Response of cable stayed and susspension bridge to moviing vehicles
v = 110 km/h
25 15 5 -5
the truck leaves the bridge
Mid-point vertical displacement (mm) -
-15 -25 -35 -45
with tuned mass damper (TMD) without tuned mass damper (TMD)
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
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: Huvudhandledare:
Docent Sven Ohlsson 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
Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden
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–
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.
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 bridgevehicle 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
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
joints in earlier bridges. For cable supported bridges and in particular long span cablestayed 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.
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 author’s 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 611), 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 crosssections 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.
It is believed that Part A presents the first study of the moving load problem of cablestayed 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 bridgevehicle 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.
State-of-the-art Review and a Simplified Analysis Method for Cable-Stayed Bridges
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 Strömsund 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  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
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 cablestayed 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 . 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 (Vägverket) 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.
Dynamic amplification factor (DAF)
Canada CSA-S6-88m OHBDC Swiss SIA-88, single vehicle Swiss SIA-88, lane load AASHTO-1989 India, IRC Germany, DIN1075 U.K. - BS5400 (1978) France LCPC D/L=0.5 France LCPC D/L=5
D/L = Dead load / Live load
3 4 5 6 7 Bridge fundamental frequency (Hz)
Dynamic amplification factors used in different national codes