[TECH]Cable Stayed Bridges

MSc DissertationSeptember 1999

Engineering SeismologyandEarthquake Engineering

Cable-Stayed Bridges -Earthquake Response and Passive Control

Guido Morgenthal

Imperial College of Science, Technology

and Medicine

Civil Engineering Department

London SW7 2BU

Cable-Stayed Bridges -Earthquake Response and Passive Control

Dissertation submitted by

Guido Morgenthal

in partial fulfilment of the requirements of the Degree of

Master of Science and the Diploma of Imperial College

in

Earthquake Engineering and Structural Dynamics

September 1999

Supervisors: Professor A. S. Elnashai, Professor G. M. Calvi

Engineering Seismology and Earthquake Engineering Section

Department of Civil Engineering

Imperial College of Science, Technology and Medicine

London SW7 2BU

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my two supervisors for this dissertation. Firstly mythanks must go to Professor A. S. Elnashai for his help and guidance throughout the year. Hislectures have laid a sound foundation for the work on this project and his constant support evenduring my stay in Italy is greatly appreciated.

Equally important, I would like to thank Professor G. M. Calvi from the Structural MechanicsSection of Università di Pavia. Through him I had the opportunity to work on a fascinatingproject and to experience a beautiful country and a lovely town at the same time. His generosityin taking time to discuss the progress of my work and his support in organising my stay wereessential for my completing the work in time.

The comments of Professor N. Priestley and the help of the other people at San Diego are alsogratefully acknowledged.

Finally and most importantly, I would like to thank my parents who are always there for me.I am grateful for their encouragement and unending support.

Introduction Page 4

TABLE OF CONTENTS

Acknowledgements 3Table of contents 4

1 INTRODUCTION 6

1.1 Preamble 61.2 Significance of long-span bridges 7

1.2.1 Impact of bridges on economy 71.2.2 The trans-European transport network 7

1.3 Recent cable-stayed bridge projects 81.3.1 Öresund Bridge, Sweden 81.3.2 Tatara Bridge, Japan 91.3.3 The Higashi-Kobe Bridge, Japan 10

2 STATE OF RESEARCH ON CABLE-STAYED BRIDGES 11

2.1 Configuration of Cable-Stayed Bridges 112.1.1 General remarks 112.1.2 Cable System 122.1.3 Stiffening Girder 142.1.4 Towers 162.1.5 Foundations 17

2.2 Nonlinearities in Cable-Stayed Bridges 172.3 Dynamic behaviour and earthquake response 19

2.3.1 General dynamic characteristics 192.3.2 Damping characteristics 222.3.3 Influence of soil conditions and soil-structure interaction effects 242.3.4 Structural control 27

3 THE RION ANTIRION BRIDGE 29

3.1 Introduction to structure and site 293.2 Description of the structure 29

3.2.1 The deck 303.2.2 The pylons and piers 313.2.3 The transition piers 323.2.4 The stay cables 323.2.5 The foundation 33

4 FINITE ELEMENT MODEL OF THE BRIDGE 34

4.1 Introduction 344.2 Description of the finite element model 34

4.2.1 The deck 344.2.2 The cables 364.2.3 The pylons and piers 364.2.4 The foundations and abutments 37

4.3 Accelerograms 394.3.1 Structural damping 42

4.4 Calibration investigations on the piers 42

Introduction Page 5

5 CHARACTERISTICS OF THE RION-ANTIRION BRIDGE 44

5.1 Static characteristics - special considerations 445.1.1 Relative displacements 445.1.2 Static push-over analyses on the pier/pylon system 45

5.2 Dynamic characteristics - modal analyses 47

6 EARTHQUAKE RESPONSE AND ITS CONTROL 51

6.1 Introduction 516.2 Investigations on basic systems 51

6.2.1 Introduction 516.2.2 Modelling assumptions 516.2.3 Results 52

6.3 Design considerations and performance criteria 556.3.1 Introduction 556.3.2 Serviceability conditions 556.3.3 Slow tectonic movements 556.3.4 Earthquake conditions 56

6.4 Devices for deck connection 606.4.1 Fuse device 606.4.2 Shock transmitter 606.4.3 Hydraulic dampers 606.4.4 Elasto-plastic isolators 61

6.5 Parametric studies on different deck isolation devices 626.5.1 Introduction 626.5.2 Analysis assumptions 626.5.3 Results 63

6.6 Conclusions 73

7 SUMMARY 76

8 REFERENCES 78

APPENDIX

Introduction Page 6

1 INTRODUCTION

1.1 Preamble

Man's achievements in Structural Engineering are most evident in the world's largest bridgespans. Today the suspension bridge reaches a free span of almost 2000m (Akashi-KaikyoBridge, Japan) while its cable-stayed counterpart can cross almost 1000m (Tatara Bridge, Japan,Normandie Bridge, France, Figure 1). Cable-supported bridges therefore play an important rolein the overcoming of barriers that had split people, nations and even continents before.

Figure 1: Normandie Bridge, France

It is evident that they are an important economical factor as well. By cheapening the supply ofgoods they contribute significantly to economical prosperity.

Cable-stayed bridges, in particular, have become increasingly popular in the past decade in theUnited States, Japan and Europe as well as in third-world countries. This can be attributed toseveral advantages over suspension bridges, predominantly being associated with the relaxedfoundation requirements. This leads to economical benefits which can favour cable-stayedbridges in free spans of up to 1000m.

Many of the big cable-stayed bridge projects have been executed in a seismically active environ-ment like Japan or California. However, very few of them have so far experienced a strongearthquake shaking and measurements of seismic response are scarce. This enforces the need foraccurate modelling techniques. Three methods are available to the engineer to study thedynamic behaviour: forced vibration tests of real bridges, model testing and computer analysis.The latter approach is becoming increasingly important since it offers the widest range of possi-ble parametric studies. However, testing methods are still indispensable for calibration purposes.

Herein the seismic behaviour of the Rion-Antirion cable-stayed Bridge, Greece, is studied bymeans of computer analyses employing the finite element method. A framework of performancecriteria is set up and within this different possible structural configurations are investigated.Conclusions are drawn regarding the effectiveness of deck isolation devices.

Introduction Page 7

1.2 Significance of long-span bridges

1.2.1 Impact of bridges on economy

Roads and railways are the most important means of transport in all countries of the world. Theyact as lifelines on which many economic components depend. Naturally rivers, canals, valleysand seas constitute boundaries for these networks and therefore considerably confine theunopposed supply of goods. They cause significant extra costs because goods have to bediverted or even shipped or flown. These extra costs can exclude economies from foreignmarkets.

It is evident that in this situation bridging the gap is worth considering. Cable-supported bridgesoffer the possibility to cross even very large distances without intermediate supports. Hence, it isonly since their development, that people can consider crossings like the Bosporus (Istanbul -Anatolia, completed 1973 and 1988), Öresund (Denmark - Sweden, to be completed 2000), theStrait of Messina (mainland Italy - Sicily, design stage finished), the Strait of Gibraltar (Spain -Morocco) or the Bering Strait (Alaska - Russia).

Of course infrastructure projects like these are costly. Countries take up high loans to affordthese road links. Cost-benefit analyses are inevitable as proof for banks. However, the numberof already executed major projects emphasises that even the exorbitant costs can be worthwhile.The bridges become an important factor for the whole region and can significantly boost theindustry on both sides of the new link.

Furthermore and equally importantly, those bridge projects can become a substantial factor inthe cultural exchange among people.

1.2.2 The trans-European transport network

The European Parliament has on the 23 July 1996 introduced plans for the development of a"trans-European transport network" ([29]). This project comprises infrastructures (roads,railways, waterways, ports, airports, navigation aids, intermodal freight terminals and productpipelines) together with the services necessary for the operation of these infrastructures.Investments of about 15 billion Euro per year in rail and road systems alone underline theremarks made in the previous section regarding the importance of transport networks and thelinks within them.

The objectives of the network were defined by the European Parliament as follows:

- ensure mobility of persons and goods;- offer users high-quality infrastructures;- combine all modes of transport;- allow the optimal use of existing capacities;- be interoperable in all its components;- cover the whole territory of the Community;- allow for its extension to the EFTA Member States, countries of Central and Eastern

Europe and the Mediterranean countries.

Introduction Page 8

Some of the broad lines of Community action concern:

- the development of network structure plans;- the identification of projects of common interest;- the promotion of network interoperability;- research and deve lopment,

with priority measures defined as follows:

- completion of the connections needed to facilitate transport;- optimization of the efficiency of existing infrastructure;- achievement of interoperability of network components;- integration of the environmental dimension in the network.

It is apparent that the connections as means of interoperation between sub-networks are one ofthe most important components within the network. Many of the currently planned majorbridges in Europe are therefore part of the network and supported by the EU. Among them arethe Öresund and Rion-Antirion Bridges which are discussed subsequently.

1.3 Recent cable-stayed bridge projects

1.3.1 Öresund Bridge, Sweden

The £1.3 billion Öresund crossing will link Denmark and Sweden from the year 2000 on. Itcomprises an immersed tunnel, an artificial island and a bridge part of which is a cable-stayedbridge (Figure 2).

Figure 2: Öresund Bridge, Sweden

For a combined road and railway cable-stayed bridge the center span of 490m (8th largest cable-stayed bridge in the world) is remarkable. A steel truss girder of dimensions 13.5x10.5m was

Introduction Page 9

employed to accommodate road and railway traffic on two levels. The concrete slab is 23.5mwide and provides space for a 4 lane motorway.

The structurally more difficult harp pattern (see section 2.1.2.1) was chosen for aestheticreasons. It should be mentioned that the struts of the girder were inclined according to the angleof the cables which is favourable from the structural as well as pleasing from the aesthetic pointof view.

The money for the project was borrowed on the international market and jointly guaranteed bythe governments of Denmark and Sweden. It will be paid back from the toll fees introduced.Being part of the trans-European transport network the link will be one of the most importantEuropean Structures carrying railway and at least 11,000 vehicles per day.

More information on the Öresund project can be found in [91].

1.3.2 Tatara Bridge, Japan

Upon completion in 1999 the Tatara Bridge will be the cable-stayed bridge with the longest freespan in the world. It is shown in Figure 3, an elevation is given in section 2.1, Figure 5. Thecenter span is 890m, supported by a semi-fan type cable system. Compared with this the sidespans with 270 and 320m are extremely short and asymmetric so that intermediate piers andcounterweights needed to be applied there.

Figure 3: Tatara Bridge, Japan

The girder is a steel box section with a streamlined shape to decrease wind forces. It is 31mwide and only 2.70m deep. To act as counterweight the deck in parts of the sidespans is made ofconcrete. At the towers the girder is kept free because of high temperature induced forces in thecase of a fixing.

In model tests it was found to be necessary to install additional damping devices for the cables.Particularly the upper ones (the longest one having a length of over 460 m - the longest staycable ever) were found to be prone to wind and rain induced vibration. Additional ropes

Introduction Page 10

perpendicular to the stay cables were installed and connected to damping devices at the deck.This yielded cable damping ratios of over 2% of critical.

The Tatara Bridge is being constructed in an area of high seismicity. It was designed for anearthquake event of magnitude 8.5 at a distance of 200km. The fundamental period of the bridgeis 7.2s being associated with a longitudinal sway mode.

All information about the Tatara Bridge were taken from [33].

1.3.3 The Higashi-Kobe Bridge, Japan

The Higashi-Kobe Bridge in Kobe City, Japan, is one of the busiest bridges in the world. As partof the Osaka Bay Route it spans the Higashi-Kobe Channel connecting two reclaimed land areas(Figure 4).

Figure 4: Higashi-Kobe Bridge, Japan

The bridge's main span is 485m with the side span being 200m each. The main girder is aWarren truss with height a of 9m. It accommodates 2 roads at the top and bottom of the trussrespectively. Both of these have three lanes, the width of the truss being 16m.

For the cable system the harp pattern was chosen. The steel towers are of the H-shape and havea height of 146.5m. These are placed on piers which are founded on caissons of size 35 (W) x32 (L) x 26.5 (H) m.

An important feature of the bridge is that the main girder can move longitudinally on all itssupports. This results in a very long fundamental period which was found to be favourable forthe seismic behaviour.

On 17 January 1995 Kobe was struck by an earthquake of magnitude 7.2. Although the Higashi-Kobe Bridge performed well in this earthquake, certain damage did occur which was reported in[44]. Important information about the soil behaviour could be obtained from this event becausethe bridge was instrumented. These will be further discussed in section 2.3.3.

State of Research on Cable-Stayed Bridges Page 11

2 STATE OF RESEARCH ON CABLE-STAYED BRIDGES

2.1 Configuration of Cable-Stayed Bridges

In this section a brief overview of the structural configuration and the load resisting mechanismsof cable-stayed bridges is given. This is necessary because they are in many ways distinctlydifferent from beam-type bridges and these differences strongly affect their behaviour understatic as well as dynamic loads. It has to be noted that herein only the current trend of design canbe described. An outline of the evolution of cable-stayed bridges and more elaborateinformation can be found elsewhere: [50], [87].

2.1.1 General remarks

Cable-stayed bridges present a three-dimensional system consisting of the following structuralcomponents, ordered according to the load path:

- stiffening girder,- cable system,- towers and- foundations.

The stiffening girder is supported by straight inclined cables which are anchored at the towers.These pylons are placed on the main piers so that the cable forces can be transferred down to thefoundation system. As an example the configuration of the Tatara Bridge is given in Figure 5.

Figure 5: Tatara Bridge, Japan, elevation

It is apparent from the picture that the close supporting points enable the deck to be very slim.Even though it has to support considerable vertical loads, it is loaded mainly in compressionwith the largest prestress being at the intersection with the towers. This is due to the horizontalforce which is applied by each of the cables. This characteristic also distinguishes the cable-stayed bridge from the suspension bridge because necessary provisions for anchoring cables aremuch more relaxed. Often cable-stayed bridges are even constructed as being self-anchored.

The particular components of this bridge type will now be discussed in more detail. However, ifmore comprehensive information are sought the reader is referred to [50] and [87].


2.1.2 Cable System

2.1.2.1 Cable patterns

The cable system connects the stiffening girder with the towers. There are essentially 3 patternswhich are used:

- fan system,- harp system and- modified fan system.

These are depicted in Figure 6. All of these patterns can be used for single as well as for doubleplane cable configurations.

Figure 6: Cable patterns in cable-stayed bridges ([50])

In the fan system all cables are leading to the top of the tower. Structurally this arrangement isusually considered the best, since the maximum inclination of each cable can be reached. Thisenables the most effective support of the vertical deck force and thus leads to the smallestpossible cable diameter.

The fan system causes severe detailing problems for the configuration of the anchorage systemat the tower. The modified fan system overcomes this problem by spreading the anchoragepoints over a certain length. If this length is small, the behaviour is not significantly altered.

The stay cables are an important part of the bracing system of the structure. It was found thattheir stiffness is highest when the cable planes are inclined from the vertical. This favours A-shaped towers with all the cables being attached to one point or line at the top.

In the harp system the cables are connected to the tower at different heights and placed parallelto each other. This pattern is deemed to be more aesthetically pleasing because no crossing ofcables occurs even when viewing from a diagonal direction (in contrast see Figure 1). However,this system causes bending moments in the tower and the whole configuration tends to be lessstable. However, excellent stiffness for the main span can be obtained by anchoring each cableto a pier at the side span as was done for the Knie Bridge, Germany ([87]).

Most of the recent cable-stayed bridges, particularly the very long ones, are of the modified fan


type with A-shaped pylons for the discussed reasons. However, there are still many variationsregarding the configuration of the abutments, piers and towers and their respective connectionwith the stiffening girder. These problems will be discussed subsequently in the light of thedynamic behaviour.

2.1.2.2 Types of cables

The success of cable-stayed bridges in recent years can mainly be attributed to the developmentof high strength steel wires. These are used to form ropes or strands, the latter usually beingapplied in cable-stayed bridges.

There are 3 types of strand configuration:

- helically-wound strand,- parallel wire strand and- locked coil strand.

Figure 7 shows these arrangements.

Figure 7: Helically-wound, parallel wire and locked coil cable strands ([50])

The first two types are composed of round wires. Helically-wound strands comprise a centrewire with the other wires being formed around it in a helical manner. They have a lowermodulus of elasticity than their parallel counterparts and furthermore experience a considerableamount of self-compacting when stressed for the first time.

