[TECH]Cable Stayed Bridges

MSc Dissertation September 1999

Engineering Seismology and Earthquake Engineering

Cable-Stayed Bridges Earthquake Response and Passive Control

Guido Morgenthal

Imperial College of Science, Technology and Medicine Civil Engineering Department London SW7 2BU


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


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 my thanks must go to Professor A. S. Elnashai for his help and guidance throughout the year. His lectures have laid a sound foundation for the work on this project and his constant support even during my stay in Italy is greatly appreciated. Equally important, I would like to thank Professor G. M. Calvi from the Structural Mechanics Section of Università di Pavia. Through him I had the opportunity to work on a fascinating project and to experience a beautiful country and a lovely town at the same time. His generosity in taking time to discuss the progress of my work and his support in organising my stay were essential for my completing the work in time. The comments of Professor N. Priestley and the help of the other people at San Diego are also gratefully acknowledged. 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 Introdu ction

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TABLE OF CONTENTS Acknowledgements Table of contents

3 4

1 INTRODUCTION 1.1 Preamble 1.2 Significance of long-span bridges 1.2.1 Impact of bridges on economy 1.2.2 The trans-European transport network 1.3 Recent cable-stayed bridge projects 1.3.1 Öresund Bridge, Sweden 1.3.2 Ta Tatara Bridge, Japan 1.3.3 The Higashi-Kobe Bridge, Japan

6 6 7 7 7 8 8 9 10

2 STATE OF RESEARCH ON CABLE-STAYED BRIDGES 2.1 Configuration of Cable-Stayed Bridges 2.1.1 General remarks 2.1.2 Cable System 2.1.3 Stiffening Girder 2.1.4 Towers 2.1.5 Foundations 2.2 Nonlinearities in Cable-Stayed Bridges 2.3 Dynamic behaviour and earthquake response 2.3.1 General dynamic characteristics 2.3.2 Damping characteristics 2.3.3 Influence of so soil co conditions an and soil-structure interaction effects 2.3.4 St Structural control

11 11 11 12 14 16 17 17 19 19 22 24 27

3 TH THE RION ANTIRION BRIDGE 3.1 Introduction to structure and site 3.2 De D escription of the structure 3.2.1 The deck 3.2.2 Th The pylons and piers 3.2.3 T Thhe transition piers 3.2.4 The stay cables 3.2.5 The foundation

29 29 29 30 31 32 32 33

4 FINITE ELEMENT MODEL OF THE BRIDGE 4.1 Introduction 4.2 Description of the finite element model 4.2.1 The deck 4.2.2 The cables 4.2.3 Th The pylons and piers 4.2.4 The foundations and abutments 4.3 Accelerograms 4.3.1 St Structural damping 4.4 Calibration investigations on the piers

34 34 34 34 36 36 37 39 42 42

Introduction

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5 CHARACTERISTICS OF THE RION-ANTIRION BRIDGE 5.1 Static characteristics - special considerations 5.1.1 Relative displacements 5.1.2 Static push-over analyses on the pier/pylon system 5.2 Dynamic characteristics - modal analyses

44 44 44 45 47

6 EARTHQUAKE RESPONSE AND ITS CONTROL 6.1 Introduction 6.2 Investigations on basic systems 6.2.1 Introduction 6.2.2 Modelling assumptions 6.2.3 Results 6.3 Design considerations and performance criteria 6.3.1 Introduction 6.3.2 Serviceability conditions 6.3.3 Slow tectonic movements 6.3.4 Earthquake conditions 6.4 Devices for deck connection 6.4.1 Fuse device 6.4.2 Shock transmitter 6.4.3 Hydraulic dampers 6.4.4 Elasto-plastic isolators 6.5 Parametric studies on different deck isolation devices 6.5.1 Introduction 6.5.2 Analysis assumptions 6.5.3 Results 6.6 Conclusions

51 51 51 51 51 52 55 55 55 55 56 60 60 60 60 61 62 62 62 63 73

7 SUMMARY

76

8 REFERENCES

78

APPENDIX

Introduction

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1 INTRODUCTION

1.1 Preamble Man's achievements in Structural Engineering are most evident in the world's largest bridge spans. Today the suspension bridge reaches a free span of almost 2000m (Akashi-Kaikyo Bridge, 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 role in 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 of goods they contribute significantly to economical prosperity. Cable-stayed bridges, in particular, have become increasingly popular in the past decade in the United States, Japan and Europe as well as in third-world countries. This can be attributed to several advantages over suspension bridges, predominantly being associated with the relaxed foundation requirements. This leads to economical benefits which can favour cable-stayed bridges in free spans of up to 1000m. Many of the big cable-stayed bridge projects have been executed in a seismically active environment like Japan or California. However, very few of them have so far experienced a strong earthquake shaking and measurements of seismic response are scarce. This enforces the need for accurate modelling techniques. Three methods are available to the engineer to study the dynamic 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 possible 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 by means of computer analyses employing the finite element method. A framework of performance criteria is set up and within this different possible structural configurations are investigated. Conclusions are drawn regarding the effectiveness of deck isolation devices.

Introduction

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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. They act as lifelines on which many economic components depend. Naturally rivers, canals, valleys and seas constitute boundaries for these networks and therefore considerably confine the unopposed supply of goods. They cause significant extra costs because goods have to be diverted or even shipped or flown. These extra costs can exclude economies from foreign markets. It is evident that in this situation bridging the gap is worth considering. Cable-supported bridges offer the possibility to cross even very large distances without intermediate supports. Hence, it is only since their development, that people can consider crossings like the Bosporus (Istanbul Anatolia, completed 1973 and 1988), Öresund (Denmark - Sweden, to be completed 2000), the Strait 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 afford these road links. Cost-benefit analyses are inevitable as proof for banks. However, the number of 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 the industry on both sides of the new link. Furthermore and equally importantly, those bridge projects can become a substantial factor in the 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 product pipelines) 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 the remarks made in the previous section regarding the importance of transport networks and the links 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

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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 development,

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 of the most important components within the network. Many of the currently planned major bridges in Europe are therefore part of the network and supported by the EU. Among them are the 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. It comprises an immersed tunnel, an artificial island and a bridge part of which is a cable-stayed bridge (Figure 2).

Figure 2:

Öresund Bridge, Sweden

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

Introduction

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employed to accommodate road and railway traffic on two levels. The concrete slab is 23.5m wide and provides space for a 4 lane motorway. The structurally more difficult harp pattern (see section 2.1.2.1) was chosen for aesthetic reasons. It should be mentioned that the struts of the girder were inclined according to the angle of the cables which is favourable from the structural as well as pleasing from the aesthetic point of view. The money for the project was borrowed on the international market and jointly guaranteed by the 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 important European 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 free span in the world. It is shown in Figure 3, an elevation is given in section 2.1, Figure 5. The center span is 890m, supported by a semi-fan type cable system. Compared with this the side spans with 270 and 320m are extremely short and asymmetric so that intermediate piers and counterweights 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 31m wide and only 2.70m deep. To act as counterweight the deck in parts of the sidespans is made of concrete. At the towers the girder is kept free because of high temperature induced forces in the case 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 stay cable ever) were found to be prone to wind and rain induced vibration. Additional ropes

Introduction

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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 an earthquake event of magnitude 8.5 at a distance of 200km. The fundamental period of the bridge is 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 part of 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 a Warren truss with height a of 9m. It accommodates 2 roads at the top and bottom of the truss respectively. 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 have a height of 146.5m. These are placed on piers which are founded on caissons of size 35 (W) x 32 (L) x 26.5 (H) m. An important feature of the bridge is that the main girder can move longitudinally on all its supports. This results in a very long fundamental period which was found to be favourable for the seismic behaviour. On 17 January 1995 Kobe was struck by an earthquake of magnitude 7.2. Although the HigashiKobe 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 because the bridge was instrumented. These will be further discussed in section 2.3.3.

State of Research on Cable-Stayed Bridges

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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 mechanisms of cable-stayed bridges is given. This is necessary because they are in many ways distinctly different from beam-type bridges and these differences strongly affect their behaviour under static as well as dynamic loads. It has to be noted that herein only the current trend of design can be described. An outline of the evolution of cable-stayed bridges and more elaborate information can be found elsewhere: [50], [87].

2.1.1 General remarks Cable-stayed bridges present a three-dimensional system consisting of the following structural components, 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 the foundation 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 compression with the largest prestress being at the intersection with the towers. This is due to the horizontal force which is applied by each of the cables. This characteristic also distinguishes the cablestayed bridge from the suspension bridge because necessary provisions for anchoring cables are much 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, if more comprehensive information are sought the reader is referred to [50] and [87].


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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 patterns which 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 double plane 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 is usually considered the best, since the maximum inclination of each cable can be reached. This enables the most effective support of the vertical deck force and thus leads to the smallest possible cable diameter. The fan system causes severe detailing problems for the configuration of the anchorage system at the tower. The modified fan system overcomes this problem by spreading the anchorage points 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 that their stiffness is highest when the cable planes are inclined from the vertical. This favours Ashaped 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 parallel to each other. This pattern is deemed to be more aesthetically pleasing because no crossing of cables 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 less stable. However, excellent stiffness for the main span can be obtained by anchoring each cable to 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


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type with A-shaped pylons for the discussed reasons. However, there are still many variations regarding the configuration of the abutments, piers and towers and their respective connection with the stiffening girder. These problems will be discussed subsequently in the light of the dynamic behaviour. 2.1.2.2 Types of cables The success of cable-stayed bridges in recent years can mainly be attributed to the development of high strength steel wires. These are used to form ropes or strands, the latter usually being applied 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 centre wire with the other wires being formed around it in a helical manner. They have a lower modulus of elasticity than their parallel counterparts and furthermore experience a considerable amount 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. It is surrounded by several layers of wedge or keystone shaped wires and finally several layers of Z- or S-shaped wires. The advantages of this type of cable are a more effective protection against 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 wind impact. Second, their modulus of elasticity is even 50% higher compared with a normal strand of same diameter. Third, they are largely insensitive to bearing pressure because of a better interaction of the individual wires. The vast majority of modern cable-stayed bridges use galvanised locked-coil wire strands. These are assembled to the large diameter cables, which are usually parallel strand cables.


