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A parametric study of the seismic response of structures fitted with coulomb friction damping mechanisms

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Title:
A parametric study of the seismic response of structures fitted with coulomb friction damping mechanisms a proposed approach to the design of friction damped structures
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Secary, Daniel W
Place of Publication:
Denver, CO
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University of Colorado Denver
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English
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184 leaves : ; 28 cm

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Subjects / Keywords:
Damping (Mechanics) ( lcsh )
Earthquake resistant design ( lcsh )
Coulomb functions ( lcsh )
Coulomb functions ( fast )
Damping (Mechanics) ( fast )
Earthquake resistant design ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 183-184).
Thesis:
Civil engineering
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Daniel W. Secary.

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|University of Colorado Denver
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|Auraria Library
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47826574 ( OCLC )
ocm47826574
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LD1190.E53 2001m .S43 ( lcc )

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Full Text
A PARAMETRIC STUDY OF THE SEISMIC RESPONSE OF STRUCTURES FITTED
WITH COULOMB FRICTION DAMPING MECHANISMS:
A PROPOSED APPROACH TO THE DESIGN OF FRICTION DAMPED STRUCTURES
by
Daniel W. Secary
B.S., University of Colorado at Denver, 1986
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2001


This thesis for the Master of Science
degree by
Daniel W. Secary
has been approved
by
John R. Mays

Date
!
I


Secary, Daniel William (M.S., Civil Engineering)
A Parametric Study of the Seismic Response of Structures
Fitted with Coulomb Friction Damping Mechanisms:
A Proposed Approach to the Design of Friction Damped Structures
Thesis directed by Assistant Professor Kevin L. Rens
ABSTRACT
This thesis presents the results of an analytical parametric study completed on the
seismic response of single-degree-of-freedom systems fitted with friction dampers. The
primary goals of the study were; 1) to gather data key to the design of friction damping
mechanisms including the number of slip cycles, the total amount of energy dissipation, and
the maximum rate of energy dissipation experienced by the dampers during strong
earthquakes; 2) to develop a method to formulate inelastic response spectra based on the peak
ground acceleration, one-second spectral acceleration, and fundamental period of the
earthquake under consideration; and 3) to evaluate the effect elastic secondary stiffness has
on the deformation response of the systems.
A secondary goal of the study was to outline a proposed method that might be used by
structural engineers in the design of friction damped structures. As part of this goal, the
correlation between the inelastic deformation of multi-degree-of-freedom systems fitted with
in


friction dampers and the corresponding displacements calculated using modal analysis
techniques and the inelastic response spectra developed in the study was evaluated.
The study consisted of a series of inelastic dynamic analyses completed on single-
degree-of-ffeedom systems with natural periods ranging from 0.25 to 2.5 seconds and elastic
secondary stiffness ratios ranging from 0 to 100 percent. The dynamic analyses were
completed using the computer program Response developed by the Author at the
University of Colorado at Denver and the recorded ground motions of five earthquakes scaled
to yield consistent peak ground accelerations of 1.0g. To evaluate the correlation between the
inelastic deformation response spectra developed in the study and the maximum
displacements and story drifts experienced by multi-degree-of-freedom structures subjected
to earthquake ground motions, inelastic time history analyses were completed for a ten-story
friction damped moment frame using the same five earthquakes considered in the parametric
study.
Four levels of friction forces were considered in the analyses completed for both the
single-degree-of-freedom systems and the friction damped moment frame. The force levels
are related to representative code prescribed elastic response spectra and elastic response
reduction factors similar to those used in current earthquake design.
T his abstract accurately represents the content of the candidate's thesis. I recommend its
publication.
Signed I
/
Kevin L. Rens
iv


ACKNOWLEDGEMENT
I would like to thank my thesis advisor, Dr. Kevin L. Rens for his help and guidance in
the completion of this thesis.
Additionally, I would like to express my sincere thanks to each of the members of my
thesis committee, Dr. Kevin L. Rens, Dr. Judith J. Stalnaker, and Dr. John Mays for the
generous gift of knowledge that they have given me through the numerous class lectures and
discussions that have made up both my undergraduate and graduate studies at the University
of Colorado at Denver.


CONTENTS
Figures..................................................................................ix
Tables...................................................................................xi
Chapter
1. Introduction..........................................................................1
2. Previous Research.....................................................................9
2.1 Introduction.........................................................................9
2.2 Pall Friction Dampers...............................................................15
2.3 Optimum Slip Force..................................................................17
2.4 Optimum Slip Force Design Spectrum..................................................21
2.5 Simplified Design Approach Proposed by Filiatrault and Cherry.......................23
2.6 Areas Requiring Additional Study....................................................25
3. Overview of Parametric Study.........................................................27
3.1 Introduction........................................................................27
3.2 Objectives and Scope of Parametric Study............................................27
3.3 Friction Damped Moment Frames.......................................................30
3.4 Solution of the Equation of Motion by Time Stepping Methods.........................37
3.4.1 Introduction......................................................................37
3.4.2 Elastic SDOF Systems..............................................................39
3.4.3 Inelastic SDOF Systems............................................................41
3.4.4 Inelastic SDOF Systems with Friction Dampers......................................46
vi


3.4.5 System Energy Content
48
3.4.6 Components of Inelastic Response.............................................49
3.5 Earthquake Ground Motions used in Study.......................................53
3.6 Code Prescribed Earthquakes...................................................59
3.7 Elastic Response Reduction Factors............................................61
3.8 Representative Code Prescribed Earthquakes....................................63
3.9 Concluding Remarks............................................................67
4. Parametric Study Results.......................................................68
4.1 Introduction..................................................................68
4.2 Mechanism Design Data ........................................................69
4.3 Inelastic Deformation Response Spectra for SDOF Systems.......................70
4.4 Effects of Secondary Stiffness................................................78
4.5 Conclusions and Discussion of Parametric Study Results........................81
5. Application of Parametric Study Results to MDOF Systems........................84
5.1 Introduction...................................................................84
5.2 Deformation Response of MDOF Systems..........................................85
5.2.1 Introduction.................................................................85
5.2.2 Elastic MDOF Systems.........................................................85
5.2.3 Response Spectrum Analysis...................................................89
5.2.4 Inelastic MDOF Systems.......................................................92
5.3 Response of a Ten-Story Friction-Damped Moment Frame..........................99
5.4 Estimated Frame Deflection and Story Drift Using Inelastic Spectra...........104
5.5 Equivalent Slip Coefficients Based on Optimum Slip Force.....................106
vii


5.6 Concluding Remarks........................................................110
6. Discussion, Conclusions and Recommendations for Further Study..............112
6.1 Discussion and Conclusions.................................................112
6.2 Recommendations for Further Study.........................................115
Appendix
A Nomenclature.................................................................117
B Users Manual for Computer Program Response................................122
C Representative Code Prescribed Response Spectra..............................137
D Slip Cycles and Energy Dissipation...........................................143
E Inelastic Deformation Response Spectra.......................................149
F Normalized Inelastic Deformation.............................................155
G Multi-Degree-of-Freedom Systems .............................................166
Bibliography...................................................................183
viii


FIGURES
1.1 System energy content during El Centro, 1940 earthquake..........................4
2.1 Response of friction dampers under cyclic loading...............................12
2.2 Pall friction damper ...........................................................16
2.3 Strain energy content of systems with and without friction dampers..............18
2.4 Slip load optimization..........................................................20
2.5 Optimum slip force design spectrum..............................................22
3.1 Friction-damped moment frame....................................................30
3.2 Idealized combined damper/framing system behavior...............................31
3.3 Force-Deformation relationship for friction damped moment frame..................33
3.4 Friction-damped moment frame at arbitrary instant in time........................35
3.5 Schematic representation of friction damped structure...........................36
3.6 Northridge earthquake, fy 0.90, (a) Inelastic deformation response
(b) Yield deformation............................................................50
3.7 Northridge earthquake, Elastic deformation response.............................51
3.8 Northridge earthquake, (a) Difference between inelastic and elastic responses
(b) Transient component of response..............................................52
3.9 Normalized Earthquake Ground Motions............................................54
3.10 Fourier Amplitude Spectra......................................................55
3.11 El Centro Deformation Response Spectrum........................................56
3.12 El Centro Velocity Response Spectrum...........................................57
IX


3.13 El Centro Pseudo-Acceleration Response Spectrum................................58
3.14 Code prescribed pseudo-acceleration response spectrum...........................59
3.15 Pseudo-velocity response spectrum from code prescribed earthquake...............60
3.16 Deformation response spectrum from code prescribed earthquake..................61
3.17 Comparison of code prescribed pseudo-acceleration response spectrum
and code prescribed design spectrum.............................................62
3.18 Simplified pseudo-acceleration response spectrum...............................63
3.19 El Centro representative code prescribed earthquake response spectra...........66
4.1 Inelastic deformation response spectrum..........................................71
4.2 Energy Equilibrium State.........................................................73
4.3 Idealized Hysteresis Loop........................................................74
4.4 Inelastic deformation response spectrum. El Centro, S|=1.42g, R=6................77
4.5 System displacement with variation of normalize slip strength and
secondary stiffness. El Centro, Tn = 1.0 second..................................79
4.6 Normalized system displacement...................................................80
5.1 Schematic diagram of MDOF friction-damped system.................................93
5.2 DAlembert free-body diagram of MDOF system mass.................................93
5.3 Ten-story friction-damped moment frame...........................................99
5.4 Comparison of estimated and actual inelastic response...........................103
5.5 Comparison of estimated and actual inelastic story drift........................104
5.6 Equivalent R values.............................................................109
x


TABLES
Table
2.1 Friction Damper Installations ........................................................13
2.2 Filiatrault and Cherry Study Parameters...............................................21
3.1 Study Parameters......................................................................30
3.2 Earthquakes used in Study.............................................................53
3.3 Representative Code Prescribed Earthquake Parameters..................................65
4.1 Slope reduction coefficient,/?........................................................76
5.1 Design shear distribution............................................................101
5.2 Parameters used for ten-story frame evaluation.......................................102
xi


1. Introduction
In recent years, Structural Engineers have become increasingly interested in the use of
passive energy absorbing mechanisms to reduce the response of structures during strong
earthquakes. Common forms of these mechanisms have included hydraulic struts or visco-
elastic material dampers, which dissipate energy through viscous damping, metallic yield
elements, which dissipate energy through inelastic deformation, and friction dampers, which
dissipate energy through coulomb friction. These mechanisms are referred to as "passive"
because they rely only on the relative motion of the structure during an earthquake to
dissipate energy. By dissipating a portion of the energy transferred to the structure during the
earthquake, these devices reduce the structure's overall response. This reduced response
translates into reduced damage to the main structural elements, resulting in both added safety
against collapse and economic gain from the reduced cost of rehabilitation of the structure
after the earthquake.
Modem building codes acknowledge that it is not economically feasible to design
structures that remain fully elastic during strong earthquakes. The basic philosophy behind
the seismic design requirements in modem building codes is that it is acceptable to design
structures that will sustain no damage to structural and non-structural components during
frequent minor earthquakes, a small amount of damage to non-structural components during
less frequent moderate earthquakes, and moderate to severe damage to both structural and
non-structural components during infrequent strong earthquakes. The design approach
emphasizes life safety and although damage to structural elements during a strong earthquake
1


is deemed acceptable, structural collapse and the potential resulting loss of life must be
avoided.
This approach to seismic design relies on yielding of key structural elements to dissipate
the excess energy transferred to the structure during strong earthquakes. During minor and
moderate earthquakes, the stresses within the main structural elements are below yield levels
and the energy transferred to the structure by the earthquake is dissipated entirely through the
equivalent viscous damping present in the structure. During strong earthquakes, the
equivalent viscous damping is not sufficient alone to dissipate the large amount of energy
transferred by the earthquake. As kinetic and strain energy levels build up, displacements
eventually exceed the elastic limits of the structure and yielding of elements occurs. In a
properly designed structure, the yielding of elements and subsequent absorption of excess
energy takes place within regions of the structure that are specially designed and constructed
to allow for large inelastic deformations. Although the yielding takes place within a
predetermined portion of the structure, the yielding of structural members results in
permanent deformations and damage that can be severe enough to render the structure
uninhabitable without some level of rehabilitation. In the case of a poorly detailed or
constructed building, collapse can occur.
During an earthquake, the general equation describing the energy content within a
structure at an arbitrary instant in time is;
Ei = Es + Ek + Ed + Ey + Em (1.1)
In Equation 1.1, the variable Ei represents the total amount of energy transferred to the
structure by the earthquake. Variables Es and EK represent the energy stored within the
structure. Es is the elastic strain energy and EK is the kinetic energy. Variables ED, Ey, and
2


Em represent the energy that is dissipated from the system. ED is the total amount of energy
dissipated through the equivalent viscous damping present in the structure, EY is the total
amount of energy dissipated through inelastic deformation of structural elements and EM is
the total amount of energy dissipated by the passive energy dissipating mechanisms present
within the structure. Passive energy absorbing mechanisms act to limit damage to the main
building framing system by minimizing or eliminating the need to dissipate energy through
yielding of structural elements.
While each of the various energy dissipating mechanisms mentioned earlier have the
ability to dissipate energy during an earthquake, the simplicity and low cost of friction
dampers make them especially attractive. Friction dampers dissipate energy through
coulomb friction by sliding a series of plates, incorporated into the lateral load resisting
system of a structure, relative to one another. The amount of energy dissipated by the
mechanism is equal to the product of the friction force developed by the mechanism and the
total cumulative distance the plates slide relative to one another. The friction force developed
within the mechanism is equal to the product of the normal compressive force holding the
plates in contact and the coefficient of friction for the materials used to fabricate the damper.
Tensioned high strength bolts passing through the plies of the damper typically provide the
normal force.
The effectiveness of friction dampers in reducing the amount of energy dissipated
through yielding of structural elements is demonstrated in Figure 1.1. The figure presents the
energy content of two single-degree-of-freedom systems subjected to the El Centro, 1940
earthquake. Figure 1.1a) shows the energy content of an inelastic system without additional
energy dissipating devices. Figure 1.1b) shows the energy content of the same system fitted
3


with friction dampers. Of key interest is the amount of energy dissipated through yielding of
structural elements. Review of the figure indicates that, for this example, the amount of
energy dissipated through yielding of structural elements was decreased by approximately 80
percent after friction dampers were incorporated.
(a)
- Strain + Kinetic ---Equivalent Viscous Damping Yield
(b)
Figure 1.1 System energy content during El Centro, 1940 earthquake, a) Inelastic system,
b) Inelastic system with friction dampers.
4


In addition to effectively dissipating excess energy, friction dampers provide two
additional features that are useful in mitigating earthquake damage to structures. The first is
the ability to limit forces transferred to structural elements during an earthquake. Because the
friction force is dependent on the damper normal force and friction coefficient rather than
displacement, the maximum force transferred through the damper during an earthquake is
well defined. In effect, the friction force specified for the damper defines the maximum force
that will be transferred through the structural bracing system during a strong earthquake.
This is in contrast to the over-strength forces that must be accounted for in current seismic
design. The second feature that makes friction dampers attractive is the ability to develop
two levels of stiffness within a structure. When lateral forces within the bracing system are
below the slip level of the friction dampers, no slippage occurs and the system acts as a
relatively rigid braced frame system. When forces in the bracing system exceed the slip level
of the dampers, slippage occurs and the stiffness of the structure is reduced. The reduction in
stiffness of the system, and subsequent shift to a longer natural period, can be effective in
reducing the lateral forces acting on the structure during strong earthquakes. Because of their
effectiveness in dissipating excess energy and these unique features, friction dampers show
great promise in providing a means for Structural Engineers to design buildings that can resist
strong earthquakes with little or no damage to the main building structure.
Currently, friction dampers have been incorporated into the design of a number of
buildings in Canada, Japan, and the United States. Typically, the designs have relied on
inelastic time history dynamic analyses to determine the distribution of slip forces to be used
throughout the lateral load resisting system of the buildings. The time histories have been
either recorded ground motions of past earthquakes or synthetically generated ground
5


motions. This method of design has a number of shortfalls; the time history analyses are time
consuming to complete, the analyses consider only a limited number of possible ground
motions, and the engineers carrying out the designs require specialized training in structural
dynamics beyond that of the typical structural engineer. Before the structural engineering
community as a whole can readily incorporate friction dampers into building designs, a
straightforward approach, which requires no greater effort than is required for current seismic
design, needs to be developed. This thesis has as one of its objectives the development of
such a design approach.
This thesis presents the results of an analytical parametric study completed on the effect
friction dampers have on the response of single-degree-of-freedom systems subjected to
earthquake ground motions. The study considers the ground motions of five different
earthquakes and has as its primary objectives:
The collection of data on characteristics key to the design of friction damping
mechanisms, including the number of slip cycles experienced, the amount of energy
dissipated, and the maximum rate of energy dissipation occurring in single-degree-of-
freedom systems of varying periods during each of the earthquakes studied.
A study of the effects damper slip force and secondary system stiffness has on the
overall displacement of the single-degree-of-freedom systems. The final objective of
this portion of the study is the development of an inelastic design spectrum that could
be used in the design of multi-degree-of-freedom friction damped structures.
A study of the correlation between the inelastic deformation of multi-degree-of-
freedom structures fitted with friction dampers and the corresponding displacements
6


calculated using modal analysis techniques and the inelastic response spectra
developed in the study.
A secondary objective of the study is the development of a general design approach,
based on current code prescribed design basis earthquakes, that can be used by Structural
Engineers in the design of friction damped structures.
This thesis begins with a general discussion in Chapter 2 of friction dampers and some of
the earlier research completed on friction dampers. The concept of an optimum level of total
friction force distributed throughout the bracing system of a structure is presented along with
a method developed by researchers to determine the optimum force. The chapter concludes
with a short discussion on areas where further research on friction dampers is needed.
Chapters 3 and 4 deal primarily with the parametric study completed on SDOF systems.
Chapter 3 presents an overview of the parametric study and a discussion of the analytical
modeling techniques used to complete the study. A representative code prescribed
earthquake is established for each of the earthquakes considered and a method to define the
damper slip force in terms of the representative code prescribed earthquake and an equivalent
slip coefficient is presented. Chapter 4 presents the results of the parametric study and the
method used to generate the inelastic response spectra for each of the earthquakes.
Chapter 5 presents the results of the study completed on MDOF systems. The results of
the inelastic dynamic analyses completed for a ten story friction damped moment frame are
presented along with a comparison of the analysis results and the deflections and story drifts
calculated using the inelastic spectra developed in the study. A method for estimating the
maximum story drifts and overall building displacements is presented.
7


Chapter 6 presents conclusions and a general discussion of the study results along with a
proposed design approach that could be used by Structural Engineers in designing friction
damped structures. The design approach is based on current code prescribed earthquakes, the
optimum slip coefficient for the structure, and the inelastic deformation response calculated
by the methods presented in Chapter 5. The chapter concludes with a discussion regarding
areas where additional study on friction dampers might be warranted.
The dynamic analyses of single-degree-of-freedom systems presented in this thesis were
completed using the personal computer based program Response developed by the Author
at the University of Colorado at Denver to study the seismic response of friction-damped
structures. A general description of the program is presented in the Users manual included
as Attachment B to this thesis. The analyses of multi-degree-of-freedom systems were
completed using the computer program Matlab and code written by the Author.
8


2. Previous Research
2.1 Introduction
In their simplest form, friction dampers can be simple bolted framing connections with
slotted holes and pre-tensioned bolts. As long as a friction force can be developed between
the plies of the connection and slippage takes place, the damper will dissipate energy.
However, because friction dampers are critical elements in the lateral load resisting system of
structures, the dampers must be capable of dependable operation and must exhibiting stable
response during a large number of slip cycles without failure. Additionally, to allow
engineers to design structures fitted with friction dampers, the response of the dampers must
be predictable.
The surface conditions of the materials used to fabricate friction dampers plays an
important role in their overall performance. Researchers have studied the performance
characteristics of simple slotted bolted connections made up of sliding plates with both steel
on steel and steel on brass sliding surfaces and found the characteristics to be erratic and
unpredictable, Grigorian et al. (1993). In an effort to develop dampers with stable and
predictable performance characteristics, a variety of damper surface treatments were studied
under static and cyclic loading conditions, Pall et al (1980). The surfaces studied included
plain mill scale, sand and grit blasted surfaces, surfaces with metalized and zinc-rich painted
finishes, and common automotive break lining materials. Of the surface treatments studied,
the automotive break lining materials proved to have the most stable and predictable
performance characteristics. The break lining materials were found to produce a nearly
9


constant friction force during damper displacement and a nearly rectangular hysteresis loop
under cyclic loading. Additionally, researchers found that the use of the break lining
materials provided stable damper performance with negligible fad after 50 cycles of loading,
Fihatrault and Cherry (1987).
Considering the untold hours of research and development expended by the automobile
industry in developing current brake lining materials, it is not surprising that the materials
provide stable and predictable performance in friction dampers for structures. Automotive
brake lining materials are well suited for use in damping mechanisms because the physical
concepts involved in both automobile braking and friction dampers are essentially the same.
Both systems function by dissipating energy through coulomb friction.
The operation of a friction damper is based on the simple concept of dry or Coulomb
friction whereby the force necessary to slide one body past another is equal to the product of
the coefficient of friction for the materials in contact and the normal force maintaining
contact of the two surfaces;
fsl=MN (2.1)
As the two surfaces are slid past one another, energy is dissipated in the form of heat. The
quantity of energy dissipated is equal to the work completed in sliding the surfaces such that;
E = fslL (2.2)
where L is equal to the distance the two surfaces slide relative to one another.
From Equation (2.2) the amount of energy dissipated by a friction damper is a function
of the friction force and the distance the damper slips. The maximum distance a damper can
slip is typically limited by the amount of interstory drift that can be tolerated by a structure.
Additionally, the displacements induced in a friction-damped structure are related to the slip
10


force specified for the friction dampers present in the structure. Because of this relationship,
efforts to maximize the amount of energy dissipated by friction dampers have concentrated
on studies of the variation of slip force specified for the damper. In an attempt to optimize
the amount of energy dissipated by friction dampers, researchers have proposed a number of
damper configurations. The various damper configurations can typically be classified as one
of two basic types; configurations that maintain a constant force throughout the full range of
damper slip, and configurations that incorporate a friction force that varies as a function of
damper displacement. In damper assemblies with varying friction forces, the variations are
typically achieved by varying the normal force between the damper friction pads and the
inner surface of cylindrical casings by using springs. The assemblies are arranged such that
an increase in damper displacement results in an increased damper friction force. Damper
assemblies that are based on a constant friction force are typically much simpler mechanisms
with few moving parts other than the plies of the damper. These dampers are arranged such
that a constant normal compressive force is maintained across the plies of the damper as they
slid relative to one another.
The theoretical responses of constant force and variable force friction dampers under
cyclic loading conditions are presented in Figure 2.1. The energy dissipated by each of the
dampers is equal to the area inside the hysteresis loop for each damper. Referring to Figure
2.1, the greater effectiveness of the constant force dampers is apparent. From the figure, it
can be seen that with the same maximum slip force and displacement occurring in both
dampers, the constant force dampers dissipate four times the energy of the varying force
damper. The effectiveness of variable force dampers can be increased by providing a
pretension force in the internal springs of the damper such that a hysteresis loop shown by the
11


dashed lines in Figure 2.1 (b) is developed. However, it can be seen that the results are less
than that of the constant force damper. The studies completed in this thesis consider only the
response of structures fitted with constant force dampers.
(a) (b)
Figure 2.1 Response of friction dampers under cyclic loading, a) Constant force damper, b) Variable
force damper.
In addition to the various damper configurations proposed, researchers have also
proposed a number of methods for incorporating friction dampers into the framing systems of
building structures. These have included unique configurations such as the installation of
dampers at the column-beam joints in moment resisting space frames to eliminate the need
for vertical bracing and to maintain the open area provided by the space frame system ,Way
(1996) however, the most common and perhaps the simplest method of incorporating friction
dampers into the design of a structure is to incorporate the dampers directly into a vertical
bracing system. The studies completed by Filiatrault and Cherry (1987 & 1990) considered
friction damped moment frames that combine both a moment resisting frame and a vertical
bracing system into the overall building structure. This type of a combined system provides
an elastic restoring force that acts to limit the overall deflection of the friction dampers and
12


help restore the structure to near its original undeformed shape. Through the various
configurations studied, researchers have shown that the methods for incorporating friction
dampers into building structures are limited only by the fundamental requirement that the
dampers be located in an area of the structure where differential movement and internal
mechanism forces can be generated during strong earthquakes.
Currently, friction dampers are gaining acceptance in the engineering community and
have been used in the new design or retrofit of a number of structures in Canada, Japan, and
the United States between the 1970s and the present. The dampers have been installed in
both steel and reinforced concrete structures varying in height from one to thirty-one stories
and have been installed in such varied structures as buildings, elevated water towers, and
supports for electrical circuit breakers. Table 2.1 presents a partial list of friction damper
installations in Canada, Japan, and the United States.
Table 2.1 Friction Damper Installations
Building/Structure Structure Type Location Damper Type Date
Gorgas Hospital - Panama Friction Dampers 1970's
McConnel Building, Concordia University library complex, Reinforced Concrete, 6 and 10 Story Bldgs Montreal, Canada Pall Friction Dampers 1987
Residential House Wood Stud, 2 Story Montreal, Canada Pall Friction Dampers 1988
Sonic Office Building Steel, 31 Story Omiya City, Japan Sumitomo Friction Dampers 1988
Asahi Beer Azumabashi Building Steel, 22 Story Tokyo,Japan Sumitomo Friction Dampers 1989
Ecole Polyvalantc Precast Concrete, 3 Story Sorel, Canada Pall Friction Damper and Pall Friction Panels 1990
Canadian Information and Travel Center Steel, 4 Story Laval, Canada Pall Friction Dampers 1992
Department of Defense Reinforced Concrete, 3 Story Ottawa, Canada Pall Friction Dampers 1992
13


Table 2.1 (Cont.)
Building/Structure Structure Type Location Damper Type Date
Canadian Space Agency Steel, 3 Story St. Hubert, Canada Pall Friction Dampers 1993
Casino de Montreal Steel, 8 Story Montreal, Canada Pall Friction Dampers 1993
Building 610, Stanford University Brick and Stucco, 1 Story Palo Alto, California Slotted Bolted Connections 1994
Hoover Building, Stanford University 2 Story Palo Alto, California Slotted Bolted Connections 1994
Maison 1 McGill Reinforced Concrete, 11 Story Montreal, Canada Pall Friction Dampers 1995
Ecole Technologie Superieure Steel Montreal, Canada Pall Friction Dampers 1995
Federal Building Reinforced Concrete, 4 Story Sherbrooke, Canada Pall Friction Dampers 1995
Desjardin Life Insurance Building Reinforced Concrete, 6 Story Quebeo, Canada Pall Friction Dampers 1995
Overhead Water Tank Steel Beaux Arts, Washington Pall Friction Dampers 1995
St. Luc Hospital Reinforced Concrete, 8 Story Montreal, Canada Pall Friction Dampers 1995
Residence Maison-Neuve Steel, 6 Story Montreal, Canada Pall Friction Dampers 1996
Hamilton Courthouse Steel, 8 Story Hamilton, Canada Pall Fiction Dampers 1996
Water Towers, University of California at Davis Steel Davis, California Pall Friction Dampers 1996
Harry Stevens Building Reinforced Concrete, 3 Story Vancouver, Canada Pall Friction Dampers 1996
Justice Headquarters Reinforced Concrete, 8 Story Ottawa, Canada Pall Friction Dampers 1996
BCBC Selkirk Waterfront Office Buildings Steel, 5 Story Victoria, Canada Pall Friction Dampers 1997
Maisons de Beaucours Reinforced Concrete, 6 Story Quebec City, Canada Pall Friction Dampers 1997
Maison Sherwin William Reinforced Concrete, 6 Story Montreal, Canada Pall Friction Dampers 1997
Constantinou, M.C., Soong, T.T., and Dargush, G F. (1998)
14


2.2 Pall Friction Dampers
Of the installations of friction dampers throughout Canada and the United States, the vast
majority have been based on the damper arrangement developed by Pall and Marsh in 1982.
Considerable studies have been carried out at the University of British Columbia in
Vancouver, British Columbia on the effectiveness of the damper arrangement at reducing the
seismic response of structures. Filiatrault and Cherry (1990), studied the effect variations in
damper slip force had on the overall response of structures fitted with the dampers. Their
studies resulted in the development of a basic design approach for structures fitted with the
damper assemblies and a method for determining the damper slip force that minimized the
seismic response of the structures. Because of the great deal of study centered on the damper
assembly, a basic description of the assembly is instructive.
The dampers, known as "Pall Friction Dampers", are well suited to flexible tension-only
bracing systems, although they can also be installed in vertical bracing systems made up of
members capable of resisting both tension and compressive loads. The basic arrangement of
the system is shown in Figure 2.2. The assembly consists of a series of links and dampers
incorporated into the intersection of vertical cross bracing members. As a frame fitted with
common tension-only bracing is displaced laterally, only the bracing members carrying
tensile loads are effective due to buckling of the relatively flexible compression members. In
a system fitted with Pall Friction Dampers, as the frame displaces laterally the bracing
members carrying tensile loads act to pull the links of the damper into a rhomboid shape.
This results in slippage of the friction dampers and a shortening of the compression braces.
15


As the motion of the structure reverses and the frame displaces in the opposite direction, the
bracing members earlier in compression now carry tensile loads and act to pull the damper
assembly into the opposite direction again resulting in slippage of the dampers and shortening
of the bracing members now carrying compressive loads. The arrangement of the system
eliminates buckling of the compression braces and allows the dampers to effectively dissipate
energy as the motion of the structure reverses. If friction dampers were simply installed into
the vertical bracing members, loads sufficient to cause slippage of the dampers could not be
developed in the slender compression members due to buckling and the overall damper
performance would be less than that developed in the Pall system. The reader is referred to
Pall and Marsh (1982) for a more thorough discussion of the system.
Figure 2.2 Pall friction damper (a) At rest; (b) Displaced configuration
The damper arrangement developed by Pall and Marsh is effective at dissipating energy
and has been extensively studied, however it is not the only arrangement possible for friction
dampers. As previously stated, researchers have proposed a number of different damper
arrangements and methods for incorporating the dampers into the framing system of
structures. The system is presented in detail only to give the reader an understanding of the
basic arrangement before presenting the results of studies completed on the system.
16


2.3 Optimum Slip Force
Filiatrault and Cherry (1990), studied the effect variations in slip force had on the overall
seismic response of structures fitted with Pall Friction Dampers. They measured the
effectiveness of the dampers through a relative performance index defined as:
RPI =
SEA U
\SEAm j
(2.3)
In equation 2.3, SEA is equal to the summation of the instantaneous strain energy present
in the friction damped structure during the duration of the earthquake and is equal to the area
under the strain energy time history plot. SEAm is equal to the summation of the
instantaneous strain energy present in an identical structure with the friction damper slip force
set equal to zero. Umax is equal to the maximum strain energy occurring at any time during
the duration of the earthquake in the friction damped structure and Umax{Q) is equal to the
maximum strain energy occurring in an identical structure with the friction force set equal to
zero.
The relative performance index provides a means of comparing the response of a fully
elastic system to that of the same system with friction dampers added and provides a measure
of the effectiveness of the friction dampers at reducing the response of the structure. A value
of 1.0 corresponds to the response of the fully elastic structure before friction dampers are
added. An RPI value less than 1.0 indicates that the response of the structure has been
decreased by the installation of the dampers. A value greater than 1.0 would indicate the
response has been increased.
17


The calculations involved in determining the relative performance index are presented by
way of an example. Figure 2.3 presents strain energy time history plots for two SDOF
systems and are representative of those that would be used to determine the relative
performance index for a structure.
Strain Energy
Figure 2.3 Strain energy content of systems with and without friction dampers
The time dependent variations of strain energy present in the friction damped and elastic
structures are determined by completing time history analyses considering the structural
properties of the systems, the slip force specified for the dampers, and the earthquake ground
motion under consideration. Based on the definition of strain energy used by the researchers,
the strain energy present in the structures at an arbitrary instant in time represents the total
amount of recoverable energy stored in the system and is equal to the sum of the kinetic and
potential energy present such that;
Es(t) = EK(0 + Es(t) (2.4)
where;
18


EK{t) = ^m{ii{t))2
(2.5)
and,
Es{t) = k(u{t))2 (2.6)
From Figure 2.3, it can be seen that Umax(0> is equal to 280 in-lb and occurs at 4.6 seconds
and Umax is equal to 147 in-lb and occurs at 2.06 seconds. The areas below each plot were
calculated yielding values for SEAm and SEA of 676 and 284 in-lb-sec respectively. Using
these values and equation (2.3), the resulting RPI value is determined to be 0.47 for the
structure and damper slip force considered.
It should be noted that the definition of strain energy used by the researchers and that
presented in Chapter 1 are different. It is assumed that the intent of the researchers is to use
the value for SEA and SEA(0) based on the total recoverable energy content of the systems. In
strict terms, the strain energy of the system is defined by Equation (2.6) and does not include
a kinetic energy component. The difference however is only applicable to the computation of
SEA and SEAm because at the peak displacement of the structure, the total stored energy is
equal to the strain energy of the system.
The example calculations presented above consider only one value for the damper slip
force. Filiatrault and Cherry considered the effect variations in the damper slip force had on
the resulting relative performance index. They completed a series of time history analyses
using the computer program DRAIN-2D, A. E. Kanaan, G. H. Powell (1975), and varying
levels of slip force to determine the level of slip force that resulted in minimum relative
performance indices for a variety of structures. Figure 2.4 presents a representative graph of
19


their findings for one such study. The graph shows the variation of the relative performance
index as the slip force is varied.
Relative Performance Index
Figure 2.4 Slip load optimization (Reproduced from Cherry, S. and Filiatrault, A., 1993)
Filiatrault and Cherry found that there was an optimum level of slip force associated
with a given structure and a predominate frequency of earthquake ground motion.
Additionally, they found that there was very little variation in the relative performance index
at slip forces within a relatively wide range near the optimum level. The researchers
concluded that the response of the subject structure was not particularly sensitive to 10 to
15 percent variations in the optimum slip load. In an earlier study completed by the same
researchers, they found that for an optimum slip load of 134 kN, there was very little
variation in the relative performance index from slip loads between 90 and 220 kN. These
results indicate that small variations in damper material properties and installed damper slip
loads would have little effect on the overall response of structures during strong earthquakes.
20


2.4 Optimum Slip Force Design Spectrum
Based on the concept of a relative performance index, Filiatrault and Cherry (1990)
completed a parametric study of the response of friction damped multi-story structures in an
attempt to develop a correlation between four key input parameters and the optimum level of
friction force. The input parameters considered in the study included the number of stories,
NS; the ratio of the braced to unbraced natural periods for the structure, Tb/Tu; the ratio of the
natural period of the braced structure to the predominant natural period of the ground motion,
Tb/Tg; and the peak ground acceleration occurring during the earthquake, a/g. The study
considered a total of 45 structures and the values listed in Table 2.2 for the input parameters.
Table 2.2 Filiatrault and Cherry Study Parameters
Parameter Value
NS 1,3,5, 10
t/tu 0.20, 0.40, 0.60, 0.80 for NS = 1 0.20, 0.50, 0.80 for NS = 3, 5, 10
T/ru 0.1 sec/T,,; 0.7sec/T; 1,4sec/Tu; 2.sec/Tu
Ag/g 0.005, 0.05, 0.10, 0.15, 0.20, 0.30, 0.40 for NS = 1 0.05, 0.10, 0.20, 0.40 for NS = 3, 5, 10
In the study, the optimum slip force for the multi-degree-of-freedom structures, V0 was
considered by the researchers to be equal to the sum of the slip forces specified for the
dampers located at each story of the structure such that;
NS
(2-7)
i~ 1
where v, is the slip force for the damper assembly located at the Ith level of the structure.
They further proposed using an equal story slip force such that;
21


(2.8)
v. =
K_
NS
The time history analyses completed in the study were based on artificial earthquake
ground motions developed specifically for each set of input parameters of peak ground
acceleration and period of earthquake ground motion. The reader is referred to Filiatrault and
Cherry (1990) for a discussion on the development of the artificial ground motions.
Based on the results of the study, Filiatrault and Cherry proposed the following equations
to define the optimum slip force:
(-1.24^-0.31)7;
+ 1.04MS + 0.43
for
ma_
T
0<^- T
(2.9a)
and,
(0.0 IMS+ 0.02)7
-125NS 0.32
Tl
ma_
(0.002 0.002Atf )r
+1.047/5 + 0.42
for
T
(2.9b)
Figure 2.5 presents Equations (2.9) in terms of an optimum slip force design spectrum.
Figure 2.5 Optimum slip force design spectrum
22


The optimum slip force design spectrum developed by Filiatrault and Cherry provides
engineers the means to determine the level of slip force that will minimize the seismic
response of a structure. The spectrum requires input values based on the properties of the
structure, the peak ground acceleration expected at the building site, and the predominant
period of the earthquake ground motion. The braced and unbraced natural periods are
calculated from the physical properties of the structure and the peak ground acceleration is
defined by the local building code. The period of the earthquake ground motion is not readily
available in most cases. To provide the engineer with a method to determine an appropriate
value, the researchers suggested that the methods proposed by Vanmarcke and Lai (1980) be
used to estimate the value as;
T =------------ \0bn * 27-0.09/?^
or,
T=------------- 5 < M, < 7 (2.11)
* 65 1.5M L
In Equations (2.10) and (2.11), REQ is equal to the distance to the epicenter of the earthquake
in km, and ML is the Richter magnitude of the earthquake.
2.5 Simplified Design Approach Proposed by Filiatrault and Cherry
Based on the optimum slip force design spectrum, Filiatrault and Cherry (1990) proposed
a simplified seismic design procedure for structures fitted with the Pall friction dampers. The
design procedure consists of the following steps:
1. The main framing system of the structure is designed as a moment frame
proportioned to carry vertical gravity loads only. It is assumed that the vertical
23


bracing system fitted with friction dampers will safely dissipate all earthquake energy
so lateral loads need not be considered in the design of the moment frame. The
natural period of the moment frame is calculated and taken as the period of the
unbraced structure Tu.
2. Vertical bracing members are selected and the natural period of the braced moment
frame Tb is calculated. The vertical bracing members are selected such that the ratio
Tt/Tu falls within the range 0.20 and 0.80. The researchers suggest proportioning the
bracing members such that the ratio is less than 0.40 if economically feasible. The
range of 0.20 to 0.80 represents reasonably practical limits and matches that used in
the parametric study.
3. The earthquake parameters ag and Tu are determined for the building site. Typically,
the value of the peak ground acceleration is available from building codes. The
researchers suggest using the equations presented by Vanmarcke and Lai to estimate
the predominate period of ground motion.
4. The optimum slip force for the entire building is calculated from the design slip force
spectrum and the slip force is distributed equally to each level of the structure.
5. The capacity of the vertical bracing members are calculated and compared to the
loads resulting from the distributed slip forces. If necessary, new vertical member
sizes are selected and steps 2 through 5 are repeated.
6. A wind load analysis is completed for the braced moment frame to verify that the
dampers will not slip during wind loading conditions. If wind loads induce damper
slip, the moment frame is modified to carry a larger portion of the wind loads and
steps 2 through 6 are repeated.
24


