Response of cable stayed and suspension bridges to moving loads

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M i d - p o i n t v e r t i c a l d i s p l a

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Response of Cable-Stayed andSuspension Bridges to Moving Vehicles

Analysis methods and practical modeling techniques

Raid Karoumi

TRITA-BKN. Bulletin 44, 1998ISSN 1103-4270ISRN KTH/BKN/B--44--SE

Doctoral Thesis

146 m 146 m335 m

v = 110 km/h

Royal Institute of TechnologyDepartment of Structural Engineering


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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


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To my wife, Lena,

to my daughter and son, Maria and Marcus,

and to my parents, Faiza and Sabah.

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i

Stockholm framlägges till offentlig granskning för avläggande av teknologie

doktorsexamen fredagen den 12 februari 1999.

Raid Karoumi 1999KTH, TS- Tryck & Kopiering, Stockholm 1999


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______________________________________________________________________

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 ofthe 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.


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______________________________________________________________________

Contents ______________________________________________________________________

Abstract i

Preface iii

General Introduction and Summary 1

Part A State-of-the-art Review and a Simplified Analysis Method for Cable-

Stayed Bridges

7

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


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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


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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 .............................. 16310.3.2.2 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


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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


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______________________________________________________________________

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 otherhand 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 recentdevelopments 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


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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 suchas 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 engineersworldwide 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.


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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 cablesupported 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 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


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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 elementcode.

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-vehicleinteraction 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


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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.


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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 dynamictraffic 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.


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Part A

State-of-the-art Review and aSimplified Analysis Method

for Cable-Stayed Bridges


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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 19 th 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 levelof 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 1.3on 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


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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 underdynamic 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


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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 movingvehicles, 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 isidealized 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 vehicleSwiss 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

D y n a m i c

a m p l i f i c a t i o n f a c t o r ( D A F )

0 1 2 3 4 5 6 7 8 9 10

2.0

1.8

1.6

1.4

1.2

1.0

Figure 1.1 Dynamic amplification factors used in different national codes [66]


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Bridge name Country Center span

(m)

Year of

completion

Girder

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

Öresund Sweden/Denmark 490 2000 Steel

Ikuchi Japan 490 1991 SteelHigashi-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


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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

Tjörn (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 SteelTalmadge Memorial USA (Georgia) 335 1990 Concrete

Rio Parana (2 bridges) Argentina 330 1978 Steel

Karnali Nepal 325 1993 Composite

Köhlbrand 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)


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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

Kärkistensalmi 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

Strömsund Sweden 183 1956 Steel


100

200

300

400

500

600

700

800

900

1000

1950 1960 1970 1980 1990 2000

Year of completion

L e n g t h o f c e n t e r s p a n ( m )

Steel girder

Composite girder

Concrete girder

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


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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 ofinteresting 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 analysisresults 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.


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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 experimentalsystem 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 simplysupported 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 theflexibility function (Green’s function) for the bridge.


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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 wereconsidered. 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 forspeeds 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 surfaceroughness 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.


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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 witha 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 vehiclemovements, 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 wereused 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


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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 numericalintegration 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


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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 aniterative 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 twospan 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


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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 foranalyzing 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.


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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 andunsprung 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

Yoshida et al.

(1971) [93]

SSB, SS slab MF, MM not considered FEM

Nagaraju et al.

(1973) [57]

3-SB

cantilever

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

superpositionTing 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


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Genin et al.

(1975) [26]

SSB,

2-SB

MF, SMS-1-1-2,

air cushion system

harmonic

sinusoidal

structural impedance

method

Ginsberg (1976)

[29]

SSB multiple

SMS-1-1-2

not considered structural impedance

methodFilho (1978) [20] SSB SMS-1-2-2 not considered FEM

Blejwas et al.

(1979) [7]

SSB MM, SMS-1-2-2 harmonic

sinusoidal

Lagrange’s eqn. with

multipliers

Chu et al. (1979)

[16]

SS girder &

truss railway

SMS-4-3-3 for

each railcar

not considered lumped mass method

Gupta et al.

(1980) [31]

SS orthotr.

plate, SSB

SMS-2-3-2 not considered vehicle braking, eccentric

loading

Ting et al.

