Seismic Response of a Full-scale 5-story Steel Frame Building Isolated... Triple Pendulum Bearings under 3D Excitations

Seismic Response of a Full-scale 5-story Steel Frame Building Isolated... Triple Pendulum Bearings under 3D Excitations
University of Nevada, Reno
Seismic Response of a Full-scale 5-story Steel Frame Building Isolated by
Triple Pendulum Bearings under 3D Excitations
A dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in
Civil and Environmental Engineering
by
Nhan Dinh Dao
Dr. Keri L. Ryan/Dissertation Advisor
August, 2012
Copyright by Nhan Dinh Dao 2012
All Rights Reserved
THE GRADUATE SCHOOL
We recommend that the dissertation
prepared under our supervision by
NHAN DINH DAO
entitled
SEISMIC RESPONSE OF A FULL-SCALE 5-STORY STEEL FRAME
BUILDING ISOLATED BY TRIPLE PENDULUM BEARINGS
UNDER 3D EXCITATIONS
be accepted in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
Keri L. Ryan, Ph.D., Advisor
Ian G. Buckle, Ph.D., Committee Member
Ahmad M. Itani, Ph.D., Committee Member
Raj Siddharthan, Ph.D., Committee Member
Faramarz Gordaninejad, Ph.D., Graduate School Representative
Marsha H. Read, Ph. D., Dean, Graduate School
August, 2012
i
Abstract
Seismic Response of a Full-scale 5-story Steel Frame Building Isolated by
Triple Pendulum Bearings under 3D Excitations
by
Nhan Dinh Dao
Keri L. Ryan, Advisor
A full-scale shake table test of a 5-story steel moment frame building was carried out as
part of a collaborative NEES/E-Defense research program. The building was tested in
three configurations: isolated with triple pendulum bearings (TPB), isolated with lead
rubber bearings combined with cross linear sliders (which is not discussed in this
dissertation), and fixed base. The test provided full-scale response data of both the
isolation system and the isolated structure and demonstrated the efficiency of the
isolation system in reducing the demands in the isolated structure. A 3-dimensional TPB
element with a general friction model that accounts for the variation of friction
coefficients on both velocity and vertical force was developed to predict the response of
individual TPB and the overall isolation system. The element accounts for both verticalhorizontal coupling behavior and bidirectional coupling of TPB. The horizontal behavior
of the element is based on a series combination of bidirectional elastic-plastic springs and
the circular gap elements. The new TPB element was verified by the full scale test data
and has been implemented in OpenSees so that it is available for general use. The
analytical model of the specimen building was also developed and validated by the test
ii
data from both the isolated base and fixed base tests. The following modeling assumption
were shown to best present the response characteristics of the tested structure: (1) beam
were modeled as nonlinear elements with resultant composite sections, (2) moment
connection were modeled using a Krawinkler panel zone model, and (3) energy
dissipation was represented by Rayleigh damping calibrated to include higher mode
effects observed in the test data, along with additional interstory dampers.
The vertical component of the excitation was shown to amplify the horizontal
response of both the fixed base and isolated base structures. This coupling effect was
small in the tested fixed base configuration relative to the isolated base configuration.
The calibrated analytical model was used to identify 3 main sources of this amplification:
(1) vertically and horizontally coupled modes of the structure, (2) the rocking of the
structure on the isolation system due to vertical flexibility of the isolators and supports
and uplift, and (3) the vertical-horizontal coupled response of friction bearings. Only the
first source of coupling was applicable to the fixed base building.
iii
To my parents and mother-in-law:
Dao Quang Than, Phan Thi Minh Tam and Tran Thi Thanh Tam
iv
Acknowledgements
I would like to specially thank Dr. Keri L. Ryan, my kind advisor, who has
mentored me during last three years. She has helped to develop and organize my
background to explore an interesting area: Structural Earthquake Engineering, which has
fascinated me since I was an undergraduate student. She has been providing excellent
conditions so that I could finish my PhD program in the best way. She gave me good
opportunities working in a fantastic research project and interacting with wonderful
people in the field. Besides the academic life, she has been also supporting me in my
personal life. I am definitely very lucky working with her.
I am very grateful for the participation of my dissertation committee members:
Foundation Professor Ian Buckle, Professor Ahmad Itani, Professor Raj Siddharthan and
Foundation Professor Faramarz Gordaninejad. It is my pleasure to have them in the
committee. Their recommendations regarding my dissertation are valuable.
This research program was funded by the National Science Foundation through
Grant No. CMMI-1113275. Additional support for tests of a triple pendulum bearing
isolation system at E-Defense was provided by NIED and Takenaka Corporation. The
isolators, connection plates, and design services were donated by Earthquake Protection
Systems. I am thankful to these sponsors for their support.
Many thanks to Japanese collaborators, E-Defense staffs, and Japanese students
helping during the test, including but not limited to: Dr. Eiji Sato, Dr. Tomohiro Sasaki,
v
Dr. Taichiro Okazaki. They had great contribution on the shake table test program. They
also gave me the good time staying in Japan.
I am very fortunate to know and interact with respected people: Professor Stephen
Mahin, Dr. Ronald Mayes, Dr. Victor Zayas. They provided very helpful advice and
instruction to my research.
I would like to take this opportunity to thank my friends, both at Utah State
University and University of Nevada – Reno, for their help during my staying in the U.S:
Emad Abraik, Camila Coria, Hartanto Wibowo, Yumei Jiang, Guoxun Tian, Siavash
Soroushian, Eric Monzon, Hamed Shotorbani, Chunli Wei. I also appreciate the help
from Vinh-Loc Tran and Tram Vo for my first trip to the U.S. To my home owner
Michael Warner: I appreciate your kindness and time for sharing good things to me.
Other friends, you are not named here but I am always thankful for your help and
encouragement.
From the bottom of my heart, I would like to express my deep gratitude to my
family. Their endless support and encouragement have helped me to overcome the most
difficult moments and finish my study in the U.S. Son and daughter: I am doing
everything for you.
vi
ƒ„Ž‡‘ˆ‘–‡–•
List of Tables .................................................................................................................... xi
List of Figures ................................................................................................................. xiv
Chapter 1: Introduction ..................................................................................................... 1
1.1
Background ...........................................................................................................1
1.2 Objectives of Research ..........................................................................................8
1.3 Organization of the Dissertation ...........................................................................8
Chapter 2: Specimen ........................................................................................................ 10
2.1 Description of Specimen .....................................................................................10
2.1.1
Basic Dimensions.........................................................................................10
2.1.2 Design Spectra and Design Criteria .............................................................12
2.1.3 Framing System ...........................................................................................12
2.1.4 Slabs .............................................................................................................15
2.1.5 Material Properties .......................................................................................16
2.2 Non-Structural Components and Contents..........................................................17
2.3 Weights................................................................................................................18
2.4 Condition of the Specimen before Testing..........................................................20
Chapter 3: Selection of Ground Motions and Design of Isolation System .................. 22
3.1 Target Spectra .....................................................................................................22
3.1.1 The U.S. Target Spectra ...............................................................................22
3.1.2 Japan Target Spectra ....................................................................................26
3.2 Selection and Scaling Ground Motions Representing the U.S Code ..................33
3.3 Selection and Scaling of Ground Motions Representing Japan Code .................39
3.4 Long Duration, Long Period, Subduction Motions .............................................41
3.5 Design of Isolation System .................................................................................44
3.5.1 Introduction ..................................................................................................44
3.5.2 Theoretical Unidirectional Multi-Stage Behavior of Triple Friction
Pendulum Bearings......................................................................................47
3.5.3 Design of the Bearings .................................................................................55
vii
3.5.4
Expected response of the isolation system to different earthquake
scenarios ..........................................................................................................
3.6 Preliminary analysis of the specimen ..................................................................61
Chapter 4: Instrumentation and Test Schedule............................................................. 65
4.1 Instrumentation....................................................................................................65
4.1.1
Load Cells ....................................................................................................65
4.1.2 Displacement Transducers ...........................................................................75
4.1.3 Accelerometers ............................................................................................78
4.1.4 Strain Gauges ...............................................................................................80
4.2 Installation of the Specimen to the Shake Table .................................................81
4.3 Test Schedule ......................................................................................................86
4.4
Table Motions .....................................................................................................88
4.5 Derived Response ................................................................................................94
4.5.1 Horizontal Displacement of the Isolation System .......................................94
4.5.2 Isolator Reactions and Initial Vertical Force of the TPBs ...........................96
4.5.3 Deriving Internal Forces of the Northeast Column Based on Strain
Data ...........................................................................................................102
4.5.4 Horizontal Acceleration and Story Drift at Geometric Center of
Floors .........................................................................................................104
Chapter 5: General Experimental Results ................................................................... 106
5.1 Responses of Isolation System to Sine-wave Excitation and Friction
Coefficients of Isolators ....................................................................................106
5.2 General Response of the Isolation System to Earthquake Motions ..................122
5.2.1 Peak Isolator Displacement........................................................................122
5.2.2 Peak Torsion of the Isolation System ........................................................125
5.2.3 Residual Isolator Displacement .................................................................129
5.2.4 Uplift ..........................................................................................................136
5.2.5 Peak Base Shear .........................................................................................137
5.3 General Response of the Specimen in the Isolated Base Configuration ...........142
5.3.1 Peak Floor Acceleration .............................................................................142
5.3.2 Peak Story Drift .........................................................................................145
viii
5.4 General Response of the Specimen in the Fixed Base Configuration...............148
5.5 Comparison of Responses to XY versus 3D excitations ...................................150
5.5.1 Isolated base ...............................................................................................151
5.5.2 Fixed base ..................................................................................................154
5.6 Comparison of Responses of the Isolated Base and the Fixed Base
Structures ..........................................................................................................156
Chapter 6: Modeling of Triple Pendulum Bearings .................................................... 164
6.1
Introduction .......................................................................................................164
6.2 Modeling of Components for Horizontal Behavior ..........................................167
6.2.1 Bi-directional Plasticity with Circular Yield Surface ................................167
6.2.2
Circular Elastic Gap Elements ...................................................................173
6.3 Modeling Vertical Behavior ..............................................................................176
6.4 Element Formulation for Horizontal Behavior .................................................177
6.4.1
Assembly of Tangent Stiffness Matrix in Horizontal Behavior ................177
6.4.2
Iterating over Triple Friction Pendulum Element ......................................184
6.4.3 Iterating over Element Group ....................................................................187
6.5 Preparation for Assembly of Element Stiffness and Force into Global
Equations...........................................................................................................190
Chapter 7: Analytical Modeling of the Building Specimen ........................................ 194
7.1 Material Models ................................................................................................196
7.2 Modeling Columns ............................................................................................199
7.3
Modeling Beams ...............................................................................................202
7.3.1
Primary Beams ...........................................................................................202
7.3.2
Secondary Beams .......................................................................................205
7.4
Modeling Panel Zones .......................................................................................206
7.5
Modeling Gravity Load and Mass .....................................................................211
7.6 Support Conditions............................................................................................213
7.7
Modeling Damping ...........................................................................................216
7.8
Adjusting Vertical Reaction ..............................................................................223
7.9
Effect of Modeling Assumption on Response of the Analytical Models ..........226
7.9.1 Effect of Frame Section and Connection Assumptions .............................226
ix
7.9.2 Effect of Damping Model ..........................................................................236
Chapter 8: Responses of the Analytical Models and Evaluation of Performance
Objectives..................................................................................................... 248
8.1 Fundamental Properties of the Model ...............................................................248
8.1.1 Modal Information .....................................................................................248
8.1.2 Pushover Curve and Strength of the Model ...............................................251
8.2 Responses of the Fixed Base Model .................................................................253
8.2.1 Acceleration Response to 80WSM and 35RRS .........................................253
8.2.2 Story Drift Response to 80WSM and 35RRS ............................................263
8.2.3.
Column Forces Response to 80WSM and 35RRS.....................................268
8.3. Responses of the Isolated Base Model ..............................................................277
8.3.1 Effect of Friction Model on the Response of the Isolation System ...........277
8.3.2.
Response of the Isolation System to 80TCU .............................................282
8.3.3.
Acceleration Response to 100TAK and 80TCU ........................................287
8.3.4.
Story Drift Response to 100TAK and 80TCU ...........................................296
8.3.5.
Column Forces Response to 100TAK and 80TCU....................................301
8.4 Checking Performance Objectives ....................................................................308
Chapter 9: Influence of Vertical Excitation on the Response of the Structure ........ 314
9.1 Identifying the Sources of the Vertical- Horizontal Coupling Effect ...............314
9.1.1 Sources from Superstructure: Vertical-Horizontal Coupling Modes ........316
9.1.2 Sources from Isolation System: Vertical Deformation/Uplift of the
Isolation System and Coupling Behavior of the TPBs ..............................326
9.2 Effect of the Vertical-Horizontal Coupling Behavior of Friction
Bearings on Responses of the Isolated Structures ............................................335
9.2.1 Rigid Structures .........................................................................................335
9.2.2 Cantilever Structures ..................................................................................342
9.2.3 General 3D Flexible Structures ..................................................................349
9.2.4 Effect of Frequency of the Vertical Response on the Horizontal
Response of the Isolated Structures...........................................................352
9.3 Effect of the Roof Steel Weights on the Horizontal Response of the
Tested Specimen ...............................................................................................363
x
Chapter 10: Conclusions and Recommendations ........................................................ 372
10.1 Conclusions .......................................................................................................372
10.1.1 Test Results ................................................................................................372
10.1.2 Analytical Modeling and Verification .......................................................373
10.1.3 Vertical-horizontal Coupling in Response of Fixed-base Structure
and Isolated-base Structure with TPBs .....................................................375
10.2 Recommendations for Future Studies ...............................................................377
References ...................................................................................................................... 380
Appendix A: Design Drawings of the Connection Assemblies................................... 385
Appendix B: Spectra of Table and Target Motions .................................................... 398
xi
‹•–‘ˆƒ„Ž‡•
Table 2-1
Actual yield and ultimate strengths of steel
Table 2-2
Estimated break down weights of the specimen
Table 3-1
ܵ௦ and ܵଵ for different hazard levels at ͷΨ damping ratio
Table 3-2
Damping coefficient factor ‫ܤ‬
Table 3-3
Selected records for the U.S code
Table 3-4
Parameters of the selected records for the U.S code
Table 3-5
Selected records for Japan code
Table 3-6
Parameters of the selected records for Japan code
Table 3-7
Parameters of the selected long duration, long period, subduction motions
Table 3-8
Design parameters of triple pendulum bearings
Table 3-9
Expected displacement, effective period and damping ratio of the design isolation
system at different earthquake levels
Table 4-1
Properties of load cells
Table 4-2
Peak responses of the isolation system from pre-test analysis
Table 4-3
Peak load cell forces from preliminary analysis of the connection assemblies
Table 4-4
Peak load cell forces from analysis of the finite element models of connection
assemblies
Table 4-5
Vertical load on each bearing at first iteration of installation
xii
Table 4-6
Vertical load on each bearing at last iteration of installation
Table 4-7
Schedule for shaking the isolated building
Table 4-8
Schedule for shaking the fixed base building
Table 4-9
Peak acceleration of target motions and table motions
Table 4-10
Initial vertical reaction at all TPBs computed from dynamic reaction
Table 5-1
Friction coefficients of isolators computed from the equivalent dissipated energy
approach
Table 5-2
Peak displacement of the isolation system for each earthquake motion
Table 5-3
Peak torsion of the isolation system subjected to each earthquake motion
excitation
Table 5-4
Maximum residual isolator displacement
Table 5-5
Number of uplift excursion
Table 5-6
Peak base shear of the isolated base structure
Table 5-7
Peak normalized horizontal force of the isolation system
Table 5-8
Peak floor acceleration of the isolated base structure
Table 5-9
Peak story drift of the isolated base structure
Table 5-10
Peak floor acceleration of the fixed base structure for each excitation
Table 5-11
Peak story drift of the fixed base structure for each excitation
Table 6-1
Parameters of theoretical series model
Table 7-1
Weight of analytical models
xiii
Table 7-2
Natural periods and damping ratios of the fixed base configuration
Table 7-3
Global damping matrix in modal coordinates system contributed by unit damping
coefficient damper in the X direction
Table 8-1
Natural periods of the fixed base configuration
Table 9-1
Modal information of the first 20 modes of the fixed base model
xiv
‹•–‘ˆ ‹‰—”‡•
Figure 1-1
Isolated structure
Figure 1-2
Deformation in fixed base and isolated base structures
Figure 1-3
Typical idealized unidirectional hysteresis behavior of isolators in horizontal
direction
Figure 1-4
Typical design spectral acceleration and displacement
Figure 1-5
Two common types of isolators
Figure 2-1
The 5-story steel moment frame specimen
Figure 2-2
Basic dimensions of the specimen
Figure 2-3
Beam, beam-to-column connection and slab
Figure 2-4
Column base
Figure 2-5
Horizontal braces at base level
Figure 2-7
Office room
Figure 2-6
Hospital room
Figure 2-8
Location of steel weights at roof
Figure 2-9
Cracks in concrete slab and rust on steel member
Figure 3-1
Acceleration spectrum developed by 2-point approach
Figure 3-2
The 5% damped U.S. acceleration spectra at the assumed site
Figure 3-3
Bedrock 5% damped acceleration spectrum for Japan code
xv
Figure 3-4
Soil amplification factor for Japan code
Figure 3-5
Zone factor for developing Japan design spectra
Figure 3-6
5% damped design acceleration spectra for Japan code
Figure 3-7
Design acceleration spectra for the U.S code and Japan code
Figure 3-8
ͷΨ damped spectra of the two horizontal components of the scaled motion
representing Service event
Figure 3-9
ͷΨ damped spectra of the two horizontal components of the scaled motion
representing DBE event
Figure 3-10
ͷΨ damped spectra of the two horizontal components of the scaled motions
representing MCE event
Figure 3-11
ͷΨ damped spectra of selected motions representing Japan code
Figure 3-12
Accelerogram of the selected long duration, long period, subduction motions
Figure 3-13
5% damped response spectra of the selected long duration, long period,
subduction motions
Figure 3-14
Isolation system tested
Figure 3-15
Triple friction pendulum bearing
Figure 3-16
Normalized backbone curve of a standard triple pendulum bearing
Figure 3-17
Five stages of sliding
Figure 3-18
Hysteresis loop of stage 1
Figure 3-19
Hysteresis loop of stage 2
xvi
Figure 3-20
Hysteresis loop of stage 3
Figure 3-21
Hysteresis loop of stage 4
Figure 3-22
Hysteresis loop of stage 5
Figure 3-23
Backbone curve of the designed bearings
Figure 3-24
Capacity curve of the isolation system vs. the demand curves at different
earthquake levels
Figure 3-25
Peak isolator vector-sum displacement from pre-test analysis
Figure 3-26
Peak story drift from pre-test analysis
Figure 3-27
Peak horizontal floor acceleration from pre-test analysis
Figure 3-28
Distribution of peak story drift and floor acceleration from pre-test analysis
Figure 4-1
Connection assembly
Figure 4-2
Load cell configuration at corner isolators
Figure 4-3
Load cell configuration at all isolators
Figure 4-4
Simplified model for strength analysis of the top connecting plate at edge
isolators
Figure 4-5
Finite element model of the connecting assembly at the center isolator
Figure 4-6
Meshing at the top face of the top connecting plate and locations of acting load
Figure 4-7
Deformation of the connecting assembly at the center isolator
Figure 4-8
Von-Misses stress contour on the top and bottom surfaces of the top connecting
plate of the connecting assembly at the center isolator
xvii
Figure 4-9
Layout of displacement transducers at base
Figure 4-10
Laser displacement transducer for measuring vertical movement of isolator
Figure 4-11
Instrumentation for measuring story drift
Figure 4-12
Layout of displacement transducers to measure story drift from stories 2 to 5
Figure 4-13
Accelerometers measuring acceleration at the top connecting plate
Figure 4-14
Layout of accelerometers at the 5 floor
Figure 4-15
Assembly of the connection assembly
Figure 4-16
Connection assemblies on the shake table
Figure 4-17
Installing the isolator to the connecting assembly
Figure 4-18
Installing the specimen to the isolation system
Figure 4-19
Bolt holes for connecting the specimen to the isolation system
Figure 4-20
Time history acceleration of 88RRS motion in the isolated base test
Figure 4-21
Ratio of table motion spectral accelerations and target motion spectral
th
accelerations: isolated base test
Figure 4-22
Ratio of table motion spectral accelerations and target motion spectral
accelerations: fixed base test
Figure 4-23
Configurations for solving displacement of the isolation system
Figure 4-24
Free body diaphragm illustrating derivation of isolator reaction
Figure 4-25
Extrapolated vs. recorded accelerations at corners of the shake table: 88RRS
excitation of the isolated structure
xviii
Figure 4-26
Effect of filtering on the recorded data of a load cell beneath the center TPB
Figure 4-27
Offsetting the dynamic vertical reaction to get the total vertical reaction at the
Center TPB: 88RRS excitation
Figure 4-28
Vertical reaction at bearings at the beginning of all simulations
Figure 4-29
Diagram illustrating the computation of axial stress on a cross section of column
members
Figure 4-30
Diagram illustrating the computation of drift at the geometric center
Figure 5-1
Sine-wave excitation
Figure 5-2
Response of the IsoS to the sine-wave excitation
Figure 5-3
Normalized force and displacement histories of the IsoS subjected to the sinewave excitation
Figure 5-4
Normalized hysteresis loops of all isolators subjected to the sine-wave excitation
Figure 5-5
Vertical force and displacement histories of the IsoW subjected to the sine-wave
excitation
Figure 5-6
Diagram for computing area of normalized hysteresis loop
Figure 5-7
Theoretical and experimental normalized hysteresis loop of all isolators subjected
to the sine-wave excitation: constant friction model
Figure 5-8
Diagram for computing friction coefficient based on zero-displacement intercept
method
Figure 5-9
Zero-displacement intercept of a data point
Figure 5-10
Dependence of friction coefficient on vertical load
xix
Figure 5-11
Dependence of friction coefficient on velocity at different vertical loads
Figure 5-12
Rate parameter at different vertical loads
Figure 5-13
Theoretical and experimental normalized hysteresis loop of all isolators subjected
to the sine-wave excitation: variable friction coefficient
Figure 5-14
Peak displacement of the isolation system for each earthquake motion relative to
scenario limits
Figure 5-15
Peak torsion of the isolation system subjected to all earthquake motion
excitations
Figure 5-16
Peak torsion vs. peak displacement of the isolation system
Figure 5-17
Peak torsion vs. peak isolator displacement in X-direction
Figure 5-18
Peak torsion vs. peak isolator displacement in Y-direction
Figure 5-19
Maximum residual isolator displacement
Figure 5-20
Residual displacement trace of the Center isolator
Figure 5-21
Displacement history of the center isolator from 100TAK to 100TAB
Figure 5-22
Displacement history of the center isolator in 100IWA
Figure 5-23
Displacement history of the center isolator in 100SCT
Figure 5-24
Displacement history of the center isolator from 80WSM to 130ELC
Figure 5-25
Residual isolator displacement vs. peak isolator displacement
Figure 5-26
Residual displacement vs. duration from peak acceleration to the end of the
ground motion
xx
Figure 5-27
Residual displacement vs. duration from peak isolator displacement to the end of
the ground motion
Figure 5-28
Peak normalized horizontal force vs. peak displacement of the Center isolator
Figure 5-29
Peak normalized horizontal force in the X-direction vs. peak displacement of the
Center isolator in the X-direction
Figure 5-30
Peak normalized horizontal force in the Y-direction vs. peak displacement of the
Center isolator in the Y-direction
Figure 5-31
Peak floor acceleration of the isolated base structure for each earthquake motion
excitation
Figure 5-32
Distribution of peak floor acceleration of the isolated base structure for each
earthquake motion excitation
Figure 5-33
Peak story drift of the isolated base structure subjected to all earthquake motion
excitations
Figure 5-34
Distribution of peak story drift of the isolated base structure subjected to all
earthquake motion excitations
Figure 5-35
Peak floor acceleration of the fixed base structure for each excitation
Figure 5-36
Peak floor acceleration of the fixed base structure for each excitation
Figure 5-37
Time-history of the acceleration at roof of the isolated base structure: 3D vs. XY
excitation
Figure 5-38
st
Time-history of the drift at 1 story of the isolated base structure: 3D vs. XY
excitation
xxi
Figure 5-39
Time-history of bending moment at column base of the NE column of the isolated
base structure: 3D vs. XY excitation
Figure 5-40
Time-history of the base shear of the isolated base structure: 3D vs. XY
excitation
Figure 5-41
Time-history of the displacement of the center isolator: 3D vs. XY excitation
Figure 5-42
Time-history of the acceleration at roof of the fixed base structure: 3D vs. XY
excitation
Figure 5-43
Time-history of the drift at 5
th
story of the fixed base structure: 3D vs. XY
excitation
Figure 5-44
Time-history of bending moment at column base of the NE column of the fixed
base structure: 3D vs. XY excitation
Figure 5-45
Response spectra of table motion, 80WSM
Figure 5-46
Response spectra of table motion, 88RRS
Figure 5-47
Response spectra of table motion, 100IWA
Figure 5-48
Peak floor acceleration, peak story drift and peak torsion drift of the isolated base
and fixed base structures subjected to 80WSM
Figure 5-49
Peak floor acceleration, peak story drift and peak torsion drift of the isolated base
and fixed base structures subjected to 88RRS
Figure 5-50
Peak floor acceleration, peak story drift and peak torsion drift of the isolated base
and fixed base structures subjected to 100IWA
Figure 6-1
Theoretical series model for multi-stage behavior of TFP
Figure 6-2
Modeling friction behavior
xxii
Figure 6-3
Numerical series model for multi-stage behavior of TPB
Figure 6-4
One-dimensional elastic-plastic model
Figure 6-5
Displacement and force diagrams of gap element
Figure 6-6
Vertical behavior of TPB
Figure 6-7
Finite element configuration for horizontal behavior of TPB
Figure 6-8
Inverse Newton – Raphson iteration
Figure 6-9
Flow chart for solving TPB element
Figure 6-10
Situation where the inverse Newton – Raphson iteration fails
Figure 6-11
Flow chart for solving Element Group
Figure 6-12
Newton – Raphson iteration for iterating over Element Group
Figure 6-13
Basis coordinate system of TPB in global coordinate system
Figure 6-14
Force diagram for computing overturning moment and torsion
Figure 7-1
Models of the specimen
Figure 7-2
Normal stress in a cross section of 1-dimensional elements
Figure 7-3
Behavior of steel material model
Figure 7-4
Behavior of concrete material model
Figure 7-5
Behavior of force-based elements and displacement-based elements
Figure 7-6
Behavior of a bending member simulated by force-based element and
displacement-based elements
Figure 7-7
Discretization of typical primary beams
xxiii
Figure 7-8
Behavior of a composite fiber section beam with and without axial restraint
Figure 7-9
Composite section behavior
Figure 7-10
Panel zone model for beam to column connection
Figure 7-11
Gusset plate and its finite element model
Figure 7-12
Panel zone model and equivalent truss of the gusset plate
Figure 7-13
Total deviation of vertical reaction for tuning vertical stiffness of isolators
Figure 7-14
Total vertical reaction of the isolated base structure subjected to 70LGP
excitation
Figure 7-15
Rayleigh damping models
Figure 7-16
Additional damper for adjusting damping of the 2
Figure 7-17
Distribution of the initial static vertical reaction at bearings
Figure 7-18
Roof acceleration of the fixed base structure subjected to 35RRS: test vs.
nd
mode in the ܺ direction
analysis with different frame models
Figure 7-19
Roof drift of the fixed base structure subjected to 35RRS: test vs. analysis with
different frame models
Figure 7-20
Peak floor acceleration of the fixed base structure subjected to 35RRS: test vs.
analysis with different frame models
Figure 7-21
Peak story drift of the fixed base structure subjected to 35RRS: test vs. analysis
with different frame models
Figure 7-22
Floor spectra at roof of the fixed base structure subjected to 35RRS: test vs.
analysis with different frame models
xxiv
Figure 7-23
Roof drift of the fixed base model subjected to 35RRS: Elastic Section vs. Fiber
Section
Figure 7-24
Roof acceleration of the isolated base structure subjected to 70LGP: test vs.
analysis with different frame models
Figure 7-25
Roof drift of the isolated base structure subjected to 70LGP: test vs. analysis with
different frame models
Figure 7-26
Peak floor acceleration of the isolated base structure subjected to 70LGP: test
vs. analysis with different frame models
Figure 7-27
Peak story drift of the isolated base structure subjected to 70LGP: test vs.
analysis with different frame models
Figure 7-28
Floor spectra at roof of the isolated base structure subjected to 70LGP: test vs.
analysis with different frame models
Figure 7-29
Roof acceleration of the isolated base structure subjected to 80TCU: test vs.
analysis with different frame models
Figure 7-30
Roof drift of the isolated base structure subjected to 80TCU: test vs. analysis with
different frame models
Figure 7-31
Peak floor acceleration of the isolated base structure subjected to 80TCU: test
vs. analysis with different frame models
Figure 7-32
Peak story drift of the isolated base structure subjected to 80TCU: test vs.
analysis with different frame models
Figure 7-33
Floor spectra at roof of the isolated base structure subjected to 80TCU: test vs.
analysis with different frame models
xxv
Figure 7-34
Roof acceleration of the fixed base structure subjected to 35RRS: test vs.
analysis with different damping models
Figure 7-35
Peak floor acceleration of the fixed base structure subjected to 35RRS: test vs.
analysis with different damping models
Figure 7-36
Roof drift of the fixed base structure subjected to 35RRS: test vs. analysis with
different damping models
Figure 7-37
Peak story drift of the fixed base structure subjected to 35RRS: test vs. analysis
with different damping models
Figure 7-38
Floor spectra at roof of the fixed base structure subjected to 35RRS: test vs.
analysis with different damping models
Figure 7-39
Roof acceleration of the isolated base structure subjected to 70LGP: test vs.
analysis with different damping models
Figure 7-40
Roof drift of the isolated base structure subjected to 70LGP: test vs. analysis with
different damping models
Figure 7-41
Peak floor acceleration of the isolated base structure subjected to 70LGP: test
vs. analysis with different damping models
Figure 7-42
Peak story drift of the isolated base structure subjected to 70LGP: test vs.
analysis with different damping models
Figure 7-43
Floor spectra at roof of the isolated base structure subjected to 70LGP: test vs.
analysis with different damping models
Figure 7-44
Roof acceleration of the isolated base structure subjected to 80TCU: test vs.
analysis with different damping models
xxvi
Figure 7-45
Roof drift of the isolated base structure subjected to 80TCU: test vs. analysis with
different damping models
Figure 7-46
Peak floor acceleration of the isolated base structure subjected to 80TCU: test
vs. analysis with different damping models
Figure 7-47
Peak story drift of the isolated base structure subjected to 80TCU: test vs.
analysis with different damping models
Figure 7-48
Floor spectra at roof of the isolated base structure subjected to 80TCU: test vs.
analysis with different damping models
Figure 7-49
Floor spectra at roof of the isolated base structure subjected to 70LGP: test vs.
analysis with and without interstory damper models
Figure 7-50
Floor spectra at roof of the isolated base structure subjected to 80TCU: test vs.
analysis with and without interstory damper models
Figure 8-1
The first 3 modes and the first vertical mode of the fixed base model
Figure 8-2
Shapes of first 3 modes in X and Y directions
Figure 8-3
Pushover curves of the fixed base model
Figure 8-4
Acceleration response in X direction of the fixed base structure subjected to
80WSM: analytical model vs. test data
Figure 8-5
Acceleration response in Y direction of the fixed base structure subjected to
80WSM: analytical model vs. test data
Figure 8-6
Acceleration response in X direction of the fixed base structure subjected to
35RRS: analytical model vs. test data
xxvii
Figure 8-7
Acceleration response in Y direction of the fixed base structure subjected to
35RRS: analytical model vs. test data
Figure 8-8
Peak floor acceleration of fixed base structure: analytical model vs. test data
Figure 8-9
Floor spectra of the fixed base structure subjected to 80WSM: analytical model
vs. test data
Figure 8-10
Floor spectra of the fixed base structure subjected to 35RRS: analytical model
vs. test data
Figure 8-11
Drift response in X direction of the fixed base structure subjected to 80WSM:
analytical model vs. test data
Figure 8-12
Drift response in Y direction of the fixed base structure subjected to 88WSM:
analytical model vs. test data
Figure 8-13
Drift response in X direction of the fixed base structure subjected to 35RRS:
analytical model vs. test data
Figure 8-14
Drift response in Y direction of the fixed base structure subjected to 35RRS:
analytical model vs. test data
Figure 8-15
Peak story drift of fixed base structure: analytical model vs. test data
Figure 8-16
Dynamic bending moment about X-axis in the NE column of the fixed base
structure subjected to 80WSM: analytical model vs. test data
Figure 8-17
Dynamic bending moment about Y-axis in the NE column of the fixed base
structure subjected to 80WSM: analytical model vs. test data
Figure 8-18
Dynamic axial force response in the NE column of the fixed base structure
subjected to 80WSM: analytical model vs. test data
xxviii
Figure 8-19
Dynamic bending moment about X-axis in the NE column of the fixed base
structure subjected to 35RRS: analytical model vs. test data
Figure 8-20
Dynamic bending moment about Y-axis in the NE column of the fixed base
structure subjected to 35RRS: analytical model vs. test data
Figure 8-21
Dynamic axial force response in the NE column of the fixed base structure
subjected to 35RRS: analytical model vs. test data
Figure 8-22
Peak dynamic forces at every section of column NE of the fixed base structure:
analytical model vs. test data
Figure 8-23
Displacement history of the center isolator subjected to 100TAK: analytical model
vs. test data
Figure 8-24
Displacement history of the center isolator subjected to 100TAB: analytical model
vs. test
Figure 8-25
Global normalized loop of the isolation system subjected to 100TAK: analytical
model vs. test data
Figure 8-26
Global normalized loop of the isolation system subjected to 100TAB: analytical
model vs. test
Figure 8-27
Energy dissipated by the isolation system during 100TAK: analytical model vs.
