Response of cable stayed and susspension bridge to moviing vehicles

v = 110 km/h

335 m

146 m

25
15
5
-5

the truck leaves the bridge

Mid-point vertical displacement (mm) -

146 m

-15
-25
-35
-45

with tuned mass damper (TMD)
without tuned mass damper (TMD)

-55
0

10

20

30

40

Time (s)

Response of Cable-Stayed and
Suspension Bridges to Moving Vehicles
Analysis methods and practical modeling techniques
Raid Karoumi

Royal Institute of Technology
Department of Structural Engineering
TRITA-BKN. Bulletin 44, 1998
ISSN 1103-4270
ISRN KTH/BKN/B--44--SE
Doctoral Thesis



Response of Cable-Stayed and Suspension
Bridges to Moving Vehicles
Analysis methods and practical modeling techniques

Raid Karoumi
Department of Structural Engineering
Royal Institute of Technology
S-100 44 Stockholm, Sweden

Akademisk avhandling

Som med tillstånd av Kungl Tekniska Högskolan i Stockholm framlägges till offentlig
granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari
1999 kl 10.00 i Kollegiesalen, Valhallavägen 79, Stockholm. Avhandlingen försvaras
på svenska.
Fakultetsopponent:
Huvudhandledare:

Docent Sven Ohlsson
Professor Håkan Sundquist

TRITA-BKN. Bulletin 44, 1998
ISSN 1103-4270
ISRN KTH/BKN/B--44--SE
Stockholm 1999



Response of Cable-Stayed and Suspension
Bridges to Moving Vehicles
Analysis methods and practical modeling techniques

Raid Karoumi

Department of Structural Engineering
Royal Institute of Technology
S-100 44 Stockholm, Sweden

_____________________________________________________________________

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


To my wife, Lena,
to my daughter and son, Maria and Marcus,
and to my parents, Faiza and Sabah.

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

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


______________________________________________________________________

Abstract
______________________________________________________________________

This thesis presents a state-of-the-art-review and two different approaches for solving
the moving load problem of cable-stayed and suspension bridges.
The first approach uses a simplified analysis method to study the dynamic response of
simple cable-stayed bridge models. The bridge is idealized as a Bernoulli-Euler beam
on elastic supports with varying support stiffness. To solve the equation of motion of
the bridge, the finite difference method and the mode superposition technique are used.

The second approach is based on the nonlinear finite element method and is used to
study the response of more realistic cable-stayed and suspension bridge models
considering exact cable behavior and nonlinear geometric effects. The cables are
modeled using a two-node catenary cable element derived using “exact” analytical
expressions for the elastic catenary. Two methods for evaluating the dynamic response
are presented. The first for evaluating the linear traffic load response using the mode
superposition technique and the deformed dead load tangent stiffness matrix, and the
second for the nonlinear traffic load response using the Newton-Newmark algorithm.
The implemented programs have been verified by comparing analysis results with
those found in the literature and with results obtained using a commercial finite
element code. Several numerical examples are presented including one for the Great
Belt suspension bridge in Denmark. Parametric studies have been conducted to
investigate the effect of, among others, bridge damping, bridge-vehicle interaction,
cables vibration, road surface roughness, vehicle speed, and tuned mass dampers.
From the numerical study, it was concluded that road surface roughness has great
influence on the dynamic response and should always be considered. It was also found
that utilizing the dead load tangent stiffness matrix, linear dynamic traffic load
analysis give sufficiently accurate results from the engineering point of view.

Key words: cable-stayed bridge, suspension bridge, Great Belt suspension bridge, bridge,
moving loads, traffic-induced vibrations, bridge-vehicle interaction, dynamic analysis,
cable element, finite element analysis, finite difference method, tuned mass damper.
–i–


– ii –


______________________________________________________________________

Preface
______________________________________________________________________

The research presented in this thesis was carried out at the Department of Structural
Engineering, Structural Design and Bridges group, at the Royal Institute of
Technology (KTH) in Stockholm. The project has been financed by KTH and the Axel
and Margaret Ax:son Johnson Foundation. The work was conducted under the
supervision of Professor Håkan Sundquist to whom I want to express my sincere
appreciation and gratitude for his encouragement, valuable advice and for always
having time for discussions. I also wish to thank Dr. Costin Pacoste for reviewing the
manuscript of this report and providing valuable comments for improvement.
Finally, I would like to thank my wife Lena Karoumi, my daughter and son, and my
parents for their love, understanding, support and encouragement.

