Passive Energy Dissipation Systems in Structural Engineering
To Dottie, my wife T. T. Soong
To Andrea, my wife G. F. Dargush
Passive Energy Dissipation Systems in Structural Engineering T. T. Soong G. F. Dargush State University of New York at Buffalo, USA
Copyright © 1997 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO19 1UD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries):
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Preface
Recent damaging earthquakes and hurricanes provided powerful reminders of how vulnerable we all are to the forces of nature. Even in an advanced industrial nation, our built environment is still quite susceptible to natural disasters. Consequently, one of the principal current challenges in structural engineering concerns the development of innovative design concepts to better protect structures, along with their occupants and contents, from the damaging effects of destructive environmental forces including those due to wind, waves and earthquakes. The traditional approach to seismic design, for example, has been based upon providing a combination of strength and ductility to resist the imposed loads. For major earthquakes, the structural design engineer relies upon the inherent ductility of conscientiously detailed buildings to prevent catastrophic failure, while accepting a certain level of structural and nonstructural damage. In many ways this traditional approach views the transient seismic disturbance as an equivalent static lateral load that must be resisted by the structure. However, by discarding this notion and considering the actual dynamic characteristics of environmental disturbances, dramatic improvements can be realized. In fact, as a result of this dynamical point of view, many new and innovative concepts of structural protection have been advanced and are at various stages of development, onewhen of which is passive energy dissipation. The basic role of passive energy dissipation devices incorporated into a structure is to absorb or consume a portion of the input energy, thereby reducing energy dissipation demand on primary structural members and minimizing possible structural damage. In recent years, serious efforts have been undertaken to develop the concept of passive energy dissipation or supplemental damping into a workable technology, and a number of these devices have been installed in structures throughout the world. The present book represents an attempt to provide the first unified treatment for passive energy dissipation systems by consolidating and synthesizing the available information. The book is directed primarily toward the professional engineer faced with structural design decisions and to
Contents Preface
ix
Acknowledgments
xi
Chapter 1- Introduction
1
References
4
Chapter 2 - Fundamentals
5
2.1 Introduction
5
2.2 Dynamic Response of Simple Structural Models
2.2.1 General Formulations 2.2.2 Free Vibration 2.2.3 Forced Vibration 2.2.4 Transient Response 2.2.5 Response with Passive Damper 2.3 Dynamic Analysis of Structural Systems
2.3.1 General Formulation 2.3.2 Modal Superposition Method 2.3.3 Damping in Traditional and Passively Damped Structures 2.3.4 Direct Time Domain Analysis 2.4 EnergyFormulations
2.4.1 SDOF Structures 2.4.2 Energy-based Design 2.4.3 MDOF Structures 2.5 Concluding Remarks References
6 6 7 8 11 14 18 18 20 23 24 25 26 30 31 32 33
Chapter 3 - Metallic Dampers 3.1 Introduction 3.2 Basic Principles
3.2.1 Plasticity 3.2.2 Viscoplasticity 3.2.3 Thermal Effects 3.2.4 Failure Theories 3.3 Macroscopic Modeling
3.3.1 Experiment-based Modeling 3.3.2 Mechanics-based Modeling 3.4 Structural Analysis 3.5 Experimental Studies 3.6 Design Considerations
35 37 37 44 48 48 50 50 51 57 59 69
3.7 Structural Implementations References
71 79
Chapter 4 - Friction Dampers
83
4.1 Introduction 4.2 Basic Principles
83 86 86 89 90
4.2.1 Solid Friction 4.2.2 Environmental Effects 4.3 Damper Behavior and Macroscopic Modeling 4.4 Analysis 4.5 Structural Experimental Studies 4.6 Design Considerations 4.7 Structural Implementations References
104 110 116 119 123
Chapter 5- Viscoelastic Dampers 5.1 Introduction 5.2 Basic Principles 5.3 Shear Storage and Shear Loss Moduli
127 127 127 131 134 138 141 141 144 144
5.3.1 Dependence on Excitation Frequency 5.3.2 Dependence on Ambient Temperature 5.3.3 Dependence on Internal Temperature 5.4 Analysis of Structures with Added VE Dampers 5.5 Experimental Studies
5.5.1 2/5-Scale Five-story Steel Frame 5.5.2 1/3-Scale Three-story Frame ational Issues 5.6 Design Consideratio ns Concrete and Implement 5.7 Structural Implementation References
152 156 159 159 161 168
Chapter 6- Viscous Fluid Dampers
171
6.1 Introduction 6.2 Basic Principles of Fluid Dynamics
171 174 175 175 176 177 181 181 186 190 199 202 218 219 225
5.6.1 Placement Strategies for VE Dampers 5.6.2 Architectural Considerations
6.2.1 Balance Laws 6.2.2 Inviscid Fluids 6.2.3 Newtonian Fluids 6.2.4 Temperature-dependent New-Newtonian Fluids 6.3 Damper Behavior and Macroscopic Modeling
6.3.1 Cylindrical Pot Fluid Dampers 6.3.2 Viscous Damping Walls 6.3.3 Orificed Fluid Dampers 6.4 Structural Analysis 6.5 Experimental Studies 6.6 Design Considerations 6.7 Structural Implementations References
Chapter 7-Tuned Mass Dampers
227
7.1 Introduction
227 229 229 231 233 237 241 245
7.1.1 Tuned Mass Dampers 7.2 Basic Principles
7.2.1 Den Hartog's Solution 7.2.2 The Case of Damped Structures 7.2.3 Other Optimization Criteria 7.2.4 Wind vs. Earthquake Loads 7.3 Structural Analysis
7.3.1 7.3.2 Elastic InelasticStructures Structures
References
247 249 249 249 255 256 257 259 260 263 264 265 268 269 270 276
Chapter 8 -Tuned Liquid Dampers
281
8.1 Introduction 8.2 Basic Principles of Operation 8.3 Damper Behavior and Macroscopic Modeling 8.4 Structural Analysis and Design 8.5 Experimental Studies 8.6 Implementational Issues 8.7 Structural Implementations References
281 285 292 299 303 305 306 315
Chapter 9 -Smart Materials
319
9.1 Introduction 9.2 Shape Memory Alloys
319 319 321 322 327 327 328 332 332 336
7.4 Experimental Studies
7.4.1 Small-scale Tests 7.4.2 Full-scale Test 7.5 Design Considerations and Implementational Issues 7.6 Structural Implementations
7.6.1 Centerpoint Tower, Sydney, Australia 7.6.2 Citicorp Center, New York and John Hancock Tower, Boston 7.6.3 Chiba Port Tower, Tokyo Bay, Japan 7.6.4 Funade Bridge Tower, Osaka, Japan 7.6.5 Steel Stacks, Kimitsu City, Japan 7.7 Related Development and Concluding Remarks
7.7.1 Nonlinear TMD and Impact Vibration Absorbers 7.7.2 Semi-Active TMD and Hybrid Mass Damper
9.2.1 Basic Principles 9.2.2 Structural Applications 9.3 Piezoelectric Materials
9.3.1 Basic Principles 9.3.2 Structural Applications 9.4 Electrorheological Fluids
9.4.1 Basic Principles 9.4.2 Structural Applications
9.5 Magnetorheological Fluids References
338 341
Appendix - Conversion Table
345
Author Index
347
Subject Index
353
graduate students and other researchers in civil engineering. It is organized to address a wide range of behavior characteristics associated with passive energy dissipation devices, ranging from basic principles to implementational issues and design. The intent is to provide the reader with not only a working knowledge of this technology but also with an added degree of understanding and maturity in order to address important practical issues related to structural applications of passive energy dissipation devices. At the same time, it should be emphasized that this entire technology is still evolving. Significant improvements in both hardware and design procedures will certainly continue for a number of years to come.
T. T. SOONG AND G. F. DARGUSH BUFFALO, NEW YORK, 1997
Acknowledgments Our work in this technical area has been supported since 1986 by the National Science Foundation and the State of New York under the auspices of the National Center for Earthquake Engineering Research. This continuing support is gratefully acknowledged. It is also a pleasure to acknowledge support received from the 3M Company, which permitted the performance of some of the experimental studies reported in this book. Preliminary chapters of the book have been used for several years in a graduate course on Passive and Active Structural Control in Civil Engineering developed and taught by the first author. We are indebted to many students in these classes who supplied new information, contributed ideas, and helped to better organize the material as presented in the present form. They include, T. P. Bucklaew, M. I. Bujakov, G. Chen, L. Ge, H. Gupta, T. Jiang, K. Kosar, R. J. Helgeson, C.D. Huang, C. Li, N. Makris, G. Pekcan, R. Rana, Y. Reichman, M. Riley, K. Shah, K. L. Shen, Z. Shen, M. Symans, P. W. Szustak, P. Tsopelas, R. Valles, and R. H. Zhang. We are also indebted to our colleagues, Professors M. C. Constantinou and A.M. Reinhorn, with whom we have collaborated and have had frequent discussions on various technical issues. Our sincere thanks go to Mrs. Carmella Gosden, who efficiently typed several drafts of this book and helped organize many aspects of the project, and to Mr. Hector Velasco, who did most of the art work. Finally, we are grateful to our wives for their help, encouragement and endurance.
1 Introduction In the design of most buildings, the primary loads that must be considered are those due to the effects of gravity. are always present and consequently be resisted throughout the lifeThese of theloads building. Typically, the variation with timemust is slow compared to characteristic times of the structure. As a result, a static idealization is quite appropriate. Furthermore, the magnitudes can be readily determined based upon self-weight and occupancy requirements. This combination of factors greatly simplifies building design, and, in fact, allowed our ancestors to design and construct impressive structures prior to the development of rational scientific principles. The simplicity o the problem permits the use of a trial-and-error approach to design, particularly if one is not unduly constrained by material and labor costs. In our modern era, resources are often severely limited. Efficient designs must be sought. Additionally, we demand protection from environmental forces, including winds, waves, and earthquakes, which are neither static nor unidirectional. For these types of loads inertial effects become important, resulting in dynamic amplification and cyclic response. Compared to gravity loads, the magnitudes are also much more difficult to predict, since the temporal and spatial scales of these phenomena are much smaller. Despite these significant differences, there is a natural tendency to treat the environmental forces with the same methods used for gravity loads. For example, wind and earthquake forces are often idealized as lateral static loads of suitable magnitude that must be resisted by the structure. Utilizing this approach, the lateral loads corresponding to wind and small earthquakes are designed to be resisted by elastic action only, while those associated with moderate or severe seismic events are permitted to damage but not collapse the structure. This philosophy has provided the basis for a number of building codes since the early twentieth century, and results have been reasonably successful. Even an approximate accounting for lateral effects will almost certainly improve building survivability. However, by considering the actual dynamic nature of environmental
disturbances, more dramatic improvements can be realized. As a result of this dynamical point of view, new and innovative concepts of structural protection have been advanced and are at various stages of development. Modern structural protective systems can be divided into three groups as shown in Table 1.1. These groups can be distinguished by examining the approaches employed to manage the energy associated with transient environmental events. Table 1.1 Structural Protective Systems Seismic
Passive Energy
Semi-active and Active
Isolation
Dissipation
Control
Elastomeric Bearings
Lead Rubber Bearings
Sliding Friction Pendulum
Metallic Dampers
Active Bracing Systems
Friction Dampers
Active Mass Dampers
Viscoelastic Dampers
Variable Stiffness or Damping Systems
Viscous Fluid Dampers
Smart Materials
Tuned Mass Dampers Tuned Liquid Dampers
The technique of seismic isolation is now widely used in many parts of the world. A seismic isolation system is typically placed at the foundation of a structure. By means of its flexibility and energy absorption capability, the isolation system partially reflects and partially absorbs some of the earthquake input energy before this energy can be transmitted to the structure. The net effect is a reduction of energy dissipation demand on the structural system, resulting in an increase in its survivability. A detailed review of seismic isolation technology is provided in the recent monograph by Skinner et al. (1993). On the other end of the spectrum as shown in Table 1.1 are semi-active and active control systems. Semi-active and active structural control is an area of structural protection in which the motion of a structure is controlled or modified by means of the action of a control system through some external energy supply. However, semi-active systems require only nominal amounts of energy to adjust their mechanical properties and, unlike fully active systems, they cannot add energy to the structure. Considerable attention has been paid to both semi-active and active structural control research in recent years, with particular emphasis on the alleviation of wind and seismic response. The technology is now at the stage where actual systems have been designed, fabricated and installed in full-scale structures. The interested reader may consult Soong (1990) and Soong and Constantinou (1994) for discussions on
many theoretical and practical aspects of semi-active and active structural control. While all these technologies are likely to have an increasingly important role in structural design, the focus of the present book is on passive energy dissipation (PED) systems, displayed in the middle column of Table 1.1 and, to a limited extent, smart materials. Research and development of passive energy dissipation devices for structural applications have roughly a 25-year history. Similar seismic isolation technology, the basic function of passive energy dissipation devicestowhen incorporated into a structure is to absorb or consume a portion of the input energy, thereby reducing energy dissipation demand on primary structural members and minimizing possible structural damage. Unlike seismic isolation, however, these devices can be effective against wind induced motions as well as those due to earthquakes. Contrary to semiactive and active systems, there is no need for an external supply of power. In recent years, serious efforts have been undertaken to develop the concept of energy dissipation or supplemental damping into a workable technology, and a number o these devices have been installed in structures throughout the world. This book is an attempt to introduce the basic concepts of passive energy dissipation, and to consolidate the available information concerning these devices, which is now scattered over many sources including technical journals, conferences proceedings, laboratory reports, and academic theses. The book is organized to address a wide range of behavior characteristics associated with passive energy dissipation devices. Each major device type, listed in the middle column of Table 1.1, is presented and discussed in an individual chapter based upon the following outline: (a) historical review; (b) basic principles; (c) mathematical modeling, theory and experiments; (d) practical considerations, design and implementation issues; and (e) structural applications. In Chapter 9, a brief account o current activities in smart materials research as related to civil engineering structural applications is presented. As shown in Table 1.1, these activities are considered to fall within the domain of semi-active systems, an extension of passive systems with certain adaptivity properties. As mentioned above, in order to fully appreciate and realize the benefits of passive energy dissipation technology, one must possess a good understanding of structural dynamics. While a detailed presentation of that subject is certainly beyond the scope o this book, an introductory account, emphasizing only the most pertinent aspects, is provided in Chapter 2. Included is a discussion of the energy concept in structural engineering, which provides a unifying principle for the theory of passive energy dissipation and a central theme for our presentation in subsequent chapters. Finally, it is pointed out that, since many references were used in the development o this book, no attempt was made to unify the units of quantities used in the text. It was felt that, to leave them in their srcinal units, easier
references to the srcinal publications could be made. For convenience, a conversion table for English-unit to SI-unit conversion is provided in the Appendix. References Skinner, R. I., Robinson, W. H., and McVerry, G. H. (1993), An Introduction to Seismic Isolation, Wiley, Chichester and New York. Soong, T. T. (1990), Active Structural Control: Theory and Practice, Longman,
London and Wiley, New York. Soong. T. T. and Constantinou, M. C., eds. (1994), Passive and Active Vibration Control in Civil Engineering, Springer-Verlag, Wien and New York.
2 Fundamentals 2.1 Introduction With the development and subsequent implementation of modern protective systems, including those involving passive energy dissipation, the entire structural engineering discipline is now undergoing major change, at least conceptually. The traditional idealization of a building or bridge as a static entity is no longer adequate. Instead, structures must be analyzed and designed by considering their dynamic behavior. It is with this in mind that we present some basic concepts related to the subject of structural dynamics. In this chapter, we concentrate on topics which are of primary importance in understanding, analyzing, and designing structures that incorporate passive energy dissipation systems. In the following section, a simple single-degree-of-freedom structural model is discussed. This represents the prototype for dynamic behavior. Particular emphasis is given to the effect of damping. As we shall see, increased damping can significantly reduce system response to time-varying this model useful for developing an understanding of dynamic disturbances. behavior, it isWhile not sufficient for is representing real structures. We must include more detail. Consequently, a multi-degree-of-freedom model is then introduced, and several numerical procedures are outlined for general dynamic analysis. A discussion comparing typical damping characteristics in traditional and passively damped structures is also included. Finally, a thorough treatment o energy formulations is provided. Essentially one can envision an environmental disturbance as an injection of energy into a structure. Design then focuses on the management of that energy. These energy concepts are particularly relevant in the discussion of passively damped structures, and will provide a recurring theme throughout the remainder of the book.
2.2 Dynamic Response of Simple Structural Models
2.2.1 GeneralF ormulation In this section, we begin with the simplest model that exhibits some of the key response characteristics of a structure subjected to dynamic loading. It is important to keep in mind that here structure. we are studying behavior of required the mathematical idealization, that of a physical Much the more detail is to properly model annot actual building. However, a number of concepts can be most easily introduced by examining these simple models. Our treatment was influenced by presentations included in Newmark and Rosenblueth (1971) and Hanson (1993). Consider the lateral motion of the basic single-degree-of-freedom (SDOF) model, shown in Fig. 2.1, consisting of a mass m, supported by springs with total linear elastic stiffness k, and a damper with linear viscosity c. This SDOF system is then subjected to a seismic disturbance characterized by a spatially uniform, time-dependent ground displacement xg(t), and to a wind load represented by p(t). The excited model responds with a lateral displacement x(t) relative to the ground. Thus, the total displacement o the idealized structure is simply xt(t) = xg(t) + x(t)
(2.1)
Both xg(t) and x(t) are assumed to be continuous, twice differentiable functions. By definition, a linear elastic spring produces a force proportional to the displacement, while the force in a linear viscous damper is proportional
Figure 2.1 SDOF Model
to the velocity. As a result, the equation of motion for this SDOF model can be written
in which a superposed dot represents differentiation with respect to time.
2.2.2 Free Vibration Assume for the moment that the mass m is displaced by an amount x0 and released with zero initial velocity at time t= 0. With both = 0 and p(t) = 0, Eq. (2.2) reduces to the case of free vibration. Furthermore, in the absence of damping (i.e., c = 0), the response is simple harmonic motion x(t) = x0coswot
(2.3)
where wo is the natural circular frequency of the undamped structure, defined as
Other quantities of interest, the natural frequency fo and the natural period To are related via
The undamped system is, of course, conservative, characterized by the perpetual exchange of potential (strain) and kinetic energies. There is no dissipation. On the other hand, experience indicates that physical systems are always dissipative to some extent. With non-zero damping, the free vibration response of the SDOF model becomes
where WD and represent the damped natural circular frequency and the nondimensional damping ratio, respectively. More specifically,
Notice that for = 0, Eq. (2.6) reduces directly to Eq. (2.3). Actually, Eq. (2.6) is only valid for underdamped systems, i.e., for < 1. However, that is exactly
the case of most interest to us. The response defined in Eq. (2.6) is plotted in Fig. 2.2 for several values of the damping ratio in order to illustrate the dramatic effect that increased viscous damping can have on the response of the SDOF system. At = 0.01 the system oscillates through nearly 37 cycles before reducing its amplitude to 0.10x0, while for = 0.20 less than 2 cycles are needed. In all cases, the damped systems asymptotically return to their undeformed state as the energy initially stored in the springs is dissipated through viscous action.
Figure 2.2 Free Vibration Response
2.2.3 F orced Vibration Turning next to problems of forced vibration, we first examine the response of the SDOF model due to harmonic loading of magnitude p0 and frequency w. It is beneficial at this stage to utilize the complex exponential form of the load. Thus, let the applied force in Eq. (2.2) take the form, p(t) = p0eiwt
(2.8)
where from Euler's formula eiwt = coswt + i sinwt
(2.9)
with i representing the imaginary unit (i.e., i = ( - 1) ). The ground acceleration is assumed zero. For a system starting from rest with (0) = 0 and x(0) = 0, the solution o 1/2
Eq. (2.2) with harmonic loading can then be written x(t) = x1ei(wt - 1) + e - wotx2ei(wdt - 2)
(2.10)
in which the amplitudes x1 and x2, and phase angles 1 and 2 are given by
with ß representing the ratio of applied frequency to natural frequency. Thus,
The two terms appearing in Eq. (2.10) correspond to the steady-state and transient components of the motion, respectively. The magnitude of the transient term diminishes with time in the same manner as did the response in free vibration. Consequently, if the harmonic load acts for a sufficiently long time, only the steadystate term will have a significant contribution. We will now assume that is the case. Then the displacement can also be written in terms of the complex frequency response function . That is, at steady-state
where
Further examination of Eq. (2.13) reveals that the magnitude of the displacement response x1 is not only time-independent, but can be written as the product of the static response (xgt = p0/k) and a dynamic amplification factor, which depends only on the frequency and damping ratios. Plots of the amplification factor ( x1/xgt) versus ß at various values of is provided in Fig. 2.3. Peak response occurs near resonance (i.e., = 1) and becomes unbounded for the undamped system. With = 0.01 the peak value of the dynamic amplification factor is approximately 50, while for = 0.20 it is roughly 2.5. Once again viscous damping is seen to have a dramatic effect on the response characteristics of the SDOF model. Note, however, that for forcing frequencies removed from the resonance condition, viscous damping has much less influence on response. For any level of damping, the static response is approached as ß 0, while for large ß the amplification
Figure 2.3 Amplification Factor for Harmonic Loading
factor becomes negligibly small since the load varies too rapidly to invoke a displacement response. It is also instructive to consider the steady-state phase relationships. Figure 2.4 illustrates the dynamic balance of the forces associated with the applied load ( p), inertia (f1), stiffness (fs), and damping (fD) in the complex plane. Of course, the actual force at any time t is the real component of the complex force shown in this Argand diagram. For the damped system, the spring force, and consequently the displacement response, lag behind the applied load by the phase angle1. Meanwhile, the damping and inertia forces, which are in turn proportional to the velocity and acceleration, lead the displacement
Figure 2.4 Damper Force Balance for Harmonic Excitation
response by phase angles of 90° and 180°, respectively. At resonance, 1 = 90° and the applied loading is exactly balanced by the force in the damper. Additional insight can be gained by examining the force-displacement curves as illustrated in Fig. 2.5 for the spring and damper. The energy stored in the spring at any time is equal to the area under the curve in Fig. 2.5a, while the energy dissipated via viscous action per cycle is equivalent to the area within the elliptical hysteresis loop shown in Fig. 2.5b. The elliptical shape is a direct result of the 90° phase difference between damper andFor displacement response, with thewithin constant amplitudethe nature of theforce motion. a given structure, the along area contained the hysteresis loop is a function of the frequency ratio, with a maximum occurring near resonance. The specific case displayed in Fig. 2.5 has ß = 0.90 and = 0.05.
Figure 2.5 Force-displacement Response for Harmonic Excitation; a) Spring, b) Damper
2.2.4 TransientResponse The environmental loads of most interest to us here are not pure harmonics, but rather the result of transient processes. With that in mind, consider the response of our SDOF model, initially at rest, to a general loading . The solution can be formally written in terms of a Duhamel integral. With zero initial displacement and velocity, this becomes
where
The unit impulse response function h(t - ) defines the displacement at time t due to a unit impulse force applied at time . For simple time dependence, the integral in Eq. (2.15) can often be evaluated analytically. For more complicated loadings, the evaluation can be performed numerically using, for example, an extended trapezoidal rule or Romberg integration (Press et al., 1992). Unfortunately, the convolution appearing in Eq. (2.15) requires increasing computational effort as time progresses. An alternate approach involves the use of the Laplace transform, since in the transform domain the convolution in Eq. (2.15) becomes simply a product. Thus, for transform parameter s, we have in which (s) and (s) are the Laplace transforms of f(t) and h(t), respectively, with the latter given in explicit form in Eq. (2.14). The desired time domain solution x(t) is then obtained by performing an inverse Laplace transform on (s). However, in practice alternative numerical approaches, such as the Newmark-beta algorithm, are often employed to solve the governing differential Eq. (2.2) directly (Bathe, 1982). As an example, consider the SDOF model with period To = 1.0s (wo = 2 rad/s) and damping ratio = 0.05, subjected to the 1940 El Centro S00E earthquake accelerogram shown in Fig. 2.6. The relative displacement (x) and total acceleration response are displayed in Figs. 2.7a and b, respectively. Meanwhile, the force-displacement curve for the damper is provided in Fig. 2.8. (The force is shown per unit mass.) Notice that the hysteresis loops that determine energy dissipated in the damper are no longer simple ellipses, since there is no longer constant amplitude cycling. Eventually,
Figure 2.6 1940 El Centro S00E Accelerogram
Figure 2.7 1940 El Centro SDOF Time History Response; a) Displacement, b) Acceleration
Figure 2.8 1940 El Centro Force-displacement Damper Response
the loops collapse to a point at the srcin as the linear structure returns to its undisturbed configuration. Plots of the maximum relative displacement and total acceleration response versus structure parameters To and are presented in Figs. 2.9a and b, respectively. These response spectra clearly illustrate the beneficial effect o increased viscous damping. At any particular structural period, an increase in viscous damping tends to diminish the displacement response. Beyond a damping ratio of 50%, however, very little additional benefit is realized.
Figure 2.9 1940 El Centro Spectral Response; a) Relative Displacement, b) Total Acceleration
2.2.5 Response with PassiveDamper The mechanism responsible for energy dissipation in all of the cases considered so far has been due to a linear viscous damper. This represents the prototype energy dissipator. However, there are many other possible mechanisms, as will be detailed in the forthcoming chapters. For example, the beneficial effects of increased energy dissipation can also be realized by incorporating yielding, frictional, or viscoelastic mechanisms. Consider the addition of a generic passive damper element into the SDOF model, as indicated in Fig. 2.10. The response of the system is now influenced by this additional element, which must be characterized in terms of a suitable macroscopic force-displacement model. Typically, this passive element will not be purely viscous in nature, but instead will provide additional stiffness perhaps mass along with dissipative mechanisms. The symbol in Fig. and 2.10 represents a generic integrodifferential operator, such that the force corresponding to the
Figure 2.10 SDOF Model with Passive Damper Element
passive device is written simply as x. This permits quite general response characteristics, including displacement, velocity, or acceleration-dependent contributions, as well as hereditary effects. The equation of motion for the extended SDOF model then becomes
with representing the mass of the passive element. Assume for illustrative purposes that the base structure has a viscous damping ratio = 0.05 andThe thatforce-displacement a simple masslessrelationship yielding device added to serve elastic-perfectly as the passive element. for thisisrate-independent plastic element, depicted in Fig. 2.11, is defined in terms of an initial stiffness and a yield force y.
Figure 2.11 Force-Displacement Model for Elastic- Perfectly Plastic Passive Element
The passively damped SDOF model is first subjected to harmonic loading defined in Eq. (2.8), and examined under steady-state conditions after all transients have dissipated. Figure 2.12 presents the amplitude of the displacement response versus forcing frequency. In order to clearly identify
Figure 2.12 Amplification Factors for Harmonic Excitation with Passive Element
the effect of the added passive damper, the normalization is accomplished in terms o the natural frequency wo = (k/m)1/2 and static response xgt = p0/k of the base structure. Thus, the uppermost curve in Fig. 2.12 is merely a reproduction of the = 0.05 curve from Fig. 2.3. The remaining two curves illustrate the effect of adding a yielding device to the system. It is evident that the peak response, near resonance, is greatly reduced due to the energy dissipation characteristics of the yielding devices. It should be noted that based upon energy considerations at resonance an effective viscous damping ratio can be calculated for our passively damped SDOF model. However, the usefulness o that linearization is limited, except for very preliminary performance estimates. In general, it is best to analyze the actual nonlinear system. Force-displacement plots for the linear spring, viscous damper, and passive damper are provided in Fig. 2.13 for the specific case with ß = 0.90 and = 0.05 to permit comparison with results for the base structure, which were plotted previously to the same scale in Fig. 2.5.
Figure 2.13 Force-displacement Response for Harmonic Excitation with Passive Element; a) Spring, b) Viscous Damper, c) Passive Damper
Response is noticeably decreased. The area contained within the loops present in Figs. 2.13b and c measures the energy dissipated per cycle in the viscous damper and passive damper, respectively. For the example considered, under steady-state conditions, the passive damper dissipates approximately 37% of the energy input to the system. Finally, the SDOF passively damped structure is analyzed for response due to the 1940 El Centro S00E ground motion. The initial stiffness of the elastoplastic passive device is specified as = k, while the yield force y is equal to 20% of the maximum applied ground force. That is, The resulting relative displacement and total acceleration time histories are presented in Fig. 2.14. Again, there is significant reduction in response compared to that of the base structure without the passive element, as shown in Fig. 2.7. Force-displacement loops for the viscous and passive dampers are displayed in Fig. 2.15. In this case, the size o these loops indicates that a significant portion of the energy is dissipated in the passive device. This tends to reduce the forces and displacements in the primary structural elements, which of course is the purpose of adding the passive device. A more detailed examination of elastoplastic devices is included in Chapter 3, which addresses metallic dampers.
Figure 2.14 1940 El Centro Time History Response for SDOF with Passive Element; a) Displacement, b) Acceleration
When the passive damper is purely viscous in nature, the response of the SDOF model is always reduced, since this corresponds precisely to an increase in the damping ratio. The behavioral trends, illustrated in Figs. 2.2, 2.3, and 2.9 can then be expected. For all other typeseffects of passive devices, careful analysis must be performed to insure that beneficial do indeed
Figure 2.15 1940 El Centro Force-displacement Response for SDOF with Passive Element; a) Viscous Damper, b) Passive Damper
result. For example, with the addition of our elastoplastic device above, we found significant improvement in the response due to the El Centro seismic signal. However, this need not be the case. Under certain conditions, the response can actually increase, even for the SDOF representation studied here. The situation is further complicated because location and orientation of passive devices within a structure can have a significant influence on their effectiveness. As a result, the simplistic treatment provided in this section is primarily conceptual. The analysis of real building structures requires a more general methodology. 2.3 Dynamic Analysis of Structural Systems
2.3.1 GeneralF ormulation In light of the preceding arguments, it becomes imperative to accurately characterize the behavior of any passive device by constructing a force-displacement model, suitable under cyclic time-dependent loading. Multiaxial representations may be required. The specific form of that model will necessarily be highly dependent on the device type. Consequently, nothing further will be said here, but appropriate models will be developed in detail in each of the following chapters. Once that model is established for a device, it must be properly incorporated into a mathematical idealization of the overall structure. As mentioned above, seldom is it sufficient to employ an SDOF idealization for an actual structure. Thus, in the present section, the formulation for dynamic analysis is extended to a multi-degree of freedom (MDOF) representation. The finite element method (e.g., Zienkiewicz and Taylor, 1989) currently
provides the most suitable basis for this formulation. From a purely physical viewpoint, each individual structural member is represented mathematically by one or more finite elements having the same mass, stiffness, and damping characteristics as the srcinal member. Beams and columns are represented by one-dimensional elements, while shear walls and floor slabs are idealized by employing two-dimensional finite elements. For more complicated or critical structural components, complete three-dimensional models can be developed, and incorporated into the overall structural model in a straightforward manner via substructuring techniques. The finite element method (FEM) actually was developed largely by civil engineers in the 1960s from this physical perspective. However, during the ensuing decades the method has also been given a rigorous mathematical foundation, thus permitting the calculation of error estimates and the utilization of adaptive solution strategies (e.g., Szabó and Babuska *, 1991). Additionally, FEM formulations can now be derived from variational principles or Galerkin weighted residual procedures. Details of these formulations is beyond our scope. However, it should be noted that numerous generalpurpose finite element software packages currently exist to solve the structural dynamics problem, including ABAQUS, ADINA, ANSYS, and MSC/NASTRAN. While none of these programs specifically address the special formulations needed to characterize passive energy dissipation devices, most permit generic user-defined elements. Alternatively, one can utilize packages geared exclusively toward civil engineering structures, such as ETABS, DRAIN, and IDARC, which in some cases can already accommodate typical passive elements. Via any of the above-mentioned methods and programs, the displacement response o the structuretheis components ultimately represented by a discrete of variables, vector which x( can considered of a generalized relativesetdisplacement t) be o dimension N. Then, in analogy with Eq. (2.2), the N equations of motion for the discretized structural system, subjected to uniform base excitation and time varying forces, can be written
where M, C, and K represent the mass, damping, and stiffness matrices, respectively, while symbolizes a matrix of operators which model the passive dampers present in the structure. Meanwhile, the vector contains the rigid body contribution of the seismic g round displacement to each degree of freedom, and p includes the forces due to aerodynamic loading. The matrix represents the mass of the passive dampers. There are several approaches that can be taken to solve Eq. (2.20). The preferred approach, in terms of accuracy and efficiency, depends upon the form of the various terms in that equation. Let us first suppose that the
passive dampers can be modeled as direct linear functions of the acceleration, velocity, and displacement vectors. That is,
Then, Eq. (2.20) can be rewritten as
in which
Equation (2.22) is now in the form of the classical matrix structural dynamic analysis problem. In the simplest case, which we will now assume, all of the matrix coefficients associated with the primary structure and the passive elements are constant. As a result, Eq. (2.22) represents a set of N linear second-order ordinary differential equations with constant coefficients. These equations are, in general, coupled. Thus, depending upon , the solution of Eq. (2.22) throughout the time range of interest could become computationally demanding. This required effort can be reduced considerably if the equation can be uncoupled via a transformation; that is, if , , and can be diagonalized. Unfortunately, this is not possible for arbitrary matrices , , and . However, with certain restrictions on the damping matrix modal coordinates accomplishes the objective, as shown below. , the transformation to
2.3.2 Modal Superposition Metho Consider the generalized eigenvalue problem associated with the undamped free vibration of our MDOF structure. That is
where wo represents an undamped natural frequency of the structure including passive elements and is the associated mode shape vector. The present undamped system will have N such natural frequencies and mode shapes labeled woi, and i, respectively, for i = 1, 2, . . . , N. Usually, the natural frequencies are ordered by increasing numerical value, with the lowest (wol)
referred to as the fundamental frequency. Additionally, the mode shapes satisfy the following orthogonality conditions
and form a complete set spanning the N-dimensional vector space. Consequently, this set provides the basis for a suitable transformation that can be applied to our srcinal system defined in Eq. (2.22). In Eqs. (2.25a) and (2.25b), superscript T indicates vector or matrix transpose. There are numerous methods available to solve the generalized eigenvalue problem defined in Eq. (2.24). The choice depends largely upon the size and structure of the matrices and . Nearly all of the structural analysis codes noted above contain efficient and robust eigenvalue extraction routines that require little user intervention. Routines are also available in the public domain through the LAPACK (Anderson, et al., 1992) implementation. For notational convenience, the natural frequencies are placed in a diagonal matrix wo. The corresponding mode shape vectors form the columns of a square matrix , which functions as the transformation matrix. Thus, any relative displacement vector x can be represented by x = y (2.26) where y is the vector of modal (or normal) coordinates. Utilizing Eq. (2.26), along with Eq. (2.25), in Eq. (2.22) leads to the following equations of motion expressed in terms of the modal coordinates,
where
In general, Eq. (2.27) still represents a coupled set of ordinary differential equations. The equations uncouple only when is also a diagonal matrix. This occurs for the case of proportional (or Rayleigh) damping, in which
for scalar constants 0 and 1. From Eqs. (2.25) and (2.29), one obtains
which is diagonal. The form of can actually be generalized to the Caughey series
while still permitting diagonalization. Equation (2.31) is seldom used to compute from a given set of j. Instead, modal viscous damping ratios i are specified, such that
with representing a diagonal matrix containing the i. With this assumed, Eq. (2.27) becomes
Since the equations are now uncoupled, we can write a scalar equation, for each mode i, as
in which
Equation (2.34) has the same form as the SDOF system we examined in the previous section. Consequently, all of the methodology and behavioral patterns discussed in that section are directly applicable to Eq. (2.34). The solution of our srcinal problem expressed in Eq. (2.22) is greatly simplified. Once Eq. (2.33) is solved, the relative displacement vector x can be determined at any time via the transformation Eq. (2.26). The major computational task in this whole process is the determination of the natural frequencies and mode shapes. Even this task is not as onerous as it first appears, since for most physical problems only a small percentage of the N modes actually participate significantly in the system response. As a result, only the structural modes within a certain frequency range need be calculated. The price paid for this simplicity is the initial restriction to system matrices with constant coefficients, and the further constraint on the damping matrix specified in Eq. (2.31). If the latter condition is relaxed, it still may be advantageous to use a modal approach. The governing equations in modal coordinates, Eq. (2.27), remain coupled. However, it is often still possible to utilize a set much smaller than N, since typically only a small portion of the undamped modes will be excited. For a more thorough treatment of the modal superposition method, the textbook by Clough and Penzien (1975) is recommended.
2.3.3 Damping in Traditionaland PassivelyDamped Structures In traditional structures, the mass and stiffness properties of various members can generally be modeled with a reasonable degree of accuracy. The damping properties are much more difficult to characterize, with significant energy dissipation attributable to the primary structural materials, to non-ideal joint behavior, as well as to various nonstructural components. experimental suggest that inherent the damping forces are nearly independent of Most frequency, which results is contrary to the frequencydependence of viscous damping forces. Additionally, it is known that damping properties of real structures are functions of amplitude. Despite these general observations, it has not been possible to adequately quantify the energy dissipation. As a result, it is quite reasonable to assume proportional damping, or its generalized form, in order to simplify the analysis. Historically, this has been the approach taken. Structural analysis computer codes and design procedures that exist today most often assume proportional damping. Modal damping ratios in the range of 1%-5% are generally employed, depending upon the type of structure. In light of the results for our SDOF model, considerable care must be exercised not to overestimate the damping capability of an actual structure, since this could lead to the development o unconservative designs. An argument can be made for the use of zero damping, unless sufficient documentation can be provided to justify a certain damping ratio. The use of passive energy dissipation technology in structures, not only allows one to reduce structuralofdamage resulting from environmental disturbances, also permits characterization the primary damping mechanisms which are now but concentrated in the passive elements. This characterization is accomplished through the development o detailed force-displacement models for the passive dampers from physical tests and analytical studies. These models can subsequently be incorporated into the overall system equations, as indicated by the term x in Eq. (2.20). Under some circumstances, the resulting equations can be reduced to the form of Eq. (2.22), or perhaps even to Eq. (2.33) for proportional damping. In the latter case, analysis and design procedures for the passively damped structure can closely follow those developed for traditional structures. An example of such a procedure is provided for viscoelastic dampers in Chapter 5. Generally speaking, however, the governing equations for a passively damped structure are not reducible to an uncoupled set such as Eq. (2.33) that are associated with proportional damping. In some cases, nonlinearities are present, while in others hereditary effects appear. In fact, from a physical standpoint, it is not necessarily optimal design awhich structure such that Eq. (2.31) is satisfied. (For example, isolated to buildings, can be particularly effective against seismic loads, dobase not utilize proportional
damping, but instead feature a flexible, highly damped mechanism at the base.) Consequently, it is important to develop alternative numerical approaches and design methodologies applicable to more generic passively damped structural systems governed by Eq. (2.20). Direct time domain numerical integration algorithms are most useful in that regard. The Newmark beta algorithm, which has been used extensively in structural dynamics, is briefly described in the following for a structure modeled by Eq. (2.22) with matrices and not necessarily constant. 2.3.4 Direct Time DomainAnalysis The implicit Newmark algorithm begins with a discretization of the time axis into intervals of duration t. At any instant, the equations of motion given in Eq. (2.22) can be recast in the form:
where the superscripts denote evaluation at time t. It is assumed that this solution is known, and that the role of the analysis is to determine a solution at time t + t. That is, we wish to solve
With thetoNewmark parameters chosen usual the trapezoidal rule is then t + in t tothe invoked relate quantities at time those at manner, time t. Thus,
After substituting Eq. (2.38) into Eq. (2.37), one obtains
where
In general, Eq. (2.39) represents a set of nonlinear equations in the variables x t+t. There are numerous iterative solution techniques that are applicable, including full, modified, and quasi-Newton methods. For the full Newton-Raphson method, one solves the linearized system
at each iteration (n). In Eq. (2.41), the effective tangent stiffness matrix and force vector can be written
respectively. The solution of Eq. (2.41) is used to update the displacements via
Iteration continues until a suitable norm of x is reduced below a specified small tolerance. For the solution of highly nonlinear problems, the inclusion of a line searching algorithm is often needed to obtain convergence. Once convergence is obtained at a particular time step, the displacement is established from , the velocity and acceleration vectors at t + t are computed from Eq. (2.38), and member forces and moments are evaluated. Then, the time parameter t is incremented, a new time step size t is selected, and the analysis proceeds to the next step by assuming or by utilizing an extrapolation algorithm. Since Eq. (2.41) requires the formation and factorization of at each iteration, in practice, the full Newton-Raphson algorithm is seldom employed. Instead, the system matrix is updated less frequently and acceleration techniques are often utilized. All major structural analysis codes include algorithms similar to the one outlined above. In many cases, the program adaptively determines appropriate time step size to maximize computational efficiency and to reduce the burden placed on the design engineer. Additional details concerning time domain numerical analysis can be found in Zienkiewicz and Taylor (1989) and Bathe (1982). It is expected that these methods will play an increasingly important role in structural design, particularly for systems employing passive energy dissipation technology. 2.4 Energy Formulations In the previous two sections, we have considered SDOF and MDOF structural systems. The primary thrust of our analysis procedures has been the
determination of displacements, velocities, accelerations, and forces. These are the quantities that, historically, have been of most interest. However, with the advent o innovative concepts for aseismic design, including base isolation systems and passive energy dissipation systems, it is important to rethink current analysis and design methodologies. In particular, a focus on energy as a design criterion is conceptually very appealing. With this approach, the engineer is concerned, not so much with the resistance to lateral but disturbance. rather, with the need to this dissipate the concept energy input intonew. the structure from the loads seismic Actually, energy is not Housner (1956) suggested an energy-based aseismic design approach even for more traditional structures several decades ago. Furthermore, the energy approach is not restricted to consideration of earthquake resistance alone, and is easily extended to incorporate dynamic effects due to wind loading. The resulting formulation is quite appropriate for a general discussion of energy dissipation in structures, since passive devices can be employed to suppress vibrations caused by both seismic and aerodynamic forces. In the next section, an energy formulation is developed for our idealized SDOF structural system, which may include one or more passive devices. The srcinal work of Housner provides the necessary basis for this simple formulation. Then, with all o the terminology firmly established, a literature review of the application of energybased aseismic concepts to traditional structures is presented. Included is a discussion of both the advantages and shortcomings of the approach. More importantly, however, the energy concept is ideally suited for application to non-traditional structures employing passive dampers, since for these systems proper energy management is a key to successful design. With that in mind, the formulation is extended to a more general multi-degree of freedom representation and certain aspects of a modern energybased approach to aseismic design are discussed.
2.4.1 SD OF Structures Consider once again the SDOF oscillator shown in Fig. 2.1 and governed by the equation of motion defined in Eq. (2.2). An energy representation can be formed by integrating the individual force terms in Eq. (2.2) over the entire relative displacement history. The result becomes EK + ED + ES = EI
where
(2.44)
where EI = EIs + EIw
(2.45f)
The individual contributions included on the left-hand-side of Eq. (2.44) represent the relative kinetic energy of the mass (EK), the dissipative energy caused by inherent damping within the structure (ED), and the elastic strain energy ( ES). The summation o these energies must balance the input energy ( EIs) imposed on the structure by the seismic event, plus the input energy (EIw) due to wind forces. Note that each of the energy terms is actually a function of time, and that the energy balance obtains at each instant throughout the duration of the loading. It is unrealistic to expect that a traditionally designed structure will remain entirely elastic during a major seismic disturbance. Instead, the design engineer relies upon the inherent ductility of structures to prevent catastrophic failure, while accepting the fact EIs) from energy that some damagethe may occur. Inofsuch case, thetoenergy the earthquake simply exceeds capacity the astructure storeinput and (dissipate by the mechanisms specified in Eqs. (2.45a-c). Once this capacity is surpassed, portions of the structure typically yield or crack. The stiffness is then no longer a constant, and the spring force in Eq. (2.2) must be replaced by a more general functional relation fS(x), which will commonly incorporate hysteretic effects. In general, Eq. (2.45c) is redefined as follows for inelastic response:
in which ES is assumed separable into additive contributions ESe and ESp, representing the fully recoverable elastic strain energy and the dissipative plastic strain energy, respectively. Figure 2.16a provides the energy response of a 0.3-scale, six-story concentrically braced steel structure as measured by Uang and Bertero (1986). The seismic input consisted of the 1978 Miyagi-Ken-Oki Earthquake signal scaled totheproduce a peak shaking table acceleration of 0.33g, which was deemed to represent damageability limit state of the model. At this level of loading, a significant portion of the energy input to the structure is dissipated, with both viscous damping and inelastic hysteretic mechanisms
having substantial contributions. If the intensity of the signal is elevated, an even greater share of the energy is dissipated via inelastic deformation. Finally, for the collapse limit state of this model structure at 0.65g peak table acceleration, approximately 90% of the energy is consumed by hysteretic phenomena, as shown in Fig. 2.16b. Evidently, the consumption of this quantity of energy has destroyed the structure.
Figure 2.16 Energy Response of Traditional Structure; a) Damageability Limit State, b) Collapse Limit State (Uang and Bertero, 1986)
From an energy perspective, then, for proper aseismic design, one must attempt to minimize the amount of hysteretic energy dissipated by the structure. There are basically two viable approaches available. The first involves designs that result in a reduction in the amount of energy input to the structure. Base isolation systems, for example, fall into that category. The second approach, and the one that provides the central theme for this text, focuses on the introduction of additional energy dissipating mechanisms
into the structure. These devices are designed to consume a portion of the input energy, thereby reducing damage to the main structure caused by hysteretic dissipation. Naturally, for a large earthquake, the devices must dissipate enormous amounts o energy. The SDOF system with a passive damper is displayed in Fig. 2.10, while the governing integrodifferential equation provided in Eq. (2.18). After integrating with respect to x, an energy balance equationis can be written EK + ED + ESe + ESp + EP = EI
(2.47)
where the energy associated with the passive dampers is
and the other terms are as previously defined. As an example of the effects of passive devices on the energy response of a structure, consider the tests of a one-third scale three-story lightly reinforced concrete framed building conducted by Lobo et al. (1993). Figure 2.17a displays the measured response of the structure due to the scaled 1952 Taft N21E earthquake signal normalized for peak ground accelerations of 0.20g. A considerable portion of the input energy is dissipated via hysteretic mechanisms, which tend to damage the primary structure through cracking and the formation of plastic hinges. On the other hand, damage is minimal with the addition of a set of viscoelastic braced dampers. The energy response of the braced structure, due to the same seismic signal, is shown in Fig. 2.17b. Notice that although the input energy has increased slightly, the dampers consume a significant portion of the total, thus protecting the primary structure.
Figure 2.17 Energy Response of Test Structure; a) Without Passive Devices, b) With Passive Devices (Lobo et al., 1993)
2.4.2 Energy-basedDesign While the energy concept, as outlined briefly in the prior section, does not currently provide the basis for aseismic design codes, there is a considerable body of knowledge that has been developed from its application to traditional structures. Housner (1956, 1959) was the first to propose an energy-based philosophy for earthquake resistant design. In collapse particular, was concerned with active limit-design toward preventing of he structures in seisinically regions.methods Housneraimed assumed that the energy input calculated for an undamped, elastic idealization of a structure provided a reasonable upper bound to that for the actual inelastic structure. Berg and Thomaides (1960) examined the energy consumption in SDOF elastoplastic structures via numerical computation, and developed energy input spectra for several strong-motion earthquakes. These spectra indicate that the amount of energy EI imparted to a structure from a given seismic event is quite dependent upon the structure itself. The mass, the natural period of vibration, the critical damping ratio, and yield force level were all found to be important characteristics. On the other hand, their results did suggest that the establishment of upper bounds for EI might be possible, and thus provided support for the approach introduced by Housner. However, the energy approach was largely ignored for a number of years. Instead, limit-state design methodologies were developed which utilized the concept o displacement ductility to construct inelastic response spectra as proposed initially by Veletsos and Newmark (1960). More recently, there has been a resurgence of interest in energy-based concepts. For example, Zahrah and Hall (1982) developed a multi-degree of freedom energy formulation and conducted an extensive parametric study of energy absorption in simple structural frames. Their numerical work included a comparison between energybased and displacement ductility-based assessments of damage, but the authors stopped short of issuing a general recommendation. A critical assessment of the energy concept as a basis for design was provided by Uang and Bertero (1988). The authors initially contrast two alternative definitions of the seismic input energy. The quantity specified in Eq. (2.45d) is labeled the relative input energy, while the absolute input energy ( EIa) is defined by
In conjunction with this latter quantity, an absolute kinetic energy ( EKa) is also required, where
The absolute energy equation corresponding to Eq. (2.47) then becomes EKa + ED + ESe + ESp + EP = EIa + EIw
(2.50)
Based upon the development of input energy spectra for a SDOF system, the authors conclude that, while both measures produce approximately equivalent spectra in the intermediate period range, EIa should be used as a damage index for short period structures, and EI is more suitable for long period structures. Furthermore, an investigation revealed that the assumption of Housner to employ the idealized elastic strain energy, as an estimate of the actual input energy, is not necessarily conservative. Uang and Bertero also studied a multi-degree of freedom structure, and concluded that the input energy spectra for a SDOF can be used to predict the input energy demand for that type of building. In a second portion of the report, an investigation was conducted on the validity of the assumption that energy dissipation capacity can be used as a measure of damage. In testing cantilever steel beams, reinforced concrete shear walls, and composite beams the authors found that damage depends upon the load path. The last observation should come as no surprise to anyone familiar with classical failure criteria. However, it does highlight a serious shortcoming for the use of the energy concept for limit design of traditional structures. As was noted above, in these structures, a major portion of the input energy must be dissipated via inelastic deformation, but damage to the structure is not determined simply by the magnitude o the dissipated energy. On the other hand, in non-traditional structures incorporating passive damping mechanisms, the energy concept is much more appropriate. The emphasis in design is directly on energy dissipation. Furthermore, since an attempt is made to minimize the damage to the primary structure, the selection of a proper failure criterion is less important.
2.4.3 MD OF Structures In Section 2.4.1, an energy formulation was presented for a SDOF system which included passive dampers. While this formulation is of interest for a general understanding of the overall behavior of a structure during a seismic or aerodynamic event, local characteristics are also quite important. In particular, the ultimate collapse of a structure can often be traced to areas of damage concentration. In order to properly account for this behavior, a more comprehensive approach is needed, which permits a better spatial representation of the structure. Consequently, a general multi-degree o freedom energy formulation is developed below, in a manner similar to that presented by Zahrah and Hall (1982) and Uang and Bertero (1986). The starting point is the equations of motion for the MDOF structure provided in Eq. (2.20). An energy representation for the structural response
can be formed by integrating the individual force components in Eq. (2.20) over the corresponding relative displacements. Once again this produces the scalar equation, EK + ED + ES + EP = EI
(2.51)
where
The terms on the left hand side of Eq. (2.51) quantify the contribution of the various forms of energy that the structure employs to resist the earthquake or wind load event. These energies can be evaluated in a straightforward manner after a solution to Eq. (2.20) is obtained via any of the methods outlined in Section 2.3. The total energy balance Eq. (2.51) is, of course, important for an overall assessment o performance. However, in addition, the energy response of each structural member can be determined by utilizing the individual element submatrices. These latter calculations permit an investigation of the flow of energy within the structure as, for example, a seismic event progresses. Potential zones of damage concentration can be easily identified, and perhaps subsequently eliminated through the use of passive dampers. With the continuing advancement in computational technology, one can envision an interactive computer-aided design process, in which the design engineer directs the development of aseismic structures by considering this energy flow. 2.5 Concluding Remarks An attempt has been made to present some fundamental concepts related to the behavior, analysis, and design of structures subjected to transient disturbances. Emphasis was placed on those aspects of particular importance to our investigation o passive energySDOF dissipation Consequently, we began by examining the behavior of the simple system,systems. as a basis
for the understanding of structural dynamics. Our presentation stressed the beneficial effects of increased damping on system performance. In order to address the analysis o actual structures, an MDOF formulation was then introduced. Solution procedures based upon modal superposition and direct time domain analysis were presented. The most appropriate method for a particular structure depends largely on the form of the damping matrix. For structures with passive energy dissipation systems, time domain methods are often required. Finally, a discussion of energy concepts was included. These energy formulations provide a useful alternative viewpoint for the design o passively damped structures. The interested reader should consult Clough and Penzien (1975) for a more complete treatment on structural dynamics and Bathe (1982) for a thorough discussion on finite element analysis of time-dependent phenomena. All of the formulations presented in this chapter tacitly assume that the time history o the seismic ground motion and the aerodynamic loading is fully known. Since this information is not actually available a priori, there is good reason to attempt to quantify the uncertainty. The characterization of earthquake and wind loading as stochastic processes is discussed, for example, in Bolotin (1984) and in Soong and Grigoriu (1993). Meanwhile, development of probabilistic finite element methods is described in Clough and Penzien (1975), Shinozuka (1987), and Cruse et al. (1990). Although widespread acceptance of these approaches has not yet occurred, techniques which admit uncertainty are likely to have an increasing importance in future structural design methodologies. Irrespective of whether formulations are posed with a deterministic or stochastic basis, there is a need for accurate mathematical models describing the force-deformation and energy dissipation characteristics of the various types of passive dampers. Fortunately, many such models already exist, and will be presented in detail in the forthcoming chapters, along with information on experimental testing and full-scale implementation. References Anderson, E. et al. (1992), LAPACK Users' Guide, SIAM, Philadelphia. Bathe, K.-J. (1982), Finite Element Procedures in Engineering Analysis, PrenticeHall, Englewood Cliffs, New Jersey. Berg, G. V. and Thomaides, S. S. (1960), Energy Consumption By Structures in Strong Motion Earthquakes, Proceedings of the Second World Conference on Earthquake Engineering, II, 681-697, Tokyo. Bolotin, V.V. (1984), Random Vibrations of Elastic Systems, Martinus Nijhof Publishers, The Hague. Clough, R. W. and Penzien, J. (1975), Dynamics of Structures, McGraw-Hill, New York.