test data
Figure 8-28
Displacement of the center isolator when the isolated base structure subjected to
80TCU: analytical model vs. test
Figure 8-29
Reactions at center bearing of the isolated base structure subjected to 80TCU:
analytical model vs. test
xxix
Figure 8-30
Hysteresis loops of the center bearing and of the isolation system when the
isolated base structure subjected to 80TCU: analytical model vs. test
Figure 8-31
Acceleration response in X direction of the isolated base structure subjected to
100TAK: analytical model vs. test
Figure 8-32
Acceleration response in Y direction of the isolated base structure subjected to
100TAK: analytical model vs. test
Figure 8-33
Acceleration response in X direction of the isolated base structure subjected to
80TCU: analytical model vs. test
Figure 8-34
Acceleration response in Y direction of the isolated base structure subjected to
80TCU: analytical model vs. test
Figure 8-35
Peak floor acceleration of fixed base structure: analytical model vs. test
Figure 8-36
Floor spectra of the isolated base structure subjected to 100TAK: analytical
model vs. test
Figure 8-37
Floor spectra of the isolated base structure subjected to 80TCU: analytical model
vs. test
Figure 8-38
Drift response in X direction of the isolated base structure subjected to 100TAK:
analytical model vs. test
Figure 8-39
Drift response in Y direction of the isolated base structure subjected to 100TAK:
analytical model vs. test
Figure 8-40
Drift response in X direction of the isolated base structure subjected to 80TCU:
analytical model vs. test
Figure 8-41
Drift response in Y direction of the isolated base structure subjected to 80TCU:
analytical model vs. test
xxx
Figure 8-42
Peak story drift of the isolated base structure: analytical model vs. test
Figure 8-43
Dynamic bending moment about X-axis in the NE column of the isolated base
structure subjected to 100TAK: analytical model vs. test
Figure 8-44
Dynamic bending moment about Y-axis in the NE column of the isolated base
structure subjected to 100TAK: analytical model vs. test
Figure 8-45
Dynamic axial force in the NE column of the isolated base structure subjected to
100TAK: analytical model vs. test
Figure 8-46
Dynamic bending moment about X-axis in the NE column of the isolated base
structure subjected to 80TCU: analytical model vs. test
Figure 8-47
Dynamic bending moment about Y-axis in the NE column of the isolated base
structure subjected to 80TCU: analytical model vs. test
Figure 8-48
Dynamic axial force in the NE column of the isolated base structure subjected to
80TCU: analytical model vs. test
Figure 8-49
Peak dynamic forces at every section of column NE of the isolated base
structure: analytical model vs. test
Figure 8-50
Peak isolator displacement of the analytical model subjected to selected motions
representing different earthquake scenarios
Figure 8-51
Peak story drift of the analytical model subjected to selected motions
representing different earthquake scenarios
Figure 8-52
Peak floor acceleration of the analytical model subjected to selected motions
representing different earthquake scenarios
Figure 8-53
Distribution of peak story drift of the analytical model subjected to selected
motions representing different earthquake scenarios
xxxi
Figure 8-54
Distribution of peak floor acceleration of the analytical model subjected to
selected motions representing different earthquake scenarios
Figure 9-1
Horizontal acceleration at roof of the fixed base structure subjected to the 88RRS
motion
Figure 9-2
Peak horizontal floor acceleration of the fixed base structure subjected to the
88RRS motion
Figure 9-3
Horizontal acceleration at roof of the isolated base structure subjected to the
70LGP motion
Figure 9-4
Peak horizontal floor acceleration of the isolated base structure subjected to the
70LGP motion
Figure 9-5
A horizontal-vertical coupling mode of the fixed base structure model
Figure 9-6
A horizontal-vertical coupling mode of the isolated base structure model
Figure 9-7
Horizontal acceleration at roof of the fixed base model subjected to the vertical
component of 88RRS motion
Figure 9-8
Peak horizontal floor acceleration of the fixed base model subjected to the
vertical component of 88RRS motion
Figure 9-9
Horizontal acceleration at roof of the isolated base model subjected to the
vertical component of 70LGP motion
Figure 9-10
Peak horizontal floor acceleration of the isolated base model subjected to the
vertical component of 70LGP motions
Figure 9-11
Horizontal acceleration at roof of the fixed base model subjected to 88RRS
motion: 3D vs. XY+Z
xxxii
Figure 9-12
Peak horizontal floor acceleration of the fixed base model subjected to 88RRS
motion: 3D vs. XY+Z
Figure 9-13
Horizontal floor spectra of the fixed base model subjected to 88RRS motion: 3D
vs. XY excitations
th
Figure 9-14
The 10 mode shape of the fixed base model
Figure 9-15
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: 3D vs. XY+Z
Figure 9-16
Peak horizontal floor acceleration of the isolated base model subjected to 70LGP
motion: 3D vs. XY+Z
Figure 9-17
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: 3D vs. Z Restrained
Figure 9-18
Peak horizontal floor acceleration of the isolated base model subjected to 70LGP
motion: 3D vs. Z Restrained
Figure 9-19
Influence of rocking on the isolation system to the horizontal response of the
isolated structure
Figure 9-20
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: Uncoupled vs. Z Restrained
Figure 9-21. Peak horizontal floor acceleration of the isolated base model subjected to 70LGP
motion: Uncoupled vs. Z Restrained
Figure 9-22
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: Uncoupled vs. Const Kz
Figure 9-23
Peak horizontal floor acceleration of the isolated base model subjected to 70LGP
motion: Uncoupled vs. Const Kz
xxxiii
Figure 9-24
Horizontal acceleration at roof of the isolated base model subjected to 88RRS
motion: Uncoupled vs. Const Kz
Figure 9-25
Peak horizontal floor acceleration of the isolated base model subjected to 88RRS
motion: Uncoupled vs. Const Kz
Figure 9-26
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: Full 3D vs. Uncoupled
Figure 9-27
Peak horizontal floor acceleration of the isolated base model subjected to 70LGP
motion: Full 3D vs. Uncoupled
Figure 9-28
Single mass isolated system with friction bearing
Figure 9-29
Two cases of excitation on the isolated single mass system
Figure 9-30
System for Numerical Example 9.1
Figure 9-31
Input acceleration components for Numerical Example 9.1
Figure 9-32
Hysteresis loops of isolation system in Numerical Example 9.1
Figure 9-33
Horizontal acceleration of the isolated structure in Numerical Example 9.1
Figure 9-34
Two cases of excitation on the isolated cantilever structure
Figure 9-35
System for Numerical Example 9.2
Figure 9-36
Hysteresis loops of isolation system in Numerical Example 9.2
Figure 9-37
Horizontal acceleration of the isolated structure in Numerical Example 9.2
Figure 9-38
Mode shapes of the isolated cantilever
Figure 9-39
Envelope mode shapes of the isolated cantilever
xxxiv
Figure 9-40
Peak horizontal acceleration at different periods of the vertical excitation
Figure 9-41
Peak acceleration distribution at different periods of vertical excitation
Figure 9-42
Total vertical reaction of the isolated structure subjected to 70LGP
Figure 9-43
Fourier spectrum of the dynamic vertical reaction of the isolated structure
subjected to 70LGP
Figure 9-44
Modes of the isolated base structure model with periods close to the period of the
first vertical mode
Figure 9-45
Distribution of the peak acceleration and the peak story drift of modes with period
close to the frequency of the first vertical mode
Figure 9-46
5% damped floor spectra of the isolated structure subjected to 70LGP
Figure 9-47
Total vertical reaction of the isolated structure subjected to 88RRS
Figure 9-48
Fourier spectrum of the dynamic vertical reaction of the isolated structure
subjected to 88RRS
Figure 9-49
5% damped floor spectra of the isolated structure subjected to 88RRS and
88RRSXY
Figure 9-50
Selected mode shapes of the isolated base model without the roof weights
Figure 9-51
Total vertical reactions of the isolated base models with and without roof weights
subjected to 70LGP
Figure 9-52
Fourier spectra of the dynamic vertical reaction of the isolated base models with
and without roof weights subjected to 70LGP
Figure 9-53
5% damped floor spectra of the isolated base model subjected to 70LGP: with
vs. without roof weights
xxxv
Figure 9-54
5% damped floor spectra ratio of the isolated base model subjected to 70LGP:
with vs. without roof weights
Figure 9-55
Total vertical reactions of the isolated base structures with and without roof
weights subjected to 88RRS
Figure 9-56
Fourier spectrum of the dynamic vertical reaction of the isolated base structures
with and without roof weights subjected to 88RRS
Figure 9-57
5% damped floor spectra of the isolated base model subjected to 88RRS: with
vs. without roof weights
Figure 9-58
5% damped floor spectra ratio of the isolated base model subjected to 88RRs:
with vs. without roof weights
Chapter 1
–”‘†— –‹‘
1.1
Background
Base isolation is an effective approach to mitigate losses caused by damage to the
structural system, nonstructural components and contents of buildings during
earthquakes. The idea of this approach is to “detach”, or “isolate”, a building/structure
from the ground using flexible devices, called isolators, (Figure 1-1) so that the
mechanical energy transmitted to the isolated building/structure from high frequency
seismic wave is reduced (Kelly, 1997). The reduction of the transmitted energy reduces
response, including deformation and absolute movement, velocity and acceleration, and
hence the damage, of the isolated building/structure. Figure 1-2 schematically compares
the response of fixed base and isolated base building.
Isolated structure
Isolator
Figure 1-1
Isolated structure
In order for isolated structures to support small horizontal loads such as wind
without significant horizontal displacement, isolators are usually produced with large
initial stiffness. The initial stiffness reduces flexibility of the isolator, and hence the
isolation effect. Typical idealized unidirectional hysteresis behavior (force vs.
2
Fixed base
structure
Isolated base
structure
Earthquake excitation
Figure 1-2
Deformation in fixed base and isolated base structures
deformation) in horizontal direction of an isolator is shown in Figure 1-3. The isolator
hysteresis loop indicates that mechanical energy is dissipated, which provides damping to
the isolated structure.
In earthquake engineering language, an isolation system lengthens the natural
period of the isolated structure, shifting it to a lower spectral acceleration region of the
design spectrum (Figure 1-4 (a)). The reduction in spectral acceleration reduces the
inertia force on superstructure, and hence reduces the damage. However, the flexibility
increases the displacement relative to the ground of the isolated structure during an
earthquake (Figure 1-4 (b)). Another important aspect of base isolation is that the modal
F
u
Figure 1-3
Typical idealized unidirectional hysteresis behavior of isolators in
horizontal direction
ܵ௔ Increasing
damping
Fixed base structure
(a)
Figure 1-4
ܵௗ Isolated base
structure
ܶ
Fixed base
structure
Increasing
damping
3
Isolated base
structure
ܶ
(b)
Typical design spectral acceleration and displacement
(a) Spectral acceleration
(b) Spectral displacement
properties of the isolated structure are modified so that the contribution of higher modes
to the response of the isolated structure due to horizontal earthquake excitation is small
(Naeim and Kelly, 1999).
Current isolators in the United State are classified into 2 basis types (Kelly, 1997;
Naeim and Kelly, 1999): elastomeric bearings and friction bearings. An elastomeric
bearing includes alternating steel and rubber layers to provide flexibility in horizontal
direction while being stiff in the vertical direction. Lead plugs are usually installed into
elastomeric bearings to provide damping and initial horizontal stiffness to the bearings.
An elastomeric bearing with a lead plug is referred to as a lead rubber bearing (Figure 1-5
(a)). Friction bearings are categorized into 2 types: flat sliders and pendulum bearings.
Pendulum bearings (Figure 1-5 (b)) consist of concave plate(s) and intermediate slider(s)
with spherical surfaces. The combination of vertical force and the spherical surfaces
provides restoring force while the friction between surfaces provides initial stiffness and
dissipates kinematic energy when sliding occurs. Pendulum bearings are classified as
single pendulum bearings, double pendulum bearings and triple pendulum bearings
(TPBs), where the name reflects the number of pendulum mechanisms the bearing can
4
(a)
(b)
Figure 1-5
Two common types of isolators
(a) Lead rubber bearing (www.dis-inc.com)
(b) Friction pendulum bearing (www.earthquakeprotection.com)
produce. Because of their multi-stage behavior, Morgan and Mahin (2011) suggested that
TPBs are ideal for performance-based design. TPBs also have capacity to provide large
displacements with smaller devices compared to single or double pendulum bearings.
Analytical models of TPBs have been studied by previous researchers. Fenz and
Constantinou (2008 (1)) and Morgan and Mahin (2011) developed a theoretical model for
unidirectional (1D) behavior of TPB based on equilibrium analysis of every component
of the bearing at different stages. Fenz and Constantinou (2008 (2)) introduced an
approach for implementing a 3-dimensional (3D) model of TPB, as an extension of their
1D model, using current elements in commercial finite element analysis software. The
effect of velocity and vertical force on friction coefficient can be accounted for in this
model depending on the software capabilities. Tsai et al. (2010) extended the equilibrium
approach to develop 1D theoretical models for friction bearings with numerous
intermediate sliders. Tsai et al. (2010) also extended the model to bidirectional (2D)
behavior and introduced a 2D plasticity model with multiple yield and bounding surfaces
to model the behavior of TPBs. Becker and Mahin (2012) developed a general TPB
model that can track the movement of individual components of a TPB. However, the
5
vertical-horizontal coupling behavior of TPBs and the variation of friction coefficients on
velocity and vertical force, as reported by Mokha et al. (1990), were not fully considered
in these studies. A case study on a bridge model isolated with single pendulum bearings
subjected to 3D earthquake motion excitations by Eroz and DesRoches (2008) shows that
the coupling behavior and the variation of friction coefficient have important roles on
response of the system.
Besides analytical studies, several experimental programs have been conducted to
investigate the response of structures isolated with TPBs, and validate the analytical
models. The shake table test of a quarter-scale, 6-story steel structure by Fenz and
Constantinou (2008(3)) provided information about the response of the TPBs and the
isolated structure. The analytically predicted response of the isolation system and low
frequencies spectral floor acceleration was validated by the test data but the analytical
peak floor acceleration and spectral floor acceleration at high frequencies significantly
differ from the test data. The comparison of the story drift was not reported. Shake table
tests of a small 3-story steel structure, not a scaled structure, (Morgan and Mahin (2011))
also provided many experimental results on the response of a structure isolated with
TPBs. The tested structure has the plane dimension of ͵Ǥ͸ͷͺ ൈ ͳǤͺʹͻ (ͳʹˆ– ൈ ͸ˆ–)
and is ͷǤʹͺ͵ (ͳ͹ˆ– െ Ͷ‹) height. A method for deriving bearing parameters based on
the experimental cyclic response of the bearings was proposed in their research. Based on
comparison between the experimental and analytical responses of the bearings subjected
to the 1D cyclic excitations, their proposed 1D model of TPBs mentioned previously
predicts the response of the bearings very well. Analytical response of the bearings and
6
the isolated structure subjected to earthquake excitation was not considered. Becker and
Mahin (2012) performed an extensive test on TPBs with various controlled-displacement
orbits to investigate the behavior of the TPBs and validate their analytical model of
TPBs. Prior to the test reported in this dissertation, no full scale test of a large structure
isolated by friction bearings had been done. To fill the gap, a full scale test of a 5-story
steel moment frame building was performed at Hyogo Earthquake Engineering Research
Center (also known as E-Defense), Japan in August 2011. Three building configurations
were tested: (1) isolated base with TPBs, (2) isolated base with lead rubber bearings
combined with linear cross sliders, and (3) fixed base. The test aimed to provide a full
scale proof of the effectiveness of the isolation systems and the data of the response of
the fixed base and isolated base buildings subjected to variety earthquake excitations.
Both responses of individual bearing and the isolation systems, such as displacement,
reaction, torsion, residual displacement, were observed. Response of the superstructure,
nonstructural components and contents in all 3 configurations was also recorded.
Response of the bearings, isolation system and superstructure in the isolated base
configuration with TPBs and the response of the structure in the fixed base configuration
are specifically reported in this dissertation.
Intuitively, the vertical excitation of earthquakes should affect the horizontal
response of a structure isolated by friction bearings since the horizontal response of
friction bearings depends on its vertical force. This effect has been experimentally and
analytically studied. Most of these studies, described below, concluded that the influence
of vertical excitation on the displacement of the isolation system is insignificant, but the
7
response of the isolated structures may be significantly affected. In the test of a vertically
rigid, single story structure (Zayas et al., 1987), the vertical excitation had little influence
on the horizontal displacement of the isolation system. In a test of a simple bridge system
(Mosqueda et al. (2004)), the horizontal response of the isolation system was almost
identical when subjected to the excitations with and without vertical component. Fenz
and Constantinou (2008(3)) stated that the effect of the vertical excitation on the peak
horizontal response of the isolation system and the isolated structure was minor.
However, the reported test data showed that the presence of vertical excitation sometimes
increased the peak floor acceleration by an approximate factor of 2, compared to the peak
floor acceleration subjected to horizontal excitation only. Morgan and Mahin (2011)
reported that the effect of vertical excitation on isolator displacement, peak base shear of
the isolated structure and total dissipated energy was very small. The vertical excitation
in this test was small. An analytical study by Lin and Tadjbakhsh (1986) showed
significant effect of the vertical excitation on the horizontal response of a single-mass
isolated by a flat slider (pure-friction isolator) subjected to both sine-waves and
earthquake excitations. A case study of an asymmetric 3D single-story structure isolated
with single pendulum bearings (Shakib and Fuladgar (2003)) suggested that the vertical
component of earthquake excitations has significant influence on structure’s response and
the torsion of the isolation system. Calvi et al. (2004) concluded that the displacement
demand of a numerical model of a bridge isolated with friction bearing was not
significantly affected by vertical excitation, but the shear, bending and torsion demand on
piers were strongly influenced. Recent analytical studies (Panchal et al., 2009; Rabiei and
Khoshnoudian, 2011) also showed the strongly increased of the horizontal response of
8
the structures isolated by friction bearings when subjected to 3D motion excitations
compared to horizontal excitation only. However, the sources and the mechanisms of the
effect of vertical excitation on horizontal response of flexible structures (both in
horizontal and vertical directions) isolated by friction bearings have not been fully
investigated and explained.
1.2
Objectives of Research
This research program aims to: (1) provide the full-scale experimental proof of
the effectiveness of TPBs in reducing demands on an isolated building subjected to
various types of earthquake motions, (2) develop and validate the analytical models of
TPBs and the isolated structure, and (3) investigate and understand the response of the
isolation system and the isolated structure. Both isolated base and fixed base
configurations of a 5-story steel moment frame specimen were tested and responses of
both isolation system and the superstructure were observed. The extensive data recorded
from the test are used to compare the response between the 2 configurations, develop and
validate analytical models, and investigate and understand the response of the isolation
devices and the structural system under earthquakes. As an important part of the
investigation, the sources and mechanism of the effect of the vertical excitation on the
horizontal response of both configurations are studied.
1.3
Organization of the Dissertation
This dissertation is organized into 10 chapters. The detailed information of the
tested specimen is described in Chapter 2. Chapter 3 presents the selection of earthquake
9
ground motions for the test and the design of the isolation system, based on the 1D multistage behavior of TPBs. The selected motions represent both the 3 earthquake levels at a
high seismicity area in the United State (Los Angeles) and the Japanese code. Chapter 4
describes the experimental program including the instrumentation for data acquisition,
assembly, test schedule and processing of sensor data to derive structural responses. The
generated table motions are also summarized in this chapter. Chapter 5 summarizes and
compares the general experimental response of both isolated base and fixed base
configurations. Responses to excitations with and without vertical component are also
compared for both configurations. Parameters of the friction coefficient model of the
bearings accounting for the variation of friction coefficient on velocity and axial force are
derived based on data from the sine wave characterization test. Chapter 6 reports the
development of a general 3D TPB element that accounts for the coupled verticalhorizontal response of the bearing, the bi-directional coupling response in horizontal
direction, and the variation of the friction coefficient on both velocity and vertical force.
The element has been programmed and implemented into the Open System for
Earthquake Engineering Simulation (OpenSees). The development of the analytical
models of the building specimen is described in detail in Chapter 7. Chapter 8 validates
the analytical model by comparing the analytical response to the experimental. Chapter 9
investigates the vertical-horizontal coupling in responses of both the fixed base and
isolated base configurations. The mechanism of the coupling effect in structures isolated
by friction bearings is analytically and numerically studied. Chapter 10 summarizes the
results of this research.
Chapter 2
’‡ ‹‡
2.1
Description of Specimen
The specimen used in this experiment program was designed by Hyogo
Earthquake Engineering Research Center (2008) and was used in a test in March 2009
(Kasai et al., 2010). The author was not involved in the design process. Hereafter is
description of the specimen for convenience.
ʹǤͳǤͳ ƒ•‹ ‹‡•‹‘•
The tested specimen was a five-story steel moment frame building with
rectangular plan (Figure 2-1). The building is ͳͲ ൈ ͳʹ in plan and approximately
ͳ͸ height with 2 bays in each direction. To make the building unsymmetrical, bay
widths in the long direction (ͳʹ) are ͹ and ͷ. Figure 2-2 shows basic dimensions
of the building.
To match the global coordinate system of the building with the conventional
global coordinate system of the shaking table, X- and Y-axis of the building are set as
shown in Figure 2-2 and Z-axis is the up-right axis.
The number of story of the specimen (5 stories) was selected because “it
represents many office building seen in Japan; it is about the tallest of the majority of
steel building stock, and; it tends to deform, if not damped, much more than taller steel
buildings under the major quake.” (Kasai et al. 2010).
11
Figure 2-1
The 5-story steel moment frame specimen
1
3
3m
5m
1
5m
3m
Y
3m
X
7m
5m
1: concrete block, size 2 x 4 x 0.18 m
2: concrete block, size 2 x 2 x 0.25 m
3: concrete block, size 0.8 x 1.5 x 0.45 m
3.85 m
3m
2
7m
(a)
Figure 2-2
Basic dimensions of the specimen
(a) Typical plan view from Floors 2 to 5
(b) Elevation view
5m
(b)
12
ʹǤͳǤʹ ‡•‹‰’‡ –”ƒƒ†‡•‹‰”‹–‡”‹ƒ
The design of the lateral system was based on Japanese level II and level III
earthquake design spectra. These 2 levels of earthquakes are described in Section 3.1.2.
Because the specimen was designed to be “value-added” building (Kasai et al
2008), whose structural components and non-structural components are protected under
major earthquakes, the story drift angle of the frame is limited to ͲǤͲͲͷ”ƒ† under level II
earthquakes. The drift angle limit for conventional frames subjected to this level of
earthquake is ͲǤͲͳ”ƒ†. The structure was also required to remain elastic without damage
when drift angle is less than ͲǤͲͳ”ƒ† (Kasai et al 2008).
ʹǤͳǤ͵ ”ƒ‹‰›•–‡
The structure of the specimen was designed and detailed according to Japanese
code and style. The framing system is a three dimensional steel moment frame where
columns are engaged in flexure about both their principal axes. The columns were made
of ͵ͷͲ ൈ ͵ͷͲ hollowed-square-section (HSS) with thickness varying from story
to story. The beams are either rolled or built-up I-section. The primary beams, which are
connected to the columns, consist of a small-section segment at the middle and two largesection segments at the ends (Figure 2-3). These 3 segments are all ͶͲͲ height and
bolted together at the approximate inflection points determined from gravity loading.
Connections between columns and beams are all fully restrained moment connections
where both flanges and web of the beam are welded to the column. Generally, the
primary beams were haunched at the ends to improve their bending strength and beam-to-
13
Corrugated deck slab
Small section segment
Connection between
segments
Concrete protecting
shear studs
Figure 2-3
Stress haunch
Big section segment
Panel zone
stiffener
Beam, beam-to-column connection and slab
column connection strength. Continuity plates were also provided to protect the panel
zones (Figure 2-3).
To connect the specimen to the shake table and to provide the stiffness so that
structure can be considered to be fixed at the base, column bases and base girders were
designed with special details. The column bases were detailed as steel boxes with
dimension of ʹ ൈ ʹ ൈ ͲǤͻ (Figure 2-4). Vertical steel walls were installed inside
the boxes as stiffeners. The base girders, which are connected to the column base by
14
(b)
(a)
(c)
Figure 2-4
Column base
(a) View from top, (b) View from bottom, (c) Stiffeners
bolts, have the same height as column bases (ͲǤͻ). Horizontal braces provide in-plan
stiffness to the base system (Figure 2-5).
15
Figure 2-5
Horizontal braces at base level
ʹǤͳǤͶ Žƒ„•
Floor slabs are composite slabs formed from ͹ͷ height corrugated steel decks
covered by ͺͲ thick normal concrete. The corrugated steel decks (Figure 2-3) are
ͳǤʹthick and oriented parallel to the Y-direction. Reinforcement for the floor slabs is
typically a single layer of ԄͳͲ̷ͳͷͲ in both directions and placed at the mid
surface of slabs.
16
The roof slabs are ͳͷͲ normal concrete slabs casted on a ͳǤʹ flat steel
deck. Reinforcement for the roof slab includes two layers of Ԅͳ͵̷ʹͲͲ
reinforcement in each direction. Note that the roof slab was nearly twice as thick as the
floor slabs, as it was designed to carry additional weight simulating the weight from a
combination of roof mounted equipment (e.g. air conditioner system or water tanks) and
a penthouse. Such weight can increase the average load carried by the roof by a factor of
2 compared to a typical floor (Kasai, 2012).
Shear studs connecting the concrete slabs to primary beams are provided at all
slabs to provide composite effect to the beams. The shear studs were covered by concrete
for protection (Figure 2-3).
ʹǤͳǤͷ ƒ–‡”‹ƒŽ”‘’‡”–‹‡•
The nominal yield strengths of steel are ʹͻͷƒ and ͵ʹͷƒ for columns and
beams, respectively. Their expected ultimate strengths are ͶͲͲƒ for columns and
ͶͻͲƒ for beams. However, coupon tests showed that actual yield and ultimate
strengths of steel vary from member to member, and the average over strength factor
relative to nominal strengths is about ͳǤʹ (Kasai et al. 2010). Table 2-1 shows the range
of observed yield and ultimate strengths of steel used for the beams and columns.
The design compression strength of normal weight concrete used for slabs is
ʹͳƒ with the expected compression strength of standard samples in the compression
test is ʹͶƒ. The concrete slabs are reinforced by SD295A grade reinforcement. The
17
Table 2-1
Actual yield and ultimate strengths of steel
Member
࣌࢟ ሺࡹࡼࢇሻ
࢛࣌ ሺࡹࡼࢇሻ
Columns
346 – 398
430 – 470
Beams
331 – 422
510 – 557
Source: Kasai et al 2010
nominal yield stress for this steel grade is ʹͻͷƒ, as shown in its designation. The
actual strength of concrete and reinforcement from the coupon test are not available.
2.2
Non-Structural Components and Contents
Nonstructural components, including an integrated system of interior walls,
suspended ceilings, and sprinkler piping were installed at 4th and 5th stories, where the
maximum acceleration was expected to occur. At the time of this writing, the response of
these nonstructural components is under investigation. The comparison of their response
in the isolated building and fixed base building configurations will also be examined.
However, the nonstructural components’ response is out of the scope of this dissertation.
For investigating the response of loose contents in the isolated and fixed base
building configurations during a variety of types of earthquake excitation, furnishings
representing a hospital room at the 4th floor (Figure 2-6) and an office room at the 5th
floor (Figure 2-7) were installed in specially designed enclosed areas. Both rooms have
the size of ʹ ൈ Ͷ and were built on the top of the concrete weight blocks on the
floors (Figure 2-2 (a)). Contents in the hospital room included a patient bed on wheels, a
dresser containing medical equipments, medical cart, storage cart, IV poles, mobile lamp,
18
Figure 2-6
Figure 2-7
Hospital room
Office room
medical bottles and boxes. Many of these items are on wheels. The office room was
furnished with desks, chairs, computer system, bookcases and a photocopy machine.
2.3
Weights
In addition to the weight of structural components, nonstructural components and
contents, concrete and steel weights were installed to simulate a realistic live load.
Concrete weights, whose typical size and position on the floors are shown in Figure 2-2
19
(a), were built in as permanent part of structure at 2nd floor to 5th floor. The weights of air
conditioner system, water tank and all other technical systems were simulated by steel
weights at roof as shown in Figure 2-8. Each weight includes either 7 or 8 steel plates
whose size is ʹǤͳ ൈ ͶǤ͵ ൈ ͲǤͲʹͷ. The weight at the roof was altered from the original
configuration in the value added steel building project (Kasai et al., 2010), specifically,
weight was removed over a part of the roof to introduce additional eccentricity.
For designing the isolation system, modeling the structure and computing inertia
force from recorded acceleration, the weight of the specimen was estimated. The weights
5m
5m
of the specimen are tabulated in Table 2-2.
Steel
weight
(8 plates)
Steel
weight
(7 plates)
Steel
weight
(8 plates)
Steel
weight
(7 plates)
7m
Figure 2-8
5m
Location of steel weights at roof
20
Table 2-2
Estimated break down weights of the specimen
unit: 
Floor
Structural
Conc.
Weight
Steel weight
Nonstructural
Total
Roof
598.786
0
535.065
19.367
1153.218
5F
477.778
257.534
0
35.541
770.853
4F
496.534
267.958
0
16.174
780.666
3F
527.892
213.092
0
41.244
782.228
2F
527.072
175.598
0
89.597
792.267
0
0
48.352
842.352
Base
(*)
794.000
Sum w/ base
3422.062
914.182
535.065
250.275
5121.584
Sum w/o
base
2628.062
914.182
535.065
201.923
4279.232
(*) Before the test, the weight of structural component at base was estimated at ʹͷ͸.
This low value did not account the weight of column bases. The total weight of the
specimen correspondent to this value was Ͷͷͺͷ.
Total estimated weight of the specimen, about ͷͳʹʹ, is well below the
maximum capacity of the shake table, which is ͳʹͲͲͲ. The actual weight of the
specimen measured from the test was ͷʹʹͲ (see Section 4.5.2).
2.4
Condition of the Specimen before Testing
The specimen was built in 2008 and used in a test in March 2009 with several
types of dampers to provide enhanced performance (Kasai et al. 2010). The specimen had
been left outdoor at E-defense’s facility since then.
Several cracks in concrete slabs formed during the March 2009 test (Kasai et al.
2010). Some of these cracks are long and deep, as shown in Figure 2-9. Steel members of
the specimen had not been painted, and thus accumulated rust after long term exposure to
21
Figure 2-9
Cracks in concrete slab and rust on steel member
weather (Figure 2-9). These degradations may affect the mechanical properties of the
specimen, but may also represent the condition of buildings after years of operation.
Chapter 3
‡Ž‡ –‹‘‘ˆ ”‘—†‘–‹‘•ƒ†
‡•‹‰‘ˆ •‘Žƒ–‹‘›•–‡
3.1
Target Spectra
For designing the isolation system and selecting ground motions, representative
target spectra representing both the U.S. and Japan seismicity design practices were
developed.
͵ǤͳǤͳ Š‡ǤǤƒ”‰‡–’‡ –”ƒ
An objective of the design of the isolation system for this test is that the structural
system and contents of the isolated building will be damage free in a maximum
considered earthquake (MCE). To meet this objective, the effective period of the isolation
system is lengthened as much as possible. Current uniform hazard spectra (UHS) were
used to develop target spectra representative of frequent, design and MCE earthquakes.
These UHS only give spectral values at periods not longer than ʹ•. To consider the
spectral values at long periods, the UHS were used as the basis for developing smoothed
spectra that resemble code design spectra through the two-point approach. In this
approach, a spectrum is developed from its values at two periods: ܶ ൌ ͲǤʹ•(representing
short-period) and ܶ ൌ ͳǤͲ•. Figure 3-1 shows the typical shape and parameters of a
pseudo acceleration spectrum developed by this approach. The following procedure for
developing this spectrum was used:
Spectral response, ܵ௔
23
ܵ௔௦
ܵ௔ଵ
ͲǤͶܵ௔௦
ܶ଴ ͲǤʹ
ܶ௦
ͳǤͲ
ܶ௅
Period, ܶሺ‫ݏ‬ሻ
Figure 3-1
Acceleration spectrum developed by 2-point approach
Step 1: Spectral values ܵ௦ and ܵଵ at ܶ ൌ ͲǤʹ• and ܶ ൌ ͳǤͲ• were obtained. These
values were determined from seismic hazard analysis and are available through USGS
(USGS).
Step 2: The values of ܵ௦ and ܵଵ from step 1 are for site class B (rock site) and
were scaled to account for soil amplification:
ܵ௔௦ ൌ ܵ௦ ൈ ‫ܨ‬௔
(3.1-1)
ܵ௔ଵ ൌ ܵଵ ൈ ‫ܨ‬௩
(3.1-2)
where site amplification factors ‫ܨ‬௔ ǡ ‫ܨ‬௩ are functions of ܵଵ ǡ ܵ௦ ǡ ܶ and ܸ௦ଷ଴ – average shear
wave velocity of the top ͵Ͳ of soil layer at the site.
Step 3: The corner periods were computed:
ܶ଴ ൌ ͲǤʹ
ܵ௔ଵ
ܵ௔௦
(3.1-3)
24
ܶ௦ ൌ
ܵ௔ଵ
ܵ௔௦
(3.1-4)
Step 4: The long-period transition period ܶ௅ for the site was determined. This
period is a regional function and can be found in ASCE (2005).
Step 5: The spectral values at all periods were determined:
If ܶ ൏ ܶ଴ :
ܵ௔ ൌ ܵ௔௦ ൬ͲǤͶ ൅ ͲǤ͸
ܶ
൰
ܶ଴
(3.1-5)
If ܶ଴ ൑ ܶ ൑ ܶ௦ :
ܵ௔ ൌ ܵ௔௦
(3.1-6)
ܵ௔ଵ
ܶ
(3.1-7)
ܵ௔ଵ ܶ௅
ܶଶ
(3.1-8)
If ܶ௦ ൏ ܶ ൑ ܶ௅ :
ܵ௔ ൌ
If ܶ ൐ ܶ௅ :
ܵ௔ ൌ
To represent high seismicity, a Los Angeles (California) location with site class D
soil conditions (ܸ௦ଷ଴ ൌ ͳͺͲȀ• to ͵͸ͲȀ•) was selected. These site assumptions were
used in previous NEES-TIPS project studies (Erduran et al., 2010; Sayani and Ryan,
2009). ܵ௔௦ and ܵ௔ଵ values at ͷΨ damping ratio for different hazard levels at this site were
developed formerly and are shown in Table 3-1 (Erduran et al. 2010; Sayani and Ryan,
2009). ܶ௅ for this site is ͺǤͲ• (ASCE, 2005).
25
Table 3-1
ܵ௦ and ܵଵ for different hazard levels at ͷΨ damping ratio
MCE
(*)
DBE
(*)
Service
ܵ௔௦ (‰)
(**)
ʹǤʹͳͳ
ͳǤͳͺͶ
ͲǤͶ͹ͷ
ܵ௔ଵ (‰)
(**)
ͳǤͳͺ͹
ͲǤ͹ͳ͵
ͲǤ͵͵͸
(*)
(*)
MCE = maximum considered earthquake, ʹΨ probability of exceedance in
ͷͲ›‡ƒ”• (ʹȀͷͲ)
DBE = design basis earthquake, ͳͲΨ probability of exceedance in ͷͲ›‡ƒ”•
(ͳͲȀͷͲ)
Service level: ͷͲΨ probability of exceedance in ͷͲ›‡ƒ”• (ͷͲȀͷͲሻ
(**)
‰ = gravity acceleration
For this testing program, the site characteristics were used only for developing the
design spectra representing high seismicity. The selected input motions may not reflect
these site conditions. Rather, the selected motions reflect various site conditions, source
mechanisms and distance (Section 3.2).
Using the above procedure and spectral values of Table 3-1, target acceleration
spectra at different hazard levels at the site were developed as shown in Figure 3-2.
Design spectra at damping ratio other than ͷΨ were computed by dividing the
spectra at ͷΨ damping by a damping coefficient factor ‫ ܤ‬given in Table 3-2.
26
Spectral acceleration, Sa (g)
2.5
MCE
DBE
Service
2
1.5
1
0.5
0
0
2
4
6
Period, T (s)
Figure 3-2
The 5% damped U.S. acceleration spectra at the assumed site
Damping coefficient factor ‫ܤ‬
Table 3-2
Effective damping ratio (Ψ)
‫ܤ‬
൑ʹ
0.8
ͷ
ͳǤͲ
ͳͲ
ͳǤʹ
ʹͲ
ͳǤͷ
͵Ͳ
ͳǤ͹
ͶͲ
ͳǤͻ
൒ ͷͲ
ʹǤͲ
The damping coefficient factor shall be based on linear
interpolation for effective damping values other than those
given
Source: ASCE, 2005
͵ǤͳǤʹ ƒ’ƒƒ”‰‡–’‡ –”ƒ
According to Pan et al. (2005), design of an isolated building should satisfy the
performance requirements for three earthquake levels:
27
o Level 1 (L-1). At this service level, the building should be fully functional
and the superstructure should behave elastically. The drift limit for this
level is less than ͳȀʹͲͲ. The design spectrum for this level is obtained by
dividing the design spectrum for a level 2 earthquake by a factor of ͷǤͲ.
o Level 2 (L-2). This level represents rare major earthquakes with a return
period of about 500 years. At this earthquake level, yielding and plastic
hinges are allowed at a few locations, but the fully plastic mechanism
must be prevented. The design spectrum for this earthquake level is
presented next.
o Level 3 (L-3 or L-2+). This additional level is used for checking the safety
margin or collapse. This level is sometimes defined as a ͷͲΨ increase in
intensity over a level 2 motion.