(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

harmoniccosine

FEM, special bridge-vehicle element

Schneider et al.

(1983) [71]

SSB MF, MM not considered used the FEM package

ADINA

Arpe (1984)

[4, 5]SSB SMS-2-4-2 not considered theoretical & experimental

study

Hino et al. (1984)

[35]

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

integration

Palamas et al.

(1985) [65]

SSB, 2-SB SMS-1-1-2 sinusoidal,

pothole

Rayleigh-Ritz method

Chu et al. (1986)

[17]

SS PC railway SMS-4-23-3 PSD lumped mass method

Honda et al.

(1986) [37]

2-SB, 3-SB, 4-

SB, 5-SB, SSB

SMS-1-2-2 PSD, bump at

entrance

1 vehicle & multiple

groups of vehicles


Olsson (1986) SSB, 2-SB, MF, MM, SMS-1-2-2, not considered FEM, special bridge-


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[64] 6-SB SMS-2-4-2,

SMS-2-6-2, SMS-2-7-2

vehicle element, vehicle

braking

Inbanathan et al.

(1987) [44]

SSB MF, MM considered FEM, PSD for interaction

forceBryja 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.

(1991) [43]

SSB SMS-2-4-2,

SMS-3-7-2

PSD traffic simulations, one and

two trucks

Olsson (1991)

[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

Wang et al.(1992) [85]

SS multigirder SMS-2-7-3,SMS-3-12-3

PSD FEM, one and two trucks

Wang et al.

(1992) [83]

SSB SMS-2-7-3,

SMS-3-12-3

bump, PSD FEM, validation of vehicle

models

Knothe et al.

(1993) [50]

review of dynamic modeling of railway track and of vehicle-track interaction

Nielsen (1993)

[58]

beam on elastic

foundation,3-D

track model

MM, SMS-1-3-2,

SMS-2-4-2, SMS-2-6-2

harmonic sinus-

oidal for rail-

head, wheelflat

railway structures,

compared theoretical and

experimental results

Saadeghvaziri

(1993) [70]

SSB,

3-SB

MF not considered used the FEM package

ADINA

Wang et al.

(1993) [86]

no bridge SMS-2-7-3,

SMS-3-12-3

bump, PSD only validation of the

vehicle models


Wang (1993) [87] SS truss

railway

SMS-4-23-3 for each

railcar

PSD lumped mass method


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Cai et al. (1994)

[12]

SSB, 2-SB moving pulsating force,

SMS-1-2-2

not considered continuum approach, mode

superposition

Chatterjee et al.

(1994) [15]

suspension

bridge

SMS-1-1-2,

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

PSD continuum approach, mode

superpositionWakui 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

Yener et al.

(1994) [92]

slab on SSB’s MF, SMS-1-3-2,

SMS-2-6-2

not considered FEM, different traffic

conditions

Chatterjee et al.

(1995) [13]

arch bridge MF not considered mixed and lumped mass

method

Green et al.

(1995) [30]

3-SB, 4-SB SMS-4-11-2 PSD, 20 mm

bump

compared leaf sprung with

air sprung vehicles

Huang 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

Humar et al.

(1995) [42]

SS orthotr.

plate

SMS-1-2-2 not considered FEM, different traffic

conditions

Lee (1995) [51] 2-SB, 3-SB,

4-SB

MF not considered beams on one-sided point

constraintsLee (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


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-


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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. Theopposite 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 author’s 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 Frýba [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 Frýba.

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, toshow 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 studythe influence of different vehicle models and the influence of different parameters,

such as vehicle speed and bridge deck surface roughness, on the dynamic response.


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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].


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– 29 –

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 inertiaeffects 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 moreeffective 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


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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.

w1(t )

v(t )

v(t )

m2

cS

m1

k S w1(t )

w2(t )

v(t )

(m1+m2) g

m1+m2

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 actsdownward on the bridge, the following equations of motion can be established

[23, 63]:

( ) ( ) ( )− + − + − + −

+ =m m g m

w

t k w w c

w

t

w

t F t S S 1 2 1

21

2 2 12 1 0

d

d

d

d

d

d (2.1)

( ) 0d

d

d

d

d

d 12122

22

2 =

−−−−−t

w

t

wcwwk

t

wm S S (2.2)

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 thedisplacements of the vehicle unsprung mass m1 and the vehicle sprung mass m2,

respectively, k S 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 m2 g . The contact force may be expressed by use of equations (2.1) and (2.2) giving:


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( ) ( ) F t m m g mw

t m

w

t = + + +1 2 1

21

2 2

22

2

d

d

d

d (2.3)

where the first term on the right-hand side is the dead weight (static part) of the contactforce and the other terms represent the inertia effects.

