CENTRO INTERNACIONAL DE METODOS NUMERICOS EN INGENIERIA

Monografías de Ingeniería Sísmica
Editor A.H. Barbat
Seismic Protection of Cable-Stayed Bridges Applying Fluid Viscous Dampers
Galo E. Valdebenito
Ángel C. Aparicio
Acknowledgments

The authors wish to thank to the Department of Geotechnical Engineering and Geosciences and the Department of Construction Engineering at Technical University of Catalonia for their help and support during the doctorate years of Mr. Galo Valdebenito. This work is inspired in the basic result of that investigation. Likewise, thank to the Faculty of Engineering Sciences and the Department of Research and Development (DID) at Universidad Austral de Chile for their help and support in this publication.

Preface

Earthquakes can be really destructive. There is no doubt. Recent seismic events have demonstrated the important effects on structures, and especially on bridges. In this sense, cable-stayed bridges are not an exception, although their seismic performance during recent events has been satisfactory. Their inherent condition as part of life-lines makes the seismic design and retrofitting of such structures be seriously considered.

Traditionally, seismic protection strategies have been based on provide enough strength and ductility. In the case of buildings or bridges with adequate supports and degrees of redundancy, that approach can be satisfactory, however, in the case of structures with few degrees of redundancy, or questionable ductility, that scheme could be inadequate, and worse, dangerous, as usually happens with cable-stayed bridges. All traditional modern strategies to design seismic structures are focused on the adequate comprehension of the mechanisms involved, in which ductility can be provided by some elements specially designed for these purposes. In these sense, strategies such as performed-based design or displacement-based design consider that well-designed structures need to dissipate enough energy by hysteresis in order to obtain economic and safe structures.

The incorporation of additional energy dissipation and isolation devices, by means of passive, active, semi-active and hybrid strategies, constitutes without doubt efficient schemes to protect structures controlling or avoiding damage, in which the energy dissipation is guaranteed through the action of external elements specially designed for those purposes. By this way, now it is possible to provide enough strength and energy dissipation capacity at the same time, avoiding damage on important structural elements, with the subsequent guaranty of the functionality, very important on life-lines, even during strong ground motions.

The present work constitutes an approach to the seismic protection of cable-stayed bridges including the incorporation of fluid viscous dampers as additional energy dissipation devices. The idea of the authors is to provide an up-to-date vision of the problem taking into account that long-period structures such as those proposed here, need to be adequately protected against strong motions, and considering that, because of their importance, an elastic behaviour is desirable. Chapter 1 describes the object to study in general terms. Chapter 2 constitutes a state-of-the-art review regarding the seismic behaviour and performance of fluid viscous dampers as external energy dissipation devices. The mechanical behaviour and technological aspects are now introduced with an energetic point of view, in which some practical applications are exposed and discussed. Chapter 3 describes the seismic response of cable-stayed bridges without external seismic protection, considering a parametric analysis in order to study the effects of the stay cable layout, stay spacing and deck level. A complete modal characterization is exposed, followed by a response spectrum analysis for comparative purposes. The effect of variations of the stay forces is analyzed, and finally, a nonlinear step-by-step analysis is performed for the critical structures, considering the velocity dependence of such bridges and the effects of far-fault and near-fault ground motions. The last Chapter exposes the seismic analysis of the selected structures including the incorporation of fluid viscous dampers as passive additional energy dissipation devices. Because of the inherent nonlinear behaviour of the structures and external devices, a mandatory nonlinear direct integration time-history analysis is performed for all the cases, in which parametric analyses are carried out in order to select the best damper parameters, and for the case of both far-fault and near-fault ground motions. In this part, comparative results are exposed with the aim to propose some practical recommendations.

Galo E. Valdebenito
Ángel C. Aparicio
Llavaneras (Barcelona), October 2009.
Contents

 Chapter 1. Introduction 1.1 Cable-Stayed Bridges and Seismic Protection 1.2 Historical Background Chapter 2. Fluid Viscous Damping Technology 2.1 General Overview 2.2 Technological Aspects 2.2.1 Historical Background 2.2.2 General Behaviour 2.2.3 Application to Bridges 2.3 Mechanical Behaviour 2.3.1 Energy Approach 2.3.2 Effect of the Damper Parameters 2.3.2.1 Damping oefficient cd 2.3.2.2 Velocity exponent N 2.3.3 Non-linear Viscous Dampers 2.3.3.1 Earthquake response 2.3.3.2 Equivalent linear viscous damping 2.3.4 Performance of Viscous Dampers During Near-Field Ground Motions 2.4 Analysis and Design Issues 2.4.1 Structural Analysis Including Viscous Dampers 2.4.2 Design Issues for Viscous Dampers 2.5 Practical Applications 2.5.1 Study Case 1: Rion-Antirion Bridge, Greece 2.5.2 Study Case 2: Tempozan Bridge, Japan Chapter 3. Seismic Response. Parametric Analysis 3.1 Introduction 3.2 Structural Modelling 3.2.1 Geometric Layout 3.2.2 Basis of Design and Actions 3.2.3 Nonlinearities 3.2.4 Modelling 3.2.4.1 Tower modeling 3.2.4.2 Deck modelling 3.2.4.3 Stay cable model 3.2.4.4 Connections and boundary conditions 3.3 Nonlinear Static Analysis Under Service Loads 3.4 Modal Analysis 3.4.1 Natural Frequencies and Modal Shapes 3.4.2 Damping 3.5 Seismic Response Analysis Applying the Response Spectrum Method 3.6 Effect of Variation of the Stay Prestressing Forces on the Seismic Response of Cable-Stayed Bridges 3.7 Seismic Response Applying Nonlinear Direct Integration Time History Analysis 3.7.1 General Considerations and Selected Models 3.7.2 Input Ground Motions 3.7.3 Importance of Velocity Spectra on the Seismic Response of Long-Period Structures 3.7.4 Seismic Response Considering Far-Fault Ground Motions 3.7.5 Seismic Response Considering Near-Fault Ground Motions 3.8 Comparative Results Chapter 4. Seismic Protection. Application of Fluid Viscous Dampers 4.1 General Considerations 4.2 Modelling of Nonlinear Fluid Viscous Dampers 4.3 Optimal Arrange of the Dampers 4.4 Modal Analysis Considering the Optimal Arrange of the Dampers 4.5 Optimal Damper Parameters 4.5.1 Parametric Analysis 4.5.2 Selection of the Damper Parameters 4.5.2.1 Far-fault ground motion 4.5.2.2 Near-fault ground motion 4.5.3 Influence of the Velocity Exponent and Damping Coefficient 4.6 Nonlinear Time-History Analysis 4.6.1 Far-Fault Ground Motion 4.6.2 Near-Fault Ground Motion 4.6.3 Specifications of the Dampers 4.7 Comparative Results and Discussion 4.7.1 Seismic Response Comparison 4.7.2 Energy Analysis Appendix A. Step-by-Step Nonlinear Time History Analysis A.1 General Considerations A.2 The Hilber-Hughes-Taylor-α Method A.3 Fast Nonlinear Analysis A.4 Recent Integration Algorithms A.5 Current Speed of Personal Computers for Nonlinear Analysis References 1 1 5 11 11 13 13 14 18 19 19 22 22 24 26 26 28 30 31 31 32 33 33 36 39 39 40 41 46 47 48 48 49 50 52 54 58 58 62 63 74 79 79 81 85 88 94 100 104 104 105 107 115 117 117 121 122 123 124 125 126 132 138 139 139 144 147 147 148 150 152 153 156

Chapter 1
Introduction
1.1 Cable-Stayed Bridges and Seismic Protection

Bridges are without a doubt attractive civil engineering works from a structural point of view. But they are not only exciting as a structure: the project, construction, maintenance, operation as well as functional, aesthetic, economic and political aspects make them extremely interesting constituting a great social event [Maldonado et al, 1998].

Suspension bridges are very interesting and useful structures because they can be used for long-spans, solving many practical problems for which is necessary to cross large distances without intermediate supports. These kinds of structures are a challenge from all points of view, due to the constant increase of the main span length demand, constituting most of the times a human whim or that competitive and insatiable desire to break goals at any price. Cable supported bridges can be divided into suspended and cable-stayed bridges, as can be appreciated in Fig. 1.1.

From a structural point of view, both types of bridges are completely different, since contrary to suspended bridges, in cable-stayed bridges the cables are prestressed. Keeping in mind functional and economical aspects, suspension bridges permit longer spans with more economical results than cable-stayed bridges [Podolny and Scalzi, 1986]. Actually, the longer main spans in cable-stayed bridges reach 900 m, although recent investigations show the feasibility and possibility of building bridges of this kind with main spans exceeding 1000 m. These studies are based on the current high standard technologies and the lightness of superstructures that use orthotropic slabs [Aschrafi, 1998; Nagai et al, 1998].

(a) Cable-Stayed bridge
(b) Suspended bridge
Fig. 1.1 Cable-Supported Bridges

In spite of the relative simplicity of bridges, the recent earthquake events of San Fernando (1971), Loma Prieta (1989), Northridge (1994), Kobe (1995) and Taiwan (1999) have shown that these systems are very vulnerable, mainly those of reinforced concrete. For that reason, is a high-priority to improve the comprehension of this phenomenon, learning from the recent earthquake lessons [Priestley et al, 1996]. These structural systems expose a few degrees of redundancy, and the collapse mechanisms should be known in detail to reach an appropriate performance. Some aspects that should be considered are: degree of redundancy of the system, soil-structure interaction, spatial variability effects, near source effects, geological faults and geotechnical aspects, bridge length effects, vertical component of motion and damping [Valdebenito, 2005]. All these aspects are explained in the references of Ghasemi (1999), Kawashima (2000); Cheung et al (2000) and Calvi (2004).

The structural analysis of a bridge depends undoubtedly on the structural modelling. Therefore, a well-done modelling is reflected in the degree of accuracy of the results. The vertiginous development of high-performance computers permits to solve more complex and large structures, testing a lot of conditions in a relatively short time. Thus, computing time will depend on the modelling used and the required accuracy for the results. Because of almost all the seismic isolators or energy dissipators experience non-linear behaviour, consideration of non-linear aspects in the analysis of the bridge – energy dissipation system is advisable. In spite of the current computer capacity and better non-linear structural analysis software, it is clear that the time and knowledge level of the designer are two serious limitations of the extensive application of these methodologies [Jara and Casas, 2002]. In fact, sometimes is preferable the use of simplified methods that show sufficiently accurate results in short time. In the case of long-span cable-stayed bridges, the problem is more complex, maybe due to the high non-linear behaviour of those structures, and hence, non-linear analysis becomes an indispensable condition, leaving aside the classical response spectrum analysis or the equivalent static analysis. Thus, a relatively complex structure can be solved by the iterative definition of the stiffness and equivalent damping.