Locked coil strands consist of three layers of twisted wire. The core is a normal spiral strand. Itis surrounded by several layers of wedge or keystone shaped wires and finally several layers ofZ- or S-shaped wires. The advantages of this type of cable are a more effective protectionagainst corrosion and more favourable properties compared with the previous arrangements.First, the density is 30% higher, thus enabling slimmer cables which are less sensitive to windimpact. Second, their modulus of elasticity is even 50% higher compared with a normal strandof same diameter. Third, they are largely insensitive to bearing pressure because of a betterinteraction of the individual wires.

The vast majority of modern cable-stayed bridges use galvanised locked-coil wire strands. Theseare assembled to the large diameter cables, which are usually parallel strand cables.


2.1.2.3 Anchorage of cables

Cables need to be anchored at the deck as well as at the towers. For each of these connectionsnumerous devices exist depending on the configuration of deck and tower as well as of thecable. Exemplary, some arrangements for tower and deck are shown in Figure 8 and Figure 9respectively.

Figure 8: Devices for cable anchorage at the tower ([87])

Figure 9: Devices for cable anchorage at the deck ([50])

Cable supports at the tower may be either fixed or movable. They are situated at the top or atintermediate locations mainly depending on the number of cables used. While fixed supports areeither by means of pins or sockets, movable supports have either roller or rocker devices.

Connections to the deck are by means of special sockets. Their configuration strongly dependson the type of cable used. Usually these sockets are threaded and a bolt is used to allowadjustments on the tension of the cable.

2.1.3 Stiffening Girder

The role of the stiffening girder is to transfer the applied loads, self weight as well as trafficload, into the cable system. As mentioned earlier, in cable-stayed bridges these have to resistconsiderable axial compression forces beside the vertical bending moments. This compressionforce is introduced by the inclined cables.

The girder can be either of concrete or steel. For smaller span lengths concrete girders areusually employed because of the good compressional characteristics. However, as the span


increases the dead load also increases, thus favouring steel girders. The longest concrete bridgethat has been constructed is the Skarnsund Bridge, Norway, with a main span of 530 m ([58]).Also composite girders have been extensively used, entering the span range above 600 m.

The shape of the stiffening girder depends on the nature of loads it has to resist. In the design ofvery long-span bridges aerodynamic considerations can govern the decision. These are beyondthe scope of this work but brief account of this issue will be given. It was shown in [41], thatbluff cross sections which have a higher drag coefficient, experience considerably highertransverse wind forces than less angular sections. Specially designed streamlined sections canalso avoid the creation of wind-turbulence at the downstream side, a phenomenon referred to asvortex-shedding. Considerable affords are therefore made to account for these circumstances.For the Tatara Bridge these were reported in [33].

There are three types of girder cross sections used for cable-stayed bridges:

- longitudinal edge beams,- box girders and- trusses.

These are shown in Figure 10.

a)

b)

c)

Figure 10: Girder cross-sections: a) simple beam arrangement (Knie Bridge,Germany), b) box section (Oberkasseler Bridge, Germany),c) truss (Öresund Bridge, Sweden) ([50])


Beam arrangements consist of a steel or concrete deck which is supported by either a steel or aconcrete beam. The beams carry the loads to the cables where they are anchored. Although easyto construct and generally efficient, beam-type girders have only a small torsional stiffnesswhich can be undesirable depending on the structural system.

Box sections possess high torsional stiffness and can be formed in a streamlined shape thusshowing best behaviour under high wind impact. However, there are numerous possible shapesand the choice depends on the distances between the supports, the desired width of the section,the type of loading and the cable pattern.

Trusses have been used extensively in the past. They possess similar torsional stiffnesses as boxsections. The aerodynamic behaviour is generally good. Trusses are of steel and thus thestiffness is high with respect to the weight. However, the high depth of the section can becriticised for aesthetic reasons. Trusses are unrivalled if double deck functionality is desired. Inthis case the railway deck can be accommodated at the bottom chord.

2.1.4 Towers

The function of the towers is to support the cable system and to transfer its forces to thefoundation. They are subjected to high axial forces. Bending moments can be present as well,depending on the support conditions. It has already been pointed out that the towers in harp-typebridges are subjected to severe bending moments. Box sections with high wall widths generallyprovide best solutions. They can be kept slender and still possess high stiffnesses.

Towers can be made of steel or concrete. Concrete towers are generally cheaper than equivalentsteel towers and have a higher stiffness. However, their weight is considerably higher and thusthe choice also depends on the soil conditions present. Furthermore, steel towers haveadvantages in terms of construction speed.

The shape of the towers is strongly dependent on the cable system and the applied loads butaesthetic considerations are important as well. Possible configurations are depicted in Figure 11.

Figure 11: Tower configurations: H-, A- and λ-shapes ([50])


While I- and H-shapes are vertical tower configurations and therefore support vertical cableplanes, A- and λ-shaped towers correspond to inclined cable planes. The influence of thesepatterns on the overall stiffness of the structure have been discussed earlier. As far as thestiffness of the tower itself is concerned, A- and λ-shapes are preferable. However, theirstructural configuration is significantly different from the I- and H-shape which can haveadverse effects on the ductility (cf. section 5.1.2).

2.1.5 Foundations

Foundations are the link between the structure and the ground. Their configuration is mainlyinfluenced by the soil conditions and the load acting.

For cable-stayed bridges often pile foundations are used, with the pier being connected to thepile cap. Various arrangements are possible and the choice mainly depends on the magnitude ofthe overturning moment.

Cable-stayed bridges often need to be founded in water. In this case caisson foundations areused. The caisson acts as a block and can be placed either on the sea bed or, again, on piles.

2.2 Nonlinearities in Cable-Stayed Bridges

Cable-stayed bridges have an inherently nonlinear behaviour. This has been revealed by veryearly studies and shall be discussed in detail here because the nonlinearity is of greatestimportance for any kind of analysis.

Nonlinearities can be broadly divided in geometrical and material nonlinearities. While the latterdepend on the specific structure (materials used, loads acting, design assumptions), geometricnonlinearities are present in any cable-stayed bridge.

Geometric nonlinearity originates from:

- the cable sag which governs the axial elongation and the axial tension,- the action of compressive loads in the deck and in the towers,- the effect of relatively large deflections of the whole structure due to its flexibility

([1], [4], [9], [50], [52], [73], [74], [75], [87], [88]).

It is well known from elementary mechanics that a cable, supported at both ends and subjectedto its self weight and an externally applied axial tension force, will sag into the shape of acatenary. Increasing the axial force not only results in an increase in the axial strain of the cablebut also in a reduction of the sag which evidently leads to a nonlinear stress-displacementrelationship. The influence of the cable sag on its axial stiffness has first been analyticallyexpressed by Ernst ([34]). If an inclined cable under its self weight is considered, an equivalentelastic modulus can be calculated as follows:


( )3

2

121

σElw

EE e

⋅+= (1)

where: Ee is the effective modulus of elasticity of the sagging cable,E is the modulus of elasticity of the cable which is taut and loaded vertically,w is the unit weight of the cable,l is the horizontal projection of the cable length andσ is the prevailing cable tensile stress.

This relationship can be easily implemented in nonlinear computer codes.

It is interesting to note that the above described cable behaviour leads to an increase in thebridge stiffness if the forces are increased. This is depicted in Figure 12 and clearly distin-guishes cable-supported structures from standard structures. They can be classified as being ofthe geometric-hardening type ([2], [4], [5]).

GeneralizedDisplacement

GeneralizedForce

CABLE-STAYED BRIDGES

Non-cable Structures

Cable Structures

Figure 12: Nonlinearities in cable-stayed bridges

Today most finite element programs offer nonlinear solution algorithms. With these it ispossible to take the above mentioned characteristics of cable-stayed bridges into account. Thenonlinear cable behaviour can be either treated utilising Ernst's formula or applying multi-element cable-formulations. This issue will be further discussed in section 2.3.1.

The nonlinear behaviour of the tower and girder elements due to axial force-bending momentinteraction is usually accounted for by calculating an updated bending and axial stiffness of theelements. Detailed descriptions of nonlinear element formulations can be found in [32], [57],[107] and elsewhere.

The overall change in the bridge geometry as third source of nonlinearity can be accounted forby updating the bridge geometry by adding the incremental nodal displacements to the previousnodal coordinates before recomputing the stiffness of the bridge in the deformed shape ([74]).


2.3 Dynamic behaviour and earthquake response

2.3.1 General dynamic characteristics

Long-span cable-supported bridges, due to their large dimensions and high flexibility, usuallyhave extremely long fundamental periods. This distinguishes them from most other structuresand strongly affects their dynamic behaviour. However, the flexibility and dynamic charac-teristics depend on several parameters such as the span, the cable system and the supportconditions. These will be discussed in detail here.

The dynamic behaviour of a structure can be well characterised by a modal analysis. The linearresponse of the structure to any dynamic excitation can be expressed as superposition of itsmode shapes. The contribution of each mode depends on the frequency content of the excitationand on the natural frequencies of the modes of the structure.

The results of modal analyses of cable-stayed bridges can be found in most of the researchpapers dealing with their seismic behaviour. In Figure 13 the first modes obtained by Abdel-Ghaffar for a model bridge in [1] are shown. The first modes of vibration have very long periodsof several seconds and are mainly deck modes. These are followed by cable modes which arecoupled with deck modes. Tower modes usually are even higher modes and their coupling withthe deck depends on the support conditions between these. The influence of different supportconditions on the mode distribution has been investigated by Ali and Abdel-Ghaffar in [9]. It isapparent from the resulting diagram shown in Figure 14 that movable supports lead to a moreflexible structure, thus shifting the graph towards longer periods. As an example, in [44] it wasmentioned by Ganev et al that the Higashi-Kobe Bridge has been deliberately designed withlongitudinally movable deck in order to shift the fundamental period to a value of low spectralamplification. The decision upon the support conditions of the deck is usually governed byserviceability as well as earthquake considerations. A restrained deck will avoid excessivemovements due to traffic and wind loading and may thus be preferred. However, in the case ofan earthquake a restrained deck will generate high forces which are applied to the pier-pylonsystem. It is thus a trade-off and often intermediate solutions are sought. Elaborate investiga-tions on possible damping solutions are discussed subsequently in this report.

Usually the modes obtained are classified in their directional properties. Thus, vertical,longitudinal, transverse and torsional modes are distinguished and the order of these wellcharacterises the bridge behaviour without the need to depict the individual mode shapes. As anexample the first 25 modes of the Quincy Bayview Bridge, US, are given in Table 1. They havebeen identified experimentally as will be discussed later.


Figure 13: First six computed mode shapes (considering one-element cablediscretisation) ([1])

Figure 14: Effect of support conditions on the natural periods ([9])

Typical for cable-stayed bridges is a strong coupling (such as bending-torsion coupling) in thethree orthogonal directions as can also be seen in Table 1. This coupled motion distinguishescable-stayed bridges from suspension bridges for which pure vertical, lateral and torsional


motions can be easily distinguished. This also implies that three-dimensional modelling isinevitable for dynamic analyses.

The importance of the cable vibration for the overall response of the bridge was pointed out byAli and Abdel-Ghaffar in [2], Abdel-Ghaffar and Khalifa in [9] and Ito in [56]. They concludedthat appropriate modelling of the cable vibrations is necessary. For this purpose multi-elementformulations for the cables are suggested. Only if the mass distribution along the cable ismodelled and associated with extra degrees of freedom the vibrational response of the cablescan be obtained. In [2] a modal analysis was performed modelling the cables with multi-elementcable discretization. It was pointed out that there is coupling between the cable and deck motioneven for the pure cable modes which suggests not to solely rely on analytical expressions fornatural frequencies of a cable alone. In [9] it was found that the natural frequencies of the cablesare strongly dependent on the cable sag as can be seen in Figure 15.

Figure 15: Effect of the sag-to-span ratio on the natural frequencies of a cable,analytical and experimental results ([9])

An analytical method for calculating natural frequencies of cable-stayed bridges has beendeveloped by Bruno and Colotti in [18] and Bruno and Leonardi in [19]. Prevailing trussbehaviour and small stay spacing have been assumed and on this basis diagrams showing themain natural frequencies depending on geometric parameters were developed. These werecompared with results from numerical analyses and a good agreement was found.

For existing bridges the modes can be obtained from vibration measurements. Ambientvibration measurements of the Quincy Bayview Bridge, US were undertaken by Wilson et al.The dynamic response of the bridge to wind and traffic excitation was measured. The resultsobtained were reported in [94] and the mode shapes are depicted in Table 1.


Table 1: Experimentally identified Modes of the Quincy Bayview Bridge([94])

It should be mentioned here, that a response spectrum analysis on the basis of a performedmodal analysis is highly questionable although differently stated in some older papers (e.g.[92]). Firstly, modal superposition is only possible for linear structural behaviour, which isusually not the case for cable-supported bridges as explained in section 2.2. Secondly, even iflinearity could be presupposed, the superposition procedure must be based on reasonableassumptions and methods like the SRSS procedure are only valid for well-spaced modes whichis, again, not necessarily the case for cable-supported structures. Hence, the level of safetyreached would not be assessable.

2.3.2 Damping characteristics

Cable-stayed bridges have inherently low values of damping. It is therefore even more importantto have accurate information about the level of damping reached by the structure. Research onthis issue has shown that generalisation of damping values is difficult because dampingcharacteristics vary significantly depending on the configuration of the bridge. Sources ofenergy dissipation in cable-stayed bridges are: material nonlinearity, structural damping such asfriction at movable bearings, radiation of energy from foundations to ground and friction withair.

There are essentially two ways in which damping is considered in most past investigations.Firstly, energy dissipation by elastic-plastic hysteresis loss can be considered. This requiresconducting a nonlinear analysis with the application of material nonlinearity and is mostimportant when special energy absorption devices are to be modelled. Secondly and mostcommonly, an equivalent viscous damping can be introduced in the system in the form of adamping matrix C. Damping in cable-stayed bridges is undoubtedly not viscous. However, it isan easy to implement and reasonable treatment of the problem as explained by Abdel-Ghaffar in[9]. Usually the damping matrix is established using a linear combination of mass and stiffnessmatrix. This is called Rayleigh damping and enables satisfying damping ratio exactly for 2modes as shown by Clough and Penzien ([27]). An established estimate of viscous dampingratio seems to be 2% for cable-stayed bridges. According to Abdel-Ghaffar ([1]) values of thislevel have been found in many measurements. Damping ratios of 2-3% have in the past been


employed by many investigators in their analyses without further discussion ([1], [2], [4], [5],[9], [10], [31], [38] , [66], [73]).

Dynamic testing of two bridge models of 3.22m length was conducted by Garevski and Severn([47], [48]). Shaking table tests were performed as well as excitation by means of electro-dynamic shakers. The results obtained are shown in Table 2. It is important to note that thedamping ratio notably depends on the mode under consideration. This is due to the differentmode shapes and the different member contributions in these. On the other hand it is obviousfrom the differences between the testing methods that experimental results should also be usedwith caution.

Model A Model BShaking table Portable shakers Shaking table Portable shakers

Modeno.

Type Frequency(Hz)

Damping(%)

Frequency(Hz)

Damping(%)

Frequency(Hz)

Damping(%)

Frequency(Hz)

Damping(%)

1 Lateral 4.28 0.45 4.24 0.42 4.16 0.43 4.17 0.342 Vertical 6.19 0.38 6.19 0.37 5.86 0.33 5.81 0.563 Vertical 8.93 0.42 9.03 0.38 8.83 0.72 8.66 0.834 Lateral 11.88 0.29 11.88 0.26 11.39 2.01 11.58 1.005 Vertical 13.65 0.44 13.65 0.42 13.75 0.56 13.81 1.23

Table 2: Experimentally measured damping (Garevski and Severn, [48])

Several studies on damping characteristics of cable stayed bridges were conducted byKawashima et al and reported in [60], [61], [62], [63] and [64]. Model oscillation tests weredone in which the damping ratio was computed from the averaged decay over 13 cycles. Apicture of the model, which represents the Meiko-nishi Bridge, Japan, is given in Figure 16.

Figure 16: Experimental Bridge Model, Kawashima et al ([64])

In the tests the damping ratio was found to be dependent on the amplitude of excitation and themode shape (cf. Figure 17) as well as on the cable pattern. Damping ratios for the fan-typebridge were in the range between 0.6-0.8% while the harp-type structure had damping ratiosbetween 1.2 and 1.5%, the higher values being for higher amplitudes of oscillation. Higherdamping ratios of the harp configuration can be attributed to larger flexural deformations of thedeck in vertical direction which leads to a higher energy dissipation.


