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CENTRO INTERNACIONAL DE METODOS NUMERICOS EN INGENIERIA

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.

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.

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].

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):

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].

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 19^th 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. 1.16 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].

1.1 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).

1.2 Technological Aspects

1.2.1 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 1800`s, 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 80`s, 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].

1.2.2 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.V^α 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.

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.

1.2.3 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 90`s. 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. 2.8 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]

1.3 Mechanical Behaviour

1.3.1 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 [D`Alembert, 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 m_s damping coefficient c_s, stiffness k_s and control force u, subject to ground acceleration is:

where ${\textstyle x(t)}$ , ${\textstyle {\overset {.}{x}}(t)}$ and ${\textstyle {\overset {..}{x}}(t)}$ 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: ${\textstyle \int _{0}^{x}m_{s}{\overset {..}{x}}(t)dx+\int _{0}^{x}c_{s}{\overset {.}{x}}(t)dx+}$${\displaystyle \int _{0}^{x}k_{s}x(t)dx+\int _{0}^{x}udx=-\int _{0}^{x}m_{s}{\overset {..}{x}}(t)dx}$ where each term is now an energy component. Thus, we can define: ${\textstyle \int _{0}^{x}m_{s}{\ddot {x}}dx=\int _{0}^{x}m_{s}{\frac {d{\dot {x}}}{dt}}dx=}$${\displaystyle \int _{0}^{x}m_{s}{\dot {x}}d{\dot {x}}={\frac {1}{2}}m_{s}{\dot {x}}^{2}=}$${\displaystyle E_{k}}$ [Eq. 2.3] ${\textstyle \int _{0}^{x}c_{s}{\dot {x}}dx=\int _{0}^{x}c_{s}{\frac {dx}{dt}}{\frac {dt}{dt}}dx=}$${\displaystyle \int _{0}^{t}c_{s}{\dot {x}}^{2}dt=E_{v}}$ [Eq. 2.4]

${\textstyle \int _{0}^{x}udx=E_{h}}$ [Eq. 2.6]

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

where:

E_k: Kinetic energy

E_v: Dissipated energy by inherent damping

E_e: Elastic strain energy

E_h: Dissipated energy by additional damping devices

E_i: 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 (E_s) has to be as low as possible to avoid damages. Therefore the value of the dissipated energy (E_d) must be great. In the term E_h 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

${\textstyle E_{s}=E_{e}+E_{k}}$ : Stored energy within structure

${\textstyle E_{d}=E_{h}+E_{v}}$ : Dissipated energy within structure

Thus:

The control force u by non-linear viscous dampers with damping coefficient c_d 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:

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).

1.3.2 Effect of the Damper Parameters

_1.3.2.1 Damping coefficient c_d

In general terms, for viscous dampers, c_d 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 c_d (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. 2.17 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.

1.3.2.2 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 c_d, 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

${\textstyle F(N)=c_{d}{\dot {x}}^{N}}$ where cd is a constant.

If c_d is constant, F is maximum if ${\textstyle {\dot {x}}^{N}}$ is maximum.

Let ${\textstyle f(N)={\dot {x}}^{N}}$ .Maximizing f:

${\textstyle f{\grave {(}}N)={\dot {x}}^{N}log{\dot {x}}=0}$ if and only if  

${\textstyle log{\dot {x}}=0}$ if and only if  ${\textstyle {\dot {x}}=1}$ which implies a constant force F = cd

Analyzing f in its domain:

(1) If ${\textstyle {\dot {x}}>1}$ then f is maximum if N is maximum, that is to say, if ${\textstyle N\rightarrow \infty }$

(2) If ${\textstyle 0\leq {\dot {x}}<1}$ then ${\textstyle f(N)={\dot {x}}^{N}}$ can be written as ${\textstyle f(N)={\left({\frac {1}{m}}\right)}^{N}={\frac {1}{m^{N}}};{\mbox{ }}m\in \mathbb {R} }$ .

Then, f is maximum if ${\textstyle m^{N}}$ is little, which implies ${\textstyle N\rightarrow 0}$ .

This analytical approach shows that the critical point is ${\textstyle {\dot {x}}=1}$ . 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 c_d = 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 c_d 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, c_d 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 c_d 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.

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 c_d, 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 c_d. 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 c_d and its units, which depends in some sense on the damping exponent.

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.

1.3.3 Non-linear Viscous Dampers

1.3.3.1 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 ≤ T_n ≤ 3 sec) and small influence in the displacement (T_n ≥ 3 sec) and acceleration (T_n ≤ 0.6 sec) sensitive regions. These aspects has the useful implication for design applications that, for a given supplemental damping ratio ξ_d, 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.

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 T_n→0, supplemental damping does not affect response because the structure moves rigidly with the ground. And as T_n→∞, 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, c_dNL, must be larger than that of the linear damper, c_dL, as given by

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 (S_v = ω₀S_d). 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].

1.3.3.2 Equivalent linear viscous damping

The energy dissipation capacity of a fluid viscous damper can be characterized by the supplemental damping ratio ξ_d 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

that is basically the same equation as 2.6. Here, T₀ = 2π/ω₀, where ω₀ is the circular frequency of the system and ${\textstyle {\overset {.}{x}}=x_{0}sin{\omega }_{0}t}$ .

Equation 2.14 can be integrated to give

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:

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

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:

where u_max and x_max 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

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

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).

1.3.4 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 < T_n/T_p < 5/3, T_n and T_p 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 T_p > 4/5T_n and they achieve less displacement reduction when T_p < 4/5T_n. 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.

1.4 Analysis and Design Issues

1.4.1 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].

1.4.2 Design Issues for Viscous Dampers

The peak force f_D0(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 β_N is a constant defined as

[Eq. 2.22]

The non-linear damper force can be computed from Eq. 2.21 if x₀ and ${\textstyle {\overset {.}{x_{0}}}}$ 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 ξ_d. However, the actual velocity ${\textstyle {\overset {.}{x_{0}}}}$ of the corresponding linear system required in Eq. 2.21 and to compute f_D0 (N=1) = c₁ ${\textstyle {\overset {.}{x_{0}}}}$ is not readily available, because the velocity spectrum is not plotted routinely. If the velocity ${\textstyle {\overset {.}{x_{0}}}}$ is replaced by the pseudo-velocity, Eq. 2.21 changes to

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 c_d 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 ξ_d, the necessary supplemental damping. Many different non-linear fluid viscous dampers can be chosen to provide the required supplemental damping ratio ξ_d. Thus, for a selected value of N:

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].

1.5 Practical Applications

1.5.1 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.

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.

1.5.2 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.

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.

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.

As a conclusion remark, it is clear that additional viscous damping to control the seismic response of cable-stayed bridges is beneficial, reducing the seismic forces on members as well as their displacements by increasing the structural damping.

1.1 Introduction

As a starting point, the seismic analysis without the incorporation of additional damping devices is presented. In order to compare the analysis results of the bridge models considering the incorporation of additional passive devices, this part is essential, and must be employed as basic configuration (reference structures).

The structural analysis begins with the nonlinear static analysis under service loading. This analysis is very important, because the assessment of some geometric nonlinearities involved with the overall change in the bridge geometry is based on the deformed shape of the structures under this static condition, being the starting point of the nonlinear time history analysis. In other words, these initial stresses and strains, including all nonlinearities available, are considered as starting point for the subsequent analyses.

Secondly, the assessment of the natural frequencies, modal shapes and damping properties in order to obtain a general dynamic characterization of the bridge models, is exposed. After that, and only as first approach, the seismic response of the cable-stayed bridges applying the response spectrum method is obtained. This analysis gives an idea about the seismic response of the structures, of course, considering a linear behaviour. Because of the inherent nonlinear behaviour of the bridges, this analysis permits a comparison of the maximum values of the structural responses for a linear condition with regard to the nonlinear time history analysis, with the aim to recommend the best structural configurations for those bridges. With this analysis, the critical bridges to be analyzed applying the direct integration nonlinear time history analysis are selected. Because of the tremendous computational effort involved in the step-by-step analysis, two comparable structures are selected for the subsequent analyses.

The last point of this chapter corresponds to the seismic response of the selected bridges considering far-fault and near-fault ground motions respectively. In this part, nonlinear time history analysis is conducted taking into account the geometric and material nonlinearities of the structures and the cable vibration effects. The input ground motions are acceleration time histories: five artificial three-orthogonal component acceleration records for far-fault analysis and five real three-orthogonal component acceleration records for near-fault analysis. The seismic response of the bridges considers the displacement, velocity and acceleration time histories and the response of the deck, cables and towers. Because of the complex nature of the nonlinear time history analysis and according to the recommendations of Eurocode 8 Part 2 [CEN, 1998b], the average of the maximum response parameters in the assessment of the structural response for the record selection is considered.

