridge cables were approximated by linear springs with equal stiffness, and solutions with two to five cables in the main span were presented. Only the main span was considered in this study, and the road surface roughness was neglected. The experimental models consisted of straight steel beams (cross section = 0.0492 cm 1.97 cm and length = 2.36 m) spliced end to end at the supports (springs) so that continuous spans of up to 23.6 m could be tested. By prestressing the bridge model, an initial flatness to within 0.2 cm under self-weight was achieved. For the interior span supports, coil springs were used. One or two linear induction motors running in a separate track above the bridge model were used to move the point load vehicle at constant speeds in the range of 1.22 m/s to 8.85 m/s. The total vehicle model weight was about 1.2 kg. Wilson et al. presented diagrams showing, for both the theoretical and the experimental models, the influence of the speed parameter on the DAF values for displacements and bending 16 moments. To show the influence of cable stiffness, diagrams with different values for the spring stiffness were also presented. The results showed good agreement between the theoretical and the experimental work. According to Wilson et al., the main reasons for the differences in the results were due to the inability of the experimental system to maintain constant speed, and the neglection of the inertia effects of the experimental transit load in the theoretical model. Wilson et al. concluded also that increasing the spring stiffness at the supports will for most cases lead to an increase in the bridge dynamic response. Rasoul (1981) [69] used the structural impedance method1 and studied the dynamic response of bridges due to moving vehicles. The bridge flexibility functions were evaluated by using a static analysis of the bridge subjected to unit loads. A simply supported beam, a continuous beam, and very simple cable-stayed bridges were studied. For the cable-stayed bridges, two different analysis methods were used, namely an approximate method using the concept of continuous beam with intermediate elastic supports, fixed pylon heads and with the cables approximated by springs, and a more exact method taken into account the effect of the axial force in the girder and the transverse displacement of the pylons by using the reduction method. Solutions with different girder damping ratios for a simple 2-D cable-stayed bridge with only two cables were presented. The traffic load was modeled as a series of vehicles traversing along the bridge. Each vehicle was modeled with a sprung mass and an unsprung mass giving a vehicle model with two degrees of freedom (2 DOF). Different traffic conditions were studied, and the effect of vehicle speed and bridge damping on DAF was presented. Rasoul concluded that bridge damping was one of the important parameters affecting the DAF, and that the DAF was considerably higher for the cables than for other elements of the bridge. Rasoul found also that for a single vehicle travelling at constant speed, the moving force solutions are good approximations of the exact solutions. The road surface roughness was totally neglected in this study. Alessandrini, Brancaleoni and Petrangeli (1984) [3] studied the dynamic response of railway cable-stayed bridges subjected to a moving train. The bridge was discretized using the finite element method, and geometric nonlinearities for the cables were considered by using an equivalent modulus of elasticity. The solution was carried 1 In this study, the equation of motion of the bridge was formulated in an integral form using the flexibility function (Greens function) for the bridge. 17 out using a direct time integration procedure (explicit algorithm). 2-D fan type cable-stayed bridges with steel deck and center spans of about 160, 260, and 412 m were adopted. Five different train lengths of 12-260 m and three different values for the mass per unit length of the train to the mass per unit length of the bridge were considered. The train was simulated using moving masses at three different speeds of 60, 120, and 200 km/h. DAF values for mid-span vertical displacement, axial force in the longest center span cable, and axial force in the anchor cables, were presented and compared with those obtained by the Italian Railways Steel Bridge Code. Alessandrini et al. concluded that, for most cases, the standard expression for DAF given in the Italian Railway Code were not admissible for cable-stayed bridges. It was also found that for speeds of up to about 120 km/h, the dynamic effects were small if not negligible. For speeds higher than 120 km/h the DAF values increase rapidly and for speeds of about 200 km/h, DAF values greater than those prescribed by the Italian Railway Code were observed. The rail surface roughness was neglected in this study. Brancaleoni, Petrangeli and Villatico (1987) [8] presented solutions for the dynamic response of a railway cable-stayed bridge subjected to a single moving high-speed locomotive. The bridge was discretized using the finite element method and geometric nonlinearities were considered in the analysis. The analysis was carried out using a direct time integration procedure (explicit algorithm). A 2-D modified fan type cable-stayed bridge with concrete deck and a main span of 150 m, was adopted. The bridge deck and the pylons were modeled using beam elements, while nonlinear cable elements with parabolic shape functions were adopted for the cables. For the bridge, a Rayleigh type damping producing 2 % of the critical on the first mode has been used. Solutions for a total train weight of about 95 tons, treated as a set of moving forces, a set of moving masses, and a four axles 6 DOF sprung mass model, were presented. Three different train speeds were considered, 60, 120, and 200 km/h. Diagrams showing the variation of DAF with speed for the three different vehicle models, and time histories for the mid-span vertical displacements, were presented. The rail surface roughness was neglected in this study. Brancaleoni et al. concluded that treating the train as a set of moving forces or moving masses results in lower DAF values for the girder bending moments and the cable axial forces, and higher DAF values for the center span vertical displacements. Brancaleoni et al. showed also that bending moment amplification factors were greater than those for cable axial forces and center span vertical displacements. The rail surface roughness was neglected in this study. 