The design approach is valid only if the structure and earthquake parameters fall within
the range of those considered in the original parametric study such that;
0.20 < < 0.80 0.05 < <20 0.005 < <0.40 NS <10
Tu T g
2.6 Areas Requiring Additional Study
The simplified design procedure proposed by Filiatrault and Cherry provides engineers
with a means of determining the optimum slip force and distribution of damper forces
throughout the structure and can be used for new design and for retrofitting existing
structures. The procedure focuses primarily on the determination of the level of friction force
required to minimize the seismic response of the structure. It does not however provide a
method for evaluating the seismic loads induced in the moment frame and furthermore does
not provide a means to evaluate the inelastic deformation characteristics of the friction-
damped structures.
Two distinct systems are present in friction-damped structures; the bracing system fitted
with dampers, and the elastic moment frame. The forces carried through the bracing system
are controlled by the slip forces specified for the associated dampers. Because of this, the
design approach proposed by Filiatrault and Cherry provides an effective means of designing
the vertical bracing system fitted with dampers. Forces carried by the elastic moment frame
however are controlled by the displaced shape of the frame. Proper design requires that the
forces induced by the lateral displacement of the frame be included. Engineers must have the
ability to estimate the overall displacement of the structure in order to determine these loads.
Additionally, engineers must have a means of estimating the displaced shape of the structure
to evaluate loads resulting from P-Delta effects and to verify that interstory drifts do not
25


become so great as to adversely effect nonstructural elements of the building. Because of
this, additional study is required regarding the deformation characteristics of friction-damped
structures.
The second area that requires additional study deals with the performance requirements
of the damper assemblies themselves. Friction dampers dissipate the input energy of an
earthquake by transferring the kinetic and strain energy present in the structure to thermal
energy. To operate in a stable manner, the dampers must be capable of dissipating this
thermal energy without overheating. Proper design of damper assemblies must therefore take
into consideration the total amount of energy to be dissipated and the maximum rate at which
the energy is to be dissipated. Because of this, additional studies are required to determine
the energy dissipating requirements of friction dampers. Additionally, to fully account for the
repeated loading conditions experienced by friction dampers, engineers must have the ability
to estimate the total number of cycles of loading expected to occur during the design
earthquake. Studies related to damper assembly design requirements should include the
number of slip cycles experienced.
26


3. Overview of Parametric Study
3.1 Introduction
In this chapter, an overview of the parametric study of friction-damped single-degree-of-
freedom systems will be presented along with the development of the analytical modeling
methods used to carry out the study. The overall goals of the study along with the structural
and earthquake parameters considered will be discussed in Section 3.2. The properties of
friction damped moment frames will be developed and the SDOF system model considered in
the study along with the equations governing the time dependent response of the frames will
be presented in Section 3.3. Section 3.4 will present the methods used in the study to solve
the equations of motion. Additionally included in Section 3.4 will be a discussion of the
energy content of the systems during earthquake loading and the components of response for
inelastic systems. Section 3.5 will present the earthquake ground motions considered in the
study. Current code prescribed earthquakes and elastic response reduction factors will be
discussed in Sections 3.6 and 3.7. The discussion will lead to the development in Section 3.8
of representative code prescribed earthquakes for each of the earthquakes considered to allow
correlations to be made between the inelastic response of the frames and the earthquake
parameters of peak ground acceleration and one-second spectral acceleration normally
specified by building codes.
3.2 Objectives and Scope of Parametric Study
The primary goal of the parametric study presented in this thesis is to develop a method
for estimating the inelastic deformation response of single-degree-of-freedom friction-
27


damped structures based on known input parameters of the structure and the earthquake under
consideration. The structural parameters considered in the study include the structures
natural period prior to slippage of the friction dampers, the slip force specified for the
dampers, and the amount of elastic secondary stiffness present in the structure during damper
slip. The parameters of the earthquake considered in the study include the peak ground
acceleration and one-second spectral acceleration associated with the earthquake. The
earthquake parameters are selected to correspond to those currently used by building codes in
defining design basis earthquakes.
The approach used to develop a method of estimating the inelastic response of friction-
damped structures is to first develop inelastic response spectra for SDOF systems with a
small amount of secondary stiffness present in the structure during damper slip. For this
portion of the study, the inelastic responses of SDOF systems with 0 and 5 percent elastic
secondary stiffness and four levels of slip force are evaluated and a correlation is developed
between the earthquake input parameters and the responses. Based on the results of this part
of the study, a method that can be used to formulate the inelastic response spectra for SDOF
systems with small amounts of secondary stiffness is presented. Once the relationship
between the earthquake input parameters and the inelastic response spectra are developed, an
evaluation of the effects of secondary stiffness is completed. In this part of the study, a series
of inelastic time history analyses are completed for the same systems considered in the first
portion of the study with secondary stiffness ratios varying between 0 and 100 percent.
Again, the correlation between the amount of secondary stiffness and system response is
developed and a method to formulate inelastic response spectra for systems of varying
secondary stiffness is presented.
28


The ultimate goal of the study of the inelastic response of SDOF systems is to develop a
method that could be used to estimate the displaced shape of multi-degree-of-freedom
friction-damped structures subjected to earthquake ground motions. The goal is to develop a
method that is based on the inelastic response spectra developed in this part of the study and
standard modal analysis techniques currently used in the analysis of multi-degree-of-freedom
systems. A method that could be used to estimate the displaced shapes of multi-degree-of-
freedom structures is presented in Chapter 5 along with a study of the correlation between the
estimated displacements and those determined by inelastic time history analyses.
An additional goal of the study of SDOF friction damped structures is the quantitative
evaluations of key parameters that affect the design of friction damping mechanisms. These
parameters include the number of slip cycles experienced by the dampers, the maximum
amount of energy dissipated, and the maximum rate that the dampers dissipate energy during
earthquakes. To determine the values associated with each of the parameters, inelastic time
history analyses are completed for SDOF systems with natural periods ranging from 0.25 to
2.5 seconds and four levels of damper slip force. For this portion of the study, only systems
with zero secondary stiffness are considered.
In all, the study consists of three parts; the determination of inelastic response spectra for
SDOF systems with small levels of secondary stiffness; the study of the effects secondary
stiffness has on the overall deformation response of the SDOF systems, and the quantitative
evaluation of the parameters affecting the design of friction damping mechanisms. The
parameters considered in the study include; the natural period of the system prior to damper
slip, TN; the amount of secondary stiffness present in the system during damper slip, defined
as the ratio of the moment frame stiffness to the total structure stiffness, (j); and the slip force
29


specified for the dampers in terms of a slip coefficient, R. Table 3.1 presents the values of
the parameters considered in each portion of the study.
Table 3.1 Study parameters
Parameter Parameter Value
Inelastic Spectra Secondary Stiffness Effects Damper Design
Tn (sec) 0 to 3 0.25,0.50, 0.75, 1.0, 1.25, 1.50, 1.75,2.0, 2.25,2.5 0.25,0.50, 0.75, 1.0, 1.25, 1.50, 1.75,2.0,2.25,2.5
0, 5 % Oto 100% 0%
R 4, 6, 9, 12 0 to CO 4,6, 9, 12
3.3 Friction-Damped Moment Frames
The damper configuration considered throughout this thesis is shown in Figure 3.1. The
system consists of a moment frame, possessing a lateral stiffness kF, fitted with a vertical
bracing system that adds an additional stiffness, kD to the system. The vertical bracing is
connected to the beam of the moment frame through a friction damper assembly. The friction
damper assembly is proportioned such that it will begin to slip when the force transferred
through the damper is equal to a specified damper slip force, ft. The slip force is equal to the
product of the normal compressive force applied across the slip planes of the damper and the
coefficient of friction of the materials used to fabricate the damper.
Figure 3.1 Friction-damped moment frame.
30


The idealized force-deformation relationship for the combined vertical bracing and
damper under cyclic loading conditions is shown in Figure 3.2. The system considered in the
figure consists only of the vertical bracing system present in the friction damped moment
frame and a damper with load-deformation properties consistent with those presented in
Figure 2.1(a). The system begins in an unloaded state at point 0 in the figure. The system
is then displaced in the positive direction. While the forces transferred through the damper
are below the slip force, the assembly responds in a linearly elastic fashion. In this region of
response, identified by the portion of the graph between points 0 and a, the damper
displacement is related directly to the force transferred through the assembly such that the
displacement at which slippage of the damper occurs can be defined as;
Figure 3.2 Idealized combined damper/framing system behavior
Once the displacement of the damper assembly exceeds the slip displacement Dsh the
force transferred through the damper assembly remains constant and equal to the slip force as
the plies of the damper slide relative to one another. During this portion of the response, the
31


force transferred through the damper assembly remains constant until the motion of the
damper is stopped at point b, at which time the damper plies become locked into position
by the friction force developed in the damper. As the direction of motion of the system
reverses, the force carried by the bracing system decreases until the bracing system returns to
its unloaded state. This condition is identified by point c on the graph. At this point, the
damper has slipped a distance equal to the segment Oc and the energy dissipated by the
damper assembly is equal to the area of the hysteresis loop bounded by OabcO. As the
motion of the system continues, the force transferred through the damper assembly again
increases until slippage takes place in the negative direction at point d. Slippage of the
damper again takes place until the motion stops at point e at which time the damper plies
again become locked into position and another unloading cycle begins for the bracing system.
The motion of the damper assembly effectively limits the lateral deflection of the vertical
bracing system to a maximum of Dsi. Displacements of the system mass greater than the slip
displacement are accompanied by slippage of the damper and a lateral displacement of the
vertical bracing system equal to Z)s!. The behavior of the vertical bracing system with the
damper assembly added is the same as that of an idealized elastic-perfectly plastic material
with the exception that yielding of the members does not occur. In a properly designed
friction damped structure, the braces are designed to carry a load somewhat higher than that
transferred through the damper assembly.
The friction-damped moment frame shown in Figure 3.1 possesses two levels of stiffness.
While the lateral displacement of the damper assemble is less than the slip displacement, the
bracing and moment frame act together and the total stiffness of the combined system is equal
32


to the sum of the moment frame stiffness, kF and the stiffness of the bracing system, kD such
that;
kT kf+ kD uD< DSi (3.2)
If the lateral displacement of the damper assembly is greater than the slip displacement of the
damper, the stiffness contributed to the overall system by the bracing system decreases to
zero and the total system stiffness is then equal to that of the moment frame alone such that;
kF = kf Ud^ D$i (3.3)
The force-displacement relationship for the combined system, assuming only motion in the
positive direction, is shown in Figure 3.3. The change in stiffness for displacements greater
than the damper slip displacement is apparent in the figure. Note that in the combined
system, if the motion of the system stops such that the damper becomes locked into position
and then reverses direction, the structure unloads along the dashed line shown in the Figure
3.3.
Figure 3.3 Force-Deformation relationship for friction damped moment frame.
It is convenient to define the ratio of the stiffness contributed by the moment frame to
the total combined system stiffness as the secondary stiffness ratio, ^ such that;
(= ki~ (3.4)
kT
33


A value of zero for the secondary stiffness ratio indicates a framing system that relies only on
the vertical bracing to resist lateral loads such as in the case of an ordinary braced frame fitted
with friction dampers. In such a system, there would theoretically be no elastic stiffness
present at displacements greater than the slip displacement of the damper assembly. In real
world structures however, there would typically be some level of secondary stiffness due to
the inherent stiffness of framing connections, interior partitions, wall panels, etc. Although
some secondary stiffness would be present, the amount would be small in comparison to the
stiffness of the system prior to slippage of the friction dampers and is neglected in the studies
completed for systems with secondary stiffness ratios equal to zero. A value of 1.0 for the
secondary stiffness ratio indicates a framing system with no vertical bracing or dampers
present. Such a system would consist only of the moment resisting frame. Values between 0
and 1.0 represent a combined system with both a moment resisting frame and a vertical
bracing system fitted with dampers.
Along with two levels of stiffness, friction-damped moment frames posses two distinct
natural periods of vibration. The natural period of vibration of the combined framing system,
prior to slippage of the dampers, is a function of the total system stiffness and is defined as;
Tb=2n
n D < A
SI
(3.5)
During the time slippage is taking place in the damper assembly, the stiffness of the
system decreases to kF and the natural period increases to;
Tu = 2 rc
uD> DSi
(3.6)
34


In terms of the secondary stiffness ratio , the ratio of the braced to unbraced natural periods
can be expressed as;
We now consider the response of the system shown in Figure 3.1 at an arbitrary instant
in time while subjected to an earthquake strong enough to cause slippage of the dampers.
Under these conditions, the system is as shown in Figure 3.4. It is assumed that the forces
transferred through the damper assembly have previously resulted in at least some slippage of
the damper. Two distinct displacements exist within the system; the displacement of the
mass and the displacement of the friction damper. The displacement of the mass is equal to
u(t) and the displacement of the damper is equal to iio(t).
u(t)
rt
Figure 3.4 Friction-damped moment frame at arbitrary instant in time.
The combined system is shown schematically in Figure 3.5 where the stiffness of the
moment frame and vertical bracing systems are replaced by individual springs with
stiffnesses of kF and kD. The specified damper slip force is included in the model of the
vertical bracing system as fS\. The viscous damping present in the system is modeled by a
dashpot with a viscous damping coefficient c. The system is shown in an unloaded state in
Figure 3.5(a) and at an arbitrary instant in time in Figure 3.5(b). A DAlembert free-body
35


diagram showing the forces acting on the mass at the time under consideration is shown in
Figure 3.5 (c).
a u.
/-cshMft-
r '-HI i (
/ \A\A l* m
y. Ke L2 id:
\V
(a)
Figure 3.5 Schematic
The individual forces acting
as;
(b) (c)
representation of friction damped structure
on the mass due to the combined framing system can be written
/rf-(0 = M(0 u^uy (3-8a)
fA) = fy u>uy (3.8b)
where fsF is equal to the force carried by the moment framing system, fy is equal to the yield
strength of the frame, and uy is equal to the yield displacement.
/,d(0 = *dd(0 UD^ltsl (3-9a)
fsD (0 = fs! UD>Ua (3'9b)
where/sD is equal to the force carried by the vertical bracing system,/ji is equal to the
specified slip force for the damper assembly, and Mst is equal to the damper slip displacement
as previously defined, and;
fD(t) = cu(t) (3.10)
where fD is equal to the viscous damping force. The inertia force acting on the mass during
the earthquake, Pt(t) can be written in terms of the earthquake ground acceleration m (/) as;
P (0 = -/ms(0
(111)
36


The equation governing the time dependent motion of the system is obtained by summing
the forces acting on the mass and applying Newtons second law of motion such that;
IF = mii(t) = P(t)-cu(t)-kFu(t)-kDuD{t) (3.12)
or with Equation (3.11) substituted for the forcing function, P(t) the equation governing the
earthquake response of the system becomes;
mii(t) = -rniig(t) cu(t) kFu{t) kDuD{t) (3-13)
Solution of Equation (3.13) yields the time dependent displacement, velocity, and
acceleration of the system mass along with the time dependent displacement of the friction
damper assembly. The time history analyses of SDOF systems completed in the parametric
study presented in this thesis are based on the model shown in Figure 3.5 and the solution of
Equation (3.13) by time stepping methods. The methods used to solve Equation (3.13) are
presented in the following section.
3.4 Solution of the Equation of Motion by Time Stepping Methods
3.4.1 Introduction
The dynamic response of structures subjected to continuous periodic forcing functions can be
determined by direct solution of Equation (3.12). This is not the case however for structures
subjected to earthquake ground motions because of the non-continuous nature of the motion and
the potential nonlinear response of the structures. To evaluate the response of structures
subjected to earthquake ground motions, it is necessary to use approximate time stepping
methods to determine the response of the structure. The methods are referred to as time stepping
because the overall response of the structure is determined by considering the response over a
series of short time steps. The response during each time step is dependent upon the initial
37


conditions of the system as determined by the response during the previous time step and
assumptions regarding either the variation of the forcing function or the system acceleration over
the time duration being considered. By completing analyses for a number of short time durations
it is possible to approximate the actual motion of the system during the earthquake.
In time stepping methods, the inertia force acting on a structure is transformed from the
continuous function given by Equation (3.11) to a series of discrete forces at specific instances in
time such that;
Pt =-mU". (3.14a)
and;
PM=-mugM (3.14b)
where iig. is equal to the ground acceleration at the Anstant in time and corresponds to the
start of the z'th time step and h is equal to the ground acceleration at the /th + l instant in time
and corresponds to the end of the zlh time step. Typically earthqauke ground accelerations are
recorded at intervals of 0.02 seconds. To determine the overall response of a system subjected
to 30 seconds of earthquake ground motion, 1500 time steps of 0.02 second duration need to
be considered.
The dynamic analyses of single-degree-of-freedom systems presented in this thesis were
completed using the PC based program Response developed by the Author as part of an
independent study at the University of Colorado at Denver. Response has the ability to analyze
the elastic and inelastic response of SDOF systems with and without friction dampers. The
reader is referred to the program users manual included in Appendix B for a detailed discussion
of the programs capabilities. In the program, the analyses of elastic systems are completed
38


using a time stepping method that incorporates an exact solution of the governing equations of
motion based on an assumed linear variation of the forcing function over each time step.
Inelastic analyses are completed using Newmark's Method with a linear variation of system
acceleration over each time step and automatic time step reductions at transitions between linear
and nonlinear regions. In addition to time step reductions, an iterative procedure is incorporated
into the program to assure convergence of accelerations within the transition time step. The
methods incorporated into the program to analyze elastic, inelastic, and inelastic systems with
friction dampers are presented in the following sections.
3.4.2 Elastic SDOF Systems
The response of linear systems is calculated in the program using the method of
interpolation of excitation presented in Chopra (1995). This method provides an exact
solution to the equations of motion governing the response of a single-degree-of-freedom
system subjected to a forcing function that varies linearly over the time interval being
considered.
The total system response during each time interval is comprised of three individual
responses; the free vibration response of the system subjected to initial displacement and
velocity alone; the forced response of the system subjected to a constant force, P. with zero
initial conditions; and the forced response of the system subjected to a ramp function which
varies from 0 at /( to (/)+1 P) at tj+], again with zero initial conditions. The reader is
referred to Chopra (1995) for a complete development of the method.
The displacement and velocity of the system at the end of each time step are equal to;
uM = Aut + But + C/> + DPm (3.15)
39


(3.16)
,.+1 =A'i/i+B'u.+ C'P. + D'P:.
i i + l
and the coefficients in equations 3.15 and 3.16 are calculated as;

A = e
£
Me
sinty^Ar + coscoDAt
(3.17a)
/
B = e~^"
£ \
sin(yDA/
\coD
(3Mb)
C = -
k
-?i-+ ?-*"A'
co At
\-2? 4
v 03 D
At JiM?
f -v r 5
sin coDAt
1 + -
24
V C0A tj
coscoDAt
> (3.17c)
k
24
\-^ + e
co At

fi42-\ 24
sxn coD At +--coscoDAt
\ conAt
co. At

(3 Mb)
A'=-e^'
co.

^^skuy^A/
(3.17e)
(
B'=e^
\
cosconAt 1 sin conAt
v
MI1
(3.17f)
J
CM
- + e
At
-C(onSt
CO.
£
Me &Me
sinryA/ + cos£ynAf
D At D
(3.17g)
D' =
kAt
.eM,A>
4
r= sm con At + cos con At
Me .
(3.17h)
The coefficients calculated by Equations (3.17) are functions of only the structural properties
of the system i.e. mass, stiffness, and damping ratio and need be calculated only once for the
analysis duration.
40


The initial values ui and iii are known from the previous time step or from the initial
conditions of the system in the case of the first time step. Additionally, the values of Pt and
Pi+] are determined from Equations (3.14) using the recorded ground acceleration for the
earthquake being considered.
The velocity and displacement at the end of each time step are determined by
substituting the initial conditions, coefficients as determined by Equations (3.17), and
earthquake forces as determined by Equations (3.14) into equations (3.15) and (3.16). The
process is repeated for each of the time steps throughout the duration of the analysis.
3.4.3 Inelastic SDOF Systems
Before developing the method used to analyze the inelastic response of moment frames
fitted with friction dampers, it is instructive to first consider the case of inelastic SDOF
systems without dampers. The model of such a system is the same as that presented in Figure
3.5 with/sD set equal to zero.
The response of inelastic systems is calculated in the program using Newmark's Method
as presented in Chopra (1995) and Clough (1993) with the added assumption of elastic-
perfectly-plastic material properties that exhibit the same force-deformation relationships
previously shown in Figure 3.2. The method is based on an assumed variation of system
acceleration over each time step and the added requirement that dynamic equilibrium is
satisfied at the start and end of each time step.
Newmark developed the following equations, which relate the velocity and displacement
of the system at the end of a time step to the initial conditions and an assumed variation in the
system's acceleration during the time step.
41


(3.18)
/. = / + [0 Tn )A/}', + (/VA/>C,
= u, + (A/),, + [(0.5 /i, iM)1 \c + [fiK (A/)2 )v,+, (3.19)
The parameters gamma and beta are weighting functions that control the variation of
velocity and displacement over the time step. The reader is referred to Clough (1993) for a
detailed discussion of equations (3.18) and (3.19).
The initial values u, and ui are known from the calculations for the previous time step or
from the initial conditions of the system in the case of the first time step. The initial
acceleration u, is calculated from Newton's second law as,
Y.F = mu, = Pgl fD fsF = -mugi cu, kFu,.
or
U- =
miigi cu- kFu,
m
(3.20)
This relationship is also valid for the acceleration at the end of the time step giving,
(3.21)
mUgM -cuM -kFuM
m
Substitution of equations (3.18) and (3.19) into equation (3.21), and rearranging terms,
provides an equation for the acceleration at the end of the time step in terms of the initial
conditions and the earthquake ground acceleration at the end of the time step,
- i cu, ~ ^{t1 Yn ]WK} kFu, kF jzi,. (At) + [0.5 - ](A/)2 u, j

(3.22)
\n + kFPN{ At)2 +ey(A/)]
Equations (3.18), (3.19), and (3.22) can be used directly to determine the response of
elastic systems and will yield results very close to those determined by Equations (3.15) and
(3.16). However, the analysis of nonlinear systems requires modifications to the equations
42


presented above to account for yielding. As the system transitions into the inelastic range,
the spring force is assumed to be equal to its yield strength and the stiffness of the system
becomes zero.
With yielding included in the analysis, the spring force is limited such that,
\fsF | fy (3-23)
The relationship between the displacement of the system, u and the spring force, ftF is
no longer a continuous linear function because of the limit imposed in equation (3.23). To
account for this limit, we define the effective spring displacement, ue^ such that,
fsF kFUeff
The effective spring displacement is calculated at the end of a time step as,
Ueffi+\ ~ Ueffi + Mi+1 Ui
(3.24)
(3.25)
To assure that the limit imposed by equation (3.23) is met, ueff is limited such that,
Two separate displacements are now defined; that of the system mass, u and that of the
spring, ueff Prior to the occurrence of yielding, the two displacements will be equal.
However, after yielding has occurred, the two may or may not be equal depending on the
amount and direction of the resulting yield cycles. The difference between the two
displacements represents the amount of yield deformation in the spring.
With the effective spring displacement included, equations (3.20) and (3.22) are modified
to,
43


(3.27)
m
and
u
mu
- cu. c
{[ Yn ](A/K } kFUcffi kF K (A/) + [-5 Pn KA/)2 Ui }
(3.28)
{m + kFfiN(Aty +cxa,(A/)]
For elastic-perfectly-plastic systems, the stiffness is assumed to be equal to kF when
| uejj | < uy and equal to zero when \ue^\= uy Therefore, two distinct regions of response
exist; the elastic region {\ueff\< uv), and the inelastic region {\ueff\= uy). Equations (3.25)
through (3.28) along with equations (3.18) and (3.19) are used directly to calculate the
response of a nonlinear system while within the elastic region.
While the system is within the inelastic region, the effective spring displacement remains
constant and the system stiffness is zero. With the constant spring displacement and zero
stiffness included, equations (3.27) and (3.28) can be reduce to,
Equations (3.25), (3.26), (3.29), and (3.30) along with equations (3.18) and (3.19) define the
response of a nonlinear system while within the inelastic region.
(3.29)
u.
m
and,
u
(3.30)
44


The two groups of equations presented above define the response of a nonlinear system
while the system is completely within the elastic region or completely within the inelastic
region during the time step. During either of these cases, the acceleration calculated at the
end of the previous time step will be equal to the acceleration calculated at the beginning of
the current time step. This is not the case as the system transitions from elastic to inelastic
regions or back again. Differences in the two calculated values arise because the stiffness of
the system changes during the transition time step making one or the other groups of
equations invalid for a portion of the time step. Error can be introduced into the calculated
response if the differences are not addressed. The reader is referred to Chopra (1995) for a
complete discussion.
Two procedures are incorporated into the program used in the parametric study to assure
convergence of accelerations. The first is a decrease in the calculation time step during the
transition period. After detection of a difference in ending and beginning accelerations, this
procedure divides the current calculation time step into smaller steps and recalculates the
response. The second procedure is iterative and is completed only for the shortened time step
during which the transition actually occurs.
In the second procedure, the ending velocity and displacement for the transition time step
are calculated using the initial acceleration iii for the time step occurring immediately after
the transition time step in place of the final acceleration /+l for the transition time step.
Because the initial acceleration ii- for the time step occurring immediately after the transition
time step is a function of the final acceleration iiM for the transition time step, iterations are
45


required to obtain convergence of the two accelerations. The steps of the procedure are as
follows:
1. The acceleration, velocity, and displacement are calculated at the end of the transition
time step using equations (3.18), (3.19) and (3.28) or (3.30) depending on whether
the system is within the elastic or inelastic region at the start of the transition time
step.
2. Using the velocity and displacement from step 1, the acceleration at the start of the
time step immediately after the transition time step are calculated using either
equation (3.27) or (3.29) again depending on the region the system is in at the end of
the transition time step.
3. Step 1 is then repeated to determine the velocity and displacement at the end of the
transition time step using the acceleration from step 2 in place of that calculated by
equation (3.28) or (3.30).
4. Steps 2 and 3 are repeated until the accelerations determined by successive
calculations in steps 1 and 2 converge to within an acceptable tolerance.
The procedure described above achieves convergence after only a few iterations. The
tolerance on convergence is set to 0.001 in/sec/sec in the program used in the study.
3.4.4 Inelastic SDOF Systems with Friction Dampers
The analysis of systems with friction dampers is completed using the method presented
above for inelastic systems with modifications to equations (3.27) through (3.30) to account
'for the additional spring included in the system. In addition to equations (3.18) and (3.19),
the equations governing the response of friction damped systems are;
UeJJ]+\s ueffis + UM Ui
(3.31)
46


(3.32)
V
(3.33)
I UcffD
(3.34)
u
mu .
s*
(3.35)
m
-ksFUeffis -k!DUefr.D-(ksF + kSD ){, (A0 + [0 5 fis , } } (3-36)
The subscript S is used to identify variables associated with the spring without dampers
and the subscript D is used for variables associated with the spring fitted with the friction
dampers. The variable usl defines the displacement at which slip occurs in the friction
dampers.
Equations (3.31) through (3.36) are used directly to calculate the response of a friction
damped structure while both the frame and friction dampers are within the elastic region. As
with equations (3.29) and (3.30) governing the nonlinear system, equations (3.35) and (3.36)
can be modified to account for the limiting force and loss of stiffness associated with either
yielding of the frame or slippage of the friction dampers. The modifications are similar to
those presented in developing equations (3.29) and (3.30) and will not be presented here.
The procedure identified above for assuring convergence of accelerations in inelastic
systems without friction dampers is also applicable to inelastic systems with friction dampers.
The procedure is completed using the same steps.
47


3.4.5 System Energy Content
The time stepping methods presented above allow the determination of the displacement,
velocity, and acceleration of SDOF systems during earthquake excitation. Results of the
analyses provide the information necessary to determine the energy content of the system
during the earthquake. The equation governing the energy content of a system during
earthquake excitation was presented as Equation (1.1) in Chapter 1 and is repeated here for
convenience.
E[ = Es + EK + ED + EY + Em
Equation (1.1) indicates that all energy imparted to the system by the earthquake is either
stored within the system in the form of strain and kinetic energy or is dissipated from the
system by means of viscous damping, structural yielding, and damper slip. The amount of
energy stored within the system is calculated at the end of each time step as;
The amount of energy dissipated from the system during each time step is calculated as;
(3.37)
(3.38)
(3.39)
Ey. = fyAu
(3.40)
EMi ~ fsi^UD
(3.41)
48


where Au and AnDare the distances the system displaces during yielding and damper slip
respectively. The cumulative energy dissipated from the system at a particular instant in time
is calculated as the sum of energy dissipated during each time step up to the time under
consideration as;
<3-42)
j=i
Ey ^ Eyj
/=!
(3.43)
(3-44)
;=i
The rate at which energy is dissipated from the system through either structural yielding or
damper slip is approximated using the values determined by Equations (3.40) and (3.41) as;
SEy __ Erj
St ~ At
and;
(3.45)
SEm Em,
St At
(3.46)
3.4.6 Components of Inelastic Response
Throughout the parametric study, the deformation characteristics of inelastic systems are
related to those of elastic systems. Because of this, it is informative to consider the
relationship between the seismic response of inelastic systems and that of corresponding
elastic systems. The response of inelastic systems can be shown to be comprised of three
components; the linear response of an identical system assuming elastic properties throughout
49


the full range of response, the resulting plastic yield deformation, and a transient component
of response resulting from the energy dissipation occurring during yielding.
The various components are identified by way of an example using a SDOF system
subjected to the Northndge earthquake. The response of the system was completed using the
time stepping method presented in section 3.4.3. For the analysis, the earthquake ground
acceleration was scaled to yield a peak acceleration of 1,0g. The natural period of the system
was chosen to be 1.0 second and damping was assumed to be 5 percent of critical. The
normalized yield strength of the system was chosen to be 90 percent of that of an elastic
system such that a single cycle of yielding occurs during the earthquake. The system
deformation response and resulting yield deformation are presented in Figure 3.6.
Inelastic System Response
t (sec)
Yield Deformation
Q
3
2
1
0
I
-2
-3
r

: i - -

j
0
I 0
I 5
2 0
2 5
3 0
t (sec )
(b)
Figure 3.6 Northridge earthquake, fy = 0.90, (a) Inelastic deformation response, (b) Yield
deformation
50


Review of Figure 3.6(b) indicates that a single yield cycle takes place at approximately 4
seconds. Yielding results in a shift of the equilibrium position of the system as is apparent in
Figure 3.6(a) during the later portion of the response. The deformation response of a
corresponding elastic system with the same structural properties and subjected to the same
ground acceleration is presented in Figure 3.7.
E lastic System Response
Figure 3.7 Northridge earthquake, Elastic deformation response
The various components of the inelastic system response become apparent by considering
the difference between the inelastic response shown in Figure 3.6(a) and the elastic response
shown in Figure 3.7. Subtracting the elastic response from the inelastic response identifies
the portion of the inelastic response that is composed of the sum of the yield deformation and
the transient component. The results of this operation are presented in Figure 3.8(a).
Subtracting the yield deformation from the combined results determined above then identifies
the transient portion of the inelastic response. The results of this operation are presented in
Figure 3.8(b).
51


The total response of the inelastic system is in this way shown to be equal to the sum of
the response of an equivalent elastic system, the yield deformation occurring in the inelastic
system, and the transient component of response due to yielding of the inelastic system.
Graphically, the individual components are identified in Figures 3.7, 3.6(b), and 3.8(b).
Inelastic Response Elastic Response
(a)
Transient Component
(b)
Figure 3.8 Northridge earthquake, (a) Difference between inelastic and elastic responses
(b) Transient component of response
52


3.5 Earthquake Ground Motions Used in Study
Five recorded earthquake ground motions were selected for use in the study. The
earthquakes considered, along with the recorded peak ground accelerations and the estimated
predominant period of ground motion associated with each of the earthquakes, are presented
in Table 3.2. The analyses completed in the parametric study considered 30 seconds of
ground motion and a 0.02 second digitizing time step for each of the earthquakes.
Table 3.2 Earthquakes used in Study
Earthquake Recorded Peak Ground Acceleration Predominant Period of Ground Motion, Tc
El Centro 1940 S00E Component 0.319g 0.85 seconds
Loma Prieta 1989 Corralitos CHAN 1: 90 Deg 0.479g 0.76 seconds
Northridge 1994 Sylmar County Hospital Parking Lot Chan 1: 90 Deg 0.843g 0.51 seconds
Olympia 1949 N86E Component 0.280g 0.60 seconds
San Fernando 1971 Pacoima Dam S74W 1,076g 0.43 seconds
Source of ground motion files: Strong Motion Database, Institute for Crustal Studies (ICS), University of
California, Santa Barbara (UCSB).
To standardize the responses obtained from each of the ground motions, the earthquakes
were normalized to yield peak ground accelerations of 1.0g by linearly scaling the recorded
ground motions. Scaled time history plots for each earthquake are presented in Figure 3.9.
Additionally, the frequency content of each earthquake is presented by way of the Fourier
Amplitude Spectra shown in Figure 3.10.
53


Ax(X) Ax(R) A*(x) Ax(|)
E I C e n t r o
<>
L o m a Priela
H- M JU



l t i t ( N orth rid S 2 2 * ) e * 5 3 9


r
t

U
IT HV\JW VlfVW
V
i



s i 15 2 2 5 3 A
<***)
O ly m p i a
. (.)
San Fernando
1
] s
2 #
2 S
3 0
( >
Figure 3.9 Normalized earthquake ground motions.
54


w*v w*v wav wav wav
E t C t tr o
L o ro a P r i e
=j=l
1
0 0 5.0 10.0 15.0 20.0 25.0
MHz)
N o
I k r id ( t
0.10
0.0 9
0.0 8
0.0 7
0.0 6
0.0 5
0.0 4
0.0 3
0.0 2
0 .0 1
0.0 0
0.10
0.09
0.0 8
0.0 7
0.0 6
0.05
0.0 4
0.0 3
0.0 2
0.0 1
0.0 0
0.10
0.0 9
0.0 8
0.07
0.0 6
0.0 5
0.04
0.0 3
0.0 2
0.0 1
0.0 0



fh, mito

.0 5 0 1 0 o .0 1 5 f (H z) 1 y ro p i a .0 2 0 .0 2 5







. AJ 4 JIU LITHIUM. 1. n


.0 5 0 1 0 San .0 15 r ( h Z ) Fernando .0 2 0 .0 2 5








Mlmk ___

.0 5 0 1 0 0 1 5 .0 2 0.0 2 5
f ( H z )
Figure 3.10 Fourier Amplitude Spectra.
55


The Fourier Amplitude Spectra provide useful information regarding the frequency
content of the earthquake ground motions and were used in the study to estimate the
predominant periods of ground motion. The spectra were calculated using a discrete fast
Fourier transform and the first 1024 digitized ground acceleration records representing 20.48
seconds of each earthquake. The reader is referred to Paz (1985) for a discussion of the
development and implementation of the discrete fast Fourier transform method.
Using the normalized time history files, elastic deformation and velocity response spectra
were generated for each of the earthquakes. The spectra are based on the response of 150
single-degree-of-freedom systems with natural frequencies ranging from 0.02 to 3.0 seconds
and were calculated using the method presented in Section 3.4.2. Plots of the deformation
and velocity response spectra developed for the EL Centro earthquake are presented in
Figures 3.11 and 3.12. The spectra represent the maximum deformation and velocity
calculated by Equations (3.15) and (3.16) for the 150 elastic systems during the full 30
seconds of earthquake ground motion. Thus, it is possible to determine the maximum
displacement or velocity for an elastic system from the spectra if the natural period of the
system is known.
Deformation Response Spectrum
El Centro, 5% Damping
Figure 3.11 El Centro Deformation Response Spectrum
56


Pseudo-Velocity Response Spectrum
Kl Centro, 5% Damping
Figure 3.12 El Centro Velocity Response Spectrum
For elastic systems, the force transferred through the spring and the system deformation
is related by Equation (3.8a). Because the system deformation is by definition less than the
yield displacement, the limit imposed in Equation (3.8a) does not apply. In this case, the
force transferred through the spring can be calculated as;
fSF(t) = kFu(t)
The maximum force transferred through the spring during the earthqauke can be related to the
spectral displacement such that;
f!FM,,=KD
Representing the maximum force in terms of the system mass, in and a peak pseudo-
acceleration yields;
A nig = kFD
where A is the spectral pseudo-acceleration in units of g. By substituting the relationship
between system mass, stiffness, and natural period into the above equation and rearranging
terms;
57


(3.47)
Equation (3.47) is a function of the spectral deformation and natural period so it is
possible to develop the pseudo-acceleration response spectrum for an earthqauke directly
from the defromation response spectrum. The pseudo-acceleration response spectrum for the
El Centro earthqauke is presented in Figure 3.13. The spectrum was calculated using the
deformation response spectrum shown in Figure 3.11 and Equation (3.47)
Pseudo-Acceleration Response Spectra
El Centro, 5% Damping
Figure 3.13 El Centro Pseudo-Acceleration Response Spectrum
The spectrum shown in Figure 3.13 is referred to as a pseudo-acceleration spectrum
because it does not present the actual maximum system acceleration as defined by Equation
(3.20). Rather the spectrum presents only the portion of the system acceleration that is
associated with the spring force.
Plots of the deformation, velocity, and pseudo-acceleration spectra for each of the
earthquakes are included in Appendix C.
58


3.6 Code Prescribed Earthquakes
Building codes such as the Uniform Building Code (1997) specify the earthquake to be
considered in the seismic design of structures by defining a design basis earthquake for the
building site. Design basis earthquakes are defined in terms of a smoothed pseudo-
acceleration response spectrum such as that shown in Figure 3.14. The response spectrum
represents an earthquake that has a ten percent probability of exceedence in a 50-year period
and is defined by two input variables; the maximum peak ground acceleration expected to
occur during the earthquake, Sg and the maximum expected pseudo-acceleration of a fully
elastic structure with a natural period of one second, S,. The spectra are generally developed
for structures with an equivalent viscous damping ratio of 5 percent and are specific to a
given site in that both the peak ground acceleration and the one-second spectral accelerations
used to define the spectrum are dependent on the soil conditions present at the site and the
proximity of the site to major earthquake faults.
Figure3.14 Code prescribed pseudo-acceleration response spectrum.
59


The magnitude and extent of the region of peak spectral acceleration is defined such that;
Ss = 2.5S o (3.48)
(3.49)

T0 = 0.2 Ts (3.50)
Note that the spectral pseudo-acceleration determined from the smoothed spectrum of
Figure 3.14 is identified as A to distinguish it from the pseudo-acceleration determined from
an actual earthquake response A as shown in Figure 3.13.
Velocity and deformation response spectra can be developed for the design basis
earthquake using the following equations, which provide the relationships between pseudo-
acceleration, pseudo-velocity, and deformation;
(3.51)
2 n
= (3.52)
2n 4 n
Figure 3.15 Pseudo-velocity response spectrum from code prescribed earthquake.
60


Figure 3.16 Deformation response spectrum from code prescribed earthquake.
An objective of the parametric study is to develop a correlation between the code
prescribed earthquake for a site, specifically the input variables of peak ground acceleration
and one-second spectral acceleration, and the corresponding inelastic response characteristics
of structures fitted with friction dampers. It is therefore necessary to determine a
representative design basis earthquake for each of the earthquakes studied. Details of the
method used to complete this determination are presented in the following Sections.
3.7 Elastic Response Reduction Factors
It was mentioned earlier that the basic philosophy behind the seismic design requirements
in building codes allows yielding of structural elements during strong earthquakes.
Consistent with this philosophy, codes do not require that structures be designed to remain
elastic during the full design basis earthquake but rather allow structures to be designed to a
reduced earthquake. This is accomplished by specifying an elastic response reduction factor,
R which is used to reduce the design level earthquake forces to magnitudes less than that
determined from the design basis earthquake response spectrum. The reduction factor is
61


based on a structures ability to absorb energy during yielding of structural elements during
strong earthquakes. Values of R vary from 2.2 to 8.5 in the Uniform Building Code (1997)
and depending on the type of structural system and materials used for construction.
A'
Design
A'
R
Figure 3.17 illustrates the concept of the response reduction factor.
(3.53)
T (sec)
Figure 3.17 Comparison of code prescribed pseudo-acceleration response spectrum and code
prescribed design spectrum.
This same concept will be used to define the slip force for the friction dampers
considered in the study. The slip displacement of the dampers will be defined as a ratio of the
deformation resulting from the design basis earthquake such that;
DSI=^- (3.54)
A
In Equation (3.54), the coefficient R is referred to as the slip coefficient. In terms of damper
slip force;
/ =
D'K
R
(3.55)
62