Figure 17: Experimentally measured damping (Kawashima and Unjoh, [61])

Because damping ratios vary with structural types, Kawashima suggested an approach in whichthe energy dissipation capacity is evaluated for each structural segment ([60], [64]). From thisthe overall damping can be calculated. The structural segments in which energy dissipation isconsidered uniform are referred to as substructures. These could be deck, towers, cables andbearing supports. Material nonlinearity is considered to be the prevailing mechanism for energydissipation. Kawashima et al reduced the problem to applying so called energy dissipationfunctions for the substructures which can either be obtained by experiments or taken from theirwork.

Kawashima and his co-workers also investigated the damping activated under earthquakeconditions. In [62] it is stated that in an earthquake considerably higher damping values arefound than the ones usually obtained from forced vibration tests. Hence, strong motion datarecorded at the Suigo Bridge, Japan, during 3 earthquakes were used to estimate dampingratios. These are found to be 2% and 0-1% of critical in longitudinal and transverse direction,respectively, for the tower, and 5% in both directions for the deck.

From their model tests of the Quincy Bayview Bridge (already mentioned in 2.3.1) Wilson et al([94]) also obtained an estimate for the range of damping of the bridge. They found the upperand lower bound of the damping ration for the first coupled transverse/torsion mode to be 2.0-2.6% and 0.9-1.8% respectively.

2.3.3 Influence of soil conditions and soil-structure interaction effects

It is well established that the soil that a structure is founded on has a significant effect on theearthquake response of this structure. This is due to three main mechanisms that are referred toas soil amplification, kinematic interaction and inertial interaction. Firstly, amplitude andfrequency content of the seismic waves are modified while propagating through the soil.Secondly, kinematic interaction means the influence that the soil would have on the movementof a massless, rigid foundation embedded in the soil. Thirdly, inertial interaction describes theeffect that the inertia of the moving structure has on the deformation of the soil. Thesecomponents cannot be further discussed here. However, brief account shall be given ofinvestigations on the importance of soil-structure interaction on the behaviour of cable-stayedbridges and possible treatments of the problem.


Well established for modelling the soil-structure interaction is the substructure approach,described by Betti et al in [16]. It deals with each part of the system (soil, foundation,superstructure) separately. The main advantage is that the analysis of each subsystem can beperformed by the analytical or numerical technique best suited to that particular part of theproblem: e.g. finite element method for the superstructure, continuum mechanics analysis forthe soil. The individual responses are combined so as to satisfy the continuity and equilibriumconditions.

The other modelling approach comprises the so called direct methods. Here, the soil is includedin the global analysis model. Different element formulations and properties can be used for soiland structure but the problem is solved as one. This imposes great importance on the boundaryconditions to be used in the model. For further treatment of this problem reference is made tothe publications by Wolf: [95], [96].

Zheng and Takeda in [106] investigated the applicability of soil-spring models for foundationsystems. Analyses on a 2-D finite element model of the soil were compared with those of thesimplified model. It was found that the simplified model shows good agreement with reality forlow frequency input motions while the errors increase for higher frequencies. In Figure 18transfer functions for both horizontal and vertical motions are shown. These results suggest thata mass-spring model would be a good approach for the analysis of long-period structures likecable-stayed bridges. However, the contribution of higher modes could be underestimated.

Figure 18: Transfer functions for horizontal and vertical component of effectiveinput acceleration at top of foundation computed by 2-D FEM andmass-spring model ([106])

Elassaly et al in [31] presented results of a case study investigating the effects of differentidealisation methods for a pile foundation system on the earthquake forces acting on thestructure. Two cable-stayed bridges were studied in this context. Firstly, a so called Winklerfoundation making use of spring and damper elements was employed. Secondly, a discretizationof the surrounding soil with plain strain elements was used as shown in Figure 19.


Figure 19: Modelling approaches for soil-pile interaction; top: Winkler model,bottom: plain strain finite element modelling ([31])

It should be emphasised that, in contrary to other investigations, the bridge superstructure wasmodelled entirely, too. This is deemed to be important, since it significantly contributes to theinteraction between soil and foundation. It could be shown in time-history analyses, thatneglecting the effects of local soil conditions and the soil-structure interaction results in a greatunderestimation of displacements of the superstructure particularly for soft soils. Under-estimation mainly occurs if the soil site has a fundamental frequency close to the one of thestructure. In terms of simplified approaches it was stated that fixed base modelling approachescan only be justified for structures founded on rock. On the other hand, the accuracy ofemploying an "equivalent stiffness" strongly depends on the conditions of the case in questionand the way in which this stiffness has been evaluated.

Betti, Abdel-Ghaffar et al ([16]) investigated the influence of both soil-structure interactioneffects and the different seismic waves on the response of a cable-stayed bridge. Analyses werecarried out using a fixed structure and a structure with the soil interaction modelled using thesubstructure method. Most interestingly it was found that inclined incoming waves, in-plane aswell as oblique, cause a rocking motion of the foundations. This behaviour underlines theimportance of including soil-structure interaction effects since a rocking motion of thefoundations has a great effect on the behaviour of the bridge piers and thus of the wholestructure.

Investigations on the response of the Higashi-Kobe Bridge (see also section 1.3.3) and thesurrounding soil during the Hyogoken-Nanbu Earthquake have been undertaken by Ganev et al([44]). Time-histories that were recorded by the downhole soil accelerometer and the surfaceaccelerometer showed clear evidence of liquefaction. Because the acceleration at the surface issmaller than the one at 34m depth and exhibits longer period motions, it was concluded that thesurface soil layers which consist of loose saturated sands had actually been liquefied during theearthquake. The measurements from the instrumented bridge could also be employed to validatenumerical approaches for analysis of soil-structure interaction. Extensive analyses usingdifferent computer codes have been conducted in this context. In the present case one of themost important issues associated with interaction analysis is the degradation of the soil stiffness.Three factors are considered to have the largest effect on this: non-linear stress-strain


dependence of the soil material, separation of soil from the structure and pore-water pressurebuildup. For explanations on the modelling approach used to take these into account the readeris referred to the paper mentioned above. Good correlation between the instrumentation data andthe numerical analyses could be achieved as can for example be seen in Figure 20.

Figure 20: Simulation of the earthquake response to the 1995 Hyogoken-NanbuEarthquake ([44])

2.3.4 Structural control

The field of vibration control has experienced an immense evolution in the past years.Fundamentally different techniques have been developed and many different devices areavailable. For extensive descriptions of these reference is made to publications like [72] and[42]. Herein, only a broad overview can be given.

Control of dynamic response can be either active or passive. While active control is dependenton external power supply, passive devices are not. Active control is essentially based onavoiding the impact of forces by modifying the vibration in a favourable way, e.g. additionalmasses are controlled so as to counteract the inertial forces of the structural members. For cable-stayed bridges special solutions like "active tendon control" (Achkire et al [6], Warnitchai et al[90]) have been developed. In this, sensors near the lower anchorage of the cables detect thegirder motion. Passed through a linear feedback operator the resulting signals are fed toservohydraulic actuators fixed at the cable ends. These actuators change the cable tension,therefore providing a time-varying force upon the girder. This mechanism is depicted inFigure 21.

Results of investigations on the effectiveness of active tendon control can be found in [84], arecent paper by Schemmann et al.


Figure 21: Active tendon control of a cable-stayed bridge ([90])

Although active control can be highly effective, the costs are obstructive and for an emergencysituation like an earthquake the dependence on electrical power is deemed to be unfavourable.

Passive control is based on energy dissipation in special devices. These devices are placed atcritical zones such as the deck-abutment and deck-tower connections to concentrate hystereticbehaviour in these specially designed energy absorbers. Inelastic behaviour in primary structuralelements of the bridge can therefore be avoided, assuring the serviceability after an earthquake.

An extensive study on possible applications of energy dissipation devices on cable-stayedbridges has been conducted by Ali and Abdel-Ghaffar ([10]). Recent applications have beenintroduced and modelling guidelines for lead-rubber bearings proposed. This paper makes clearthat particularly for cable-stayed bridges no standardised passive control solutions can bedeveloped. Many solutions are generally possible and their applicability depends on thebehaviour characteristics of the bridge as well as on the design aim. However, it is stated thatthe new trend in cable-stayed bridge design is to have the main deck fixed to neither towers norpiers but to support them elastically by means of dampers, cables and links. The use of theseelastic supports makes it possible to control the natural period of vibration, and accordingly arevery effective in reducing the dynamic forces and, consequently, the size of the towers and thefoundations.

The Rion Antirion Bridge Page 29

3 THE RION ANTIRION BRIDGE

3.1 Introduction to structure and site

The Rion-Antirion Bridge will cross the Gulf of Corinth near Patras in western Greece. It is partof the country's new west axis, a major national transport project. This highway will connectKalamata in the south, Patras, Rion, Antirion and Igoumenitsa in the north. Being linked withtwo other major axes it forms part of a new high-performance transport network which is also apart of the trans-European network described in section 1.2.2. The objective is to establish directaccess of all major urban centers of Greece (Patras, Athens, Lamia, Larissa, Thessaloniki) fromthe developing neighbouring countries in the Balkan region, the other European countries andthe east. Vital is also the connection of the major harbours to the network.

The main bridge of the Rion-Antiron crossing is a cable-stayed bridge. It is located in anexceptional environment consisting of high water depths and rather weak soil deposits as will befurther discussed in 3.2.5. Additionally, the seismic activity in the area is severe which makesthe design even more challenging. The final solution will be described in detail subsequently.

3.2 Description of the structure

The main part of the Rion-Antirion Bridge is a continuous multi-cable-stayed bridge, supportedby four large pylon/pier structures named M1, M2, M3 and M4 resting on the sea bed. Doublecantilevers are built from each of the pylons. Final junctions are then made between the centralcantilevers and the outer parts are extended towards the transition piers so that the final spanlengths are 286, 3x560 and 286 m. The transition piers are connected to approach viaducts.

The bridge has two 9.50m wide carriageways, separated by a 0.50m wide central separator witha double safety fence and bounded by lateral crash barriers.

The main structural configuration can be seen in Figure 22.

Figure 22: Rion-Antirion Bridge, elevation of main bridge ([81])


3.2.1 The deck

The bridge deck is a composite steel-concrete structure (depicted in Figure 23) consisting of:

- two main longitudinal steel girders, 2.20m high, spaced at 22.10m. These beams areI-shaped plate girders with variable sections, the maximum width of the lowerflange being 2.00m.

- Transverse I-shaped plate girders spaced at 4.06m. Both longitudinal and transversebeams include stiffeners to avoid local buckling of the steel plates.

- A top reinforced concrete slab connected with the girders by steel studs. Theconcrete grade is C60/75. The slab thickness is generally 25cm, increased to up to40cm above the girders.

The overall slab width is 27.20m including two 1.95m wide cantilevers. The central partbetween the main steel girders is precast to ensure best concrete quality.

Figure 23: Rion-Antirion Bridge, deck cross section ([81])

The deck is supported by steel stay cables. These are anchored directly above the web of themain girders.

It should be noted that the deck is only supported by the stay cables at the main pylons. Nobearings are provided to the deck by the piers. However, isolation and dissipation devices areplanned to be placed between the deck and the pylon base in transverse direction. These are tolimit the relative displacements between the deck and the piers in the case of a severeearthquake.


3.2.2 The pylons and piers

The four pylons, providing the anchorage for the stay cables, are composed of four reinforcedconcrete legs, joined at their top level to a composite steel-concrete pylon head. The bases of thelegs are rigidly restrained in a prestressed concrete pylon base. The total height of the pylons is113m above the 3.50m thick pylon base.

The pylon legs are square shaped concrete box girders, the outer section of these being 4.0x4.0mand the minimum wall thickness being 60cm. The pylon head is 35.0m high and has a squarehollow section of 8.0x8.0m.

The pylon base is composed of four prestressed concrete beams, 3.50m high and 6.00m wide.These beams form a square grid of 40.0x40.0m. They provide a structural junction between thepier head and the pylon legs and also constitute the anchorage for the isolating devices betweendeck and pier.

An elevation of the pylon/pier system is shown in Figure 24.

Figure 24: Rion-Antirion Bridge, tower ([81])


The main piers are reinforced concrete structures, consisting of the following parts:

- the "pier head": a spatial structure of height 15.28m and varying thickness, joiningthe square pylon base to the octagonal pier shaft,

- the "pier shaft" with its base 3m above mean sea level. Because of the variableheight of the deck the length of this pier shaft is different for the four piers.

- The "cone" allows the adaptation to the actual soil level at each pier site. Hence, thelength of the cones varies for the certain piers. They have an external diameter of26.0m at the top and of 38.0m at the base.

- The roughly cylindrical "footing" consists of a system of walls and slabs for stabilityreasons and has a diameter of 90m. The external wall is cylindrical and has athickness of 0.80m.

The footing and the very first meters of the cone are constructed in a dry dock and towed out toa wet dock where the cone walls are completed. The pier shaft slab and the upper pier arecompleted on site after the base has been immersed to its final position. The various compart-ments of the stiffener ring as well as the hollow central part are filled by a concrete or a gravelpier ballast adjusted to ensure the stability of the pier at any time.

3.2.3 The transition piers

The transition piers act as junction between the high bridge and the approach bridge. Theirdesign has not been finalised yet. However, the following conditions will apply:

- vertically, the deck lays on fixed bearings with anti-lifting devices,

- longitudinally, sliding bearings are provided,

- in the transverse direction, the conditions of the main piers are likely to be adopted.Isolation and energy dissipation devices are provided for improving the earthquakebehaviour.

3.2.4 The stay cables

The two planes of stay cables are arranged in the semi-fan pattern. Each of the spans issupported by 4 sets of 23 cables. The horizontal spacing of cables along the deck is 12.174m.The cables have increasing numbers of parallel strands towards the mid-span. Each strand has anarea of 150mm2.

At the pylon head the cables are anchored with typical threaded anchorages that permitadjustment if required.


3.2.5 The foundation

The bridge is founded on a deep soil strata of weak alluvions, the bedrock being approximately800m below the sea bed level. It was therefore found to be necessary to reinforce the top soillayers with steel inclusions consisting of driven 25m long steel tubes of 2m diameter and 20cmthickness as explained in [28].

The bridge also has to accommodate possible fault movements which lead to horizontaldisplacements of one part of the bridge with respect to the other. The tectonic movement and thethus caused expansion has been discussed in a paper by Ambraseys and Jackson ([13]). Usingrecords of earthquakes in Central Greece since 1694 they found an average extension rate of theGulf of Corinth of 11mm/year.

Finite Element Model of the Bridge Page 34

4 FINITE ELEMENT MODEL OF THE BRIDGE

4.1 Introduction

For purposes of static and dynamic analysis a finite element model of the Rion-Antirion Bridgewas set up for use in the program system ADINA ([7]). It is depicted in Figure 25. As summary,Table 3 lists the mass distribution within the model.

Figure 25: Rion-Antirion Bridge, finite element model

M1[kt]

M2[kt]

M3[kt]

M4[kt]

Deck 18.4 18.8 18.5 18.4Cable Stays 1.4 1.4 1.4 1.4Pylons 11.5 11.5 11.5 11.5Piers 149.1 157.5 159.4 121.9Structural mass 180.4 189.2 190.8 153.2

Table 3: Mass distribution of the bridge

The bridge was modelled in full taking into account all major structural components and theircharacteristics. Since the properties could be taken from design documents the bridge modelreflects an actual structure designed to all code requirements which was not the case for allearlier investigations on cable-stayed bridges.

The accuracy of finite element analyses naturally depends on the assumptions made for settingup the model. A description of the model and these assumptions will thus be givensubsequently.

4.2 Description of the finite element model

4.2.1 The deck

As described in 3.2.1 the deck of the Rion-Antirion Bridge is a composite member consisting ofsteel girders and a concrete slab. It had to be modelled such as to behave correctly in bendingand torsion on one hand and to resemble the inertia effects correctly on the other hand.

The finite element model of the bridge deck is depicted in Figure 26.


Figure 26: Finite element model of the deck

A modelling approach suggested by Wilson and Gravelle in [93] was utilised. This comprisesintroducing a single central spine of linear elastic beam elements that has the actual bending andtorsional stiffness of the deck. These stiffnesses were evaluated by establishing an equivalentsteel cross section, thus taking into account the concrete contribution. The cross section of thedeck is not uniform along the bridge which was taken into account while setting up the deckspine elements. The torsional stiffness is the most difficult one to evaluate. The deck will haveboth pure and warping torsional stiffnesses. As an approximation, the deck cross section (shownin Figure 23) was considered to be a thin-walled open C-shaped section.