The acceleration design spectrum at ͷΨ damping ratio for Level 2 earthquake is
defined as:
ܵ௔ ሺܶሻ ൌ ܼ ൈ ‫ܩ‬௦ ሺܶሻ ൈ ܵ଴ ሺܶሻ
where:
ܶ
= natural period of structure
ܵ଴ ሺܶሻ = default bedrock spectrum at ͷΨ damping ratio
‫ܩ‬௦ ሺܶሻ = surface soil layer amplification factor
ܼ
= seismic zone factor
The bedrock acceleration spectrum at ͷΨ damping ratio is defined as:
(3.1-9)
28
͵Ǥʹ ൅ ͵Ͳܶ
ͺǤͲ
ܵ଴ ሺܶሻ ൌ ൞ ͷǤͳʹ
ܶ
ˆ‘”
ܶ ൏ ͲǤͳ͸‫ݏ‬
ˆ‘” ͲǤͳ͸‫ ݏ‬൑ ܶ ൑ ͲǤ͸Ͷ‫ݏ‬
ˆ‘”
ܶ ൒ ͲǤ͸Ͷ‫ݏ‬
ൢሺ݉Ȁ‫ ݏ‬ଶ ሻ
(3.1-10)
Figure 3-3 shows the bedrock acceleration response spectrum at ͷΨ damping
ratio.
Local soil amplification factor ‫ܩ‬௦ ሺܶሻ is a function of soil type and period. The
expression of ‫ܩ‬௦ ሺܶሻ is defined as follows (Otani et al., 2002):
‫ܩ‬௦ ሺܶሻ ൌ
ܶ
‫ܩ‬௦ଶ
ˆ‘”
‫ۓ‬
ͲǤͺܶ
ଶ
ۖ
ۖ
‫ ܩ‬െ ‫ܩ‬௦ଶ
ۖ‫ܩ‬௦ଶ ൅ ௦ଵ
ሺܶ െ ͲǤͺܶଶ ሻ ˆ‘”
ۖ
ͲǤͺሺܶଵ െ ܶଶ ሻ
‫۔‬
‫ܩ‬௦ଵ
ۖ
ۖ
ۖ ‫ ܩ‬൅ ‫ܩ‬௦ଵ െ ͳǤͲ ൬ͳ െ ͳ ൰
ۖ ௦ଵ
ͳ
ܶ ͳǤʹܶଵ
‫ە‬
ͳǤʹܶଵ െ ͲǤͳ
ܶ ൑ ͲǤͺܶଶ
ͲǤͺܶଶ ൏ ܶ ൑ ͲǤͺܶଵ
ˆ‘”
ͲǤͺܶଵ ൏ ܶ ൑ ͳǤʹܶଵ
ˆ‘”
ͳǤʹܶଵ ൏ ܶ
4
6
Spectral acceleration, S0 (g)
1
0.8
0.6
0.4
0.2
0
0
2
Period, T (s)
Figure 3-3
Bedrock 5% damped acceleration spectrum for Japan code
(3.1-11)
29
where:
ܶଵ = predominant period of surface soil layers for the first mode
ܶଶ = predominant period of surface soil layers for the second mode
‫ܩ‬௦ଵ = surface soil layer amplification factor at ܶଵ
‫ܩ‬௦ଶ = surface soil layer amplification factor at ܶଶ
If detailed analysis is not available, the following expressions of ‫ܩ‬௦ ሺܶሻ can be
used:
•
For soil type I (including rock, stiff sand gravel and pre-Tertiary deposits):
‫ܩ‬௦ ൌ
•
ͳǤͷ
‫ۓ‬
ۖͲǤͺ͸Ͷ
‫ܶ ۔‬
ۖ
‫ͳ ە‬Ǥ͵ͷ
ˆ‘”
ܶ ൏ ͲǤͷ͹͸‫ݏ‬
ˆ‘” ͲǤͷ͹͸‫ ݏ‬൑ ܶ ൏ ͲǤ͸Ͷ‫ݏ‬
ˆ‘”
(3.1-12)
ͲǤ͸Ͷ‫ ݏ‬൑ ܶ
For soil type III (alluvium layer mainly consisting of humus and mud
whose depth is greater than ͵Ͳ, or filled land less than 30 years old
whose depth exceeds ͵) and soil type II (anything other than soil type I
and soil type III):
‫ۓ‬
ۖ
ͳǤͷ
ˆ‘”
ܶ
‫ܩ‬௦ ൌ ͳǤͷ ͲǤ͸Ͷ ˆ‘”
‫۔‬
ۖ
ˆ‘”
‫݃ ە‬௩
ܶ ൏ ͲǤ͸Ͷ‫ݏ‬
ͲǤ͸Ͷ‫ ݏ‬൑ ܶ ൏ ͲǤ͸Ͷ
ͲǤ͸Ͷ
݃௩
‫ݏ‬
ͳǤͷ
݃௩
‫ ݏ‬൑ ܶ
ͳǤͷ
݃௩ ൌ ʹǤͲ͵ for soil type II and ݃௩ ൌ ʹǤ͹Ͳ for soil type III.
(3.1-13)
30
Local soil amplification factor ‫ܩ‬௦ ሺܶሻ for the three soil type determined from
Equations (3.1-12) and (3.1-13) are shown in Figure 3-4. Soil amplification factors at
short periods are the same for all soil types. At long periods, softer soils have larger
amplification factors as expected.
The seismic zone factor ܼ is a regional seismicity factor varying from ͲǤ͹ to ͳǤͲ
as shown in Figure 3-5. The majority of the country including the major urban areas of
Tokyo and Osaka fall under the largest seismicity factor ܼ ൌ ͳǤͲ.
Acceleration design spectra at ͷΨ damping ratio for Japan are shown in Figure 36. In these spectra, the zone factor ܼ ൌ ͳǤͲ and soil type II were assumed.
Design spectra at any damping ratio are computed by multiplying the ͷΨ damped
spectrum by a factor ‫ܨ‬௛ :
‫ܨ‬௛ ൌ
ͳǤͷ
ͳǤͲ ൅ ͳͲ݄௘௤
(3.1-14)
where ݄௘௤ is equivalent damping ratio.
Soil amplification factor, G
s
3
Soil type I
Soil type II
Soil type III
2.5
2
1.5
1
0
2
4
6
Period, T (s)
Figure 3-4
Soil amplification factor for Japan code
31
Source: http://iisee.kenken.go.jp/net/seismic_design_code/japan/fig-japan1.htm
Figure 3-5
Zone factor for developing Japan design spectra
Design acceleration spectra for both the U.S. code and Japan code at ͷΨ and
ʹͲΨ damping are all shown in Figure 3-7 for comparison. Prior to design, the effective
damping ratio of the isolation system is unknown, but ʹͲΨ was taken as an estimate.
Because the Japan spectrum is subjected to greater long period amplification due to soil
effects, the spectrum of Japan L-2 at ͷΨ damping (Figure 3-7) is close to that of the U.S
32
Spectral acceleration, Sa (g)
1.5
L-2
L-1
1
0.5
0
0
2
4
6
Period, T (s)
Figure 3-6
5% damped design acceleration spectra for Japan code
2.5
2
Spectral acceleration, Sa (g)
a
Spectral acceleration, S (g)
2.5
The U.S. MCE
Japan L-2
1.5
The U.S. DBE
The U.S. Service
Japan L-1
1
0.5
0
0
2
4
Period, T (s)
(a)
Figure 3-7
6
2
1.5
The U.S. MCE
The U.S. DBE
Japan L-2
The U.S. Service
Japan L-1
1
0.5
0
0
2
4
6
Period, T (s)
(b)
Design acceleration spectra for the U.S code and Japan code
(a) 5% damping, (b) 20% damping
DBE at short periods, but at long periods the spectrum of Japan L-2 is close to the
spectrum of the U.S. MCE. However, the two codes use different factor to accounting for
damping so that at ʹͲΨ damping the Japan L-2 spectrum is close to the U.S DBE
spectrum at long periods, while the Japan L-2 spectrum is significantly smaller than the
U.S DBE spectrum at short periods. In both cases, the U.S Service spectrum is much
33
larger than the Japan L-1 spectrum. These comments are valid for the assumed location
and site class.
It should be noted that the U.S code requires isolation system being designed for
MCE event and Japanese code requires that an isolation system shall be designed base on
L-2 earthquakes.
3.2
Selection and Scaling Ground Motions Representing the U.S Code
In the U.S, current practice for selecting ground motions requires that appropriate
ground motions shall be selected from events having parameters consistent with those
that control the MCE event (ASCE, 2005). In case there are not enough recorded motions
satisfying this requirement, simulated motions shall be used.
For purposes of this experiment, the selected ground motions may not reflect the
seismicity and soil conditions of the assumed site. Instead, motions were intentionally
selected that reflect different source mechanisms, site conditions and distances. However,
these motions were scaled to match the target spectrum at the site, which represents high
seismicity.
To select motions representing MCE event, recorded motions with high
acceleration (peak horizontal acceleration larger than ͲǤͷ‰), and rich in long period
content (to challenge the isolation system) were selected. These motions were then scaled
so that their acceleration response spectra match the target spectrum in an approximate
way. The method of scaling is discussed below. Motions whose scale factors are close to
1 were given preference for selection. For evaluating the effect of vertical excitation on
34
the response of the system, preference was also given to motions with large vertical
acceleration. Our philosophy was to use real, strong, un-scaled, long period motions to
represent the MCE event.
Motions representing design and service levels were selected from recorded
motions having parameters (including magnitude, fault distance and source mechanism)
consistent with the assumed site. Scaling factors were assigned to the selected motions so
that their acceleration response spectra match the target spectra at the site.
Several methods of scaling ground motions have been proposed. ASCE (2005)
requires that selected motions be scaled such that the square root of sum squares (SRSS)
of the components, averaged over all records, does not fall below ͳǤ͵ times the design
spectrum by more than ͳͲΨ in the period range from ͲǤͷܶ஽ to ͳǤʹͷܶெ , where ܶ஽ and
ܶெ are effective periods of the isolation system corresponding to isolator displacement at
the DBE and MCE. Somerville et al. (1997) used a single scaling factor that minimized
the weighted sum of square difference between the target spectra and the average spectra
of the two horizontal components at selected periods to scale all three components of the
selected motion. The weights were ͲǤͳ, ͲǤ͵, ͲǤ͵ and ͲǤ͵ for periods at ͲǤ͵•, ͳǤͲ•, ʹǤͲ•
and ͶǤͲ•. Shome et al. (1998) proposed to scale the motion so that its response spectrum
matches the target spectrum at the first mode period of the building. This method was
used for predicting the median response of structures using nonlinear analysis.
In this study, the three components of each record were scaled by a common
factor that minimized the least square error between ͳǤ͵ times ͷΨ damped target
spectrum and the SRSS of the ͷΨ spectra of the two horizontal components from
35
ܶଵ ൌ ͲǤͷܶ௘௙௙ to ܶଶ ൌ ͳǤʹͷܶ௘௙௙ , where ܶ௘௙௙ is the effective period of the isolation
system. Because ܶ௘௙௙ was not known prior to design of the isolation system, these scale
factors were determined iteratively. As shown in Section 3.5.4, ܶ௘௙௙ ൌ ͶǤʹ͵•, ͵Ǥ͵͹• and
ʹǤͳͺ• for MCE, DBE and Service level events, respectively.
The scaling factor can be found as follows.
Let:
ͳǤ͵ܵ௔ = ͳǤ͵ times target spectrum
ܵௌோௌௌ = square root of sum squares of spectra of the two horizontal
components
‫ = ܨ‬scaling factor
The following function is to be minimized:
்మ
݂ሺ‫ܨ‬ሻ ൌ න ሺ‫ ܨ‬ൈ ܵௌோௌௌ െ ͳǤ͵ܵ௔ ሻଶ ݀ܶ
(3.2-1)
்భ
To minimize, evaluate the derivative of ݂ሺ‫ܨ‬ሻ with respect to ‫ ܨ‬and set equal to
zero:
்మ
݂݀ሺ‫ܨ‬ሻ
ൌ ʹ න ܵௌோௌௌ ሺ‫ ܨ‬ൈ ܵௌோௌௌ െ ͳǤ͵ܵ௔ ሻ݀ܶ ൌ Ͳ
݀‫ܨ‬
்భ
்మ
்మ
݂݀ሺ‫ܨ‬ሻ
ଶ
ൌ ʹ‫ ܨ‬න ܵௌோௌௌ
݀ܶ െ ʹ න ܵௌோௌௌ ͳǤ͵ܵ௔ ݀ܶ ൌ Ͳ
݀‫ܨ‬
்భ
்భ
(3.2-2)
்
‫ܨ‬ൌ
మ
‫ܵ ்׬‬ௌோௌௌ ͳǤ͵ܵ௔ ݀ܶ
భ
்
మ ଶ
݀ܶ
‫ܵ ்׬‬ௌோௌௌ
భ
Integrals in Equation (3.2-3) were evaluated numerically.
(3.2-3)
36
Based on the criteria and scaling method described above, 6 motions representing
different earthquake levels were selected. The basic parameters of these motions are
summarized in Tables 3-3 and 3-4. Figures 3-8 to 3-10 plot the SRSS spectral
acceleration of the scaled bidirectional components of motions against target spectra of
different earthquake levels scaled by 1.3. Based on the scaling procedure, the area of the
SRSS scaled spectrum above the target spectrum does not necessarily equal the area
below the target spectrum.
Table 3-3
Selected records for the U.S code
Event
Earthquake
Date
M
Service
Imperial
Valley
1979/15/10
6.53
DBE
Northridge
1994/01/17
6.69
Loma
Prieta
1989/10/18
6.93
Northridge
1994/01/17
6.69
Tabas
1978/09/16
7.35
Chi-Chi
1999/09/20
7.62
MCE
Station
CDMG 11369
Westmorland
Fire Sta (WSM)
DWP 77 Rinaldi
Receiving Sta.
(RRS)
UCSC 16
LGPC
(LGP)
DWP 74 Sylmar
- Hospital Sta.
(SYL)
9101 Tabas
(TAB)
CWB 99999
TCU065
(TCU)
Hypocenter
distance (km)
Vs30
(m/s)
53.71
193.70
20.62
282.30
25.42
477.70
24.24
440.50
55.54
766.80
27.85
305.90
Notes: Records will be referred to by the abbreviations in bold
hereafter.
37
Table 3-4
Motion
Imperial
Valley WSM
Northridge RRS
Loma Prieta
- LGP
Northridge SYL
Tabas TAB
Chi-Chi TCU
(+)
Parameters of the selected records for the U.S code
Component
090
180
UP
228
318
UP
000
090
UP
090
360
UP
LN
TR
UP
E
N
V
Duration
ሺ‫ݏ‬ሻ
40
40
40
19.91
19.91
19.91
25.005
25.005
25.005
40
40
40
32.84
32.84
32.84
90
90
90
(+)
(+)
Peak A
ሺܿ݉Ȁ‫ ݏ‬ଶ ሻ
168.57
206.85
244.11
809.52
477.26
818.47
947.92
575.99
869.19
593.01
827.28
525.28
819.93
835.58
675.42
798.35
591.33
267.28
Peak V
ሺܿ݉Ȁ‫ݏ‬ሻ
23.47
31.01
8.70
160.13
74.51
43.53
108.55
47.04
68.78
78.11
129.37
18.82
97.72
121.23
44.42
126.22
78.82
77.07
(+)
Peak D
ሺܿ݉ሻ
13.05
20.26
4.18
29.67
26.92
10.05
65.75
24.46
65.01
16.98
31.79
9.49
39.06
95.16
16.52
92.62
60.77
53.72
Scale
Factor
0.80
0.80
0.80
0.88
0.88
0.88
(*)
1.09
(*)
1.09
(*)
1.09
(*)
1.22
(*)
1.22
(*)
1.22
(*)
1.03
(*)
1.03
(*)
1.03
(*)
0.89
(*)
0.89
(*)
0.89
The peak values here are of the un-scaled motions.
(*)
The scale factors shown here were developed to match the target response spectra.
The actual factors applied in the test differed from these due to external considerations.
See schedule of tests in Section 4.3 for the actual factors applied in the test program.
0.4
0.2
0
0
Figure 3-8
Teff
0.6
1.3 × Service
0.80 × WSM - SRSS
0.80 × WSM - X
0.80 × WSM - Y
1.25 T eff
0.8
0.5 T eff
Spectral acceleration, Sa (g)
1
1
2
Period, T (s)
3
4
ͷΨ damped spectra of the two horizontal components of
the scaled motion representing Service event
38
2
1.5
1
0.5
0
0
Figure 3-9
1
2
3
Period, T (s)
1.25 T eff
1
2
4
6
0
0
2
4
6
5
2
1
1.25 T eff
3
0.5 T eff
0.5 T eff
3
1.3 × MCE
0.89 × TCU- SRSS
0.89 × TCU - X
0.89 × TCU - Y
4
1.25 T eff
4
Teff
1.3 × MCE
1.03 × TAB- SRSS
1.03 × TAB - X
1.03 × TAB - Y
Teff
5
Spectral acceleration, Sa (g)
Teff
2
0.5 T eff
3
1
0
0
1.3 × MCE
1.22 × SYL- SRSS
1.22 × SYL - X
1.22 × SYL - Y
4
1.25 T eff
Teff
0.5 T eff
Spectral acceleration, Sa (g)
3
2
5
5
1.3 × MCE
1.09 × LGP- SRSS
1.09 × LGP - X
1.09 × LGP - Y
4
0
0
4
ͷΨ damped spectra of the two horizontal components of
the scaled motion representing DBE event
5
2
Teff
0.5 T eff
Spectral acceleration, Sa (g)
2.5
1.25 T eff
1.3 × DBE
0.88 × RRS - SRSS
0.88 × RRS - X
0.88 × RRS - Y
3
1
2
4
Period, T (s)
6
0
0
2
4
Period, T (s)
Figure 3-10
ͷΨ damped spectra of the two horizontal components
of the scaled motions representing MCE event
6
39
3.3
Selection and Scaling of Ground Motions Representing Japan Code
According to the Japanese code, all isolated structures are analyzed and designed
for 3 standard ground motions in analysis. These motions are: El Centro 1940, Taft 1952
and Hachinohe 1968. For Japanese design, these single component motions are scaled
such that their peak velocity equals ͲǤͷȀ•. In addition to this set of standard motions, 3
other motions must be included to account for local site effects (Pan et al., 2005).
In this experimental program, only one of the three standard motions, the El
Centro record of the Imperial Valley 1940 earthquake, was included. To represent local
site effects, two motions recorded during the 1995 Kobe earthquake were selected (JMA
99999 KJMA station and CUE 99999 Takatori station). The KJMA record is commonly
considered for in the practical design in Japan, and the Takatori record is a strong near
fault record.
The parameters of the three Japan motions are shown in Tables 3-5 and table 3-6.
Their response spectra are compared to the ͷΨ damped Japanese design spectrum in
Figure 3-11. Observe that the response spectrum of the scaled El Centro motion is much
lower than the L2 spectrum at periods longer than ͳ•. For the Takatori motion, high
spectral acceleration spans a wide range of periods and the SRSS spectrum of the
Takatori motion is always larger than the L2 spectrum for periods less than Ͷ•. The
SRSS of KJMA motion is slightly smaller than L2 spectrum at periods ranging from ʹ•
to ͵Ǥͷ•, but larger than the L2 spectrum for periods less than ʹ•. The expected effective
period at the peak displacement was ͵Ǥʹ• for the KJMA motion. This peak displacement
was estimated from the pre-test analysis (Section 3.6).
40
Table 3-5
Selected records for Japan code
Earthquake
Date
M
Imperial
Valley
1940/05/19
6.95
Kobe
1995/01/16
6.90
Kobe
1995/01/16
6.90
Station
Hypocenter
distance (km)
Vs30
(m/s)
15.69
213.40
25.58
312.00
22.19
256.00
USGS 117 El
Centro Array
#9 (ELC)
JMA 99999
KJMA (KJM)
CUE 99999
Takatori
(TAK)
Notes: Motions will be referred to by the abbreviations in bold hereafter.
Table 3-6
Motion
Imperial
Valley ELC
Kobe - KJM
Kobe - TAK
Component
180
270
UP
000
090
UP
000
090
UP
Parameters of the selected records for Japan code
Duration
ሺ‫ܿ݁ݏ‬ሻ
40
40
40
48
48
48
40.96
40.96
40.96
Peak A
ሺ݈݃ܽሻ
306.94
210.68
201.28
805.72
587.14
336.25
599.79
603.82
266.46
Peak V
ሺܿ݉Ȁ‫ݏ‬ሻ
29.69
29.63
10.55
81.30
74.35
38.31
127.19
120.73
16.02
Peak D
ሺܿ݉ሻ
12.98
21.93
8.41
17.71
19.93
10.29
35.78
32.74
4.47
Scale
Factor
1.30
1.30
1.30
1.00
1.00
1.00
1.00
1.00
1.00
41
5
Spectral acceleration, Sa (g)
Spectral acceleration, Sa (g)
5
L2
1.3×ELC - SRSS
1.3×ELC - X
1.3×ELC - Y
4
3
2
1
0
0
2
4
6
L2
TAK - SRSS
TAK - X
TAK - Y
4
3
2
1
0
0
2
4
6
Period, T (s)
Period, T (s)
a
Spectral acceleration, S (g)
5
L2
KJM - SRSS
KJM - X
KJM - Y
4
3
2
1
0
0
2
4
6
Period, T (s)
Figure 3-11
3.4
ͷΨ damped spectra of selected motions representing Japan code
Long Duration, Long Period, Subduction Motions
In addition to the selected motions described in the previous sections, 3 long-
duration, long-period, subduction motions were selected for evaluating friction isolation
system. These selected motions include: (1) the motion recorded at Communication
Center station (SCT) during September 19, 1985 Mexico city earthquake, (2) the motion
recorded at Iwanuma station during March 11, 2011 Tohoku earthquake, and (3) the
simulated Sannomaru motion. Only the horizontal components of these motions were
used for the test program. The accelerograms of these components are shown in Figure 3-
42
12, and their basic parameters are given in Table 3-7. Their response spectra are shown in
Acceleration, a (g)
Figure 3-13.
Time, t (s)
Figure 3-12
Accelerogram of the selected long duration, long period, subduction motions
43
Table 3-7
Motion
Mexico SCT
Tohoku IWA
Simulated –
SAN
Figure 3-13
Parameters of the selected long duration, long period, subduction motions
Component
EW
NS
EW
NS
EW
NS
Duration
ሺ‫ܿ݁ݏ‬ሻ
80.02
80.02
300
300
327.68
327.68
Peak A
ሺ݈݃ܽሻ
168.00
98.90
353.23
410.69
185.87
165.83
Peak V
ሺܿ݉Ȁ‫ݏ‬ሻ
61.10
31.70
51.16
78.00
49.44
51.52
Peak D
ሺܿ݉ሻ
21.50
13.90
30.14
33.26
23.00
16.50
Scale
Factor
1.00
1.00
1.00
1.00
1.00
1.00
5% damped response spectra of the selected long duration, long period,
subduction motions
The SCT motion is known to be rich in long period components due to the soft
soil conditions at the site. The shear wave velocity (ܸ௦ଷ଴ ) at the site is around ͹ͲȀ•
44
(Stone et al., 1987). The recorded motion has a dominant frequency component at a
period at around ʹ• (Figure 3-13).
The Iwanuma motion recorded from the devastating Tohoku earthquake is a long
motion with bracket duration (duration between the first and last excecdances of ͲǤͲͷ‰
acceleration) of ͳͶͻ•. This motion is also rich in lower frequency content with a peak at
around ͳǤ͹• in response spectra (Figure 3-13).
The synthesized Sannomaru motion is an extremely long motion with total record
time exceeding ͵ʹͲ•. The bracket duration is ͳ͵͸•. The spectral acceleration is mild at
short periods, but significant over a wide range with a peak at around ͵ǤͲ•. This motion
was also used in a full scale test of a 4-story reinforced concrete hospital building at EDefense (Sato et al., 2011).
3.5
Design of Isolation System
͵ǤͷǤͳ –”‘†— –‹‘
The isolation system designed for the test included 9 identical TPBs. Each bearing
was installed beneath a column base as shown in Figure 3-14. At the beginning of the
Figure 3-14
Isolation system tested
45
test, the maximum and minimum static vertical loads on the bearing were ͺ͸ͺ and
ͶͷͲ while the average static load on bearings was ͷͻͷȀ„‡ƒ”‹‰, which is close to
the assumed value for the design of the bearing (͸ͲͲ).
The general composition of a TPB is presented in Figure 3-15. The bearing
consists of an inner slider that can slide between two articulated sliders with spherical
surfaces, which in turn can slide between top and bottom concave plates. Friction
between surfaces resists small horizontal loads without movement and dissipates energy
during sliding between surfaces that occurs under large horizontal load. The curvature of
the surfaces combined with vertical load provides a restoring force when horizontal
displacement occurs.
TPBs produced by Earthquake Protection System Incorporation (EPS Inc.) have
geometrical parameters and friction coefficients designed such that the bearings can
provide different pendulum mechanisms under different earthquake levels. A first
pendulum mechanism is formed by the inner slider and the two articulated sliders. The
spherical radius ܴଵ of the inner slider is small, leading to relatively large stiffness, and
the friction coefficient ߤଵ between these surfaces is low so that sliding occurs under small
earthquakes. A second pendulum mechanism is formed by the lower articulated slider
and bottom concave plate. The friction coefficient ߤଶ between these surfaces is selected
to engage sliding under moderate earthquakes. This pendulum mechanism has small
stiffness since the spherical radius ܴଶ of the concave plate is large. A third pendulum
mechanism is formed by the upper articulated slider and top concave plate. The friction
coefficient ߤଷ between these surfaces is high so that the sliding only occurs during large
46
(a)
݀ଷ
݀ଵ
Inner
slider
Articulated
ߤଷ
slider
݄ଷ
݄ଶ
ܴଷ
ʹ݄ଵ
ܴଶ
݀ଶ
Convex
plate
ܴଵ
Concave
plate
Articulated
slider
ߤଵ
ߤଶ
(b)
Figure 3-15
Triple friction pendulum bearing
(a) 3D view
(b) Section view and basic parameters
earthquakes. The displacement limit of the lower articulated slider is large enough so that
both articulated sliders can slide in the third pendulum mechanism, which provides very
small horizontal stiffness to limit acceleration response in superstructure. The friction
47
coefficients ߤ௜ between surfaces and displacement limits ݀௜ of the sliders can be selected
such that the bearing can provide stiffening stages to reduce the displacement demand of
isolation system or slow the movement of superstructure prior to reaching the overall
displacement limit in an extreme earthquake. In summary, the response of a TPB is
determined by the following design parameters: friction coefficients between surfaces ߤ௜ ,
displacement limits ݀௜ and effective pendulum lengths ‫ܮ‬௜ ൌ ܴ௜ െ ݄௜ ..
Relative to single and double pendulum bearings, TPBs provide a larger
displacement capacity without increasing the size of the bearings, thus providing a more
cost effective design. Second, they can produce multi-stage behavior that can be utilized
to optimize the performance at different earthquake levels. The theoretical behavior of
triple friction pendulum was described in detail by Fenz and Constantinou (2008(1)) and
Morgan and Mahin (2011). The next section summarizes theoretical rate-independent
unidirectional multi-stage behavior of TPBs as developed by these authors. The
theoretical multi-stage behavior forms the basis of the design.
͵ǤͷǤʹ Š‡‘”‡–‹ ƒŽ‹†‹”‡ –‹‘ƒŽ—Ž–‹Ǧ–ƒ‰‡‡Šƒ˜‹‘”‘ˆ”‹’Ž‡ ”‹ –‹‘
‡†—Ž—‡ƒ”‹‰•
Consider a bearing with characteristic dimensions and friction coefficients shown
in Figure 3-15. Let the effective radii of spherical surfaces be:
ଵ ൌ ଵ െ Šଵ Ǣଶ ൌ ଶ െ Šଶ Ǣ ଷ ൌ ଷ െ Šଷ Ǣ
(3.5-1)
The unidirectional backbone curve for horizontal force-deformation of a
generalized TPB can be divided into 5 stages as shown in Figure 3-16. Note that in this
48
݂ൌ
‫ܨ‬
ܹ
݂଺‫כ‬
݂ହ‫כ‬
ͳ
݂ସ‫כ‬
݂ଷ‫ כ‬ൌ ߤଷ
݂ଶ‫ כ‬ൌ ߤଶ
݂ଵ‫ כ‬ൌ ߤଵ
ͳ
ͳ
݇ଵ ൌ
ʹ‫ܮ‬ଵ
ͳ
‫ݑ‬ଶ‫כ‬
Figure 3-16
݇ଶ ൌ
ͳ
‫ܮ‬ଵ ൅ ‫ܮ‬ଶ
ͳ
݇ଷ ൌ ͳ
‫ܮ‬ଶ ൅ ‫ܮ‬ଷ
‫ݑ‬ସ‫כ‬
‫ݑ‬ଷ‫כ‬
ͳ
݇ସ ൌ
݇ହ ൌ
ͳ
ʹ‫ܮ‬ଵ
ͳ
‫ܮ‬ଵ ൅ ‫ܮ‬ଷ
‫ݑ‬ହ‫כ‬
‫ כ଺ݑ‬ൌ ‫ݑ‬௟௜௠௜௧
‫ݑ‬
Normalized backbone curve of a standard triple pendulum bearing
figure, the normalized horizontal force ݂ ൌ ‫ܨ‬Ȁܹ is used and the curve is called
normalized backbone curve. The bearing dimensions and friction coefficients must
satisfy the following:
‫ܮ‬ଵ ൏ ‫ܮ‬ଶ ൌ ‫ܮ‬ଷ
(3.5-2a)
ߤଵ ൏ ߤଶ ൏ ߤଷ
(3.5-2b)
݀ଵ ൐ ሺߤଷ െ ߤଵ ሻ‫ܮ‬ଵ
(3.5-2c)
ߤସ ൏
݀ଵ
൅ ߤଵ
‫ܮ‬ଵ
(3.5-2d)
The constraint on the pendulum lengths (Equation 3.5-2a) must be satisfied so
that the bearing is activated with high stiffness at small earthquakes and lower stiffness at
49
moderate earthquakes. Constraint on the friction coefficients (Equation 3.5-2b)
guarantees the sequence of sliding, engaging in this order: inner slider, lower articulated
slider, and upper articulated slider. Constraint of Equation (3.5-2c) guarantees that the
displacement limits are reached on articulated sliders before the inner slider to activate
the stiffening stage at the end of the backbone curve. The constraint of Equation (3.5-2d)
guarantees that sliding occurs on all surfaces before the bearing goes into a stiffening
stage.
Five stages of sliding are shown in Figure 3-17. Explanations of these stages are
given next.
i)
Stage 1: The inner slider slides between the two articulated sliders.
When the horizontal force exceeds the friction force between the inner slider and
the lower articulated slider, equal to the friction force between the inner slider and upper
articulated slider, the inner slider will slide between the two articulated sliders.
The normalized stiffness of the backbone curve in this stage is:
݇ଵ ൌ
ͳ
ʹ‫ܮ‬ଵ
(3.5-3)
The maximum displacement and normalized force of this stage are:
‫ݑ‬ଶ‫ כ‬ൌ ʹ‫ܮ‬ଵ ሺߤଶ െ ߤଵ ሻ
(3.5-4)
݂ଶ‫ כ‬ൌ ߤଶ
(3.5-5)
Finally, the hysteresis loop of stage 1 shown in Figure 3-18.
50
Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Note: The surfaces marked with “x” are surfaces where sliding happens
Figure 3-17
Five stages of sliding
݂
ʹ݂ଵ‫כ‬
݂ଵ‫כ‬
݇ଵ
‫ݑ‬
Figure 3-18
Hysteresis loop of stage 1
51
ii)
Stage 2: The inner slider slides on the upper articulated slider and the lower
articulated slider slides on the bottom concave plate.
When the normalized horizontal force ݂ exceeds ݂ଶ‫ כ‬ൌ ߤଶ , relative sliding will
occur between the articulated slider and bottom concave plate. Sliding on the contact
surface between the inner slider and the upper articulated slider also continues sliding
between the upper slider and the convex plate has not commenced. The inner slider must
slide to increase the inclination to balance the increase of horizontal force. The inner
slider stops sliding on the lower articulated slider, because the increasing horizontal force
is accommodated by the inclination of the inner slider and lower articulated slider on the
bottom concave plate rather than the sliding of the inner slider on the articulated slider. In
other words, the increasing horizontal force in this stage is provided from the increased
inclination of the inner slider. On the top part of the inner slider, this inclination comes
from the sliding between the inner slider and the upper articulated slider. On the bottom
part of the inner slider, the inclination is provided from the sliding between the lower
articulated slider and the bottom concave plate.
The normalized stiffness of the backbone curve in this stage is:
݇ଶ ൌ
ͳ
‫ܮ‬ଵ ൅ ‫ܮ‬ଶ
(3.5-6)
The maximum displacement and normalized force of this stage is:
‫ݑ‬ଷ‫ כ‬ൌ ‫ܮ‬ଵ ሺߤଶ ൅ ߤଷ െ ʹߤଵ ሻ ൅ ‫ܮ‬ଶ ሺߤଷ െ ߤଶ ሻ
(3.5-7)
52
݂ଷ‫ כ‬ൌ ߤଷ
(3.5-8)
The hysteresis loop of stage 2 is shown in Figure 3-19.
݂
ʹ݂ଵ‫כ‬
݇ଶ
݂ଶ‫כ‬
݂ଵ‫כ‬
݇ଵ
ʹ݂ଶ‫כ‬
‫ݑ‬
Figure 3-19
iii)
Hysteresis loop of stage 2
Stage 3: The lower articulated slider slides on the bottom concave plate and the
upper articulated slider slides on the top concave plate.
When the horizontal force exceeds the friction force between the upper articulated
slider and the top concave plate, sliding between the components occurs. At this stage,
the lower articulated slider continues sliding on the bottom concave plate, but the sliding
between the inner slider and the upper articulated slider stops. The increasing horizontal
force is provided by the increased inclination of the inner slider caused by the sliding of
the two articulated sliders on the two concave plates at the top and the bottom.
The normalized stiffness of the backbone curve in this stage is:
݇ଷ ൌ
ͳ
‫ܮ‬ଶ ൅ ‫ܮ‬ଷ
(3.5-9)
53
The maximum displacement and normalized force of this stage is:
݀ଶ
‫ݑ‬ସ‫ כ‬ൌ ‫ݑ‬ଷ‫ כ‬൅ ൬ ൅ ߤଶ െ ߤଷ ൰ ሺ‫ܮ‬ଶ ൅ ‫ܮ‬ଷ ሻ
‫ܮ‬ଶ
(3.5-10)
݀ଶ
‫ܮ‬ଶ
݂ସ‫ כ‬ൌ ߤଶ ൅
(3.5-11)
The hysteresis loop of stage 3 is shown in Figure 3-20.