1.2 Structural Modelling

The seismic response analysis of the concrete cable-stayed bridges takes into consideration eight 3-D symmetric bridge models for an adequate study. The chosen bridges were taken from the specialized literature, and specifically, from Walter’s Bridges [Walter, 1999] including the recommendations of Aparicio and Casas (2000) and Priestley at al (1996). The examination is based on a symmetric multi-stay reference cable-stayed bridge, having two pylons, double-plane cable layout and a main span length of about 200 m. Two stay cable layouts were selected: fan-type and harp-type. The semi-harp pattern was rejected because this typology is an intermediate pattern, and both harp and fan patterns are enough for an adequate analysis. The main span lengths of the bridges are 217 m and 204.60 m, depending on the stay spacing. In this sense, long-span cable-stayed bridges have experienced adequate performance during recent earthquake events, and it is expected that short-to-medium spans bridges show a worse seismic performance, mainly if near-source effects are considered. That is the main reason to select the proposed span lengths. Moreover, the deck pattern considers two cases: a slab-type deck and a hollow-box type deck. The first one, due to its inherent flexibility, considers a stay spacing of 6.20 m. In the second case, 12.40 m - stay spacing is considered. The selected tower, for all cases, is a concrete frame-type tower, with deck levels of 30 and 60 m from bottom. The height of the towers is 81 m and 111 m respectively. All these dimensions were taken from Walter’s recommendations.

The structural analysis considers the application of two codes: RAM Advanse [RAM International, 2003] and SAP2000 [Computers & Structures, 2007]. The first software is employed to generate the whole geometry of the bridges, the 3-D meshing, the structural modelling and to carry out the static analysis. Also, this tool is employed to obtain the general dynamic characterization, such as natural frequencies and modal shapes, and the seismic analysis applying the design response spectrum method. This structural analysis package is very easy to use, with an excellent graphical interface. Some specific characteristics in relation to the structural modelling and analysis such as intuitive member selection, advanced automatic data generation, realistic rendering of 3-D structures, rigid links, tension – only elements (cable elements), P – Δ effects, eccentricity due to offset columns and beams, automatic self-weigh calculation and dynamic analysis using response spectra are considered by this software. The second tool is employed for the non-linear time history analysis of the critical bridges previously selected. To use this software, all available data is exported from RAM ADVANSE using [. DXF] extension files. This software can be used for the nonlinear seismic analysis of the bridges considering cable element modelling, incorporation of nonlinear fluid viscous dampers as link elements and the incorporation of all material and geometric nonlinearities available. Also, some specific bridge analysis tools are incorporated in the last version of this software.

Regarding the bridge modelling, the analysis is carried out considering the use of beam and cable elements for all the bridges. The deck is modelled using a single spine to avoid the use of shell elements, with the incorporation of transverse rigid-links to simulate the anchor of cables. In fact, the use of beam elements can be more useful to assess forces on members, with clear graphical results and a considerable decrease of the computing time, especially when non-linear time history analysis is applied. Moreover, the non-linear analysis takes into consideration the geometric non-linearities that are present in almost all cable-stayed bridges. These non-linearities due to high compressions in the deck and pylons are considered by the axial – bending interaction. Non-linear behaviour of cables is considered by a multi-element cable formulation (tension-only elements), in order to take into account the spatial vibrations of them. Likewise, non-linearities due to large displacements in the overall geometry are considered too. Spatial variability is not considered because of the main span length of the bridges (of about 200 m), that not recommends this effect according to Eurocode 8.

1.2.1 Geometric Layout

The longitudinal layout of the cable stays is one of the fundamental items in the design of cable-stayed bridges. It influences, in fact, not only the structural performance of the bridge, but also the method of erection and the economics.

The main longitudinal layouts are the harp pattern, semi-harp pattern and fan pattern. Whilst the harp pattern is not the best from the static or economic point of view, it is attractive because of its undeniable aesthetics advantages [Walter, 1999]. Also, this structural configuration can be very interesting from a dynamic point of view. The fan pattern brings all the stays together to the top of the pylons, being used in several recent structures. Some important advantages, according to Walter (1999) are: the total weight of the cables needed is substantially below that for a harp pattern; horizontal force introduced by the cable in the deck is less; longitudinal bending moments of the pylons remains moderate; movements of the deck due to changes in temperature can be absorbed by conventional expansion joints placed across the abutments, if the horizontal connection between the pylons and the deck is freed; flexibility of the structure is favourable where horizontal movements of the deck take place and increases the stability against seismic activity. However, the Achilles` heel of the fan pattern solution lies in the design and construction of the heads of the pylon, which are heavily stressed.

Regarding the stay spacing, it can be said that the use of wide spacing between the stays may still be imposed today. The multiple-stay has numerous advantages: the large number of elastic supports leads to moderate longitudinal bending in the deck; the individual cables are smaller than in a structure with concentrated staying, which simplifies their installation and anchoring; replacement of stays is relatively simple, which is an important aspect to consider mainly when structural damage occurs after an important earthquake for example. Of course, when it is a matter of major bridges, with spans of some hundred metres, the multiple-stay is the only possible solution. The maximum spacing of the stays depends of several parameters, in particular on the width and stiffness of the deck. In fact, as general rule, spacings between 6 m and 25 m are adopted for concrete decks [Walter, 1999].

For this study, eight cases were considered for the chosen typologies. Fig. 3.1 (a) shows the case of a fan pattern with stay spacing of 6.2 m, 30 m – deck level and 81 m – tower height. Fig. 3.1 (b) is similar to Fig. 3.1 (a), but it considers a tower height of 111 m and 60 m – deck level. Figs. 3.1 (c) and (d) are similar to Figs. 3.1 (a) and (b) respectively, but the stay spacing is 12.4 m. The main span length is 217 m for all cases, with 93 m span length at the approximation spans. The relationship (l/L) = 0.43 implies rigid pylons in this case. Moreover, the relationship (H/L) = 0.23 implies a good selection for the deck level and the towers in conjunction with the whole geometry, according to the design recommendations [Aparicio and Casas, 2000].

The geometry for the harp pattern can be seen in Figs. 3.2 (a), (b), (c) and (d). In the first case, 6.2 m stay spacing, 30 m – deck level and 81 m – tower height are considered. The second case is similar to the previous one, but with 60 m – deck level and 111 m – tower height. Figs. 3.2 (c) and (d) show similar conditions to the previous cases, with 12.4 m – stay spacing. In these cases the relationship (l/L) = 0.48 implies rigid pylons, and (H/L) = 0.25 is adequate according to the design recommendations [Aparicio and Casas, 2000]. The main span length for this configuration is 204.6 m.

The extreme spacings of 6.2 m and 12.4 m were selected according to the deck type. In fact, 6.2 m – stay spacing is adequate for flexible decks, as long as 12.4 m –stay spacing is a good length for rigid decks, as the hollow-box type.

Fig. 3.1 Longitudinal Layouts for Fan Pattern (dimensions in metres)

Fig. 3.2 Longitudinal Layouts for Harp Pattern (dimensions in metres)

In the transverse direction, the majority of existing structures consist of two planes of cables, generally on the edge of the structure. However, several bridges have been successfully built recently with only one central plane of cables. In principle, it is quite possible to envisage solutions using three or more planes, with the aim of reducing the cross-sectional forces when the deck is very wide, but this possibility has been rarely exploited [Walter, 1999].

Fig. 3.3 Transverse Configuration

In the case of central suspension systems, the torsional moments to which such a system is subjected call for a rigid deck. In fact, when dealing with bridges which are very wide or which have large spans, central suspension leads to excessive torsion moments. Likewise, placing the pylons in the centre of the carriageway means inevitably increasing the width of the deck. This may prove a ruling disadvantage in the field of very large span structures, which require pylons of considerable height and thus width at the base.

In the case of cable-stayed bridges with vertical lateral suspension, the stays ensure a rigid connection between the pylons and deck. The vertical suspension does not give rise to any clearance problem above the deck. The width of the deck depends on the minimum distance required for the arms of the pylons. Of course, it is possible to reduce this one by placing these arms outside the deck, as occurs in this research (Fig. 3.3). On the other hand, the vertical lateral suspension seems to be an advantage from a seismic point of view, maybe due to its better stability, mainly when the earthquake occurs in the transverse direction. Although the stiffness and stability of the structure can be further improved by the use of A-frame pylons, the inclined suspension can give rise to certain clearance problems in the transverse direction. Also, the erection of A-frame pylons is generally more complicated than that of vertical pylons. Lateral suspension with this kind of pylons is particular suited for bridges of very large spans, where aerodynamic stability becomes all-important [Walter, 1999]. As can be appreciated in Fig. 3.4, and taking into account the above mentioned, a frame- type pylon with lateral suspension was selected. Its geometric simplicity and better comprehension of the structural and non-linear seismic characteristics is one of the main reasons to consider this typology in this study. T

Two deck levels were chosen for this study: 30 m and 60 m. The first height was selected as general rule. The second one was selected to take into consideration free height below the deck for navigation. Of course, those deck levels are a different structural and seismic condition for the bridges. In fact, in the second case due to its inherent flexibility, natural periods will be larger than the first case. This can be a special condition for the supports, and for that reason, a fixed hinge support between the deck and the pylons is selected for the bridges.