Traditional seismic control strategies are based on the modification of stiffness, mass or geometric properties of the structure, reducing inertial forces and displacements caused by an earthquake. Thus, in the current design is necessary to permit controlled structural damage by the ductility provided, with the aim of avoiding too conservative designs and expensive costs. In other words, in the current philosophy, a structure with energy dissipation capacity is required, more than a resistant structure against all events. Although it is certain that traditional strategies for the seismic protection of bridges have progressed in the last years, for appropriate bridge strength and to assure a satisfactory behaviour for different intensity levels, development of special vibration control devices has given origin to a new path in seismic engineering. In general terms, instead of provide more strength, is more attractive to reduce internal forces and displacements through special isolation systems or energy dissipation devices. This energy distribution means that the seismic energy proceeding from the subsoil is distributed to different structural components and thus significant energy accumulation is avoided.

Amongst the existent control systems on bridges, passive strategies are well accepted because of their low comparative price, simple installation and maintenance as well as their great reliability and better theoretical and technological development [Jara and Casas, 2002]. Active, semi-active and hybrid systems seems to be an excellent strategy for the seismic control of structures, however, a lack of regulations and uncertainty regarding their real performance under strong ground motions are important limitations for their application. Without a doubt, there is a very promising future, mainly with semi-active and hybrid systems because of their incomparable advantages, although now their use is very limited, not been properly tested on real structures with real earthquakes. Thus, the general approach reducing the seismic demand of structures, more than trying to increase their strength or deformation capacity with appropriate criteria, is without a doubt an advantageous seismic protection system. These new seismic control strategies are conceived for the reduction of the seismic demand, and the appropriate application of this approach leads to systems that behave elastically during great earthquakes, on the contrary of a traditional design, where high energy dissipation capacity by controlled damage is needed. Passive control systems convert the kinetic energy of the system into heat, transferring it among different vibration modes. They do not require additional external energy for their operation, constituting their main advantage. In general terms, these systems operate elastically during great earthquakes, permitting structural functionality conditions after the event. Because of their low cost, high efficiency and low maintenance, they are additional seismic protection systems widely used in the world. Passive control systems can be classified as follows (Table 1.1):

Table 1.1 Passive Seismic Control Systems [Adapted from Valdebenito and Aparicio, 2005]
 Base Isolation Energy Dissipators Seismic Connectors Resonant Dampers 1. Rubber Bearings (RB) 1. Metallic Yield Dampers (MD) 1. Shock transmission Units TU) 1. Tuned Mass dampers (TMD) 2. High Damping Rubber Bearings (HDR) 2. Friction Dampers (FD) 2. Displacement Control Devices (DCD) 2. Tuned Liquid Dampers (TLD) 3. Lead Rubber Bearings (LRB) 3. Viscoelastic Dampers (VE) 3. Rigid Connection Devices (RCD) 4. Rubber Bearings with Additional Energy Dissipation 4. Fluid Viscous Dampers (VF) 5. Sliding Bearings (SB) 5. Lead Extrusion Dampers (LED) 6. Shape Memory Alloy (SMA)

Base isolation and dissipation result in decreasing the energy applied to the system and the transformation from energy to heat. This is also designated as energy approach, which especially takes into account the energy character of the seismic event. In the seismic isolation, the structure is separated from the subsoil, automatically limiting the energy that affects the structure, which is considerably reduced. As a result, the natural period is increased, which causes a considerable reduction of the structural acceleration during seismic events. Depending on the installed type of isolator, they do not only guarantee the vertical load transmission but also the restoring capacity during and after a seismic event.

 Fig. 1.2 (a) Energy Dissipation of a Traditional Bridge, (b) Energy Dissipation of a Seismic Isolated Bridge [Adapted from Jara and Casas, 2002] Fig. 1.2 (a) exposes a traditionally designed bridge, in which the seismic energy is dissipated by damage at the plastic zones (plastic hinges). For the above-mentioned, an adequate ductility to dissipate the earthquake energy is required. Fig. 1.2 (b) shows the case of an isolated bridge with rubber bearings. In this situation, inertial forces on the pylon are reduced, and the inelastic energy dissipation during severe earthquakes is achieved by hysteretic deformation of the supports [Jara and Casas, 2002].

Base isolation systems and seismic connectors applied to bridges have been properly tested and used for more than 20 years, and there is a lot of documentation and experience regarding to this. In relation to energy dissipation systems, the use of fluid viscous dampers can be the future for the application to large structures such as long-span cable-stayed bridges, mainly due to their high capacity, robustness, and good results of recent investigations.

 Fig. 1.3 Minimized Seismic Energy Penetration by Seismic Isolation and Energy Dissipation It seems that additional damping devices are clearly adequate considering the current high standards and technology, and in conjunction with isolation, produce the best possible seismic protection, mainly if the structural system is not velocity-dependent. On one hand, isolation reduces the spectral acceleration (demand), and on the other hand, fluid viscous dampers dissipate input energy avoiding structural damage (Fig. 1.3). A good state-of-the-art in relation to supplemental energy dissipation can be found in the work of Soong and Spencer (2002).

In the case of cable-stayed bridges, their seismic behaviour has been, in general terms, very satisfactory, maybe due to their great flexibility. In spite of the above-mentioned, comprehension of their behaviour is very complex being appropriate and promising to consider special systems of additional seismic protection. On those structures, these additional systems have been applied basically to control vibrations on cables due to the effect of the wind and rain (rain - wind vibration), to solve aerodynamic problems on unstable and complex structures and for the seismic retrofit of existing bridges. Now, application of these devices for the control of seismic actions begins to be used with more frequency; not only on the cables to mitigate the cable-deck interaction [Macdonald and Georgakis, 2002] but also to isolate the superstructure, as can be appreciated in the recently inaugurated Rion-Antirion Bridge (Fig. 1.4), in the Gulf of Corinthian, Greece [Infanti et al, 2004].

Fig. 1.4 Rion-Antirion Bridge, Greece [from [1] www.aecom.com]]

Design of almost all cable-stayed bridges located at moderate-to-high seismicity zones is more complex than design of conventional bridges. Bridge design regulations and modern previsions have been developed in general terms and for standard bridges, in order to provide safe and economical structures. As general design philosophy, it is accepted the important request of having structural damage but permitting emergency communications for a not frequent severe earthquake. For the new cable-stayed bridges, code previsions cannot be applicable, being necessary the urgent improvement of regulations and general recommendations for the seismic design of these bridges, based on numeric, experimental or full-scale testing investigations. Also, the lack of information about the real performance of these bridges during strong earthquakes increases the uncertainty in terms of an appropriate design [Abdel-Ghaffar, 1991]. In fact, according to Eurocode 8 Part 2 [CEN, 1998b], cable-stayed bridges are classified as special bridges, aspect that implies that these regulations need to be considered only as general recommendations. At the moment, existent regulations with regard to passive systems are limited to seismic isolation and energy dissipation devices, without the incorporation of hybrid, active or semi-active systems. Design specifications for bridges with LRB systems, published by the New Zealand Ministry of Works and Development in 1983, were the first regulations about bridges with special seismic protection based on isolation and energy dissipation systems. Later, in the 90s, official recommendations for the first time in USA [1991, 2000], Italy (1991), Japan (1996), and Europe through Eurocode 8 [CEN, 1998a, 1998b] were published.

1.2 Historical Background

The early stayed bridges used chains or bars for the stays. The advent of various types of structural cables, with their inherent high carrying capacity and ease of installation, led engineers and contractors to replace the chains and bars [Podolny and Scalzi, 1986].

 Fig. 1.5 Löscher-type Timber Bridge [Courtesy of the British Constructional Steelwork Association, Ltd] The concept of a bridge partially suspended only by inclined stays is credited to C.J. Löscher, a carpenter from Fribourg, Switzerland who built a completely timber bridge including stays and tower in 1784, with a span of 32 m. (Fig. 1.5).

Cable-stayed bridges might have become a conventional form of construction had it not been for the bad publicity that followed the collapses of two bridges: the 79 m pedestrian bridge crossing the Tweed River near Dryburgh-Abbey (England) in 1818; and the 78 m long bridge over the Saale River near Nienburg, Germany, in 1824 [Podolny and Scalzi, 1986]. The famous French engineer, Navier, discussed these failures with his colleagues, and his adverse comments are assumed to have condemned the stay-bridge concept to relative obscurity. Whatever the reason, engineers turned to the suspension bridge, which was also emerging, as the preferred type of bridge for river crossings.

The principle of using stays to support a bridge superstructure returned with the works of John Roebling. The Niagara Falls Bridge (Fig. 1.6), the Old St. Clair Bridge in Pittsburgh (USA), the Cincinnati Bridge over the Ohio River (USA) and the Brooklyn Bridge (Fig. 1.7) in New York (USA) are good examples.

 Fig. 1.6 Niagara Falls Bridge [Courtesy of the Niagara Falls Bridge Commission] Fig. 1.7 Brooklyn Bridge [from [2] www.elclubdigital.com]]

It should be noted that the stays used by Roebling in his suspension bridges were used as an addition to the classical suspension bridge with the main catenary cable and its suspenders. During Roebling’s time the suspension bridge concept was suffering with failures resulting from wind forces. He knew that by incorporating the diagonal stays he could minimize the susceptibility of his structures to adverse wind loading. However, it is not clear whether he used the two suspension systems compositely.

Towards the end of the 19th century, the success of these hybrid structures – part suspension, part stayed – resulted in a slowing down of the use of structures supported exclusively by inclined rods. However, it was not until 1899 that the French engineer A. Gislard further advanced the development of stayed bridges by the introduction of a new system of hangers, at the same time economic and sufficiently rigid [Walter, 1999]. The system was characterized by the addition of cables intended to take up the horizontal components of the forces set up by the stays. This arrangement cancels out any compressive forces in the deck and thus avoids deck instability.

 Fig. 1.8 The Bridge over the Donzère Canal, France [photo: J. Kerisel] Surprisingly, the first “modern” cable-stayed bridges were built in concrete by Eduardo Torroja in the 1920s (Tampul aqueduct) and by Albert Caquot in 1952 (Donzère Canal Bridge, Fig. 1.8). However, the real development came from Germany with papers published by Franz Dischinger and with the famous series of steel bridges crossing the river Rhine, as the Oberkassel Bridge, in Düsseldorf, Germany (Fig. 1.9). Fig. 1.9 Oberkassel Bridge, Düsseldorf, Germany Fig. 1.10 Maracaibo Bridge, Venezuela [from en.structurae.de]

The increasing popularity of this new type of structure with German engineers slowly extended to other countries. Thus, the Italian architect R. Morandi designed several cable-stayed bridges in reinforced and prestressed concrete. His most outstanding work is the bridge on Lake Maracaibo, Venezuela, built in 1962 (Fig. 1.10).