18 Walther (1988) [80] performed experimental study on a cable-stayed bridge model with slender deck to determine the dynamic displacements produced by the passage of a 250 kN vehicle at different speeds. The bridge model, which was equipped with rails and a launching ramp, represented a 3 span modified fan type cable-stayed bridge with a 200 m main span and about 100 m side spans. The deck and the two A-shaped pylons were made of reinforced microconcrete, while piano cord wires with a diameter of 2 to 3 mm were used for the cables. The scale adopted was 1/20 giving a total length of about 20 m for the bridge model and a model vehicle weight of 62.5 kg. Different model vehicle speeds from 0.6 to 3.8 m/s (corresponds to real vehicle speeds of about 10 to 61 km/h) were used, and tests with and without a plank in the main span were undertaken to simulate different road surface conditions. Time histories for mid-span vertical displacements were presented, for centric and eccentric vehicle movements, with or without a plank, and for fixed joint and free joint at mid-span. Based on measured data, vertical accelerations were calculated and a study of physiological effects (human sensitivity to vibrations) was undertaken. Walther concluded that from the physiological effects point of view, the structure could be considered acceptable to tolerable depending on the road surface condition. The maximum DAF value for mid-span vertical displacement was found to be 1.3. Walther found also that placing a joint at the center of the bridge deck only give very local effects and have little influence on the global dynamic behavior of the model. Indrawan (1989) [45] studied the dynamic behavior of Rama IX cable-stayed bridge in Bangkok due to an idealized single axle vehicle travelling over the bridge at constant speeds. The 450 m main span, modified fan type, single plane, cable-stayed bridge, was modeled in 2-D. The dynamic response was analyzed using the finite element method and mode superposition technique, including only the first 10 modes of vibration. All analyses were carried out in the frequency domain and time domain responses were calculated using the fast Fourier transform (FFT) technique. The bridge deck and pylons were modeled using beam elements while truss elements were used for the cables. When evaluating the stiffness of each cable, the cable sag was considered by using an equivalent tangent modulus of elasticity. Time histories showing cable forces, mid-span vertical displacements, and pylon tops horizontal displacements, were presented for different types of vehicle models moving over a smooth surface, a rough surface, and a bumpy surface, at speeds of 36 to 540 km/h. The single axle vehicle was modeled as a constant force, an unsprung mass, and a sprung mass (1 DOF system). For the sprung mass vehicle model the assumed natural frequency and damping ratio were 1.39 Hz and 3.5 % respectively. The inertial effect 19 in the vehicle due to bridge vibrations was totally neglected by the author. The road surface roughness was generated from a power spectral density function (PSD) (the same as the one used here in sec. 2.3). Since Rama IX bridge is equipped with tuned mass dampers (TMD) to suppress wind induced oscillations, a comparison was made between the dynamic response with and without the presence of a TMD. The TMD was assumed to be installed at mid-span and tuned to the first flexural mode of vibration. Indrawan found that the TMD was very effective in reducing the vibration level of cables anchored in the vicinity of the mid-span. But he suggested that, instead of using TMDs, viscous dampers should be installed in all cables to more effectively increase the fatigue life of the cables. The analysis results showed also that the DAF increases with increasing vehicle speed and can for bumpy surface reach very high values. Khalifa (1991) [49] carried out an analytical study on two cable-stayed bridges with main spans of 335 m and 670 m. The 3 spans cable-stayed bridges were of the double plane modified fan type, and were modeled in 3-D and discretized using the finite element method. The dynamic response was evaluated using the mode superposition technique, where each equation was solved adopting the Wilson- numerical integration scheme. The linear dynamic analysis, based on geometrically nonlinear static analysis (see Figure 1.3), was conducted using the deformed dead load tangent stiffness matrix. The effect of including cable modes on the overall bridge dynamics was investigated by discretizing each cable of the longer bridge as one element and as eight equal elements. The dynamic response was evaluated for a single moving vehicle and a train of vehicles moving in one direction or in both directions. The vehicles, travelling with constant speeds of about 43 to 130 km/h over a smooth and a rough surface, were approximated using a constant moving force model and a sprung mass model. For the sprung mass vehicle model the assumed natural frequency and damping ratio were 1 or 3 Hz and 3 %, respectively. The road surface roughness was generated from a power spectral density function (PSD) (the same as the one used here in sec. 2.3). Diagrams showing the influence of bridge damping ratio, cable vibrations, vehicle model type, vehicle speed, number of vehicles, traffic direction, and deck condition, on the bridge dynamic response, were presented. A stress-life fatigue analysis was also conducted to estimate the virtual cable life under continuous moving traffic loads. Khalifa found that the fatigue