݂
݂ଷ‫כ‬
݂ଶ‫כ‬
݂ଵ‫כ‬
ʹ݂ଵ‫כ‬
݇ଷ
݇ଶ
݇ଵ
ʹ݂ଶ‫כ‬
݇ଵ
݇ଶ
ʹ݂ଷ‫כ‬
‫ݑ‬
݇ଷ
Figure 3-20
iv)
Hysteresis loop of stage 3
Stage 4: The inner slider slides on lower articulated slider and the upper
articulated slider slides on the top concave plate.
At this stage, the lower articulated slider reaches its displacement limit and cannot
slide further on the bottom concave plate. This forces the inner slider to slide on the
lower articulated slider, increasing the inclination of the inner slider so that the bearing
can support the increasing horizontal force. The upper articulated slider has not reached
the limit yet and relative sliding does not occur between the inner slider and the upper
articulated slider.
54
The normalized stiffness of the backbone curve in this stage is:
݇ସ ൌ
ͳ
‫ܮ‬ଵ ൅ ‫ܮ‬ଷ
(3.5-12)
The maximum displacement and normalized force of this stage is:
݀ଷ
݀ଶ
‫ݑ‬ହ‫ כ‬ൌ ‫ݑ‬ସ‫ כ‬൅ ൬ ൅ ߤଷ െ െ ߤଶ ൰ ሺ‫ܮ‬ଵ ൅ ‫ܮ‬ଷ ሻ
‫ܮ‬ଷ
‫ܮ‬ଶ
݂ହ‫ כ‬ൌ ߤଷ ൅
(3.5-13)
݀ଷ
‫ܮ‬ଷ
(3.5-14)
The hysteresis loop of stage 4 is shown in Figure 3-21.
݂
݇ସ
݂ସ‫כ‬
݂ଶ‫כ‬
݂ଵ‫כ‬
݇ଷ
݇ଵ
ʹ݂ଵ‫כ‬
݇ଵ
݇ଶ
ʹ݂ସ‫כ‬
݂ସ‫ כ‬െ ʹ݂ଶ‫כ‬
݇ଶ
‫ݑ‬
݇ଷ
Figure 3-21
v)
Hysteresis loop of stage 4
Stage 5: The inner slider slides between the two articulated sliders.
At this stage, the upper articulated slider reaches its displacement limit and cannot
go further on the top concave plate. As the result, the inner slider is forced to slide on the
upper articulated slider to provide the increasing inclination to support the increasing of
55
the horizontal force. When the inner slider contacts the displacement restraining rings of
the articulated sliders, the bearing reaches its displacement limit and cannot go further
without damage.
Normalized stiffness of the backbone curve in this stage is:
݇ହ ൌ
ͳ
ʹ‫ܮ‬ଵ
(3.5-15)
The maximum displacement and normalized force of this stage is:
‫ כ଺ݑ‬ൌ ʹ݀ଵ ൅ ݀ଶ ൅ ݀ଷ ൅
݂଺‫ כ‬ൌ ߤଵ ൅
‫ܮ‬ଵ
‫ܮ‬ଵ
݀ଷ െ ݀ଶ
‫ܮ‬ଷ
‫ܮ‬ଶ
݀ଵ ݀ଷ
൅
‫ܮ‬ଵ ‫ܮ‬ଷ
(3.5-16)
(3.5-17)
For standard bearings where ݀ଶ ൌ ݀ଷ , combining Equations (3.5-2a) and (3.5-16) yields:
‫ כ଺ݑ‬ൌ ʹ݀ଵ ൅ ʹ݀ଶ . This is the displacement limit of the bearing. The hysteresis loop of
stage 5 is shown in Figure 3-22.
Numerical model of triple friction pendulum bearings extended to bi-direction
with vertical-horizontal coupling behavior is described in details in Chapter 6.
͵ǤͷǤ͵ ‡•‹‰‘ˆ–Š‡‡ƒ”‹‰•
As shown in previous section, the behavior of a typical triple friction pendulum
bearing depends on 9 independent parameters: 3 friction coefficients ߤ௜ , 3 pendulum
lengths ‫ܮ‬௜ and 3 displacement limits ݀௜ . In the design process these parameters are
selected based on the design criteria including maximum base displacement, maximum
56
݂
ʹ݂ଵ‫כ‬
݂ହ‫כ‬
݇ସ
݂ସ‫כ‬
݂ଷ‫כ‬
݇ଷ
݂ଶ‫כ‬
݂ଵ‫כ‬
݇ଵ
݇ଵ
݇ଶ
݂ସ‫ כ‬െ ʹ݂ଶ‫כ‬
݇ଶ
ʹ݂ଷ‫ כ‬െ݂ହ‫כ‬
‫ݑ‬
݇ଷ
Figure 3-22
Hysteresis loop of stage 5
base shear, peak story drift and peak floor acceleration. The design process is a trial and
error process.
Parameters of the first pendulum mechanism (e.g. ߤଵ and ‫ܮ‬ଵ ) are selected to
control the floor acceleration subjected to frequent earthquakes. Friction coefficients ߤଶ ,
ߤଷ and pendulum length ‫ܮ‬ଶ (=‫ܮ‬ଷ ) of the second and third pendulum mechanism are
chosen to control floor acceleration and story drift in large earthquakes. This length can
be chosen such that the period of the third pendulum mechanism is ͹‫ ݏ‬or longer. The
friction coefficient ߤଷ is also selected to limit the displacement demand to the desired
capacity. The displacement limits of sliders are selected to provide the desired
displacement capacity of isolation system and conditions in Equation (3.5-2).
Friction coefficients of TPBs in early projects were selected in the proportion of
ߤଵ ǣ ߤଶ ǣ ߤଷ ൌ ͳǣ ͷǣ ͳͲ. Recently, EPS Inc. found that the proportion of ߤଵ ǣ ߤଶ ǣ ߤଷ ൌ ͳǣ ͵ǣ ͸
57
gives better performance in the isolated building (Zayas, 2012). This proportion is the
typical selection for current triple friction pendulum bearings.
The isolation system for the test was designed and provided by EPS Inc. The
sizes, including spherical radii, of top and bottom plates of the bearings were selected
using similar criteria for recent EPS projects. Friction coefficients ߤଶ ൌ ߤଷ ൌ ͺΨ were
selected to accommodate the displacements caused by selected ground motions TCU and
TAB, which were expected to cause the largest displacement. The spherical radius of
inner slider and friction coefficient ߤଵ were determined through iterative dynamics
analysis of a simple model to limit floor acceleration to ͲǤ͵ͷ‰ and story drift below
ͲǤͷΨ. Table 3-8 summarizes the design parameters of the bearings used for the test.
The theoretical backbone curve of the designed bearing is shown in Figure 3-23.
Because effective length ‫ܮ‬ଶ ൌ ‫ܮ‬ଷ , friction coefficients ߤଶ ൌ ߤଷ and displacement limits
݀ଶ ൌ ݀ଷ so that stage 2 and stage 4 in the general case in Figure 3-16 are collapsed. This
can be easily observed since ‫ݑ‬ଶ‫ כ‬ൌ ‫ݑ‬ଷ‫ כ‬, ݂ଶ‫ כ‬ൌ ݂ଷ‫ כ‬,‫ݑ‬ସ‫ כ‬ൌ ‫ݑ‬ହ‫ כ‬, ݂ସ‫ כ‬ൌ ݂ହ‫( כ‬Equations (3.5-4),
(3.5-5), (3.5-7), (3.5-8), (3.5-10), (3.5-11), (3.5-13) and (3.5-14)). The displacement limit
of the bearings is ͳǤͳ͵݉ at base shear coefficient of ͲǤʹ͹ͷ.
Table 3-8
Design parameters of triple pendulum bearings
ߤଵ ൌ ͲǤͲʹ
ܴଵ ൌ Ͷͷ͹
݄ଵ ൌ ͵ͺ
݀ଵ ൌ ͷͳ
‫ܮ‬ଵ ൌ Ͷͳͻ
ߤଶ ൌ ͲǤͲͺ
ߤଷ ൌ ͲǤͲͺ
ܴଶ ൌ ͵ͻ͸ʹ
ܴଷ ൌ ͵ͻ͸ʹ
݄ଶ ൌ ͳͳͶ
݄ଷ ൌ ͳͳͶ
݀ଶ ൌ ͷͳͶ
݀ଷ ൌ ͷͳͶ
‫ܮ‬ଶ ൌ ͵ͺͶͺ
‫ܮ‬ଷ ൌ ͵ͺͶͺ
Length is in 
58
݂
ͲǤʹ͹ͷ
ܶ ൌ ͳǤͺͶ•
ͲǤʹͳͶ
ͳ
ͳ
ͺͶ
ܶ ൌ ͷǤͷ͹•
ͳ
ͲǤͲͺ
ͲǤͲʹ ͳ
ͳ
͹͹Ͳ
ͳ
ͺͶ
ͷ
Figure 3-23
ͳͲͺ ͳͳ͵ ‫ݑ‬ሺ ሻ
Backbone curve of the designed bearings
͵ǤͷǤͶ š’‡ –‡†”‡•’‘•‡‘ˆ–Š‡‹•‘Žƒ–‹‘•›•–‡–‘†‹ˆˆ‡”‡–‡ƒ”–Š“—ƒ‡
• ‡ƒ”‹‘•
As described in Section 3.2, effective periods of the isolation system at different
earthquake levels were determined to scale ground motions to match the target response
spectra representing these earthquake levels. Given the hysteresis loop of the isolation
system (Figures 3-18 to 3-22) and a ͷΨ damped target response spectrum, the expected
displacement ‫ܦ‬௘௫௣ , effective period ܶ௘௙௙ and effective damping ratio ߞ௘௙௙ of the isolation
system subjected to motions represented by the spectrum were estimated following these
steps:
Step 1: Assume a trial effective period ܶ௧௥௜௔௟ , damping ratio ߞ௧௥௜௔௟ .
Step 2: Compute damping coefficient factor ‫ ܤ‬from Table 3-2.
59
Step 3: Compute displacement ‫ܦ‬௘௫௣ from the response spectrum with ‫ ܤ‬and
ܶ௧௥௜௔௟ .
Step 4: Determine the effective stiffness of isolation system ‫ܭ‬௘௙௙ and area ‫ܣ‬௘௙௙ of
the hysteresis loop at displacement ‫ܦ‬௘௫௣ .
Step 5: Compute the effective period and damping ratio:
‫ܯ‬
ܶ௘௙௙ ൌ ʹߨඨ
‫ܭ‬௘௙௙
ߞ௘௙௙ ൌ
‫ܣ‬௘௙௙
ଶ
ʹߨ‫ܭ‬௘௙௙ ‫ܦ‬௘௫௣
(3.5-18)
(3.5-19)
Step 6: If ܶ௘௙௙ ؆ ܶ௧௥௜௔௟ then stop. Otherwise set ܶ௧௥௜௔௟ ൌ ܶ௘௙௙ ǡ ߞ௧௥௜௔௟ ൌ ߞ௘௙௙ and
repeat from Step 2 until convergence is achieved.
Expected displacements, effective periods and effective damping ratios of the
designed isolation system at different earthquake levels computed from these steps are
shown in Table 3-9. The effective periods of isolation system are ʹǤͳͺ‫ ݏ‬at the Service
level and ͶǤʹ͵‫ ݏ‬at the MCE level so that floor acceleration response was expected to be
low. The long period also leads to large displacement (͹͹Ǥͳ  at MCE level). The
effective damping ratios at the service level and DBE level are about the same but the
effective damping ratios at the MCE level is much lower. This does not appear to be
effective in term of dissipating energy to reduce displacement of isolation system during
extreme earthquakes. This problem exists in any conventional isolation system.
60
Table 3-9
Expected displacement, effective period and damping ratio of the design
isolation system at different earthquake levels
Earthquake level
‫ܦ‬௘௫௣ ሺ ሻ
ܶ௘௙௙ ሺ•ሻ
ߞ௘௙௙ ሺΨሻ
Service
10.2
2.18
34
DBE
32.8
3.37
36
MCE
77.1
4.23
26
Figure 3-24 presents the capacity curves of the designed isolation system and
demand curves at different earthquake levels. In this figure, the target response spectra at
different earthquake levels are plotted in format of ܵ஽ versus ܵ஺ and represent demand
curves. Each point in a demand curve indicates spectral displacement and spectral
acceleration at a single period. The demand curves with higher damping ratio are shifted
toward the origin. In this coordinate system, the normalized backbone curve of the
isolation system is also plotted and represents the capacity curve. The intersection point
Figure 3-24
Capacity curve of the isolation system vs. the demand curves at different
earthquake levels
61
where capacity meets demand shows the expected peak response of the system. One can
see that the expected peak responses of the designed isolation system at the 3 selected
earthquake levels are well below the maximum capacity of the system.
3.6
Preliminary analysis of the specimen
Preliminary analysis of the specimen with both isolated base and fixed base
configurations subjected to selected motions was done to check the performance
objective of the design and predict the response of the specimen during the test. The
nonlinear analysis was performed in OpenSees, with a model similar to that described in
Chapters 6 and 7. However, the described model contains several refinements that have
been completed since the test. In particular, the preliminary analytical model used an
uncoupled bearing model in which there was no coupling between vertical and horizontal
behavior. Thus, the vertical load on the bearings was the assumed static load and
unchanged during simulation. The friction coefficients of the bearings were the design
values, which were found later to be smaller than the values measured from the test. The
vertical stiffness of supports in the model was selected equal to the vertical stiffness of
bearings, which correspondent vertical frequency of ͵Ͳ œ. The structure damping model
was stiffness proportional damping calibrated to ʹǤͷΨ damping ratio at the 1st structural
mode (period of ͲǤͶ•). The weight of this model was lower than the actual weight
measured from test (Table 2-2). The investigation on the lateral-vertical coupling effect
in response of isolated structures (Chapter 9) later shows that some of these factors have
significant influence on responses of the superstructure. Revised analysis with both
selected motions and table motions are in Section 8.4.
62
Figures 3-25 to 3-28 shows peak responses of the pre-test model subjected to
selected records. Scale factors of input motions were the actual factors used in the test
(Table 4-7). The result shows that peak isolator displacements are all below the
displacement limit of bearings. Many of these peaks fall between the expected
displacements of DBE level and MCE level. Peak displacements caused by TAB and
TCU were expected to exceed the displacement of MCE level.
Peak story drift from all simulation is well below the limit value set in design.
This criterion is easy to satisfy since the specimen is very stiff. Horizontal acceleration at
floors is also below the performance criterion. Note that the peak floor acceleration
subjected to the SCT motion is larger than the peak ground acceleration (Table 3-7). This
could happen because of the resonant of the isolation system to the dominant frequency
1.2
Bearing limit
1
MCE
0.8
0.6
Ground motion
Figure 3-25
Peak isolator vector-sum displacement from pre-test analysis
SAN
IWA
SCT
TAK
KJM
ELC
TCU
TAB
SYL
0
Service
LGP
0.2
DBE
RRS
0.4
WSM
Peak isolator displacement, dmax (m)
component of the motion.
63
0.6
Design limit
Peak drift, δmax (%)
0.5
0.4
0.3
0.2
SCT
IWA
SAN
SCT
IWA
SAN
TAK
KJM
ELC
TCU
TAB
SYL
LGP
WSM
0
RRS
0.1
Ground motion
Figure 3-26
Peak story drift from pre-test analysis
0.4
Design limit
0.3
0.2
TAK
KJM
ELC
TCU
TAB
SYL
LGP
0
RRS
0.1
WSM
Peak acceleration, Amax (g)
0.5
Ground motion
Figure 3-27
Peak horizontal floor acceleration from pre-test analysis
64
Roof
5
5
3
2
Floor
Design limit
Story
4
3
2
Design limit
4
Base
1
0
0.1
0.2
0.3
Peak drift, δ
0.4
(%)
max
(a)
Figure 3-28
0.5
0.6
Ground
0
0.2
0.4
0.6
Peak acceleration, A
max
0.8
(g)
1
(b)
Distribution of peak story drift and floor acceleration from pre-test analysis
(a) Peak story drift
(b) Peak horizontal floor acceleration
Chapter 4
•–”—‡–ƒ–‹‘ƒ†‡•– Š‡†—Ž‡
4.1
Instrumentation
Six-hundred-and-fifteen sensor channels were used for measuring the table
motion and the responses of structural and nonstructural components. Sampling
frequency of all channels was ͳͲͲͲ‫ݖܪ‬. Four-hundred-and-one sensors measuring the
table motion and structural component responses were classified into 4 types:
1. Sensors for measuring force: load cells (219 channels).
2. Sensors for measuring displacement: displacement transducers (32 channels).
3. Sensors for measuring acceleration: accelerometers (110 channels).
4. Sensors for measuring strain: strain gauges (40 channels).
All recorded data of the structural responses, except the load cell data which is
explained later, were filtered using Butterworth lowpass filter with cut off frequency of
ʹͷ‫ ݖܪ‬before using. Following is the detail instrumentation of these sensor types.
ͶǤͳǤͳ ‘ƒ†‡ŽŽ•
a) Instrumentation of Load Cells
Load cells were used for measuring reaction of every isolator. Several load cells
were installed beneath each isolator in a connection assembly that consists of a steel plate
on the top connecting to the isolator, a steel plate on the bottom connecting to the shake
table and the load cells connecting the 2 steel plates (Figure 4-1). The number and
66
configuration of load cells for each isolator were decided based on the number of
available load cells, the maximum expected load on the isolator, the size and bolt pattern
of the isolator, and the bolt pattern of the shake table. Two types of load cells were
available for use at E-Defense at the testing time. The number of these types and their
properties are shown in Table 4-1. Peak displacement and reaction of isolators from
preliminary analysis are summarized in Table 4-2.
Isolator
Top connecting
plate
Load cell
Shake table
Figure 4-1
Table 4-1
Bottom
connecting
plate
Connection assembly
Properties of load cells
Type
Number
(units)
Height
(mm)
Vertical
capacity
(kN)
Horizontal
capacity
(kN)
Vertical
stiffness
(kN/mm)
Horizontal
stiffness
(kN/mm)
A
44
180
400
250
8500
2400
B
32
195
700
400
14000
3500
67
Table 4-2
Peak responses of the isolation system from pre-test analysis
Peak isolator
displacement
(cm)
Peak isolator
lateral reaction
(kN)
Center isolator
Edge isolators
Corner isolators
99
244
1889
1221
832
Peak isolator vertical reaction (kN)
Configurations of load cells for isolators at the 4 corners are shown in Figure 4-2.
Note that the connection assembly was flipped so that the load cells are seen to be on the
top of the plate connecting the load cells and the isolator. Configurations of load cells at
all isolators are in Figure 4-3. The hexagonal configuration of load cells was selected to
minimize the number of load cells and reduce the deformation of the connecting plates.
This configuration also possesses similar stiffness in any horizontal direction. The
distance between load cells was selected based on the preliminary analysis of the load
distributed to load cells from the peak isolator responses given in Table 4-2. In this
analysis, the peak responses were conservatively assumed to occur at the same time. The
connecting plates were assumed rigid in this preliminary analysis. According to the
analysis, maximum load transferred to load cells are summarized in Table 4-3. From this
table, the peak axial load and lateral load in Type A load cells are ͵͵ͳ݇ܰ and is ͵ͷ݇ܰ,
respectively. These peaks value in Type B load cells are ͷ͹ͳ݇ܰ and ͵ͷ݇ܰ. These
values are well below the limits in Table 4-1.
b) Design and Analysis of the Connection Assemblies
The thickness of the connecting plates, including the top plate which connects the
load cells to the isolator and the bottom plate which connects the load cells to the shake
table, were selected based on the required thickness for bolting and preliminary analysis
68
of the strength of the plates. In this simplified strength analysis, schematically shown in
Figure 4-4, the plates were considered as one-dimensional elements fixed at properly
faces. Load applied on the elements were computed from the reaction of the load cells,
the reaction from isolator which reduces the effect of the reaction of the load cells was
neglected for conservatism. According to this simplified analysis, the required thickness
of the plates is ͺ͹݉݉. The thickness of ͻͷ݉݉ was chosen for conservatism. The
connecting plates were produced by milling plate steel having thickness of ͳͲͳǤ͸݉݉
(Ͷ݄݅݊ܿ݁‫)ݏ‬. The milling process helped to level the surface. Design drawing of the
connection assemblies is in Appendix A.
Load cell
Plate connecting
load cells and
isolator
Figure 4-2
Load cell configuration at corner isolators
69
Plate connecting
load cells to table
(a)
6 load cell B
900
350
Plate connecting
isolator to load cells
3 load cell A
Plate connecting
load cells to table
(b)
Plate connecting
isolator to load cells
6 load cell B
750
Load cell A
Plate connecting
load cells to table
(c)
6 load cell A
350
900
Plate connecting
isolator to load cells
3 load cell A
Figure 4-3
Load cell configuration at all isolators
(a) Center isolator, (b) Edge isolators, and (c) Corner isolators
70
Table 4-3
Peak load cell forces from preliminary analysis of the connection assemblies
Isolator
Peak load
(kN)
Center
Edge
Corner
Load cell A
Load cell B
Load cell A
Load cell B
Load cell A
Load cell B
Vertical
331
571
174
492
271
NA
Lateral
27
27
35
35
27
NA
(a)
Figure 4-4
(b)
Simplified model for strength analysis of the top connecting plate
at edge isolators
(a) Connecting plate, (b) Simplified 1D model
Finite element models of the connecting assemblies were developed in SAP2000
v14 for refined checking of the strength and deflection. In this finite element model
(Figure 4-5), connecting plates were modeled by 8-node solid elements. The meshing of
the top surface of the top connecting plate is shown in Figure 4-6. The caps at the two
ends of the load cells were modeled by thick-shell elements and connected together by an
elastic spring element representing the stiffness of the load cells (Table 4-1). The
71
Top connecting
plate
Bottom
connecting plate
Load cell
Compression only
supports
Figure 4-5
Finite element model of the connecting assembly at the center isolator
72
C
A
Figure 4-6
B
Meshing at the top face of the top connecting plate and locations of acting load
assembly is fixed at bolt holes connecting the bottom connecting plate to the shake table.
Compression only supports, modeled by nonlinear friction isolator elements, were also
added to the bottom surface of the bottom connecting plate at the load cell locations for
representing the support of the shake table. The initial model had these compression only
supports everywhere on the bottom surface but it encountered the convergence problem.
Load transferred to the connecting assembly was done through the bolt holes of the top
connecting plate where the isolator is connected. This load was computed from the peak
isolator responses listed in Table 4-2. Peak displacement of isolators was taken equal to
the displacement limit (ͳǤͳ͵݉) for conservatism. Note that this displacement is evenly
shared by the top and bottom parts of the isolator so that the location of the axial load
acting to the compound is limited to an area within a radius of ͲǤͷͷ͸݉ from the center
of the top connecting plate of the compound. Several locations of the acting load, shown
73
in Figure 4-6 for center and corner assemblies, were investigated. Location B caused
largest deformation to the top connecting plate (Figure 4-7). Largest out-of-plan bending
deformation of the connecting plates is ͲǤ͹͸݉݉ȀͳͲͲͲ݉݉. The vertical displacement
of the acting load is ͲǤ͹ͺͺ݉݉, corresponding to the vertical stiffness of ʹͶ‫ܰܯ‬Ȁܿ݉.
Smaller deformation and larger vertical stiffness are desirable but we were forced to
accept these values because thicker plate steel was not available and the number of load
cells was limited. Von-Mises stress contour on the top and bottom surfaces of the top
connecting plate is shown in Figure 4-8. The Von-Mises stress is smaller than yield stress
of steel (ʹͶͺ‫ )ܽܲܯ‬at most place, except at the bolt holes, where the stress concentration
happens because of the concentrated load and/or the modeling assumption. The local
concentrated stress is unimportant because the nonlinear behavior will redistribute the
stress after yielding. Though not shown here, the stress in the bottom connecting plate is
much smaller than the stress in the top connecting plate. Peak forces of load cells from
analysis of the finite element model are given in Table 4-4. These values are within the
Figure 4-7
Deformation of the connecting assembly at the center isolator
74
(a)
(b)
‫ܽܲܯ‬
Figure 4-8
Von-Misses stress contour on the top and bottom surfaces of the top
connecting plate of the connecting assembly at the center isolator
(a) Top surface, (b) Bottom surface
75
Table 4-4
Peak load cell forces from analysis of the finite element models
of connection assemblies
Isolator
Peak load
(kN)
Center
Edge
Corner
Load cell A
Load cell B
Load cell A
Load cell B
Load cell A
Load cell B
Vertical
419
590
43
523
267
NA
Lateral
22
60
31
62
42
NA
limit values given in Table 4-1, except the axial load in load cell A at center compound,
which exceeds the limit by 4.75Ψ.
ͶǤͳǤʹ ‹•’Žƒ ‡‡–”ƒ•†— ‡”•
Displacement transducers were used to measure the displacement of isolation
system and story drift. Figure 4-9 shows the layout of displacement transducers at base
level for measuring the displacement of the isolation system. Three transducers each were
installed at the column bases at the North side and East side to measure the displacement
in X and Y directions, respectively. Three unparallel transducers are needed to determine
the translation and rotation of the isolation system; other 3 were selected for redundancy.
From the changes in length of the strings of these transducers, the horizontal
displacement of the base and every isolator can be determined (see Section 4.5.1). Figure
4-9 also presents locations of laser displacement transducers for measuring the vertical
displacement at the Southeast, Northeast and Northwest isolators. The laser transducers
were attached to the column base through a small truss as shown in Figure 4-10. The
width of the reflection plate is larger than twice the displacement limit of the isolation
76
Y
X
S
SE
SW
Column
base
Base
girder
Isolator
Load cell
C
E
W
Connecting
plate
N
NE
Displacement
transducer
Laser
Displacement
transducer
Figure 4-9
NW
Reflection
disk
N
Layout of displacement transducers at base
system (ʹ ൈ ͳǤͳ͵݉) so that the vertical displacement can be measured at any horizontal
displacement. The vertical displacement of an isolator is estimated as the average vertical
displacement of the 2 laser transducers attached to the column base it supports.
Laser displacement transducers were also used for measuring story drift. Each
sensor was attached to a truss built on the concrete mass block on bottom floor and its
reflecting plate was attached to the top floor as shown in Figure 4-11. A pair of
transducers measures the relative displacement between the two floors in each direction
at 2 locations (Figure 4-12). Using a rigid floor diaphragm assumption, 3 unparallel
77
Column
base
Laser
transducer
Reflection
panel
Figure 4-10
Laser displacement transducer for measuring vertical movement of isolator
Reflecting
plate
Top floor
Support
truss
Bottom
floor
Figure 4-11
Instrumentation for measuring story drift
Laser
transducer
78
Laser transducer
Support
truss
Figure 4-12
Layout of displacement transducers to measure story drift from stories 2 to 5
displacement transducers are needed for determining relative displacement between the
adjacent floors. An additional displacement transducer at each story for redundancy. The
layout of the 4 displacement transducers was added at every story from story 2 to story 5
is consistent with Figure 4-12. At the first story, the 4 displacement transducers were
installed at the Southeast and Northwest columns.
ͶǤͳǤ͵ ‡Ž‡”‘‡–‡”•
Three triaxial accelerometers were installed to measure the 3 components of
acceleration at the 4 corners of the shake table. Beside acceleration at these locations,
acceleration at the center of the shake table was also measured by permanent sensors
integrated into the shake table control system. The measured acceleration at the center of
79
the table includes 6 components (3 translational components and 3 rotational
components).
Two uniaxial accelerometers were also installed at the top connecting plate of the
connecting assemblies to measure the horizontal acceleration at these plates (Figure 413). The recorded acceleration is used for deriving isolators’ force. The load cells
described in the previous section do not measure the isolator force but the force just
beneath the top connecting plate. These forces differ by the inertia force of the top
connecting plate and the bottom part of the isolator. Since the total mass separating these
2 locations is large (about Ͷ‫ݏ݊݋ݐ‬, depending on location) and the expected acceleration
is also large (approximate to the table acceleration, about ͳ݃), the inertia force is
significant compared to the isolator force and should be accounted for.
Floor accelerations (2 horizontal and vertical components) were measured
through 3 triaxial accelerometers installed at the Southeast, Northeast and Northwest
corners of every floor. These triaxial accelerometers were attached to the face of the
bottom section of the columns. Vertical acceleration at other locations on the floor slab
Isolator
Accelerometers
Top connecting plate
Load cell
Bottom connecting plate
Shake table
Figure 4-13
Accelerometers measuring acceleration at the top connecting plate
80
was also recorded. Figure 4-14 shows the layout of accelerometers at the 5th floor, a
typical layout for all floors. The vertical accelerometers were attached to the bottom of
the slabs.
ͶǤͳǤͶ –”ƒ‹ ƒ—‰‡•
Strain gauges were used to measure strain in the Northeast column at every story.
The original purpose of these strain gauges was to observe the vertical propagation of
strain wave during impact when the isolation system reaches to its displacement limit.
The axial strain at the middle of the 4 sides of a column section was measured. The axial
stress (or normal stress) at these locations can be computed, and from these stresses, the
internal forces at the section can be determined (see Section 4.5.3). Because the
Vertical
accelerometer
Triaxial
accelerometer
Figure 4-14
th
Layout of accelerometers at the 5 floor
81
transverse load due to inertia force in the column is small, the distribution of the internal
forces in the column is almost linear. To determine this linear distribution, the values of
internal force at 2 locations per story are required. These 2 location need to be far enough
apart so that the extrapolated straight lines of the internal forces are reliable. The 2
locations also need to be far from the floor so that the Euler-Bernoulli’s plane crosssection assumption is valid. Combining these requirements, sections at approximately
ͳȀͶ and ͵ȀͶ of the clear length of the column segment were selected for measuring the
axial strain.
4.2
Installation of the Specimen to the Shake Table
In the isolated base configuration, the load cells was first installed to the top
connecting plate, the hexagonal plate, of the connection assembly through the bolt holes
drilled in advance (Figure 4-2). The bottom plate was then added to the assembly (Figure
4-15). The connection assemblies were then flipped and installed to the shake table
(Figure 4-16) before the isolators were installed (Figure 4-17). Finally, the specimen was
craned by two 400-ton-cranes and installed to the isolation system (Figure 4-18). The
specimen was connected to the isolation system through the bolt holes in the column
bases (Figure 4-19), which were drilled before the installation process.
82
Figure 4-15
Figure 4-16
Assembly of the connection assembly
Connection assemblies on the shake table
83
Figure 4-17
Figure 4-18
Installing the isolator to the connecting assembly
Installing the specimen to the isolation system
84
Bottom face of
the column base
Bolt hole for
connecting the
specimen to
the isolation
system
Figure 4-19
Bolt holes for connecting the specimen to the isolation system
The behavior of each friction isolator and the isolation systems depends on the
applied vertical load. The vertical load on an isolator is expected to be proportional with
the mass of the tributary area of the isolator so that the isolation system produces the best
performance. This condition can be obtained if the specimen had been constructed
directly on the isolation system (similar to the construction process in reality). However,
the specimen had been built and settled outdoors more than 2 years before tested so that
its base was warped and the distribution of vertical load on all isolators was not even
close to the expected values. Table 4-5 shows the measured vertical load on all isolators
when the specimen was settled on the isolation system and the expected vertical load on
the isolators based on the analysis of the pre-test analytical model. The vertical load on
the center isolator was almost zero and the discrepancy between expected load and the
measured load was very large. The specimen was then reset several times for adjusting
the vertical load on isolators. At each iteration, shimming plates were installed between
85
Table 4-5
Vertical load on each bearing at first iteration of installation
Bearing
^
E
^
E
^t
t
Et
Actual load
(kN)
ϳϵϰ
ϭϳϯ
ϭϭϭϴ
ϭϬϯϱ
ϳ
ϱϵϵ
ϲϳϯ
ϵϬ
ϴϳϳ
Expected
(*)
load (kN)
ϰϴϬ
ϴϬϯ
ϰϳϭ
ϳϯϱ
ϭϮϭϲ
ϳϱϮ
ϭϲϳ
ϰϭϰ
ϯϯϬ
Difference
(%)
ϲϱ͘ϰϰ
ϰϬ͘ϵϬ
Ͳϵϵ͘ϰϮ
Ͳϳϴ͘ϰϱ ϭϯϳ͘ϲϬ
ͲϮϬ͘ϯϲ ϯϬϯ͘ϵϬ Ͳϳϴ͘Ϯϳ ϭϲϱ͘ϵϴ
(*)
The expected load was linearly scaled such that the total vertical load matches the
measured load
the isolators and column base plates to adjust the distribution of loads on the isolators.
Table 4-6 shows the measured vertical load on the isolators after the final iteration. The
discrepancy between the actual load and the expected load was still large but much better
than the discrepancy before adjusting.
In the fixed base configuration, the specimen was connected directly to the shake
table through anchor bolts.
Table 4-6
Vertical load on each bearing at last iteration of installation
Bearing
^
E
^
E
^t
t
Et
Actual load
(kN)
ϰϮϮ
ϴϱϵ
ϲϰϰ
ϳϵϯ
ϱϬϱ
ϲϬϱ
ϰϰϯ
ϳϭϭ
ϱϭϬ
Expected
(*)
load (kN)
ϰϵϭ
ϴϮϭ
ϰϴϮ
ϳϱϮ
ϭϮϰϰ
ϳϳϬ
ϭϳϭ
ϰϮϰ
ϯϯϳ
Difference
(%)
Ͳϭϰ͘Ϭϵ
ϰ͘ϱϳ
ϯϯ͘ϳϮ
ϱ͘ϰϴ
Ͳϱϵ͘ϰϭ
ϲϳ͘ϳϭ
ϱϭ͘ϭϮ
(*)
ͲϮϭ͘ϰϭ ϭϱϵ͘ϳϳ
The expected load was linearly scaled such that the total vertical load matches the
measured load
86
4.3
Test Schedule
Schedule for shaking the isolated base building spanned 3 days with 21
simulations. Two of these simulations were sin-wave excitations, while the others were
earthquake excitations using motions selected in Chapter 3. The sine-wave excitations
were for determining the properties of the isolation system, which is discussed in detail in
Section 5.1. These sine-wave excitations used the same input with different scale factors.
Some of the earthquake motion excitations were horizontal excitation only. The largest
earthquake motions were ramped up in several simulations. The damage was inspected at
the end of every shaking day and after the 88RRS simulation, which caused significant
damage to non-structural components and disruption to contents. Table 4-7 summarizes
the schedule for shaking the isolated base building where the shaded simulations were 3component excitations and the simulations in the same color other than black used the
same input with different scale factors.
The shaking of the fixed base building, completed in 1 day, included 5 earthquake
excitations, with 3D white noise excitation for system identification preceding and
following every earthquake simulation. Unidirectional white noise excitations were also
applied at the beginning and end of the test day. Because the nonstructural components
were inspected and partially repaired after every earthquake excitation, the properties
were assessed before and after these repairs. Table 4-8 shows the shaking schedule for
the fixed base building. In this table, the earthquake simulations are shaded and the
motion in red used the same input with different scale factors.