Cruse, T. A., Unruh, J. F., Wu, Y.-T., and Harren, S. V. (1990), Probabilistic Structural Analysis for Advanced Space Propulsion Systems, Journal o Engineering for Gas Turbines and Power, 112, 251-260. Hanson, R. D. (1993), Supplemental Damping for Improved Seismic Performance, Earthquake Spectra, 9(3), 319-334. Housner, G. W. (1956), Limit Design of Structures to Resist Earthquakes, Proceedings of the World Conference on Earthquake Engineering, 5-1 - 5-13,
Earthquake Engineering Research Center, Berkeley, California. Housner, G. W. (1959), Behavior of Structures During Earthquakes, Journal of the Engineering Mechanics Division, ASCE, 85(EM4), 109-129. Lobo, R. F., Bracci, J. M., Shen, K. L., Reinhorn, A.M., and Soong, T. T. (1993), Inelastic Response of R/C Structures with Viscoelastic Braces, Earthquake Spectra, 9(3), 419-446. Newmark, N.M. and Rosenblueth, E. (1971), Fundamentals of Earthquake Engineering, Prentice Hall, Englewood Cliffs, New Jersey. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992), umerical Recipes in Fortran, Cambridge University Press, Cambridge, UK. Shinozuka, M. (1987), Stochastic Mechanics, I, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York. Soong, T. T. and Grigoriu, M. (1993), Random Vibration of Mechanical and Structural Systems, P T R Prentice-Hall, Englewood Cliffs, New Jersey. Szabó, B. and Babuska *, I. (1991),Finite Element Analysis, John Wiley and Sons, New York. Uang, C. M. and Bertero, V.V. (1986), Earthquake Simulation Tests and ssociated Studies of a 0.3 Scale Model of a Six-Story Concentrically Braced Steel Structure, Report No. UCB/EERC-86/10, Earthquake Engineering Research Center, Berkeley, California. Uang, C. M. and Bertero, V.V. (1988), Use of Energy as a Design Criterion in Earthquake Resistant Design, Report No. UCB/EERC-88/18, Earthquake Engineering Research Center, Berkeley, California. Veletsos, A. S. and Newmark, N.M. (1960), Effect of Inelastic Behavior on the Response of Simple Systems to Earthquake Motions, Proceedings of the Second World Conference on Earthquake Engineering, II, 895-912, Tokyo. Zahrah, T. F. and Hall, W. J. (1982), Seismic Energy Absorption in Simple Structures, Structural Research Series No. 501, University of Illinois, Urbana, Illinois. Zienkiewicz, O. C. and Taylor, R. L. (1989), The Finite Element Method, Volumes 1 and 2, Fourth Edition, McGraw-Hill, London.
3 Metallic Dampers 3.1 Introduction One of the most effective mechanisms available for the dissipation of energy, input to a structure during an earthquake, is through the inelastic deformation of metallic substances. In traditional steel structures, aseismic design relies upon the post-yield ductility of structural members to provide the required dissipation. However, the idea o utilizing separate metallic hysteretic dampers within a structure to absorb a large portion of the seismic energy began with the conceptual and experimental work by Kelly et al. (1972) and Skinner et al. (1975). Several of the devices considered by those researchers included torsional beam, flexural beam, and U-strip dampers as shown schematically in Fig. 3.1.
Figure 3.1 Metallic Damper Geometries (Skinner et al, 1975; a) Torsional Beam, b) Flexural Beam, c) U-strip
In order to effectively include these devices in the design of an actual structure, one must be able to characterize their expected nonlinear force-displacement behavior under arbitrary cyclic loads. Özdemir (1976) was the first to consider this modeling problem. He utilized analogies with existing elastoplastic and viscoplastic constitutive theories to develop appropriate forms for the force-displacement relationships. Additionally, Özdemir detailed efficient numerical algorithms for computing the response of structures with by metallic dampers Shortly subjected to general time-dependent loading, such as that caused an earthquake. thereafter, Bhatti et al. (1978) employed that methodology to study the response of structures that utilized torsion bar dampers in conjunction with a seismic base isolation system. During the ensuing years, considerable progress has been made in the development o metallic dampers. For example, many new designs have been proposed, including the X-shaped and triangular plate dampers displayed in Fig. 3.2. Alternative materials, such as lead and shape-memory alloys, have been evaluated. Numerous experimental investigations have been conducted to determine performance characteristics o individual devices and laboratory test structures. As a result of this ongoing research program, several commericial products have been developed and implemented in both new and retrofit construction projects. In particular, a number of existing structures in New Zealand, Mexico, Japan, Italy, and the United States now include metallic dampers as a means for obtaining improved seismic resistance. a)
b)
Figure 3.2 Metallic Dampers; a) X-shaped Plate Damper (Courtesy of CounterQuake Corp.), b) Triangular Plate Damper (Tsai et al., 1993)
In order to fully appreciate the behavior of metallic energy dissipators, one must first consider the inelastic response of metals under time-dependent cyclic loading. This is addressed in some detail in the following section, which deals primarily with the definition of constitutive models. These models are then used in Section 3.3 to develop appropriate force-displacement relationships for the metallic dampers. Once the forcedisplacement model is established, the design engineer must incorporate that information in theconsiderable analysis of the overall task istodiscussed in Section As noted above, effort has structure. also beenThis devoted experimental studies3.4. at both the component and structural level. Results from several of these studies are examined in Section 3.5. The focus then shifts to design and implementation. An overview of some of the key aspects of design associated with a decision to include metallic dampers in a construction project is provided in Section 3.6. The chapter then concludes with a description of several existing structures that have either been newly designed or retrofitted with metallic energy dissipators for enhanced seismic protection. 3.2 Basic Principles Despite obvious differences in the geometric configuration of the devices displayed in Figs. 3.1 and 3.2, the underlying dissipative mechanism in all cases results from the inelastic deformation of a metal. Usually that metal is mild steel, although sometimes lead or more exotic metal alloys are employed. Ideally, one would hope to develop a model of any metallic damper starting from the micromechanical theory of dislocations, which must ultimately determine the inelastic response. However, since that direct physical approach from first principles is not yet feasible, one normally accepts a phenomenological description based upon observation of behavior at the macroscopic level. A mathematically consistent framework, such as plasticity or viscoplasticity theory, is then constructed to reproduce that behavior and to predict response under very general conditions. In the following subsections, we begin by presenting some o the experimental observations most relevant to our study of inelastic deformation in metallic dampers and then discuss several associated theories that have been developed. This treatment is necessarily incomplete. More detail can be found in the texts by Fung (1965), Mendelson (1968), and Shames and Cozzarelli (1992).
3.2.1 Plasticity Consider first the behavior of a cylindrical metal rod, with initial length L0 and crosssectional area A0, subjected to uniaxial tension as shown in Fig. 3.3. It is assumed that the load P is incremented slowly to insure the validity
Figure 3.3 Cylindrical Rod Subjected to Uniaxial Tension
of a quasistatic approximation. Typical stress-strain curves are displayed in Fig. 3.4. In these diagrams, the abscissa represents the conventional strain , while the nominal stress n is plotted on the ordinate-axis. Referring to Fig. 3.3,
The curve in Fig. 3.4a is characteristic of most metals. At loads corresponding to nominal stress less than the yield stress y, the response of the specimen is fully elastic with n proportional to . In this range, the initial state O is fully recoverable with removal of the applied load, and there is no energy dissipation. On the other hand, when the nominal stress exceeds the yield stress (i.e., beyond point Y on the curve), irreversible plastic deformation occurs in conjunction with inelastic energy dissipation. Consider for a moment the state labeled B. It is useful to partition the total strain at B into elastic (el) and inelastic (in) contributions, as indicated in the diagram. Thus, = el + in
(3.2)
in which
E representing the elastic modulus. The energy, or more precisely with the energy density, is measured by the area under the stress-strain curve from O to B. Part of that energy is recoverable. However, the remainder, associated with the inelastic strain in and identified by the shaded portion
of Fig. 3.4a, is dissipative. A significant portion of that dissipative energy is converted into heat. In this strain hardening regime, successive increments of stress produce correspondingly greater increments of strain, until the point M is reached associated with the maximum load that the tensile specimen can withstand. Beyond M the specimen becomes unstable. Localization phenomena commence with the appearance of a necking region in which three-dimensional states of stress are present. Ultimately failure occurs at point X.
Figure 3.4 Nominal Stress - Conventional Strain Diagrams
As noted above, this description applies to most metals. However, annealed mild steel and some other alloys behave as depicted in Fig. 3.4b. In this case, the response is similar to that previously discussed, except in the region just beyond first yield at YU. In these materials, there is an abrupt drop in stress from point YU to YL, corresponding to the upper and lower yield stress, respectively. This phenomenon and the ensuing stressstrain plateau is caused by the formation and propagation of Lüders bands. It should be noted, however, that strain hardened mild steel does not exhibit this behavior, but instead follows that depicted in Fig. 3.4a. The final stress-strain curve, shown in Fig. 3.4c, is typical of a brittle metal, such as cast iron. Obviously, this type of material is not a good candidate for use in metallic dampers, since very little energy dissipation occurs before fracture. The definitions employed in Eq. (3.1) are perfectly adequate when the length and area of the specimen does not change significantly from the initial values. However, for higher strain levels, the natural strain e and true stress are
more appropriate measures, where
Replotting Fig. 3.4a in these coordinates leads to the curve displayed in Fig. 3.5. Notice that now the true stress increases with natural strain all the way to rupture. More importantly, if one were to test the specimen instead in compression, a nearly identical -e curve would result. On the other hand, a plot of n versus for compression would not reproduce Fig. 3.4a much beyond the yield stress.
Figure 3.5 True Stress - Natural Strain Diagram
Over the years, numerous mathematical models have been introduced to idealize the stress-strain curves, including the elastic-perfectly plastic model, the elastic-linear strain hardening model, and the Ramberg-Osgood model shown in Fig. 3.6. The last o these models can be written
with material constants E, k, and n. Employing Eqs. (3.2) and (3.3) and assuming that all nonlinearity is inelastic, one can alternatively write
Thus, the Ramberg-Osgood model essentially establishes a power law relationship between stress and inelastic strain, a condition which is approximately satisfied during monotonic loading experiments with a variety of metals. However, Eq. (3.5) is not adequate for describing response to arbitrary cyclic loading in which the state of stress is dependent upon not just the current strain but rather the entire prior history.
Figure 3.6 Stress-Strain Mathematical Models; a) Elastic-Perfectly Plastic, b) Elastic-Linear Strain Hardening, c) Ramberg-Osgood
Consequently, in order to develop models for metallic dampers, we must extend our discussion to behavior under load reversals that involve excursions into the inelastic range. For these cases, response is path dependent. Consider the cylindrical specimen first loaded in tension past yield at Y to a point B in the strain hardening range, and then gradually unloaded as indicated in Fig. 3.7. The unloading branch of the curve is parallel to the initial loading curve, indicating purely elastic response. However, i unloading continues sufficiently into the compression range, yielding will again take place at the point labeled Y'. The stress associated with Y', i.e. , is dependent upon the prior amount of strain hardening. This is known as the Bauschinger effect. As useful simplifications, isotropic hardening theories establish
while in kinematic hardening theories
From experiments, one finds that Y' is somewhere in-between, suggesting a combination of isotropic and kinematic theories is more realistic. Many ad hoc schemes have been proposed to extend the Ramberg-Osgood model to handle general cyclic conditions. These schemes tendthe to require the stipulation of manyplasticity special rules. A more satisfying approach involves formulation of a consistent theory that permits the development of a rational constitutive model.
Figure 3.7 Cyclic Stress-Strain Response
All of the foregoing discussion concerned uniaxial loading. Naturally, inelastic deformation often involves more complicated conditions. In those instances, the theory must be generalized to three dimensions utilizing stress ( ) and strain () tensors. This generalization is complicated by experimental evidence which indicates that inelastic deformation destroys the inherent isotropy of the material. A metal, such as steel, with zero residual stress, typically possesses a uniform yield stress for loading in any direction. However, once that metal is loaded into the strain hardening regime in a certain direction, its yield stress will vary with direction (i.e., yield stress will not be the same in longitudinal andmodel transverse directions). Another characteristic of thatthemust be incorporated into a valid pertains to the relative incompressibility plastic portion of the response. The material does remain compressible, however, due to the recoverable elastic contribution to any deformation. In order to accommodate all of the above observations, while maintaining sel consistency, plasticity theories are incremental in nature and contain three key items; namely, a yield criterion, flow rule, and hardening rule. The yield criterion (or yield surface in stress space) is used to determine if yielding has occurred for a given increment of load. A simple example involves the von Mises yield surface, in which
where J2 is the second invariant of the deviatoric stress. For F < 0, the response is elastic. Yielding occurs when F = 0 and (F/) > 0. In that case, a flow rule is needed to determine the incremental inelastic strains ( din).
For an associative flow rule, normally valid for metal plasticity,
with proportionality factor d. The hardening rule completes the picture by incorporating some combination of isotropic and kinematic hardening to determine the new yield surface in stress space. A particularly successful metal plasticity theory began with the nested yield surface models of Mröz (1967), and the subsequent twosurface development by Krieg (1975) and Dafalias and Popov (1975). These models not only account for the Bauschinger effect, but also permit a smooth transition from the elastic to elasoplastic range. Consequently, two-surface models have found wide application in the computational mechanics literature. It is worthwhile at this point to examine a specific two-surface formulation that will later be applied to model the response of metallic dampers. In this model, two distinct, but nested, yield surfaces are first defined in stress space. The inner or loading surface, which separates the elastic and inelastic response regimes, is characterized by its center and radius represented by the back stress b and inner yield strength , respectively. On the other hand, the outer or bounding surface, which completely contains the smaller inner surface, is always centered at the srcin of stress space with radius equal to a variable outer yield strength . Translation of the inner surface corresponds to kinematic hardening, while expansion of the outer surface produces isotropic hardening. Fig. 3.8 illustrates these geometrical quantities for the uniaxial case. The yield criteria, flow rules, and hardening rule are established to ensure that the state of stress always lies on or within both surfaces, that all transitions during loading are smooth, and that infinitesimal strain cycles do not cause anomalous behavior. The model, which requires the determination of six material parameters ( E, , , , , n), is defined in Table 3.1. Note that corresponds to the initial value of , while a superposed dot represents differentiation with respect to time or to a pseudotime loading parameter.
Figure 3.8 Two Surface Plasticity Model - Uniaxial Case
Table 3.1 Uniaxial Two-Surface Model Definition
The constitutive model presented in Table 3.1 is next used, as an illustration, to represent ASTM A36 structural steel. The elastic modulus was equated with the usual handbook cyclic value,data while the remaining parameters (1985). were established from stabilized presented by Cofie five and Krawinkler A comparison of the stress-strain response obtained from the two-surface model and the experimental data is displayed in Fig. 3.9. Notice from that figure that the stabilized cyclic curves do not exhibit a yield plateau. Of course, the model presented above represents only one particular formulation. For a further discussion of metal plasticity within the context of finite element structural analysis, the interested reader may consult Zienkiewicz and Taylor (1989).
3.2.2 Viscoplasticity In the previous subsection on plasticity, we have ignored all time dependency in the formulation of the constitutive model. Plastic flow is assumed to occur instantaneously compared to the time variation of the applied load. This
Figure 3.9 Cyclic Stress-Strain Modeling (Stabilized Response for A36 Steel)
is reasonable for steel at approximately room temperature deforming under moderate strain rates. It is not appropriate for lead under similar conditions, nor for steel at high temperature or under very high strain rates. In the latter cases, creep and relaxation phenomena must be considered. Creep is characterized by increasing strain with time for a constant stress, while relaxation signifies a continual reduction in stress with time for a material under constant strain. Initially separate sets of equations were introduced to model the time-independent plastic strains and the time-dependent creep strains occurring in metals. However, since the underlying physical mechanism for both involves the motion of dislocations, it is desirable to incorporate both effects into a unified theory. An edited volume by Miller (1987) provides an excellent introduction to the unified creep-plasticity state-variable models. Discussion of these models is beyond our scope, however one particular formulation developed earlier by Özdemir is relevant to our examination of metallic dampers. Based upon dislocation theories of solid state physics, Özdemir (1976) proposed the following uniaxial model suitable for the study of constitutive behavior
in which and b are the relaxation time and back stress, respectively, while the parameter n is an odd integer. The drag stress d is a material constant. Equation (3.10a) is written in a form that emphasizes the individual elastic and inelastic contributions to the total strain rate. For the special case of b = 0 and n = 1, Eq. (3.10a) reduces to the classical Maxwell model of viscoelasticity. However, here we are more b isinclusion interested in thewhile viscoplastic models for n > as 1. The back stressThe an internal obtained state variable, the quantity - b is denoted an overstress. o these stresses
permits the modeling of kinematic hardening. Consequently, both are used extensively in viscoplastic theories (Krempl, 1987). However, an additional equation is then needed to determine the evolution of the internal state variable. For the Özdemir ratedependent model the evolution equation can be written
with material parameter . The two Eqs. (3.10a) and (3.10b) completely define the response for a variety of rate-dependent materials subjected to general non-monotonic time-dependent uniaxial loading. Notice that there is no need to check a yield criterion nor to consider any special rules for loading or unloading. This last feature suggests that the viscoplastic models may also be useful for rateindependent plastic analysis, if a suitable limiting form can be obtained. With that in mind, Özdemir derived the following differential equations:
in which is now a constant controlling the slope of the - curve in the inelastic range. In constructing the final form of Eqs. (3.11), absolute values were introduced in the equations given by Özdemir to permit n to assume any positive real value. The sgn function in Eqs. (3.11) returns the sign of the argument. Comparing Eqs. (3.11) with (3.10), one finds that rate-independence is achieved by selecting a variable relaxation time, such that
The integration of either Eqs. (3.10) or (3.11) in time is required for determination o material response. This is most easily accomplished numerically using, for example, a high order Runge-Kutta formula with adaptive step size (Press et al., 1992). As an illustration of the behavior predicted by Eq. (3.11), consider the response shown in Fig. 3.10 due to a sinusoidal variation of enforced strain. Notice from Fig. 3.10a that the stress is indeed independent of the strain rate. Furthermore, it is evident from Fig. 3.10b that the parameter n controls the sharpness of the knee, while a determines the slope in the inelastic range as indicated in Fig. 3.10c. For n , the drag stress d becomes equal to the usual yield stress y, and the response approaches the elastic-linear strain hardening idealization shown
Figure 3.10 Özdemir Rate-independent Model; a) Variation of Strain Rate, b) Variation of Sharpness of Knee, c) Variation of Strain Hardening
in Fig. 3.6b. Additionally, with = 0 the elastic-perfectly plastic model of Fig. 3.6a is recovered. An extension of the loading Özdemirwhich modelmay is needed, in order to address situations involving multiaxial occur inhowever, some metallic dampers. Graesser and Cozzarelli (1991) developed such an extension and then applied the resulting rateindependent model to study biaxial cyclic response of structural steel. This extended Özdemir model and the two-surface plasticity model, discussed in the previous subsection, are both applicable for mild steel near room temperature under moderate strain rates. However, the structure of the Özdemir rate-independent model is closely related to early versions of endochronic plasticity theory (Valanis, 1971), and therefore this model suffers from the same theoretical defects present in the initial Valanis formulations (e.g., Rivlin, 1981). Although stable results are typically obtained, some caution is advised. In a series of experiments, Chang and Lee (1987) found measureable differences in the response of annealed structural steel due to strain rates above 10 -4/s. Figure 3.11 illustrates the typical behavior under uniaxial cyclic straining. The effect during monotonic loading was even more pronounced.
For example, the apparent yield strength increased 27% due to an increase in strain rate from 10-5/s to 10-2/s. Consequently, for steel dampers involving high strain rates, it may be appropriate to utilize a rate-dependent model, such as that presented in Eq. (3.10) or one of the established formulations discussed in Krempl (1987). For lead dampers, the use of a rate-dependent model is even more important.
Figure 3.11 Experimental Strain Rate Dependence
3.2.3 ThermalE ffects In the vicinity of room temperature, the mechanical properties of structural steel are both consistent and stable. This behavior is, of course, one of the reasons for its frequent selection as a building material. During a major earthquake, the structural steel within a metallic damper will typically well into energy the inelastic As mentioned previously, a significant portion cycle of the dissipated will be range. converted into heat, thus elevating the temperature of the surrounding metal. The amount o temperature rise can be estimated by considering an energy balance. The energy dissipated represents a heat source, while conduction and convection processes seek to redistribute that heat energy. The result will depend upon the material, geometry, strain amplitude, and strain rate. However, for reasonable steel damper designs, it is not expected that the temperature increase will significantly alter the mechanical properties of the device. On the other hand, thermal effects may be important in lead dampers, since the behavior of that metal is much more sensitive to moderate increases beyond room temperature.
3.2.4 Failure Theories Metallic dampers are utilized in structures to enhance the overall energy dissipation capacity. Under the action of strong earthquakes, these dampers
employ continuum damage mechanics (CDM) theories, which postulate the existence of internal state variables related to damage. Evolution equations, similar in form to Eqs. (3.10), are then introduced to determine the accumulation of damage. A brie discussion on low-cycle fatigue is provided in Hertzberg (1983), while further details on CDM can be found in Maugin (1992) and in the review article by Krajcinovic (1989). 3.3 Macroscopic Modeling Having discussed the mechanical behavior of metals at the constitutive level, the emphasis now shifts to the characterization of the overall metallic damper response. In the development of suitable force-displacement models, there are at least a couple different approaches that can be taken. The first approach, to be discussed in the following subsection, involves the direct use of experimental data obtained from component testing of the damper. The basic form of the force-displacement model is first selected, and then the model parameters are determined via a curve fitting procedure. In the second approach, the force-displacement model is instead constructed from an appropriate constitutive relationship for the metal by applying the principles o mechanics. This latter approach can often provide additional insight into the behavior of the device, while reducing the requirements for component testing.
3.3.1 E xperiment-basedModeling The first serious attempt to develop a rational force-displacement relationship for a metallic damper cantorsion be found in the work conducted by Özdemir (1976). The focus in that study was on beam dampers, however the formulations developed have more general applicability. The response of any metallic damper is a function of its geometry and the mechanical characteristics of the metal from which it is manufactured. Consequently, it is quite logical to utilize force-displacement models that have a form similar to those employed for constitutive modeling of that metal. This is exactly the approach taken by Özdemir. The structure of his damper models is based upon those frequently used in state-variable viscoplasticity. For example, consider the torsion beam damper illustrated previously in Fig. 3.la. Özdemir developed the following model for that device:
with damper force P, displacement , and the internal variable B representing a backforce. The damper parameters K0, P0, n, and were selected to
provide a best fit with experimental results for a displacement-controlled sinusoidal loading. The model was subsequently verified by comparisons involving random displacement-controlled excitations, and found to be in good overall agreement for moderate displacement magnitudes. The amount of energy dissipated by the passive device modeled by Eqs. (3.15) can be determined easily at any instant via numerical integration. Extensions of Eqs. (3.15) are also possible in order to model more complicated cyclic responses. In particular, Özdemir that include the deterioration o yield strength, elastic modulus, andexamined hardeningmodels characteristics by introducing additional internal state variables. These extended versions, with perhaps slight variations, have direct applicability to many of the metallic dampers presently in existence, although some care is still needed to ensure that stable results are obtained for rate-independent formulations. Most of the recent metallic damper modeling efforts reported in the literature have employed more simplistic representations. For example, one finds numerous application of elastic-perfectly plastic or Ramberg-Osgood idealizations, and also many attempts to convert the resulting inelastic behavior into equivalent linear viscous damping. However, it is felt that a more rigorous approach, such as that utilized by Özdemir, is much more appropriate for the design of critical components involved in life safety.
3.3.2 Mechanics-basedModeling The modeling approach outlined in the previous subsection requires experimental data for each damper configuration and size. There is no direct link between model parameters and damper geometry. Alternatively, a force-displacement relationship can be developed from a constitutive model of the metal, along with a geometric description of the device, by employing the laws of mechanics. The geometric description may require a finite element idealization, or a simple strength of materials representation may be adequate. As an example, let us consider the application of this procedure to the triangular plate metallic damper depicted previously in Fig. 3.2b. This device, which consists of identical triangular structural steel plates positioned in parallel, is typically installed within a frame bay between a chevron brace and the overlying beam, as indicated in Fig. 3.12. The base of each triangular plate is welded into a rigid base plate to approximate a fixed end condition, while a slotted pin connection is employed at the apex to ensure relatively free movement in the vertical direction. As a result of this configuration, the damper primarily resists horizontal forces P, associated with an interstory drift , via uniform flexural deformation of the individual plates. Thus, it is appropriate to examine a single cantilevered plate of thickness h, length L, and base width w0, subjected to a load P/N applied at its free end as detailed
Figure 3.12 Triangular Plate Damper Within Structural Frame
Figure 3.13 Mathematical Model for Triangular Plate Device; a) Geometric Definition, b) Beam Idealization
in Fig. 3.13. Note that coordinate axes x, y, z are defined on the undeformed midsurface of the plate. The force-displacement relationship for the damper can be readily established for infinitesimal, elastic response. For that case, the classical Euler-Bernoulli beam theory is valid. A quasistatic formulation is adopted by ignoring inertia of the plate. Then, at any cross-section, the moment equilibrium equation can be written:
With the cancellation of the term (L - x) from both sides of Eq. (3.16), it is evident that the stress is independent of position along the beam axis. After applying the unidirectional elastic constitutive relationship = E
(3.17)
and the kinematic condition = ky
(3.18a)
one finds that the curvature k is constant along the entire length of the plate, with
Substituting Eqs. (3.17), (3.18a), and (3.18b) in Eq. (3.16) produces the following force-displacement model:
This simple result is provided in Tsai et al. (1993) for the triangular plate damper, and has been found to agree reasonably well with experiments. However, the above model pertains only to the elastic response of the damper. Very little can be inferred from Eq. (3.19) concerning the energy dissipation characteristics of the device. Information of that type, which is vital for proper aseismic design, requires examination of the inelastic response. As a first approximation, the kinematic assumptions employed in Eqs. (3.18) can be retained, while replacing Eq. (3.17) by a rate-independent inelastic constitutive model. For the constitutive model defined in Table 3.1, the governing equations can be written in the following rate form:
Unfortunately the resulting force-displacement relationship can no longer be expressed analytically as in the elastic case. Instead, the necessary integrals can be evaluated numerically with standard gaussian quadrature formulae utilized for the spatial integration specified in Eq. (3.20) and an adaptive Runge-Kutta algorithm invoked to solve the set of first order differential equations. Under certain conditions (e.g. major earthquakes), the interstory drift may become comparable in magnitude to the damper plate length L. In these situations, the effects o finite deformation on damper response cannot be ignored. Either the present beam model can be extended to include these effects or a finite element analysis can be performed using a general purpose computer code, such as ABAQUS. The former approach, which essentially involves writing the equation of moment equilibrium in the deformed configuration, is briefly summarized in the following. First an intrinsic , n coordinate system is introduced, as shown in Fig. 3.13b, which deforms with the beam axis. Also defined in that diagram is , the angle between the xaxis and the tangent at . As a simplification, we will continue to ignore the effects o shear deformation and the axial extension of the midsurface. Then, strictly from kinematic considerations,
With finite deformation, the curvature k is no longer constant along the length of the plate. Consequently, at each instant of time, a nonlinear boundary value problem (BVP) must be solved to determine the unknown curvature function. This BVP requires satisfaction of the moment equilibrium equation
which now varies with . In addition, the free end deflection condition y(L) =
(3.25)
must be accommodated. In establishing the stress in Eq. (3.24), the constitutive model defined in Table 3.1 can still be employed. An approximate numerical solution of the BVP can be obtained by introducing a small number of nodes along the beam axis, and employing the method of collocation. The solution consists of the nodal values of the is, curvature andhistory the total force Pdue for to a specified enforced displacement . The response of course, dependent the material nonlinearity. As an illustrative example, consider the triangular plate damper studied by Tsai et al. (1993), with N = 8, h = 36.1mm, L = 304mm, and w0 = 133.3mm. Experimental results are presented in Fig. 3.14a for a cyclically increasing displacement-controlled component test. The pseudoangle in that diagram represents /L. For the numerical simulation, constitutive model parameters were determined from the data presented by Cofie and Krawinkler (1985) for cyclic response of structural steel. The resulting forcedisplacement response prediction for the cyclic loading is provided in Fig. 3.14b, plotted to the same scale as the experimental data. Notice that the shapes of the hysteresis loops are quite similar, and that the estimated force in the damper at = 0.30 is within 10% of the measured value. Also of interest is the increasing stiffness of the device that becomes apparent for > 0.20. This is due to the effects of finite deformation, which become ever more significant as y increases.
Figure 3.14 Triangular Plate Metallic Damper; a) Experimental (Tsai et al., 1993), b) Numerical
In addition to the determination of the force-displacement response, the numerical model provides information concerning inelastic strain histories that can be used to estimate fatigue life. Consequently, some tentative calculations
can be made for the triangular plate damper. From an analysis with a constant cyclic displacement amplitude of approximately 55mm corresponding to = 0.18, the maximum inelastic strain range present in the damper is approximately in = 0.04, which occurs at the fixed end of the triangular plate. An estimate of fatigue life of the plate for this type of loading can then be made by using the Coffin-Manson relationship defined in Eq. (3.13). For a low carbon steel with fatigue ductility parameters = 0.103 and c = -0.384 (Boller and Seeger, 1987), the predicted life is approximately 35 cycles. Estimates for additional levels of displacement amplitude are provided in Table 3.2. The performance of the metallic device within a structure subjected to seismic loading is naturally of more importance, and because the triangular plates are welded along the base, fatigue data for the weldment should be employed for a more accurate evaluation. However, the simple approach defined above appears to provide at least a reasonable order-of-magnitude guide to anticipated life that could be useful in preliminary damper design. Table 3.2 Tentative Fatigue Life Estimates for Triangular Plate Damper 0
in
Nf
0.12
0.029
82
0. 18
0.040
35
0.24
0.058
13
0.30
0.078
6
A more detailed account of this mechanics-based modeling approach for the triangular plate damper is provided in Dargush and Soong (1995), while Tsai and Tsai (1995) also address the behavior of this device with a related mechanics-based approach. However, it should be emphasized that the above model development and subsequent calculations were intended primarily for illustrative purposes. The methodology presented should be applicable to a wide variety of the metallic dampers that have been developed, including in particular the X-shaped plate dampers. A more rigorous investigation o any of these dampers would involve a finite element analysis including geometric nonlinearity and a three-dimensional constitutive model, written in terms of a co-rotational stress rate. Additional material and component tests would also be required to determine reliability and variability of experimental data. In any case, after establishing a mechanics-based representation for the damper, numerical experiments can be conducted to determine parameters in a simplified force-displacement relationship such as Eqs. (3.15), or the model can be used directly in the analysis of the overall structure. Both options are
discussed in the following section. Before shifting to that topic, let us remark that the mechanics-based approach permits one to learn much more about the behavior of a metallic damper than is contained in the force-displacement hysteresis loops. This additional information can be used to improve damper design and to obtain a better understanding of its performance within a structural system. 3.4 Structural Analysis In Section 2.2.5,plastic the response a single degree of freedom withthat, an elastic-perfectly passiveofdamper was examined in some(SDOF) detail. Itstructure was found by providing a means for energy dissipation, the damper could substantially reduce the steady-state response near resonance and the transient response due to a particular seismic signal. For those analyses an Ozdemir model of Eq. (3.15) was employed to simulate an elastic-perfectly plastic idealization by setting n = 25 and = 0.0. (An alternative elastoplastic model was also used for confirmation.) Careful examination o Fig. 2.12 reveals anomalous behavior in the range ß < 0.5. Of course, counterintuitive response is often observed in nonlinear systems, and provides a strong argument against the use of so-called equivalent linearizations. In this case, the irregularities are due to the abrupt change in stiffness of the passive damper at yield. A more important aspect of the SDOF analyses that should be emphasized with regard to metallic dampers relates to the predicted response under earthquake excitation. For an SDOF system, the addition of a purely viscous device always reduces response caused by an input signal. However, while the insertion of metallic dampers into the SDOF system generally reduces response by dissipating a portion of the energy, the response may actually increase for some specific seismic inputs. For example, consider the system with the elastic-perfectly plastic damper subjected to the 1940 El Centro S00E ground motion presented in Fig. 2.6. Plots of maximum relative displacement and total acceleration response versus elastic structural period To are shown in Fig. 3.15, where
It is clear from these diagrams that there exist ranges of To in which the elastoplastic damper is either ineffective or even detrimental under this particular ground motion. The above results for the simple SDOF system certainly suggest that detailed nonlinear transient dynamic anaylses are required to properly evaluate the effectiveness of any real structure employing metallic dampers for enhanced seismic protection. As mentioned in Chapter 2, the finite element
Figure 3.15 SDOF Response to 1940 El Centro S00E; a) Relative Displacement, b) Total Acceleration
method (FEM) currently provides the most suitable framework for a multi-degree o freedom (MDOF) analysis of an overall structure. The modified Newton-Raphson time domain approach, detailed in Section 2.3.4, is directly applicable for structures that include metallic dampers. This type of algorithm is typically available in most major FEM computer programs. Many of these programs also permit user-defined structural elements. In that case, force-displacement relationships, such as those defined by Eqs. (3.15) or Eqs. (3.20)-(3.22), can be easily incorporated into the finite element model. Additionally, major codes include substructuring facilities, which permit separation o linear and nonlinear degrees of freedom (dof). The effective tangent stiffness matrix for the linear dof can be formed and eliminated once at the beginning of the analysis. The solution at each time step then involves the formation and decomposition of a relatively small system containing only the nonlinear
dof. This type of partitioning was utilized by Ozdemir (1976) for analysis of structures incorporating torsional beam dampers. Subsequently, the same methodology was used by Bhatti et al. (1978) to develop an optimal design approach for a base isolation system, which included the torsional beam damper. The method of feasible directions was employed in the latter work for the optimization of several simple structural systems. In light of the considerable advances in computing capabilities and optimization software that has occurred during the intervening years, it would be interesting to apply a similar approach for the design of metallic dampers positioned within the superstructure. Recent applications of nonlinear structural analysis to building frames incorporating metallic dampers can be found in Xia and Hanson (1992), Jara et al. (1993), and Tsai et al. (1993). The first two references consider X-shaped plate dampers, while the last examines triangular plate dampers. All three works utilize the DRAIN-2D computer code (Kanaan and Powell, 1973) for the analyses. As a final note in this section, it should be mentioned that there are several attractive alternatives to the time-domain algorithm outlined in Section 2.3.4. One of these alternatives involves rewriting the governing equations in state-space form by introducing both displacement and velocity as primary unknowns, as is often done in the control literature (Soong, 1990). This permits the use of very accurate and efficient first-order differential equation solvers. While such algorithms are typically not currently available in the major FEM programs, Inaudi and de la Llera (1992) have developed a research code using this formulation. 3.5 Experimental Studies In the previous three sections, we have presented a theoretical foundation for the consideration of metallic damper performance. However, from a review of the literature, it is apparent that much greater effort has been expended on experimental testing of metallic dampers and test structures. In fact, it is safe to say that most investigators have employed a design-by-experiment approach. Since it is impossible to review all of the available work, we will instead focus on a few of the more comprehensive experimental studies. The first of these to be considered is described in Kelly et al. (1972) and Skinner et al. (1975). The former work introduced the idea of using metallic dampers for the express purpose of dissipating seismic energy in a structural system. The authors then examined in detail the performance of three such devices, involving torsional or flexural deformation of steel. Emphasis was placed on component testing. Devices were tested in an Instron machine towere determine cyclic force-displacement fatigueInlife. Analytical expressions also developed in a first attempt toresponse quantifyand behavior.
the latter paper, the devices shown previously in Fig. 3.1 were investigated, along with a cantilever plate damper, using similar component tests and simple analytical models. While, at this stage, the dampers were not yet mature, all four types appeared to have the potential for use in aseismic design of actual structures. Recently, Aguirre and Sánchez (1992) have tested U element dampers very similar to that depicted in Fig. 3.1c. Included in their investigation was force-displacement response, fatigue life, and temperature rise of the specimen during cyclic loading. Robinson Greenbank (1976) developed and tested extrusion for use as and energy dissipators. Two such devices appearseveral in Fig.lead 3.16. Typicaldevices forcedisplacement hysteresis loops are shown in Fig. 3.17. The nearly rectangular shape indicates that these devices tend to maximize energy dissipation for a given force and displacement amplitude. Furthermore, the nature of the design is such that fatigue is not a major concern, since lead is hot worked at room temperature. The test results displayed in Fig. 3.17 were obtained at very low frequencies (i.e., f < 0.004 Hz). However, additional experiments conducted at more appropriate frequencies suggest that the devices are only moderately strain rate dependent. The continuation of this research has led to the development of a lead shear damper, intended for commercial application, as discussed in Monti and Robinson (1996).
Figure 3.16 Lead Extrusion Energy Absorbers (Robinson and Greenbank, 1976); a) Constricted Tube Design, b) Bulged Shaft Design
Figure 3.17 Lead Extrusion Energy Absorbers Force-Displacement Loops at 1cm/min (Robinson and Greenbank, 1976); a, b) Typical Constricted Tube Response, c, d) Typical Bulged Shaft Response
Currently, the flexural plate steel dampers have become the predominant form for metallic devices, and a number of experimental programs have been conducted at university laboratories. Bergman and Goel (1987) reported on the cyclic testing of Xshaped and V-shaped plate dampers manufactured by Bechtel Corporation. Geometric details for the X-shaped damper are provided in Fig. 3.18, while the experimental setup using a single story full-scale building test frame is displayed in Fig. 3.19. A total o three dampers were tested under constant amplitude displacement controlled cycling at 0.33 Hz. For each specimen, seven different levels of displacement amplitude ranging up to approximately 1.5in. were applied for ten cycles each. Cycling at the highest level then continued until failure occurred. Force-displacement curves for the X-shaped damper at displacement amplitudes of 0.42in., 1.04in. and 1.56in. are reproduced in Fig. 3.20. Energy dissipated per cycle can, of course, be calculated from these hysteresis loops, and is provided in the report. The fatigue life of the specimen was 44 cycles at an amplitude of 1.56in. Meanwhile, the V-shaped damper proved to be a less favorable design configuration in terms of both energy dissipation and durability. In concluding, the authors
mention that axial shortening of the device, which occurs at large horizontal displacement, must be considered in design of the adjacent structural members and connections.
Figure 3.18 X-shaped Device Geometry (Bergman and Goel, 1987)
Figure 3.19 X-shaped Device Experimental Setup (Bergman and Goel, 1987)
Figure 3.20 Force-Displacement Response of X-shaped Device (Bergman and Goel, 1987); Displacement Amplitude a) 0.42in, b) 1.04in, c) 1.56in
A more comprehensive experimental program was performed at the University o California at Berkeley and reported in Whittaker et al. (1991). The investigation, which was sponsored by Bechtel Power Corporation and CounterQuake Corporation, focused on the evaluation of particular X-shaped plate dampers known as Added Stiffness And Damping (ADAS) elements. Dimensional characteristics of a typical ADAS element are shown in Fig. 3.21. This design is very similar to X-plate damper considered by Bergman and Goel (1987). However, in Whittaker et al. (1991) both cyclic component tests and earthquake simulator tests were performed. For determination of the mechanical characteristics of the ADAS elements, an In-Plane Testing Frame was employed using horizontal displacement control and vertical force control. The resulting experimental force-displacement response for a seven plate ADAS element is displayed in Fig. 3.22 for displacement amplitudes of approximately 0.45in., 1.5in., and 2.2in.
Figure 3.21 ADAS Element Geometry (Whittaker et al., 1991)
Figure 3.22 Force-Displacement Response of ADAS Elements (Whittaker et al., 1991); Displacement Amplitude a) 0.45in, b) 1.5in, c) 2.2in
Based upon the similarities in design, one would expect that the hysteresis loops obtained by Bergman and Goel (1987) and Whittaker et al. (1991) should also have similarities. Comparing Figs. 3.20a and 3.22a, we find that to be the case for small amplitude cycling. However, at larger amplitudes the response contains significant differences, as can be seen by comparing Figs. 3.20c and 3.22b. This difference at large displacement amplitudes is due to axial effects associated with finite deformation. In Bergman and Goel (1987), axial deformation is resisted by the supporting frame members, while in the study by Whittaker et al. (1991) there is no vertical constraint. For ADAS deflections that are likely to occur in severe earthquakes, there is coupling between lateral and axial response, which has not been considered. Even without axial constraint, effects of geometric nonlinearity will eventually become evident as the displacement amplitude is increased. For example, the apparent stiffening of the device in Fig. 3.22c is due to finite deformation, not strain hardening of the material. (Recall that the stabilized response of steel under cyclic loading follows the curve displayed in Fig. 3.9. There is no plateau.) It should be emphasized that this discussion does not imply that ADAS elements are unsuitable at large deflection, but rather that the effects of finite deformation should be included in the assessment of a design. As mentioned above, Whittaker et al. (1991) also examined the performance of ADAS elements in a test structure under seismic excitation. A schematic of the three story, single bay Ductile Moment Resisting Space Frame (DMRSF), having plan dimensions 12ft. by 6ft. is shown in Fig. 3.23. With the six ADAS elements in place, the entire structure is referred to as ADAS-3. The natural periods and damping ratios for DMRSF and ADAS-3 are detailed in Table 3.3. Notice that there is a significant shift in period of the first mode due to the stiffness of theattributed chevron solely bracingtoand Meanwhile, the increase in added damping ratio was the ADAS ADASelements. framing. For the seismic simulation, three different records were utilized including a soft soil synthetic denoted THSSR, 1985 Chile (Llolleo) N10E, and 1940 El Centro S00E. Table 3.4 provides a comparison of the response of the structure without (DMRSF) and with (ADAS-3) the metallic dampers due to the Chilean earthquake with a maximum ground acceleration of 0.13g. Selected results are displayed graphically in Fig. 3.24, which clearly indicates the beneficial effects of the ADAS dampers in this case. The passively damped structure was also subjected to El Centro at 0.33g, THSSR at 0.40g, and Chile at 0.56g. Hysteresis loops for the ADAS dampers during the El Centro signal are shown in Fig. 3.25, while Fig. 3.26 contains the corresponding energy plot for the entire structure. From the latter figure, it is evident that the ADAS dampers dissipate a significant portion of the input seismic energy.
Figure 3.23 ADAS Test Structure (Whittaker et al., 1991) Table 3.3 Dynamic Characteristics of Test Structure(Whittaker et al., 1991) Free Vibration Mode DMRSF ADAS-3
Ti (sec)
1st
2nd
0.74
0.22
i (%)
2.1
1.3
Ti (sec)
0.47
017
i (%)
3.4
1.7
Figure 3.24 Response Comparison 1985 frame] Chile (Llolleo) N10E at 0.13g [solid = MRF; dashed =for ADAS (Whittaker et al., 1991)
Table 3.4 Chile 0.1g Response Envelopes for Test Structure (Whittaker et al., 1991) a) DMRSF response Roof/3 Floor/Story
3/2
2/1
Max
Min
Max
Min
Max
Min
Lateral Displacement (in)
2.21
-2.17
1.69
-1.68
0.89
-0.90
Time (sec)
8.38
6.44
8.38
6.45
6.83
6.46
Interstory Drift (in) Interstory Drift Index (%)
0.54 0.84
-0.52 0.81
0.83 1.30
-0.80 1.25
0.89 1.11
-0.90 1.13
Time (sec)
8.38
6.42
8.38
6.43
6.83
6.46
Story Shear (k)
12.4
-12.0
21.3
-20.6
25.0
-25.3
Story Shear/Total Wt (%)
13.5
13.0
23.1
22.4
27.2
27.4
Time (sec)
8.38
6.42
8.38
6.43
6.84
6.43
Inertia Force (k)
12.0
-12.4
9.3
-9.2
8.0
-7.3
Time (sec)
6.42
8.38
6.46
5.27
18.13
5.26
Overturn. Moment (k-in)
795
-769
2159
-2085
4145
-4104
Time (sec)
8.38
6.42
8.37
6.42
8.37
6.42
Min
Max
b) ADAS-3 response Roof/3 Floor/Story
Max
Min
3/2 Max
2/1 Min
Lateral Displacement (in)
0.71
-0.81
0.56
-0.67
0.32
-0.37
Time (sec)
12.43
14.92
12.44
14.92
12.45
14.92
Interstory Drift (in)
0.17
-0.20
0.25
-0.29
0.32
-0.37
Interstory Drift Index (%)
0.27
0.31
0.39
0.45
0.40
0.46
Time (sec)
13.01
12.73
13.00
14.92
12.45
14.92 -20.5
Story Shear (k)
8.7
-9.5
14.0
-16.2
17.6
Story Shear/Total Wt (%)
9.2
10.0
14.7
17.0
18.5
21.5
13.00
12.72
12.43
14.92
12.44
14.92
Time (sec) Inertia Force (k)
9.5
-8.7
7.9
-6.3
5.1
-4.7
Time (sec)
12.72
13.00
14.92
12.44
12.78
14.65
Overturn. Moment (k-in) Time (sec)
555 13.00
-606 12.72
1466 13.00
-1570 14.92
2798 12.43
-3211 14.92
Some results of experiments on the triangular plate damper conducted by Tsai et al. (1993) have already been discussed in Section 3.3.2. The first portion of their experimental program involved cyclic testing of the damper, referred to as a TADAS device in that paper. Various plate dimensions and cyclic load patterns were included. A typical force-displacement response
Figure 3.25 ADAS Hysteresis Loops for 1940 El Centro S00E at 0.33g (Whittaker et al., 1991)
Figure 3.26 Energy Time Histories for 1940 El Centro S00E at 0.33g (Whittaker et al., 1991)
for the TADAS device was presented previously in Fig. 3.14a for a loading program that involved increasing displacement amplitude. Note the similarities in these hysteresis loops compared to those obtained for the X-shaped dampers. This is, o course, expected since the mechanism is the same. However, in the TADAS device there are no axial forces developed at large displacement due to the slotted pin connection at the apex, except for a portion attributable to friction. Tsai et al. (1993) also performed pseudodynamic testing of the two story steel frame shown in Fig. 3.12. Experimental results for the first and second floor displacements are displayed in Fig. 3.27 for a simulation of the 1940 El Centro S00E earthquake. A significant reduction is obtained with the inclusion of the TADAS elements.
Figure 3.27 Pseudodynamic Response of Test Frame (Tsai et al., 1993)
3.6 Design Considerations In order to utilize metallic dampers within a structural system, it is necessary to formulate design guidelines and procedures, based upon knowledge gained from theoretical and experimental studies. For ADAS elements, the parametric numerical investigations by Xia and Hanson (1992) are aimed in that direction. The ADAS devices are first characterized by an elastic-linear hardening model having elastic stiffness Kd, yield displacement y, and dimensionless hardening ratio. Then key structural parameters are defined. The parameter B/D denotes the ratio of bracing stiffness Kb to device stiffness Kd, while SR represents the ratio of the brace-damper K toconstant assemblage that of the bare structural story KK.d Since B /D and SR stiffness are assumed for corresponding a given passively damped structure, and Kboth b are chosen to be proportional to Kf for each story. Two different ten story frames were a
f
considered in the DRAIN-2D analyses, which included three earthquake signals. Numerous values of B/D, SR, y, and the hardening ratio were examined. The results were then used to develop the design criteria and methodology that is presented in the more detailed report by Xia et al. (1990). Essentially the design procedure is based upon the Uniform Building Code criteria for a dual system, in which the moment frame and brace-damper assemblages primarily resist vertical and horizontal loads, respectively. The procedure is largely geared for establishment of the parameters B/D, SR, and y via hand calculation. A design example is also provided in Xia et al. (1990). A related approach is defined in Tsai et al. (1993) for structural design with TADAS devices. The force-displacement diagram in Fig. 3.28 defines the relevant parameters for a given story, assuming that the yield displacement of the TADAS devices ( y) is less than that of the bare structural frame ( yf) Paraphrasing those authors, the design procedure can be outlined as follows: 1. Define the site-specific service level design earthquake.
2. Select a suitable SR value based upon the fundamental period of the frame. 3. Proportion the bare frame, and compute the lateral stiffness Kf and yield displacement yf of each story. 4. Compute TADAS and brace assembly stiffness Ka = (SR)Kf
(3.27)
and TADAS yield displacement
for the selected values of the parameters SR, and U. Design the TADAS device for each story based upon y and Kd, where
5. Perform lateral force analysis of the structure with TADAS elements in place for the service level design earthquake. Repeat steps 3 and 4 as necessary to meet the design requirements. 6. Perform capacity design checks for all members. Assume that the ultimate force generated in the TADAS device is 1.5 Pu, where
Additional details on the process, along with a design example for a twenty story, three bay planar frame, can be found in Tsai et al. (1993). An alternative design approach based upon the concept of equivalent viscous damping has also been outlined in Scholl (1993). While the methods proposed in the above references are useful in the near term, all are based on the notion of design as a manual process, with perhaps some use of elastic analyses at various steps. However, in light of the complicated nature of the metallic devices, the superstructure, and the seismic signal, it would seem more appropriate to view design as a computational process. Effort would then be directed toward constructing realistic mathematical models of devices and superstructures, and on developing the necessary software tailored for the solution of the resulting optimal structural design problems. This could also be tied in with the ongoing efforts to establish building code requirements for passively damped systems which are discussed in Whittaker et al. (1993).
Figure 3.28 Force-Displacement Model for TADAS Frame System (Tsai et al., 1993)
3.7 Structural Implementations After gaining confidence in the performance of metallic dampers based primarily on experimental evidence, a number of researchers practitioners have proceded to implement these devices in full-scale structures.andThe earliest implementations o metallic dampers in structural systems occurred in New Zealand. A number of these interesting applications are reported in Skinner et al. (1980). The first implementation involved the use of a torsion beam steel damper for the piers of the Rangitikei Bridge. Transverse flexibility of the bridge is afforded by permitting rocking and uplift of the piers, while damping is provided by the first-generation metallic dampers. Details are shown in Fig. 3.29. Meanwhile, tapered plate cantilever devices were employed at the base of a chimney at Christchurch and for the Dunedin Motorway Overbridge. Six 300kN flexural beam dampers were used in the Cromwell Bridge, while lead extrusion dampers found application in two sloping highway bridges in Wellington. Additional implementations for metallic dampers as components in base isolation systems are discussed in Skinner et al. (1980). Many other base isolation applications have appeared; however, in the remainder o this chapter, we will concentrate on four examples in which metallic dampers are utilized as passive energy theofsuperstructure of a building. All three four utilize ADAS elements fordissipators the seismicwithin upgrade an existing structure. The first of these occurred in Mexico City and are reported by Martinez-Romero (1993).
Figure 3.29 Details of Rangitikei Bridge (Skinner et al., 1980)
The thirteen story Izazaga #38 - 40 building is shown in Fig. 3.30. This is a reinforced concrete frame with brick infilled end walls constructed in the late 1970s. The building sustained moderate damage in the 1985 Mexico City earthquakes. An upgrade afterward was unsuccessful, since further damage occurred during 1986 and 1989 seismic attacks. Passive energy dissipation technology was selected for a second retrofit. For this project approximately 250 ADAS dampers were installed in 1990 in outer frame bays to permit continued building operation during construction. A picture of a typical installation of a brace-damper subassembly is provided in Fig. 3.31. Structural analysis indicated that the fundamental periods in the principal directions were reduced from 3.82s and 2.33s to 2.24s and 2.01s with the addition of the ADAS elements. Calculations also determined a 40% reduction in interstory drift, while retaining the same base shear coefficient. Nonlinear, time history DRAIN-2DX analyses were used to verify the final design. The second Mexican application involves retrofit of the six story Cardiology Hospital Building, built in the 1970s and damaged in the 1985 earthquakes. The seismic upgrade of the reinforced concrete frame structure, completed in 1990, features a series o eighteen external buttresses connected to the building via a total of ninety ADAS dampers. A front view with the buttresses in-place is presented in Fig. 3.32. This particular design permitted minimal interference with hospital operation during retrofit construction. Nonlinear, time history DRAIN-2D analysis was again employed extensively in the redesign process. However, in this case, the addition of the buttresses and ADAS dampers reduced both base shears and interstory drift.
Figure 3.30 Izazaga #38-40 Building (Martinez-Romero, 1993)
Figure 3.31 Brace-Damper Assembly in Izazaga #38-40 Building (Martinez-Romero, 1993)
Figure 3.32 Cardiology Hospital Building with Exterior Buttresses and ADAS Dampers (Martinez-Romero, 1993)
In 1992, the Reforrna #476 buildings, which contain key operations of the Mexican Institute of Social Security (IMSS), underwent major seismic upgrade. This cluster o three ten-story buildings constructed in 1940 is listed in the National Register o Classical Buildings. An exterior view is provided in Fig. 3.33. While the buildings sustained some significant damage during
Figure 3.33 IMSS Reforma Building (Martinez-Romero, 1993)
the 1957 earthquake, the more recent seismic events of 1985 had only minor effect. Through the years, the foundation has been improved to alleviate problems o differential settlement. Nevertheless, it was felt that the buildings
remained vulnerable to future severe earthquakes. The most recent retrofit, discussed by Martinez-Romero (1993), involves the use of ADAS dampers with chevron bracing in forty frame bays throughout the three structures. An example is shown in Fig. 3.34. The buildings were analyzed in detail using the nonlinear DRAIN-2D program, along with four earthquake ground motions. Two-dimensional analysis was performed on thirteen different cross-sections. In order to assess the suitablity of the revised design, comparisons made foronly the the barebraces. frame,The thecalculated structure response with braces and ADAS dampers, and were a design using at one section o the central building is enumerated in Table 3.5 for one of the earthquakes. The roo displacements and interstory drifts are significantly reduced for the ADAS design. However, base shear, critical column force, and roof acceleration all
Figure 3.34 Brace-Damper IMSS Reforma Building Retrofit Scheme From Outside (Martinez-Romero, 1993)
increased, although not as much as would occur with the addition of only the bracing. In the final design, a floor diaphragm and several interior columns were strengthened to accommodate the elevated force levels. As in the other Mexican projects, the retrofit was completed while the complex remained in operation. It should be noted that in addition to describing the details of these three retrofit projects, Martinez-Romero (1993) relates many worthwhile observations concerning design and construction issues.
Table 3.5 Calculated Earthquake Response for IMSS Reforma Building (Martinez-Romero, 1993) Deformations Displacement at Roof
Story Drift (cm)
Story Drift Ratio
Building Condition
(cm)
Max.
Avg.
Max.
Avg.
Bare Frame
30.9
5.9
3.1
0.0147
0.0077
Frame w/ADAS
25.8
3.5
2.6
0.0088
0.0064
Frame w/Elastic Bracing
32.1
4.5
3.2
0.0113
0.0080
Forces Force in Critical
Roof
Base Shear
Base Shear
Column (Tons)
Acceleration
(Tons)
Coefficient
Comp.
Bare Frame
455
0.11
Frame w/ADAS
652
0.13
Frame w/Elastic Bracing
680
0.14
Building Condition
Ten.