The international development of this bridge type began in the 1970s, but a very big step forward took place in the 1990s, when cable-stayed bridges entered the domain of very long spans which was previously reserved for suspension bridges. As examples, the Barrios de Luna Bridge – also called the Fernandez Casado Bridge – in Spain (430 m, 1983, Fig. 1.11); the Yang Pu Bridge in Shangai, China (602 m, 1993, Fig. 1.12); the Normandie Bridge in Le Havre, France (856 m, 1994, Fig. 1.13) and the Tatara Bridge in Japan (890 m, 1998, Fig. 1.14). It is extremely interesting to analyse the progress in the world record for cable-stayed bridges, since it provides keys to understand the evolution of their design (Fig. 1.15).

The recently inaugurated Millau Bridge in the Tarn Valley, France, is one of the world’s famous multi-span cable-stayed bridge, with 342 m main span length and 343 m height for the highest pylon. This also called “bridge over the clouds” is one of the more interesting French engineering works at the present (Fig. 1.16). In the same way, the new Sutong Bridge, in Nantong, China (inaugurated in 2008), is considered the longest cable-stayed bridge of the world, with a main span length of 1088 m, and surpassing the Japanese record reached with the Tatara Bridge (Fig. 1.17).

 Fig. 1.11 Barrios de Luna Bridge, Spain [from en.structurae.de] Fig. 1.12 Yang Pu Bridge, China [photo: M. Virlogeux] Fig. 1.13 Normandie Bridge, France [from fr.structurae.de] Fig. 1.14 Tatara Bridge, Japan [from [3] www.answers.com]]

Fig. 1.15 Evolution of Record Spans for Cable-Stayed Bridges [Virlogeux, 1999]
 Fig. 1.16 Millau Bridge, France 312px Fig. 1.17 Sutong Bridge, Nantong, China

Although the use of energy dissipation devices began as an attempt to control the cable vibration on cable-stayed and suspension bridges, very common on those structures due to the inherent low damping of the cable system, the inclusion of additional seismic protection, with the introduction of passive and active energy dissipation devices, has just begun. In this sense, the use of fluid viscous dampers in the recently inaugurated Rion-Antirion Bridge (Greece) is an exceptional opportunity to test in situ, with a real structure in a high-seismicity zone, those devices (Fig. 1.18). The deck of this multi-span cable-stayed bridge is continuous and fully suspended from four pylons (total length of 2252 meters). Its approach viaducts comprise 228m of concrete deck on the Antirion side and 986m of steel composite deck on the Rion side. The Main Bridge seismic protection system comprises fuse restraints and viscous dampers of dimensions heretofore never built. The same act in parallel, connecting the deck to the pylons. The restrainers of the Rion Antirion Bridge were designed as a rigid link intended to withstand high wind loads up to a pre-determined force. Under the reaction of the design earthquake, fuse restrainers will fail and leave the dampers free to dissipate the earthquake-induced energy acting upon the structure. The Approach Viaducts were seismically isolated utilizing elastomeric isolators and viscous dampers [Infanti et al, 2004].

 Fig. 1.18 Rion-Antirion Viscous Dampers [Courtesy of FIP Industriale, Italy] Another interesting application of passive/active devices is to retrofit existent bridges. After important earthquake events, or adjusting the seismic behaviour of existent structures in accordance with new codes and specifications, many bridges need to be retrofitted. For cable-stayed bridges, it seems to be impractical to reinforce structural members, and it will be more simple and efficient to conduct the bridge retrofit by using isolation systems if the system is proved to be feasible [Lai et al, 2004].

The recent application of active (i.e. hybrid, semi-active) systems on cable-stayed bridges is very limited. Actually, a benchmark structural problem for cable-stayed bridges was defined in order to provide a test bed for the development of strategies for the seismic control of those structures. The problem is based on the new cable-stayed bridge that spans the Mississippi River: the Bill Emerson Memorial Bridge, in Cape Girardeau, Missouri, USA. [Dyke et al, 2002].

 Fig. 1.19 Dongting Lake Bridge, China Real applications of active systems to cable-stayed bridges are limited only to aerodynamic structural control of the stays. In this sense, the recent application of Magnetorheological Dampers on the Dongting Lake Bridge over the Yangtze River in the southern central China (Fig. 1.19) is the first known application of those devices to control the rain-wind vibration. The installation finished in June 2002 [Chen et al, 2003].

Chapter 2
Fluid Viscous Damping Technology
2.1 General Overview

Structures situated on seismic areas must be designed to resist earthquake ground motions. A fundamental rule regarding the seismic design of structures, express that higher damping implies lower induced seismic forces. For conventional constructions, the induced earthquake energy is dissipated by the structural components of the system designed to resist gravity loads. It is well known that damping level during the elastic seismic behaviour is generally very low, which implies not much dissipated energy. During strong ground motion, energy dissipation can be reached through damage of important structural elements, and considering only the resulting response forces within the structure due to an earthquake leads to massive structural dimensions, stiff structures with enormous local energy accumulation and plastic hinges. This strengthening method combined with usual bearing arrangements permits plastic deformations by way of leading to yield stress and cracks. In this sense, structural repair after an important seismic event is generally very expensive, the structure is set temporarily out of service and sometimes a lot of damaged structures must be demolished [Alvarez, 2004].

General concepts for appropriate protection of structures against earthquakes do not exist, as every structure is quite unique and requires individual considerations. Earthquakes are often interpreted in terms of deformations and acting forces induced upon the structure. As a consequence, there is a tendency to think only about increasing the strength of the structure. Actually, forces and displacements are nothing but a mere manifestation of seismic attacks and do not in fact represent their very essence. An earthquake is actually an energy phenomenon and the forces causing stresses in the structure are the final effect of that event.

In recent years, other strategies have been developed to reduce the seismic response of the structures using additional passive devices. A passive control system may be defined as a system which does not require an external power source for operation and utilizes the motion of the structure to develop the control forces, as a function of the response of the structure at the location of the passive control system, according to Fig. 2.1.

 Fig. 2.1 Block Diagram of Passive Control System [Symans and Constantinou, 1999] A passive control system may be used to increase the energy dissipation capacity of a structure through localized discrete energy dissipation devices located either within a seismic isolation system or over the height of the structure. Such systems may be referred to as supplemental energy dissipation systems [Symans and Constantinou, 1999]

Passive supplemental damping strategies, including base isolation systems, viscoelastic dampers and tuned mass dampers are well understood and are widely accepted by the engineering community as a means for mitigating the effects of dynamic loading on structures. In this sense, energy dissipation systems can be considered as an important passive strategy in which the objective of these devices is to absorb a significant amount of the seismic input energy, thus reducing the demand on the structure by means of the relative motion within the passive devices which, in turn, dissipate energy. In general terms, these devices are not part of the structural system that resists gravity loads, constituting an external system that can be easily replaced after a strong earthquake. Of course, in this case the structural functionality is not affected as well as the stability of the structure, with a low replacement cost of such devices compared with repair or service interruption costs.

Additional damping devices dissipate energy by means of yielding, friction, Viscoelastic action or fluid flow through orifices [Soong and Dargush, 1997; Constantinou, 2003]. In this sense, fluid viscous dampers constitutes one of the well accepted energy dissipation systems by the scientific and engineering community, being considered as additional damping system in this work, as was previously explained and justified. These systems are capable of dissipate an important amount of energy during strong ground motions as well as to control long period displacements. These dampers are basically comprised of a cylinder filled with silicone fluid (oil or paste) and a piston that divides it into two chambers and is free to move in both directions. In case of sudden movements, due to earthquakes or other dynamic actions like braking, wind, etc., lamination of silicone fluid occur through an appropriate hydraulic circuit and leads to energy dissipation. In case of slow displacements, due to structure thermal expansion, such flow is obstructed, so that during normal service the behaviour is substantially rigid, acting like a shock absorber. Because of those advantages, utilization of this technology permits to take full advantage of the strength of structural elements, because it is possible to maximize energy dissipation reaching the maximum level of force that the structure can sustain, without exceeding it. As a consequence, structural elements remain in the elastic field also during high intensity earthquakes.

Actually, manufacture of fluid viscous dampers permits to design such devices for a wide range of specific requirements of velocity and force, constituting a good choice for implementation on new and existing facilities. Those devices are properly tested at specific laboratories, especially when they are applied on important structures or they are required for special conditions. In this sense, manufacturers such as Alga s.P.a. (Milano, Italy); FIP Industriale s.P.a. (Selvazzano, Italy), Taylor Devices, Inc. (New York, USA), Maurer Söhne (München, Germany), Mageba (Bülach, Switzerland) or Nanjing Damper Technology Engineering Co. Ltd (Nanjing, China) design and manufacture a wide variety of such systems.

Today an increasing number of applications of energy dissipation devices on bridges for the control of seismic displacements and energy dissipation is taking place. The more common solution is, probably, the use of linear / non-linear viscous dampers, permitting an adequate control of the displacements avoiding an increase of the structural internal forces and the increase of stiffness for piers and abutments [Jerónimo and Guerreiro, 2002].

The new tendencies regarding the seismic analysis and design of fluid viscous dampers capture the frequency dependence of such devices [Singh et al, 2003]; the earthquake response of non-linear fluid viscous dampers [Peckan et al, 1999; Lin and Chopra, 2002]; the seismic performance and behaviour of these devices during near-field ground motion [Tan et al, 2005; Xu et al, 2007] and the performance-based design of viscous dampers [Kim et al, 2003; Li and Liang, 2007]. A state-of-the-art review can be found in the works of Lee and Taylor (2001) and Symans et al (2008).

2.2 Technological Aspects
2.2.1 Historical Background

As with many other types of engineered components, the requirements, needs and available funds from the military allowed rapid design evolution of fluid dampers to satisfy the needs of armed forces. Early fluid damping devices operated by viscous effects, where the operating medium was sheared by vanes or plates within the damper. Designs of this type were mere laboratory curiosities, since the maximum pressure available from shearing a fluid is limited by the onset of cavitation, which generally occurs at between 0.06 N/mm2 and 0.1 N/mm2, depending on the viscosity of the fluid. This operating pressure was so low that for any given output level, a viscous damper was much larger and more costly than other types [Taylor, 1996].