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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 connections numerous devices exist depending on the configuration of deck and tower as well as of the cable. Exemplary, some arrangements for tower and deck are shown in Figure 8 and Figure 9 respectively.

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 at intermediate locations mainly depending on the number of cables used. While fixed supports are either 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 depends on the type of cable used. Usually these sockets are threaded and a bolt is used to allow adjustments 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 traffic load, into the cable system. As mentioned earlier, in cable-stayed bridges these have to resist considerable axial compression forces beside the vertical bending moments. This compression force is introduced by the inclined cables. The girder can be either of concrete or steel. For smaller span lengths concrete girders are usually employed because of the good compressional characteristics. However, as the span


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increases the dead load also increases, thus favouring steel girders. The longest concrete bridge that 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 of very long-span bridges aerodynamic considerations can govern the decision. These are beyond the scope of this work but brief account of this issue will be given. It was shown in [41], that bluff cross sections which have a higher drag coefficient, experience considerably higher transverse wind forces than less angular sections. Specially designed streamlined sections can also avoid the creation of wind-turbulence at the downstream side, a phenomenon referred to as vortex-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])


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Beam arrangements consist of a steel or concrete deck which is supported by either a steel or a concrete beam. The beams carry the loads to the cables where they are anchored. Although easy to construct and generally efficient, beam-type girders have only a small torsional stiffness which can be undesirable depending on the structural system. Box sections possess high torsional stiffness and can be formed in a streamlined shape thus showing best behaviour under high wind impact. However, there are numerous possible shapes and 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 box sections. The aerodynamic behaviour is generally good. Trusses are of steel and thus the stiffness is high with respect to the weight. However, the high depth of the section can be criticised for aesthetic reasons. Trusses are unrivalled if double deck functionality is desired. In this 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 the foundation. 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-type bridges are subjected to severe bending moments. Box sections with high wall widths generally provide 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 equivalent steel towers and have a higher stiffness. However, their weight is considerably higher and thus the choice also depends on the soil conditions present. Furthermore, steel towers have advantages in terms of construction speed. The shape of the towers is strongly dependent on the cable system and the applied loads but aesthetic considerations are important as well. Possible configurations are depicted in Figure 11.

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


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While I- and H-shapes are vertical tower configurations and therefore support vertical cable planes, A- and λ-shaped towers correspond to inclined cable planes. The influence of these patterns on the overall stiffness of the structure have been discussed earlier. As far as the stiffness of the tower itself is concerned, A- and λ-shapes are preferable. However, their structural configuration is significantly different from the I- and H-shape which can have adverse 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 mainly influenced by the soil conditions and the load acting. For cable-stayed bridges often pile foundations are used, with the pier being connected to the pile cap. Various arrangements are possible and the choice mainly depends on the magnitude of the overturning moment. Cable-stayed bridges often need to be founded in water. In this case caisson foundations are used. 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 very early studies and shall be discussed in detail here because the nonlinearity is of greatest importance for any kind of analysis. Nonlinearities can be broadly divided in geometrical and material nonlinearities. While the latter depend on the specific structure (materials used, loads acting, design assumptions), geometric nonlinearities 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 subjected to its self weight and an externally applied axial tension force, will sag into the shape of a catenary. Increasing the axial force not only results in an increase in the axial strain of the cable but also in a reduction of the sag which evidently leads to a nonlinear stress-displacement relationship. The influence of the cable sag on its axial stiffness has first been analytically expressed by Ernst ([34]). If an inclined cable under its self weight is considered, an equivalent elastic modulus can be calculated as follows:


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E

E e =

(1)

2

1+ where: E e E w l

σ

(w ⋅ l ) E 12σ 3 is the effective modulus of elasticity of the sagging cable, is the modulus of elasticity of the cable which is taut and loaded vertically, is the unit weight of the cable, is the horizontal projection of the cable length and 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 the bridge stiffness if the forces are increased. This is depicted in Figure 12 and clearly distinguishes cable-supported structures from standard structures. They can be classified as being of the geometric-hardening type ([2], [4], [5]). Generalized Force

CABLE-STAYED BRIDGES Non-cable Structures

Cable Structures

Generalized Displacement

Figure 12: Nonlinearities in cable-stayed bridges Today most finite element programs offer nonlinear solution algorithms. With these it is possible to take the above mentioned characteristics of cable-stayed bridges into account. The nonlinear cable behaviour can be either treated utilising Ernst's formula or applying multielement 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 moment interaction is usually accounted for by calculating an updated bending and axial stiffness of the elements. 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 for by updating the bridge geometry by adding the incremental nodal displacements to the previous nodal coordinates before recomputing the stiffness of the bridge in the deformed shape ([74]).


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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, usually have extremely long fundamental periods. This distinguishes them from most other structures and strongly affects their dynamic behaviour. However, the flexibility and dynamic characteristics depend on several parameters such as the span, the cable system and the support conditions. These will be discussed in detail here. The dynamic behaviour of a structure can be well characterised by a modal analysis. The linear response of the structure to any dynamic excitation can be expressed as superposition of its mode shapes. The contribution of each mode depends on the frequency content of the excitation and 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 research papers dealing with their seismic behaviour. In Figure 13 the first modes obtained by AbdelGhaffar for a model bridge in [1] are shown. The first modes of vibration have very long periods of several seconds and are mainly deck modes. These are followed by cable modes which are coupled with deck modes. Tower modes usually are even higher modes and their coupling with the deck depends on the support conditions between these. The influence of different support conditions on the mode distribution has been investigated by Ali and Abdel-Ghaffar in [9]. It is apparent from the resulting diagram shown in Figure 14 that movable supports lead to a more flexible structure, thus shifting the graph towards longer periods. As an example, in [44] it was mentioned by Ganev et al that the Higashi-Kobe Bridge has been deliberately designed with longitudinally movable deck in order to shift the fundamental period to a value of low spectral amplification. The decision upon the support conditions of the deck is usually governed by serviceability as well as earthquake considerations. A restrained deck will avoid excessive movements due to traffic and wind loading and may thus be preferred. However, in the case of an earthquake a restrained deck will generate high forces which are applied to the pier-pylon system. It is thus a trade-off and often intermediate solutions are sought. Elaborate investigations 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 well characterises the bridge behaviour without the need to depict the individual mode shapes. As an example the first 25 modes of the Quincy Bayview Bridge, US, are given in Table 1. They have been identified experimentally as will be discussed later.


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Figure 13: First six computed mode shapes (considering one-element cable discretisation) ([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 the three orthogonal directions as can also be seen in Table 1. This coupled motion distinguishes cable-stayed bridges from suspension bridges for which pure vertical, lateral and torsional


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motions can be easily distinguished. This also implies that three-dimensional modelling is inevitable for dynamic analyses. The importance of the cable vibration for the overall response of the bridge was pointed out by Ali and Abdel-Ghaffar in [2], Abdel-Ghaffar and Khalifa in [9] and Ito in [56]. They concluded that appropriate modelling of the cable vibrations is necessary. For this purpose multi-element formulations for the cables are suggested. Only if the mass distribution along the cable is modelled and associated with extra degrees of freedom the vibrational response of the cables can be obtained. In [2] a modal analysis was performed modelling the cables with multi-element cable discretization. It was pointed out that there is coupling between the cable and deck motion even for the pure cable modes which suggests not to solely rely on analytical expressions for natural frequencies of a cable alone. In [9] it was found that the natural frequencies of the cables are 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 been developed by Bruno and Colotti in [18] and Bruno and Leonardi in [19]. Prevailing truss behaviour and small stay spacing have been assumed and on this basis diagrams showing the main natural frequencies depending on geometric parameters were developed. These were compared with results from numerical analyses and a good agreement was found. For existing bridges the modes can be obtained from vibration measurements. Ambient vibration 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 results obtained were reported in [94] and the mode shapes are depicted in Table 1.


Table 1:

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

It should be mentioned here, that a response spectrum analysis on the basis of a performed modal 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 is usually not the case for cable-supported bridges as explained in section 2.2. Secondly, even if linearity could be presupposed, the superposition procedure must be based on reasonable assumptions and methods like the SRSS procedure are only valid for well-spaced modes which is, again, not necessarily the case for cable-supported structures. Hence, the level of safety reached would not be assessable.

2.3.2 Damping characteristics Cable-stayed bridges have inherently low values of damping. It is therefore even more important to have accurate information about the level of damping reached by the structure. Research on this issue has shown that generalisation of damping values is difficult because damping characteristics vary significantly depending on the configuration of the bridge. Sources of energy dissipation in cable-stayed bridges are: material nonlinearity, structural damping such as friction at movable bearings, radiation of energy from foundations to ground and friction with air. 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 requires conducting a nonlinear analysis with the application of material nonlinearity and is most important when special energy absorption devices are to be modelled. Secondly and most commonly, an equivalent viscous damping can be introduced in the system in the form of a damping matrix C . Damping in cable-stayed bridges is undoubtedly not viscous. However, it is an 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 stiffness matrix. This is called Rayleigh damping and enables satisfying damping ratio exactly for 2 modes as shown by Clough and Penzien ([27]). An established estimate of viscous damping ratio seems to be 2% for cable-stayed bridges. According to Abdel-Ghaffar ([1]) values of this level have been found in many measurements. Damping ratios of 2-3% have in the past been


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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 electrodynamic shakers. The results obtained are shown in Table 2. It is important to note that the damping ratio notably depends on the mode under consideration. This is due to the different mode shapes and the different member contributions in these. On the other hand it is obvious from the differences between the testing methods that experimental results should also be used with caution.