The contact force for the moving mass model will be F t m m g w

t ( ) ( )

d

d= + +

1 2

21

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 linemay 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)

& ( )w t y

xv

y

t

r

xv1 = + +

∂∂

∂∂

∂∂

(2.4b)

&& ( )w t y

v y

x t v

y

xa

y

t

r v

r

xa1

2

22

2 2

2

2

222= + + + + +

∂

∂

∂∂ ∂

∂∂

∂

∂

∂

∂

∂∂

(2.4c)

where & ( )w t 1 and && ( )w t 1 denote the unsprung mass vertical velocity and acceleration,

respectively, v and a the vehicle velocity and acceleration in the longitudinal direction,

respectively, y x , t ( ) the bridge vertical displacement, and r ( ) 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


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– 32 –

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 themore 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 lengthof 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


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• 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 velocityof 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]:

( )( ) ( )

( )( ) ( ) E I

y x,t

xk x y x,t m

y x,t

t x x F t g g

g

v

∂

∂

∂

∂ δ

4

4

2

2+ + = − − (2.5)

where δ is the Dirac delta function, E g the modulus of elasticity, I g the moment ofinertia, m g 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 sheardeformation 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:

( ) ( )

( ) ( )

y t y t

x y L t

y L t

x0 0

00 0 0

2 2, ,

,, , ,

,= = = =

2 2∂

∂

∂

∂ (2.6a-d)

and the initial conditions are:

( ) ( )

00,

,00, =∂

∂=

t

x y x y (2.7a,b)


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2.2.3 Spring stiffness

Using the notations of Figure 2.2, the stiffness of the spring idealizing cable i

is given by [75]:

k E A

Lic i i i

i

=sin2 α

(2.8)

Denoting the allowable cable stress by σa , the dead load and the live load per unitlength by q q g qand , the cross-sectional area of cable i is given by [9]:

( ) A

q q si

g q

a i

= +

σ αsin (2.9)

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 x E E

x E

x L

ic

cc

= =

+

≤ ≤

112

022 2

3

γ

σo

(2.10)

where E c is the modulus of elasticity for the straight cable, γ c the specific weight ofthe cable material, and σ

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 σo is here setequal to σ g , which is the tension stress due to dead load q g and is given by [9]:

σ σ g a g

g q

q

q q=

+( ) (2.11)


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– 35 –

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:

( ) ( ) ( )k x E x q q H x

H

x L

c g q

a

= +

+

≤ ≤

σ

1

1

022

(2.12)

The horizontal force on the pylon top due to the tensile force F i in the main span cable

i is:

( ) ( )T F

q q s q q s

H xi i i g q

i

g q

= =

+

=

+

cos tanα α (2.13)

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:

( ) A xq q

H x

g q

ao

o

= +

σ αcos (2.14)

( ) A A x xq q

H

L L

g q

aotot

oo

2

= = +

∫ d cos

/

0

2

8σ α (2.15)

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 F i in cable i are:

∆ L F L

E A

i io

o

o o o

=cos

cos

αα

(2.16)

b L F L

E A

i i= =∆ o

o

o

o o2

ocos

cos

cosαα

α (2.17)


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– 36 –

δi b x

H = (2.18)

The internal force in cable i due to the vertical displacement δi = 1, and the springstiffness are:

F E A H

L xi

i

i( )cos

cosδ α

α= =1

2 1o o o

o

(2.19)

k F E A H

L xia i i

i

= ==( ) sin cosδ α α1 2 2

1

1o o o

o2

(2.20)

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:

( ) ( )

k x E L H q q

L x x

L

a s g q

a

= +

≤ ≤

o

o2

σ

10

2 (2.21)

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:

bT H

E I

i p

p p

Documents

Response of cable stayed and suspension bridges to moving loads