In the late 1800s, applications for dampers arose in the field of artillery, where a high performance device was needed to attenuate the recoil of large cannons. After extensive experimentation, the French Army incorporated a unique (and “top-secret”) fluid damper into the design of their 75 mm gun. These first fluid damper designs used inertial flows, where oil was forced through small orifices at high speeds, in turn generating high damping forces. This allowed the damper to operate at relatively high operating pressures, in the 20 N/mm2 range. The output of those devices was not affected by viscosity changes of the fluid, but rather by the specific mass of the fluid, which changes only slightly with temperature. Thus, the technology of fluid inertial dampers became widespread within the armies and navies of most countries in the 1900 – 1945 period.

During the World War II, the emergence of radar and similar electronic systems required the development of specialized shock isolation techniques. During the Cold War period, the guided missile became the weapon of choice for the military, and the fluid inertial damper was again turned to by the military as the most cost effective way of protecting missiles against both conventional and nuclear weapon detonation. In these cases, transient shock from a miss near weapons detonation can contain free field velocities of 3 to 12 m/s, displacements of up to 2000 mm, and accelerations up to 1000 times gravity. For that reason, extremely high damping forces were needed to attenuate these transient pulses on large structures, and fluid inertial dampers became a preferred solution to these problems [Taylor, 1996].

With the end of the Cold War in the late 80s, much of this fully developed defence technology became available for civilian applications. In this context, demonstration of the benefits of damping technology on structures could take place immediately, using existing dampers and the seismic test facilities available at U.S. university research centres. In this sense, application of fluid viscous dampers as part of seismic energy dissipation systems was experimentally and analytically studied, being validated by extensive testing on one-sixth to one-half scale building and bridge models in the period 1990 – 1993 at the Multidisciplinary Centre for Earthquake Engineering Research (MCEER), located on the campus of the State University of New York at Buffalo in USA. Thus, implementation of fluid viscous damping technology began relatively swiftly, with wind protection usage beginning in 1993, and seismic protection usage beginning in 1995 [Taylor and Duflot, 2002].

2.2.2 General Behaviour

Fluid viscous dampers operate on the principle of fluid flow through orifices. A stainless steel piston travels through chambers that are filled with silicone oil. The silicone oil is inert, non flammable, non toxic and stable for extremely long periods of time. The pressure difference between the two chambers cause silicone oil to flow through an orifice in the piston head and seismic energy is transformed into heat, which dissipates into the atmosphere. This associated temperature increase can be significant, particularly when the damper is subjected to long-duration or large-amplitude motions [Makris 1998; Makris et al, 1998]. Mechanisms are available to compensate for the temperature rise such that the influence on the damper behaviour is relatively minor [Soong and Dargush, 1997]. However, the increase in temperature may be of concern due to the potential for heat-induced damage to the damper seals. In this case, the temperature rise can be reduced by reducing the pressure differential across the piston head (e.g., by employing a damper with a larger piston head) [Makris et al, 1998]. Interestingly, although the damper is called a fluid viscous damper, the fluid typically has a relatively low viscosity (e.g., silicone oil with a kinematic viscosity on the order of 0.001 m2 /s at 20°C). The term fluid viscous damper is associated with the macroscopic behaviour of the damper which is essentially the same as that of an ideal linear or nonlinear viscous dashpot (i.e., the resisting force is directly related to the velocity). Generally, the fluid damper includes a double-ended piston rod (i.e., the piston rod projects outward from both sides of the piston head and exits the damper at both ends of the main cylinder). Such configurations are useful for minimizing the development of restoring forces (stiffness) due to fluid compression [Symans et al, 2008]. The force/velocity relationship for this kind of damper can be characterized as F = C.Vα where F is the output force, V the relative velocity across the damper; C is the damping coefficient and α is a constant exponent which is usually a value between 0.1 and 1.0 for earthquake protection, although at the present time some manufactures begin to apply dampers with very low damping coefficients, typically in the order of 0.02. Fluid viscous dampers can operate over temperature fluctuations ranging from –40°C to +70°C, and they have the unique ability to simultaneously reduce both stress and deflection within a structure subjected to a transient. This is because a fluid viscous damper varies its force only with velocity, which provides a response that is inherently out-of-phase with stresses due to flexing of the structure [Taylor and Duflot, 2002].

Fluid velocity is very high in the piston head so the upstream pressure energy converts almost entirely to kinetic energy. When the fluid subsequently expands into the full volume on the other side of the piston head it slows down and loses its kinetic energy into turbulence. There is very little pressure on the downstream side of the piston head compared with the full pressure on the upstream side of the piston head. This difference in pressures produces a large force that resists the motion of the damper. Viscous dampers, when correctly designed and fabricated, have zero leakage and require no accumulator or external liquid storage device to keep them full of fluid. They have nearly perfect sealing. In a correctly designed and fabricated viscous damper there is nothing to wear out or deteriorate over time so there is no practical limit on expected life. Warranty periods of 35 years are common [Lee and Taylor, 2001]. Fig. 2.2 shows a general view of a fluid viscous damper, and Fig. 2.3 shows fluid viscous dampers for a high-speed railway bridge in Spain.

 Fig 2.2 General view of a Fluid Viscous Damper [Courtesy of FIP Industriale s.P.a., Italy] Fig. 2.3 Fluid Viscous Dampers for De Las Piedras-High Speed Railway Bridge, Spain [Courtesy of Maurer Sönhe, Germany]

Fig. 2.4 exposes a schematic of a typical fluid viscous damper showing its elements, which are described next.

 Fig. 2.4 Typical Viscous Damper [Lee and Taylor, 2001] The piston rod is machined from high alloy steel stainless steel and then highly polished. This high polish provides long life for the seal. The piston rod is designed for rigidity as it must resist compression buckling and must not flex under load, which would injure the seal.

The cylinder contains the working fluid and must withstand the pressure loading when the damper operates. Cylinders are usually made from seamless steel tubing and are sometimes machined from steel bars. Proof pressure is generally 1 - 5 times expected internal pressure for the maximum credible seismic event.

Structural applications require a fluid that is fire-resistant, non-toxic, thermally stable and that will not degrade with age. Under current OSHA (Occupational Safety & Health) guidelines this means a flash point of at least 200°F. Silicone fluid is often used as it has a flash point over 650°F and is cosmetically inert, completely non-toxic and one of the most thermally stable fluids available.

The seal must provide a service life of at least 35 years without replacement. As dampers often sit for long periods without use, the seal must not exhibit long-term sticking or allow fluid seepage. The dynamic seal is made from high-strength structural polymer to eliminate sticking or compression set during long periods of inactivity. Acceptable materials include Teflon®, stabilized nylon and members of the acetyl resin family. Dynamic seals made from structural polymers do not age, degrade or cold flow over time.

The piston head attaches to the piston rod and effectively divides the cylinder into two separate pressure chambers. This space between the outside diameter of the piston and in the inside diameter of the cylinder forms the orifice. Very often the piston head is made from a different material than the cylinder to provide thermal compensation. As the temperature rises the annulus between the piston head and the cylinder shrinks to compensate for thinning of the fluid.

The damper shown in Fig. 2.4 uses an internal accumulator to make up for the change in volume as the rod strokes. This accumulator is either a block of closed-cell plastic foam or a movable pressurized piston, or a rubber bladder. The accumulator also accommodates thermal expansion of the silicone fluid.

Viscous dampers add energy dissipation to a structure, which significantly reduces response to earthquakes, blasts, wind gusts and other shock and vibration inputs. A value of 30% of the critical damping ratio is a practical upper limit for combined viscous and structural damping. Around 25% of this is viscous damping and the remaining 5% is structural damping [Lee and Taylor, 2001]. This provides a 50% reduction in structural response compared with the same structure without viscous dampers. Note that the addition of viscous dampers does not change the period of the structure. This is because viscous damping is 90 degrees out of phase with the structural forces. Fig. 2.5 shows a typical plot of base shear against interstorey drift, taken from a laboratory test, according to Lee and Taylor (2001).

 Fig. 2.5 Typical Plot of Base Shear Against Interstorey Drift [Lee and Taylor, 2001] Fig. 2.6 Base Shear Against Interstorey Drift with Added Dampers [Lee and Taylor, 2001]

Note that the hysteresis loop is very flat and thin as there is only 5% damping. Figure 2.6 shows a plot of the same structure with the same input only this time with added viscous damping. Note that interstorey drift is 50% less and that the hysteresis curve is much fuller. In this case, 20% of added linear damping to the structure increased its earthquake resistance compared to that of the same structure without added damping. The area inside the hysteresis loop is the same as in Fig. 2.5. It is theoretically possible to provide enough viscous damping to completely prevent plastic hinging. This provides a totally linear structure. Economically, it is best to retain some plastic hinging as this results in the least overall cost. Viscous dampers still limit interstorey drift sufficiently to provide immediate occupancy after a worst-case event. They also limit and control the degree of plastic hinging and greatly reduce base shear and interstorey shear [Lee and Taylor, 2001]. Only as comparative purpose Table 2.1 shows equivalent damping coefficients for different structures and components. It is clear that an enormous amount of energy can be dissipated with the implementation of seismic dampers, reaching the largest values of dissipated energy. Of course, with those quantities, structural damping in the case of cable-stayed bridges may represent no more than 3% of the additional damping provided by the dampers, that is to say, a negligible amount.

Table 2.1 Comparison of Equivalent Damping Coefficients ξ of Different Structures and Components [Courtesy of Maurer Sönhe, Germany]
 Structural Component Damping ratio (ξ) Steel bridge 0.02 Concrete bridge 0.05 Elastomeric bearing 0.05 – 0.06 High damping rubber bearing 0.16 – 0.19 Lead rubber bearing and friction pendulum 0.30 – 0.40 Fluid viscous dampers Up to 0.60

In terms of the efficiency, the damping coefficient ξ relates to the efficiency η according to:

[Eq. 2.1]

This ends up in a maximum efficiency η = 96% for fluid viscous dampers.

As a summary, the overall characteristics of fluid viscous dampers include:

• During service conditions the device is not pre-tensioned and the fluid is under insignificant pressure
• An extra-low damping exponent, such as those proposed from some manufacturers, provides maximum and well-defined force to a certain limit. No structural damages due to higher damping forces occur even in case the vibration frequency exceeds the expected value.
• With the current technology, velocity ranges from 0.1 mm/sec to 1500 mm/sec or even more can be reached for fluid viscous dampers, which implies wide-variety of applications.
• Maximum response force is given within tenths of second, so structural displacements and vibrations can be more effectively minimized.
• Automatic volume compensation of the fluid caused by temperature changes without pressure increase inside the devices. Any compensation containers are located inside.
• No maintenance works necessarily. Visual inspection can be recommended during the period bridge inspections. Depending on the accumulated displacements and displacement velocities the service life can be reach up to 40 years.
• With the current development, the devices are not prone to leaking
• Range of operating temperatures varies from -40ºC to +70ºC.
• Non-toxic, not inflammable and not ageing fluids are applied.
2.2.3 Application to Bridges

Decks for viaducts and long-span bridges require adequate expansion joints for large displacements under service conditions to absorb the effects of creep and thermal expansion. A common structural layout used in Europe, consists of continuous deck supported by POT devices [Priestley et al, 1996]. By this way, the idea of employing devices with an insignificant response under long-period displacements and at the same time, capable of dissipating much induced seismic energy, was developed.