Mode no. 1 2 3 4 5

Type Lateral Vertical Vertical Lateral Vertical

Table 2:

Model A Model B Shaking table Portable shakers Shaking table Portable shakers Frequency Damping Frequency Damping Frequency Damping Frequency Damping (Hz) (%) (Hz) (%) (Hz) (%) (Hz) (%) 4.28 0.45 4.24 0.42 4.16 0.43 4.17 0.34 6.19 0.38 6.19 0.37 5.86 0.33 5.81 0.56 8.93 0.42 9.03 0.38 8.83 0.72 8.66 0.83 11.88 0.29 11.88 0.26 11.39 2.01 11.58 1.00 13.65 0.44 13.65 0.42 13.75 0.56 13.81 1.23

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

Several studies on damping characteristics of cable stayed bridges were conducted by Kawashima et al and reported in [60], [61], [62], [63] and [64]. Model oscillation tests were done in which the damping ratio was computed from the averaged decay over 13 cycles. A picture 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 the mode shape (cf. Figure 17) as well as on the cable pattern. Damping ratios for the fan-type bridge were in the range between 0.6-0.8% while the harp-type structure had damping ratios between 1.2 and 1.5%, the higher values being for higher amplitudes of oscillation. Higher damping ratios of the harp configuration can be attributed to larger flexural deformations of the deck in vertical direction which leads to a higher energy dissipation.


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Figure 17: Experimentally measured damping (Kawashima and Unjoh, [61]) Because damping ratios vary with structural types, Kawashima suggested an approach in which the energy dissipation capacity is evaluated for each structural segment ([60], [64]). From this the overall damping can be calculated. The structural segments in which energy dissipation is considered uniform are referred to as substructures. These could be deck, towers, cables and bearing supports. Material nonlinearity is considered to be the prevailing mechanism for energy dissipation. Kawashima et al reduced the problem to applying so called energy dissipation functions for the substructures which can either be obtained by experiments or taken from their work. Kawashima and his co-workers also investigated the damping activated under earthquake conditions. In [62] it is stated that in an earthquake considerably higher damping values are found than the ones usually obtained from forced vibration tests. Hence, strong motion data recorded at the Suigo Bridge, Japan, during 3 earthquakes were used to estimate damping ratios. 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 upper and lower bound of the damping ration for the first coupled transverse/torsion mode to be 2.02.6% 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 the earthquake response of this structure. This is due to three main mechanisms that are referred to as soil amplification, kinematic interaction and inertial interaction. Firstly, amplitude and frequency 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 movement of a massless, rigid foundation embedded in the soil. Thirdly, inertial interaction describes the effect that the inertia of the moving structure has on the deformation of the soil. These components cannot be further discussed here. However, brief account shall be given of investigations on the importance of soil-structure interaction on the behaviour of cable-stayed bridges and possible treatments of the problem.


Page 25

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 be performed by the analytical or numerical technique best suited to that particular part of the problem: e.g. finite element method for the superstructure, continuum mechanics analysis for the soil. The individual responses are combined so as to satisfy the continuity and equilibrium conditions. The other modelling approach comprises the so called direct methods. Here, the soil is included in the global analysis model. Different element formulations and properties can be used for soil and structure but the problem is solved as one. This imposes great importance on the boundary conditions to be used in the model. For further treatment of this problem reference is made to the publications by Wolf: [95], [96]. Zheng and Takeda in [106] investigated the applicability of soil-spring models for foundation systems. Analyses on a 2-D finite element model of the soil were compared with those of the simplified model. It was found that the simplified model shows good agreement with reality for low frequency input motions while the errors increase for higher frequencies. In Figure 18 transfer functions for both horizontal and vertical motions are shown. These results suggest that a mass-spring model would be a good approach for the analysis of long-period structures like cable-stayed bridges. However, the contribution of higher modes could be underestimated.

Figure 18: Transfer functions for horizontal and vertical component of effective input acceleration at top of foundation computed by 2-D FEM and mass-spring model ([106]) Elassaly et al in [31] presented results of a case study investigating the effects of different idealisation methods for a pile foundation system on the earthquake forces acting on the structure. Two cable-stayed bridges were studied in this context. Firstly, a so called Winkler foundation making use of spring and damper elements was employed. Secondly, a discretization of the surrounding soil with plain strain elements was used as shown in Figure 19.


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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 was modelled entirely, too. This is deemed to be important, since it significantly contributes to the interaction between soil and foundation. It could be shown in time-history analyses, that neglecting the effects of local soil conditions and the soil-structure interaction results in a great underestimation of displacements of the superstructure particularly for soft soils. Underestimation mainly occurs if the soil site has a fundamental frequency close to the one of the structure. In terms of simplified approaches it was stated that fixed base modelling approaches can only be justified for structures founded on rock. On the other hand, the accuracy of employing an "equivalent stiffness" strongly depends on the conditions of the case in question and the way in which this stiffness has been evaluated. Betti, Abdel-Ghaffar et al ([16]) investigated the influence of both soil-structure interaction effects and the different seismic waves on the response of a cable-stayed bridge. Analyses were carried out using a fixed structure and a structure with the soil interaction modelled using the substructure method. Most interestingly it was found that inclined incoming waves, in-plane as well as oblique, cause a rocking motion of the foundations. This behaviour underlines the importance of including soil-structure interaction effects since a rocking motion of the foundations has a great effect on the behaviour of the bridge piers and thus of the whole structure. Investigations on the response of the Higashi-Kobe Bridge (see also section 1.3.3) and the surrounding 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 surface accelerometer showed clear evidence of liquefaction. Because the acceleration at the surface is smaller than the one at 34m depth and exhibits longer period motions, it was concluded that the surface soil layers which consist of loose saturated sands had actually been liquefied during the earthquake. The measurements from the instrumented bridge could also be employed to validate numerical approaches for analysis of soil-structure interaction. Extensive analyses using different computer codes have been conducted in this context. In the present case one of the most 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


Page 27

dependence of the soil material, separation of soil from the structure and pore-water pressure buildup. For explanations on the modelling approach used to take these into account the reader is referred to the paper mentioned above. Good correlation between the instrumentation data and the 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-Nanbu Earthquake ([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 are available. 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 dependent on external power supply, passive devices are not. Active control is essentially based on avoiding the impact of forces by modifying the vibration in a favourable way, e.g. additional masses are controlled so as to counteract the inertial forces of the structural members. For cablestayed 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 the girder motion. Passed through a linear feedback operator the resulting signals are fed to servohydraulic 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 in Figure 21. Results of investigations on the effectiveness of active tendon control can be found in [84], a recent paper by Schemmann et al.


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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 emergency situation 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 at critical zones such as the deck-abutment and deck-tower connections to concentrate hysteretic behaviour in these specially designed energy absorbers. Inelastic behaviour in primary structural elements 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-stayed bridges has been conducted by Ali and Abdel-Ghaffar ([10]). Recent applications have been introduced and modelling guidelines for lead-rubber bearings proposed. This paper makes clear that particularly for cable-stayed bridges no standardised passive control solutions can be developed. Many solutions are generally possible and their applicability depends on the behaviour characteristics of the bridge as well as on the design aim. However, it is stated that the new trend in cable-stayed bridge design is to have the main deck fixed to neither towers nor piers but to support them elastically by means of dampers, cables and links. The use of these elastic supports makes it possible to control the natural period of vibration, and accordingly are very effective in reducing the dynamic forces and, consequently, the size of the towers and the foundations.

The Rion Antirion Bridge

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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 part of the country's new west axis, a major national transport project. This highway will connect Kalamata in the south, Patras, Rion, Antirion and Igoumenitsa in the north. Being linked with two other major axes it forms part of a new high-performance transport network which is also a part of the trans-European network described in section 1.2.2. The objective is to establish direct access of all major urban centers of Greece (Patras, Athens, Lamia, Larissa, Thessaloniki) from the developing neighbouring countries in the Balkan region, the other European countries and the 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 an exceptional environment consisting of high water depths and rather weak soil deposits as will be further discussed in 3.2.5. Additionally, the seismic activity in the area is severe which makes the 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, supported by four large pylon/pier structures named M1, M2, M3 and M4 resting on the sea bed. Double cantilevers are built from each of the pylons. Final junctions are then made between the central cantilevers and the outer parts are extended towards the transition piers so that the final span lengths 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 with a 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])


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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 are I-shaped plate girders with variable sections, the maximum width of the lower flange being 2.00m. - Transverse I-shaped plate girders spaced at 4.06m. Both longitudinal and transverse beams include stiffeners to avoid local buckling of the steel plates. - A top reinforced concrete slab connected with the girders by steel studs. The concrete grade is C60/75. The slab thickness is generally 25cm, increased to up to 40cm above the girders. The overall slab width is 27.20m including two 1.95m wide cantilevers. The central part between 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 the main girders. It should be noted that the deck is only supported by the stay cables at the main pylons. No bearings are provided to the deck by the piers. However, isolation and dissipation devices are planned to be placed between the deck and the pylon base in transverse direction. These are to limit the relative displacements between the deck and the piers in the case of a severe earthquake.


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3.2.2 The pylons and piers The four pylons, providing the anchorage for the stay cables, are composed of four reinforced concrete legs, joined at their top level to a composite steel-concrete pylon head. The bases of the legs are rigidly restrained in a prestressed concrete pylon base. The total height of the pylons is 113m above the 3.50m thick pylon base. The pylon legs are square shaped concrete box girders, the outer section of these being 4.0x4.0m and the minimum wall thickness being 60cm. The pylon head is 35.0m high and has a square hollow 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 the pier head and the pylon legs and also constitute the anchorage for the isolating devices between deck and pier. An elevation of the pylon/pier system is shown in Figure 24.