Each of the spine beam elements has a length of approximately 12m, spanning from one cableanchor location to the next. At these nodes two rigid links were placed on either sideperpendicular to the spine to attach the cable elements, thus achieving the proper offset of thecables with respect to the centre of inertia of the spine. Using the rigid link option allows thedisplacements of the "slave" node to be expressed in terms of the displacements of the "master"node, thus not introducing additional degrees of freedom into the model.

Since the spine beam does not allow for the torsional inertia effects of the real bridge additionallumped masses were attached on either side of the central spine. By placing these below thelevel of the spine the difference between the centre of stiffness and the centre of mass can beaccounted for. This produces coupling between the torsional and the transverse motions of thedeck.

The modelling approach for the deck is shown in Figure 27.

z

Imx

M

Truss element

Lumped mass(position changes depending

on section properties)

Rigid links

YSpine

Figure 27: Theoretic modelling of the deck


4.2.2 The cables

As has already been mentioned in 2.2 and 2.3.1 the modelling of the cables is a difficult issuebecause nonlinearities arise from the cable sag. The stiffness therefore changes with the appliedload. However, in this study one linear truss element without stiffness in tension was employedfor each of the cables as shown in Figure 28. Taking into account the cable sag and thusnonlinear cable behaviour by means of an equivalent stiffness would have been extremelytedious since every cable would have had to be associated with a different force-displacementrelationship because of the changing inclination and length of the cables. This would have alsoconsiderably increased the computation time as would have done utilising multi-elementdiscretisation because of introducing new degrees of freedom.

It should be mentioned that linear elastic elements have also been used by Wilson et al in [93].Even though the authors suggested that the error is not significant it is clear that this approachcan lead to considerable inaccuracies.

F

dFigure 28: Employed cable behaviour

A prestress was applied to all the cables in order to ensure small deformations of the deck whenthe self weight is applied. The bridge was modelled picking up the geometry from the designdrawings. Since these show the as-built configuration the application of the self-weight to thestructure has to be taken into account. In reality the cables are prestressed according to a priorcalculation so that the final shape is correct. Accordingly, in the present study the prestress wasadjusted so as to have as small as possible a deflection when the self-weight is applied.

4.2.3 The pylons and piers

Modelling of the pylons and piers was by means of beam elements. As an example, the model ofpylon M3 is shown in Figure 29.

The piers were modelled with a single set of beam elements. The change in the cross-sectionalong the cone was taken into account. However, in preliminary studies the stiffness of thecantilever was found to be inaccurate and a calibration analysis using a refined model wasconducted as described in section 4.4.

The pylon legs have been connected to the piers using rigid links. This was done because thepier head was deemed to be extremely stiff in terms of relative rotations between the pier topand the base of the pylon. The pylon legs unify at the pylon head which was divided into beamelements as long as the distance between the cable anchorage points, thus readily allowing forthe connection of the cable elements.


Figure 29: Finite element model of a pylon

An important feature for the present study was the damping system connecting the deck and thepylon base. To accommodate these, rigid links were placed between the top of the pier andpoints at the level of the deck on either side of it. This enabled the damping devices only to actin the transverse direction because they have no component in longitudinal and vertical directionas can also be seen in Figure 29.

4.2.4 The foundations and abutments

The interaction between soil and structure was modelled using linear damped springs in thevertical direction and nonlinear undamped springs in the horizontal directions. Nonlinear springswere also applied for the bending rotational degrees of freedom.

As was already mentioned in 3.2.5, the Rion-Antirion Bridge is to be built on very weak soildeposits. Extensive investigations on the soil properties were conducted by the designers andequivalent spring stiffnesses, including the contribution of the piles, were available from these.To retain the advantage of having a model close to reality it was decided to make use of theseproperties. As will be explained subsequently, there was also concern that the nonlinearities inthe soil could play an important role in the behaviour of the bridge, thus making it necessary totake them into account. As an example, the characteristics of the nonlinear springs at pier M3are shown in Figure 30. The linear spring's properties are:

- vertical stiffness: kv=23⋅108 N/m2

- torsional stiffness: kt=51⋅108 Nm/rad.


Pier M3, Force-Displ. x,y

0

100

200

300

400

500

600

0.00 0.20 0.40 0.60 0.80 1.00Displacement [m]

Fo

rce

[MN

]

Pier M3, Moment-Rot. xx, yy

0

5000

10000

15000

20000

0.000 0.002 0.004 0.006 0.008 0.010Rotation [rad]

Mo

men

t [M

Nm

]

Figure 30: Characteristics of nonlinear soil springs at pier M3

The modelling approach employed for the base of the piers is depicted in Figure 31. Asexplained, nonlinear springs with the behaviour adopted from design documents were applied inhorizontal direction. These were identical in both directions at each pier, but different from pierto pier. In vertical direction linear springs were applied. The rotational springs, not shown inFigure 31, have nonlinear behaviour and are the same for the two bending and different for thetorsional degree of freedom.

y-acc

lgr.

x-acclgr.

z-ac

clgr

.

F

d

F

d

F

d

Rotational springsnot shown here

Pier

Figure 31: Modelling of the foundation

To the end point of the translational springs the accelerograms were applied, thus exciting thestructure via the ground springs.


4.3 Accelerograms

Synthetic accelerograms have been used in the present study that were derived from actualearthquake records using a special procedure to fit them to a predefined spectrum. These spectrathat are referred to as KME spectra have been agreed upon with the client, the Greekgovernment.

The spectrum compatible accelerograms have been derived by the designers and are for thepurpose of structural analysis. The real earthquake records that have been used as basis are listedin Table 4.

No Reference/name PGD[m]

PGV[m/s]

PGA[g]

A/V

1 San Fernando 0.724 0.960 0.479 0.502 San Fernando 0.881 0.981 0.516 0.533 Borrego 0.588 0.859 0.522 0.614 Lower California 0.701 0.978 0.586 0.605 Long Beach 0.873 1.003 0.497 0.506 Kern County 1.020 0.777 0.567 0.737 Kern County 0.900 0.860 0.600 0.708 Kern County 0.778 0.883 0.554 0.639 Kern County 0.952 0.904 0.565 0.6310 Hyogo-ken Nanbu 0.804 1.064 0.554 0.52

Table 4: Basis earthquake records for KME spectrum

The last column shows the A/V ratio. This is a good measurement for the demand of the record.Records with low A/V ratios (smaller than unity), as have all of the ones used here, tend to havehigher spectral accelerations in the long period range. Hence, the records can be expected toimpose high demand on a long period structure like a cable-stayed bridge.

The records obtained were scaled so as to represent a return period of 2000 years, thus imposinga very high demand on the structure. This means that a design earthquake was considered whichis only appropriate for a very important structure.

The accelerograms and their response spectra are shown in Figure 32 through Figure 37. It isapparent from the response spectra that the records impose a high demand on a long periodstructure.


Accelerogram h1 (longitudinal direction)

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0 20 40 60 80 100

Time [s]

Acc

eler

atio

n [

m/s

2]

Figure 32: Accelerogram, longitudinal direction (PGA=0.44g)

Response spectrum, h1 (longitudinal dir.)

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

0 1 2 3 4 5 6 7 8 9 10Period [s]

Sp

ectr

al a

cc. [

m/s

2]

Figure 33: Response spectrum, longitudinal direction

Accelerogram h2 (transverse direction)

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0 20 40 60 80 100

Time [s]

Acc

eler

atio

n [

m/s

2]

Figure 34: Accelerogram, transverse direction (PGA=0.37g)


Response spectrum, h2 (transverse dir.)

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 1 2 3 4 5 6 7 8 9 10Period [s]

Sp

ectr

al a

cc.

[m/s

2]

Figure 35: Response spectrum, transverse direction

Accelerogram v (vertical direction)

-4.00-3.00-2.00-1.000.001.002.003.004.00

0 10 20 30 40 50 60

Time [s]

Acc

eler

atio

n [m

/s2]

Figure 36: Accelerogram, vertical direction (PGA=0.28g)

Response spectrum, v (vertical dir.)

0.00

0.20

0.40

0.60

0.80

1.00

0 1 2 3 4 5 6 7 8 9 10Period [s]

Sp

ectr

al a

cc.

[m/s

2]

Figure 37: Response spectrum, vertical direction


4.3.1 Structural damping

A structural damping of 3% applied as Rayleigh damping has been used for all analyses. Thiswas chosen according to many earlier investigations as discussed in section 2.3.2. A deck modeand a tower mode with high mass participation have been used to calculate the Rayleighcoefficients.

4.4 Calibration investigations on the piers

Calibration investigations for the simplified beam model of the piers were conducted in order toensure stiffness conditions compatible with reality. It was this deemed necessary because of theexceptional pipe-type cross-section of the piers with diameters of more than 30m. Such astructure can be expected to show considerable shear effects and therefore a refined model wasset up in order to assess the influence of these.

Figure 38 shows the model which was established using shell elements. On this static analyseswere performed applying a unit force and moment to the top of the pier.

Figure 38: Refined model of the pier

The displaced shape under a unit load at the top of the pier is shown in Figure 39. Thedisplacements resulting were compared with those arrived at by using pure bending theory beamelements with the correct cross-section. As was expected, the refined model showed largerdeflections. These could be reached with the simplified model by using shear modelling andcalibration was done by calculating the correct shear area. This enabled modelling the actualshear behaviour in a simplified fashion. The shear area found to be adequate was by a factor of2.3 larger than the actual cross-sectional area which suggests that the structure undergoesconsiderable distortion.


Figure 39: Deformation under unit load

Characteristics of the Rion-Antirion Bridge Page 44

5 CHARACTERISTICS OF THE RION-ANTIRION BRIDGE

5.1 Static characteristics - special considerations

5.1.1 Relative displacements

The Rion-Antirion Bridge is crossing the Gulf of Corinth. Being seismically active it is atectonic zone undergoing extension. The degree of extension was investigated by Ambraseys etal in [13]. It is obvious that the extension of the Gulf leads to relative displacements applied to abridge crossing it.

In the context of this report it was deemed particularly important to know about the transversedisplacements that the deck undergoes in the case of relative displacements between the piers.These strongly depend on the stiffness distribution within the bridge and the boundaryconditions, meaning the way in which the deck is restrained.

Two cases were considered here: a strike-slip fault located in the middle span causing transverserelative displacements between the piers and, for completeness, a normal fault causinglongitudinal extension of the bridge.

The deformed shape of the bridge for transverse displacements in the fault of 40 cm is shown inFigure 40.

Figure 40: Deformation in plan under relative transverse displacements,full and half system

The analysis was done for a deck freely movable in longitudinal as well as in transversedirection. This is realistic, since a rigid shear connection is not planned for the bridge in ordernot to impose additional forces on the piers in the case of tectonic movements. It is apparentfrom the figure, that the largest relative displacements between deck and pylons occur at thetowers adjacent to the mid-span. These are about one third of the absolute relative displacementin the fault and should be considered in further investigations on bridge deck deformations asreported upon in section 6.

Also, the response of the bridge to an extension of the strait was investigated. A relativedisplacement of 2m between the two middle towers causes a deformation as shown in Figure 41.The bridge deck is lifted upwards in this case.


Figure 41: Deformation in elavation under relative longitudinal displacements,full and half system

The maximum vertical displacement of the deck occurs at mid-span. It has a value of almost 2.5times the absolute displacement between the towers. This shows that in the case of tectonicmovements adjustments in the cable tensions could be necessary.

5.1.2 Static push-over analyses on the pier/pylon system

Static push-over analyses on the bridge pylons were performed by a private consultancy andreported in [23]. These are given here because they will later be utilised to investigate the bridgetower's behaviour.

Analyses were performed for forces pushing at heights of 78 and 92m above the pier head intransverse and diagonal direction, the results being shown in Figure 43. The displacements inthis diagram are relative displacements between the point of pushing and the pier top excludingthe displacements caused by the rotations at the pier top. Referring to Figure 42 these aredisplacements d5.

Figure 42: Definitions for local displacements within the pylon


Push-over analyses at 92m and 78m

0

20

40

60

80

100

120

0.0 0.2 0.4 0.6 0.8 1.0 1.2Deflection [m]

Fo

rce

[MN

]92m: transverse

92m: diagonal

78m: transverse

78m: diagonal

Figure 43: Nonlinear force-displacement relationship for a bridge pylon,obtained from static push-over analysis

The results of the push-over analyses shall not be discussed herein. Figure 44 gives anexplanation of the significant events which induce changes in stiffness.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Deflection (m)

0

20,000

40,000

60,000

80,000

100,000

120,000

For

ce (

kN)

0

4,000

8,000

12,000

16,000

20,000

24,000

Force (kip

s)

0 8 16 24 32 40Deflection (in)

Force-Deflection Curves

92m : Transverse

78m: Transverse

axial tension cracking at top

axial tension cracking along entire leg

hinge: tension leg (top)

hinge: tension leg (bottom)

steel strain = 0.06: tension leg (top)

hinge:tension leg(top)ax = 45%

hinge:compressionleg (top)ax = 49%

hinge:tension leg(bottom)ax = 49%

steel strain = 0.06tension leg (top)ax = 50%

ax = 76%

ax = 68%

ax = 65%

ax = 62%

ax = 57%

Figure 44: Explanation of events in push-over analysis


It should be mentioned here that the results of the analysis with the pushing force at 92m havebeen used for the subsequent investigations. Hence, relative displacements within the pylon aremeasured with respect to the 92m point.

5.2 Dynamic characteristics - modal analyses

Modal analyses have been conducted on the bridge model in order to investigate the basicdynamic characteristics of the bridge. Different combinations of boundary conditions for thedeck have been considered as described in Table 5. The piers have been fixed at the base in allcases.

model x-direction (longitudinal) y-direction (transverse)1 free free2 free fixed3 fixed free4 fixed fixed

Table 5: Boundary conditions for the deck considered in modal analyses

Table 6 shows the first 20 modes for the model 2 which has a deck freely movable inlongitudinal and fixed in transverse direction. The modal mass participation is given as well as adescription of the nature of the modes. The fundamental mode with period 7.5s is a verticalmode as depicted in Figure 45. The first modes are all deck modes of various shapes. These canbe grouped into vertical, transverse and torsional modes, the first ones of each being shown inFigure 45, Figure 46 and Figure 47 respectively.

ModeNo.

T Modalmass x

Modalmass y

Modalmass z

Nature Dir.

[s] [%] [%] [%]1 7.53 9.84 0.00 0.00 1. vert. x2 6.51 0.00 0.00 0.00 2. vert. z3 5.99 0.00 0.14 0.00 1. trans. y4 5.25 0.08 0.00 0.00 3. vert. x5 4.77 0.00 0.00 0.00 2. trans. y6 4.41 0.00 0.00 0.04 4. vert. z7 3.99 0.00 11.92 0.00 3. trans. y8 3.43 15.64 0.00 0.00 5. vert. x9 2.76 0.00 0.00 5.50 6. vert. z10 2.61 0.01 0.00 0.00 7. vert. x11 2.48 0.00 0.04 0.00 1. tors. y12 2.48 0.00 0.00 0.72 8. vert. z13 2.44 0.00 2.89 0.00 2. tors. y14 2.38 0.00 0.00 0.00 3. tors. y15 2.37 0.00 0.00 0.00 4. tors. y16 2.35 0.01 0.00 0.00 9. vert. x17 2.34 0.00 1.84 0.00 5. tors. y18 2.33 0.00 0.00 0.00 6. tors. y19 2.32 0.00 0.87 0.00 7. tors. y20 2.21 0.00 0.00 0.00 8. tors. y

Table 6: First 20 modes for model 2 (free in x-dir., fixed in y-dir.)


Figure 45: Fundamental mode, model 2

Figure 46: Mode 3, model 2

Figure 47: Mode 11, model 2

The distribution of the modal periods is shown in Figure 48. Only the first 8 modes have periodsabove 3s. Below 3s many closely spaced modes follow.

Period distribution, model 2

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 100 200 300 400 500 600Mode number

Per

iod

[s]

Figure 48: Distribution of modal periods, model 2 (free in x-dir., fixed in y-dir.)

Because the deck has only a small mass compared to the whole structure (cf. Table 3) the modalmasses of the deck modes are relatively small. This particularly applies to the higher deckmodes. The accumulated modal mass in longitudinal direction is shown in Figure 50. In this thetower modes which contribute significantly can be located clearly. The first tower mode is mode261. The tower mode with the highest modal mass participation of 8.2% in longitudinaldirection is depicted in Figure 49.