Some manufacturers differentiate the type of damper according to the motion of the device in the presence of slow displacements. In this case, as for example when thermal expansion occurs, in the OTP type the fluid flows from one chamber to the other with minimum opposition (normally smaller than 10% of the maximum force), while in the OP type such a flow is obstructed, so that during normal service the behaviour is substantially rigid [see the scheme of the typical application of viscous dampers on bridges in Fig. 2.7].

Application of fluid viscous dampers to bridges have been used since middle 90s. Although these devices may be applied to any kind of structures, their application is easier and more effective in bridges. One of the problems in the use of such devices is that the analysis of the dynamic behaviour becomes more elaborated and difficult than the analysis of a bridge with its seismic resistance based on the ductile capacity of the piers [Virtuoso et al, 2000]. Figs. 2.8 and 2.9 show some examples of application of fluid viscous dampers to bridges.

An important aspect to consider is that, if there is some available stiffness and resistance in the connection between the deck and the piers/towers or abutments, it is possible to obtain optimised solutions without inducing significant forces in the structure. That stiffness as the advantage of guaranteeing recentering capability after an earthquake can be used to improve the structure behaviour under other actions [Virtuoso et al, 2000].

Fig. 2.7 Typical Application of Viscous Dampers in Bridges [Courtesy of FIP Industriale s.P.a., Italy]
 Fig. 2.8 Fluid Viscous Dampers at G4-Egnatia Motorway Bridge, Greece [Courtesy of Maurer Sönhe, Germany] Fig. 2.9 850 kN Capacity Damper for the Chun-Su Bridge, South Korea [Courtesy of FIP Industriale s.P.a., Italy]

2.3 Mechanical Behaviour
2.3.1 Energy Approach

An earthquake is an energy phenomenon and therefore this energy character should be considered to achieve the best possible seismic protection for the structure. Without seismic protection system, the seismic energy is entering the structure very concentrated at the fixed axis. By means of shock transmission units the entering energy is distributed to several spots within the structure. In this case the energy input into the structure is still in same magnitude like without those devices, but now the energy is spread over the entire structure in more portions. By implementing additional energy dissipation capability, less energy is entering the structure, with the consequent response mitigation.

The principles of physics that govern the effects of dissipation on the control of dynamic phenomena were studied more than two centuries ago [DAlembert, Traité de Dynamique, 1743]. Nonetheless, their practical application has come about much later and within a much different time-frame in several sectors of engineering. As was previously exposed, the sector that was the first to adopt such damping technology was the military [France, 1897], followed by the automobile industry. In 1956 Housner already suggested an energy-based design of structures. Kato and Akiyama (1975) and Uang and Bertero (1990) made a valuable contribution to the development of the aspects of an energy-based approach, which presently meets with great concensus.

The dynamic equation of a single-degree-of-freedom structure with mass ms damping coefficient cs, stiffness ks and control force u, subject to ground acceleration is:

[Eq. 2.2]

where , and are the displacement, velocity and acceleration responses respectively. The involved parameters are clearly explained in Fig. 2.10, which shows a simplified scheme for a single-degree-of-freedom system. Of course, each term in Eq. 2.2 is a force.

 Fig. 2.10 Complex Bridge Structure Explained with a Simplified Single Oscillation Mass Integrating Eq. 2.2 with respect to x: where each term is now an energy component. Thus, we can define: [Eq. 2.3] [Eq. 2.4]

[Eq. 2.5]

[Eq. 2.6]

[Eq. 2.7]

An energy balance equation can be proposed in terms of the above defined:

[Eq. 2.8]

where:

Ek: Kinetic energy

Ev: Dissipated energy by inherent damping

Ee: Elastic strain energy

Eh: Dissipated energy by additional damping devices

Ei: Induced energy in the structure.

The concept of energy approach (Fig. 2.11) easily explains the energy terms involved in Eq. 2.8. The amount of structural stored energy (Es) has to be as low as possible to avoid damages. Therefore the value of the dissipated energy (Ed) must be great. In the term Eh energy dissipated by hysteretic or plastic deformation may be included; however this part must be kept low, as this way of energy dissipation causes structural yielding and cracks. For that reason, the drastic increase of the value of the energy of additional damping devices is the final opportunity to control the energy balance of the structure.

 Fig. 2.11 Concept of Energy Approach Considering the Energy Exchange Between Structure and Environment

: Stored energy within structure

: Dissipated energy within structure

Thus:

[Eq. 2.9]

The control force u by non-linear viscous dampers with damping coefficient cd is expressed as

[Eq. 2.10]

In Eq. 2.10, the exponent N controls the damper nonlinearity and has typical values in the range of 0.10 to 1.0 for seismic applications. For the special case of N = 1, Eq. 2.10 represents the force applied by linear viscous dampers. In the case of N = 0, Eq. 2.10 changes to a friction damper as follows:

[Eq. 2.11]
 Fig. 2.12 Plot of Force Against Velocity for Several Values of Damping Exponent N Typical values of the exponent N for the interval 0.1 – 2 are plot in Fig. 2.12. According to this, when the value of N is lower than one, the curve has a strong force increase for low velocity values and small force increase for high velocities. In these cases there is a large amount of energy dissipated in each cycle. In the case of high values of N, the curve has a strong increase for high values of velocity, aspect that can be dangerous because of the excessive forces developed at the dampers.

Linear damping is easy to analyze and can be handled by most software packages. Also, linear damping is unlike to excite higher modes in a structure. Another advantage of linear damping is that there is very little interaction between damping forces and structural forces. Structural forces peak when damping forces are zero as well as damping forces peak when structural forces are zero. Between these points there is a gradual transfer of force [Lee and Taylor, 2001].

Applying the force – velocity relationship expressed in Eq. 2.10 to Eq. 2.6 results:

[Eq. 2.12]

which represents the dissipated energy for a non-linear fluid viscous damper. The hysteretic behaviour of fluid viscous dampers can be plotted and shown in Fig. 2.13.

 Fig. 2.13 Hysteresis Loops for Linear and Non-linear Fluid Viscous Dampers [Lee and Taylor, 2001] In Fig. 2.13, non-linear damping with a low exponent shows much more rectangular hysteresis curve and the damping forces tend more to superimpose on the structural forces. In addition, non-linear damping can possibly excite higher modes in a structure.

In the case of a linear damper, the hysteresis loop is a pure ellipse. In this case it is clear that the dissipated energy is lower than the case of a non-linear damper for similar conditions. As example, Fig. 2.14 shows typical Force – Displacement hysteretic curves of a non-linear viscous damper according to prototype tests carried out by FIP Industriale Laboratories (Italy).

Fig. 2.14 Force – Displacement Hysteretic Diagram of a Viscous Damper, N = 0.15 [Courtesy of FIP Industriale, s.P.a., Italy]
2.3.2 Effect of the Damper Parameters
2.3.2.1 Damping coefficient cd

In general terms, for viscous dampers, cd does not affect the shape of the hysteretic force – displacement cycle; however, an increase of the value of this parameter increases the energy dissipation capacity and the maximum force in the device [Guerreiro, 2006]. In this sense, the work carried out by Virtuoso et al (2000) studies the modelling of the seismic behaviour of bridges with added viscous dampers, analyzing the effects of the constant cd (here called C). To allow an analysis on the influence of that parameter on the structural response, values of the constant C between 0.10 and 10 were considered, since those values, together with the values considered for the parameter N (here called α), will cover forces corresponding to seismic coefficient varying from 1% to 50% of the weight. In this study a set of five artificial accelerograms compatible with the response spectrum defined in Eurocode 8 – Part 2 [CEN, 1998b] with a peak ground acceleration of 0.30g, type B soil and 30 sec total duration of the series, were used. Two extreme cases were considered: a solution without elastic stiffness (deck totally free over the piers) and a solution with elastic stiffness (low stiffness elastic connection between the piers and the deck). Also, in this research the configurations of the force – velocity relation curves were presented for different values of α, corresponding to the linear branch, which, were defined by the origin and the point corresponding to 10% of the maximum velocity and force corresponding to the defined seismic action and obtained without the consideration of the linear branch. Figs. 2.15 and 2.16 show maximum forces and displacements in the viscous damper without and with elastic stiffness respectively. They show that solution involving a higher displacement control always lead to higher force levels in the device. It is also possible to observe that the more efficient solutions, with better displacement control for the same force level, generally corresponds to low α values. Likewise, for device solutions with low values of the constant C, the elastic stiffness of the structure has an important contribution on the displacement control. It is important to notice that the contribution of the elastic force is out of phase with the one transmitted by the devices, what means that, in a solution of this type there is always a force restraining the movement of the deck. The problem is that the forces transmitted to the structure must be controlled to limit the contribution of the piers to values lower than their elastic limit.

Fig. 2.15 Maximum Forces and Displacement in the Viscous Dampers – Without Elastic Stiffness [Virtuoso et al, 2000]
Fig. 2.16 Maximum Forces and Displacement in the Viscous Dampers – With Elastic Stiffness [Virtuoso et al, 2000]

Fig. 2.17 presents the forces in the structure corresponding to the device solutions considered in the study and whose results, in terms of devices response, were represented in Fig. 2.16. The results show that, for low C values the forces transmitted to the structure are important and higher than the corresponding forces in the devices. For C values higher than 1 a significant reduction on the displacements is verified. For the device solutions in this range there is no influence of the elastic stiffness in the device forces. From these results, it can be concluded that the best solutions corresponds to devices with C values higher than 1.

The contribution of the elastic spring is irrelevant for the forces in the device and conducts to some negligible reduction in the displacements when C > 1. From these results, consideration of the elastic stiffness of the structure is not important for the displacement control of the deck.

 Fig. 2.17 Maximum Forces in the Structure – Solution with Dampers and Elastic Stiffness [Virtuoso et al, 2000] The presence of the elastic force transmitted by the piers can be important to recover the initial position of the deck after an earthquake and to provide a minimum stiffness for slow movements of the deck.