life of stays cables were relatively very short if they were subjected to extreme vibrational stresses resulting from a continuous fluctuating heavy traffic. The results also showed that the magnitude of the dynamic response was influenced by the bridge damping ratio, the type of vehicle model, and 20 the roughness of the bridge deck. The author recommended discretizing each cable into small elements when calculating the dynamic response due to environmental and service dynamic loads. Wang and Huang (1992) [84] studied the dynamic response of a cable-stayed bridge due to a vehicle moving across rough bridge decks. The vehicle was simulated by a nonlinear vehicle model with 3-axles and seven degrees of freedom. A 2-D modified fan type cable-stayed bridge with concrete deck and a main span of 128 m, was adopted. The bridge deck roughness was generated using PSD functions. The dynamic response was analyzed using the finite element method and the geometric nonlinear behavior of the bridge due to dead load was considered. The equation of motion for the vehicle was solved using the fourth-order Runge-Kutta integration scheme, and an iterative procedure with mode superposition technique was used for solving the equation of motion for the bridge. Wang et al. concluded that the mode superposition procedure used was effective and involved much less computation, because accurate results of the bridge dynamic response could be obtained based on solving only 8 to 12 equations of motion of the bridge. Wang et al. noted that the DAF of all components of the bridge were generally less than 1.2 for very good road surface, but increased tremendously with increasing road surface roughness. High values of DAF were noted at the girder near the pylons and at the lower ends of the pylons and piers, but comparatively small DAF values were noted at the girder adjacent to the mid-span of the bridge. Miyazaki et al. (1993) [55] carried out an analytical study on the dynamic response and train running quality of a prestressed concrete multicable-stayed railway bridge planned for future use on the high-speed Shinkansen line. For the analysis, the simulation program DIASTARS, developed at the Japanese Railway Technical Research Institute, was used. The railway track and the bridge structure were modeled using the finite element method. In this study, a 2-D and a 3-D bridge models of a two span cable-stayed bridge, were used. The 2-D bridge model together with a simple 12 cars train model consisting of only constant forces were used to evaluate the dynamic response of the bridge, while the 3-D bridge and the 3-D train model were used to evaluate the train running quality. The 3-D Shinkansen train model consisted of 12 cars where each car consisted of a body, two bogies, and four wheelsets giving 23 DOF. The track was assumed to be directly placed on the bridge deck surface, and the rail surface roughness was neglected. The 3-D bridge deck was modeled by 3-D beam elements connected to the cables through transversely extended rigid beams. In the 21 study, a comparison was also made with the design value of DAF specified in the Japanese Design Standards for Railway Concrete Structures. Miyazaki et al. presented diagrams showing the speed, 0-400 km/h, influence on the DAF for the deck and pylons bending moments, deck and pylons shear forces, deck and pylons axial forces, and axial forces in cables. For the vehicle, diagrams were presented showing wheel load variations and vertical car body accelerations. Miyazaki et al. concluded that the examined PC cable-stayed bridge had a satisfactory train running quality (acceptable riding comfort). For the different bridge members, the authors recommended different values for the coefficient included in the DAF expression in the Japanese design standard. Chatterjee, Datta and Surana (1994) [14] presented a continuum approach for analyzing the dynamic response of cable-stayed bridges. The effects of the pylons flexibility, coupling of the vertical and torsional motion of the bridge deck due to eccentric vehicle movement, and the roughness of the bridge surface, were considered. The vehicle was simulated using a vehicle model with 3 DOF and 3-axles. A PSD function was used to generate the road surface roughness and mode superposition technique was adopted for solving the equation of motion of the bridge. Chatterjee et al. investigated the influence of vehicle speed, eccentrically placed vehicle, spacing between first and second vehicle axles, and bridge damping ratios on the dynamic behavior of a double-plane harp type cable-stayed bridge with roller type cable-pylon connections and a main span of 335 m. Chatterjee et al. concluded that pylon rigidity and the nature of cable-pylon connection have significant effect on the natural frequencies of vertical vibration, but no effect on those of torsional vibration. Chatterjee et al. noted that idealizing the vehicle as a constant force leads to overestimation of the DAF compared to the sprung mass model. The same conclusion was found when assuming that there is no eccentricity in the vehicle path. And finely, it was noted that increasing the axle spacing of the vehicle, or not including the roughness of the bridge surface, decreases the DAF values. 1.2.2 Research on other bridge types The dynamic effects of moving vehicles on bridges have been investigated by various researchers, using bridge and vehicle models of varying degrees of sophistication. 