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
Ã‚Â£
Me
sinty^Ar + coscoDAt
(3.17a)
/
B = e~^"
Ã‚Â£
\
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.
Ã‚Â£
Me &Me
sinryA/ + cosÃ‚Â£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
\nRÃ‚Â£,
2 +
R
-4
U2
2_
R
nR2$)
(4.6)
In Equation (4.6) Ã‚Â£ 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
2nÃ‚Â£, 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 Ã‚Â£>,' in place of ut(t), Equation (5.2) becomes;
d;=i>,.Ã‚Â£>; (5.5)
89