Stockholm, January 1999
Raid Karoumi

– iii –


– iv –


______________________________________________________________________

Contents
______________________________________________________________________

Abstract

i

Preface

iii

General Introduction and Summary

Part A

1

1.3

7

Review of previous research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1

Research on cable-stayed bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.2

Research on other bridge types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

General aims of the present study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Vehicle and Structure Modeling
29
2.1 Vehicle models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2

2.3
3

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

9
Introduction
1.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2

2

1

Bridge structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1

Major assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.2

Differential equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3

Spring stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bridge deck surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Response Analysis
43
3.1 Dynamic analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1

Eigenmode extraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

–v–


3.1.2
3.2
4

5

Response of the bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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

83
Conclusions and Suggestions for Further Research
5.1 Conclusions of Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2

Suggestions for further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography of Part A

87

Part B

97

6

7

Refined Analysis Utilizing the Nonlinear Finite Element Method

Introduction
99
6.1 General ......................................................................................................... 99
6.2

Cable structures and cable modeling techniques ....................................... 101

6.3

General aims of the present study .............................................................. 103

Nonlinear Finite Elements
105
7.1 General ....................................................................................................... 105
7.2

7.3

Modeling of cables ..................................................................................... 106
7.2.1

Cable element formulation............................................................ 107

7.2.2

Analytical verification................................................................... 111

Modeling of bridge deck and pylons.......................................................... 113

– vi –


8

Vehicle and Structure Modeling
117
8.1 Vehicle models........................................................................................... 117
8.2

Vehicle load modeling and the moving load algorithm ............................. 121

8.3

Bridge structure .......................................................................................... 123

8.4
9

8.3.1

Modeling of damping in cable supported bridges......................... 123

8.3.2

Bridge deck surface roughness...................................................... 126

Tuned vibration absorbers.......................................................................... 127

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
9.2

Nonlinear dynamic analysis .......................................................... 138

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 bridgevehicle interaction and cable modeling.................................. 165

10.3.2.3 Speed and bridge damping effect ........................................... 166
10.3.2.4 Effect of surface irregularities at the bridge entrance ............ 167
10.3.2.5 Effect of tuned vibration absorbers ........................................ 168

– vii –


11 Conclusions and Suggestions for Further Research
181
11.1 Conclusions of Part B................................................................................. 181
11.1.1 Nonlinear finite element modeling technique............................... 181
11.1.2 Response due to moving vehicles ................................................. 182
11.2 Suggestions for further research................................................................. 184
A

Maple Procedures
187
A.1 Cable element ............................................................................................. 187
A.2 Beam element ............................................................................................. 188

Bibliography of Part B

189

– viii –


______________________________________________________________________

General Introduction and Summary

______________________________________________________________________

Due to their aesthetic appearance, efficient utilization of structural materials and other
notable advantages, cable supported bridges, i.e. cable-stayed and suspension bridges,
have gained much popularity in recent decades. Among bridge engineers the
popularity of cable-stayed bridges has increased tremendously. Bridges of this type are
now entering a new era with main span lengths reaching 1000 m. This fact is due, on
one hand to the relatively small size of the substructures required and on the other
hand to the development of efficient construction techniques and to the rapid progress
in the analysis and design of this type of bridges.
Ever since the dramatic collapse of the first Tacoma Narrows Bridge in 1940, much
attention has been given to the dynamic behavior of cable supported bridges. During
the last fifty-eight years, great deal of theoretical and experimental research was
conducted in order to gain more knowledge about the different aspects that affect the
behavior of this type of structures to wind and earthquake loading. The recent
developments in design technology, material qualities, and efficient construction
techniques in bridge engineering enable the construction of lighter, longer, and more
slender bridges. Thus nowadays, very long span cable supported bridges are being
built, and the ambition is to further increase the span length and use shallower and
more slender girders for future bridges. To achieve this, accurate procedures need to
be developed that can lead to a thorough understanding and a realistic prediction of the
structural response due to not only wind and earthquake loading but also traffic
loading. It is well known that large deflections and vibrations caused by dynamic tire
forces of heavy vehicles can lead to bridge deterioration and eventually increasing
maintenance costs and decreasing service life of the bridge structure.
The recent developments in bridge engineering have also affected damping capacity of
bridge structures. Major sources of damping in conventional bridgework have been
largely eliminated in modern bridge designs reducing the damping to undesirably low
levels. As an example, welded joints are extensively used nowadays in modern bridge
designs. This has greatly reduced the hysteresis that was provided in riveted or bolted