22 A review of the early work on the dynamic response of structures under moving loads was presented in the paper by Filho [20]. For a thorough treatment of the analytical methods used for problems of moving loads with and without mass in both structures and solids, see the excellent book by Frba [23]. In this book, analysis of sprung and unsprung mass systems moving along a beam covered with elastic layer of variable stiffness and surface irregularities, were presented. The dynamics of railway bridges and railway vehicle modeling are described in the book by Frba [24] and the book by Garg and Dukkipati [25]. Interesting research was also presented by Olsson, see Table 1.2, where he derived a structure-vehicle finite element by eliminating the contact degrees of freedom of the vehicle. The stiffness and damping matrices thus became time-variant and non-symmetric. Previous investigations on the dynamic response of other bridge types subjected to moving loads are summarized in Table 1.2 below. Author(s) Bridge type Vehicle model Surface Other remarks like roughness analysis methods used etc. function Hillerborg (1951) SSB SMS-1-1-2 not considered theoretical & experimental [34] study Hirai et al. (1967) suspension MF, moving pulsating not considered theoretical & experimental [36] bridge force study Veletsos et al. 3-SB cantilever SMS-3-3-2 not considered lumped mass method (1970) [77] , SSB Yoshida et al. SSB, SS slab MF, MM not considered FEM (1971) [93] Nagaraju et al. 3-SB MF, SMS-1-1-2 not considered continuum approach, mode (1973) [57] cantilever superposition Ting et al. (1974) SSB MM not considered structural impedance [72] method Table 1.2 Previous investigations on the dynamic response of other bridge types subjected to moving loads. SMS-x-y-z=sprung mass system with x-axles, y degrees of freedom, and in z dimensions, MF=moving force, MM=moving mass, SSB=simply supported beam, x-SB=x span beam, SS xx=simply supported xx, FEM=finite element method 23 Genin et al. SSB, MF, SMS-1-1-2, harmonic structural impedance (1975) [26] 2-SB air cushion system sinusoidal method Ginsberg (1976) SSB multiple not considered structural impedance [29] SMS-1-1-2 method Filho (1978) [20] SSB SMS-1-2-2 not considered FEM Blejwas et al. SSB MM, SMS-1-2-2 harmonic Lagranges eqn. with (1979) [7] sinusoidal multipliers Chu et al. (1979) SS girder & SMS-4-3-3 for not considered lumped mass method [16] truss railway each railcar Gupta et al. SS orthotr. SMS-2-3-2 not considered vehicle braking, eccentric (1980) [31] plate, SSB loading Ting et al. SSB MF, MM, not considered review, different analysis (1980,1983) SMS-1-2-2 procedures and vehicle [73, 74] models Hayashikawa et SSB, 2-SB, MF not considered eigen stiffness matrix al. (1981) [32] 3-SB method Hayashikawa et suspension MF not considered continuum approach, mode al. (1982) [33] bridge superposition Mulcahy (1983) SS orthotr. SMS-2-4-3, 10 mm bump finite strip method, vehicle [56] plate SMS-3-7-3 braking Olsson (1983, SSB MF, MM, harmonic FEM, special bridge- 1985) [63, 62] SMS-1-2-2 cosine vehicle element Schneider et al. SSB MF, MM not considered used the FEM package (1983) [71] ADINA Arpe (1984) SSB SMS-2-4-2 not considered theoretical & experimental [4, 5] study Hino et al. (1984) 1-SB cantilever SMS-1-1-2 not considered FEM, direct time [35] integration Palamas et al. SSB, 2-SB SMS-1-1-2 sinusoidal, Rayleigh-Ritz method (1985) [65] pothole Chu et al. (1986) SS PC railway SMS-4-23-3 PSD lumped mass method [17] Honda et al. 2-SB, 3-SB, 4- SMS-1-2-2 PSD, bump at 1 vehicle & multiple (1986) [37] SB, 5-SB, SSB entrance groups of vehicles Table 1.2 (continued) Olsson (1986) SSB, 2-SB, MF, MM, SMS-1-2-2, not considered FEM, special bridge- 24 [64] 6-SB SMS-2-4-2, vehicle element, vehicle SMS-2-6-2, SMS-2-7-2 braking Inbanathan et al. SSB MF, MM considered FEM, PSD for interaction (1987) [44] force Bryja et al. suspension multiple MF not considered random highway traffic (1988) [10] bridge Diana et al. suspension SMS-4-23-3 for each not considered FEM, different traffic (1988) [19] bridge railcar conditions Coussy et al. SSB SMS-2-2-2 PSD continuum approach, mode (1989) [18] superposition Wang (1990) [81] SS PC railway SMS-4-23-3 for each PSD influence of ramp/ bridge railcar track stiffness Hwang et al. SSB SMS-2-4-2, PSD traffic simulations, one and (1991) [43] SMS-3-7-2 two trucks Olsson (1991) SSB MF not considered compared analytical [61] solution with FEM Wang et al. SS truss SMS-4-23-3 for each PSD lumped mass method (1991) [82] railway railcar Huang et al. continuous SMS-3-12-3 PSD FEM, one and two trucks (1992) [39] multigirder Wang et al. SS multigirder SMS-2-7-3, PSD FEM, one and two trucks (1992) [85] SMS-3-12-3 Wang et al. SSB SMS-2-7-3, bump, PSD FEM, validation of vehicle (1992) [83] SMS-3-12-3 models Knothe et al. review of dynamic modeling of railway track and of vehicle-track interaction (1993) [50] Nielsen (1993) beam on elastic MM, SMS-1-3-2, harmonic sinus- railway structures, [58] foundation,3-D SMS-2-4-2, SMS-2-6-2 oidal for rail- compared theoretical and track model head, wheelflat experimental results Saadeghvaziri SSB, MF not considered used the FEM package (1993) [70] 3-SB ADINA Wang et al. no bridge SMS-2-7-3, bump, PSD only validation of the (1993) [86] SMS-3-12-3 vehicle models Table 1.2 (continued) Wang (1993) [87] SS truss SMS-4-23-3 for each PSD lumped mass method railway railcar 25 Cai et al. (1994) SSB, 2-SB moving pulsating force, not considered continuum approach, mode [12] SMS-1-2-2 superposition Chatterjee et al. suspension SMS-1-1-2, PSD continuum approach, mode (1994) [15] bridge SMS-3-3-2, SMS-3-6-3 superposition Wakui et al. describes a computer program developed using FEM and mode superposition to solve the (1994) [78] dynamic interaction problem between high speed railway vehicles, each of SMS-4-31-3, and railway structures Yener et al. slab on SSBs MF, SMS-1-3-2, not considered FEM, different traffic (1994) [92] SMS-2-6-2 conditions Chatterjee et al. arch bridge MF not considered mixed and lumped mass (1995) [13] method Green et al. 3-SB, 4-SB SMS-4-11-2 PSD, 20 mm compared leaf sprung with (1995) [30] bump air sprung vehicles Huang et al. thin walled SMS-3-12-3 PSD FEM (1995) [40] box-girder Huang et al. hor. curved SMS-3-12-3 PSD FEM, one and two trucks (1995) [41] I-girder Humar et al. SS orthotr. SMS-1-2-2 not considered FEM, different traffic (1995) [42] plate conditions Lee (1995) [51] 2-SB, 3-SB, MF not considered beams on one-sided point 4-SB constraints Lee (1995) [52] SSB rigid wheel not considered unknown wheel nominal motion, FEM Paultre et al. arch, box