(cm/sec2)
317
58
178
477
272
230
556
340
340
The final implementation of metallic dampers considered in this chapter involves the retrofit of the Wells Fargo Bank building in San Francisco, California (USA) discussed in Perry et al. (1993). The building is a two story nonductile concrete frame structure srcinally constructed in 1967 and subsequently damaged by the 1989 Loma Prieta earthquake. The voluntary upgrade by Wells Fargo utilized chevron braces and ADAS damping elements. More conventional retrofit schemes were rejected due to an inability to meet the performance objectives while avoiding foundation work. A plan view of the second floor including upgrade details is provided in Fig. 3.35. A total of seven ADAS devices were employed, each with a yield force of 150kips. Numerous linear and nonlinear analyses were used in the retrofit design process. For example, the threedimensional model shown in Fig. 3.36 provided elastic modes and response spectrum analysis results for the srcinal structure from the program SAP90. Further threedimensional response spectrum analyses, using an approximate equivalent linear representation for the ADAS elements, furnished a basis for the redesign effort. The final design was verified with DRAIN-2D nonlinear time history analyses. A comparison of computed response before and after the upgrade is contained in Fig. 3.37. The numerical results indicted that the revised design was stable and that all criteria were
met. In addition to the introduction of the bracing and ADAS dampers, several interior columns and a shear wall were strengthened. The entire project, including design, permit approval, and construction were completed within a six month period, in time for a February 1992 opening of the bank branch. Perry et al. (1993) also describe the permit approval process, and suggest that the ADAS elements have many potential applications for both new and retrofit projects involving either concrete or steel framed structures.
Figure 3.35 Wells Fargo Bank Building Retrofit Details (Perry et al., 1993)
Figure 3.36 Structural Analysis Model for Wells Fargo Bank Building (Perry et al., 1993)
Figure 3.37 Comparison of Computed Results for Wells Fargo Bank Building (Perry et al., 1993)
References Aguirre, M. and Sánchez, A. R. (1992), Structural Seismic Damper, J. Structural Engineering, ASCE, 118(5), 1158-1171. Bergman, D. M. and Goel, S.C. (1987), Evaluation of Cyclic Testing of Steel-Plate Devices for Added Damping and Stiffness, Report No. UMCE 87-10, The University of Michigan, Ann Arbor, MI. Bhatti, M. A., Pister, K. S. and Polek, E. (1978), Optimal Design of an Earthquake Isolation System, Report No. UCB/EERC-78/22, University of California, Berkeley, CA. Boller, Chr. and Seeger, T. (1987), Material Data for Cyclic Loading, Part A: Unalloyed Steels, Elsevier, Amsterdam. Chang, K. C. and Lee, G. C. (1987), Strain Rate Effect on Structural Steel Under Cyclic Loading, J. Engrg. Mech., ASCE, 113(9), 1292-1301. Cofie, N. G. and Krawinkler, H. (1985), Uniaxial Cyclic Stress-Strain Behavior of Structural Steel, J. Engrg. Mech., ASCE, 111(9), 1105-1120. Dafalias, Y. F. and Popov, E. P. (1975), A Model of Nonlinearly Hardening Materials for Complex Loading, Acta Mechanica, 21, 173-192. Dargush, G. F. and Soong, T. T. (1995), Behavior of Metallic Plate Dampers in Seismic Passive Energy Dissipation Systems, Earthquake Spectra, 11(4), 545-568. Dowling, N.E. (1972), Fatigue Failure Predictions for Complicated Stress-Strain Histories, J. aterials, JMLSA, 7(1), 71-87. Fung, Y. C. (1965), F oundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ. Graesser, E. J. and Cozzarelli, F. A. (1991), A Multidimensional Hysteretic Model for Plastically Deforming Metals in Energy Absorbing Devices, Technical Report NCEER-91-0006, National Center for Earthquake Engineering Research, Buffalo, NY. Hertzberg, R. W. (1983), Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, New York, NY. Inaudi, J. A. and de la Llera, J.C. (1992), Dynamic Analysis of Nonlinear Structures Using State-Space Formulation and Partitioned Integration Schemes, Report UCB/EERC-92/18, University of California, Berkeley, CA. Jara, J. M., Gómez-Soberón, C., Vargas, E. and González, R. (1993), Seismic Performance of Buildings with Energy Dissipating Systems, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 663-673. Kanaan, A. E. and Powell, G. H. (1973), DRAIN-2D - A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Sturctures, Report No. UCB/EERC 73-06, University of California, Berkeley, CA. Kelly, J. M., Skinner, R. I. and Heine, A. J. (1972), Mechanisms of Energy Absorption in Special Devices for Use in Earthquake Resistant Structures, Bull. N. Z. Soc. Earthquake Engrg., 5(3), 63-88. Krajcinovic, D. (1989), Damage Mechanics, Mechanics of Materials, 8, 117-197. Krempl, E. (1987), Models of Viscoplasticity: Some Comments on Equilibrium (Back) Stress and Drag Stress, Acta Mechanica, 69, 25-42. Krieg, R. D. (1975), A Practical Two Surface Plasticity Theory, J. Applied Mechanics, ASME, E42, 641-646. Martinez-Romero, E. (1993), Experiences on the Use of Supplemental Energy Dissipators on Building Structures, Earthquake Spectra, 9(3), 581-625. Maugin, G. (1992), The Thermomechanics of Plasticity and Fracture, Cambridge University Press, Cambridge, UK.
Mendelson, A. (1968), Plasticity: Theory and Application, MacMillan, New York, NY. Miller, A. K., ed. (1987), Unified Constitutive Equations for Creep and Plasticity, Elsevier Applied Science, London, UK. Monti, M.D. and Robinson, W. H. (1996), A Lead Shear Damper Suitable for Reducing the Motion Induced by Wind and Earthquake, Proceedings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico. Mröz, Z. (1967), On the Description of Anisotropic Work Hardening, J. Mech. Phys. Solids, 15, 163-175. Ozdemir, H. (1976), Nonlinear Transient Dynamic Analysis of Yielding Structures, Ph.D. Dissertation, University of California, Berkeley, CA. Perry, C. L., Fierro, E. A., Sedarat, H. and Scholl, R. E. (1993), Seismic Upgrade in San Francisco Using Energy Dissipation Devices, Earthquake Spectra, 9(3), 559-579. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992), Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge, UK. Rivlin, R. S. (1981), Some Comments on the Endochronic Theory of Plasticity, Int. J. Solids Struct., 17, 231-248. Robinson, W. H. and Greenbank, L. R. (1976), An Extrusion Energy Absorber Suitable for the Protection of Structures During an Earthquake, Earthquake Engineering and Structural Dynamics, 4, 251-259. Scholl, R. E. (1993), Design Criteria for Yielding and Friction Energy Dissipators, Proc. TC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 485-495. Shames, I. H. and Cozzarelli, F. A. (1992), Elastic and Inelastic Stress Analysis, PrenticeHall, Englewood Cliffs, NJ. Skinner, R. I., Kelly, J. M. and Heine, A. J. (1975), Hysteresis Dampers for EarthquakeResistant Structures, Earthquake Engineering and Structural Dynamics, 3, 287-296. Skinner, R. I., Tyler, R. G., Heine, A. J. and Robinson, W. H. (1980), Hysteretic Dampers for the Protection of Structures from Earthquakes, Bull. N. Z. Soc. Earthquake Engrg., 13(1), 22-36. Soong, T. T. (1990), Active Structural Control: Theory and Practice, Wiley, New York, NY. Tsai, K. C., Chen, H. W., Hong, C. P. and Su, Y. F. (1993), Design of Steel Triangular Plate Energy Absorbers for Seismic-Resistant Construction, Earthquake Spectra, 9(3), 505-528. Tsai, C. S. and Tsai, K. C. (1995), TPEA Device as Seismic Damper for High-Rise Buildings, J. Engrg. Mech., ASCE, 121(10), 1075-1081. Valanis, K. C. (1971), A Theory of Viscoplasticity without a Yield Surface, Arch. Mech., 23, 517-534. Whittaker, A. S., Bertero, V.V., Thompson, C. L. and Alonso, L. J. (1991), Seismic Earthquake Spectra Testing of Steel Plate Energy Dissipation Devices, , 7(4), 563-604. Whittaker, A., Aiken, I., Bergman, D., Clark, P., Cohen, J., Kelly, J. and Scholl, R. (1993), Code Requirements for the Design and Implementation of Passive Energy Dissipation Systems, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 497-508. Xia, C. and Hanson, R. D. (1992), Influence of ADAS Element Parameters on Building Seismic Response, J. Structural Engineering, ASCE, 118(7), 1903-1918.
Xia, C., Hanson, R. D. and Wight, J. K. (1990), A Study of ADAS Element Parameters and Their Influence on Earthquake Response of Building Structures, Report No. UMCE 87-10, The University of Michigan, Ann Arbor, MI. Zienkiewicz, O. C. and Taylor, R. L. (1989), The Finite Element Method, McGraw-Hill, London, UK.
4 Friction Dampers 4.1 Introduction In the previous chapter, we examined devices that dissipate energy via the inelastic deformation of metals. The mechanism involved can be categorized as one form of internal friction. On the other hand, the present chapter will focus on dampers that utilize the mechanism of solid friction to provide the desired energy dissipation. Thus, we will consider the friction that develops between two solid bodies sliding relative to one another. Processes of this type are prevalent in nature and have also been employed in many engineered systems. For example, solid friction plays an important role in the overall control of tectonic movement and earthquake generation. On a much smaller scale, friction is also used in automotive brakes as a means to dissipate the kinetic energy of motion. Based primarily upon an analogy to the automotive brake, Pall et al. (1980) began the development of passive frictional dampers to improve the seismic response of structures. The objective is to slow down the motion of buildings 'by braking rather than breaking' (Pall and Marsh, 1982). There has been considerable progress during the intervening years, and a number of devices have been developed. Several different types of friction dampers are illustrated in Fig. 4.1. The Limited Slip Bolted (LSB) joint srcinated by Pall et al. (1980) is shown in Fig. 4.1a. It is intended for seismic control of large panel structures. The LSB design incorporated brake lining pads between steel plates in order to provide a consistent force-displacement response. Fig. 4.1b displays an alternative design proposed by Pall and Marsh (1982) for application in conjunction with cross-bracing in framed structures. Once again, brake lining pads are utilized for the sliding surfaces. As will be discussed in Section 4.7, modern versions of these devices have already been implemented in a number of structures in Canada. Two more recent uniaxial friction devices are shown in Figs. 4.1c and d. The first of these is a Sumitomo friction damper that has found application in Japan (Aiken and Kelly, 1990). The copper alloy friction pads slide along the inner surface of the cylindrical
steel casing. The required normal force is provided through the action of the spring against the inner and outer wedges. Fig. 4.1d presents the somewhat more sophisticated Energy Dissipating Restraint (EDR) described in Nims et al. (1993a). In this design, dissipation occurs on the interface between bronze friction wedges and the steel cylinder wall. The combination of wedges, stops, and internal spring produces a frictional force proportional to the relative displacement of the device ends. Meanwhile, Fig. 4.1e shows a typical Slotted Bolted Connection (SBC) intended for application in concentrically braced frames. Several versions have appeared in the literature. FitzGerald et al. (1989) employ all structural steel components, while Grigorian et al. (1993) advocate inclusion of brass insert plates. In both cases, Belleville washers are used to maintain initial bolt tensions.
Figure 4.1 Friction Dampers; a) Limited Slip Bolt Joint (Pall et al., 1980), b) X-braced Friction Damper (Pall and Marsh, 1982)
Figure 4.1 Friction Dampers; c) Sumitomo Friction Damper (Aiken and Kelly, 1990), d) Energy Dissipating Restraint (Nims et al., 1993a), e) Slotted Bolted Connection (FitzGerald et al., 1989)
In order to obtain a better appreciation for the behavior of these and other friction dampers, we begin in Section 4.2 with a brief presentation of basic principles. These principles are then used to some extent in Section 4.3 to construct the macroscopic force-displacement models for the devices. The models are, in turn, required for proper analysis and design, which inevitably involves the consideration of nonlinearities. Section 4.4 provides a discussion of procedures applicable for overall structural analysis. Despite a long history of interest in frictional behavior and its apparent simplicity, purely analytical predictions are not yet possible. There is a need for physical experiments at both the component and structural level. A number o experimental studies employing friction dampers within test structures are described in Section 4.5, while Section 4.6 contains additional information important for design. As mentioned above, several full-scale structural implementations of friction dampers have already occurred. These are discussed in further detail in Section 4.7 at the end of the chapter. 4.2 Basic Principles
There are numerous forms of friction that can be effectively used to mitigate damage to structures during environmental disturbances. All of the devices to be discussed in the present chapter employ solid sliding friction as their basic dissipative mechanism. Thus, in friction dampers, irrecoverable work is done by the tangential force required to slide one solid body across the surface of another. One objective, of course, is to maximize the energy dissipation. Consequently, there is no desire to introduce a hydrodynamic lubricating layer on the interface. The contacting surfaces are generally intended to remain dry during operation. In accordance with this, our presentation o basic principles will begin with a discussion of dry, sliding, solid friction. More detailed information, along with numerous references, can be found in review articles by Tabor (1981) and Larsen-Basse (1992), and in the classical textbooks by Bowden and Tabor (1950, 1964). It is naturally of paramount importance that a consistent, predictable frictional response be maintained throughout the life of the damper. However, as will be discussed, this response depends to a considerable extent on surface conditions, which may in turn be affected by environmental factors. An additional subsection is included to briefly address corrosion and related environmental issues.
4.2.1 Solid F ri ction The scientific study of dry friction has a long history dating to the illustrious work o daVinci, Amontons, and Coulomb. The basic theory is founded upon the following hypotheses, which were initially inferred from physical
experiments involving planar sliding of rectilinear blocks: 1. The total frictional force that can be developed is independent of the apparent surface area of contact. 2. The total frictional force that can be developed is proportional to the total normal force acting across the interface. 3. For the case of sliding with low relative velocities, the total frictional force is independent of that velocity. As a result of these assumptions, at the instant of impending slippage or during sliding itself, one can write F = µN
(4.1)
where F and N represent the frictional and normal forces, respectively, and p is the coefficient of friction. Since it is frequently observed that the coefficient of friction is somewhat higher when slippage is imminent than it is during sliding, separate static (µs) and kinetic (µk) coefficients are often introduced. In either case, the frictional force F acts tangentially within the interfacial plane in the direction opposing the motion or impending motion. In order to extend the theory to more general conditions, involving non-uniform distributions or non-planar surfaces, these basic assumptions are often abstracted to the infinitesimal limit. Thus, total forces are replaced by surface tractions, and the generalization of Eq. (4.1) becomes t = µn
(4.2)
in terms of the tangential t and normal n, tractions. This form is also useful for determining the nominal contact stresses that are often required for proper design. Note that an integration of Eq. (4.2) over a planar contact area recovers Eq. (4.1). The concept of Coulomb friction, as described above, provides the theoretical basis for most of the work that has appeared concerning friction dampers. However, it should be emphasized that frictional processes are seldom all that simple. In practice the Coulomb theory is only approximately true. Furthermore, although the coefficient of friction µ, which appears in Eqs. (4.1) and (4.2), is often assumed to be a constant for a given pair of contacting materials, this is not always the case. For example, the value of µ at any instant depends not only upon the selection of sliding materials, but also on the present condition of the sliding interface. Since surfaces are often the site of numerous ongoing physical and chemical processes, the coefficient of friction associated with an interface may actually vary considerably over time. Many bimetallic interfaces are particularly susceptible to this behavior.
Although considerable gaps still remain in our understanding of frictional phenomena, it is appropriate to briefly consider several key aspects of the more contemporary approach. In particular, the modern theory of solid dry friction focuses on identification of the true area of contact, the mechanisms involved in interfacial bonding, and the localized inelastic deformation that occurs in the contact region. Upon detailed examination, one finds that natural and engineered surfaces are not smooth at the microscopic level, but rather contain irregularities, often categorized as waviness roughness. These topography irregularitiesand arethe typically present characteristics over a wide range o scales. Asand a result of this initial deformational of the materials, true contact occurs only through the interaction of surface asperities, as depicted in Fig. 4.2a. The diagram suggests that the true area of contact between two mating surfaces differs substantially from the apparent contact area. Researchers have found that a variety of topographical models involving conical, pyramidal, and hemispherical asperities, undergoing elastic or plastic deformation, produce true contact areas roughly proportional to the normal force. This, of course, is in general agreement with the Coulomb theory. In any case, it becomes clear that any detailed investigation of frictional behavior must account for the actual surface irregularities instead of the naive idealization of perfectly smooth Euclidean objects. When true contact does occur directly between metals, adhesive bonds form across the interface often producing coefficients of friction µ > 1. However, adhesion provides a significant contribution primarily for the contact of clean metals in a high vacuum. Actually, the schematic presented in Fig. 4.2a is representative of this situation. For more applications, surface and debris particles also be presenttypical at the engineering interface as indicated in Fig. 4.2b.films In particular, oxide layers may readily form under atmospheric conditions. These layers generally prevent the development o adhesive bonds. However, their presence also greatly complicates the investigation o frictional processes, since the mechanical characteristics of oxide films is not well understood. The final aspect of the modern theory concerns the local deformational processes that occur in the vicinity of the interface. These processes involve the elastic, elastoplastic, and possibly viscoplastic response of the asperities, surface films, debris particles, and substrata. This can be viewed as a continuum mechanics problem. However the geometry, as depicted in Fig. 4.2, is now quite complex. Thermal processes associated with inelastic dissipation are activated. Additionally, brittle or ductile failure of oxide films can significantly affect the frictional strength of the interface through the establishment of direct metal-to-metal contact. It is obviously difficult to analyze all o these factors. In order to obtain a more tractable problem, both geometric and constitutive simplifications are usually introduced. For
example, a typical simulation involves the indentation and sliding of a rigid wedge over a rigid-perfectly plastic half-space. This, along with other related contact mechanics solutions, are presented in the recent monograph by Johnson (1987). These solutions provide a means for estimating the effect of key material and geometric parameters on the frictional response, but neglect many relevant factors.
Figure 4.2 Schematic of Frictional Contact; a) Clean Metallic Surfaces, b) Metallic Surfaces with Oxide Films and Debris Particles
This modern mechanistic approach to solid friction has led to an improved qualitative understanding of the process, however a quantitative assessment of frictional response from first principles is not yet possible. More importantly, since there is still no theory for sliding friction comparable to the well-established theory of metal plasticity, there is a need for much more reliance on physical testing. We will return to this issue later in the chapter, however now attention shifts briefly to the potential effects o environmental factors on frictional systems.
4.2.2 E nvironmental E ffects During slippage, localized heating of the constituent materials will definitely occur as energy is dissipated along the interface. In some cases, these thermal effects may alter the frictional response by causing material softening or by promoting oxidation. However, for the type of sliding systems typically encountered in friction dampers, it is unlikely that system response will be sensitive to the relatively small variations in ambient temperature that can be anticipated at any site.
Of more concern are physiochemical processes, often triggered by atmospheric moisture or contaminents, that may occur in the interfacial region. These processes may change the physical and chemical character of the surfaces, and consequently produce a significant impact on the frictional response. The formation of oxide layers or scale on the exposed surfaces of metals in air has already been discussed. In more aggressive environments, corrosion may pose a problem. Both crevice corrosion and bimetallic corrosion are relevant to our present discussion. In the former case, the geometry of the component of prime importance. Exposed cathodic due to a surplus of oxygen, iswhile inaccessable regions (e.g.,surfaces threadedbecome joints, clamped surfaces) having much less dissolved oxygen become anodic and thus corrode. On the other hand, bimetallic corrosion refers to situations involving direct electrical contact between two dissimilar metals within a corrosive atmosphere. For example, the corrosion rates for mild steel can be greatly enhanced when placed in contact with copper or brass. While these basic corrosive mechanisms are well understood and discussed at length in Shreir et al. (1994), quantitative predictions of the kinetics are not yet possible. One must resort to physical testing to determine the extent of corrosion expected in a given situation and to ascertain its potential effect on the frictional characteristics of a sliding system. 4.3 Damper Behavior and Macroscopic Modeling It should be clear from the preceding discussion that the development of macroscopic force-displacement models for friction dampers must depend to a considerable extent on the results of physical testing. With this in mind, the present section contains a review of both the pertinent experimental data and the corresponding mathematical models that have been formulated for friction dampers. Consideration is given to all o the devices illustrated in Fig. 4.1. Pall et al. (1980) began their development of friction dampers by conducting static and dynamic tests on a variety of simple sliding elements having different surface treatments. The goal was not necessarily to obtain maximum energy dissipation, but rather to identify a system that possesses a consistent, predictable response. For these tests contact was maintained between the faying surfaces by pretensioning 12.7mm diameter high strength bolts. The resulting load-displacement response under monotonic loading is shown in Fig. 4.3, while Fig. 4.4 details the hysteretic behavior under constant amplitude displacement-controlled cyclic loading.
Figure 4.3 Load-displacement Response of Limited Slip Bolted Joints (Pall et al., 1980)
Figure 4.4 Hysteresis Loops of Limited Slip Bolted Joints (Pall et al., 1980)
Of the systems considered by Pall et al. (1980), the maximum static slip load was obtained for metalized surfaces. However, the cyclic response was quite erratic, with considerable stick-slip associated with the transition from static to kinetic frictional response. Similar cyclic behavior was obtained for contact between steel plates with mill scale or sand blasted surfaces. Zincrich painted surfaces and polyethylene coatings produced a smoother cyclic response, however degradation of the slip load occurred. On the other hand, as is apparent from the figures, the systems containing heavy duty brake lining pads inserted between steel plates did provide a consistent, predictable response. Experimental results from a number of other frictional materials were reported more recently by Anagnostides and Hargreaves (1990). It is perhaps not surprising that brake lining materials would perform well in the dynamic tests. After all, these materials have been developed over a great number o years in the automotive industry specifically to provide reliable frictional response. However, the use of brake lining pads does not guarantee suitable performance. For example, Tyler (1985) presented test results for a brake lining damper that exhibited significant stick-slip. Mechanical design of the damper is also important. In particular, bolt tensions must be maintained and various geometric tolerances must be controlled. Based upon the behavior obtained by Pall et al. (1980) and illustrated in Fig. 4.4e, characterization of their simple brake lining frictional system in terms of an elasticperfectly plastic model is quite appropriate. The model employed to simulate the behavior of their Limited Slip Bolted (LSB) joint is shown in Fig. 4.5a, b. Notice that the model sketched in those diagrams includes a stiff bearing stage for displacements beyond the slip length. In order tomodel quantify response, mathematical must also be provided. A suitable hysteretic is defined in aTable 4.1, wheremodel the parameters Ps and b represent the slip load and the total displacement at first contact with the bearing surface, respectively. Additionally, K0 is the elastic stiffness, while the stiffness in bearing is K2. The model, which assumes zero stiffness
Figure 4.5 Macroscopic Model for Limited Slip Bolted Joints (Pall et al., 1980)
Figure 4.5 Macroscopic Model for Limited Slip Bolted Joints (continued); c) Parameter Definition and Sample Result
during slippage (i.e., K1 = 0), is specified in rate form in order to properly address arbitrary loading-unloading histories. (The superposed dot represents differentiation with respect to time or a pseudotime loading parameter.) A typical result for displacement controlled cycling at two different amplitudes is displayed in Fig. 4.5c, where all of the model parameters are detailed.
Subsequently, Pall and Marsh (1982) proposed a system in which the braces in a moment resisting frame incorporated frictional devices. These devices utilize the same heavy duty brake lining pads discussed above. In a typical X-braced system, the braces are designed to buckle at relatively low compressive loads. As a result, the braces contribute only when subjected to tension. By installing uniaxial friction elements within each brace, slippage would only occur in the tensile direction and very little energy dissipation would result during cyclic loading. However, the special damper mechanism, devised by Pall and Marsh (1982) and depicted in Fig. 4.1b, permits much more effective
operation. During cyclic loading, the mechanism tends to straighten buckled braces and also enforces slippage in both tensile and compressive directions. Initially Pall and Marsh (1982) used a simple elastoplastic model to represent the behavior of this X-braced friction damper. However, Filiatrault and Cherry (1987) determined that this is only valid if the device slips during every cycle, and if that the slippage is always sufficient to completely straighten any buckled braces. Otherwise, the Pall-Marsh model overestimates the energy dissipation. To remedy this situation, Filiatrault Cherryis(1987) proposed moreEach detailed macroscopic model for the device. A and schematic provided in Fig.a 4.6. member of the bracing-damper system is represented by elements reflecting its individual axial and bending characteristics. Thus, the structural braces are assumed to yield in tension, but buckle elastically in compression. The device links are permitted to yield in both tension and compression, while the sliding brake pads are represented by a hysteretic model corresponding to the experimental results obtained by Pall et al. (1980). Bending stiffness is included to maintain stability of the damper mechanism.
Figure 4.6 Refined Model for X-braced Friction Damper (Filiatrault and Cherry, 1987)
In addition to developing the refined model, Filiatrault and Cherry (1987) conducted some physical experiments. In particular, they examined the response of the X-braced damper, with heavy duty asbestos brake lining pads, subjected to cyclic displacementcontrolled loading along one diagonal. Results are shown in Fig. 4.7a. Notice that, although a consistent response is still maintained over a total of fifty cycles with a displacement amplitude of 15mm, the hysteresis loops are no longer perfectly rectangular. The notches
appearing in the two corners of the loops were attributed to the difference between the diameters of the bolts and bolt holes. In fact, the results shown in the figure were obtained after inserting steel bushings in the corner holes and slots of the device. From this investigation, it is clear that seemingly minor fabrication details can affect the overall performance of the friction damper. A hysteretic model can also be developed for this imperfect Coulomb friction element, which essentially involves multilevel bearing stops. The model requires the P1, 1, K1) beyond those required for the single specification three parameters (i.e., level friction of element. Additionally, an internal variable , representing position within the inner slip length, is introduced. Table 4.2 contains an explicit definition of the model, while typical results are displayed in Fig. 4.7b. (The shape of the virgin loading curve depends upon the initial value assumed for .) Although the model is somewhat complicated, it is suitable for examining the response of the X-braced damper friction element under arbitrary loading.
Figure 4.7 Hysteresis Loops for X-braced Friction Damper; a) Experimental (Filiatrault and Cherry, 1987), b) Hysteretic Model
As mentioned previously, several simple friction damper designs have been proposed in the recent literature. For example, Roik et al. (1988) discuss seismic control o structures through the use of three-stage friction-grip elements. Both concrete-steel and steel-steel grip elements are considered. A test specimen for the latter type is detailed in Fig. 4.8a, while the hysteresis loops obtained during a displacement-controlled testing procedure are reproduced in Fig. 4.8b. The authors note that a consistent response is obtained without the complexity or cost of including brake lining materials, provided proper care is exercised in the joint design. For example, nonlinear disc springs (or Belleville washers) are employed to maintain bolt prestress. It
should be noted, however, that these results are for displacement amplitudes of less than 1mm. Additionally, a close comparison reveals cycle-to-cycle variability, and unlike an ideal Coulomb friction element, the slip load is amplitude dependent. Putting aside the details of the measured response, Roik et al. (1988) then propose that friction elements with varying degrees of stiffness, prestress, and allowable slip be combined in order to obtain a smooth transition from the sticking state to the slipping state as illustrated in Fig. 4.9. This model subsequently provides the basis for an investigation of anticipated structural response.
Figure 4.8 Steel-Steel Friction Grip Connection (Roik et al., 1988); a) Test Specimen, b) Hysteresis Loops
Figure 4.9 Hypothetical Three-Stage Stiffening Element (Roik et al., 1988)
Another simple conceptual design, the Slotted Bolted Connection (SBC), shown in Fig. 4.1e, was investigated by FitzGerald et al. (1989). Experimental results obtained under sizeable displacement controlled loading are provided in Fig. 4.10. In State 1, the gusset slips relative to the channel plates, while in State 2 there is also slippage between the channel and cover plates. The model defined previously in Table 4.2 can be utilized to simulate this behavior for the purposes of structural analysis. It should be noted that the consistent experimental response present in Fig. 4.10 is attributed by the authors to the inclusion of Belleville spring washers. On the other hand, Grigorian et al. (1993) also employ Belleville washers in an SBC design, but obtain more erratic behavior as indicated in Fig. 4.11. Clearly this latter type of response from a critical structural component must be deemed unacceptable. Additionally, as discussed previously, considerable care must be exercised in selecting materials for use in a bimetallic interface.
Figure 4.10 Typical Load Deformation Diagram for Slotted Bolted Connections (FitzGerald et al., 1989)
Figure 4.11 Cyclic Response of Slotted Bolted Connections (Grigorian et al., 1993)
The uniaxial friction damper illustrated in Fig. 4.1c, manufactured by Sumitomo Metal Industries Ltd., utilizes a slightly more sophisticated design. The pre-compressed internal spring exerts a force that is converted through the action of inner and outer wedges into a normal force on the friction pads. These copper alloy friction pads contain graphite plug inserts, which provides dry lubrication. This helps to maintain a consistent coefficient of friction between the pads and the inner surface of the steel casing. Aiken and Kelly (1990) indicate that the response of these dampers is extremely regular
and repeatable with rectangular hysteresis loops. Furthermore, the effect of loading frequency and amplitude, number of cycles, or ambient temperature on damper response was reported to be insignificant. Although the supportive component test data is not available in the literature, it would seem that a simple elastic-perfectly plastic hysteretic model, defined above in Table 4.1, is appropriate for structural analysis involving this device. The final friction damper to be considered is the Energy Dissipating Restraint (EDR) manufactured by Fluor Daniel, Inc.since and this detailed in Fig. Superficially design is similar to the Sumitomo concept, device also 4.1d. includes an internalthe spring and wedges encased in a steel cylinder. However, there are several novel aspects of the EDR that combine to produce very different response characteristics. A detailed presentation of the design and its performance is provided in Nims et al. (1993a). As indicated in Fig. 4.1d, the EDR utilizes steel compression wedges and bronze friction wedges to transform the axial spring force into a normal pressure acting outward on the cylinder wall. Thus, the frictional surface is formed by the interface between the bronze wedges and the steel cylinder. Internal stops are provided within the cylinder in order to create the tension and compression gaps that are illustrated in Fig. 4.1d. Consequently, unlike the Sumitomo device, the length of the internal spring can be altered during operation, providing a variable frictional slip force. Typical experimental hysteretic behavior is displayed in Fig. 4.12 for three different configurations. Figure 4.12a represents the response obtained with zero gaps and zero spring preload. Triangular shaped hysteresis loops result indicating slip force proportional to the device displacement. With non-zero spring preload and very large gaps, the device acts as a standard Coulomb damper as indicated in Fig. 4.12b. The model presented previously in Table 4.1 is obviously applicable for this second case. Finally, with a non-zero preload, but no initial gaps, the flag-shaped hysteresis loops of Fig. 4.12c are obtained. Clearly, from Fig. 4.12, the repsonse characteristics of the EDR are quite different from those of the other friction dampers. Consequently, it is appropriate to briefly discuss the underlying mechanics. Consider the case associated with Fig. 4.12a having zero gaps and zero preload. This configuration corresponds to that depicted schematically in Fig. 4.13, except with G1 = 0. In its initial state, due to a zero internal spring force, there exists no normal contact pressure acting between the wedges and casing. However, once a force P is applied in either tension or compression, the spring, with stiffness Ks, is compressed and frictional resistance results. Let the spring displacement be represented by s, while the overall displacement of the device is , which includes deformation of the rod and connections r. Thus, = s + r
If the stiffness of the rod and connections equals K3, then
(4.3)
Figure 4.12 Experimental Data for EDR (Richter et al., 1990)
Figure 4.13 EDR Configuration Schematic (Nims et al., 1993a)
P = K3r = K1
(4.4a, b)
where K1 is the effective overall stiffness of the device during initial loading. Furthermore, the spring force becomes Ps = Kss
(4.5a)
and the corresponding frictional resistance during slippage can be written
Pf = Kss
(4.5b)
The positive factor a, which is less than one for practical designs, incorporates the geometric and Coulomb friction effects involved in transforming the action of the spring force through the wedges into a frictional resistance. For slippage during loading, equilibrium requires that P = Ps + Pf
(4.6)
Consequently, from Eqs. (4.3)-(4.6), one obtains the following expression for the effective stiffness:
This is simply the stiffness of a system featuring a parallel combination of internal spring and frictional elements, (1 + )Ks, in series with the rod/connection spring K3. Upon subsequent unloading of the device, the frictional force reduces immediately and further slippage is prevented. Thus, for the initial stage of unloading, the spring displacement remains constant. The stiffness of the device is then simply equal to the stiffness of the rod and connections K3. Eventually, as the applied force P is reduced, a level is reached at which slippage occurs in the unloading direction. In this regime, the frictional force now opposes the action of the internal spring. As a result, the effective stiffness of the device becomes
Note that in this configuration, the EDR device is self-centering. In the absence o external force, the internal spring will return to its initial zero preload state. A detailed description of an appropriate hysteretic model is presented in Table 4.3. Results obtained from the model for displacement controlled cyclic loading are shown in Fig. 4.14a. It should be noted that this model also permits the simulation of response due to arbitrary loadings, which may include partial unloading-reloading cycles. Some care is required, however, in the numerical implementation, particularly for 1. An extended version can also be developed to incorporate the effects of initial preload, indicated in Fig. 4.12c, by including an internal variable corresponding to s. While a detailed presentation is not included here, typical numerical results obtained from the model are provided in Fig. 4.14b. The device in this configuration is again selfcentering.
Figure 4.14 Hysteretic Model for EDR; a) No Gaps, No Prelaod, b) No Gaps, Finite Preload
Before shifting attention to the overall analysis of structures that incorporate friction dampers, it is perhaps appropriate to mention a couple of aspects of damper response that have not been addressed adequately in the literature. In reviewing the available published component test data, one finds that nearly all dampers have been tested at a single, relatively low frequency. Classical Coulomb behavior is assumed, but not demonstrated. In particular, there have been no systematic attempts to investigate the effects of amplitude and frequency on response. Additionally, an examination of the consistency of damper response after long periods of inactivity has not been reported for any of the proposed dampers. Consequently, some care is needed in adopting one o the models presented above in Tables 4.1-4.3, since all assume Coulomb damping with a constant coefficient of friction. 4.4 Structural Analysis After a hysteretic model has been validated for a particular friction damper that model can be readily incorporated in an overall structural analysis. Although some attempts have been made to introduce the concepts of equivalent viscous damping (e.g., Scholl, 1993), in general, a full nonlinear time domain analysis is required. The finite element methodologies outlined in Section 2.3.4 are directly applicable, and in fact have been used in a number of detailed numerical investigations involving friction dampers. The present section contains a review of the more prominent efforts, which attempt to highlight the benefits of incorporating various frictional devices into structural systems. As a part of their initial work, Pall et al. (1980) performed parametric studies on a hypothetical panelized apartment building incorporating Limited Slip Bolted (LSB) oints. The nonlinear analysis utilized DRAIN-2D (Kanaan and Powell, 1973) to determine the response due to the 1940 El Centro S00E seismic ground motion scaled to various intensities. The primary structure was assumed to rest on a rigid foundation and remain elastic with 5% critical viscous damping, while the nonlinear model illustrated in Fig. 4.5b was used for the LSB joints. Design parameters varied in the analyses included number of building stories, LSB slip load, slot length, and joint stiffness. The last item was found to have little effect on response. However, sufficient slot length was needed to prevent the sudden increase in forces associated with bearing contact of the LSB joints. Both building height and slip load had a major influence on seismic performance. For example, Fig. 4.15a and b present the maximum wall normal stress at the base and the maximum displacement at the top, respectively, as a function of those two parameters. Ratios of less than unity indicate enhanced performance for the friction-damped structure. It is apparent that significant improvement is possible for the 15 and 20 story buildings, while the addition of LSB joints is not beneficial for the stiffer five
and ten-story models. Unfortunately, this behavior is difficult to generalize, because the response is strongly influenced by the frequency content of the seismic signal.
Figure 4.15 Numerical Results of LSB Joint for El Centro Ground Motion, Ratio of LBS Jointed Wall to Elastically Jointed Wall (Pall et al., 1980); a) Normal Stress, b) Deflection
Pall and Marsh (1982) presented similar numerical trends for a tenstory steel frame supported on a rigid foundation, again subjected to the 1940 El Centro earthquake. Results for three configurations, consisting of a moment resisting (MR) frame, a crossbraced moment resisting (BMR) frame, and a friction damped braced (FDB) frame, were compared. The DRAIN-2D analysis included consideration of tensile yielding and compressive buckling of primary frame members, along with their simple hysteretic friction damper model. Zero viscous damping was assumed for the structure in all
three configurations. The resulting deflection and column force envelopes are reproduced in Fig. 4.16. The addition of friction dampers results in significant reductions in interstory drifts. Reduced column shear and bending moment are also apparent. On the other hand, axial forces are greater than those obtained for the moment resisting frame. For the particular case considered, plastic hinges form in beam members of the MR and BMR designs, while the primary members in the FDB with optimal slip loads remain elastic. The authors note, however, that results obtained for a single seismic record may not be conclusive.
Figure 4.16 Numerical Results of X-braced Friction Damper for El Centro Ground Motion (Pall and Marsh, 1982)
Filiatrault and Cherry (1987) considered several different earthquake inputs in their combined numerical and experimental investigation of a three-story, one-third scale steel FDB frame. In the numerical DRAIN-2D analysis, the model was subjected to 1940 El Centro S00E at 0.52g, 1966 Parkfield N65E at 0.52g, Newmark-Blume-Kapur artificial signal at 0.30g, and band-limited white noise at 0.50g. Optimal slip loads for the friction dampers were determined via a parametric study, and found to be reasonably independent of the seismic signal. Results were then compared with those obtained for the MR frame and BMR frame configurations, which correspond to the zero and infinite slip load cases, respectively. Two different friction damper models were considered in the analyses; the simple Pall-Marsh model of Fig. 4.5b, and the refined model depicted in Fig. 4.6. Although the latter more realistic model produced a significantly different response, the FDB frame still performed better than the traditional MR and BMR frames. For example, typical column shear force envelopes are illustrated in Fig. 4.17.
Figure 4.17 Numerical Column Shear Force Envelopes for Three-story Frame (Filiatrault and Cherry, 1987)
Subsequently, Filiatrault and Cherry (1988) conducted a comparative numerical study of conventional, friction-damped, and base-isolated BMR steel frame structures. A typical ten-story building, with cross bracing in the even stories, was employed for the comparison. Optimal friction device slip loads were determined using DRAIN-2D for the 1940 El Centro S00E signal. In a similar manner, design parameters were established for the lead-rubber hysteretic bearing base isolation system. Mathematical models of all three systems were then subjected to signals from the 1977 Bucharest and 1985 Mexico City earthquakes, which have a very strong low frequency content. Damage predicted for the three structural configurations is identified in Fig. 4.18. The damage ratio (DR), included in that diagram, is defined as the ratio of the number o yielded members to total number of members. Both base isolated (BIBMR) and friction damped (FDB) frames performed well for the El Centro earthquake. However, only the FDB design was effective for
the remaining two disturbances. As the authors note, this suggests that the proposed friction damped system may offer new opportunities in earthquake resistant design. Despite this observation, it is difficult to draw general conclusions concerning the relative merits of FDB and BIBMR designs. It can be argued that the BIBMR system was simply not designed to respond favorably to strong low frequency content earthquakes. In that regard, this study by Filiatrault and Cherry (1988) highlights the importance of optimizing structural performance, not for a single seismic signal, but rather for the entire range of earthquake inputs that can be expected to occur at a particular site. The computational effort required for such an approach is not beyond that available with modern engineering workstations.
Figure 4.18 Predicted Damage in Ten-story Frame (Filiatrault and Cherry, 1988)
In other related analysis work, Baktash and Marsh (1987) conducted a comparative study of the performance of friction damped braced frames (FDBF) and eccentric braced frames (EBF). The authors found that while both systems were adequate, the FDBF design produced smaller deflections, accelerations, and forces. An innovative friction damped timber shear wall concept was examined by Filiatrault (1990) and also found to be effective in reducing seismic response. More recently, numerical and theoretical investigations on the response of structural systems incorporating devices have appeared. al. (1993a) conducted parametric studies on EDR idealized SDOF systems with Nims addedetfrictional elements that produce either the flag-shaped or triangular-shaped hysteresis loops discussed in Section 4.3. The parameters considered include the frequencies of the braced and unbraced structure, unbraced damping ratio , EDR device initial slip load P1 as a percentage of the structural weight, and amplitude and frequency content of the seismic signal. Fig. 4.19 provides a typical set of results for devices with flag-shaped hysteresis loops. In those plots the acceleration response, due to the El Centro and Zacatula earthquakes, is plotted versus unbraced structural frequency with
Figure 4.19 Numerical Response of SDOF Structure with EDR (Nims et al., 1993a)
unbraced damping ratio = 0.05. The frequency of the braced structure is assumed, in all cases, to equal twice the unbraced frequency. With reference to Fig. 4.14b, the device stiffness ratio K1/K3 = 10. From Figs. 4.19a, c, it is apparent that the addition o frictional devices reduces response, except for structures in the low frequency range. (Results for an unbraced structure with = 0.10 are also included in those figures to illustrate the effects of adding purely viscous damping mechanisms.) The variation o response with initial slip load is quantified in Figs. 4.19b, d. The abovesteel EDRMDOF results are for an SDOF system. However, the response of a six-story, 0.3-scale structure, subjected to the El Centro and Zacatula ground motions, has been reported in Nims et al. (1993b). Comparisons, made with the corresponding unbraced and conventionally braced frames, indicate that the friction dampers effectively reduce displacements, while maintaining comparable acceleration levels. Nonlinear analysis methods have been employed out of necessity in all of the efforts referred to in this section to describe the behavior of structural systems that incorporate friction dampers. In recent work, Inaudi et al. (1993a) have developed some interesting and potentially useful methods to approximate the response of systems that include EDR devices with triangular-shaped hysteresis loops. The authors note that although these frictional elements are nonlinear and the principle of superposition does not apply, the response is scale invariant. Thus, if a deformation history is scaled by a certain factor, the forces in the device are multiplied by that same factor. This permits the effective use of the techniques of harmonic linearization and statistical linearization (Inaudi et al., 1993a, b) to develop estimates of the actual nonlinear response. In general, very good results are obtained, indicating that these methods may be suitable for preliminary design calculations. It should be noted, however, that the use o discontinuous memory functions within a force-displacement model can lead to difficulties under arbitrary excitations which include partial unloading-reloading cycles. 4.5 Experimental Studies As noted previously, the lack of well-developed theories necessitates significant reliance upon physical testing to establish the suitability of friction dampers. Component level testing has been reviewed in some detail in Section 4.3. Consequently, in the present section, the focus is on experimental studies at the structural level. The first such investigation was conducted by Filiatrault and Cherry (1987), who evaluated the performance of the cross-braced friction dampers. Two identical threestory, one-third scale steelbetween frame structures a manner that Natural readily permitted transformation MR, BMRwere and fabricated FDB frameinconfigurations. frequencies and damping ratios were
measured at low amplitude for all three configurations. Then the structures were mounted on a shaking table and subjected to a series of earthquake records with varying magnitude and frequency content. In general, the authors found that the friction-damped structures responded significantly better than the MR or BMR designs for high intensity signals. For example, beam bending moment envelopes for the 1952 Taft earthquake scaled to a peak acceleration of 0.60g are displayed in Fig. 4.20a. The moments are lowest for the FDB frame on all three Meanwhile,well deflections shown in Fig. 4.20b. whichfloors. agree reasonable with theare experimental results, are (Numerical also includedpredictions, in both o these plots.) Third floor acceleration for the same earthquake scaled to a magnitude o 0.90g is depicted in Fig. 4.21. Once again, the FDB design produces the lowest response. For this particular earthquake, the friction dampers dissipate over 90% of the input energy, and effectively protect the primary structure from damage.
Figure 4.20 Response of Three-Story Test Frame for Taft 0.60g (Filiatrault and Cherry, 1987)
Figure 4.21 Measured Third Floor Accelerations for Taft 0.90g Input (Filiatrault and Cherry, 1987)
The experimental program designed by Filiatrault and Cherry (1987), and described briefly above, was well conceived and quite comprehensive. However, one minor point must be made concerning the use of full scale cross-braced friction devices in a onethird scale structure. Even though slip loads were adjusted to appropriate levels, this scaling imbalance reduces the contribution of any geometric nonlinearities that may be associated with the device. It is difficult to determine the significance of that contribution without further testing or detailed modeling. An additional experimental study on the cross-braced dampers was conducted by Aiken et al. (1988). A three-bay, nine-story, one-quarter scale steel structure was extensively tested on an earthquake simulator, in both MR and FDB frame configurations. The dissipating elements in the FDB design utilized a brake lining pad/stainless steel frictional interface. Low amplitude natural frequencies and damping ratios were first determined for both configurations. The fundamental frequencies were 2.0 Hz and 2.23 Hz for the MR and FDB frames, respectively, while the corresponding damping ratios were calculated at 2.4% and 5.6%. The structure was subjected to a total of ten different seismic signals with varying magnitude. As expected, for a given earthquake, an increase in magnitude was found to increase the effectiveness of the friction dampers. For example, the ratio of roof-to-ground acceleration dropped from 3.1 to 2.0 as the peak ground acceleration of an El Centro time history was elevated from 0.30g to 0.84g. At that highest intensity, the FDB structure had an estimated equivalent damping ratio of 32%. In order toscaling compare performance of the MR and FDB frames, Aiken aetquasi-resonance al. (1988) used temporal of the 1985 Mexico City ground motion to achieve response for each configuration. The predominant
signal frequency was adjusted to coincide with the fundamental frequency of the structure. For the MR frame, a peak ground acceleration of 0.25g then produced a maximum displacement of 3.1 in., while a 0.65g peak acceleration resulted in a displacement of only 2.8 in. for the friction-damped frame. Furthermore, in the latter case, approximately 70% of the input energy was dissipated by the frictional devices. The primary structural members remained in the elastic range. More recently, using the same nine-story test frame, Aiken and Kelly (1990) performed aelements combined experimental and numerical investigation ofcomprehensive uniaxial Sumitomo friction installed in conjuction with chevron bracing. A test program was conducted, involving free vibration, random excitation, and pulse loadings, along with a total of fourteen different seismic ground motions. Only a brief summary can be provided here, but all test results are thoroughly documented in the referenced report. The authors note that the performance of the friction dampers was outstanding. The hysteresis loops indicated very consistent, nearly ideal Coulomb behavior throughout the duration of the test program. For example, the response at a number of friction damper locations, for the El Centro ground motion at 0.712g peak acceleration, are reproduced in Fig. 4.22. Approximately 60% of the input energy is dissipated in the dampers. Meanwhile, the corresponding floor
Figure 4.22 Response of Sumitomo Dampers in Nine-Story Frame for El Centro 0.712g Input (Aiken and Kelly, 1990)
accelerations, displacements, and interstory drifts are presented in Fig. 4.23, along with those for the conventional MR and BMR frames. It is apparent from these profiles that the FDB structure nearly attains the benefits of the reduced displacement response o the BMR frame, combined with the acceleration response of the MR frame. The report also contains details of a numerical study and comparisons with responses obtained using viscoelastic dampers in the same structural frame.
Figure 4.23 Overall Response Comparison in Nine-Story Frame for El Centro 0.712g Input (Aiken and Kelly, 1990)
The final experimental study, to be discussed in this section, concerns the behavior o the EDR device. Richter et al. (1990) describe results from a series of over 400 shaking table tests conducted on a small, 6ft. high, three-story steel frame structure. Although a number of different internal in EDR wereresponse considered, in all cases, two devices were mounted eachdamper story. configurations Typical hysteretic of the dampers, configured with no gaps and an initial slip load of 200lbs., is shown in Fig. 4.24 for a scaled Zacatula ground motion. The flag-shaped loops are well-defined and quite consistent. Cumulative energy time histories are provided in Fig. 4.25a and b for Zacatula and El Centro signals, respectively. In both cases, the frictional devices dissipate over 90% of the total input energy. However, as noted by the authors, hysteretic mechanisms do not respond quickly to sudden impulses. This is evident in these energy response curves. Additionally higher modes were sometimes excited due to sudden stiffness changes associated with the frictional devices. Despite these limitations, the EDR device consistently provided reductions in displacements and interstory drifts, and increased the effective damping ratio of the test structure.
Figure 4.24 EDR Response in Three-story Test Frame for Zacatula Ground Motion (Richter et al., 1990)
Figure 4.25 Cumulative Energy Time History for EDR Damped Three-story Test Frame (Richter et al., 1990)
4.6 Design Considerations Several aspects of the design problem have already been discussed. For example, in Section 4.4, a number of works were reviewed in which the optimal device slip loads were determined by direct enumeration, using multiple nonlinear analyses. As the structural systems become more complicated and the number of unknown parameters increases, this type of approach becomes less attractive. Furthermore, there is no guarantee that the resulting design will indeed satisfy all of design criteria or code requirements. Obviously, there is a need to develop more systematic design methodologies. Filiatrault and Cherry (1990) present an approach, intended for practicing engineers, based upon development of design slip-load spectra. In previous work (Filiatrault and Cherry, 1989), the authors developed a streamlined friction-damped braced frame analysis program (FDBFAP) to provide a tool for efficient determination of optimal slip loads. This program includes a detailed model for cross-braced friction dampers, but assumes that all remaining structural members remain elastic. Additionally, the program computes a relative performance index (RPI) that provides an estimate of the response of the friction damped structure compared to that of the corresponding unbraced frame. More specifically, the RPI is an average of the peak and timeintegrated strain energy responses. Optimum slip load is then assumed to be associated with the minimum value of RPI. The program FDBFAP is used in Filiatrault and Cherry (1990) to perform an extensive series of analyses to establish the functional relationship between design slip load and the key parameters that define a particular structure. The proposed relationship can be written: Vs = f(Tb/Tu, Tg/Tu, N)mag
(4.9)
where Vs is the total slip shear load in all friction dampers in the N-story structure, m is the total structural mass, while Tb, Tu, and Tg represent the braced, unbraced, and predominant ground periods, respectively. The parameter ag is the design peak ground acceleration. A total of 45 different structural configurations, ranging from one to ten stories, were examined for various values of the parameters Tb/Tu, Tg/Tu and ag. Each structure was subjected to a series of five different artificially generated earthquakes. Results were then curve fit to produce bilinear design slip load spectra for each particular set of values for Tb/Tu and N. A typical spectrum is illustrated in Fig. 4.26. The design procedure for framed structures proposed by the authors can be summarized as follows: Step 1 Design the unbraced frame by considering all loads, except those associated with significant earthquakes.
Step 2 Determine Tu, and choose cross braces such that Tb< 0.4Tu, if possible. Estimate ag and Tg for the particular building site. Step 3 Verify that the parameters Tb/Tu, Tg/Tu, ag/g, and N fall within the range o the parametric studies. If not, perform detailed dynamic analysis using FDBFAP to determine the optimal slip loads. Step 4 Construct the design slip load spectrum for the particular values of Tb/Tu and N. Step 5 Utilize the design slip load spectrum to determine Vs and distribute this slip shear load uniformly to all stories. Step 6 Distribute the story slip shear load to the friction dampers in each story, and from geometric considerations, determine the device slip loads. Step 7 Check to insure that wind loads do not cause devices to slip. If necessary, modify the unbraced frame design and return to Step 2, or modify slip loads and perform detailed dynamic analysis. Step 8 Check to insure that cross braces do not yield in tension before slipping occurs. If necessary modify design. A detailed design example is also provided in Filiatrault and Cherry (1990). Although not stated by the authors, it is necessary to conduct a detailed nonlinear analysis of the final design in order to verify that all members are properly designed.
Figure 4.26 Typical Design Slip-Load Spectrum (Filiatrault and Cherry, 1990)
While the above procedure is appropriate for the present traditional design office, with the widespread availability of engineering workstations, it should be possible to employ more automated approaches for structural design. In fact, more than one decade ago, Austin and Pister (1985) developed an optimization-based design approach for friction damped braced frames. With this approach, the engineer explicitly specifies the criteria that must be satisfied. For example, member bending moments are constrained to limit yielding, axial loads are restricted to prevent buckling, and structural drift limitations are also imposed. The authors consider the response to gravity loads alone and in conjuction with a moderate or severe earthquake. A method of feasible directions is used for the optimization method. This ensures that an acceptable design is obtained at each iteration. As an example, Austin and Pister (1985) examine the design of a tenstory friction damped steel frame structure. Included are some preliminary analyses to determine initial member sizes and a condensed list of potentially active constraints. The latter was necessary to reduce computing effort. The optimization program DELIGHT.STRUCT was then used to find an improved design which attempted to minimize volume, primary member hysteretic energy, and interstory drifts. The result was not only an improved design, but additionally the sensitivity of various design parameters was explored. This problem was analyzed on a VAX 11/780 computer. Each design iteration required over 4hr. of run time. This would appear to be prohibitive. However, in 1997, the same analysis would require only 2min. of run time on a modern engineering workstation. From all indications, this trend will continue well into the next century. This suggests that an optimization-based design approach is quite feasible, even for the small design office, provided that the proper software is developed. Before closing this section, the other design issue that must be raised concerns the predictability and reliability of friction damper performance. If one considers the design procedure devised by Filiatrault and Cherry (1990), it is clear that the integrity of the entire structure during a major earthquake is dependent to a large degree on the performance of the friction dampers. These dampers function as critical elements in the aseismic protective system. Consequently, their performance must be established not only at the time of installation, but also throughout their intended design life. Since there are no reliable theories upon which to base an assumption of consistent frictional response, this must be established by physical testing. There is an urgent need to conduct such tests.
4.7 Structural Implementations In recent years, there have been several commercial applications of friction dampers aimed at providing enhanced seismic protection of new and retrofitted structures. This activity is primarily associated with the use of Pall friction devices in Canada and Sumitomo friction dampers in Japan. The Pall friction dampers, shown in Fig. 4.27, are updated versions of the cross-braced devices described previously. As documented in Pall and Pall (1996), several variations of these devices have been installed in thirteen buildings; six retrofits and seven new facilities. Several of these applications are described below in more detail.
Figure 4.27 Pall Friction Dampers (Pall et al., 1993)
The McConnelofLibrary of the Concordia University Montreal, consists o two buildings six and ten stories interconnected by aingalleria. An Canada, exterior view of the structure is provided in Fig. 4.28. The application of friction dampers to this structure is discussed in Pall and Pall (1993). A total of 143 dampers were employed. Interestingly, the architects chose to expose sixty of the dampers to view due to their aesthetic appeal. A typical example is illustrated in Fig. 4.29. A series of nonlinear DRAIN-TABS (Guendeman-Israel and Powell, 1977) analyses were utilized to establish the optimum slip load for the devices, which ranges from 600-700kN depending upon the location within the structure. For the three-dimensional time-history analyses, artificial seismic signals were generated with a wide range of frequency contents and a peak ground acceleration scaled to 0.18g to represent expected ground motion in Montreal. Under this level excitation, an estimate of the equivalent damping ratio for the structure with frictional devices is approximately 50%. In addition, for this library complex, the use o the friction dampers resulted in a net saving of 1.5% of the total building cost. The authors note that higher savings would be expected in more seismically vulnerable regions.