–1–


joints in earlier bridges. For cable supported bridges and in particular long span cablestayed bridges, energy dissipation is very low and is often not enough on its own to
suppress vibrations. To increase the overall damping capacity of the bridge structure,
one possible option is to incorporate external dampers (discrete damping devices such
as viscous dampers and tuned mass dampers) into the system. Such devices are
frequently used today for cable supported bridges. However, it is not believed that this
is always the most effective and the most economic solution. Therefore, a great deal of
research is needed to investigate the damping capacity of modern cable supported
bridges and to find new alternatives to increase the overall damping of the bridge
structure.
To consider dynamic effects due to moving traffic on bridges, structural engineers
worldwide rely on dynamic amplification factors specified in bridge design codes.
These factors are usually a function of the bridge fundamental natural frequency or
span length and states how many times the static effects must be magnified in order to
cover the additional dynamic loads. This is the traditional method used today for
design purpose and can yield a conservative and expensive design for some bridges
but might underestimate the dynamic effects for others. In addition, design codes
disagree on how this factor should be evaluated and today, when comparing different
national codes, a wide range of variation is found for the dynamic amplification factor.
Thus, improved analytical techniques that consider all the important parameters that
influence the dynamic response, such as bridge-vehicle interaction and road surface
roughness, are required in order to check the true capacity of existing bridges to
heavier traffic and for proper design of new bridges.
Various studies, of the dynamic response due to moving vehicles, have been conducted
on ordinary bridges. However, they cannot be directly applied to cable supported
bridges, as cable supported bridges are more complex structures consisting of various
structural components with different properties. Consequently, more research is

required on cable supported bridges to take account of the complex structural response
and to realistically predict their response due to moving vehicles. Not only the
dynamic behavior of new bridges need to be studied and understood but also the
response of existing bridges, as governments and the industry are seeking
improvements in transport efficiency and our aging and deteriorating bridge
infrastructure is being asked to carry ever increasing loads.

–2–


The aim of this work is to study the moving load problem of cable supported bridges
using different analysis methods and modeling techniques. The applicability of the
implemented solution procedures is examined and guidelines for future analysis are
proposed. Moreover, the influence of different parameters on the response of cable
supported bridges is investigated. However, it should be noted that the aim is not to
completely solve the moving load problem and develop new formulas for the dynamic
amplification factors. It is to the author’s opinion that one must conduct more
comprehensive parametric studies than what is done here and perform extensive
testing on existing bridges before introducing new formulas for design.
This thesis contains two separate parts, Part A (Chapter 1-5) and Part B (Chapter 611), where each has its own introduction, conclusions, and reference list. These two
parts present two different approaches for solving the moving load problem of
ordinary and cable supported bridges.
Part A, which is a slightly modified version of the licentiate thesis presented by the
author in November 96, presents a state-of-the-art review and proposes a simplified
analysis method for evaluating the dynamic response of cable-stayed bridges. The
bridge is idealized as a Bernoulli-Euler beam on elastic supports with varying support
stiffness. To solve the equation of motion of the bridge, the finite difference method
and the mode superposition technique are used. The utilization of the beam on elastic
bed analogy makes the presented approach also suitable for analysis of the dynamic
response of railway tracks subjected to moving trains.

In Part B, a more general approach, based on the nonlinear finite element method, is
adopted to study more realistic cable-stayed and suspension bridge models
considering, e.g., exact cable behavior and nonlinear geometric effects. A beam
element is used for modeling the girder and the pylons, and a catenary cable element,
derived using “exact” analytical expressions for the elastic catenary, is used for
modeling the cables. This cable element has the distinct advantage over the
traditionally used elements in being able to approximate the curved catenary of the real
cable with high accuracy using only one element. Two methods for evaluating the
dynamic response are presented. The first for evaluating the linear traffic load
response using the mode superposition technique and the deformed dead load tangent
stiffness matrix, and the second for the nonlinear traffic load response using the
Newton-Newmark algorithm. Damping characteristics and damping ratios of cable
supported bridges are discussed and a practical technique for deriving the damping
–3–