2.3.2.2 Velocity exponent N

The damping exponent N represents the essence of the non-linear behaviour of fluid viscous dampers. On the contrary of the damping coefficient cd, this parameter does not affect the size of the hysteretic force-displacement cycle and for that reason incidence of this parameter on the seismic response is not decisive as occurs with the damping coefficient, aspect enlarged in 2.3.2.1. Changes in the N-exponent imply changes in the shape of the hysteretic force-displacement cycle, as was explained in Fig. 2.13. Low damping exponents tend to expose rectangular force-displacement hysteresis, as well as linear behaviour implies more elliptical force-displacement hysteresis cycles. The more practical incidence of the N-exponent relates with the damper forces, depending on the relative velocities.

If we consider the force at the dampers F as a function of the exponent N, we can write

where cd is a constant.

If cd is constant, F is maximum if is maximum.

Let .Maximizing f:

if and only if

if and only if which implies a constant force F = cd

Analyzing f in its domain:

(1) If then f is maximum if N is maximum, that is to say, if
(2) If then can be written as .

Then, f is maximum if is little, which implies .

This analytical approach shows that the critical point is . Being the damper velocities larger than 1, the maximum damper forces are obtained for high values of the damping exponent, on the contrary of the case where the damper velocities are lower than 1, in which the maximum damper forces are obtained when N is close to zero, that is to say, for non-linear dampers. Graphically, the above-mentioned can be clearly exposed in Fig. 2.18. Fig. 2.18 exposes variations of the dampers forces with the velocity exponent N for some common damper velocities. From these results, it is necessary to be cautious if velocity pulses are considered in the presence of linear dampers or dampers with N > 1. Likewise, similar considerations are necessary to take into account if non-linear dampers are considered in the presence of low velocities.

 Fig. 2.18 Plot of Damper Forces as Function of the N-exponent for Several Velocities and cd = 10 MN/(m/s)N These results suggest that non-linear fluid viscous dampers can be more suitable for high velocities, as usually happens in the presence of near-fault ground motions; on the contrary of the case of low-to-moderate velocities (far-fault ground motions), in which dampers with higher velocity exponent seems to be more adequate.

The consideration of the damper parameters must be taken carefully, because sometimes interpretation of the results could be confused. In this sense, tables 2.2 and 2.3 show similar conditions of the damper parameters. Table 2.2 exposes the forces for a damper with cd = 2015 kN/(mm/sec)N, and design velocity of 300 mm/sec for three different values of N. The obtained forces increase when N increases. On the right, in Table 2.3, the same situation is represented; however, the units have been changed. Because of the units for the damping coefficient cd depend on the value of N it is necessary to be cautious with the change of units. This transformation implies a change of the values for the damping coefficients. Now, cd increases as a result of this change, which implies that the damper forces obviously increase. Notice that in Table 2.2 damper velocities are higher than 1.0, which implies that for a constant value of cd damper forces increase as the damper exponent increases, as was previously explained. In Table 2.3, damper forces increase because the damping coefficient increases, regardless the damper exponent increases and the damper velocity is lowers than 1.0.

Table 2.2 Damper Forces for Three Different Damper Exponents and cd in kN/(mm/sec)N
 N Cd [kN/(mm/sec)N] Velocity [mm/sec] F [kN] 0.015 2015 300 2195 0.15 2015 300 4740 0.25 2015 300 8386

Table 2.3 Damper Forces for Three Different Damper Exponents and cd in kN/(m/sec)N
 N Cd [kN/(m/sec)N] Velocity [m/sec] F [kN] 0.015 2235 0.3 2195 0.15 5679 0.3 4740 0.25 11331 0.3 8386

A similar situation is exposed in tables 2.4 and 2.5. Table 2.4 shows the damper forces considering the same damper exponents, a constant value of 2235 kN/(m/sec)N for cd, and 0.3 m/sec for the damper velocity. In this case, the damper forces decrease as the damper exponent increases. This situation is in agreement with results shown in Fig. 2.18 because now the damper velocity is lower than 1. In Table 2.5, the same situation is represented, but now the units have been changed to [kN/(mm/sec)N] for cd. This change implies that the values of the damping coefficient decrease, which implies that the damper forces decrease regardless the damper velocity is higher than 1. Results of this analysis show that influence of the damper exponent N on the damper forces is in relation with relative velocities of the dampers, being one the critical value. It is not possible to formulate valid conclusions only considering the damper exponent and the damper velocity, as some manufacturers propose. It is necessary to take into account the damping coefficient cd and its units, which depends in some sense on the damping exponent.

Table 2.4 Damper Forces for Three Different Damper Exponents and cd in kN/(m/sec)N
 N Cd [kN/(m/sec)N] Velocity [m/sec] F [kN] 0.015 2235 0.3 2195 0.15 2235 0.3 1866 0.25 2235 0.3 1654

Table 2.5 Damper Forces for Three Different Damper Exponents and cd in kN/(mm/sec)N
 N Cd [kN/(mm/sec)N] Velocity [mm/sec] F [kN] 0.015 2015 300 2195 0.15 793 300 1866 0.25 397 300 1654

Commonly, use of fluid viscous dampers limits the damping exponent N between 0.1 and 1.0 for seismic applications. Recently, some manufacturers propose the application of extra-low damping exponents, using values in the order of 0.02 or lower. As was previously explained, using damping exponents close to zero implies an almost constant response force for the damper, aspect that can be useful for situations involving high damper velocities or velocity pulses, as usually happens in the presence of near-fault ground motions. Fig. 2.19 shows the Damper Force – Velocity relation for an extra-low damping exponent damper. In this case, 0.015 damping exponent was selected according to practical applications of some manufacturers. For damper velocities higher than 0.7-1.0 m/sec, the damper responses with an almost constant force, well defined to a certain limit. This special characteristic can be very positive to control peak responses when high velocities are demanding the damper, however, this selection cannot be an efficient solution for earthquakes inducing low-to-moderate velocities.

 Fig. 2.19 Extra-low Damping for Viscous Damper with N=0.015 In case of application of extra-low damping exponents, no structural damages occur even in case the earthquake was more severe than expected, and the structure can be easily calculated with this maximum response force, independent from velocity. This allows designers to model the dampers with a bilinear force-displacement relation, characterized by a force independent on the displacement.

2.3.3 Non-linear Viscous Dampers
2.3.3.1 Earthquake response

Numerous experimental and analytical investigations have focused on linear fluid viscous dampers, because they can be modelled simply by a linear dashpot. While being effective in reducing seismic demands on the structure, linear viscous dampers may develop excessive damper forces in applications where large structural velocities can occur, as for example in long period structures subjected to intense ground shaking, especially in the near-fault region. Recently, some researchers and earthquake engineering professionals have begun to focus on fluid viscous dampers exhibiting non-linear Force-Velocity relationship because of their ability to limit the peak damper force at large structural velocities while still providing sufficient supplemental damping [Lin and Chopra, 2002; Symans et al, 2008].

In the last years, some analytical and experimental investigations have been conducted regarding the dynamic response of fluid viscous dampers, and especially, with non-linear dampers. In order to verify the behaviour and constitutive laws, prototype viscous dampers have been tested at National Center for Earthquake Engineering Research in Buffalo, USA [Seleemah and Constantinou, 1997]; University of Florence [Terenzi, 1999] and University of California at Berkeley [Infanti et al, 2003].

Although the mean response spectra for deformation, relative velocity, and total acceleration are affected very little by damper non-linearity, the influence increases at longer periods and for smaller values of the non-linearity parameter (here called α). Fig. 2.20 shows as example, the mean response spectra for deformation, relative velocity and total acceleration for elastic single-degree-of-freedom systems and considering 20 ground motions. If the ratio of responses r for α=0.35 and 1 are plotted for three response quantities, as shown in Fig. 2.21, clearly, then damper non-linearity has essentially no influence on system response in the velocity-sensitive spectral region (0.6 ≤ Tn ≤ 3 sec) and small influence in the displacement (Tn ≥ 3 sec) and acceleration (Tn ≤ 0.6 sec) sensitive regions. These aspects has the useful implication for design applications that, for a given supplemental damping ratio ξd, the response of systems with non-linear fluid viscous dampers can be estimated to a sufficient degree of accuracy by analysing the corresponding linear viscous system (α = 1). Likewise, damper non-linearity has very little influence on the deformation, velocity and acceleration time histories of the system (Fig. 2.22), but affects the damper force significantly, primarily near the response peaks, as was previously explained.

Fig. 2.20 Mean Response Spectra for (a) Deformation, (b) Relative Velocity, and (c) Total Acceleration for SDF Systems with ξ = 5% and Supplemental Damping ξd = 0, 5, 15 and 30% due to Non-linear Fluid Viscous Dampers with Different α Values [Lin and Chopra, 2002].

Regarding the influence of supplemental damping, as expected, it reduces the structural response, with greater reduction achieved by the addition of more damping (Fig. 2.20). As Tn→0, supplemental damping does not affect response because the structure moves rigidly with the ground. And as Tn→∞, supplemental damping again does not affect the response because the structural mass stays still while the ground underneath moves. The response reduction is significant over the range of periods considered. Moreover, the reduction in responses is essentially unaffected by damper non-linearity in the velocity-sensitive region and only weakly dependent in the acceleration and displacement sensitive regions (Fig. 2.21).

 Fig. 2.21 Influence of Damper Non-linearity on Mean Peak Responses, r: Deformation, Relative Velocity, and Total Acceleration for Systems with ξd = 30% [Lin and Chopra, 2002]. Fig. 2.22 Response History for Deformation of a SDF System (Tn = 1 sec, ξ = 5%) with ξd = 15% [Lin and Chopra, 2002].

It is important to say that for a given force and displacement amplitude, the energy dissipated per cycle for a nonlinear fluid damper is larger, by a factor λ/π (where λ is a parameter whose value depends exclusively on the velocity exponent), than that for the linear case and increases monotonically with reducing velocity exponent (up to a theoretical limit of 4/π=1.27 which corresponds to a velocity exponent of zero); however, the additional energy dissipation afforded by the nonlinear dampers is minimal. For a given frequency of motion, ω, and displacement amplitude, x0, to dissipate the same amount of energy per cycle, the damping coefficient of the nonlinear damper, cdNL, must be larger than that of the linear damper, cdL, as given by

[Eq. 2.13]

As an example, for a frequency of 1.0 Hz and displacement amplitude of 5 cm, the damping coefficient of a nonlinear damper with velocity exponent of 0.5 must be approximately three times larger than that of a linear damper to dissipate the same amount of energy per cycle. Conversely, if nonlinear dampers are used to limit the damper force, a reduction in energy dissipation capacity as compared to the case of linear dampers would be accepted [Symans et al, 2008].