Figure 49: Mode 313, model 2, tower M4

These results show clearly that the contribution of the higher modes is important and needs to betaken into account. However, also first modes contribute considerably, causing susceptibility tolong period excitation.

Modal mass participation, model 2

0.0010.00

20.0030.00

40.0050.00

60.0070.00

80.0090.00

100.00

0 100 200 300 400 500 600Mode number

Acc

um

ula

ted

mas

s [%

]

Figure 50: Accumulated modal mass participation, model 2 (free in x-dir., fixedin y-dir.)

In order to compare the behaviour of the different models analysed, the period distributions ofthese are shown jointly in Figure 51.


Period distributions, all models

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Mode No.

Per

iod

[s]

Model1

Model2

Model3

Model4

Figure 51: Distribution of modal periods, all models

For models 1 and 3 which have a free deck in transverse direction, the fundamental mode is atransverse sway mode of period 18.6 s. Given that the period of a pendulum is (G. Galilei)

gl

Tpendulum π2= (2)

the fundamental period of the bridge can be easily verified, knowing that the vertical distancebetween pylon top and deck is 95 m:

s6.19m95

2 ==g

Tpendulum π . (3)

The actual deck is just a bit stiffer than an equivalent pendulum system.

It is apparent from Figure 51 that the first modes are mainly affected by the movability of thedeck in transverse direction. Fixing the deck in longitudinal direction shifts the perioddistribution downwards, but only by a small amount.

Earthquake Response and its Control Page 51

6 EARTHQUAKE RESPONSE AND ITS CONTROL

6.1 Introduction

To evaluate the earthquake response characteristics of a cable-stayed bridge is a complex issue.In the present study investigations have been focussed on the transverse behaviour of the Rion-Antirion bridge.

The aim was to firstly understand the influence of the bridge deck and its connection to thepiers. Two basic cases were studied at first:

- deck freely movable,- deck rigidly connected to the piers.

To be able to comment on the acceptability of a bridge response, a framework of design criteriawas developed in the next step.

Within the framework established investigations on possible isolation devices for the bridgedeck have been conducted. The objective was to better understand the impact that devices withdifferent properties have on the bridge response to find a realistic solution.

6.2 Investigations on basic systems

6.2.1 Introduction

In the context of later evaluating possible isolation devices for the bridge deck, investigationswere started by considering two basic cases: a deck completely free in transverse direction and adeck rigidly connected to the pier top (shear key). This allowed for a better understanding of thebehaviour characteristics and critical parameters.

Also, the influence of the components of the earthquake motion was studied by applying thecomponents described in section 4.3 in different combinations.

6.2.2 Modelling assumptions

The finite element model introduced in section 4 was used. The following conditions apply to allanalyses conducted:

- dynamic time-history analysis, Newmark implicit integration scheme (δ=0.5, α=0.25),- time steps ∆t=0.02s, time domain t=82s,- geometric nonlinearity considered,- nonlinear soil springs as described in 4.2.4,


- deck free in x-direction (as in actual bridge),- two models considered in terms of y-connection of the deck as shown in Table 7.

Time steps of 0.02s enable capturing of modes up to a period of 0.126s which corresponds tomodes in the order of 440. Hence, the modal sum of the modal masses properly represented bythe analyses is approximately 80%. Higher values could only be attained by using considablyshorter time stepping and thus increasing the computation time. Inaccuracies may result butthese will be confined to the tower response.

model y (transverse)-direction1 free2 fixed

Table 7: Boundary conditions for the deck

Model 1 was analysed with combinations of ground motion components as shown in Table 8while model 2 was only analysed with all components acting.

case x-component y-component z-component1 x2 x x3 x x4 x x x

Table 8: Combinations of ground motion components considered for model 1

6.2.3 Results

From the analyses the maxima in the time domain were obtained for certain parameters. For thecase with all ground motion components acting some of these are listed in Table 9. Alldisplacements regarding pier behaviour were monitored at tower M3. This tower was chosenbecause it has the longest piers as well as the weakest soil and is therefore supposed to deliverupper bound solutions.

Dir. Model 1(free deck)

Model 2(fixed deck)

Rel. difference[%]

Deck at M3 - ground x 0.57 0.57 0Deck at M3 - ground y 0.81 0.47 72Deck in field 3 - ground x 0.57 0.57 0Deck in field 3 - ground y 0.93 1.66 78d5, rel. dspl. in pylon,(cf. Figure 42)

y 0.44 0.33 33

Pylon top M3 - ground x 0.52 0.52 0Pylon top M3 - ground y 2.41 1.90 27Soil deformation x 0.40 0.40 0Soil deformation y 0.50 0.30 67Pier M3 - deck M3 x 0.63 0.62 2Pier M3 - deck M3 y 0.95 0.00 inf.

Table 9: Max. displacements [m] for models 1 and 2


An important value is the maximum relative displacement between the deck and the pier whichis 95cm for a free deck (cf. the plot in Figure 52). However, it can easily be seen from Table 9that fixing the deck in transverse direction does not only affect the relative displacementsbetween deck and pier. The absolute displacements in the mid-span are decreased by almost80% and even the displacements of the pylon top are decreased.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0. 10. 20. 30. 40. 50. 60. 70. 80. 90.

Figure 52: Transverse displacements [m] of the deck relative to the pier M3,model 1 (free deck)

For the case of a rigid connection the maximum force applied on the pier by the deck has beenfound to be 63.7MN (cf. the plot in Figure 53). Knowing this, it is most interesting to note thatthe transverse deformation in the soil springs is by almost 70% smaller for the rigid deck model.Although a very high transverse force is applied to the pier top, the displacements in the soil aredecreased. This points out, that by fixing the deck the whole structure is changed in its dynamiccharacteristics.

0. 10. 20. 30. 40. 50. 60. 70. 80. 90.

*1

07

-8.

-6.

-4.

-2.

0.

2.

4.

6.

Figure 53: Force [N] in connection deck-pier, model 2 (fixed deck)

The relationship between the force applied to the pier and the displacement of the deck relativeto the pier obtained herein is depicted in Figure 54 and will be an important framework for thefurther considerations in section 6.5.


No connection / rigid connection

0.00

0.20

0.40

0.60

0.80

1.00

0 20 40 60 80Max. force in connection deck-pier [MN]

Max

. dsp

l. d

eck-

pie

r [m

]

Figure 54: Relationship force-displacement for junction deck-pier

A further important result concerns the deformation of the pylon. The relative displacementwithin the pylon is given in Table 9 as d5, which is the displacement excluding the influence ofthe rotation at the pier base as depicted in Figure 42. The deformation in the pylon is 25% largerfor the case of a free deck. This imposes as higher demand on the pylon (cf. the results of push-over analyses reported in 5.1.2).

The influence of the three earthquake motion components has been investigated on the modelwith the free deck (model 1). The aim was to find out whether performing analyses with all 3earthquake components can be regarded as being on the safe side which is important for furtherstudies and general design considerations.

Table 10 shows results of maximum displacements obtained for analyses with variouscombinations of ground motion components. In this "=0" means that the respective componentis not present and vice versa. The y-component was present in all cases since transversedisplacements were of main interest.

x=0 x=1z=0 0.57 0.76y-dspl. deck-ground in span 2z=1 0.57 0.75z=0 0.00 0.57x-dspl. deck-ground at pier M3z=1 0.00 0.57z=0 0.66 0.80y-dspl. deck-ground at pier M3z=1 0.67 0.81z=0 0.61 0.96y-dspl. deck-ground in span 3z=1 0.59 0.93z=0 2.40 2.41y-dspl. pylon top M3 - goundz=1 2.40 2.41z=0 0.00 0.40x-dspl. in soil at pier M3z=1 0.00 0.40z=0 0.50 0.50y-dspl. in soil at pier M3z=1 0.50 0.50

Table 10: Displacements [m] for various combinations of earthquakecomponents, y-component is always present, model 1 (free deck)


Firstly, the vertical (z-) component appears to be of minor importance thus suggesting to benegligible as far as transverse displacements are concerned. However, it should be noted thatnonlinear cable effects arising from the cable sag have not been taken into account. The stiffnessof the cable is unaltered until the strain reaches zero. In reality the effect of cable sag wouldalready reduce the stiffness of the cables for tensile strains thus probably leading to highertransverse displacements in the case of an upwards moving deck. Hence, the results obtainedherein should be used with caution.

The longitudinal (x-) component seems to have a much larger impact on the transversedisplacements. While the tower behaviour remains almost unchanged, the deck displacementsare increased by up to 50%. This can be partly attributed to geometric nonlinearities. However,investigations on the cable behaviour have shown that several exterior cables of the respectivecable groups reach zero tensile strains for the peak longitudinal displacements. For longitudinaldisplacement in the opposite direction cables on the other side of the tower go slag. This, ofcourse, significantly decreases the transverse stiffness of the cable groups and apparently givesrise to high transverse displacements.

Concluding, the use of all earthquake components leads to upper bound solutions and was thusemployed in all analyses discussed subsequently.

6.3 Design considerations and performance criteria

6.3.1 Introduction

In the preceding section results have been presented for two basic systems in terms ofdisplacements and forces. It is of course inevitable to derive criteria for the assessment of theseresults. Whether deck displacements of 95cm are acceptable needs to be discussed as well aswhat forces can be applied to the towers by deck restrainers. These issues will be looked intosubsequently. The aim is to present a set of criteria that could also be applicable to other bridgestructures.

6.3.2 Serviceability conditions

Usually, the dynamic displacements under frequent loading such as wind and traffic should belimited to very small values. Under serviceability conditions large displacements cannot betolerated because they can jeopardise the traffic safety and speed. Psychological effects onpeople play an important role in this issue.

For the case of the Rion-Antirion it was decided to seek a solution which provides a fixed deckunder serviceability conditions, thus avoiding any transverse displacements.

6.3.3 Slow tectonic movements

Tectonic movements can cause relative displacements between certain parts of the structure.This has already been discussed for the case of the Rion-Antirion Bridge in section 5.1.1.


If the deck is fixed in the transverse direction at the piers, relative transverse displacementsbetween piers cause permanent forces on the deck as well as on the towers. This favours deckconnection solutions that can accommodate certain relative displacements as described in 6.4.

6.3.4 Earthquake conditions

6.3.4.1 Displacement limits

Pounding of the deck against the towers

Since the pylons are fragile members that are designed to resist high forces, pounding of thedeck against the pylon legs should be avoided under any circumstances. If the deck is smashedagainst the pylon leg, extremely high impact forces act on these. Since pylon legs are usuallyhollow sections designed for bending and axial forces, local buckling can occur and jeopardisethe overall stability of the pylons.

In the case of the Rion-Antirion Bridge the clear distance between the deck and the pylon legs isabout 80 cm on either side. This should, however, not be the design limit for deck displacementssince the space can be increased in the vicinity of the pylons.

Deformations in the soil

Under earthquake conditions high deformations in the underlying soil can occur. For structureson particularly weak soil there will be concern as to how large the deformations will be duringan earthquake because the degree of plastisation affects the post-earthquake performance of thesoil bed.

The soil investigations for the Rion-Antirion Bridge project have not yet been finalised. Limitscan therefore not be defined. However, displacements in excess of approximately 40cm could becritical.

6.3.4.2 Force limits

Forces in the pylons

In section 5.1.2 static push-over analyses on the pylons of the Rion-Antirion Bridge werepresented. It is clear from these that the deformation capacity of the pylons is limited and thatyielding in the pylons is an important design factor.

As an approximation, the results of the static push-over analyses can be used to comment on thedisplacements obtained from dynamic linear-elastic analyses. There is of course a discrepancybetween the displacements because the linear-elastic analysis neglects changes in stiffness, butas a rough approximation this still enables to take nonlinearities into account.

For the dynamic analyses a wall thickness of 60cm was used for the pylon leg box section.Figure 44 shows, that for such a system first cracking occurs at a relative transverse displace-ment of 30cm, first hinging occurs at 38cm. These are upper bound values while conditions aremuch more unfavourable for diagonal loading. This rises the question of the interaction of the x-and y-components of the pylon deformation. For the case of the free deck which has in 6.2.3been identified as most unfavourable, these components are plotted in time in Figure 55.


Although the x-deformations are smaller, the peaks of these can occur at the same time as forthe y-direction and thus increase the danger for the pylon. The closer the resultant is to an angleof 45° the more unfavourable is the situation. This is because the diamond-shaped pylons haveless ductility and deformation capacity in the diagonal direction. This should be taken intoaccount when interpreting the results.

Comparison of components of d5 (free deck)

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.100.20

0.30

0.40

0 10 20 30 40

Time [s]

Dis

pla

cem

ent

[m]

d5-y

d5-x

Figure 55: Free deck model, x- and y-components of d5 (dspl. in the pylonexcluding rotation)

Forces applied by the deck

If the deck is somehow connected to the piers, forces will be applied to the towers by the deck.These depend on the type of connection and the upper bound of the force has been evaluated insection 6.2.3 as 64MN.

It is obvious that the force that is applied to the towers by the deck should be kept at small aspossible. High interaction forces could not only impose a higher structural demand on the piersbut also increase the demand on the foundation. However, the piers themselves are usuallystrong members and the maximum acceptable force may depend on other considerations. Forexample, if damping devices are used between deck and pier, the force might have to be limitedfor stability reasons of the pistons.

6.3.4.3 Acceleration limits - sliding of cars

Even during an exceptional event like an earthquake, the safety of the cars passing the bridgeshould be ensured. This not only means to avoid the collapse of the structure but also to makesure that cars are not driven down from the bridge. Usually bridges are equipped with crashbarriers that can also serve in the case of an earthquake. However, it is interesting to know ifthere is actually the possibility of the cars starting to slide on the road. Therefore, a method toinvestigate this has been developed here and is discussed subsequently. It should be mentioned,however, that the implications of this study are not at all straightforward and that furtherresearch would be necessary to evaluate the safety of the cars in the case of an earthquake.Herein it has neither been inspected for how long and far the cars are sliding and when thewheels will get grip again, nor what corrective actions could be employed by the driver duringsuccessive sliding events.


The situation of a car moving on a displaced bridge deck is depicted in Figure 56. Rotations ofthe deck about the x-axis are taken into account as well as transverse and vertical accelerationsof the deck.

xy

z

G

N

ah

av

R

GN

H1

H

α

Figure 56: Conditions for sliding considerations

The force that has to be transmitted via friction between the street and the tyre is composed ofthe component of the weight in the direction of the deck and the component of the horizontalinertia force in the direction of the deck:

cos1 αhMaHH += (4)

where 1cos ≈α for small angles and

αα tansin1 eqeq GGH ≈= (5)

and xθα −=tan (6)

because positive rotations about the x-axis mean inclinations in negative y-direction.

In these Geq is an equivalent weight taking the influence of the vertical acceleration of the deckinto account. Because G is reduced for a deck moving downwards, Geq is given as:

)( veq agMG += (7)

where M mass of the car,av vertical acceleration of the deck,ah horizontal acceleration of the deck andG=Mg weight of the car.

This yields

hxv MaagMH ++−= θ)( . (8)


The friction resistance against sliding can be written as a simple Coloumb-type relationship:

NcR = (9)

where N force normal to the deckc friction coefficient between tyre and street

with )(cos veqeq agMGGN +=≈= α . (10)

The limit state of sliding can thus be expressed as:

xvhv agacag θ)()( +−=+ . (11)

To calculate the safety against sliding the following expression can be employed:

xvh

vs aga

cagθ

γ)(

)(+−

+= . (12)

The value γs and its development in time can give an impression about the likelihood of carssliding on the bridge deck.

The friction coefficient c can be up to 1.0 for optimum conditions. For a wet road surface typicalvalues are 0.6-0.7. A value of 0.6 has been employed in this study.

Analyses on the value γs have been carried out for the basic systems discussed in section 6.2.The time-histories of this parameter have been monitored in the middle of spans 2 and 3 and atthe tower M3. The absolute minima obtained are given in Table 11.

free deck,x,y,z components

free deck,x,y components

(no vertical motion)

fixed deck,x,y,z components

0.95 1.01 0.21

Table 11: Minimum values of sliding-safety (span 2, tower M3, span 3considered), basic systems; input earthquake components used areshown

Apparently the accelerations in the case of a fixed deck are such that sliding is much more likelyto occur. A free deck seems to be favourable. Also the vertical (z-) component seems has a smallunfavourable influence. The safety is increased if no vertical motion is present.

Further discussion of the results can be found in section 6.5.