(a) A-type Pylon

(b) B-type Pylon

The main dimensions for the towers were selected according to the recommendations of Walter [1999] and Aparicio and Casas (2000). Special seismic considerations according to the recommendations of Priestley et al (1996) were applied too, considering an elastic seismic behaviour of the structures according to the general guidelines of Eurocode 8 Part 2 [CEN, 1998b] for this special kind of bridges. For the A-type pylon, the critical mechanical slenderness is λ = 124.7, which is a reasonable value for the arms. The dimensions of the cross-beams are enough, and are in relation with the whole geometry of the structure. For the B-type pylon, the critical mechanical slenderness is λ = 128, so the geometric dimensions of the arms are enough to guarantee an adequate geometric stability. Likewise, dimensions of the cross-beams are adequate. Nevertheless, the whole dimensions of the pylons (pre-design) were checked for service loads.

Regarding the decks, two 13 m - width concrete decks are proposed: slab-type and hollow-box type, as can be appreciated in Figs. 3.5 (a) and (b). The main geometric characteristics (height, area A, shear areas A_V, torsion constant J_T and inertias I) can be seen in Table 3.1. Y and Z are the horizontal and vertical axis respectively.

(a) Slab-Type Deck

(b) Hollow-box Type Deck

Fig. 3.5 Proposed Decks for the Analysis (dimensions in metres)

The cross-section areas for the slab-type and the hollow-box type are the same, which implies the same weight. This point is very important, because the main difference between both typologies is the inertia. The slab-type deck considers a very low inertia. The slenderness ratio in this case is h/L = 0.40/217 = 1/543 for the fan pattern, and h/L = 0.40/204.6 = 1/512 for the harp pattern, which implies a very slender deck. Of course, in this case bending moments are expected to be low. The hollow-box type deck shows a high inertia. The slenderness ratio is h/L =2.25/217=1/96 for the fan pattern, and h/L = 2.25/204.6 = 1/91 for the harp pattern, which implies a rigid deck. These differences involve the use of 6.2 m stay-spacing for the flexible deck (slab-type), and 12.4 m stay-spacing for the rigid deck (hollow-box type).

1.2.2 Basis of Design and Actions

Materials and their mechanical properties have been chosen according to the general specifications and regulations for bridge design, taking into account seismic considerations [Priestley et al, 1996; Walter, 1999; Ministerio de Fomento, 2000; Aparicio and Casas, 2000].

For the seismic design, high strength concrete is employed, with strengths between 22.5 Mpa and 45Mpa, avoiding higher strengths because of their less ductility. For decks and pylons on all the bridges, a prestressed concrete HP-40 (Table 3.2) is employed according to the Spanish regulation EHE [Ministerio de Fomento, 2000]. Regarding the steel for reinforced concrete, welding steel B-400-SD with special characteristics for ductility is employed according to Ministerio de Fomento (2000). Its mechanical properties are shown in Table 3.3.

The stays have been considered applying parallel-strand cables (Fig. 3.6). The employed steel is Y 1860 S7 according to the EHE instruction [Ministerio de Fomento, 2000]. For the non-linear analysis, the mechanical properties of the stays were considered taking into account a multi-cable formulation, with the incorporation of tension - only elements. The main properties of the stays (Table 3.4) have been obtained from the reference of Walter (1999).

Regarding the design of the stays, consideration of strength and fatigue factors according to the recommendations of Aparicio and Casas (2000) were applied in order to satisfy strength requirements as well as fatigue design criteria.

Fig. 3.6 Parallel-Strand Cables

Table 3.4 Mechanical Properties of the Stays Mechanical Property Value Nominal diameter ( ΦN) 0.6” Nominal area (AΦ) 140 mm2 Modulus of elasticity (E) 190000 MPa 0 – 2% proof stress ( f0.02) 1700 MPa Ultimate tensile strength ( fu) 1900 MPa Volumetric Weight (ρ) 78.5 kN/m3 Poisson’s Ratio ( ν) 0.3 Thermal expansion coeff ( α) 1.1x10-5 (1/0C)

For definition of the actions in this research, the criteria of the Dirección General de Carreteras de España [Ministerio de Fomento, 1998] and the specific regulations of Eurocode 8 - Part 2 [CEN, 1998b] regarding the seismic action on bridges, were considered. The main objective of this research is to study the seismic response of cable-stayed bridges, and that is the reason why some actions are not considered. In fact, some prestressing forces, rheological, thermic or brake loads are not considered. It means that these actions are less important with regard to dead loads, live loads and of course the seismic action.

In this investigation, the bridges were considered with a medium importance and normal design traffic. These considerations involve a seismic importance factor γ_I = 1.00 according to Eurocode 8 Part 2 [CEN, 1998b], and a live load factor ψ₂₁ = 0, according to Eurocode 1 Part 3 [CEN, 1998c]. By this way, to study the seismic response of the bridges, the only considered actions are the permanent loads (q_PL), the stay prestressing forces (q_SPL) and the seismic action of course (q_E). These considerations are reasonable because the permanent loads of a cable-stayed bridge may contribute 80 – 90% to total bridge loads [Ren and Obata, 1999].

For the above mentioned, to combine the loads on the bridge models, it is necessary to add each action: q_PL + q_SPL + q_E.

1.2.3 Nonlinearities

Nonlinearities can be broadly divided into geometric and material nonlinearities. Material nonlinearities depend on the specific structure (materials used, loads acting, design assumptions). Although it is certain that the elastic-plastic effect tends to reduce the seismic response of long-span cable-stayed bridges [Ren and Obata, 1999], material nonlinearities depend highly on the characteristics of the input earthquake records. In general terms, cable-stayed bridges experience very long periods, and for that reason formation of plastic hinges at the supports can be difficult. In fact, EC8 – 2 [CEN, 1998b] recommends for a well-designed cable-stayed bridge a behaviour factor q = 1, that is to say, an elastic seismic behaviour. Moreover, because of the high axial forces on the pylons, ductility of them can be questionable, and due to the importance of such structures, it is preferable an elastic behaviour of the materials, without formation of plastic hinges at the pylons. That is the main reason why the inelastic behaviour is not considered in this research. In this sense, dimensions and some special considerations for the selected bridge typologies take into account an elastic seismic behaviour of the materials. However, material nonlinearities due to the presence of additional viscous dampers as well as the tension-only nonlinear effect of the cables are considered.

Moreover, geometric nonlinearities are present in any cable-stayed bridge, and they originate from [Nazmy and Abdel-Ghaffar, 1990; Morgenthal, 1999]:

• The non-linear cable sag effect due to the inclined cable stays which governs axial elongation and the axial tension;

• The non-linear axial force and bending moment interaction for the tower and longitudinal girder elements;

• The non-linear behaviour due to the geometry change caused by the large displacements on the whole structure.

Those nonlinearities are especially considered in this research, because in some sense they govern the behaviour of this kind of structures, as was explained before. Geometric nonlinearity can be considered on a step-by-step basis in nonlinear static and direct integration time history analysis, and incorporated in the stiffness matrix for linear analysis.

1.2.4 Modelling

1.2.4.1 Tower modelling

Fig. 3.7 Modelling of the Towers

The bridge towers are represented by three-dimensional portal frames, as shown in Fig. 3.7. The tower legs and struts were modelled using linear elastic beam elements based on gross cross-section properties. Inertia factors Ief/Ig equal to 0.70 for the legs and 0.50 for the struts, where Ief is the effective moment of inertia and Ig is the moment of inertia of the gross cross-section, were considered to take into account a modified stiffness of the frames, according to recommendations of Priestley et al (1996). Moreover, because of the finite cross-sectional dimensions of the members and the significant length of the overlaps of the connecting elements, rigid links for the strut – leg connection were considered. This approximation allows an adequate modelling of the encasement of the struts into the tower legs.

The stiffening effect of the rigid links for the modelling of the leg-strut connection (also called End Offsets) is accounted by specification of rigid-end factors, which give the fraction of each end offset that is assumed to be rigid for bending and shear deformation. In the case of the portal frames used for the towers, a value of 0.5 for the rigid-end factor was used.

For the adequate consideration of the vibrational characteristics of the towers, the distribution of lumped masses shown in Fig. 3.7 was considered. The masses were obtained from the elements using the mass density of the materials and the volume of the elements. These uncoupled masses are equal for each of the three translational degrees-of-freedom. In the case of the code RAM Advanse, these masses need to be explicitly added in the selected joints. In SAP2000, these masses are automatically added at the end joints of the elements. However, it was necessary to place additional masses to take into account an adequate distribution of lumped masses in order to have a more accurate dynamic model. The lumped masses involves no mass coupling between degrees of freedom at a joint or between different joints

1.2.4.2 Deck Modelling

As was previously mentioned, the decks of the bridges are of two kinds: a slab-type deck for a stay spacing of 6.2 m and a hollow box-type deck for a stay spacing of 12.40 m. To simplify the computing process, the decks were modelled using a single spine passing through the centroid of the cross-sections, and applying linear beam elements. To simulate the exact stiffness and masses of the decks, an accurate definition of the geometry of the cross sections was developed with the computational code RAM Advanse and also exported to SAP2000. That was possible modifying the internal code in RAM Advanse, in order to generate the desired sections. A 3-D rendering of the deck models can be seen in Fig. 3.8.