87
Table 4-7
Date
(dd/mm/yy)
17/08/11
18/08/11
19/08/11
Schedule for shaking the isolated building
Scale factor
Simulation
name
Motion
12:01:46
65SIN
12:49:54
Time
X
Y
Z
Sine-wave
0.65
0.00
0.00
100SIN
Sine-wave
1.00
0.00
0.00
13:42:20
80WSM
WSM
0.80
0.80
0.80
14:30:21
130ELC
ELC
1.30
1.30
1.30
15:20:16
88RRS
RRS
0.88
0.88
0.88
17:16:16
100SYL
SYL
1.00
1.00
1.00
17:48:56
50TAB
TAB
0.50
0.50
0.50
11:35:31
70LGP
LGP
0.70
0.70
0.70
12:25:40
50TCU
TCU
0.50
0.50
0.00
13:55:30
70TCU
TCU
0.70
0.70
0.00
14:31:59
100IWA
IWA
1.00
1.00
0.00
15:45:46
100SAN
SAN
1.00
1.00
0.00
16:34:58
100TAK
TAK
1.00
1.00
1.00
17:05:03
100KJM
KJM
1.00
1.00
1.00
11:29:55
88RRSXY
RRS
0.88
0.88
0.00
12:16:55
80TCU
TCU
0.80
0.80
0.00
13:08:07
80TAB
TAB
0.80
0.80
0.80
14:02:19
90TAB
TAB
0.90
0.90
0.00
14:50:46
100TAB
TAB
1.00
1.00
0.00
15:28:19
100SCT
SCT
1.00
1.00
0.00
16:19:03
115TAK
TAK
1.15
1.15
1.00
Damage
inspection
Yes
Yes
Yes
Yes
88
Table 4-8
Date
(dd/mm/yy)
31/08/11
4.4
Schedule for shaking the fixed base building
Scale factor
Simulation
name
Motion
10:19:52
100WHT1
10:30:02
Time
X
Y
Z
White noise
1.00
0.00
0.00
100WHT2
White noise
0.00
1.00
1.00
10:38:32
100WHT3
White noise
1.00
1.00
1.00
10:50:35
80WSM
WSM
0.80
0.80
0.80
11:02:50
100WHT4
White noise
1.00
1.00
1.00
12:06:31
100WHT5
White noise
1.00
1.00
1.00
12:18:47
35RRSXY
RRS
0.35
0.35
0.00
12:28:02
100WHT6
White noise
1.00
1.00
1.00
13:37:34
100WHT7
White noise
1.00
1.00
1.00
13:51:20
35RRS
RRS
0.35
0.35
0.35
14:03:01
100WHT8
White noise
1.00
1.00
1.00
15:12:50
100WHT9
White noise
1.00
1.00
1.00
15:24:53
88RRS
RRS
0.35
0.35
0.88
15:33:51
100WHT10
White noise
1.00
1.00
1.00
17:07:04
100WHT11
White noise
1.00
1.00
1.00
17:22:33
70IWA
IWA
0.70
0.70
0.00
17:35:28
100WHT12
White noise
1.00
0.00
0.00
17:43:12
100WHT13
White noise
0.00
1.00
0.00
17:52:47
100WHT14
White noise
1.00
1.00
1.00
Damage
inspection
Yes
Yes
Yes
Yes
Yes
Table Motions
Peak accelerations of target motions and table motions are listed in Table 4-9.
Target motions are the selected ground motions in Chapter 3 scaled by the scaled factors
given in Table 4-7 and Table 4-8. Table motions are motions generated by the shake
89
Table 4-9
Peak acceleration of target motions and table motions
Peak ܽ௑ ሺ݃ሻ
Peak ܽ௒ ሺ݃ሻ
Peak ܽ௓ ሺ݃ሻ
Difference ሺΨሻ
Fixed base test
Isolated base test
Run
Target
Table
Target
Table
Target
Table
X-dir
Y-dir
Z-dir
80WSM
0.171
0.169
0.135
0.147
0.174
0.140
-1.23
8.64
-19.38
130ELC
0.278
0.293
0.408
0.484
0.263
0.261
5.39
18.62
-0.72
88RRS
0.427
0.586
0.730
1.213
0.722
1.241
37.33
66.25
71.91
100SYL
0.601
0.674
0.869
1.145
0.519
0.543
12.25
31.78
4.59
50TAB
0.450
0.585
0.418
0.463
0.327
0.357
29.88
10.71
9.01
70LGP
0.415
0.445
0.391
0.628
0.641
0.687
7.15
60.70
7.18
50TCU
0.408
0.453
0.304
0.278
0.000
0.015
11.06
-8.53
70TCU
0.571
0.648
0.425
0.378
0.000
0.027
13.50
-11.10
100IWA
0.364
0.409
0.418
0.580
0.000
0.031
12.35
38.80
100SAN
0.190
0.231
0.167
0.161
0.000
0.020
21.90
-3.54
100TAK
0.747
0.789
0.619
0.922
0.288
0.259
5.61
48.94
-9.98
100KJM
0.595
0.680
0.822
0.893
0.340
0.408
14.29
8.70
19.92
88RRSXY
0.427
0.532
0.730
1.194
0.000
0.098
24.56
63.61
80TCU
0.653
0.747
0.486
0.418
0.000
0.034
14.45
-14.11
80TAB
0.720
0.870
0.670
0.836
0.523
0.593
20.73
24.88
90TAB
0.810
0.930
0.753
1.011
0.000
0.102
14.81
34.21
100TAB
0.901
0.995
0.837
1.139
0.000
0.120
10.46
36.07
100SCT
0.171
0.177
0.101
0.106
0.000
0.017
3.69
5.14
115TAK
0.859
0.936
0.712
1.088
0.288
0.278
9.01
52.72
-3.37
80WSM
0.171
0.219
0.135
0.175
0.174
0.136
28.31
29.47
-21.79
35RRSXY
0.170
0.201
0.290
0.398
0.000
0.011
18.55
37.14
35RRS
0.170
0.201
0.290
0.406
0.287
0.350
18.14
39.83
21.85
88RRS
0.170
0.228
0.290
0.409
0.722
1.062
34.45
41.14
47.22
70IWA
0.255
0.270
0.292
0.373
0.000
0.013
6.25
27.43
13.26
90
table. The peak acceleration of the table motions are compared to the peak acceleration of
the target motions in Table 4-9. The shake table generally amplified the motions in term
of peak values. The largest amplification occurred during the 88RRS simulation of the
isolated base test. The time history acceleration of the 3 components of this motion is
plotted in Figure 4-20. The tremendous amplification occurred at the peak part of the
pulse in the Y-direction and follows by a large amplification in the Z-direction.
Response spectra at ͷΨ damping ratio of the target motions and table motions are
shown in Appendix B. The ratio between these spectra at period ranging from ͲǤͲͳ‫ ݏ‬to
5‫ ݏ‬is plotted in Figures 4-21 and 4-22. At periods longer than ͲǤ͹‫ݏ‬, the response
spectrum amplitudes of the table motions and target motions are not much different. At
shorter periods, the response spectrum amplitude of the table motions is generally larger
than that of the target motions. The isolation system was mainly working at periods
longer than ͳ‫ ݏ‬in horizontal direction so that it should not be significantly affected by the
difference between the table motions from the target motions. However, the responses of
the superstructure may be affected because of the participation of the higher modes. One
can observe that the shake table tends to amplify the period components at around ͲǤʹ‫ݏ‬
more strongly than components at neighbor periods. This period may be a natural period
of the shake table.
91
1
0.5
X
Acceleration, a (g)
Table motion
Target motion
0
-0.5
-1
0.5
Y
Acceleration, a (g)
1
0
-0.5
-1
-1.5
1.5
Z
Acceleration, a (g)
1
0.5
0
-0.5
-1
0
Figure 4-20
5
10
Time, t (s)
15
20
Time history acceleration of 88RRS motion in the isolated base test
92
S
AxTable
/S
AxTarget
2
1.5
1
0.5
/S
0.5
AyTable
1
AzTable
/S
AzTarget
0 -2
10
2
S
-1
10
10
0
10
1
1.5
S
AyTarget
0 -2
10
2
-1
10
10
0
10
1
1.5
1
0.5
0 -2
10
-1
10
10
0
10
1
Period, T (s)
Figure 4-21
Ratio of table motion spectral accelerations and target motion spectral
accelerations: isolated base test
93
SAxTable/SAxTarget
2
1.5
1
0.5
SAyTable/SAyTarget
0 -2
10
2
0
10
1
10
1.5
1
0.5
0 -2
10
2
SAzTable/SAzTarget
-1
10
-1
0
10
10
1
10
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
Period, T (s)
Figure 4-22
Ratio of table motion spectral accelerations and target motion spectral
accelerations: fixed base test
94
4.5
ͶǤͷǤͳ
Derived Response
‘”‹œ‘–ƒŽ‹•’Žƒ ‡‡–‘ˆ–Š‡ •‘Žƒ–‹‘›•–‡
The transducers measuring horizontal displacement and rotation of the isolation
system in Figure 4-9 does not directly measure the X, Y and rotation components of
displacement of the isolation system, but actually measure the change in length of their
strings. If the displacement of the isolation system is much smaller than the original
length of the transducers, the X and Y displacement components of each isolator
approximately equal the change in length of the correspondent transducers (small
displacement assumption). However, the displacement of the isolation system was large
and the small displacement assumption is inapplicable. An algorithm to solve the
displacement of the isolation system and the isolators accounting for the large
displacement was developed (Sato and Okazaki, 2011). Accordingly, the algorithm is
described as follows.
From the original and displaced configurations of the isolation system in Figure 423, the coordinates ܺ஺ᇲ ǡ ܻ஺ᇲ ǥ of displaced nodes A’, B’, D’, F’, G’ and H’ are:
ܺ஺ᇲ ൌ ȟܺ ൅ ܺ஺ ‘• ߶ െ ܻ஺ •‹ ߶
ܻ஺ᇲ ൌ ȟܻ ൅ ܺ஺ •‹ ߶ ൅ ܻ஺ ‘• ߶
ܺ஻ᇲ ൌ ȟܺ ൅ ܺ஻ ‘• ߶ െ ܻ஻ •‹ ߶
ܻ஻ᇲ ൌ ȟܻ ൅ ܺ஻ •‹ ߶ ൅ ܻ஻ ‘• ߶
ܺ஽ᇲ ൌ ȟܺ ൅ ܺ஽ ‘• ߶ െ ܻ஽ •‹ ߶
ܻ஽ᇲ ൌ ȟܻ ൅ ܺ஽ •‹ ߶ ൅ ܻ஽ ‘• ߶
ܺிᇲ ൌ ȟܺ ൅ ܺி ‘• ߶ െ ܻி •‹ ߶
ܻிᇲ ൌ ȟܻ ൅ ܺி •‹ ߶ ൅ ܻி ‘• ߶
ܺீᇲ ൌ ȟܺ ൅ ܺீ ‘• ߶ െ ܻீ •‹ ߶
ܻீᇲ ൌ ȟܻ ൅ ܺீ •‹ ߶ ൅ ܻீ ‘• ߶
ܺு ᇲ ൌ ȟܺ ൅ ܺு ‘• ߶ െ ܻு •‹ ߶
ܻு ᇲ ൌ ȟܻ ൅ ܺு •‹ ߶ ൅ ܻு ‘• ߶
(4.5-1)
95
A
a
SW
S
SE
‫ܮ‬௒
Y
b
E
B
a)
C
W
N
NW
‫ܮ‬௑
D
d
NE
G
F
H
g
f
h
X
a
C
ȟܺ
‫ܮ‬஺
ȟܺ
Ԅ
C’
A’
Y
b
C’
‫ܮ‬஻
b)
B’
d
H’
‫ܮ‬஽
G’
D’
‫ܮ‬ி
f
Figure 4-23
‫ܮ‬ு
‫ீܮ‬
F’
g
X
h
Configurations for solving displacement of the isolation system
(a) Original configuration (b) Displaced configuration
96
where ȟܺǡ ȟܻand ߶ are displacement components at the center bearing of the isolation
system with sign convention shown in Figure 4-23 (b); ܺ஺ ǡ ܻ஺ are coordinates of the
original point A, and so on.
From the displaced configuration in Figure 4-23 (b):
ሺܺ஺ᇲ െ ܺ௔ ሻଶ ൅ ሺܻ஺ᇲ െ ܻ௔ ሻଶ ൌ ‫ܮ‬ଶ஺ ሺܺ஻ᇲ െ ܺ௕ ሻଶ ൅ ሺܻ஻ᇱ െ ܻ௕ ሻଶ ൌ ‫ܮ‬ଶ஻ ሺܺ஽ᇲ െ ܺௗ ሻଶ ൅ ሺܻ஽ᇲ െ ܻௗ ሻଶ ൌ ‫ܮ‬ଶ஽ ଶ
ଶ
ଶ
ଶ
൫ܺிᇲ െ ܺ௙ ൯ ൅ ൫ܻிᇲ െ ܻ௙ ൯ ൌ ‫ܮ‬ଶி (4.5-2)
൫ܺீᇲ െ ܺ௚ ൯ ൅ ൫ܻீᇲ െ ܻ௚ ൯ ൌ ‫ܮ‬ଶீ ሺܺு ᇲ െ ܺ௛ ሻଶ ൅ ሺܻு ᇲ െ ܻ௛ ሻଶ ൌ ‫ܮ‬ଶு where ܺ௔ ǡ ܻ௔ are coordinate of node a; ‫ܮ‬஺ is the distance between a and A’, measured by
the transducer.
Substituting Equation (4.5-1) into Equation (4.5-2) leads to a system of 6
nonlinear equations to solve for 3 unknown ȟܺǡ ȟܻand ߶. The system of equations can be
solved using lsqnonlin command, which is for solving nonlinear least-squares (nonlinear
data-fitting) problems, in Matlab. After solving for ȟܺǡ ȟܻand ߶, the coordinate of the
displaced isolators can be determined by equations similar to those of Equation (4.5-1).
Subtracting the displaced coordinates by the original coordinates gives the displacement
components of the isolators.
ͶǤͷǤʹ •‘Žƒ–‘”‡ƒ –‹‘•ƒ† ‹–‹ƒŽ‡”–‹ ƒŽ ‘” ‡‘ˆ–Š‡•
The X, Y and Z components of the recorded dynamic force from all load cells of a
TPB were added to get the X, Y and Z components of the dynamic reaction at the load
cells level. This reaction was then modified by the inertia forces of the connecting plate
97
and the bottom concave plate of the bearing to get the dynamic reaction at the TPB level.
From the free body diaphragm in Figure 4-24, the relationships between the dynamic
reaction components at the TPB level ܴ௑ , ܴ௓ and the dynamic reaction components at the
load cell level ܴ௖௑ ǡ ܴ௖௓ are:
ܴ௑ ൌ ܴ௖௑ െ ݉௖ ܽ௖௑
(4.5-3)
ܴ௓ ൌ ܴ௖௓ െ ݉௖ ܽ௖௓
(4.5-4)
where ݉௖ ǡ ܽ௖௑ and ܽ௖௓ are mass, horizontal acceleration and vertical acceleration of the
compound including top connecting plate and the bottom concave plate of the TPB.
These reactions are dynamic reactions so that the participation of the gravity load is not
Colum base
ܴ௓ ܴ௑ ܴ௓ Top concave plate of
the TPB
ܴ௑ Bottom concave plate
of the TPB
ܽ௖௓ ݉௖ ܽ௖௑ ܽ௖௑ ݉௖ ܽ௖௓ ܴ௖௑ Top connecting
plate
ܴ௖௓ Figure 4-24
Free body diaphragm illustrating derivation of isolator reaction
98
included in the equations. Note that the reactions ܴ௑ and ܴ௓ in Equations (4.5-3) and
(4.5-4) are actually at the top surface of the bottom concave plate of the TPB.
The vertical acceleration at the top connecting plate was not measured but
computed from the measured acceleration of the shake table assuming that the load cells
are vertically rigid. The vertical acceleration of the shake table at every TPB was
extrapolated from the measured acceleration at the center of the shake table including the
effect of roll and pitch components. The validity of these extrapolated accelerations was
checked by comparing the extrapolated acceleration at the 4 corners of the shake table to
the recorded acceleration at these locations (Figure 4-25). The accelerations recoded from
88RRS test, which has the largest vertical acceleration, was used in this comparison. The
comparison shows very little difference between the extrapolated accelerations and the
recorded accelerations so that the validity of the extrapolated acceleration is confirmed.
The recorded data of the load cells used in this modification process was not
filtered. The raw data was used because the filtering process dramatically alters the
recorded data when the TPB is uplifted (Figure 4-26). During uplift, the TPB does not
support the superstructure and the vertical force ‫ܨ‬௅௓ in every load cell beneath this TPB is
constant as explained below. After uplifting, the vertical force in the load cell increases
suddenly because of the vertical impact and causes a very sharp corner in the time history
of the vertical force. This sharp corner cannot be captured by a finite number of harmonic
functions (generated by filtering process), which causes the filtered data to differ
significantly from the unfiltered data.
99
AZ (g)
1.5
SE corner
0
Recorded
Extrapolated
-1.5
AZ (g)
1.5
SW corner
0
-1.5
AZ (g)
1.5
NE corner
0
-1.5
AZ (g)
1.5
NW corner
0
-1.5
1
1.5
2
Figure 4-25
2.5
3
3.5
Time, t (s)
4
4.5
5
5.5
6
Extrapolated vs. recorded accelerations at corners of the shake table:
88RRS excitation of the isolated structure
600
Unfiltered
Filtered
500
Force, FLZ (kN)
400
300
200
100
0
-100
-200
3.7
Figure 4-26
33%
3.75
3.8
3.85
3.9
Time, t (s)
3.95
4
4.05
4.1
Effect of filtering on the recorded data of a load cell beneath the center TPB
100
Static vertical force at all load cells was measured before the test series. The load
cells were zeroed before the first simulation so that they measured only the dynamic force
variation during the simulations. The measurement was continuous from test to test, but
the sampling process was only done during the excitation. This means that the measuring
of the next test started from the measuring at the end of the previous test and any
redistribution of vertical loads on the bearings were reflected as offsets in the vertical
forces at the beginning of each new simulation. The static vertical force of the load cells
before the test series was used to compute the initial vertical reaction of the TPBs. The
total vertical reaction of a TPB was then computed as the summation of the initial vertical
reaction and the dynamic vertical reaction. However, the initial vertical reaction
computed from this approach was found to be unreliable because of the installation
process, including shimming, bolt fastening processes… For instance, the total vertical
reaction read after settling the specimen on the isolation system was ͷ͵͸͸݇ܰ (Table 4-5)
while the total vertical reaction read at the end of the installation (Table 4-6) was
ͷͶͻʹ݇ܰ. As an alternative, the initial static vertical reaction ܴ௓ǡ௜௡௜௧ of a TPB was
derived by offsetting the dynamic vertical reaction of that TPB such that the vertical
reaction during uplift equals zero (Figure 4-27). This approach was possible because
uplift occurred in every TPB at least once during the test series.
The initial vertical reactions at all TPBs computed from this approach are shown
in Table 4-10. The total weight of the specimen, excluding the weight of TPBs (͵Ͳ݇ܰȀ
݄݁ܽܿሻ, computed from these initial vertical reactions is ͷʹʹͲ݇ܰ. The static vertical load
101
2000
1500
RZ (kN)
1000
500
0
-500
-1000
0
ܴ௓ǡ௜௡௜௧
1
2
3
4
5
Time, t (s)
6
7
8
9
10
7
8
9
10
Offset
2500
2000
Z
R (kN)
1500
1000
500
0
-500
0
1
Figure 4-27
Table 4-10
2
3
4
5
Time, t (s)
6
Offsetting the dynamic vertical reaction to get the total vertical reaction
at the Center TPB: 88RRS excitation
Initial vertical reaction at all TPBs computed from dynamic reaction
Unit: (kN)
SE
449
S
790
SW
467
E
860
C
486
W
490
NE
650
N
607
NW
554
102
at all bearings at the beginning of all simulations is shown in Figure 4-28. Observe that
the static vertical load at a bearing was redistributed at every simulation.
Vertical reaction, R z (kN)
900
SE
E
NE
S
C
N
SW
W
NW
800
700
600
500
400
300
2
4
Figure 4-28
6
8
10
12
Test
14
16
18
20
22
24
Vertical reaction at bearings at the beginning of all simulations
ͶǤͷǤ͵ ‡”‹˜‹‰ –‡”ƒŽ ‘” ‡•‘ˆ–Š‡‘”–Š‡ƒ•–‘Ž—ƒ•‡†‘–”ƒ‹ƒ–ƒ
Axial stress ߪ௭ at any point (‫ݔ‬ǡ ‫ )ݕ‬in a cross section of a column member (Figure
4-29) is determined from bending moments ‫ܯ‬௫ ǡ ‫ܯ‬௬ and axial force ܰ௭ as follows:
‫ݕ‬
Measured
strain
ሺ‫ݔ‬ǡ ‫ݕ‬ሻ
‫ܯ‬௫
‫ݔ‬
‫ܯ‬௬
Figure 4-29
Diagram illustrating the computation of axial stress on a cross section
of column members
103
ߪ௭ ൌ
‫ܯ‬௬
‫ܯ‬௫
ܰ௭
‫ݕ‬൅
‫ݔ‬൅
‫ܫ‬௫
‫ܫ‬௬
‫ܣ‬
(4.5-5)
where ‫ܫ‬௫ ǡ ‫ܫ‬௬ and ‫ ܣ‬are the moment of inertia about ‫ ݔ‬െ and ‫ ݕ‬െaxes and the area of the
section, respectively. The positive bending moment convention is shown in Figure 4-29.
The axial force is positive if it applies tension to the section.
Axial strain ߳௭ is computed from the axial stress using Hook’s law with the
Young modulus ‫( ܧ‬assuming ߳௬ ൌ ߳௫ ൌ Ͳ):
߳௭ ൌ
‫ܯ‬௬
ߪ௭ ͳ ‫ܯ‬௫
ܰ௭
ൌ ቆ ‫ݕ‬൅
‫ݔ‬൅ ቇ
‫ܫ ܧ ܧ‬௫
‫ܫ‬௬
‫ܣ‬
(4.5-6)
This equation is valid in the elastic range, which is applicable to the column sections
because strain was measured far from the column’s end (about 1.8 times the section’s
height) and the column did not experience significant plasticity.
Based on Equation (4.5-6), 3 components of the internal force (‫ܯ‬௫ ǡ ‫ܯ‬௬ ǡ ܰ௭ ) at a
section can be solved if axial strains ߳௭ଵ ǡ ߳௭ଶ ǡ ߳௭ଷ at 3 different locations on that section
are known:
‫ݕ‬ଵ
‫ۍ‬
‫ܫ ێ‬௫
ͳ ‫ݕێ‬ଶ
‫ܫ ێ ܧ‬௫
‫ێ‬
‫ݕێ‬ଷ
‫ܫ ۏ‬௫
‫ݔ‬ଵ
‫ܫ‬௬
‫ݔ‬ଶ
‫ܫ‬௬
‫ݔ‬ଷ
‫ܫ‬௬
ͳ
‫ې‬
‫ۑܣ‬
߳௭ଵ
ͳ ‫ܯ ۑ‬௫
ቐ‫ܯ‬௬ ቑ ൌ ൝߳௭ଶ ൡ
‫ۑܣ‬
߳௭ଷ
‫ܯ ۑ‬௭
ͳ‫ۑ‬
‫ےܣ‬
(4.5-7)
As mentioned before, the axial strain of a section was measured at 4 different
locations on the section (Figure 4-29). These locations were grouped into 4 different
104
groups with 3 sensor locations per group. The internal forces of the section were
computed as the average internal forces from the 4 groups.
ͶǤͷǤͶ
‘”‹œ‘–ƒŽ ‡Ž‡”ƒ–‹‘ƒ†–‘”›”‹ˆ–ƒ– ‡‘‡–”‹ ‡–‡”‘ˆ Ž‘‘”•
As shown in Figure 4-14, horizontal acceleration of a floor was measured at the
SE, NW and NE corners of the floor. The horizontal acceleration at the geometric center
of the floor ܽ௫஼ ǡ ܽ௬஼ were determined by interpolating the acceleration from these
corners. Specifically:
ܽ௫஼ ൌ
ͳ ܽ௫ௌா ൅ ܽ௫ோ
൬
൅ ܽ௫ேௐ ൰
ʹ
ʹ
ܽ௬஼ ൌ
ͳ ܽ௬ேௐ ൅ ܽ௬ோ
൬
൅ ܽ௬ௌா ൰
ʹ
ʹ
(4.5-8)
(4.5-9)
where ܽ௫ௌா ǡ ܽ௬ௌா are ܺ െ and ܻ െcomponents of the horizontal acceleration at the SE
corner, and so on.
The story drift in X- and Y-direction at the geometric center were also be
interpolated from the measured story drift at the 2 locations shown in Figure 4-12. For
instance, the story drift ߜ௫஼ in the X-direction at the geometric center (Figure 4-30) are
extrapolated from the story drift in the X-direction at the SE and NW corners ߜ௫ௌா , ߜ௫ேௐ
as follow:
ߜ௫஼ ൌ ߜ௫ௌா ൅
‫ܮ‬ଵ
ሺߜ
െ ߜ௫ௌா ሻ
‫ܮ‬ଶ ௫ேௐ
(4.5-10)
105
Y
Laser transducer
X
Geometric
center
ߜ௫ௌா
ߜ௫஼
‫ܮ‬ଵ
Figure 4-30
ߜ௫ேௐ
‫ܮ‬ଶ
Diagram illustrating the computation of drift at the geometric center
Chapter 5
‡‡”ƒŽš’‡”‹‡–ƒŽ‡•—Ž–•
5.1
Responses of Isolation System to Sine-wave Excitation and Friction
Coefficients of Isolators
A unidirectional sine-wave excitation was applied to the isolated base building for
determining the hysteresis loop of the isolation system subjected to cyclic loads. The
period of the sine-wave excitation was selected such that it does not resonate with any
component of the system. The amplitude of the excitation was selected strong enough to
drive the isolation system to moderate displacement, which is about a half of the
displacement limit of the system.
As described in Section 3.5.3, the backbone curve of the designed isolators
includes 3 stages. The hysteresis loop of the designed isolators can be fully determined if
the inner loop, which represents the first pendulum mechanism caused by the sliding of
the inner slider, and the outer loop representing the second pendulum mechanism caused
by the sliding of the articulated sliders, are specified. To excite both these two loops in a
single simulation, the sine-wave excitation was designed with a step increase in
amplitude. Numerically, two sine-waves were connected together and connected at the
zero amplitude phases at the beginning and end by fifth order polynomials so that the
whole motion is continuous in displacement, velocity and acceleration. Figure 5-1 shows
displacement, velocity and acceleration of this target motion.
2
Acc., a (cm/s ) Vel., v (cm/s)
Disp., u (cm)
107
50
0
-50
100
0
-100
200
0
-200
0
5
10
Figure 5-1
15
Time, t (s)
20
25
30
Sine-wave excitation
Figure 5-2 shows time-history response of displacement and forces of the isolator
beneath the South column (IsoS for short) when the isolated base building was subjected
to the sine-wave motion applied in the ܺ direction. As expected, 2 levels of isolator
displacement amplitude are observed. In the larger-amplitude oscillation, both inner
slider and articulated sliders are excited, while only inner slider responses are excited in
the smaller-amplitude oscillation. The vertical load in this isolator is not constant but
varies, due to the effect of overturning. The maximum and minimum vertical load on this
isolator was ͳͲͷͳ and ͵ͺͶ, respectively. A higher frequency component at about
͵ œ can also be observed in this response. Modal analysis of the analytical model of the
isolated base configuration (Chapter 8) shows that this corresponds to the frequency of
the 2nd mode (or the 1st structural mode) in both directions.
The combined plot of displacement and forces in Figure 5-2 (d) shows that the
variation of the horizontal force does not correspond to the variation of the displacement.
Force, F Z (kN)
Force, F X (kN)
Disp., u X (cm)
108
50
(a)
0
-50
200
(b)
0
-200
2000
(c)
1000
0
1
uX/50, cm
(d)
0
-1
(F - 790)/500, kN
F /100, kN
-2
0
Z
X
5
10
15
Time, t (s)
20
25
30
Figure 5-2
Response of the IsoS to the sine-wave excitation
(a) Displacement history, (b) Horizontal load history
(c) Vertical load history, (d) Combined responses
This happens because the normalized force is not linearly proportional to displacement
and the vertical load varies. The normalized force and displacement histories are plotted
together in Figure 5-3. One can see the sudden change in the normalized force when the
isolator sustains a motion reversal the maximum or minimum, causing the friction force
to reverse as the motion direction. This change is more evident in the larger-amplitude
oscillation phase than in the smaller-amplitude oscillation phase, because the small
friction coefficient of inner slider is overshadowed by the contribution of the stiffness of
this pendulum mechanism to the restoring force. After this sudden change, the magnitude
of the normalized force gradually increases while the isolator moves to the peak in the
109
opposite direction. At the larger-amplitude phase, the rate of change is larger when the
isolator moves to the positive direction than when it moves to the negative direction,
where the axial load is larger. The increase of friction coefficient due to the decrease in
vertical load causes this difference. This effect produces the stiffening stage on the
normalized hysteresis loop when the isolator moves to the positive direction as shown in
Figure 5-4. In this figure, all isolators on the South side (on the top row) stiffen when the
isolated structure moves to the positive direction. The reverse phenomenon happens to
the isolators on the North side. For isolators E, C and W, the axial load increases when
the isolators pass the zero displacement (Figure 5-5) so that slight pinching is observed in
these isolators. The breakaway friction, which is the increased friction when sliding starts
(Mokkha et al, 1990), and the effect of velocity on friction coefficient also affect the
shape of the normalized hysteresis loops. The dependence of friction coefficient on axial
load and velocity is investigated in detail later.
0.3
f
0.2
X
u /200, cm
X
0.1
0
-0.1
-0.2
0
5
Figure 5-3
10
15
Time, t (s)
20
25
Normalized force and displacement histories of the IsoS
subjected to the sine-wave excitation
30
110
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
Norm. force, f X
-0.2
-50
-0.1
SE
0
50
-0.1
S
-0.2
-50
0
50
-0.2
-50
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
-0.2
-50
-0.1
E
0
50
0
0
-0.1
C
-0.2
-50
SW
50
W
-0.2
-50
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
50
0
50
Unreliable data
-0.1
-0.2
-50
-0.1
NE
0
50
Figure 5-4
-0.1
N
-0.2
-50
0
Disp., uX (cm)
50
NW
-0.2
-50
0
50
Normalized hysteresis loops of all isolators
subjected to the sine-wave excitation
1
0.5
0
-0.5
u /50, cm
X
-1
0
(FZ - 490)/500, kN
5
Figure 5-5
10
15
Time, t (s)
20
25
Vertical force and displacement histories of the IsoW
subjected to the sine-wave excitation
30
111
The area of the outer loop of all bearings decreases with increasing number of
cycles. This may be the result of the increasing of temperature on sliding surfaces due to
the overlapping movement. This effect is again small on the inner loops because of the
small friction and large stiffness.
Friction coefficients of isolators were estimated from their normalized hysteresis
loops from the sine-wave excitation test. Two approaches were used for estimating the
friction coefficient. The first approach is based on equivalent dissipated energy. The
second approach uses the zero-displacement intercept method suggested by Morgan and
Mahin (2011).
In the equivalent dissipated energy approach, friction coefficients are computed
such that the theoretical normalized loop has the same area as that of the experimental
normalized loop in one cycle at a given displacement. Accordingly, the friction
coefficients of inner slider and articulated slider are computed based on the parameters of
inner and outer loops shown in Figure 5-6. From Figure 5-6 (a), the area of the inner
loop, ‫ܣ‬௜௡௡௘௥ , is:
‫ܣ‬௜௡௡௘௥ ൌ ʹߤଵ ሺ‫ݑ‬௠௔௫ െ ‫ݑ‬௠௜௡ ሻ
(5.1-1)
Or:
ߤଵ ൌ
‫ܣ‬௜௡௡௘௥
ʹሺ‫ݑ‬௠௔௫ െ ‫ݑ‬௠௜௡ ሻ
(5.1-2)
112
݂
(a)
ߤଵ
‫ݑ‬௠௜௡
‫ݑ‬௠௔௫
െߤଵ
‫ݑ‬
݇ଵ
‫ݑ‬௠௜௡
‫ݑ‬௠௔௫
‫ݑ‬ଶ
ʹ‫ݑ‬ଶ
Figure 5-6
ʹߤଶ
݇௕
ʹሺߤௗ െ ߤଵ ሻ
ߤଵ ͳ
ͳ
‫ݑ‬
ʹሺߤଶ െ ߤௗ ሻ
ߤଶ
ߤௗ
ʹߤଵ
݂
(b)
Diagram for computing area of normalized hysteresis loop
(a) Inner loop, (b) Outer loop
The area of the outer loop (Figure 5-6 (b)) is:
ͳ
‫ܣ‬௢௨௧௘௥ ൌ ʹߤௗ ሺ‫ݑ‬௠௔௫ െ ‫ݑ‬௠௜௡ ሻ െ ʹ ʹ‫ݑ‬ଶ ʹሺߤௗ െ ߤଵ ሻ
ʹ
(5.1-3)
The intercept normalized force ߤௗ can then be determined from Equation (5.1-3):
ߤௗ ൌ
‫ܣ‬௢௨௧௘௥ െ Ͷ‫ݑ‬ଶ ߤଵ
ʹሺ‫ݑ‬௠௔௫ െ ‫ݑ‬௠௜௡ െ ʹ‫ݑ‬ଶ ሻ
On the other hand, ߤௗ is also related to ‫ݑ‬ଶ :
(5.1-4)
113
ߤௗ ൌ ߤଶ െ ݇௕ ‫ݑ‬ଶ
(5.1-5)
Combining Equation (5.1-4) and Equation (5.1-5), friction coefficient ߤଶ of articulated
slider can be determined:
ߤଶ ൌ
‫ܣ‬௢௨௧௘௥ െ Ͷ‫ݑ‬ଶ ߤଵ
൅ ݇௕ ‫ݑ‬ଶ
ʹሺ‫ݑ‬௠௔௫ െ ‫ݑ‬௠௜௡ െ ʹ‫ݑ‬ଶ ሻ
(5.1-6)
Given the experimental areas ‫ܣ‬௜௡௡௘௥ and ‫ܣ‬௢௨௧௘௥ of the inner and outer normalized
hysteresis loops, the maximum and minimum displacements ‫ݑ‬௠௔௫ and ‫ݑ‬௠௜௡ of each loop,
and the normalized stiffnesses ݇ଵ and ݇௕ of the inner and outer loops (defined in Section
3.5.2), friction coefficients of the inner and articulated sliders are estimated from
Equation (5.1-2) and Equation (5.1-6), respectively. Because ‫ݑ‬ଶ is dependent on ߤଶ so
that Equation (5.1-6) is not an explicit equation of ߤଶ , the friction coefficient ߤଶ is
determined iteratively.
Table 5-1 shows friction coefficients of 8 isolators computed from the equivalent
dissipated energy approach. The isolator beneath the Northwest column was omitted from
these computations because of the unreliable data (Figure 5-4). Data in this table shows
that friction coefficients decrease after each cycle, which confirms the observation made
earlier. Theoretical normalized hysteresis loops with average friction coefficients of these
isolators are plotted together with experimental loops in Figure 5-7. These theoretical
loops were generated by pseudo-static analysis of a TPB elements (described in Chapter
6) subjected to recorded displacement histories. Obviously, this constant-friction model
cannot capture stiffening and other effects of axial force and velocity variation.
