Response of cable stayed and susspension bridge to moviing vehicles

The aim of this work is to study the moving load problem of cable supported bridges using different analysis methods and modeling techniques.. In Part B, a more general approach, based o

Response of Cable-Stayed and

Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques

Royal Institute of Technology

Department of Structural Engineering

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

Fakultetsopponent: Docent Sven Ohlsson

Huvudhandledare: Professor Håkan Sundquist

TRITA-BKN Bulletin 44, 1998

ISSN 1103-4270

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

To my wife, Lena,

to my daughter and son, Maria and Marcus,

and to my parents, Faiza and Sabah

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari 1999

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

supervision of Professor Håkan Sundquist to whom I want to express my sincere

appreciation and gratitude for his encouragement, valuable advice and for always

having time for discussions I also wish to thank Dr Costin Pacoste for reviewing the

manuscript of this report and providing valuable comments for improvement

Finally, I would like to thank my wife Lena Karoumi, my daughter and son, and my

parents for their love, understanding, support and encouragement

Stockholm, January 1999

Raid Karoumi

Contents

Abstract i Preface iii

Cable-Stayed Bridges

7

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

3.1.2 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

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

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

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

joints in earlier bridges For cable supported bridges and in particular long span 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

cable-as viscous dampers and tuned mcable-ass 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 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

6-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

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

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

It is believed that Part A presents the first study of the moving load problem of stayed bridges where this simple modeling and analysis technique is utilized For Part

cable-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

Part A

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

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 [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

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

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

Bridge fundamental frequency (Hz)

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

Bridge name Country Center span

(m)

Year of completion

Girder material

Pont de Normandie France 856 1995 Steel

Qingzhou Minjiang China (Fuzhou) 605 1996 Composite

Tsurumi Tsubasa Japan 510 1994 Steel

Ting Kau Hong Kong 475 1997 Steel

Annacis Island Canada 465 1986 Composite

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

Kap Shui Mun Hong Kong 430 1997 Composite

Vasco da Gama Portugal 420 1998 unknown

Yuanyang Hanjiang China (Hubei) 414 1993 Concrete

Meiko-Nishi Ohashi Japan 405 1986 Steel

S:t Nazarine France 404 1975 Steel

Table 1.1 Major cable-stayed bridges in the world

Dame Point USA (Florida) 396 1989 Concrete

Houston Ship Channel USA (Texas) 381 1995 Composite

Luling, Mississippi USA 372 1982 Steel

Sunshine Skyway USA (Florida) 366 1987 Concrete

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

East Huntington USA 274 1985 Concrete

River Waal Holland 267 1974 Concrete

Theodor Heuss Germany 260 1958 Steel

Oberkassel Germany 258 1975 Steel

Chaco/Corrientes Argentina 245 1973 Concrete

Pasco Kennewick USA 229 1978 Concrete

Jinan Yellow River China (Shandong) 220 1983 Concrete

Figure 1.2 Span length increase of cable-stayed bridges in the last fifty years

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, TMD’s, 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 Tjörn 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

linear s

tatic

nonli

near static

eigenvalue

r dy

namicGeneralized force

linear dynamic

nonl

inea

r dy

namic

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

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

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

out using a direct time integration procedure (explicit algorithm) 2-D fan type 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

cable-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

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

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 TMD’s, 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-Θ 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

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

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

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 Frýba [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 Frýba [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

roughness function

Other remarks like analysis methods used etc

Hillerborg (1951)

[34]

SSB SMS-1-1-2 not considered theoretical & experimental

study Hirai et al (1967)

[36]

suspension bridge

MF, moving pulsating force

not considered theoretical & experimental

study Veletsos et al

(1970) [77]

3-SB cantilever , SSB

SMS-3-3-2 not considered lumped mass method

MF, SMS-1-1-2 not considered continuum approach, mode

superposition Ting et al (1974)

[72]

SSB MM not considered structural impedance

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

Genin et al

(1975) [26]

SSB, 2-SB

MF, SMS-1-1-2, air cushion system

harmonic sinusoidal

structural impedance method

not considered lumped mass method

(1980,1983)

[73, 74]

SSB MF, MM,

SMS-1-2-2

not considered review, different analysis

procedures and vehicle models

Hayashikawa et

al (1981) [32]

SSB, 2-SB, 3-SB

MF not considered eigen stiffness matrix

method Hayashikawa et

al (1982) [33]

suspension bridge

MF not considered continuum approach, mode

superposition Mulcahy (1983)

[56]

SS orthotr

plate

SMS-2-4-3, SMS-3-7-3

10 mm bump finite strip method, vehicle

braking Olsson (1983,

1985) [63, 62]

SSB MF, MM,

SMS-1-2-2

harmonic cosine

FEM, special vehicle element Schneider et al

[35]

1-SB cantilever SMS-1-1-2 not considered FEM, direct time

integration Palamas et al

(1988) [10]

suspension bridge

multiple MF not considered random highway traffic

Diana et al

(1988) [19]

suspension bridge

SMS-4-23-3 for each railcar

not considered FEM, different traffic

conditions Coussy et al

(1989) [18]

SSB SMS-2-2-2 PSD continuum approach, mode

superposition Wang (1990) [81] SS PC railway SMS-4-23-3 for each

railcar

PSD influence of ramp/ bridge

track stiffness Hwang et al

[61]

SSB MF not considered compared analytical

solution with FEM Wang et al

(1991) [82]

SS truss railway

SMS-4-23-3 for each railcar

PSD lumped mass method

Huang et al

(1992) [39]

continuous multigirder

SMS-3-12-3 PSD FEM, one and two trucks

MM, SMS-1-3-2, SMS-2-4-2, SMS-2-6-2

harmonic oidal for rail- head, wheelflat

sinus-railway structures, compared theoretical and experimental results Saadeghvaziri

(1993) [70]

SSB, 3-SB

MF not considered used the FEM package

(1994) [15]

suspension bridge

SMS-1-1-2, SMS-3-3-2, SMS-3-6-3

PSD continuum approach, mode

superposition Wakui et al

(1994) [78]

describes a computer program developed using FEM and mode superposition to solve the dynamic interaction problem between high speed railway vehicles, each of SMS-4-31-3, and railway structures

(1995) [13]

arch bridge MF not considered mixed and lumped mass

method Green et al

(1995) [40]

thin walled box-girder

SMS-3-12-3 PSD FEM

Huang et al

(1995) [41]

hor curved I-girder

SMS-3-12-3 PSD FEM, one and two trucks

4-SB

MF not considered beams on one-sided point

constraints Lee (1995) [52] SSB rigid wheel not considered unknown wheel nominal

motion, FEM Paultre et al

(1995) [67]

arch, box girder

ambient & controlled traffic

dynamic bridge testing

Yang et al (1995)

[90, 91]

SSB, 3-SB, 5-SB

MF, MM, SMS-1-2-2, SMS-3-6-2

PSD FEM, special

bridge-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-