Figure 4.28 McConnel Library at Concordia University in Montreal (Pall and Pall, 1993)
Figure 4.29 Exposed Friction Damper in McConnel Library Galleria (Pall and Pall, 1993)
The application of friction dampers to the main headquarters building of the Canadian Space Agency Complex near Montreal is described in Vezina et al. (1992) and Pall et al. (1993). An aerial view of the entire
complex is shown in Fig. 4.30. The headquarters building is a three-story steel framed structure clad with aluminum panels. Since it contains sensitive equipment and instrumentation, additional protection is required from the potentially damaging effects of earthquakes. Three-dimensional nonlinear dynamic analyses were conducted by the authors using DRAIN-TABS to determine the suitability of employing friction dampers. Input included several artificially generated seismic signals scaled to produce a peak ground acceleration of 0.18g. Comparison with traditional braced and unbraced configurations indicated a superior performance for the friction damped design. Based upon these results, a total of 58 frictional devices were specified, each with a slip load of 500kN. The cross-braced frame bays with friction dampers are distributed evenly throughout the building, as illustrated for the first story in Fig. 4.31. Several dampers were intentionally exposed for viewing purposes.
Figure 4.30 Aerial View of Canadian Space Agency Complex (Pall et al., 1993)
Figure 4.31 Ground Floor Plan of Main Building in Canadian Space Agency (Pall et al., 1993)
The three school buildings at Ecole Polyvalante near Montreal, pictured in Fig. 4.32, were damaged in the 1988 Saguenay earthquake. The srcinal structure, built in 1967, consists of precast concrete beams and columns with welded connections. Floor and roof panels are also precast concrete. The result is a structure with little lateral resistance. In fact, analysis indicated that the existing structure could not withstand an earthquake with a peak acceleration of 0.05g. Two different retrofit schemes were considered; a conventional method using concrete shearwalls, and an approach employing friction dampers. The latter was chosen because it resulted in a 40% reduction in retrofitting cost and a 60% reduction in construction time. Based upon results from nonlinear time domain dynamic analyses, a total of 64 friction dampers were specified, along with 388 friction-based panel connectors. The project was completed during the summer break in 1990. Further details are provided in Pall and Pall (1993).
Figure 4.32 Exterior View of Ecole Polyvalante (Pall and Pall, 1993)
The eight story steel structure, shown in Fig. 4.33, was srcinally constructed by the French government for EXPO'67 held in Montreal. After the exhibition, the building was donated to the city, and in 1992 a decision was made by Lotto-Quebec to rehabilitate the structure to house 'Casino de Montreal.' Preliminary analysis indicated that the building would no longer satisfy the seismic code requirements, which had been updated since the time of the srcinal construction. Both conventional and friction-damped retrofit designs were considered. However, the former required considerable pile foundation work, which is expensive and time consuming. The friction-damped alternative employed 32 devices throughout the structure, based upon results from nonlinear dynamic analysis. In this case, a three-dimensional analysis was essential due to the eccentricities of the srcinal structure. Additional information on this retrofit project can be found in Pasquin et al. (1994).
Figure 4.33 North View of Casino de Montreal (Pasquin et al., 1994)
Three building projects in Japan, involving Sumitomo friction dampers, are briefly described in the report by Aiken and Kelly (1990). The first is the 31-story steel frame Sonic Office Building in Omiya City, constructed in 1988. A total of eight 22kip dampers were utilized on each story, primarily to reduce the effect of ground-borne vibration and small earthquakes. A similar motivation led to the use of four 22kip dampers per level in the Asahi Beer Azumabashi Building in Tokyo. This 22-story braced steel frame structure was completed in 1989. The third project mentioned involves the use of Sumitomo friction dampers as part of a base isolation system for a six-story reinforced concrete structure in Tokyo. References
Aiken, I.D. and Kelly, J. M. (1990), Earthquake Simulator Testing and Analytical Studies of Two Energy-Absorbing Systems for Multistory Structures, Report No. UCB/EERC-90/03, University of California, Berkeley, CA. Aiken, I.D., Kelly, J. M. and Pall, A. S. (1988), Seismic Response of a Nine-Story Steel Frame with Friction Damped Cross-Bracing, in Report No. UCB/EERC88/17, University of California, Berkeley, CA. Anagnostides, G. and Hargreaves, A. C. (1990), Shake Table Testing on an Energy Absorption Device for Steel Braced Frames, Soil Dyn. Earthquake Engrg., 9(3), 120-140. Austin, M. A. and Pister, K. S. (1985), Design of Seismic-Restraint FrictionBraced Frames, J. Struct. Engrg., ASCE, 111(12), 2751-2769. Baktash, P. and Marsh, C. (1987), Damped Moment-Resistant Braced Frames: A Comparative Study, Can. J. Civ. Engrg., 14, 342-346. Bowden, F. P. and Tabor, D. (1950), The Friction and Lubrication of Solids, Part I, Clarendon Press, Oxford. Bowden, F. P. and Tabor, D. (1964), The Friction and Lubrication of Solids, Part II, Clarendon Press, Oxford.
Filiatrault, A. (1990), Analytical Predictions of the Seismic Response of Friction Damped Timber Shear Walls, Earthquake Engrg. Struct. Dyn., 19, 259-273. Filiatrault, A. and Cherry, S. (1987), Performance Evaluation of Friction Damped Braced Frames Under Simulated Earthquake Loads, Earthquake Spectra, 3(1), 57-78. Filiatrault, A. and Cherry, S. (1988), Comparative Performance of Friction Damped Systems and Base Isolation Systems for Earthquake Retrofit and Aseismic Design, Earthquake Engrg. Struct. Dyn., 16, 389-416. Filiatrault, A. and Cherry, S. (1989), Efficient Numerical Modelling for Seismic Design of Friction Damped Braced Steel Plane Frames, Canadian J. Civ. Engrg., 16(3), 211-218. Filiatrault, A. and Cherry, S. (1990), Seismic Design Spectra for Friction-Damped Structures, J. Struct. Engrg., 116(5), 1334-1355. FitzGerald, T. F., Anagnos, T., Goodson, M., and Zsutty, T. (1989), Slotted Bolted Connections in Aseismic Design for Concentrically Braced Connections, Earthquake Spectra, 5(2), 383-391. Grigorian, C. E., Yang, T. S. and Popov, E. P. (1993), Slotted Bolted Connection Energy Dissipators, Earthquake Spectra, 9(3), 491-504. Guendeman-Israel, R. and Powell, G. H. (1977), DRAIN-TABS- A Computerized Program for Inelastic Earthquake Response of Three Dimensional Buildings, Report No. UCB/EERC 77-08, University of California, Berkeley, CA. Inaudi, J. A., Nims, D. K. and Kelly, J. M. (1993a), On the Analysis of Structures With Energy Dissipating Restraints, Report No. UCB/EERC 93-13, University of California, Berkeley, CA. Inaudi, J. A., Kelly, J. M. and To, C. W. S. (1993b), Statistical Linearization Method in the Preliminary Design of Structures with Energy Dissipating Devices, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 509-520. Johnson, K. L. (1987), Contact Mechanics, Cambridge University Press, Cambridge, UK. Kanaan, A. E. and Powell, G. H. (1973), DRAIN-2D - A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Structures, Report No. UCB/EERC 73-06, University of California, Berkeley, CA. Larsen-Basse, J. (1992), Basic Theory of Solid Friction, in Friction, Lubrication and Wear Technology, ASM International Handbook Committee, American Society for Metals, Materials Park, Ohio. Nims, D. K., Richter, P. J. and Bachman, R. E. (1993a), The Use of the Energy Dissipating Restraint for Seismic Hazard Mitigation, Earthquake Spectra, 9(3), 467-489. Nims, D. K., Inaudi, J. A., Richter, P. J. and Kelly, J. M. (1993b), Application of the Energy Dissipating Restraint to Buildings, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 627-638. Pall, A. S. and Marsh, C. (1981), Friction-Damped Concrete Shearwalls, Amer. Concrete Inst. J., 78(3), 187-193. Pall, A. S. and Marsh, C. (1982), Response of Friction Damped Braced Frames, J. Struct. Div., ASCE, 108(ST6), 1313-1323. Pall, A. S., Marsh, C. and Fazio, P. (1980), Friction Joints for Seismic Control of Large Panel Structures, J. Prestressed Concrete Inst., 25(6), 38-61. Pall, A. S. and Pall, R. (1993), Friction-Dampers Used for Seismic Control of New and Existing Buildings in Canada, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 675-686.
Pall, A., Vezina, S., Proulx, P. and Pall, R. (1993), Friction-Dampers for Seismic Control of Canadian Space Agency Headquarters, Earthquake Spectra, 9(3), 547557. Pall, A. S. and Pall, R. (1996), Friction-Dampers for Seismic Control of Buildings: A Canadian Experience, Proceedings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico. Pasquin, C., Pall, A. and Pall, R. (1994), High-Tech Seismic Rehabilitation o Casino de Montreal, Structures Congress ASCE, 1994. Richter, P. J., Nims, D. K., Kelly, J. M. and Kallenbach, R. M. (1990), The EDR Energy Dissipating Restraint, A New Device for Mitigation of Seismic Effects, Proc. 1990 Structural Engineers Assoc. of Calif. (SEAOC) Convention, Lake Tahoe. Roik, K., Dorka, U. and Dechent, P. (1988), Vibration Control of Structures Under Earthquake Loading by Three-Stage Friction-Grip Elements, Earthquake Engrg. Struct. Dyn., 16, 501-521. Scholl, R. E. (1993), Design Criteria for Yielding and Friction Energy Dissipators, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2,
485-495. Shreir, L. L., Jarman, R. A., and Burstein, G. T., eds. (1994), Corrosion, Butterworth-Heineman, Oxford, UK. Tabor, D. (1981), Friction - The Present State of Our Understanding, J. Lubr. Tech., ASME, 103, 169-179. Tyler, R. G. (1985), Test on a Brake Lining Damper for Structures, Bull. N. Z. Soc. Earthquake Engrg., 18(3), 280-284.
Vezina, S., Proulx, P., Pall, R. and Pall, A. (1992), Friction-Dampers for Aseismic Design of Canadian Space Agency, Tenth World Conference on Earthquake Engineering, Madrid, 4123-4128.
5 Viscoelastic Dampers 5.1 Introduction The application of viscoelastic materials to vibration control can be dated back to the 1950s when it was first used on aircraft as a means of controlling vibration-induced fatigue in airframes (Ross et al., 1959). Since that time, it has been widely used in aircrafts and aerospace structures for vibration reduction (Morgenthaler, 1987; Gehling, 1987). Its application to civil engineering structures appears to have begun in 1969 when 10,000 viscoelastic dampers were installed in each of the twin towers of the World Trade Center in New York to help resist wind loads (Mahrnoodi, 1969 Mahmoodi et al., 1987; Caldwell, 1986). This was followed by several similar applications to tall buildings against wind, both in the U.S. and abroad. Some of these applications will be highlighted in Section 5.7. Seismic applications of viscoelastic dampers have a more recent srcin. For seismic applications, larger damping increases are usually required in comparison with those required for mitigation of wind-induced vibrations. Furthermore, energy input into the structure is usually spread over a wider frequency range, requiring more effective use of the viscoelastic materials. Extensive analytical and experimental studies in the seismic domain have led to the first seismic retrofit of an existing building using viscoelastic dampers in the U.S. in 1993, which will also be briefly discussed in Section 5.7. 5.2 Basic Principles Viscoelastic materials used in structural application are typically copolymers or glassy substances which dissipate energy when subjected to shear deformation. A typical viscoelastic (VE) damper is shown in Fig. 5.1 which consists of viscoelastic layers bonded with steel plates. When mounted in a structure, shear deformation and hence energy dissipation takes place when the structural vibration induces relative motion between the outer steel flanges and the center plate.
Figure 5.1 Typical Damper Configuration
Under a sinusoidal load with frequency w, the shear strain (t) and the shear stress (t) of a linear viscoelastic material oscillate at the same frequency w but in general out-ofphase. They can be expressed by (Zhang et al., 1989) (t) = osinwt, (t) = osin(wt + )
(5.1)
where, as shown in Fig. 5.2,o, and o are, respectively, the peak shear strain and peak shear stress, and 6 is the lag angle. For a given o, both o and are functions of w.
Figure 5.2 Stress and Strain under Sinusoidal Load
The shear stress can also be written as (t) = o[G'(w)sinwt + G''(w)coswt]
where
(5.2)
The quantity osin wt in Eq. (5.2) can be replaced by (t) as given by Eq. (5.1), giving
With sin wt given by Eq. (5.1) and using the identity sin 2wt+cos2wt =1, we have the stress-strain relationship
which defines an ellipse as shown in Fig. 5.3, whose area gives the energy dissipated by the viscoelastic material per unit volume and per cycle of oscillation. It is given by
Figure 5.3 Plot of Stress Versus Strain
It is seen from Eq. (5.5) that the first term of the shear stress is the in-phase portion with G'(w) representing the elastic modulus, and the second term or the out-of-phase portion represents the energy dissipation component. This is seen more clearly if Eq. (5.2) is rewritten in the form
which is valid under harmonic motion since in that case (t) = owcoswt. The quantity G"(w)/w is the damping coefficient of the damper material. The equivalent damping ratio is
Accordingly, G'(w) is defined as the shear storage modulus of the viscoelastic material, which is a measure of the energy stored and recovered per cycle; and G"(w) is defined shear loss modulus which gives a measure of the energy dissipated per cycle. as loss factor, , defined ,by Thethe
is also often used as a measure of energy dissipation capacity of the viscoelastic material. As seen from Eq. (5.8), the damping ratio becomes
Another useful expression for the stress-strain relationship can be derived using complex notation. Equation (5.1) can be written as (t) = oeiwt, (t) = oei(wt+) (5.11) and (t) = G*(w) (t)
(5.12)
where G*(w) is the complex modulus of the viscoelastic material with
and It is seen that the two moduli, G'(w) and G"(w), or G'(w) and , determine the dynamic behavior of the linear viscoelastic material in shear under time harmonic excitation. These moduli are not only functions of the excitation frequency w, but also functions o the ambient temperature and, sometimes, the shear strain. Thus, a constitutive modeling of the viscoelastic material consisting of (t), (t) and their time derivatives must first be established. Another factor which may be important in evaluating the performance of VE dampers is the variation of internal temperature within the material during operation since energy is dissipated in the form of heat, giving rise to temperature rise in the viscoelastic material.
5.3 Shear Storage and Shear Loss Moduli As discussed above, the shear storage modulus and shear loss modulus of a viscoelastic material are generally functions of excitation frequency (w), ambient temperature (T), shear strain (), and material temperature (). One of the expedient ways of estimating their dependence on these parameters is to perform experiments on viscoelastic specimens over representative ranges of these variables. A number of such tests have been carried out over the last few years. In one series o these tests (Chang et al., 1993a), three types of viscoelastic dampers with configurations shown in Fig. 5.1 and distinguished by dimensions and types of the viscoelastic material were used. They are designated as Type A, B and C, respectively. Types A and B dampers are made of similar VE materials but different in damper dimensions. Type C damper is made from a different VE material. Table 5.1 lists the area, thickness and volume of each type of the dampers. At least three dampers from each group were studied experimentally. Table 5.1 Viscoelastic Damper Dimensions Type
Area(in2)
Thickness (in)
Volume (in3)
A
1.0 × 1.5 = 1.50
0.20
0.30
B
2.0 × 1.5 = 3.00
0.30
0.90
C
6.0 × 3.0 = 18.0
0.15
2.70
Type A dampers were first tested under six different ambient temperatures (21°C, 24°C, 32°C, 36°C and 40°C). At each temperature, six tests were conducted at frequencies of 0.1 Hz, 1.0 Hz, 2.0 Hz, 3.0 Hz, 3.5 Hz and 4.0 Hz, respectively, for up to fifty cycles of deformation in three different strain ranges (5%, 20% and 50%). Next, Type B and Type C dampers were tested at constant 5% strain at frequencies of 3.0 Hz, 3.5 Hz and 4.0 Hz at five different ambient temperatures (25°C, 30°C, 34°C, 38°C and 42°C). Finally, Type B dampers were tested at three more strains (15%, 25% and 50%) at 24° C to simulate the effect of damper strain on the energy dissipation capacity of the VE dampers under medium to strong earthquake ground motions. The force-deformation curves of the three types of dampers subjected to sinusoidal excitations with frequency of 3.5 Hz and 5% damper strain at two ambient temperatures are shown in Figs. 5.4a-5.4f. All of the hysteresis loops are fairly rounded in shape, indicating that the dampers can effectively dissipate energy. It is seen from these figures that the damper stiffness and the amount of energy dissipation in one cycle decrease for all types of dampers with increasing ambient temperature. The loss factors, however, remain more
or less constant for each type of the dampers regardless of the change in ambient temperature. Comparisons of damper properties among the three types of dampers are listed in Table 5.2.
Figure 5.4 Comparison of Force-Deformation Relationships (5% Strain, 3.5 Hz)
From Table 5.2, it may be concluded that the Type C damper is less sensitive to the change of ambient temperature. The percent reductions in energy dissipation capacity due to the change in ambient temperature from 24°C to 42°C are 73%, 71% and 60%, respectively, for Types A, B and C dampers. The lower temperature sensitivity of the Type C damper can also be observed in the reduction rates of the damper stiffness, which are 70%, 68% and 34% respectively, for Types A, B and C dampers.
Table 5.2 VE Damper Properties at 3.5 Hz and 5% Strain Damper
Temp.
G'
G"
Type
(°C)
(psi)
(psi)
A
B
21
402.8
436.7
1.08
24
305.0
344.5
1.13
28
228.4
275.1
1.20
32
169.0
198.2
1.17
36
120.7
130.7
1.08
40
91.4
92.0
1.01
25
251.1
301.3
1.20
30
187.8
223.5
1.19
34
136.9
161.5
1.18
38
110.9
122.0
1.10
42
89.8
94.3
1.05
25
28.2
24.6
0.87
30
23.1
18.1
0.78
34
21.0
15.0
0.71
38
17.6
11.6
0.65
42
15.6
9.8
0.62
C
From the descriptions above, it is clear that one has to take into account the effect o ambient excitationThe frequency an effective design viscoelastic dampers temperature in structural and applications. damper for properties are also, to a of certain degree, dependent on the number of loading cycles and the range of deformation, especially under large strain due to temperature increase within the damper material. Test results of typical Type B dampers under the excitation frequencies of 1 Hz and 3 Hz, ambient temperatures of 24° and 36° and damper strains of 5% and 20% are listed in Table 5.3. It can be seen that the damper properties remain somewhat constant and independent o strain (below 20%) for each temperature and frequency. Table 5.3 Typical Damper Properties Temp.
Freq.
Strain
G'
G"
(°C)
(Hz)
(%)
(psi)
(psi)
24
1.0
5
142
170
1.20
24
1.0
20
139
167
1.20
24 24
3.0 3.0
5 20
272 256
324 306
1.19 1.20
36
1.0
5
59
67
1.13
36
1.0
20
58
65
1.12
36
3.0
5
108
119
1.10
36
3.0
20
103
112
1.09
As mentioned above, these test results can be used to arrive at empirical formulas for G' and G" as functions of these parameters based on, for example, regression analysis. For example, the empirical formulas for G' for the three types of dampers used in this study were obtained as follows:
where c1, c2 and c3 are proportional constants. The formulas given in Eq. (5.15) for Type A and Type C dampers were derived based on the average of first twenty cycles of damper deformation with an average strain o 5%. For Type B dampers, the damper strain y was included in the equation to account for the various ranges of the damper strain due to large excitations. It is, of course, more desirable to derive constitutive relationships for viscoelastic materials based on the theory of thermo-viscoelasticity. One type of the constitutive modeling following this approach is developed below.
5.3.1 Dependenceon E xcitation F requenc Constitutive models that have been proposed for viscoelastic materials include the Maxwell model, the Kelvin-Voight model and complex combinations of these elementary models. An attractive feature of these models is their simplicity, and they work for many solid materials whose properties do not show significant variation with respect to temperature and frequency. The major drawback, however, is that the evaluation of a large number of derivative terms, acting on stress and strain, is required in order to characterize the frequency- and temperature-dependent shear storage and shear loss moduli for most VE materials (Bagley and Torvik, 1983). As a result, conventional structural dynamic analysis methods, in which the VE dampers usually contribute to modification of stiffness and damping matrices, may not yield accurate results for the characteristics of the VE dampers and the behavior of structures with added VE dampers. The concept of fractional derivatives was first employed by (Gemant, 1938) to describe such VE material moduli. In the last few years, fractional calculus has been used in the development of the force-displacement relationship for elastomeric bearings and viscous-fluid dampers (Koh and Kelly, 1990; Makris and Constantinou, 1991). More recently, this concept was used in modeling VE dampers (Tsai and Lee, 1993, Kasai et al., 1993). However, it is found that the some discrepancy exists between the VE damper models proposed based on this concept and the test data, especially when they are used to predict the
damper moduli over a wide frequency range. A more consistent model based on the Boltzmann superposition principle is presented below, focusing first on the dependence of G' and G" on the excitation frequency. The general constitutive relation between stress and strain for polymeric materials, known as the Boltzmann's superposition principle (Ferry, 1980; Rosen, 1982), is
or
if (t) = 0 for t < 0, which is true in most cases. In the above, G(t) is the stress relaxation modulus, defined as the ratio of stress to strain at constant deformation, and is usually determined from test data for a given material. For our purposes, one of the pertinent expressions for G(t) (Williams, 1964) is
where Ge is the rubbery modulus, Gg is the glassy modulus, to is the relaxation time, and a is a constant giving the slope of the relaxation curve through the transition region between glassy and rubbery behavior. The stress relaxation modulus G(t) as given in Eq. (5.18) predicts a bounded modulus for all non-negative time and has been found to be reasonably for most VE materials if a issmoothly not restricted to be anthe integer. Its initial value isaccurate the glassy modulus, which then approaches rubbery modulus with increasing time. The relationship between G(t) and G'(w) and G"(w) can be found by taking the Laplace transform of Eq. (5.17). Since where [ ] denotes Laplace transform, Eq. (5.17) gives
For > -1, where () represents the gamma function. Therefore, the Laplace transform of the relaxation modulus is
Upon substituting Eq. (5.22) into Eq. (5.20), one obtains
If one lets s = iw, G*(iw) is the complex modulus whose real and imaginary parts give, respectively, G'(w) and G"(w) as indicated in Eq. (5.13). They have the forms
and
Equations (5.18), (5.24) and (5.25) permit the determination of the model parameters Ge, Gg, to and from a combination of tests in the time domain, such as stress relaxation tests, and in the frequency domain, such as sinusoidal tests. These parameters can be first estimated based on the data obtained from the stress relaxation tests. Their accuracy can then be checked by the data obtained from the sinusoidal tests (Shen and Soong, 1995). As an example, Fig 5.5 shows a set of stress relaxation tests performed on a set o viscoelastic dampers at about 26°C in which the maximum strain was 20% and strain rates were 0.1, 0.2, 0.5, and 1.0 in/sec, respectively. The
Figure 5.5 Stress Relaxation Tests at Different Strain Rates
time axis of all the curves are shifted so that they have the same srcin. It is interesting to note that the curves for different strain rates constitute parts of the same "real" stress relaxation curve. With this approximation, one can begin with a set of constants, Ge, Gg, and to, as their initial values in an iterative procedure if the measured curves are included in the curve generated by these constants. These dampers were also tested under sinusoidal excitations at prescribed frequencies, temperatures and strains. Figure 5.6 shows the first twenty cycles of the hysteresis loops at the of 3.0 and 10%obtained strain atinabout the same temperature. These data can be frequency used to check theHz constants the stress relaxation tests. In this case, the parameters are estimated to be, at 26°C,
It can be seen in Fig. 5.7 that the predicted moduli are very close to the test results available within the frequency range from 1.0 to 3.0 Hz.
Figure 5.6 Typical Hysteresis Loop Results from a Sinusoidal Test
Figure 5.7 Comparison of Storage and Loss Moduli Between Simulation and Test
It is noted that Eqs. (5.24) and (5.25), while offering a good estimate of the properties of the VE material, can be further simplified for engineering purposes. It is seen that when the relaxation time is very small as in this case, cos( /2+wto) and sin(/2+wto) remain almost unchanged within a certain frequency range. Therefore, the variations o G'(w) and G''(w) with respect to frequency fall into straight lines if log-log scale is used. This observation provides a significant simplification for estimating the frequency dependence of the VE material properties. Basically, at a given temperature, only two damper tests at different frequencies are required in order to identify the straight line relationship.
5.3.2 Dependenceon Ambient Temperatur In order to determine the dependence of G'(w) and G"(w) on the ambient temperature, the method of reduced variables can be used (Ferry, 1980). This method affords a convenient simplification in separating the two principal variables, frequency and temperature, on which the VE material properties depend, and expressing these properties in terms of a single function of each, whose form can be experimentally determined. By using this method, the temperature dependence of the VE properties can be obtained by plotting
where T, in °K, is the ambient temperature of interest, To is an arbitrarily selected reference temperature, in °K, at which the measurements are made, p is the density o the VE material, and ßT is a shift factor of time or frequency, which is determined experimentally. The factor poTo/pT is very close to unity if T is not significantly different from To. For the same VE material tested above, Fig. 5.8 shows log ßT versus temperature from 21°C to 40°C. The regression expression of this relation is a straight line with log ßT = -0.0561 + 1.218 T
(5.28)
for 21°C < T <40°C. The dependence of G'(w) and G"(w) on the ambient temperature can be estimated using Eqs. (5.27). On the log-log scale, Figs. 5.9 and 5.10 show that the temperature effect is a simple shift of the G'(w) and G"(w) curves over the frequency range from 0.5 Hz to 8.0 Hz for this case.
Figure 5.8 Regression of Scale Factor, ßT
Figure 5.9 Simplified Expression for Shear Storage Modulus
Figure 5.10 Simplified Expression for Shear Loss Modulus
5.3.3 Dependenceon I nternal Temperatur The effect of temperature rise within the VE material when it is subjected to shear deformation can also be taken into account using the method of reduced variables. In this case, the external work needs to be converted into heat in terms of temperature and the relaxation modulus in Eq. (5.18) needs to be updated incrementally using Eqs. (5.27) and (5.28). The internal temperature, , within the VE material due to the mechanical work done by the damper can be calculated from the heat transfer equation (Kasai et al., 1993)
where cv is the specific heat of the VE material, p is the mass density, and k is thermal conductivity. The spatial variation of the temperature is assumed to occur in the zdirection across the VE layer. Experimental results and finite element analyses indicate, however, this transient heat conduction term is small (Lai, 1991) and Eq. (5.29) can be approximated by
where T is the ambient temperature. Thus, one can account for the temperature rise within the VE material in G'(w) and G"(w) by replacing T by T + (t) in Eqs. (5.27) and (5.28). The relaxation modulus, G(t), can be updated at each time increment by the reduction factors poTo/p(T + ) and T+.
In structural applications, one is interested in the temperature rise within the VE material over a loading episode. Field observations and laboratory experiments have shown that, during each wind and earthquake loading cycle, this transient temperature increase is typically less than 10°C and has a minor effect on the performance of VE dampers. In summary, assuming VE dampers undergo moderate strain (< 20%), the shear storage and loss moduli can be considered as functions of only the excitation frequency w and the ambient temperature T. 5.4 Analysis of Structures with Added VE Dampers Based the development Section 5.3, it isthe seen from that, at a given ambientupon temperature and underinmoderate strain, stress in Eq. a VE(5.7) material is linearly related to the strain and strain rate under harmonic motion. For a viscoelastic damper such as that shown in Fig. 5.1 with total
shear area A and total thickness h, the corresponding force-displacement relationship is
where
Thus, unlike metallic or friction dampers, a linear structure with added VE dampers remains linear with the dampers contributing to increased viscous damping as well as lateral stiffness. This feature represents a significant simplification in the analysis o viscoelastically damped structures (Zhang et al., 1989; Zhang and Soong, 1992). For a single-degree-of-freedom structure such as the one shown in Fig. 2.10 and oscillating at frequency w, the term x in Eq. (2.18) due to the addition of a viscoelastic damper with shear area A and thickness h takes the form where is the position factor; = cos2 if the VE damper is installed diagonally at an inclination angle with the horizontal as shown in Fig. 5.11. Equation (5.33) strictly applies only under harmonic motion at frequency w. However, it represents a reasonable approximation for more general motions within a limited frequency band, provided that and are nearly constant throughout that band.
Figure 5.11 The Five-story Test Structure
The extension of this analysis procedure to multi-degree-of-freedom structural systems is straightforward if proportional damping is assumed. The equations of motion for the modal displacements are decoupled in this case, and modifications to the modal damping and stiffness due to addition of VE dampers can be obtained by following the modal strain energy method (Ungar and Kerwin, 1962; Johnson and Kienholz, 1982; Soong and Lai, 1991). The ith modal damping ratio due to added VE dampers can be calculated as
where (wi) is the loss factor of the VE material at the modal frequency wi of the srcinal structure, Ei is the ith modal strain energy of the system with dampers and Eu is the energy stored in the viscoelastic dampers. These energies are calculated from
where i is the ith mode shape vector associated with wi, K is the srcinal stiffness matrix without added dampers, and is the stiffness matrix attributed to the added dampers. Equation (5.34) can be written as
The modified ith modal frequency is
where M is the mass matrix of the structure. If the change of the mode shapes due to added dampers can be neglected, Eq. (5.36) can be further simplified to (Chang et al., 1993a)
where wi is the ith modal frequency corresponding to the srcinal structure. Analytical calculations structures with added dampers have carried out response for several steel and for concrete structures. Some VE of these results willbeen be discussed along with experimental results presented in the next section.
5.5 Experimental Studies In order to assess seismic applicability of viscoelastic dampers, extensive experimental programs have been designed and carried out for steel frames in the laboratory (Ashour and Hanson, 1987; Su and Hanson, 1990; Lin et al., 1991; Fujita et al., 1992; Kirekawa et al., 1992; Aiken et al., 1993; Bergman and Hanson, 1993; Chang et al., 1993b; Chang et al., 1995), for lightly reinforced concrete frames in the laboratory (Foutch, et al., 1993; Lobo et al., 1993; Chang et al., 1994; Shen et al., 1995), and for a full-scale steel frame structure in the field (Chang et al., 1993a; Lai et al., 1995). In what follows, two of these experiments are described and their results summarized.
5.5.1 2/5-Scale Five-story Steel F rame The test structure in this case is a 2/5-scale five-story steel frame of a prototype structure. Its overall dimensions are 52 in x 52in in plan and 224in in height as shown in Fig. 5.11 (Chang et al., 1993a). A lumped mass system simulating the dynamic properties of the prototype structure was accomplished by adding steel plates at each floor level. The weight at each floor is 1.27 kips for the first four floors and 1.31 kips for the top floor. All the girder-to-column joints are fully welded as rigid connections. The calculated fundamental frequency of the test structure without added dampers is 3.1 Hz and the first-mode damping ratio used in the analysis program is 1.0% o critical. A pair of properly designed dampers was diagonally placed on the model structure at each floor as shown in Fig. 5.11. The responses of the test structure with and without added VE dampers and with precisely controlled ambient temperatures were studied experimentally under simulated white noise and earthquakes of varying intensities. The natural frequency of the model structure without added dampers is about 3.1 Hz, while the natural frequency of the viscoelastically damped model structure lies between 3.2 Hz and 3.7 Hz, depending on the ambient temperature. The VE dampers were designed to increase the damping ratio of the model structure to about 15% of critical at room temperature of about 25°C without significantly changing the structure's natural frequency. As the ambient temperature of VE dampers increases, the VE material becomes softer and their efficiency decreases. Figures 5.12a-5.12b show the effect of ambient temperature on the structure's natural frequency and damping ratio. Table 5.4 summarizes the effectiveness of the VE dampers under various ambient temperatures (Chang et al., 1992). A 0.12g white noise excitation was used as the input motion for the comparison to eliminate the fluctuation in the input frequencies. Figures 5.13a5.13c show displacement time histories at the roof of the model structure without added dampers
(w = 3.1 Hz) and of the model structure with added dampers at ambient temperatures o 25°C (w = 3.6 Hz) and 42°C (w = 3.26 Hz), respectively, under 0.12g Hachinohe earthquake. The damping ratios corresponding to these cases are given in Fig. 5.12b. It is seen that, even at 42°C, due to extra damping provided by the VE dampers, the viscoelastically damped structure still achieves a significant reduction in seismic response as compared to the no damper case. Similar observations can be made on story drifts and floor accelerations at all floor levels. The temperature increase during ground shaking in the VE material in this study was insignificant (below 2°C).
Figure 5.12 2/5-Scale Five-story Steel Frame; a) Measured Natural Frequency, b) Measured Damping Ratio
Figure 5.13 Roof Displacement Time Histories; (a) No Damper Added, (b) with Dampers at 25°C, (c) with Dampers at 42°C
Table 5.4 Summary of Dynamic Response Reduction Percentage under 0.12g White Noise Excitation No Maximum
Floor
Damper
Response
Level
Reference
With Dampers (% Reduction of No-Damper Case) T=25°C
T=30°C
T=34°C
T=38°C
T=42°C
Relative
5
0.696
81.9
81.3
76.4
70.7
66.7
Floor
4
0.588
83.3
80.1
75.0
69.9
66.7
Disp. (in)
3 2
0.484 0.328
84.3 83.5
80.8 79.6
75.6 75.6
71.1 71.6
68.6 69.5
1
0.116
76.7
73.3
70.7
65.5
63.8
Interstory
5
0.152
78.3
82.2
78.3
75.0
73.0
Drift
4
0.164
86.0
84.1
78.7
73.2
70.1
(in)
3
0.178
82.0
82.0
74.2
68.5
65.2
2
0.214
85.0
82.7
77.1
72.4
68.2
1
0.116
76.7
73.3
70.7
65.5
63.8
Maximum
5
1.222
85.7
85.1
81.1
77.0
73.9
Floor
4
0.856
82.5
80.5
76.8
72.5
69.4
Accel.
3
0.988
86.5
85.2
81.5
78.7
77.8
(g)
2
0.948
84.6
84.7
84.7
80.1
78.4
1
0.658
74.6
75.2
77.4
75.2
76.4
Shaking table tests were also carried out on the viscoelastically damped structure at room temperature (25°C) under the time scaled El Centro earthquake with a peak acceleration of 0.6g. Numerical studies using an inelastic analysis program DRAIN-2D (Kanaan and Powell, 1973) showed that, without added VE dampers, the model structure would undergo inelastic deformation under this strong ground motion. Therefore, under such severe ground motions, only analytical studies were conducted on the model structure without added dampers. The inelastic analysis results are used to assess the effectiveness of VE dampers under strong earthquake ground motions. Figures 5.14a-5.14b show displacement time histories at the roof of the model structure without and with added VE dampers under the 0.6g El Centro earthquake ground motion. The natural frequencies of the structure without and with added VE dampers are about 3.1 Hz and 3.7 Hz, respectively. It can be seen that VE dampers provide significant extra damping to the structure so that the structure behaved elastically and the seismic response was greatly reduced. Similar observations can be made for story drifts and floor accelerations at all floor levels. The effectiveness of VE dampers under the scaled 0.6g El Centro and Hachinohe earthquake ground motions are summarized in Table 5.5.
Figure Added 5.14 (Simulated), (b) with Dampers (Measured) Displacement Time History at Roof; (a) No Damper
Table 5.5 Summary of Dynamic Response under 0.60g El Centro and Hachinohe Earthquakes With Dampers No Damper (Inelastic Analysis)
(% Reduction of No-Damper Case)
Maximum
Floor
El Centro
Hachinohe
El Centro
Hachinohe
Response
Level
(0.60g)
(0.60g)
(0.60g)
(0.60g)
Relative
5
2.150
3.490
0.766
0.823
(64.4)
(76.4)
0.665 (66.6)
0.719 (77.8)
Floor Disp. (in)
4
1.990
3.240
3
1.650
2.700
2 1 Interstory
5-4
1.110 0.390 0.207
1.630 0.470 0.310
Drift (in)
4-3
0.365
0.599
3-2
0.598
1.100
2-1 1-0
0.721 0.394
1.185 0.470
0.529
0.579
(67.9)
(78.6)
0.346
0.382
(68.8)
(76.6)
0.143
0.148
(63.3)
(68.5)
0.104
0.111
(49.8)
(64.2)
0.137
0.146
(62.5)
(75.6)
0.187
0.201
(68.7)
(81.7)
0.214
0.234
(70.3)
(80.3)
0.143
0.148
(63.7)
(68.5)
Figures 5.15a-5.15d show the envelope curves of the lateral displacement, interstory drift, cumulated story shear, and overturning moment of the model structure with and without added dampers under the 0.6g El Centro earthquake. It can be seen that adding VE dampers to the structure reduces not only the deformation but also the base shear and overturning moment even when the structure without added dampers behaves inelastically. The VE dampers dissipate a significant amount of seismic input energy to prevent structure from undergoing inelastic deformation. Similar results were obtained the for the 0.6g Hachinohe earthquake.
Figure 5.15 Response Envelopes; (- - - - -) with Dampers, () without Dampers
The equivalent structural damping of the model structure with added VE dampers can be estimated by using the modal strain energy method as discussed in Section 5.4. Figures 5.16a-5.16b show the comparisons of the measured damping ratios and those predicted by Eqs. (5.36) and (5.38), respectively, at various ambient temperatures. As can be seen, the modal strain energy method predicts the dynamic characteristics of the viscoelastically damped structure very well.
Figure 5.16 Measured and Predicted 1st Modal Damping Ratios; a) Eq. (5.36), b) Eq. (5.38)
Using the predicted damping ratios as the modal damping ratios in the linear dynamic analysis, one can simulate response time histories of the model structure. Figures 5.17a5.17b show good agreement between the experimental and analytical displacement time histories at the roof of the structure with added VE dampers at 34°C under the 0.12g Hachinohe earthquake. Even under the 0.6g El Centro earthquake, good correlation can be obtained when the responses are compared as shown in Fig. 5.18 and Fig. 5.14b. Thus, the seismic response of viscoelastically damped structures can be reliably estimated using conventional linear dynamic analysis methods as developed in Section 5.4.
Figure 5.17 Displacement Time History at Roof; (a) Experimental, (b) Analytical
Figure 5.18 Predicted Displacement at Roof under 0.6g El Centro Earthquake
5.5.2 1/3-Scale Three-story ConcreteFram In this study, the seismic response of a scaled reinforced concrete structure using VE dampers as a means for seismic retrofit was investigated. Unlike steel structures, the seismic response of a reinforced concrete structure is by and large inelastic, which is often accompanied by permanent deformation and damage. The addition of viscoelastic dampers in this case can dissipate energy at the early stages of cracking of the concrete elements and reduce the development of damage. With proper selection of dampers, this damage be substantially even eliminated. The quantification of the influence of can viscous and elastic reduced stiffnessorproperties of dampers during the inelastic response of reinforced concrete structures was the subject of this investigation. A one-third scale model of a three-story lightly reinforced concrete framed building was tested under simulated base motions using a shaking table (Foutch et al., 1993; Lobo et al., 1993; Chang et al., 1994). The structure was tested using a series o simulated ground motions obtained from the scaled 1952 Taft earthquake, N21E component, normalized for peak ground accelerations of 0.05g, 0.20g, and 0.30g, representing minor, moderate, and severe ground motions. The structure, which was previously damaged in shaking table tests, was retrofitted by adding viscoelastic diagonal braces in the interior bay of each frame as shown in Fig 5.19. The viscoelastic dampers were similarly positioned as in the steel frame case. Some selected experimental results under the 0.2g Taft earthquake are summarized here for one type of dampers tested from the viewpoint of energy dissipation. The interstory drifts and story shears in the columns are substantially reduced at all floors as indicated in Table 5.6. While the deformations are reduced approximately three times, the shear forces are reduced only twice. These forces are much smaller than the ultimate strength of the columns; moreover, they are smaller than their yielding strengths. A sample set of force-deformations at the first floor as shown in Fig. 5.20 indicates that the column forces and deformations are substantially reduced, while most of the energy dissipation (area of hysteretic loops) is transferred from the columns to the viscoelastic dampers. Although some inelastic deformations are experienced by the columns in the presence of the viscoelastic braces, the column response is substantially improved. The VE dampers alter the overall energy balance as shown in Fig. 5.21. For the 0.2g Taft earthquake used in the experiment (Figs. 5.21a-5.21b), the total input energy is increased. However, the added VE dampers dissipate a majority of this energy, leaving only a small amount of hysteretic energy to be dissipated by the structural members. Similar energy calculations under some other earthquakes are also shown in Fig. 5.21, showing that the overall energy input may vary depending on the match between the structural frequencies and the earthquake frequency content.
Figure 5.19 Details of the R/C Model with Viscoelastic Braced Dampers
Table 5.6 Maximum Measured Story Response for 0.2g Taft Excitation Interstory Drifts (in)
Column Story Shears (kips)
First
Second
Third
First
Second
Third
Without Dampers
0.656
0.388
0.167
20.63
16.20
10.71
With Dampers
0.194
0.147
0.066
7.68
5.71
4.19
Figure 5.20 Force-deformation Curves under 0.2g Taft Earthquake
This series of experiments reveals that retrofitting reinforced concrete frames using viscoelastic dampers can reduce the overall response, but more importantly, can reduce the risk of developing a damaging mechanism near collapse. In particular, the hysteretic energy dissipation is transferred from the load bearing elements, such as the columns or beams, to non-load bearing devices that are not essentially damaged.
Figure 5.21 Contributions of Energy Terms
5.6 Design Considerations and Implementational Issues One of the fundamental requirements in structural design is to reliably predict the designed structural response under specified loading conditions. Current state-of-thepractice permits the engineers to correctly analyze the structures they design, provided all the design parameters are properly given. In designing structures with added VE dampers, the most important design parameter is the damping ratio. By properly incorporating the modal strain energy method into the design flow chart, design o structures with added VE dampers can be accomplished with minimum modifications to the current design practice (Chang et al., 1993b, c). Like many other design problems, the design of viscoelastically damped structures is in general an iterative process. First, an analysis of the structure without added dampers should be carried out. Then the required damping ratio becomes the primary design parameter for adding VE dampers to the structure. The design will normally contain the following steps which may continue to update the structural properties after each design cycle: (a) determine structural properties of the building and perform structural analysis; (b) determine the desired damping ratio; (c) select desirable and available damper locations in the building; (d) select damper stiffness and loss factor; (e) calculate the equivalent damping ratio using the modal strain energy method; and (f) perform structural analysis using the designed damping ratio. When steps (e) and (f) satisfy the desired damping ratio and the structural performance criteria, the design is complete. Otherwise, a new design cycle will proceed which may lead to new structural properties, damper locations or damper dimensions and properties. A general flow chart of the design procedure is given in Fig. 5.22. It can be seen that this design procedure falls into the traditional design procedure except for the determination of the required damping ratio and the selection of damper stiffness and loss factor. In general, the required damping ratio can be estimated by using the response spectra of the design earthquake with various damping ratios. The selection of damper stiffness and loss factor can be a trial and error procedure. They can also be determined based on the principle that the added stiffness due to the VE dampers be proportional to the story stiffness of the structure. This is obtained by modifying the modal strain energy method for each story as:
where is the target damping ratio and i and ki are, respectively, the damper stiffness and the structural story stiffness without added dampers at the ith
story. For a VE material with known G' and G'' at the design frequency and temperature, the area of the damper, A, can be determined from, as seen from the first of Eqs. (5.32),
The thickness of the VE material, h, can be determined from the maximum allowable damper deformation to insure that the maximum strain in the VE material is lower than the ultimate value. In this design procedure, the structure is assumed to be linear elastic. If inelastic deformation is allowed in the structure, the demand in VE damping can be reduced and a modified procedure has to be used.
Figure 5.22 VE Damper Design Flow Chart
As a design example, consider the 2/5-scale steel frame model structure considered in Section 5.5.1. The design earthquakes are 0.6g El Centro and Hachinohe earthquakes and the design requirements are (a) story drift is less than 0.5%, (b) structure remains elastic under the design earthquakes, and (c) operating temperature is 25°C. (a) Analysis of the Structure. Analytical results using DRAIN-2D show that plastic hinges will form over the structure under the specified design earthquakes. The story drifts under the scaled 0.6g Hachinohe and El Centro earthquakes are 2.5% and 1.53%, respectively. In order to achieve the design requirement, VE dampers will be needed. (b) Determination of the Required Damping Ratio. The required damping ratio in general can be determined from the response spectra of the design earthquakes. In this example, it is determined that an equivalent structural damping ratio of 15% will be the initial goal. (c) Select Desirable and Available Damper Locations. VE dampers can be placed in any available locations which allow shear deformations to occur within the dampers. In this example, dampers will be placed as diagonal braces. The angle between the bracing members and the floor is 42.1° except for the first floor. (d) Design the VE Dampers. Equation (5.39) can be used to calculate the damper stiffness with an assumed value of the loss factor. Since the inclination angle is 42.1°, the damper stiffness is equal to /cos2 in this example. The typical story stiffness without dampers, ki, is 14.73 kip/in. Assuming = 1.1, the damper stiffness at a typical story, ki, is 5.02 kip/in based on Eq. (5.39). The thickness of the VE material, h, can be determined from the maximum allowable damper deformation to insure that the maximum strain in the VE material is smaller than the maximum allowable value. In this example, the maximum damper deformation is 0.005 x 47 x cos = 0.174 in. If the maximum damper strain of 60% is allowed, the damper thickness h is determined as 0.3 in. The damper properties can be determined based on one-third of that of the maximum damper strain, or 20%. In this example, G' = 250 psi. Finally, if two VE layers are used, the area of the VE dampers can be obtained from Eq. (5.40) as A = h/2G' = 3.0 in2. (e) Estimate the Structural Damping Ratio. Following the modal strain energy method with the damper properties corresponding to 25°C, 20% strain and 3.5 Hz, the damping ratio of the viscoelastically damped structure is about 15%. If the calculated damping ratio at this stage is lower than the required value, more dampers or larger dampers may have to be used. (f) Perform Dynamic Analysis using the Designed Damping Ratio. In this example, it shows that the structure behaves elastically and the maximum
story drift is less than 0.5%. Since all the performance criteria are satisfied, the design is complete.
5.6.1 PlacementStrategies for VE Dampers A topic which has not been touched upon in the development of the design procedure is one of damper placements. It is clear that physical constraints exist which influence the choice of damper locations. It is also clear that the way in which the dampers are distributed in a structure may have significant effect on their effectiveness. A damper placement strategy is discussed in Ashour and Hanson (1987) for structures modeled as a uniform shear beam. The optimal locations of the dampers are found to conform with the pattern of distribution which will result in maximizing the first mode's damping ratio. In another approach (Zhang and Soong, 1992), a sequential optimization procedure is proposed in which a performance index is maximized at each step in order to determine the optimal location of a damper in a sequence. The basic idea is to place each damper at a location experiencing the largest relative displacement between attachment points of the damper. The advantage of adding dampers at optimal locations can be shown by estimating the number of dampers that can be saved to achieve the same effect as compared with the case of non-optimal placement. Using a ten-story structure as an example, Fig. 5.23 shows one of such comparisons. These two sets of curves are, respectively, the maximum displacements and maximum story drifts at the top floor plotted against the numbers of VE dampers added at optimal and at uniformly distributed locations. It is seen from Fig. 5.23b, for example, that the same story drift of approximately 0.62 in can be achieved by using four optimally located dampers as compared with nine uniformly spaced dampers.
5.6.2 Architectural Considerations Aside from engineering considerations, architectural concerns may lead to necessary modifications to the method of installation of VE dampers to structural frames. This is particularly relevant for damper installation in interior bays. While diagonal installation, such as that shown in Fig. 5.11, is a logical choice in order to maximize relative displacements between attachment points of a VE damper, other configurations are possible in order to conform with architectural requirements. Figure 5.24 shows several installation options in a case where VE dampers are installed in interior bays o a slab-column nonductile concrete frame structure. Each arrangement can generate required shear deformation in the VE material while satisfying certain specified architectural constraints.
Figure 5.23 Examples of Maximum Response as Function of Number of Dampers; a) Maximum Displacement, b) Maximum Interstory Drift
Figure 5.24 Installation Options for VE Dampers
5.7 Structural Implementation As described in Section 5.1, the first applications of VE dampers to structures were for reducing acceleration levels, or increasing human comfort, due to wind. In 1969, VE dampers were installed in the twin towers of the World Trade Center in New York, NY, as shown in Fig. 5.25a, as an integral part of the structural system. They were designed to assist the tubular steel frame in limiting wind-induced building vibrations to levels below human perception. (Mahmoodi, 1969; Mahmoodi et al., 1987). The selection, quantity, shape and location of the dampers were chosen based on the dynamics of the towers and required damping to achieve the performance objectives. There are about 10,000 VE dampers in each tower, which are evenly distributed throughout the structure from the 10th to the 110th floor. As shown in Fig. 5.25b, they are located between the lower chords of the horizontal trusses and the columns of the outside wall. The towers have experienced a number of moderate to severe wind storms over the last twenty-five years. The observed performance of the VE dampers has been found to agree well with theoretical values. After hurricane Gloria in 1978, the total damping of the building was calculated and found to be in the range of 2.5% to 3% o critical. The aging characteristics of the VE dampers have also been found to be excellent.
(a) The World Trade Center, NY
(b) Damper Installation
Figure 5.25 Damper Installation in the TradeSt.Center, New York (Courtesy of the 3MWorld Company, Paul, MN)
In 1982, VE dampers were incorporated into the Columbia SeaFirst Building in Seattle as shown in Fig. 5.26a against wind-induced vibrations (Keel and Mahmoodi, 1986). The design called for 260 dampers to be located alongside the main diagonal members in the building core as shown in Fig. 5.26b, c. The addition of VE dampers to this building was calculated to increase its damping ratio in the fundamental mode from 0.8% to 6.4% for frequent storms and to 3.2% at design wind. Similar applications o VE dampers were made to the Two Union Square Building in Seattle, as seen in Fig. 5.27a, in 1988. In this case, 16 large VE dampers were installed parallel to four columns in one floor as shown in Fig. 5.27b.
(a) The Columbia SeaFirst Building
(b) Damper Installation
Figure 5.26 Damper Installation in the Columbia SeaFirst Building, Seattle (Courtesy of the 3M Company, St. Paul, MN)
Figure 5.27 Damper Installation in the Two Union Square Building, Seattle (Courtesy of the 3M Company, St. Paul, MN)
As mentioned in Section 5.1, seismic applications of VE dampers to structures began only recently. A seismic retrofit project using VE dampers began in 1993 for a 13-story Santa Clara County building in San Jose, CA (Crosby et al., 1994). Situated in a high seismic risk region, the building was built in 1976. As shown in Fig. 5.28, it is approximately 64 meters in height and nearly square in plan, with 51m x 51m on typical upper floors. The exterior cladding consists of full-height glazing on two sides and metal siding on the other two sides. The exterior cladding, however, provides little resistance to structural drift. The equivalent viscous damping in the fundamental mode is less than 1% of critical. The building has been extensively instrumented, providing invaluable response data obtained during a number of past earthquakes. A plan for seismic upgrade of the building was developed, in part, when the response data indicated large and longduration response, including torsional coupling,
to even moderate earthquakes. Initially, three different types of devices were considered. They were the steel-yielding ADAS device, the friction-slip energy dissipation restraints, and the VE dampers. The VE dampers were chosen primarily because they provided the structure with significantly increased damping for frequent low-level ground shaking, as well as for larger seismic events.
Figure 5.28 The Santa Clara County Building, San Jose, CA
The final design called for installation of two dampers per building face per floor level as shown in Fig. 5.29, which would increase the equivalent damping in the fundamental mode of the building to about 17% of critical, providing substantial reductions to building response under all levels of ground shaking. A typical damper configuration is shown in Fig. 5.30.
Figure 5.29 Location of VE Dampers (Courtesy of the Crosby Group, Redwood City, CA)
Figure 5.30 Damper Configuration (Courtesy of the Crosby Group, Redwood City, CA)
The structure shown in Fig. 5.31 represents the first application of viscoelastic damper technology to a reinforced concrete structure for seismic upgrade. Located in San Diego, the building is a three-story structure with its plan and elevation shown in Fig. 5.32. Its lateral load resisting system consists of 8-inch reinforced concrete perimeter walls. There is a 4-inch separation joint at the center of the building, in the north/south direction. The second and third floor systems are 10-1/4 and 9-inch flat slabs, respectively, both with column capitals. The foundation consists of reinforced concrete wall footings at the exterior long wall, with all columns resting on the 6-inch slab on grade, which has 6 × 6 inch mesh of No. 6 wire for reinforcement. Results of extensive seismic evaluations show that the structure is not of sufficient quality to withstand the expected seismic loads. Appropriate reduction of interstory drifts was the primary objective of seismic upgrade and, based on nonlinear dynamic analysis, a total of 64 dampers were required to meet seismic demands on the structural elements in this case. As shown in Fig. 5.33 each damper consists of four damper units incorporated into the structure in a K-brace configuration.