114
Table 5-1
Isolator
Friction coefficients of isolators computed from the
equivalent dissipated energy approach
Inner slider (ߤଵ ǡ Ψ)
Cycle
Cycle
Average
2
3
(5)
(3)
(4)
Cycle
1
(6)
Articulated slider (ߤଶ ൌ ߤଷ ǡ Ψ)
Cycle
Cycle
Cycle
Average
2
3
4
(10)
(7)
(8)
(9)
(1)
Cycle
1
(2)
S
1.66
1.56
1.52
1.58
10.93
9.77
9.01
8.46
9.54
C
1.45
1.39
1.39
1.41
11.23
10.44
9.76
9.35
10.19
N
1.97
1.84
1.78
1.86
9.73
9.59
9.16
8.91
9.35
SE
1.46
1.39
1.37
1.41
12.99
11.89
11.29
10.81
11.74
E
1.43
1.36
1.31
1.37
10.45
9.34
8.62
8.11
9.13
NE
2.02
1.92
1.83
1.92
10.20
9.41
8.75
8.42
9.19
SW
1.51
1.45
1.45
1.47
11.86
10.45
9.73
9.09
10.28
W
2.03
1.89
1.85
1.92
10.86
10.17
9.51
9.17
9.93
NW
NA
NA
NA
NA
NA
NA
NA
NA
NA
Average
1.69
1.60
1.56
1.62
11.03
10.13
9.48
9.04
9.92
115
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
Norm. force, f X
-0.2
-50
-0.1
SE
0
50
-0.2
-50
-0.1
S
0
50
-0.2
-50
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
-0.2
-50
-0.1
E
0
50
-0.2
-50
50
-0.2
-50
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
-0.1
-0.1
-0.2
-50
Figure 5-7
NE
0
50
-0.2
-50
N
0
Disp., uX (cm)
0
-0.1
C
0
SW
50
-0.2
-50
50
W
0
50
Theory
Test
Unreliable data
NW
0
50
Theoretical and experimental normalized hysteresis loop of all isolators
subjected to the sine-wave excitation: constant friction model
In the zero-displacement intercept method (Morgan and Mahin, 2011), the friction
coefficient ߤଵ and zero-displacement intercept normalized force ߤௗ are computed as (see
Figure 5-8):
ߤଵ ൌ ‫ܪ‬ଵ Ȁʹ
(5.1-7)
ߤௗ ൌ ‫ܪ‬ଶ Ȁʹ
(5.1-8)
Friction coefficient ߤଶ can be computed from ߤௗ using Equation (5.1-5). This method
works well if the friction coefficients are constant and the normalized hysteresis is
piecewise linear. If friction coefficients change, ‫ܪ‬ଵ and ‫ܪ‬ଶ can be interpreted as heights
116
݂
ߤௗ
‫ܪ‬ଶ
‫ܪ‬ଵ
b
a
‫ݑ‬
െߤௗ
Figure 5-8
Diagram for computing friction coefficient based on zero-displacement
intercept method
of the loops at a certain displacement (zero) and cannot represent the height of the whole
loop. In such a case, the method can be extended to estimate friction coefficient at every
data point.
As depicted in Figure 5-9, the zero-displacement intercept normalized force ߤௗ at
any data point coming from the third pendulum mechanism, the mechanism caused by
sliding of articulated sliders on top and bottom concave plates, can be computed. Friction
coefficient ߤଶ ሺൌ ߤଷ ሻ of the articulated sliders can then be computed from Equation (5.15). This extended method does not work for the inner loop when the initial displacement
is not zero. As shown in Figure 5-8, when the isolator starts moving from non-zero initial
displacement, the zero-displacement intercept of the inner loop differs from the friction
coefficient ߤଵ . However, the outer loop, an envelope loop, does not depend on the initial
117
݂
ߤௗଶ
ߤௗଵ
1
2
‫ݑ‬
Figure 5-9
Zero-displacement intercept of a data point
displacement. For example, no matter if the isolator starts moving from a or b (Figure 58), it produces the same outer loop given the same displacement amplitude and isolator
properties.
In the sin-wave excitation test, all isolators started from a non-zero initial
displacement caused by the previous simulation, so that their friction coefficients ߤଵ
cannot be computed by the zero-displacement intercept method. Luckily, as mentioned
earlier, the normalized force of the inner loop is dominated by the restoring force and the
change in friction coefficient on the normalized hysteresis loop can be neglected. In other
words, the inner loop can be sufficiently reproduced using a constant friction coefficient
ߤଵ estimated by the equivalent dissipated energy assumption. Comparison of the
theoretical loops and experimental loops in Figure 5-7 supports this conclusion.
118
Friction coefficient ߤଶ at selected data points of the first outer cycle of the 8
bearings was computed. The data points were restricted to time steps where the
movement of bearings was uncomplicated by the reversal movement. The first outer
cycle was selected for the practical reason that during a single simulation (as well as real
earthquakes) the displacement trace of one bearing is not expected to repeat on the same
route, except at small displacement, where sliding only occurs to the inner slider. The
friction coefficient then be plotted against vertical load and sliding velocity to find the
fitted relationships between sliding velocity, vertical load and friction coefficient.
Mokha et. al. (1990) investigated several effects on friction coefficient of Teflon
surface and concluded that sliding velocity and bearing pressure have significant
influence on the friction coefficient. The effect of these factors on friction coefficient can
be approximately expressed by following equation (Constantinou et. al., 1990):
ߤ ൌ ߤ௠௔௫ െ ሺߤ௠௔௫ െ ߤ௠௜௡ ሻ ‡š’ሺെܽ‫ݑ‬ሶ ሻ
(5.1-9)
where ߤ௠௔௫ and ߤ௠௜௡ are friction coefficients at very high and low velocity, ܽ is a rate
parameter dependent on bearing pressure, and ‫ݑ‬ሶ is sliding velocity.
Relationship between pressure, or vertical force, and friction coefficient at a
constant velocity was not explicit proposed by these authors. However, Bowden and
Tabor (1964) suggested that friction coefficient ߤ relates to vertical force ܹ by:
ߤ ൌ ‫ ܹܣ‬௡ିଵ
(5.1-10)
where ‫ ܣ‬is a constant and ݊ is a coefficient less than unity, which may not be constant
over a very wide load range. The test data from Mokha et. al. (1988) supports this form of
119
relationship and indicates that ݊ depends on sliding velocity; that is higher velocity leads
to larger values of ݊.
Figure 5-10 shows the relationship between friction coefficients of the outer loops
and vertical load in the bearing at low and high sliding velocity, along with fitted curves.
Velocity less than ͲǤͲʹȀ• was characterized as low while velocity greater than
ͲǤʹͷȀ• was characterized as high. The fitted equations follow Equation (5.1-10) are
also presented in the figure. The least squares method was used for finding the fitted
curves.
Relationship between the friction coefficient and velocity at vertical loads ܹ of
͵ͲͲ േ ͷͲ, ͶͲͲ േ ͷͲ,…, and ͳͲͲͲ േ ͷͲ is shown in Figure 5-11. The least
squares fitted curves following Equation (5.1-9) are also presented. At very low vertical
load, the friction coefficient does not saturate but rather reduces when sliding velocity
0.15
Friction coefficient, μ
-0.38
μ=17.239 W
0.1
-0.34
μ=8.701 W
0.05
0
0
Figure 5-10
Slow friction
Fast friction
Fitted curves
2
4
6
8
Vertical load, W (N)
10
12
5
x 10
Dependence of friction coefficient on vertical load
120
becomes very large. This requires further study on friction coefficient of TPB at high
sliding speed and low vertical load to improve understanding of TPBs.
Much effort in the past has gone into predicting and modeling the velocity effect
on the friction coefficient, when in fact the influence of axial load on friction coefficient
is shown here to be much more significant than velocity (compare Figures 5-10 and 511). This should be the case for large scale structures subjected to strong motions,
especially for buildings with large height to width ratios, which may not be represented
by the majority of scaled structures tested in the past. In structures tested at full scale, the
velocity is often high enough so that it is above the threshold at which the friction
coefficient varies. On the other hand, strong overturning of buildings with large height to
width ratio causes strong fluctuation of vertical force in individual bearings.
The rate parameter ܽ in the equation representing the relationship between friction
coefficient and velocity is plotted against vertical load ܹ in Figure 5-12. This figure
suggests that the relationship between ܽ and ܹ can be approximately represented by a
parabolic curve:
ܽ ൌ ߙ଴ ൅ ߙଵ ܹ ൅ ߙଶ ܹ ଶ
(5.1-11)
where ߙ଴ ǡ ߙଵand ߙଶ are fitted constants.
Given relationship between friction coefficient and vertical load at slow and fast
velocities (Equation (5.1-10), Figure 5-10) and relationship between rate parameter and
vertical load (Equation (5.1-11), Figure 5-12), friction coefficient at any velocity and
vertical load can be determined using these equations in combined with Equation (5.1-9).
121
0.15
0.15
0.1
μ=0.1416-0.0231e
0.1
-22.92v
-11.32v
μ=0.1275-0.0196e
0.05
0.05
ܹ ൌ ͵ͲͲ േ ͷͲ݇ܰ
0
0
0.1
0.2
0.3
0.4
0
0.15
0.15
0.1
0.1
μ=0.1185-0.0175e
Friction coefficient, μ
ܹ ൌ ͶͲͲ േ ͷͲ݇ܰ
0
0.1
0.05
0
0.15
ܹ ൌ ͸ͲͲ േ ͷͲ݇ܰ
0.1
0.2
0.3
μ=0.1035-0.0140e
0.4
0
0
0.15
-11.64v
0.1
0.1
0.05
0.05
0.1
0
0.15
0.1
0.2
μ=0.0947-0.0120e
0.3
0.4
0
-2.88v
0
0.1
0.05
0.3
0.3
0.4
0
0.4
0
Velocity, v (m/s)
-16.69v
ܹ ൌ ͳͲͲͲ േ ͷͲ݇ܰ
ܹ ൌ ͻͲͲ േ ͷͲ݇ܰ
Figure 5-11
0.2
μ=0.0898-0.0110e
0.05
0.2
0.4
0.15
-11.20v
0.1
0.1
0.3
ܹ ൌ ͺͲͲ േ ͷͲ݇ܰ
0.1
0
0.2
μ=0.0987-0.0129e
ܹ ൌ ͹ͲͲ േ ͷͲ݇ܰ
0
0.4
-6.01v
ܹ ൌ ͷͲͲ േ ͷͲ݇ܰ
0
0.3
μ=0.1098-0.0154e
-8.92v
0.05
0
0.2
0.1
0.2
0.3
Dependence of friction coefficient on velocity at different vertical loads
0.4
122
Rate parameter, a (s/m)
30
a = 54.411 - 0.141×10-3×W
+ 0.102 ×10-9×W2
25
20
15
10
5
0
0
Figure 5-12
Data
Fitted curve
2
4
6
8
Vertical load, W (N)
10
12
5
x 10
Rate parameter at different vertical loads
Theoretical loops with this general friction model and the experimental loops of
the 8 bearings subjected to the sine-wave excitation are compared in Figure 5-13. The
theoretical loops match well with the first cycles of the outer loop of the experimental
hysteresis loops, which were used to construct the friction model. The theoretical loops
can especially capture the pinching behavior of the experimental loop (see the isolators at
N and W).
5.2
General Response of the Isolation System to Earthquake Motions
ͷǤʹǤͳ ‡ƒ •‘Žƒ–‘”‹•’Žƒ ‡‡–
Table 5-2 shows peak displacement over all isolators of the isolation system for
each of the 19 earthquake motion excitations. The smallest and largest peak isolator
displacement in a single simulation were ͳͷǤͻ  and ͹ͲǤʹ , respectively. The X- and
Y-coordinate of the peak displacements are listed in Table 5-2, and plotted in scatter
format in Figure 5-14. The expected displacements at different earthquake scenarios,
123
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
Norm. force, f X
-0.2
-50
-0.1
SE
0
50
-0.2
-50
-0.1
S
0
50
-0.2
-50
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.1
-0.2
-50
-0.1
E
0
50
-0.2
-50
50
0.2
0.2
0.1
0.1
0.1
0
0
0
-0.2
-50
-0.1
NE
0
50
-0.2
-50
0
Disp., uX (cm)
0
50
50
Theory
Test
Unreliable data
-0.1
N
50
W
-0.2
-50
0.2
-0.1
0
-0.1
C
0
SW
-0.2
-50
NW
0
50
Figure 5-13
Theoretical and experimental normalized hysteresis loop of all isolators
subjected to the sine-wave excitation: variable friction coefficient
which were computed based on the design friction coefficients of the isolators, are also
plotted in this figure. The peak isolator displacement for most simulations falls between
the expected displacements of the DBE scenario and the MCE scenario. As shown in last
section, the observed friction coefficients during the test were higher than the design
friction coefficients. Thus the peak displacements due to TCU and TAB motions, for
which the isolation system was designed, were not as large as expected. Recall that the
pre-test analysis predicted TCU and TAB motions to cause the displacement of the
isolation system to exceed the expected displacement of the MCE scenario.
124
Table 5-2
Peak displacement of the isolation system for each earthquake motion
Number
Run
Peak disp. (cm)
Disp. X at peak
(cm)
Disp. Y at peak
(cm)
1
80WSM
15.900
-15.510
3.498
2
130ELC
20.456
-18.238
9.263
3
88RRS
42.705
-22.883
36.057
4
100SYL
51.673
-22.159
-46.681
5
50TAB
27.542
-18.944
-19.992
6
70LGP
49.391
-3.093
-49.294
7
50TCU
34.299
31.038
-14.597
8
70TCU
51.029
49.779
-11.227
9
100IWA
38.358
7.775
-37.562
10
100SAN
33.346
23.040
24.107
11
100TAK
56.008
-54.008
14.834
12
100KJM
27.324
13.514
-23.748
13
88RRS2D
36.752
-13.304
34.260
14
80TCU
53.230
52.501
-8.781
15
80TAB
43.210
-36.232
-23.544
16
90TAB
53.159
-46.066
-26.530
17
100TAB
70.190
69.876
-6.633
18
100SCT
37.586
-34.866
-14.038
19
115TAK
61.314
-59.555
14.581
125
Limit
Disp. Y, u Y (cm)
100
MCE
50
DBE
Service
0
-50
-100
-100
Figure 5-14
0
100
Disp. X, uX (cm)
200
Peak displacement of the isolation system for each earthquake motion
relative to scenario limits
ͷǤʹǤʹ ‡ƒ‘”•‹‘‘ˆ–Š‡ •‘Žƒ–‹‘›•–‡
Friction isolation systems are known for their ability to reduce the torsional
displacement of the isolated structure. The restoring force, friction force and tangent
stiffness in a friction isolator are proportional to the weight (or mass) that it carries.
Because of this, in an isolation system having identical friction isolators, the distribution
of horizontal support stiffness is proportional to the distribution of mass of superstructure
and the distribution of reactions in the isolators is proportional to the distribution of
inertia force applied to the isolated structure. This is expected to eliminate the torsional
movement of the isolated structure subjected to ground motion excitations. The
experimental data from this test will be shown to confirm this advantage of the friction
isolation system.
As mentioned before, the specimen is asymmetrical due to different bay widths in
Y-direction and the placement of isolators beneath all the columns extends the
126
asymmetry to the isolation system. The asymmetry is exaggerated by introducing the
asymmetrically configured supplemental steel weight at the roof. Despite this asymmetry,
the torsional movement of the isolation system was small during the test. Table 5-3
summarizes peak torsion and the peak discrepancy in displacement across all isolators for
each earthquake motion. The peak torsion angle was ͲǤͷͶͳ ൈ ͳͲିଷ ”ƒ† and the peak
difference in displacement between isolators was ͺǤ͵ , which occurred during response
to 80TCU motion. The peak torsion of the isolation system in each excitation is plotted as
a bar graph in Figure 5-15.
For a typical isolation system with resisting force and stiffness independent of the
weight carried, the peak torsion is expected to increase along with the increasing of peak
isolator displacement. This is not the case for the friction isolation system. As shown in
Figure 5-16, Figure 5-17 and Figure 5-18, no correlation is observed between the peak
displacement and the peak torsion. Thus the torsion appears to be a small accidental
effect rather than a true system asymmetry. The accidental torsion may be from the
variation of friction coefficients between bearings because of the uneven distribution of
vertical force on the bearings. It may also come from the unexpected distribution of static
vertical force on bearings due to the warping of the specimen base (Section 4.2).
127
Table 5-3
Peak torsion of the isolation system subjected to each
earthquake motion excitation
Number
Run
Peak torsion (rad)
Max difference in
isolator disp. (cm)
1
80WSM
0.00416
6.370
2
130ELC
0.00508
7.105
3
88RRS
0.00449
7.010
4
100SYL
0.00416
6.430
5
50TAB
0.00470
7.314
6
70LGP
0.00339
4.450
7
50TCU
0.00311
4.840
8
70TCU
0.00377
5.649
9
100IWA
0.00343
5.357
10
100SAN
0.00383
5.710
11
100TAK
0.00504
7.735
12
100KJM
0.00215
3.082
13
88RRS2D
0.00319
4.374
14
80TCU
0.00541
8.338
15
80TAB
0.00375
5.826
16
90TAB
0.00464
7.226
17
100TAB
0.00540
7.821
18
100SCT
0.00503
7.460
19
115TAK
0.00301
4.578
128
x 10
4
115TAK
100SCT
100TAB
90TAB
80TAB
80TCU
88RRS2D
100KJM
100TAK
100SAN
100IWA
70TCU
50TCU
70LGP
50TAB
130ELC
100SYL
0
88RRS
2
80WSM
Peak torsion, φmax (rad)
-3
6
Excitation motion
Figure 5-15
Peak torsion of the isolation system subjected to all
earthquake motion excitations
-3
Peak torsion, φmax (rad)
6
x 10
5
4
3
2
0
10
20
30
40
Peak iso. disp., u
max
Figure 5-16
50
(cm)
60
70
80
Peak torsion vs. peak displacement of the isolation system
-3
Peak torsion, φmax (rad)
6
x 10
5
4
3
2
0
10
20
30
40
Peak iso. disp. X, u
Xmax
Figure 5-17
50
(cm)
60
70
Peak torsion vs. peak isolator displacement in X-direction
80
129
-3
Peak torsion, φmax (rad)
6
x 10
5
4
3
2
0
10
20
30
40
Peak iso. disp. Y, u
Ymax
Figure 5-18
50
(cm)
60
70
80
Peak torsion vs. peak isolator displacement in Y-direction
ͷǤʹǤ͵ ‡•‹†—ƒŽ •‘Žƒ–‘”‹•’Žƒ ‡‡–
Maximum residual displacement over all isolators for each earthquake motion is
summarized in Table 5-4 and by bar graph in Figure 5-19. The maximum residual
displacement of the isolation system during the test was ͳͲǤͺ , which is ͻǤ͸Ψ of the
displacement limit. One may observe from this figure that the residual displacement is
accumulated from 100IWA to 80TCU and from 80TAB to 100TAB. However, the
residual displacement trace of the center bearing in Figure 5-20 shows that the residual
displacement does not accumulate in any specific direction, so that the increasing of
residual displacement may just be an accident. The comparison of isolator displacement
between analysis and test data later (Chapter 8) shows that in the response to 80TCU and
100TAB, the analysis and test displacement histories are offset at the beginning (because
the analysis histories start from zero displacement) but aligned after the displacements
reach to a large pulse. This means that the initial displacement does not affect the residual
displacement in some cases and the residual displacement was not accumulated. The
displacement history of the center isolator from 100TAK to 100TAB is shown in Figure
130
5-21. The increasing of residual from 80TAB to 100TAB is purely from the increasing of
the motion amplitude.
Table 5-4
Maximum residual isolator displacement
Number
Run
Residual disp.
(cm)
Residual disp. X
(cm)
Residual disp.Y
(cm)
1
80WSM
6.746
-6.348
2.282
2
130ELC
5.649
-5.003
2.624
3
88RRS
3.742
3.678
0.686
4
100SYL
2.521
1.846
1.717
5
50TAB
3.867
2.673
-2.794
6
70LGP
7.661
1.263
-7.557
7
50TCU
5.945
-2.693
5.299
8
70TCU
10.341
-2.535
10.025
9
100IWA
0.960
-0.752
-0.595
10
100SAN
1.771
1.499
0.942
11
100TAK
3.022
1.647
2.533
12
100KJM
4.605
4.402
1.353
13
88RRS2D
6.565
5.854
2.970
14
80TCU
10.817
-3.698
10.165
15
80TAB
5.922
5.445
2.329
16
90TAB
8.241
8.158
1.171
17
100TAB
10.196
10.049
1.720
18
100SCT
1.598
1.341
-0.870
19
115TAK
3.970
1.850
3.512
15
10
Figure 5-19
Maximum residual isolator displacement
15
10
5
Start
0
-5
-10
-15
-15
-10
Figure 5-20
-5
0
5
Residual disp. X, uresX (cm)
10
Residual displacement trace of the Center isolator
15
115TAK
100SCT
100TAB
90TAB
80TAB
80TCU
88RRS2D
100KJM
100TAK
100SAN
100IWA
70TCU
50TCU
70LGP
50TAB
100SYL
88RRS
Excitation motion
Residual disp. Y, u resY (cm)
0
130ELC
5
80WSM
Residual disp., u res (cm)
131
132
Disp. X,uX (cm)
100
100TAB
100TAK
50
100KJM
88RRSXY 80TCU
80TAB
90TAB
0
-50
-100
Disp. Y,uY (cm)
50
0
-50
0
50
Figure 5-21
100
150
200
Time,t (s)
250
300
350
400
Displacement history of the center isolator from 100TAK to 100TAB
Figures 5-22 and 5-23 show the displacement history of the center isolator in
100IWA and 100SCT, respectively. These two long far-field subduction motions tend to
re-center the bearing. However, the re-centering process can only happen if the motion is
strong enough. As can be seen from Figure 5-8, if the bearing starts moving from ܾ and
the motion is not strong enough to excite the articulated sliders, the hysteresis loop is
limited in the inner loop and the bearing can never be centered, no matter how long the
motion is. This problem is evident for the bearings whose inner friction coefficient is
much smaller than the outer friction coefficients. This comment is confirmed by
observing the displacement history of the center isolator subjected to 80WSM and
130ELC (Figure 5-24). The residual displacement caused by the sin-wave excitation,
prior to 80WSM, does not seem to be reduced after these two earthquake excitations.
133
Disp. X,uX (cm)
20
0
-20
-40
Disp. Y,uY (cm)
20
0
-20
-40
0
20
40
Figure 5-22
60
80
100
Time,t (s)
120
140
160
180
200
80
90
Displacement history of the center isolator in 100IWA
Disp. X,uX (cm)
50
0
-50
Disp. Y,uY (cm)
20
0
-20
-40
0
10
Figure 5-23
20
30
40
50
Time,t (s)
60
70
Displacement history of the center isolator in 100SCT
134
Disp. X,uX (cm)
20
10
0
-10
-20
Disp. Y,uY (cm)
10
0
-10
-20
0
10
Figure 5-24
20
30
40
50
Time,t (s)
60
70
80
90
100
Displacement history of the center isolator from 80WSM to 130ELC
Maximum residual displacement is plotted against the peak isolator displacement
in Figure 5-25. The plot shows that there is not any correlation between the peak isolator
displacement and the residual displacement. Figure 5-26 and Figure 5-27 show the plots
of the residual displacement against the duration since the peak table acceleration or the
peak isolator displacement happens to the end of the record. The end of the record is
defined as the last instant when the exciting acceleration exceeds ͲǤͲͳ‰. The figures
show no correlation between the residual displacement and these durations except that at
very long records (100IWA, 100SCT, 100SAN), the residual displacement is small (as
presented earlier). This supports the thought that the aftershocks of an earthquake can
help re-centering friction bearings after the main shock. However, the aftershocks may
not be long and strong enough to help as explained. Careful investigation is required to
make any conclusion about this issue.
135
Residual disp, u res (cm)
15
10
5
0
10
20
30
40
Peak disp., u
max
Figure 5-25
50
(cm)
60
70
80
Residual isolator displacement vs. peak isolator displacement
Residual disp., u res (cm)
15
10
5
0
0
50
100
150
200
Duration from peak acc. to end, T
(s)
250
300
EndA
Figure 5-26
Residual displacement vs. duration from peak acceleration to the end of the
ground motion
Residual disp., u res (cm)
15
10
5
0
0
50
100
150
Duration from peak disp. to end, T
EndD
Figure 5-27
200
250
(s)
Residual displacement vs. duration from peak isolator displacement to the
end of the ground motion
136
ͷǤʹǤͶ ’Ž‹ˆ–
The TPB used in this experiment have no tensile resistance and the isolators could
be uplifted when subjected to overturning and/or the strong vertical excitation. In TPB,
the horizontal movement of the isolator generates vertical displacement because of the
curvature of the concave plates. This causes difficulty in measuring the uplift
displacement of the isolators using displacement transducers. Alternatively, the uplift of
the isolators was determined by examining the vertical reaction. Any instant when the
vertical reaction reduced to zero was recognized as an occurrence of uplift. Table 5-5
summarizes the number of uplift occurrence of every isolator during each simulation.
Every bearing uplifted at least once during the test series, and all but the Southwest
isolator uplifted during the excitation of 88RRS. Further investigation indicated that the
total vertical reaction subjected to this excitation was almost zero, synonymous with near
total uplift of the whole building for a duration of about ͲǤͳ• during the 88RRS
excitation. At the beginning of the test series, uplift was restricted mainly to the Center
isolator. After 88RRSXY excitation, the center of the uplift shifted to the South isolator.
The concentration of uplift on some bearings may come from the warping of the base of
the specimen described in Chapter 4. Recall that at the installation process, the load on
the Center bearing was almost zero before shimming. The detail of the initial reaction
during the test was presented in Section 4.5.2.
137
Table 5-5
Number of uplift excursion
Bearing
Number
Run
S
C
N
SE
E
NE
SW
W
NW
1
80WSM
0
0
0
0
0
0
0
0
0
2
130ELC
0
0
0
0
0
0
0
0
0
3
88RRS
6
30
7
7
7
4
0
4
4
4
100SYL
0
2
1
1
1
0
0
1
2
5
50TAB
0
2
0
0
0
0
0
0
0
6
70LGP
0
6
0
0
0
0
0
1
0
7
50TCU
0
0
0
0
0
0
0
0
0
8
70TCU
0
0
0
0
0
0
0
0
0
9
100IWA
0
0
0
0
0
0
0
0
0
10
100SAN
0
0
0
0
0
0
0
0
0
11
100TAK
0
1
3
0
0
0
0
0
1
12
100KJM
0
0
0
0
0
0
1
0
0
13
88RRS2D
0
0
0
0
0
0
0
0
0
14
80TCU
12
0
0
0
0
0
0
0
0
15
80TAB
37
11
6
0
0
0
0
0
0
16
90TAB
12
0
0
0
0
0
0
0
1
17
100TAB
16
0
0
0
0
0
0
0
0
18
100SCT
10
0
0
0
0
0
0
0
0
19
115TAK
26
0
0
0
0
0
0
0
0
ͷǤʹǤͷ ‡ƒƒ•‡Š‡ƒ”
Peak vector-sum base shear, peak base shear in X- and Y-direction of the isolated
base building are presented in Table 5-6. The maximum base shear over all simulation is
138
Table 5-6
Peak base shear of the isolated base structure
Number
Run
Vector-sum
peak base
shear (kN)
Peak base
shear X (kN)
Peak base
shear Y (kN)
1
80WSM
607.06
603.39
362.48
2
130ELC
737.47
672.44
568.86
3
88RRS
1555.4
1147.2
1269.1
4
100SYL
994.81
727.23
930.69
5
50TAB
755.19
697.34
592.9
6
70LGP
902.69
430.87
900.77
7
50TCU
672.54
671.79
565.35
8
70TCU
798.38
794.97
690.08
9
100IWA
755.86
623.61
755.38
10
100SAN
714.84
679.9
592.39
11
100TAK
1119.1
985.63
720.4
12
100KJM
890.29
701.4
750.86
13
88RRS2D
896.3
695.28
703.61
14
80TCU
841.32
828.53
702.96
15
80TAB
1093.6
1009.6
837.93
16
90TAB
1063.3
1043.1
638.79
17
100TAB
1130.1
1102.2
646.93
18
100SCT
739.95
709.21
629.3
19
115TAK
1162.8
1093.9
744.55
ͳͷͷͷǤͶ, corresponding to a base shear coefficient of ͲǤʹͻͳ, observed during 88RRS
motion excitation. The observed base shear coefficient was larger than the normalized
force of the isolation system at limit displacement, which is ͲǤʹ͹ͷ. The base shear
coefficient was increased due to the effect of vertical excitation. Because the vertical
139
excitation causes a fluctuation in vertical reaction, the base shear and the base shear
coefficient (normalized with respect to static weight) change even when the normalized
force is unchanged.
To assess the base shear in term of normalized force, the peak normalized force in
X and Y and vector-sum was computed and summarized in Table 5-7. The normalized
forces at every time step was computed by dividing the base shear by the instantaneous
total vertical reaction, and the peak normalized force subjected to each excitation was the
maximum value of the normalized force throughout the response. These peak normalized
force in X, Y and vector-sum are plotted against the peak displacements of the Center
isolator in Figures 5-28 to 5-30, along with the theoretical normalized backbone curves of
the isolation system. The “Design backbone curve” uses the designed friction coefficient
and the “Test backbone curve” uses the friction coefficient computed from the first loop
using equivalent dissipated energy of the sine-wave test (Section 5.1). The scatter of test
data matches well with the “Test backbone curve”. The discrepancy between the
backbone curve and the test data points results from bidirectional coupling and the
variation of friction due to vertical load and velocity mentioned before. Note that any
initial displacement should affect the peak normalized force only if the hysteresis loop is
limited in the inner loop, as explained in Section 5.1 (Figure 5-8). This explains why the
discrepancy between the test data and theoretical loop for peak displacement limited to
the first stage of sliding is larger than for peak in the second stage of sliding (Figure 5-29
and Figure 5-30). The outlier point in these figures is from the 88RRS simulation, where
140
as described earlier almost the whole building was uplifted for a short duration.
Normalization by a small denominator (small vertical reaction) produced this outlier.
Table 5-7
Peak normalized horizontal force of the isolation system
Number
Run
Vector-sum
peak
normalized
force
1
80WSM
0.113
0.112
0.066
2
130ELC
0.126
0.118
0.097
3
88RRS
0.365
0.328
0.318
4
100SYL
0.184
0.132
0.174
5
50TAB
0.130
0.117
0.104
6
70LGP
0.159
0.079
0.159
7
50TCU
0.124
0.124
0.106
8
70TCU
0.148
0.146
0.129
9
100IWA
0.138
0.119
0.138
10
100SAN
0.133
0.127
0.111
11
100TAK
0.167
0.157
0.127
12
100KJM
0.166
0.152
0.125
13
88RRS2D
0.163
0.132
0.128
14
80TCU
0.156
0.154
0.132
15
80TAB
0.169
0.168
0.110
16
90TAB
0.212
0.212
0.109
17
100TAB
0.220
0.220
0.115
18
100SCT
0.138
0.131
0.117
19
115TAK
0.201
0.196
0.127
Peak
normalized
force X
Peak
normalized
force Y
141
Norm. force, f
0.4
0.3
0.2
Test
Design backbone curve
Test backbone curve
0.1
0
0
Figure 5-28
20
40
60
Iso. disp., u (cm)
80
100
120
Peak normalized horizontal force vs. peak displacement of the Center isolator
Norm. force X, fX (kN)
0.4
0.3
0.2
Test
Design backbone curve
Test backbone curve
0.1
0
0
20
40
60
Iso. disp. X, u (cm)
80
100
120
X
Figure 5-29
Peak normalized horizontal force in the X-direction vs. peak displacement of
the Center isolator in the X-direction
142
Norm. force Y, fY (kN)
0.4
0.3
0.2
Test
Design backbone curve
Test backbone curve
0.1
0
0
20
40
60
Iso. disp. Y, u (cm)
80
100
120
Y
Figure 5-30
5.3
Peak normalized horizontal force in the Y-direction vs. peak displacement of
the Center isolator in the Y-direction
General Response of the Specimen in the Isolated Base Configuration
ͷǤ͵Ǥͳ ‡ƒ Ž‘‘” ‡Ž‡”ƒ–‹‘
Peak acceleration at geometric center of all floors of the isolated specimen is
listed in Table 5-8 and plotted in Figure 5-31 and Figure 5-32. The peak values are vector
sum values of the X- and Y-components. The peak floor acceleration from test did not
meet the performance objective of ͲǤ͵ͷ‰, which was met by the pre-test analysis. The big
difference between the test acceleration and the expected acceleration came from the
amplification of horizontal acceleration due to vertical excitation (see Chapter 9), which
was not captured in the pre-test analytical model. Recall that the test data was lowpass
filtered with cut off frequency of ʹͷ œ. The cut off frequency has significant effect on
the peak acceleration in some cases: smaller cut off frequency produces smaller peak
acceleration. Comparison between acceleration response to 88RRS (which includes 3
components in the excitation) and 88RRSXY (which includes only horizontal
components in the excitation) shows that the existence of vertical excitation affects the
143
Table 5-8
Peak floor acceleration of the isolated base structure
Unit: g
Floor
Number
Simulation
Base
2
3
4
5
Roof
1
80WSM
0.144
0.138
0.133
0.151
0.146
0.150
2
130ELC
0.323
0.242
0.313
0.327
0.255
0.332
3
88RRS
0.864
0.925
0.945
1.147
0.903
0.845
4
100SYL
0.535
0.371
0.475
0.628
0.410
0.598
5
50TAB
0.346
0.282
0.310
0.310
0.236
0.329
6
70LGP
0.425
0.347
0.423
0.407
0.270
0.422
7
50TCU
0.161
0.148
0.134
0.138
0.160
0.180
8
70TCU
0.188
0.158
0.153
0.163
0.183
0.226
9
100IWA
0.269
0.207
0.183
0.178
0.224
0.338
10
100SAN
0.227
0.161
0.156
0.152
0.160
0.222
11
100TAK
0.618
0.395
0.546
0.555
0.376
0.659
12
100KJM
0.642
0.327
0.431
0.505
0.531
0.660
13
88RRSXY
0.334
0.316
0.247
0.239
0.339
0.384
14
80TCU
0.181
0.162
0.158
0.160
0.170
0.206
15
80TAB
0.847
0.401
0.751
0.786
0.387
0.770
16
90TAB
0.377
0.237
0.337
0.363
0.227
0.387
17
100TAB
0.441
0.264
0.395
0.419
0.242
0.460
18
100SCT
0.183
0.159
0.156
0.153
0.157
0.183
19
115TAK
0.694
0.409
0.575
0.596
0.362
0.686
response of the isolated structure dramatically. The distribution of the peak floor
acceleration at small acceleration is almost constant throughout the height, as expected
distribution for isolated structures when the participation of higher modes is small. At
large acceleration, the distribution of the peak floor acceleration follows a pattern of
144
small values at 2nd and 5th floors and larger at other floors. Analysis of the revised model
subjected to selected motions (Section 8.4) also follows these distribution trends. The
theory on vertical-horizontal coupling behavior in Chapter 9 explains the distribution and
the amplification of horizontal acceleration due to vertical excitation.
1
115TAK
100SCT
100TAB
90TAB
80TAB
80TCU
88RRSXY
100KJM
100TAK
100SAN
100IWA
70TCU
50TCU
70LGP
50TAB
100SYL
88RRS
0
130ELC
0.5
80WSM
Peak acc., A max (g)
1.5
Excitation motion
Figure 5-31
Peak floor acceleration of the isolated base structure for
each earthquake motion excitation
Roof
5
Floor
4
3
2
Base
Table
0
Figure 5-32
0.2
0.4
0.6
0.8
Peak acc., Amax (g)
1
1.2
Distribution of peak floor acceleration of the isolated base structure
for each earthquake motion excitation
1.4
145
ͷǤ͵Ǥʹ ‡ƒ–‘”›”‹ˆ–
Table 5-9 presents peak story drift at geometric center of all stories of the isolated
specimen for each all earthquake excitations, computed as the maximum of the drifts in X
and Y directions. The peak story drift over all stories throughout the test was generally
less than ͲǤ͵Ψ, except in the response to 88RRS, where the peak story drift of ͲǤͶͺ͹Ψ
was observed at the 2nd story. Recall that the drift was measured indirectly through a
small truss built on the concrete mass block on the floor slab (Section 4.1.2). When
subjected to vertical excitation, the truss could rotate in the vertical direction due to the
vertical deflection of the slab and caused the relative displacement between the top of the
truss and the top floor. This undesired displacement was picked up by the drift sensor.