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


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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, joining the square pylon base to the octagonal pier shaft, - the "pier shaft" with its base 3m above mean sea level. Because of the variable height 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, the length of the cones varies for the certain piers. They have an external diameter of 26.0m at the top and of 38.0m at the base. - The roughly cylindrical "footing" consists of a system of walls and slabs for stability reasons and has a diameter of 90m. The external wall is cylindrical and has a thickness of 0.80m. The footing and the very first meters of the cone are constructed in a dry dock and towed out to a wet dock where the cone walls are completed. The pier shaft slab and the upper pier are completed on site after the base has been immersed to its final position. The various compartments of the stiffener ring as well as the hollow central part are filled by a concrete or a gravel pier 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. Their design 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 earthquake behaviour.

3.2.4 The stay cables The two planes of stay cables are arranged in the semi-fan pattern. Each of the spans is supported 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 an area of 150mm2. At the pylon head the cables are anchored with typical threaded anchorages that permit adjustment if required.


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3.2.5 The foundation The bridge is founded on a deep soil strata of weak alluvions, the bedrock being approximately 800m below the sea bed level. It was therefore found to be necessary to reinforce the top soil layers with steel inclusions consisting of driven 25m long steel tubes of 2m diameter and 20cm thickness as explained in [28]. The bridge also has to accommodate possible fault movements which lead to horizontal displacements of one part of the bridge with respect to the other. The tectonic movement and the thus caused expansion has been discussed in a paper by Ambraseys and Jackson ([13]). Using records of earthquakes in Central Greece since 1694 they found an average extension rate of the Gulf of Corinth of 11mm/year.

Finite Element Model of the Bridge

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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 Bridge was 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

Deck Cable Stays Pylons Piers Structural mass Table 3:

M1 [kt] 18.4 1.4 11.5 149.1 180.4

M2 [kt] 18.8 1.4 11.5 157.5 189.2

M3 [kt] 18.5 1.4 11.5 159.4 190.8

M4 [kt] 18.4 1.4 11.5 121.9 153.2

Mass distribution of the bridge

The bridge was modelled in full taking into account all major structural components and their characteristics. Since the properties could be taken from design documents the bridge model reflects an actual structure designed to all code requirements which was not the case for all earlier investigations on cable-stayed bridges. The accuracy of finite element analyses naturally depends on the assumptions made for setting up the model. A description of the model and these assumptions will thus be given subsequently.

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 of steel girders and a concrete slab. It had to be modelled such as to behave correctly in bending and 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.


Page 35

Figure 26: Finite element model of the deck A modelling approach suggested by Wilson and Gravelle in [93] was utilised. This comprises introducing a single central spine of linear elastic beam elements that has the actual bending and torsional stiffness of the deck. These stiffnesses were evaluated by establishing an equivalent steel cross section, thus taking into account the concrete contribution. The cross section of the deck is not uniform along the bridge which was taken into account while setting up the deck spine elements. The torsional stiffness is the most difficult one to evaluate. The deck will have both pure and warping torsional stiffnesses. As an approximation, the deck cross section (shown in 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 cable anchor location to the next. At these nodes two rigid links were placed on either side perpendicular to the spine to attach the cable elements, thus achieving the proper offset of the cables with respect to the centre of inertia of the spine. Using the rigid link option allows the displacements 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 additional lumped masses were attached on either side of the central spine. By placing these below the level of the spine the difference between the centre of stiffness and the centre of mass can be accounted for. This produces coupling between the torsional and the transverse motions of the deck. The modelling approach for the deck is shown in Figure 27. Truss element

z Imx

Rigid links

Y Lumped mass

Spine

(position changes depending on section properties)

Figure 27: Theoretic modelling of the deck

M


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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 issue because nonlinearities arise from the cable sag. The stiffness therefore changes with the applied load. However, in this study one linear truss element without stiffness in tension was employed for each of the cables as shown in Figure 28. Taking into account the cable sag and thus nonlinear cable behaviour by means of an equivalent stiffness would have been extremely tedious since every cable would have had to be associated with a different force-displacement relationship because of the changing inclination and length of the cables. This would have also considerably increased the computation time as would have done utilising multi-element discretisation 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 approach can lead to considerable inaccuracies.

F

d Figure 28: Employed cable behaviour A prestress was applied to all the cables in order to ensure small deformations of the deck when the self weight is applied. The bridge was modelled picking up the geometry from the design drawings. Since these show the as-built configuration the application of the self-weight to the structure has to be taken into account. In reality the cables are prestressed according to a prior calculation so that the final shape is correct. Accordingly, in the present study the prestress was adjusted 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 of pylon M3 is shown in Figure 29. The piers were modelled with a single set of beam elements. The change in the cross-section along the cone was taken into account. However, in preliminary studies the stiffness of the cantilever was found to be inaccurate and a calibration analysis using a refined model was conducted as described in section 4.4. The pylon legs have been connected to the piers using rigid links. This was done because the pier head was deemed to be extremely stiff in terms of relative rotations between the pier top and the base of the pylon. The pylon legs unify at the pylon head which was divided into beam elements as long as the distance between the cable anchorage points, thus readily allowing for the connection of the cable elements.


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Figure 29: Finite element model of a pylon An important feature for the present study was the damping system connecting the deck and the pylon base. To accommodate these, rigid links were placed between the top of the pier and points at the level of the deck on either side of it. This enabled the damping devices only to act in the transverse direction because they have no component in longitudinal and vertical direction as 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 the vertical direction and nonlinear undamped springs in the horizontal directions. Nonlinear springs were 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 soil deposits. Extensive investigations on the soil properties were conducted by the designers and equivalent 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 these properties. As will be explained subsequently, there was also concern that the nonlinearities in the soil could play an important role in the behaviour of the bridge, thus making it necessary to take them into account. As an example, the characteristics of the nonlinear springs at pier M3 are shown in Figure 30. The linear spring's properties are: - vertical stiffness: k v=23⋅108 N/m2 8 - torsional stiffness: k t=51 ⋅10 Nm/rad.


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Pier M3, Force-Displ. x,y

600 500 ] 400 N M [ 300 e c r 200 o F100

0 0.00

20000

0.20

0.40 0.60 Displacement [m]

0.80

1.00

Pier M3, Moment-Rot. xx, yy

] m N15000 M [ t 10000 n e m 5000 o M

0

0.000

0.002

0.004

0.006

0.008

0.010

Rotation [rad]

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. As explained, nonlinear springs with the behaviour adopted from design documents were applied in horizontal direction. These were identical in both directions at each pier, but different from pier to pier. In vertical direction linear springs were applied. The rotational springs, not shown in Figure 31, have nonlinear behaviour and are the same for the two bending and different for the torsional degree of freedom.

Pier

F d

d

x-acclgr.

d

F

r. l g c a c y -

F

. r g l c c a z

Rotational springs not shown here

Figure 31: Modelling of the foundation To the end point of the translational springs the accelerograms were applied, thus exciting the structure via the ground springs.


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4.3 Accelerograms Synthetic accelerograms have been used in the present study that were derived from actual earthquake records using a special procedure to fit them to a predefined spectrum. These spectra that are referred to as KME spectra have been agreed upon with the client, the Greek government. The spectrum compatible accelerograms have been derived by the designers and are for the purpose of structural analysis. The real earthquake records that have been used as basis are listed in Table 4. No

Reference/name

1 2 3 4 5 6 7 8 9 10

San Fernando San Fernando Borrego Lower California Long Beach Kern County Kern County Kern County Kern County Hyogo-ken Nanbu

Table 4:

PGD [m] 0.724 0.881 0.588 0.701 0.873 1.020 0.900 0.778 0.952 0.804

PGV [m/s] 0.960 0.981 0.859 0.978 1.003 0.777 0.860 0.883 0.904 1.064

PGA [g] 0.479 0.516 0.522 0.586 0.497 0.567 0.600 0.554 0.565 0.554

A/V 0.50 0.53 0.61 0.60 0.50 0.73 0.70 0.63 0.63 0.52

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 have higher spectral accelerations in the long period range. Hence, the records can be expected to impose 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 imposing a very high demand on the structure. This means that a design earthquake was considered which is only appropriate for a very important structure. The accelerograms and their response spectra are shown in Figure 32 through Figure 37. It is apparent from the response spectra that the records impose a high demand on a long period structure.