6.4 Devices for deck connection

6.4.1 Fuse device

Fuse devices are used to restrict displacements up to a certain force acting. They can be used tosuppress the displacements under serviceability conditions and to limit the forces during anearthquake. The general behaviour is depicted in Figure 57.

The fuse acts as a shear key under loads that correspond to wind and traffic impact. If forcesexceed a certain limit, the fuse device is broken and gives the deck free. It can be combined withother devices so as to hand the earthquake behaviour law of the connection over to anothermechanism like a damping device.

d

Ffuse rupture point

Figure 57: Fuse device behaviour

6.4.2 Shock transmitter

A shock transmitter is used if very slow movements within the structure shall not cause anyforces. It can thus be successfully applied if slow tectonic movements need to be accommodatedby the bridge.

The behaviour of a shock transmitter is similar to a damping device in the sense that the force inthe device depends on the velocity. While extremely slow movements do not cause any forces,faster movements like wind and traffic impact are responded to by suppressing thedisplacements almost rigidly.

6.4.3 Hydraulic dampers

Hydraulic dampers, filled with oil, silicone or a mixture of these have the following behaviourlaw:

αvCF ⋅= . (13)


Hence, the force output depends on the velocity. Figure 58 shows the influence of the parameterα on the behaviour.

Constitutive laws for dampers - F =3MN*v αα

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6Velocity v [m/s]

Fo

rce

F [

MN

]

alpha=0.1

alpha=0.2

alpha=0.3

alpha=0.4

Figure 58: Hydraulic damping device behaviour

Although hydraulic dampers do not have a maximum reaction force, the increase in forcebecomes very small for high velocities. This is more pronounced for small values of α as can beseen in Figure 58. Hence, there is actually a limitation on the force that the damping device canreact with.

The energy absorbed by the hydraulic device is mainly released as heat. There is always alimiting temperature up to which the correct behaviour is ensured. It is thus important to knowabout the energy that is dissipated in the device. However, it is difficult to calculate thetemperature thus caused.

Prototype tests on the devices to be used are necessary in order to:

- confirm the constitutive law and energy dissipation,- verify symmetry in compression and tension,- verify leakage (fatigue),- prove consistency of response between two units.

For devices which could be used for the bridge discussed herein full-scale dynamic testing is notpossible. Usually tests are then performed for the maximum stroke and reduced speed on onehand and for maximum force and speed at a reduced stroke on the other hand.

6.4.4 Elasto-plastic isolators

These devices follow a simple elasto-plastic behaviour law which is depicted in Figure 59.


d

F

device yield point

Figure 59: Behaviour of elasto-plastic isolator

Elasto-plastic isolators show linear elastic response up to a yield point. The plastic behaviourcan be either ideally plastic or hardening. These dampers can be made of steel as investigated inthis report and the yield point is easily adjusted by designing the device accordingly. If steelwith a hardening branch is used the hysteresis loops are arrived at by using the constitutive lawfrom Figure 59 and kinematic hardening. Reduction of energy dissipation capacity due toBauschinger effect may be considered.

6.5 Parametric studies on different deck isolation devices

6.5.1 Introduction

The studies in the previous sections have indicated that there is room for considerationsregarding possible damping devices for the deck of the Rion-Antirion Bridge. For a free deckdisplacements during an earthquake are rather large (0.95m) so that pounding could occur.Particularly if tectonic movements preceded, these add to the dynamic displacements.Transverse displacements can thus considerably exceed 1m. Also, deformations in the soil arelarge for a free deck (0.50m). Perhaps most importantly, the deformation demand on the pylonsis very large in the case of a free deck so that yielding can occur. On the other hand, a fullyrestrained deck applies high forces to the towers (64MN). Also, the safety of cars against slidingon the surface is reduced.

The aim of the studies was to investigate the influence of different isolation devices on theresponse of the bridge and to shed light on the question which device properties are of majorinfluence and whether there is an optimum solution.

6.5.2 Analysis assumptions

Hydraulic dampers were considered as well as elasto-plastic devices. The cases that have beenanalysed are compiled in Table 12 for hydraulic and in Table 13 for steel dampers. The forces Cand Fy are given as absolute forces per pier. These can in reality of course be distributed overseveral dampers.


Model No C per pier[MN]

α[-]

Hydr#1 3.0 0.2Hydr#2 6.0 0.1Hydr#3 6.0 0.2Hydr#4 6.0 0.3Hydr#5 6.0 0.4Hydr#6 12.0 0.1Hydr#7 12.0 0.2Hydr#8 12.0 0.3Hydr#9 12.0 0.4Hydr#10 18.0 0.2Hydr#11 18.0 0.4Hydr#12 24.0 0.2Hydr#13 24.0 0.4

Table 12: Analysed hydraulic damping systems

Model No Fy per pier[MN]

Eel

[N/mm2]Epl

Stl#1 10.0 210,000 0.05Eel

Stl#2 15.0 52,500 0.05Eel

Stl#3 15.0 105,000 0.05Eel

Stl#4 15.0 210,000 0.05Eel

Stl#5 20.0 52,500 0.05Eel

Stl#6 20.0 52,500 0Stl#7 20.0 210,000 0.05Eel

Stl#8 20.0 210,000 0Stl#9 30.0 210,000 0

Table 13: Analysed elasto-plastic isolation systems

Further modelling assumptions are similar to the analyses reported in 6.2.2:

- dynamic time-history analysis, Newmark implicit integration scheme (δ=0.5, α=0.25),- time steps ∆t=0.02s, time domain t=82s,- geometric nonlinearity considered,- nonlinear soil springs as described in 4.2.4,- deck free in x-direction (as in actual bridge),- all 3 earthquake components were applied.

6.5.3 Results

Herein only the most important results obtained can be presented. For a comprehensivesummary of all the values, the reader is referred to the appendix of this report.


6.5.3.1 Hydraulic dampers

Table 14 shows the maximum values of certain displacements that were obtained in theanalyses. As can be seen from these, the relative displacements between deck and pier areconsiderably reduced as result of using damping devices. Also, as Figure 60 shows exemplary,the cyclic amplitudes are decreased. Change in sign does only occur once. There are residualdisplacements after the earthquake which diminish very slowly.

α=0.1 α=0.2 α=0.3 α=0.4Relative displacement deck-pier at M3 [m]C=3.0MN 0.75C=6.0MN 0.77 0.75 0.70 0.71C=12.0MN 0.77 0.60 0.70 0.66C=18.0MN 0.67 0.60C=24.0MN 0.60 0.53Absolute displacement pylon top M3 - ground [m]C=3.0MN 2.19C=6.0MN 2.02 2.04 2.05 2.20C=12.0MN 2.02 1.92 2.05 2.06C=18.0MN 2.00 2.01C=24.0MN 1.92 1.96Relative displacement in the pylon d5 (without rot.)C=3.0MN 0.31C=6.0MN 0.27 0.28 0.28 0.32C=12.0MN 0.28 0.31 0.28 0.28C=18.0MN 0.27 0.28C=24.0MN 0.28 0.30Relative displacement in the soil spring [m]C=3.0MN 0.45C=6.0MN 0.41 0.41 0.41 0.45C=12.0MN 0.41 0.40 0.41 0.41C=18.0MN 0.41 0.41C=24.0MN 0.40 0.41

Table 14: Maximum displacements [m] in time, hydraulic dampers

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 60.

Figure 60: Transverse displacements [m] of the deck relative to the pier M3,model 1 (hydraulic dampers #7, C=12MN, α=0.2)


Figure 61 shows the hysteresis loops that the damping device undergoes. There is, apparently,high energy dissipation in the device. Particularly the first cycles are very large. Since themaximum velocity is greater than 1m/s the forces in the damper exceed the value of C. For thecase below the maximum force is 13.1MN. The maximum forces in the device for the othercases are given in Table 15.

Hysteresis for the damping device, C=12 MN, alpha=0.2 (0-40s only)

-15

-10

-5

0

5

10

15

-0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20

Displacement [m]

Fo

rce

[MN

]

Figure 61: Hysteresis loops of the damping device (hydraulic dampers #7,C=12MN, α=0.2)

α=0.1 α=0.2 α=0.3 α=0.4C =3.0MN 3.5C =6.0MN 6.4 6.9 7.3 7.9C =12.0MN 12.9 13.1 14.6 15.3C =18.0MN 20.1 22.1C =24.0MN 26.3 28.0

Table 15: Maximum forces [MN] in the damping device, hydraulic dampers

This rises the question of the relationship between the displacements that the isolation devicesustains and the forces that are thus applied on the pier. These are depicted in Figure 62 for allinvestigated hydraulic damping devices. Clearly, there is a tendency towards smallerdisplacements for higher forces applied. However, the influence is surprisingly small. Increasingthe maximum force and therefore the demand on the piers by a factor of 8 leads to a reduction indisplacements by just 30%. Even more interestingly, there seem to be solutions that are moreeffective than others. For example, the isolator 12MNv0.1 reaches a max. displacement of 77cmat a force of 12.9MN while the 12MNv0.2 reduces the displacements to 60cm at an almost equal13.1MN.


Relationship max. force - max. displacement

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 5 10 15 20 25 30Max. force in the damper

Max

. rel

. dis

pla

cem

ent

dec

k -

pie

r

Figure 62: Relationships max. force [MN] - max. displacement [m] in theisolation device, hydraulic dampers

In Table 14 also the maximum values of d5, the relative transverse displacement in the pylonwithout the rotation contribution (cf. Figure 42) are given. These correspond to the demand onthe pylons and are thus of particular interest. Figure 63 shows the displacement response of thebridge pylon M3, giving the absolute displacement within the pylon as well as d5. Apparently,the rotation contributes significantly to the displacement response which can also be seen fromthe corresponding displaced shape for the maximum displacement in Figure 64. Also, the towerseems to mainly respond in its fundamental mode. Similar results have been obtained for othertimes as well as for all other cases studied.

Displacements in the pylon(hydraulic dampers, C=12MN, alpha=0.2)

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0 10 20 30 40 50 60

Time [s]

Dis

pla

cem

ents

[m]

Abs. Dspl. Pylon top-pylon base

Displacement d5 (without rot.)

Figure 63: Relative displacements [m] in the pylon M3, including andexcluding rotations (hydraulic dampers #7, C=12MN, α=0.2)


Displaced shape of tower at time of max d5

0

50

100

150

200

250

-0.50 0.00 0.50 1.00 1.50 2.00

Figure 64: Deformed shape of tower M3 at time of max d5 (displacement in thepylon without rotation), hydraul. dampers #7, C=12MN, α=0.2

The maximum values of d5 obtained for the certain isolation devices are rather similar and thedifferences seem not to be correlated to any particular parameter. Importantly, they are allsmaller than the displacements for the free deck as well as for the fixed deck system. This meansthat an essential improvement regarding the demand on the pylon can be achieved by employingisolation devices between deck and tower. However, for displacements of approximately 30cmslight cracking in the tension legs may occur according to the results given in section 5.1.2.

Displacements in the soil are hardly affected by the damping device used. Deformations areapproximately 41cm for most of the hydraulic devices.

The value of safety against sliding of cars γs as derived in 6.3.4.3 has been studied for 5 cases.The value has been monitored in the middle of the second and third span as well as at tower M3.The absolute minimum values are shown in Table 16 and exemplary the time-history of γs attower M3 is plotted in Figure 65.

α=0.1 α=0.2 α=0.3 α=0.4C =3.0MN 1.09C =6.0MN 0.78C =12.0MN 0.58C =18.0MN 0.61C =24.0MN 0.40

Table 16: Minimum values of sliding-safety (span 2, tower M3, span 3considered), hydraulic devices


0

1

2

5

4

3

10 20 30 40 50 60 70 80 90

Figure 65: Sliding safety factor of deck at tower M3 in time, hydraul.dampers #7, C=12MN, α=0.2

In all cases apart from the weakest damper the factor becomes smaller than unity and thereforeaccelerations are so that sliding can actually occur. It can be inferred from the table that strongerdampers cause an increase in the accelerations and therefore increase the danger of cars loosinggrip with the surface. The results also well correspond to the values obtained for the casesstudied in 6.2. For the case of a free deck the minimum safety is 0.95 and for a fixed deck thefactor is reduced to 0.21.

It should be noted, that for the hydraulic dampers all the minima obtained and given in Table 16correspond to the deck at the tower M3, the point where the dampers act.

As can be seen from the time-history in Figure 65, the peaks are very short. Hence, onlymomentary sliding occurs and corrective actions by the driver should be possible. This pointsout that the danger for the traffic is not as big as could be concluded from the numbers.

6.5.3.2 Elasto-plastic isolators

Elasto-plastic isolation devices have been studied as outlined in section 6.5.2. For the post-yieldbranch hardening has been used as well as perfectly plastic behaviour. Hardening, if employed,is kinematic.

A summary of displacement results is shown in Table 17.

Firstly, the displacements between bridge deck and pier, meaning the displacements in thedampers, seem to be well correlated to the yield strength and initial stiffness of the isolators.Higher yield forces as well as higher initial stiffnesses reduce the displacements. Also, isolatorswith a hardening behaviour show smaller displacements then corresponding perfectly plasticdevices.

A displacement time-history is exemplary shown in Figure 66. The corresponding force-displacement diagram is plotted in Figure 67. Obviously, the displacement response is quitedifferent from the one for hydraulic dampers (cf. Figure 60). The hysteresis loops clearly showthe behaviour law employed and energy dissipation is particular large for the first two cycles.


The residual displacements after the earthquake are also smaller than for the hydraulic device.

Fy=10MNEpl=0.05Eel

Fy=15MNEpl=0.05Eel

Fy=20MNEpl=0.05Eel

Fy=20MNEpl=0

Fy=30MNEpl=0

Relative displacement deck-pier at M3 [m]Eel=0.525⋅105 0.73 0.60 0.65Eel=1.05⋅105 0.66Eel=2.1⋅105 0.70 0.59 0.51 0.61 0.60Absolute displacement pylon top M3 - ground [m]Eel=0.525⋅105 2.12 2.13 2.10Eel=1.05⋅105 2.14Eel=2.1⋅105 2.20 2.14 2.13 2.07 2.10Relative displacement in the pylon d5 (without rot.)Eel=0.525⋅105 0.31 0.31 0.31Eel=1.05⋅105 0.31Eel=2.1⋅105 0.31 0.31 0.31 0.31 0.30Relative displacement in the soil spring [m]Eel=0.525⋅105 0.44 0.45 0.45Eel=1.05⋅105 0.44Eel=2.1⋅105 0.44 0.44 0.44 0.44 0.43

Table 17: Maximum displacements [m] in time, elasto-plastic isolators

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0 10 20 30 40 50 60 70 80

Figure 66: Transverse displacements [m] of the deck relative to the pier M3,elasto-plastic isolators #1, Fy=10MN, Eel=2.1E5 N/mm2, Epl=0.05Eel


Hysteresis for the damping device,Fy=10MN, Eel=2.1E5, Epl=0.05Eel (0-40s only)

-30

-20

-10

0

10

20

-0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40

Displacement [m]

Fo

rce

[MN

]

Figure 67: Hysteresis loops of the device, elasto-plastic dampers #1, Fy=10MN,Eel=2.1E5N/mm2, Epl=0.05Eel

The maximum forces in the isolators are given in Table 18. The relationship between themaximum force and the maximum displacement for all the dampers is depicted in Figure 68.Leaving the perfectly plastic isolators aside, the relationship is unambiguous. On the otherhand, the perfectly plastic devices can be favourable, as is obvious from the diagram.Particularly a very stiff non-hardening isolator considerably reduces the displacements whilehaving the same maximum force. It is this because of the higher energy dissipation capacitycompared to a hardening device with the same maximum force.

Fy=10MNEpl=0.05Eel

Fy=15MNEpl=0.05Eel

Fy=20MNEpl=0.05Eel

Fy=20MNEpl=0

Fy=30MNEpl=0

Eel=0.525⋅105 20.3 25.3 20.0Eel=1.05⋅105 24.6Eel=2.1⋅105 24.3 32.3 39.5 20.0 30.0

Table 18: Maximum forces [MN] in the damping device, elasto-plastic devices

Relationship max. force - max. displacement

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35 40 45

Force in damper [MN]

Max

. rel

. dis

pla

cem

ent d

eck

- pie

r

hardening

perf. plast.

Figure 68: Relationships max. force [MN] - max. displacement [m] in theisolation device, elasto-plastic isolators


In terms of deformations of the pylon the displacements d5 (relative displacement between pylontop and pylon base excluding rotational contribution) are given in Table 17. Figure 69exemplary shows a time-history of the d5 values. Again, it is apparent that the rotation at the pierhead strongly contributes to the displacements at the top as can also be seen from the deformedshape of the pylon shown in Figure 70.