Defining the damper slip force in terms of the design basis earthquake allows linear scaling of
the response of friction-damped structures. The response quantities of deformation, velocity,
and pseudo-acceleration become a function of a given earthquake and peak ground
acceleration. If the response quantities are desired for the same earthquake scaled to a
different peak ground acceleration, the results can simply be scaled by the same amount used
to scale the earthquake ground motion.
3.8 Representative Code Prescribed Earthquakes
To allow correlations to be made between the response of the SDOF systems and the
parameters used to define design basis earthquakes in building codes, representative code
prescribed earthquake design spectra were developed for each of the earthquakes considered.
A smoothed pseudo-acceleration response spectrum as shown in Figure 3.18 was considered
as representative.
Figure 3.18 Simplified pseudo-acceleration response spectrum.
The spectrum was modified from that presented in Figure 3.14 by eliminating the varying
acceleration region for periods less than T0 and thus extending the constant acceleration
region from 0 to Ts seconds. This modification was incorporated because an increase in
63


natural period for structures associated with this region results in increased acceleration and
friction dampers in structures during strong earthquakes results in an increased natural period,
the use of the higher acceleration over this region was deemed appropriate.
The spectrum shown in Figure 3.18 is defined by two distinct regions; the region
extending from 0 to Ts seconds, and the region extending from Ts to 3.0 seconds. In each of
these regions the pseudo-acceleration, pseudo-velocity, and deformation are related to the
initial code prescribed input variables of peak ground acceleration, Sg and 1.0-second spectral
acceleration, S| by the following relationships:
thereby increased loading. Because the decreased stiffness associated with slippage of
(3.56)
Region 1: T < Ts
A'- S,
(3.57)
con 2 n 2 n
(3.58)
(3.59)
Region 2: Tn > Ts
(3.60)
A'Tng _
2n 2 n
(3.61)
(on 4 n1 An2
(3.62)
64


By normalizing the earthquake ground motions, the value of Sg for each of the
earthquakes was set equal to 1.0. Region 2 of the spectrum defines the constant velocity
range of response for the earthquake. The value of the 1.0-second spectral acceleration, Si for
each of the earthquakes was determined by considering the maximum spectral velocity, V,
occurring between 1.0 and 3.0 seconds over the constant velocity range of response and
relating the 1.0-second spectral acceleration to the velocity by:
2 nV'
g
(3.63)
The values of V] determined from the velocity response spectra along with the resulting
values of Sg and Si used to represent the code prescribed earthquake ground motion for each
site are presented in Table 3.3.
Table 3.3 Representative Code Prescribed Earthquake Parameters
Tg (sec) V| (in/sec) Sg(g) s,(g)
El Centro 0.85 87.10 1.00 1.42
Loma Prieta 0.76 69.80 1.00 1.14
Northridge 0.51 95.25 1.00 1.55
Olympia 0.60 72.02 1.00 1.17
San Fernando 0.43 48.08 1.00 0.78
The representative smoothed deformation, pseudo-velocity, and pseudo-acceleration
spectra along with the actual spectra generated for the El Centro ground motion are presented
in Figure 3.19. Spectra for the five earthquake ground motions considered in the study are
included in Appendix C. These smooth spectra represent the design basis earthquakes for
each of the sites and will be used as a basis to define the slip displacements considered in the
remainder of the study.
65


D (in) V (in/sec) A (g)
Pseudo-Acceleration Response Spectra
El Centro, SI = 1.42g, 5% Damping
Pseudo-Velocity Response Spectra
El Centro, SI = l-42g, 5% Damping
Deformation Response Spectra
El Centro, SI 1.42g, 5% Damping
Tn (sec)
Figure 3.19 El Centro representative code prescribed earthquake response spectra.
66


3.9 Concluding Remarks
The representative earthquakes developed in this chapter form the basis of the parametric
study of friction-damped single-degree-of-freedom systems presented in Chapter 4. The
earthquakes were developed based on normalized earthquake ground motions with peak
ground accelerations of 1,0g. By defining the damper slip force as a function of a slip
coefficient and the response spectrum representative of the earthquake ground motions, the
resulting responses become a linear function of the peak ground acceleration of the
earthquake. This allows direct linear scaling of the results obtained in the study. Thus, for
earthquakes with the same ratio of one-second spectral acceleration to peak ground
acceleration, the study results can be directly scaled. This allows the results of the study to be
applied to earthquakes with varying peak ground accelerations.
The one-second spectral accelerations used to define the representative design basis
earthquakes were selected based on the peak spectral velocity occurring at periods greater
than one second. It can be seen from a review of the response spectra presented in Appendix
C that this approach in selecting representative parameters resulted in smoothed spectra that
is exceeded at periods less than one second. The amount the smoothed spectra are exceeded
is generally small, however, in the case of the Loma Prieta earthquake, the degree to which
the smoothed spectrum is exceeded is substantial. The selection of the predominant period of
ground motion from the Fourier amplitude spectra also requires some judgment as the
earthquakes cannot be represented by a single harmonic forcing function. In developing
conclusions from the results presented in Chapter 4, these points should be kept in mind.
67


4. Parametric Study Results
4.1 Introduction
In this chapter, the results of the parametric study of friction-damped single-degree-of-
freedom systems are presented. Included are the results of the quantitative evaluation of the
parameters affecting the design of friction damping mechanisms and the results of the study
of the inelastic deformation response characteristics of friction-damped SDOF systems. The
results of the quantitative evaluation of the parameters affecting the design of friction
damping mechanisms are presented in Section 4.2 and Appendix D. The results of the study
of the inelastic deformation response characteristics of SDOF systems are presented in
Sections 4.3 and 4.4. Section 4.3 presents the results of the study completed on systems with
small amounts of secondary stiffness. The section presents a method to develop smoothed
inelastic deformation response spectra for systems with small amounts of secondary stiffness
along with the assumptions used in developing the method. A comparison between the
inelastic response spectra generated for systems subjected to the normalized ground motions
considered in the study and that estimated by the smoothed spectrum developed in Section
4.3 is included in Attachment D. Section 4.4 presents the results of the study completed on
the effect secondary stiffness has on the inelastic response of SDOF systems and presents a
method to develop deformation response spectra for systems with varying amounts of
secondary stiffness. A comparison between the inelastic response of systems with secondary
stiffness ratios varying from 0 to 100 percent and that estimated by the response spectra
developed in this chapter is presented in Appendix F. The chapter closes with Section 4.5
68


where a discussion and statement of the conclusions drawn from the study of SDOF systems
fitted with friction dampers is presented.
4.2 Mechanism Design Data
The quantitative evaluation of the parameters affecting the design of friction damping
mechanisms was completed by considering the inelastic response of a series of friction-
damped SDOF systems with natural periods ranging from 0.25 to 2.5 seconds and four levels
of damper slip force defined by slip coefficients of 4, 6, 9, and 12. The system responses
were calculated using the time stepping methods for inelastic systems presented in Section
3.4 and the normalized earthquake ground motions developed in Chapter 3. Only systems
with zero secondary stiffness were considered in the quantitative evaluation of the parameters
affecting the design of friction-damping mechanisms.
Three parameters were evaluated; the number of slip cycles experienced by the dampers,
the maximum amount of energy dissipated, and the maximum rate that the dampers
dissipated energy during each of the earthquakes considered. In the evaluations, a damper
slip cycle was defined as an occurrence of damper slip in one direction from the time slippage
was initiated to the time the damper velocity reached zero. The total number of slip cycles
was determined from the response of the systems during the full 30 seconds of ground motion
considered. The amount of energy dissipated by slippage of the friction dampers during each
time step was calculated by Equation (3.41) and the total amount of energy dissipated during
the duration of the earthquakes was calculated by Equation (3.44). The maximum rate that
the damper dissipated energy during the earthquake was calculated by Equation (3.46) and
considered the maximum amount of energy dissipated during any single time step over the
full 30 seconds of ground motion.
69


The results of this portion of the study are presented graphically in Appendix D. The
results of the evaluation of the total amount of energy dissipated and the maximum rate that
energy is dissipated are presented in terms of system mass. In the analyses completed, the
mass of the system was taken to be equal to 1.0 slug. The results presented in Appendix D
can be scaled linearly for systems with masses other than that considered in the study.
4.3 Inelastic Deformation Response Spectra for SDOF Systems
The development of a method to formulate inelastic deformation response spectra was
based on the results of a series of inelastic time history analyses completed for friction-
damped SDOF systems with natural periods ranging from 0.04 to 3.0 seconds and subjected
to the five normalized earthquake ground motions developed in Chapter 3. The analyses
were completed using the time stepping methods for inelastic systems fitted with friction
dampers presented in Section 3.4 and considered secondary stiffness ratios of 0 and 5 percent
and damper slip forces defined by slip coefficients of 4, 6, 9, and 12.
The goal of this potion of the study was the development of deformation response spectra
for friction-damped SDOF systems with small amounts of secondary stiffness. The
controlling parameters for the earthquake were taken as the peak ground acceleration, the 1-
second spectral acceleration, and the predominant period of ground motion. The controlling
parameters for the SDOF systems were taken as the natural period, the damper slip
coefficient, and the secondary stiffness ratio of the systems.
In general terms, the inelastic deformation response of a friction-damped SDOF system,
D" can be described as a function of the five input parameters as;
ir = G(Sg,Sl,Tg,R,t) (4.1)
70


In the case of systems with small amounts of secondary stiffness, the inelastic response
D0 can be described as;
D0=G(Sg,Sl,Tg,R) (4.2)
Based on the results of the analyses completed in this portion of the study, the smoothed
inelastic deformation response spectrum presented in Figure 4.1 was developed in terms of
the four input parameters included in Equation (4.2).
Figure 4.1 Inelastic deformation response spectrum
Similar to the elastic spectrum used to define the code prescribed earthquake, the
spectrum shown in Figure 4.1 is defined by two distinct regions; the region extending from 0
to Ts seconds, and the region extending from Ts to 3.0 seconds. In each of these regions the
inelastic deformation is defined by the following relationships:
Region 1: Tn < Ts
+ (4.3)
4tt Ts
and
Region 2: 7 > Ts
71


(4.4)
D;=|4[i + /?(rv-i)]
4tt~
The variables Ts and p are defined as;
7's'=7<(U67lr) <4-5)
p=\-~
R
\nR£,
2 +
R
-4
U2
2_
R
nR2$)
(4.6)
In Equation (4.6) £ is the equivalent viscous damping ratio for the system and was taken as 5
percent in the study.
By review of the elastic deformation response spectrum shown in Figure 3.18 and
Equation 3.62, it can be seen that the inelastic response of SDOF systems with natural periods
greater than Ts is defined in terms of a modified elastic response spectrum. In developing
the spectrum shown in Figure 4.1 two modifications of the elastic spectrum have been
incorporated; the first is a decrease in the slope of the line defining the deformation of
systems with longer periods, and the second is a shift of the line defining the longer period
deformations to the left. The decrease in slope results from the additional energy dissipated
by the friction dampers and the shift of the line results from changes in the natural period of
the system during damper slip.
The slope of the portion of the graph for natural periods greater than Ts was established
by considering the relationship between the deformations of elastic and inelastic systems
subjected to harmonic forcing functions. For an elastic system subjected to a harmonic
72


forcing function P(t) = P0 sin ^ it can be shown that over a period of time the system will
achieve steady state conditions with a constant amplitude of vibration. In steady state
conditions, the energy input by the harmonic force is equal to the energy dissipated through
viscous damping where the input energy can be written as;
E! =miaPs,m(j> (4.7)
and the energy dissipated by viscous damping can be written as;
ED=2^ kul (4.8)
coN
From Equations (4.7) and (4.8) it can be seen that the energy input during a single cycle
of vibration is directly proportional to the displacement amplitude of the system and the
energy dissipated by viscous damping during a single cycle of vibration is proportional to the
square of the displacement amplitude. Thus for an elastic system subjected to a harmonic
forcing function, equilibrium of the input and dissipated energy can only occur at a single
amplitude of vibration. As the force is applied to the system, the amplitude of vibration
increases until equilibrium is obtained between input energy and the dissipated energy.
Figure 4.2 shows the relationship between the input and dissipated energy for the system.
73


The relationship presented in Figure 4.2 is valid only for elastic systems where viscous
damping provides the only means of removing energy from the system. In the case of a
friction-damped system, energy is dissipated both by viscous damping and by slippage of the
friction damper. To determine the amount of energy that is dissipated by a friction damper
during a single cycle of steady state vibration, the idealized hysteresis loop shown in Figure
4.2 will be considered. From the figure, the system can be seen to displace from the initial
equilibrium point a maximum distance D" during each cycle. Additionally, the system can
be seen to oscillate about a new equilibrium point with a maximum displacement of u0 where;
u^\(D" + Ds) (4.9)
Figure 4.3 Idealized Hysteresis Loop
The energy dissipated through a single' cycle of vibration is equal to the area inside the
hysteresis loop and can be written as;
E], = 2kDD's,{D" D'sl)
(4.10)
74


The amount of energy dissipated through viscous damping in the friction-damped system
is taken to be equal to the square of the ratio of the inelastic to elastic displacement
amplitudes times the energy dissipated through viscous damping in the elastic system such
that;
E"d=Ed
( y
u
\u0 J
(4.11)
Taking the maximum displacement of the elastic system to be equal to that defined by the
smoothed elastic deformation response spectrum shown in Figure 3.15 and substituting
Equation (4.9), the energy dissipated through viscous damping can be written as;
(D" + D'J
ed=ed
4 D'
(4.12)
The amount of energy input to the inelastic system is taken to be equal to the ratio of the
inelastic to elastic displacement amplitudes times the energy input to the elastic system.
Again relating the maximum displacement of the elastic system to the smoothed elastic
deformation response spectrum and substituting Equation (4.9) the input energy is written as;
e;=e,
d"+d:,
2D'
(4.13)
Equilibrium is achieved between the input and dissipated energy when;
F" F" + F"
^ l nD + nsl
(4.14)
or
e.SDEl
2D'
{D' + Vj
4D'~
+ 2kDD'sl(D" D'sl)
(4.15)
The input energy E; is equal to the energy dissipated by viscous damping ED allowing
Equation (4.15) to be written as;
75


2n£, kul
co.
d+d:<
ID'
= 2n^-^ku\
co.
4 D'2
+ 2 kDD'sl{D"-D'sl) (4.16)
Defining the ratio of the inelastic to elastic displacement amplitudes as;
D"
P~ (417)
stating the slip displacement in terms of the elastic code prescribed displacement and the slip
coefficient;
and maximizing the response by assuming the frequency co of the forcing function to be
equal to the natural frequency of the system coN yields the slope reduction coefficients defined
by Equation (4.6) from Equation (4.16). Table 4.1 provides the values of the slope reduction
coefficients for the four slip coefficients considered in the study.
Table 4.1 Slope reduction coefficient, /?
R P
4 0.386
6 0.346
9 0.369
12 0.453
The inelastic deformation of friction-damped SDOF systems as defined by Equation
(4.16) is based on the assumption that the natural period of the elastic system and that of the
friction-damped system during damper slip are the same. The inelastic deformation response
spectrum presented in Figure 4.1 however is based on the natural period of the system prior to
damper slip. Because of this, Equation (4.16) is not used directly to calculate the inelastic
76


deformation. The equation is modified by incorporating a constant that shifts the response to
a natural period associated with the system prior to damper slip. Because of the greater
stiffness present in the system prior to slippage of the damper, the line describing the inelastic
response is shifted toward a lower period. The amount of shift was determined by
examination of the inelastic response spectra generated during the study and noting that the
deformations of elastic and inelastic systems are approximately equal for systems with
natural periods near 1.0 second. Because of this, Equation (4.4) was developed to yield
inelastic deformations equal to elastic deformations at 1.0 second.
The extent of the first region of the inelastic spectrum defined by Ts was determined by
examination of the inelastic response spectra generated during the study. A comparison of
the smoothed inelastic deformation response spectrum developed using the representative
code prescribed earthquake parameters for the El Centro earthquake presented in Table 3.3
and the inelastic response spectra calculated using the time stepping methods presented in
Section 3.4 is presented in Figure 4.4. A comparison of the smooth and actual spectra
developed for each of the earthquakes and slip coefficients is presented in Appendix E.
Figure 4.4 Inelastic deformation response spectrum. El Centro, S]=1.42g, R=6
77


4.4 Effects of Secondary Stiffness
The evaluation of the effects secondary stiffness has on the deformation response of
friction-damped SDOF systems was based on a series of inelastic time history analyses
completed for systems with natural periods ranging from 0.25 to 2.5 seconds and normalized
damper slip strength and secondary stiffness ratios ranging from 0 to 100 percent. Figure 4.5
presents the results of one such evaluation complete for a system with a natural period of 1.0
second subjected to the normalized El Centro earthquake ground motion. In all, 50 such
evaluations were completed using the 5 normalized ground motions and 10 values of natural
period. For each evaluation, 10,000 individual system analyses were completed and the
maximum system deformation occurring during the earthquake was determined.
Figure 4.5 shows the maximum system deformation as a ratio of the maximum
deformation of an elastic system with the same natural period occurring during the
earthquake as the two parameters of normalized slip force and secondary stiffness ratio are
varied. The secondary stiffness ratio is defined by Equation (3.4). The normalized slip
strength is defined as the ratio of the damper slip force to the maximum force developed in an
elastic system such that;
/ =
fs,
kDD0
(4.18)
or by substituting Equation (3.55) the normalized slip force can be written as;
/ =
D
RD0
where D0 is the maximum elastic system deformation.
(4.19)
78


DIDn
Normalized Slip
Strength (%)
100
Secondary
Stiffness (%)
Figure 4.5 System displacement with variation of normalize slip strength and secondary stiffness.
El Centro, Tn = 1.0 second.
The responses of the systems presented in Figure 4.5 are approximated at the limits of
zero and 100 percent secondary stiffness by the smoothed elastic and inelastic deformation
response spectra shown in Figures 3.16 and 4.1. The deformation of systems with secondary
stiffness ratios of 100 percent is equal to that of a fully elastic system regardless of the
normalized slip strength specified. In this case, the maximum deformations of the systems
are calculated by either Equation (3.59) or (3.62) depending on which region of the spectrum
the system falls. At the opposite side of the graph, the deformations of the systems with
secondary stiffness ratios of zero are dependent on the level of slip force specified for the
damper. The level of slip force is defined by the slip coefficient and the deformation is
calculated by either Equation (4.3) or (4.4) again depending on which region of the spectrum
the system falls.
79


For systems with secondary stiffness ratios between the two limits of zero and 100
percent, the deformation can be assumed to be a function of the deformations occurring at the
limits. In this portion of the study, the deformations are assumed to be a linear function of
the limiting deformations such that;
D" = D' + {\-)D0 (4.20)
Figure 4.6 presents a comparison of the actual maximum displacement occurring in a
series of systems with natural periods of 1.0 second and secondary stiffness ratios varying
from zero to 100 percent and the inelastic displacements estimated by Equation (4.20). The
system displacements were taken from Figure 4.5 considering a normalized slip strength
corresponding to a slip coefficient of 4. In Figure 4.6, the horizontal line corresponding to a
value of 1.0 identifies an exact match between the actual and estimated displacements. From
the figure, it can be seen that for all values of secondary stiffness a conservative displacement
is estimated by Equation (4.19). The results for the evaluations completed in this portion of
the study are presented in Appendix F.
2.0
1.8
1.6
1.4
z 1.2
S i .o
= 0.8
0.6
0.4
0.2
0.0
0 10 20 30 40 50 60 70 80 90 100
Secondary Stiffness (%)
Figure 4.6 Normalized system displacement.
Normalized Displacement
El Centro, Tn =1.0 Sec,R = 4
80


4.5 Conclusions and Discussion of Parametric Study Results
The results of the parametric study completed on the deformation response characteristics
of friction-damped single-degree-of-freedom system demonstrate the feasibility of
developing inelastic response spectra based on the parameters used to define the design basis
earthquake for a given site. The parameters necessary to formulate the inelastic spectra
include the earthquake parameters of peak ground acceleration, one-second spectral
acceleration, and predominant period of ground motion and the system parameters of natural
period, secondary stiffness, and damper slip force. For systems with small secondary
stiffness ratios, it was shown that inelastic deformation response spectra could be developed
from the code prescribed parameters used to define the design basis earthquake for a given
site. It was further shown that the inelastic deformation of systems with secondary stiffness
ratios between 0 and 100 percent could be calculated as a function of the elastic and inelastic
response spectra developed for the earthquake.
In general, the inelastic responses of the friction-damped systems were found to be larger
than the corresponding elastic response at natural periods less than one second and smaller at
periods greater than one second. Additionally, from review of the inelastic response spectra
presented in Appendix E it can be seen that variations in the slip coefficient have a greater
effect on the maximum response of short period structures than on the maximum response of
long period structures. Because the response of short period friction-damped structures are
more sensitive to variations in the damper slip force, greater care is required during the
construction phase of these structures to assure that the proper slip force is achieved.
81


Because the damper slip force is a function of the earthquake under consideration, the
results of the deformation response study are a linear function of the peak ground acceleration
of the earthquake. Therefore, the results can be directly scaled when considering earthquakes
of different peak ground accelerations.
Although the results of the quantitative evaluation of the parameters affecting the design
of friction damping mechanisms are presented in raw form, several points can be made about
the results. For all earthquakes studied the number of slip cycles experienced by the systems
is inversely proportional to the natural period of the systems. Long period structures
experienced fewer slip cycles than did the short period structures. With few exceptions, the
number of slip cycles experienced by structures with longer periods and constant slip
coefficients was nearly constant. During every earthquake studied, the number of slip cycles
experienced by structures with natural periods greater than 0.75 seconds was less than 50. In
some cases, shorter period structures experienced a total number of slip cycles in excess of
120. In systems with constant natural periods, the number of slip cycles varied in proportion
to the slip coefficient selected for the damper. The evaluation regarding the amount and rate
of energy dissipation indicates that the rate energy is dissipated by the dampers in systems
with constant natural periods is inversely proportional to the slip coefficient used.
Additionally, the results of the evaluation indicate that the amount and rate of energy
dissipation is maximum for structures with natural periods that are near the predominant
period of ground motion.
The results of the quantitative evaluation regarding energy dissipation are also a function
of the peak ground acceleration of the earthquake under consideration. From Equations
(3.37), (3.38), and (3.39), it can be seen that the amount of energy dissipated is a function of
82


the square of the system response. Therefore, the results can be scaled by the square of the
ratio of the earthquake peak ground accelerations when considering other earthquakes. The
number slip cycles remains constant for earthquakes of varying peak ground accelerations.
83


5. Application of Parametric Study Results to MDOF Systems
5.1 Introduction
In this chapter, the results of the study completed on multi-degree-of-freedom systems
are presented. The chapter presents an evaluation of the correlation between the response of
a ten-story friction-damped moment frame estimated using the inelastic response spectra
developed in Chapter 4 and the corresponding response determined by inelastic time history
analyses. Section 5.2 presents background information regarding dynamic analysis
techniques for MDOF systems. The section begins with the presentation of the method of
modal superposition used in the analysis of elastic systems along with a discussion leading to
the development of the response spectrum analysis method. Additionally included in Section
5.2 is a discussion of the method used in the study to calculate the inelastic response of the
moment frame using time stepping methods similar to those presented in Chapter 3 for SDOF
systems. Section 5.3 presents the properties of the friction-damped moment frame considered
in the study along with the approach used to determine the distribution of stiffness and
damper slip forces throughout the frame. Additionally, the section presents the parameters
considered for the friction-damped moment frame and the corresponding values considered in
the analyses. Section 5.4 presents the modal analysis method used to estimate the inelastic
deformation of the frame. Comparisons of the maximum deformations calculated by inelastic
time history analyses and the corresponding estimated responses are presented in Attachment
G. Additionally included in the chapter is a method to determine the slip coefficient in terms
of the optimum slip force discussed in Chapter 2 presentation of the method is made in
84


Section 5.5. The chapter closes with Section 5.6 where a discussion and conclusions drawn
from the study are presented.
5.2 Deformation Response of MDOF Systems
5.2.1 Introduction
The response of multi-degree-of-freedom systems subjected to earthquake ground
motions can be estimated using time stepping methods similar to those used to calculate the
response of SDOF systems. The response of elastic MDOF systems can be shown to be equal
to the summation of the responses of the systems individual vibrational modes. It can be
further shown that the vibrational modes of a MDOF system are directly related to the
distribution of mass and stiffness in the system. For elastic MDOF systems, the response of
each mode can be calculated as a function of the response of an equivalent elastic SDOF
system and the natural properties of the structure. Two methods that can be used to calculate
the elastic response of MDOF systems are presented in Section 5.2.2.
The response of inelastic MDOF systems can be calculated using methods similar to
those used to calculate the response of inelastic SDOF systems. The method used to calculate
the response of the ten-story friction-damped moment frame is presented in Section 5.2.3.
5.2.2 Elastic MDOF Systems
The dynamic response of elastic multi-degree-of-freedom systems can be shown to be
comprised of a series of independent modal responses where each modal response is
determined from the natural properties of the system and the forcing function used to excite
the system. It can be further shown that the independent response of each mode is a function
of the response of an equivalent SDOF system and the natural properties of the mode under
consideration. Because of these relationships, the response of elastic MDOF systems can be
85


calculated as the summation of the responses of the individual modes of the system. This
method of dynamic analysis is referred to as modal superposition.
The time varying deformation response of elastic MDOF systems subjected to earthquake
ground motion can be calculated using the method of modal superposition presented in
Chopra (1995) as;
u(0=Xr>''(r) (51)
l =
where the vector u(r) describes the time varying displacement of each of the ./V degrees of
freedom used to define the system. Equation (5.1) defines the overall response of the MDOF
system as the summation of the responses of the individual modes of the system where the
response of the ilh mode is calculated as;
(0 (5-2)
Equation (5.2) defines the response of the ih mode of the system in terms of the natural
properties of the system, as used to calculate the quantity rj.ip,., and the time varying response
tii(t) of an equivalent SDOF system with a natural period and viscous damping ratio the same
as that of the ilh mode of the MDOF system. The vector ip,, describes the displaced shape
associated with the i'h mode of vibration and is determined from the solution of the
eigenvalue problem;
kr

In Equation (5.3), kr is the total structure stiffness matrix and is defined as;
86


-iKA) 0 0 0
-%*K) ik -k )Mk -ik) v /: izs V n its 0 0
0 -(K+K) (k+kJMkn+kJ . 0 0
0 0 0 ..Oc -+k )Mk +k) -(k +k) \ rsi DMS \ FS oy v FM DMS
0 0 0 .. -Qnk) V FM DM/
where kFi represents the lateral stiffness of the moment frame at the i'h level of the structure
and ^represents the lateral stiffness of the bracing system fitted with friction dampers again
at the i'h level. The structure mass matrix m is defined as;
w 0 0 ... 0 0 '
0 vv 0 ... 0 0
i 0 0 vv ... 0 0
8 ... ...
0 0 0 ... 0
0 0 0 ... 0 w,
where w, is the total weight lumped at the i'h level of the structure and g is the acceleration
of gravity. Solution of Equation (5.3) establishes the N natural mode shapes

corresponding natural frequencies co for the MDOF system. The reader is referred to
Chopra (1995) for a complete discussion of the formulation of Equation (5.3) and the
various methods available to obtain the solution of the equation. The factor T,. is a
function of the natural properties of the system (i.e. mass and stiffness matrices) and is
equal to;
(5.4)
87


where ipf is the transpose of the i'h mode shape of the system obtained from Equation (5.3)
and the vector 1 represents the effect the earthquake ground acceleration has on the masses
lumped within the system. The vector 1 is known as the influence vector and is formulated
by considering the displacements of the system masses resulting from a static application of a
unit ground displacement in the direction earthquake ground motions are being considered.
In the case of a planer system subjected to horizontal earthquake ground motion, the values of
the influence vector associated with horizontal degrees of freedom will be equal to 1.0
indicating that earthquake inertial forces will act along these degrees of freedom. The
remaining values of the influence vector will be associated with vertical degrees of freedom
and will therefore be equal to zero. The influence vector effectively eliminates the inertial
acceleration of system masses along degrees of freedom that are perpendicular to the
direction of earthquake excitation.
The individual vibrational modes of the MDOF system can be shown to respond
dynamically in a manner similar to that of an SDOF system. The time varying response tq(r)
in Equation (5.2) is the response of an elastic SDOF system with the same natural period and
damping ratio as that of the i,h mode of the MDOF system and can be calculated using the
time stepping methods presented in Section 3.4.
Thus, from Equation (5.1) it can be seen that the time varying response of an elastic
MDOF system subjected to earthquake ground motion is a function of the natural properties
of the system and the dynamic responses of a series of SDOF systems. The natural properties
of the system are defined by the distribution of mass and stiffness as identified by the system
88


mass and stiffness matrices. The SDOF system responses are determined by the same time
stepping methods used to complete the evaluations carried out in Chapter 4.
5.2.3 Response Spectrum Analysis
The method of modal superposition presented in the previous section provides the time
varying response of elastic MDOF systems subjected to a specific earthquake ground motion.
However, during the design process, the actual time varying earthquake ground motions that
the structure might be subjected to during its useful life are not known. Because of this,
structural engineers typically rely on the design basis earthquake specified for a given site to
determine the seismic response to be considered in the design of a structure. Additionally,
the maximum system response is typically of more interest to structural engineers than the
actual time varying response. Because of these reasons, seismic analyses of structures are
typically completed by considering a combination of maximum modal responses calculated
using the peak SDOF system responses defined by the code prescribed design basis
earthquake. This method of analysis is referred to as a response spectrum analysis.
To estimate the maximum expected deformation using a response spectrum analysis, the
maximum response of each independent mode of an elastic MDOF system is first calculated
by Equation (5.2) with the value D\ determined from Figure 3.16 substituted in place of the
time varying response f(r). The maximum independent modal responses are then combined
using one of a number of methods developed to approximate the actual response of the
system. With the substitution of £>,' in place of ut(t), Equation (5.2) becomes;
d;=i>,.£>; (5.5)
89


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A PARAMETRIC STUDY OF THE SEISMIC RESPONSE OF STRUCTURES FITIED WITH COULOMB FRICTION DAMPING MECHANISMS : A PROPOSED APPROACH TO THE DESIGN OF FRICTION DAMPED STRUCTURES by Daniel W. Secary B S., University of Colorado at Denver, 1986 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 2001

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This thesis for the Master of Science degree by Daniel W. Secary has been approved by Kevin L. Ren s Jf -s--a; D ate

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Secary, Daniel William (M S Civil Engineering) A Parametric Study of the Seismic Response of Structures Fitted with Coulomb Friction Damping Mechanisms : A Proposed Approach to the Design of Friction Damped Structures Thesis directed by Assistant Professor Kevin L. Rens ABSTRACT This thesis presents the results of an analytical parametric study completed on the seismic response of single-degree-of-freedom systems fitted with friction dampers. The primary goals of the study were ; 1) to gather data key to the design of friction damping mechanisms including the number of slip cycles the total amount of energy dissipation and the maximum rate of energy dissipation e x perienced by the dampers during strong earthquakes ; 2) to develop a method to formulate inelastic response spectra based on the peak ground acceleration one-second spectral acceleration, and fundamental period of the earthquake under con s ideration ; and 3) to evaluate the effect elastic secondary stiffness has on the deformation re s pon s e of the sys tems. A secondary goal of the s tudy wa s to outline a propo s ed method that might be u s ed by s tructural engine er s in the de sign of friction d ampe d s tructures. A s part of this g oal the corr e lation between the inela s tic deformation of multi-degree-of-freedom s y s tems fitted with Ill

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friction damper s and the co rre s ponding di s placement s calculated using modal analysis techniques and the inelastic re spo n se spectra developed in the study was evaluated The study consisted of a series of inela s tic dynamic analyses comp leted on single degree-of-freedom sys tems with natural period s ranging from 0.25 to 2 5 seconds and elastic seco ndary stiffness ratios ranging from 0 to 100 percent. The dynamic a nalyses were completed using the computer program "Response" developed b y the Author at the University of Colorado at D e nver and the recorded ground motions of five earthquakes scaled to y ield consistent peak ground accelerations of l.Og To evaluate the correlation between the inelastic deformation re s pon se s pectra de ve loped in the s tudy and th e maximum displacements and s tory drifts experienced by multi-degree-of-fr ee dom s tructure s subjected to ea rthquake ground motions, inelastic time hi s tory analyses were comp leted for a tens tory f riction damped moment frame u s ing the sa me five earthquakes considered in the parametric stu dy Four leve ls of friction forces were con s idered in the anal yses comp l eted for both the sing l edegre e-of-freedo m sys t ems and the friction d am p e d mom e nt frame. The force level s are r e lat e d to representative code pre scri b e d e la s tic re s pon se s pectra and ela s tic r es ponse reduction factor s s imil a r to those u se d in current earthquake d es i gn. This abstract a ccurat e l y repres e nt s th e content of th e candidate's thesis. I r eco mmend it s publication. Signed IV

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ACKNOWLEDGEMENT I would like to thank my thesis advisor, Dr. Kevin L. Rens for his help and guidance in the completion of this thesis Additionally, I would like to express my sincere thanks to each of the members of my thesis committee Dr. Kevin L. Rens Dr. Judith 1. Stalnaker, and Dr. John Mays for the generous gift of knowledge that they have given me through the numerous class lectures and discussions that have made up both my undergraduate and graduate studies at the University of Colorado at Denver.

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CONTENTS Figures .... .... .... ... . . . .......... ............... ....... . .... ..... ................ . .... ..... . . .............. ............ ... ix Ta ble s . . . .... ..... . .... .... . ... ........... .... .... . ....... .... ........ .... ....... ... ............ . ........... ....... ........... xi C hapter I. Introduction ..... .......... .... ...... . ......... . ........... ......... ... . .......... . .... . . .... ......... ... . ........... .... ... I 2 Pre v iou s Re sea rch .................. ......... .... ..... ... ...... .... . ....... ............ ......... . ........... . . ........ . 9 2.1 Introduction ..... ..... .............. ....... . .......................... .... . ...... ...... . ............ .... ........ . ... .... . 9 2.2 Pall Friction Damper s . ............... .... .... . ......... ... .... .... .... . .... .... ....... ..... ....... . ...... ......... 15 2 3 Optimum Slip Force . .... ..... .......... ... ....... .... .... .......... .... . .... .... .......... ....... . .... . ......... .. 17 2.4 Optimum Slip Force De sign Spectrum ... ............. . .... . .... .... .... ... .... .......... ... . ..... .... ... 21 2.5 Simplified D es ign Approach Propo se d b y Filiatrault and Cherry ... . . ........ ... ........ ..... ...... 23 2 6 Areas R eq uirin g Additional Study ...... ...... ... . .... ... ....... . . ..... ..... .... . ............. .... ...... . .... . 25 3. Overview ofParametric S tud y ............ ...... ........... ...... . ......... ... ....... . ...... . .... ...... ..... . .... 27 3.1 Introduction ..... ................ ... .... ............ .... .... ..... ...... ....... . ... ...... .............. ... .... .... ....... 27 3 2 Objectives and Scope ofParametric Study . ............... .... .... .... . .... . . ..... . ........... .... ...... 27 3.3 Friction Damped Moment Frames ...... ......... .... ....... ............ ............... . ..... ....... . ... .... ..... ... 3 0 3.4 Solution of th e Equation of Motion b y Time Stepping M e thods . . . . . ......... ........ ......... 37 3 .4.1 Introduction ........... .......... . ..... . ......... .... ........... ... .... . . . ... . . .... . ............. ........... ...... 3 7 3.4 2 Elastic SDOF Systems . ...... ....... . ... .... ... ..... ... ........... ..... .... . . . . ....... ....... ..... ... ... . .... . 39 3 .4. 3 Inelastic S DOF Systems ........ ........... ... ... .... . .... . . .... . .... ....... ................. . . ..... ........ . .41 3.4.4 Inelastic S DOF Systems with Friction Dampers . ... . ... ...... ... . . ......... ........ ...... ... . . ... 46 V I

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3.4.5 System Energy C ontent ...... ........ ... ................................. ... ............ ...... .. ... ...... ........ . .48 3.4 6 Components of Inelastic Response .......... .... ....... . .......... ......................... ... ........ ......... .49 3.5 Earthquake Ground Motion s used in Study ... ... ... ......... ............ . ... ................ ..... ......... ... 53 3.6 C ode Prescribed Earthquakes .... . ............ ... .... ........ ...... . .............................. ................ 59 3.7 Elastic Response Reduction Factors ........ ...... ................................... . .............................. 61 3 8 Representative Code Prescribed Earthq u akes ........... ....................... .... .......................... ... 63 3.9 C oncluding Remarks ......... .................. ... . ................... . .................. ........... ................. . ... 67 4 Parametric Study Results ......... ............ ........ ...................... .......... ... ..................... ... ..... .... 68 4.1 Introduction ................................................ ........ ....... ........................................ ......... ..... 68 4.2 Mechanism Design Data .... ......... .................... ............. ......................... .................. ...... 69 4.3 Inelastic Deformation Response Spectra fo r SDOF Systems .............................. ............. 70 4.4 Effects of Secondary Stiffne ss .... ...... ........... ...... ...................................... .... .... ............... 78 4 5 Co nclu sio ns and Discus s ion of Parametric Study Results .......... ............................ ..... ..... 81 5 Appl ication of Parametric Study Results to MDOF Systems .............. ............................. 84 5.1 Introduction ................................ ................. ..... ................... ......................................... ... 8 4 5.2 Deformation Response of MDOF S yste m s ...................... ...... .............................. ....... ...... 85 5 .2. 1 Introduction ..................................................................... ... ............................................. 85 5 .2.2 Elastic MDOF Systems ........ ......... .... ........... ............................ ......... ........ ........... .... ..... 85 5.2 3 Response Sp ec trum Analysis ... ..................... ............... .................... ...... ....................... 8 9 5.2.4 Inela s tic MDOF Systems ..................................... ......................................................... 92 5 3 Re s ponse of a Ten-Story Friction-Damped Moment Frame ............ ................................ 99 5.4 Estimated Frame Deflection and Story Drift Using In e la s tic Spectra ............................. 104 5.5 Equivalent Slip Coefficients Ba se d on Optimum Slip Force ...................................... ..... 106 VII

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5.6 Concluding Remarks ... .................................... ........ .... ..................... .... .... ..... ............ .... ... 110 6 Discuss i on Conclusion s and Recommendations for Further Study ....................... ......... 112 6 1 Discussion and Conclusions .............. .................... .......................... ..... ............ ........ ....... 112 6 2 Recommendations for Further Study .................................................... .... ...................... 115 Appendix A Nomenclature .......... ............. ....... ...... ....................... ........... .... ...... ... . ................ .. ...... ........ l17 BUs er s Manual for Computer Program Response .... .... ...... ...... .......... ...... ....... .... ........ .. 122 C Representative Code Prescribed Respon s e Spectra .... ........ .. ......... .................................... 13 7 D Slip Cycles and Energy Dissipation .... .... .... .... .... .... .......... ...... ...... ........ ........ .. .................. 143 E Inela s tic Deformation Re s ponse Spectra .... .. ........................................................... ........... l49 F Normalized Inelastic Deformation .............. ...... ...... .... ............ ... .......................... ....... ..... 155 G Multi-Degree-of-Freedom Systems ........ ............................................ .......................... ... 166 Bibliography .... . ...... . ..... .... ............. . ....... . .... .......... . ..... ... .... ......... ......... . ................. 183 VIII