matrix from modal damping ratios, is presented. Among other things, the effectiveness
of using a tuned mass damper to suppress traffic-induced vibrations and the effect of
including cables motion and modes of vibration on the dynamic response are
investigated.
To study the dynamic response of the bridge-vehicle system in Part A and B, two sets
of equations of motion are written one for the vehicle and one for the bridge. The two
sets of equations are coupled through the interaction forces existing at the contact
points of the two subsystems. To solve these two sets of equations, an iterative
procedure is adopted. The implemented codes fully consider the bridge-vehicle
dynamic interaction and have been verified by comparing analysis results with those
found in the literature and with results obtained using a commercial finite element
code.
The following basic assumptions and restrictions are made:
• elastic structural material

• two-dimensional bridge models. Consequently, the torsional behavior caused by
eccentric loading of the bridge deck is disregarded
• as the damage to bridges is done mostly by heavy moving trucks rather than
passenger cars, only vehicle models of heavy trucks are used
• simple one dimensional vehicle models are used consisting of masses, springs, and
viscous dampers. Consequently, only vertical modes of vibration of the vehicles
are considered
• it is assumed that the vehicles never loses contact with the bridge, the springs and
the viscous dampers of the vehicles have linear characteristics, the bridge-vehicle
interaction forces act in the vertical direction, and the contact between the bridge
and each moving vehicle is assumed to be a point contact. Moreover, longitudinal
forces generated by the moving vehicles are neglected.
Based on the study conducted in Part A and B, the following guidelines for future
analysis and practical recommendations can be made:
• for preliminary studies using very simple cable-stayed bridge models to determine
the feasibility of different design alternatives, the approach presented in Part A can
–4–


be adopted as it is found to be simple and accurate enough for the analysis of the
dynamic response. However, for analysis of more realistic bridge models where
e.g. exact cable behavior, nonlinear geometric effects, or non-uniform crosssections are to be considered, this approach becomes difficult and cumbersome.
For such problems, the finite element approach presented in Part B is found to be
more suitable as it can easily handle such analysis difficulties
• for cable supported bridges, nonlinear static analysis is essential to determine the
dead load deformed condition. However, starting from this position and utilizing
the dead load tangent stiffness matrix, linear static and linear dynamic traffic load
analysis give sufficiently accurate results from the engineering point of view
• it is recommended to use the mode superposition technique for such analysis
especially if large bridge models with many degrees of freedom are to be analyzed.

For most cases, sufficiently accurate results are obtained including only the first 25
to 30 modes of vibration
• correct and accurate representation of the true dynamic response is obtained only if
road surface roughness, bridge-vehicle interaction, bridge damping, and cables
vibration are considered. For the analysis, realistic bridge damping values, e.g.
based on results from tests on similar bridges, must be used
• care should be taken when the dynamic amplification factors given in the different
design codes and specifications are used for cable supported bridges, as it is not
believed that these can be used for such bridges. For some cases it is found that
design codes underestimate the additional dynamic loads due to moving vehicles.
Consequently, each bridge of this type, particularly those with long spans, should
be analyzed as made in Part B of this thesis. For the final design, such analysis
should be performed more accurately using a 3D bridge and vehicle models and
with more realistic traffic conditions
• to reduce damage to bridges not only maintenance of the bridge deck surface is
important but also the elimination of irregularities (unevenness) in the approach
pavements and over bearings. It is also suggested that the formulas for dynamic
amplification factors specified in bridge design codes should not only be a function
of the fundamental natural frequency or span length (as in many present design
codes) but also should consider the road surface condition.

–5–


It is believed that Part A presents the first study of the moving load problem of cablestayed bridges where this simple modeling and analysis technique is utilized. For Part
B of this thesis, it is believed that this is the first study of the moving load problem of
cable-stayed and suspension bridges where results from linear and nonlinear dynamic
traffic load analysis are compared. In addition, such analyses have not been performed
earlier taking into account exact cable behavior and fully considering the bridgevehicle dynamic interaction.
Most certainly this study has not provided a complete answer to the moving load

problem of cable supported bridges. However, the author hopes that the results of this
study will be a help to bridge designers and researchers, and provide a basis for future
work.