Displacements in the pylon(steel dampers, Fy=10MN, Eel=2.1E5, Epl=0.05Eel)

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0 20 40 60 80

Time [s]

Dis

pla

cem

ents

[m] Abs. Dspl. Pylon top-pylon base

Displacement d5 (without rot.)

Figure 69: Relative displacements [m] in the pylon M3, including andexcluding rotations, elasto-plastic isolators#1, Fy=10MN,Eel=2.1E5N/mm2, Epl=0.05 Eel

Displaced shape of tower M3at time of max d5

0.00

50.00

100.00

150.00

200.00

250.00

-0.50 0.00 0.50 1.00 1.50 2.00 2.50

Figure 70: Deformed shape of tower M3 at time of max d5 (displacement in thepylon without rotation), elasto-plastic isolators#1, Fy=10MN,Eel=2.1E5N/mm2, Epl=0.05 Eel


For the isolation devices studied the deformations in the pylon are very uniform with a value of31cm. The selection of the device parameters is therefore not governed by this consideration.The displacements in the pylon are considerably reduced with respect to the case of a free deckand only slight cracking in the tension legs needs to be expected.

The deformations in the soil springs are larger than for the hydraulic dampers studied and thecase of a fixed deck. However, with approximately 44cm they are still smaller than for the caseof a free deck.

Also for the case of the elasto-plastic dampers the factor of safety against sliding of cars γs hasbeen investigated. The absolute minima found are given in Table 19, an exemplary time-historyplot is shown in Figure 71.

Fy=10MNEpl=0.05Eel

Fy=15MNEpl=0.05Eel

Fy=20MNEpl=0.05Eel

Fy=20MNEpl=0

Fy=30MNEpl=0

Eel=0.525⋅105 0.86 0.74Eel=1.05⋅105 0.73Eel=2.1⋅105 1.02 0.61 0.45

Table 19: Minimum values of sliding-safety (span 2, tower M3, span 3considered), elasto-plastic devices

0 10 20 30 40 50 60 70 80 90

1

2

5

4

3

Figure 71: Sliding safety factor of deck at tower M3 in time, elasto-plasticisolators#4, Fy=15MN, Eel=2.1E5N/mm2, Epl=0.05 Eel

As has already been found for the hydraulic dampers, sliding of cars can occur. Also for theelasto-plastic dampers there is a trend towards lower safety for stronger dampers which can beattributed to higher accelerations of the deck. Again, sliding does only occur momentarily andshould thus not impose a high danger on the traffic.


6.6 Conclusions

The earthquake behaviour of the Rion-Antirion Bridge has been studied for various configura-tions. The cases of a transversely free and a restrained deck have been considered as well aspossible isolation devices.

In the light of the set of performance criteria described in 6.3 the following can be concludedfrom the results obtained for earthquake conditions.

A configuration with a free deck is unfavourable because

- relative displacements between deck and pier are too large, particularly if transversetectonic movements precede;

- deformations in the soil are very large;

- the deformation demand on the pylon is very large; hinging occurs.

The case of a fixed deck (shear key) is unfavourable because

- high forces are applied from the deck on the pier;

- accelerations of the deck are such that sliding of cars can occur.

Damping devices can be favourable because

- relative transverse displacements of the deck can be reduced; however, theefficiency of the devices varies

- the deformation demand on the pylon is considerably reduced by any device studied

- deformations in the soil are reduced with respect to a free deck configuration;hydraulic devices are more efficient

- forces on the pier are reduced with respect to a fixed connection and can be adjustedby choosing the appropriate device

- accelerations of the deck can be reduced with some dampers; the likelihood of carssliding on the deck can thus be reduced.

Table 20 shows a comparison of possible solutions for the deck connection. It is apparent thatdamping devices provide a good solution with results lying either between a free and a fixeddeck or even being more favourable than both as for the deformation demand on the pylon.


Free deck Hydraulicdampers,

C=12MN, α=0.2

Elasto-plasticisolators,Fy=20MN,

Eel=52500N/mm2,Epl=0.05Eel

Fixed deck

Max. transverse dspl.deck-pier [m]

0.95 0.60 0.60 0.00

Max. transv. dspl. d5 inthe pylon (excl. rot.) [m]

0.44 0.31 0.31 0.33

Max. transverse dspl. inthe soil spring [m]

0.50 0.40 0.45 0.30

Force applied on the pierby the deck [MN]

0.0 13.1 25.3 63.7

Safety against sliding [-] 0.95 0.58 0.74 0.21

Table 20: Comparison of possible deck configurations

If an optimal solution in terms of deck displacements and pier forces is sought, Figure 72 givesan interesting relationship. The devices studied seem to lie close to the straight line between theresults for a free and a restrained deck. This means that the damping device can to a certainextent be adjusted to the response that is required. If lower deck displacements are sought higherforces on the pier need to be accommodated and vice versa. However, there are more and lesseffective solutions as has already been pointed out in section 6.5. While displacement reductionseems to depend on the energy dissipation capacity of the isolator the maximum force iscorrelated to the strength of the device.

Comparison, all cases

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50 60 70Max. force deck-pier [MN]

Max

. re

l. d

isp

l. d

eck-

pie

r [m

]

free deck

fixed deck

hydraulic dampers

elasto-plastic isolators

Figure 72: Comparison of all considered deck configurations

As was explained earlier, for the design of the deck configuration serviceability conditions alsoneed to be considered. If wind induced deck movements shall be avoided, a fuse device can beemployed as explained in 6.4.1. Also, the impact of slow tectonic movements needs to be takeninto account. In the case of the Rion-Antirion Bridge these are required so as not to cause


additional forces in the structure. Therefore, a so called shock transmitting device as describedin section 6.4.2 needs to be employed.

If all the above mentioned requirements are taken into account, a possible deck configurationfeaturing a fuse, a shock transmitter and isolation devices could look as shown in Figure 73 andFigure 74.

Figure 73: Possible deck isolation system in plan

Figure 74: Possible deck isolation system in elevation

Summary Page 76

7 SUMMARY

The main objective of this dissertation was to study the seismic behaviour and performance ofcable-stayed bridges. To this end, investigations on the Rion-Antirion Bridge structure wereconducted employing the finite element method. With the three-dimensional model and theanalysis code used it was possible to take into account all major member characteristics andboundary conditions as well as geometric nonlinearities.

As a first step static analyses were conducted on the bridge subjected to relative displacementsbetween the piers to study the impact of tectonic movements on the structure. It was found thatthe flexibility of the bridge which is provided by a freely movable deck acts favourably in termsof induced forces. The bridge deck responds by considerable relative displacements with respectto the pylons which can, however, be disadvantageous if deck displacements need to be limited.Displacements of preceded tectonic movements add to the dynamic displacements during anearthquake and can thus increase the danger for pounding of the deck against the pylon legs.

Modal analyses were performed to investigate the basic dynamic characteristics of the bridge.Four different cases in terms of deck connectivity were considered and the followingconclusions can be drawn:

- The fundamental mode has a very long period. It is either a transverse (18.6s) or avertical deck (7.5s) mode depending on whether the deck is free in the transversedirection or not. Several well spaced long period modes follow succeeded by manyclosely spaced modes below 3s.

- Tower modes have a high mass participation. These are beyond mode no. 100 whichpoints out the importance of higher mode contribution.

- For the shape and the natural frequency of the long period modes the boundaryconditions of the deck are most influential.

To study the seismic behaviour of the bridge a framework of performance criteria was set upconsidering the following parameters:

- displacements: relative deck displacements, soil deformations;

- forces: forces in the pylon (results of static push-over analyses were utilised), forceson the piers from the deck connection system;

- accelerations of the deck: sliding of cars on the road surface was considered.

Dynamic time-history analyses were performed to study the bridge response to a designearthquake. A set of significant response parameters was monitored and commented on. Firstly,the basic structural configurations with a free and a fully restrained deck were investigated.

Summary Page 77

Then it was analysed whether the seismic performance can be improved by employing isolationdevices between the deck and the pylon base. To this end, parametric studies with 13 hydraulicdampers and 9 elasto-plastic devices were conducted. The following was found:

- Relative displacements of a free deck are very large. These can be limited by usingisolation devices. However, this gives rise to additional forces applied on the piers.A straightforward relationship was found between these forces and the maximumdisplacements. By choosing the appropriate isolation device it is thus possible toadjust the bridge response as desired. It was, however, also found that, depending onthe energy dissipation capacity with respect to the maximum force, the effectivenessof the devices varies.

- The deformation demand on the pylon was found to be considerably reduced by anyof the damping devices. Only slight cracking on the legs needs to be expected in thatcase.

- Soil deformations are largest for the system with a free deck and smallest for arestrained deck. Isolation devices provide an intermediate solution.

- Generally, a free deck is favourable in terms of deck accelerations. A restrained deckexperiences very high accelerations and sliding of cars can occur. Damping devicescan prove to be favourable. It was found, however, that sliding of cars does onlyoccur momentarily and further investigations are needed as to what possible conse-quences are.

In the light of the framework of performance criteria it can be concluded that by applying deckisolation devices the earthquake behaviour of cable-stayed bridges can be significantlyimproved. An optimum performance with these passive devices can be obtained by balancingthe reduction in forces along the bridge against tolerable displacements. It was also shownwhich device properties provide the most efficient solutions.

Further investigations are necessary to substantiate the results obtained. Parametric studies onthe soil properties and the input motion were beyond the scope of this work. Also, modellingapproaches for the foundation and the cables could be improved by enhanced soil-structureinteraction modelling and nonlinear cable models respectively. Most importantly, it should benoted, that only one particular structure has been studied. Since every cable-stayed bridge is anindividual structure with respect to all its characteristics, also the effect of changes in geometryshould be looked into. Not only have these an influence on the basic dynamic properties but alsothe effectiveness of isolation devices as proven herein could be altered.

References Page 78

8 REFERENCES

[1] Abdel-Ghaffar, A.M., "Cable-stayed bridges under seismic action",Cable-Stayed Bridges - Recent Developments and Their Future, Ito,M. (ed.), Elsevier Science Publishers, 1991, pp. 171-192

[2] Abdel-Ghaffar, A.M., M. Khalifa, "Importance of Cable Vibration inDynamics of Cable-Stayed Bridges", Journal of EngineeringMechanics, Vol. 117, pp. 2571-2589

[3] Abdel-Ghaffar, A.M., S.F. Masri, A.-S.M. Niazy, "Seismicperformance evaluation of suspension bridges", Proceedings of the10th World Conference on Earthquake Engineering, Madrid, 1992,pp. 4845-4850

[4] Abdel-Ghaffar, A.M., A.S. Nazmy, "3-D Nonlinear SeismicBehaviour of Cable-Stayed Bridges", Journal of StructuralEngineering, Vol. 117, pp. 3456-3476, 11/1991

[5] Abdel-Ghaffar, A.M., A.S. Nazmy, "Nonlinear seismic response ofcable-stayed bridges subjected to nonsynchronous support motions",Proceedings of 9th World Conference on Earthquake Engineering,Tokyo-Kyoto, 1988, Vol. 6, pp. 483-488

[6] Achkire, Y., A. Preumont, "Active tendon control of cable-stayedbridges", Earthquake Engineering and Structural Dynamics, Vol. 25,pp. 585-597, 1996

[7] ADINA System 7.2, 1998, ADINA R&D, Inc., 71 Elton Avenue,Watertown, MA 02472, USA

[8] Ali, H.M., A.M. Abdel-Ghaffar, "Modelling of Rubber and LeadPassive-Control Bearings for Seismic Analysis", Journal ofStructural Engineering, Vol. 121, pp. 1134-1144, 1995

[9] Ali, H.M., A.M. Abdel-Ghaffar, "Modelling the nonlinear seismicbehaviour of cable-stayed bridges with passive control bearings",Computers & Structures, Vol. 54, No. 3, pp. 461-492, 1995

[10] Ali, H.M., A.M. Abdel-Ghaffar, "Seismic energy dissipation forcable-stayed bridges using passive devices", Earthquake Engineeringand Structural Dynamics, Vol. 23, pp. 877-893, 1994

References Page 79

[11] Alireza, R., G. Amin, "An investigation into the effect of earthquakeon bridges", Proceedings of the 10th World Conference onEarthquake Engineering, Madrid, 1992, pp. 4763-4766

[12] Ambraseys, N.N., J.J. Bommer, "Attenuation relations for use inEurope: An overview", "Fifth SECED Conference - EuropeanSeismic Design Practice", Elnashai (ed.), Balkema, 1995, pp. 67-74

[13] Ambraseys, N.N., J.A. Jackson, "Seismicity and strain in the gulf ofCorinth (Greece) since 1694, Journal of Earthquake Engineering,Vol. 1, No. 3, 1997, pp. 433-474

[14] Anderson, E., S.A. Mahin, "A displacement-based design approachfor seismically isolated bridges", Seismic Design Methodologies forthe Next Generation of Codes, Fajfar, P., H. Krawinkler (eds.),Balkema, 1997, pp. 383-394

[15] Aschrafi, M., "Comparative Investigations of Suspension Bridgesand Cable-Stayed Bridges for Spans Exceeding 1000m", Long-Spanand High-Rise Structures, IABSE Symposium, Kobe, 1998, pp. 447-452

[16] Betti, R., A.M. Abdel-Ghaffar, A.S. Niazy, "Kinematic soil-structureinteraction for long-span cable-supported bridges", EarthquakeEngineering and Structural Dynamics, Vol. 22, pp. 415-430, 1993

[17] Broderick, B.M., A.S. Elnashai, B.A. Izzuddin, "Observations on theeffect of numerical dissipation on the nonlinear dynamic response ofstructural systems", Engineering Structures, Vol. 16, 1994, pp. 51-62

[18] Bruno, D., V. Colotti, "Vibration Analysis of Cable-Stayed Bridges",Structural Engineering International, pp. 23-28, 1/94

[19] Bruno, D., A. Leonardi, "Natural Periods of Long-Span Cable-Stayed Bridges", Journal of Bridge Engineering, Vol. 2, 1997, pp.105-115

[20] Caetano, E., Alvaro Cunha, C.A. Taylor, "Dynamic analysis of acable-stayed bridge: Correlation with experimental results on thephysical model and on the prototype", Seismic Design Practice intothe Next Century, Booth (ed.), Balkema, 1998, pp. 363-370

[21] Caetano, E., A. Cunha, "Experimental analysis of coupled cable-deck motions in cable-stayed bridges", Proceedings of the 11th worldconference on Earthquake Engineering, 1996, Paper No. 913

References Page 80

[22] Caetano, E., A. Cunha, J. Macdonald, C. Taylor, "Experimentalanalysis of the effect of cable vibrations on the dynamic behaviour oftwo cable-stayed bridges", Proceedings of the 11th Europeanconference on Earthquake Engineering, 1998

[23] "Calibration Seismic Analysis of the Rion-Antirion Bridge",SEQAD Consulting Engineers, San Diego, 1999

[24] Calvi, G.M., "Seismic design of bridges in Europe", Fifth SECEDConference - European Seismic Design Practice, Elnashai (ed.),Balkema, 1995, pp. 35-42

[25] Calvi, G.M., A. Pavese, "Conceptual design of isolation systems forbridge structures", Journal of Earthquake Engineering, Vol. 1, No. 1(1997), pp. 193-218

[26] Calvi, G.M., A. Pavese, "Displacement based design of buildingstructures", Fifth SECED Conference - European Seismic DesignPractice, Elnashai (ed.), Balkema, 1995, pp. 127-132

[27] Clough, R.W., J. Penzien, "Dynamics of Structures", 2nd edition,McGraw-Hill, 1993

[28] Combault, J., P. Morand, "The Exceptional Structure of the RionBridge in Greece", Long-Span and High-Rise Structures, IABSESymposium, Kobe, 1998, pp. 495-499

[29] "Community guidelines for the development of the trans-Europeantransport network", Decision No 16, 92/96/EC, European Parliamentand the Council, 23 July 1996

[30] Dumanoglu, A.A., J.M.W. Brownjohn, R.T. Severn, "Seismicanalysis of the Fatan Sultan Mehmet (Second Bosporus) suspensionbridge, Earthquake Engineering and Structural Dynamics, Vol. 21,pp. 881-906, 1992

[31] Elassaly M., A. Ghali, M.M. Elbadry, "Influence of soil conditionson the seismic behabiour of two cable-stayed bridges", CanadianJournal of Civil Enginering, Vol. 22, pp. 1021-1040, 1995