Figure 5.31 Building 116, Naval Supply Facility, San Diego
Figure 5.33 Damper Configuration
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Gehling, R.N. (1987), Large Space Structure Damping Treatment Performance: Analytic and Test Results, Role of Damping in Vibration and Noise Control, ASME, NY, 93-100. Gemant, A. (1938), On Fractional Differentials, Philosophical Magazine, 25, 540-549. Johnson, C.D. and Kienholz, D. A. (1982), Finite Element Prediction of Damping in Structures with Constrained Viscoelastic Layers, AIAA Journal, 20(9), 1284-1290. Kanaan, A. E. and Powell, G. H. (1973), Drain-2D - A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Structures, Report No. UCB/EERC 73-06, University of California, Berkeley, CA. Kasai, K., Munshi, J. A., Lai, M. L. and Maison, B. F. (1993), Viscoelastic Damper Hysteretic Model: Theory, Experiment and Application, Proc. ATC-17-1 on Seismic Isolation, Energy Dissipation, and Active Control, 2, 521-532. Keel, C. J. and Mahmoodi, P. (1986), Designing of Viscoelastic Dampers for Columbia Center Building, Building Motion in Wind, (eds., N. Isyumov and T. Tschanz), ASCE, NY, 66-82. Kirekawa, A., Ito, Y. and Asano, K. (1992), A Study of Structural Control using Viscoelastic Material, Proc. 10th World Conf. Earthquake Engrg., 2047-2054, Balkema, Rotterdam. Koh, C. G. and Kelly, J. M. (1990), Application of Fractional Derivatives to Seismic Analysis of Base-isolated Models, Earthquake Engineering and Structural Dynamics, 19(2), 229-241. Lai, M. L. (1991), Private Communication. Lai, M. L., Chang, K. C., Soong, T. T., Hao, D.S. and Yeh, Y. C. (1995), Full-scale Viscoelastically Damped Steel Frame, ASCE Journal of Structural Engineering, 121(10), 14431447. Lin, R. C., Liang, Z., Soong, T. T. and Zhang, R. H. (1991), An Experimental Study on Seismic Structural Response with Added Viscoelastic Dampers, Engineering Structures, 13, 75-84. Lobo, R. F., Bracci, J. M., Shen, K. L., Reinhorn, A.M. and Soong, T. T. (1993), Inelastic Response of R/C Structures with Viscoelastic Braces, Earthquake Spectra, 9(3), 419-446. Mahmoodi, P. (1969), Structural Dampers, ASCE J. of the Structural Division, 95(8), 1661-1672. Mahmoodi, P. and Keel, C. J. (1986), Performance of Structural Dampers for the Columbia Center Building, Building Motion in Wind, (eds., N. Isyumov and T. Tschanz) ASCE, NY, 83106. Mahmoodi, P., Robertson, L. E., Yontar, M., Moy, C. and Feld, I. (1987), Performance of Viscoelastic Dampers in World Trade Center Towers, Dynamic of Structures, Structures Congress '87, Orlando, FL. Makris, N. and Constantinou, M. C. (1991), Fractional-Derivative Maxwell Model for Viscous Dampers, ASCE Journal of Structural Engineering, 117(9), 2708-2724. Morgenthaler, D. R. (1987), Design and Analysis of Passive Damped Large Space Structures, Role of Damping in Vibration and Noise Control, ASCE, NY, 1-8. Rosen, S. L. (1982), Fundamental Principles of Polymeric Materials, John Wiley, New York, NY. Ross, D., Ungar, E. E., and Kerwin, E.W. (1959), Damping of Plate Flexural Vibrations by Means of Viscoelastic Laminar, Structural Damping, (ed., Ruzicka, E. J.), ASME, NY. Shen, K. L. ASCE and Soong, T. (1995), Modeling Viscoelastic Applications, , 121(6), 694-701.Dampers for Structural JournalT. of Engineering Mechanicsof
Shen, K. L., Soong, T. T., Chang, K. C. and Lai, M. L. (1995), Seismic Behavior of Reinforced Concrete Frame with Added Viscoelastic Dampers, Engineering Structures, 17(5), 372-380. Soong, T. T. and Lai, M. L. (1991), Correlation of Experimental Results with Predictions of Viscoelastic Damping for a Model Structure, Proc. Damping '91, San Diego, CA. Su, Y-F. and Hanson, D. (1990), Seismic Response of Building Structures with echanical Damping Devices, Report No. UMCE 90-02, University of Michigan, Ann Arbor, MI. Tsai, C. S. and Lee, H. H. (1993), Application of Viscoelastic Dampers to HighRise Buildings, ASCE Journal of Structural Engineering, 119(4), 1222-1233. Ungar, E. E. and Kerwin, Jr., E. M. (1962), Loss Factor of Viscoelastic Systems in Terms of Energy Concepts, Journal of American Acoustic Society, 34, 954-957. Williams, M. L. (1964), Structural Analysis of Viscoelastic Materials, AIAA Journal, 2(5), 785-808.
Zhang, R. H., Soong, T. T. and Mahmoodi, P. (1989), Seismic Response of Steel Frame Structures with Added Viscoelastic Dampers, Earthquake Engineering and Structural Dynamics, 18, 389-396. Zhang, R. H. and Soong, T. T. (1992), Seismic Design of Viscoelastic Dampers for Structural Applications, ASCE Journal of Structural Engineering, 118(5), 13751392.
6 Viscous Fluid Dampers 6.1 Introduction The previous three chapters have described passive dampers that dissipate energy based upon various forms of inelastic deformation. Metallic, friction, and viscoelastic dampers all utilize the action of solids to enhance the performance of structures subjected to transient environmental disturbances. However, as will be discussed in the present chapter, fluids can also be effectively employed in order to achieve the desired level of passive control. In fact, the concept of a fluid damper for general shock and vibration mitigation is well-known (Harris and Crede, 1976). One prominent example is, of course, the automotive shock absorber. Significant effort has been directed in recent years toward the development of viscous fluid dampers for structural applications, primarily through the conversion o technology from the military and heavy industry. In this chapter, several of the most promising design concepts will be examined in detail, along with pertinent theoretical and experimental developments. One case, straightforward is patterned directly after the classical dashpot. In this dissipationdesign occursapproach via conversion of mechanical energy to heat as a piston deforms a thick, highly viscous substance, such as a silicon gel. Fig. 6.1a depicts a particular damper manufactured by GERB Vibration Control, which can be designed to provide vibration control in piping networks (Schwahn and Delinic, 1988) or for use as components in seismic base isolation systems (Huffmann, 1985; Makris and Constantinou, 1990). As indicated in the diagram, ribs and other geometric details are sometimes included in the piston design to enhance performance. The axisymmetric configuration provides motion, and hence energy dissipation, in all six degrees o freedom. While these devices could also be deployed within the superstructure, an alternative, and perhaps more effective, design concept involves the development of the viscous damping wall (VDW) illustrated in Fig. 6.lb (Arima et al., 1988). In this design, developed by Sumitomo Construction Company, the piston is simply a steel plate constrained to move in its plane
within a narrow rectangular steel container filled with a viscous fluid. For typical installation in a frame bay, the piston is attached to the upper floor, while the container is fixed to the lower floor. Relative interstory motion shears the fluid and thus provides energy dissipation. By incorporating a sufficient number of VDW panels within the structural frame, a significant increase in the damping level can be achieved (Miyazaki and Mitsusaka, 1992).
Figure 6.1 Viscous Liquid Dampers; a) Cylindrical Pot GERB Damper (Makris and Constantinou, 1991), b) Viscous Damping Wall (Miyazaki and Mitsusaka, 1992)
Both of the devices discussed above accomplish their objectives through the deformation of a viscous fluid residing in an open container. In order to maximize the energy dissipation density of these devices, one must employ materials with large viscosities. Typically, this leads to the selection of materials that exhibit both frequency and temperature dependent behavior. Thus, many of the issues that were addressed in Chapter 5 will again be relevant here. There is, however, another class of fluid dampers that rely instead upon the flow o fluids within a closed container. In these designs, the piston acts now,
not simply to deform the fluid locally, but rather, to force the fluid to pass through small orifices. As a result, extremely high levels of energy dissipation density are possible. However, a correspondingly high level of sophistication is required for proper internal design of the damper unit. A typical Taylor Devices fluid damper for seismic application is illustrated in Fig. 6.2a (Constantinou et al., 1993; Constantinou and Symans, 1993b). This cylindrical device contains a compressible silicone oil which is forced to flow via the action of a stainless steel rod with with aa bronze The head includes to a state-of-the-art fluidic control orificepiston design passivehead. bimetallic thermostat compensate for temperature changes. In addition, an accumulator is provided to compensate for the change in volume due to rod positioning. High strength seals are required to maintain closure.
Figure 6.2 Viscous Liquid Dampers; a) Taylor Devices Fluid Damper (Constantinou et al., 1993), b) Jarret Elastomeric Spring Damper (Pekcan et al., 1995)
These uniaxial devices, which were srcinally developed for military and harsh industrial environments, have recently found application in seismic base isolation systems as well as for supplemental damping during seismic and wind-induced vibration. Another fluid damper featuring orifice flow to achieve energy dissipation is shown in Fig. 6.2b. This device, manufactured by Jarret, utilizes a pressurized compressible silicone-based elastomer to provide additional structural stiffness and damping. Recent scale experiments suggest that the device is suitable for application in aseismic designmodel (Pekcan et al., 1995). Although the four dampers illustrated in Figs. 6.1 and 6.2 all depend upon the flow o fluids to achieve energy dissipation, each device is distinct in terms of geometric design and material performance. In order to obtain a better understanding of their anticipated behavior within a structural system, one can examine the internal flow fields from a continuum fluids dynamics viewpoint. This approach is adopted in Section 6.2, where some of the relevant basic principles of fluid mechanics are provided. Since the working fluids in most of these dampers do not typically exhibit classical Newtonian response, a presentation of appropriate constitutive models is also included. Unfortunately, this section is a little more mathematical than previous chapters. Those readers interested primarily in macroscopic performance may wish to proceed directly to Section 6.3, where overall damper behavior is considered. The objective of this latter section is to provide meaningful macroscopic models for the dampers. Mechanicsbased approaches are discussed, along with those based directly upon measurements from component level analysis physical in testing. The6.4, resulting models are utilized for overall structural Section while macroscopic structure level experiments are detailed in Section 6.5. Of course, in order to apply viscous fluid dampers in actual structures, design procedures must be developed. Progress in that area is reviewed in Section 6.6. Finally, in Section 6.7, several recent full-scale structural implementations of fluid dampers are examined. 6.2 Basic Principles of Fluid Dynamics In presenting the basic principles relevant to the performance of viscous fluid dampers, two different aspects will be considered. The first aspect involves a presentation of the governing differential equations for a fluid element. This essentially is a characterization of the balance laws of mass and momentum for a generic fluid. Secondly appropriate constitutive models will be introduced. As mentioned previously, the fluids employed in these dampers typically have some memory (i.e., exhibit frequency dependence). Consequently, emphasis will be placed on viscoelastic rheological models. The thermal sensitivity of these materials will also be addressed, while the application of these principles and models will be examined in Section 6.3.
6.2.1 Balance Laws Consider the isothermal flow of a fluid represented in terms of its mass density p, velocity vector vi, and stress tensor ij at each fixed point xi in space and at each time t. With this Eulerian viewpoint, the governing differential equations can be written (Landau and Lifshitz, 1959; Batchelor, 1967; Bird et al., 1987):
where Eqs. (6.1) and (6.2) represent the conservation of mass and momentum, respectively. In writing these equations, standard Cartesian tensor notation has been used. Summation is implied by repeated indices, a superposed dot represents a partial time derivative, and indices to the right of a comma indicate partial derivatives with respect to spatial coordinates. A concise summary of this notational system is provided in Shames and Dym (1985). The first term on the left side of these equations represents the time rate of change o the mass and momentum densities at the fixed point xi, while the second term measures the convection of those same quantities past that point. The appearance of two separate terms is a consequence of adopting the Eulerian framework. In three-dimensional space, the general flow field is described in terms of 13 unknowns (p, vi, ij). Equations (6.1) and (6.2) represent a set of four partial diffential equations. Conservationfor ofstress angular momentum furnishes that reduce to the requirement tensor symmetry (i.e., ij three = ij).more Thus,equations six additional relationships must be provided to close this system. Various constitutive models that accomplish that task by relating stresses to velocities will be discussed in the following subsections. Before examining those models, it should be noted that for dampers involving significant changes in temperature during operation, the flow becomes non-isothermal and consideration should also be given to the conservation of energy. In this chapter, we will limit the theoretical formulations to the isothermal case, which is generally a reasonable approximation for fluid dampers. However, the effects of changes in ambient temperatures will be considered.
6.2.2 I nviscid F luids This subsection will address the simplest of models, which involves the flow of an inviscid fluid. In this case, the stress tensor can be completely defined in terms of a scalar pressure p. Thus,
where ij represents the Kronecker delta. Substituting Eq. (6.3) into Eq. (6.2) produces
This is Euler's equation, which was derived by Euler in the mid-18th century. Equation (6.4), together with the equation of continuity Eq. (6.1), govern the flow field. When a fluid can be idealized as incompressible, the density is constant in both space and time. Consequently, Eq. (6.1) reduces to a requirement for a divergence-free velocity field. That is, = 0 vjj
(6.5)
Equations (6.4) and (6.5) govern the flow of an inviscid, incompressible fluid. If in addition the flow field is irrotational, then a velocity potential can be defined such that vi = ,i
(6.6)
with satisfying the Laplace equation. Thus, = 0,jj
(6.7)
and the determination of the flow field is greatly simplified.
6.2.3 NewtonianF luids Real fluids, of course, have at least some viscosity, and the above idealizations are only valid under a very limited set of circumstances. More generally, the stress tensor can be written in terms of the pressure plus a fluid stress tensor ij that is a function of the deformation field. Thus, ij = -ijp + ij
(6.8)
The most primitive constitutive model incorporating viscous effects is that for a Newtonian fluid. In this case,
assuming the usual Stokes' hypothesis. For an incompressible fluid from Eq. (6.5), this constitutive relationship simplifies to the following form: ij = µ(vij + vj,i)
(6.10)
Substituting Eqs. (6.8) and (6.9) into Eq. (6.2) produces the Navier-Stokes equations governing the behavior of an incompressible, viscous fluid, which can be written:
Consider now the response of this fluid to an infinitesimal harmonic excitation
with velocity amplitude and circular frequency w. Then in this case, the nonlinear convective term in Eq. (6.11) is negligible, and the following equation written in terms of velocity and pressure amplitudes obtains:
For sufficiently small frequencies, the remaining inertial term is also negligible. As a result, the response becomes quasistatic with equilibrium satisfied at each instant o time. Furthermore, one finds that this quasistatic response is frequency-independent as a consequence of adopting the classical Newtonian model. Returning to the definitions introduced for viscoelastic materials in Section 5.2, the complex shear modulus G* for a Newtonian fluid is simply G*(w) = iwµ
(6.14)
Thus, the storage and loss shear moduli become G' = 0 and G'' = wµ, respectively. -dependent 6.2.4 Temperature Non-NewtonianFluids While this behavior is quite appropriate for many fluids, including water at moderate frequencies, the liquids that are generally employed in fluid dampers exhibit frequency dependent viscosities. Over the years, numerous constitutive models have been proposed for such viscoelastic fluids. A particularly effective class is based upon the generalization of classical models to incorporate fractional derivative operators. This was first suggested by Gemant (1936). Subsequently, Bagley and Torvik (1983) employed fractional calculus models to analyze the response of viscoelastic structures. More recently, Makris and Constantinou (1991) applied fractional derivative Maxwell models to represent the behavior of viscous fluid dampers, and then extended those models by incorporating complex order derivatives (Makris and Constantinou, 1993).
Consider, for example, the complex-derivative Maxwell constitutive model that has been used to characterize a particular polybutane fluid over a broad frequency and temperature range (Makris et al., 1995). At some reference
temperature T0, under the assumption of infinitesimal incompressible deformation, the proposed model can be written:
where = 1 + i2, v = v1 + iv2, and µ = µ1 + iµ2 are complex-valued material parameters. As indicated, both and µ are functions of temperature. The symbol dv/dtv denotes a generalized derivative of order v with respect to time (Oldham and Spanier, 1974). For v = 1, Eq. (6.15) reduces to the classical Maxwell model, which may be adequate to model the frequency dependence over a moderate range. The storage modulus G' and loss modulus G" corresponding to Eq. (6.15) can be obtained by performing a Fourier transform. Using the relationship for the Fourier transform of a generalized derivative (Oldham and Spanier, 1974; Makris, 1992)
one obtains
with
In Eq. (6.17), and extract the real and imaginary part of their argument, respectively. The material parameters are then obtained from experimental data by employing a nonlinear regression algorithm in the complex space, with an additional causality constraint to ensure non-negative phase lag. At the zero frequency limit for the complex-derivative Maxwell model, the causality considerations force µ2/µ1 < 0. For the polybutane fluid, this constraint is active. Consequently, the parameter µ becomes real-valued and equal to the zero shear rate viscosity of the fluid. The optimal parameter values at T0 = 293K are then µ = 9.30kPa-sec, = 0.0536 + i0.0147sec, and v = 0.540 - i0.072. Having established the material parameters in the complex-derivative Maxwell model, the frequency dependence is completely characterized at the reference temperature To. It remains to determine the temperature dependence of the response. For a large class o polymers, the shape of the dynamic moduli versus frequency curves is similar when evaluated at various temperatures. This led to the development of the method o reduced variables (Ferry, 1980), which was discussed previously in Section 5.3.2. In order to
combine data taken at different temperatures into a single master curve, a frequency shift function ßT is employed, where For the polybutane fluid in the temperature range of interest, one finds from experimental data that the expression
is adequate with c = 0.043K-1 for T0 = 293K. Then a reduced dynamic modulus can be defined at circular frequency w, such that
A plot of the reduced dynamic modulus versus reduced angular frequency, ßTw, is shown in Fig. 6.3 for the polybutane fluid. This experimental data, obtained at three different temperatures, clearly delineates a pair of master curves for the storage and loss moduli, and thus validates the use of the method of reduced variables for this material. Also displayed in that figure are the dynamic shear moduli obtained from the complexderivative Maxwell model. The correlation is quite good over a broad frequency range. The only noticeable deviation occurs in the storage modulus at low frequency. However, in that range, the loss modulus is an order of magnitude greater, thus reducing the significance of the deviation. Finally, by combining Eqs. (6.19) and and (6.21), one can obtainthean bounds expression for the dynamic moduli at any(6.18), frequency temperature within of the experiments. For the complex-derivative Maxwell model, this can be written with µ(T) and (T) representing the zero shear rate viscosity and relaxation time at temperature T, where (T) = ßT(T0) (6.23) In the time domain, the constitutive model can now be expressed in the general form:
For infintesimal harmonic excitation under isothermal conditions at temperature T, the governing equations become
with
The above model has been developed for a particular polybutane fluid that is used within a cylindrical pot damper similar to that shown in Fig. 6.1a. The same methodology can no doubt be employed to develop constitutive models for several other materials that are presently in use in fluid dampers. In some cases, however, the simple Newtonian model may be sufficient throughout the frequency of interest, while in other applications a nonlinear constitutive model may be required.
Figure 6.3 Dynamic Modulus for Polybutane Fluid (Makris et al., 1995)
6.3 Damper Behavior and Macroscopic Modeling In order to properly design structures incorporating any of the fluid dampers illustrated in Figs. 6.1 and 6.2, macroscopic models of their behavior must be available. As mentioned above and in previous chapters, two alternative approaches are available. One approach relies upon physical testing of individual damper units to construct the macroscopic model. Most of the work on viscous fluid dampers has utilized this approach. The alternative is to develop mechanics-based models starting from suitable constitutive relations for the fluid and a representation of the geometric characteristics of the device. A mixture of both approaches will be provided in the following presentation, which will address each of the four fluid damper types described in the introduction.
6.3.1 Cylindrical Pot Fluid Dampers Consider first the cylindrical pot viscous fluid damper shown in Fig. 6.4, which contains the polybutane fluid modeled in Section 6.2. The damper consists of inner and outer stationary cylinders of radii r3 and r2, respectively, with a moving hollow cylindrical piston of intermediate radius r1. The viscoelastic liquid occupies the space between the inner stationary cylinder and the piston as well as the space between the moving piston and the outer cylindrical boundary of the damper. With reference to Fig. 6.4, the geometric dimensions of GERB damper Type RHY-40/V50/H50 are: r1 = 0.084m, r2 = 0.150m, r3 = 0.018m, H1 = 0.230m, H2 = 0.060m, and H = 0.370m.
Figure 6.4 Cylindrical Pot GERB Damper (Makris et al., 1995)
For harmonic input, the dynamic stiffness of the damper is the ratio of the amplitude o the resulting force P0 to the imposed displacement x0. Accordingly, the dynamic stiffness of the device is K1+iK2 = P0/x0, where K1 and K2 are the storage and loss stiffness, respectively. A complete description of K1 and K2 as a function of the frequency can be used to characterize damper behavior under general time-dependent excitation. A series of dynamic tests on the viscoelastic fluid damper have been conducted to measure response.(lateral) In eachmotion test, theofdamper was subjected to eitherThe vertical (axial) orthis horizontal specificpiston amplitude and frequency. tests were completed one after the other at eight different frequencies with an ambient temperature T0 = 20°C (293K). The storage and loss stiffnesses of the damper were then determined from records of the force needed to maintain the imposed motion by assuming linear steady-state behavior. Typical force-displacement loops obtained from this device are shown in Fig. 6.5. Meanwhile, the corresponding dynamic stiffnesses are plotted in Figs. 6.6a and 6.6b for the axial and lateral response, respectively.
Figure 6.5 GERB Damper Force-Displacement Hysteresis Loops
This is adequate for design purposes, however, each geometric configuration must be tested. Alternatively, one can develop a mechanics-based approach to determine the dynamic stiffness. The linearity of the governing equations, presented in Eqs. (6.25) and (6.26), makes the boundary element method (e.g., Banerjee, 1994) an attractive approach for the numerical solution of this problem. The development of the method for this application is beyond the scope of the present chapter, however details can be found in Makris et al. (1993b, 1995). Essentially one uses the governing equations defined by Eqs. (6.25)-(6.27), along with an appropriate reciprocal theorem, to derive a
Figure 6.6 GERB Damper Dynamic Stiffness (Makris et al., 1995); a) Axial, b) Lateral
boundary integral representation, which can be written:
where the integration is performed over the bounding surface S of the damper fluid. In the above equation, v is the velocity, while t represents the surface traction. The point y is the location at which the velocity is determined, whereas x identifies the integration point which, according to Eq. (6.28), varies all along the surface of the body. The kernel functions G and F are derived from the infinite space fundamental solutions o the governing equations. It should be noted Eq. (6.28) involves surface velocities and tractions only. Volume integration has been completely eliminated. Consequently, a boundary element analysis requires a mesh only on the surface S. Numerical results obtained from the developed boundary element method for the viscous fluid damper are also displayed in Fig. 6.6. The correlation between the experimental data and numerical predictions is quite good throughout most of the frequency range 0-30Hz. At the higher end of that range, the loss stiffness is overpredicted by approximately 20%. While errors of
this magnitude may not be significant for design calculations, it is interesting to identify their srcin. It appears that the as-built dampers do not dissipate as much energy as theoretically possible under idealized linear behavior. This may be due to imperfect contact between the fluid and damper, to nonisothermal straining, or to localized finite deformation effects for which the present method does not account. As noted above, experiments were conducted for both axial and lateral motion of the piston. However, the damper construction permits movement in all six degrees o freedom, including andmodes rocking motion. The boundary element was also applied to thesetorsional additional to complete the characterization of method the damper. Numerical results are presented in Fig. 6.7.
Figure 6.7 GERB Damper Torsional and Rocking Dynamic Stiffness (Makris et al., 1995)
Both physical experiments and mechanics-based models provide values of the dynamic stiffnesses at discrete frequencies. A macroscopic model is still needed that is more suitable for inclusion in an overall structural analysis. This problem is addressed in Makris and Constantinou (1991) and Makris et al. (1993a), where the following fractional derivative Maxwell force-displacement model is proposed:
with P as the force applied to the piston and x as the resulting piston displacement. Furthermore, the damper parameters C0, , and v represent the zero-frequency damping coefficient, the relaxation time, and the order of fractional derivative, respectively. In Makris and Constantinou (1991) the damper parameters are determined directly from experimental data, while in Makris et al. (1993a) the parameters are estimated from material data and a simplified analytical model. A typical result is shown in Fig. 6.8. It should
Figure 6.8 Macroscopic Model for GERB Damper Response (Makris and Constantinou, 1991)
be noted that for sufficiently low frequencies, inertia effects are unimportant, and one finds that the macroscopic parameters and v coincide with those obtained from constitutive modeling. Thus, only Co is a function of the damper geometry.
6.3.2 Viscous Damping Walls A similar methodology could also be developed to model viscous damping walls (Arima et al., 1988), shown in Fig. 6.1b. Assuming that the fluid is incompressible and frequency-dependent, the response of the VDW to harmonic excitation is governed by Eqs. (6.25) and (6.26), with the form of µ* established on the basis of viscometric flow experiments. However, in this case, the fluid is confined to flow within a small gap between two large parallel plates. Consequently, the flow is predominantly onedimensional as indicated in Fig. 6.9, and governing equations reduce to the single equation:
where is now the fluid velocity in the x-direction and the excitation V exp(iwt) represents the relative velocity between the inner and outer plates.
Figure 6.9 Viscous Damping Wall Flow Schematic
As indicated in Fig. 6.1b, spacers are used to maintain a uniform gap h throughout the motion. Furthermore, assuming that the pressure gradients are negligible for this shear dominated flow, and that the usual no-slip boundary conditions apply, an analytical solution can be easily obtained. The velocity profile through the gap becomes
where k = (iwp/µ*)1/2. Typically for seismic applications, since reduces to the linear profile
, the above
independent of frequency, and the traction developed on each face becomes
Consequently, for a VDW panel of length L and height H, the frequency-domain forcedisplacement response can be written:
where x0 = -iV/w. Then the storage and loss stiffness are simply
respectively. Alternatively, the response can be defined in terms of K 1 and the damping coefficient C = K2/w. The above formulation is based upon an assumption of isothermal flow involving infinitesimal deformation. Under seismic loads, since the gap h is typically on the order of 1cm, relatively large shear strains may occur. In such cases, it may be appropriate to extend the formulation to include the effects of finite deformation by using convective stress and strain measures within the constitutive model (e.g., Bird et al., 1987). While the presentation above has concentrated on a mechanics-based formulation, most of the VDW development work that has appeared in the literature has involved physical testing. Typical hysteresis loops plotting force versus relative displacement from Arima et al. (1988) are shown in Fig. 6.10 for harmonic excitation at 0.98Hz. Notice that significant energy dissipation occurs. Close examination reveals a dependency o response on amplitude. This is more clearly seen in Fig. 6.11, which presents the variation of an equivalent damping constant as a function of the displacement amplitude. Based upon these experimental results, and those briefly described in Miyazaki and Mitsusaka (1992), an empirical non-Newtonian macroscopic model has been developed for the VDW. This model is presented in detail in Reinhorn and Li (1995). This most recent report also presents data from an extensive set of component tests conducted on two different panels, approximately 1.0m x 1.0m and 1.0m x 0.8m. Typical hysteresis loops obtained at room temperature (22°C) are shown in Fig. 6.12, while results for the larger panel are also summarized in Table 6.1. Notice from the tabular results at 4.0Hz that the response is somewhat amplitude dependent.
Figure 6.10 VDW Hysteresis Loops at 0.98Hz (Arima et al., 1988)
Figure 6.11 VDW Damping Versus Amplitude (Arima et al., 1988)
Figure 6.12 VDW Force-Displacement Response (Reinhorn and Li, 1995)
Table 6.1 Summary of Component Tests on 1m x 1m Viscous Damping Walls (Reinhorn and Li, 1995) Storage
Damping
Frequency
Amplitude
Stiffness
Coefficient
(Hz)
(in)
(k/in)
(k-sec/in)
0.1
0.90
0.00
11.88
0.5 1.0
0.48 0.26
7.20 17.27
7.34 5.82
2.0
0.11
26.48
4.55
3.0
0.07
30.33
3.83
4.0
0.06
24.58
3.50
4.0
0.08
30.08
3.27
4.0
0.11
35.55
3.13
4.0
0.13
36.34
3.05
5.0
0.06
26.88
2.99
10.0
0.01
38.44
2.03
6.3.3 Orificed Fluid Dampers Both the cylindrical pot viscous dampers and the viscous damping walls have a relatively low energy dissipation density. This implies that a significant number o massive dampers will be needed to effectively control the response of a large structure. Alternatively, the concept of fluid orificing can be used to gain mechanical advantage. The resulting dampers typically feature much higher energy dissipation densities. However, the price paid for this increase is the requirement for a more sophisticated internal design. For the purposes of illustration, let us consider the hypothetical cylindrical damper with internal radius R, whose cross-sectional design is shown in Fig. 6.13. The piston head is characterized by its radius Rp and axial length Lp. The chambers are filled with an incompressible viscous fluid having viscosity µ and density p. Meanwhile, the piston rod is assumed to move in the axial direction with velocity V, forcing the fluid through the annular passage of width h = R - Rp and thus producing a pressure differential across the head. Assume that an accumulator is present to compensate for the volumetric changes associated with the piston rod. In general, the force-deformation response of this damper will depend upon all of these geometric and material parameters. orderhead to simplify problem, only the annular region formed between the In piston and the the inner cylinderconsider wall. Assuming that h « R and ignoring end effects, one can idealize the behavior as planar uniaxial flow. Then the Navier-Stokes equations, defined in Eq. 6.11, reduce to the following single equation
Figure 6.13 Hypothetical Orificed Fluid Damper
governing flow within the annular region:
where v represents the axial velocity of the fluid within the annular region. (Again the indices following commas represent differentiation with respect to spatial coordinates. However, these indices do not vary, and hence summation is no longer implied by repeated subscripts.) Let us consider two limiting cases. The first applies for low viscosity fluids µ, relatively large gaps h, short flow passages Lp or high velocity flow. For this case, an inviscid fluid idealization is appropriate, and Eq. (6.36) becomes simply For low frequencies, the time-dependent term also becomes negligible, and one is left with pvv,z = -p,z (6.38) which is the one-dimensional steady version of Euler's equation. Integrating both sides of Eq. (6.38) along the flow length Lp, one obtains Bernoulli's equation
where p and v1 represent the pressure differential across the piston head and the exit velocity, respectively. In Eq. (6.39), the entrance velocity has been assumed to be zero. After invoking the conservation of mass, one obtains the following relationship between overall damper force PI and velocity V: PI = CIV2 (6.40)
where
Notice from Eq. (6.40) that the damper force is proportional to the velocity squared, and that in this limiting case there is no energy dissipation. The damper force is inertial, generated exclusively by accelerating the fluid through the annular passage. This velocity squared relationship is generally not desirable for structural applications because of the dramatic increase in forces at higher velocities. At the other end of the spectrum, let us examine the case resulting from the use of high viscosity fluids, small gaps, long flow passages or low velocity flows. In this situation, the convective terms in Eq. (6.36) can be ignored, along with the variations of v with z. Then, for low frequency excitation, Eq. (6.36) reduces to the form µv,rr = p,z (6.42) After applying no-slip boundary conditions and invoking the conservation of mass, the damper response can be written: PV = CVV
(6.43)
where
Thus, one obtains the case of a linear viscous damper, with energy dissipation occurring due to strictly viscous action within the annular passage. Taking the ratio of the inertial and viscous limiting cases produces
where Re = pVh/µ is a nondimensional Reynolds number. For µ » 1, the damper response will be largely inertial with force proportional to V2. On the other hand, a nearly viscous damper will result from designs with µ « 1. Of course, in any case, the velocity of the fluid within the annular passage may be quite large, and the corresponding Mach number must be kept low or else compressibility effects should be examined. Additionally, it should be emphasized that the above arguments are for illustrative purposes. In an actual orificed fluid damper design, other details including those associated with end effects must be considered. The features of the fluid damper shown moreinternal complex, involving theTaylor flow ofDevices a compressible silicone oil in Fig. 6.2a are much
Figure 6.14 Schematic of Fluidic Control Orifice Design (Constantinou and Symans, 1993b)
through specially designed passages located in and around the piston head. A schematic of the fluidic control orifice design is provided in Fig. 6.14. By changing the configuration of that orifice, nearly power law force-velocity behavior is obtained for the damper response (Constantinou et al., 1993). That is,
where is a predetermined exponent. Usually for seismic application, 1, producing linear response. Mathematical modeling of the this orificed fluid damper is developed further in Constantinou and Symans (1993b). Typical experimentally measured forcedisplacement loops are shown in Fig. 6.15. Notice that for this particular fluid damper, the temperature dependence is much less dramatic. Furthermore, a classical Maxwell model is adequate to capture the frequency dependence throughout the frequency range of interest. There is no need to include non-integer order derivatives, and Eq. (6.29) reduces to the following:
with real parameters and C0 representing the relaxation time and zero frequency damping coefficient, respectively. Comparisons of the model with the experimentally determined storage stiffness, damping coefficient and phase angle are displayed in Fig. 6.16. The relaxation time A = 0.006s is quite small. This indicates that, below a cut-of frequency of approximately 4Hz, Eq. (6.47) can be simplified further by neglecting the second term on the left hand side which becomes insignificant. One is then left with the linear, purely viscous dashpot model
In this case, the subsequent structural analysis becomes greatly simplified and standard analysis packages are directly applicable.
Figure 6.15 Orificed Fluid Damper Force-Displacement Response (Constantinou and Symans, 1993b)
Figure 6.16 Orificed Fluid Damper Response (Constantinou and Symans, 1993b)
Next, consider the fluid device shown in Fig. 6.2b that utilizes the orificed flow of a pressurized compressible silicone-based elastomer in order to enhance stiffness and damping (Pekcan et al., 1995). Figure 6.17 presents typical force-displacement loops obtained experimentally for the device. Until the internal preload is overcome, the device remains quite stiff. However, beyond that level of axial force, the piston compresses the elastomer and forces it to flow through the orifice provided between the piston head inand inner casing. Thus, one bothloading, a stiffness damping component thethe response. Upon removal of obtains the extenal the and device tends toward its undeformed state
Figure 6.17 Force-Displacement Response of a Pressurized Fluid Damper (Pekcan et al., 1995)
due to the initial internal elastomer pressure. The following mathematical model was developed in Pekcan et al. (1995) to describe the response under general conditions:
Py, K1 and K2 represent the damper static prestress force, the initial damper where stiffness and the elastomeric damper stiffness, respectively. Meanwhile, the remaining model parameters C, xmax and define the
damping constant, the maximum device stroke and the velocity exponent, respectively. A typical result obtained with Eq. (6.49), provided in Fig. 6.18, is in general agreement with the experimental response.
Figure 6.18 Response of a Pressurized Fluid Damper Based Upon Analytical Model (Pekcan et al., 1995)
A similar fluid restoring device was employed by Tsopelas and Constantinou (1994) to provide damping and re-centering capability for a base isolation system. This doubleacting preloaded device is illustrated in Fig. 6.19. The resistance is provided by a combination of effects, including the preload due to initial presssurization, the device stiffness associated with the compressibility of the silicone oil, seal friction, and damping due to the passage of fluid through the orifices. The following mathematical model was developed to represent the
Figure 6.19 Preloaded Fluid Device (Tsopelas and Constantinou, 1994)
macroscopic response:
with the four individual terms on the right-hand-side corresponding, in order, to the mechanisms defined above. In this equation, P0 is the preload, K0 is the stiffness, Pmin is the seal friction at zero displacement, and Pd is the fluid damping force. Additionally, one needs the evolution equation for the internal variable Z,
along with the following definitions:
Model parameters include P0, K0, Pmin, Pdp, Pdm, xy, , ß, , 0, 1, p and m. Some comparisons between analytical and experimental response, obtained by Tsopelas and Constantinou (1994), using this model are shown in Fig. 6.20. The correlation is quite good during both static and dynamic testing. Several additional fluid dampers have been examined in the recent literature. O particular note is the analytical and experimental work detailed in Niwa et al. (1995) on a mineral oil-based damping device. The classical Maxwell model of Eq. (6.47) was employed to represent the behavior of the damper under sinusoidal excitation.
Figure 6.20 Force-Displacement Response of Preloaded Fluid Device (Tsopelas and Constantinou, 1994)
6.4 Structural Analysis A number of mathematical models were introduced in the previous section to describe the force-displacement response of fluid dampers with different levels of accuracy and sophistication. In the present section, various approaches are discussed for the incorporation of these models into an overall structural analysis. The simplest case, o course, corresponds to the linear viscous model defined in Eq. (6.48) for the orificed fluid damper. When this approximation
is valid, the analysis simplifies greatly, and the methods outlined in Chapter 2 are directly applicable. Of course, this is of significant benefit, since most of the general purpose structural analysis computer codes (e.g., SAP, DRAIN, ABAQUS, ADINA) already include all of the required capabilities. For analyses in which the primary structure remains elastic and the dampers are reasonably well distributed throughout the superstructure, the modal methods discussed in Section 2.3.2 become attractive. Otherwise, direct time domain approachforof the Section 2.3.4 models can be defined employed. The time domainthemethods are also appropriate nonlinear in Eqs. (6.49) and (6.50) for the preloaded fluid dampers. The remaining models discussed in the previous section are viscoelastic. One approach for these models, defined by Eqs. (6.29), (6.34) and (6.47), is to utilize the modal strain energy method as presented in Section 5.4 for viscoelastic solid dampers. A second alternative was developed in Constantinou and Symans (1993b) for dampers employing the classical Maxwell models of Eq. (6.47). This latter formulation can be derived starting from Eq. (2.20). Assuming that the forces due to the passive devices are x = fr and that = 0, one can write
For building models that employ one degree of freedom per floor, the passive damper force vector fr contains the horizontal components of damper forces acting on the individual floors. That is,
where nj is the number of identical dampers installed in the jth story and Pj is the corresponding horizontal force from a single damper. For the classical Maxwell model under consideration,
with j representing the placement angle from the horizontal for the jth floor dampers. After performing a Fourier transform on Eqs. (6.54)-(6.56), one obtains
where the tilde symbolizes a Fourier transformed quantity, and
represents the dynamic stiffness of the structure. The frequency dependent coefficient matrix in Eq. (6.57) can be readily obtained from
with
The time history response can then be obtained by utilizing the discrete Fourier transform method (e.g., Press et al., 1992). In essence, one solves Eq. (6.57) for independently at a number of frequencies, and an inverse fast Fourier transform algorithm is subsequently employed to reconstruct the time domain response x(t). The cylindrical pot fluid damper and viscous damping wall models can also be accommodated with this approach. One only needs to replace Eq. (6.60) with the appropriate frequency domain macroscopic model. A time history analysis can also be conducted by solving the set of first-order differential equations (Constantinou and Symans, 1993b):
directly in the time domain. From Eq. (6.56), the various matrices and vectors in Eq. (6.61) are defined as follows:
Furthermore, complex eigenvalues and eigenvectors can be determined by solving the generalized eigenvalue problem (A + µB)z = 0
(6.63)
The natural frequencies and damping ratios are then determined from
Clearly from the above brief discussion, there are many different approaches that can be adopted for the analysis of structures with passive fluid dampers, although in some cases typical structural analysis software packages do not presently provide these capabilities, except through user-defined formulations. On the other hand, for the linear viscous orificed damper, the structural analysis is straightforward. Even in that case, however, it must be remembered that generally bracing is added to the structure in conjuction with the passive fluid dampers. It is naturally quite important to include the stiffness associated with this bracing in the overall structural model. 6.5 Experimental Studies In this section, several experimental studies demonstrating the effectiveness of viscous fluid dampers will be reviewed. Here attention will be directed toward those investigations in which the dampers are positioned within the superstructure. Additional studies in which the dampers are used as part of a base isolation system for buildings and bridges (e.g., Huffmann, 1985; Constantinou et al., 1993; Tsopelas and Constantinou, 1994) will not be detailed. A comprehensive experimental program for the linear viscous orificed fluid damper, illustrated in Fig. 6.2a, was conducted by Constantinou and Symans (1993b). The program centered around the performance of a three-story 1:4 scale steel frame, which was subjected to a series of 66 simulation tests involving five different earthquake motions. Tests were performed on the bare structure itself, and on the structure with two, four or six fluid dampers installed as shown in Fig. 6.21. In some tests, the second and third stories were rigidly braced to simulate the response of a single story structure. The total mass in the three-story configuration was approximately 2900kg, with uniform distribution to the individual stories. Dynamic properties of the structure are provided in Table 6.2. Each individual fluid damper had a weight of 10N, and a length of 280mm with a ±51mm stroke. constant At roomC0 temperature, using and the the classical Maxwell model of Eq. (6.56), the damping = 15.5Nsec/mm relaxation time = 0.006sec.
Figure 6.21 Damper Configurations for Three-story Steel Test Structure (Constantinou and Symans, 1993b)
Selected experimental results for the structure in the single-story configuration are presented in Table 6.3, while those for the full three-story structure are displayed in Table 6.4. Peak accelerations, story shear forces and interstory drifts obtained for scaled El Centro and Taft signals are shown in Fig. 6.22. Notice that all three quantities are reduced substantially with the introduction of the passive fluid dampers. These reductions are primarily due to an increased ability to dissipate seismic energy, and not to a shift in frequency. Figure 6.23 provides energy time histories for the structure in the single-story configuration subjected to the Taft motion. Notice that fluid dampers, when present, dissipate a vast majority of the absolute input energy. Comparisons were also made between the experimental data and results from a numerical analysis that employed the classical Maxwell model of Eq. (6.56) to represent the behavior of the fluid dampers. Typical results
Table 6.2 Dynamic Properties of Steel Test Structure under Elastic Conditions (Constantinou and Symans, 1993b) Single-story Configuration: Number
Frequency Damping
of Dampers
Mode
(Hz)
Ratio (%)
0 2
1 1
2.00 2.04
0.55 28.4
4
1
2.10
57.7
Three-story Configuration: Number
Frequency Damping
of Dampers
Mode
(Hz)
Ratio (%)
0
1
2.00
1.7
2
6.60
0.8
3
12.2
0.3
1
2.03
9.9
2
6.88
14.7
3
12.34
5.0
1
2.11
17.7
2
7.52
31.9
3
12.16
11.3
1
2.03
19.4
2
7.64
44.7
3
16.99
38.0
2
4
6
Table 6.3 Summary of Experimental Results for Single-story Steel Test Structure (Constantinou and Symans, 1993b) Excitation El Centro 33.3% Taft 100%
Number
Peak Base Shear
Peak Drift (%)
of Dampers
Weight
Height
0
0.228
1.842
2
0.087
0.500
0
0.224
1.740
2
0.129
0.724
4
0.115
0.504
Table 6.4 Summary of Experimental Results for Three-story Steel Test Structure (Constantinou and Symans, 1993b) Number Excitation El Centro 50%
Taft 100%
Peak Story
Peak
of Orificed
Peak
Shear Force
Story Drift
Fluid Dampers
Acceleration (g)
Weight
Height
0
0.585
0.295
1.498
2
0.301
0.196
0.865
4
0.282
0.159
0.660
6
0.205
0.138
0.510
0
0.555
0.255
1.161
2
0.271
0.150
0.696
4
0.246
0.130
0.638
6
0.178
0.120
0.463
(%)
Figure 6.22 Experimental Results for Three-story Steel Test Structure (Constantinou and Symans, 1993b)
obtained for the first story total damper force are presented in Fig. 6.24 for three different earthquake motions. The correlation is quite good, suggesting that a reasonable model has been developed for both the dampers and the primary structure. Furthermore, Constantinou and Symans (1993b) conclude that the inclusion of viscous fluid dampers generally reduced drifts by 30% to 70%, while also reducing story shear forces by 40% to 70%.
Figure 6.23 Energy Time History Response for Steel Test Structure in Single-story Configuration (Constantinou and Symans, 1993b)
Figure 6.24 Comparison of Experimental and Analytical Results for Three-story Steel Test Structure (Constantinou and Symans, 1993b)
A second more recent study involved the application of this same type of orificed fluid dampers to a 1:3 scale three-story lightly reinforced concrete frame, tested under the action of a series of scaled earthquake motions (Reinhorn et al., 1995). The fluid dampers were positioned within the diagonal braces in the middle bay of each of the three stories as shown in Fig. 6.25. Dynamic characteristics of the structure with and without the dampers are provided in Table 6.5. Notice that the addition of the fluid dampers significantly increased the effective damping of the structure from 3% to 16% under low amplitude excitations. Top story ratio displacement and acceleration response for the El Centro PGA 0.3g test are provided in Fig. 6.26, while peak deformations and forces obtained from several different tests are presented in Table 6.6. Typically, displacements are significantly reduced. However, accelerations and total forces transmitted to the foundation are only minimally affected, and in some cases may increase slightly. Figure 6.27 illustrates the beneficial effect of the fluid dampers in reducing both interstory drift and column shear. The seismic energy input to the structure increases due to the stiffening associated with the added bracing as indicated in Fig. 6.28. This is more than compensated for by the ability of the fluid dampers to dissipate a large portion of the overall energy, thus minimizing damage to the primary structural members. A comparison of the experimental results with those obtained from a nonlinear time history analysis using the computer code IDARC was also made, and excellent correlation was obtained using the Maxwell model for the viscous fluid dampers.
Figure 6.25 Orificed Fluid Damper Locations for Three-story Reinforced Concrete Test Structure (Reinhorn et al., 1995)
Table 6.5 Dynamic Characteristics of Reinforced Concrete Test Structure (Reinhorn et al., 1995) Low Amplitude Testing (El Centro S00E) Fundamental
Strong Motion Testing (El Centro S00E) Fundamental
Frequency
Damping
Frequency
Damping
Configuration
(Hz)
Ratio (%)
(Hz)
Ratio (%)
Without Dampers
1.61
3
1.31
5
With Orificed Fluid Dampers
1.88
16
1.61
28
With Viscous Damping Walls
4.08
50
3.70
46
Figure 6.26 Experimental Response of Three-story Reinforced Concrete Test Structure Under El Centro 0.3g Motion; a) Top Story Displacement, b) Top Story Acceleration (Reinhorn et al., 1995)
Table 6.6 Summary of Experimental Results for Three-story Reinforced Concrete Test Structure (Reinhorn et al., 1995) Orificed Excitation El Centro 0.3g Taft 0.2g
Fluid
First Story
Base Shear
Dampers
Drift (in)
(kips)
No
0.86
19.17
Yes
0.47
16.92
No
0.62
15.17
Yes
0.28
14.28
Figure 6.27 Experimental Response Three-story Concrete Under El Centro 0.3gofMotion; a, b) Reinforced Without Dampers; c, Test d, e) Structure With Orificed Fluid Dampers (Reinhorn et al., 1995)
Figure 6.28 Energy Response of Three-story Reinforced Concrete Test Structure Under El Centro 0.3g Motion (Reinhorn et al., 1995)
A significant amount of experimental work has been performed on the viscous damping wall concept. Arima et al. (1988) studied the response of a scaled five-story steel building model and a full-scale four-story test frame on an earthquake simulator. A typical result for the former structure is
shown in Fig. 6.29, which illustrates the reduction in acceleration levels with the inclusion of the passive damping walls. More recently, Reinhorn and Li (1995) have conducted an investigation of viscous damping walls within the same 1:3 scale threestory reinforced concrete frame utilized for the orificed fluid dampers. Figure 6.30 provides a schematic of the various structural configurations examined in that study. Details on the dynamic characteristics of this three-story structure before and after installation of the damping walls are included in Table 6.5. Notice that both the stiffness and damping are significantly increased. Top story displacement and acceleration response for the El Centro PGA 0.3g test are displayed in Fig. 6.31. Generally deformations are significantly reduced due to the enhancement of both damping and stiffness. However, the latter contributes to an increase in accelerations and to the forces transmitted to the foundation. Selected results are summarized in Table 6.7. Once again the passive dampers consume a significant proportion of the input seismic energy, as indicated in Fig. 6.32.
Figure 6.29 Experimental Response of Five-story Building Model With and Without Viscous Damping Walls (Arima et al., 1988)
Figure 6.30 Viscous Damping Wall Locations for Three-story Reinforced Concrete Test Structure (Reinhorn and Li, 1995)
Figure 6.31 Experimental Response of Three-story Reinforced Concrete Test Structure Under El Centro 0.3g Motion; a) Top Story Displacement, b) Top Story Acceleration (Reinhorn and Li, 1995)
Table 6.7 Summary of Experimental Results for Three-story Reinforced Concrete Test Structure (Reinhorn and Li, 1995) Number of Excitation El Centro 0.3g
Taft 0.2g
First Story Total
Viscous
First Story
Column Shear
Damper Walls
Drift (in)
(kips)
0
0.67
14.65
2
0.22
5.14
4
0.25
7.02
6
0.23
7.20
0
0.69
14.82
6
0.11
4.40
Figure 6.32 Energy Response of Three-story Reinforced Concrete Test Structure Under El Centro 0.3g Motion (Reinhorn et al., 1995)
The last experimental program to be discussed in this section concerns the preloaded orificed damper shown in Fig. 6.2b. This work, which is detailed in Pekcan et al. (1995), again examined the response of the aforementioned three-story reinforced concrete frame building. Dampers were incorporated in diagonal bracing added to the central bay of the structure. Several different configurations were examined, including cases in which dampers were placed in all stories, the first two stories, and only the first story. Dynamic properties of the structure are summarized in Table 6.8. A total of 17 shaking table tests were conducted using four different scaled earthquake motions. Typical maximum response values are provided in Table 6.9. Significant reductions
in interstory drifts and moderate reductions in base shears result from introduction o the pressurized fluid dampers. Fig. 6.33 provides a comparison of the response of the structure in various configurations to the Taft 0.2g motion. Also shown in that figure are analytical predictions obtained by using the damper model from Eq. (6.49). Excellent correlation is obtained. The energy response for that same motion is displayed in Fig. 6.34. Once again, although the input energy increases with the addition of fluid dampers, the damage to the primary structural members is significantly reduced as indicated by the decrease in hysteretic energy dissipation. Table 6.8 Dynamic Properties of Reinforced Concrete Test Structure with Pressurized Fluid Dampers (Pekcan et al., 1995) Fundamental
Damping
Frequency
Ratio
(Hz)
(%)
Without Dampers
1.42
8.9
Dampers on First Story
2.02
17.
Dampers on First and Second Stories
2.71
22.
Dampers on All Stories
2.76
23.
Configuration
Figure 6.33 First Story Displacements for Three-story Reinforced Concrete Test Structure Under Taft 0.2g Motion With Pressurized Fluid Dampers (Pekcan et al., 1995)
Table 6.9 Maximum Responses for Three-story Reinforced Concrete Test Structure with Pressurized Fluid Dampers (Pekcan et al., 1995) Number of Excitation El Centro 0.3g
Taft 0.2g
Pressurized
First Story Drift/
First Story Shear/
Fluid Dampers
Story Height
Total Weight
0
0.024
0.176
2 4
0.013 0.015
0.108 0.120
6
0.017
0.141
0
0.019
0.144
2
0.012
0.105
4
0.009
0.090
6
0.007
0.093
Figure 6.34 Energy Response of Three-story Reinforced Concrete Test Structure Under Taft 0.2g Motion (Pekcan et al., 1995)
6.6 Design Considerations The design of a structure that utilizes viscous fluid dampers for enhanced energy dissipation, of course, relies heavily on the ability to perform the appropriate structural analyses. A number of different approaches were discussed previously in Section 6.4. This analysis is considerably simplified for cases involving the linear orificed fluid damper, which can usually be modeled as a purely viscous damper. If the dampers are reasonably well distributed throughout the structure, design reduces primarily to the selection of the desired level of the critical damping ratio. However, since bracing is generally required, structural stiffness is also affected. As a result, design methodologies akin to those described in Section 5.6 are relevant when the response o the primary structural members and dampers remain in the linear range. Different methodologies are needed for design in the presence of nonlinearities. Several rational, simplified nonlinear design procedures are currently under development. For example, preliminary ideas are presented in Reinhorn et al. (1995), based upon the construction of a composite demand spectrum relating peak inertia forces (or accelerations) to peak displacements. With this approach, the structural capacity is estimated from a monotonic nonlinear pushover analysis. In detailed design, consideration must be given to the actual load transfer paths. For example, Reinhorn and Li (1995) discuss the additional column axial forces that develop due to the insertion of viscous damping walls. As illustrated in Fig. 6.35, bending moments generated by the damping walls are balanced by axial forces that accumulate for in-line arrangements. Staggered positioning of the walls can be beneficial in loads minimizing this with effect. Similar shouldConstantinou be given to and the column axial associated other types consideration of passive dampers. Symans (1993a) discuss the potential benefits of purely viscous devices, which produce additional column axial forces that are out-of-phase with the column bending moments. This provides improved margins of safety as determined from a column axial forcemoment interaction analysis. In general, however, existing structures that are retrofitted with passive dampers may also require strengthening of their columns. Reliability is, of course, of paramount importance. The mechanical simplicity o cylindrical pot dampers and viscous damping walls is attractive in this regard. However, the physical properties of the thick viscoelastic fluids, utilized in these devices, are often quite sensitive to temperature changes. In the more sophisticated orificed devices, the working fluids are typically less thermally-sensitive. Furthermore, some include passive controls to compensate for temperature changes. A critical issue in the orificed damper designs concerns the durability of the high strength seals, particularly in those devices that must maintain pressurization throughout their design life.
Figure 6.35 Load Transfer Paths With Viscous Damping Walls (Reinhorn and Li, 1995)
6.7 Structural Implementations As indicated in the introduction, fluid dampers have been successfully used for many years heavy industrial andAmilitary However, of a discussion of these is beyondin the present scope. numberapplications. of implementations fluid dampers have occurred in connection with various types of isolation systems. For example, Chiarugi et al. (1986) reported that orificed fluid dampers were first applied to a bridge in Italy in 1974. Subsequently, Grenier (1991) presented an overview of several additional Italian bridge applications of viscoelastic fluid dampers designed for compression-only performance. Tezcan and Civi (1979) discuss applications in which cylindrical pot fluid dampers are used in conjunction with helical steel springs for the seismic isolation of structures. The spring and damper system is placed at the foundation-superstructure interface to form a base isolation system as indicated in Fig. 6.36. The concept relies upon the ability o the springs to shift the natural frequency of the system away from the dominant frequency of the ground motion. The viscous dampers are then used to reduce vibration amplitudes. These dampers typically provide 20-30% of critical damping, and in some cases, the overall isolation system is capable of reducing forces in the structure by an order of magnitude compared to fixed base systems.
Figure 6.36 Base Isolation System With Helical Springs and Cylindrical Pot Fluid Dampers (Huffmann, 1985)
More recently, full-scale structural implementations of the Taylor Devices orificed fluid dampers have taken place, and by mid-1996 this technology was incorporated in more than a dozen civil engineering structures for the alleviation of wind and seismic vibrations. For example, following a study of wind loads acting on the light standards at Rich Stadium on the outskirts of Buffalo, New York, a total of twelve 50kN fluid damperswall, werethereby installed in 1993. The dampers connectand theenhancing light poles the stadium parapet significantly reducing vibrations thetofatigue life o the anchor bolts. Several additional projects involving these orificed fluid dampers are described in Taylor and Constantinou (1996), and briefly summarized in the following. In Sacramento, California, Taylor Fluid Dampers were employed for seismic mitigation o the new three-story Pacific Bell North Area Operations Center, shown under construction in Fig. 6.37 and completed in 1995. A total of 62 dampers each with a capacity of 130kN and a stroke of ±50mm were utilized within the steel braced frame. Figure 6.38 illustrates a typical installation of the fluid dampers. These damper-chevron brace assemblies are distributed throughout the structure, and sufficient supplemental damping is provided to maintain an elastic frame response even under a maximum level earthquake. This level of protection is necessary because the building serves as the central '911' facility for Northern California, and consequently must remain operational after seismic events.