ambient & controlled dynamic bridge testing (1995) [67] girder traffic Yang et al. (1995) SSB, 3-SB, MF, MM, SMS-1-2-2, PSD FEM, special bridge- [90, 91] 5-SB SMS-3-6-2 vehicle element Table 1.2 (continued) 1.3 General aims of the present study In all the aforementioned studies on the dynamic behavior of cable-stayed bridges, authors either used very simple vehicle models, or very complicated and time- 26 consuming vehicle and bridge models. In [21, 22, 89], the vehicle was modeled as a constant moving force, neglecting the vehicle inertial effects, and in [69, 3, 8, 55], the road (or rail) surface roughness was neglected and only the elastic displacements of the bridge, caused by the varying position of the vehicle, were considered. The opposite assumption was made in [45], where the bridge elastic displacements were neglected and only the excitation caused by the road surface roughness was considered. Of course, the assumptions made by those authors are acceptable, if for example the vehicle is travelling at low speed, the road surface is smooth, and the vehicle mass to the bridge mass ratio is low. The vehicle inertial effects, the road surface roughness, and the bridge displacements were considered in [49]. However, the formulations for the coupling equations (equations (2.4a-c) in section 2.1) are, according to the authors opinion, incorrect. Only the models developed in [84, 14] are believed to be general and handle the bridge-vehicle contact problem correctly. On the other hand, the vehicle models used are very complicated and, as Frba [24] pointed out, very detailed and complicated vehicle models are unnecessary, if the main purpose is to study the bridge dynamic response. In the work presented here, the most detailed vehicle model used consists of two degrees of freedom, as this is adequate for large span bridges, according to Frba. The main aims of this study are as follows: to develop a general but simple analysis tool which fully consider the bridge- vehicle interaction, including all inertial terms, in evaluating the dynamic response of bridges subjected to moving vehicles to investigate on the applicability of the beam on elastic bed analogy and the finite difference method for dynamic analysis of cable-stayed bridges. Moreover, to show that the proposed simplified analysis method, which uses the finite difference method and the mode superposition technique for dynamic response evaluation, is very efficient and is easy to implement and understand to analyze the dynamic response of simple cable-stayed bridge models and to study the influence of different vehicle models and the influence of different parameters, such as vehicle speed and bridge deck surface roughness, on the dynamic response. 27 For this purpose a computer code has been developed using the MATLAB language [53], where the dynamic interaction between the bridge and the vehicle is included by utilizing an iterative scheme. Time histories and dynamic amplification factors are presented as functions of a limited set of parameters for quite simple but representative bridge and vehicle models. The implemented code has been verified by comparing analysis results with those obtained using the commercial finite element code ABAQUS [1]. Special emphasis is put on verification of the proposed model and on investigating the effects of local and global irregularities on the dynamic response. Part of this work was presented earlier at the 15th Congress of IABSE, Copenhagen, 1996 [48]. 28 Chapter

______________________________________________________________________ Vehicle and Structure Modeling _____________________________________________________________________ _ 2.1 Vehicle models Heavy vehicles consist of several major components, such as tractors, trailers and suspension systems, and can be modeled by a set of lumped masses, springs and dampers. As illustrated in Figure 2.1, the vehicle models used in this study include a moving force model, a moving mass model, and a sprung mass model with two degrees of freedom. The moving force model (constant force magnitude) is sufficient if the inertia forces of the vehicle are much smaller than the dead weight of the vehicle. For a vehicle moving along a straight path at a constant speed, these inertia effects are mainly caused by bridge deformations (bridge-vehicle interaction) and bridge surface irregularities. Hence factors that are believed to contribute in creating vehicle inertia effects include: high vehicle speed, flexible bridge structure, large vehicle mass, small bridge mass, stiff vehicle suspension system and large surface irregularities. In the present study, the adopted sprung mass model is a one-axle vehicle model of a real multi-axle vehicle. This model is acceptable, when the bridge span is considerably larger than the vehicle axle base [24], as the case is for cable supported bridges. The author believes that the use of simplified models may be more effective in identifying correlation between the governing bridge-vehicle interaction parameters and the bridge response. Very detailed vehicle models are unnecessary and will not bring any great advantage, when the main purpose is to study the dynamic response of bridges. Heavy roadway vehicles generate most of their dynamic wheel loads in two distinct frequency ranges [30]: body-bounce and pitch motions at 1.5-4 Hz and wheel-hop 29 motion at 8-15 Hz. This explains the increase of some of the specified DAF in Figure 1.1, for bridges with a fundamental frequency in the range of 1 to 5 Hz. v( t) w 2( t) m 2 v( t) v( t) w 1( t) w 1( t) ( m k c S S 1+ m 2) g m 1 m 1+ m 2 Moving force model Moving mass model Sprung mass model Figure 2.1 Vehicle modeling Considering the sprung mass model, shown in Figure 2.1, and denoting the contact force between the bridge and the vehicle by F( t) , defined positive when it acts downward on the bridge, the following equations of motion can be established [23, 63]: 2 ( d w d w d w m + m 1 2 1 1 2 ) g m + k 1 ( w w ) + c F( t S S ) 0 (2.1) d t 2 2 1 d t d t + = d2 2 w w