Because of this, the drift data may not be reliable during the excitations with very strong
vertical component such as the 88RRS motion. The significant difference in story drift
response to 88RRS and 88RRSXY can also be observed. Peak story drift of all simulation
is plotted in Figure 5-33, and their distribution throughout the height is plotted in Figure
5-34. There does not seem to be any obvious distribution trend of the peak story drift,
however, two main distribution trends can be detected depending on the amplitude of the
drift. At small story drift, the peak story drift is smaller at the upper stories. At large story
drift, the distribution of the peak story drift is similar to that at small story drift, except
that the 3rd story drift is smallest. The first trend corresponds to the distribution of story
drift when the vertical excitation is small and the second trend corresponds to the drift
distribution subjected to motions with large vertical component due to the participation of
higher modes, which will be explained in Chapter 9.
146
Table 5-9
Peak story drift of the isolated base structure
Units: %
Story
Number
Run
1
2
3
4
5
1
80WSM
0.092
0.103
0.084
0.069
0.052
2
130ELC
0.133
0.135
0.110
0.090
0.086
3
88RRS
0.278
0.487
0.405
0.333
0.305
4
100SYL
0.214
0.233
0.218
0.176
0.230
5
50TAB
0.152
0.146
0.139
0.150
0.120
6
70LGP
0.220
0.231
0.199
0.164
0.181
7
50TCU
0.139
0.143
0.137
0.118
0.092
8
70TCU
0.140
0.174
0.149
0.134
0.093
9
100IWA
0.177
0.190
0.195
0.153
0.131
10
100SAN
0.132
0.149
0.135
0.122
0.098
11
100TAK
0.215
0.230
0.201
0.246
0.193
12
100KJM
0.218
0.267
0.274
0.260
0.235
13
88RRSXY
0.189
0.195
0.167
0.190
0.118
14
80TCU
0.177
0.155
0.168
0.147
0.118
15
80TAB
0.212
0.211
0.183
0.235
0.245
16
90TAB
0.224
0.191
0.178
0.167
0.144
17
100TAB
0.225
0.201
0.184
0.185
0.147
18
100SCT
0.195
0.160
0.167
0.133
0.117
19
115TAK
0.225
0.240
0.210
0.265
0.205
147
0.4
0.3
0.2
115TAK
100SCT
100TAB
90TAB
80TAB
80TCU
88RRSXY
100KJM
100TAK
100SAN
100IWA
70TCU
50TCU
70LGP
50TAB
100SYL
88RRS
130ELC
0
80WSM
0.1
Excitation motion
Figure 5-33
Peak story drift of the isolated base structure subjected to
all earthquake motion excitations
5
4
Story
Peak drift, δmax (%)
0.5
3
2
1
0
0.05
0.1
0.15
0.2
0.25
Peak drift, δ
max
Figure 5-34
0.3
(%)
0.35
0.4
0.45
Distribution of peak story drift of the isolated base structure
subjected to all earthquake motion excitations
0.5
148
5.4
General Response of the Specimen in the Fixed Base Configuration
Peak floor acceleration of the fixed base structure are summarized in Table 5-10
and plotted in Figure 5-35. As expected, the fixed base configuration amplifies the
acceleration response and the distribution of peak floor acceleration follows a typical 1st
mode shape, though not strictly. The peak floor acceleration subjected to 35RRSXY
(which includes 35% of the horizontal components of the RRS motion) and 35RRS
(which consists of all 3 components of the RRS motion scaled down to 35%) is similar,
but the peak floor acceleration subjected to 88RRS (which includes 35% of the horizontal
components and 88% of the vertical component of the RRS motion) is notably larger than
the peak floor acceleration from these two simulation. This difference comes from the
effect of vertical excitation.
Table 5-10
Peak floor acceleration of the fixed base structure for each excitation
Unit: g
Floor
Number
Run
Base
2
3
4
5
Roof
1
80WSM
0.227
0.303
0.383
0.463
0.463
0.544
2
35RRSXY
0.397
0.578
0.796
0.929
0.967
1.015
3
35RRS
0.402
0.606
0.794
0.915
0.967
1.061
4
88RRS
0.402
0.733
0.862
0.982
1.190
1.219
5
70IWA
0.374
0.468
0.653
0.811
0.928
1.127
149
Roof
5
1
4
Floor
Peak acc., A max (g)
1.5
0.5
Excitation motion
Figure 5-35
70IWA
88RRS
35RRS
35RRSXY
2
80WSM
0
3
Base
Table
0
0.5
1
Peak acc., Amax (g)
1.5
Peak floor acceleration of the fixed base structure for each excitation
Table 5-11 shows peak story drift of all stories of the fixed base structure, which
are also plotted in Figure 5-36. The maximum story drift of ͲǤͻͲ͹Ψ was observed at the
2nd story during 35RRS, which was also the maximum story drift of the specimen during
the 42 simulations of the 6-day test program (including 3 days of the isolated base with
TPBs test, 2 days of isolated base with lead rubber bearings test, and 1 day of the fixed
base test). The general trend of the peak story drift distribution of the specimen in the
fixed base configuration is that the story drift is moderate at the first story, largest at the
2nd story and reduces in each upper story. Similar to the peak floor acceleration, the peak
story drift of the 88RRS is generally larger than the peak story drifts of the 35RRSXY
and 35RRS, except at the first 2 stories, where they are about the same.
150
Table 5-11
Peak story drift of the fixed base structure for each excitation
Unit: %
Story
Number
Run
1
2
3
4
5
1
80WSM
0.321
0.360
0.311
0.279
0.203
2
35RRS2D
0.738
0.896
0.727
0.602
0.413
3
35RRS
0.750
0.907
0.737
0.619
0.421
4
88RRS
0.744
0.892
0.792
0.653
0.486
5
70IWA
0.661
0.850
0.760
0.663
0.455
5
0.8
4
Story
0.6
0.4
Excitation motion
Figure 5-36
5.5
70IWA
88RRS
35RRS
0
35RRSXY
0.2
80WSM
Peak drift, δmax (%)
1
3
2
1
0
0.2
0.4
0.6
Peak drift, δmax (%)
0.8
1
Peak floor acceleration of the fixed base structure for each excitation
Comparison of Responses to XY versus 3D excitations
As shown in previous sections, the existence of vertical excitation affects the peak
responses of both isolated base and fixed base structures. This section examines the effect
of vertical excitation on the time history responses of the structure. Thorough
investigation and explanation of the effect of the vertical excitation on the responses of
the isolated base and fixed base structures is presented in Chapter 9.
151
ͷǤͷǤͳ •‘Žƒ–‡†„ƒ•‡
Figures 5-37 to 5-41 compare time history responses of the isolated base structure
subjected to the horizontal components of the 88RRS excitation (named “XY excitation”)
and the 3 components of the 88RRS excitation (named “3D excitation”). The vertical
excitation introduced the high frequency component in floor acceleration, story drift,
internal force and base shear responses. The frequency of this component was
approximately ͸ œ, which is close to the frequency of the first vertical mode of the
superstructure. The high frequency component amplified the peak responses to the 3D
excitation compare to the peak responses to the XY excitation. The high frequency
component did not appear in the isolator displacement, and the vertical excitation had no
apparent influence on the isolator displacement, except that the movement of the isolator
did not damp out as quickly in the 3D excitation as in the XY excitation (Figure 5-41).
Acc. X, a X (g)
1
3D excitation
XY excitation
0.5
0
-0.5
Acc. Y, a Y (g)
-1
1
0.5
0
-0.5
-1
0
1
Figure 5-37
2
3
4
5
Time, t (s)
6
7
8
9
Time-history of the acceleration at roof of the isolated base structure:
3D vs. XY excitation
10
Drift. Y, δY (%)
Drift. X, δX (%)
152
0.2
0.1
0
-0.1
-0.2
3D excitation
XY excitation
0.2
0.1
0
-0.1
-0.2
0
1
Moment Y, MY (kNm)
Moment X, M X (kNm)
Figure 5-38
2
3
4
5
Time, t (s)
6
7
8
9
10
st
Time-history of the drift at 1 story of the isolated base structure:
3D vs. XY excitation
200
3D excitation
XY excitation
100
0
-100
-200
200
100
0
-100
-200
0
1
Figure 5-39
2
3
4
5
Time, t (s)
6
7
8
9
Time-history of bending moment at column base of the NE column
of the isolated base structure: 3D vs. XY excitation
10
Base shear Y, V BY (kN) Base shear X, V BX (kN)
153
3D excitation
XY excitation
1000
0
-1000
1000
0
-1000
0
1
2
Figure 5-40
3
4
5
Time, t (s)
6
7
8
9
10
Time-history of the base shear of the isolated base structure:
3D vs. XY excitation
Disp. X, u X (cm)
40
3D excitation
XY excitation
20
0
-20
Disp. Y, u Y (cm)
-40
40
20
0
-20
-40
0
5
10
15
20
Time, t (s)
Figure 5-41
Time-history of the displacement of the center isolator:
3D vs. XY excitation
25
154
ͷǤͷǤʹ ‹š‡†„ƒ•‡
Time histories of acceleration, story drift and internal force in the NE column of
the fixed base structure subjected to XY excitation and 3D excitation of the RRS motion
are compared. The XY excitation included the 2 horizontal components of the RRS
motion with the scale factor of ͵ͷΨ. The horizontal components of 3D excitation were
identical to those of the XY excitation, but its vertical excitation was ͺͺΨ of the vertical
component of the RRS motion. The comparison of acceleration at roof in Figure 5-42 and
the story drift at 5th story in Figure 5-43 show that the high frequency component exists
and amplifies the peak response to the 3D excitation, especially in the Y-direction.
However, the amplification in the fixed base structure is much smaller than the
amplification in the isolated base structure. The comparison of the internal forces at
column base of the NE column in Figure 5-44 shows little difference between responses
to 3D and XY excitations.
155
Acc. X, a X (g)
1
3D excitation
XY excitation
0.5
0
-0.5
-1
Acc. Y, a Y (g)
1
0.5
0
-0.5
-1
0
1
Figure 5-42
2
3
4
5
Time, t (s)
6
7
8
9
10
Time-history of the acceleration at roof of the fixed base structure:
3D vs. XY excitation
Drift. X, δX (%)
0.3
3D excitation
XY excitation
0.15
0
-0.15
Drift. Y, δY (%)
-0.3
0.5
0
-0.5
0
1
Figure 5-43
2
3
4
5
Time, t (s)
th
6
7
8
9
Time-history of the drift at 5 story of the fixed base structure:
3D vs. XY excitation
10
Moment Y, MY (kNm)
Moment X, M X (kNm)
156
500
3D excitation
XY excitation
0
-500
500
0
-500
0
1
Figure 5-44
5.6
2
3
4
5
Time, t (s)
6
7
8
9
10
Time-history of bending moment at column base of the NE column of
the fixed base structure: 3D vs. XY excitation
Comparison of Responses of the Isolated Base and the Fixed Base Structures
To show the effectiveness of the isolation system in reducing the demands on
superstructure, the responses of the isolated base and the fixed base structures subjected
to 3 different earthquake motion excitations were compared. The WSM motion is a small
amplitude motion representing frequent earthquakes, where the isolation system slides
mostly on the high stiffness inner sliders of the TPB. The RRS motion has a strong
vertical component, which was observed to reduce the effectiveness of the isolation
system. The IWA motion, which is a long duration motion rich in long period component
(has a peak at around ͳǤͺ• in its response spectra, Figure 3-13), may also have the
possibility of resonating with the longer period system. Because of the safety concern,
these motions were not applied with the same scale factor to the two systems. The scale
factors of these motions in the isolated base configuration were ͺͲΨ, ͺͺΨ and ͳͲͲΨ,
157
respectively, while the scale factors for the fixed base configuration were ͺͲΨ, ͵ͷΨ and
͹ͲΨ. For comparison purposes, the responses of the fixed base configuration to RRS and
IWA are linearly scaled up by the factors of ͺͺȀ͵ͷ and ͳͲͲȀ͹Ͳ, which is valid if the
system remains linear. Figures 5-45 to 5-47 show the ͷΨ damped response spectra of the
scaled table motions. The comparison shows that the response spectra between the input
motions of the two configurations are generally similar. The comparison of peak floor
acceleration and peak story drift of the 2 systems in Figure 5-48 to 5-50 shows that the
isolation system significantly reduces the demand in the superstructure. Observe that the
scaled-up peak story drift of the fixed base configuration is larger than 1% (the expected
yield drift of the specimen, see Section 8.1.2) in RRS and IWA, so that the specimen
should have had the nonlinear response if have been subjected to these scaled-up
motions. The nonlinear response would have produced larger story drift to the fixed base
structure than the scale-up drift in Figures 5-49 and 5-50.
158
Spectral acc. X, S
AX
(g)
0.8
Isolated-base
Fixed-base
0.6
0.4
0.2
0 -2
10
-1
0
10
10
1
10
Spectral acc. Y, S
AY
(g)
0.8
0.6
0.4
0.2
0 -2
10
-1
0
10
10
1
10
0.4
Spectral acc. Z, S
AZ
(g)
0.5
0.3
0.2
0.1
0 -2
10
-1
0
10
10
Period, T (s)
Figure 5-45
Response spectra of table motion, 80WSM
1
10
159
2
Isolated-base
Fixed-base
Spectral acc. X, S
AX
(g)
2.5
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
2
Spectral acc. Y, S
AY
(g)
2.5
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
Spectral acc. Z, S
AZ
(g)
4
3
2
1
0 -2
10
-1
0
10
10
Period, T (s)
Figure 5-46
Response spectra of table motion, 88RRS
1
10
160
2
Spectral acc. X, S
AX
(g)
Isolated-base
Fixed-base
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
(g)
2.5
Spectral acc. Y, S
AY
2
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
Spectral acc. Z, S
AZ
(g)
0.08
0.06
0.04
0.02
0 -2
10
-1
0
10
10
Period, T (s)
Figure 5-47
Response spectra of table motion, 100IWA
1
10
161
Roof
4
Story
4
3
Isolated
Fixed
2
0.2
0.4
Peak acc., A
max
3
2
1
0
0.6
(g)
0.1
0.2
Peak drift, δ
max
0.3
(%)
0.4
Isolated
Fixed
5
4
Story
Floor
5
Base
0
Isolated
Fixed
5
3
2
1
0
0.5
Peak torsion drift, θ
max
Figure 5-48
1
(rad/m)
1.5
-4
x 10
Peak floor acceleration, peak story drift and peak torsion drift of the isolated
base and fixed base structures subjected to 80WSM
162
Roof
5
4
4
Story
Floor
5
3
1
Peak acc., A
max
5
2
(g)
3
2
Isolated
Fixed
2
Base
0
Isolated
Fixed
3
1
0
0.5
1
1.5
Peak drift, δ
(%)
2
2.5
max
Isolated-base
Fixed-base
Story
4
3
2
1
0
Figure 5-49
2
Peak torsion drift, θ
4
max
6
(rad/m) x 10-4
Peak floor acceleration, peak story drift and peak torsion drift of the isolated
base and fixed base structures subjected to 88RRS
163
Roof
4
Story
4
3
Isolated
Fixed
2
0.5
1
Peak acc., A
max
1.5
(g)
3
2
2
1
0
0.5
Peak drift, δ
max
1
(%)
1.5
Isolated-base
Fixed-base
5
4
Story
Floor
5
Base
0
Isolated
Fixed
5
3
2
1
0
Figure 5-50
2
Peak torsion drift, θ
4
max
6
(rad/m) x 10-4
Peak floor acceleration, peak story drift and peak torsion drift of the isolated
base and fixed base structures subjected to 100IWA
Chapter 6
‘†‡Ž‹‰‘ˆ”‹’Ž‡‡†—Ž—‡ƒ”‹‰•
6.1
Introduction
The normalized unidirectional multi stage behavior of TPB mentioned in Figure
3-16 can be obtained using a combination of spring elements, friction elements and gap
elements parallel and series together as shown in Figure 6-1 (Fenz and Constantinou
2008 (2)). The relationship of the parameters of these elements (in Figure 6-1) and the
geometric parameters of bearings (Section 3.5.2) are shown in Table 6-1. Each element
group in this figure represents a pendulum mechanism described in Section 3.5.1. Linear
springs with stiffness ݇௜ represent the linear restoring behavior due to the curvature of the
spherical surfaces and vertical force. Friction elements with friction coefficient ݂௬௜
represent the friction behavior between surfaces. Gap elements with gap distance ‫ீݎ‬௜
represent the displacement limit of the inner slider and articulated sliders. Note that the
model in Figure 6-1 is a normalized model so that stiffness and forces of all elements are
normalized values (divided by the vertical force ܹ). To account for the influence of
velocity on the variation of friction coefficients, the modified rate parameters ܽത௜ for
friction elements in the series model were introduced (Fenz and Constantinou 2008 (2)).
These modified parameters ܽത௜ relate to the rate parameters ܽ௜ of friction surfaces of TPB
according to:
ܽതଵ ൌ
ܽଵ
‫ܮ‬ଶ
‫ܮ‬ଷ
Ǣܽതଶ ൌ
ܽଶ Ǣܽതଷ ൌ
ܽ
ʹ
‫ܮ‬ଶ െ ‫ܮ‬ଵ
‫ܮ‬ଷ െ ‫ܮ‬ଵ ଷ
(6.1-1)
165
݇ଵ ൌ
‫ܨ‬
ܹ
ͳ
‫ܮ‬௘௙௙ଵ
݂௬ଵ ൌ ߤଵ
݇ଶ ൌ
ͳ
‫ܮ‬௘௙௙ଶ
݂௬ଶ ൌ ߤଶ
݇ଷ ൌ
ͳ
‫ܮ‬௘௙௙ଷ
‫ܨ‬
ܹ
݂௬ଷ ൌ ߤଷ
‫ீݎ‬ଵ ൌ ‫ݑ‬തଵ
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
‫ீݎ‬ଶ ൌ ‫ݑ‬തଶ
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
‫ீݎ‬ଷ ൌ ‫ݑ‬തଷ
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૚
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૛
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૜
Figure 6-1
Theoretical series model for multi-stage behavior of TFP
Table 6-1
Parameters of theoretical series model
Elastic model
Friction model
‫ܮ‬௘௙௙ଵ ൌ ʹ‫ܮ‬ଵ
݂௬ଵ ൌ ߤଵ
‫ܮ‬௘௙௙ଶ ൌ ‫ܮ‬ଶ െ ‫ܮ‬ଵ
݂௬ଵ ൌ ߤଶ
‫ݑ‬തଶ ൌ ሺͳ െ ‫ܮ‬തଵ Ȁ‫ܮ‬തଶ ሻ݀ଶ
‫ܮ‬௘௙௙ଷ ൌ ‫ܮ‬ଷ െ ‫ܮ‬ଵ
݂௬ଵ ൌ ߤଷ
‫ݑ‬തଷ ൌ ሺͳ െ ‫ܮ‬തଵ Ȁ‫ܮ‬തଷ ሻ݀ଷ
ሺ‫כ‬ሻ
Gap model
‫ݑ‬തଵ ൌ ‫ݑ‬௟௜௠௜௧ െ ‫ݑ‬തଶ െ ‫ݑ‬തଷ
ሺ‫כ‬ሻ
‫ݑ‬௟௜௠௜௧ ൌ ʹ݀ͳ ൅ ݀ʹ ൅ ݀͵ ൅ ‫ ͵݀ ͳܮ‬Ȁ‫ ͵ܮ‬െ ‫ ʹ݀ ͳܮ‬Ȁ‫ʹܮ‬
In practice, friction behavior is usually presented by an elastic/perfectly-plastic
model with very high initial stiffness (Figure 6-2). Combining the elastic component and
the friction component in the model from Figure 6-1 and replacing friction model by
elastic/perfectly-plastic model will give the normalized model shown in Figure 6-3.
Existing elements and materials in structural analysis software can be used to
build the model in Figure 6-3. To construct this model, the software must have a bi-linear
plastic model and a gap model. For bi-directional behavior, a bi-directional circular yield
surface model and a circular gap model are required. A circular gap behavior can be
166
Norm. force
Norm. force
ߤ
ߤ
݇ஶ
Disp.
Disp.
െߤ
െߤ
(b)
(a)
Figure 6-2
Modeling friction behavior
(a) Actual friction behavior
(b) Equivalent friction behavior with very large initial stiffness
Element 1
݂ଵ
Element 3
݂ଷ
݇
݇ଵଵ ଶଵ
Element 5
݂ହ
݇
݇ଵଷ ଶଷ
݇
݇ଵହ ଶହ
‫ܨ‬
ܹ
‫ܨ‬
ܹ
݇ଶ
݇଺
݇ସ
‫ீݎ‬ସ
‫ீݎ‬ଶ
‫଺ீݎ‬
Element 2
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
Element 4
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
Element 6
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૚
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૛
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૜
݇ଵଵ ǡ ݇ଵଷ ǡ ݇ଵହ ǡ ݇ଶ ǡ ݇ସ ǡ ݇଺ are very large
Figure 6-3
Numerical series model for multi-stage behavior of TPB
obtained by a circular arrangement of of 1-D gap elements. A friction model is required
to get the vertical-horizontal coupling behavior where the horizontal behavior is
dependent on vertical force. However, at the moment, not many software has the friction
model that accounts for the variation of friction coefficient on both velocity and vertical
force. The convergence problem also usually occurs to this approach of modeling,
167
especially when the elastic stiffness of the element becomes very large. To solve these
problems, a TPB element that accounts for the vertical-horizontal coupling behavior and
the variation of friction coefficient due to both velocity and vertical force is developed in
this chapter. This TPB element combines all elements (from Element 1 to Element 6) of
the model in Figure 6-3 into a single element. The TPB element was programmed in C++
programming language and implemented in OpenSees. The source code is also available
in the OpenSees website. The following describes the algorithm for this element in detail.
6.2
Modeling of Components for Horizontal Behavior
͸ǤʹǤͳ ‹Ǧ†‹”‡ –‹‘ƒŽŽƒ•–‹ ‹–›™‹–Š‹” —Žƒ”‹‡Ž†—”ˆƒ ‡
Bi-directional plasticity model with circular yield surface behavior can be
obtained by extending the one-dimensional plasticity model described by Simo and
Hughes (1998). The model used in the TPB element is solved in the normalized domain
with normalized force and stiffness. This bi-directional plasticity model with kinematic
hardening can be described as follows:
i.
Force-displacement relationship:
௣
‫ݑ‬௫
‫ݑ‬௫
݂௫
൜ ൠ ൌ ݇ଵ ቆቄ‫ ݑ‬ቅ െ ቊ ௣ ቋቇ
݂௬
௬
‫ݑ‬௬
(6.2-1)
where ݂௫ ǡ ݂௬ are components of the force vector; ݇ଵ is the initial stiffness; ‫ݑ‬௫ ǡ ‫ݑ‬௬ are
௣
௣
components of the displacement vector; ‫ݑ‬௫ ǡ ‫ݑ‬௬ are components of the plastic
displacement vector. Equivalent quantities reflecting these components in a onedimensional model are shown in Figure 6-4.
168
Force
݂
Current state
݂௒ Current
back stress
‫ݍ‬
݇ଵ ͳ
‫ݑ‬௣ Figure 6-4
ii.
‫ݑ‬
Displacement
One-dimensional elastic-plastic model
Yield function:
‫ݍ‬௫
݂௫
‫ ܨ‬ൌ ฯ൜ ൠ െ ቄ‫ ݍ‬ቅฯ െ ݂௒ ൑ Ͳ
݂௬
௬
(6.2-2)
where ‫ݍ‬௫ ǡ ‫ݍ‬௬ are components of back stress and ݂௒ is the yield strength (Figure 6-4).
iii.
Associated flow rule:
ቊ
௣
݊௫
‫ݑ‬ሶ ௫
௣ ቋ ൌ ߛǤ ቄ݊ ቅ
௬
‫ݑ‬ሶ ௬
where ݊௫ ǡ ݊௬ are components of the normal unit vector of the current yield surface.
iv.
Kinematic hardening law:
(6.2-3)
169
݊௫
‫ݍ‬௫ሶ
൜‫ ݍ‬ሶ ൠ ൌ ߛǤ ‫ܪ‬Ǥ ቄ݊ ቅ
(6.2-4)
௬
௬
where ‫ ܪ‬is the kinematic hardening modulus and ߛ is a consistency parameter satisfying
the Kuhn – Tucker complementary condition and consistency conditions as follows.
v.
Kuhn – Tucker complementary conditions:
ߛ ൒ Ͳǡ
vi.
‫ ܨ‬൑ Ͳǡ
ߛǤ ‫ ܨ‬ൌ Ͳ
(6.2-5)
Consistency condition:
ߛǤ ‫ܨ‬ሶ ൌ Ͳǡ
݂݅‫ ܨ‬ൌ Ͳ
(6.2-6)
A numerical return mapping algorithm for state determination of a rate
independent plasticity model at any time step was proposed by Simo and Hughes (1998).
The following is an extended algorithm for bi-directional rate independent plasticity with
kinematic hardening, where the behavior in each direction is coupled through a circular
yield surface.
i.
Given a trial displacement ሼ‫ݑ‬௫
‫ݑ‬௬ ሽ்௡ାଵ at step ݊ ൅ ͳ, compute a trial stress
௧௥௜௔௟
assuming elastic response and evaluate the yield function ‫ܨ‬௡ାଵ
:
௣
‫ݑ‬௫
‫ݑ‬௫
݂௫ ௧௥௜௔௟
൜ ൠ
ൌ ݇ଵ ൭ቄ‫ ݑ‬ቅ
െ ቊ ௣ቋ ൱
݂௬ ௡ାଵ
௬ ௡ାଵ
‫ݑ‬௬
௡
(6.2-7)
170
‫ݍ‬௫
ߦ௫ ௧௥௜௔௟
݂௫ ௧௥௜௔௟
൜ ൠ
ൌ൜ ൠ
െ ቄ‫ ݍ‬ቅ
ߦ௬ ௡ାଵ
݂௬ ௡ାଵ
௬ ௡
௧௥௜௔௟
‫ܨ‬௡ାଵ
ii.
ߦ௫ ௧௥௜௔௟
ൌ ብ൜ ൠ
ብ െ ݂௒
ߦ௬ ௡ାଵ
(6.2-8)
(6.2-9)
௧௥௜௔௟
Elastic condition: If ‫ܨ‬௡ାଵ
൑Ͳ
Update state:
ሺǤ ሻ௡ାଵ ൌ ሺǤ ሻ௧௥௜௔௟
௡ାଵ
(6.2-10)
Evaluate elastic tangent modulus:
‫ ்ܭ‬ൌ ൤
iii.
݇ଵ
Ͳ
Ͳ
൨
݇ଵ
(6.2-11)
௧௥௜௔௟
Plastic condition, using return mapping: If ‫ܨ‬௡ାଵ
൐Ͳ
Compute normal vector:
݊௫
ͳ
ߦ௫ ௧௥௜௔௟
ቄ݊ ቅ ൌ
Ǥ൜ ൠ
௬
ԡߦԡ௧௥௜௔௟ ߦ௬
௡ାଵ
௡ାଵ
(6.2-12)
Compute consistency parameter:
௧௥௜௔௟
‫ܨ‬௡ାଵ
ȟߛ ൌ
݇ଵ ൅ ‫ܪ‬
Update state:
(6.2-13)
171
݂௫
݂௫ ௧௥௜௔௟
݇
൜ ൠ
ൌ൜ ൠ
െ ȟɀǤ ൤ ଵ
݂௬ ௡ାଵ
݂௬ ௡ାଵ
Ͳ
௣
ቊ
‫ݑ‬௫
௣ቋ
‫ݑ‬௬
ൌቊ
௡ାଵ
݊௫
Ͳ
൨ Ǥ ቄ݊ ቅ
௬
݇ଵ
௣
݊௫
‫ݑ‬௫
௣ ቋ ൅ ȟɀǤ ቄ݊ ቅ
௬
‫ݑ‬௬
(6.2-14)
(6.2-15)
௡
‫ݍ‬௫
‫ݍ‬௫
݊௫
ቄ‫ ݍ‬ቅ
ൌ ቄ‫ ݍ‬ቅ ൅ ߂ߛǤ ‫ܪ‬Ǥ ቄ݊ ቅ
௬ ௡ାଵ
௬ ௡
௬
(6.2-16)
Elastoplastic tangent modulus:
‫ ்ܭ‬ൌ ൤
݇௫௫
݇௬௫
݇௫௬
൨
݇௬௬
(6.2-17)
The terms of elastoplastic tangent modulus matrix can be determined as:
݇௫௫
߲݂௫
߲݂௫௧௥௜௔௟
߲ሺȟߛǤ ݊௫ ሻ
ൌ൜
ൠ
ൌቊ
ቋ
െ ݇ଵ
߲‫ݑ‬௫
߲‫ݑ‬௫ ௡ାଵ
߲‫ݑ‬௫ ௡ାଵ
(6.2-18)
߲݂௫௧௥௜௔௟
ቋ
ൌ ݇ଵ
߲‫ݑ‬௫ ௡ାଵ
(6.2-19)
߲ሺȟߛǤ ݊௫ ሻ ߲ȟߛ
߲݊௫
ൌ
Ǥ ݊௫ ൅
Ǥ ȟߛ
߲‫ݑ‬௫
߲‫ݑ‬௫
߲‫ݑ‬௫
(6.2-20)
ቊ
From Equations (6.2-7), (6.2-8), (6.2-9) and (6.2-13):
߲ȟߛ
݇ଵ
ߦ௫ ௧௥௜௔௟
ൌ
Ǥ൬
൰
߲‫ݑ‬௫ ݇ଵ ൅ ‫ ܪ‬ԡߦԡ ௡ାଵ
From Equations (6.2-7), (6.2-8) and (6.2-12):
(6.2-21)
172
ଶ
௧௥௜௔௟
݇ଵ Ǥ ൫ߦ௫ǡ௡ାଵ
൯
߲݊௫
݇ଵ
ൌ
െ
ଷ
߲‫ݑ‬௫ ԡߦԡ௧௥௜௔௟
൫ԡߦԡ௧௥௜௔௟ ൯
௡ାଵ
(6.2-22)
௡ାଵ
Substitute Equations (6.2-12), (6.2-21) and (6.2-22) into Equations (6.2-20):
݇ଵ
ߦ௫ ௧௥௜௔௟
߲ሺȟߛǤ ݊௫ ሻ
ൌ
Ǥ ൝ቆ൬
൰
ቇ
ԡߦԡ ௡ାଵ
߲‫ݑ‬௫
݇ଵ ൅ ‫ܪ‬
ଶ
ଶ
௧௥௜௔௟
௧௥௜௔௟
൫ߦ௫ǡ௡ାଵ
൯
‫ܨ‬௡ାଵ
൅
Ǥ
൭ͳ
െ
ଶ ൱ൡ
௧௥௜௔௟
ԡߦԡ௡ାଵ
൫ԡߦԡ௧௥௜௔௟ ൯
(6.2-23)
௡ାଵ
Substitute Equations (6.2-19) and (6.2-23) into Equations (6.2-18):
݇௫௫
݇ଵଶ
ߦ௫ ௧௥௜௔௟
ൌ ݇ଵ െ
Ǥ ൝ቆ൬
൰
ቇ
ԡߦԡ ௡ାଵ
݇ଵ ൅ ‫ܪ‬
ଶ
ଶ
௧௥௜௔௟
௧௥௜௔௟
൯
൫ߦ௫ǡ௡ାଵ
‫ܨ‬௡ାଵ
൅
Ǥ
൭ͳ
െ
ଶ ൱ൡ
௧௥௜௔௟
ԡߦԡ௡ାଵ
൫ԡߦԡ௧௥௜௔௟ ൯
(6.2-24)
௡ାଵ
Similarly,
௧௥௜௔௟ ଶ
݇௬௬
ߦ௬
݇ଵଶ
ൌ ݇ଵ െ
Ǥ ቐ൭ቆ
ቇ
൱
ԡߦԡ
݇ଵ ൅ ‫ܪ‬
௡ାଵ
ଶ
௧௥௜௔௟
௧௥௜௔௟
൫ߦ௬ǡ௡ାଵ
൯
‫ܨ‬௡ାଵ
൅
Ǥ
൭ͳ
െ
൱ൡ
௧௥௜௔௟ ଶ
ԡߦԡ௧௥௜௔௟
൫ԡߦԡ
൯
௡ାଵ
(6.2-25)
௡ାଵ
௧௥௜௔௟
݇௫௬ ൌ ݇௬௫
௧௥௜௔௟
൫ߦ௫ Ǥ ߦ௬ ൯௡ାଵ
݇ଵଶ
‫ܨ‬௡ାଵ
ൌെ
Ǥ
Ǥቆ
െ ͳቇ
݇ଵ ൅ ‫ ܪ‬൫ԡߦԡ௧௥௜௔௟ ൯ଶ ԡߦԡ௧௥௜௔௟
௡ାଵ
௡ାଵ
(6.2-26)
173
When implemented into the TPB element, the yield strength ݂௒ of the bidirectional plasticity model, which equals to the friction coefficient, is updated at every
time step considering velocity and vertical force effects as shown in Equations (5.1-9) to
(5.1-11) of Section 5.1.
͸ǤʹǤʹ ‹” —Žƒ”Žƒ•–‹ ƒ’Ž‡‡–•
Two nodes connected by a circular elastic gap element can freely move with
respect to each other whenever the distance between them in any direction does not
exceed the restraint radius, ‫ ீݎ‬. When the relative distance meets or exceeds ‫ ீݎ‬, these two
nodes shall be subjected to the elastic restoring with elastic modulus ‫ܭ‬. When ‫ ܭ‬goes to
infinity, the element becomes a circular gap where the distance between the two nodes
can never exceed ‫ ீݎ‬.
Let the local coordinate system of this element be ‫ ݕݔ‬lying in the horizontal
plane. The origin of this coordinate system is attached to node ݅. Let ‫ ݎ‬be a distance
vector pointing from node ݅ to node ݆. The restoring force vector and tangent stiffness
matrix of the element in its local coordinate system are as follows:
Case 1: ȁ‫ݎ‬ȁ ൑ ‫ ீݎ‬:
൜
݂݀௫
ൠൌ
݂݀௬
Ͳ Ͳ
ቂ
ቃ
ᇣᇤᇥ
Ͳ Ͳ
௧௔௡௚௘௡௧௦௧௜௙௙௡௘௦௦௠௔௧௥௜௫
Ǥ൜
݀‫ݔ‬
ൠ
݀‫ݕ‬
(6.2-27)
where ݀‫ ݔ‬and ݀‫ ݕ‬are components of incremental displacement in the ‫ ݔ‬and ‫ ݕ‬direction;
݂݀௫ and ݂݀௬ are components of incremental restoring force in the ‫ ݔ‬and ‫ ݕ‬directions.