Figure 6.37 Pacific Bell North Area Operations Center Under Construction (Courtesy of Taylor Devices, Inc., North Tonawanda, NY)
Figure 6.38 Damper Installation in the Pacific Bell North Area Operations Center (Courtesy of Taylor Devices, Inc., N. Tonawanda, NY)
A seismic retrofit of the Woodland Hotel, an historic masonry structure in Woodland, California, also employed Taylor Fluid Dampers. The four-story structure, srcinally constructed in 1927, is shown in Fig. 6.39. In this case, a total of sixteen 450kN dampers were utilized with chevron bracing for the project completed in 1995. Another recent project involved the installation of a total of 40 orificed fluid dampers in the 35-story building in downtown Boston, Massachusetts, displayed in Fig. 6.40. Each damper has a capacity of 670kN and a stroke of ±25mm. This 1996 retrofit is intended to improve the comfort of occupants during moderate to high wind conditions by significantly increasing the overall damping of the structure.
Figure 6.39 Woodland Hotel (Courtesy of Taylor Devices, Inc., North Tonawanda, NY)
Figure 6.40 28 State Street, Boston, MA (Courtesy of Taylor Devices, Inc., North Tonawanda, NY)
As mentioned previously, these fluid devices are also appropriate for application as components in base isolation systems. The newly-constructed San Bernardino County Medical Center in California uses 186 nonlinear Taylor Fluid Dampers, each with a capacity of 1460kN and a stroke of ±600mm, to enhance energy dissipation in the rubber bearing base isolation system. An overall sketch of the five building complex is provided in Fig. 6.41. Meanwhile, Fig. 6.42 shows a layout of the damper-base isolation system assembly, along with a more detailed schematic of the individual dampers.
Figure 6.41 Architectural Drawing of San Bernardino County Medical Center (Courtesy of Taylor Devices, Inc., North Tonawanda, NY)
Figure 6.42 Damper-Base Isolation System Assembly for San Bernardino County Medical Center (Courtesy of Taylor Devices, Inc., North Tonawanda, NY)
Viscous damping walls were recently used for aseismic protection in the newly constructed SUT-Building in Shizuka City, Japan, as described in Miyazaki and Mitsusaka (1992). A perspective view of the structure is shown in Fig. 6.43. The building consists of a 78m 14-story above-ground steel frame structure, along with a two-story reinforced concrete basement. The interior design includes a music hall, video theater, and banquet facilities. A total of 170 damping walls are employed within the steel frame to elevate the effective damping ratio of the building to approximately 27% at 20°C. A schematic illustrating the locations of the walls is provided in Fig. 6.44. Notice the staggered arrangement. Based upon time history dynamic analysis, the damping walls reduced response by 70-80%. Consequently, no damage is expected for level-2 earthquakes having a maximum input velocity of 50cm/s.
Figure 6.43 Perspective View of SUT-Building in Shizuka City, Japan (Courtesy of Sumitomo Construction Co., Ltd., Tokyo, Japan)
Figure 6.44 Viscous Damping Wall Locations in SUT-Building (Miyazaki and Mitsusaka, 1992)
References Arima, F., Miyazaki, M., Tanaka, H. and Yamazaki, Y. (1988), A Study on Buildings with Large Damping Using Viscous Damping Walls, Ninth World Conf. Earthquake Engrg., Tokyo, V, 821826. Bagley, R. L. and Torvik, P. J. (1983), Fractional Calculus - A Different Approach to the Analysis of Viscoelastically Damped Structures, AIAA Journal, 21(5), 742-748. Banerjee, P. K. (1994), The Boundary Element Methods in Engineering, McGraw-Hill, London, UK. Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge. Bird, R. B., Armstrong, R. C. and Hassager, O. (1987), Dynamics of Polymeric Liquids, John Wiley, New York. Chiarugi, A., Melegari, A. and Spinelli, P. (1986), On the Dynamic Behavior of Bridge Dampers Under Seismic Excitation, Proc. Eighth European Conf. on Earthquake Engrg., 5, Lisbon, 73-80. Constantinou, M. C., Symans, M.D., Tsopelas, P. and Taylor, D. P. (1993), Fluid Viscous Dampers in Applications of Seismic Energy Dissipation and Seismic Isolation, Proc. ATC 17-1 on Seismic Isolation, Energy Dissipation and Active Control, 2, 581-591. Constantinou, M. C. and Symans, M.D. (1993a), Seismic Response of Structures with Supplemental Damping, Struct. Design Tall Bldgs., 2, 77-92. Constantinou, M. C. and Symans, M.D. (1993b), Experimental Study of Seismic Response of Buildings with Supplemental Fluid Dampers, Struct. Design Tall Bldgs., 2, 93-132. Ferry, J. D. (1980), Viscoelastic Properties of Polymers, John Wiley, New York. Gemant, A. (1936), A Method of Analyzing Experimental Results Obtained from Elastoviscous Bodies, Physics, 7(8), 311-317. Grenier, A. (1991), Seismic Protection of Bridges by the Viscoelastic Technique, Proc . Third World Congress Joints and Bearings, Toronto, Canada, 1205-1223. Harris, C. M. and Crede, C. E. (1976), Shock and Vibration Handbook, 2nd Ed., McGraw-Hill, New York. Huffmann, G. K. (1985), Full Base Isolation for Earthquake Protection by Helical Springs and Viscodampers, Nuclear Engrg. Design, 84(2), 331-338. Landau, L. D. and Lifshitz, E. M. (1959), Fluid Mechanics, Pergamon Press, Oxford. Makris, N. (1992), Theoretical and Experimental Investigation of Viscous Dampers in Applications of Seismic and Vibration Isolation, Ph.D. Dissertation, State University of New York at Buffalo, NY. Makris, N. and Constantinou, M. C. (1990), Viscous Dampers: Testing, Modeling and Application in Vibration and Seismic Isolation, Technical Report NCEER-90-0028, National Center for Earthquake Engineering Research, Buffalo, NY. Makris, N. and Constantinou, M. C. (1991), Fractional Derivative Model for Viscous Dampers, J. Struct. Engrg., ASCE, 117, 2708-2724. Makris, N. and Constantinou, M. C. (1993), Models of Viscoelasticity with Complex Order Derivatives, J. Engrg. Mech., ASCE, 119(7), 1453-1464. Makris, N., Constantinou, M. C. and Dargush, F. (1993a), Analytical Model of Viscoelastic Fluid Dampers, J. Struct. Engrg ., ASCE , 119(11),G. 3310-3325. Makris, N., Dargush, G. F. and Constantinou, M. C. (1993b), Dynamic Analysis of Generalized Viscoelastic Fluids, J. Engrg. Mech., ASCE, 119(8), 1663-1679.
Makris, N., Dargush, G. F. and Constantinou, M. C. (1995), Dynamic Analysis o Viscoelastic Fluid Dampers, J. Engrg. Mech., ASCE, 121(10), 1114-1121. Miyazaki, M. and Mitsusaka, Y. (1992), Design of a Building with 20% or Greater Damping, Tenth World Conf. Earthquake Engrg., Madrid, 4143-4148. Niwa, N., Kobori, T., Takahashi, M., Hatada, T., Kurino, H. and Tagami, J. (1995), Passive Seismic Response Controlled High-Rise Building With High Damping Device, Earthquake Engrg. and Struct. Dynamics, 24, 655-671. Oldham, K. B. and Spanier, J. (1974), The Fractional Calculus, Academic Press, New York. Pekcan, G., Mander, J. B. and Chen, S. S. (1995), The Seismic Response of a 1:3 Scale Model R. C. Structure with Elastomeric Spring Dampers, Earthquake Spectra, 11(2), 249-267. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992), umerical Recipes in FORTRAN, Cambridge University Press, Cambridge, UK. Reinhorn, A.M. and Li, C. (1995), Experimental and Analytical Investigation o Seismic Retrofit of Structures with Supplemental Damping, Part III: Viscous Damping Walls, Technical Report NCEER-95-0013, National Center for Earthquake Engineering Research, Buffalo, NY. Reinhorn, A.M., Li, C. and Constantinou, M. C. (1995), Experimental and nalytical Investigation of Seismic Retrofit of Structures with Supplemental Damping, Part I: Fluid Viscous Damping Devices, Technical Report NCEER-95-
0001, National Center for Earthquake Engineering Research, Buffalo, NY. Schwahn, K. J. and Delinic, K. (1988), Verification of the Reduction of Structural Vibrations by Means of Viscous Dampers, Seismic Engineering, ASME, Pressure Vessel and Piping Conf., Pittsburgh, PA, 144, 87-95. Shames, I. H. and Dym, C. L. (1985), Energy and Finite Element Methods in Sturctural Mechanics, Hemisphere Publishing, New York. Taylor, D. P. and Constantinou, M. C. (1996), Fluid Dampers for Applications o Seismic Energy Dissipation and Seismic Isolation, Proceedings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico. Tezcan, S. and Civi, A. (1979), Reduction in Earthquake Response of Structures by Means of Vibration Isolators, Proc. Second U.S. National Conf. on Earthquake Engrg., Earthquake Engrg. Research Institute, Stanford Univ., California, 433-442. Tsopelas, P. and Constantinou, M. C. (1994), NCEER-Taisei Corporation Research Program on Sliding Seismic Isolation Systems for Bridges: Experimental and Analytical Study of a System Consisting of Sliding Bearings and Fluid Restoring Force/Damping Devices, Technical Report NCEER-94-0014, National
Center for Earthquake Engineering Research, Buffalo, NY.
7 Tuned Mass Dampers 7.1 Introduction The objective of incorporating a tuned mass damper into a structure is basically the same as those associated with metallic dampers and other energy dissipation devices discussed in the preceding chapters, namely, to reduce energy dissipation demand on the primary structural members under the action of external forces. This reduction, in this case, is accomplished by transferring some of the structural vibrational energy to the tuned mass damper (TMD) which, in its simplest form, consists of an auxiliary mass-spring-dashpot system anchored or attached to the main structure. The modern concept of tuned mass dampers for structural applications has its roots in dynamic vibration absorbers studied as early as 1909 by Frahm (Frahm, 1909; Den Hartog, 1956). A schematic representation of Frahm's absorber is shown in Fig. 7.1, which consists of a small mass m and a spring with spring stiffness k attached to the main mass M with spring stiffness K. Under a simple harmonic load, one can show that the main mass M can be kept completely stationary when the natural frequency (k/m) o the attached absorber is chosen to be (or tuned to) the excitation frequency.
Figure 7.1 Undamped Absorber and Main Mass Subject to Harmonic Excitation (Frahm's Absorber)
Den Hartog (Ormondroyd and Den Hartog, 1928) first studied the theory of undamped and damped dynamic vibration absorbers in the absence of damping in the main system, who developed their basic principles and the procedure for proper selection o absorber parameters. The damping in the main system was included in the analysis o dynamic vibration absorbers developed by Bishop and Welbourn (1952). Subsequent to the above work, an optimization procedure was devised by Falcon et al. (1967) to obtain minimum peak response and maximum effective damping in the main system. Jennige andboth Frohrib (1977) numerically evaluated to control bending and torsional modes in aa translatory-rotary building structure.absorber Ioi andsystem Ikeda (1978) developed empirical formulas for correction factors of these optimum absorber parameters in the case of light damping in the main system. Randall et al. (1981) presented design charts for these parameters when the effect of damping in the main system is taken into account. Warburton and Ayorinde (1980) further tabulated optimum values of the maximum dynamic amplification factor, tuning frequency ratio, and absorber damping ratio for specified values of the mass ratio and the main system damping ratio. In order to increase the effectiveness of an absorber to reduce maximum dynamic response of the main system, investigators have attempted to widen the tuning frequency range by introducing nonlinear absorber springs. Roberson (1952) considered the dynamic response of a main system supporting a dynamic absorber on a linear plus cubic spring (i.e., Duffing type spring) without damping. He defined a 'suppression band' as a frequency band between the resonance peaks over which the normalized system amplitudetoisbeless thanwider unity.than This the nonlinear absorber wasmain clearly demonstrated much thatband for aforlinear absorber. Pipes (1953) considered a hardening spring with a hyperbolic sine characteristic, concluding that the effect of nonlinearity in the spring is to prevent the existence o sharp resonant peaks and to introduce odd harmonic components of relatively small amplitude in the motion of the absorber and main system. To improve the performance of dynamic vibration absorbers, Snowdon (1960) investigated the behavior of a solid-type absorber in reducing the main system response. It was demonstrated that the dynamic absorber using a material with a stiffness proportional to frequency and a constant damping factor can reduce the resonant vibration of the main system considerably. Its performance is clearly superior to that of the spring-dashpot type absorber. Srinivasan (1969) analyzed parallel damped dynamic vibration absorbers, i.e., a second undamped tuned mass is added in parallel with an absorber. The main system would remain stationary in this case when the damper frequency is exactly tuned to the excitation frequency. However, the suppression band decreased in absorber this case.showing Snowdon alsoelement, studiednamely alternative absorbers such as aistriple-element that,(1974) if a third a
second spring, is added in series with the damper, a 15% to 30% reduction in the main system amplitude can be achieved. However, this reduction is quite sensitive to the tuning frequency, which will affect the absorber performance in practice.
7.1.1 TunedMass Dampers Much of the early development as described above has been limited to the use o dynamic absorbers in mechanical engineering systems in which one operating frequency is in resonance with a machine's fundamental frequency. Building structures, however, are subjected to environmental loads, such as wind and earthquakes, which possess many frequency components. The performance of a dynamic vibration absorber, frequently referred to as tuned mass damper (TMD), in complex multidegree-of-freedom and usually damped building structures is expected to be different. Much research and development work in the past two decades has thus been directed to the study of TMD effectiveness when operating in such a vibrational environment. An account of the theory and practice of TMDs in structural applications is given in the following sections.
7.2 Basic Principles Consider the response of a single-degree-of-freedom (SDOF) structural system subjected to a vibratory force f(t) as shown in Fig. 7.2a. The response of the structural system can be reduced under some circumstances by adding a secondary mass, or a TMD, which has motion relative to the system as shown in Fig. 7.2b. The equations o motion for the structure/TMD system are, using the notation given in Fig. 7.2b,
where y1(t) is the displacement of the structural system relative to its base and z(t) is the relative displacement of the added mass with respect to the structure. The damping coefficients and stiffnesses are denoted by c and k for the added mass and C and K for the structural system, respectively. The external force on the structure is represented by (t), while g(t) equals zero for wind excitation and equals µf(t) for earthquake loading, µ = m/M being the mass ratio. Summation of Eqs. (7.1) and (7.2) leads to
Figure 7.2 Models of SDOF Structure and TMD
It is seen that the net effect of the added small mass ( m) on the structure, aside from a slight decrease in natural frequency and a slight increase in external force from f(t) to (t) + g(t), is the addition of a 'force term' . When f(t) is considered as a harmonic force or a stationary random input, Eq. (7.3) can be rewritten in the form o energy or power balance as
in which < > is the mathematical expectation for the case of stochastic input and the time average in one cycle for the case of harmonic excitation. When the steady-state response y1(t) is concerned, . Equation (7.4) is thus changed into the simple power balance equation:
in which C< > is the dissipated power due to structural damping and < > is the input power from external force which is always positive. As indicated in Section 7.1, the power flow (m < >) from the structural system to the secondary mass plays a basic role in the use of TMDs in structures and thus is an appropriate measure of TMD effectiveness, i.e., the larger the power flow, the smaller the mean-square velocity response of the structural system. Maximum power flow is obtained when the relative displacement of the secondary mass lags that of the structural system by a phase angle of 90°. In this case, the relative acceleration of the secondary mass is in phase with the velocity response in the structural system and the power flow is equivalent to an effective dissipative power which increases the total effective damping (Ceq) in the structural system as shown in Fig. 7.2c, where
It is worth noting here that input power from the external force varies with TMD parameters. However, the variation is small compared to the power flow and, in fact, the magnitude of the input power may decrease to produce a change which is favorable to TMD efficiency. 7.2.1 Den Hartog's Solution An appreciation of TMD efficiency in reducing structural response can be gained by following the basic development of Den Hartog for the simple case when the structural system is undamped (C = 0) and is subject to a sinusoidal excitation with frequency w (f(t) = Posinwt and g(t) = 0). In this procedure, the dynamic effect of a TMD is measured in comparison with the static deflection produced by the maximum force applied statically to the structure. For a sinusoidal excitation with frequency w, the static deflection is yst = Po/K, while the dynamic amplification factor for an undamped structural system, R, is
where = w/ws = forced frequency ratio = wa/ws = frequency ratio (natural frequencies)
= k/m = squared natural frequency of TMD = K/M = squared natural frequency of structural system a = c/cc = c/2mwa = damping ratio of TMD
The amplification factor is a function of the four essential variables: µ, a, , and ß. Figure 7.3 shows a plot of R as a function of the frequency ratio ß for = 1 (tuned case), µ = 0.05, and for various values of TMD damping ratio a. Observe what happens with increased damping in TMD. Without structural damping, the response amplitude is infinite at two resonant frequencies of the combined structure/TMD system. When the TMD damping becomes infinite, the two masses are virtually fused to each other and the result is a SDOF system with mass 1.05M so that the amplitude at resonant frequency becomes infinite again. Therefore, somewhere between these extremes there must be a value of a for which the peak becomes a minimum.
Figure 7.3 Amplification Factor as Function of ß (µ = 0.05, = 1)
An objective in adding the TMD is to bring the resonant peak of the amplitude down to its lowest possible value so that smaller amplifications over a wider frequency bandwidth, with ß close to unity, can be achieved. There are two points ( P and Q) on the R vs. ß plots in Fig. 7.3 at which R is independent of damping ratio a and the minimum peak amplitude can be obtained by first properly choosing to adjust these two fixed points to reach equal heights. The optimum frequency ratio following this procedure is determined as
which gives the amplitude at P or Q:
A good estimate for opt can be determined as the average of two values
which make the fixed points P and Q maxima on an R vs. ß plot, respectively (Brock, 1946), giving
From Eq. (7.9) one can easily observe that an increase in damper mass always reduces the maximum amplification factor for the optimally-designed TMD. This maximum amplification and optimum absorber parameters are summarized in Table 7.1 for variety of excitations and response quantities of interest. In Cases 5 and 6 in Table 7.1,a white noise excitation of spectral density So is assumed. Minimizing mean-square displacement < > with respect to and a by setting < > / = 0 and < > /a = 0 led to the results presented in Table 7.1 for these two cases.
7.2.2 The Case of Damped Structures Equation (7.7) for R can also be established when damping is present in the structural system; however, invariant points P and Q as in the undamped structure case no longer exist. Hence, one must resort to numerical means in order to determine the optimum values of a and a. An approach taken by Randall et al. (1981) is to optimize a and a by minimizing the higher of the two peaks in the amplification curve (Fig. 7.3) using numerical search procedures. The optimum values of and a for small values of s can also be formulated empirically (Ioi and Ikeda, 1978). The results are
where s, is the damping ratio of the primary mass. Tolerance or permissible error ranges attached to the above equations are found to be less than 1% for 0.03 < µ < 0.40 and 0.0 < s < 0.15, which are the ranges of practical interest. A frequency locus method proposed by Thompson (1981) also describes optimization of and a for a damped primary structure. In this procedure, is optimized by numerical methods and a can be analytically determined once a is known. A detailed analysis was carried out by Warburton (1982) to determine optimum damper parameters for both harmonic and random excitations, with random excitation being applied as a force to the structure (as in the case of wind) or as base acceleration (as in the case of earthquake). Low damping of the main structure was again assumed in this analysis.
For harmonic excitations, the optimization criterion was to minimize response amplitude R, given by
where j is case number referring to a particular case as given in Table 7.2 on the previous page, Aj and Bj are given in the same table, and
The optimum values of a and (a can again be found by setting Rj/ and Rj/, equal to zero and solving the resulting equations. However, these equations are generally nonlinear in and a, and require numerical solutions. As an example, optimum absorber parameters for a lightly damped SDOF structure are given in Table 7.3 for several values of µ and s for the case in which random white noise acceleration with spectral density So is applied to Table 7.3 Optimum Absorber Parameters Attached to Damped SDOF Structure (Warburton, 1982) µ
0.01
0.03
0.1
0.2
Ropt
opt
opt
10.138
0.9876
0.04981
0.01
7.743
0.9850
0.04981
0.02
6.205
0.9819
0.04981
0.05 0.1
3.798 2.249
0.9704 0.9436
0.04982 0.04982
0
6.058
0.9636
0.08566
0.01
5.110
0.9592
0.08566
0.02
4.424
0.9545
0.08566
0.05
3.109
0.9380
0.08567
0.1
2.036
0.9032
0.08569
0
3.602
0.8861
0.1527
0.01
3.285
0.8789
0.1527
0.02
3.014
0.8714
0.1528
0.05
2.399
0.8468
0.1529
0.1
1.765
0.7991
0.1531
0
2.865
0.7906
0.2097
0.01
2.680
0.7815
0.2098
0.02
2.516
0.7721
0.2099
0.05
2.113
0.7421
0.2103
0.1
1.649
0.6862
0.2112
s
0
the base and the response amplitude to be optimized is . It is noted that, for each value of µ, the optimized response is reduced ass increases; however, opt and , opt are relatively insensitive to s over the range of s -values tabulated. Warburton (1982) also provided SDOF analogies to multi-degree-of-freedom (MDOF) systems so that the results given in Table 7.1 can be used in the case of MDOF systems by replacing a MDOF system by an equivalent SDOF system using these analogies.
7.2.3 Other OptimizationCriteria Den Hartog's optimization technique as explained in Section 7.2.1 was developed by minimizing displacement of the structure. Displacement essentially determines safety and integrity of a structure under external excitations. Since large accelerations of a structure under excitations produce detrimental effects on functionality of nonstructural components, base shear, and occupant comfort, minimizing structural acceleration can also be a viable optimization criterion. Similarly, many other criteria are possible and have been considered by different researchers. Some of these are listed below. (a) Minimum displacement of main structure (Den Hartog, 1956; Thompson, 1981; Jacquot and Hoppe, 1973; Fujino and Abe, 1993). (b) Maximum dynamic stiffness of main structure (Falcon et al., 1967). (c) Maximum effective damping of combined structure/TMD system (Luft, 1979). (d) A mixed criterion involving frequency tuning using minimum displacement criterion and TMD damping determination using maximum effective damping criterion (Luft, 1979). (e) Minimum travel of damper mass relative to main structure (Luft, 1979). (f) Minimum velocity of main structure (Warburton, 1982). (g) Minimum acceleration of main structure (Ioi and Ikeda, 1978; Warburton, 1982). (h) Minimum force in main structural frame (Warburton, 1982). It is clear that some of the above criteria overlap with each other and have similarities. Several of them are developed below in more detail. Consider first the case of minimizing acceleration response of the structure. Due to the fact that high frequency components are involved more in acceleration than in displacement, the optimum stiffness and damping of the TMD are expected to be different for these two criteria. When a sinusoidal force f(t) is exerted on the structural system as shown in Fig. 7.2b, the acceleration (ÿ1) of the structure is proportional to the corresponding displacement (y1). The proportionality constant is the square o excitation
frequency, i.e., ÿl = w2yl. A dimensionless acceleration amplification can then be formulated as
We observe from Eq. (7.16) that acceleration amplification is equivalent to displacement amplification with magnitude proportional to the square of the excitation frequency. Empirical formulas for optimum stiffness and damping of a TMD to minimize acceleration response have been presented by Ioi and Ikeda (1978) as
These empirical equations have errors of less than 1% in the same range of µ and s, as Eqs. (7.11) and (7.12). For building design where both displacement and acceleration are restricted to certain levels, Eqs. (7.11) and (7.17) and Eqs.(7.12) and (7.18) are at odds. In order to evaluate their inconsistencies, the values for and are compared in Fig. 7.4 and and are compared in Fig. 7.5 when p varies for a specific structural damping ratio s. It can be observed that the optimum damping ratios for displacement and acceleration reduction are consistent while the frequency ratios show a large discrepancy. Brock (1946), in his note on Den Hartog's srcinal work, calculated optimum damping for the case when the damper mass is exactly tuned (which is not optimal) to the excitation frequency. For this case, the optimum TMD damping is given by
Also, if the damper has only damping and mass and no stiffness (Lanchester type damper), then optimal damping is given as
Another optimization criterion can be the maximization of effective damping of the combined structure/TMD system. A closed form expression for this effective damping of an undamped structure is given by (Luft, 1979)
Maximizing eff with respect to and a by taking partial derivatives, setting them to zero, and neglecting higher order terms of µ, we obtain
and
By mixing optimization criteria we can arrive at even simpler expressions. Suppose one optimizes a under the minimum displacement criterion (Eq. 7.8) and a under the minimum effective damping criterion, the optimum a is given by (Luft, 1979)
Another optimization criterion can be travel of the damper mass relative to the structure. root-mean-square of the damper mass is proportional to , oneSince may select a value for displacement a larger than what is optimum from other points o view, i.e.,
The corresponding effective damping ratio for the structure/TMD system is (1 2/2)eff. Thus, increased TMD damping only has a minor effect on eff and damper displacement can easily be reduced without reducing eff significantly. Optimization of µ has also been considered (Lee, 1990). However, µopt takes values much larger than those considered practical and hence does not appear to be useful in practice.
Comparison of
Figure 7.4 and (Ioi and Ikeda, 1978)
Figure 7.5 Comparison of
and
(Ioi and Ikeda, 1978)
7.2.4 Wind vs. E arthquake Loads Under earthquake-type excitations, the effective forces acting on the structure and TMD are proportional to their masses. In such a case, the external force on the damper mass, g(t), is equal to µf(t). The performance of TMD can still be improved by increasing the mass ratio as presented in Fig. 7.6. However, this extra force always reduces the TMD efficiency for any mass ratio because g(t) is in phase with f(t) in this case as shown in Fig. 7.7. The optimum frequency and damping ratio for the case without structural damping, given in Eqs. (7.8) and (7.10), are now shifted from opt and opt to new values as expressed in Table 7.1 (Case 3). It is to be noted that wind and earthquake loads are stochastic in nature. While these loads are nonstationary, an analysis of a structure/TMD system under white noise excitation will shed more light on the effectiveness of TMD with regard to its application in wind and earthquake engineering. When f(t) is a white noise, the steady-state displacement y1(t) is stationary and its variance can be obtained by integrating its power spectral density over an infinite frequency range (Soong and Grigoriu, 1993). The optimum damper parameters of an undamped structure for minimizing the displacement variance are given in Table 7.1 (Cases 5 and 6) for both wind and earthquake type excitations. As has been pointed out earlier, for a damped structure, it is not easy to obtain a closed-form solution for the optimum damper parameters, but it can be studied numerically (Wiesner, 1979; Wirsching and Campbell, 1974; Warburton, 1982).
Figure 7.6 Amplification Factor as Function of µ and ß
Figure 7.7 Amplification Factor as Function of g(t) and ß
As another illustration, the optimum TMD frequency and damping ratio for a damped structure are presented as functions of the mass ratio in Figs. 7.8 and 7.9, respectively. Included here are the results from the Den Hartog optimum tuning based on the sinusoidal solution of an undamped structure along with the Wiesner tuning and Wirsching and Campbell tuning for wind and earthquake white noise solutions o damped structures, respectively. Since these curves represent optimum tuning under different conditions, one can expect some differences among these results. However, for small mass ratios, these differences are tolerable in practical design (Sladek and Klingner, 1983).
Figure 7.8
aopt as Function of µ (Sladek
and Klingner, 1983)
Figure 7.9 opt as Function of µ (Sladek
and Klingner, 1983)
Under random excitations, TMD effectiveness is often described by an effective damping ratio which is defined as the damping ratio of a SDOF system in Fig. 7.2c whose mean-square displacement is equal to that of the structure/TMD system as shown in Fig. 7.2b. This definition is equivalent to that given in Eq. (7.6) when the mass ratio is small. Figure 7.10 shows the effects of variation in p and the finite-band ( B) of a random load on the optimized maximum effective damping of the structure/TMD system (Tanaka and Mak, 1983). The two extreme cases ofB and B = 0.05 represent white noise and an almost harmonic excitation, respectively. In all cases, the predominant excitation frequency was chosen to coincide with the natural frequency of the structural system. As shown in the figure, for a given µ, the narrower the bandwidth B, the higher the observed maximum effective damping. Furthermore, the maximum effective damping increases with mass ratio, and is more sensitive to this change when the excitation bandwidth is narrower. Figure 7.11 also demonstrates that the maximum effective damping occurs when the TMD is tuned close to the structural frequency, but not necessarily at = 1.0. Hence, it is important to tune the damper to the correct frequency in order to achieve its maximum effectiveness. The effect of a shift in excitation frequency on the maximum effective damping is significant only when the bandwidth is narrow as is seen in Fig. 7.12. As bandwidth B becomes narrower, the optimum TMD damping tends to occur at a lower mass ratio as shown in Fig. 7.11. The results when
B = 0.05 seem to be contradictory to those for optimum damping in the case o
harmonic excitation as shown in Fig. 7.9. In fact, these results are obtained based on two different optimization criteria; the former to minimize the peak values on the resonance curve and the latter to minimize the resonance value at a tuning frequency only. As a result, the optimum damping consistently increases with µ for harmonic excitations, but not for the case illustrated in Fig. 7.11.
7.10 eq as Function of µ Figure and B (Tanaka and Mak, 1983)
Figure 7.11 opt as Function of µ and B (Tanaka and Mak, 1983)
Figure 7.12 eq as Function of Forced Frequency Ratio and B ( = 0.98) (Tanaka and Mak, 1983)
7.3 Structural Analysis Once the optimal parameter values for the TMD are found, Eqs. (7.1) and (7.2) can be used for structural response analysis. These equations are valid, of course, only for single-degree-of-freedom structural systems. Since most building structures are multidegree-of-freedom systems, a more general form of the equations of motion for a structure/TMD system under, for example, earthquake loading as shown in Fig. 7.13 has the vector-matrix form
where N is the number of degrees of freedom of the primary structure, is the displacement of the ith mass relative to the ground, and z(t) is the displacement of the TMD with respect to the top floor. Additionally, e is a vector of l's and is the ground acceleration. The structural analysis thus needs to be carried out in the ( N + 1)dimensional space under general conditions. Let us note that a TMD can only be tuned to a single structural frequency. It is thus expected that its effectiveness is the greatest when an N-degree-of-freedom structure oscillates around a predominant mode. Consider the case where the response vector y(t) can be approximately represented by a single
Figure 7.13 Model of Multi-degree-offreedom Structure and TMD
coordinate yN (the displacement at the floor where the damper is attached) and a mode shape 1, i.e., y = 1yN (7.28) Substituting Eq. (7.28) into Eq. (7.26) and premultiplying Eq. (7.26) by equation of motion (7.26) becomes
, the
where is the modal mass; and , 1 and wl being the damping ratio and natural frequency of the first mode of the structure; and . The participation factor is expressed as
By comparing Eqs. (7.27) and (7.29) with Eqs. (7.2) and (7.1), one can find that first modal representation of a MDOF structure is exactly the same as a SDOF structure, except that the modal mass, stiffness and damping are employed instead of physical parameters in the SDOF case. The mass ratio (µ) in this case is defined as .
Under wind-type loading, g(t) = 0 and TMD effectiveness when attached to one floor o an MDOF structure is exactly the same as in the SDOF system discussed before. For earthquake-type excitations, however, the relation between g(t) and f(t) is
which is different from that in the SDOF case. However, this has a minor influence on the behavior of a TMD. When the TMD is situated at the top floor of a building structure, every element of the modal vector 1 is less than or equal to unity. The participation factor 1 is thus always greater than unity which improves the TMD effectiveness in the sense that the effect of g(t) on the damper is decreased. On the other hand, if the damper is installed at a lower floor, the mass ratio µ is reduced at a greater rate than is the participation factor, and therefore TMD effectiveness is expected to diminish.
7.3.1 E lastic Structures When the first mode of a MDOF structure dominates the structural response, a response reduction of the MDOF structure using TMD under earthquake-type loading can be easily achieved. Wirsching and Campbell (1974) investigated the minimum first-mode response as well as optimum absorber parameters for one-story, five-story and ten-story structures. Results show that dimensionless response ratio yN/o, the ratio of rootmean-square (RMS) top-floor displacement with and without TMD, increases little as the number of stories increases. The optimum TMD stiffness becomes less sensitive to structural damping and mass ratio when the number of stories increases and the optimum TMD damping is insensitive to structural damping even for one-story buildings. In their paper, mass ratio was defined as the ratio of the added mass and story mass of a uniform building. If this mass ratio is defined as the modal mass ratio, the above conclusions become even more obvious, i.e., all the previously discussed parameters are practically independent of the number of stories. This confirms the effectiveness of TMD in reducing the first-mode response of multi-story structures and does not preclude the use of TMD design when the structural damping cannot be identified accurately. The optimum RMS response ( yN/o) of the structure under random excitation at the base and the corresponding relative displacement RMS (z/o) of TMD for various structural damping ratios are shown in Fig. 7.14 from which one can observe that yN/o decreases at a slow rate but z/o decreases substantially as p increases within the practical range. Additionally, yN/o and z/o both increase as structural damping increases.
Figure 7.14 yN/o and z/o as Functions of µ
While the first-mode response of a structure with TMD is proved to be substantially reduced, the higher mode response may in fact increase as the number of stories increases. For earthquake-type excitations, it has been demonstrated that, for shear structures up to 12 floors, the first mode response contributes more than 80% to the total motion (Wirsching and Yao, 1970). However, for a taller building on a firm ground, higher modal response may be a problem which needs further study. An attempt has been made by Chowdhury et al. (1987) to evaluate the higher-mode effect on TMD efficiency. A 25-story shear-flexure uniform building was considered for a sensitivity study of peak response ratio at the top-floor to the modal frequency ratio with and without damper. It was demonstrated that the higher-mode response ratio may be greater than unity while the first-mode response is always less than unity around the tuning frequency. Unfortunately, since optimal damper parameters were not used, it is difficult to draw conclusions from these results regarding the applicability of TMDs to control seismic response of structures. A numerical investigation of a tall building with a TMD was conducted by Kawaguchi et al. (1992). Time histories of a generalized wind force were generated from a selected wind spectra, and a 12-story building was represented by its first three modes at frequencies 0.71, 1.82, and 2.93 Hz. The TMD was optimized via Eqs. (7.8) and (7.10) to suppress the first-mode response by assuming an undamped structure. As expected, numerical results indicated that the TMD effectively reduced response near the tuning
frequency, but had virtually no effect on vibrations of longer periods or on higher mode vibrations. Xu and Kwok (1992) used a transfer matrix formulation to examine the frequency domain response of soil-structure - TMD linear systems subjected to wind loading. Both alongwind turbulence and crosswind wake excitation were modeled as stochastic processes. Two high rise structures were investigated. Results show that, for moderate to stiff soils, the TMD should be tuned to the natural frequency of the soil-structure system rather than to that of theineffective. structure alone. forTMD soft soils, properly tuned damper proved largely In allHowever, cases, the had even little aeffect on mitigating higher mode response.
7.3.2 I nelastic Structures For earthquake resistant design, a structure is permitted to experience inelastic deformation under strong earthquakes. Attempts to reduce structural response in this case using TMD were made by a few investigators. When the structure experiences elastoplastic deformations, the 'frequency' of the system decreases so that the TMD may lose part of its effectiveness due to detuning effect. In Kaynia et al. (1981), two quantities were introduced to measure the effectiveness o TMD in the case of inelastic structures, i.e., the ratio between cumulative yielding ductility in a 2-DOF and in a SDOF system and the ratio between the associated ductility ratios. The latter is actually the same as the peak response ratio. The parameters of the SDOF elastic-perfectly plastic system consist of mass M, frequency of small-amplitude response ws, damping ratio s, and yielding displacement c; whereas the mass damper is identified as before by parameters µ, , and a. For several combinations of ws and yc, the sensitivity of cumulative yielding ductility ratio and ductility ratio to dimensionless damper parameters was investigated by using a set of historical earthquakes. The preliminary study showed that these two ratios have a weak dependence on the aforementioned parameters similar to that of peak response ratio in an elastic structure. 7.4 Experimental Studies
7.4.1 Small-scale Tests Several small-scale tests have been carried out in wind tunnels in order to assess the effectiveness of TMDs applied to tall buildings under wind loadings. In Isyumov et al (1975), a 2-DOF aeroelastic model was studied in a wind
tunned to investigate the structural behavior of a proposed tall building in a turbulent wind environment. Each DOF represented the fundamental sway mode of vibration in either the X- or Y-direction for the building under investigation. The TMD installed on the building model consisted of a mass of 1% of the averaged generalized modal mass in each of the sway modes of vibration. The initial damping ratio in the damper was 56%, which is near the optimum damping for this mass ratio. However, actual construction of the TMD model deviated from optimal tuning. Figure 7.15 indicates the model and TMD responses to an initial displacement, showing the TMD effectiveness on structural response (McNamara, 1977). The average hourly peak deflection and acceleration at the top of the tower were predicted and presented in Figs. 7.16 and 7.17 for different return periods. As can be seen, an optimum TMD with a 1% mass ratio can reduce responses more than the TMD as tested. The effective damping in this case is approximately 2% of critical, which is consistent with the value shown in Fig. 7.15. Also shown in Figs. 7.16 and 7.17 are the optimum TMD design results with mass ratio equals to 2% and 4%, which demonstrate further response reduction of response with increase in the mass ratio. Tanaka and Mak (1983) demonstrated damper effectiveness using a 1:1000 scaled aeroelastic model consisting of a rigid box of light material with a strain gauged lea spring at the base. The model was mounted on a turntable so that wind direction could be changed relative to the model as shown in Fig. 7.18a.
Figure 7.15 Tower and TMD Displacements (Isyumov et al., 1975)
Figure 7.16 Average Hourly Peak Deflections for Different TMD Configurations (McNamara, 1977)
Figure 7.17 Average Hourly Peak Accelerations for Different TMD Configurations (McNamara, 1977)
The leaf spring was designed to be adjustable to measure the model response in X and directions separately. As a TMD model, a simple pendulum consisting of an aluminum block and a thin wire was secured to the top cover of the building model. The proper stiffness and damping was provided to the TMD by adhering a foam tape to the wire. The schematic drawing of the prototype CAARC building is shown in Fig. 7.18b. The model was tested once without, then with, the TMD system. The effectiveness of the damper in suppressing dynamic response of the building is clearly demonstrated in Figs. 7.19 and 7.20. The reduction in magnitude was significant, generally in the range of 30% to 60%. The reduced velocityVr is defined as VH/wD (wX = wY = 0.2 Hz) in the and Y directions, respectively, in which VH is the horizontal wind speed at a particular elevation.
Figure 7.18 Model and Schematic of CAARC Building (Tanaka and Mak, 1983)
Figure 7.19 Typical Responses With and Without TMD (Tanaka and Mak, 1983)
Figure 7.20 RMS Responses With and Without TMD (Tanaka and Mak, 1983)
More recently, Xu et al. (1992) studied torsional response of 1:400 scaled models, with and without a TMD, in a series of wind tunnel tests. A wooden rectangular prism model represented a building of 193m in height, with plan dimensions of 76m by 36m. The TMD was mounted in a transparent plastic enclosure on the top of the model. When properly tuned, the TMD was found to be effective in reducing torsional vibration caused by eccentricities between the centers of mass, elasticity, and instantaneous aerodynamic pressure. However, the importance of properly characterizing wind excitation mechanisms, building size and shape, and surrounding terrain was emphasized. 7.4.2 Full -scale Tes Under a temporary control arrangement, the TMD system installed in the Citicorp Center, New York, was employed to initiate the building motion and then respond to it upon the cut-off of control power. The building frequency and damping in each direction were identified at varying amplitudes of building motion up to 0.01g. The TMD performance was then assessed by damping out the self-excited building motion in each direction. A portion of the building acceleration and TMD mass relative displacement time histories were recorded during the latter test and shown in Fig. 7.21 (Wiesner, 1979). It is evident that the relative displacement lags the building acceleration by a phase angle of 90°. The effective building damping described in Eq. (7.6) was increased to 4% of critical as compared to 1% measured without TMD action.
Figure 7.21 Building Acceleration and TMD Displacement Time Histories (Wiesner, 1979)
The performance of TMDs in many other buildings as well as in the Citicorp Center are being monitored to provide a wealth of TMD performance data. Some of these findings are described in Section 7.6 under structural implementations.
7.5 Design Considerations And Implementational Issues Based upon the development presented above, a design procedure for an optimum TMD for a building structure oscillating at a predominant, say, the first fundamental frequency can be developed. As the first step, the first modal representation of the structure (ws, s) and the maximum acceptable response level y(max or ÿmax) need to be specified as described in Fig. 7.22. Based upon the design response spectrum prescribed in existing codes and provisions, effective damping of the representative SDOF system can be estimated, i.e., the effective damping ratio which satisfies
whichever is larger, can be used for the determination of optimum TMD parameters. Here, Sy(ws,eq) and Sÿ(ws,eq) are, respectively, the response spectra of displacement and acceleration at natural frequency w,and damping ratio eq. The required mass ratio p can be then estimated by (Luft, 1979)
The optimum tuning frequency ratio and damping of the TMD can be estimated from Figs. 7.8 and 7.9. The final step in designing the TMD is to check that the selected TMD parameters give the building response which is in the range of predetermined response threshold. Otherwise, the preliminary design has to be refined using an iterative procedure.
Figure 7.22 Design Flow Chart
A number of practical restraints, however, must be observed in the engineering design of a TMD system. First and foremost is the amount of added mass which can be practically placed on the top story of a building. Once the mass is specified, the TMD effectiveness is determined and the percentage of response reduction is set. TMD travel relative to the building is another important design parameter, which can be appreciably reduced by the selection of proper TMD damping or by choosing a proper damper configuration. However, large TMD movements often need to be accommodated for a reasonable response reduction of the building. Another major engineering problem associated with a sliding mass arrangement is to provide a very low friction bearing surface so that the mass can respond to the building movement at low levels o excitation. This becomes more critical when TMD functions as an additional damper to improve occupant comfort. Finally, cost is an issue which must be addressed in the evaluation of a TMD for a specific application. Involved in cost estimation are: Design and system engineering Mass damper supporting system and accessories, e.g., pressure balanced support bearings for a sliding mass damper Mass damper restraint system usually including anti-yaw torque box or equivalent hardware, over-travel snubber system with reaction guides and forces Building preparation and strengthening to support a large amount of actuator and spring force Enclosure of space occupied by the entire TMD system Roof elevator access and hoisting capability during installation Maintenance and operational cost 7.6 Structural Implementations A number of TMDs have been installed in tall buildings, bridges, towers, and smoke stacks for response control against primarily wind-induced external loads. In terms o TMD configurations, there is also a large variety as indicated in Fig. 7.23. The sliding mass block has been employed, for example, in the Citicorp Center discussed in Section 7.4.2. For a simple pendulum type TMD as shown in Fig. 7.23a, the vibration period only depends on the pendulum arm length which often requires large space to match the resonant frequency. In order to be flexible in selecting the pendulum length, one may modify this basic pendulum in such a way that the pendulum period depends on both the pendulum length and mass. For example, the pendulum arm can be designed as a rigid bar with counterweights (Grossman, 1990) or a two-mass damper can be used, with one mass on theasbuilding pendulum (Kumagai Gumi Co.,sliding Ltd., 1991) shown floor and the other acting as a
in Figs. 7.23b, c. Their natural frequencies can be flexibly adjusted to tune with the structural frequency in these cases. However, these arrangements are much more difficult to implement than a simple pendulum, especially when their masses increase. In addition, the pendulum period is elongated by the participation of the counterweight, which diminishes effectiveness of the TMD system. The practical application of such arrangements could be limited for the above reasons. Alternative solutions for space-limited applications include the tuned roller pendulum damper proposedHeavy by Okumura Co.Ltd. (1991) andasthe multistage proposed(TRD) by Mitsubishi Industries, (1991), shown in Fig. pendulum 7.23d, e, respectively. The height of the TRD equipment is smaller or comparable with that of a simple pendulum TMD even when the predominant natural frequency of a building is lower than 0.2-0.25 Hz. The multi-stage pendulum can even produce a lower natural frequency within a practical size. A two-stage pendulum can reduce the vertical space requirement to 1/2 and keep the same horizontal occupied space. In what follows, a few of these structural applications are described in some detail.
Figure 7.23 Various Forms of TMD
7.6.1 CenterpointTower, Sydney, Australi It appears that the first structure in which a TMD was installed is the Centerpoint Tower in Sydney, Australia (ENR, 1971; Kwok and MacDonald, 1987). The structure consists of a 150-ft-high office building topped with a 700-ft-high tower as shown in Fig. 7.24a. Serving as the building water and fire protection supply, the tower's water tank in conjunction with hydraulic shock absorbers was incorporated into the design o the TMD to reduce wind-induced motions. The tank, 7-ft deep and 7-ft wide with a capacity 7.24b. of 35,000 gal hangs from the top radial members of the turret as shown in Fig. A 40-ton secondary mass was later installed on the intermediate anchorage ring to further increase damping in the second mode, which resulted in increases in the damping level from 1.0% to 1.2% and from 0.4% to 1.5% in the first and second mode, respectively. Results of acceleration measurements showed that the wind-induced acceleration response was reduced by 40% to 50% (Kwok, 1984; Kwok and MacDonald, 1990). Other examples of structures having pendulum-type TMDs include the 102m steel antenna mast atop the 553m CN Tower in Toronto, Canada, which has two doughnut-shaped TMDs to reduce the second and fourth modes of vibration, and the 157m Crystal Tower in Osaka, Japan, which also makes use of water storage tanks at the top of the structure as pendulum TMDs.
Figure 7.24 TMD in Centerpoint Tower (Kwok and MacDonald, 1987)
7.6.2 Citicorp Center, New York and J ohn H ancock Tower, Boston One of only two buildings in the U.S. equipped with a TMD, the 960-ft Citicorp Center (Fig. 7.25) has a distributed mass of about 60,000 tons, a first bending mode lateral natural frequency of about 0.16 Hz, and a damping ratio of 1% in its resonant response to wind gust force. The Citicorp TMD, as shown in Fig. 7.26, is installed on the 63rd floor of the building. At this elevation, the building can be represented by a simple modal mass o approximately 20,000 tons, to which the TMD is attached to form a 2-DOF system as represented by Fig. 7.2b. The TMD system specifications are given in Table 7.4. Tests and actual observations have shown that the TMD produces an approximate effective damping of 4% compared to the 1% srcinal structural damping, which can reduce the building acceleration level by about 50%.
Figure 7.25 Citicorp Center, New York
Figure 7.26 TMD in Citicorp Center (Petersen, 1980; 1981)
Table 7.4 Tuned Mass Dampers in John Hancock Tower, Boston and Citicorp Center, New York
Typical floor size Floor area Building height Building modal weight Building period 1st mode
John Hancock
Citicorp Center
Boston, MA
New York, NY
(ft)
343 × 105
160 × 160
(sq ft)
36,015
25,600
(ft)
800
920
(tons)
47,000
20,000 6.25
(sec)
7.00
Design wind storm
(years)
100
30
Mass block weight
(tons)
2×300
373
Mass block size
(ft)
18×18×3
30×30×8
Mass block material
(type)
lead/steel
concrete
TMD/AMD stroke
(ft)
±6.75*
44.50*
Max spring force
(kips)
135
170
Max actuator force
(kips)
50
50
Max hydraulic supply
(gms)
145
190
Max operating pressure
(psi)
900
900
(g)
.002
.003
Operating trigger - acceleration Max power
(HP)
120
160
Equivalent damping
(%)
4.0%
4.0%
* Including overtravel
Involved in the design of the Citicorp TMD system are the linear gas spring, pressure balance supporting system, control actuator, power supply and electronic control (Petersen, 1980; 1981). The passive spring was provided by a pair of opposed trunnion mounted and pneumatically precharged cylinders with connected piston rods. The 373ton concrete block was supported on twelve pressure balanced bearings, each with a 22in diameter. The individual bearings were hydraulically coupled together into three groups, each group containing its own height control value. Fluid can flow between individual bearings to accommodate floor surface irregularities. The overall friction coefficient achieved due to the pressure balanced bearing installation was about 0.003, requiring about 2400 lbs drive force to compensate for friction losses. The biaxial motion of the mass block was controlled by two double-acting hydraulic actuators. Each control and spring actuator subassembly was mounted in a single linear guide assembly for each axis of the TMD. The mass block was restrained from rotational motion about its vertical axis (yaw) by a torsion box assembly mounted on the floor and connected to the mass block by two radius rods. The same design principles were followed in the development of the TMD for installation in the John Hancock Tower, Boston (ENR, 1975). In this case, however, the TMD consisted of two 300-ton mass blocks as shown schematically in Fig. 7.27. They move in phase to provide lateral response control and out-of-phase for torsional control. A comparison of the TMD and structural parameters of this system with those of the Citicorp TMD system is given in Table 7.4.
Figure 7.27 Dual TMD System in John Hancock Tower
7.6.3 Chiba Port Tower, TokyoBay, J apan In Japan, the first TMD was installed in Chiba Port Tower, a 125m high steel structure. It has a rhombus plan with a side length of 15m. The TMD, as shown in Fig. 7.28, consists of a mass and two frames overlapping at right angle. The mass can move in X or Y direction but cannot rotate. Each frame has coil springs and a damping device. Pertinent values of the TMD and structural parameters are given in Table 7.5 (Kitamura et al., 1988; Mataki et al., 1989; Ohtake et al., 1992). Table 7.5 TMD and Structural Parameters, Chiba Port Tower (Kitamura, et al., 1988) X-direction
Structure:
1st mode effective weight (tons)
Y-direction
1200
Period: 1st mode (sec)
2.25
2nd mode (sec)
0.51
Damping ratio (%) TMD:
Weight(tons)
2.70 0.57 0.005
10.0
15.4
Period (sec)
2.24
2.72
Spring constant (tons/cm)
0.080
0.084
Friction force (tons)
0.045
0.045
Damping ratio (%)
0.15
Figure 7.28 TMD in Chiba Port Tower (Ohtake et al., 1992)
Figure 7.29 shows the wind induced root-mean-square acceleration response under Typhoon 8719 (before tuned) and under strong wind Run 880205 (after tuned). The response due to Run 880205 is seen to have been reduced by 40% in theX-direction and by 50% in the Y-direction approximately.
Figure 7.29 RMS and With Accelerations TMD (MatakiWithout et al., 1989)
7.6.4 F unade Bridge Tower, Osaka, Japan Wind tunnel tests of Funade Pedestrians Bridge tower shows that galloping may occur at a wind speed of about 10m/s. The SDOF equivalent weight of the tower is 8.09 tons with a fundamental frequency of 0.835 Hz. The air damper type TMD developed for the structure is shown in Fig. 7.30. It is composed of a circular cylinder, a disc weight of 97.4kg (p = 1.2%) and coil springs. The TMD damping results from the passing o air through the narrow space between the disc weight and the cylinder. Fine adjustments in the damping coefficient could be made by varying the number of holes passing through the disc (Ueda et al., 1992). Results of damped free vibration test are shown in Fig. 7.31. The logarithmic decrement increases from 0.046 to 0.26, phase lag is about 105 degrees, and the ratio o motion amplitudes of TMD and tower is about 6. The resonance frequency of the coupled motion is 0.86 Hz.
Figure 7.31 Damped Free Vibrations Without and With TMD (Ueda et al., 1992)
7.6.5 Steel Stacks, Kimitsu City, Japan Two identical, 120m-high steel stacks were constructed in early 1980s (Ueda et al., 1992). These are situated 200m apart and one of these stacks is schematically shown in Fig. 7.32. Before the TMD was installed, the onset velocity for vortex induced oscillation was about 20m/s. Therefore, both stacks have often been subjected to vortex induced oscillation. The pendulum type TMD, installed in one of the stacks to suppress wind induced oscillations, has a 5-ton ring shaped weight where a permanent magnet is used as a damping device (Fig. 7.33). The TMD was found to be effective in all directions. The resonant motion curve of this system, as theoretically expected, is shown in Fig. 7.34.
Figure 7.32 Outline of Steel Stack, Kimitsu City (Ueda et al., 1992)
Figure 7.33 Ring-shaped TMD on Stack A (Ueda et al., 1992)
Figure 7.34 Resonant Response Curves of Stack A (Ueda et al., 1992)
Some of the damped free vibration test results are shown in Fig. 7.35, where the logarithmic decrement increases from 0.01 to 0.25. Later, both stacks were observed simultaneously. In four months, the largest vibration amplitude of the stack without TMD was 22cm in contrast to 2.7cm of the stack with TMD. The effectiveness of TMD was thus confirmed and a TMD was also installed in the other stack. Figure 7.36 shows that, in the small amplitude range less than 10mm, the motions are the same, and the TMD begins functioning when the stack amplitude exceeds 10mm. The ratio of the TMD and stack motion amplitudes is about 3 with a phase lag of about 150 degrees.
Figure 7.35 Damped Free Vibrations of Stack A (Ueda et al., 1992)
Figure 7.36 Comparisons of Stack and TMD Displacements (Ueda et al., 1992)
7.7 Related Development and Concluding Remarks It is seen that almost all TMD applications have been made towards mitigation of windinduced motion. Seismic effectiveness of TMDs, which has been discussed briefly in the preceding sections, remains to be an important issue. While studies to date have not produced conclusive results, it can be generally stated that, under earthquake-type loading, TMDs are not as effective as for wind-type loading due to the following reasons:
(a) The frequency bandwidth of an earthquake excitation is not only wider than that of wind load but also richer in high frequency content so that high modes of a building structure are usually excited and the first mode representation of the structure is not adequate. Conventional TMD, tuned to the fundamental frequency of the structure, could suppress little or even amplify the dynamic response o higher modes and therefore may fail to reduce the total response under these conditions; (b) As pointed by a few the first peak in the response cannot be easilyoutreduced dueinvestigators, to the fact that TMD passively responds history to the structural movement and then reversely mitigates the response of the structure by vibrating out-of-phase with the structural movement. These concerns have resulted in a number of recent developments of more innovative concepts, some of which are described below.