w m k w w c S S = 2 ( 2 1) d d 2 1 0 (2.2) d 2 t d t d t Equations (2.1) and (2.2) are the dynamic equilibrium equations for the unsprung mass and the sprung mass, respectively. Referring to Figure 2.1, w ( t 1 ) and w ( t 2 ) are the displacements of the vehicle unsprung mass m 1 and the vehicle sprung mass m 2, respectively, kS the stiffness of the linear spring connecting the two masses, cS the damping coefficient of the viscous damper, and g the acceleration of gravity. It should be noted that w ( t 2 ) is measured from the equilibrium position under the dead weight m 2 g. The contact force may be expressed by use of equations (2.1) and (2.2) giving: 30 2 d w 2 d w F( t) = ( m + m 1 2 1 2 ) g + m + m 1 (2.3) d t 2 2 d t 2 where the first term on the right-hand side is the dead weight (static part) of the contact force and the other terms represent the inertia effects. d2 w The contact force for the moving mass model will be F( t) = ( m + m ) g + 1 1 2 , d t 2 and for the moving force model F = ( m + m ) g 1 2 . Assuming that the vehicle never loses contact with the bridge (that is F( t) > 0), and that the deformation between the unsprung mass center and the bridge deck center line may be neglected, the following coupling equations for the point of contact, x ( t) = x ( t v ) (see Figure 2.2), must be fulfilled [20, 63]: w t = y( x t ,t ) + r( x t 1( ) ( ) ( )) (2.4a) y y r & w ( t) = v + + v 1 (2.4b) x t x 2 y 2 2 2 2 y y y r 2 r & w ( t) = v + 2 v + a + + v + a 1 (2.4c) x 2 x t x t 2 x 2 x where & w ( t) 1 and & w ( t) denote the unsprung mass vertical velocity and acceleration, 1 respectively, v and a the vehicle velocity and acceleration in the longitudinal direction, respectively, y ( x , t ) the bridge vertical displacement, and r( x) the surface irregularity function. The first term on the right-hand side of equation (2.4c) represent the influence of the bridge deck curvature (centripetal acceleration), the second term the influence of Coriolis acceleration, and the fourth term the influence of the acceleration of the point of contact in the vertical direction. 2.2 Bridge structure For the present study, the fan-shaped self or earth anchored cable-stayed bridge scheme shown in Figure 2.2 is adopted. To make the presentation of the model more 31 clear, the derivation of the equations in this section will be presented including only the main span of the bridge as shown in Figure 2.2, and assuming that the stiffening girder, having a uniform mass and flexural rigidity, is simply supported at the pylons. Of course the developed computer code is very general and capable of handling the more realistic case including side spans, suspended or not suspended, and as many supports as needed. Figure 2.2 Idealized vehicle in contact with a cable-stayed bridge 2.2.1 Major assumptions The following assumptions are made: multicable system with small stays spacing compared to the bridge length negligible cable mass the cables are idealized as vertical springs continuously distributed along the length of the stiffening girder according to the usual erection procedures, the bridge in its initial configuration under dead load is free from bending moments, while only axial forces are present 32 cable forces under dead load are so adjusted that all displacements remain zero axial girder forces have negligible effect on the frequencies and mode shapes and are therefore neglected only in-plane flexural behavior of the bridge is considered. The torsional behavior caused by eccentric loading of the bridge deck is disregarded in this study bridge damping is small and therefore neglected when the vehicle enters the bridge, the vertical deflection and the vertical velocity of the moving vehicle are assumed to be zero. 2.2.2 Differential equation of motion The governing equation of motion for vertical vibration of the bridge at any section of the stiffening girder (idealized as a Bernoulli-Euler beam on elastic supports) is given by [23]: 4 y( x,t) 2 y x,t E I + k( x) y( x,t) ( ) + m = g ( x xv) F( t g g ) (2.5) x 4 t 2 where is the Dirac delta function, Eg the modulus of elasticity, I g the moment of inertia, mg the mass per unit length, and k( x) the spring stiffness (to study ordinary beam type bridges k( x) is set to zero). The effects of rotatory inertia and shear deformation are neglected as the cross-sectional dimensions of the stiffening girder are small in comparison with its length and the higher vibration modes are not significantly excited. The boundary conditions are: 2 2 ( 0, , 0, ) y( t) y L t = 0, = 0, y( L, t) ( ) y t = 0, = 0 (2.6a-d) x 2 x 2 and the initial conditions are: y( y x x,0) ( ,0) = , 0 = 0 (2.7a,b) t 33 2.2.3 Spring stiffness Using the notations of Figure 2.2, the stiffness of the spring idealizing cable i is given by [75]: c E A k i i i = sin2 i (2.8) Li Denoting the allowable cable stress by a , the dead load and the live load per unit length by q and q g q , the cross-sectional area of cable i is given by [9]: ( q + q g q ) s A = i (2.9) sin a i Due to its own dead weight, a stay cable actually takes the shape of a curved line, rather than a straight one, between the two anchorage points. When the cable tension increases, the sag decreases, and the apparent axial stiffness of the inclined cable increases. In the present study, the cable geometric nonlinearity, due to the change of the sag and shape under varying stresses (forces), is approximately taken into account by introducing the following equivalent tangent modulus of elasticity [27, 9, 84]: E L E( x) = E c = (2.10) 2 2 0 x i x 2 c + 1 E 3 c 12o where Ec is the modulus of elasticity for the straight cable, c the specific weight of the cable material, and o the initial tensile stress in the cable. As cable forces caused by the vehicle load are small when compared to those created by dead load, the starting equlibrium configuration under dead load is used [9, 84, 14] and o is here set equal to g , which is the tension stress due to dead load qg and is given by [9]: q = g g a (2.11) ( q + g qq ) 34 After