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FIGURES 1.1 Sy s tem energy content during El Centro 1940 earthquake . ........ .... ....... ... ... .... ... ....... .. .. .4 2 1 Response of friction dampers under cyclic loading .... ......................... ..... .......... ..... ... ...... 12 2 2 Pall friction damper .......... . ...... . . .... . . ......... ..... ................ ....... ....... ........... ........ ..... . .... 16 2 3 Strain energy content of systems with and without friction dampers ....................... ........ 18 2.4 Slip load optimi z ation ..... .... ........ ......................... . . ... . ........... ...... . ........... ... . .... .... .... 20 2 5 Optimum slip force design spectrum ............ .... ............. ... .. ....... ........ ... ... ................... ... 22 3 1 Friction-damped moment frame .................. ...... ............. .... ... . ........ ........ . ...... .... ...... . .... 30 3 2 Idealized combined damper / framing sy stem behavior ....... ................ ..... ... .. .. ..... . .... ........ 31 3.3 Force-Deformat i on relationship for friction damped moment frame .................. . ........ . . 33 3.4 Friction-damped moment frame at arbitrary instant in time ....... .......... ... .. ......... . . ......... .. 35 3 5 Schematic repre s entation of friction damped s tructure .... ................... . .......... ..... .......... 36 3 6 Northridge earthquake Jy = 0.90 (a) Inela s tic deformation respon s e (b) Yield deformation ....... .... .......... .... ......... . ....... . . ... ...... .... .... .... . ..... .... ..... ........... ...... .. 50 3 7 Northridge earthquake, Ela s tic deformation re s ponse .......... . ......... . ..... ............. .......... .... 51 3 8 Northridge earthquake (a) Difference between inela s tic and e l a s tic response s (b) Transi e nt component of re s pon s e ..... . . ..... . ...... .......... . . ............. ...... .... ......... ... ........ 52 3.9 Normali z ed Earthquake Ground Motion s ... ....... .... ......... . ............. .......... . . . .............. .... 54 3 .10 Fourier Amplitude Spectra ......... . ....... ... ............... ....... ......... ............ ................. .......... 55 3 .11 El C entro Deformation Re s ponse Spectrum ................... ....... ........ ............................. ... 56 3.12 E l C entro Velocity Response Spectrum ................ ...... .............. ................. ... ............... ... 57 I X

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3.13 El Centro Pseudo-Acceleration Re sp on se Spectrum ................. . ......................... ..... ...... 58 3.14 Code prescribed pseudo-acceleration response spectru m ................................................ 59 3.15 Pseudo-velocity response spectrum from code prescribed earthquake ............................ 60 3.16 Deformation response spectrum from code prescribed earthquake .. ............................... 61 3.17 Comparison of code prescribed pseudo-acceleration respon se s pectrum and code prescribed design spectrum .......... .......... ................................ ........ ........ .......... 62 3.18 Simplified pseudo-acceleration respon se spectrum ........................ .......... .......... ............ 63 3.19 E l Centro representative code prescribed earthquake response spectra ........................... 66 4 1 Inela s tic deformation response spectrum .............................. ........ .... .. .... .... ...................... 71 4.2 Energy Equi librium State .................................................................................................. 73 4.3 Idealized Hysteresis Loop .... ................ .... ..... ..... .............. .......... ...... ........................... ...... 74 4.4 Inelastic deformation response spectru m E l Centro, S1= 1.42g, R = 6 ................................ 77 4.5 System displacement with variation of normali z e s lip strength and secondary s tiffness. El Centro, Tn = 1 0 sec ond .............................. .. ............................... 79 4.6 Nom1ali ze d sys tem displacement ................ .... ............ ..... ................. .... ................... ....... 80 5.1 Schematic diagram of MDOF friction-damped sys tem .... .... ........... .......... ........................ 93 5.2 D'Al e mbert free-body diagram ofMDOF syste m ma ss .......... .... .................................. .... 93 5.3 Tens tory friction-damped moment frame .................... .... .............................. ............ ....... 99 5.4 Compari so n of estimated and actual inela s tic re s pon se .............................................. .... 103 5.5 Comparison of estimated and actua l inelastic story drift.. ............................................... 104 5.6 Equival ent R values ......... .... .... ........ ..... . ........ .......... ................. ...... ................................ 109 X

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TABLES Table 2 1 Friction Damper Installations ........................................................................................... 13 2 2 Filiatrault and Cherry Study Parameters ............................................................................ 21 3.1 Study Parameters ...................................................................................................... ............................... 30 3.2 Earthquakes used in Study ................ ...... .... .... ............... ... ............................... ...... ............ 53 3.3 Representative Code Prescribed Earthquake Parameters .................... 00 ...... 00 .................... 65 4.1 Slope reduction coefficient fJ .... oo .......... oo ...... oo ...... oo .. oo ................ oooo .. oo ....... oooo .......... 00 ...... 76 5.1 Design shear distribution .......... 00000000 ....................................... 0000 .. 0000 00 00 ...... 00 .................. 101 5 2 Parameters used for ten-story frame evaluation ooooooooooo ............................... oo .. oo .............. 102 XI

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1. Introduction In recent years, Structural Engineers have become increasingly interested in the u se of passive energy absorbing mechanisms to reduce the response of structures during strong earthquakes. Common forms of the se mechanisms have included hydraulic struts or visco e lastic material dampers, which dissipate energy through viscous damping metallic y ield e lements, which dis sipate energy through inelastic deformation, and friction dampers which dissipate energy through coulomb friction. The se mechani s ms are referred to as "passive" because they rely only on the relative motion of the struc ture during an earthquake to dissipate energy. By dissipating a portion of the energy transferred to the structure during the eart hquake these devic es reduce the struc ture' s overall respon se. This reduc ed response translates into reduced damage to the main structura l elements resulting in both added safety agai nst collapse and economic ga in from the reduced cost of rehabilitation of the structure after the earthquake. Modern building codes acknowledge that it is not econom i cally feasible to design str ucture s that remain fully elastic durin g strong earthquakes. The basic philosophy behind the seismic design requirements in modem building co de s i s that it is acceptable to design s tructures that will s u stain no damage to structu ral and nonstructura l components durin g frequent minor earthquakes, a s mall amo unt of damage to non -structura l compone nts durin g le ss frequent moderate earthquakes, and mod erate to seve r e damage to both structura l and non-structural components during in frequent strong eart hquak es. The de s ign approach emphasizes life safety and a lthough damage to s tructural elements during a strong eart hquake

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is d eemed acce ptable structu ral collapse a nd the potential r esulting loss of life must b e avoided This approach to seismic de s ign reli es on yielding of k ey s tructural elements to dis s ipat e the excess energy transferred to the structure during stro n g earthquakes. During minor and moderate earthquakes, the s tr esses within the main struc tur a l e l eme nt s are b e lo w y ield le vels a nd the energy transferred to the str uctur e b y the earthq uak e is dissipated e ntir e ly through the e qui va lent visco u s d amping pr ese nt in the str u c ture During strong earthquak es, the equiva lent visco u s d amping i s not s u fficie nt a lon e to dissipate the large amount of energy transferred by the earthquake. As kinetic a nd s train e n e r gy l evels build up displacem e nt s eventually exceed the elast ic limits of the structure and y i e ldin g of e l ements occurs In a properly des igned s tructure the yielding of elements a nd s ub se qu e nt absorption of excess energy takes place within regions of the str u cture that are s pecially designed and constructed to a llo w for l arge inelastic deformations Although th e yie ldin g takes plac e w ith in a predetermined portion of the s tructure th e yielding of s tru c tur a l m e mb e r s results i n permanent deformations and damage that ca n be severe e nou g h to render the structure uninhabitable wit hout some le ve l of rehabilitation. In th e case of a poorly d eta il e d or constructed building collapse ca n occur. During a n earthqu ake the genera l eq u ation describing the energy co nt e nt withi n a s tructure at a n arbitrary instant in time is ; (1.1) In Equation 1 1 the variable E1 represent s the total amoun t of energy transferred to the s tructure b y the earthquake Variables Es and EK repr ese nt the energy s tored within the s tructure E s is the elastic stra in e n ergy and EK is the kin etic e n ergy Variables E0 Ev, and 2

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EM represent the energy that is dissipated from the system E0 is the total amount of energy dissipated through the equivalent viscous damping present in the structure, Ev is the total amount of energy dissipated through inelastic deformation of structural elements and EM is the total amount of energy dissipated by the passive energy dissipating mechanisms present within the structure Passive energy absorbing mechanisms act to limit damage to the main building framing system by minimizing or eliminating the need to dissipate energy through yielding of structural elements. While each of the various energy dissipating mechanisms mentioned earlier have the ability to dissipate energy during an earthquake, the simplicity and low cost of friction dampers make them especially attractive. Friction dampers dissipate energy through coulomb friction by sliding a series of plates, incorporated into the lateral load resisting system of a structure relative to one another. The amount of energy dissipated by the mechanism is equal to the product of the friction force developed by the mechanism and the total cumulative distance the plates slide relative to one another. The friction force developed within the mechanism is equal to the product of the normal compressive force holding the plates in contact and the coefficient of friction for the materials used to fabricate the damper. Tensioned high strength bolts passing through the pi ies of the damper typically provide the normal force. The effectiveness of friction dampers in reducing the amount of energy dissipated through yielding of structural elements is demonstrated in Figure 1.1. The figure presents the energy content oftwo single-degree-of-freedom systems subjected to the El Centro, 1940 earthquake. Figure 1 1 a) shows the energy content of an inelastic system without additional energy dissipating devices. Figure 1.1 b) shows the energy content of the same system fitted 3

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with friction dampers. Of key interest is the amount of energy dissipated through yielding of s tru c tural el e ment s R evie w of the figure indicate s that for thi s example the amount of ener gy dis sipated through y ielding of structural element s wa s decreased by approximately 80 percent after friction damp e r s were incorporated 1200 1000 800 g 600 . .P --= 400 200 ,//: 0 0 5 10 15 20 25 30 Time (se c ) -Strain+ Kinetic ---Equivalent Visc ous Damping Yield 1200 1000 800 ..0 1 600 r;f ""l 400 0 200 1 0 0 5 10 15 20 25 30 Time (sec) -Strain+ Kinetic --Equivalent Viscous Damping -Yield M echanis m Figure 1.1 S y s t e m e n e r gy cont e nt duri n g El Ce ntr o, 1 9 4 0 ea rth q u a k e a) Ine l as t i c sys t e m b) Ine l astic sys t e m with friction d a mp e rs. 4 (a) (b)

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In addition to effectively dissipating excess energy, friction dampers provide two additional features that are useful in mitigating earthquake damage to structures. The first is the ability to limit forces transferred to structural elements during an earthquake Because the friction force is dependent on the damper normal force and friction coefficient rather than displacement the maximum force transferred through the damper during a n earthquake is well defined In effect, the friction force specified for the damper defines the maximum force that will be tran s ferred through the structural bracing system during a strong earthquake. This is in contrast to the over-strength forces that must be accounted for in current seismic design The second feature that makes friction dampers attractive is the ability to develop two levels of s tiffness within a structure When l ateral forces within the bracing system are below the slip level of the friction dampers, no slippage occurs and the system acts as a relatively rigid braced frame system. When forces in the bracing system exceed the slip l evel of the dampers slippage occurs and the stiffness of the structure i s reduced. The reduction in stiffness of the system and subsequent shift to a lon ger natural period, can be effective in reducing the lateral forces acting on the structure during strong earthquakes Because of their effectiveness in dissipating excess energy and these unique feature s, friction dampers show great promise in providing a means for Structura l Engineers to de s ign buildings that can resist strong earthquakes with little or no damage to the main building structure. Currently friction dampers have been incorporated into the design of a number of buildings in Canada Japan, and the United States Typically the designs have relied on inelastic time history dynamic ana l yses to determine the distribution of s lip forces to be used throughout the lateral l oad resisting system of the buildings The time histories have been either recorded ground motions of past earthquakes or synthetically generated ground 5

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motions. This method of design has a number of shortfalls; the time history analyses are time consuming to complete the analyses consider only a limited number of possible ground motions, and the engineers carrying out the designs require speciali zed training in structural dynamics beyond that of the typical structural engineer. Before the structural engineering community as a whole can readily incorporate friction dampers into building designs, a straightforward approach, which requires no greater effort than is required for current seismic design needs to be developed. This thesis has as one of its objectives the development of such a design approach This thesis presents the results of an analytical parametric study completed on the effect friction dampers have on the response of single-degree-of-freedom systems subjected to earthquake ground motions. The study considers the ground motions of five different earthquakes and has as its primary objectives : The col lection of data on characteristics key to the design of friction damping mechanisms including the number of slip cycles experienced, the amount of energy dissipated and the maximum rate of energy dissipation occurring in single-degree-of freedom systems of varying periods during each of the earthquakes studied. A study of the effects damper slip force and secondary system stiffne s s has on the overall displacement of the single-degree-of-freedom systems. The final objective of this portion of the study is the development of an inelastic design spectrum that could be used in the design of multi-degree-of-freedom friction damped structures. A study of the correlation between the inelastic deformation of multi-degree-of freedom structures fitted with friction dampers and the corresponding displacements 6

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calculated using modal analysis techniques and the inelastic response spectra developed in the study. A secondary objective of the study is the development of a general design approach, based on current code prescribed design basis earthquakes that can be used by Structural Engineers in the design of friction damped structures. This thesis begins with a general discussion in Chapter 2 of friction dampers and some of the earlier research completed on friction dampers. The concept of an optimum level of total friction force distributed throughout the bracing system of a structure is presented along with a method developed by researchers to determine the optimum force. The chapter concludes with a short discussion on areas where further research on friction dampers is needed. Chapters 3 and 4 deal primarily with the parametric study completed on SDOF systems. Chapter 3 presents an overview of the parametric study and a discussion of the analytical modeling techniques used to complete the study. A repre se ntative code prescribed earthquake is established for each of the earthquakes considered and a method to define the damper slip force in terms of the representative code prescribed earthquake and an equivalent slip coefficient is presented. Chapter 4 presents the results of the parametric study and the method used to generate the inelastic response s pectra for each of the earthquakes. Chapter 5 presents the results of the study completed on MDOF systems. The results of the inelastic dynamic analyses completed for a ten story friction damped moment frame are presented along with a comparison of the analys is results and the deflections and story drifts calculated using the inelastic spectra developed in the study. A method for estimating the maximum story drifts and overall building displacements is presented. 7

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Chapter 6 presents conclusions and a general discussion of the study results along with a proposed de s ign approach that could be used by Structural Engineers in designin g friction damped structures. The design approach is based on current code prescribed earthquakes, the optimum slip coefficient for the s tructure and the inelastic deformation response calculated by the method s presented in C hapter 5. The chapter concludes with a di sc us si on regarding area s where additional study on friction dampers might be warranted. The dynamic analyses of single degree-of-freedom systems pre se nted in this the s is were completed u s ing the per so nal computer based program Respon s e developed b y the Author at th e University of Co lorado at Denver to study the seismic response of friction-damped str uctures. A general de scr iption of the program is pre se nted in the User's manual included a s Attachment B to this thesis The analyses of multi-degree-of-freedom systems were completed u sing the computer program Mat.lab and code written by the Author. 8

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2. Previous Research 2.1 Introduction In their simplest form, friction dampers can be simp le bolted framing connections with slotted hole s and pre-tensioned bolts As long as a friction force can be developed between the plies of the connection and s lippa ge t akes place the damper will dissipate energy. However becau se friction dampers are critica l e l ements in the l ateral load resi s ting system of structures th e dampers must be capable of d ependab le operation and must exhibiting stab le response during a large number of s lip cyc les wit hout failure Additionally, to allow engineers to de sign s tructure s fitted with friction damper s, the response of the damper s must be predictable. The s urface conditions of the materials u se d to fabricate friction dampers plays an important role in their overall performance Researchers have studied the performance characteristics of s imple s lott e d bolted connections made up of s liding plates with both stee l on s teel and steel on brass sliding surfaces and found the characteristics to be erratic and unpredictable Grigorian eta!. (1993) In an effort to deve l op dampers wit h stable and predictabl e performance characteristics, a variety of damper s urfa ce treatments were s tudi ed under s tatic and cyclic loading conditions Pall et al (1980). The surfaces s tudied included plain mill s cale sand and grit blasted surfaces, surfaces with metalized and zinc -ri ch painted finis h es, and common automotive break lining materials Of the surface treatment s s tudied the automotive break lining materials proved to have the most s table and predictable performance characteristics The break lining material s were found to produce a nearly 9

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constant friction force during damper dis placement and a nearly rectangular hy steres is loop under cyclic loading Additionally researchers found that the u se of the break linin g mat eria ls provided s table damper performance with negligible fad after 50 cycles of loadin g Filiatrault and Cherry (1987). Co n si derin g the untold hour s of re sea rch and de ve lopment expended by th e automobile indu s try in de v eloping c urr e nt brake linin g material s, it is not su rpri sing that the materials provide s table and predict a bl e performance in friction damp e rs for s tructures. Automotive brak e lining material s are well suited for u se in dampin g mechani s m s becau se th e physical co nc e pt s in vo lved in both automobile brakin g and friction damper s are essentially the same Both sys tem s function by dis s ipating energy throu g h co ulomb friction The operation of a friction damp er is ba se d on th e si mple concept of dry or Co ulomb friction whereby the force nece ssa ry to s lide one bod y pa s t another i s equal to th e produ c t of th e coefficient of friction for the mat eria l s in contac t an d the normal force m ai ntainin g contact of the two s urfa ces; = j.iN (2.1) As the two s ur faces are s lid pa s t one another e n e r gy is di ss ipat e d in the form of h eat. The quantity of energy di ss ipated i s equal to th e wo rk comp l e t e d in s lidin g the surface s s uch that ; E = fs,L (2 2) w h ere Lis eq u a l to th e di s tance the two surfaces s lid e relative to one another. F rom E quation (2.2) the amount of e n e r gy di ssi p ate d by a friction d a mp e r i s a function of t h e friction force and the di s tance the damper s lip s The m axi mum distanc e a damper can slip i s typically limited by the a mount of int e r s tory drift that can be tol e rated by a s tru c tur e. Additio nall y, the dis pl a cem e nt s induced in a frict ion-d a mp ed s tructure are related to the s lip 10

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force specified for the friction dampers pre se nt in th e struct ure Becau se of thi s relationship effo rt s to maximi ze the amount of energy diss ipated by friction dampers have concentrated on studies of the variation of slip force s pecified for the damper. In an attempt to optimi z e the amount of energy dissi pated by fric tion damper s researchers ha ve propo se d a numb e r of d a mper configurations The various damper configurations can typically be cla ss ified as one of two basic types ; configuration s that maintain a con s tant force throughout the f ull rang e of damper slip, and configurations that incorporate a friction force that varies a s a function of damper displacement. In damper assemblies wit h varying friction force s the variations are typically achieved by varying the normal force betwe e n the damper friction p a d s and the inner surface of cylindrical casings b y using s prings The assemblies are arranged s uch that an increa se in damper dis placem e nt re s ult s in an incr ease d damper friction force Damper assemblies that are ba s ed on a constant friction force are typically much s impler mechani s m s with few movin g part s other than t h e plies of the damper These damper s are a rr a n ge d s u c h that a constant normal compressive force is maintained across the pli es of the damper as they s lid r e l a tive to one a noth er. The theoreti ca l r es pon ses of c on s t a nt force a nd va riable force friction damp e r s under cyclic lo ad ing conditions are pre se nt e d in Figure 2.1. The energy dissi pated b y eac h of th e dampers i s equa l to th e area insid e th e h ysteres i s loop for eac h damp er. R e ferrin g to Figure 2 1 the greater effectiveness of the con s t ant force damper s i s apparent. From the figure it ca n b e see n that w ith the sa m e max imum sli p force a nd dis plac e m e nt occurring in both damp e rs, the constant force damper s di ssi pat e four times the e ner gy of the varying force damp e r. The effectiveness of variable force d ampe r s ca n be increa se d b y providin g a pretension force in the interna l s prings of the damp e r s uch that a h ys t e resi s loop s hown by the ll

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dashed lines in Figure 2.1 (b) i s developed However it can be seen that the re su lts are less than that of the constant force damper. The studies completed in this thesis consider only the response of structures fitted with constant force dampers a f Sf ( Spring with f51 force ---L .. L ---(a) (b) Figure 2.1 Re s pon se of friction d am p e r s under c yc lic loading a) Co n s tant force damper, b) Variable force damper. In addition to the various damper configurations proposed researchers have also proposed a number of methods for incorporating friction dampers into the framing systems of building struc tures These have included unique configurations suc h as the in stalla tion of dampers at the column-beam joints in moment resisting space frames to eliminate the need for ve rtical bracing and to maintain the open area provided by the space frame system Wa y ( 1996) however the most common and perhaps the simplest method of incorporating friction dampers into the design of a structure i s to incorporate the damper s directly into a vertical bracing sys tem. The s tudies completed by Filiatrault and Cherry (1987 & 1990) considered friction damped moment frames that combine both a moment re sisti ng frame and a vertical bracing syste m into the overall building struc ture This type of a combined system provides an elastic restoring force that acts to limit the overall deflection of the friction dampers and 12

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help restore the structure to near its original undeforrned shape. Through the various configurations studied, researchers have shown that the methods for incorporating friction dampers into building structures are limited only by the fundamental requirement that the dampers be located in an area of the structure where differential movement and internal mechanism forces can be generated during strong earthquakes. Currently friction dampers are gaining acceptance in the engineering community and have been used in the new de s ign or retrofit of a number of structures in Canada Japan and the United States between the 1970's and the present. The dampers have been installed in both s teel and reinforced concrete structures varying in height from one to thirty-one stories and have been installed in such varied structures as buildings elevated water towers and s upports for electrical circuit breakers Table 2.1 presents a partial list of friction damper installations in Canada Japan and the United States Table 2.1 Frictio n D ampe r Ins t allatio n s Building/Structure Structure T y p e Locat io n D ampe r Type Date Go r gas H osp i tal P a n a m a F rictio n Damp ers 1 970's M cCon n e l Building, Co n cor d ia R einforced M on tr ea l Cana d a P all Frictio n Damp e r s I 987 U n ive r s i ty l ib ra ry co mpl ex, Co n c r e t e 6 a nd I 0 S t ory Bldgs Resident ial H ouse Wood S t ud, 2 Mo ntr ea l Ca n ada P all Fri ctio n Da m pe r s 1 988 S t ory Sonic Office B uil d i ng Stee l 3 I S t ory Omiya City, Japan Su mit omo Frictio n D a m pers 1 988 Asa h i B ee r Azum a bash i Stee l 22 S t ory T okyo, J a p a n Sum it o m o Frictio n Dampe r s 1 989 Bu ildi ng Eco l e P o l yva l a nt e Pr ecas t Con c r e t e, So r e l Can a d a P all Fric t io n D a mp e r I 990 3 S t ory a n d Pal l Frictio n P a n e l s Ca nadi a n Inform atio n and S t ee l 4 S t o ry Lava l Can a d a P all Fricti o n Damp e r s I 992 Tr ave l Ce nter D epa rtm ent of D efe nse R einfo r ced Otta w a, Ca n a d a P all Frictio n D a mp e r s I 992 Co n c r e t e, 3 Story 13

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Table 2.1 (Cont.) Building/Structure Structure Type Location Damper Type Date Ca nadian Space Agency S t ee l 3 Story St. H ube rt Ca n ada Pall Frictio n Dampers 1 993 Casino de Montr ea l S t ee l 8 Story Montreal, Cana da P all Fric tion Damp e r s 1 99 3 B uilding 610, Sta n ford Brick and Stucco, P alo A lt o California Slotted Bo lt ed Connec tion s 1 99 4 Univers ity 1 Story Hoover Building Stanford 2 Story P a lo Alto, Califo rnia S l otted Bolt e d Connectio ns 19 9 4 University Mai so n I McGill R einfo rced Montreal, Canada Pall Friction Dampers 1 99 5 Co n cre t e, II Story Eco l e Techno I ogie Superieure Stee l Montrea l Ca n ada Pall Friction Dampers 1995 Federa l Building R einforce d She rb rooke, C anad a Pall Friction Damper s 1 995 Concre te, 4 Story De sja rdin Life Insurance Reinforced Quebe(} Canada P all Frictio n Damper s 1 995 Buildin g Concret e, 6 Story Ove rh ea d Water Ta nk Stee l B e a u x Arts P all Friction D ampers 1 995 Washingto n St. Luc Hos pit a l Reinforced Montreal Canada Pall Friction Dampers 1 995 Con c r e t e, 8 Story Residence Maison-Neuve Stee l 6 S t ory Montreal Canada P all Friction Dampers 1 996 H amilto n Courthouse S t ee l 8 S t ory Hamilton Ca n ada Pall Fiction Dampers 1996 Wate r Towers Unive r s ity of S t ee l Dav i s, Ca lif o rnia Pall Friction Damper s 1 996 Cal ifornia at Davis H arry Stevens Building Reinforced Vancouver, Canada Pall Friction D ampe r s 1 996 Co n crete, 3 S t ory Justic e H eadquarters R ein forced Ottawa, Ca nada Pall Friction Damp e r s 1 996 Co n c r ete, 8 S t ory BCBC Se lkirk Wate r front S t ee l 5 S t ory Victoria, Canada Pall Friction Damper s 1 997 Office Buildings Mai s on s de Be auco ur s Reinforced Qu ebec City, Ca nada Pall Friction Damper s 1 997 Co ncr e t e, 6 S t ory Mai so n S h e r win William Reinforced Montreal Ca nada Pall Friction Dampers 1 997 Conc r ete, 6 S t ory Con s t antinou, M C Soong, T.T. and D argus h G F ( 1998) 1 4

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2.2 Pall Friction Damper s Of the in s t a llation s of friction damper s throughout Canada and the United States, the vast majority have been based on the damper arrange m ent developed by Pall and Mars h in 1982 Considerable s tudies have been carried out at the University of Briti s h Columbia in Vancouver, British Columbia on the effectiveness of the damper arrangement at reducing the se i s mic respon se of structures. Filiatrault and C herry ( 1990), studied the effect va riations in damper slip force had on the overall respon se of s tructures fitted with the dampers. Their st udies resulted in the development of a basic design approach for structures fitted with the damper assemblies and a method for determining the damper slip force that minimized the seismic response of the structures. Becau se of the great deal of s tudy centered on the damper assemb l y, a basic description of the assembly is instructive The dampers known as "Pall Friction Dampers" are well s uited to flexible tension-only bracing systems a lthou gh they can also be in s talled in vert i ca l bracing systems made up of members capable of resisting both tension and compressive loads The bas ic arrangeme nt of the sys tem is s hown in Figure 2 .2. The assemb l y consists of a se ri es of hnks and dampers incorporated into the inter sec tion of vertical cross bracing member s. As a frame fitted with common tension only bracing is displaced laterally only the br a cing membe rs carryi n g tensile loads are effective due to buckling of the relatively flexible compression members. In a system fitted with Pall Friction Dampers as the frame dis plac es laterally the bracing members carrying tensile loads act to pull the link s of the damper into a rhomboid s hape. This results in slippage of the friction dampers and a shortening of the compression braces. 1 5

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As the motion of the structure reverses and the frame displaces in the opposite direction, the bracing members earlier in compression now carry tensile loads and act to pull the damper assembly into the opposite direction again resulting in slippage of the dampers and shortening of the bracing members now carrying compressive loads The arrangement of the system eliminates buckling of the compression braces and allows the dampers to effectively dissipate energy as the motion of the structure reverses If friction dampers were simply installed into the vertical bracing members, loads sufficient to cause slippage of the dampers could not be developed in the slender compression members due to buckling and the overall damper performance would be less than that developed in the Pall system. The reader is referred to Pall and Marsh (1982) for a more thorough discussion of the system. Damp e r Mechanism (a) (b) Figure 2 2 Pall friction damper (a) At re st; (b) Dis placed confi g uration The damper arrangement developed by Pall and Marsh is effective at dissipating energy and has been extensively studied however it is not the only arrangement possible for friction dampers. As previously stated, researchers have proposed a number of different damper arrangements and methods for incorporating the dampers into the framing system of structures. The system is presented in detail only to give the reader an understanding of the basic arrangement before presenting the results of studies completed on the system 16

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2.3 Optimum Slip Force Filiatrault and Cherry (1990) s tudied the effect v ariations in slip force had on the overall seismic respon s e of structures fitted with Pall Friction Dampers. They measured the effectiveness of the dampers through a relative performance index defined as: RPI=}_( SEA + Umax J 2 SEA{O) U max(O) (2.3) In equation 2.3 SEA is equal to the summation of the instantaneous strain energy present in the friction damped structure during the duration of the earthquake and is equal to the area under the strain energy time history plot. SEA(o) is equal to the summation of the instantaneous strain energy present in an identical structure with the friction damper slip force set equa l to zero Umax is equa l to the maximum strain energy occurring at any time during the duration of the earthquake in the friction damped structure and Umax(O) is equal to the maximum strain energy occurring in an identical structure with the frictio n force set equa l to z ero The relati ve performance index provides a means of comparing the response of a fully elastic system to that of the same system with friction dampers added and provides a measure of the effectiveness of the friction dampers at reducing the response of the s tructure A value of 1.0 corresponds to the response of the fully elastic s tructure before friction dampers are added. An RPI value less than 1 0 indi cates that the response of the structure has been decreased by the installation of the dampers A value greater than 1 0 would indicate the re s ponse has been increased 17

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The calculations involved in determining the relative performance index are presented by way of an example. Figure 2.3 presents strain energy time his tory plots for two SDOF systems and are representative of those that would be used to determine the relative performance index for a structure. Strain Energy 300 .--------.--------.--------.--------.--------,--------. 250 200 ISO 100 0 10 I 5 20 25 30 t (sec) !-Without Damper---With Damper I Figure 2.3 Strain energy co nt ent of syste m s with a nd without friction dampers The time dependent va riations of strain energy present in the friction damped and elastic s tructure s are determined by completing time history ana l yses considering the structura l properties of the systems, the s lip force specified for the dampers and the earthquake ground motion under consideration. Ba se d on the definition of s train energy used by the researchers the strain e nergy present in the structures at an arbitrary instant in time represents the total amount of recoverable energy stored in the system and is equal to the sum of the kinetic and potential energy present s uch that ; (t) = E K (t) + E s (t) (2.4) where; 1 8

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(2 5) and, (2.6) From Figure 2.3, it can be seen that Umax(O) is equal to 280 in-lb and occurs at 4.6 seconds and Umax is equal to 147 in-lb and occurs at 2.06 seconds. The areas below each plot were calculated yielding values for SEA(o ) and SEA of 676 and 284 in-lb sec respectively. Using these values and equation (2.3), the resulting RPI value is determined to be 0.47 for the structure and damper slip force considered. It should be noted that the definition of strain energy used by the researchers and that presented in Chapter 1 are different. It is assumed that the intent of the researchers is to use the value for SEA and SEA(o) based on the total recoverable energy content of the systems In strict terms, the strain energy of the system is defined by Equation (2.6) and does not include a kinetic energy component. The difference however is only applicable to the computation of SEA and SEA(o ) because at the peak displacement of the structure, the total stored energy is equal to the strain energy of the system. The example calculations presented above consider only one value for the damper slip force. Filiatrault and Cherry considered the effect variations in the damper slip force had on the resulting relative performance index They completed a series of time history analyses using the computer program DRAlN-2D, A. E. Kanaan G H Powell (1975), and varying levels of slip force to determine the level of slip force that resulted in minimum relative performance indices for a variety of s tructures. Figure 2.4 presents a representative graph of 19

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their findings for one such study. The graph shows the variation of the relative performance index as the slip force is varied. Rel ative Perfo rman ce Index I 0.9 0.8 0.7 0.6 0.4 0.3 0.2 0.1 0 0 45 90 1 35 180 225 Slip Load (kN) Fig ur e 2.4 Slip load optimi z ation (Reproduced from Cherry, S. a nd Filiatrault A ., 1 993) Filiatrault and Cherry found that there was an optimum level of slip force associated with a given structure and a predominate frequency of earthquake ground motion Additionally they found that there was very little variation in the relative performance index at slip forces within a relatively wide range near the optimum level. The researchers concluded that the re s ponse of the subject structure was not particularly sensitive to l 0 to 15 percent variations in the optimum slip load In an earlier study completed by the same researchers, they found that for an optimum s lip load of 134 kN there was very little variation in the relative performance index from slip load s between 90 and 220 kN. The se results indicate that small variations in damper material properties and installed damper s lip load s would have little effect on the overall response of structures during strong earthquakes. 20

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2.4 Optimum Slip Force Desig n Spectrum B ase d on t h e co ncept of a re lati ve per forma nce ind ex, Filiatrault and C h erry ( 1990) co mpleted a parametric stu d y of the re sponse of friction d a mped multi -s tory structu re s in an a ttempt to d eve lop a correlation betw ee n four key input p a rameter s and t he optimum l eve l of friction force The input parameters considered in the s tudy included the numb er of stories, NS ; the ratio of the braced to unbraced natur a l period s for the s tructure T t/ T u; the r a tio of the natural period of th e brac e d s tructure to the predomin a nt natural period of th e ground motion T t/T g ; and the peak ground acceleration o cc urrin g during the earthquake aglg. The study co n s id e red a total of 45 struc ture s and t he value s lis ted in Table 2.2 for the input p ar a meter s Ta bl e 2.2 Fili a trault and Che rr y Study Param ete r s P a r amete r Value NS I 3 5 10 Ttlfu 0.20 0.40, 0.60, 0.80 for S=I 0.20 0.50 0 .80 for NS = 3 5 I 0 T./Tu 0.1 sec/Tu ; 0.7sec/Tu; 1.4 sec/ T u ; 2 sec / T u A.jg 0 005 0 05, 0 1 0, 0 1 5 0 20 0 30 0 40 for NS = I 0.05, 0.1 0 0 20, 0 40 for NS = 3 5, I 0 In th e s tud y, the optimum s lip force for the multi-de greeof-fr ee dom s tructur es, V0 wa s co n sidere d b y the re searc h ers to be e qual to the s um of th e s lip forces s p ecifie d for th e dampers lo cate d at each story of the structu r e s uch that; (2.7) where v; i s the s lip forc e for the damper assembly lo ca t e d at the t h leve l of the struc ture They further propo se d usin g an equal s tory s lip for ce s uch tha t ; 21

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v o v = -' NS (2. 8) The time history analyses co mpl eted in the study were based on artific ial earthquake ground motions developed specifically for each set of input parameters of p eak grou nd acce l eration an d period of earthquake ground motion. The reader is referred to Filiatrault and Cherry (1990) for a discussion on the development of the artificia l groun d motions Based on the results of the stu d y Filiatrault a nd Cherry proposed the following equations to define the optimum slip force: [(-1.24NS-03I}T, l + l.04NS + 0.43 T v o T g T u for (2. 9a) mag T,, T,, and v o [ (0 OINS + 0 02}T, l 1 25NS 0 32 T b T u mag T u (0. 002 -0 002NS)T T + g + 1 04NS + 0.42 for __.!_ > I (2 9b) T,, T u Figure 2 5 presents Equations (2 .9) in terms of an optimum slip force design spectrum T, Figure 2.5 Optimum s l ip force desi g n s pectrum 22 15 .. ...