(a) Slab-type deck

(b) Hollow-box type deck

Most of the nodes were arranged corresponding to the anchor locations of the cables. The cable anchorages and the deck spine were connected by massless rigid links placed horizontally at 90^o to the spine (Fig. 3.9). These rigid links were idealized by linear springs, with all the degrees-of-freedom fixed.

Fig. 3.9 Deck Modelling

Also, translational and rotational lumped masses were considered in the centroid of the cross-sections, with the same spacing of the anchorages (Fig. 3.9). Specifically, rotational lumped masses were added to take into account the torsional modes due to eccentricity of the cross-section of the hollow box type deck. Translational masses consider the action of the permanent load including the effect of the dead load. Moreover, because of the decks are made of prestressed concrete, an inertia factor equal to 1.0 was considered.

As for the non-structural components, such as parapets and beacons, their contribution to the structural rigidity is expected to be quite insignificant and therefore is ignored in the modelling. Likewise, since the cross sections of the deck are rigid (especially the hollow-box type deck), the corresponding warping constants are very large. Consequently, no cross-sectional warping is anticipated.

1.2.4.3 Stay cable model

In a first stage, truss-elements for the cables were applied to compute the prestressing forces on the stays. The strategy of modelling stay cables using truss-elements has been widely used, in which the nonlinear behaviour of the cables is accounted by linearizing the cable stiffness using the concept of an equivalent modulus of elasticity [Ernst, 1965]. However, prediction of the tri-dimensional vibration of the cables is not possible, aspect that can be important for an accurate dynamic analysis, especially if the cable-deck interaction plays an important role. For that reason, this strategy was only considered for the evaluation of the cable forces, through the nonlinear static analysis applying the code RAM Advanse.

For all the subsequent analyses, cables were modelled using a multi-element cable formulation (tension-only elements), employing 5-node isoparametric cable elements [Ali and Abdel-Ghaffar, 1995; Förars et al, 2000], formulation that is considered in the structural codes applied.

For an inclined cable, the force-deformation relation is expected to be non-linear due to cable tension and cable sagging. The cable element uses an elastic catenary formulation to represent the behaviour of a slender cable under its own self-weight, temperature and strain loading. This highly nonlinear behaviour inherently includes tension-stiffening and large-deflection effects, that is to say, slack and taut behaviour. Although it is possible to model the cable as a series of straight elements to consider material nonlinearity or complicated loadings, the catenary formulation is better suited to most applications. In the case of the formulation here employed, it was necessary to determine the undeformed length of the cable, using a shape calculator. This point is very important, because the relationship between the undeformed length and the cord length (the distance between the two end joints) is extremely critical in determining the behaviour of the cable; although recently some researchers have proposed the use of cable models applying iterative numerical techniques using the finite difference method, in which the initial cable geometry is not required, being internally computed [Girija Vallabhan, 2008]. In simple terms, when the undeformed length is longer than the chord length, the cable is slack and has significant sag. When the undeformed length is shorter than the chord length, the cable is taut and carries significant tension with little sag (Fig. 3.10).

If the undeformed length of the cable is shorter than the chord length at the beginning of a nonlinear analysis, tension will immediately exist in the cable and iteration may be required to bring the structure into equilibrium before any load is applied, as always happens with cable-stayed bridges. With regard to the initial geometry of the cables, a lot of ways to calculate it are available, as for example: specifying the undeformed length, either absolute or relative to the chord length; specifying the maximum vertical sag, measured from the chord to the cable; specifying the tension at either end of the cable, etc. In the case of cable-stayed bridges, because of the high stay prestressing forces, it is advisable to specify the undeformed length as the initial geometry of the cables, as was applied in this work. Of course, the shape calculated here may not actually occur during any analysis case, because only the cable length is determined. The deformed shape of the cable and the tensions it carries will depend upon the loads applied and the behaviour of the structure during analysis.

With regard to the modelling of the cable forces applied in SAP2000, the prestress of the stays was considered applying negative axial deformation loads, which cause a decrease of the undeformed length of the cables, implying an internal tension. This axial deformation was calculated for each cable of the bridge models, according to the specified cable tensions, the elasticity modulus, the undeformed length and the area of the cables. In the case of the code RAM Advanse, the prestressing forces of the stays were directly applied. The self-weight load was automatically included with the previous data of mass density and cable geometry.

For the mass distribution along the cables, it was necessary a discretization of the cable elements in order to obtain lumped masses to provide accurate results from the dynamic analyses including the spatial vibration of the cables. Of course, the mass contributed by the cable element is lumped at the end joints, with no inertial effects considered within the element itself. For that reason, discretization in multi-cable elements was mandatory to capture full dynamics of the cable itself. To decide the number of segments necessary for this purpose, a preliminary modal analysis was conducted for the models AB4 and AR4. Those structures were selected because they consider the same deck type, deck level and stay spacing, being differentiated in the stay cable layout. The cables for each bridge were divided into 4 and 12 segments, and a modal analysis using eigenvectors was performed. Tables 3.5 and 3.6 show the first periods obtained for the cables considering both discretizations.

Results of this analysis showed that differences for both bridges are very similar. In both cases, the lowest periods were obtained for a discretization with 12 segments of the cables. Likewise, differences up to 2.3% were obtained for the first periods of the cables and the same magnitude of the differences were obtained for higher order modes of the cables. This implies that a minimum of four segments is adequate to capture full cable behaviour, although more segments imply more accurate results. For that reason, 12 segments were employed in this research.

The influence of the stay cable vibration on the response of the bridge is either ignored or considered by approximate procedures. The transverse vibration of the stay cables, which can be significant in some cases, is usually neglected. The common practice of modelling the cables by a single truss element is inadequate for seismic response calculations because it essentially precludes the lateral cable vibration modes. In fact, cable vibration effects are found to be significant for seismic response calculations, particularly when the cable fundamental frequencies are overlapping with the first few frequencies of the bridge. In this case, the equivalent modulus method cannot in any way account for the cable vibration effects [Tuladhar et al, 1995]. Only if the mass distribution along the cable is modelled and associated with extra degrees of freedom, the vibrational response of the cables can be obtained. The investigations of Förars et al (2000) and Cheng and Lau (2002) are a good approach regarding the cable vibration effects on the seismic response of cable-stayed bridges.

Another aspect that is interesting is the convergence conditions using cable elements. Models with cable elements will usually converge better if a large number of iterations in the analyses are allowed. In the case of SAP2000, at least 25 iterations are recommended [Computers & Structures, 2007]. Convergence behaviour is generally improved by using fewer elements in the cable object, and by applying larger load increments. In this research, accurate results and convergence were reached with 60 iterations for the static nonlinear analyses. For the nonlinear direct integration time history analyses, up to 180 iterations were necessary depending on the bridge model and the input record, aspect enlarged in Chapter 4.

The stay cable design considers the employ of parallel strand cables. Nominal diameters were obtained with the described multi-cable formulation. This preliminary design work considered the bridges in their operational status, and the inclusion of the permanent loads and the prestressing forces of the stays. Design of the stays took into account the strength criteria and recommendations for the dimensioning on a fatigue basis [Walter, 1999; Aparicio and Casas, 2000]. A safety factor γ₁ = 1.30 and a behaviour factor γ₂ = 1.25 were considered, that is to say, an allowable stress σ_u = 0.45 f_u, where f_u is the ultimate tensile strength of the stays.

294px 

1.2.4.4 Connections and boundary conditions

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 [Iemura and Pradono, 2003]. The influence of different support conditions on the mode distribution has been investigated by Ali and Abdel-Ghaffar (1995) and Tuladhar and Dilger (1999). Movable supports lead to a more flexible structure, and of course, the decision upon the support conditions of the deck is usually governed by serviceability as well as earthquake considerations. A restrained deck will avoid excessive movements due to traffic and wind loading, however, in the case of an earthquake a restrained deck will generate high axial forces which are applied to the pier-pylon system. Elastic supports for the deck at the towers give very low deck displacements and deck bending moments compared to pinned or fixed connections. Roller supports also cause the bridge have very low first longitudinal direction mode frequency, indicating that the bridge is very flexible in that direction.

Fig. 3.11 Modelling of the Tower-deck Connection

In the present investigation, connection between deck and towers is supported by the lower strut through vertical rigid links with pinned bearings at the end-joints, in the connection with the deck (Fig. 3.11). These rigid links were idealized using linear massless springs with all the directional degrees-of-freedom fixed. For the abutment-to-deck connection, sliding bearings were used, in order to permit free longitudinal displacements of the structure due to normal expansions, and free rotations about the transverse axes (Fig. 3.12).

The towers were founded to bedrock and their bases were treated as being fixed in all degrees-of-freedom at the piers, as usually happens on cable-stayed bridges. This affirmation implies that soil-structure interaction is not considered here, as was previously explained. For situations involving large amplitude–large force response, such as during an earthquake, it may be appropriate to reconsider the realistic nature of these boundary conditions as well as the permissible degrees-of-freedom of the deck-tower bearings [Wilson and Gravelle, 1991].