[32] Elnashai, A.S., "Advanced Finite Element Analysis", Lecture Notes,Imperial College London, 1998

[33] Endo, T., T. Iijima, A. Okukawa, M. Ito, "The technical challenge ofa long cable-stayed bridge - Tatara Bridge", Cable-Stayed Bridges -Recent Developments and Their Future, Ito, M. (ed.), ElsevierScience Publishers, 1991, pp. 417-436

References Page 81

[34] Ernst, H.J., "Der E-Modul von Seilen unter Berücksichtigung desDurchhängens", Bauingenieur, Vol. 40, 1965, pp. 52-55

[35] Fajfar, P., H. Krawinkler, "Seismic design methodologies for thenext generation of codes", Seismic Design Practice into the NextCentury, Booth (ed.), Balkema, 1998, pp. 459-466

[36] Fan, L., S. Hu, W. Yuan, "Nonlinear seismic response analysis oflong-span cable-stayed bridge", Proceedings of the 10th WorldConference on Earthquake Engineering, Madrid, 1992, pp. 4815-4820

[37] Filiatrault, A., R. Tinawi, B. Massicotte, "Damage to Cable-StayedBridge during 1988 Saguenay Earthquake. I: Pseudostatic Analysis",Journal of Structural Engineering, Vol. 119, pp. 1432-1449, 5/1993

[38] Filiatrault, A., R. Tinawi, B. Massicotte, "Damage to Cable-StayedBridge during 1988 Saguenay Earthquake", Journal of StructuralEngineering, Vol. 119, pp. 1450-1463, 5/1993

[39] Fleming, J.F., E.A.Egeseli, "Dynamic behaviour of a cable-stayedbridge", Earthquake Engineering and Structural Dynamics, Vol. 8,pp. 1-16, 1980

[40] Fleming, J.F., J.D. Zenk, B. Wethyavivorn, "Seismic analysis ofcable-stayed bridges", Proceedings of the 8th World Conference onEarthquake Engineering, San Francisco, 1984, Vol. 5, pp. 207-214

[41] Frandsen, J., A. McRobie, "Comparison of Numerical and PhysicalModels for Bridge Deck Aeroelasticity", IABSE Symposium Kobe1998

[42] Friedland, I.M., M.C. Constantinou (ed.), Proceedings of the U.S.-Italy Workshop on Seismic Protective Systems for Bridges

[43] Fouad, N. A.: "Rechnerische Simulation der klimatisch bedingtenTemperaturbeanspruchungen von Bauwerken - Anwendung aufBeton-Kastenträgerbrücken und -Sandwichwände", Fraunhofer IRBVerlag, 1998

[44] Ganev, T., F. Yamazaki, H. Ishizaki, M. Kitazawa, "Responseanalysis of the Higashi-Kobe Bridge and surrounding soil in the1995 Hyogoken-Nanbu Earthquake", Earthquake Engineering andStructural Dynamics, Vol. 27, 1998, pp. 557-576

References Page 82

[45] Garevski, M., V. Mitrovski, "Dynamic behaviour of cable-stayedbridges", Proceedings of the 8th World Conference on EarthquakeEngineering, San Francisco, 1984, Vol. 5, pp. 199-205

[46] Garevski, M., T. Paskalov, "Application of FEM in modelling ofcable-stayed bridges", Proceedings of the 8th European Conferenceon Earthquake Engineering, Lisbon, 1986, Vol. 3, pp. 6.9/9-6.9/13

[47] Garevski, M.A., R.T. Severn, "Damping and response measurementon a small-scale model of a cable-stayed bridge", EarthquakeEngineering and Structural Dynamics, Vol. 22, pp. 13-29, 1993

[48] Garevski, M.A., R.T. Severn, "Dynamic analysis of cable stayedbridges by means of 3D analytical and physical modelling",Proceedings of the 10th World Conference on EarthquakeEngineering, Madrid, 1992, pp. 4809-4814

[49] Gentile, C., F. Martinez Y Cabrera, "Dynamic investigation of arepaired cable-stayed bridge", Earthquake Engineering andStructural Dynamics, Vol. 26, 1997, pp. 41-59

[50] Gimsing, N.J., "Cable supported bridges - concept & design", JohnWiley, 2nd edition, 1998

[51] Gregory, I.H., A.H. Muhr, "Design of elastic anti-seismic bearings",Fifth SECED Conference - European Seismic Design Practice,Elnashai (ed.), Balkema, 1995, pp. 479-486

[52] Gupta, S.P., Kumar, A., "A study on dynamics of cable stayed bridgeincluding foundation interaction", Proceedings of the 8th EuropeanConference on Earthquake Engineering, Lisbon, 1986, Vol. 5, pp.8.3/9-8.3/16

[53] Gupta, S., A. Kumar, "Dynamic response of cable stayed bridgeincluding foundation interaction effect", Proceedings of 9th WorldConference on Earthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6,pp. 501-506

[54] Hodhod, O., J.C. Wilson, "Characteristics of the seismic response ofa cable-stayed bridge tower", Proceedings of the 10th EuropeanConference on Earthquake Engineering, Vienna, 1994, Vol. 3, pp.2069-2074

[55] Hu, S., W. Yuan, L. Fan, "Non-linear seismic response analysis oflong-span suspension bridge", Proceedings of the 10th EuropeanConference on Earthquake Engineering, Vienna, 1994, pp. 2057-2074

References Page 83

[56] Ito, M., "Design practices of Japanese steel cable-stayed bridgesagainst wind and earthquake effects", Proceedings of theInternational Conference on Cable-Stayed Bridges, Bangkok, 1987,Vol. 1, pp. 15-22

[57] Izzuddin, B.A., "Nonlinearities in plain frames", Lecture Notes,Imperial College London, 1998

[58] Karoumi, R., "Response of Cable-Stayed and Suspension Bridges toMoving Vehicles – Analysis methods and practical modelingtechniques", Doctoral Thesis, TRITA-BKN Bulletin 44, Departmentof Structural Engineering, Royal Institute of Technology,Stockholm, 1998

[59] Kawano, K., K. Furukawa, "Random seismic response analysis ofsoil cable-stayed bridge interaction", Proceedings of 9th WorldConference on Earthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6,pp. 495-500

[60] Kawashima, K., S. Unjoh, "Damping characteristics of cable stayedbridges", Proceedings of the 10th World Conference on EarthquakeEngineering, Madrid, 1992, pp. 4803-4808

[61] Kawashima, K., S. Unjoh, "Seismic behaviour of cable-stayedbridges", Cable-Stayed Bridges - Recent Developments and TheirFuture, Ito, M. (ed.), Elsevier Science Publishers, 1991, pp. 193-212

[62] Kawashima, K., S. Unjoh, Y. Azuta, "Analysis of DampingCharacteristics of a Cable Stayed Bridge Based on Strong MotionRecords", Structural Engineering/ Earthquake Engineering (JapanSociety of Civil Engineers), Vol. 7, No. 1, 4/1990, pp. 169-178

[63] Kawashima, K., S. Unjoh, Y.-I. Azuta, "Damping characteristics ofcable-stayed bridges", Proceedings of 9th World Conference onEarthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6, pp. 471-476

[64] Kawashima, K., S. Unjoh, M. Tunomoto, "Estimation of DampingRatio of Cable-Stayed Bridges for Seismic Design", Journal ofStructural Engineering, Vol. 119, pp. 1015-1031, 4/1993

[65] Khalil, M.S., "Seismic analysis and design of the skytrain cable-stayed bridge", Canadian Journal of Civil Enginering, Vol. 23, pp.1241-1248, 1995

References Page 84

[66] Kitazawa, M., K. Nishimori, J. Noguchi, I. Shimoda, "Earthquakeresistant design of a long-period cable-stayed bridge", Proceedingsof the 10th World Conference on Earthquake Engineering, Madrid,1992, pp. 4797-4802

[67] Konok-Nukulchai, W., P.K.A. Yiu, D.M. Brotton, "MathematicalModelling of Cable-Stayed Bridges", Structural EngineeringInternational, Vol. 2, 1992, pp. 108-113

[68] Loo, Y.-C., G. Iseppi, "Nonlinear effects of cable sag - a case study",Proceedings of the International Conference on Cable-StayedBridges, Bangkok, 1987, Vol. 1, pp. 289-303

[69] Lubkowski, Z.A., J.M. Tandy, T.F. Piepenbrock, M.R. Willford,"Non-linear dynamic soil-structure interaction analysis of a deepbasement embedded in soft soil", Seismic Design Practice into theNext Century, Booth (ed.), Balkema, 1998, pp. 167-174

[70] Makropoulos, K.C., D. Diagourtas, "The Corinthian Gulf (Greece)strong-motion databank", Fifth SECED Conference - EuropeanSeismic Design Practice, Elnashai (ed.), Balkema, 1995, pp. 317-322

[71] Mylonakis, G., A. Nikolaou, "Soil-pile-bridge interaction: kinematicand inertial effects. Part I: soft soil", Earthquake Engineering andStructural Dynamics, Vol. 26, 1997, pp. 337-359

[72] Naeim, F., J.M. Kelly, "Design of Seismic isolated structures: fromtheory to practice", John Wiley, 1999

[73] Nazmy, A.S., A.M. Abdel-Ghaffar, "Effects of ground motionspatial variability on the response of cable-stayed bridges",Earthquake Engineering and Structural Dynamics, Vol. 21, pp. 1-20,1992

[74] Nazmy, A.S., A.M. Abdel-Ghaffar, "Non-linear earthquake-responseanalysis of long-span cable-stayed bridges: theory", EarthquakeEngineering and Structural Dynamics, Vol. 19, pp. 45-62, 1990

[75] Nazmy, A.S., A.M. Abdel-Ghaffar, "Non-linear earthquake-responseanalysis of long-span cable-stayed bridges: applications",Earthquake Engineering and Structural Dynamics, Vol. 19, pp. 63-76, 1990

[76] Nuti, C., "Seismic analysis of isolated bridges", Proceedings of the10th World Conference on Earthquake Engineering, Madrid, 1992,pp. 4893-4896

References Page 85

[77] Officer, P., "Response of Long-Span Cable-Supported Bridges toSeismic Excitation", MSc Thesis, Imperial College, 1998

[78] Pacheco, B.M., Y. Fujino, A. Sulekh, "Estimation Curve for ModalDamping in Stay Cables with Viscous Damper", Journal ofStructural Engineering, Vol. 119, pp. 1961-1979, 6/1993

[79] Parvez, S.M., M. Wieland, "Earthquake behaviour of continousmulti-span cable-stayed bridge", Proceedings of 9th WorldConference on Earthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6,pp. 477-482

[80] Priestley, M.J.N., G.M. Calvi, "Concepts and procedures for directdisplacement-based design", Seismic Design Methodologies for theNext Generation of Codes, Fajfar, P., H. Krawinkler (eds.), Balkema,1997, pp. 171-181

[81] Rion-Antirion Bridge, Design drawings and technical documentation

[82] Saafan, S.A., "Nonlinear Behaviour of Structural Plane Frames",Proceedings American Society of Civil Engineers, Vol. 89, 1963, pp.557-579

[83] Saiidi, M.S., E.M. Maragakis, T. Isakovic, M. Randall,"Performance-based design of seismic restrainers for simply-supported bridges", Seismic Design Methodologies for the NextGeneration of Codes, Fajfar, P., H. Krawinkler (eds.), Balkema,1997, pp. 395-406

[84] Schemmann, A.G., H.A. Smith, "Vibration control of cable-stayedbridges", parts 1 and 2, Earthquake Engineering and StructuralDynamics, Vol. 27, 1998, pp. 811-824, pp. 825-843

[85] Sethia, M.R., P. Krishna, A.S. Arya, "Model tests of a cable-stayedbridge", Proceedings of the International Conference on Cable-Stayed Bridges, Bangkok, 1987, Vol. 2, pp. 927-938

[86] Simoes, L.M.C., J.H.I.O. Negrao, "Comparison between modal andstep-by-step approaches in the optimization of cable-stayed bridgessubjected to seismic loads", Proceedings of the 11th worldconference on Earthquake Engineering, 1996, Paper No. 1881

[87] Troitsky, M.S., "Cable-stayed bridges: theory and design", 2ndedition, 1988

References Page 86

[88] Tuladhar, R., D.M. Brotton, "A computer program for non-lineardynamic analysis of cable-stayed bridges under seismic loading",Proceedings of the International Conference on Cable-StayedBridges, Bangkok, 1987, Vol. 1, pp. 315-326

[89] Vaz, C.T., A. Rito, R.T. Duarte, "Seismic studies of the Arade rivercable-stayed bridge", Proceedings of 9th World Conference onEarthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6, pp. 507-512

[90] Warnitchai, P., Y. Jujino, B.M. Pacheco, R. Agret, "An experimentalstudy on active tendon control of cable-stayed bridges", EarthquakeEngineering and Structural Dynamics, Vol. 22, pp. 93-111, 1993

[91] Webpage of "Öresundkonsortiet", the contractor for theÖresundproject: www.oeresundkonsortiet.com

[92] Wethyavivorn, B., J.F. Fleming, "Three dimensional seismicresponse of a cable-stayed bridge", Proceedings of the InternationalConference on Cable-Stayed Bridges, Bangkok, 1987, Vol. 1, pp.387-398

[93] Wilson, J.C., W. Gravelle, "Modelling of a cable-stayed bridge fordynamic analysis", Earthquake Engineering and StructuralDynamics, Vol. 20, pp. 707-721, 1991

[94] Wilson, J.C., T. Liu, W. Gravelle, "Ambient vibration and seismicresponse of a cable-stayed bridge", European EarthquakeEngineering, Vol. 5, 1991, pp. 9-15

[95] Wolf, J.P., "Dynamic Soil-Structure Interaction", 1985, Prentice-Hall Publishers

[96] Wolf, J.P., "Soil-Structure Interaction Analysis in the TimeDomain", 1988, Prentice-Hall Publishers

[97] Woo, G., "Long period earthquake risk in Europe", Fifth SECEDConference - European Seismic Design Practice, Elnashai (ed.),Balkema, 1995, pp. 59-65

[98] Wyatt, T.A., "The dynamic behaviour of cable-stayed bridges:fundamentals and parametric studies", Cable-Stayed Bridges -Recent Developments and Their Future, Ito, M. (ed.), ElsevierScience Publishers, 1991, pp. 151-170

References Page 87

[99] Wyllie, L.A., "Seismic design in California with the newmillennium", Seismic Design Practice into the Next Century, Booth(ed.), Balkema, 1998, pp. 59-62

[100] Yamanobe, S., T. Takeda, T. Ichinomiya, A.S. Cakmak, "Seismicsafety of prestressed concrete cable-stayed bridges", Proceedings ofthe 10th World Conference on Earthquake Engineering, Madrid,1992, pp. 4821-4826

[101] Yamasaki, Y., T. Ikeda, "Cable Supported Bridge under Movementof Foundation due to Earthquake", Long-Span and High-RiseStructures, IABSE Symposium, Kobe, 1998, pp. 403-408

[102] Yiu, P.K.A., D.M. Brotton, "Mathematical modelling of cable-stayedbridges for computer analysis", Proceedings of the InternationalConference on Cable-Stayed Bridges, Bangkok, 1987, Vol. 1, pp.249-260

[103] Yokoyama M., S. Tanaka, M. Iwano, "Analytical study on seismicbehaviour of cable-stayed concrete bridge", Proceedings of 9th WorldConference on Earthquake Engineering, Tokyo-Kyoto, 1988, Vol. 6,pp. 489-494

[104] Yokoyama, K., S. Unjoh, K. Tamura, T. Moritani, "EarthquakeProtective Design for Super-Long-Span Bridges", Long-Span andHigh-Rise Structures, IABSE Symposium, Kobe, 1998, pp. 173-178

[105] Zhang, X.-L., X.-Y. Yan, "A method for evaluating earthquakeresistant behaviour of bridge", Proceedings of the 10th WorldConference on Earthquake Engineering, Madrid, 1992, pp. 4827-4831

[106] Zheng, J., T. Takeda, "Effects of soil-structure interaction on seismicresponse of PC cable-stayed bridge", Soil Dynamics and EarthquakeEngineering, Vol. 14, pp. 427-437, 1995

[107] Zienkiewicz, O.C., R.L. Taylor, "The Finite Element Method", 4th

edition, Vol. 1,2, 1991, Mc Graw Hill

Documents

[TECH]Cable Stayed Bridges