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The optimum slip force design spectrum developed by Filiatrault and Cherry provides engineers the means to determine the level of slip force that will minimize the seismic response of a structure. The spectrum requires input va lu es based on the properties of the structure the peak ground acce l eration expected at the building site, and the predominant period of the earthquake ground motion. The braced and unbraced natural periods are calcu l ated from the physical properties of the structure and the peak grou nd accelera tion is defined by the local building code. The period of the earthquake ground motion is not readily avai l ab l e in most cases. To provide the eng ineer with a method to determine an appropriate value, the researchers suggested that the methods proposed by Vanmarcke and Lai (1980) be used to estimate the value as; T = 2n 8 270.09REQ 10km s R s 160km (2.10) or 27r T =---8 65 7 .5M L (2. 11) In Equations (2.1 0) and (2.11 ), REQ i s equal to the distance to the epicenter of the earthquake in km, and ML is the Richter magnitude of th e earthquake. 2.5 Simplified Design Approach Proposed by Filiatrault and Cherry Base d on the optimum slip force des ign spectrum, Filiatrault and Cherry (1990) proposed a simplified seismic de sign procedure for structures fitted with the Pall friction dampers. The design procedure consists of the following steps: 1. The main framing sys tem of the structure is de s igned as a moment frame proportioned to carry vertical gravity loads on l y. It is assumed that the vertical 23

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bracing syste m fitted wit h friction d ampers wi ll safe l y di ssipate a ll earthquake energy so l a t era l lo ads n eed not b e considered in the design of the moment frame. The natural p er iod of the moment frame i s calculated and taken as the period of the unbra ce d s tructur e Tu. 2. Vertical br aci n g membe r s are selected and the natural period of the braced moment frame T b is calculated. The vertica l bracing members are se lected such that the ratio Tt!Tu falls within th e ran ge 0 .2 0 and 0.80 The researchers s uggest proportioning the bracing members s uch that the ratio i s less than 0.40 if economically feasible The range of 0.20 to 0 8 0 represents reasonably prac tical limit s and matche s that u se d in the parametric study 3 The earthquake parameters a g and T u are d e termin e d for the building s ite Typically, th e va lue of the peak ground acceleration is available from build i n g codes The re searc h ers s u ggest using the equations presented b y Vanmarcke and Lai to estima t e the pre dominate p eriod of gro und motion. 4 The optimum s lip force for the entire building is ca lcul ated from the design s lip force spectrum an d the slip force is distributed e quall y to eac h l eve l of the struc ture 5. The ca p acity of the vertica l bracing members are calculated and c omp a r e d to the loads resulting from the distributed s lip forces. I f n ecessary, new ver tic a l memb e r s i zes a r e selected a nd steps 2 through 5 a r e r epea ted. 6 A wind loa d a n a l ysis is completed for the brac e d moment frame to verify th a t the dampers will not s lip durin g w ind loading conditions. If w ind load s indu ce damper sli p the moment frame i s modified to carry a larger portion of the w ind load s and steps 2 throu g h 6 are repeated. 24

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The design approach is valid only if the structure and earthquake parameters fall within the range of those considered in the original parametric study such that; T 0.20:::; __!!_:::; 0 .80 T,, T 0.05:::; _!__:::; 20 T,, 2.6 Areas Requiring Additional Study a 0 .005:::; _!_:::; 0.40 g NS :o;IO The simplified design procedure proposed by Filiatrault and Cherry provides engineers with a mean s of determining the optimum slip force and distribution of d amper forces throughout the structure and can be u se d for new design and for retrofitting existing structures. The procedure focuses primarily on the determination of the l eve l of friction force required to minimize the seismic r esponse of the structure. It does not however provide a method for evaluating the seismic loads induced in the moment frame and furthermore does not provide a means to evaluate the inela s tic deformation characteristics of the frictiondamped structures Two distinct systems are pre se nt in friction-damped structures; the bracing system fitted with dampers, and the elastic moment frame. The forces carried through the bracing system are controlled by the s lip forces specified for the associated dampers Because of this the design approach propo se d by Filiatrault and Cherry provides an effective means of designing the vertical bracing syste m fitted with dampers. Forces carried by the e l astic moment frame however are controlled by the displaced shape of the frame Proper design requires that the forces induced by the lateral displacement of the frame be included. Engineers must have the ability to estimate the overall displacement of the s tructure in order to determine the s e load s Additionally, engineers must have a means of estimating the displaced shape of the structure to evaluate loads re s ulting from P-Delta effects and to verify that interstory drifts do not 25

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become so great as to adversely effect nonstructural elements of the building. Because of this, additional study is required regarding the deformation characteristics of friction-damped structures The second area that requires additional study deals with the performance requirement s of the damper assemblies themselves Friction dampers dissipate the input energy of an earthquake by transferring the kinetic and strain energy present in the structure to thermal energy To operate in a s table manner the dampers must be capable of dissipating this thermal energy without overheating. Proper design of damper assemblies must therefore take into consideration the total amount of energy to be dissipated and the maximum rate at which the energy i s to be dissipated Because of this, additiona l studies are required to determine the energy dissipating requirements of friction dampers. Additionally, to fully account for the repeated loading conditions experienced by friction dampers, engineers must have the ability to estimate the total number of cycles of loading expected to occur during the design earthquake Studies related to damper assembly design requirements should include the number of slip cycle s experienced 26

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3. Overview of Parametric Study 3.1 Introduction In this chapter, an overview of the parametric study of friction-damped single-degree-of freedom systems will be presented along with the development of the ana lytical modeling methods used to carry out the study. The overall goals of the study along with the structural and earthquake parameters considered will be discussed in Section 3.2 The properties of friction damped moment frames will be developed and the SDOF system model considered in the study along with the equations governing the time dependent response of the frames will be presented in Section 3 3. Section 3.4 will present the methods used in the study to solve the equations of motion. Additionally included in Section 3.4 will be a discussion of the energy content of the systems during earthquake loading and the components of response for inelastic systems. Section 3 5 will present the earthquake ground motions considered in the study. Current code prescribed earthquakes and elastic response reduction factors will be discussed in Sections 3 6 and 3 7. The discus s ion will lead to the development in Section 3.8 of representative code prescribed earthquakes for each of the earthquakes considered to allow correlations to be made between the inelastic response of the frames and the earthquake parameters of peak ground acceleration and one-second spectral acceleration normally specified by building codes 3.2 Objectives and Scope of Parametric Study The primary goal of the parametric study presented in this thesis is to develop a method for estimating the inelastic deformation response of single-degree-of-freedom friction-27

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damped s tructures ba se d on known input paramet ers of the s tructure and the earthquake under consideration. The s tructural parameters con s idered in the s tud y include the s tructures n a tural period prior to sli ppage of the friction damp ers, the s lip force specified for the damper s and the amount of elastic sec ondary st i ff n ess present in the structure during damper s lip The parameters of the earthquake considered in the study include the peak ground acceleration and one-second s p ec tr a l acce leration associated with the earthquake. The earthquake parameters are se l ected to corre s pond to those currently used by building code s in defining d esign bas i s earthquakes The approach used to develop a method of estimating the inelastic respon se of friction d ampe d struc tur es i s to first develop inelastic response s pectra for SDOF sys tem s with a s mall amount o f sec ondary stiffness pre se nt in the s tructure durin g damp e r slip For thi s portion of the s tud y the inela s tic responses of S DOF sys t e m s with 0 and 5 percent elastic seco nda ry stiffness and four l evels of s lip force are evaluated and a correlation i s de ve loped b etwee n the earthquake input par a meter s and the res pon ses. Ba se d on the r es ult s of thi s part of the stu d y a method that can be u se d to formulate the inelastic response spectra for SDOF systems w ith small amounts of sec ondary s tiffn ess i s presented Once th e r e lationship between the eart hqu ake input p a r a m eters and the inelastic r es pon se spect ra are developed a n evaluation of the effects of secondary stiffne ss is com pleted. In this part of th e s tudy a series of inelastic time history ana ly ses a r e compl e t e d for the same sys t e m s considered in the first portion of the s tud y wit h sec ondary stiffness r atios varyi n g bet wee n 0 and I 00 p er cent Again the correlation b etwee n the amount of seco ndary stiffness and system r esp onse i s d eve lop e d and a method to formulate inelastic re s ponse spectra for sys tem s of varying seco ndary stiffness is presented. 28

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The ultimate goal of the study of the inelastic response of SDOF systems is to develop a method that could be used to estimate the displaced shape of multi-degree-of-freedom friction-damped structures subjected to earthquake ground motions The goal is to develop a method that is based on the inelastic response spectra developed in this part of the study and standard modal analysis techniques currently used in the analysis of multi-degree-of-freedom systems. A method that could be used to estimate the displaced shapes of multi-degree-of freedom structures is presented in Chapter 5 along with a study of the correlation between the estimated displacements and those determined by inelastic time history analyses. An additional goal of the study of SDOF friction damped structures is the quantitative evaluations of key parameters that affect the design of friction damping mechanisms These parameters include the number of slip cycles experienced by the dampers, the maximum amount of energy dissipated and the maximum rate that the dampers dissipate energy during earthquakes To determine the values associated with each of the parameters, inelastic time history analyses are completed for SDOF systems with natural periods ranging from 0 .25 to 2.5 seconds and four levels of damper slip force. For this portion of the study, only systems with zero secondary stiffness are considered In all, the study consists of three parts; the determination of inelastic response spectra for SDOF systems with small levels of secondary stiffness ; the study of the effects secondary stiffness has on the overall deformation response of the SDOF systems and the quantitative evaluation of the parameters affecting the design of friction damping mechanisms. The parameters considered in the study include; the natural period of the system prior to damper slip, T N ; the amount of secondary stiffness present in the system during damper slip, defined as the ratio of the moment frame stiffness to the total structure stiffness,; and the slip force 29

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specified for the dampers in terms of a slip coefficient, R. Table 3.1 presents the values of the parameters considered in eac h portion of the study. Tab l e 3.1 Study parameters Parameter Value P a ram e ter Inela s tic Sg_ectra SecondaryStiffness Effects Dampe r Design T N (sec) 0 to 3 0 25, 0 50, 0 .75, 1.0 1.25 0 25, 0 50 0 75, 1 0 1 .25, 1 50, I 75, 2.0 2 25, 2.5 1.50, I. 75, 2.0 2.25, 2.5 0, 5% 0 to 100 % 0% R 4 6 9 12 Oto 00 4 6, 9, 12 3.3 Friction-Damped Moment Frames The damper configuration considered throughout this thesis is shown in Figure 3.1. The system consists of a moment frame possessing a lateral st iffness kF, fitted with a vertica l bracing system that adds an additional stiffness, k0 to the system. The vertical bracing i s connected to the beam of the moment frame through a friction damper assembly The friction damper assembly is proportioned such that it wi ll begin to slip when the force transferred through the damper is equa l to a specified damper slip force ,fs1 The slip force is equa l to the product of the normal compressive force applied across the slip planes of the damper and the coefficient of friction of the material s used to fabricate the damper. Moment Frame, kF Friction Damper,.fs, Figure 3.1 Friction-damped mom ent frame. 30

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The idealized force -defo rmation relationship for the combined vertical bracing and damper under cyclic loading conditions is shown in Figure 3 2. The system considered in the figure consists only of the vertical bracing system present in the friction damped moment frame and a damper with load-deformation properties consistent with those presented in Figure 2.1 (a) The system begins in an unloaded state at point "0" in the figure. The system is then displaced in the positive direction. While the forces transferred through the damper are below the slip force the assembly responds in a linearly elastic fashion. In this region of re s pon se, identified b y the portion of the graph between points "0" and "a", the damper displacement is related directly to the force transferred through the assembly such that the displacement at which slippage of the damper occurs can be defined as; D =fs t sf k D fo Figure 3.2 Ideali z ed combined damper /fra min g system behavior (3.1) Once the di s plac eme nt of the damper assembly excee ds th e sli p displacement D51, the force transferred through the damper assembly remains constant and equal to the s lip force as the plies of the damper sli de relative to one another. During this portion of the response, the 31

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force transferred through the damper assembly remains constant until the motion of the damper is stopped at point "b", at which time the damper plies become locked into position by the friction force developed in the damper. As the direction of motion of the system reverses, the force carried by the bracing system decreases until the bracing system returns to its unloaded state. This condition is identified by point "c" on the graph. At this point, the damper has slipped a distance equa l to the segment "Oc" and the energy dissipated by the damper assembly is equal to the area of the hysteresis loop bounded by "OabcO". As the motion of the system continues, the force transferred through the damper assembly agai n increases until slippage takes place in the negative direction at point "d". Slippage of t h e damper again takes place until the motion stops at point "e" at which time the damper plies again become locked into position and another unloading cycle begins for the bracing system. The motion of the damper assembly effectively limits the l ateral deflection of the vertica l bracing system to a maximum of D ,1 Displacements of the system mass greater than the slip displacement are accompanied by slippage of the damper and a lateral displacement of the vertical bracing system equal to D ,1 The behavior of the vertical bracing system with the damper assembly added is the same as that of an idealized elastic-perfectly plastic material with the exception that yielding of the members does not occur. In a properly designed friction damped structure the braces are designed to carry a load somewhat higher than that tran s ferred through the damper assembly. The friction-damped moment frame shown in Figure 3 1 possesses two levels of stiffness. While the lateral displacement of the damper assemble is less than the slip displacement, the bracing and moment frame act together and the total stiffness of the combined system is equal 32

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to the s um of the moment fra me s tiffne ss, kF and th e stiffuess of the bra cing system, k0 s uch that; (3.2) I f the lateral displace m e nt of the damper assembly i s greater than the s lip di s pla ce ment of the damper the s tiffn ess contributed to th e overall syste m by the bracin g syste m de creases to zero a nd the total sys tem stiffuess i s then e q ua l to that of th e moment f rame alone s uch that ; (3. 3) T he force-displacement r e lation s hip for the combined system, ass umin g only motion in the positive direction i s s hown i n Figure 3.3. T h e c h a n ge in st iffuess for displacements greate r than the damper sli p dis pl aceme nt i s apparent in th e fig ure Note th a t in the combined sys tem, if the motion of the sys tem s t ops s uch that the damper becomes locked into position an d then r eve r ses dir ec tion the s tructure unlo a d s along the d as h e d line s hown in the Figure 3.3. r / ,.. / / / / / / / / / / D Figure 3.3 Force-Deformation relation s hip for friction damp ed moment frame. It is co n venien t to define th e ratio of th e stiffuess contributed by the mom e nt frame to th e t ota l com bin e d system s tiffness as the secon da ry st iffn ess r at io s u c h that; (3.4) 33

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A value of zero for the secondary stiffuess ratio indicates a framing system that relies only on the vertical bracing to resist lateral loads such as in the case of an ordinary braced frame fitted with friction dampers In such a system, there would theoretically be no elastic stiffness present at displacements greater than the slip displacement of the damper assembly. In real world structures however there would typically be some level of secondary stiffness due to the inherent stiffness of framing connections, interior partitions, wall panels, etc. Although some secondary stiffness would be present, the amount wou ld be small in comparison to the stiffness of the system prior to slippage of the friction dampers and is neglected in the st udi es completed for systems with secondary stiffness ratios equal to zero. A value of 1.0 for the s econdary stiffne s s ratio indicates a framing system with no vertical bracing or dampers present. Such a system would consist only of the moment resisting frame Values between 0 and 1 0 represent a combined system with both a moment resisting frame and a vertical bracing sy s tem fitted with dampers Along with two levels of stiffness friction-damped moment frames posses two distinct natural period s ofvibration. The natural period ofvibration of the combined framing system, prior to slippage of the dampers, is a function of the total system s tiffne s s and is defined as ; u o < D s t (3.5) During the time slippage is taking place in the damper assembly the stiffness of the system decrease s to k F and the natural period increa s es to ; (3.6) 34

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In terms of the secondary stiffness ratio, the ratio of the braced to unbraced natural periods ca n be expressed a s; (3. 7) We now consider the response of the system shown in Figure 3.1 at an arbitrary instant in time while subjected to an earthquake strong enough to cause slippage of the dampers. Under these conditions the system is as shown in Figure 3.4. It is assumed that the forces transferred through the damper assembly have previou s ly resulted in at least some s lipp age of the damper. Two distinct displacements exist within the system; the displacement of the mass and the displacement of the friction damp er. The displacement of the ma ss i s equal to u(t) and the displacement of the damper is equal to u0(t) u(t) ti u o (t) Figure 3.4 Friction-d a mp e d mome nt frame a t a rbitr a r y ins tant i n time. The combined sys t e m is s hown sc hematically in Figure 3.5 where the stiffness of the mom e nt frame and vert ical bracing systems are replaced by individual s pring s with s tiffn esses of k F and k0 The specifi e d damper s lip force i s included in the model of the vertical bra c ing sys tem asfs1 Th e viscous damping present in the system is modeled b y a da s hpot with a viscous dampin g coefficient c The sys tem is shown in an unloaded s tate in Figure 3 5(a) and at an arbitrary in s tant in time in Figure 3.5(b). AD' Alembert free-body 35

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diagram showing the forces acting on the mass at the time under consideration is shown in Figure 3 5 (c) P(t ) fo(t) f s F (t) FBD (a) (b) (c) Fig u re 3 5 Sch e mati c re pre s entation of friction damped s tructure The individual forces acting on the mass due to the combined framing system can be written as; f s F (t) = kFu(t) h F (t) = J y u > u y (3.8a) (3.8b) where.fs F i s equal to the forc e carried by the moment framing sys tem,/y is equal to the yield s tr e n g th of the frame, and u y i s equal to the yie l d displacement. fsv (t) = k D u D (t) f.v (l) = J., (3.9a) (3.9b) w here.fs0 i s equal to the force carried by the vertical bracing syste m,.fs1 is equal to the s pecified s lip force for the damper assembly and u5 1 i s equal to the damper slip di s placement as pre v iou s ly defin e d and ; fv (t) = cu(t) (3.10) where /0 is equal to the viscous d a mping force The inertia force acting on the mass during the earthquake, Pg(t) can be written in terms ofthe earthquake ground acceleration iig (t) as ; (t) = mii g (t) (3. 11) 36

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The equation governing the time dependent motion of the system is obtained by summing the forces acting on the mass and applying Newton's second law of motion such that; (3. 12) or with Equation (3.11) subs tituted for the forcing function, P(t) the equation governing the earthquake response of the system becomes ; (3.13) Solution of Equation (3. 13) yields the time dependent displacement velocity and acce leration of the system mass along wit h the time dependent displacement of the friction damper assembly. The time history analyses of SDOF systems completed in the parametric s tudy pre sente d in this thesis are based on the model s hown in Figure 3.5 and the solution of Equation (3.13) by time stepping methods. The methods u se d to solve Equation (3. 13) are presented in the following sec tion. 3.4 Solution of the Equation of Motion by Time Stepping Methods 3.4.1 Introduction The dynamic response of str ucture s s ubjected to conti nuou s periodic forcing functions can be detel11llned by direct solu tion of Equation (3 12) This is not the case however for structures s ubjected to eart hquake ground motion s becau se ofthe non-continuous nature of the motion and the potential nonlinear response of the str ucture s. To evaluate the response of struc tures s ubjected to earthquake ground motions, it is nece ssa ry to u se approximate time stepping methods to determine the re s pon se of the structu re. The methods are referred to as time steppi ng because the overall re sponse of the structure is determined by considering the response over a se ries of s hort time steps. The response during each time step is dependent upon the initial 37

PAGE 49

conditions of the system as determined by the response during the previous time step and assumptions regarding either the variation of the forcing function or the system acce l eration over the time duration being considered. By comp leting ana l yses for a number of short time durations it is possible to approximate the actual motion of the system during the earthquake. In time stepping methods, the inertia force acting on a structure is transformed from the continuous function given by Equation (3.11) to a series of discrete forces at specific instances in time such that; P =-mii I gj (3.14a) and ; P = -mii t+l Ei+l (3.14b) w h ere iig; is equa l to the ground acceleration at the ithinstant in time and corresponds to the start of the i1 h time step and ii is equal to the ground acceleration at the /h + 1 instant in time g l + l and corresponds to the end of the i1h time step Typically earthqauke gro und accelera tion s are recorded at intervals of 0 02 seconds To determine the overall response of a system subjected to 30 seconds of earthquake ground motion, 1500 time steps of 0 02 second duration need to be considered The dynamic analyses of single-degree-of-freedom systems presented in this thesis were comp leted using the PC based program "Response" developed by the Author as part of an independent study at the University of Colorado at Denver. "Response" has the ability to analyze the elastic and inelastic response of SDOF systems with and without friction dampers. The reader is referred to the program user's manual included in Appendix B for a detailed discussion of the program's capabilities In the program, the analyses of elastic systems are comp l eted 38

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using a time stepping method that incorporate s an exact s olution of the governing e quations of motion based on an assumed linear variation of the forcing function over each time step. Inela s tic analyses are completed u sing Newmark's Method with a linear variation of system acceleration over each time step and automatic time s tep reduction s at tran s ition s between linear and nonlinear re g ions In addition to time s tep reductions an iterative procedure i s incorporated into the program to assure convergence of accelerations within the transition time s tep The m et hod s incorporated into the program to analyze elastic, inelastic a nd inela s tic syste m s with friction damper s are presented in the followin g sec tions. 3.4.2 Elast i c SDOF Systems The res pon se of linear sys tem s is calculated in th e program u sing the method of int e rpol at ion of excitation pre se nted in C hopra ( 1995) This method provides an exact so lution to the e quation s of motion g overnin g the res pon se of a si n g le-degre e-of-free dom sys t e m s ubjected to a forcing function that varies line arly over the time interval bein g co nsidered. The total sys tem r es pon se during each tim e interv a l is comprised of thr ee indi vid ual r espo n ses ; the free v ibration re s pon se of the syste m s ubj e cted to initi a l di s pla ce m e nt and ve l ocity a lone; the forced response of the sys t e m s ubj ec t e d to a constant force P; w ith ze ro initial conditions; and the fo rc e d re s pon se of th e syste m s ubj ecte d to a ramp f unction which varie s fro m 0 a t t; to (P; +1 -P;) at t i + l agai n wit h zero initial con dition s The r eade r i s r eferre d to C hopra (1995) for a complete dev e lopment of the m e thod The dis plac e m e nt and velocity of the sys t e m at the end of each time s t e p are e qual to ; (3.15) 39

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(3 16 ) and the coefficient s in eq u atio n s 3.15 and 3. 16 are calcula t ed as ; (3. 1 7a) (3.17 b ) (3. 1 7c) D = -1---+ e -,;w.ll sin m !1t +--cos m !1t 1 [ 1 )] k (J), /1/ (J)D /1/ D (1)11/1/ D (3.17d) (3.17e) (3.17f) (3. 1 7g) (3.17h) The coeffic i ents calculated b y E quation s (3. 1 7) are function s of only th e s tru c tur a l prop ert ies of the sys tem i.e ma ss sti ffness, and d a mpin g ratio a nd ne e d be calculated only once for the a n a l ysis dur at ion 40

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The initial values U ; and il; are known from the pre v iou s time s tep or from the initial conditions of the system in the case of the first time step. Additionally the values of P; and f>;+1 are determined from Equations (3.14) using the recorded ground acceleration for the earthquake being considered. The velocity and displacement at the end of each time step are determined by substituting the initial conditions, coefficients as determined by Equations (3. 17), and earthquake forces as determined by Equations (3.14) into equations (3.15) and (3.16). The process is repeated for each of the time steps throughout the duration of the analysis. 3.4.3 In elas tic S DO F Sys t e m s Before developing the method used to analyze the inelastic response of moment frames fitted with friction dampers it is instructive to first consider the case of inelastic SDOF systems without dampers. The model of such a system is the same as that presented in Figure 3.5 with/so set equal to zero. The response of inelastic systems is calculated in the program using Newmark's Method as presented in Chopra (1995) and Clough (1993) with the added assumption of elastic perfectly-plastic material properties that exhibit the same force-deformation relationships previou sly s hown in Figure 3.2. The method is based on an assumed variation of system acceleration over each time step and the added requirement that dynamic equilibrium i s satisfie d at the s tart and end of each time s tep Newmark developed the followin g e quation s, which r e late the velocity and disp l acement of the system at the end of a time step to the initial condition s and an assumed variation in the system's acceleration during th e time step. 41

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(3 .18) (3 .19) The parameters gamma and beta are weighting functions that control the variation of velocity and displacement over the time step. The reader is referred to Clough (1993) for a detailed discussion of equations (3 .18) and (3 19) The initial values u; and u; are known from the calcu lations for the previous time step or from the initial conditions ofthe system in the case of the first time step. The initial acceleration ii.; is calculated from Newton's second law as or mUgi-CU;kFUi u,. = m (3. 20) This relationship is also valid for the acceleration at the end of the time step giving, (3 .21) m Substitution of equations (3. 18) and (3.19) into equation (3.21), and rearranging terms, provides an equation for the acceleration at the end of the time step in term s of the initial conditions and the earthquake ground acceleration at the end of the time s tep Equations (3.18), (3. 19) and (3.22) can be u se d directly to determine the response of elastic systems and will yield results very close to those determined by Equations (3.15) and (3 .16). However the analysis of nonlinear systems requ i res modification s to the equations 42

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pre s e nted a b ove t o acco unt for yie ldin g As th e sys t e m tra n s ition s into th e ine la s tic ran ge t he s p r i n g f o rce is ass um e d t o b e e qu a l t o i t s yield s tr e n gt h a nd th e s tiffn ess of th e sys t e m b eco m es ze r o. With y ie ldin g included in th e anal ys i s th e s prin g i s limited s uch th a t ( 3.23 ) The relat i on s hip between the di s pl a c e ment of the system u and the sprin g force f s F i s no lon g er a continuous linear function becau s e o f the limit impos e d in equation (3. 2 3 ) To a cc ount for thi s limit we define the e f fecti v e s prin g dis plac e m e nt ueff s uch that f s F = k F uef! ( 3 .24) The effective spring displacement is calculated at the end of a time step a s, (3.25) To as s ure that the limit imposed by equation (3.23) is met u eff is limited s uch that ( 3 .26 ) T w o s eparate displacement s a re now defin e d ; that of the s ystem mass u and that of the s p r in g ueff. Prior to th e occurr e nc e o f yieldin g th e two displac e m e nt s will b e e qual. How e ver afte r y i e ldin g h a s o c c u rred th e two m ay o r m ay not b e e qu a l d e p e nd i n g o n t h e amount a n d d irectio n of th e re s ult i n g yie ld cyc l e s T h e d iffere n ce b e t wee n th e two dis p la cement s repre s ent s t h e amount of y ie ld defom1a t ion in t he s pr i ng W ith th e effective sprin g dis p lace m e n t incl u de d e q ua tion s (3.20) a nd (3.22) are modifi e d t o 4 3

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(3.27) m and .. = -miigi+l -cuic {[t-r N K6t)u"Jk F u cffi-kF ) + [ 0 5 ,BN K6t Yii; } (3 28 ) u,+l [m+ k F,BN (6t)2+crN (6t)] For elastic-perfectly-pla s tic systems the stiffness is a s sumed to be equal to k F when lu eff I < u>' and equal to zero when lu eff I= u r Therefore two distinct regions of respon s e exist ; the elastic region ( lueff I < u>' ) and the inelastic region ( lueff I= u>' ) Equation s (3 .25) through (3 .28) along with equations (3 18) and (3 .19) are used directly to calculate the response of a nonlinear system whi l e within the elastic region. While the system is within the inelastic region, the effective spring displacement remains constant and the system stiffness is zero. With the constant spring displacement and zero stiffness included equations (3.27) and (3.28) can be reduce to mii -cit -f (lueffil) g t I )' u e ffi (3. 29) m and ui+l = [m+cy(6t)] ( 3.3 0) Equ a tion s (3.25) ( 3 .26) (3 29) and (3 3 0) a lon g with equations ( 3 18) a nd ( 3 19) defin e the r es p o n s e o f a n o nline a r system whil e within the ine la s tic re g ion. 44

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The two gro up s of equations presented above defin e the respon se of a nonlin ear syste m while the sys tem i s completely within th e e l ast ic re g ion or completely w ithin the inelastic region during the time step During either of these ca ses, the acceleration calculated at the end of the previo u s time step will b e equal to the acceleration calculated at the beginning of the current time s tep Thi s i s not the case as the system tran s ition s from elastic to inela s tic regions or back again. Differenc es in the two calculated values arise because the s tiffness of the system changes during the transition time step making one or the other groups of equations invalid for a portion of the time step. Error can be introduced into the calculated respon se if the difference s are not addressed The reader is referred to Chopra (1995) for a complete discussion. Two procedures are incorporated into the program used in the parametric study to assure convergence of accelerations The first is a decrease in the calculation time step during the transition period After detection of a difference in ending and beginning accelerations this procedure divides the current calculation time step into smaller s teps and recalculates the re s pon se. The second procedure i s iterati ve and i s completed only for the s hortened time step durin g which the transition actually occurs. In the seco nd procedur e, the e ndin g ve lo c i ty a nd di sp lacement for th e transition time step a r e calculated u s in g the initial acceleration U ; for the tim e ste p occurring immediat e l y after the tran s ition tim e s t e p in pl ace of th e final acce l e ration iii+ l for th e transition tim e ste p Becau s e the initial acce l e rati o n ii; for the t ime s tep occurring immed i ate l y after the transition time ste p i s a function of th e fina l acce l eratio n ui+l for the transition tim e ste p iter atio n s are 45

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required to obtain convergence of th e two acce l erations The s tep s of the procedur e are as follows: 1. Th e acceleration, ve locity and di s placement are calculated at the end of the transition time s tep u s ing equations (3. 1 8), (3.19) and (3.28) or (3.30) dependin g on w heth e r the system is within the e l astic or inelastic region at the start of the tran s ition time s tep 2. Usin g the velocity and displacement from step 1 the acceleration at the start of the time step immediately after the tran s ition time step are calculated u s ing either equation (3.27) or (3.29) again depending on the region the system is in at the end of the transition time step. 3. Step 1 is then repeated to determine the velocity and displacement at the end of the transition time step using the acceleration from step 2 in place of that calculated by equation (3.28) or (3.30) 4. Steps 2 and 3 are repeated until the accelerations determined by successive calculations in s tep s 1 and 2 converge to within an acceptabl e tolerance The procedure de sc ribed above achieves convergence after onl y a few iteration s. T h e tolerance on convergence is se t to 0.001 in/sec /se c in the program u se d in the s tud y. 3.4.4 Ine l astic SDOF Systems with Friction Dampers The ana l ysis of sys t ems wit h fric tion damper s i s comp lete d u sing the m e thod prese nt ed above for inela st ic sys tems with modification s to eq u ations (3.27) through (3.30) to acco unt for the additional sp rin g in c lud ed in the system. In addition to equations (3.18) and (3.19) the e qu a tion s gove rning the re spon se of fric tion damp e d syste m s are ; (3.31) 46

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(3.32) ( 3.33) (3.34) (3.35) m The subscript S is u se d to identify var iables as s ociated with the s pring without damper s and the subscript D is used for variables associated with the s pring fitted with the friction dampers The variable u SJ defines the dis placement at which s lip occurs in the friction dampers Equations (3.31) through (3.36) are used directly to calculate the response of a friction damped s tru c ture while both the frame and friction damp ers are within the elastic region As with equations (3. 29) and (3.30) governing the nonlinear system, equations (3.35) and (3.36) can be modifi e d to account for the limitin g force and lo ss of stiffness associated wit h either yie ldin g of the frame or slippage of the friction dampers. The modifications are simil ar to those pr esente d in d evelopi n g e quati ons (3.29) and (3.30) and w ill not b e presented h e re. The procedure identified above for ass urin g convergence of acce l erations in inela stic sys tem s witho ut friction damper s i s also applicable to inelastic systems with frictio n dampers. The procedure i s completed u sing the sam e s t e p s 47

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3.4.5 System E nergy Content The time stepping methods presented above allow the determination of the di splacement, velocity and acceleration of SDOF systems during earthquake excitation. Results of the analyses provide the information necessary to determine the energy content of the sys tem during the earthquake. The equation governing the energy content of a system during earthquake excitation was presented as Equation ( 1 1) in Chapter 1 and is repeated here for convemence. Equation (1.1) indicates that all energy imparted to the system by the earthquake is either stored within the system in the form of strain and kinetic energy or is dissipated from the system by means of viscous damping structural yielding, and damper slip The amount of energy stored within the system is calculated at the end of each time step a s; 1 2 1 2 E = k (u ) + k (u ) Si 2 sF ef!i+l S 2 sD ef!i+ I D (3. 37) E 1 2 Ki =2mui+l (3.38) The amount of energy dissipated from the system during each time step i s calculated as ; it + ti. -[ ], E Di = c I + I 2 I 8.1 (3.39) (3.40) (3 .41) 4 8

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whe r e u and !'J.u0 are the dis tances th e system displaces durin g yieldi n g a nd damp e r s lip r espect i ve ly. The cumulative energy dissipated from the syste m at a particular in s tant in tim e i s calcu l ated as th e s um of e ner gy diss ip ate d durin g each tim e s tep up to the tim e under co n s ideration as ; ; ED= I EDj j = l ; Ey = I Elj j = l ; EM= I EMj j = l (3.42) (3 .43) (3.44) The rate at which energy is dissipated from the system through either structural yielding or damper slip is approximated using the values determined by Equations (3.40) and (3.41) as ; and ; t5EM ::::: EMi t5t -!'J.t 3.4.6 Components of Inelastic Respon se (3.45) (3.46) Thro u ghout the parametric s tud y the d efo rm at i on c h a r ac t eris tic s of inela s tic sys tem s are related to those of e l astic syste ms. Becau se of thi s, it i s informative to cons id er the relationship b e tw ee n the seismic r espo n se of inela st i c s ystems a nd th at of co rr esponding elastic sys tems. The response of ine l astic syste m s can be s h o wn t o be comprise d of thr ee compo n e nts; the linear re s ponse of an identical sys t em assuming ela s tic prop erties throu g hout 49

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the full range of r es ponse, the re s ultin g pla s tic yield deformation and a transient component of re s ponse re s ulting from the energy di ssi p at ion occurrin g durin g y ielding The variou s components are identified by way of an example using a SDOF system s ubj ecte d to the Northridge earthquake. The re s ponse of the system was completed u s in g the time s t e ppin g method presented in section 3.4.3 For the analys is the earthquake ground acceleration wa s scaled to yield a peak acceleration of l .Og. The natural period of the system was chosen to be 1 0 second and damping was assumed to be 5 percent of critical. The normalized yield strength of the system was chosen to be 90 perc ent of that of an elastic system such that a single cycle of yielding occurs during the earthquake. The system deformation response and resulting yield deformation are presented in Figure 3 6. e "" 0 = 0 15 I 0 0 I 0 -I 5 I l 3 A I \J I Inelastic System Response A A A V V\ v V'--.JVV\. 'V\/'V v I 0 I 5 2 0 25 30 1 (sec) Y idd 0 eform at i o n I 0 I 5 20 2 5 30 1 (sec) Figure 3.6 o rthridge earthquake f y = 0 90 (a) Ine l astic defom1 ation r es p o n se, (b) Yi e l d deform a tion 50 (a) (b)

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Review of Figure 3 6(b) indicat es th at a single yield cyc le take s pla ce at approximately 4 seco nd s Yielding re s ults in a s hift of the eq uilibrium po s ition of th e sys tem as i s apparent in Figure 3 6(a) durin g the later portion of the re s pon se. The deform ation response of a correspo ndin g e la s tic sys tem with the same str uctural properti es and s ubjected to the sa me gro und accelerat i on is presented in Figure 3. 7. E la stic Syst e m Response 1 5 I 0 0 c I AA 1\ h f\f\f\ A A vv v v v v v -5 -I 0 I 5 0 5 I 0 1 5 20 25 30 t (sec) F i g ur e 3 .7 Northrid ge ea rthqu ake, Elastic deform a tion re spo n se The va riou s components of the inelastic system re s pon se b ecome apparent b y considering the difference between the inelastic re s pon se shown in Figure 3.6(a) and the e la stic response s hown in Figure 3 7. Subtracting the e la stic response from th e inela s tic re s pons e identifies the portion of the inelastic r espo n s e th a t is co m pose d of th e s um of th e yield d efo rm a tion a nd th e transie nt component. The re s ult s of this operation are pre se nt e d in Figu r e 3.8(a). S ubtracting th e y ield d eformat ion from th e combine d r esults d e t em1ine d a bove th en identifies the transient portion of the inel as tic re s pon se. Th e result s of thi s operation are pre se nt ed in Figure 3 8 (b) 5 1

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The total r es pon se of the inelastic sys t e m i s in this way s hown to be equal to the s um of the r es p onse of an equivalent e l astic sys tem th e y ield d efo rm a tion occurring in the in e l a stic sys tem and the transient component of re s pon se due to yielding of the i nela s tic sys tem G raphi cally the indi vi dual co mpon ents are id e ntified in Figures 3.7, 3 6(b) and 3.8(b) Inelastic Response-Elasti c Resp o nse 0 0 I \ A {\ {\ v VVV\ V'V -2 -3 0 5 I 0 I S 20 25 t ( sec) Transient Component 0 vvvv v -I 2 3 0 5 I 0 I 5 10 25 1 (sec) F i g ur e 3.8 Northrid g e eart h quake (a) Difference betwee n ine l ast i c and e l a stic r espo n ses (b) Tran s ient component of res p o n se 52 (a) 30 (b) 30

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3.5 Earthquake Ground Motions Used in Study Five recorded earthquake ground motions were selected for use in the study. The earthquakes considered, along with the recorded peak ground accelerations and the estimated predominant period of ground motion associated with each of the earthquakes, are presented in Table 3 2. The analyses completed in the parametric study considered 30 seconds of ground motion and a 0.02 second digitizing time step for each of the earthquakes. Table 3.2 Earthquakes used in Study Earthquake R ecor ded Peak Ground Predominant Period of Acceleration Ground Motion T. El Centro 1940 0 319g 0 85 seco nd s SOOE Component Lorna Prieta 1989 0.479g 0 76 seco nds Corralitos CHAN I : 90 De g Northrid ge 1994 0 .8 43 g 0 .51 seconds Sylmar County Hospital Parking Lot Chan I : 90 Deg Olympia 1949 0 280g 0 60 seco nd s N86E Component San Fern a nd o 197 1 1.076 g 0.43 seco nd s Pacoima Dam-S74W Source of gro und motion file s : Strong Motion Databa se, Ins titute for Crustal Studies ( I CS), Unive r sity of Ca l ifornia, Santa Barbar a (UCSB). To standar di ze the re s ponses obtained from each of the ground motions the earthquakes were norm alize d to yield peak gro und accelerations of l.O g by lin ear l y scaling the recorded ground motions Scaled time hi s tory plots for each earthquake are pre se nted in Figure 3.9. Additionally, the frequency content of each earthquake is pre sented by way of the Fourier Amplitude Spectra shown in Figure 3.10. 53

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E I C en t r o I z I 0 . . . . 3 .z . t l . ... -Ill-II I """ 'II/I II"UIMI 1Wrr'I'I'IY' 'IVIIIV 1' ,.,., "I . . .1 I 0 I z 10 I JO t ( tft') Loma Prieta I z I . I . . 3 z . z n 'IIIIU ._, w ' I . I .z I .z I . 0 I . . . 3 z . . z 4 II MM ll'lAIIIIJIIi .Ill\ "VI . -1. 1 I 0 I z 10 I " JO San I z 0 . . ' ' la. - 0 :t 0 z 0 WI . 0 1 I . I z It I' JO t (cc) F i gure 3.9 ormal i z ed ea rthquake ground motions. 54

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[ I C t n 1 r o 0 I 0 0 9 0 8 0 0 1 B 0 o 6 0 0 s < 0 .0" 0 0 J 0 0 2 0 0 I 0 0 0 m iJIIA lfllllj IU, II UINII lllil l't/1 IU" 111''1 v.. 'IY 111 .. rrJT"J" 0 0 s 0 1 0 0 I S .0 2 0 o 2 s o r ( H z) l om :a P r it 1 :a 0 I 0 0 0 9 0 0 8 0 o 1 B 0 0 6 M 0 5 < o 4 0 o 3 0 o 2 0 0 1 0 0 0 n, n 111\ .tid!. rv .,, v 0 0 s .o I 0 .0 I S 0 2 0 0 2 s 0 r < u z ) Nortkrldc;e 0 I 0 0 0 9 0 0 8 0 0 7 B 0 o 6 0 0 s 0 .0. 0 .0) 0 0 1 0 o l 0 o 0 an AI. IWinl '''IINI 11.1 v nv 0 0 s o I 0 0 I 5 0 2 0 0 2 s 0 r ( 11 z) 0 l y m p i:a 0 1 0 0 0 9 0 0 8 0. 0 1 B 0 0 6 M 0 0 s < 0 0 4 0 0) 0 0 2 0 0 I 0 0 0 I . .II .. 1ft ... I .II'M I .Ill M'll'l MaJMI.ollV!.IH -... ,,' 0 o s .0 I 0 0 I S 0 2 0. 0 2 s 0 r < H z) 0.1 0 0 0 9 0 0 8 0 0 1 B 0 6 w 0 5 < 0 0" 0 J 0 l 0 0 I 0 0 0 .M '""' "'' .... "' 0 0 s 0 I 0 0 I S 0 2 0. 0 2 s 0 r ( 11 z ) F igure 3.10 Fourier Amplitude Spectra. 55

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The Fourier Amplitude S p ec tra pr ovide useful information rega rdin g the frequency co nt e nt of the ea rthquak e gro und moti o n s a nd we r e u se d in the study to estimate the predominant p e riod s of ground motion The spectra were calculated u sing a disc rete fast Fourier tran sfo rm and the first 1024 digit i ze d ground acceleration records representin g 20.48 sec ond s of eac h earthquake. The reader i s referred to Pa z ( 1985) for a disc us s ion of the d eve lopment and implementation of the di s crete fast Fourier transform method. Usin g the normali ze d time history files, elastic deformation and velocity response spectra were generated for each of the earthquakes. The s pectra are based on the response of 150 single-degree-of-freedom systems with natural frequencies ranging from 0 02 to 3.0 seconds and were calculated using the method presented in Section 3.4.2 Plots of the deformation and velocity response spectra developed for the EL Centro earthquake are presented in Figures 3.11 and 3.12. The spectra represent the maximum deformation and velocity calculated by Equations (3. 15) and (3. 16) for the 150 elastic systems during the full30 seconds of earthquake ground motion. Thus it is possible to determine the maximum displacement or velocity for an elastic sys tem from the spectra if the natural period of the sys tem is known. 45 40 35 30 I 25 0 20 I 5 10 D eformation Response Spectrum El Centro, S' Y Damping r Tn (uc) F igure 3.11 El Centro Deform ati o n Re s pon s e Spec trum 56 ____........_._ / / / v

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y l > 140 120 100 80 60 40 20 / / ;J / Pseudo-Velocity Response Spectrum El Centro. sy., Damping \J \ Tn (stc) Figure 3.12 El Centro Velocity Re s ponse Spectrum -/ "'-..... For elastic systems, the force transferred through the spring and the system deformation is related by Equation (3.8a). Because the system deformation is by definition less than the yield displacement, the limit imposed in Equation (3.8a) does not apply. In this case, the force transferred through the spring can be calculated as; The maximum force transferred throu gh the spring during the earthqauke can be related to the spectral di splacement such that ; Represe nting the maximum force in term s of the syste m mass m and a peak p se udoacceleration yields ; where A is the spectral pseudo-acceleration in unit s of g. By substituting the relation s hip between system ma ss, stiffness and natural period into the above equation and rearranging terms ; 57

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(3.47) Equation (3 .4 7) is a function of the spectral deformation and natural period so it is possible to develop the pseudo-acceleration response spectrum for an earthqauke directly from the defromation response spectrum. The p se udo-acceleration response spectrum for the El Centro earthqa uk e is presented in Figure 3.13. The spectrum was calculated using the deformation response spectrum shown in Figure 3.11 and Equation (3 .4 7) 3 5 3.0 \ ,N vv \ JV : 2 5 2 0 <( 1.5 1.0 0 5 0 0 P seudo-Accelerati o n Response S p ectra E l Centro, 5% Damping \/\-_ v "\ Tn (s) Figure 3.13 El Centro P se udo -Accelerat ion Response Spectrum -The spectrum shown in Figure 3.13 i s referred to as a pseudo-acceleration spectrum because it does not pre sent the actual maximum system acceleration as defined by Equation (3.20). Rather the spectrum presents only the portion of the system acceleration that i s associa t e d with the s prin g force Plot s of the deform ation, velocity and pseudo-acceleration spectra for each of the eart hquake s are included in Appendix C. 58