For the incorporation of viscous dampers, it is important to replace the existing pinned bearings by roller supports plus these special devices at the deck-tower connection to reduce seismic forces, absorb large seismic energy, and reduce the response amplitudes considering pure viscous damping. Additionally, energy-absorbing devices may also be put between the deck-ends and the abutments. In order to show some of the complete 3-D finite element models of the bridges, Fig. 3.13 exposes a general structural view of the bridges AB1, AB4, AR1 and AR4.

(a) AB1 pattern	(b) AB4 pattern
(c) AR1 pattern	(d) AR4 Pattern

1.3 Nonlinear Static Analysis under Service Loads

As a first step of the nonlinear seismic analysis, a nonlinear static analysis under gravity loads and stay prestressing forces was carried out. Nonlinearities in this stage include the stay cable sag effect, axial force-bending moment interaction, large displacements effect and the material nonlinearity due to the tension only formulation of the cable elements. Of course, the nonlinear static analysis was performed for each bridge model, considering the actualized data of the stay prestressing forces.

Mathematically, nonlinear static analysis does not always guarantee a unique solution. Inertial forces in dynamic analysis and in the real world limit the path a structure can follow. But this is not true for static analysis, particularly in unstable cases where strength is lost due to material or geometrical nonlinearity. In this sense, small changes in properties or loading can cause large changes in nonlinear response [Computers & Structures, 2007].

With regard to the nonlinear solution control, the specified combination of loads is applied incrementally, with as many steps as necessary to satisfy equilibrium. The nonlinear equations are solved iteratively in each step, requiring re-forming and re-solving the stiffness matrix, until the solution converges. The parameters to control the iterations include the maximum total steps, which are the maximum number of steps allowed in the analysis, including saved and intermediate sub-steps. This parameter permits to control over how long the analysis will run, considering that the length of time it takes to run a nonlinear static analysis is approximately proportional to the number of steps. For all the bridge models in this research, the maximum total steps per stage were 200. The maximum null steps occur when iteration does not converge and a smaller step size is required. An excessive number of null steps indicate that the solution is stalled due to catastrophic failure or numerical sensitivity. In this investigation, the maximum null steps were 50. The number of iterations allowed in a step before the use of smaller sub-steps (maximum iterations per step) depends on the nature of the employed elements in the analysis. In the case of frame elements, normally 10 iterations is enough, however in the case of cable structures, more iterations are necessary. In the static analysis presented here, the convergence was guaranteed selecting 60 maximum iterations per step. With regard to the iteration convergence tolerance (a comparison of the magnitude force error acting on the structure), accurate results were obtained using a value of 1x10^-6. In this sense, it is necessary to explain that the smaller values of this parameter are necessary when large-displacement problems are solved, how occurs in this investigation.

Fig. 3.14 Location of Measured Displacements

Fig. 3.14 shows the location of the measured displacements on the bridges. ∆1-L corresponds to the longitudinal displacement of the tower-top, and ∆1-Tcorresponds to the transverse displacement of the same point. ∆3-V is the vertical displacement of the deck at the mid-span, and finally, ∆4-Lcorresponds to the longitudinal displacement of the deck-ends.

M_max-tower corresponds to the maximum bending moments of the towers; M_max-deck2is the maximum in-plane bending moment of the deck at the tower-deck connection; M_max-deck3 is the maximum in-plane bending moment of the deck at the mid-span; N_max-tower corresponds to the maximum compressive force of the tower legs; N_max-deck corresponds to the maximum compressive force of the deck and N_max-cable is the maximum axial force of the cables.

Table 3.8 summarizes the maximum main displacements and Table 3.9 summarizes the maximum main forces for each bridge.

^a Positive value implies a displacement toward the mid-span

^b Negative value implies a descending

^a At the tower base ^c In the bridge plane ^e Out-of-plane

^b At the tower-deck level ^d At the upper strut level - Implies compression

Values of displacements and forces shown in Tables 3.8 and 3.9 are in accordance with the structures. Although a static parametric analysis is not the aim of this investigation, it can be interesting to observe that maximum displacements at the deck-ends are associated with the harp pattern. Because of the loads and geometric symmetry of the structures, transverse deflections at the tower-deck level as well as the longitudinal and transverse deflections of the decks at the mid-span are zero.

Regarding the internal forces, it can be observed in general terms that maximum bending moments of the towers occur for the tallest structures (for both in-plane and out-of-plane directions). Likewise, because of the longer dimensions of the towers, maximum compressive forces of the tower legs occur for the bridges AB2, AB4, AR2 and AR4. Of course, maximum in-plane bending moments and axial forces of the towers occur at their base. With regard to the internal forces of the decks, it can be appreciated that maximum in-plane bending moments are associated with the longest stay spacing, aspect that is obvious because the stays work as elastic supports. In the same way, maximum bending moments take place for the harp pattern with the longest stay spacing at the tower-deck connection, and maximum axial forces of the decks occur for the harp pattern with the shortest stay spacing at the tower-deck connection. In general terms, an increase of the stay spacing increases the bending moments of the deck, but decreases the compressive forces. The maximum compressive forces and bending moments of the decks take place in the vicinity of the tower-deck connection, because of the presence of the fixed hinges. This issue can be very important in a seismic analysis, and it can be a sign for the location of additional energy dissipation devices in order to mitigate excessive axial forces. With regard to the cable forces, maximum values occur for the longest back stays. It is interesting to observe that the highest cable forces are associated with the fan pattern, and it is clear that longer stay spacing involves higher cable forces. In other words, the worse cable condition occurs for the back stays in the presence of fan pattern with the longest stay spacing. A detailed static parametric analysis can be found in the reference of Walter (1999).

As example, Figs. 3.15 and 3.16 show some results of the static analysis under service loads for the bridges AB1 and AR4. These bridges were chosen because they can be considered as extreme cases. Because of the loads and geometric symmetry of the structures, only one-half is shown.

(a) In-plane bending moments of the deck	(b) Axial forces of the deck
(c) In-plane bending moments of the pylon	(d) Axial forces of the pylon

(a) In-plane bending moments of the deck	(b) Axial forces of the deck
(c) In-plane bending moments of the pylon	(d) Axial forces of the pylon

Nonlinearities associated to large-displacements and axial force-bending moment interaction involved in the static analysis, are less important than those nonlinearities in the dynamic analysis. Because of the rapid change of the forces during an earthquake, those geometric nonlinearities can be very important in this situation. In this sense, Table 3.10 shows a brief comparison of the static response for AB4 bridge, considering only cable-sag effect, cable sag plus P-Δ effects and cable sag plus P-Δ plus large-displacement effects. Displacements are shown in [cm], forces in [kN] and moments in [MN.m].

From Table 3.10, it is clear that in this case P-Δ and large-displacement effects are not very significant on the longitudinal deck displacements. More important differences can be found for the bending moments of the deck, with differences up to 30% comparing the case of only cable sag effect and the case including all the geometric nonlinearities. Differences for the axial force of the towers, axial force of the deck and axial force of the cables are less sensitive, which implies that P-Δ and large-displacement effects are not very important in such cases. However, it can be supposed that those differences could be more important on longer bridges, and of course, in the nonlinear dynamic analyses. For that reason it is important to consider all those nonlinearities in the static case as starting point for the nonlinear seismic analyses [Abdel-Ghaffar, 1991; Morgenthal, 1999].

1.4 Modal Analysis

1.4.1 Natural Frequencies and Modal Shapes

The dynamic response of a structure can be well characterized by a modal analysis. Because of the complex nature of these structures and their seismic response, a two-dimensional analysis is not adequate to capture the three-dimensionality of the system, and for that reason, a 3D analysis is always recommended for the nonlinear static/dynamic analysis of cable-stayed bridges, and of course, for a modal analysis.

As first approach of the seismic response of these structures, a characterization of natural frequencies, modal shapes, modal participation factors as well as damping mechanism is highly recommended, as was explained in the state-of-the-art review. In order to obtain the modified stiffness matrix considering all nonlinearities available, a nonlinear static analysis was performed first using the proposed finite element modelling. The dynamic free vibration analysis is then performed based on the deformed configuration, and considering as previous stage the nonlinear static analysis. As a consequence, the modal analysis can be used as the basis for modal superposition in the response-spectrum analysis.

A lot of strategies to find the natural frequencies have been proposed. The Eigenvector analysis determines the undamped free-vibration mode shapes and frequencies of the system, solving the generalized eigenvalue problem ${\textstyle [-{\omega }^{2}M+K]\phi =0}$ , where M is the diagonal mass matrix, K is the stiffness matrix, ω² is the diagonal matrix of eigenvalues, and Φ is the matrix of the corresponding eigenvectors (modal shapes). This problem can be solved using the classical tools of linear algebra, or applying approximate strategies to speed-up the solution, as for example the Stodola-Vianella method.