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Accelerogram h1 (longitudinal direction) 6.00 ] 2 4.00 s / m [ 2.00 n o i t 0.00 a r e -2.00 0 l e c c -4.00 A

20

40

60

80

100

-6.00

Time [s]

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

Response spectrum, h1 (longitudinal dir.) 16.00 ] 14.00 2 s / 12.00 m [ . c 10.00 c a 8.00 l a r 6.00 t c e 4.00 p S 2.00

0.00 0

1

2

3

4 5 6 Period [s]

7

8

9

10

Figure 33: Response spectrum, longitudinal direction

Accelerogram h2 (transverse direction) 6.00 ] 2 4.00 s / m [ 2.00 n o i t 0.00 a r e -2.00 0 l e c c -4.00 A

20

40

60

80

-6.00

Time [s]

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

100


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Response spectrum, h2 (transverse dir.) 12.00 ] 2 10.00 s / m [ 8.00 . c c a 6.00 l a r t 4.00 c e p 2.00 S

0.00 0

1

2

3

4 5 6 Period [s]

7

8

9

10

Figure 35: Response spectrum, transverse direction

Accelerogram v (vertical direction) 4.00

] 2 3.00 s / m 2.00 [ n 1.00 o i t 0.00 a r e -1.00 0 l e c -2.00 c A-3.00

10

20

30

40

50

-4.00

Time [s]

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

Response spectrum, v (vertical dir.) 1.00 ] 2 s / m [ . c c a l a r t c e p S

0.80 0.60 0.40 0.20 0.00 0

1

2

3

4 5 6 Period [s]

7

Figure 37: Response spectrum, vertical direction

8

9

10

60


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

4.4 Calibration investigations on the piers Calibration investigations for the simplified beam model of the piers were conducted in order to ensure stiffness conditions compatible with reality. It was this deemed necessary because of the exceptional pipe-type cross-section of the piers with diameters of more than 30m. Such a structure can be expected to show considerable shear effects and therefore a refined model was set up in order to assess the influence of these. Figure 38 shows the model which was established using shell elements. On this static analyses were 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. The displacements resulting were compared with those arrived at by using pure bending theory beam elements with the correct cross-section. As was expected, the refined model showed larger deflections. These could be reached with the simplified model by using shear modelling and calibration was done by calculating the correct shear area. This enabled modelling the actual shear behaviour in a simplified fashion. The shear area found to be adequate was by a factor of 2.3 larger than the actual cross-sectional area which suggests that the structure undergoes considerable distortion.


Figure 39: Deformation under unit load

Page 43

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 a tectonic zone undergoing extension. The degree of extension was investigated by Ambraseys et al in [13]. It is obvious that the extension of the Gulf leads to relative displacements applied to a bridge crossing it. In the context of this report it was deemed particularly important to know about the transverse displacements 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 boundary conditions, meaning the way in which the deck is restrained. Two cases were considered here: a strike-slip fault located in the middle span causing transverse relative displacements between the piers and, for completeness, a normal fault causing longitudinal extension of the bridge. The deformed shape of the bridge for transverse displacements in the fault of 40 cm is shown in Figure 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 transverse direction. This is realistic, since a rigid shear connection is not planned for the bridge in order not to impose additional forces on the piers in the case of tectonic movements. It is apparent from the figure, that the largest relative displacements between deck and pylons occur at the towers adjacent to the mid-span. These are about one third of the absolute relative displacement in the fault and should be considered in further investigations on bridge deck deformations as reported upon in section 6. Also, the response of the bridge to an extension of the strait was investigated. A relative displacement of 2m between the two middle towers causes a deformation as shown in Figure 41. The bridge deck is lifted upwards in this case.


Page 45

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 2.5 times the absolute displacement between the towers. This shows that in the case of tectonic movements 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 and reported in [23]. These are given here because they will later be utilised to investigate the bridge tower's behaviour. Analyses were performed for forces pushing at heights of 78 and 92m above the pier head in transverse and diagonal direction, the results being shown in Figure 43. The displacements in this diagram are relative displacements between the point of pushing and the pier top excluding the displacements caused by the rotations at the pier top. Referring to Figure 42 these are displacements d 5.

Figure 42: Definitions for local displacements within the pylon


Page 46

Push-over analyses at 92m and 78m 120

92m: transverse

100

92m: diagonal 78m: transverse 78m: diagonal

80 ] N M [ e 60 c r o F

40

20

0 0.0

0.2

0.4

0.6 0.8 Deflection [m]

1.0

1.2

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 an explanation of the significant events which induce changes in stiffness. Deflection (in) 0

8

16

24

32

40

120,000 steel strain = 0.06: tension leg (top)

ax = 76%

24,000 100,000

ax = 68% ax = 65% ax = 62%

hinge: tension leg (bottom) hinge: tension leg (top)

20,000

axial tension cracking along entire leg

80,000

) N k ( e c r o F

ax = 57%

axial tension cracking at top hinge: compression leg (top) ax = 49%

60,000

hinge: tension leg (bottom) ax = 49%

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

12,000

hinge: tension leg (top) ax = 45%

40,000

16,000

Force-Deflection Curves

8,000

92m : Transverse 78m: Transverse

20,000

4,000

0

0 0.0

0.2

0.4

0.6

0.8

1.0

Deflection (m)

Figure 44: Explanation of events in push-over analysis

1.2

F o r c e ( k i p s )


Page 47

It should be mentioned here that the results of the analysis with the pushing force at 92m have been used for the subsequent investigations. Hence, relative displacements within the pylon are measured 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 basic dynamic characteristics of the bridge. Different combinations of boundary conditions for the deck have been considered as described in Table 5. The piers have been fixed at the base in all cases. model 1 2 3 4 Table 5:

x-direction (longitudinal) free free fixed fixed

y-direction (transverse) free fixed free fixed

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 in longitudinal and fixed in transverse direction. The modal mass participation is given as well as a description of the nature of the modes. The fundamental mode with period 7.5s is a vertical mode as depicted in Figure 45. The first modes are all deck modes of various shapes. These can be grouped into vertical, transverse and torsional modes, the first ones of each being shown in Figure 45, Figure 46 and Figure 47 respectively. Mode No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Table 6:

T [s] 7.53 6.51 5.99 5.25 4.77 4.41 3.99 3.43 2.76 2.61 2.48 2.48 2.44 2.38 2.37 2.35 2.34 2.33 2.32 2.21

Modal mass x [%] 9.84 0.00 0.00 0.08 0.00 0.00 0.00 15.64 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00

Modal mass y [%] 0.00 0.00 0.14 0.00 0.00 0.00 11.92 0.00 0.00 0.00 0.04 0.00 2.89 0.00 0.00 0.00 1.84 0.00 0.87 0.00

Modal mass z [%] 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 5.50 0.00 0.00 0.72 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Nature Dir. 1. 2. 1. 3. 2. 4. 3. 5. 6. 7. 1. 8. 2. 3. 4. 9. 5. 6. 7. 8.

vert. vert. trans. vert. trans. vert. trans. vert. vert. vert. tors. vert. tors. tors. tors. vert. tors. tors. tors. tors.

x z y x y z y x z x y z

x y y y

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


Page 48

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 periods above 3s. Below 3s many closely spaced modes follow. Period distribution, model 2 8.00 7.00 6.00 ] s 5.00 [ d o 4.00 i r e P 3.00

2.00 1.00 0.00 0

100

200

300 400 Mode number

500

600

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 modal masses of the deck modes are relatively small. This particularly applies to the higher deck modes. The accumulated modal mass in longitudinal direction is shown in Figure 50. In this the tower modes which contribute significantly can be located clearly. The first tower mode is mode 261. The tower mode with the highest modal mass participation of 8.2% in longitudinal direction is depicted in Figure 49.


Page 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 be taken into account. However, also first modes contribute considerably, causing susceptibility to long period excitation. Modal mass participation, model 2 100.00 ] % [ s s a m d e t a l u m u c c A

90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0

100

200

300

400

500

600

Mode number

Figure 50: Accumulated modal mass participation, model 2 (free in x-dir., fixed in y-dir.) In order to compare the behaviour of the different models analysed, the period distributions of these are shown jointly in Figure 51.


Page 50

Period distributions, all models 20.0 18.0 16.0

Model1

14.0

Model2

] s 12.0 [ d o 10.0 i r e P 8.0

Model3 Model4

6.0 4.0 2.0 0.0 1

2

3

4

5

6

7 8 9 Mode No.

10

11 12

13

14

15

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 a transverse sway mode of period 18.6 s. Given that the period of a pendulum is (G. Galilei) T pen dul um = 2π

l

(2)

g

the fundamental period of the bridge can be easily verified, knowing that the vertical distance between pylon top and deck is 95 m: T pen dul um = 2π

95m g

= 19.6s .

(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 the deck in transverse direction. Fixing the deck in longitudinal direction shifts the period distribution 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 RionAntirion bridge. The aim was to firstly understand the influence of the bridge deck and its connection to the piers. 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 criteria was developed in the next step. Within the framework established investigations on possible isolation devices for the bridge deck have been conducted. The objective was to better understand the impact that devices with different 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, investigations were started by considering two basic cases: a deck completely free in transverse direction and a deck rigidly connected to the pier top (shear key). This allowed for a better understanding of the behaviour characteristics and critical parameters. Also, the influence of the components of the earthquake motion was studied by applying the components 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 all analyses 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,


Page 52

- 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 to modes in the order of 440. Hence, the modal sum of the modal masses properly represented by the analyses is approximately 80%. Higher values could only be attained by using considably shorter time stepping and thus increasing the computation time. Inaccuracies may result but these will be confined to the tower response. model 1 2 Table 7:

y (transverse)-direction free fixed Boundary conditions for the deck

Model 1 was analysed with combinations of ground motion components as shown in Table 8 while model 2 was only analysed with all components acting. case 1 2 3 4 Table 8:

x-component x x

y-component x x x x

z-component x x

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 the case with all ground motion components acting some of these are listed in Table 9. All displacements regarding pier behaviour were monitored at tower M3. This tower was chosen because it has the longest piers as well as the weakest soil and is therefore supposed to deliver upper bound solutions. Dir. Deck at M3 - ground Deck at M3 - ground Deck in field 3 - ground Deck in field 3 - ground d5, rel. dspl. in pylon, (cf. Figure 42) Pylon top M3 - ground Pylon top M3 - ground Soil deformation Soil deformation Pier M3 - deck M3 Pier M3 - deck M3 Table 9:

x y x y y

Model 1 (free deck) 0.57 0.81 0.57 0.93 0.44

x y x y x y

0.52 2.41 0.40 0.50 0.63 0.95

Model 2 Rel. difference (fixed deck) [%] 0.57 0 0.47 72 0.57 0 1.66 78 0.33 33 0.52 1.90 0.40 0.30 0.62 0.00

Max. displacements [m] for models 1 and 2

0 27 0 67 2 inf.