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3.6 Co d e Prescribed Earthquakes Building codes such as the Uniform Building Code (1997) specify the earthquake to be considered in the se i s mic design of structures by defining a design basis earthquake for the building site. Design ba s i s ea rthquak es are defined in terms of a smoot h ed pseudoacceleration response spectrum such as that shown in Figure 3.14 The response spectrum represents an earthquake that has a ten percent probability of exceedence in a 50-year period and is defined by two input variables; the maximum peak ground acceleration expected to occur durin g the earthquake S8 and the maximum expected pseudo-acceleration of a fully elastic structure with a natural period of one second, S 1 The s pectra are generally developed for structures with an equivalent viscous damping ratio of 5 percent and are specific to a given site in that both the peak ground acceleration and the one-second spectral accelerations used to define the s pectrum are dependent on the soil conditions present at the site and the proximity of the site to major earthquake faults. A' (g) I I I I I I _ _ I I I I I I I I I I I I I I T o T5 1.0 T,, T n ( sec) F i gure3.14 Code prescribed pseudo-acce l e r at i on re s pon se spec t rum 59

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The magnitud e and extent of the region of p eak spectra l acceleration is defined s uch that ; S s = 2.5S g (3.48) T_ (3.49) s s s To = 0.2Ts (3 .50) Note that the spectral pseudo-acceleration determined from the smoothed spectrum of Figure 3.14 is identified as A to distinguish it from the pseudo-acceleration determined from an actual earthquake response A as shown in Figure 3.13. Velocity and deformation response spectra can be developed for the design basi s earthquake using the following equations which provide the relationships between pseudoacce l eration, pseudo velocity, an d d eformation ; D vrN Agr ; ------27[ 47r2 V' T s 1.0 V'= Slg 27r Tn (sec) F i g u re 3.15 Pseudo-velocit y response s pectrum from code prescribed earthquake. 60 (3 51) (3. 52)

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D' Ts 1.0 T., (sec) Figure 3.16 Deformation re s ponse spectrum from code prescribed earthquake. An objective of the parametric study is to develop a correlation between the code prescribed earthquake for a site, specifically the input variables of peak ground acceleration and one-second spectral acceleration, and the corresponding inelastic response characteristics of structures fitted with friction dampers. It is therefore necessary to determine a representative design basis earthquake for each of the earthquakes studied. Details of the method used to complete this determination are presented in the following Sections. 3.7 Elastic Response Reduction Factors It was mentioned earlier that the bas ic philosophy behind the seismic design requirements in building codes allows yielding of structural elements during strong earthquakes. Consistent with this philosophy, codes do not require that structures be designed to rem ai n e l ast i c during the full design bas i s earthq u ake but rather allow structures to be designed to a reduced earthquake. This is accomplished by spec ifyin g an elastic response reduction factor R which is u sed to reduce the design level earthquake forces to magnitudes l ess than that determined from the design ba sis earthquake re s ponse s pectrum. The reduction factor is 61

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based on a structures ability to absorb energy durin g yielding of structura l elements during s trong earthquakes. Value s ofR vary from 2 .2 to 8.5 in the Uniform Building Code (1997) and depending on the type of structura l system and materials u sed for construction A' A'Design = R Figure 3.17 illustrates the concept of the respon se reduction factor. A' (g) A' From Elastic Spectrum (Design Basis Earthq uak e) Tn (sec) (3.53) Figure 3.17 Comparison of code prescribed pseudo-acceleration response spectrum and code prescribed de sign s pectrum This same concept will be used to define the slip force for the friction damper s considere d in the s tudy The slip di sp la cement of the dampers will be defined as a ratio of the deformation resulting from the design basi s earthquake s uch that ; D D s,= -R (3. 54) In Eq uati on (3. 54) the coefficient R i s r e ferred to as th e s lip co e fficient. In t erms of damper s lip force ; J: = D'k0 sf R (3 55) 62

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Defin in g the damper s lip force in terms of the design basis earthquake a l lows lin ear scaling of the response of friction-damped structures. The response quantities of deformation velocity and pseudo-acceleration b ecome a function of a given earthquake and peak ground acceleration I f the response quantities are desired for the same earthquake scaled to a different peak gro und acceleration the resu l ts can simp ly be sca l ed by th e same amo unt u sed to scale the earthq uake ground motion. 3.8 Representative Code Prescribed Earthquakes To allow correlations to be made between the response of the SDOF systems and the parameters u se d to define design basis earthquakes in buildin g codes representative code prescribed earthquake design spectra were developed for each of the earth quake s considered. A s moothed pseudo-acceleration response spectrum as shown in Figure 3. 1 8 was considered as representative. A' ( g ) s s -+-----,., ' ' S 1 ---------1----, ' ' ' ' T 1.0 r T (sec) F i gure 3.18 S implifi e d p seudo-acce l e r atio n r es p o n se s p ec trum The spectrum wa s modified from that pre s ented in Figure 3 14 by eliminatin g the varying acceleration region for periods less than T0 and thu s exte ndin g the constant acceleration region from 0 toTs seconds This modification was incorporated becau s e an increa s e in 6 3

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natural period for structures associated wit h thi s region results in increased acceleration and thereby increased load ing. Because the decreased stiffness associated with slippage of friction dampers in structures during strong earthquakes re s ults in an increased natural period the u se of the higher acceleration over this region was deemed appropriate. The spectrum shown in Figure 3.18 is defined by two distinct regions; the region extending from 0 toTs seconds, and the region extending from Ts to 3.0 seconds. In each of these regions the pseudo-acceleration pseudo-velocity, and deformation are related to the initial code prescribed input variables of peak ground acceleration, Sg and 1 .0-seco nd spectral acceleration, S1 by the following relationships: V' A'g D'------2 con con Region 1: T, ::;; I; A'= Ss V'= A'g = A'T,g = Ssg T con 2tr 2tr n Region 2: T,, > 7'. T,, V'= A'g = A'T,,g = S1g co" 2tr 2tr 64 (3.56) (3.57) (3.58) (3 .59) (3.60) (3.61) (3.62)

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By normali zing the earthquake ground motion s, the value of S g for each of the earthquakes was set equal to 1.0. Region 2 of the spectrum define s the constant velocity ran ge of response for the earthquake. The value of the 1.0seco nd s pectral acceleration S, for each of the earthquakes was determined b y considering the maximum spectra l velocity, V 1 occurring between 1.0 and 3.0 seconds over the constant velocity range of response and relating the 1 0 second spectral acceleration to the velocity by: 2nY' S,=--g (3.63) The values ofV1 determined from the velocity response spectra along with the resulting values of S g and S1 used to represent the code prescribed earthquake ground motion for each site are presented in Table 3.3. Tab l e 3.3 Representative Code Pr esc ribed Earthquake Parameter s T g (sec) V1 (in/sec) S g (g) s, (g) El Centro 0.85 87.10 1.00 1.42 Lorna Prieta 0.76 69.80 1.00 1.14 Northrid ge 0.51 95.25 1.00 1.55 Olympia 0 60 72.02 1.00 1.17 San Fernando 0.43 4 8.08 1.00 0.78 The repre se ntativ e smoothed defom1ation pseudo-velocity, an d pseudo-acceleration spect r a along with the actua l spectra generated for the E l Centro ground motion are pre se nted in Figure 3 .19 Spectra for the five earthquake gro und motions considered in th e study are in c lud e d in Appendix C. These smoot h spectra represent th e de s ign ba s i s earthquakes for eac h of the s it es and will be u se d as a basis to define the s lip disp lacem ents considered in the remainder of the s tudy. 65

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: < li ., > = 0 3 5 3 0 2 5 2 0 ( ._. -' .!V 1.5 . P seudo-Accelera tion Response S p ectra El Centro, S l = 1.42g. 5 % D amping ............... """ .. 1.0 0 .5 -c.::.-.:..:-:..:.:_--_,_,_, .. -=:::+::::::::::===---,..,.,d 0 140 120 100 ..... 80 / 6 0 / : 40 j( 20 4 5 40 35 30 25 20 I S I 0 0 Tn (sec) 1----A -A' I Pseudo-Velocity Response S p ectra El Centro, S l = 1.42g 5 % Damping ... .... ....... Tn (sec) 1 ---v -vl Deformation Re s pon se Spectra El Centro, St = 1.42g 5 % Damping --._ ... Tn (sec) 1-----o o l Figure 3.19 El Centro r epresen t ative code prescribed earthquake respo nse spectra 66

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3.9 Co n cl ud i n g R ema rks The repre se ntative earthquakes developed in thi s chapter form the basis of the parametric study of friction-damped single-degree-of-freedom systems presented in Chapter 4 The earthquakes were developed ba se d on normali ze d earthquake ground motion s with peak ground accelerations of l.Og By defining the damper slip force as a function of a slip coefficient and the response spectrum representative of the earthquake ground motions the resulting responses become a linear function of the peak ground acceleration of the earthquake. This a ll ows direct linear scaling of the results obtained in the study. Thus, for earthquakes with the same ratio of one-second spectral acceleration to peak ground acceleration, the study results can be directly scaled. This allows the results of the study to be applied to earthquakes with varying peak ground accelerations. The one second spectral accelerations used to define the representative design basis earthquakes were selected based on the peak spectral velocity occurring at periods greater than one second. It can be seen from a review of the response spectra presented in Appendix C that this approach in selecting representative parameters resulted in smoothed spectra that is exceeded at periods less than one second. The amo unt the smoot hed spectra are exceeded i s ge nerall y small, however in the case of the Lorn a Prieta earthquake, the degree to which the smoothed s pectrum is exceeded i s su b sta ntial. The selection of the pr e domin a nt period of grou nd motion from the Fourier amplitude s p ectra also requires some judgment as the earthquakes cannot be represented by a single hannonic forcing function. In developing conclusions from the results pre sen ted in Chapter 4 these point s shou l d be kept in mind 67

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4. Para m etr i c S tud y R es ult s 4.1 In t r od u ct i o n In this chap ter the results of the parametric stu d y of friction-damped single-degree-of freedom syste ms are presented. Included are the results of the quantitative evaluation of the parameters affecting the design of friction damping mechani s m s and the results of the study of the inela stic deformation response characteristics of friction-damped SDOF systems. The results of the quantitative evaluation of the parameters affecting the design of friction damping mechanisms are presented in Section 4.2 and Appendix D. The result s of the study of the inelastic deformation response characteristics of SDOF systems are presente d in Sections 4 3 and 4.4. Section 4.3 presents the results of the study comp l eted on systems with small amounts of secondary stiffness. The section presents a method to develop smoothed inelastic deformation response spectra for systems with small amounts of secondary stiffness along with the assumptions u se d in developing the method. A comparison between the inelastic response spectra generated for systems s ubjected to the normalized ground motions considered in the study and that estimated by the s moothed spectrum developed in Section 4.3 is included in Attachment D. Section 4.4 pr esents the results of the study com pl eted on the effect secondary stiffness ha s on the inelastic response of SDOF systems and present s a method to develop de formatio n re sponse spectra for systems with varying amounts of secondary stiff n ess A comparison between the inelast i c response of systems with secondary s tiffne ss ratio s varying from 0 to 100 percent and that estimated by the re s pon se spectra developed in this c hapt er is presented in Appendix F. The chapter closes with Section 4.5 68

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where a disc u ss ion a nd s tat e m e nt of the conclu s ions dr awn from the stud y of SDOF sys t e m s fit t ed with friction dampers is pr ese nt ed. 4.2 Mechanism Design Data The quantitative evaluation of the parameters affec tin g the d es i gn of fri ction damping mechanisms was completed by considering the inelastic response of a series of frictio n damped SDOF sys t e m s with natural p e riod s r a n ging from 0.25 to 2.5 seco nd s a nd four l eve l s of damp e r s lip force d efi ned b y s lip coefficients of 4 6 9 and 12. The sys t em responses we r e calculated u s in g the tim e ste ppin g methods for inela s tic systems prese nted in Sectio n 3.4 and the normali ze d earthquake ground motion s developed in Chapter 3. Onl y sys tems with ze ro sec ondary stiffness were con s idered in the quantitative evaluation of the parameter s affecting the design of friction-damping mechanisms Three parameters were evaluated; the number of slip cycles experienced by the d a mpers, the maximum amount of energy dis s ipated and the maximum rate that the damper s dissipated energy during each of the earthquakes considered. In the evaluations, a damp er s lip cycle was defin e d as an occurrence of damp e r s lip in on e dir ection from the time s lipp age was initiated to the tim e the damper ve locity r eac h e d ze ro The total numb er of slip cyc l es was determined from th e r es pon se of the sys tem s durin g the full 30 seco nd s of gro und motion considered. The amount of ene r gy dis s ipated by slippage of t h e frict ion damper s during each time step was calculated by Equation (3.41) and the total amount of energy dissipate d durin g the duration of the earthquakes was calculated by Equation (3 .44). The maximum rate that the d am per dissipated energy during the eart h quake was calculated by Eq u ation (3.46) and co n side r e d the maximum a mount of energy dissi pated during any sing l e time step ove r the full 30 seconds of ground motion. 69

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The re s ults of thi s portion of the study are pre s ented grap hicall y in Appendix D. T h e resu lt s of the evaluation of the t ota l amount of energy dissipated and the ma x imum rate that energy i s di ss ipated are pre se nt ed in term s of system ma ss. In the ana l yses completed the mass of the syste m was taken to be equal to 1 0 s l ug. The res ult s prese nt ed in Appe ndi x D ca n be sca l ed lin early for sys t e m s with masses other th a n that conside r e d in th e study. 4.3 Inelastic Deformation Response Spectra for SDOF Systems The development of a method to formulate inelastic deformation respon se spectra was based on the resu lt s o f a ser i es of inela stic time hi story analyses completed for friction damped SDOF syste m s with natural periods ranging from 0 04 to 3.0 seco nd s and s ubj ected to the five normalized earth quak e ground motion s d eve lop ed in Chapter 3. The analyses were completed using the time stepping method s for inela s tic systems fitted with friction damper s presented in Section 3.4 and considered secon da ry stiffness ratio s of 0 and 5 percent a nd damper sli p force s d efi ned by slip coefficients of 4 6 9 and 12. The goa l of this potion of the study was the d eve lopm e nt of deformation respon se spectra for friction-damped SDOF systems with sma ll amounts of secon d ary st iffn ess The co ntrollin g parameters for the earthquake were taken as th e peak gro und acceleratio n the 1-sec ond s pectral acceleration a nd the predominant p e riod of gro und motion The controlling parameter s for the SDOF sys t ems were taken as the natura l p eriod, t h e d am per sli p coefficient and the secondary s tif f n ess rat i o of the sys t e ms. In genera l terms the inelastic deformation r esponse of a frictio n-damp ed SDOF sys t e m D" ca n b e d esc ribed as a funct ion of the five inpu t para m eter s as ; (4.1) 70

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In the case of systems with s mall amounts of secondary s tiffne ss th e inela s tic r e ponse can be de sc ribed as ; (4 2) Base d on the re s ult s of th e analyses completed in thi s portion of the st ud y the smoothed in e la s tic deformation response spectrum presented in Figure 4 1 was de ve loped in term s of the four input parameters included in Equation (4 2) r ; Tn (sec) Figure 4.1 Inelastic deformation res pon se spectrum Similar to the elastic spectrum u sed to define the code prescribed earthquake, the spectrum shown in Figure 4 1 is defined by two distinct r egions; th e region extending from 0 to r ; seconds, and the region extending from r ; to 3 0 secon d s. In each of these regions the in e la s tic deformation i s defined by the followin g relationship s : Reg ion 1 : T,, :::;; r ; (4.3) and R egio n 2 : T,, > r ; 7 1

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= [1 + fJ(T.v -1)] (4.4) The variables r ; and fJ are defined as; (4.5) fJ = 1 _!_ ( 1 + + _!_ (-4 2 + 2 -4(-1 -_4_) R 2 R R2 R (4.6) In Equation (4.6) i s the equiva l e nt visco u s d am pin g ratio for the system and was t a k en as 5 percent in the stu d y. By review of the elastic deformation respon se spectrum shown in Figure 3. 1 8 and E quation 3.62, it can b e seen that the inelastic response of SDOF systems with natural p eriods greater than r; is defin e d in term s of a modified e la stic r es pon se spec trum In developing the spectrum s hown in Figure 4 1 two modification s of th e elastic spec trum have been incorporated ; the first is a decrease in the s lo pe of th e line d efining the d eformation of sys tem s w ith l o n ge r p e riod s, and the seco nd i s a s hift of th e lin e defining th e lon ge r p er iod deformation s to the l eft. The decrease in s l ope re s ults from the additiona l energy di ss ipated by the friction damper s and the s hift of the l ine results from changes in the natural p er iod of the system during damp e r s lip The s lope of the portion of the graph for natural periods greater than r ; was establis hed by con s id ering the relation s hip between the d efo rmation s of elastic a nd in e l astic systems s ubj ected to harmonic forcing functions. For an e l ast i c sys tem s ubjected to a harmonic 72

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forcing function P(t) = ?0 sin it can be s hown that over a period of time the s y stem will achieve steady state conditions with a con s tant amplitude of vibration. In stead y s tate conditions the energy input by the harmonic force is equal to the energy dissipated through viscous damping where the input energy can be written as ; (4.7) and the energy dissipated by viscous damping can be written as; (4 .8) From Equations (4.7) and (4.8) it can be seen that the energy input during a single cycle of vibration is directly proportional to the displacement amplitude of the system and the energy dissipated by viscous d amping during a single cycle of vibration is proportional to the square of the displacement amplitude Thus for an elastic system subjected to a harmonic forcing function equilibrium of the input and d issipated energy can only occur at a single amplitude of vibration. force is applied to the system, the amplitude of vibration increases until equilibrium is obtained between input energy and the dissipated energy Figure 4 2 shows the relationship between the input and dissipated energy for the system E E o u a u Figure 4.2 Ener gy Equilibrium Sta t e 73

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The relation s hip presented in Figure 4 2 is valid only for ela s tic systems where viscous damping pro v id es the onl y means of removing energy from the system. In the ca se of a friction-damped system, energy i s dis s ipated both by viscous damping and by slippage of the friction damper. To determine the amount of energy that is dissipated b y a friction damper during a s ingle cycle of steady state vibration the ideali zed hystere s i s loop shown in Figure 4.2 w ill be considered. From the figure, the system can be seen to displace from the initial equilibrium point a maximum distance D" during each cycle. Additionall y, the s ys tem can be seen to oscillate about a new equilibrium point with a maximum displacement of where ; 1 ( ) U0 =2 D +D,1 (4 9) D" -lsi Figure 4.3 I dealized H ys t eresis Loop The energy diss ipated thr o u gh a sing le' c yc l e of vibratio n i s equal to the area inside the h ysteresis loop and can be written as; E" = 2k D (D" D' ) sf D sl sf ( 4.1 0) 74

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The amount of energy dissipated through viscous damping in the friction-damped system is taken to be equal to the square of the ratio of the inelastic to elastic di s placement amplitudes times the energy dissipated through viscous damping in the elastic system such that; E; llo ( 4 11) Taking the maximum displacement of the elastic system to be equal to that defined by the smoothed elastic deformation response spectrum shown in Figure 3 .15 and substituting Equation (4.9), the energy dissipated through viscous damping can be written as; (4.12) The amount of energy input to the inelastic system is taken to be equal to the ratio of the inelastic to elastic displacement amplitudes times the energy input to the elastic system. Again relating the maximum displacement of the elastic system to the smoothed elastic deformation response spectrum and substituting Equation (4.9) the input energy is written as ; (4 13) Equilibrium i s achieved between the input and dissipated energy when; E" = E" + E" I D sf (4 14) or E D" + D;, = E (D" + D;, y + 2k D' (D" D' ) I 2D' D 4 D'1 D sf sf ( 4.15) The input energy 1 is equal to the energy di ss ipated by viscous damping 0 allowing Equation ( 4 15) to be written a s; 75

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2Tr;: .!!!._ ku2 + sf = 2tr;: .!!!._ ku 2 -sf + 2k D' (D"-D' ) [D" D' l [(D" D' )"] ':> (i)N 0 2D' ':> (i)N 0 4D'2 D s f sf ( 4.16) Defining the ratio of the inelastic to elastic displacement amplitudes as; ( 4.1 7) stating the slip di splacement in terms of the elastic code prescribed displacement and the slip coefficient; D' =D' sf R and maximizing the response by assuming the frequency w of the forcing function to be equal to the natural frequency of the system wN yields the slope reduction coefficients defined by Equation ( 4.6) from Equation ( 4.16). Table 4 1 provides the values of the slope reduction coefficients for the four slip coefficients considered in the study. Table 4.1 Slope reduction coefficient, f3 R f3 4 0.386 6 0.346 9 0 369 12 0453 The inela stic deformation of friction-damped SDOF systems as defined by Equation ( 4 16) i s based on the assumption that the natural period of the elastic system and that of the friction-damped system during damper slip are the same. The inela s tic deformation respons e spectrum presented in Figure 4.1 however is based on the natural period of the system prior to damper slip Because of thi s, Equation ( 4.16) is not used directl y to calculate the inelastic 76

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deformation. The equation i s modified by incorporating a constant that shifts the re s ponse to a natural period as s ociated with the system prior to damper slip. Because of the greater stiffness present in the system prior to slippage of the damper the line describing the inelastic res pon se is shifted toward a lower period The amount of shift was determined b y examination of the inela s tic r es pon s e spectra generated during the study and noting that the deformation s of elastic and inelastic systems are approximatel y equal for systems with n a tural periods near 1.0 second. Because of this, Equation ( 4.4) was developed to yield inelastic deformations equal to elastic deformations at 1.0 second. The extent of the first region of the inelastic spectrum defined by r ; was determined by examination of the inelastic response spectra generated during the study A comparison of the smoothed inelastic deformation response spectrum developed using the representative code prescribed earthquake parameters for the El Centro earthquake presented in Table 3.3 and the inelastic response spectra calculated using the time stepping methods presented in Section 3.4 is presented in Figure 4.4. A comparison of the smooth and actual spectra de v eloped for each of the earthquakes and slip coefficients is pre s ented in Appendix E 35 30 25 20 <=-0 I 5 I 0 0 0 3 T n (sec) -o;. Sec ondary Stiffness s;. Secondary Stiffness --D F i g u re 4.4 In e l astic deforma tion respon s e spectrum. El Cen tr o S1= 1 .4 2g, R = 6 77

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4.4 Effects of Secondary Stiffness The evaluation of the effects secondary stiffness has on the deformation response of friction-damped SDOF systems was based on a series of inelastic time history analyses completed for systems with natural periods ranging from 0.25 to 2.5 seconds and normalized damper slip strength and secondary stiffness ratios ranging from 0 to 100 percent. Figure 4.5 presents the results of one such evaluation complete for a system with a natural period of 1.0 second subjected to the normalized El Centro earthquake ground motion. In all, 50 such evaluations were completed using the 5 normalized ground motions and 10 values of natural period For each evaluation, 10,000 individual system analyses were completed and the maximum system deformation occurring during the earthquake was determined. Figure 4.5 shows the maximum system deformation as a ratio of the maximum deformation of an elastic system with the same natural period occurring during the earthquake as the two parameters of normalized slip force and secondary stiffness ratio are varied. The secondary stiffness ratio is d efined by Equation (3.4). The normalized slip strength is defined as the ratio of the damper slip force to the maximum force developed in an ela s tic s ystem such that; ( 4 18) or b y s ub s titutin g Equation (3.55) the normali z ed s lip force can be written as; ( 4 19) where D o i s the ma x imum elastic system deformation 7 8

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0 10 20 30 40 50 Nonnalized Slip 60 70 Strength (%) 80 90 100 90 80 100 Second ary Stiffness(%) 2 .00 1.75 1.50 1.25 D I D0 1.00 0.75 0 .50 F i g ur e 4.5 System displacement with variation ofnonnalize slip strength and secondary stiffness. E l Centro, Tn = 1 0 second. The responses of the systems presented in Figure 4.5 are approximated at the limits of zero and 100 percent secondary stiffness by the smoothed elastic and inelastic deformation response spectra shown in Figures 3.16 and 4.1. The deformation of systems with secondary st iffness ratios of 1 00 percent is equal to that of a fully elastic system regardless of the normalized slip stre n gt h specified. In this case, the maximum deformat ions of th e systems are calculated by either Equation (3.59) or (3.62) depending on which region of the spectrum the system falls. At the opposite side of th_e graph, the deformations of the systems with secondary stiffness ratio s of zero are dependent on the level of slip force specified for the damper. The level of slip force is defined by the slip coefficient and the deformation is calculated by either Equation ( 4.3) or ( 4.4) again depending on which re g ion of the spectrum the system falls. 79

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For systems with secon dary stiffness ratios between the two limits of zero and 100 percent the deformation can be assumed to be a function of the deformations occurring at the limits. In thi s portion of the study, the deformations are assumed to be a linear function of the limitin g deformations such that ; D" = D' + (1(4.20) Figure 4.6 presents a comparison of the actual maximum displacement occurring in a series of systems with natural periods of 1 0 second and secondary stiffness ratios varying from zero to 100 percent and the inelastic displacements estimated by Equation ( 4 20). The system disp lacements were taken from Figure 4 5 considering a normalized slip strength corresponding to a slip coefficient of 4 In Figure 4 .6, the horizontal line corresponding to a value of 1 0 identifies an exact match between the actual and estimated displacements. From the figure, it can be seen that for all values of secondary stiffness a conservative displacement is estimated by Equation ( 4.19). The results for the evaluations completed in this portion of the study are presented in Appendix F 2.0 1.8 1.6 1.4 0 1.2 1.0 0 0.8 0.6 0.4 0.2 0 0 0 I 0 20 Normalized Displacement El Centro, Tn =1.0 Sec, R = 4 30 40 50 60 Secondary Stiffness(%) Figure 4.6 onnalized system displacement. 80 70 80 90 100

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4.5 Co nclu s ions and Di sc ussion of Parametric Study Results The results of the parametric study completed on the deformation response characteristics of friction-damped single-degree-of-freedom sys t em demonstrate the feasibility of developing inelastic response spectra based on the parameters used to define the design basis earthquake for a given site. The parameters nece ssary to formulate the inelastic spectra include the earthquake parameters of peak ground acceleration one-second spectra l acceleration, and predom inan t period of ground motion and the system parameters of natural period, secondary stiffness, and damper s lip force. For systems with small secondary stiffness ratios it was shown that inelastic deformation respon se spectra could be developed from the code prescribed parame t ers used to define the d esign basis earthquake for a given s ite It was further shown that the inelastic deformation of systems with secondary stiffness ratios between 0 and 100 percent could be calculated as a function of the elastic and inelastic response spectra develop ed for the earthq uake In general, the inelastic responses of the friction-damped systems were found to be larger than the corresponding elastic r esponse at n atural periods l ess than one second and sma ller at periods greater than one second. Additionally, from review of the inela st ic response spectra presented in Appendix E it can be seen that variations in the slip coefficient have a greater effect on the maximum re sponse of short period struc ture s than on the m aximum re spo n se of l ong period structures. B eca use the response of short period friction-damped structures are more sensitive to variations in the damper slip force greater care is required during the construction phase of these structures to a ss ure that the proper slip force is achieved. 81

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Because the d amper slip force is a function of the earthquake under consideration the re s ults of the deformation response study are a linear function of the peak ground acceleration of the earthquake. Therefore, the results can be directly scaled when considering earthquakes of different peak ground accelerations Although the results of the quantitative evaluation of the parameter s affecting the de sign of friction damping mechanisms are pre sented in raw form, several points can be made about the results. For all earthquakes studied the number of slip cycles experienced by the systems is inversely proportional to the natural period of the systems. Long period structures experienced fewer slip cycles than did the short period structures With few exceptions, the number of slip cycles experienced by structures with longer periods and constant slip coefficients was nearly constant. During every earthquake studied, the number of slip cycles experienced by structures with natural periods greater than 0.75 seconds was less than 50. In some ca ses, shorter period structures experienced a total number of slip cycles in excess of 120 In systems with constant natural periods, the number of slip cycles varied in proportion to the slip coefficient selected for the damper. The evaluation regardin g the amount and rate of energy dissipation indicates th at the rate energy is dissipated by the dampers in systems with constant natural periods i s inversely proportional to the slip coefficient u s ed Additionally the results of the e va luation indicate that the amount and rate o f energy dissipa tion is ma x imum for structures natural periods tha t are near the predominant period of ground motion The re s ults of the quantitative evaluation regarding energy di ss ipation are als o a function of the peak ground acceleration ofthe earthq u ake under consideration. From Equations (3.37) (3.38) and (3.39) it can b e see n that the amount of ener gy dis s ipated i s a function of 82

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the sq uare of the system response Therefore the results can be scaled by the square of the ratio of the earthquake peak ground accelerations when considering other earthquakes. The number slip cycles remains constant for earthquakes of varying peak ground accelerations. 83

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5. Application of Parametric Study Results to MDOF Systems 5.1 Introduction In this chapter the result s of the study completed on multi-degree-of-freedom systems are pre se nted. The chapter presents an evaluation of the correlation between the re s ponse of a ten-story friction damped moment frame estimated using the inela st ic re s ponse spectra developed in Chapter 4 and the corresponding response determined by inelastic time history analyses. Section 5.2 presents background information regarding dynamic analysis techniques for MDOF systems. The section begins with the presentation of the method of modal superposition used in the analysis of eJastic systems along with a di scussion leading to the development of the response spectrum analysis method. Additionally included in Section 5.2 is a discussion of the method u sed in the study to calculate the inelastic response of the moment frame using time stepping methods similar to those presented in Chapter 3 for SDOF systems. Section 5.3 presents the properties of the fiiction-damped moment frame considered in the study along with the approach used to determine the distribution of stiffness and damper slip force s throu g hout the frame. Additionally, the section presents the parameters cons idered for the friction-damped moment frame and the corresponding values considered in the analyses. Section 5.4 pre sents the modal analysis method used to estimate the inela stic deformation of the frame. Comparisons of the maximum deformation s calculated by inelastic time history analyses and the corresponding estimated response s are presented in Attachment G. Additionally included in the chapter is a method to determin e the s lip coefficient in term s of the optimum slip force di scusse d in Chapter 2 pre se nt ation of the method i s made in 84

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Section 5.5 The chapter closes with Section 5.6 where a discussion and conclusions drawn from the study are presented. 5 2 D e formati o n Resp o n se o f M DO F Sys t e m s 5.2.1 Intro du ct i o n The response of multi-degree-of-freedom systems subjected to earthquake ground motions can be estimated using time stepping methods similar to those used to calculate the response of SDOF systems. The response of elastic MDOF systems can be shown to be equal to the summation of the responses of the system's individual vibrational modes. It can be further shown that the vibrational modes of a MDOF system are directly related to the distribution of mass and stiffuess in the system For elastic MDOF systems, the response of each mode can be calculated as a function of the response of an equivalent elastic SDOF system and the natural properties of the structure. Two methods that can be used to calculate the elastic response ofMDOF systems are presented in Section 5.2.2. The response of inelastic MDOF systems can be calculated using methods similar to those used to calculate the response of inelastic SDOF systems. The method used to calculate the response of the ten-story friction-damped moment frame is presented in Section 5.2.3. 5 2 2 E l astic MD O F Systems The dynamic response of elastic multi-degree-of-freedom systems can be shown to be comprised of a series of independent re sponses where each modal re spon se i s determined from the natural properties of the system and the forcing function u se d to excite the system. It can be further shown that the independent respon se of each mode is a function of the respon se of an equivalent SDOF system and the natural properties of the mode under consideration Becau se of the se relation s hips, the re spo n se of elastic MDOF systems can be 85

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calculated as the summation of the respon ses of the individual modes of the system. This method of dynamic analysis is referred to as modal superposition. The time varying deformation response of elastic MDOF systems subjected to earthquake ground motion can be calculated using the method of modal superposition presented in Chopra (1995) as ; N u(t)= Ir;q>;u; (t) (5.1) i:;l where the vector u(t )describes the time varying displacement of each of theN degrees of freedom used to define the system Equation (5.1) defines the overall response of the MDOF system as the summation of the responses of the individual modes of the system where the response of the i1h mode is calculated as; (5.2) Equation (5.2) defines the response of the i1h mode of the system in terms of the natural properties of the system, as used to calculate the quantity r ;q>;, and the time varying response u ; (t) of an equivalent SDOF system with a natural period and viscous damping ratio the same as that of the i1h mode of the MDOF system. The vectorq>; d escribes the displaced shape associated with the / h mode of vibration and i s determined from the solution of the ei ge nvalue problem ; (5.3) In Equation (5. 3) k r i s the total structure stiffness matrix and i s defined as; 86

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Ck, -tk, )-+%" -tk,J 0 0 0 -(k Ck, -tkJ --%, 0 0 0 --i&, Ck, -tkJ 0 0 I) 0 0 0 Ck?o> -*oJ-t{frn; -*a) --(frFM -*o.J 0 0 0 -*oJ Ck71 -tk,) where kF; represents the lateral stiffness of the moment frame at the i'" level of the structure and kD;represents the lateral stiffness of the bracing system fitted with friction dampers again at the i'" level. The structure mass matrix m is defined as; w, 0 0 0 0 0 w 0 0 0 1 0 0 m=w 0 0 g 0 0 0 w,.._. 0 0 0 0 0 w where W; is the total weight lumped at the i'" level of the structure and g is the acceleration of gravity. Solution of Equation (5.3) establishes theN natural mode shapes (j) and corresponding natural frequencies (J) for the MDOF system The reader is referred to Chopra (1995) for a complete discussion ofthe formulation of Equation (5.3) and the various methods available to obtain the solution of the equation. The factor 1; is a function of the natural properties of the system (i.e. ma ss and stiffness matrices) and is equal to ; (5.4) 87

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where is the transpose of the i'h mode shape of the system obtained from Equation (5. 3) and the vector I represent s the effect the earthquake ground acceleration has on the masses lumped within the system The vector I is known as the influence vector and is formulated b y considering the displacements of the system masses resulting from a static application of a unit ground displacement in the direction earthquake ground motions are being considered. In the case of a planer system subjected to horizontal earthquake ground motion, the values of the influence vector associated with hori z ontal degrees of freedom will be equal to 1 0 indicating that earthquake inertial forces will act along these degrees of freedom. The remaining values of the influence vector will be associated with vertical degrees of freedom and will therefore be equal to zero. The influence vector effectively eliminates the inertial acceleration of system masses along degrees of freedom that are perpendicular to the direction of earthquake excitation. The individual vibrational modes of the MDOF system can be shown to respond dynamically in a manner similar to that of an SDOF system. The time varying response tt;{t) in Equation (5. 2) is the response of an elastic SDOF system with the same natural period and damping ratio as that of the /h mode of the MDOF system and can be calculated using the time stepping methods presented in Section 3.4 Thus from Equation (5.1) it can be seen that the tim e varying r esponse of an elastic MDOF system subjected to earthquake ground motion i s a function of the natural propertie s of the system a nd the dynamic re s pon ses of a serie s of SDOF systems. The natural propertie s of the system are d efi ned by the distribution of ma ss and stiffness as identified by th e sy stem 88

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mass and stiffness matrices. The SDOF system responses are determined by the same time stepping methods used to comp lete the evaluations carried out in Chapter 4 5.2.3 Respon se S p ect rum A nal ys i s The method of modal superposition presented in the previous section provides the time varying response of elastic MDOF systems subjected to a specific earthquake ground motion. However, during the design process, the actual time varying earthquake ground motions that the structure might be subjected to during its useful life are not known. Because of this, structural engineers typically rely o n the design basis earthq u ake specified for a given site to determine the seismic response to be considered in the design of a structure. Additionally, the maximum system response is typically of more interest to structural engineers than the actual time varying response. Because of these reasons, seismic analyses o f structures are typically completed by considering a combination of maximum modal res p onses calculated using the peak SDOF system responses defined by the code p rescribed design basis earthquake. This method of a n alysis is r eferred to as a respo n se spectrum analysis. To estimate the maximum expected deformation using a response spectrum analysis, the maximum response of each independent mode of an elastic MDOF system is first calculated by Equation (5.2) with the valueD; determined from Figure 3.16 substituted in place of the time varying response u ; (t) The maximum independent modal responses are then combined using one of a number of method s develop-ed to approximate the actual response of the system. With the substitution of D ; in place of u;{t), Equation (5. 2) becomes ; (5.5) 89

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where the vector n ; represents the maximum deformation response of the i1 h mode expected to occur during the design basis earthquake If the peak response n ; associated with all of the modes of the system were to occur simultaneously an accurate estimate of the system response could be determined by directly summing the individual modal responses of the system such that ; N n = Ir;
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spaced frequencies. The CQC method wa s developed to overcome the limitation s associated with the u se of the SRSS method when analyzing the response of structures with close l y spaced modal frequencies. The m e thod considers the relationship betwe e n each of the individual modal responses throu g h the u se of a correlation coefficient such that ; (5.8) On e po ss ible form of the correlation coefficient is given by Chopra (1995) as ; (5.9) where; c; is the viscous damping ratio assumed for each of the modes of the system and w ; and w j are the natural frequencies of the / h and/h modes The reader is referred to Chopra (1995) for a more detailed discus si on of the methods used to combine ind ivi du a l modal responses and additional expressions a vai lable to calculate the correlation coefficients used in Equation (5.8). For structures with regular distributions of st i ffness and ma ss, the seismic r espo n se is typically governed by the fundamental mode of the structure. Because of thi s, building codes allow the results of a re sponse s p ectrum ana l ysis t o be sca l e d such that th e r esu lt ing b ase shear is adjusted to match that determined for an e quivalent SDOF system with a ma ss equal to the total m ass of the s tructure and po ssessing the sa me natural p e riod as the fundamental mod e of the structure. The b ase s h ear co rre spo ndin g to the d esign ba sis earthquake i s 9 1

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calculated as the product of the total structure weight Wand the pseudo-acceleration such that ; V =WA Btt.rc (5. 10) where the pseudo-acceleration A. is determined from Figure 3.14 As discussed in Section 3 7, the response corresponding to the design basis earthquake is further reduced for use in design by dividing the base shear determined by Equation (5. 10) by an elastic response reduction factor such that; V =WA Des1g n R (5.11) The results of the response spectrum analysis are scaled to yield a structure base shear equal to that given in Equation (5.11). From the previous discussion, it should be apparent that the response spectrum analysis method provides only an estimate of the actual response of the structure during earthquake excitation The accuracy of the results is governed by the method used to combine the individual modal responses of the structure and the degree to which the results are scaled. The method does however provide structural engineers with a tool to quickly and easily estimate the maximum response of structures and has gained wide acceptance in the structural engineering community 5.2.4 Inelastic MDOF Systems A schematic diagram of a MDOF fiction-damped moment frame i s shown in Figure 5.1. The figure presents the schematic diagram of a three story frame where the stiffness of the moment frame and vertical bracing at the /h level are represented by k s F j and k s o j respectively. Additionally the damper slip force i s defined for the/h level a sfsu and the vis cous damping 92

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associated with the /h level is represented by a dashpot with a viscous damping coeeficient of F i g ur e 5. 1 Schematic diagram of MDOF friction-damped system A D'Alembert free-body diagram ofthe system mass located at thejth level ofthe frame at an arbitrary instant in time during an earthquake is shown in Figure 5.2 ... p l)j fv (t) j .m f.F (t) j F B D F i g u r e 5.2 D' Alembert free-body diagram of MDOF system mass The forces acting on the /h level mass can be written in terms of the relative displacements and velocities of the levels above and below the mass as ; fsF (t) j = kFj (u j (t)U jl (t))-kFj+l (u j + l (t)-U j(t)) (5.12) f.v (t) j = k0j (u /t) u jl (t)) -k0j+l (u j+l (r) -u j (r)) (5. 13) f v (I) j = C j (ti j (t)Li j l (I))C j+l (:i j+l (I) -Li j (t)) (5. 14) Additionally, the forcing function P(t) j can be written in term s of the earthqauke ground acceleration as; P(t) j = P g (t) j = -m jii g (t) (5 .15) 9 3

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In vector form the equations can be written as; (5.16) (5.17) f D (t) = CU(t) (5 .18) and; (5.19) where I is an Nx1 unity vector. In equations (5.16) and (5.17), the stiffuess matrices k,F and k,0 are the components of the total stiffuess matrix k T associated with the moment frame and vertical bracing system alone. k sF is the stiffuess matrix of the moment frame prior to yielding and is defined as; (kFl +kF2) -kF2 0 0 0 -k F2 (kF2 +kF3) -k F3 0 0 0 -k (kF3 +kF4) 0 0 ksF= F3 0 0 0 (kFN-1 +kF N ) -kFN-1 0 0 0 -kFN-1 kFN k ,0 is the stiffuess matrix of the vertical bracing prior to yielding defined as; (k D I +kD J -kD2 0 0 0 -kD2 (kD2 +kD3) -kD3 0 0 0 kD3 (kD3 +kD4) 0 0 k = sD 0 0 0 (kDN-1 +kDN) -kDN-1 0 0 0 -kDN-1 kDN 94