A last aspect to consider regarding the earthquake response of non-linear fluid viscous dampers, is that the earthquake-induced force in a non-linear viscous damper can be estimated from the damper force in a corresponding system with linear viscous damping, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo-velocity as this approximation introduces a large error in the damper force. In fact, if spectral pseudo-velocities are used, they are based on design displacements (Sv = ω0Sd). It is well known that effectiveness of non-linear viscous dampers is highly dependent on operating velocities, being necessary to have reliable estimates of the true velocity in the device [Pekcan et al, 1999; Lin and Chopra, 2002].

2.3.3.2 Equivalent linear viscous damping

The energy dissipation capacity of a fluid viscous damper can be characterized by the supplemental damping ratio ξd and its non-linearity by the parameter N; and it is found that the structural response is most effectively investigated in terms of these parameters because they are dimensionless and independent, and the structural response varies linearly with the excitation intensity [Lin and Chopra, 2002]. In this sense, a system with non-linear dampers is usually replaced by an equivalent linear system, with its properties determined using different methods: equalling the energy dissipated in the two systems [Jacobsen, 1930; Fabunmi, 1985]; equalling power consumption in the two systems [Pekcan et al, 1999]; replacing the non-linear viscous damping by an array of frequency and amplitude-dependent linear viscous model [Rakheja and Sankar, 1986]; random vibration theory [Caughey, 1963; Roberts, 1976], and more recently, applying closed-form formulas based on probabilistic concept to obtain fundamental modal damping ratio without carrying out structural analysis [Lee et al, 2004].

Thus, equalling the energy dissipated in a vibration cycle of the non-linear system to that of equivalent viscous system [Pekcan et al, 1999] and considering equation 2.10:

Soong and Constantinou (1994) have shown that the work done (dissipated energy) in one cycle of sinusoidal loading can be written as

[Eq. 2.14]

that is basically the same equation as 2.6. Here, T0 = 2π/ω0, where ω0 is the circular frequency of the system and .

Equation 2.14 can be integrated to give

[Eq. 2.15]

where Г( ) is the gamma function.

The equivalent (added) damping is calculated by equating equation 2.15 and the energy dissipated in equivalent viscous damping:

4πξdω0Es = Wd [Eq. 2.16]

in which strain energy Es = kx02/2. Solving Eq. 2.16 for equivalent damping ratio:

[Eq. 2.17]

where M is the mass of the system, and x0 the amplitude of harmonic motion at the undamped natural frequency ω0.

Of course, the additional damping that the passive system introduces to the structure can be obtained by its energy dissipation capacity in each hysteretic cycle. This dissipated energy for each cycle, can be obtained calculating the area of the cycle in the force – displacement relationship of the viscous damper. Thus, for a selected cycle, it is possible to assess the equivalent damping ratio as follows:

[Eq. 2.18]

where umax and xmax are the maximum force and maximum displacement at the damper respectively.

Pekcan et al (1999), proposed a simple method for making the transformation from the non-linear damper behaviour to equivalent viscous damping. They explain that for velocity-dependent systems such as viscous dampers, consideration of the rate of energy dissipation – that is power (rather than energy) – becomes more important in seeking the equivalent linear properties for these systems. The proposed equivalent damping is

[Eq. 2.19]

Given the customary definition of damping ratio (ξ) obtained from c = 2ξω0M, equation 2.19 can be expressed as follows:

[Eq. 2.20]

This formulation, also called power equivalent approach, predicts higher damping values compared with energy based method (Eq. 2.17). This difference is greater for low values of N, and for that reason it is necessary to be cautious using any of the above formulations for small N powers (N<0.1), since the mechanism of the devices changes from viscous (velocity dependent) to Coulomb friction type (when N tends to zero).

2.3.4 Performance of Viscous Dampers During Near-field Ground Motions

Near-field earthquakes are characterized by short duration pulses of long period with large peak ground velocities and accelerations. It has been observed from recent earthquake records that motions in the fault-normal direction contain destructive long-period pulses with high peak ground velocities, aspect that negatively affects long-period structures such as cable-stayed bridges. A lot of approaches to model these pulses have been recently proposed [Makris, 1997; He, 2003; Mavroeidis et al, 2004].

Linear viscous dampers have been found to perform well during mild to moderate earthquakes. However, the force demand on linear dampers during pulse-type excitations may be excessive, leading to device capacity saturation and larger force demands on structural components. Non-linear viscous dampers may be more suitable in such situations because of their inherent force saturation capability at high velocities.

The recent investigation by Xu et al (2007) on the performance of passive energy dissipation systems during near-field ground motions, shows that both linear and non-linear viscous dampers with 25% supplemental damping ratio are effective in achieving more than 40% displacement reduction when 3/5 < Tn/Tp < 5/3, Tn and Tp being structural and excitation periods, respectively. Non-linear viscous dampers can yield additional 10% reductions in displacement and input energy over those by linear dampers when Tp > 4/5Tn and they achieve less displacement reduction when Tp < 4/5Tn. Likewise, performance of viscous dampers depends on their absorbability of instantaneous input energy. If a damper cannot dissipate the input energy instantaneously, even through it may have an excellent ability to dissipate the total input energy, the structure may still undergo damage due to instantaneous accumulation of input energy during an earthquake.

Although the damper non-linearity does not significantly influence the displacement response (As was demonstrated in the research by Lin and Chopra, 2002), in general terms non-linear viscous dampers are more advantageous than linear dampers in reducing peak structural displacements and peak input energies when a structure is subjected to pulse-type excitation with pulse period longer than the natural period of the structure.

2.4 Analysis and Design Issues
2.4.1 Structural Analysis Including Viscous Dampers

The first step in the analysis is to find out how added damping affects the structure. This is generally done with a simple stick model with one node for each storey. Adding global damping to the stick model provides a good indication of how damping elements can benefit the structure. The analyst will then construct a simple two-dimensional model of the structure. In this model the dampers are entered as discrete elements. At this point there are a number of variables to play with: force capacity of the dampers, location and number of dampers, damper coefficient and damper exponent. The analyst has the task of finding the best solution. This is generally a trial-and-error process but there are some general guidelines. It is always best to minimize the number of dampers and the number of bays that use dampers. Also, it is known from experience that approximately 20%–30% of critical damping is a desirable range, and that 5% of this can be structural, leaving 15%–25% for viscous damping. So the first objective of the analyst is to determine the smallest possible number of dampers to provide approximately 20% critical damping without overloading either the beams or the columns. Also, it is always best to start with linear dampers and then find out what happens with nonlinear dampers after the locations, number and characteristics of the dampers have been fairly well determined [Lee and Taylor, 2001].

Note that analysis of a structure with dampers always involves a step-by-step time-history simulation. Sometimes a time-history is not available for a particular location but a shock spectrum is. In this case, a time-history can be arrived at by going through a library of time histories, comparing their shock spectra with the specified shock spectrum at the site and selecting the one that fits best. Likewise, with the current computer capability, a detailed non-linear time history analysis to satisfy individual requirements can be applied. Some advantages of a non-linear time history analysis include: more exact determination of structural displacements, more accurate assessment of the seismic response forces acting onto the device and structure, exact evaluation of real structural safety factors and possible economical benefits due to savings in design.

Finally, it can be important to say that in the present days exist good and powerful computing tools that permit to solve non-linear structures equipped with linear/non-linear energy dissipation devices such as fluid viscous dampers. Commercial computing codes such as ANSYS [Ansys Inc, 2005] or SAP2000 [Computers & Structures, 2007] include the option of applying non-linear energy dissipation devices. However, modelling of some damping elements (e.g. dampers with temperature-dependent or frequency-dependent) can be more challenging or, in some cases, not possible with a given program. When the modelling of such behaviour is not possible, the expected response may be bounded by analyzing the structure over a range of behaviours. Fortunately, for majority of fluid viscous dampers actually manufactured, properties are largely independent with respect to frequency and temperature [Symans et al, 2008].

2.4.2 Design Issues for Viscous Dampers

The peak force fD0(N) in the non-linear fluid viscous damper with known non-linear parameter N can be expressed as

[Eq. 2.21]

where V = ω0x0 is the spectral pseudo-velocity for the SDF system; c1 is the damping coefficient of the linear system and βN is a constant defined as

[Eq. 2.22]

The non-linear damper force can be computed from Eq. 2.21 if x0 and are known. They can be estimated as the peak values of deformation and relative velocity of the corresponding linear system. Equation 2.21 is almost exact in the velocity-sensitive region of the spectrum, overestimates the damper force in the acceleration-sensitive region (by at most 15%); and underestimates in the displacement-sensitive region (by at most 7%). Moreover, the accuracy of Eq. 2.21 deteriorates slightly with the increase of the equivalent damping ξd. However, the actual velocity of the corresponding linear system required in Eq. 2.21 and to compute fD0 (N=1) = c1 is not readily available, because the velocity spectrum is not plotted routinely. If the velocity is replaced by the pseudo-velocity, Eq. 2.21 changes to

[Eq. 2.23]

The resulting estimate of the damper force is not accurate, which increase with the system’s period, damper non-linearity and supplemental damping ratio. Thus, velocity should not be approximated by the pseudo-velocity [Lin and Chopra, 2002].

Another important point regarding the design of non-linear fluid viscous dampers is how to select the properties cd and N to satisfy a design requirement. As was previously explained, the structural deformation is essentially unaffected by the damper non-linearity parameter N and it is essentially the same as that for the corresponding linear system. The total damping capacity that must be provided in the system to limit the deformation of a linear system to a design value can be determined directly from the design spectrum. Subtracting the inherent damping in the structure from the total damping required gives ξd, the necessary supplemental damping. Many different non-linear fluid viscous dampers can be chosen to provide the required supplemental damping ratio ξd. Thus, for a selected value of N:

[Eq. 2.24]

where M is the mass of the system and D is the allowable deformation.

Fig. 2.21 suggests that the selected damper defined by Eq. 2.24 should satisfy the design constraint reasonably well. Also, the structural deformation should be very close to the allowable value in the velocity-sensitive region, less than the allowable value in the acceleration–sensitive spectral region, but exceed slightly the allowable value in the displacement-sensitive spectral region [Lin and Chopra, 2002].

2.5 Practical Applications
2.5.1 Study Case 1: Rion-Antirion Bridge, Greece

Amongst long-span cable-stayed bridges that incorporate additional passive seismic protection, the recently inaugurated Rion – Antirion Bridge in the Gulf of Corinth, Greece, is one of the most interesting bridges located in a high seismicity zone generated by active local faults.

The structure is a concrete multi-span double-plane semi-harp type cable-stayed bridge, with a continuous floating deck, as can be appreciated in Fig. 2.23. The 2252 m-long bridge is divided into three spans of 560 m and two of 286 m (Fig. 2.24). A general description of the structure and the basic aspects regarding the design and construction can be found in the works of Combault et al (2000), Teyssandier (2002) and Teyssandier et al (2003).