The Ritz-vector analysis is another strategy to solve this problem, in which the analysis seeks to find modes that are excited by a particular loading. In this sense, it has been demonstrated [Wilson et al, 1982] that dynamic analyses based on a special set of load-dependent Ritz vectors yield more accurate results than the use of the same number of natural mode shapes. The reason why the Ritz vectors yield good results is that they are generated by taking into account the spatial distribution of the dynamic loading, whereas the direct use of the natural mode shapes neglects this very important information. In addition, the Ritz-vector algorithm automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction and static correction due to higher-mode truncation. The spatial distribution of the dynamic load vector serves as a starting load vector to initiate the procedure. The first Ritz vector is the static displacement vector corresponding to the starting load vector. The remaining vectors are generated from a recurrence relationship in which the mass matrix is multiplied by the previously obtained Ritz vector and used as the load vector for the next static solution.

In this research, it was found that differences regarding the application of eigenvectors or Ritz-vector analysis were absolutely negligible, with differences of about 0.5%. In spite of the low differences, the modal analysis was performed applying the Ritz-vector Analysis, according to the algorithm detailed in Wilson (1985). The total number of modes was selected to reach at least 90% of the effective translational mass, in which more than 300 modes were necessary, although a total of 30 natural periods for each bridge were obtained that range between 0.237 and 2.938 sec. Depending on the relative amplitudes of the modal shapes, these modes were classified into the following groups: vertical modes (V), transverse modes (Tr), longitudinal modes (L) and torsional modes (Tor) for the deck and towers. All the dynamic properties were obtained applying the structural code RAM Advanse [RAM International, 2003]. For all the analyses, the predominant period of a structure corresponds to the natural period associated with the highest translational mass, sometimes different from the fundamental period, usually associated with the first natural frequency.

Longest periods for the cables can be observed in the bridges AB1 and AB2 (2.51 sec) as well as the shortest periods correspond to the bridges AR3 and AR4 (1.12 sec). The main differences on the determination of the predominant periods of the cables come from the mass m and the prestressing forces T.

Tables 3.12 and 3.13 show natural periods and nature of the modal shapes for the first 15 modes considering the fan and harp pattern respectively. Table 3.14 shows modal shapes of the bridges for the first three modes, as well as Tables 3.15 and 3.16 show modal participation factors for the fan and harp pattern respectively. The longest fundamental periods correspond to the bridges AR2, AR4, AB4 and AB2 with values of 2.938, 2.845, 2.804 and 2.694 sec respectively. Those results are obvious because they are the tallest bridges, that is to say, the more flexible structures. For those cases, longitudinal/vertical motion of the deck governs; with the highest periods for the harp pattern; and modal participation factors for the fundamental periods over 70%. For the smallest bridges, fundamental periods of 2.020, 1.992, 1.806 and 1.801 sec, associated to bridges AR3, AR1, AB1 and AB3 were obtained respectively. In this case, modal shapes are governed by longitudinal/vertical oscillations of the deck, for the harp pattern; as long as transverse oscillations govern the modal shapes of the fan pattern bridges. Regarding the modal participation factors associated with the fundamental periods for the smallest bridges, values over 55% were observed. Comparing fundamental periods for tall and small bridges, it is easy to see that the highest periods were obtained with the harp pattern bridges. In other words, for these support conditions, harp pattern structures are more flexible than the fan pattern bridges. Analyzing the incidence of the stay spacing, fundamental period of AB1 bridge is slightly longer than period of AB3 bridge; and fundamental period of AB4 bridge is longer than period of AB2 bridge. For the harp pattern, fundamental period of AR3 bridge is longer than period of AR1 bridge; as long as fundamental period of AR2 bridge is longer than period of AR4 bridge. As a result, an increase of the longitudinal stiffness if the stay spacing decreases is not evident and significant.

If the regression analysis provided by Kawashima et al (1993), is applied here, an average predominant period of 1.76 sec is obtained. This value, associated with vertical bending vibrations, is relatively close to the average fundamental period computed with the modal analysis for the smallest bridges (T = 1.90 sec), which implies 7.4% difference.

From Tables 3.12 and 3.13 it is evident that the close spacing of the natural periods is a vibrational characteristic of these bridges, in accordance with the observations of Abdel-Ghaffar (1991). In this sense, some modal participation factors (Tables 3.15 and 3.16) are very close, indicating strong modal coupling, a very important characteristic of the dynamic behaviour of cable-stayed bridges.

It is interesting to observe that the first ten modes for all bridges are associated with periods that range between 0.4 and 2.94 sec, that is to say, those structures can be more affected by velocity than acceleration or displacements according to Eurocode 8 [CEN, 1998a]. In fact, this code explains that velocity-sensitive region corresponds to periods in the range between 0.4<T<3 sec. On the other hand, for structures equipped with additional damping devices, the reduction in responses is essentially unaffected by damper nonlinearity in the velocity-sensitive region. For that reason, cable-stayed bridge models of this work can be more affected by velocity, with no incidence of the damper nonlinearity on the seismic response. These aspects are necessary to be taken into account in the analysis of the bridges.

From Tables 3.12, 3.13 and 3.14 it is clear, for all bridges, that first modes are associated with the deck. They are followed by tower or cable modes, depending on the bridge configuration. In this sense, some cable modes are overlapped with the deck modes, especially on bridges AB1, AB2 and AR1, aspect that can imply some incidence of the cable vibrations on the seismic response of the bridges, according to Tuladhar et al (1995).

From Tables 3.15 and 3.16, it is clear that some important higher order modes are present, especially on bridges AB1, AB3, AR1 and AR3, that is to say, the smallest structures. Because of the relatively low vertical modal participation factors, importance of the vertical motion on the seismic response seems to be not very significant, although some important vertical effects can occur in the presence of near-source earthquakes.

Another interesting observation can be proposed with regard to the modal shapes related to the deck torsion. The deck torsion is coupled with the longitudinal motion of the tower legs, with no evidence of pure torsion for the first 15 modes and independently on the deck type, stay spacing, stay cable layout and deck level. This implies that torsion generated by the eccentricity of the cross-section of the hollow-box type deck can be ignored.

	Mode 1	Mode 2	Mode 3
AB1	T = 1.806 sec	T = 1.663 sec	T = 1.585sec
AB2	T = 2.694 sec	T = 2.539 sec	T = 2.064 sec
AB3	T = 1.801 sec	T = 1.798 sec	T = 1.689 sec
AB4	T = 2.804 sec	T = 2.662 sec	T = 2.156 sec
AR1	T = 1.992 sec	T = 1.900 sec	T = 1.733 sec
AR2	T = 2.938 sec	T = 2.568 sec	T = 2.130 sec
AR3	T = 2.020 sec	T = 1.795 sec	T = 1.772 sec
AR4	T = 2.845 sec	T = 2.535 sec	T = 2.080 sec

1.4.2 Damping

Damping on cable-stayed bridges is low, of about 2%, according to Morgenthal (1999). Kawashima and Unjoh (1991) found that critical damping ratio was dependent with the excitation amplitude and modal shape, aspect that makes the damping estimation very complex. Of course, this approach is not by the side of this research, and constant values for the damping ratio are suggested for each bridge, depending on the modal shape. In this sense, the approximation by Kawashima and Unjoh (1991), in which the critical damping ratio for the main modes is correlated with the natural frequencies of the bridges, can be applied here. Table 3.17 shows results of this formulation. Here, ξ^BV, ξ^BH and ξ^T are the critical damping ratios for vertical bending oscillations, transverse bending oscillations and torsional oscillations, respectively.

As can be seen, critical damping ratios are similar for all bridges and strongly dependent on the considered mode. Lowest values of the critical damping ratio correspond to torsional oscillations (≈ 0.50%). They are followed by vertical bending oscillations (≈ 0.80%). The highest values of the critical damping ratio correspond to transverse bending oscillations, with an average value of 1.70%. Anyway, all the damping values are very low, and very different from the classical 5% of the critical damping adopted in the design codes. Likewise, it can be observed that damping associated with vertical bending, transverse bending and torsional oscillations depend on the deck level, that is to say, the bridge flexibility. In this sense, the lowest damping ratios are experienced with the tallest structures.

Finally, it is important to say that results of the damping analysis are in accordance with real measures on bridges, as can be seen in the works of Garevski and Savern (1992, 1993); Yamaguchi and Manabu (1997) and Atkins and Wilson (2000).

1.5 Seismic Response Analysis Applying the Response Spectrum Method

The response spectrum method for the seismic analysis of structures is a useful and powerful tool, well implemented in the current seismic regulations. This methodology computes the maximum seismic response of a structure using modal superposition, on the basis of a modal analysis previously performed, and applying an elastic design response spectrum as seismic input. Although this strategy is questionable in the case of structures with nonlinear seismic behaviour, it can be applicable as first approach for the seismic analysis, in order to compare the maximum seismic responses with those obtained from the nonlinear time history analysis considering similar conditions of soil and effective ground acceleration.