Page 53

An important value is the maximum relative displacement between the deck and the pier which is 95cm for a free deck (cf. the plot in Figure 52). However, it can easily be seen from Table 9 that fixing the deck in transverse direction does not only affect the relative displacements between deck and pier. The absolute displacements in the mid-span are decreased by almost 80% and even the displacements of the pylon top are decreased. 0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0 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 been found to be 63.7MN (cf. the plot in Figure 53). Knowing this, it is most interesting to note that the 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 are decreased. This points out, that by fixing the deck the whole structure is changed in its dynamic characteristics. 6.

4.

2.

0. *

-2.

-4.

-6.

-8. 0.

10.

20.

30.

40.

50.

60.

70.

80.

90.

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 relative to the pier obtained herein is depicted in Figure 54 and will be an important framework for the further considerations in section 6.5.


Page 54

No connection / rigid connection 1.00

] m [ r 0.80 e i p k 0.60 c e d . l p 0.40 s d . x 0.20 a M

0.00 0

20

40

60

80

Max. force in connection deck-pier [MN]

Figure 54: Relationship force-displacement for junction deck-pier A further important result concerns the deformation of the pylon. The relative displacement within the pylon is given in Table 9 as d 5, which is the displacement excluding the influence of the rotation at the pier base as depicted in Figure 42. The deformation in the pylon is 25% larger for the case of a free deck. This imposes as higher demand on the pylon (cf. the results of pushover analyses reported in 5.1.2). The influence of the three earthquake motion components has been investigated on the model with the free deck (model 1). The aim was to find out whether performing analyses with all 3 earthquake components can be regarded as being on the safe side which is important for further studies and general design considerations. Table 10 shows results of maximum displacements obtained for analyses with various combinations of ground motion components. In this "=0" means that the respective component is not present and vice versa. The y-component was present in all cases since transverse displacements were of main interest. y-dspl. deck-ground in span 2 x-dspl. deck-ground at pier M3 y-dspl. deck-ground at pier M3 y-dspl. deck-ground in span 3 y-dspl. pylon top M3 - gound x-dspl. in soil at pier M3 y-dspl. in soil at pier M3

z=0 z=1 z=0 z=1 z=0 z=1 z=0 z=1 z=0 z=1 z=0 z=1 z=0 z=1

x=0 0.57 0.57 0.00 0.00 0.66 0.67 0.61 0.59 2.40 2.40 0.00 0.00 0.50 0.50

x=1 0.76 0.75 0.57 0.57 0.80 0.81 0.96 0.93 2.41 2.41 0.40 0.40 0.50 0.50

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


Page 55

Firstly, the vertical (z-) component appears to be of minor importance thus suggesting to be negligible as far as transverse displacements are concerned. However, it should be noted that nonlinear cable effects arising from the cable sag have not been taken into account. The stiffness of the cable is unaltered until the strain reaches zero. In reality the effect of cable sag would already reduce the stiffness of the cables for tensile strains thus probably leading to higher transverse displacements in the case of an upwards moving deck. Hence, the results obtained herein should be used with caution. The longitudinal (x-) component seems to have a much larger impact on the transverse displacements. While the tower behaviour remains almost unchanged, the deck displacements are 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 respective cable groups reach zero tensile strains for the peak longitudinal displacements. For longitudinal displacement in the opposite direction cables on the other side of the tower go slag. This, of course, significantly decreases the transverse stiffness of the cable groups and apparently gives rise to high transverse displacements. Concluding, the use of all earthquake components leads to upper bound solutions and was thus employed 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 of displacements and forces. It is of course inevitable to derive criteria for the assessment of these results. Whether deck displacements of 95cm are acceptable needs to be discussed as well as what forces can be applied to the towers by deck restrainers. These issues will be looked into subsequently. The aim is to present a set of criteria that could also be applicable to other bridge structures.

6.3.2 Serviceability conditions Usually, the dynamic displacements under frequent loading such as wind and traffic should be limited to very small values. Under serviceability conditions large displacements cannot be tolerated because they can jeopardise the traffic safety and speed. Psychological effects on people 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 deck under 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.


Page 56

If the deck is fixed in the transverse direction at the piers, relative transverse displacements between piers cause permanent forces on the deck as well as on the towers. This favours deck connection 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 the deck against the pylon legs should be avoided under any circumstances. If the deck is smashed against the pylon leg, extremely high impact forces act on these. Since pylon legs are usually hollow sections designed for bending and axial forces, local buckling can occur and jeopardise the overall stability of the pylons. In the case of the Rion-Antirion Bridge the clear distance between the deck and the pylon legs is about 80 cm on either side. This should, however, not be the design limit for deck displacements since 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 structures on particularly weak soil there will be concern as to how large the deformations will be during an earthquake because the degree of plastisation affects the post-earthquake performance of the soil bed. The soil investigations for the Rion-Antirion Bridge project have not yet been finalised. Limits can therefore not be defined. However, displacements in excess of approximately 40cm could be critical. 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 were presented. It is clear from these that the deformation capacity of the pylons is limited and that yielding 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 the displacements obtained from dynamic linear-elastic analyses. There is of course a discrepancy between the displacements because the linear-elastic analysis neglects changes in stiffness, but as 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 displacement of 30cm, first hinging occurs at 38cm. These are upper bound values while conditions are much more unfavourable for diagonal loading. This rises the question of the interaction of the xand y-components of the pylon deformation. For the case of the free deck which has in 6.2.3 been identified as most unfavourable, these components are plotted in time in Figure 55.


Page 57

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

Comparison of components of d5 (free deck) 0.40 d5-y

0.30

] m0.20 [ t 0.10 n e m0.00 e c-0.10 0 a l p s-0.20 i D

d5-x

10

20

30

40

-0.30 -0.40 -0.50

Time [s]

Figure 55: Free deck model, x- and y-components of d 5 (dspl. in the pylon excluding 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 in section 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 as possible. High interaction forces could not only impose a higher structural demand on the piers but also increase the demand on the foundation. However, the piers themselves are usually strong members and the maximum acceptable force may depend on other considerations. For example, if damping devices are used between deck and pier, the force might have to be limited for 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 bridge should be ensured. This not only means to avoid the collapse of the structure but also to make sure that cars are not driven down from the bridge. Usually bridges are equipped with crash barriers that can also serve in the case of an earthquake. However, it is interesting to know if there is actually the possibility of the cars starting to slide on the road. Therefore, a method to investigate 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 further research 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 the wheels will get grip again, nor what corrective actions could be employed by the driver during successive sliding events.


Page 58

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

z y

G

N H H 1

α

ah

G

x

av

R

N

Figure 56: Conditions for sliding considerations The force that has to be transmitted via friction between the street and the tyre is composed of the component of the weight in the direction of the deck and the component of the horizontal inertia force in the direction of the deck: H = H 1 + Mah cos α

where

and

(4)

cos α ≈ 1 for small angles and H 1 = Geq sin α ≈ Geq tan α

(5)

tan α = −θ x

(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 deck into account. Because G is reduced for a deck moving downwards, Geq is given as: Geq = M ( g + a v )

(7)

where M

mass of the car, av vertical acceleration of the deck, ah horizontal acceleration of the deck and G=Mg weight of the car.

This yields H = − M ( g + a v )θ x + Ma h .

(8)


Page 59

The friction resistance against sliding can be written as a simple Coloumb-type relationship: (9)

R = Nc

where N

force normal to the deck friction coefficient between tyre and street

c

with

N = Geq cos α ≈ Geq = M ( g + a v ) .

(10)

The limit state of sliding can thus be expressed as: ( g + a v )c = a h

− ( g + av )θ x

.

(11)

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

=

( g + a v )c . a h − ( g + a v )θ x

(12)

The value γ s and its development in time can give an impression about the likelihood of cars sliding on the bridge deck. The friction coefficient c can be up to 1.0 for optimum conditions. For a wet road surface typical values are 0.6-0.7. A value of 0.6 has been employed in this study. Analyses on the value γ s 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 at the tower M3. The absolute minima obtained are given in Table 11. free deck, x,y,z components 0.95

free deck, x,y components (no vertical motion) 1.01

fixed deck, x,y,z components 0.21

Table 11: Minimum values of sliding-safety (span 2, tower M3, span 3 considered), basic systems; input earthquake components used are shown Apparently the accelerations in the case of a fixed deck are such that sliding is much more likely to occur. A free deck seems to be favourable. Also the vertical (z-) component seems has a small unfavourable influence. The safety is increased if no vertical motion is present. Further discussion of the results can be found in section 6.5.


Page 60

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 to suppress the displacements under serviceability conditions and to limit the forces during an earthquake. 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 forces exceed a certain limit, the fuse device is broken and gives the deck free. It can be combined with other devices so as to hand the earthquake behaviour law of the connection over to another mechanism like a damping device.

F

fuse rupture point

d

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 any forces. It can thus be successfully applied if slow tectonic movements need to be accommodated by the bridge. The behaviour of a shock transmitter is similar to a damping device in the sense that the force in the 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 the displacements almost rigidly.

6.4.3 Hydraulic dampers Hydraulic dampers, filled with oil, silicone or a mixture of these have the following behaviour law: F = C ⋅ v . α

(13)


Page 61

Hence, the force output depends on the velocity. Figure 58 shows the influence of the parameter α on the behaviour. Constitutive laws for dampers -

α α

F =3MN*v

8 alpha=0.1

7

alpha=0.2

6

] N 5 M [ F

alpha=0.3 alpha=0.4

4

e c r o 3 F

2 1 0 0

1

2

3 Velocity v [m/s]

4

5

6

Figure 58: Hydraulic damping device behaviour Although hydraulic dampers do not have a maximum reaction force, the increase in force becomes very small for high velocities. This is more pronounced for small values of α as can be seen in Figure 58. Hence, there is actually a limitation on the force that the damping device can react with. The energy absorbed by the hydraulic device is mainly released as heat. There is always a limiting temperature up to which the correct behaviour is ensured. It is thus important to know about the energy that is dissipated in the device. However, it is difficult to calculate the temperature 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 not possible. Usually tests are then performed for the maximum stroke and reduced speed on one hand 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.