7.7.1 NonlinearTMD and I mpact Vibration Absorbers Regarding the first problem, a nonlinear TMD may improve the performance under earthquake excitations. As mentioned in Section 7.1, Roberson (1952) considered the dynamic response of a structural system supporting a dynamic absorber with a linear plus cubic spring without damping. The suppression band in the nonlinear case was found to be wider than that in the linear case, indicating that the linear plus cubic spring absorber would suppress structural response excited by a wider range of frequency components. This is superior to the conventional linear TMD in reducing the system response under excitations such as those due to earthquakes. However, the performance of a nonlinear TMD attached to a complex structure in which a few modes are activated needs to be further demonstrated before it can be applied in practice. Carter and Lin (1961) further studied the steady-state behavior of a nonlinear dynamic vibration absorber attached to a nonlinear structural system. They concluded that the combination of softening absorber spring and hardening structural spring gives the widest suppression band. Such a model, however, is not appropriate in earthquake-resistant design since structural systems usually experience degradation to a certain degree under earthquake excitations. Fortunately, the combination of hardening absorber spring and softening structural spring also gives a wider suppression band compared to the linear TMD as discussed by Roberson (1952). An alternative solution might be the use of an impact vibration absorber. This is defined as a system that has the added mass supported by neither a spring nor a damper but constrained to move unidirectionally in a container attached to the structural system. Its movement is limited by the bounds of the container as shown in Fig. 7.37. The vibration amplitude of the structural system is reduced by the momentum transfer between the structure and the
added mass and by the conversion of mechanical energy into noise and heat. It was reported by Hunt (1979) that the maximum reduction in dynamic response can be achieved at all excitation frequencies as the impact vibration absorber is synchronized to the excitation frequency. The adverse impact of the damper on the acceleration response of a structural system, however, should be noted due to impulsive impacts.
Figure 7.37 Impact Vibration Damper
An added mass attached to a structural system by viscoelastic materials can help to mitigate the seismic response of the structure in the same way that is accomplished with a nonlinear TMD or impact vibration absorber. A rubber like material has been used in a TMD by Snowdon (1960) to show an increase in the suppression band compared to viscous damping under harmonic loads. Performance of multi-stage rubber bearing TMDs in tall structures has also proven to be satisfactory. Another approach designed to overcome the frequency-related limitations of TMDs is to use more than one TMD in a given structure, each tuned to a different dominant frequency. The concept of multiple tuned mass dampers together with an optimization procedure was proposed by Clark (1988). Since then, a number of studies have been conducted on the behavior of MTMDs connected in parallel to the main system (Xu and Igusa, 1992; Yamaguchi and Harnpornchai, 1993; Abe and Fujino, 1994). Recently, a doubly-tuned mass damper (DTMD), consisting of two masses connected in series to the structure was proposed (Setareh, 1994). In this case, two different loading conditions were considered: harmonic excitation and zero-mean white-noise random excitation, and the efficiency of DTMDs on response reduction was evaluated. Analytical results show that DTMDs are more efficient than the conventional single mass TMDs over the whole range of total mass ratios, but are only slightly more efficient than TMDs over the practical range of mass ratios (0.01-0.05).
7.7.2 Semi -Active TMD and Hybrid Mass Damper Another promising approach to reducing the seismic response of a structural system is the application of a semi-active (SA) control concept. This will retain the benefits of an optimized active system but not pay the power and
system complexity penalties associated with it (Hrovat et al., 1983). For a simple structure subjected to a periodic force consisting of four harmonic components, Hrovat et al. (1983) reported that the performance of a semiactive TMD was found to be very similar to a fully active device. This device significantly reduced the RMS values o building displacement and acceleration as well as TMD stroke. A hybrid mass damper or a TMD with active capability is another option to reduce the vibration induced by wind and earthquakes. In fact, this type of system has been implemented in a number of tall system buildings in recent (SoongINTES et al., building 1994). An example is a hybrid mass damper installed in theyears Sendagaya in Tokyo in 1991. As shown in Fig. 7.38, the mass damper was installed atop the 11th floor and consists of two masses to control transverse and torsional motions of the structure while hydraulic actuators provide the active control capabilities. The top view of the control system is shown in Fig. 7.39 where, as shown in Fig. 7.40, ice thermal storage tanks are used as mass blocks so that no extra mass is introduced. The masses are supported by multi-stage rubber bearings intended for reducing the control energy consumed in the AMD and for insuring smooth mass movements (Higashino and Aizawa, 1993).
Figure 7.38 Sendagaya INTES Building (Courtesy of Takenaka Corp., Japan)
Figure 7.39 Top View of AMD
Figure 7.40 Elevation of AMD
Sufficient data were obtained for evaluation of the AMD performance when the building was subjected to strong wind on March 29, 1993, with peak instantaneous wind speed of 30.6 in/sec. An example of the recorded time histories is shown in Fig. 7.41 giving both the uncontrolled and controlled states. Their Fourier spectra using samples of 30-second durations are shown in Fig. 7.42, again showing good performance in the low frequency range but not in the region exceeding 3 Hz. The response at the fundamental mode was reduced by 18% and 28% for translation and torsion, respectively. Similar performance characteristics were observed during a series of earthquakes recorded between May 1992 and February 1993.
Figure 7.41 Response Time Histories (March 29, 1993)
Figure 7.42 Response Fourier Spectra (March 29, 1993)
The 160m 34-story Hankyu Chayamachi building (Fig. 7.43) in Osaka, Japan, also uses an active TMD system. In this case, the heliport at the roof top is utilized as the moving mass of the AMD, which weighs 480 tons and is about 3.5% of the weight of the tower portion. The layout of AMD devices is shown in Fig. 7.44. The heliport is supported by six multi-stage rubber bearings. The natural period of the rubber and heliport system was set to 3.6 seconds, slightly lower than that of the building (3.8 seconds). The AMD mechanism used here has the same architecture as that of Sendagaya INTES, namely, scheme of the digital controller, servomechanism and the hydraulic design, except that two actuators of 5-ton thrusts each are attached in horizontal orthogonal directions. Torsional control is not considered here.
Figure 7.43 Hankyu Chayamachi Building (Courtesy of Takenaka Corp., Japan)
Acceleration time histories during a recent typhoon are shown in Fig. 7.45 with corresponding Fourier spectra given in Fig. 7.46. Since the building in this case oscillated primarily in its fundamental mode, significant reductions in acceleration levels were observed.
Figure 7.44 Layout of AMD
Figure 7.45 Acceleration Time Histories
Figure 7.46 Acceleration Fourier Spectra; a) x-direction
Figure 7.46 Acceleration Fourier Spectra; b) y-direction
References Abe, M. and Fujino, Y (1994), Dynamic Characteristics of Multiple Tuned Mass Dampers and Some Design Formulas, Earthquake Eng. Struct. Dyn., 23(8), 813836. Bishop, R. E. D. and Welbourn, D. B. (1952), The Problem of the Dynamic Vibration Absorber, Engineering, Lond., 174, 769. Brock, J. E. (1946), A Note on the Damped Vibration Absorber, J. Appl. Mech., 13(4), A-284. Carter, W. J. and Lin, F. C. (1961), Steady-state Behavior of Nonlinear Dynamic Absorber, J. Appl. Mech., 60-WA-14, 67-70. Chowdhury, A. H., Iwuchukwu, M.D. and Garske, J. J. (1987), The Past and Future of Seismic Effectiveness of Tuned Mass Dampers, Structural Control, (Leipholz, H. H. E., ed.), Martinus Nijhoff Publishers, 105-127. Clark, A. J. (1988), Multiple Passive Tuned Mass Damper for Reducing Earthquake Induced Building Motion, Proc. 9th Wld. Conf. Earthquake Eng., 5, Japan, 779-784. Den Hartog, J. P. (1956), Mechanical Vibrations, 4th Edition, McGraw-Hill, NY. ENR (1971), Tower Cables Handle Wind, Water Tank Dampens It, Engineering ews-Record, Dec. 9, 23. ENR (1975), Hancock Tower Now to Get Dampers, Engineering News-Record, Oct. 30, 11. Falcon, K. C., Stone, B. J., Simcock, W. D. and Andrew, C. (1967), Optimization of Vibration Absorbers: A Graphical Method for Use on Idealized Systems with Restricted Damping, J. Mech. Eng. Science, 9, 374-381. Frahm, H. (1909), Device for Damped Vibrations of Bodies, U.S. Patent No. 989958, Oct. 30, 1909.
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Ohtake, K., Mataki, Y., Ohkuma, T., Kanda, J. and Kitamura, H. (1992), Full-scale Measurements of Wind Actions on Chiba Port Tower, J. Wind Engineering and Industrial Aerodynamics, 41-44, 2225-2236. Okumura Corporation (1991), R&D on Base Isolation System and Vibration Control System, Seismic Isolation and Response Control for Nuclear and Non-nuclear Structures, SMiRT 11, Tokyo, Japan, 125-132. Ormondroyd, J. and Den Hartog, J.P. (1928), The Theory of the Dynamic Vibration Absorber, Trans. ASME, APM-50-7, 9-22. Petersen, N. R. (1980), Design of Large Scale TMD, Structural Control, (Leipholz, H. H. E., ed.) North Holland, 581-596. Petersen, N. R. (1981), Using Servohydraulics to Control High-rise Building Motion, Proceeding o ational Convention on Fluid Power, Chicago, 209-213. Pipes, L. A. (1953), Applied Mathematics for Engineers and Physicists, Chapter 8, McGraw-Hill, NY. Randall, S. E., Halsted, D. M. and Taylor, D. L. (1981), Optimum Vibration Absorbers for Linear Damped Systems, J. Mech. Des., ASME, 103, 908-913. Roberson, R. E. (1952),Synthesis of a Non-linear Dynamic Vibration Absorber, J. Franklin Inst., 254, 205-220. Setareh, M. (1994), Use of the Doubly-Tuned Mass Dampers for Passive Vibration Control, Proceedings of the First World Conference on Structural Control, Los Angeles, CA, 1, WP4-12 WP4-21. Sladek, J. R. and Klingner, R. E. (1983), Effect of Tuned Mass Dampers on Seismic Response, J. Struct. Eng., ASCE, 109(8), 2004-2009. Snowdon, J.C. (1960), Steady-state Behavior of the Dynamic Absorber, J. Acoust. Soc. Am., 31(8), 1096-1103. Snowdon, J.C. (1974), Dynamic Vibration Absorbers That Have Increased Effectiveness, J. Eng. or Ind., ASME, Paper No. 74-DE-J, 940-945. Soong, T. T. and Grigoriu, M. (1993), Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, NJ. Soong, T. T., Reinhorn, A.M., Aizawa, S. and Higashino, M. (1994), Recent Structural Applications of Active Control Technology, J. Structural Control, 1(2), 5-21. Srinivasan, A. V. (1969), Analysis of Parallel Damped Dynamic Vibration Absorbers, J. of Eng. for Ind., Trans. ASME, 91, 282-287. Tanaka, H. and Mak, C. Y. (1983), Effect of TMD on Wind Induced Response of Tall Buildings, Journal of Wind Engineering and Industrial Aerodynamics, 11, 357-368. Thompson, A. G. (1981), Optimum Tuning and Damping of A Dynamic Vibration Absorber Applied to a Force Excited and Damped Primary System, J. Sound Vib., 77, 403-415. Ueda, T., Nakagaki, R. and Koshida, K. (1992), Suppression of Wind Induced Vibration by Dynamic Dampers in Tower-like Structures, J. of Wind Engineering and Industrial Aerodynamics, 41-44, 1907-1918. Warburton, G. B. and Ayorinde, E. O. (1980), Optimum Absorber Parameters for Simple Systems, Earthquake Eng. Struct. Dyn., 8, 197-217. Warburton, G. B. (1982), Optimal Absorber Parameters for Various Combinations of Response and Excitation Parameters, Earthquake Eng. Struct. Dyn., 10, 381-401. Wiesner, K. B. (1979), TMD to Reduce Building Wind Motion, ASCE Spring Convention, Boston, MA. Wirsching, P. H and Yao, J. T. P. (1970), Modal Response of Structures, J. Struct. Div., ASCE, 96(4), 879-883.
Wirsching, P. H. and Campbell, G. W. (1974), Minimal Structural Response under Random Excitation Using the Vibration Absorbers, Earthquake Eng. Struct. Dyn., 2, 303-312. Xu, K. and Igusa, T. (1992), Dynamic Characteristics of Multiple Substructures with Closely Spaced Frequencies, Earthquake Eng. Struct. Dyn., 21, 1059-1070. Xu, Y. L. and Kwok, K. C. S. (1992), Wind Induced Response of Soil-StructureDamper System, 8th Int. Conf. Wind Eng., ed. A. G. Davenport et al., London, Ontario, Canada, 2057-2068. Xu, Y. L., Kwok, K. C. S. and Samali, B. (1992), Torsion Response and Vibration Suppression of Wind Excited Buildings, 8th Int. Conf. Wind Eng., ed. A. G. Davenport et al., London, Ontario, Canada, 1997-2008. Yamaguchi, H. and Harnpornchai, N. (1993), Fundamental Characteristics o Multiple Tuned Mass Dampers for Suppressing Harmonically Forced Oscillations, Earthquake Eng. Struct. Dyn., 22, 51-62.
8 Tuned Liquid Dampers 8.1 Introduction The concept of the dynamic vibration absorber for passive control of structures was presented in the previous chapter. In tuned mass dampers (TMD), typically a solid concrete or metal block acts as the secondary mass, although in some cases a deep tank filled with water serves the same purpose. Additional springs and dampers are used to attach this secondary mass to the primary structure, and to provide the restoring and dissipative mechanisms needed to tune the system for near-optimal response under various types of dynamic excitations. Figure 8.1a is a schematic of the prototype TMD attached to an SDOF structure. Many examples of structures with TMDs were discussed in Chapter 7.
Figure 8.1 Dynamic Vibration Absorbers; (a) General, (b) Tuned Sloshing Damper
In the present chapter, attention shifts to another class of dynamic vibration absorber, namely tuned liquid dampers (TLD), in which liquids are used to provide all of the necessary characteristics of the secondary system. One particular type of TLD, the tuned sloshing damper is illustrated in Fig. 8.1b. In this case, the liquid not only supplies the required secondary mass, but also the damping through viscous action primarily in the boundary layers. Meanwhile gravity provides the necessary restoring mechanism. As a result, the secondary system has characteristic frequencies that can be tuned for optimal performance, exactly as with TMDs. In fact, the prototype dynamic vibration absorber, shown in Fig. 8.2, that was proposed by Frahm in the early 1900s was actually the forerunner of the modern tuned liquid damper. Frahm tuned the frequency of motion of the water in the two interconnected tanks to the fundamental rolling frequency of a ship in order to successfully reduce this component of motion. These Frahm anti-rolling tanks were installed on several large German ocean liners (Den Hartog, 1956). More recently, the same concept was used in nutation dampers for space satellites to reduce long period libration motion (Carrier and Miles, 1960; Alfriend, 1974). Figure 8.3 provides a schematic of a typical design using a ring partially filled with liquid.
Figure 8.2 Frahm Anti-Rolling Tanks for Ship Applications (Den Hartog, 1956)
The idea of applying tuned liquid dampers to reduce vibrations in civil engineering structures began in the mid-1980s. Bauer (1984a, b) suggested the use of a rectangular container completely filled with two immiscible fluids to dampen response through the motion of the interface. This concept is depicted in Fig. 8.4a for reduction of windinduced motion. Welt and Modi (1989a, b) were also among the first to suggest the use of a TLD in buildings to reduce overall response during strong wind or earthquakes. A few of the proposed devices studied by Welt and Modi are shown in Fig. 8.4b. All o these operate on the principle of liquid sloshing. Due to the geometric similarities, these devices are sometimes referred to as nutation dampers. Several other types of tuned liquid dampers have also been proposed. The simplest of these, known
Figure 8.3 Nutation Damper for Satellite Applications (Fujino et al., 1988)
as the tuned liquid column damper is shown in Fig. 8.4c (Xu et al., 1992). This device consists of a partially-filled tube containing several internal orifices. The fundamental frequency of the system is dependent only on the length of the column of water, while damping is provided by flow through the orifices. A number of recent structural applications, employing dampers similar to those illustrated in Fig. 8.4, will be discussed later in this chapter.
Figure 8.4 Tuned Liquid Dampers for Structural Applications; (a) Immiscible Fluids Damper (Bauer, 1984b)
Figure 8.4 Tuned Liquid Dampers for Structural Applications; (b) Nutation Dampers (Modi et al., 1995), (c) Liquid Column Damper (Xu et al., 1992)
Much of the theoretical basis for TLDs has already been presented. A brief overview o the relevant fundamentals of fluid mechanics was provided in Section 6.2, while the formulations needed for properly tuning dynamic vibration absorbers have been discussed in Section 7.2 in connection with tuned mass dampers. However, unlike tuned mass dampers, which often respond in a purely linear manner, the behavior o TLDs is generally highly nonlinear due to the fluid motion. Consequently significant research effort has been directed toward both the analytical and experimental characterization of their behavior.
The following two sections provide a review of this work. Section 8.2 concentrates on the generic problem involving the response of a shallow fluid layer subjected to horizontal excitation. Then, in Section 8.3, extensions of these models suitable for direct application to TLDs will be examined, along with experimental studies o damper response. After obtaining a better appreciation for the behavior of tuned liquid dampers, attention shifts to the impact of TLDs on overall structural performance. Section 8.4 and 8.5 present results obtained from numerical and experimental studies, respectively. Although the mathematical description of TLD response is difficult, structural implementation is often quite simple. Some additional implementation issues are noted in Section 8.6. The final section provides details of several recent full-scale implementations in Japan. 8.2 Basic Principles of Operation The basic principles involved in applying a tuned liquid damper (TLD) to reduce the dynamic response of structures is quite similar to that discussed in Chapter 7 for the tuned mass damper (TMD). In particular, a secondary mass is introduced into the structural system and tuned to act as a dynamic vibration absorber. In the case o TMDs, the secondary system typically behaves in a linear manner and can thus be defined in terms of constant parameters m, k and c as indicated in Fig. 8.1a or equivalently in terms of constant mass, frequency and damping ratios µ, and a, respectively, as was done in Section 7.2. For the design of TMDs incorporated into linear SDOF structures, optimal values of these parameters can be established depending upon the type of excitation and the structural response to be minimized. Results for undamped and damped SDOF elastic structures are summarized in Tables 7.1 and 7.2, respectively. However, unlike TMDs, generally the response of tuned liquid dampers is highly nonlinear due either to liquid sloshing or the presence of orifices. As a consequence o this inherent nonlinearity, the design criteria presented in the previous chapter are not directly applicable. The response will be amplitude dependent, even for primary structures that remain within the elastic regime. In order to develop more appropriate criteria, one needs to first understand and quantify the behavior of TLDs. Considerable research effort has been focused in that direction. Most of that work has been related to characterizing response of sloshing-type TLDs, such as those shown in Fig. 8.4b, and has largely involved physical experimentation, although several mathematical models have also been developed. Early work by Housner (1957, 1963) considered only linearized response. Several more recent models, which are applicable under the assumption of moderate amplitude sloshing with no wave breaking, represent extensions of the classical theories by Airy and Boussinesq for shallow water gravity waves of finite amplitude. The classical
theories pertain to the formation of progressive waves in an inviscid fluid layer o constant depth h and infinite extent as shown in Fig. 8.5, and are discussed in detail by Mei (1983). On the other hand, a model for the response of a fluid layer confined within a tank of finite length, due to horizontal base excitation, that was employed by Lepelletier and Raichlen (1988) is described below.
Figure 8.5 Progressive Waves in Water of Constant Depth
Consider the two-dimensional idealization of a rectangular tank of length L and width b, as shown in Fig. 8.6, containing water of nominal depth h. The following differential equations governing motion of the fluid can be written in terms of the wave elevation (x, t) and the depth-averaged horizontal velocity u(x, t):
where commas denote partial differentiation with respect to the spatial subscripts coordinatefollowing x and a superposed dot indicates a partial time derivative. Additionally, in Eq. (8.1), v, w, g and represent the kinematic viscosity of the fluid, a characteristic frequency of the fluid motion, the
Figure 8.6 Tuned Liquid Damper Geometric Definition
gravitational acceleration and the horizontal acceleration of the tank base, respectively. Notice that nonlinearities appear in the second terms in both equations. Meanwhile, the fourth term in Eq. (8.1b) involving provides dispersion, and the fifth term models dissipation associated with the boundary layer on the tank bottom and side walls. Dissipation due to free surface contamination is also included via the parameter C, which lies in the range from 0 to 2 and is usually assigned a unit value (Miles, 1985). The relative contributions of nonlinearity and dispersion can be determined by considering the dimensionless Stokes parameter defined by Us = /h3, where represents the wave amplitude and is the horizontal wavelength. For small values o Us, dispersion dominates, while for large values of the Stokes parameter the response becomes significantly nonlinear. The latter regime is characterized by the appearance o progressive solitary waves and frequency bifurcations near resonance of the liquid. These resonant frequencies can be determined approximately by considering the dispersion relations for linearized response without dissipation. The resulting natural frequency for the k-th mode can be written:
Lepelletier and Raichlen (1988) solved Eqs. (8.1a, b) numerically using a finite element approach, and also conducted physical experiments to better understand the response under harmonic base excitation at frequency w. Figure 8.7 shows the time evolution o surface waves computed for the sloshing mode with = 2L for three different values o the Stokes parameter. It should
Figure 8.7 Computed Free Surface Profiles for Sloshing Mode = 2L; (a) Us = 10, (b) Us = 100, (c) Us = 1000 (Lepelletier and Raichlen, 1988)
be noted that a linearized theory, corresponding to Us = 0, would produce a standing wave with a single node at x/L = 0.5. The nonlinear result for Us = 10 is evidently quite similar. However, as the Stokes parameter increases to Us = 100, the effects o nonlinearity become apparent. A progressive wave now forms having asymmetry about the nominal fluid level. With Us increased further to 1000, this solitary wave is clearly defined as it progresses across the length of the tank. The variation of relative wave extrema at the right-hand edge of the tank are presented w1 defined by Eq. (8.2). O in Fig. 8.8 excitation near only the first course, a for linear model frequencies would predict a resonance single peak. However, in both the experiments and with the nonlinear theory, a primary peak occurs at w/w1 1.06, while a secondary peak appears at w/w1 0.97. Notice that the positive peaks near resonance greatly exceed the negative extrema in concurrence with the surface profiles of Fig. 8.7 that correspond to the propagation of solitary waves. Typical time histories of the free surface response at x = L are shown in Fig. 8.9 for frequencies near w1 with Us 360. At w/w1 = 1.04, a single solitary wave appears traveling with a speed given approximately by its shallow water limiting value of (gh)1/2.
Figure 8.8 Variation of Relative Wave Extrema at x = L (Lepelletier and Raichlen, 1988)
Figure 8.9 Free Surface Time Histories Near Lowest Resonance; (a) Experiments, (b) Nonlinear Theory (Lepelletier and Raichlen, 1988)
At somewhat lower frequencies such as w/w1 = 0.96, the single solitary wave becomes unstable early in the transient process. A bifurcation then occurs
and a second solitary wave appears. The amplitude of this second wave grows with time until the response corresponds to the second natural mode in which two solitary waves propagate along the surface with the same speed. Thus, the second mode is excited at a driving frequency that is approximately one-half of its natural frequency. This behavior is apparently a consequence of the nonlinearities combined with low dispersion. These nonlinearities also cause the beating phenomenon that is noticeable in the figure. For larger values of Us, additional higher modes are excited. In all cases, Lepelletier and Raichlen obtained a good correlation between the physical experiments and the finite element solutions, as can be seen from the comparisons in Figs. 8.8 and 8.9. Similar numerical and experimental results have been obtained by other researchers. For example, Shimizu and Hayama (1987) developed a formulation from first principles by assuming incompressible, inviscid and irrotational flow with constant pressure on the free surface. As a result, for two dimensional flow, the governing equations of continuity and motion can be written from Eqs. (6.5) and (6.4), respectively, in the form
where u and w are the horizontal and vertical fluid velocities, respectively, p is the density and p is the pressure. Again the horizontal base acceleration is , while g represents the vertical gravitational acceleration. A velocity potential 1 is then introduced in analogy with the linear theory, such that
where k is a wave number. By utilizing these relationships, Eqs. (8.3) and (8.4) can be integrated through the fluid layer from the bottom z = -h to the surface z = . After some manipulation, the results can be written
where = tanh[kh]/(kh)
(8.8a)
= tanh[k(h + )]/tanh[kh]
TH = tanh[k(h + )]
(8.8b) (8.8c)
and us is the horizontal velocity on the surface. The above equations are valid for arbitrary wave number k. Shimizu and Hayama (1987) establish k based upon the first mode response of the corresponding linear problem. Unfortunately, this destroys the dispersive character of the solutions. In order to restore some dispersion, the authors employ a finite difference method the grid spacingthat selected such thatunder the numerical dispersion matches the with physical dispersion is expected infinitesimal conditions. The authors compare results from this finite difference approach with those obtained via physical experiments on a rectangular tank. Response time histories of surface elevation near first resonance are provided in Fig. 8.10, while plots of the relative wave extrema are presented in Fig. 8.11. It should be noted that a damping term was also included in the finite difference formulation, presumably to enhance the correlation. As a result, the numerical algorithm of Shimizu and Hayama (1987) provides quite accurate simulations in the range of behavior considered in Figs. 8.10 and 8.11. Furthermore, the response is consistent with that obtained by Lepelletier and Raichlen (1988).
Figure 8.10 Free Surface Time Histories Near Lowest Resonance; (a) Experiment With w/w1 = 1.00, (b) Calculation With w/w1 = 1.00, (c) Experiment With w/w1 = 1.05, (d) Calculation With w/w1 = 1.05 (Shimizu and Hayama, 1987)
Figure 8.11 Variation of Relative Wave Extrema (Shimizu and Hayama, 1987)
In the above shallow water approximations, the dimensionality of the governing equations is reduced by integrating or averaging through the depth of the fluid. Alternatively, one can directly model the entire fluid domain by utilizing, for example, either a finite difference or finite element formulation. Su et al. (1982) present the application of a volume-of-fluid finite difference method for free surface flows (Hirt and Nichols, 1981) to the nonlinear sloshing problem. More recently, Huerta and Liu (1988) developed an arbitrary Lagrangian-Eulerian finite element method to study viscous flows with FE large free surface Okamoto and to Kawahara (1990) used a Lagrangian approach with motions, a velocitywhile correction method address similar problems. Formulations such as these could be used to extend the range of validity o the numerical simulations for liquid sloshing. 8.3 Damper Behavior and Macroscopic Modeling In the previous section, formulations were presented for investigating the general problem of fluid sloshing in rectangular tanks. The application of these models to tuned liquid dampers is now considered, along with pertinent physical experiments conducted on individual damper units. Sun et al. (1989) extended the approach utilized by Shimizu and Hayama (1987) by including a rational approximation of the damping present in sloshing-type TLDs undergoing small amplitude oscillations. Thus, Eqs. (8.7a) and (8.7b) are assumed to govern the response, except that the latter equation is modified to include a contribution from viscous damping. In order to
establish this damping term, the flow field is idealized as shown in Fig. 8.12 having a thin viscous boundary layer of thickness hb along the bottom of the rectangular tank. The following equations are then obtained after integrating through the thickness:
with
and kinematic viscosity v. The first term on the right side of Eq. (8.9b) obviously now represents the dissipative contribution. This damping has a significant effect on the response near resonance and, therefore, must be included. By considering the nature o the flow within the boundary layer and the energy dissipation per cycle, the form of can be established as
where w is the frequency of the liquid motion. This term includes the contribution from viscous dissipation along the side walls and the free surface contamination. There is no consensus on the actual value to be assumed for the constant Cd in Eq. (8.11). Sun et al. (1989) set Cd = 8/(37), Fujino et al. (1992) determine that , while Koh et al. (1994) argue that Cd = 1. Notice that the formulation of Lepelletier and Raichlen (1988) has a similar damping term in Eq. (8.lb). In any case many assumptions are involved in these derivations and correlations with experimental data are ultimately needed to establish the appropriate value.
Figure 8.12 Profile of the Horizontal Liquid Velocity Through Depth (Sun et al., 1989)
In order to validate the above formulation, Sun et al. (1989) conducted a series o shaking table experiments on rectangular tank TLDs undergoing harmonic horizontal base excitation. The typical unit had a length L = 59cm, width b = 33.5cm and still water level h = 3cm. From the linear theory, the TLD consequently had a first sloshing mode frequency fw = wl/(2) = 0.46Hz. In addition to examining the free surface profile as a function of time, Sun et al. (1989) also measured the base shear force and computed the energy dissipated per cycle. Time histories of the normalized base shear are shown in Fig. 8.13 for various frequencies near the first resonance, while the corresponding force-displacement hysteresis loops are displayed in Fig. 8.14 for a base displacement amplitude of A = 0.25cm. A good correlation is
Figure 8.13 Base Shear Force Response (Fujino et al., 1992)
obtained between the experiments and simulated results. The steady response is characterized in Fig. 8.15, which presents plots of the normalized wave extrema, shear force amplitudes and energy dissipation per cycle versus frequency near the first resonance. Once again good correlation is obtained for base amplitudes of A = 0.5cm and below. However, significant deviations occur for A = 1.0cm. At that level o excitation, the wave height often exceeds the still water level, and wave breaking begins to occur. Since the models do not consider the additional dissipation associated with breaking waves, the simulations tend to significantly overpredict the wave heights and base shear forces, as is evident from Fig. 8.15d. A completely analogous investigation was presented more recently in Sun et al. (1995a) for pitching motions.
Figure 8.14 Typical Damper Force-Displacement Response (Fujino et al., 1992)
Koh et al. (1994) considered the response of rectangular tank TLD units subjected to earthquake ground motions. Physical and numerical experiments were conducted for two different tank lengths (59cm, 90cm), several liquid depths from 3cm to 8cm, and three different earthquake records scaled to various amplitudes. For the simulations, the model defined in Eqs. (8.9) and (8.11) was employed with Cd = 1 and the frequency w established from the instantaneous up-crossing rate of the surface waves. The amplitudes of the earthquake signals were limited to prevent wave breaking, which is not accounted for in the numerical model. A typical result is shown in Fig. 8.16 for response due to the El Centro motion. Generally the correlation between the numerical solutions and the physical experiments was good. The theory tended to predict somewhat larger free surface displacements and a greater
effect from higher modes. Perhaps the level of damping used in the model was not sufficient. As noted above, at higher levels of excitation, the surface waves break and the previous formulations become invalid. However, in some applications, TLDs must operate in that range, which is characterized by higher levels of damping. Sun et al. (1992) proposed an extension of the above formulation to empirically model the effects o wave breaking. The authors introduce two additional coefficients Cda and Cfr in Eq. (8.9b) such that the governing equations become
Figure 8.15 Steady Frequency Response in Terms of Relative Wave Extrema, Shear Forces and Energy Dissipation Per Cycle (Fujino et al., 1992)
Figure 8.15 Steady Frequency Response in Terms of Relative Wave Extrema, Shear Forces and Energy Dissipation Per Cycle (Fujino et al., 1992)
For small amplitude waves (i.e., < h), these two coefficients are unity, while for higher amplitudes Cfr models the shift in natural sloshing frequency and Cda compensates for the additional damping that is associated with breaking. Based upon a series of parametric experiments, Sun et al. (1992) determined that the following values are appropriate for these dimensionless coefficients
Figure 8.16 Time Histories of Free Surface Elevation for Scaled El Centro Motion (Koh et al., 1994)
Generally this correction was found to enhance the correlation of the model for higher amplitude response. An illustration of the performance of this modified formulation will be presented later in the chapter. All of (1989a, the aboveb)studies concerned rectangular tank TLDs. On the other hand, Welt and Modi examined the behavior of torus-shaped sloshing dampers by using both analytical and experimental techniques. For the analytical investigation, a perturbation method is applied to solve the nonlinear free surface equation within a potential flow formulation. These formulations are, of course, quite involved due to the problem geometry and consequently will not be presented here. However, Welt and Modi (1989a) did obtain approximate solutions under both nonresonant and resonant conditions that suggest the use of long and slender tank geometries with shallow water heights for efficient damper performance. As noted earlier, several other types of TLDs have been proposed, including the tuned liquid column damper (TLCD) shown in Fig. 8.4c. The following simple macroscopic model of this damper was developed by Saokaet al. (1988) where, referring to Fig. 8.4c, x is the elevation of the liquid in the tube, y
is the horizontal displacement of the tube, A is the cross-sectional area, B is the horizontal dimension of the tube and L is its total length. Additionally, p, and g are the liquid mass density, the coefficient of head loss determined by flow through the orifices and the gravitational acceleration, respectively. Notice that Eq. (8.14) is nonlinear due to the presence of the damping term. Under infinitesimal excitation, the natural frequency of this simplified TLCD model can be written
The model was later verified by Sakai et al. (1989) through a series of experiments. 8.4 Structural Analysis and Design Once a tuned liquid damper model is established, the effectiveness of incorporating the TLD in a structure can be studied. As noted previously, the results follow those associated with tuned mass dampers. However, the inherent nonlinearity of TLD response complicates the picture somewhat, since stiffness and damping are both frequency and amplitude dependent. Design methodologies must also take these characteristics into account. A number of investigators have examined this problem of TLD-structure interaction under various types of excitation. For example, Chaiseri et al. (1989) considered the response of SDOF structures incorporating rectangular tank TLDs subjected to time harmonic excitations. Their mathematical model consisted of the governing equation for an SDOF structure
along with the TLD model of Eq. (8.9). In Eq. (8.9), ws, s and ms represent the natural frequency, critical damping ratio and mass of the structure, respectively. Furthermore, s is the structural displacement, F is the TLD base shear force and Fe is the external excitation force with frequency f. Numerical solutions were compared with experimental data obtained for a variety of design configurations and excitation amplitudes. The specific structure had mass ms = 168kg, damping s = 0.0032 and a natural frequency fs = 0.91Hz. Two different TLD configurations were considered with L = 25cm and L = 32cm. For each TLD tank design, the water height was selected to minimize the steady-state response at f/fs = 1 for an excitation level that caused a bare structure response amplitude x0 = 1cm. The water levels were thus set at h = 2.1cm and h = 3.6cm, respectively. The corresponding mass ratios p were 0.01 and 0.087. Typical results obtained from a frequency sweep in the neighborhood of first resonance are shown in Fig. 8.17
from the more recent paper by Fujino et al. (1992). Significant reduction in the displacement response is obtained for this lightly damped structure. For this type and level of excitation, the TLD performs as an effective dynamic vibration absorber. Additionally, reasonably consistent results are produced by the mathematical model. For higher amplitude vibration, wave breaking occurs and the TLD model of Eq. (8.9) is no longer valid.
Figure 8.17 Frequency Response of Structure With TLD (Fujino et al., 1992)
As noted in Section 8.3, Sun et al. (1992) extended the TLD to empirically account for wave breaking by (8.12) introducing two be additional coefficients. The resulting TLD model expressed in Eq. can also combined with Eq. (8.16) to approximate the response of the structural system at higher levels of excitation. Fig. 8.18 provides the results obtained from both the model and physical experiments. Once again the excitation is a simple time harmonic force with frequency around that associated with the first structural resonance. The experiment indicated the presence of breaking waves, and clearly from Fig. 8.18 the modified model provides a significantly improved correlation for this case. More recently, the performance of TLDs for the reduction of seismic response has been considered. Koh et al. (1994) examined TLD-structure interaction under earthquake excitation. Equations (8.9) and (8.16) were again employed, and a tower or tall building was idealized as an SDOF structure with fs = 0.2Hz and s = 0.03. A synthetic seismic signal, generated from a Clough-Penzien spectrum, was employed in the analysis and the amplitude was chosen such that it would cause a bare structure displacement o 10cm. TLDs with mass ratio p. = 0.04, length L = 400cm and water height h = 26.5cm were included such that fw = fs. Numerical
results are shown in Fig. 8.19. The response of an MDOF structure with multiple TLDs was also investigated in Koh et al. (1995). However, the modal approach adopted is strictly valid only in the linear response range for both the structure and the dampers. More generally, direct time domain methods are needed to assess the performance o structures utilizing TLDs for seismic response reduction. Furthermore, the amplitude dependence of the response of structures incorporating tuned liquid dampers must be carefully examined when considering TLDs for seismic applications.
Figure 8.18 Frequency Response of Structure With TLD (L = 39cm, h = 3cm, µ = 1.05%, x0 = 5cm) for Large Amplitude Excitation (Sun et al., 1992)
Figure 8.19 Time Histories of Structural Displacement Under Seismic Excitation With and Without TLDs (Koh et al., 1994)
For design purposes, there is a strong desire to simplify the nonlinear equations through some type of linearization procedure. Xu et al. (1992) and Balendra et al. (1995) take this approach for tuned liquid column dampers by modifying the nonlinear damping term in Eq. (8.14) for the damper. Both papers focus on the vibration control of towers and tall buildings under wind excitation. An examination of the stochastic response o structures that incorporate generic linearized dynamic vibration absorbers is addressed in Kareem (1983) and Kareem and Sun (1987). It should be noted thatoptimal if the design dampercriteria behavior can be7 linearized as a reasonable approximation, then the of Chapter becomes applicable. Sun et al. (1995b) detailed a TMD analogy for the nonlinear sloshing tuned liquid dampers. The interface force between the damper and the structure was conceptualized as the force exerted by a virtual mass m, and a virtual dashpot c,. For a linear TMD, the frequency-dependent values of m, and c, can be obtained from the following analytical expressions assuming simple harmonic motion with frequency w:
where
However, for TLDs, the virtual mass and damping quantities are also amplitudedependent. Consequently, Sun et al. (1995b) performed a series of experiments to determine effective values for m, a and wa that could be used in Eqs. (8.17) and (8.18) for sloshing dampers. Typical results for rectangular and circular tanks are shown in Fig. 8.20 for three different levels of time harmonic excitation. Good fits are obtained in some, but not all, cases. No attempt was made to use the effective values for the prediction of the response under more general time-dependent loading.
Figure 8.20 Virtual Mass mv and Damping cv of TLD for Linear TMD Analogy (Sun et al., 1995b)
8.5 Experimental Studies Many of the experimental studies concerning the performance of TLDs incorporated in structures havebeen beenconsidered conducted in in conjunction modeling efforts,there and are consequently have already the previouswith section. However, a couple additional investigations that should be mentioned, including the parametric studies on free vibration response of circular tank sloshing dampers reported by Fujino et al. (1988). Although this study actually led to the work by Sun and colleagues already described in some detail, it is useful to review the major findings. Fujino et al. (1988) tested cylindrical shaped containers on a steel platform to simulate a flexible tower or a building. The platform had a natural period Ts = 2s and sufficient mass to permit the use of prototype size dampers with mass ratios µ 0.01. As a result, there was no need to satisfy similarity conditions associated with scaled experiments. In the tests, the structure was given an initial displacement A and then released from rest. The change in logarithmic decrement due to the presence of the TLD was used to measure performance. The parametric study examined the effect of TLD-to-structure frequency ratio a, initial displacement amplitude A, mass ratio µ, kinematic viscosity o the liquid v, diameter of the tank a, roughness of the bottom tank surface, and the gap between the liquid surface and the tank roof.
Results show that, for small amplitude oscillations, the added damping is highly dependent on the ratio of sloshing to structure frequencies, with maximum at 1. This, of course, is consistent with the linear theory of dynamic vibration absorbers and provides the basis for tuning. For larger amplitudes, the additional damping in the system is reduced, and is much less dependent on the frequency ratio due to the nonlinearities of liquid sloshing. In this range, tuning is much less critical, and small deviations in water height have little importance. Generally, the added damping is highly dependent on amplitude A. Several additional findings by Fujino et al. (1988) are of interest. The authors found that the use of high viscosity liquids was not necessarily preferable. In many of the viscous fluid devices discussed in Chapter 6, the level of damping provided was proportional to the viscosity of the liquid, and consequently high viscosity materials were desired. However, for TLDs there is an optimal level of absorber damping that will maximize the overall effective damping of the structural system. Thus, in the present case, higher viscosity is not always better. Increasing the roughness of the bottom surface may have similar effects. The particular implementation considered by Fujino et al. (1988), utilizing small hemispheres along the bottom, did not improve performance. The inclusion of a roof on the damper units was also found to have little effect and was not recommended, since more consistent results were obtained without the roof. The paper also includes a description and classification of surface wave patterns generated during these free vibration tests. The other major experimental program that has not yet been discussed was conducted by Modi and Welt over a number of years beginning in the mid-1980s. Their analytical work on partially filled torus-shaped sloshing dampers was mentioned in Section 8.3. The corresponding experimental investigation was detailed in Welt and Modi (1989b). Sinusoidal excitation was provided to a series of tank configurations with varying inner and outer diameters, water height, and cross-sectional shape. In some cases, internal baffles and perforated tubes were also present. Both the amplitude and frequency of the forcing function were varied over a wide range. For these tests, the structure was essentially rigid in order to focus on the damper characteristics. The authors presented results in terms of added mass and damping ratios, which characterize the magnitude and phase of the sloshing force. The former quantity is permitted to assume positive or negative values. Overall good agreement with the theory was found when the amplitude of oscillations was small. In that case, the results showed that the damping ratio is very sensitive to the frequency ratio, with a maximum at about 1.0, where a reversal in sign of the added mass was observed. This, of course, is a feature of the resonant condition. For an increase in the amplitude of oscillation, a decrease in damping was noted, consistent with other studies.
In a previous investigation, Modi and Welt (1987) tested structural models with a small-scale torus shaped damper in wind tunnels. Using both laminar and turbulent flows, several models with square or circular cross section were subjected to vortex induced resonance and galloping instability. The experiments showed a significant reduction of the induced oscillations in the models. A recent paper by Modi et al. (1995) provides a good overview of their work with emphasis on their experimental programs. 8.6 Implementational Issues As discussed earlier, tuned liquid dampers operate on the same basic principles as tuned mass dampers. However, some of the drawbacks of TMD systems are not present in TLDs. Due to the simple physical concepts on which the restoring force is provided in TLDs, no activation mechanism is required. Therefore, maintenance cost is minimized. Additionally, the mechanism activating a TMD must be set to a certain threshold level of excitation, while TLD systems are at all times active, avoiding problems due to an inadequate activation system. Although the mathematical theory involved in accurately describing motion of a fluid in the container may be quite complicated, the hardware requirements are simple and installation requirements are minimal. Each damper, in general, consists of a polypropylene tank, that may be commercially available, with several shallow layers o water. Compare this, for example, to the requirements for the TMD system shown in Fig. 7.26. Furthermore, maintenance of the TLD system is practically nonexistent. Due to simplicity of the installation, sloshing type TLDs may be added to existing buildings, even for temporary use if desired. Due to the nature of the system, a small error may be expected when measuring the still water level, which is the parameter that controls the value of the fundamental sloshing frequency. Yet, another important advantage over TMD is that, for large amplitudes o oscillation, the system is not very sensitive to the actual frequency ratio between primary and secondary systems. Note that for small amplitudes of oscillation, proper tuning of the system may considerably influence the response. Therefore, the induced error due to measuring of the water height will not significantly modify the response during strong vibrations. For structures with different fundamental frequencies in the two major directions, tuning may be accomplished by using rectangular tanks. With an adequate selection o the plan dimensions of the tank both fundamental frequencies may be tuned. Care should be taken in this situation since the theory was developed for tanks subjected only to a in unidirectional excitation.a circular For structures the same fundamental frequencies the principal directions, tank maywith be used.
8.7 Structural Implementations In practice, tuned liquid dampers have been used primarily for suppressing windinduced vibrations of tall structures. In comparison with tuned mass dampers, the advantages associated with TLDs include low initial cost, virtually maintenance free operation and ease of frequency tuning. It appears that the initial TLD applications have taken place primarily in Japan. Examples of TLD-controlled structures include the Nagasaki Airport Tower, installed in 1987, the Yokohama Marine Tower, also installed in 1987, the Shin-Yokohama Prince Hotel, installed in 1992, and the Tokyo air traffic control towers at Haneda and Narita Airports, installed in 1993. The Nagasaki Airport Tower Tuned Liquid Damper, described in Tamura et al. (1995), was actually only a temporary installation in 1987 intended to measure the effectiveness of the TLD concept for reducing wind-induced vibrations. The airport itself was constructed in 1974 on an island in Omura Bay, and the 42m high, steel framed control tower is shown in Fig. 8.21. The tower has first bending natural frequencies of 1.07Hz in both lateral directions with corresponding critical damping ratios of 0.9%. A total of 25 multilevel cylindrical tank dampers were installed with 12 in the control room and the remainder in the staircase landings in the upper portion o the tower as indicated in Fig. 8.21. Each damper contained seven individual layers with diameter 0.38m, height 0.07m and water depth 0.048m. The total mass of the TLD was 38kg, providing a mass ratio p = 0.015 in the fundamental modes. The dampers were nearly tuned to the structure, having a first sloshing resonance at 1.02Hz. In order to assess the performance of the TLD, measurements were collected for a one month period before and after installation. Wind speed and direction, and tower acceleration and displacement were monitored. Figure 8.22 displays data from a damped free vibration test on the tower. Results for the tower before installation of the dampers are shown in Fig. 8.22a. This was used to determine the critical damping ratios for the structure. Figures 8.22b and 8.22c depict the response after installation of seven and 25 damper units, respectively. In both cases, overall damping is enhanced with the addition of the TLD units. A comparison of across-wind displacement response of the tower with and without the TLD is shown in Fig. 8.23. Although only a limited amount of data is available, some improvement in performance is noticeable. However, the beating phenomenon that is particularly apparent in Fig. 8.22c suggests that there was insufficient damping in the TLD for the level of secondary mass provided. Additional tuning of the TLD design may have significantly improved the across-wind performance.
Figure 8.21 Nagasaki Airport Tower (Tamura et al., 1995)
Figure 8.22 Damped Free Oscillations of Nagasaki Airport Tower; (a) Without TLD, (b) With 7 TLD Vessels, (c) With 25 TLD Vessels (Tamura et al., 1995)
Figure 8.23 Across-Wind Response of Nagasaki Airport Tower (Tamura et al., 1995)
A similar TLD was installed that same year in the landmark Yokohama Marine Tower shown in Fig. 8.24. The tower is a 100m high, steel trussed structure having a fundamental frequency of 0.55Hz along with a damping ratio of 0.6%. In this application, detailed in Tamura et al. (1995), 39 ten-layer sloshing dampers were positioned in the lighthouse on the top of the tower, and installed as shown in Fig. 8.25. The individual dampers have a radius of 0.25m and each layer is filled with water to a height of 0.021m. This produces an overall mass ratio of approximately 1% in the fundamental mode. Free vibration tests indicated that the damping ratio of the tower increased to 4.5% with the addition of the TLD. Figure 8.26 presents the root-meansquare (RMS) acceleration response with and without the TLD in both the along-wind and across-wind directions, while typical time series comparisons are shown in Fig. 8.27. A clear improvement in response is observed with the addition of the TLD.
Figure 8.24 Yokohama Marine Tower (Tamura et al., 1995)
Figure 8.25 TLD Vessels Installed on the Yokohama Marine Tower (Tamura et al., 1995)
Figure 8.26 RMS Acceleration Response of Yokohama Marine Tower; (a) Along Wind, (b) Across-Wind (Tamura et al., 1995)
Figure 8.27 Time Histories of Across-Wind Acceleration Response of Yokohama Marine Tower; (a) Without TLD, (b) With TLD (Tamura et al., 1995)
In 1992, a TLD was installed in the new 150m high Shin-Yokohama Prince Hotel. This project, involving 30 multilevel vessels, is described in Wakahara et al. (1992) and Tamura et al. (1995). However in this case a different style damper was employed with internal protrusions to enhance energy dissipation. An individual damper, illustrated in Fig. 8.28, is 2m in diameter with nine water layers, each having a depth of 0.124m. This provides a mass ratio of nearly 1% and tunes the sloshing resonant frequency to 0.31Hz, which is within a few percent of the fundamental frequency of the hotel. The damper units are positioned around the periphery of the upper floor as indicated in Fig. 8.29.
Figure 8.28 TLD Vessel for Installation on the ShinYokohama Prince Hotel (Tamura et al., 1995)
Figure 8.29 TLD Vessels Installed on the ShinYokohama Prince Hotel (Tamura et al., 1995)
The result, displayed in Fig. 8.30, represents a significant reduction in RMS accelerations for both along-wind and across-wind response, particularly at higher wind speeds. The addition of the TLD reduces the RMS acceleration below the ISO minimum perception level at 0.31Hz for the one-year recurrence wind speed of 26m/s.
Figure 8.30 RMS Acceleration Response of the Shin-Yokohama Prince Hotel; (a) Along Wind, (b) Across-Wind (Tamura et al., 1995)
Two recent applications of tuned liquid dampers to airport towers are detailed in Tamura et al. (1992, 1994, 1995). The TLD installed in the 78m air-traffic control tower at Haneda Airport consists of about 1400 vessels containing water, floating particles and a small amount of preservatives. The vessels, pictured in Fig. 8.31, are shallow circular cylinders 0.6m in diameter and 0.125m in height. These individual vessels are then stacked in six layers on steel-framed shelves as shown in Fig. 8.32. The total mass of the TLD
Figure 8.31 TLD Vessel for Installation on the Haneda Airport Control Tower (Courtesy of Y. Tanura, Tokyo Institute of Polytechnics, Japan)
Figure 8.32 TLD Vessels Installed on the Haneda Airport Control Tower (Tamura et al., 1995)
is approximately 3.5% of the first-mode generalized mass of the tower and its sloshing frequency is optimized at 0.743Hz. Floating hollow cylindrical polyethylene particles were added in order to optimize energy dissipation through an increase in surface area together with collisions between particles. The performance of the TLD has been observed during several storm episodes. In one of such episode with a maximum instantaneous wind speed of 25m/s, the observed temporal variation of the RMS acceleration and damping ratio of the tower are given in Fig. 8.33, showing that the damping ratio in the cross-wind direction (x-direction) reached the maximum value o 7.6% before the RMS acceleration recorded its maximum. Figure 8.34 shows a comparison of the x-direction RMS accelerations of the tower with and without TLD, indicating that the TLD reduced the response to about 60% of the RMS acceleration response of the tower without the TLD.
Figure 8.33 Time Variation of RMS Response of the Haneda Airport Control Tower; (a) Acceleration, (b) Damping Ratio (Tamura et al., 1995)
Tuned liquid dampers can also be beneficial during construction to stabilize various portions of a structure. Consider the application of TLDs to the Ikuchi Bridge tower described in Ueda et al. (1992). Based upon wind tunnel tests, it was decided to suppress vortex-induced oscillation of the 120m tall freestanding tower by using sloshing dampers. The free standing towers have a natural frequency of 0.25Hz and a very low damping ratio estimated from free vibration tests to equal 0.2%. In an attempt to reduce tower displacements from 400mm to approximately 20mm, two 5m by 1.3m rectangular tank dampers were employed. The tanks contain a number of internal cross poles in order to increase energy dissipation. Free vibration tests were used to establish an optimal water height of 0.80m. This provided an effective damping ratio o approximately 3.7%. Thus, in this case, the tuned sloshing damper approach met the design objectives.
Figure 8.34 RMS Acceleration Response of the Haneda Airport Control Tower for Near AcrossWind Excitation (Tamura et al., 1995)
References Alfriend, K. T. (1974), Partially Filled Viscous Ring Nutation Damper, J. Spacecraft, 11(7), 456-462. Balendra, T., Wang, C. M. and Cheong, H.F. (1995), Effectiveness of Tuned Liquid Column Dampers for Vibration Control of Towers, Engineering Structures, 17(9), 668-675. Bauer, H.F. (1984a), Oscillations of Immiscible Liquids in a Rectangular Container: A New Damper for Excited Structures, Journal of Sound and Vibration, 93(1), 117-133. Bauer, H.F. (1984b), New Proposed Dynamic Vibration Absorbers for Excited Structures, Vibration Damping Workshop Proceedings, Lynn Rogers (ed.), DD1 DD27. Carrier, G. F. and Miles, J. W. (1960), On the Annular Damper for a Freely Precessing Gyroscope, J. Appl. Mech., 27, 237-240. Chaiseri, P., Fujino, Y., Pacheco, B. M. and Sun, L. M. (1989), Interaction o Tuned Liquid Damper and Structure: Theory, Experimental Verification and Application, Structural Engineering/Earthquake Engineering, JSCE, 6(2), 273s282s. Den Hartog, J.P. (1956), Mechanical Vibrations, McGraw-Hill, New York. Fujino, Y., Pacheco, B. M., Chaiseri, P. and Sun, L. M. (1988), Parametric Studies on Tuned Liquid Damper (TLD) Using Circular Containers by Free Oscillation Experiments, Structural Engineering/Earthquake Engineering, JSCE, 5(2), 381s391s.
Fujino, Y., Sun, L., Pacheco, B. M. and Chaiseri, P. (1992), Tuned Liquid Damper (TLD) for Suppressing Horizontal Motion of Structures, J. Engrg. Mech., 118(10), 2017-2030. Hirt, C. W. and Nichols, B. D. (1981), Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comp. Phys., 39,201-225. Housner, G. W. (1957), Dynamic Pressures on Accelerated Fluid Containers, Bull. Seism. Soc. Am., 47, 15-35. Housner, G. W. (1963), The Dynamic Behavior of Water Tanks, Bull. Seism. Soc. Am., 53, 381387. Huerta, A. and Liu, W. K. (1988), Viscous Flow with Large Free Surface Motion, Comp. Meth. Appl. Mech. Engrg., 69, 277-324. Kareem, A. (1983), Mitigation of Wind Induced Motion of Tall Buildings, J. Wind Eng. Ind. Aerodyn., 11, 273-284. Kareem, A. and Sun, W.-J. (1987), Stochastic Response of Structures with Fluid-Containing Appendages, J. Sound Vibr., 119, 389-408. Koh, C. G., Mahatma, S. and Wang, C. M. (1994), Theoretical and Experimental Studies on Rectangular Liquid Dampers Under Arbitrary Excitations, Earthquake Engrg. Struct. Dynamics, 23, 17-31. Koh, C. G., Mahatma, S. and Wang, C. M. (1995), Reduction of Structural Vibrations by MultipleMode Liquid Dampers, Engineering Structures, 17(2), 122-128. Lepelletier, T. G. and Raichlen, F. (1988), Nonlinear Oscillations in Rectangular Tanks, J. Engrg. ech., ASCE, 114, 1-23. Mei, C.C. (1983), The Applied Dynamics of Ocean Waves, John Wiley & Sons, New York. Miles, J. W. (1985), Resonantly Forced, Nonlinear Gravity Waves in a Shallow Rectangular Tank, Wave Motion, 7, 291-297. Modi, V. J. and Welt, F. (1987), Vibration Control Using Nutation Dampers, International Conference on Flow Induced Vibrations, R. King (ed.), BHRA, London, 369-376. Modi, V. J., Welt, F. and Seto, M. L. (1995), Control of Wind-Induced Instabilities Through Application of Nutation Dampers: A Brief Overview, Engineering Structures, 17(9), 626-638. Okamoto, T. and Kawahara, M. (1990), Two-dimensional Sloshing Analysis by Lagrangian Finite Element Method, Int. J. Num. Meth. Fluids, 11, 453-477. Sakai, F., Takaeda, S. and Tamaki, T. (1989), Tuned Liquid Column Damper - New Type Device for Suppression of Building Vibrations, Proc. Int. Conf. on Highrise Buildings, Nanjing, China, 926-931. Saoka, Y., Sakai, F., Takaeda, S. and Tamaki, T. (1988), On the Suppression of Vibrations by Tuned Liquid Column Dampers, Annual Meeting of JSCE, JSCE, Tokyo. Shimizu, T. and Hayama, S. (1987), Nonlinear Response of Sloshing Based on the Shallow Water Wave Theory, JSME Int. J., 30, 806-813. Su, T. C., Lou, Y. K., Flipse, J. E. and Bridges, T. J. (1982), A Nonlinear Analysis of Liquid Sloshing in Rigid Containers, Department of Transportation, DOT/RSPA/DMA-50/82/1. Sun, L. M., Fujino, Y., Pacheco, B. M. and Isobe, M. (1989), Nonlinear Waves and Dynamic Pressures in Rectangular Tuned Liquid Damper: Simulation and Experimental Verification, Struct. Engrg./Earthquake Engrg., JSCE, 6, 251s-262s. Sun, L. M., Fujino, Y., Pacheco, B. M. and Chaiseri, P. (1992), Modelling of Tuned Liquid Damper (TLD), J. Wind Eng. Ind. Aerodyn., 41-44, 1883-1894.