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The matrix c defines the system damping For the analyses of MDOF systems completed in thi s thesis Ra yleig h dampin g was considered resulting in a damping matrix that is proportional to both the mass and total stiffness of the systems such that; The coefficients a0 and a1 are defined in Chopra (1995) as; and; 2(J);(J) j ao=.;-----':.._ (J); +(J)j 2 Ql =.;-(J); +(J)j (5. 20) (5.21) (5.22) where .; is equal to the damping ratio assigned to the i1 h and/' modes and (J); and (J) j are the natural frequencies of the i'h and/h modes determined by Equation (5.3). For the analyses completed on MDOF systems, 5 percent damping was assumed for the second and fourth modes. The time varying deformation of the system shown in Figure 5 1 can be calcuated using a modified form of the time stepping equations presented in Section 3.4.3 for SDOF systems. By replacing the variables u, u, and ii in Newmark's equations with the equivalent vectors associated with MDOF systems, Equations (3 .18) and (3 19) can be rewritten as; Ui+l = U ; +[(l-yN)6t]ii; +(r N6t)ui+ l (5.23) u i + l = u ; +(6t)u; +[(o.5-fiNX6tYp; +[PN(6tYpi + l (5.24) Through the use of Newton's second law and by making the same substitutions considered in Section 3.4 3 the equation defining the accelerations of the system masses at the beginning of a time step can be written as; 95

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(5.25) additionally, by s ub stituting Equations (5.23) and (5. 24) into Equation (5. 25) the equation defining the acceleration s of the system masses at th e e nd of the t i me step can b e written as ; iii + = { iigi+lmi-cui c{[l -r N k T ui-kT )ti i + [0.5,B N y iii}} (5.26) where (5.27) Equations (5. 23) (5. 24), and (5. 26) can be u se d directly to det e rmine the time varying response of elastic MDOF systems subjected to earthqauke ground motion and will yield results consistent with those determined by Equation (5.1). In the case of systems with elements transitioning between the elastic and inelastic regions the equations require modifications similar to those made for SDOF systems to account for yielding of the elements As the individual elements transition between the elastic and inelastic regions, the overall system stiffuess matrix requires modifications to account for the changes in stiffue ss As in the analysi s of SDOF systems, it is assumed th at the stiffness of an element redu ces to ze ro durin g yielding or during damper slip The method u sed to account for yielding of elements is the same as that u sed for SDOF systems. Formulating effective spri n g and damper disp lacem ent vec tors s uch th at the effective displacement at the end of a time step i s calcuated as ; u effi+ls (j) = u f!is (j) + [u i+l (j)-u i (j)] -[ui + l (j -1)-u i (j -I)] "ffi+ID (j) = ffiD (j) + [ui +l(j) ui(J)] -[ui+l (j -1) ui(j -1)] and applying the lim its suc h that; 96 (5.28) (5.29)

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I "ffsi (J) u y i (J) (5.30) and ; I U ffo (j) U S f (j) (5 31) the forces actin g on the system masses due to the displaced shape of the moment frame and bracing can be calculated as; fsF = k sF U effs (5.32) and (5.33) By substituting Equations (5.32) and (5.33) into Equations (5 .25) and (5 .26), the systems accelerations at the start and end of a time step can be calculated as; (5.34) and, iii+! = {iigi+lml-cui c{[I-r N X.0.t )iii}fsFi +l fsDi+l"" (5.35) The stiffness matrix k r used in Equation (5.35) and to calculate the effective mass matrix in Equation (5.27) is modified from its original form to account for member yielding and damper slip by zeroing out the appropriate terms in the total structure stiffness matrix. For elastic systems and inela s tic systems with elements that remain entirely within either the ela s tic or inelastic regions during the entire time step, the acceleration s calculated at the start of a time step by Equation (5.34) will equal tho s e calculated at the end of the previous time step by Equation (5.35) A s with SDOF sys tems, i f yielding or damper slip occ urr s 9 7

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during a time step the accelerations calculated by Equation (5.34) will not be equal to that calculated by Equation (5.35) Errors in the calculated response of inelastic systems will occur if the difference in the two values are not addressed Similar to the program used to calculate the response of inelastic SDOF systems the program used to calcuate the response of inelastic MDOF systems incorporates two procedures to assure convergence of accelerations The first is a reduction in the duration of the time steps when yielding is detected. The program compares the accelerations calculated at the start of a time step with the accelerations calculated at the end of the previous time step. If the two do not match, indicating yielding or damper slip has occurred, the response over the full time step is recalculated as a series of shorter time step responses. In the evaluations completed, the full time step duration was divided into 20 equal shorter time steps. The second procedure is iterative and is completed only for the shorter time step during which yielding or damper slip takes place. In the second procedure the velocity and displacement at the end of the transition time step are calculated using the accelerations determined by Equation (5.34) at the start of the time step immediately following the transition time step in place of the accelerations calcuated at the end of the transition time step by Equation (5. 35) The iterative procedure is the same as that discussed for SDOF systems in Section 3.4.3 with the exception that a fixed number of 20 iterations were u s ed rather the convergence criteria u s ed in the evaluation of SDOF systems. 9 8

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5.3 Response of a Ten-Story Friction-Damped Moment Frame The study of the correlation between the inelastic deformation of MDOF friction-damped moment frames estimated by standard modal analysis techniques and the inelastic response spectra developed in Chapter 4 and th e deformations determined by inelastic time history analyses is based on the friction-damped moment frame shown in Figure 5.3 Figure 5.3 Ten-story friction damped moment frame 99

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The frame is assumed to have a constant story height and is furthermore assumed to carry a uniform distribution of load such that a constant load w can b e lumped to each level. The total weight of the frame W is equal to the summation of the w eig hts of each individual level or lOw. The distribution of story stiffness was selected such that a c o ns tant story drift would occur under seismic loading conditions as determined by the e q uivalent lateral force procedure recommended by the Building Seismic Safety Coun ci l ( 1997). The equivalent lateral force procedure approximates the lateral force distributi o n associated with the fundamental mode of the structure The distribution of equivalent lateral forces is defined as; (5.36) where Fx is the lateral seismic inertial force acting at level x of the structure and the coefficient Cvx is calculated as; (5.37) The exponential term kin Equation (5. 37) is defined in term s of th e fundamental period of the structure such that the term is taken equal to 1 for structures w i t h f undamental periods less than or equal to 0.5 second and 2 for structures with fundament a l periods greater than or equ a l to 2 5 seconds. For structures with fundam e ntal period s b etw een the limit s of 0.5 and 2 5 second s, the Buildin g Sei s mic Safety Council (1997) s t a t e:, tha t either a value of2 may be u s ed or a value ma y be determined by linear interpolation b et wee n the two limit s. With k = 1, the fundamental mode shape is appro x imated by a stra igh t line. With k = 2 the fundamental mode shape is approximated b y a parabola w ith its vertex at the ba s e of the 100

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structure. For the evaluations completed in thi s thesis, the value of k was taken to be equal to 1.375. The distribution of stiffness was chosen to match the dis tribution of story shears calculated for the frame Table 5 1 pre se nts the calculations necessary to determine the distribution of stiffness in acc ordance with the equivalent lateral force procedure assuming a constant story height and a uniform distribution of story masse s. Table 5.1 Desi g n shear dis tribution Story h w, w,h./ Cvx v x 10 10 1.0 23.7 0 .212 0.212 0 212 9 9 1.0 20. 5 0 .183 0 .395 0 395 8 8 1.0 17.4 0 .156 0.551 0 .551 7 7 1.0 14.5 0 .130 0.681 0 .681 6 6 1.0 11.7 0 .105 0 786 0 786 5 5 1.0 9 1 0 .082 0 .867 0 .867 4 4 1.0 6 7 0 060 0 927 0 927 3 3 1.0 4 5 0 040 0 .968 0 968 2 2 1.0 2 6 0 .023 0 .991 0 .991 1 1 1.0 1 0.009 1.000 1.000 Table 5.1 presents the total relative stiffness of each story kr x as a ratio of the total first level story stiffness. The actual total stiffness of each story is then calculated as; (5.38) In the evaluations carried out in this thesis, the term K was selected to yield a fundamental period of 1.0 second for the frame as determined by Equation (5.3) The distribution of the total story stiffness to the moment frame and the damper bracin g is determined by the sec ond ary stiffness ratio se lect ed for the fr_ame s uch that the tot a l s tiffne ss dis tributed to the moment frame is calculated as; kFx = krx (5. 39) and the total stiffness distr ibuted to the bracin g system fitted with the friction damping mechani s ms is calculated as; 101

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(5.40) The distribution of damper slip forces throughout the frame are taken to be equal to the st ory shears determined by the equivalent lateral force procedure divided by the slip coefficient selected for the frame such that; v r -v lslx-Bns. R (5 .41) where the seismic base shear is calculated from the elastic pseudo-acceleration response spectrum developed for the design basis earthquake for the site such that ; (5.42) In Equation (5.42), the elastic pseudo-acceleration is calculated from Equation (3.57) or (3.60) depending on whether the fundamental period of the structure is greater than or less than Ts. Because the fundamental period of the frame was chosen to be 1.0 second, the evaluations were based on the total frame base shear calculated by Equation (3.60) In all, the evaluation of the frame displacements consisted of 80 inelastic time history analyses using the five normalized earthquake ground motions considered in the parametric study of SDOF systems and the parameters presented in Table 5.2 The analyses were completed using the time stepping method presented in Section 5.2.4 for inelastic MDOF systems and code written specifically for the evaluation on the computer program Matlab (1996) Tab l e 5.2 P arameters u se d for t en-story frame eva l uation Para meter Parameter Value T N ( s ec) 1 0 5, 20 50 % R 4 6 9, 12 102

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For each of th e analyses completed the m aximum di s pl acement of each level of the frame was taken as th e maximum displacement that occurred at the level at any time durin g the full duration of the earthquake. The maximum story drift was then calculated as the differen ce between two adjacent levels Figure s 5.4 a nd 5.5 present the maximum di sp l aceme nt s and story drift occurring durin g each of the earthquakes w hen a secondary stiffness ratio of 20 percent and a slip coefficient of 9 were considered The upp e r and lower bounds of estimated defle cti ons as det e rmined by the method s presented in the following section are also included in the figure. The upper bound estimate represents th at calculated for the Northridge earthquake. The lower bound e stima te represents that calcula te d for the San Fernando earthquake. The results obtained from each of the analyses completed are presented in Appendix G along with the corresponding estimated values. 11 10 9 8 7 >. 6 ... 0 Vi 5 4 3 2 0 0 7 14 21 D(in) -+-ElCentro -+--Lorrn Prieta -tr-Nortfuidge -a)'Il1lia ---+-San Fernando + Bound --+--Upper Bound F i gu r e 5.4 Compariso n of estimated a nd actual inela s tic r espo n se 20% seco nd ary stiffn ess, R = 9 103

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11 1 0 9 8 7 5 4 3 2 1 -,...._ 0 2 3 4 5 D ( m ) 6 7 8 9 10 0 El Centro 0 Lorm Prieta t. Nortluidge X O yrrpa X San Fernando LO\'Iei"Bowd LPPer Bound Figure 5.5 Comparison o f estimated and actual inelastic story drift 20% secondary stiffness, R = 9 5 .4 E stimat e d Frame Deflection and Story Drift Us in g Inel a stic Spectra The method u sed t o estimate th e inelastic deformation of the ten-story frame is b ased on the inelastic response spectra developed in Chapter 4 for SDOF systems and the response spectrum analysis method presented in Section 5.2 .3. The general equation governing the displaced shape of inelastic MDOF systems is developed from Equation ( 4.19) as; (5 .43) where D" is a vector containing the maximum displacement s of each of the degrees of freedom u se d to define the structure. By review of Equation (5.43) it can be seen that D is a function ofboth the elastic and inel as tic d e formation of the frame with the contribution of each determined by the secondary stiffness ratio Additionally, it can be seen that the 104

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equation governing the inelastic response of SDOF systems has been modified to include scale factors for both the elastic and inelastic components. The elastic component D' i s calculated by Equation (5.7) u sing the SRSS method for combining individual elastic modal responses as; The component is additionally modified by a scale factor SF such that the first level shear is made equal to the base shear calculated for the structure by Equation (5.42). The elastic scale factor can be written as; (5.44) where kTl is the total lateral stiffness of the first level and o; is the first level displacement determined by Equation (5.7). The inelastic component is calculated by Equation (5.6) using the absolute sum of the individual inelastic modal responses as; N =I !,


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where s 1 is a vector containing the portion of th e mass lumped to each degree of freedom th a t participates in the fundamental mode of vibration and ; m = the structure mass matrix q> 1 = the fundamental mode shape of the structure, and f1 is determined from Equation (5.4) The total mass participating in the fundamental mode is determined by summin g the participating ma ss es from each level o f the frame such that ; N ml =Isli i=l (5.46) where s 1 ; is the mass associated with the i1 h row of the vector s 1 The total mass of the structure is equal to the total weight W divided by the acceleration of gravity. Substitution of the above relationships provides the final form of the equation used to estimate the maximum inelastic deformation of the frame; (5.47) The maximum deformation estimated by Equation (5.47) for each of the analyses completed is pre sente d in Appendix G. 5.5 Equivalent Slip Coefficients Based on Optimum Slip Force The optim um level of s lip force di s tributed throu g hout friction-damped moment frames was determined by Fili atra ult and Cherry (1990) to be a function of the natural periods of th e structure both b efo re and after damp er sli ppa ge h as occurred, the number of stories pre sen t in the structure, and th e peak ground acceleration and predominant period of ground motion 106

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associated with the earthquake under consideration The equations proposed b y the researchers for the optimum slip force are included in this the s i s as Equations (2 9a) and (2.9b) Because the inelastic deformation respon s e spectra developed in Chapter 4 is presented in terms of a slip coefficient R, it is instructive to develop a relationship between the slip coefficient and the optimum slip force determined by Equations (2 9) The derivation begins by considering the total distributed slip force resulting from the procedure presented in Section 5 3 for specifying the slip force to be used at each level in the moment frame. The slip force for the damper at level x in the moment frame was calculated by Equation (5.41) to be ; The total slip force V0 distributed throughout the frame is equal to the summation of each of the individual damper slip forces such that; v NS v ""v o R L.-"' x=l (5.48) The general form of Equation (5.48) is obtained by substitution of Equation (5.42) and stating the pseudo-acceleration for the structure in terms of Equations (3.57) and (3. 60) such that ; s w NS V =-5-""V 0 R L.-X x = l (5.49a) ( 5 .49b ) Rearranging Equations (5.49) provides the equivalent slip coefficient in term s of the total di s tributed slip force ; 107

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S W NSR=-s-Iv V O x=l x (5. 50a) (5.50b) Equations (5.50) can be used along with Equations (2. 9) to directly determine the value of R that corresponds to the optimum level of slip force. The Building Seismic Safety Council (1997) provides the following equation for approximating the natural period of a structure based on the number of stories present ; Tb =0.1NS (5.51) Additionally using Equation (3. 7) to define the unbraced natural period the following approximate relationship can b e developed ; T = 0.1NS u j (5.52) Using Equations (5.51) and (5. 52) and the predominate period of ground motion Tg from Table 3.3, the optimum slip force defined by Equations (2.9) can be calculated for each of the earthquakes considered in the study in terms of the number of stories present in the structure, the amount of secondary stiffness present, and the slip coefficient specified. Substitution of the optimum slip force determined in this fashion into Equations (5.50) yield the relationships shown in Figure 5.6 Figure 5.6 presents the variation of optimum slip force for each of the e a rthquakes considered in the study in of the three parameters of the number of stories present in the structure, the secondary stiffness ratio, and the slip coefficient. 108

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l 5 l 0 I 5 a: I 0 0 0 0 0 I 0 0. 2 0 0 J 0 0 4 0 0 .50 Sttondary Stlfrntss Ratio L om :a PrItt a l 5 l 0 I 5 a: II. "-"-I 0 0 0 0 0 t 0 0 1 0 0 3 0 0 .4 0 0 .50 Stcoadary StiUJitSS Ratio Nortbrld&t a: l 0 \\. ..-::! .. -----I 5 I 0 ...... 0 .0 0 0. J 0 0 .z 0 0 ) 0 0 4 0 0 .s. S tcodary Stirftu Ratio Olympl2 l 5 l 0 a: --"\..._---I 5 I 0 .... 0. 0 0 0 I 0 0 1 0 0 J 0 0 4 0 0 s 0 San Fernando l 5 l 0 I 5 I 0 0 0 0 0 I 0 0 l 0 0 ) 0 0 4 0 0 .s. Secondary Stirfntss Ratio I -Ns1 N S 3 N S 5 -N S 7 S 9 N S 1 0 Figure 5.6 Equivalent R values 109

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5.6 Co n cluding Remarks The re su lts obtained in Chapter 5 and presented in Appendix G for multi-degree-of freedom systems are based on normalized peak ground accelerations of 1 0g. The results can be linearly scaled to obtain displacements occurring during earthquakes with different peak gro und accelerations. Additionally, the respon ses calculated in the study represent an ultimate condition that is comparable to the maximum responses calculated with deflection amplification factors in current building codes. Thus, the overall structure dis placement and story drift results presented in Appendix G, when scaled to the peak ground acceleration specified for the design basis earthquake are directly comparable to the amplified displacements calculated in accordance with current building codes. The Building Seismic Safety Council (1997) recommends allowable story drift ratios for steel frame structures ranging from 0 010 to 0.025 times the story height. The actual allowable drift ratio depends on the structure height and the seismic use group the structure is assigned to A ss uming a typical story height of fourteen feet yields allowable story drifts of 1 68 to 4 20 inches From review of the story drift results presented in Appendix G and considering that the actual peak ground accelerations for the earthquakes ranged from 0.28g to 1.076g, it can be see n that friction-damped structures can be designe d to resi s t earthquakes with relati ve l y high peak gro und accelerat ion s witho ut exceeding co.de prescribed s tory drift ratios. Review of the results presented in Appendix G s how s that the deformations are controlled by both the secondary stiffness ratio and slip coefficient selected for the structure. 110

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Generally, the deformation s increase a s th e s lip coefficient i s increa sed or as the sec ondary s tiffnes s ratio i s decreased The ultimate base shear occurring in friction-damped structures can be written in term s of the deformation s of the two dis tinct systems that make up the frame From Equ at ion ( 5.41 ), the first level friction damper slip force is set equal to the base shear corresponding to the design ba sis earthquake div ided by the slip coefficient. Because the damper effectively limits the shear that can be transferred by the vertical bracing the component of the ba se shear as s ociated with the vertical bracing i s equal to th a t given by Equation (5.41) or ; V = VBau D R (5. 53) The shear transferred by the moment frame is displacement dependent and assuming the frame remains elastic can be written in terms of the maximum first level story drift and the stiffness of the first level of the frame as ; (5. 54) where D"(l) is the maximum first level deformation. The ultimate base shear transferred to the foundation is then equal to the summation of the two components as ; (5.55) 111

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6. Discussion Conclusions, and Recommendations for Further Study 6.1 Discussion and Conclusions The results of the studies pre se nted in this thesis have demonstrated the feasibility of using inelastic response spectra and standard modal analysis techniques to estimate the deformation response of multi-degree-of freedom friction-damped moment frames subjected earthquake ground motion The results of the analytical parametric study completed on friction-damped single-degree-of-freedom systems lead to the development of a method that can be used to formulate inelastic deformation response spectra from the parameters defining the code prescribed design basi s earthquake. The parameters necessary to formulate the spectra include the earthquake parameters of peak ground acceleration one-second spectral acceleration, and predominant period of ground motion and the structural parameters of natural period damper slip coefficient, and secondary stiffness ratio. It was shown that the response of single-degree-of-freedom systems with varying amounts o f secondary stiffness can be estimated with reasonable accuracy as a function of the elastic and inelastic response spectra corresponding to the design basis earthquake. The applicability of the inelastic spe ctra in estimating the response of multi-degree-of-freedom systems was d e monstrated by comp a ring the inelastic respon se of a ten-story friction-damped frame determined by time history analyses with the inelastic deforma'tion estimated by th e m ethod develop ed in this thesis. The study of the quantit ative evaluation of the parameters affecting the design of friction damping mechanisms provides a ba sis for determining approximate values that can be u se d in developing criteria for the de s ign of damper mechani sms. 112

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The results presented in this thesis can be u se d by struc tural engineers to estimate the maximum deformation response that would be expected to occur in friction damped structures during the design basis earthquake specified for a given building site. Based on the results of the studies, the following approach is proposed for the design of friction damped moment frames: 1 Complete a preliminary design of the moment frame considering only gravity loads. Based on the preliminary member sizes, calculate the unbraced natural period of the system Tu as that of the moment frame alone. 2. Define the design basis earthquake for the building site. Determine code prescribed earthquake parameters of peak ground acceleration and one-second spectral acceleration. Estimate the predominant period of earthquake ground motion by the equations proposed by Vanmarcke and Lai (1980) Equations proposed by the researchers for estimating the predominant period of ground motion in terms of the distance from the epicenter of the earthquake or the Richter magnitude of the earthquake are presented in this thesis as Equations (2.1 0) and (2.11) in Chapter 2. 3. Determine the distribution of lateral seismic forces either by an elastic spectrum analysis or by the equations proposed by the Building Seismic Safety Council ( 1997) to approximate the lateral force associated with the fundamental mode of the structure. The equations proposed by the Building Seismic Safety Council (1997) are presented in this th esis as Equations (5.36) and (5.37) in Chapter 5 Estimate the fundamental period of the structure T b and determine the base shear for the structure based on the code prescribed earthquake An estimate of the natural period of 113

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structures with a regular distribution of mass and stiffness can be made by Equation (5.51) Scale the distribution of lateral seismic forces such that the ba se shear matches that corresponding to the design basis earthquake. 4. Determine the optimum total slip force V0 in accordance with the optimum slip force design spectrum proposed by Filiatrault and Cheny (1990). The optimum slip force design spectrum is described by Equations (2.9a) and (2.9b). Calculate the value of the slip coefficient corresponding to the optimum slip force by Equations (5.50). Check to assure the structure and earthquake parameters used in design fall within the range considered in the study completed by Filiatrault and Cherry (1990) The optimum slip force design spectrum is valid only if the parameters fall within the range considered in the original study as discussed in Section 2.5. 5. Determine the damper slip force distribution for the structure from Equation (5.41) using the equivalent slip coefficient corresponding to the optimum slip force calculated in the previous step. 6. Select vertical bracing members to safely cany the damper slip forces determined in the Step 5. Member selection should include an appropriate factor of safety. 7. Based on the combined moment frame and vertical bracing systems, calculate the fundamental period of the structure and verify the base shear determined in Step 3 is appropriate for the final period of the structure. If nece ssary, repe at Steps 3 through 7. 8 Calculate elastic and inelastic deformation response spectra for the site based on the design b asis earthquake parameters and the expected predominant period of earthquake ground motion The elastic response spectrum is determined from Figure 114

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3.14 and Equation (3.52). The inelastic response spectrum is calculated by Equations (4 3) and (4.4). 9. Determine the maximum inelastic deformation of the structure considering both the moment frame and vertical bracing to be effective in resisting lateral forces. The deformation is calculated using the elastic and inelastic response spectra developed in the previous step and standard modal analysis techniques in accordance with the method developed in Section 5.4 The displaced shape of the structure is estimated by Equation (5.47) 10. Determine the forces induced in the moment frame by the lateral displacement of the structure as determined by Step 9. Verify that the moment frame can safely carry gravity loads a l ong with the loads induced by the lateral displacement. 11. Verify that interstory drift and overall structure displacement is within code prescribed limits If necessary, evaluate P-delta effects 12. Complete a wind load analysis to determine if wind loads will induce damper slip at any level of the structure. If necessary, modify the stiffness of the moment frame and/or decrease the slip coefficient to eliminate wind induced damper slip. 6 2 Reco m me n dations for F u rt h e r St u dy Several areas are recommended for further research the first of which is an expanded parametric study of friction-damped sy s tem s with a greater number of earthquakes included. The study could include the same steps presented in this thesis but with more earthquakes included to better refine the approach used to formulate the inelastic respon s e spectra and to better define the parameters affecting the design of friction damping mechanisms. 115

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Additional research is required regarding the response of friction-damped MDOF systems. Res earch is required to better evaluate the effectiveness of the method proposed in this thesis for estimating the inelastic response ofMDOF systems. The research cou l d include a parametric study consisting of number ofMDOF systems with varying periods damper slip coefficients, and secondary stiffness ratios to determine the correlation between the actual and estimated displacements for a range of friction-damped MDOF systems. The studies could also evaluate the correlation between the number of slip cycles experienced and energy dissipated in SDOF and MDOF systems. As a final i t em additional research is required to evaluate the effect variations in the distribution of the total damper slip force has on the overall response of friction-damped MDOF systems. The distrib u tion assumed in this thesis was based on the lateral forces associated with the fundamental mode of the structure. The research could evaluate the effectiveness of this method in terms of the total energy dissipated and whether or not all dampers present in the structure are effective in dissipating energy. The research could include a study of the number of slip cycles maximum rate of energy dissipation and total energy dissipated by each damper within the structure for a variety of slip force distributions 116

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APPENDIX A Nomenclature 117

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fJ A A' A" c D D' D" D" 0 E econdary st iffne ss ratio S lop e r ed u ct ion factor Parameter u se d in N ew mark method Duration o f time s tep Friction coefficient Par a m ete r u se d in Newmark m et hod Damped natural frequency = w NFf Natural frequency Viscous damping ratio Peak earthquake ground acce l eration Acceleration response Acceleration response from smoothed spectrum Elastic systems Acceleration response from smoothed s pectrum Inelastic systems Viscous damping coefficient Vertical distribution factor Deformation response Ma xi mum deformation re s ponse Deformation respon se from smoothed s pectrum Elastic systems Deformation response from s moothed s pectrum Inelastic systems D efo m1ation re s pon se from s moothed s pectrum Inela stic syste m s with no secondary stiffne ss D a mper s lip dis placement Energy Energy diss ipated through v i sc ou s dampin g Total sys t e m input e n ergy Kinetic e ner gy Energy dissi pat e d b y p ass iv e e ner gy diss ipatin g m ec h anisms Energy at equilibrium s tate 118

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E s E' s E" sf E v J f o J,, , fr fr g k m N NS p P g REQ R RPI Strain energy Total re cove rable system e n e r gy Energy di ss ipat e d by damper s lip Energy dis s ipated throu g h yie ld ing Force Vi scous damping force Spring force -Damper Spring force Frame Damper slip force Normalized damper slip force Yield strength Normali zed yield strength Lateral seismic force acting at le vel x of structure Gravitational acceleration System stiffness Lateral stiffness of vertical bracing fitted with friction dampers Lateral stiffness of moment frame Total lateral stiffness of friction-damped moment frame Cumulative distance of damper slip System mass Earthquake Ri c ht e r m agn itud e Normal force d eve lop ed in friction damping mechanism Number of s t ories prese nt in s tru c tur e In s tant a n eo u s force In ertia force du e to ground acce l erat i on Di stance to earthquake epi center Slip coefficient, E l astic respon se reduction factor R e l ative p erfonna n ce index 119

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SEA T' s u ueff u u g U y Umax(O) V ; v Peak ground acceleration One second spectral acceleration Peak spectral acceleration Area below the strain energy time history plot Friction-damped system Area below the strain energy tim e history plot Equivalent elastic system Base shear sca le factor Elastic systems Base shear sca l e factor Inelastic systems Natural period of friction-damped structure prior to damper slip Predominant period of earthquake ground motion Natural period Period corresponding to the extent of the peak spectra l region Beginning Period corresponding to the extent of the peak spectral region End Period corresponding to the extent of the short period region Inelastic systems Natural period of friction-damped structure during damper slip Displacement of system mass Effective displacement Displacement corresponding to energy equilibrium state Elastic systems Displacement corresponding to energy equilibrium state Inelastic systems Velocity of system mass Acceleration of system mass Earthquake ground acceleration Damper dis placement Damp e r slip dis placement Yield dis placement Maximum instantaneous strain energy Friction-damped sys t e m Maximum instantaneous strain energy Equivalent elastic s y s tem Damper slip force at/'' l evel in s tructure Velocity response 120

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V D estg n V' V" w (j) i c D' D" D" 0 m u U eJJ u Ba se shea r calculated from de sign basis earthquake De sign base s hear Velocity response from smoothed spectrum Elastic systems Velocity response from smoothed spectrum Inelastic systems Optimum slip force Total structure weight lh d h z mo e s ape System mass matrix Estimated maximum displacement Elastic MDOF systems Estimated maximum displacement Inelastic MDOF systems Estimated maximum displacement Inelastic systems with no secondary stiffness Viscous damping force vector Spring force vector Damper Spring force vector Frame Total structure stiffness matrix Moment frame stiffness matrix Vertical bracing s tiffness matrix Influence vector Structu re mas s matri x Earthquake induced inertia forc e vec tor i'h mod e mass distribution System mass displace ment vector Effective displacement vector System velocity vector System acce l eration vec tor 121

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APPENDIXB User's Manual for Co mputer Program Re sponse" 1 22

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Response Seismic Response Program Version 1.0 User's Manual Seismic response of linear and nonlinear single-degree-of-freedom systems. University of Colorado at Denver Prepared by Daniel W. Secary December 11, 1998 CE 5840 Ind ependent Study 123

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Table of Contents 1 INTRODUCTION .............. ...... .......... ........ .......... ...... ...... .............. .......................... 125 2 PROGRAM CAP ABILITIES ...................................................................................... 126 2.1 GENERAL ............................................................................................ .... ...................... 126 2.2 LINEAR ANALYSES .............. .............................. ...................................................... .. 126 2.3 NONLINEAR ANALYSES ......... ................................... ....... ....................................... ..... 126 2.4 ANALYSIS RESULTS ..................................................................................................... 126 3 PROGRAM EXECUTION ......................................... ................ ....... ..... ......... ............ 128 3.1 GENERAL .................................................. .. ..... ...................... .. ..................................... 128 3.2 FILE ............... .... ............................. .... .......... ................ . ..................................... ....... 128 3.2.1 Open Earthquake Data File ............................ ..... ................. ... .... .......................... 128 3 2.2 Save Chart Data to File ......................................................................................... 129 3.2.3 Printer Setup ........................................ ................................................................. 129 3.2.4 Print ......... ...... ..... ................ ...... ........................................................................... 129 3.2.5 Exit. ................................................................................. .... .............. ......... ..... ...... 129 3.3 STRUCTUR .. .......................................................................................................... ...... 129 3.3 1 Structure Options ................................................................ ................................. 129 3 3.2 Define Structure .................................................................................................... 129 3 4 ANALYSIS ................... ....... .. ............. ... .... .............................................................. .... 131 3. 4 1 Analysis Options ........................................................ ........ ............................... ... 131 3 4.2 System Response ............................ ....... ................... ........ ..................... .............. 133 3 4 3 Response Spectrum ......................... ..................................................................... 133 3.4.4 Perform Analysis .......... .............. ...... ..... ....... ...................................................... 134 3.5 VIEW .................. .... ........... ............... ...... .. .... .............................................................. 134 3.5.1 Analysis Sumrnary ...... .............................................................. ................. .......... 134 3 6 HELP ........................ ......................... .... .... .... ........... ................... .. ....... ........... ...... ..... .. 134 3.6.1 About Response ......................................................................... .......................... 134 4 EARTHQUAKE DATA FILES ....... ...... . ....................... . ........ ............................ .... 104 5 REFERENCES FOR RESPONSE USER'S MANUAL .......................................... .... 105 124

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1. INTRODUCTION Response is an interactive Windows based program capable of calculating the dynamic response of linear and nonlinear single-degree-of-freedom systems subjected to earthquake ground accelerations As an additional feature the program has the ability to perform seismic analyses of systems which incorporate friction dampers as a means of additional energy dissipation Both individual system responses and earthquake response spectra can be calculated. The analysis of linearly elastic systems is completed using a time stepping method which incorporates an exact solution of the governing equations of motion based on an assumed linear variation of the forcing function over each time step. Nonlinear analyses are completed using Newmark's Method with automatic time step reductions at transitions between linear and nonlinear regions. In addition to time step reductions, an iterative procedure is incorporated to assure convergence of accelerations within the transition time step. The user can specify the number of segments to be used within transition regions and the gamma and beta parameters u sed in the Newmark Method. Recorded time histories of earthquake ground accelerations are read b y Response from user prepared files. Calculation time steps smaller than the digitized time step used to record the earthquake can be specified by the user. Additionally, the user can specify a peak ground acceleration to be used for linear scaling of the recorded earthquake ground motion. Analysis results are displayed in a series of graphs which provides an effective means for comparison of information regarding the structure's response including displacement, velocity, base shear yield and slip cycles, energy dissipation and slippage of friction dampers Additionally a summary of the analysis results can be displayed. Available re s ults for response s pectra calculations include deformation velocity, and pseudo acceleration and can be completed for both linear and nonlinear systems In addition to analysis results, the earthquake ground acceleration can be displayed The u se r can save the data used to generate each of the graphs as a text file to allow further processing, or print hard copie s of the graphs and results s ummary Response was written as part of an independent stu d y completed under the direction of Dr. Kevin Rens at th e University of Co lor ado at D e nver. The program was written to provide the mean s of s tudying the effects of friction dampers on the seismic response of struct ures. 1 25

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2. PROGRAM CAPABILITIES 2.1 General Response can be used to calculate the dynamic re spo nse of linear single-degree-of freedom systems and nonlinear single degree-of-freedom systems with and without friction dampers Linear systems are modeled as a classical single-degree-of-freedom system including mass stiffness, and viscous damping. Nonlinear systems incorporate the same model but with the assumption of elastoplastic material properties. The model used for friction damped structures is the same as that used for nonlinear systems with an additional spring included to model the portion of the structure fitted with dampers The total stiffness of the structure is represented by the sum of the stiffness of both springs. The force-displacement properties used for modeling friction dampers is the same as that used for modeling elastoplastic material properties. 2.2 Linear Analyses The response of linear systems is calculated using a piecewise exact solution of the equations of motion governing the response of a single-degree-of-freedom system subjected to a forcing function which varies linearly over the time interval. The method is presented in Section 5 .2 of Chopra [1]. 2.3 Nonlinear Analyses The response of nonlinear systems, both with and without friction dampers is calculated using a step-by-step procedure based on the Newmark Method presented in Section 5.4 of Chopra [ 1] and Section 7.5 of Clough and Penzien [2]. For increased accuracy of the calculated re s pon se, the time step duration at transitions between linear and nonlinear regions is reduc e d and an iterative procedure is incorporated to assure convergence of accelerations. The procedure assures that the accel eration ca l culated at the end of each time step matches the acceleration calculated at the beginning of the next time step. The tolerance on convergence is set to 0.0001 in/sec / sec. 2.4 Analysis Results Analysis results are displayed on screen or printed in a series of graphs. The following can be displayed and printed: Earthquake Ground Acceleration 126 \

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Structure Di s pl ace ment Struct ur e Velocity Total Ba se Shear Frame Shear Damper Shear Dissipated Energy Yield Cycles Yield Energy Slip Cycles Slip Energy Damper Slip Deformation Response Spectrum Velocity Response Spectrum Pseudo-Acceleration Response Spectrum Analysis Summary 4 0 10 2.0 1.0 r YeldE""'!'Y r Sipc,des r Si!lE""'!IY r Oompe x fon) 0 1 .0 -2.0 rA-Specllaj r J -10 lh A n A A vv {\ IV w El Centro 1940 Strudure Displacement 3 .67 in Maximum ,A A n A A r v vw v v AAfl vvvv r I r P 10 15 20 Time (sec) El Ceooo. 1 S40 To allow further pro cessi n g of re s ult s, all chart data can be saved to disk. 127 h A (\ n 11\VVVv 25 30 Nooline.o
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3. PROGRAM EXECUTION 3.1 General Analyses can be completed after input of an earthquake time hi story and the struc tural parameter s defining the sys tem to be analyzed All input i s accomplished through the use of windows accessed by pull down menus as described below. Analysis results along with the ground acceleration used in the analysis, are displayed in a series of graphs within the main window of the program. The u se r can tog gle between graphs by selecting the appropriate option button to the left of the graph. Only the option buttons which correspond to available analysis results will be highlighted. In addition to the graphs a summary of the analysis results can be displayed. All graphs displayed on the screen and the summary of analy sis results can be printed. Data associated with each of the graphs can be saved to disk as a text file for further processmg The format of the following sections of this manual correspond to the locations of the pull down menus of the program's main window and provide the user information regarding the input of required parameters and the execution of the program. 3.2 File 3.2.1 Open Earthquake Data File Input of an earthquake time hi story is accompli s hed by reading a text file containing the recorded data from disk. To input a file select Op e n Earthquake D ata File under the File menu, se lect the appropriate file from the Open bo x and click Op e n This of course assumes an appropriately formatted file has been created. Inform ation on creating your own earthquake data file and the required file format is presented in Appendix A. The peak ground acceleration for the se le cte d earthquake and the calculation time step used in completing the analysis can be specifie d by the u ser. See Sectio n 3.4.1 for information on eart hqu ake sca ling and adju s ting the calculation time ste p After an earthquake data file is opened a graph of the ground acce l era tion will be dis pla y ed in the main window The title specifie d in the earthq uak e data file and th e current calcu lati on time s tep will be dis pla y ed below the gra ph s in the left portion of the status bar. 1 28

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3.2.2 Save Chart Data to File To save the data associated with the currently displayed graph, select Sa ve Chart Data to File under the Fil e menu. Input the appropriate file name and location in the save box and click Save Saved files can be viewe d usin g a text editor or can be read into a s pread sheet program 3.2.3 Printer Setup Select Printer Setup to change the current printer settings. 3.2.4 Print To print the graph currently displaye d in the main window, select Print under the File menu To print the analysis summary, click the Print button on the Anal ysis Summary window 3.2.5 Exit Select Exit to quit Response 3.3 Structure Input of structural parameters i s accomplished with the use of the Structure Options and Define Structur e windows as discussed below. 3.3.1 Structure Options The Structure Option s window is disp l ayed by selecting Structure Options under the Structure menu Structural parameter s can be input in the form of the structure's natural period or a s a combination of the structure's weight and s tiffness. Additionally, th e u ser can s pecify whether nonlinear material propertie s and friction dampers are to be included in the structura l model. Dampin g i s inp ut as th e percentage of critical damping. 3.3.2 D e fin e Structure Input of structural properties i s accomplished by selecting Define Structure under the Structure menu A window wiiJ be displayed with input bo xes appropri ate for the type of s tru c tur e a nd method of input defined in the S tru cture Option s w indow 129