substituting (2.9) into (2.8), the following equation can be established for the spring stiffness per unit length due to the elongation of the cables in the main span: E( x) + c ( q q g q ) 1 L k ( x) = (2.12) 2 0 x H a x 2 1 + H The horizontal force on the pylon top due to the tensile force Fi in the main span cable i is: ( q + q + g q ) s ( q q g q ) s T = F cos = = x i i i (2.13) tan H i Neglecting the stiffness of the pylon, the area of the anchor cable per unit length of the main span and the total area of each anchor cable can be expressed as: q + q A = o( x) g q x (2.14) H cos a o L/2 q + q 2 tot g q L A = A d = o o( x) x (2.15) H 8 0 a cos o In equation (2.15), the dead weight of the side spans are not included because the side spans are not considered in this derivation. The elongation of the anchor cable, the horizontal displacement of the pylon top, and the vertical displacement in the main span at joint i, due the force Fi in cable i are: F L L i o i = cos o (2.16) E A cos o o o L F L cos b o i o i = = (2.17) 2 coso E A o o cos o 35 x = i b (2.18) H The internal force in cable i due to the vertical displacement = i 1, and the spring stiffness are: 2 ( = ) 1 E A H cos 1 F i = o o o i (2.19) L cos x o i ( i = ) 1 2 2 a F sin E A H cos 1 k i i = = o o o i (2.20) 1 L 2 o x After substituting (2.14) into (2.20), the following equation can be established for the spring stiffness per unit length due to the elongation of the anchor cables: E L H + a o s ( q q g q ) 1 L k ( x) = 0 x (2.21) L 2 x 2 a o Where E o is evaluated according to equation (2.10). Referring now to the pylons, a horizontal force at the top gives the horizontal displacement: T H 3 b i p = (2.22) 3 E I p p Thinking of the pylon as a fictitious anchor cable having the fictitious area Ap , the force Ti gives the horizontal displacement: L T L b o i = = o (2.23) cos 2 o E A o p cos o Equating both equations (2.22) and (2.23) for b, we find that the fictitious cable area is: 36 3 E I L A p p = o p (2.24) E H 3 2 o p cos o To approximately include the effect of the pylons, equation (2.21) is modified to give: E L H + tot o + a s ( q q g q ) A A o p 1 L k ( x) = 0 x (2.25) L 2 A tot x 2 a o o And finally, the resulting spring stiffness that includes the effect of the elongation of the cables in the main span, the elongation of the anchor cables, and the stiffness of the pylons is: 1 L k( x) = 0 x (2.26) 1 + 1 2 k c( x) ka( x) The pylons shortening effect is very small and is therefore not considered in equation (2.26). Figure 2.3 shows the spring stiffness, k( x), for a fan-shaped cable-stayed bridge with L=150 m, and Figure 2.4 shows the fixed pylon top curve in Figure 2.3 but with the suspended side spans included. 2.0 )2 fixed pylon top N/m free pylon top 6 1.5 free pylon top, I p =0 ) (10( x k 1.0 0.5 Spring stiffness, 0.0 0 25 50 75 100 125 150 x (m) 37 Figure 2.3 Spring stiffness for a fan-shaped cable-stayed bridge with L=150 m, H=30 m, Hp=60 m, Ec =2.11011 N/m2, Ep=0.351011 N/m2, Ip=25 m4, a =720106 N/m2, qg=12104 N/m, qq=5104 N/m, and c =9104 N/m3 2.0 )2 N/m6 1.5 ) (10( xk 1.0 0.5 Spring stiffness, 0.0 0 50 100 150 200 250 x (m) Figure 2.4 Spring stiffness for a fan-shaped cable-stayed bridge (fixed pylon top) with the same input data as in Figure 2.3 but with the side spans, Ls=50 m, included Equations (2.1-2.5) and (2.26), together with the boundary and initial conditions, define the analytical model of the problem. 2.3 Bridge deck surface roughness The characteristics of road surface roughness is expressed generally by the power spectral density (PSD) function which is assumed from a stationary normal probability process with a zero mean value [38]. The PSD, S() , of road surface roughness is approximated generally by an exponential function [38, 46] as: S( ) = a n (2.27) in which is the wave number, a is the spectral roughness coefficient, and n is the spectral roughness exponent. 38 In the present study, the bridge road surface roughness is simulated using the onesided PSD function: 35.8 10 4 0 < < 0.05 S() = (2.28) 0 0 . 107 10 4 1 9.4 0.05 < < 1 presented in [38], and shown in log-log by the bold line in Figure 2.5. The exponent n and the coefficient a were calculated by the least square method using measurements on 56 bridges. The random surface roughness profile is assumed to be the sum of series of sinusoidal waves and is generated by the following formula [49]: M r( x) = si ( n x + i i i ) (2.29) i=1 where i is the amplitude of the sinusoidal wave, i is a random phase angle with uniform distribution in the interval {0, 2 } , and i is the i th wave number within the specified PSD interval and is given by: = + i min ( i )1 (2.30) in which the wave number increment is defined as: = max min (2.31) M where M is the total number of wave number increments in the interval between the minimum wave number min and the maximum wave number max in the defined spectra. The unit of the wave number is (m-1) and the unit of the spectrum is (m3). The amplitude i is related to the PSD function S( i ) by: 2 S( i = i ) (2.32) 2 39 10 )3 Analytical m-4 Simulated 1 ) (10 0.1 0.01 PSD of surface roughness, S( 0.001 0.01 0.1 1 Wave number, (1/m) Figure 2.5 Power spectral density of road surface roughness 0.02 0.01 0 -0.01 -0.02 Surface roughness (m) 0 50 100 150 Distance along the bridge (m) Figure 2.6 Simulated road surface roughness profile for a bridge with L=150 m 40 The simulated road surface roughness profiles are different depending on the random phase angles i , used in equation (2.29). Figure 2.6 shows a sample of the simulated random road surface roughness profile used in this study for bridges with L=150 m. To compare the analytical PSD, equation (2.28), with the PSD of the sample shown in Figure 2.6, a power spectrum analysis of the sample is performed using the following relationship [83]: 1 S () = F 2 sim (2.33) N where F is the length N fast Fourier transform (FFT) of the sample, and S sim ( ) is the resulting PSD estimate for the sample. Using the symmetry property of real FFTs, the spectral estimates