Sun, L. M., Fujino, Y. and Koga, K. (1995a), A Model of Tuned Liquid Damper for Suppressing Pitching Motions of Structures, Earthquake Engrg. Struct. Dynamics, 24, 625-636. Sun, L. M., Fujino, Y., Chaiseri, P. and Pacheco, B. M. (1995b), The Properties o Tuned Liquid Dampers Using a TMD Analogy, Earthquake Engrg. Struct. Dynamics, 24, 967-976. Tamura, Y., Koshaka, R. and Modi, V. J. (1992), Practical Application of Nutation Damper for Suppressing Wind-Induced Vibrations of Airport Towers, J. Wind Eng. Ind. Aerodyn., 41-44, 1919-1930. Tamura, Y., Shimada, K., Sasaki, A., Koshaka, R. and Fujii, K. (1994), Variation of Structural Damping Ratios and Natural Frequencies of Tall Buildings During Strong Winds, Proc. 9th Int. Conf. on Wind Engineering, New Delhi, India, 3, 1396-1407. Tamura, Y., Fujii, K., Ohtsuki, T., Wakahara, T. and Koshaka, R. (1995), Effectiveness of Tuned Liquid Dampers Under Wind Excitations, Engineering Structures, 17(9), 609-621. Ueda, T., Nakagaki, R. and Koshida, K. (1992), Suppression of Wind-Induced Vibration by Dynamic Dampers in Tower-like Structures, J. Wind Eng. Ind. erodyn., 41-44, 1907-1918. Wakahara,ofT.,a Ohyama, T. and Fujii,Tuned K. (1992), Wind-Induced Vibration Tall Building Using LiquidSuppression Damper, J.ofWind Eng. Ind. erodyn., 41-44, 1895-1906. Welt, F. and Modi, V. J. (1989a), Vibration Damping Through Liquid Sloshing: Part I - A Nonlinear Analysis, Proc. Diagnostics, Vehicle Dyn. and Special Topics, ASME, Design Engineering Division (DE), 18-5, 149-156. Welt, F. and Modi, V. J. (1989b), Vibration Damping Through Liquid Sloshing: Part II - Experimental Results, Proc. Diagnostics, Vehicle Dyn. and Special Topics, ASME, Design Engineering Division (DE), 18-5, 157-165. Xu, Y. L., Samali, B. and Kwok, K. C. S. (1992), Control of Along Wind Response of Structures by Mass and Liquid Dampers, J. of Engrg. Mech., ASCE, 118(1), 20-39.
9 Smart Materials 9.1 Introduction As indicated in Chapter 1, passive energy dissipation constitutes only one of the possible structural control techniques; considerable research and development efforts have also been expanded on active control technology involving sensing and actuation based upon a feedback scheme. Nestled between passive and active structural control technologies is an emerging area of research addressing the possible use of innovative or ''smart" materials for sensing and control purposes. This class of smart materials can be incorporated into structural members or system components as embedded sensors and actuation elements, capable of modifying structural behavior in response to external stimuli. Since some of these smart materials provide an alternative, and potentially attractive, means of energy dissipation in structural systems, they are briefly introduced here as potential members of a broader family of energy dissipation devices. The most prominent materials that have been examined as actuation devices in recent years are shape memory alloys, piezoelectric elements, electrorheological fluids and, more recently, magnetorheological fluids. These topics are briefly discussed in this chapter in the context of civil engineering structural control. Let us note that, while research in these areas has been active in the direction of finding applications in aerospace structures and mechanical systems such as land vehicles, weapon systems and robotics, their application potential to the motion control of civil engineering structures against environmental loads remains to be realistically assessed from the standpoint of cost as well as technical feasibility. 9.2 Shape Memory Alloys The existence of shape memory properties in certain alloys has been known since 1932 when it was first observed in a Gold-Cadmium (AuCd) alloy exhibiting a "rubberlike" behavior (Ölander, 1932). The shape memory effect (SME) of an alloy is generally referred to its ability to undergo reversible
and diffusionless transformation between two crystalline phases known as austenite, the high-temperature phase of the alloy, and martensite, the low-temperature crystalline phase. This micromechanical transition process is illustrated in Fig. 9.1 when the shape memory alloy (SMA) is cyclically loaded. Figure 9. la shows SMA behavior at ambient temperature T < TM, where TM is the temperature below which the microstructure is strictly martensitic. The stress-strain behavior in this temperature range is characterized by hysteresis loops similar to that exhibited by conventional steel. However, unlike plastically deforming metals in which the resulting hysteresis is due to a dislocation glide mechanism, the SMA exhibits a reversible phase stress-induced transformation (Duerig et al., 1990). Figure 9.1b shows SMA behavior at ambient temperature T > TA, where TA is the transition temperature to the austenite state, representing the associated superelastic hysteresis loop which ideally provides a hysteretic effect and has zero residual strain upon unloading. This superelastic SMA behavior results from the elastic loading of a stable austenitic parent phase up to a threshold stress whereupon a stressinduced transformation from austenite to martensite takes place. This transformation process occurs at a significantly reduced modulus, thus giving the appearance of a yield point. As deformation proceeds, the volume of martensite within the microstructure increases and the path of the stress-strain curve follows a stress plateau. As the microstructure becomes fully martensitic, further straining will cause the martensite to be loaded elastically at a modulus lower than that of elastic austenite but much higher than that of the phase transition portion of the loading curve. Since the martensite is stable only due to the presence of the applied stress, a reverse transformation takes place upon loading, but at a lower stress plateau. Ideally, after full unloading, the material returns to its srcinal undeformed geometry. This remarkable process yields the hysteretic effect with zero residual strain and hence the term "superelasticity." High temperature applications of the SMA display linear elastic behavior with no hysteresis as shown schematically in Fig. 9.1c. The material properties of SMAs which are of interest in structural applications are the martensitic hysteretic behavior as shown in Fig. 9.1a and the superelastic behavior as shown in Fig. 9.1b. Some promising characteristics of these two modes of SMA behavior include high stiffness for small strain levels (elastic loading), reduced stiffness for intermediate levels of strain (due to formation and/or reorientation of martensite), and high stiffness at large levels of strain (elastic loading of martensite). Also, since the superelastic state ideally displays a hysteretic effect with zero residual strain, an energy absorbing device made from this material would theoretically provide a self-centering mechanism. Other attractive properties associated with SMAs include their insensitivity to environmental temperature changes when properly heat treated, and their excellent fatigue and corrosion resistance properties.
Figure 9.1 Schematic Stress-Strain Curves of Shape Memory Alloys; a) at temperature T < TM: Martensitic hysteresis, b) at temperature T > TA: superelasticity, c) at high temperature: elasticity (Graesser and Cozzarelli, 1991)
9.2.1 Basic Principles While a large of literature exists dealing with constitutive laws forbehavior shape memory materials, for body our considerations, characterization of SMA material can be accomplished by extending Özdemir's uniaxial model described in Chapter 3 in our examination of metallic dampers (Graesser and Cozzarelli, 1991). As given by Eqs. (3.11a, b) with - b > 0, the Özdemir's model is described by
where all mathematical symbols are defined in Chapter 3. One notes that, on dividing Eq. (9.la) by , the slope of the - curve is given by
It is seen that the slope is a constant during purely linear elastic loading and unloading (i.e., when ( - b)/ d << 1). The Özdemir model can be modified to simulate SMA behavior by altering the expression for backstress b as given by Eq. (9.1b) (Graesser and Cozzarelli, 1991). They have shown that it is possible to describe various aspects of the SMA behavior by modifying Eq. (9.1b) to the form
where fT, a and c are material constants, u(x) is the unit step function, and erf( x) is the error function defined by
The second term is added to the Özdemir model in order to contribute to the backstress on the descending branch of the hysteresis in a way that will allow for SMA stressstrain descriptions. Calculations based upon the SMA constitutive law as described by Eqs. (9.1a) and (9.3) show that symmetric hysteresis loops can be generated for martensitic hysteresis (T < TM) and superelastic (T > TA) types of behavior. Also, the physical constants in the model, which represent the elastic modulus, initial axial yield and inelastic modulus, were reproduced in the computations. Extensions of the onedimensional SMA model to three dimensions have also been carried out (Graesser and Cozzarelli, 1994). Experimental characterization testing was also undertaken to evaluate the mechanical characteristics of SMA under cyclic conditions (Graesser and Cozzarelli, 1991; Aiken et al., 1992). The work reported in Graesser and Cozzarelli (1991) involved strain controlled cyclic loading tests of Nitinol specimens, a binary alloy of titanium and nickel. Varying levels of strain and strain rate were used in the experiments, and a heat treatment was applied such that a hysteretic material response close to that o superelasticity was attained. In comparison with the SMA constitutive model developed above, experimental results show that the SMA model was able to predict the energy absorbing capacity of the Nitinol specimens to within 35% over the range of strain and strain rate values tested.
9.2.2 Structural Applications Some analyses and small-scale experiments have been carried out with respect to the application of shape memory alloys to civil engineering structural control. In Aiken et al. (1992),was thestudied possible use of Nitinol as a passive energy dissipation device for structures experimentally by incorporating
small loops of Nitinol wire into diagonal braces in a three-story model structure. The structural model was six feet in height and weighed 3000 lb. The results of the testing included two types of behavior of special interest: large strain behavior and cyclic superelastic behavior. Figure 9.2 shows the large strain behavior of Nitinol, which has three distinct phases. Initially, the material was stiff and elastic, then at 2% strain the crystalline structure changed. Here the Nitinol became softer while still remained elastic, this corresponds to the nearly horizontal region of the hysteresis loop. This plateau continued until 6% strain was reached, then as the deformation continued, the additional deformation caused dislocation and permanent deformation in the crystalline structure, causing an increase in the stiffness. The advantage of this behavior is that, for low levels of seismic excitation, the structure behaves elastically; for moderate earthquakes, the Nitinol will dissipate large amounts of energy while remaining elastic and, for large earthquakes, the structure will stiffen and again dissipate large amounts of energy.
Figure 9.2 Large Strain Behavior of Nitinol (Aiken et al., 1992)
Figure 9.3 shows the behavior that can be achieved during a moderate earthquake. The Nitinol wire loops were preloaded so that the hysteretic behavior was confined to the flat region of the hysteretic curve. During cyclic stretching and relaxation in the diagonal braces, the increase in the tensile force caused a change in the crystalline structure of the Nitinol and, when the tensile force decreased, the Nitinol reverted to its srcinal crystalline structure. This cyclical change in crystalline structure dissipated a substantial amount of earthquake input energy. Significantly, the Nitinol did not sustain any dislocations in its crystalline structure to dissipate this energy and thus a Nitinol energy dissipator could perform in this controlled elastic manner for a
large number of seismic cycles. This behavior is in contrast to that of normal metals which must be permanently deformed to dissipate large amounts of energy. We also note that this superelastic material is capable of sustaining strains in the range of 5% to 7%, while normal steels can only sustain strains of less than 0.2% without permanent deformation.
Figure 9.3 Superelastic Behavior (Aiken et al., 1992)
The implementation of a tuned mass damper using Nitinol as an energy dissipation mechanism was studied analytically and experimentally by Inaudi et al. (1993). The TMD consisted of a mass attached to a lightly damped structure with prestressed Nitinol cables. The cables were installed in the direction perpendicular to the motion o the points of attachment - both to the mass damper and to the structural system. This geometric configuration along with prestressing in the Nitinol cables introduced a resistance scheme which can be approximated by a bilinear relationship between the force acting on the TMD and the relative displacement between the mass damper and its support, leading to a triangular hysteresis loop. Results of this study show that this TMD scheme can improve the dynamic performance of the structure with regard to its maximum deformation. However, no performance comparison with conventional TMDs was made and TMD parameters were not optimized in this study. An experimental study using Cu-Zn-Al shape memory dampers installed as diagonal braces on a model structure was carried out by Witting and Cozzarelli (1992). The model structure was the same 2/5-scale steel frame structure described in Section 5.5.1 used for the viscoelastic damper study. As a result, the performance of these SMA dampers could be compared with that of the viscoelastic dampers. A torsion bar design of the SMA damper as shown in Fig. 9.4 was used and
the structure was subjected to several simulated ground motions supplied by a shaking table. Figures 9.5 and 9.6 show the maximum floor displacements and accelerations under, respectively, the simulated El Centro and Quebec ground motions scaled to 0.06g. These results demonstrate the sensitivity of damper effectiveness to the type o ground motions used. This observation can be partially explained by the fact that the SMA damper stiffness decreases with larger material deformation, causing a decrease in the natural frequency of the structure. It is thus expected to be more effective against earthquakes with energy concentrated over the low-frequency range. Since the Quebec earthquake has higher high-frequency content, the dampers did not perform as well in this case as in the El Centro case. Test results also showed that the SMA dampers were not as effective as viscoelastic dampers operating under similar conditions.
Figure 9.4 Torsional Bar Design (Witting and Cozzarelli, 1992)
Figure 9.5 Maximum Floor Response for 0.06g El Centro (Witting and Cozzarelli, 1992)
Figure 9.6 Maximum Floor Response for 0.06g Quebec (Witting and Cozzarelli, 1992)
No large-scale structural tests involving the use of SMA energy dissipation devices have been carried out to date and, as indicated in Section 9.1, cost will also be an important factor in assessing potential applicability of SMA materials to civil engineering structural control. Nitinol, for example, can be priced as high as $150/lb, which would be prohibitive in structural applications. 9.3 Piezoelectric Materials Piezoelectricity, an as electromechanical property coupling electric fields,into wasa discovered as early 1880 by the Curie brothers (Cady, elastic 1964). and When integrated structural member, a piezoelectric material generates an electric charge or voltage in response to mechanical forces or stresses. This phenomenon is called the direct iezoelectric effect . Conversely, upon applying an electric charge or voltage to the material, it induces mechanical stress or strain, producing the so-called converse iezoelectric effect. Both the direct and converse piezoelectric effects are of interest in the field of structural sensing and control since the direct piezoelectric effect can be used for sensing and the converse piezoelectric effect can be used for control. As a result, novel piezoelectric devices have been invented and applied to vibration sensing and control of aerospace systems, robotics, micro-mechanical systems and, more recently, structural elements such as beams, plates, trusses, and shell structures (e.g., Bailey and Hubbard, 1985; Tzou, 1992). In comparison with other types of sensing and actuation devices, the advantages of using piezoelectric actuators and sensors include their effectiveness over a wide frequency range, simplicity, reliability, compactness, and lightweight.
9.3.1 Basic Principles As a demonstration of its actuation mechanism through the converse piezoelectric effect, Fig. 9.7 shows a typical configuration of piezoelectric layers acting on a beam. As an actuator, the application of an electric field to the electrode surfaces of the layers induces a tension or compression strain or stress in the layers. This adds a pair o bending moments to the beam for vibration control purposes. Let the applied voltage be V, the induced extensional strain is = (d31/hp)V
(9.5)
where hp is the thickness of each piezoelectric layer and d31 is the extension piezoelectric constant. The strain results in a longitudinal stress given by = (d31/hp)EpV
(9.6)
where Ep is the Young's modulus of the piezoelectric material. For a cantilever beam with width b and depth hm as shown in Fig. 9.8a, the actuator acts as
a concentrated force couple with force magnitude F = bhp
(9.7)
and a bending moment M(x) = d31Epb(hp + hm)V
(9.8)
as shown in Fig. 9.8b. These basic relationships can be extended to the dynamic analysis of a structural element under the action of the converse piezoelectric effect.
Figure 9.7 General Layout of a Piezoelectric Actuator
9.3.2 Structural Applications At present, there are three types of piezoelectric actuators used in vibration control. The first type, also the most popular, is that of distributed actuators for controlling distributed-parameter systems, such as beams, plates, shells and structural frames. Some of the first studies were done on beams with piezoelectric layers covering the whole length of the beam (Bailey and Hubbard, 1985; Burke and Hubbard, 1987,1988). As a result of these
Figure 9.8 Cantilever Beam with Piezoelectric Layers
investigations, a set of guidelines was proposed for the design of nonuniform spatial layer distributions. Later, interests were directed to the study of plate and shell structures with piezoelectric layers acting as actuators or sensors. Laminated piezoelectric plate and shell theories were developed based on Love's theory and Hamilton's principle (Tzou and Gadre, 1989; Lee and Moon, 1989,1990; Tzou, 1992), although experimental verification was limited to beam-type structures. The second type of piezoelectric actuators is of segmented design where the actuators are discontinuously bonded on or embedded in structural elements. Crawley and Luis (1987) conducted one of the first studies on segmented piezoelectric actuators. Based on a uniform strain assumption, they derived the static and dynamic analytical models and carried out three different experiments for cantilever beams with either bonded or embedded segmented actuators. Crawley and Anderson (1990) further developed more accurate models of the interaction between induced strain actuators and onedimensional structures, including two analytical models and one numerical model. The actuation strain produced by piezoelectric materials was characterized in greater depth. The third type consists of discrete actuators for controlling discretized-parameter structures, such as trusses with node masses. Descriptions of typical discrete actuator and sensor design and control experiments can be found in
Shibata et al. (1992), Edberg et al. (1992), and Preumont et al. (1992). Recent research in the development of piezoelectric control technology may be grouped into the following areas: First, because of complexity associated with piezoelectric actuator/sensor-structure theory, many investigators have utilized numerical models, especially finite element approach, to implement control formulations of various laminated piezoelectric elements or structures. Baz and Poh (1988), Crawley and Anderson (1990) and Hanagud et al. (1992) developed finite element formulas for evaluating single control using segmented actuators. andout Tseng (1990), Ha et al. (1992), andbeam Chandrashekhara and Agarwal (1993)Tzou carried finite element analyses for laminated or composite plates and shells. Second, an important issue in the design of segmented actuators is optimal placement and sizing. Devasia et al. (1993) proposed three performance measures to find simultaneous placement and sizing o distributed piezoelectric actuators in simply supported beams and discussed their relative effectiveness. Main et al. (1994) derived different cost functions for determining the optimal placement and thickness of embedded and surface-mounted piezoelectric actuators in beams as well as plates. Chattopadhyay and Seeley (1994) established a multiobjective design optimization procedure for control design using surface-boned piezoelectric actuators and applied the procedure to multistory frame control. Third, as shown in Fig. 9.7, piezoelectric layers can be used as either actuators or sensors or both. Recent developments have focused on combining both actuations into a single piece of a layer segment. Typical theoretical derivations and experimental verifications on self-sensing piezoactuators can be found in Dosch et al. (1992) and Garcia et al. (1992). Fourth, an improvement in damper design by replacing the cover of traditional constrained passive damping treatments by a piezoelectric layer has shown an enhanced control performance. This new treatment has been referred to as intelligent constrained layer (ICL). Agnes and Napolitano (1993), for example, added a piezoelectric layer to a viscoelastic damping layer to provide an ICL damper for vibration reduction. Baz (1993) proposed a mathematical model resulting in a sixthorder ordinary differential equation governing bending vibration control of an EulerBernoulli beam through ICL. In addition, Baz and Ro (1993) studied vibration control of beams with ICL partially covering the beams. At the same time, Shen (1993,1994) investigated Euler-Bernoulli beams and isotropic plates controlled through the ICL treatment. Although considerable progress has been made in research and applications o piezoelectric control technology, its implementability to large-scale civil engineering structures remains to be examined. A major limitation is that a very high voltage, of the order of several hundred or even thousand volts, may be needed in order to generate an effective control action for large-scale structures. Consider, for example, horizontal vibration control of a column with top mass 1000 kg under a horizontal earthquake excitation at
the base by using a piezoelectric distributed actuator as shown in Fig. 9.9. Let piezoelectric layers cover continuously both side surfaces of the column. By means of a generalized single-degree-of-freedom approximation, one can derive the piezoelectric actuator control force acting on the top mass as
in which u(t) is the control force, b and L are the width and length of the column, and h is the thickness of the column. Under a 1/3-scaled El Centro earthquake input, it isc estimated that, in order to reduce the uncontrolled top mass displacement by 30%, a maximum control voltage of 1275 volts is needed; if the required reduction is 50%, the maximum control voltage will increase to 2340 volts.
Figure 9.9 Column with Piezoelectric Layers
9.4 Electrorheological Fluids Electrorheological (ER) materials, which are mostly fluids, are characterized by their ability to undergo dramatic reversible increases in resistance to flow when subjected to an electric field. While one can characterize piezoelectric materials discussed in the preceding section as controllable solids through the application of an electric field, one can consider ER materials as controllable fluids through the same means. The potential of ER fluids in applications as control devices was first recognized by Winslow (1949) because of their ability to provide simple, efficient, and rapid-response to electronicmechanical interfaces. Until recently, the most common ER fluids consisted o nonducting particles suspended in nonconducting liquids with significant amounts of an activator, such as water, adsorbed onto the particles (Block and Kelly, 1988). Referred to as ''wet" or hydrous ER fluids, these fluids were limited in applications due to their inherent need of adsorbed water, causing possible instabilities. The discovery o anhydrous ER materials in the late 1980s, however, alleviated some of these concerns and led to a significant increase in ER fluid research (Filisko and Radzilowski, 1990) with attendant development of ER devices. Early ER devices include clutches and brakes (Winslow, 1949) and valves (Phillips, 1969). More recent examples of ER devices are engine mounts (Duclos, 1987; Petek et al., 1988), shock absorbers (Duclos, 1988), robotic devices (Gandhi et al., 1989), and structural vibration dampers (Stevens et al., 1984; Morishita and Mitsui, 1992; Morishita and Ura, 1993; Carlson et al., 1995; Makris et al., 1996). In most of these applications, the ER fluid is used either to adjust a pressure gradient (flow mode) or to provide momentum transfer (shear mode). Examplesareofexamples flow mode devices include shock absorbers and dampers. Brakes and clutches of shear mode devices.
9.4.1 Basic Principles The exact causes of the ER effect as described above are not completely understood, but it has been attributed to induced polarization of the dispersed particles of a colloid or the dissolved phase of a solution (Block and Kelly, 1988). This polarization process is manifested when yield stresses are developed in the fluid, which are strongly dependent on the electrical field. Fig. 9.10a illustrates qualitatively the stress-strain behavior, giving ultimate values of yield stress y and yield strain y (Gavin and Hanson, 1994). At pre-yield strains, ER materials are basically linearly viscoelastic. Yield strains typically decrease with the electric field and yield stresses increase roughly as squares of the electric field. In fully developed flow, an applied yield stress is resisted by a yielding component, y, and a viscous component, which is roughly linear in the shear
strain rate as shown in Fig. 9.10b. This behavior conveniently models the energy dissipation mechanism of ER fluids and their highly controllable nature makes them attractive in the development of efficient damping devices.
Figure 9.10 Stress-strain Behavior of ER Materials (Gavin and Hanson, 1994)
The behavior of ER fluids under steady-state flow as described above motivated the use of the Bingham constitutive model for ER fluids (Stanway et al., 1985,1987; Gavin and Hanson, 1994), i.e.,
where is the viscosity of the fluid. The exponent of power laws relating the yield stress y to the electrical field E ranges from 1.2 to 2.5 for typical ER fluids. Typical values of y range from 2 to 5 kPa for electric fields on the order ot 3 to 5 kV/mm at 25°C. Other properties of ER fluids will be discussed in the next section when they are compared with magnetorheological fluids. Based on the electrorheological model as described by Eq. (9.10), the hysteretic model relating the force generated by a typical ER damper, F, to the piston velocity, , is given by (Stanway et al., 1985, 1987).
where co is the damping coefficient, fc is the frictional force related to the fluid yield stress, and fo is an offset in F to account for the non-zero mean observed in the measured force due to the presence of the accumulator. Thus, the Bingham model consists of a Coulomb friction element placed in parallel with a viscous element as shown in Fig. 9.11.
Figure 9.11 Bingham Model for ER Fluids (Stanway et al., 1985,1987)
Several other hysteretic models have also been proposed. Gamota and Filisko (1991), for example, proposed an extension of the Bingham model, consisting of the Bingham model in series with a linear solid model. Another, proposed by Spencer et al. (1996a), makes use of the versatility of the Bouc-Wen model (Wen, 1976) in its ability to exhibit a wide variety of hysteretic behavior. It is schematically shown in Fig. 9.12. The force predicted by this model is given by
where the evolutionary variable z is governed by
, hysteresis ß, A and nloops In the above, are the parameters whose values control the shape of the forBouc-Wen the yieldingmodel element.
Several small scale ER devices (Ehrgott and Masri, 1992; Gavin and Hanson, 1994; Makris et al., 1996) and one large scale ER device (Gavin
Figure 9.12 Mechanical Model with Bouc-Wen Element (Spencer et al., 1996a, b)
and Hanson, 1994) were designed, constructed, and tested to explore their structural control potential. For the ER device tested by Gavin and Hanson (1994) as shown in Fig. 9.13, Fig. 9.14 illustrates the range of the hysteresis loops obtained from constant E tests under a sinusoidal input. The sinusoidal period in this case is 1.2 sec and the gap between the device plunger and the box is 0.96 mm. Figure 9.15 shows the ER damper tested by Makris et al. (1996), which consists of a main cylinder and a piston rod that pushes an ER fluid through a stationary annular duct. The proposed damper is compact and can potentially produce relatively large forces.
Figure An ER Test9.13 Device (Gavin and Hanson, 1994)
Figure 9.14 Experimental ER Hysteresis Loops at Constant Voltages [E = 0.0 kV/mm () and E = 2.50 kV/mm (- - -)] (Gavin and Hanson, 1994)
Figure 9.15 ER Damper with Annular Duct (Makris et al., 1996)
9.4.2 Structural Applications With the mechanical models described above, response of structures equipped with ER dampers can be analyzed. The crucial question remains as to whether ER dampers can be developed to generate the amount of controllable forces required for control of large civil engineering structures. The answer to this question must await a better understanding of the complex and intrinsic nonlinear behavior of ER dampers on the one hand and hardware development on the other. At the present, only small-scale experiments have been carried out. Ehrgott and Masri (1992) designed and built an ER damper for application of active pulse vibration control of a small-scale test structure subjected to random excitation. The structural has a naturalthefrequency 4.2 Hz and was the test set-up isand illustrated in Fig. 9.16. Inmodel the experiments, measuredofacceleration monitored its zero crossing is searched for. Upon such an occurrence, a 5V digital pulse was generated by the IBM PC which produced an 11,000V pulse output from the high voltage transformer. This high voltage pulse in turn activated the ER fluid halting the motion o the damper auxiliary mass, thus producing an out-of-phase force on the structure. Figure 9.17 shows the acceleration zero crossings followed by the 5V digital pulse and then the power supply high voltage response. This figure shows the time lags between each pair of these events which can hinder activating the device at the optimal time. A random excitation with bandwidth of 0-15 Hz was applied to the model structure. As shown in Fig. 9.18, a reduction of 20 to 60% in the displacement of the structure was achieved. Power requirements for the device were modest, requiring a peak power of 7 watts and an average of only 1.3 watts over a typical test run.
Figure 9.16 Test Set-up (Ehrgott and Masri, 1992)
Figure 9.17 Calibration Test of Digital and High Voltage Response to Acceleration Signal; 5-Hz Steady State Acceleration Signal (), 5-Volt Digital Control Pulse and the Power Supply High Voltage Response in kV (- - -) (Ehrgott and Masri, 1992)
Figure 9.18 Structure Displacement Response; ER Damper Off (), ER Damper On (- - -) (Ehrgott and Masri, 1992)
9.5 Magnetorheological Fluids Magnetorheological (MR) fluids are the magnetic analog of ER fluids where reversible increases in resistance to flow is caused by the application of a magnetic field to the fluids. While ER fluids have been studied over several years for possible civil engineering applications, the study of MR fluids for these applications has only been undertaken recently (Spencer et al., 1996a, b; Dyke et al., 1996). In what follows, only a brief account of this development is presented. The discovery and development of MR fluids can be dated back to the 1940s (Rabinow, 1948). Like ER fluids, MR fluids usually consist of non-colloidal suspensions o polarizable particles. A comparison of properties of typical ER and MR fluids is given in Table 9.1. In comparison with ER devices, MR devices offer a number of attractive features, including high yield strength, low viscosity, and stable hysteretic behavior over a broader temperature range (Carlson et al., 1995; Dyke et al., 1996). As in the case of ER fluids, the stress-strain behavior of MR fluids is often described by the Bingham viscoplastic model as given by Eq. (9.10). In order to study the mechanical behavior of an MR damper for structural applications, a prototype MR damper was evaluated by Spencer et al (1996a, b). The damper, as shown in Fig. 9.19, consists of micron-size, magnetically-soft iron randomly dispersed in a hydrocarbon oil along with additives that promote homogeneity and inhibit gravitational settling. The magnetic field is applied perpendicular to the direction of fluid flow. The response o the MR damper
due to a 2.5 Hz sinusoid with an amplitude of 1.5 cm is shown in Fig. 9.20 for four constant voltage levels (0.0V, 0.75V, 1.5V and 2.25V). It is observed from Fig. 9.20a that the damper force is not centered at zero due to the presence of an accumulator in the damper. Another notable feature can be seen from Fig. 9.20c. For large positive velocities, the damper force varies linearly with velocity; however, at low velocities, a rapid roll-off of the force occurs in which the force-velocity relationship is no longer linear. Spencer et al. (1996a, 1996b) found that the mechanical model depicted in Fig. 9.12 describes the damper behavior quite well. A comparison between the predicted responses and the corresponding experimental data is provided in Fig. 9.21 for the case where the applied voltage was 1.5V. Table 9.1 Comparison of Properties of Typical ER and MR Fluids (Carlson, et al., 1995) Property
ER Fluid
MR Fluid
Yield Strength
2-5 kPa
50-100kPa
(Field)
(3-5 kV/mm)
(150-250 kA/m)
Field limited by breakdown
Field limited by saturation
Viscosity
0.2-0.3 Pa-s
0.2-0.3 Pa-s
(no field)
@ 25°C
@ 25°C
Operating
+10 to + 90°C (ionic, DC)
- 40 to + 150°C
Temperature
-25 to + 125°C (non-ionic, AC)
(limited by carrier fluid)
Current Density
2-15 mA/cm2 (4kV/mm, 25°C) (x 10 -x100 @ 90°C)
can energize with permanent magnets
Specific Gravity Ancillary Materials
1-2.5
3-4
Any (conductive surfaces)
Iron/Steel
Figure 9.19 Schematic of an MR Damper (Spencer et al., 1996a, b)
Figure 9.20 Experimentally Measured Force for 2.5 Hz Sinusoidal Excitation with 1.5 cm Amplitude (Spencer et al., 1996a, b)
Figure 9.21 Comparison Between Predicted and Experimentally Obtained Responses for Proposed Model (Spencer et al., 1996a, b)
Simulation studies using simple structural models have been carried out which demonstrate that MR devices can be effective in reducing structural responses under simulated earthquake inputs. These results and their implications to full-scale structural control using MR dampers await experimental verification and further research and development. References
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Appendix
Conversion Table: English Units to SI Units To convert from
To
Multiply by
Acceleration foot/second (ft/sec )
metre/second2 (m/sec2)
2.54×10-2*
foot (ft )
metre (m )
9.2903× 10-
inch2(in2)
metre2 (m2)
6.4516× 10-4*
2
inch/second (in/sec )
2
2
3.048×10-1*
metre/second (m/sec )
2
Area
Density pound mass/inch (Ibm/in )
kilogram/metre (kg/m3) 3
3
2.7680x10
kilogram/metre (kg/m )
1.6018x10
British thermal unit (Btu)
joule (J)
1.0544×103
foot-pound force (ft-lbf)
joule (J)
1.3558
kilowatt-hour (kw-h)
joule (J)
3.60×106*
kip (1000 bf)
newton (N)
4.4482×10
pound force (lbf)
newton (N)
4.4482
foot (ft)
metre (m)
3.048×10-1*
inch (in)
metre (m)
2.54×10- *
slug (lbf-sec2/ft)
kilogram (kg)
1.4594 × 10
ton (2000 Ibm)
kilogram (kg)
9.0718×10
watt (W)
2.2597×10-2
watt (W)
7.4570×10
atmosphere (std) (14.7 Ibf/in2)
newton/metre2 (N/m2 or Pa)
1.0133×105
pound/inch2 (lbf/in2 or psi)
newton/metre2 (N/mn2 or Pa)
6.8948×103
pound mass/foot (lbm/ft ) Energy, work
Force
Length
Mass
Power foot-pound/minute (ft-lbf/min) horsepower (550 ft-lbf/sec) Pressure, stress
*Exact value
To convert from
To
Multiply by
Velocity foot/minute (ft/min)
metre/second (m/sec)
5.08×10- *
foot/second (ft/sec)
metre/second (m/sec)
3.048×10-
metre2/second (m2/sec)
9.2903×10-2
pascal-second (Pa-sec)
1.4882
pascal-second (Pa-sec)
4.788 × 10
Viscosity foot2/second (ft2/sec) pound-mass/foot-second (Ibm/ft-sec) pound-force-second/foot2 (lbf-sec/ft2) *Exact value
uthor Index A
Abe, M., 237, 270 Agarwal, A., 330 Agnes, G. S., 330 Aguirre, M., 60 Aiken, I.D., 83, 85, 99, 112, 113, 114, 123, 144, 322, 323, 324 Aizawa, S., 271 Alfriend, K. T., 282 Anagnostides, G., 92 Anderson, E., 21 Anderson, E. H., 329, 330 Arima, F., 171, 186, 187, 188, 211, 212 Ashour, S. A., 144, 159 Austin, M. A., 118 Ayorinde, E. O., 228 B
Babuska *, I., 19 Bagley, R. L., 134, 177 Bailey, Baktash,T.,P.,327, 109328 Balendra, T., 302 Banerjee, P. K., 182 Batchelor, G. K., 175 Bathe, K-J., 12, 25, 33 Bauer, H.F., 282, 283 Baz, A., 330 Berg, G. V., 30 Bergman, D. M., 61, 62, 63, 65, 144 Bertero, V.V., 27, 28, 30, 31 Bhatti, M. A., 36, 59 Bird, R. B., 175, 187 Bishop, R. E. D., 228 Block, H., 332 Boller, Chr., 49, 56 Bolotin, V.V., 33 Bowden, F. P., 86 Brock, J. E., 233, 238 Burke, S. E., 328
C
Cady, W. G., 327 Caldwell, D. B., 127 Campbell, G. W., 241, 242, 247 Carlson, J. D., 332, 338, 339 Carrier, G. F., 282 Carter, W. J., 269 Chaiseri, P., 299 K., 330 Chandrashekhara, Chang, K. C., 47, 131, 143, 144, 152, 156 Chattopadhyay, A., 330 Cherry, S., 94, 95, 107, 108, 110, 111, 112, 116, 117, 118 Chiarugi, A., 219 Chowdhury, A.H., 248 Civi, A., 219 Clark, A. J., 270 Clough, R. W., 22, 33 Cofie, N. G., 44, 55 Constantinou, M. C., 2, 134, 171, 172, 173, 177, 185, 193, 194, 195, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 218, 220 Cozzarelli, F. A., 37, 47, 321, 322, 324, 325, 326
Crawley, E. F., 329, 330 Crede, C. E., 171 Crosby, P., 163 Cruse, T. A., 33 D
Dafalias, Y. F., 43 Dargush, G. F., 56 de la Llera, J.C., 59 Delinic, K., 171 Den Hartog, J.P., 227, 228, 231, 237, 238, 242, 282 Devasia, S., 330 Dosch, J. J., 330 Dowling, N.E., 49 Duclos, T. G., 332 Duerig, T. W., 320 Dyke, S. J., 338
Dym, C. L., 175 E
Edberg, D. L., 330 Ehrgott, R. C., 334, 336, 337, 338 F
Falcon, 228, 237178 Ferry, J.K. D.,C., 135, 138, Filiatrault, A., 94, 95, 107, 108, 109, 110, 111, 112, 113, 116, 117, 118 Filisko, F. E., 332, 334 FitzGerald, T. F., 84, 85, 98, 99 Foutch, D. A., 144, 152 Frahm, H., 227 Frohrib, D. A., 228 Fujino, Y., 237, 270, 282, 294, 295, 296, 297, 300, 303, 304 Fujita, S., 144 Fung, Y. C., 37 G
Gadre, M., 329 Gamota, D. R., 334 Gandhi, M. V., 332 Garcia, E., 330 Gavin, H. P., 332, 333, 334, 335 Gehling, R.N., 127 Gemant, A., 134, 177 Goel, S.C., 61, 62, 63, 65 Graesser, E. J., 47, 321, 322 Greenbank, L. R., 60, 61 Grenier, A., 219 Grigorian, C. E., 84, 98, 99 Grigoriu, M., 33, 241 Grossman, J. S., 257 Guendeman-Israel, R., 119 H
Ha, S. K., 330 Hall, W. J., 30, 31 Hanagud, S., 330 Hanson, R. D., 6, 59, 69, 144, 159, 332, 333, 334, 335
Hargreaves, A. C., 92 Harnpornchai, N., 270 Harris, C. M., 171 Hayama, S., 290, 291, 292 Hertzberg, R. W., 49, 50 Higashino, M., 271 Hirt, C. W., 292 Hoppe, D. L., 237 Housner, G. W., 26, 30, 285 Hrovat, D., 271 Hubbard, J. E., 327, 328 Huerta, A., 292 Huffmann, G. K., 171, 202, 220 Hunt, J. B., 270 I
Igusa, T., 270 Ikeda, K., 228, 233, 237, 238, 240 Inaudi, J. A., 59, 110, 324 Ioi, T., 228, 233, 237, 238, 240 Isyumov, N., 249, 250 J
Jacquot, R. G., 237 Jara, J. M., 59 Jennige, R. L., 228 Johnson, C.D., 143 Johnson, K. L., 89 K
Kanaan, A. E., 59, 104, 147
Kareem, A., 302 Kasai, K., 134, 141 Kawaguchi, A., 248 Kawahara, M., 292 Kaynia, A.M., 249 Keel, C. J., 162
Kelly, J. M., 35, 59, 83, 85, 99, 113, 114, 123, 134, 332 Kerwin, E. M., Jr., 143 Kienholz, D. A., 143 Kirekawa, A., 144 Kitamura, H., 263 Klingner, R. E., 242, 243 Koh, C. G., 134, 293, 295, 298, 300, 301 Krajcinovic, D., 50 Krawinkler, H., 44, 55 Krempl, E., 46, 48 Krieg, R. D., 43 Kwok, K. C. S., 249, 259 L
Lai, M. L., 141, 143, 144 Landau, L. D., 175 Larson-Basse, J., 86 Lee, C. K., 329 Lee, G. C., 47 Lee, H. H., 134 Lee, J., 239 Lepelletier, T. G., 286, 287, 288, 289, 290, 291, 293 Li, C., 187, 189, 190, 212, 213, 214, 218, 219 Lifshitz, E. M., 175 Lin, F. C., 269 Lin, R. C., 144 Liu, W. K., 292 Lobo, R. F., 29, 144, 152 Luft, R. W., 237, 239, 256 Luis, J., 329 M
MacDonald, P.A., 259 Mahmoodi, P., 127, 161, 162 Main, J. A., 330 Mak, C. Y., 243, 244, 245, 250, 253, 254 Makris, N., 134, 171, 172, 177, 178, 180, 181, 182, 183, 184, 185, 332, 334, 335, 336 Marsh, C., 83, 84, 93, 94, 105, 106, 109 Martinez-Romero, E., 71, 73, 74, 75, 76 Masri, S.F., 334, 336, 337, 338 Mataki, Y., 263, 264
Maugin, G., 50 McNamara, R. J., 250, 251, 252 Mei, C.C., 286 Mendelson, A., 37 Miles, J. W., 282, 287 Miller, A. K., 45 Mitsui, J., 332 Mitsusaka, Y., 172, 187, 223, 224 Miyazaki, M., 172, 187, 223, 224 Modi, V. J., 282, 284, 298, 304, 305 Monti, M.D., 60 Moon, F. C., 329 Morgenthaler, D. R., 127 Morishita, S., 332 Mröz, Z., 43 N
Napolitano, K., 330 Newmark, N.M., 6, 30 Nichols, B. D., 292 Nims, D. K., 84, 85, 100, 101, 109, 110 Niwa, N., 198 O
Ohtake, K., 263 Okamoto, T., 292 Ölander, A., 319 Oldham, K. B., 178 Ormondroyd, J., 228 Özdemir, H., 36, 45, 50, 51, 57, 59, 321, 322 P
Pall, A. S., 83, 84, 90, 91, 92, 93, 94, 104, 105, 106, 119, 120, 121, 122
Pall, R., 119, 120, 122 Pasquin, C., 122 Pekcan, G., 173, 174, 196, 197, 215, 216, 217 Penzien, J., 22, 33 Perry, C. L., 76, 77, 78 Petek., N. K., 332 Petersen, N. R., 261, 262 Phillips, R. W., 332 Pipes, L. A., 228 Pister, K. S., 118 Poh, S., 330 Popov, E. P., 43 Powell, G. H., 59, 104, 119, 147 Press, W. H., 12, 46, 201 Preumont, A., 330 R
Rabinow, J., 338 Radzilowski, L. H., 332 Raichlen, F., 286, 287, 288, 289, 290, 291, 293 Randall, S. E., 228, 233 Reinhorn, A.M., 187, 189, 190, 208, 209, 210, 211, 212, 213, 214, 215, 218, 219 Richter, P. J., 101, 114, 115 Rivlin, R. S., 47 Ro, J., 330 Roberson, R. E., 228, 269 Robinson, W. H., 60, 61 Roik, K., 96, 97, 98 Rosen, S. L., 135 Rosenblueth, E., 6 Ross, D., 127 S
Sakai, F., 299 Sánchez, A. R., 60 Saoka, Y., 298 Scholl, R. E., 70, 104 Schwahn, K. J., 171 Seeger, T., 49, 56 Seeley, C. S., 330 Setareh, M., 270
Shames, I. H., 37, 175 Shen, I. Y., 330 Shen, K. L., 136, 144 Shibata, S., 330 Shimizu, T., 290, 291, 292 Shinozuka, M., 33 Shreir, L. L., 90 Skinner, R. I., 2, 35, 59, 71, 72 Sladek, J. R., 242, 243 Snowdon, J.C., 228, 270 Soong, T. T., 2, 33, 56, 59, 136, 142, 143, 159, 241, 271 Spanier, J., 178 Spencer, B.F., Jr., 334, 338, 339, 340 Srinivasan, A. V., 228 Stanway, R., 333 Stevens, N. G., 332 Su, T. C., 292 Su, Y-F., 144 Sun, L. M., 292, 293, 294, 295, 296, 297, 300, 301, 302, 303 Sun, W-J., 302 Symans, M.D., 173, 193, 194, 195, 200, 201, 202, 203, 204, 205, 206, 207, 218 Szabó, B., 19 T
Tabor, D., 86 Tamura, Y., 306, 307, 308, 309, 310, 311, 312, 313, 314, 315 Tanaka, H., 243, 244, 245, 250, 253, 254 Taylor, D. P., 220 Taylor, R. L., 18, 25, 44 Tezcan, S., 219 Thomaides, S. S., 30 Thompson, A. G., 233, 237 Torvik, P. J., 134, 177 Tsai, C. S., 56, 134 Tsai, K. C., 36, 55, 56, 59, 67, 68, 69, 70, 71 Tseng, C.I., Tsopelas, P.,330 197, 198, 199, 202
Tyler, R. G., 92 Tzou, H. S., 327, 329, 330 U
Uang, C. M., 27, 28, 30, 31 Ueda, T., 264, 265, 266, 267, 268, 314 Ungar, E. E., 143 Ura, T., 332 V
Valanis, K. C., 47 Veletsos, A. S., 30 Vezina, S., 120 W
Wakahara, T., 311 Warburton, G. B., 228, 233, 234, 235, 236, 237, 241 Welbourn, D. B., 228 Welt, F., 282, 298, 304, 305 Wen, Y. K., 334 Whittaker, A., 63, 64, 65, 67, 68, 70 Wiesner, K. B., 241, 242, 255 Williams, M. L., 135 Winslow, W. M., 332 Wirsching, P. H., 241, 242, 247, 248 Witting, P. R., 324, 325, 326 X
Xia, C., 59, 69 Xu, K., 270 Xu, Y. L., 249, 255, 283, 284, 302 Y
Yamaguchi, H., 270 Yao, J. T. P., 248 Z
Zahrah, T. F., 30, 31 Zhang, R. H., 128, 142, 159 Zienkiewicz, O. C., 18, 25, 44
Subject Index A
ABAQUS, 19, 54, 200 Active Control, 2 ADAS Damper, 63, 71, 72, 75, 76 ADINA, 19, 200 Ambient temperature, 138 ANSYS, 19 Asahi Beer Azumabashi Building, 123 Austenite, 320 B
Base isolation, 2, 123, 202, 219, 222 Bauschinger effect, 41, 43 Bernoulli's equation, 191 Bimetallic corrosion, 90 Bimetallic interface, 90, 98 Bingham constitutive model, 333 Boltzmann superposition principle, 135 Boundary element method, 182 Brake lining materials, 92 C
CAARC Building, 253 Canadian Space Agency Complex, 120 Cardiology Hospital Building, 72 Casino de Montreal, 122 Centerpoint Tower, 259 Chiba Port Tower, 263 Chile (Llolleo) 1985 earthquake, 65 Citicorp Center, 255, 257, 260 CN Tower, 259 Coefficient of friction, 87 Coffin-Manson relationship, 49, 56 Columbia SeaFirst Building, 162 Complex frequency response function, 9
Complex modulus, 130, 136, 177 Complex shear modulus, 130, 136, 177 Corrosion, 90 Coulomb friction, 86 Cross-braced friction damper, 83, 93, 94, 110, 116 Crystal Tower, 259 Cylindrical pot damper, 171, 181, 219 D
Damping ratio, 231 Damping ratio, 7, 202 Direct time domain analysis, 24 Dispersion, 287, 291 Doubly-tuned mass damper, 270 DRAIN, 19, 59, 69, 72, 75, 76, 104, 105, 107, 119, 121, 147, 200 Dry friction, 86 Dynamic amplification factor, 9, 10, 231 Dynamic stiffness, 201 E
Ecole Polyvalante, 122 Eigenvalue problem, 20, 202 El Centro 1940 earthquake, 12, 65, 68, 104, 105, 107, 109, 112, 113, 114, 147, 150, 203, 208, 212, 295, 325, 331 Elastic modulus, 38, 129
Elastoplastic model, 15 Electrorheological (ER) fluid, 332 Energy Dissipating Restraint, 84, 100, 114 Energy formulation, 25 ETABS, 19 Euler equations, 176, 191 F
Fatigue, 49, 59 Finite difference method, 291, 292 Finite element method, 19, 33, 54, 56, 57, 287, 292 Flow rule, 43 Forced frequency ratio, 9, 231
Fractional derivative, 134, 177 Frahm's absorber, 227, 282 Frequency ratio, 9, 231 Friction damper models, elastic perfectly plastic, 92 with bearing stops, 93, 96 Friction damper, 83 Funade Bridge Tower, 264 G
GERB Fluid Damper, 171, 181, 219 Glassy modulus, 135 H
Hachinohe earthquake, 145, 147, 150 Haneda Airport Control Tower, 313 Hankyu Chayamachi Building, 273 Hardening, 41 Hybrid mass damper, 270 I
IDARC, 19, 208 Ikuchi Bridge 'rower, 314 Impact vibration absorber, 269 Impulse response function, 11, 12 Inelastic strain, 38 Intelligent constrained layer (ICL), 330 Internal temperature, 141 Isotropic hardening, 41 Izazaga #38-40 Building, 72 J
Jarret Elastomeric Spring Damper, 173, 196, 215 John Hancock Tower, 260 K
Kelvin-Voight model, 134 Kinematic hardening, 41 L
Lanchester damper, 238
LAPACK, 21 Lead extrusion damper, 60 Limited slip bolted joint, 83, 91, 104 Loma Prieta 1989 earthquake, 76 Loss factor, 130 Loss modulus, 130, 178 Loss stiffness, 182, 187 M
Mach number, 192 Magnetorheological (MR) fluid, 338 Martensite, 320 Mass ratio, 229 Maxwell model, classical, 45, 134, 193, 198, 200, 202, 208 complex derivative, 177 fractional derivative, 177, 185 McConnel Library, 119 Metallic dampers, 35 Mexican Institute of Social Security, 74 Mexico City 1957 earthquake, 75 Mexico City 1985 earthquakes, 72, 75, 112 Miyagi-Ken-Oki 1978 earthquake, 27 Modal damping ratio, 143 Modal strain energy, 143 Modal superposition method, 20 Mode shapes, 20 MSC/NASTRAN, 19 Multi degree of freedom (MDOF) model, analysis, 58, 143, 200, 245 definition, 18 energy formulation, 31
Multi-stage pendulum, 258 Multiple tuned mass damper (MTMD), 270 N
Nagasaki Airport Tower, 306
Natural frequency, damped, 7 tuned liquid damper, 287, 299 undamped, 7, 20 with Maxwell elements, 202 Natural period, 7, 57 Naval Supply Facility, 166 Navier-Stokes equations, 177, 190 Newmark algorithm, 24 Newton-Raphson method, 25, 58 Newtonian fluid, 176 Nitonol, 322 Non-Newtonian fluid, 177 Nutation damper, 283 O
Orificed fluid damper, 173, 190, 192, 199, 202, 208, 215, 219 Orthogonality conditions, 21 Ozdemir model, 45, 50 P
Pacific Bell North Area Operations Center, 220 Pall Friction Damper, 119, 120, 122 Palmgren-Minor rule, 49 Parkfield 1966 earthquake, 107 Passive Energy Dissipation, 2 Piezoelectric effect, converse, 327 direct, 327 Piezoelectric material, 327 Plasticity models, elastic perfectly plastic, 15, 41, 51, 92 Ramberg-Osgood, 41, 51 Two-surface, 43 Plasticity theory, 37 Progressive waves, 286 Ramberg-Osgood, 40, 51 Rangitikei Bridge, 71 Rayleigh damping, 21 Re-centering, 102, 197, 320 Reduced variables, method of, 138, 141, 178 Reforma #476 Buildings, 74
Relaxation time, 135, 179, 193 Reynolds number, 192 Rich Stadium, 220 Rubbery modulus, 135 Runge-Kutta formula, 46 S
SAP, 76, 200 Saguenay (Quebec) 1988 earthquake, 122, 325 San Bernardino County Medical Center, 222 Santa Clara County Building, 163 Seismic Isolation, 2, 123, 202, 219, 222 Self-centering, 102, 197, 320 Semi-active Control, 2 Sendagaya INTES Building, 271 Shallow water gravity waves, 285 Shape memory alloy, 319 Shape memory effect (SME), 319 Shear loss modulus, 130, 131 Shear storage modulus, 130, 131 Shin-Yokohama Prince Hotel, 311 Single degree of freedom (SDOF) model, definition, 6 energy formulation, 26 forced vibration, 8 free vibration, 7 transient response, 11 with passive damper, 14, 57 Sloshing damper, 284, 292, 302, 306, 308, 311, 313, 314 Slotted bolted connection, 84, 98 Smart material, 319 Solitary wave, 288
Sonic Office Building, 123 Statistical linearization, 110 Stochastic processes, 33, 230, 241, 302 Stokes parameter, 287 Storage modulus, 130, 178 Storage stiffness, 182, 187
Strain, elastic, 38 inelastic, 38 natural, 39 Stress relaxation modulus, 135 Sumitomo Friction Damper, 83, 99, 113, 119, 123 Superelastic hysteresis loop, 320 Suppression band, 228 SUT-Building, 223 T
TADAS, 67, 69 Taft 1952 earthquake, 29, 111, 152, 203, 210, 214, 216 Taylor Devices Fluid Damper, 173, 192, 220, 221, 222 Time domain analysis, 24 Triangular plate damper, 36, 51, 67 Tuned liquid column damper, 283, 298 Tuned liquid damper, 281 Tuned mass damper (TMD), 227, 229, 281, 285, 302, 305, 324 Tuned roller pendulum damper (TRD), 258 Two Union Square Building, 162 U
Uniaxial friction damper, 83, 99, 113, 119, 123 Units, conversion, 345 V
Viscoelastic (VE) damper, 127 Viscoplasticity, 44 Viscous Damping Wall, 171, 186, 211, 218, 223 Viscous boundary layer, 293 Viscous fluid damper, 171 W
Water waves, 285 Wave breaking, 295, 296 Wells Fargo Bank Building, 76 Woodland Hotel, 221 World Trade Center, 127, 161 X
X-braced friction damper, 83, 93, 94, 110, 116
X-shaped plate damper, 36, 56, 61, 63, 68, 71, 72, 75 Y
Yield surface, 42 Yokohama Marine Tower, 308 Z
Zacatula earthquake, 109,179 114 Zero shear rate viscosity, #
28 State Street, 221 3M Viscoelastic Damper, 127, 161, 162, 163, 166