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Nonlinear material behavior is defined by the normali ze d yield strength specified in the Define Stru cture window. The normalized y ield strength i s equal to the ratio in percent of a system's yield displacement to the maximum di s placement of a corresponding elastic system subjected to the sa me earthquake ground motion. Elastoplastic material properties are assumed. The distribution of the structure's total stiffness between elements with and without friction dampers is specified within the Structure Option s window. The Damper Stiffness Ratio is defined as the ratio, in percent, of the stiffness of bracing fitted with friction dampers to the total structure stiffness. This allows analysis of structures with a combination of framing systems. The nonlinear behavior of friction dampers is defined by the normalized slip strength specified in the Define Structure window. The normalized slip strength is equal to the ratio in percent, of the friction damper slip displacement to the maximum displacement of a corresponding elastic system subjected to the same earthquake ground motion. After sli p page has occurred, the friction force is assumed constant. r El Ceooo, 19-40 E l Cen t r o 1940 Ground Acceleration -.3 1 9 g M
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Normalized Yield Strength Percentage of Total Stiffness from Friction Damped Bracing Normalized Slip Strength Analysis Type 3.4 Analysis The type of analysis to be completed (System Response or Response Spectrum) along with the analysis parameters are specified using the Analysis Options window and the selections available und er the Analysis menu. 3.4.1 Analysis Options The Analysis Options window is displayed by selecting Analysis Options under the Analysis menu [ AewomeSpecto r De!O!'fiiW..n ( r Pf.r .. El Centlo. 19-40 El Centro. 1940 Ground Acceleration mum 1 5 Time (se c ) 20 25 30 The user can specify a reduction in the time step duration to be used while the system transitions between linear and nonlinear regions. The reduction in time step duration is accomplished by dividing the current calculation time s tep by the Number of Additional Calculations specified by the user. To specify the Number of Additional Calculations select Analy sis Options under the Analysis menu and input the number of calculation s in the appropri a t e box The time s tep u se d durin g the tr a n s ition will be equal to the current calcu lation time s tep divided b y the Number of Additiona l 131

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Calculations The reduction in time step is applicable to the periods when the system transitions from linear to nonlinear and nonlinear to linear regions. Regardless of the amount of reduction specified for the time ste ps at tran sitio n regions, an iterative procedure is incorporated wh ich assures that the acceleration calculated at the end of each time step matches the acceleration calculated at the beginning of the next time step within 0 .0001 in/sec / sec. This value cannot be changed by the user. The parameters used in the Newmark Method can be specified in the Analysis Opt ions window Specia l cases of the Newmark Method include the average acceleration method and the linear acceleration method for which gamma and beta are set to the following values : 1 1 Average acceleration method: y =-, fJ =-2 4 1 1 Linear acceleration method: y =-, fJ =-2 6 The program default values for gamma and beta correspond to the linear acceleration method. It is important to note that the linear acceleration method is stable only /).t 0.551 while the average acceleration method is stable for any /).t. T, The user can specify the u se of a calculation time step which is shorter than the recorded time step identified in the earthquake data file The shorter time step is incorporated by dividing the time step identified in the earthquake data file by the number of recorded time step partitions specified by the user To specify a shorter time step, select Analysis Options under the Analysis menu and input the number of Recorded Time Step Partitions to be u sed. The calculation time step will be set equal to the recorded time step identified in the earthquake data file divided by the number of Recorded Time Step Partitions. The calculation time step is modified only when a new earthquake data file i s opened. To scale a recorded earthquake time hi story, select Anal ysis Options under the Analysis menu After checking the box titled Scale Earthquake PGA input the value the earthquake peak ground acceleration i s to be sca ling to in the appropriate box or accept the default value of 1.0. The sca le value must be set prior to readin g the file from disk. All variables input within the Analysis Options window can be reset to the default values by se l ecting the reset button at the bottom of the window. 132

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3.4.2 System Response To calc ulat e the re spo n se of an individual system se l ect S y st e m R esponse under the Analysi s menu. After the earthquake and structural parameters have been s pecified select P e r form Anal y sis to complete the calculation After the analysis has b een completed a graph of the structure displacement will be displayed in the main window grap h Depending on th e type of anal ysis comp l e t e d the appropriate option button s will b e highlighted or dimmed The choice of System R esponse and R esponse Spec trum are mutually exclusive. A checkmark w111 appear to the left of th e currently selected analysis type. Calc ulation s w111 be performed based on the selec ted analysis type until the selection is changed. 3.4.3 Response Spectrum To calculate a re s ponse spectrum select R esponse Spectrum under the Analysis menu After the earthquake and appropriate struc tural parameters ha ve been specified select P erfo rm Analysis to complete the calculation. The re s pon se spectra will be created using 150 individual syste m respon ses with p erio d s varying between 0 .02 and 3.0 seco nds. Sbucturo Ao--l r (""\tljoef._v r fo!:,)!f11!1-.eSilt'm r r r [n:ar:.cl!CdE.neiJ!' r I r \oe!JF."""" l ';I!;;, CCi5 r I [ R....,.,.. Spectw---, r. peto,m.,.""""IOrj I r Velody r P oeudo Aeceiefo
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3.4.4 Perform Analysis Select Perform Analysis to initiate calculation of the system response or respo n se spectrum. Be fore the analysis can be completed, both the earthq u ake and structural parameters to b e u se d in the analysis must b e defined 3.5 View 3.5.1 Analysis Summary A summary of the current analy sis results can be displa yed by se lecting View Analysis Summary under the View menu The summary reports the maximum calculated va lu es for a number of response variab le s To print the summary, click the print button at the bottom of the summary window 3.6 Help Ta.IB-stw. F .... s-. r: o..._s._ r YooldCjde< (' YooldEnorw r Slipc:,dot r SlipE,..sw ro..._slip rRe>ponseSpecbo I r DelOtion r Veloci\v r Pseudo Acceleoized Yoold Sbongth 100 0 Elastic Displocement 4.437 Yoold Oioplocemenl 37 Yoeld 26.273 Ductiity A olio: 0 .828 Ne) : 615. o......,;ng r ..... sw fn-1>/unit ..... t zo s Numbel ol Yield Cycles: 0 Yoold Ene/<.nt mast): 37 89 M axirrun Rate ol Energy Oits.pahon :n4. 795 ', Print I I ent lit .... ... II II I II II i I A f\ A A(\ uv vvvv -;OK I 20 25 30 1me tsec) Select About R esponse to display information regarding the curren t ver sio n of the program 1 34

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4. Earthquake Data F iles Response will read text files containing the recorded gro und accelerations for an earthquake in the format presented below. Accelerations can be in units of inches / sis, em/s is, or as a ratio of the acceleration of gravity g The digitized time step must be constant for all steps but may be of any positiv e value. In addition to ground acceleration da ta input of a title identifying the earthq uak e file, the digitized time step in seconds, and the units used to record the earthquake ground acce l erations are required. The required format for earthquake data files to be read by Response is a is follows: El Centro 1940 0.02 g 0.00630 0.00364 0.00000 Title specified by user Time step in seconds Units of ground acceleration (g, in, or em) First value of ground acceleration Intermediate value of ground acceleration ... -Final value of ground acceleration The file must be saved as a text file with a .txt extension to allow the fil e to be displayed in the Windows input box. Linear scali ng of earthquake time histories contained in the data file can be implemented during execution of Response. Additionally the calculation time step can be set to a value smaller than that u se d in the data file. See 3.4.1 Analysis Options for information. The analysis duration will be th e s ame as the total duration of the input earthquake ground motion i .e. the analysis will not include a period of free vibration after th e gro und motion has stopped. To allow for a period of free vibration, the earthquake file can be modified to include a period of zero ground acceleration at the end of the file. Recorded ground acceleration data for a number of eart hquak es are available from s ource s over the World Wide Web One s uch source i s the Strong Motion Database developed a t the Ins titute for Crusta l Studies (ICS) University of California, Santa Barbara (UCSB) which can be accessed at http: // quake crustal.ucsb.edu/scec / smdb / smdb.html. Recorded ground accelerations for several earthquakes were obtained from this source and are included on the installation disks for R espo n se Becau se the format of d ownloaded fil es cannot be read directly reformat ting i s required. 135

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5. REFERENCES FOR RESPONSE USER'S MANUAL [ 1] Anil K. Chopra "Dynamics of Structures Theory and Applications to Earthquake Engineering" 1995 Prentice -Hall, Inc [2] R.W. Clough and Joseph Penzien "Dynamics of Structures" Second Edition 1993, McGraw-Hill Inc 136

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APPENDIXC Representative Code Prescribed Response Spectra 137

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PseudoAcce ler ation Respon se Spectra E l Centro. S l = 1.42g 5% Damping 3 5 c-----------------------.-----------------------.-----------------------. 3.0 2 5 .. .[:) .. \-. .. < 1.0 0 5 140 120 100 u 80 / -"! '. = :::. 60 > .-/ 40 j( 20 / 0 45 4 0 35 30 25 0 20 1 5 1 0 0 -Tn (s) 1 ..... A -A'I Pseudo-Ve l ocity Response Spectra El Centro, S l = 1.42g 5 % Damping ... ..... Tn (sec) 1.... v -vl Deform a t ion Response Spectra E l Centro, S l = 1.42g 5 % Damping -------Tn (sec) 1 .... o-ol 1 38

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: < u .. > 3 5 3.0 2 5 2 0 1.5 1.0 ,.'A 0 5 0 0 t40 120 100 80 60 / 40 /:-'' 20 ?;J 0 Pseud o-Acce leration Response Spectra Lorna Prieta, S I = 1 .14g 5 % Damping ... -_,_______ Tn (sec) IA -A'I Pseud o-Ve locit y Response Spectra Lorna Prieta, Sl = 1.14g, 5 % D amping ,. Tn (sec) ..... v--vl Deformation Respon se Spectra Lorna Prieta, S l = 1.14g 5 % Damping -----45 .----------------------.----------------------,---------------------, 40 30 = --0 20 -::---Tn (sec) ..... o -ol 1 39

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3 5 3 .0 2 5 2 0 "-: / < 1.5 "' t.O t 0 .5 0 0 t40 120 100 u 80 e. 60 > 40 // 20 0 0 45 4 0 35 30 25 -0 2 0 1 5 1 0 0 0 .r-, \ / <7 : \ !/ P seudo-Acceleration Response Spectra Northridge S l = I.S Sg 5 % Damping Tn (sec) 1-A -A' I Pseudo-Ve locity Response Spectra Northridge, Sl = l.SSg, 5 % Damping Tn (sec) 1-----v--vl Deformation Response Spectra orthridge, Sl = 1.5Sg 5 % Damping __.-::--/.. ... ---Tn (s ec ) 1--o -ol 140

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P se ud o-A c c eler a tion Resp o nse S p ectra Ol y mpia,Sl ; 1.17g,S% D amping 3 5 .-----------------------.-----------------------.-----------------------. 3 0 +-----------------------+----------------------+----------------------1 2.5 +-,-!-;\_ -,f.:.,'_,....., ...... _'\.r----+------+--------j _____ ............... u u -'e ;. > = '=Q 1.0 0 5 .,-:-_ ,...,. __ .,.. __ .,._ ;p;p;;=d 140 120 100 8 0 6 0 40 2 0 0 45 40 35 30 25 20 I S 1 0 0 / ;. /\,,..--_: \/ ,. # : c Tn (sec) I A A Pseudo-V eloc i ty Respons e S p ectra Olympia, Sl = 1.17g 5 % D amping Tn (s ec) 1-v--vl Deformation Re s pon se Spectra Oly mpi a, S l = 1.17g, 5 % D amping --!----" ----Tn (sec) I D D'I 1 4 1 ------: ........

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< ::: -"! g > 3 5 J O 2 5 2 .0 \ .. \ ... "' ... 1.5 1.0 0 5 0 0 t40 120 100 80 60 ........ : 40 / /".' 'J 20 /1'' P seudo-Acce l e r a tion R espo nse S 1 >ectra San Fernando. S l : 0 78g 5%, D am pin g -.. Tn (sec) 1----A -A'I Pseudo-Ve locit y Response Spectra San Fernando, Sl: 0 78g 5 % Damping ..... Tn (sec ) 1---v -vl Deform at ion Re s pon se S p ectra San Fernando, S l : 0 78g 5 % D amping 45 .----------------------.----------------------,----------------------, 4 0 35 30 25 c 1 0 .. 7. .. 0 Tn (sec) I o-o l 142

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APPENDIXD Slip Cycles and Energy Dissipation 143

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"' "' .. E c = e. w 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0.25 0 .50 Number of Slip Cycle s El Centro Tn (sec) Energy Dissipated through Damper Slip El Centro 0.75 1.00 1.25 1.50 1.75 Tn (sec) Maximum Rate of Slip Energy Dissipation El Centro 2 .00 2 .25 2 .50 9000 ,-----.-----.----,-----.-----.-----,-----,-----.----. -;;;-8000 +----f---!-""----+----+----+----+---+---t----J 7000 : 5 v ......... 4000 +-___.--=-..,__--:-, 3000 e. w -r 0 0.25 0 .50 0.75 1.00 1.25 1.50 1.75 2 .00 2 .25 2 .50 Tn (sec) I-R= 4 R = 6 R = 9 >
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"' "' (j >. u ..... 0 .... "' "" E z "' "' .. E -c 140 120 100 80 60 40 20 10000 9000 8000 7000 6000 Number of S lip Cycles Lorna Prieta Tn (sec) Energy Dissipated through Damper Slip Lorna Prieta 5000 4000 . 3000 w 2000 1000 0 0.25 0.50 9000 8000 7000 E I -6000 I = 5000 I / '-' 4000 / "' I / =f 3000 / 2000 -w 1000 0 0.25 0.50 0 .75 1.00 1.25 1.50 1.75 Tn (sec) Maximum Rate of Slip Energy Dissipation Lorna Prieta --------0.75 1.00 1.25 1.50 1.75 Tn (sec) 2.00 2 .00 I--R= 4 R = 6 ---R = 9 >
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140 120 .. 100 u ,.., u 80 ... 0 .... 60 .. .t:> E 40 z 20 0 .25 0 .50 10000 9000 8000 "' "' 7000 "' E 6000 -c 5000 4000 c 3000 w 2000 1000 0 0.25 0 .50 0 .75 N umber o f S lip Cycle s orthridg e 1.00 1.25 1.50 Tn (sec) 1.75 Energy Dissipated through Damper Slip Northridge ..,.,. 0 .75 1.00 1.25 1.50 1.75 Tn (sec) Maximum Rate of Slip Energy Dissipation Northridge 2.00 2 .25 2.50 . :......,. ::_:.: .. :.2 .00 2 .25 2.50 9000,-----,----.-----.----.-----.----,-----,----,-----, 8000 7000 c 6000 ; 5000 ] 4ooo 3000 2000 w 1000 0 0 .25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Tn (sec) R=6 R = 9 >
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"' "' <:i u ... 0 ... .. .l:l E :::1 z "' "' .. E ... c :::1 w 140 t 120 .. \ 100 1 \ ' \ .. 80 \ 60 \ Number of S lip Cycles Ol y mpia 40 \ \ ::-. :-.-.;"-. \ -20 0 0 25 0.50 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 .25 0.50 ... -....... . : :-.--0 75 1.00 1.25 1.50 1.75 Tn (sec) Energy Dissipated through Damper Slip Ol y mpia 0 75 1.00 1.25 1.50 1.75 Tn (sec) Maximum Rate of S lip Energy Dissipatio n Ol ympia 2 00 2 25 2.50 2.00 2.25 2.50 90 00.-----.-----.----.-----.-----.-----.----.-----.-----. 8000 7000 E ... 60 00 L = 5000 "' :::1 () / / -""" "' 4000 "' / / / """ 3000 / 2-2000 ':. w 1000 0 0 25 0.50 0 75 1.00 1.25 1.50 1.7 5 2.00 2.25 2.50 Tn (sec) I-R= 4 R=6 _ ,.__ R = n l 147

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"' "' .. E -;: :0 -; w "' "' E = <> -s w Number of S lip Cycle s Sa n Fe rnando 0 0.25 0.50 10000 9000 8000 7000 6000 5000 4000 3000 2000 0.75 1.00 1.25 1.50 1.75 Tn (sec) Energy Dis sipated through Damper Slip San Fernando 2 00 2 25 1000 . :'!:"7 0 0 25 0 .50 9000 8000 7000 6000 5000 4000 3000 2000 / /-1000 -;:....-. 0 0 25 0 .50 0 .75 1.00 1.2 5 1.50 1.75 Tn (sec) Maximum Rate of S lip Energy Diss ipati on San Fernando 2 .00 2 25 . . ::;:._.,t:; 0 .75 1.00 1.25 !.50 1.75 2 00 2.25 Tn (sec) 1 R = 4 -R = 6 -- -R = 9 -><- R = 121 148 2 .50 2.50 2 .50

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APPENDIXE Inelastic Deformation Response Spectra 1 49

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1 5 0

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'2 c c c c 35 30 25 20 I 5 I 0 s 0 35 30 2S 2 0 I S I 0 s 0 3S 30 2S 20 IS I 0 s 0 35 30 25 20 1 5 I 0 s 0 0 0 0 0 Lom a Prieta S I -1.14 g R = 4 '"'>. -....... Tn (sec) Lom :a Prie t a S I I .14 g R = ----....... 2 T n (sec) Lorna Prieta S I 1.14g,R 9 ----=....-T n (sec) Lom a Prie t a S l = 1.1 4 g R = 1 2 --T n (sec) 1--o;., Seconda r y S t iffness 5 % Sec ondary Stiffness --D 1 5 1

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35 3 0 2 5 2 0 c I S I 0 35 30 25 20 Q 15 1 0 5 0 35 30 25 20 Q I 5 I 0 5 0 3 5 3 0 2 5 2 0 Q I 5 I 0 s 0 0 0 0 0 .....-/ ..,-.&'" ....... .......:::: 1--0% S econdary North ridge S I = I 5 5 g, R = 4 -.... ...... -:;./ T 11 ( sec) North ridge S I = 1.5 5 g, R 6 r-2 3 T n (sec) Northridge S I = 1.55g, R -9 -3 T11( sec) North ridge S l I 5 S g R -I 2 3 T 11 (sec) S tiff11 ess ... s ; Secondary Stiffness --D" !5 2

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35 30 25 I 20 0 15 I 0 5 0 0 35 30 25 20 I 5 I 0 5 0 35 30 25 '2 2 0 0 15 I 0 35 30 2 5 = '=-20 0 I 5 I 0 5 0 0 0 ,t-Olympia S l 1 1 7g, R T n (sec) Olympia Sl 1 17g,R = 6 /'... T n (sec) Olympia S l = 1.17g,R 9 2 / . ......__,. _ .... __ .. T n (sec) Olympi a S l = 1 .17g, R = 12 __,.---T n (sec) ,.-:-:. 3 -3 1--oe;. Secondary Stiffness se;. Secondary Sriffness --D" !53

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35 30 25 2. 20 c I 5 I 0 5 0 0 35 30 25 20 c I 5 I 0 5 0 0 35 30 25 20 c I 5 I 0 5 0 0 35 30 25 20 c I 5 I 0 5 0 0 -;,.--San Fern:Jndo S I T n (sec) San Fernando S I 0 .78g,R6 T n (sec) San Fernando S I T n (sec) San Fernando S l = 0 .78g, R = 12 T n (sec) 1--0% Secondary Stiffness 5 % Seconda r y Stiffness --D 154 3 3

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APPENDIXF Normali z ed Inelastic Deformation 155

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c 0 c 0 c 0 2.0 I 8 I 6 I .4 I 2 I 0 0 8 0 6 0 4 0.2 0 0 2 0 I 8 1.6 1.4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2 0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 4 0 2 0 0 2 0 1.8 1.6 1.4 I 2 I .0 0.8 0 6 0 4 0.2 0.0 . ... 0 I 0 --.. 0 I 0 .,.......,. ............ 0 I 0 ./'-::..--0 I 0 20 ........ 20 20 b. 20 Normalized Displacement Tn = 0.25 Sec R 4 -30 40 50 60 70 80 90 I 0 0 Secondary Stiffness(%) Normalized Displaceme n t T n = 0 .25 Sec R 6 .---30 40 50 60 70 80 90 I 00 Secondar y Stiffness Normalized Displacement Tn 0 .25Sec, R 9 .---..... .. 30 40 50 60 70 80 90 I 00 Seconda r y Stiffness (/.) orm alized Displacement Tn = 0 .25 Sec, R = I 2 -. 30 40 so 60 70 80 90 I 0 0 Sec ondary Stiffness(%) l-EI Centro-Lorna Prie t a ..... Northridge O l ympia --San Fernando I !56

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Q 0 0 0 2.0 1.8 1.6 I 4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2.0 I 8 I 6 1.4 I .2 1.0 0 8 0 6 0 4 0 2 0.0 2 0 1.8 1.6 1.4 1.2 1.0 0 8 0 6 0 4 0 2 0.0 2 0 1.8 1.6 I .4 I .2 1.0 0 8 0 6 0 4 0.2 0 0 0 "' 0 ..:_-.. .. 0 ... 0 1 E I Cen t r o I 0 2 0 ........ --I 0 20 / ...-I 0 2 0 ./' ... I 0 2 0 Normalized Dis p lace m ent T n = 0 .50 Sec, R 4 30 40 s o 60 S e c o ndar y Stiffness c ; ) Normalized Displacement T n = 0 .50 Sec, R = 6 .. ...... 30 40 50 60 S econdar y S tiffness(% ) Normalized Displacement T n 0 5 0 Sec, R 9 ..-...... ............... 30 4 0 5 0 6 0 Seco n d a r y Stiffness ("/o) N o rm a lized Dis p l a c e m ent T n = 0.50 S e c R = I 2 .-................ ............... ....... . 30 4 0 5 0 6 0 tiffness ( % ) L o rna Prie t a 1 5 7 ... 70 80 90 100 ...... ;n .. 70 80 90 1 0 0 ,...-........ 70 8 0 90 I 0 0 -...... 7 0 8 0 90 100 San F ernando I

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Normalized Displace m ent T n = 0.75 Sec, R = 4 ----2.0 I 8 I .6 I 4 I 2 I 0 0.8 0.6 0 4 0.2 0 0 ............. ......... ................................. 2.0 I 8 1.6 1.4 1.2 1.0 0.8 0.6 0 4 0 2 0 0 2 0 1.8 1.6 1.4 1.2 1.0 0 8 0 6 0.4 0 2 0.0 2.0 1.8 1.6 I 4 1.2 1.0 0.8 0 6 0.4 0.2 0 0 0 I 0 20 I!. ""'-. 0 10 20 0 I 0 20 .... 0 1 0 20 l E I Centro -Lorna 30 40 50 60 Secondary Stiffness (;.) Normalized Displacement T n 0 .75 Sec, R a 6 / _.. _.. 30 40 50 60 Secondary S tiffness(%) ormalized Displacement Tn0 .75 Sec,R9 .-""' ...... 30 40 50 60 Secondar y Stiffness(%) Normalized Displacem ent Tn = 0.75 Sec, R = I 2 ..... 30 40 50 60 Secondary S tiffness("/,,) 70 80 ---70 80 ----""' 70 80 --70 80 Prieta .... Northridge O i y meia --San 158 90 I 00 90 100
PAGE 170

159

PAGE 171

Q 2i e 2.0 I 8 I 6 I 4 I 2 1.0 0 8 0 6 0 4 0.2 0 0 2.0 1.8 1.6 1.4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2 0 1.8 1.6 I .4 1.2 1.0 --0 I 0 20 0 I 0 20 Normalized Displacement ...... ............. 30 40 50 60 Secondary Stiffness(%) ormalized Displacement T n -1.2 5 Sec, R = 6 30 40 50 60 Secondary Stiffness(%) Normalized Displacement To I .2 5 Sec, R 9 ...................... ..................... ................ 70 70 f-. ........ c 0.8 ......... 0 6 0 4 0 2 0 0 0 I 0 20 30 40 50 60 70 Secondary Stiffness(%) Normalized Displacement Tn = I 2 5 Sec, R = I 2 2 0 1.8 I .6 1.4 Q 1.2 ....... ................. .................. .. ................. 1.0 .. .......... 2i 0.8 0.6 0.4 0 2 0 0 0 I 0 20 30 40 50 60 70 Secondary Stiffness(%) l-EI Centro -Lorna Prieta orthridge O lympia 1 60 80 90 I 0 0 80 90 I 00 80 90 I 0 0 80 90 100 --San Fernando

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.0 8 .6 4 I 2 1.0 0.8 0 6 0 4 0 2 0 0 2.0 1.8 I .6 I 4 1.2 1.0 0 8 0.6 0 4 0 2 0.0 2.0 I 8 1.6 I .4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2 0 1.8 I 6 1.4 1.2 I 0 0.8 0 6 0 4 0.2 0 0 0 I 0 .............. 0 I 0 ................. I 0 . 0 I 0 20 20 Normalized Displacemenl Tn 1 .50 Sec, R 4 30 40 50 60 Secunda r y S liffness ("/.) ormalized Displacement Tn =1.50 Sec, R 6 30 40 50 60 Secondary Sliffness (%) Normalized D isplacem en I T n -1.5 0 Sec, R 9 70 70 .. ......................... 20 30 40 50 60 Secondary Sliffness ("/o) Normalized Displacemenl Tn =1.50 Sec, R = 12 70 80 80 80 .. .................................................. 20 30 40 50 60 70 80 90 I 00 90 I 00 90 I 0 0 90 I 0 0 E I Cenlr o -Lom a Priela orlhridge O i y m ia -San Fernando 161

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Norma li zed Displacem ent Tn = I 7 5 Sec, R 4 2 0 I 8 I 6 I .4 1.2 I .0 .............. 0 8 0 6 0.4 0.2 0 0 0 I 0 2 0 30 40 50 60 7 0 80 9 0 I 0 0 Secondary Stiffness ("/o) Normalized Displacement Tn I 7 5 Sec, R -6 2.0 1.8 1.6 1.4 1.2 1.0 ........... ... ... 0 8 ....... 0 6 0 4 0.2 0 0 0 I 0 20 30 40 50 60 70 80 90 100 Secondary Stiffness(%) Normalized Displacement To -1.75 Sec, R = 9 2 0 1.8 1.6 1.4 e 1.2 1.0 .......... ................. .................... .. ............. c 0 8 ....... 0 6 0 4 0.2 0 0 0 I 0 20 30 40 50 60 70 80 90 I 0 0 Secondary Stiffness ("/o) Normalized D i splacemen 1 T n = I 7 5 Sec, R = I 2 2 0 1.8 1.6 1.4 0 I 2 ............... ................. ........... I 0 ... Q 0 8 Eo; ........ ... 0 6 0 4 0 2 0 0 0 I 0 20 30 40 so 60 70 80 90 I 00 Secondary Stiffness(%) l-EI Centro -Lorna P r ie I a .... Northridge Oiymeia --San Fernando j 162

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Q 0 Q 0 Q 0 2 0 I 8 I 6 I 4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2.0 1.8 I 6 I 4 1.2 1.0 0 8 0.6 0 4 0.2 0 0 2.0 1.8 1.6 1.4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2 0 1.8 1.6 1.4 1.2 1.0 0.8 0 6 0.4 0.2 0 0 0 I 0 20 ::... ................... 0 I 0 20 "':-t. ............ 0 I 0 20 .. 0 I 0 20 Normalized Displa c ement T n 2.0 Sec. R 4 A ... .... AA.&AA&Aolo &&&AA&& & & A&A&&A&AA 30 40 so 60 Seconda r y Stiffness(%) Normalized Dis p lacement T n 2 0 Sec, R 6 30 40 50 60 Secondary Stiffness(%) Normalized Displacement T n 2.0 Sec, R 9 30 40 so 60 Secondary Stiffness(%) Normalized Displacement T n 2 0 Sec, R I 2 30 40 so 60 Seconda r y Stiffness(% ) 70 80 90 I 00 70 80 90 I 00 70 80 90 I 00 70 80 90 I 00 l E I Centro Lorna Prie t a orthridge Oly mpia -San Fernando I 163

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164

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0 0 0 0 2 0 I 8 I 6 1.4 I 2 I 0 0 8 0 6 0 4 0 2 0 0 2 0 1.8 I 6 I .4 1.2 1.0 0 8 0 6 0 4 0 2 0 0 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0 6 0.4 0 .2 0 .0 2 0 I 8 1.6 1.4 I 2 I 0 0.8 0 6 0 .4 0 2 0 0 0 I 0 20 ""' 0 10 20 .... ... 0 I 0 20 I 0 20 orm21lized Displacement Tn = 2 .50 Sec, R = 4 30 40 50 60 Seconda r y Stiffness(%) Normalized Displacement T n 2.50 Sec, R = 6 30 40 50 60 Secondary Stiffness(%) Normalized Displacement Tn2.50 Sec R 9 30 40 50 60 Secondary Stiffness(%) Normalized Displacement Tn2 .50 Sec, R = 1 2 ............... 30 40 50 60 Secondary Stiffness(% ) 70 70 70 7 0 l E I Centro -Lorna Prie ta ...... Northridge -Olympia 165 80 90 100 80 90 100 80 90 I 00 80 90 I 0 0 a n Fernando I

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APPENDIXG Multi-Degree-of-Freedom Systems 166

PAGE 178

11 10 9 8 7 c 6 0 V5 5 4 3 2 0 11 10 9 8 7 6 V5 5 4 3 2 1 0 7 0 2 3 0 % Secorrllry Sli1fm.'<; 14 D(m) MlXimmStory Dift, R = 4 0 %Secorrllry 4 5 D(m) 1 67 6 7 8 21 9 --e--B Gntro ----+-Lam Priw _...._Sal Fem:rd> + lbn:l -l-Rn-lbn:l 10 DBGntro Ol.aml'riW XO)npa x Sal Fcmanoo

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11 10 9 8 7 c 6 0 Vi 5 4 3 2 0 11 10 9 8 7 5 4 3 2 1 0 7 -+-*-X f------x -:----< 0 2 3 14 D(m) IAift, R = 4 4 5 D(m) 1 68 6 7 21 8 9 ---e--EJ cmro --Lam Prieta ---ir-Nrtlridge ---a}nlia . + . Ibn! ---e--10 (] El Gntro 0 Lam Prieta 6 Nrtllridge XO)olia Bourrl

PAGE 180

It 10 9 8 7 c 6 0 .... rJl 5 4 3 2 0 ]] 10 9 8 7 c 0 6 V5 5 4 3 2 I 0 7 f----+-x -0 2 3 14 D(m) MmmmStory llift, R = 4 4 5 D(m) 169 6 7 8 9 21 -e--El Ortro --lam Priela -*-O)npa ---+--Sal Fmwm .. + . lol.er lbnl ---+--LWer lbnl 10 0 ElOrtro 0 laml"rieea X 0 ) n1U X San Fcrnan00 lol.er lbnl l.Werlbnl

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g) % Serormry Sliflixs<; c 0 .... .... 0 0 7 11 10 9 8 7 6 5 4 -3 2 0 2 3 14 D(m) MIXimmStoly lAift, R =4 4 5 D(in) 1 70 6 21 7 8 9 ----El Centro -+-lam Pricta ---tr-NrtiTidge --San Fernarxkl . + .. lol.a-lbnl --+-l.An-lbnl 10 [] El Centro 0 laml'ricta "Nrthridgc X O)rrpia X S an Femarrll lol.a-Bo.nl l.An-lbnl

PAGE 182

R = 6 0 % Sronlary Sl:iffuss 11.---------------.--------------,---------------. c 0 ;.-. '-.8 V) 7 14 21 D(m) MIXimmStory llift, R = 6 0 % Sronlary Sl:iffuss 11 10 9 8 7 6 5 4 3 2 0 2 3 4 5 6 7 8 9 D(m) 171 -e--E)QIIIru ---Lmal'rieta ---+---SalMnanb .. + .. loMr Ibn! -+--lRa"lhnl DElOnlro 0 Lmal"rieta XO)npa X Sal fUnanOO lhnl 10

PAGE 183

II 10 9 8 7 c 6 0 iJ) 5 4 3 2 0 11 10 9 8 7 c 0 6 iJ) 5 4 3 2 I 0 )E ,__. I 0 2 7 3 MtximmJ:lsriammt., R = 6 5 %Secorxlary Stilfre;s 14 D(m) MIXinunStory Drift, R = 6 5 %Secorxlary Stilfre;s 4 5 D(m) 172 6 7 21 8 9 --a-El Omu ---------Sal Fana:m + Lo.wr lhnl -+-l.Wer lhnl 10 oEJOmu 0 Lmnl"rieta X O ) rrpia X San Fcmanb Lo.wr lhnl t l'wfrlhnl

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II 10 9 8 7 c 6 0 Vi 5 4 3 2 0 11 10 9 8 7 c 0 6 Vi 5 4 3 2 0 7 &I--0 2 3 R = 6 20 % Sronlary 14 D(m) MOOmmStory llift, R = 6 20 %Sronlary 4 5 D(m) 173 6 7 8 21 9 -e--El Cmlro -+-Imu Prieta ----l:r-NrtiYidge ---+---San Feman00 + lbnJ --tjJJn-lbnl 10 0 El Ccm-o 0 lmuPrieta 6Nrtlvidgc X 0 )Jl1)ia X San Feman00 + I..o.\O"Ibnl LRn-lb.IKI

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:{1 % Secorrlary Stiffness 11.---------------.---------------.--------------. 10+--------------+-x 7 -1-------------x c 6 +----------x t m'f---+-----JS,._----1----------------l 0 11 10 9 8 7 7 14 D(m) MlximmStory Dift, R = 6 :{1% Srontary Stiffness 2 1 c 0 6 Vl 5 4 3 2 I 0 I 2 3 4 5 D(m ) 1 74 6 7 8 9 --e--El Centro ---+-lam Prieta --a)"llia ---'1---San FemanOO -+ -l.tMEr lhnl ---t.Wer lhnl 10 0 El Centro OlamPricta ANrthridge X San Fcmanoo + WIEr Brurd t.Werlhnl

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0 %Sronlary 10 v -----: ; /JJv 7 j 1 lfl 11 10 9 8 7 5 4 3 2 I 0 0 2 I ... 1 A1L 7 .-.!( 3 14 D(m) MlximmStory llift, R=9 O%Seanlary 4 5 D(m) 175 6 7 21 8 9 -it--El Onlro ---+--lam Prietl ---ir-Nrtmdge ---1---San FCI113flli> ....... l..ootT lhnl -+-l.Re' llo.nl 10 DEl Centro 0 lam Prieta t:.l'b1hridge X San Fernarrl> l.ol,..-Brurd l.Re'Banl

PAGE 187

5% Seanlary Stiflil!ss 11.--------------,--------------.--------------, c 0 11 10 9 8 7 6 5 4 3 2 0 0 7 {1--2 3 14 D(m) Dift, R = 9 5 % Seaxrluy Stiflil!ss 4 5 D(m) 1 76 6 21 7 8 9 ----&---El Centro -+--Lam Prieta -----+r-N:rthridge -....-San Fem31d:l -+ -l..ol\0" Drum ---+-t.Wer Bruxl 10 0 EJCentro 0 Lam Prieta t,. Nrthridge X San Fem31d:l l..ol\0" Drum tt.Werllruxl

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11 10 9 8 7 c 6 0 Vl 5 4 3 2 0 11 10 9 8 7 6 Vl 5 4 3 2 I 0 7 6 0 2 3 20% Sroxduy Stiffixss 14 D(m) IXift, R=9 20 % Sronlary Stiffixss 4 5 D(m) 177 6 7 8 21 9 ---e-E]Qrou ----+-lam Prieta ---tr-----1(-0)rrp3 --San + -l.ol>er Botnl -----t.Wer Botnl 10 o El Centro 0 l.aml'rieta XO)rrpa X San l.ol>er Botnl t.WerBotnJ

PAGE 189

g) % Seroriiary StifJress 11.---------------.----------------.---------------, c 6 +--------* 0 c 0 7 14 21 D(m) MIXirrunStory llift, R = 9 g) 11 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 D(in) 178 ---e--EJ Gntro --+--Lam PriEta ---b-Nrtmdge ----*""-O)rrpi a _____,..___San Fernarm .. + . l.oowr lbnJ ---e--LpPer lbnJ c EJ Ortro 0 laml'rictl f, N:nhridge X0)rrpia X San Femard> + Bo.nt LpPerlbnl 10

PAGE 190

11 10 9 8 7 c 6 0 ciJ 5 4 3 2 0 11 10 9 8 7 6 ciJ 5 4 3 2 I 0 7 1-:.---l( -lt-8 0 2 3 MlximmllsJjacmmt, R = U 0 % Seanlary Stiflieo;s 14 D(m) MlximmSory lAift, R = U 0 % Seanlary Stiflieo;s 4 5 D(in) 1 79 6 7 8 21 9 --e--EJQnlro ---lam Prieta --ISan FI'J11:W1d> . ... . Lo..a-Jhn:t -+-lRB" Jhn:t 10 0 EIGntro 0 Lam Prieta X O)rrpia X San FI'J11:W1d> Lo..a-Jhn:t LiJ!rrlbnl

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5 % SeCOirllry Stiflirss 11.---------------,--------------.---------------. c 0 11 10 9 8 7 ---6 7 o--7( 14 D(m) MOOrrunStory Ikift, R = U 5 % Secolrllry 21 c 0 6 5 4 3 2 I 0 ,_______. 2 .,. -= 3 4 5 D(m) 180 6 7 8 9 --itEl Centro ---+-Lam Prieta ---&----Nrthridge -San Fem:nkJ + lhnl -+-lj:Jper lhnl 10 o El Centro 0 lam Prieta !J.Nrthridge X O ) rrpi a :1: San F e m anoo + lhnl lj:Jper lhnl

PAGE 192

20% Secmlary Sti1lnss 7 c 0 11 10 9 8 7 6 5 4 3 2 I I 0 7 -6 2 3 14 D(m) Mu:immStory llift, R = U 20 %Secmlary Sti1lnss 4 5 D(m) 1 8 1 6 7 21 8 9 ---e--El Om-o ----+---Lam Prid2 -lE-O)npa ----+----Fmr.nn . + .. l1Mer lhnl --+--LAe-lhnl I 10 cEJQm-o 0 Laml'rida XO)npa tl!Merlhnl LAe-lhnl

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11 10 9 8 7 c 6 0 (;j 5 4 3 2 0 11 10 9 8 7 6 (;j 5 4 3 2 0 7 ,. 0 2 3 14 D ( m ) MmmmStory llift, R = U 4 5 D(m) 1 8 2 6 7 8 21 9 ---&--El Gntro -+-----l1ml Prieta -lr-Nrthridge -a)npa ----+---S:vl FemanOO .. + .. l.oi>Er Ibn! --LW>-Ibnl 10 [] ElGntro 0 l1ml Prieta llNrthridge XO)npa X S:vl Fern:ni> l.oi>Er Ibn! l lW>-Ibnl

PAGE 194

BIBLIOGRAPHY Building Seismic Safety Council (1997) NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures. Federal Emergency Management Agency. Cherry, S. and Filiatrault A. (1993). Seismic Response Control of Buildings Using Friction Dampers. Earthquake Spectra Vol. 9 No.3. Chopra, A. K., ( 1995) Dynamics of Structures -Theory and Applications to Earthquake Engineering. Prentice Hall Clough, R. W., and Penzien, J. (1993) Dynamics of Structures. McGraw-Hill, Inc. Constantinou, M. C., Soong, T. T. and Dargush G. F (1998) Passive Energy Dissipation Systems for Structural Design and Retrofit. Multidisciplinary Center for Earthquake Engineering Research. Filiatrault A. and Cherry, S (1987). Performance Evaluation ofFriction Damped Braced Frames Under Simulated Earthquake Loads. Earthquake Spectra, Vol. 3 No. 1. Filiatrault, A. and Cherry S (1990). Seismic Design Spectra for Friction-Damped Structures. Journal of Structural Engineering. Vol. 116. Grigorian, C. E., Yang T.S. and Popov, E. P (1993) Slotted Bolted Connection Energy Dissipators. Earthquake Spectra Vol. 9 No.3. International Conference of Building Officials (1997) Uniform Building Code. Kanaan A. E ., Powell G H (1975). Drain-2D, A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Structures. University of California Berkeley. Matlab (1996). Student Edition, Version 5.0.0.4073. The Mathwork s Inc Pall A. S., Marsh C., and Fazio, P (1980). Friction Joints for Seismic Control of Large Panel Structures. Journal ofPrestressed Concrete Institute Vol. 25, No.6. Paz, M (1985). Structural Dynamic s Theory and Computation. Van Nos trand Reinhold Company, Inc 183

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Yanmarcke, E H., and Lai S. P. (1980) Strong Motion Duration and RMS amplitude of Earthquake Records. Bulletin of the Sei s mological Society Of America, 70(4). Way D (1996) Friction-Damped Moment-Re s isting Frames. Earthquake Spectra Vol. 12, No.3. 184