The bridge was designed to resist seismic events of 0.48g – peak ground acceleration, and tectonic motion for two consecutive pylons up to 2 m at any direction. That was possible by using an energy dissipation system, connecting the deck with each pylon, which limited their motion during the occurrence of a strong earthquake, while it dissipated energy. The basic aspects of the seismic design of the bridge include a response design spectrum that corresponds to a 2000 years return period (Fig. 2.25) with a peak spectral acceleration equal to 1.20g.

Fig. 2.23 General View of the Rion-Antirion Bridge [Infanti et al, 2004]
Fig. 2.24 Longitudinal Geometry of the Rion-Antirion Bridge [Teyssandier et al, 2003]
 The damping system consisted in fuses and viscous dampers acting in parallel, connecting the deck with the pylons in the transverse direction. The fuses were designed to work as rigid connections to resist low-to-moderate intensity earthquakes as well as high wind loads. For the design earthquake, the fuses were calculated to fail allowing energy dissipation through the fluid viscous dampers. Fig. 2.25 Response Design Spectrum [Combault et al, 2000]

For the dynamic analysis, a 3D-finite element model was used for the whole structure, taking into consideration important aspects, such as [Combault et al, 2000]:

• Non-linear hysteretic behaviour of the reinforced soil
• Possible sliding of the pylon bases on the gravel beds precisely adjusted to the accompanying vertical force
• Non-linear behaviour of the reinforced concrete of the pylon legs (including cracking and stiffening of concrete due to confinement)
• Non-linear behaviour of the cable-stays
• Non-linear behaviour of the composite deck (including yielding of steel and cracking of the reinforced concrete slab)
• Second order effects

Fig. 2.26 shows the isolation system in the Antirion approach viaduct and Fig. 2.27 shows the fuse restraint general configuration.

 Fig. 2.26 Isolation System in the Antirion Approach Viaduct [Infanti et al, 2004] Fig. 2.27 Fuse Restraint [Infanti et al, 2003]

On each pylon, four viscous dampers of 3500 kN - reaction capacity each and damping constant C = 3000 kN/(m/s)0.15 were installed. The fuses consider a reaction capacity of 10500 kN. For the transition pylons, the same previous dampers were used, but in conjunction with structural fuses of 3400 kN - reaction capacity each.

The seismic performance and the energy dissipation requirements were evaluated applying non-linear time history analysis of a 3D model of the structure. However, the seismic design hypotheses and the real behaviour of the devices could be verified with a full-scale testing. Fluid Viscous Damper Prototype tests were performed at the laboratory of the University of California – San Diego (USA), and the Fuse Restraints were tested at the FIP Industriale Testing Laboratory (Italy). The full-scale testing of the seismic devices is explained by Infanti et al (2003, 2004). In their works, they show the methodology, implementation and results of the full-scale testing. Figs. 2.28 and 2.29 show the full-scale damper testing and a view of the Fuse Restraint Testing during fatigue test respectively.

Another interesting aspect included in the Rion-Antirion Bridge, is the addition of anti-seismic deviators that work as dampers for the stay-cable vibration mitigation. Although hydraulic dampers are also used on cable-stayed and suspended bridges to reduce rain-wind vibration, they have a clear anti-seismic purpose as was commented in the previous pages. Under certain circumstances, cable vibrations can modify the global seismic response of the bridge, introducing energy in higher order vibration modes. In this sense, Lecinq et al (2003) gives a description of the alternatives to increase the damping on stay-cables, and explain the anti-seismic deviators employed in the Rion-Antirion Bridge. Fig. 2.30 shows a render view of an internal hydraulic damper used for cable vibration mitigation of cable-stayed bridges. Fig. 2.31 shows an external damper for cable vibration used in the Normandy Bridge (France).

 Fig. 2.28 Full-Scale Viscous Damper Prototype Testing [Infanti et al, 2004] Fig. 2.29 Fuse Element During Fatigue Test [Infanti et al, 2004]

 Fig. 2.30 Internal Hydraulic Damper [Lecinq et al, 2003] Fig. 2.31 External Hydraulic Damper on Normandy Bridge [Lecinq et al, 2003]

In his MSc Thesis, Morgenthal (1999) carries out a detailed research on the seismic behaviour of the Rion-Antirion Bridge. He describes the bridge and exposes analytical modelling using finite elements to study the seismic control strategies incorporating different seismic protection devices, such as structural fuses, hydraulic dampers, seismic connectors and elasto-plastic isolators. Finally, a parametric analysis of the seismic behaviour of different deck isolation devices is exposed.

2.5.2 Study Case 2: Tempozan Bridge, Japan

Due to severe damage to bridges caused by the Hyogo-ken-Nanbu earthquake in 1995, very high ground motion was required according to the bridge design specifications set in 1996 [Japan Road Association, 1996], in addition to the relatively frequent earthquake motion specifications by which old structures were designed and constructed. Hence, seismic safety of cable-stayed bridges that were built prior to that specification was reviewed, and seismic retrofit was performed. In order to study the effectiveness of passive control to the seismic retrofit of a cable-stayed bridge, a numerical analysis on a model of a cable-stayed bridge was carried out. An existing cable-stayed bridge with fixed-hinge connections between deck and towers was modelled and its connections were replaced by isolation bearings and dampers. The isolation bearings were assumed to be of the elastic and hysteretic type. The dampers were linear and variable. The objective was to increase the damping ratio of the bridge by using passive control technologies. The chosen bridge model was the Tempozan Bridge, located in Osaka, Japan.

The Tempozan Bridge, built in 1988, is a three-span, continuous steel, cable-stayed bridge situated on reclaimed land. It crosses the mouth of the Aji River in Osaka, Japan. The total length of the bridge is 640 m with a centre span of 350 m, while the lengths of side spans are 170 and 120 m (Figs. 2.32 and 2.33).

 Fig. 2.32 Tempozan Bridge [from en.structurae.de] Fig. 2.33 Side View of the Tempozan Bridge [Iemura and Pradono, 2003]

The main towers are A-shaped to improve the torsional rigidity. The cable in the superstructure is a two-plane, fan pattern, multicable system with nine stay cables in each plane. The bridge is supported on a 35-m-thick soft soil layer and the foundation consists of cast-in-place reinforced concrete piles of 2-m diameter. The main deck is fixed at both towers to resist horizontal seismic forces. The bridge is relatively flexible with a predominant period of 3.7 sec. As to the seismic design in transverse direction, the main deck is fixed at the towers and the end piers [Iemura and Pradono, 2003].

If the deck is connected with very flexible bearings to the towers, the induced seismic forces will be kept to minimum values but the deck may have a large displacement response. On the other hand, a very stiff connection between the deck and the towers will result in lower deck displacement response but will attract much higher seismic forces during an earthquake, which is the case of the original bridge structure. Therefore, it is important to replace the existing fixed-hinge bearings with special bearings or devices at the deck-tower connection to reduce seismic forces, absorb large seismic energy, and reduce the response amplitudes. Additionally, energy-absorbing devices may also be put between the deck-ends and piers. However, because doing so will attract a relatively large lateral force to the piers, this is avoided for this bridge at present.

The original structure system has fixed-hinge connections between the towers and the deck and roller connections between the deck-ends and piers, so that the deck longitudinal movement is constrained by the towers (Fig. 2.34a). For the retrofitted bridge, isolation bearings and dampers connect the deck to the towers (Fig. 2.34b). The cables were modelled by truss elements, the towers and deck were modelled by beam elements, and the isolation bearings were modelled by spring elements. The moment–curvature relationship of the members was calculated based on sectional properties of members and material used.

Fig. 2.34 Bridge Models [Iemura and Pradono, 2003]

The first modes of the structures are interesting because these modes have the largest contribution to the longitudinal motion of the bridge. The first mode shape of the original structure is shown in Fig. 2.35a. The natural period (T) of this mode is 3.75 sec, which is close to the design value for the bridge (3.7 sec). This first mode has effective modal mass as a fraction of total mass of 84%. For the retrofitted structure, the stiffness of bearings was an important issue as large stiffness produces large bearing force and makes any energy-absorbing device work ineffectively in these connections (Fig. 2.35b). However, very flexible connections produce large displacement response. Therefore, based on a study on a simplified model of the bridge under seismic motions, a bearing stiffness that produced a retrofitted main period (T) 1.7 times the original main period was chosen. This bearing stiffness makes the energy-absorbing devices work well in reducing seismic-induced force and displacement. The main natural period of the retrofitted bridge (T′) then becomes 6.31 sec and the effective modal mass as a fraction of total mass is 92%. It is clear from the figures that smaller curvatures were found at the towers and the decks of the retrofitted structure than were found in those of the original structure.

The models were analyzed by a commercial finite element program [Prakash and Powell, 1993] which produces a piecewise dynamic time history using Newmark’s constant average acceleration (β= 1/4) integration of the equations of motion, governing the response of a nonlinear structure to a chosen base excitation. The input earthquake motions were artificial acceleration data used in Japan for design for soft soil condition, according to the 1996 Seismic Design Specifications of Highway Bridges [Japan Road Association, 1996]. The data are intended as Type I (inter-plate type). Table 2.6 shows the seismic response effects due to different kinds of bearings and dampers: fixed-hinge bearings for the original bridge model, elastic bearings, elastic bearings plus viscous dampers, and hysteretic bearings for the retrofitted bridge model.

Fig. 2.35 First Mode Shape of Original (a) and Retrofitted (b) Structures [Iemura and Pradono, 2003]

From the table, if only elastic bearings are used for seismic retrofit, the sectional forces can be reduced to about 40% of the original ones; however, the displacement response increases to 176% of the original one. By adding viscous dampers to the elastic bearings, the sectional forces can be reduced to about 25% of the original ones and the displacement response is reduced to 63% of the original one. So the viscous dampers plus bearings work to reduce the seismic response of the retrofitted bridge. The structural damping ratio is calculated as 35%. If hysteretic bearings are used for seismic retrofit, the sectional forces are reduced to about 29% of the original ones and the displacement response is reduced to 67% of the original one. Equivalent structural damping ratio is calculated as 13.1% by using pushover analysis to obtain a hysteretic loop at the main mode. The hysteretic bearings are modelled by bilinear model and the second stiffness of the hysteretic bearings is 0.03 times the initial stiffness and produces a first mode natural period of 6.31 sec.

Table 2.6 Maximum Earthquake Responses and Damping Ratios in Longitudinal Direction [Iemura and Pradono, 2003]
 Retrofitted Structure Items Original Structure Elastic Bearings Elastic Bearings + Viscous Dampers Hysteretic Bearings Deck displacement