Page 62

F device yield point

d Figure 59: Behaviour of elasto-plastic isolator Elasto-plastic isolators show linear elastic response up to a yield point. The plastic behaviour can be either ideally plastic or hardening. These dampers can be made of steel as investigated in this report and the yield point is easily adjusted by designing the device accordingly. If steel with a hardening branch is used the hysteresis loops are arrived at by using the constitutive law from Figure 59 and kinematic hardening. Reduction of energy dissipation capacity due to Bauschinger 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 considerations regarding possible damping devices for the deck of the Rion-Antirion Bridge. For a free deck displacements 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 are large for a free deck (0.50m). Perhaps most importantly, the deformation demand on the pylons is very large in the case of a free deck so that yielding can occur. On the other hand, a fully restrained deck applies high forces to the towers (64MN). Also, the safety of cars against sliding on the surface is reduced. The aim of the studies was to investigate the influence of different isolation devices on the response of the bridge and to shed light on the question which device properties are of major influence 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 been analysed are compiled in Table 12 for hydraulic and in Table 13 for steel dampers. The forces C and F y are given as absolute forces per pier. These can in reality of course be distributed over several dampers.


Model No Hydr#1 Hydr#2 Hydr#3 Hydr#4 Hydr#5 Hydr#6 Hydr#7 Hydr#8 Hydr#9 Hydr#10 Hydr#11 Hydr#12 Hydr#13

C per pier

[MN] 3.0 6.0 6.0 6.0 6.0 12.0 12.0 12.0 12.0 18.0 18.0 24.0 24.0

Page 63

α

[-] 0.2 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.2 0.4 0.2 0.4

Table 12: Analysed hydraulic damping systems Model No Stl#1 Stl#2 Stl#3 Stl#4 Stl#5 Stl#6 Stl#7 Stl#8 Stl#9

F y per pier

E el

[MN] 10.0 15.0 15.0 15.0 20.0 20.0 20.0 20.0 30.0

[N/mm ] 210,000 52,500 105,000 210,000 52,500 52,500 210,000 210,000 210,000

2

E pl

0.05 E el 0.05 E el 0.05 E el 0.05 E el 0.05 E el 0 0.05 E el 0 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 comprehensive summary of all the values, the reader is referred to the appendix of this report.


Page 64

6.5.3.1 Hydraulic dampers Table 14 shows the maximum values of certain displacements that were obtained in the analyses. As can be seen from these, the relative displacements between deck and pier are considerably 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 residual displacements after the earthquake which diminish very slowly. α=0.1

α=0.2

α=0.3

α=0.4

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

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

0.0

-0.2

-0.4

-0.6

-0.8 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)


Page 65

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 the maximum velocity is greater than 1m/s the forces in the damper exceed the value of C . For the case below the maximum force is 13.1MN. The maximum forces in the device for the other cases are given in Table 15. Hysteresis for the damping device, C=12 MN, alpha=0.2 (0-40s only) 15 10 ] N M [ e c r o-0.70 F

5 0 -0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

-5 -10 -15

Displacement [m]

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

C =3.0MN C =6.0MN C =12.0MN C =18.0MN C =24.0MN

α=0.1

6.4 12.9

α=0.2

3.5 6.9 13.1 20.1 26.3

α=0.3

α=0.4

7.3 14.6

7.9 15.3 22.1 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 device sustains and the forces that are thus applied on the pier. These are depicted in Figure 62 for all investigated hydraulic damping devices. Clearly, there is a tendency towards smaller displacements for higher forces applied. However, the influence is surprisingly small. Increasing the maximum force and therefore the demand on the piers by a factor of 8 leads to a reduction in displacements by just 30%. Even more interestingly, there seem to be solutions that are more effective than others. For example, the isolator 12MN v 0.1 reaches a max. displacement of 77cm at a force of 12.9MN while the 12MN v 0.2 reduces the displacements to 60cm at an almost equal 13.1MN.


Relationship max. force - max. displacement

0.90

r e i p k c e d t n e m e c a l p s i d . l e r . x a M

Page 66

0.80 0.70 0.60 0.50

0.40 0.30 0.20 0.10 0.00 0

5

10

15

Max. force in the damper

20

25

30

Figure 62: Relationships max. force [MN] - max. displacement [m] in the isolation device, hydraulic dampers In Table 14 also the maximum values of d 5, the relative transverse displacement in the pylon without the rotation contribution (cf. Figure 42) are given. These correspond to the demand on the pylons and are thus of particular interest. Figure 63 shows the displacement response of the bridge pylon M3, giving the absolute displacement within the pylon as well as d 5. Apparently, the rotation contributes significantly to the displacement response which can also be seen from the corresponding displaced shape for the maximum displacement in Figure 64. Also, the tower seems to mainly respond in its fundamental mode. Similar results have been obtained for other times as well as for all other cases studied. Displacements in the pylon (hydraulic dampers, C=12MN, alpha=0.2) 2.00 Abs. Dspl. Pylon top-pylon base

1.50 ] m 1.00 [ s t n 0.50 e m 0.00 e c a l p -0.50 s i D-1.00

Displacement d5 (without rot.)

0

10

20

30

40

50

60

-1.50 -2.00

Time [s]

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


Page 67

Displaced shape of tower at time of max d5 250

200

150

100

50

0 -0.50

0.00

0.50

1.00

1.50

2.00

Figure 64: Deformed shape of tower M3 at time of max d 5 (displacement in the pylon without rotation), hydraul. dampers #7, C =12MN, α=0.2 The maximum values of d 5 obtained for the certain isolation devices are rather similar and the differences seem not to be correlated to any particular parameter. Importantly, they are all smaller than the displacements for the free deck as well as for the fixed deck system. This means that an essential improvement regarding the demand on the pylon can be achieved by employing isolation devices between deck and tower. However, for displacements of approximately 30cm slight 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 are approximately 41cm for most of the hydraulic devices. The value of safety against sliding of cars γ s 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 γ s at tower M3 is plotted in Figure 65. α=0.1 C =3.0MN C =6.0MN C =12.0MN C =18.0MN C =24.0MN

α=0.2

α=0.3

α=0.4

1.09 0.78 0.58 0.61 0.40

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


Page 68

5

4

3

2

1

0

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 therefore accelerations are so that sliding can actually occur. It can be inferred from the table that stronger dampers cause an increase in the accelerations and therefore increase the danger of cars loosing grip with the surface. The results also well correspond to the values obtained for the cases studied in 6.2. For the case of a free deck the minimum safety is 0.95 and for a fixed deck the factor is reduced to 0.21. It should be noted, that for the hydraulic dampers all the minima obtained and given in Table 16 correspond 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, only momentary sliding occurs and corrective actions by the driver should be possible. This points out 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-yield branch 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 the dampers, 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, isolators with a hardening behaviour show smaller displacements then corresponding perfectly plastic devices. A displacement time-history is exemplary shown in Figure 66. The corresponding forcedisplacement diagram is plotted in Figure 67. Obviously, the displacement response is quite different from the one for hydraulic dampers (cf. Figure 60). The hysteresis loops clearly show the behaviour law employed and energy dissipation is particular large for the first two cycles.


Page 69

The residual displacements after the earthquake are also smaller than for the hydraulic device. F y=10MN F y=15MN F y=20MN F y=20MN E pl =0.05 E el E pl =0.05 E el E pl =0.05 E el E pl =0

Relative displacement deck-pier at M3 [m] 0.73 0.60 E el=0.525⋅105 5 0.66 E el=1.05⋅10 5 0.70 0.59 0.51 E el=2.1⋅10 Absolute displacement pylon top M3 - ground [m] 2.12 2.13 E el=0.525⋅105 5 2.14 E el=1.05⋅10 5 2.20 2.14 2.13 E el=2.1⋅10 Relative dis lacement in the lon d5 (without rot.) 0.31 0.31 E el=0.525⋅105 5 0.31 E el=1.05⋅10 5 0.31 0.31 0.31 E el=2.1⋅10 Relative displacement in the soil spring [m] 0.44 0.45 E el=0.525⋅105 5 0.44 E el=1.05⋅10 5 0.44 0.44 0.44 E el=2.1⋅10

F y=30MN E pl =0

0.65 0.61

0.60

2.10 2.07

2.10

0.31 0.31

0.30

0.45 0.44

0.43

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

0.40 0.20 0.00 0

10

20

30

40

50

60

70

80

-0.20 -0.40 -0.60 -0.80

Figure 66: Transverse displacements [m] of the deck relative to the pier M3, elasto-plastic isolators #1, F y=10MN, E el=2.1E5 N/mm2, E pl=0.05 E el


Page 70

Hysteresis for the damping device, Fy=10MN, Eel=2.1E5, Epl=0.05Eel (0-40s only) 20 10 ] N M [ e-0.80 c r o F

-0.60

-0.40

-0.20

0 0.00 -10

0.20

0.40

-20 -30 Displacement [m]

Figure 67: Hysteresis loops of the device, elasto-plastic dampers #1, F y=10MN, E el=2.1E5N/mm2 , E pl =0.05 E el The maximum forces in the isolators are given in Table 18. The relationship between the maximum 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 other hand, the perfectly plastic devices can be favourable, as is obvious from the diagram. Particularly a very stiff non-hardening isolator considerably reduces the displacements while having the same maximum force. It is this because of the higher energy dissipation capacity compared to a hardening device with the same maximum force. F y=10MN F y=15MN F y=20MN F y=20MN E pl =0.05 E el E pl =0.05 E el E pl =0.05 E el E pl =0 E el=0.525⋅105 E el=1.05⋅105 E el=2.1⋅105

20.3 24.6 32.3

24.3

25.3

20.0

39.5

20.0

F y=30MN E pl =0

30.0

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

Relationship max. force - max. displacement 0.8 r 0.7 e i p k 0.6 c e d t 0.5 n e m e 0.4 c a l p s 0.3 i d . l e r 0.2 . x a M0.1

hardening perf. plast.