Design response spectra considered in this research were obtained from Eurocode 8 [CEN, 1998a, 1998b]. This code was selected because it considers specific recommendations and the definition of the response spectra for bridges. The structural parameters involved with the definition of the response spectra consider a medium importance for the bridges and an elastic seismic behaviour (behaviour factor q equal to 1.0). The structures are founded on bedrock, and the considered maximum effective ground acceleration is 0.5g for the horizontal component, and 0.35g for the vertical component, where g is the gravity acceleration. These values are representative for structures located in high seismicity areas founded on bedrock, as usually happens in the subduction zone of the Mexican coast (Pacific ocean) [CFE, 1993]; several areas of the California coast [AASHTO, 1994, section 3.10]; and some areas of Japan [Japan Road Association, 1996, section 6.3]. The vertical component was assessed as a function of the horizontal one, according to Eurocode 8. Critical damping ratio of 1.7% was selected according to the modal analysis of the bridges previously exposed. The damping correction factor η, for values different from the classical 5%-critical damping ratio, was applied in this case. Table 3.18 summarizes the definition of the considered parameters. Thus, the design acceleration response spectra, for both horizontal and vertical components, were obtained according to Fig. 3.17. Likewise, Fig. 3.18 shows the design velocity response spectra, which is of interest in the formulation of the artificial accelerograms.

294px Fig. 3.17 Design Acceleration Response Spectra

Fig. 3.18 Design Velocity Response Spectra

With regard to the modal superposition, CQC modal combination rule was applied because of the strong modal coupling that cable-stayed bridges experience. The bridge models were analyzed for each load condition, in which the seismic components were combined applying the 30% rule according to Eurocode 8. All the analyses were performed applying the structural code RAM Advanse [RAM International, 2003].

The analysis results show some differences regarding the longitudinal displacements of the shortest towers (bridges AB1, AB3, AR1 and AR3), aspect that is not obvious for the case of the tallest bridges. In fact, for the shortest towers, maximum displacements can be obtained for the bridges AR1 and AB3. In the same way, longitudinal displacements of the tower of AB1 bridge are larger than longitudinal displacements of the tower of AR3 bridge; however, not very interesting conclusions can be formulated according to the above mentioned, as can be seen in Figs. 3.19 and 3.20. In the case of the 111 m height-towers, differences regarding the longitudinal displacements of the towers are negligible (Fig. 3.19). Of course, maximum longitudinal displacements of the towers are obtained for the tallest bridges, with maximum values at the tower-top of about 40 cm for the tallest towers, and 30 cm for the shortest ones. Likewise, it can be appreciated that maximum longitudinal displacements at the tower-top for AB3 bridge are larger than maximum longitudinal displacements at the tower-top of AB1 bridge. In the case of AB4 bridge, maximum top displacements are larger than displacements of AB2 bridge. A similar situation can be observed with the maximum longitudinal displacements at the tower-top of the bridges AR3, AR1, AR4 and AR2; concluding that bridges with longer stay spacing experience an increase of the longitudinal displacements of the tower-top. Following with the longitudinal displacements of the towers, a brief comparison between the analysis under service loads and the seismic displacements can be observed in Figs. 3.21 and 3.22 (plots of longitudinal displacements using logarithmic scale), in this case, for the extreme cases AB1 and AR4. In both situations is clear the tremendous difference between the static and seismic longitudinal displacements of the towers, with seismic displacements more than 78 times the static displacements (AB1 bridge) and 50 times for the AR4 bridge. Of course, similar conditions can be found with the rest of the cases. It is interesting to note that maximum longitudinal seismic displacements occur at the tower-top, aspect that is not always certain for the static condition under service loads.

Fig. 3.19 Maximum Seismic Longitudinal Displacements for 81 m - height Towers

Fig. 3.20 Maximum Seismic Longitudinal Displacements for 111 m - height Towers

Fig. 3.21 Maximum Tower Longitudinal Displacements for AB1 Bridge

Fig. 3.22 Maximum Tower Longitudinal Displacements for AR4 Bridge

With regard to the maximum vertical seismic displacements of the decks, more interesting observations can be formulated. Because of the differences with the flexural stiffness of the decks, the analysis was carried out considering the slab-type deck (bridges AB1, AB2, AR1 and AR2) and the hollow-box type deck (bridges AB3, AB4, AR3 and AR4) separately. In the first case, maximum vertical displacements are concentrated in the extreme spans (in the first third-length) and the vicinity of the main-span centre (see Fig. 3.23), distribution that can be very different from the static condition (Fig. 3.25). Maximum values of the vertical displacements vary from 33 to 40 cm, depending on the bridge model. It is clear that maximum vertical displacements are obtained for the AR1 model, followed by the bridges AR2, AB1 and AB2 respectively. Fig. 3.25 exposes a comparison between the seismic and service loading condition (in this case, the extreme condition AB1), with differences in the seismic condition up to 35 times the service condition, as can be appreciated.

In the case of the hollow box-type deck (stay spacing of 12.4 m), the displacement distribution is very different from the slab-type deck, with maximum values concentrated in the half-length of the extreme spans, and the third-length of the main span, as can be seen in Fig. 3.24. Maximum displacements vary from 23 to 33 cm, depending on the bridge model. Likewise, maximum values of deck displacements are obtained for AR3 bridge followed by the bridges AR4, AB3 and AB4 respectively (see Fig. 3.24). Fig. 3.26 exposes a comparison between the seismic and static deck displacements under service loads, for the extreme condition AR4, showing differences for the seismic condition more than 20 times the service displacements.

Fig. 3.23 Maximum Vertical Seismic Displacements – Slab-type Deck

Fig. 3.24 Maximum Vertical Seismic Displacements – Hollow box-type Deck

Fig. 3.25 Maximum Vertical Deck Displacements for AB1 Bridge

Fig. 3.26 Maximum Vertical Deck Displacements for AR4 Bridge

Of course, a comparison between vertical displacements of the slab-type deck and the hollow box-type deck is not possible because they involve different stay spacing; however, it is clear that maximum values and maximum differences between the seismic and static condition under service loads are obtained with the slab-type deck because of the low flexural stiffness. In this sense, it seems to be that the deck displacement distribution can be more affected by the stay spacing than the deck type. Regarding the influence of the stay cable layout, it is evident that vertical deck displacements for the harp pattern are higher than displacements for the fan pattern, with some influence of the deck level; showing that bridges with the shortest deck level experience larger deck displacements than bridges with the highest deck level.

The comparison of internal forces shows interesting conclusions. The analysis of the compressive forces of the tower legs exposes variations for the maximum values (base) of the shortest towers with differences no greater than 5%. The results (Fig. 3.27) show important differences of the maximum values over the deck level (30 m), and increasing with the altitude. In fact, at the tower-top, differences up to 95% can be found. Also, Fig. 3.27 exposes that maximum compressive forces are obtained for AB1 bridge, followed by AR1 bridge (below the deck level) and AB3 (above the deck level). The lowest compressive forces are obtained for AR3 bridge. In a similar situation, the analysis of the compressive forces of the tower legs for the tallest bridges shows variations of the maximum values with negligible differences below the deck level (see Fig. 3.28), and more important variations above the deck level (60 m). In this case, differences up to 93% can be found at the tower-top. Below the deck level, maximum compressive forces are obtained for AB2 bridge, and minimum compressive force for AB4 bridge, on the contrary above the deck level, for which the maximum compressive forces are obtained for AB4 bridge, and the minimum values for AR4 bridge. The above mentioned implies that distribution of the seismic compressive forces depends on the deck level, with more important differences above that level. In this sense, axial forces on AB1 bridge are higher than axial forces on AR1 bridge; axial forces on AB3 bridge are higher than forces on AR3 bridge; axial forces on AB2 bridge are higher than that of AR2 bridge; and axial forces on AB4 bridge are higher than that of AR4 bridge above the deck level. As a conclusion, in general terms bridges with fan pattern show an increase of the seismic axial forces of the towers compared with the harp pattern. On the other hand, not important conclusions can be formulated with regard to the influence of the stay spacing on the compressive forces of the towers.

Fig. 3.27 Envelope of Maximum Seismic Compressive Forces for 81 m-Height Towers

204px Fig. 3.28 Envelope of Maximum Seismic Compressive Forces for 111 m-Height Towers

Of course, maximum compressive forces of the towers are obtained at their bases, with maximum average values of 90 MN and 158 MN for the 81 m-height towers and 111 m-height towers respectively. Figs. 3.29 and 3.30 show a comparison for compressive forces between the service loading condition and the seismic condition for the extreme cases AB1 and AR4. In these plots, it is clear the influence of the vertical component of the seismic motion on the axial forces of the tower legs, below the deck level, and mainly on AR4 bridge. In both situations, an increase of the maximum compressive forces of about 40% can be obtained, comparing the seismic and service loading cases.

The seismic analysis of the decks shows a similar distribution of the axial forces regarding the analysis under service loads. In fact, in both situations maximum compressive forces occur at the tower-deck connection, as can be appreciated in Figs. 3.31 to 3.34. Fig. 3.31 exposes the envelope of axial forces of the decks associated to bridges AB1, AB3, AR1 and AR3 (bridges with the shortest towers), as well as Fig. 3.32 exposes the envelope of axial forces of the decks for the rest of the cases.