corresponding to negative wave numbers are removed and a compensation is made for them in the positive side. The resulting one-sided PSD estimate for the sample in Figure 2.6 is indicated by the fine solid line in Figure 2.5. The bridge-vehicle interaction forces can now be calculated including the effect of road surface roughness by using the simulated profile of the bridge road. 41 42 Chapter

______________________________________________________________________ Response Analysis _____________________________________________________________________ _ 3.1 Dynamic analysis 3.1.1 Eigenmode extraction Before analyzing the response of the bridge to moving vehicles it is important to study the free vibration of the bridge model. The free vibration analysis is here an essential first step in obtaining the forced vibration response of the bridge and the parameters that affect the free vibration also affect the response to moving vehicles. The differential equation governing the free vibration of the bridge model is obtained by setting the right-hand side in equation (2.5) equal to zero, giving: 4 y( x,t) 2 y x,t + k + = g g ( x) y( x,t) ( ) E I m 0 (3.1) x 4 g t 2 The displacement function y( x, t) can be expressed as a product of two functions, one involving only the space coordinate x, called the mode shape function or eigenfunction z( x) , and the other one involving the variable time and called the generalized coordinates ( t) . Then when the bridge vibrates in its i th natural bending mode, the displacement at any location varies harmonically with time and can be expressed as [88]: y ( x, t) = z ( x) ( t) = z ( x)( a cos t + b sin t i i i i i i i i ) (3.2) 43 where i is the i th circular frequency of the free motion of the bridge. By expressing as in equation (3.2) and substituting into equation (3.1) gives: i ( t ) d 4 zi( x) E I + k( x) z 2 = 0 (3.3) 4 i ( x) m z g i i ( x g g ) d x Dividing the bridge model into n equal segments each of length h, giving n-1 unknowns, and replacing the derivative in equation (3.3) with its finite difference approximation, the following eigenvalue problem is obtained2: A Z = Z (3.4) where c 4 1 0 1 4 c 2 4 1 1 4 c 4 1 0 3 1 A = O O O O O , = O (3.5a,b) 1 4 c 0 3 4 1 n1( n1 x n ) 1 1 4 c 2 4 0 1 4 c 1 ( n1 x n )1 z , L z 11 1 ,n1 Z = [z , z , L, z 1 2 n1 ] = M M (3.5c) zn11 , L zn1 ,n1( n1 x n )1 4 g m h k 2 ( x 1) k 4 ( x 2 , 3 , K) = , = 5 + and = 6 + 4 i i c h c 1 2 , h (3.5d,f) E 3,K g Ig EgIg EgIg where i is the mode number and column i in Z gives the i th mode shape. The first term in equation (3.5e) for c 1 must be changed from 5 to 7, if fixed end supports are assumed. If rotational springs with stiffness Km are introduced at end supports, the first term mentioned above is evaluated as: 6+(-1)/(+1), where = K h/(2 E I m g g). 2 If nothing else is mentioned, bold upper case letters are used for matrices and bold lower case letters for vectors. The transpose of a matrix or a vector is denoted with the superscript T. 44 Mode shapes and corresponding circular frequencies are now obtained numerically by solving equation (3.4). 3.1.2 Response of the bridge Using mode superposition technique, the solution of equation (2.5) can be written as: y( x, t) = z ( x) i ( t i ) (3.6) i=1 Substituting equation (3.6) into equation (2.5) gives: ( E I z IV + k( x) z + m z & ) = ( x xv) F g g i i i i g i i (3.7) i=1 Further, both sides of the above equation are multiplied with eigenmode z j and integrated3 over the length of the bridge. Using the orthogonality properties of the eigenfunctions and setting the normalization constant4 equal to unity, the following equation is obtained (see ref. [88] for more details on derivation): 2 F & + = i i i zi ( xv ) (3.8) mg where the contact force, F, is time depended and therefore evaluated at each time step using the expressions given in section 2.1. When dividing the bridge model into n equal segments each of length h and setting the time step to t = h / v , which corresponds to having the vehicle load applied only at the segment joints, the term z ( x i v ) in equation (3.8) will be the value in the i th eigenvector z i and the row number corresponding to the point of contact (i.e. where the contact force is applied). L 3 The properties of the Dirac function gives ( x x ) z ( x) d x= zi( x v i v ) 0 L 4 2 Normalization constant is defined as z ( x) d x T i or in vector form z z i i 0 45 The solution of the simple second order differential equation in (3.8) is: F z + 2 o & i ( xv ) ( Fzi( xv) m i i g ) = + cos t o i + sin t i (3.9) 2 m 2 m i i i g i g i where and & o i o i are the initial values for the step and can, when using the orthogonality and normalization relationships, be evaluated as [88]: T = o i i z yo (3.10a) & T = o i i z & yo (3.10b) where y and y o & o are the bridge vertical displacement and velocity vectors at the previous time step. Substituting into equation (3.6), the bridge displacement vector at each time step is obtained from the sum: s Fz x F z x 2 m & i ( v ) ( i( v)+ i o i g ) y = i o z + cos t + sin i i i t (3.11) 2 2 i 1 m = m i g i g i where s is the number of included modes, s n 1. The expressions for the vertical velocity and acceleration vectors for the bridge are obtained, by differentiating equation (3.11), as: s ( Fz 2 i ( xv ) + o i i mg ) & y = z & sin t + cos i i o i i t (3.12) i 1 = i mg s ( Fz 2 i ( xv ) + i o i mg ) & y = z & cos t sin i i i o i i t (3.13) m i=1 g 46 Using the MATLAB language [53], a computer code has been developed for analysis of the dynamic response. The dynamic interaction between the bridge and the vehicle is included by utilizing the iterative scheme shown in Figure 3.1. The recalculation of bridge displacements is repeated in each step until convergence is obtained. In this study, the following convergence criterion is considered: max( y y j j1 ) Tolerance (3.14) max( y j ) where y j is the j th estimate for the bridge displacement vector at the current time step. For the results presented in this study, the tolerance was set to 5 10 6 . The mo