–6–


Part A

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

–7–


–8–


Chapter
______________________________________________________________________

Introduction
______________________________________________________________________

1.1

General

Studies of the dynamic effects on bridges subjected to moving loads have been carried

out ever since the first railway bridges were built in the early 19th century. Since that
time vehicle speed and vehicle mass to the bridge mass ratio have been increased,
resulting in much greater dynamic effects. In recent years, the interest in traffic
induced vibrations has been increasing due to the introduction of high-speed vehicles,
like the TGV train in France and the Shinkansen train in Japan with speeds exceeding
300 km/h. The increasing dynamic effects are not only imposing severe conditions
upon bridge design but also upon vehicle design, in order to give an acceptable level
of comfort for the passengers.
Modern cable-stayed bridges with their long spans are relatively new and have been
introduced widely only since the 1950, see Table 1.1 and Figure 1.2. The first modern
cable-stayed bridge was the 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
and dynamic response analysis of cable-stayed bridges due to earthquake and wind
loading, received, and have been receiving most of the attention, while only few

–9–


studies, see section 1.2.1, have been carried out to investigate the dynamic effects of
moving loads on cable-stayed bridges. However, with increasing span length and
increasing slenderness of the stiffening girder, great attention must be paid not only to
the behavior of such bridges under earthquake and wind loading but also under
dynamic traffic loading as well.

The dynamic response of bridges subjected to moving vehicles is complicated. This is
because the dynamic effects induced by moving vehicles on the bridge are greatly
influenced by the interaction between vehicles and the bridge structure. The important
parameters that influence the dynamic response are (according to previous research
conducted in this field, see section 1.2):
• vehicle speed
• road (or rail) surface roughness
• characteristics of the vehicle, such as the number of axles, axle spacing, axle load,
natural frequencies, and damping and stiffness of the vehicle suspension system
• the number of vehicles and their travel paths
• characteristics of the bridge structure, such as the bridge geometry, support
conditions, bridge mass and stiffness, and natural frequencies.
For design purpose, structural engineers worldwide rely on dynamic amplification
factors (DAF), which are usually related to the first vibration frequency of the bridge
or to its span length. The DAF states how many times the static effects must be
magnified in order to cover additional dynamic loads resulting from the moving traffic
(DAF is usually defined as the ratio of the absolute maximum dynamic response to the
absolute maximum static response). Because of the simplicity of the DAF expressions
specified in current bridge design codes, these expressions cannot characterize the
effect of all the above listed parameters. Moreover, as these expressions are originally
developed for ordinary bridges, it is believed that for long span bridges like cablestayed bridges the additional dynamic loads must be determined in more accurate way
in order to guarantee the planned lifetime and economical dimensioning.
Figure 1.1 shows the variation of the DAF with respect to the fundamental frequency
of the bridge, recommended by different standards [66]. For cases where the DAF was
related to the span length, the fundamental frequency was approximated from the span
length. It is apparent from Figure 1.1 that the national design codes disagree on the
– 10 –


evaluation of the dynamic amplification factors, and although the specified traffic

loads vary in these codes, this does not explain such a wide range of variation for the
DAF. In the Swedish design code for new bridges, the Swedish National Road
Administration (Vägverket) includes the additional dynamic loads, due to moving
vehicles, in the traffic loads specified for the different types of vehicles. This gives a
constant DAF that is totally independent on the characteristics of the bridge. For
bridges like cable-stayed bridges that are more complex and behave differently
compared to ordinary bridges, this approach can lead to incorrect traffic loads to be
used for designing the bridge.
This part of the thesis presents a state-of-the-art review and a simplified analysis
method for evaluating the dynamic response of cable-stayed bridges. The bridge is
idealized as a Bernoulli-Euler beam on elastic supports with varying support stiffness.
To solve the equation of motion of the bridge, the finite difference method and the
mode superposition technique are used. The utilization of the beam on elastic bed
analogy makes the presented approach also suitable for analysis of the dynamic
response of railway tracks subjected to moving trains.

Dynamic amplification factor (DAF)

2.0

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

1.8

1.6

D/L = Dead load / Live load

1.4

1.2

1.0
0

Figure 1.1

1

2

3
4
5
6
7
Bridge fundamental frequency (Hz)

8

9

Dynamic amplification factors used in different national codes [66]

– 11 –

10