186px Fig. 3.29 Envelope of Maximum Tower Axial Forces for AB1 Bridge

Fig. 3.30 Envelope of Maximum Tower Axial Forces for AR4 Bridge

Fig. 3.31 Envelope of Seismic Axial Forces for Decks – 81 m Tower-Height

288px Fig. 3.32 Envelope of Seismic Axial Forces for Decks – 111 m Tower-Height

Maximum values of the compressive forces on the decks vary from 32 MN (AB3 bridge) to 48 MN (AR2 bridge), that is to say, with a difference of 33%. Those values are very high due to the fixed hinge condition of the tower-deck connection, and they are in accordance with the explanations given before, constituting a potential point for the location of additional energy dissipation devices. Likewise, for all cases, a tension-zone at the mid-span of the decks can be appreciated, with values lower than 6 MN.

Some interesting appreciations can be formulated comparing the stay cable layout and the stay spacing with regard to the axial forces of the decks, according to Figs. 3.31 and 3.32. In general terms, axial forces on AR1bridge are higher than deck axial forces of AB1bridge; and axial forces of AR3 bridge are higher than axial forces of AB3 bridge. For bridges with 60 m-deck level, axial forces of AR2 bridge are higher than axial forces of AB2 bridge; and axial forces of AR4 bridge are higher than axial forces of AB4 bridge. This analysis implies that seismic axial forces of decks associated to bridges with harp pattern are higher than axial forces of decks for the fan pattern. In the same way, comparing the effect of the stay spacing, deck axial forces of AB1 bridge are higher than axial forces of AB3 bridge; axial forces of AR1 bridge are higher than axial forces of AR3 bridge; axial forces of AB2 bridge are higher than that of AB4 bridge; and finally, axial forces of AR2 bridge are higher than that of AR4 bridge, which implies that bridges with longer stay spacing show lower seismic axial forces for their decks. This situation is similar to the one observed for the static condition under service loads, and of course, this comparisons are only approaches, exposing a general tendency, as can be seen in Figs. 3.31 and 3.32. The highest axial forces for the decks are obtained for bridges AR1 and AR2, and the lowest axial forces correspond to bridges AB3 and AB4; that is to say, the worse condition for the axial forces of the decks are obtained with the harp pattern and the shortest stay spacing. As a comparison, Figs. 3.33 and 3.34 expose the envelope of axial forces taking into account the seismic and service loading cases, for two extreme conditions: bridges AB1 and AR4. It is easy to see that very important differences between both loading conditions can be observed, with differences in their maximum values up to 33% (AB1 bridge) and 28% (AR4 bridge). In this sense, and comparing with axial forces of the towers, more important differences on the determination of the maximum seismic axial forces are obtained for the tower legs.

Plots of bending moments are more complex and it can be not very clear to superimpose those diagrams. For that reason, in-plane bending moments of decks and towers for each bridge were represented separately, as can be seen in Figs. 3.35 and 3.36. The analysis of bending moments for the towers shows a similar behaviour for all cases. As usually happens, maximum bending moments were obtained at the base of the tallest towers, with maximum moments varying from 403 to 496 MN.m (AR4 bridge), which means a difference of 19%. For the case of 81 m-tower height, maximum bending moments of the towers vary from 178 MN.m (AR1 bridge) to 337.6 MN.m (AB1 bridge), which implies a difference of 48%. Comparing the effect of the stay cable layout on the bending moments of the towers, maximum tower moments of AB1 bridge are higher than maximum tower moments of AR1 bridge; maximum moments of AB3 bridge are higher than moments of AR3 bridge; maximum moments of AB2 bridge are higher than that of AR2 bridge; however, maximum moments of AB4 bridge are lower than that of AR4 bridge. According to the previous ideas, it seems to be that maximum tower bending moments for bridges with fan pattern are higher than maximum tower bending moments for bridges with harp pattern.

Regarding the influence of the stay spacing on the tower bending moments, it is easy to see that maximum tower moments of AB1 bridge are higher than maximum tower moments of AB3 bridge; maximum moments of AB2 bridge are higher than maximum moments of AB4 bridge; however, maximum moments of AR1 bridge are lower than that of AR3 bridge; and maximum moments of AR2 bridge are lower than that of AR4 bridge. This behaviour implies that for fan pattern bridges, an increase of the stay spacing decreases the maximum tower bending moments, on the contrary of the harp pattern, for which an increase of the stay spacing implies an increase of the maximum bending moments of the towers.

(a) Tower moments of AB1 bridge	(b) Deck moments of AB1 bridge
(c) Tower moments of AB3 bridge	(d) Deck moments of AB3 bridge
(e) Tower moments of AR1 bridge	(f) Deck moments of AR1 bridge
(g) Tower moments of AR3 bridge	(h) Deck moments of AR3 bridge

The analysis of in-plane deck bending moments shows a comparable behaviour depending on the stay spacing. Because of the longer stay spacing of bridges with hollow box-type deck, higher deck moments are obtained in this situation. In the seismic case, maximum deck bending moments occur in the central area of the main span, with an important modification of the curvature that makes significant variations of the moments. They are followed close by very high moments near the deck-ends, on the contrary of the static condition under service loads, for which maximum deck moments occur at the tower-deck connection.

The seismic response for decks with the longest stay spacing shows a more uniform distribution of the deck moments compared with bridges with the shortest stay spacing, for which maximum moments near the mid-span and near the deck-ends are very concentrated. For the longest stay spacing, maximum deck moments vary from 45.6 MN.m (AR4 bridge) to 62.2 MN.m (AB3 bridge), which implies a difference of 27%. For the shortest stay spacing, maximum deck moments vary from 7.5 MN.m (AR1 bridge) to 10.8 MN.m (AB1 bridge), which implies a difference of 30%, that is to say, the same magnitude order for moment difference can be obtained. The influence of the stay cable layout on the seismic response of the decks, shows for 6.2 m stay spacing, that maximum deck moment of AB1 bridge is higher than maximum deck moment of AR1 bridge; and maximum deck moment of AB2 bridge is higher than that of AR2 bridge. For 12.4 m stay spacing, maximum deck moment of AB3 bridge is higher than that of AR3 bridge; and maximum deck moment of AB4 bridge is higher than moment of AR4 bridge.

(a) Tower moments of AB2 bridge	(b) Deck moments of AB2 bridge
(c) Tower moments of AB4 bridge	(d) Deck moments of AB4 bridge
(e) Tower moments of AR2 bridge	(f) Deck moments of AR2 bridge
(g) Tower moments of AR4 bridge	(h) Deck moments of AR4 bridge

As a conclusion, maximum deck moments for the fan pattern are higher than maximum deck moments for bridges with harp pattern. Comparing the influence of the deck level on the seismic moments of the decks, maximum deck moment of AB1 bridge is higher than that of AB2 bridge; maximum deck moment of AB3 bridge is higher than moment of AB4 bridge; maximum deck moment of AR3 bridge is higher than that of AR4 bridge; however maximum deck moment of AR1 bridge is lower than maximum moment of AR2 bridge. It seems to be that bridges with the lowest deck level show higher maximum bending moments of the decks.

As a summary, Tables 3.19 and 3.20 expose maximum static (service loads) and seismic displacements as well as maximum main forces for each bridge respectively. Location of the control points for displacements is represented in Fig. 3.14, with the same parameters of Table 3.8, plus Δ_3-T (maximum transverse displacement at the mid-span). The explanations of the parameters of the maximum main forces are the same considered in Table 3.9.

^a Positive value implies a displacement toward the mid-span

^b Negative value implies a descending

From Table 3.19, it is interesting to observe that maximum seismic displacements correspond to transverse deflections of the deck at the mid-span, with values that range between 44 and 63 cm. Of course, those maximum transverse displacements are associated with the highest deck level, and are difficult to control due to the flexibility of the bridges in that direction. The deflection analysis of the towers shows values of the maximum displacements in the same magnitude order for the longitudinal and transverse direction. Those displacements are slightly minor in the transverse direction. Regarding the longitudinal seismic displacements at the deck-ends, maximum value corresponds to AB2 bridge (28.3 cm); and the minimum displacement corresponds to AR3 bridge (16.7 cm), which implies a difference of 41%. Seismic displacements are 12 times the static displacements (average), which can be adequately controlled with the addition of fluid viscous dampers. Likewise, in general terms, maximum longitudinal displacements at the deck-ends for the fan pattern are longer than that of the harp pattern.

From Table 3.20, maximum in-plane tower bending moments were obtained considering the earthquake acting in the longitudinal direction, and of course, in-plane bending moments are lower when the earthquake acts in the transverse direction. In this sense, analysis of out-of-plane tower seismic moments shows that tower moments when the earthquake acts in the transverse direction are lower than in-plane tower moments when the earthquake acts in the longitudinal direction, which implies that the worse analysis condition corresponds to the earthquake acting in the longitudinal direction. Maximum out-of-plane tower moments were obtained for bridges AB4 and AR2 (347.8 MN.m and 347.9 MN.m respectively); and minimum out-of-plane moments were obtained for AR1 bridge (186 MN.m). Comparing in-plane and out-of-plane tower moments, it is easy to see that in-plane bending moments are larger than out-of-plane moments; and differences between in-plane and out-of-plane tower moments vary from 7% (AR1 bridge) to 35% (AB3 bridge) . These conclusions implies that seismic design of the structural elements are governed by in-plane internal forces, and for that reason, only longitudinal seismic effects are considered in this part of the investigation.