0 0

5

10

15

20

25

30

35

40

45

Force in damper [MN]

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


Page 71

In terms of deformations of the pylon the displacements d 5 (relative displacement between pylon top and pylon base excluding rotational contribution) are given in Table 17. Figure 69 exemplary shows a time-history of the d 5 values. Again, it is apparent that the rotation at the pier head strongly contributes to the displacements at the top as can also be seen from the deformed shape of the pylon shown in Figure 70. Displacements in the pylon (steel dampers, Fy=10MN, Eel=2.1E5, Epl=0.05Eel) 2.50 2.00 ] 1.50 m [ s 1.00 t n e 0.50 m e c 0.00 a l p -0.50 s i D

Abs. Dspl. Pylon top-pylon base Displacement d5 (without rot.)

0

20

40

60

80

-1.00 -1.50 -2.00

Time [s]

Figure 69: Relative displacements [m] in the pylon M3, including and excluding rotations, elasto-plastic isolators#1, F y=10MN, E el=2.1E5N/mm2 , E pl =0.05 E el Displaced shape of tower M3 at time of max d5 250.00

200.00

150.00

100.00

50.00

0.00 - 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 d 5 (displacement in the pylon without rotation), elasto-plastic isolators#1, F y=10MN, E el=2.1E5N/mm2 , E pl =0.05 E el


Page 72

For the isolation devices studied the deformations in the pylon are very uniform with a value of 31cm. 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 deck and 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 the case of a fixed deck. However, with approximately 44cm they are still smaller than for the case of a free deck. Also for the case of the elasto-plastic dampers the factor of safety against sliding of cars γ s has been investigated. The absolute minima found are given in Table 19, an exemplary time-history plot is shown in Figure 71. F y=10MN F y=15MN F y=20MN F y=20MN E pl =0.05 E el E pl =0.05 E el E pl =0.05 E el E pl =0 E el=0.525⋅105 E el=1.05⋅105 E el=2.1⋅105

0.86 0.73 0.61

1.02

F y=30MN E pl =0

0.74 0.45

Table 19: Minimum values of sliding-safety (span 2, tower M3, span 3 considered), elasto-plastic devices 5

4

3

2

1

0

10

20

30

40

50

60

70

80

90

Figure 71: Sliding safety factor of deck at tower M3 in time, elasto-plastic isolators#4, F y=15MN, E el=2.1E5N/mm2, E pl=0.05 E el As has already been found for the hydraulic dampers, sliding of cars can occur. Also for the elasto-plastic dampers there is a trend towards lower safety for stronger dampers which can be attributed to higher accelerations of the deck. Again, sliding does only occur momentarily and should thus not impose a high danger on the traffic.


Page 73

6.6 Conclusions The earthquake behaviour of the Rion-Antirion Bridge has been studied for various configurations. The cases of a transversely free and a restrained deck have been considered as well as possible isolation devices. In the light of the set of performance criteria described in 6.3 the following can be concluded from 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 transverse tectonic 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, the efficiency 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 adjusted by choosing the appropriate device - accelerations of the deck can be reduced with some dampers; the likelihood of cars sliding on the deck can thus be reduced. Table 20 shows a comparison of possible solutions for the deck connection. It is apparent that damping devices provide a good solution with results lying either between a free and a fixed deck or even being more favourable than both as for the deformation demand on the pylon.


Page 74

Free deck

Max. transverse dspl. deck-pier [m] Max. transv. dspl. d 5 in the pylon (excl. rot.) [m] Max. transverse dspl. in the soil spring [m] Force applied on the pier by the deck [MN] Safety against sliding [-]

Hydraulic dampers,

Elasto-plastic isolators,

C =12MN, α=0.2

Fy=20MN,

Fixed deck

2

E el =52500N/mm , E pl =0.05 E el

0.95

0.60

0.60

0.00

0.44

0.31

0.31

0.33

0.50

0.40

0.45

0.30

0.0

13.1

25.3

63.7

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 gives an interesting relationship. The devices studied seem to lie close to the straight line between the results for a free and a restrained deck. This means that the damping device can to a certain extent be adjusted to the response that is required. If lower deck displacements are sought higher forces on the pier need to be accommodated and vice versa. However, there are more and less effective solutions as has already been pointed out in section 6.5. While displacement reduction seems to depend on the energy dissipation capacity of the isolator the maximum force is correlated to the strength of the device.

Comparison, all cases ] 1.00 m 0.90 [ r e 0.80 i p k 0.70 c e d 0.60 . l p 0.50 s i d 0.40 . l e r 0.30 . x 0.20 a M

free deck fixed deck hydraulic dampers elasto-plastic isolators

0.10 0.00 0

10

20

30 40 50 Max. force deck-pier [MN]

60

70

Figure 72: Comparison of all considered deck configurations As was explained earlier, for the design of the deck configuration serviceability conditions also need to be considered. If wind induced deck movements shall be avoided, a fuse device can be employed as explained in 6.4.1. Also, the impact of slow tectonic movements needs to be taken into account. In the case of the Rion-Antirion Bridge these are required so as not to cause


Page 75

additional forces in the structure. Therefore, a so called shock transmitting device as described in section 6.4.2 needs to be employed. If all the above mentioned requirements are taken into account, a possible deck configuration featuring a fuse, a shock transmitter and isolation devices could look as shown in Figure 73 and Figure 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 of cable-stayed bridges. To this end, investigations on the Rion-Antirion Bridge structure were conducted employing the finite element method. With the three-dimensional model and the analysis code used it was possible to take into account all major member characteristics and boundary conditions as well as geometric nonlinearities. As a first step static analyses were conducted on the bridge subjected to relative displacements between the piers to study the impact of tectonic movements on the structure. It was found that the flexibility of the bridge which is provided by a freely movable deck acts favourably in terms of induced forces. The bridge deck responds by considerable relative displacements with respect to 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 an earthquake 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 following conclusions can be drawn: - The fundamental mode has a very long period. It is either a transverse (18.6s) or a vertical deck (7.5s) mode depending on whether the deck is free in the transverse direction or not. Several well spaced long period modes follow succeeded by many closely spaced modes below 3s. - Tower modes have a high mass participation. These are beyond mode no. 100 which points out the importance of higher mode contribution. - For the shape and the natural frequency of the long period modes the boundary conditions of the deck are most influential. To study the seismic behaviour of the bridge a framework of performance criteria was set up considering the following parameters: - displacements: relative deck displacements, soil deformations; - forces: forces in the pylon (results of static push-over analyses were utilised), forces on 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 design earthquake. 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 isolation devices between the deck and the pylon base. To this end, parametric studies with 13 hydraulic dampers 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 using isolation devices. However, this gives rise to additional forces applied on the piers. A straightforward relationship was found between these forces and the maximum displacements. By choosing the appropriate isolation device it is thus possible to adjust the bridge response as desired. It was, however, also found that, depending on the energy dissipation capacity with respect to the maximum force, the effectiveness of the devices varies. - The deformation demand on the pylon was found to be considerably reduced by any of the damping devices. Only slight cracking on the legs needs to be expected in that case. - Soil deformations are largest for the system with a free deck and smallest for a restrained deck. Isolation devices provide an intermediate solution. - Generally, a free deck is favourable in terms of deck accelerations. A restrained deck experiences very high accelerations and sliding of cars can occur. Damping devices can prove to be favourable. It was found, however, that sliding of cars does only occur momentarily and further investigations are needed as to what possible consequences are. In the light of the framework of performance criteria it can be concluded that by applying deck isolation devices the earthquake behaviour of cable-stayed bridges can be significantly improved. An optimum performance with these passive devices can be obtained by balancing the reduction in forces along the bridge against tolerable displacements. It was also shown which device properties provide the most efficient solutions. Further investigations are necessary to substantiate the results obtained. Parametric studies on the soil properties and the input motion were beyond the scope of this work. Also, modelling approaches for the foundation and the cables could be improved by enhanced soil-structure interaction modelling and nonlinear cable models respectively. Most importantly, it should be noted, that only one particular structure has been studied. Since every cable-stayed bridge is an individual structure with respect to all its characteristics, also the effect of changes in geometry should be looked into. Not only have these an influence on the basic dynamic properties but also the effectiveness of isolation devices as proven herein could be altered.

References

Page 78

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[2]

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[3]

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[4]

Abdel-Ghaffar, A.M., A.S. Nazmy, "3-D Nonlinear Seismic Behaviour of Cable-Stayed Bridges", Journal of Structural Engineering, Vol. 117, pp. 3456-3476, 11/1991

[5]

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[7]

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[8]

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[9]

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Aschrafi, M., "Comparative Investigations of Suspension Bridges and Cable-Stayed Bridges for Spans Exceeding 1000m", Long-Span and High-Rise Structures, IABSE Symposium, Kobe, 1998, pp. 447452

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[20]

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

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[22]

Caetano, E., A. Cunha, J. Macdonald, C. Taylor, "Experimental analysis of the effect of cable vibrations on the dynamic behaviour of two cable-stayed bridges", Proceedings of the 11 th European conference on Earthquake Engineering, 1998

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