174
Case 2: ȁ‫ݎ‬ȁ ൐ ‫ ீݎ‬:
Suppose that node ݆ moves from ‫ ܣ‬to ‫ܣ‬Ԣ as shown in Figure 6-5(a). The nodal
forces when node ݆ is at ‫ ܣ‬and ‫ܣ‬Ԣ are ݂ and ݂Ԣ, respectively. From Figure 6-5(b):
ሬሬሬሬԦ ൌ ݂
ሬሬሬԦᇱ െ ݂Ԧ ൌ ሬሬሬሬሬሬԦ
݂݀ଵ ൅ ሬሬሬሬሬሬԦ
݂݀ଶ
݂݀
(6.2-28)
Projecting Equation (6.2-28) into the ‫ ݔ‬and ‫ ݕ‬directions and applying the Hook’s
law to express force in terms of displacement and stiffness:
൜
݂݀௫ ൌ െ݇Ǥ ‫ ீݎ‬Ǥ ݀ߙǤ ‫ ߙ ݊݅ݏ‬൅ ݇Ǥ ݀‫ݔ‬
݂݀௬ ൌ ݇Ǥ ‫ ீݎ‬Ǥ ݀ߙǤ ‘• ߙ ൅ ݇Ǥ ݀‫ݕ‬
‫ܣ‬Ԣ
݀‫ݔ‬
‫ݕ‬
݀ߙ
‫ீݎ‬
݂݀ଶ
݀‫ݕ‬
‫ܣ‬
‫ݎ‬
(6.2-29)
݂
‫ܣ‬ଶ
‫ܣ‬ଵ
݂Ԣ
݂݀ଵ
ߙ
ߙԢ
݅
‫ݔ‬
(a)
Figure 6-5
݅
(b)
Displacement and force diagrams of gap element
(a) Displacement diagram
(b) Force diagram
175
Assuming small displacements, the angle ݀ߙ can be computed as:
݀ߙ ൌ
‫ܣܣ‬ଵ െ ‫ܣ‬ଵ ‫ܣ‬ଶ ݀‫ݔ‬Ǥ •‹ ߙԢ െ ݀‫ݕ‬Ǥ ‘• ߙԢ
ൌ
‫ݎ‬
‫ݎ‬
(6.2-30)
Substituting Equation (6.2-30) into Equation (6.2-29) and applying the small
displacement assumption (ߙ ᇱ ൎ ߙ ᇱ ൅ ݀ߙ ൌ ߙ):
‫ீݎ‬
‫ீݎ‬
݂݀௫ ൌ ‫ ܭ‬ቀͳ െ •‹ଶ ߙቁ ݀‫ ݔ‬൅ ‫ ߙ ‹• ܭ‬Ǥ ‘• ߙ Ǥ ݀‫ݕ‬
‫ݎ‬
‫ݎ‬
൞
‫ீݎ‬
‫ீݎ‬
ଶ
݂݀௬ ൌ ‫ ߙ ‹• ܭ‬Ǥ ‘• ߙ Ǥ ݀‫ ݔ‬൅ ‫ ܭ‬ቀͳ െ
‘• ߙቁ ݀‫ݕ‬
‫ݎ‬
‫ݎ‬
(6.2-31)
Equation (6.2-31) can be rewritten in the matrix form:
݂݀௫
൜ ൠൌ
݂݀௬
݇௫௫ ݇௫௬
൤
൨
݇௬௫ ݇ᇧᇥ
௬௬
ᇣᇧᇧᇤᇧ
݀‫ݔ‬
൜ ൠ
݀‫ݕ‬
(6.2-32)
௧௔௡௚௘௡௧௦௧௜௙௙௡௘௦௦௠௔௧௥௜௫
where:
݇௫௫ ൌ ݇ ቀͳ െ
݇௫௬ ൌ ݇௬௫ ൌ ݇
‫ீݎ‬
•‹ଶ ߙቁ
‫ݎ‬
‫ீݎ‬
•‹ ߙ Ǥ ‘• ߙ
‫ݎ‬
݇௬௬ ൌ ݇ ቀͳ െ
‫ீݎ‬
‘• ଶ ߙቁ
‫ݎ‬
Equations (6.2-27) and (6.2-32) give the tangent stiffness matrix of the element
for any relative displacement ‫ ݎ‬between the nodes.
176
6.3
Modeling Vertical Behavior
The vertical response of the TPB element is modeled as linear elastic with a
different stiffness in tension and compression (Figure 6-6). Theoretically, the tension
stiffness is zero if the bearing is not constrained against uplift. However, if the uplift
occurs to all TPBs and the tension vertical stiffness of the TPB element is zero, the global
stiffness matrix is indeterminate and the governing equation cannot be solved. To solve
this numerical problem, the tension stiffness of the TPB elements should be set to a very
small number.
The procedure for computing the vertical stiffness and force is:
Step 1: Obtain the trial vertical displacement ‫ݑ‬௩௧௥௜௔௟ .
Step 2:
If ‫ݑ‬௩௧௥௜௔௟ ൑ Ͳ, then ‫ܭ‬௩ ൌ ‫ܭ‬௩௖
Else ‫ܭ‬௩ ൌ ‫ܭ‬௩௧
Step 3: ‫ܨ‬௩ ൌ ‫ܭ‬௩ ൈ ‫ݑ‬௩௧௥௜௔௟
‫ܨ‬௩
ͳ
‫ܭ‬௩௧
‫ݑ‬௩
‫ܭ‬௩௖
ͳ
Figure 6-6
Vertical behavior of TPB
177
6.4
Element Formulation for Horizontal Behavior
In the horizontal behavior, the normalized stiffness matrix and normalized force
vector are solved first. These normalized stiffness and force are then multiplied by the
vertical force computed from the previous section to get the actual tangent stiffness and
force vector. The horizontal tangent stiffness and force vector of the TPB element
computed this way are dependent on the fluctuation of the vertical force and the verticalhorizontal coupling behavior of the TPB element is captured.
͸ǤͶǤͳ ••‡„Ž›‘ˆƒ‰‡––‹ˆˆ‡••ƒ–”‹š‹ ‘”‹œ‘–ƒŽ‡Šƒ˜‹‘”
To assemble the normalized tangent stiffness matrix in horizontal behavior, TPB
is treated as a system with 4 nodes and 6 elements as shown in Figure 6-7. Nodes 3 and 4
are internal to the element and are only considered in the element procedure. The basis
coordinate system of TPB is attached to node 1, and displacements of all other nodes are
relative displacement to this node. Assume that the mass of TPB is lumped to nodes 1
and 2 so that there is not any external load applied to nodes 3 and 4. Hereafter is the
assembling process to assemble the sub-element’s normalized tangent stiffness matrix
into the TPB’s normalized tangent stiffness matrix.
Node 1
Ele. 1
Node 3
Ele. 3
Node 4
Ele. 5
Node 2
ሺ‫ݑ‬ଷ௫ ǡ ‫ݑ‬ଷ௬ ሻ
ሺ‫ݑ‬ସ௫ ǡ ‫ݑ‬ସ௬ ሻ
ሺ‫ݑ‬ଶ௫ ǡ ‫ݑ‬ଶ௬ ሻ
Ele. 2
Ele. 4
Ele. 6
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ሺ‫ݑ‬ଵ௫ ǡ ‫ݑ‬ଵ௬ ሻ
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૚
Figure 6-7
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૛
ࡱ࢒ࢋ࢓ࢋ࢔࢚ࡳ࢘࢕࢛࢖૜
Finite element configuration for horizontal behavior of TPB
178
ሼ‫ݑ‬ଵ௫
Relationship
between
‫ݑ‬ଵ௬
‫ݑ‬ସ௬ ሽ் in the basis coordinate system of the TPB and the
ǥ ‫ݑ‬ସ௫
deformation ሼ‫ݑ‬௫
‫ݑ‬௬ ሽ்
௘௟௘Ǥ௜
the
nodal
degree
of
freedoms
(DOFs)
of the sub-element ݅ in its basis coordinate system is:
‫ݑ‬௫
ቄ‫ ݑ‬ቅ
ൌ ࢇ࢏ሺଶൈ଼ሻ ሼ‫ݑ‬ଵ௫
௬ ௘௟௘Ǥ௜
‫ݑ‬ଵ௬
ǥ ‫ݑ‬ସ௫
‫ݑ‬ସ௬ ሽ்
(6.4-1)
െͳ Ͳ Ͳ
ܽଵሺଶൈ଼ሻ ൌ ቂ
Ͳ െͳ Ͳ
Ͳ ͳ Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ
ቃ
Ͳ Ͳ
(6.4-2)
െͳ Ͳ Ͳ
ܽଶሺଶൈ଼ሻ ൌ ቂ
Ͳ െͳ Ͳ
Ͳ ͳ
Ͳ Ͳ
Ͳ Ͳ Ͳ
ቃ
ͳ Ͳ Ͳ
(6.4-3)
Ͳ
ܽଷሺଶൈ଼ሻ ൌ ቂ
Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
െͳ Ͳ ͳ Ͳ
ቃ
Ͳ െͳ Ͳ ͳ
(6.4-4)
Ͳ
ܽସሺଶൈ଼ሻ ൌ ቂ
Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
െͳ Ͳ ͳ Ͳ
ቃ
Ͳ െͳ Ͳ ͳ
(6.4-5)
Ͳ
ܽହሺଶൈ଼ሻ ൌ ቂ
Ͳ
Ͳ ͳ Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ
Ͳ Ͳ
െͳ Ͳ
ቃ
Ͳ െͳ
(6.4-6)
Ͳ
ܽ଺ሺଶൈ଼ሻ ൌ ቂ
Ͳ
Ͳ ͳ Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ
Ͳ Ͳ
െͳ Ͳ
ቃ
Ͳ െͳ
(6.4-7)
where:
Tangent stiffness matrix of TFP:
଺
்
்݇ி௉ሺ଼ൈ଼ሻ ൌ ෍ ܽ௜ሺ଼ൈଶሻ
Ǥ ݇௘௟௘Ǥ௜ሺଶൈଶሻ Ǥ ܽ௜ሺଶൈ଼ሻ
(6.4-8)
௜ୀଵ
where ݇௘௟௘Ǥ௜ሺଶൈଶሻ is the tangent stiffness of the element ݅ in its basis coordinate system.
179
The expansions of terms in ்݇ி௉ሺ଼ൈ଼ሻ are as following. Note that the matrix is
symmetric so that ்݇ி௉ሺ଼ൈ଼ሻ ሺ݅ǡ ݆ሻ ൌ ்݇ி௉ሺ଼ൈ଼ሻ ሺ݆ǡ ݅ሻ.
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡͳሻ ൌ ݇௘௟௘Ǥଵ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥଶ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡʹሻ ൌ ݇௘௟௘Ǥଵ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥଶ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡ͵ሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡͶሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡͷሻ ൌ െ݇௘௟௘Ǥଵ ሺͳǡͳሻ െ ݇௘௟௘Ǥଶ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡ͸ሻ ൌ െ݇௘௟௘Ǥଵ ሺͳǡʹሻ െ ݇௘௟௘Ǥଶ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡ͹ሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺͳǡͺሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡʹሻ ൌ ݇௘௟௘Ǥଵ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥଶ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡ͵ሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡͶሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡͷሻ ൌ െ݇௘௟௘Ǥଵ ሺͳǡʹሻ െ ݇௘௟௘Ǥଶ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡ͸ሻ ൌ െ݇௘௟௘Ǥଵ ሺʹǡʹሻ െ ݇௘௟௘Ǥଶ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡ͹ሻ ൌ Ͳ
180
்݇ி௉ሺ଼ൈ଼ሻ ሺʹǡͺሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡ͵ሻ ൌ ݇௘௟௘Ǥହ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥ଺ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡͶሻ ൌ ݇௘௟௘Ǥହ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥ଺ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡͷሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡ͸ሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡ͹ሻ ൌ െ݇௘௟௘Ǥହ ሺͳǡͳሻ െ ݇௘௟௘Ǥ଺ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͵ǡͺሻ ൌ െ݇௘௟௘Ǥହ ሺͳǡʹሻ െ ݇௘௟௘Ǥ଺ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͶǡͶሻ ൌ ݇௘௟௘Ǥହ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥ଺ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͶǡͷሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺͶǡ͸ሻ ൌ Ͳ
்݇ி௉ሺ଼ൈ଼ሻ ሺͶǡ͹ሻ ൌ െ݇௘௟௘Ǥହ ሺͳǡʹሻ െ ݇௘௟௘Ǥ଺ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͶǡͺሻ ൌ െ݇௘௟௘Ǥହ ሺʹǡʹሻ െ ݇௘௟௘Ǥ଺ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͷǡͷሻ
ൌ ݇௘௟௘Ǥଵ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥଶ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥଷ ሺͳǡͳሻ
൅ ݇௘௟௘Ǥସ ሺͳǡͳሻ
181
்݇ி௉ሺ଼ൈ଼ሻ ሺͷǡ͸ሻ
ൌ ݇௘௟௘Ǥଵ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥଶ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥଷ ሺͳǡʹሻ
൅ ݇௘௟௘Ǥସ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͷǡ͹ሻ ൌ െ݇௘௟௘Ǥଷ ሺͳǡͳሻ െ ݇௘௟௘Ǥସ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺͷǡͺሻ ൌ െ݇௘௟௘Ǥଷ ሺͳǡʹሻ െ ݇௘௟௘Ǥସ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͸ǡ͸ሻ
ൌ ݇௘௟௘Ǥଵ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥଶ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥଷ ሺʹǡʹሻ
൅ ݇௘௟௘Ǥସ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͸ǡ͹ሻ ൌ െ݇௘௟௘Ǥଷ ሺͳǡʹሻ െ ݇௘௟௘Ǥସ ሺͳǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͸ǡͺሻ ൌ െ݇௘௟௘Ǥଷ ሺʹǡʹሻ െ ݇௘௟௘Ǥସ ሺʹǡʹሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͹ǡ͹ሻ
ൌ ݇௘௟௘Ǥଷ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥସ ሺͳǡͳሻ ൅ ݇௘௟௘Ǥହ ሺͳǡͳሻ
൅ ݇௘௟௘Ǥ଺ ሺͳǡͳሻ
்݇ி௉ሺ଼ൈ଼ሻ ሺ͹ǡͺሻ
ൌ ݇௘௟௘Ǥଷ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥସ ሺͳǡʹሻ ൅ ݇௘௟௘Ǥହ ሺͳǡʹሻ
൅ ݇௘௟௘Ǥ଺ ሺͳǡʹሻ
182
்݇ி௉ሺ଼ൈ଼ሻ ሺͺǡͺሻ
ൌ ݇௘௟௘Ǥଷ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥସ ሺʹǡʹሻ ൅ ݇௘௟௘Ǥହ ሺʹǡʹሻ
൅ ݇௘௟௘Ǥ଺ ሺʹǡʹሻ
Incremental equilibrium equation of TPB in its basis coordinate system:
݇ଵଵ ݇ଵଶ ݇ଵଷ ݇ଵସ ݇ଵହ ݇ଵ଺ ݇ଵ଻ ݇ଵ଼ ݀‫ݑ‬ଵ௫
݂݀ଵ௫
‫ۍ‬
‫ې‬
݇ଶଶ ݇ଶଷ ݇ଶସ ݇ଶହ ݇ଶ଺ ݇ଶ଻ ݇ଶ଼ ‫ݑ݀ۓ‬ଶ௫ ۗ ‫݂݀ۓ‬ଵ௬ ۗ
‫ێ‬
‫ۖۑ‬
ۖ
݇ଷଷ ݇ଷସ ݇ଷହ ݇ଷ଺ ݇ଷ଻ ݇ଷ଼ ‫ݑ݀ۖ ۑ‬ଶ௫ ۖ ݂ۖ݀ ۖ
‫ێ‬
ۖ݀‫ ۖ ۖ ݑ‬ଶ௫ ۖ
‫ێ‬
݇ସସ ݇ସହ ݇ସ଺ ݇ସ଻ ݇ସ଼ ‫ۑ‬
ଶ௬
ൌ ݂݀ଶ௬
‫ێ‬
݇ହହ ݇ହ଺ ݇ହ଻ ݇ହ଼ ‫ݑ݀۔ ۑ‬ଷ௫ ۘ ‫ۘ Ͳ ۔‬
‫ێ‬
‫ۑ‬
݇଺଺ ݇଺଻ ݇଺଼ ‫ݑ݀ۖ ۑ‬ଷ௬ ۖ ۖ Ͳ ۖ
‫ێ‬
ۖ
ۖ
݇଻଻ ݇଻଼ ‫ݑ݀ۖ ۑ‬ସ௫ ۖ ۖ Ͳ ۖ
‫ێ‬
଼଼݇ ‫ݑ݀ە ے‬ସ௬ ۙ ‫ۙ Ͳ ە‬
‫ۏ‬ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ
(6.4-9)
௞೅ುಳሺఴൈఴሻ
The developed normalized tangent stiffness matrix of the TPB element
்݇௉஻ሺ଼ൈ଼ሻ is an ͺ ൈ ͺ matrix considering all internal DOFs. However, as an element in an
analysis program, the normalized tangent stiffness matrix of TPB in horizontal direction
in its basis coordinate system for returning to the global procedure is only a ʹ ൈ ʹ matrix.
Thus, static condensation is required to eliminate internal DOFs, reducing the ͺ ൈ ͺ
matrix to a ʹ ൈ ʹ matrix.
Partition equation (6.4-9):
൤
࢑࢚࢚
࢑૙࢚
࢑࢚૙ ࢊ࢛࢚
ࢊࢌ
൨൜
ൠ ൌ ቄ ࢚ቅ
࢑૙૙ ࢊ࢛૙
૙
(6.4-10)
From the second equation of Equation (6.4-10):
ࢊ࢛૙ ൌ െ࢑ି૚
૙૙ Ǥ ࢑૙࢚ Ǥ ࢊ࢛࢚
Substitute Equation (6.4-11) into the first equation of Equation (6.4-10):
(6.4-11)
183
൫࢑࢚࢚ െ ࢑࢚૙ Ǥ ࢑ି૚
૙૙ Ǥ ࢑૙࢚ ൯ࢊ࢛࢚ ൌ ࢊࢌ࢚
(6.4-12)
Expansion form of Equation (6.4-12) is:
‫ۍ‬
‫ێ‬
‫ێ‬
‫ۏ‬
ᇱ
݇ଵଵ
ᇱ
݇ଵଶ
ᇱ
݇ଶଶ
ᇱ
݇ଵଷ
ᇱ
݇ଶଷ
ᇱ
݇ଷଷ
ᇱ
݂݀ଵ௫
݀‫ݑ‬ଵ௫
݇ଵସ
‫ۗ ݂݀ۓ‬
ᇱ ‫ۗ ݑ݀ۓ ې‬
݇ଶସ ‫ۑ‬
ଵ௬
ଵ௬
ൌ
ᇱ
݀‫ݑ‬
݂݀
݇ଷସ ‫ ۔ ۑ‬ଶ௫ ۘ ‫ ۔‬ଶ௫ ۘ
ᇱ
݇ସସ
‫ݑ݀ە ے‬ଶ௬ ۙ ‫݂݀ە‬ଶ௬ ۙ
(6.4-13)
Note that ݀‫ݑ‬ଵ௫ ൌ ݀‫ݑ‬ଵ௬ ൌ Ͳ since node 1 is the origin of the basis coordinate
system of the TPB element, the last 2 equations of Equation (6.4-13) become:
ᇱ
݀‫ݑ‬ଶ௫
݂݀ଶ௫
݇ᇱ
݇ଷସ
൤ ଷଷ
ᇱ
ᇱ ൨ ൜݀‫ ݑ‬ൠ ൌ ൜݂݀ ൠ
݇ସଷ
݇ᇧᇥ
ଶ௬
ଶ௬
ᇣᇧ
ᇧᇤᇧ
ସସ
(6.4-14)
௞೅ುಳሺమൈమሻ
்݇௉஻ሺଶൈଶሻ is the normalized horizontal tangent stiffness matrix of the TFP element
in its basis coordinate system. This normalized tangent stiffness matrix relates to the
actual tangent stiffness matrix as:
்݇௉஻ሺଶൈଶሻ ൌ
‫்ܭ‬௉஻ሺଶൈଶሻ
‫ܨ‬௩
(6.4-15)
where ‫்ܭ‬௉஻ሺଶൈଶሻ is the actual horizontal tangent stiffness of the TPB element in its basis
coordinate system. From Equation (6.4-15), ‫்ܭ‬௉஻ሺଶൈଶሻ can be computed from
்݇௉஻ሺଶൈଶሻ as follows:
‫்ܭ‬ி௉ሺଶൈଶሻ ൌ ‫ܨ‬௩ ൈ ்݇ி௉ሺଶൈଶሻ ൌ ‫ܨ‬௩ ൈ ൤
ᇱ
݇ଷଷ
ᇱ
݇ସଷ
ᇱ
݇ଷସ
‫ܭ‬ଵଵ
ᇱ ൨ ൌ ൤‫ܭ‬
݇ସସ
ଶଵ
‫ܭ‬ଵଶ
൨
‫ܭ‬ଶଶ
(6.4-16)
184
Similarly, the actual horizontal force vector ‫்ܨ‬௉஻ሺଶൈଶሻ can be computed from
normalized horizontal force vector ்݂௉஻ሺଶൈଶሻ as:
݂௫
‫ܨ‬௫
‫்ܨ‬ி௉ሺଶൈଵሻ ൌ ‫ܨ‬௩ ൈ ்݂௉஻ሺଶൈଶሻ ൌ ‫ܨ‬௩ ൈ ൜ ൠ ൌ ൜‫ ܨ‬ൠ
݂௬
௬
(6.4-17)
͸ǤͶǤʹ –‡”ƒ–‹‰‘˜‡””‹’Ž‡ ”‹ –‹‘‡†—Ž—Ž‡‡–
The TPB element was developed as a displacement-based element in which the
tangent stiffness matrix and force vector are computed based on the given trial
displacement. The inverse Newton – Raphson iteration is required for determining these
quantities at each trial displacement. The iteration process is schematically shown in
Figure 6-8 and follows the flow chart in Figure 6-9. The iterative procedure in this flow
chart solves for the tangent stiffness matrix ࡷ and force vector ࡲ given the trial
deformation ࢛௧௥௜௔௟ and the converged state (ܿ‫ )ݒ݊݋‬of the last time step. Knowledge of the
converged state includes normalized tangent stiffness matrix ࢑௖௢௡௩ , normalized force
vector ࢌ௖௢௡௩ , and deformation ࢛௖௢௡௩ of the TPB element, as well as normalized tangent
stiffness matrix ࢑௖௢௡௩ǡ௚௥ and deformation vector ࢛௖௢௡௩ǡ௚௥ of the 3 element groups. The
plastic deformation ࢛௣ǡ௖௢௡௩ǡ௘௟௘ and back stress ࢗ௖௢௡௩ǡ௘௟௘ of the bi-linear plasticity circular
yield surface elements are also required for solving these elements.
185
f
݇ଵ ൌ ݇௖௢௡௩
݇ଶ
݂ଶ
߂݂ଶ
݂ଷ
߂݂ଵ
݂௖௢௡௩
‫ݑ‬௖௢௡௩
ǻ‫ݑ‬ଵ
‫ݑ‬ଶ
‫ݑ‬௧௥௜௔௟ ‫ݑ‬ଷ
ǻ‫ݑ‬ଷ
ǻ‫ݑ‬ଶ
Figure 6-8
Inverse Newton – Raphson iteration
‫ݑ‬
186
Begin TPB element
࢛௧௥௜௔௟ ǡ ࢛௖௢௡௩
࢑௖௢௡௩ ǡ ࢌ௖௢௡௩
࢛௖௢௡௩ǡ௚௥ଵ ǡ ࢛௖௢௡௩ǡ௚௥ଶ ǡ ࢛௖௢௡௩ǡ௚௥ଷ
࢑௖௢௡௩ǡ௚௥ଵ ǡ ࢑௖௢௡௩ǡ௚௥ଶ ǡ ࢑௖௢௡௩ǡ௚௥ଷ
࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥଵ ǡ ࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥଷ ǡ ࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥହ
ࢗ௖௢௡௩ǡ௘௟௘Ǥଵ ǡ ࢗ௖௢௡௩ǡ௘௟௘Ǥଷ ǡ ࢗ௖௢௡௩ǡ௘௟௘Ǥହ
݅ ൌ Ͳǡ ܹ ൌ ݇௩ ‫ݑ‬௩௧௥௜௔௟
࢑ଵ ൌ ࢑௖௢௡௩ ǡ ࢌଵ ൌ ࢌ௖௢௡௩ ǡ ࢛ଵ ൌ ࢛௖௢௡௩
݅ ൌ݅൅ͳ
ઢ࢛௜ ൌ ࢛௧௥௜௔௟ െ ࢛௜ ǡઢࢌ௜ ൌ ࢑௜ ઢ࢛௜
Call Element Group 2, get:
Call Element Group 1, get:
ሺ࢑௘௟௘Ǥଷ ሻ௜ ǡ ൫࢛௣ǡ௘௟௘Ǥଷ ൯ ǡ ሺࢗ௘௟௘Ǥଷ ሻ௜
௜
ሺ࢑௘௟௘Ǥଵ ሻ௜ ǡ ൫࢛௣ǡ௘௟௘Ǥଵ ൯ ǡ ሺࢗ௘௟௘Ǥଵ ሻ௜
௜
ሺ࢑௘௟௘Ǥଶ ሻ௜ ǡ ൫࢛௚௥ଵ ൯ ǡ ൫࢑௚௥ଵ ൯
௜
௜
࢑௜ାଵ ൌ ෍ሺ࢑௘௟௘ ሻ௜ ǡ ࢛௜ାଵ ൌ ෍൫࢛௚௥ ൯
ࢌ௜ାଵ ൌ ࢌ௜ ൅ ઢࢌ௜
‫ ݎݎܧ‬ൌ ԡ࢛௧௥௜௔௟ െ ࢛௜ାଵ ԡ
N
ሺ࢑௘௟௘Ǥସ ሻ௜ ǡ ൫࢛௚௥ଶ ൯ ǡ ൫࢑௚௥ଶ ൯
௜
Call Element Group 3, get:
ሺ࢑௘௟௘Ǥହ ሻ௜ ǡ ൫࢛௣ǡ௘௟௘Ǥହ ൯ ǡ ሺࢗ௘௟௘Ǥହ ሻ௜
௜
௜
ሺ࢑௘௟௘Ǥ଺ ሻ௜ ǡ ൫࢛௚௥ଷ ൯ ǡ ൫࢑௚௥ଷ ൯
௜
Y
‫ ݎݎܧ‬൑ ܶ‫݈݋‬
௜
௜
ࡲ ൌ ܹǤ ሺࢌሻ௜ାଵ ǡ ࡷ ൌ ܹǤ ሺ࢑ሻ௜ାଵ
Return ࡲǡ ࡷ
Update:
࢑௖௢௡௩ ൌ ࢑௜ାଵ ǡ ࢌ௖௢௡௩ ൌ ࢌ௜ାଵ ǡ ࢛௖௢௡௩ ൌ ࢛௧௥௜௔௟
࢛௖௢௡௩ǡ௚௥ଵ ൌ ൫࢛௚௥ଵ ൯ ǡ ࢛௖௢௡௩ǡ௚௥ଶ ൌ ൫࢛௚௥ଵ ൯ ǡ ࢛௖௢௡௩ǡ௚௥ଷ ൌ ൫࢛௚௥ଷ ൯
௜
௜
௜
௜
௜
௜
࢑௖௢௡௩ǡ௚௥ଵ ൌ ൫࢑௚௥ଵ ൯ ǡ ࢑௖௢௡௩ǡ௚௥ଶ ൌ ൫࢑௚௥ଶ ൯ ǡ ࢑௖௢௡௩ǡ௚௥ଷ ൌ ൫࢑௚௥ଷ ൯
࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥଵ ൌ ൫࢛௣ǡ௘௟௘Ǥଵ ൯௜ ǡ ࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥଷ ൌ ൫࢛௣ǡ௘௟௘Ǥଷ ൯௜ ǡ ࢛௣ǡ௖௢௡௩ǡ௘௟௘Ǥହ ൌ ൫࢛௣ǡ௘௟௘Ǥହ ൯௜
ࢗ௖௢௡௩ǡ௘௟௘Ǥଵ ൌ ሺࢗ௘௟௘Ǥଵ ሻ௜ ǡ ࢗ௖௢௡௩ǡ௘௟௘Ǥଷ ൌ ሺࢗ௘௟௘Ǥଷ ሻ௜ ǡ ࢗ௖௢௡௩ǡ௘௟௘Ǥହ ൌ ሺࢗ௘௟௘Ǥହ ሻ௜
End TPB element
Figure 6-9
Flow chart for solving TPB element
187
The inverse Newton-Raphson iterating procedure detailed above works well in
most cases. However, for the TPB element, which has a stiffening stage at the end of the
backbone curve, the iteration can fail when the initial stiffness is very large. As shown in
Figure 6-10, when the displacement changes from zero to ‫ݑ‬௧௥௜௔௟ , the iteration procedure
does not converge as it cycles infinitely between the two large displacement stiffnening
stages. To solve this problem, the TPB element was implemented with a provision that
the incremental displacement is divided into smaller substeps when convergence is not
reached after the maximum number of allowed iterations.
݂
߂݂ଷ
݇ଵ
߂݂ସ
߂݂ଵ
݇ଷ ൌ ݇ହ
‫ݑ‬௧௥௜௔௟
߂݂ଶ
‫ݑ‬
݇ଶ ൌ ݇ସ
ǻ‫ݑ‬ଵ
ǻ‫ݑ‬ଷ
ǻ‫ݑ‬ହ
Figure 6-10
ǻ‫ݑ‬ଶ
ǻ‫ݑ‬ସ
Situation where the inverse Newton – Raphson iteration fails
͸ǤͶǤ͵ –‡”ƒ–‹‰‘˜‡”Ž‡‡– ”‘—’
The flow chart in Figure 6-9 requires the tangent stiffness of all sub-elements
(࢑௚௔௣ ǡ ࢑௣௟௔௦௧௜௖ ) and deformation of all Element Groups, ࢛௚௥ , for updating the
188
deformation and tangent stiffness of the TPB element. The tangent stiffness of the subelements for each Element Group and its deformation can be determined from the flow
chart in Figure 6-11. The input at the element group level includes the stiffness ࢑௚௥ǡ௖௢௡௩
and the displacement ࢛௚௥ǡ௖௢௡௩ of the element group; the plastic deformation ࢛௣ǡ௖௢௠௩ǡ௘௟௘
and the back stress ࢗ௖௢௡௩ǡ௘௟௘ of the bi-linear plasticity element belonging to the element
group from the converged state of the last time step; the normalized force vectors ࢌ௜ and
ઢࢌ௜ passed from the TPB element level. The new values of plastic deformation ‫ݑ‬௣ǡ௘௟௘ and
back stress ‫ݍ‬௘௟௘ are also determined and returned to the TPB element level for updating
state. This procedure is the Newton-Raphson iteration procedure and demonstrated in
Figure 6-12.
189
Begin Element Group
݂௜ ǡ ȟ݂௜
‫ݑ‬௖௢௡௩ǡ௚௥ ǡ ݇௖௢௡௩ǡ௚௥
‫ݑ‬௣ǡ௖௢௡௩ǡ௘௟௘ ǡ ‫ݍ‬௖௢௡௩ǡ௘௟௘
݂௧௥௜௔௟
݆ൌͲ
ൌ ݂௜ ൅ ȟ݂௜ ǡ ൫݂௚௥ ൯ ൌ ݂௜
ଵ
൫݇௚௥ ൯ ൌ ݇௖௢௡௩ǡ௚௥ ǡ ൫‫ݑ‬௚௥ ൯ ൌ ‫ݑ‬௖௢௡௩ǡ௚௥
ଵ
ଵ
݆ ൌ݆൅ͳ
ିଵ
൫ȟ݂௚௥ ൯ ൌ ݂௧௥௜௔௟ െ ൫݂௚௥ ൯ ǡ ൫ȟ‫ݑ‬௚௥ ൯ ൌ ൫݇௚௥ ൯ ൫ȟ݂௚௥ ൯
௝
௝
Solving circular elastic gap element,
get ൫݇௘௟௘ǡ௚௔௣ ൯ ǡ ൫݂௘௟௘ǡ௚௔௣ ൯
௝
௝
௝
௝
Solving bilinear plasticity element, get:
൫݇௘௟௘ǡ௣௟ ൯௝ ǡ ൫݂௘௟௘ǡ௣௟ ൯௝ ǡ ൫‫ݑ‬௣ǡ௘௟௘ ൯௝ ǡ ሺ‫ݍ‬௘௟௘ ሻ௝
൫݇௚௥ ൯
௝ାଵ
N
‫ ݎݎܧ‬൑ ܶ‫݈݋‬
Y
௝
ൌ ൫݇௘௟௘ǡ௚௔௣ ൯ ൅ ൫݇௘௟௘ǡ௣௟ ൯௝
൫‫ݑ‬௚௥ ൯
൫݂௚௥ ൯
௝
௝ାଵ
௝ାଵ
ൌ ൫‫ݑ‬௚௥ ൯ ൅ ൫ȟ‫ݑ‬௚௥ ൯
௝
ൌ ൫݂௘௟௘ǡ௚௔௣ ൯ ൅ ൫݂௘௟௘ǡ௣௟ ൯௝
௝
‫ ݎݎܧ‬ൌ ቛ݂௧௥௜௔௟ െ ൫݂௚௥ ൯
௝ାଵ
Return ൫݇௘௟௘ǡ௣௟ ൯௝ ǡ ൫‫ݑ‬௣ǡ௘௟௘ ൯௝ ǡ ሺ‫ݍ‬௘௟௘ ሻ௝ ǡ ൫݇௘௟௘ǡ௚௔௣ ൯ ǡ ൫‫ݑ‬௚௥ ൯
௝
௝ାଵ
ǡ ൫݇௚௥ ൯
End Element Group
Figure 6-11
௝
Flow chart for solving Element Group
௝ାଵ
ቛ
190
f
݇ଵ
݇ଶ
݂௧௔௥௚௘௧
߂݂௦ଶ
߂݂௦ଵ
݂௦ଶ
݂௦ଵ
‫ݑ‬௦ଵ
Figure 6-12
6.5
ȟ‫ݑ‬௦ଵ
‫ݑ‬௦ଶ
ȟ‫ݑ‬௦ଶ
‫ݑ‬௦ଷ
‫ݑ‬
Newton – Raphson iteration for iterating over Element Group
Preparation for Assembly of Element Stiffness and Force into Global
Equations
The horizontal and vertical stiffness and force of the TPB element are combined
considering the basis coordinate system DOFs:
‫்ܭ‬ி௉ሺଷൈଷሻ
‫ܭ‬ଵଵ
‫ܭ‬
ൌ ൥ ଶଵ
Ͳ
‫்ܨ‬ி௉ሺଷൈଵሻ
‫ܭ‬ଵଶ
‫ܭ‬ଶଶ
Ͳ
‫ܨ‬௫
ൌ ቐ‫ܨ‬௬ ቑ
‫ܨ‬௩
Ͳ
Ͳ൩
‫ܭ‬௩
(6.5-1)
(6.5-2)
A transformation matrix from the basis coordinate system DOFs to the global
coordinate DOFs (Figure 6-13) is defined as:
191
െͳ Ͳ
ܽ ൌ ൥ Ͳ െͳ
Ͳ
Ͳ
Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ
െͳ Ͳ Ͳ Ͳ
ͳ Ͳ
Ͳ ͳ
Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
ͳ Ͳ Ͳ
Ͳ
Ͳ൩
Ͳ
(6.5-3)
Tangent stiffness matrix of TFP element in global coordinate system:
۹ ்ி௉ሺଵଶൈଵଶሻ ൌ ்ܽ ൈ ‫்ܭ‬௉஻ሺଷൈଷሻ ൈ ܽ
‫ܭ‬ଵଵ
‫ܭۍ‬
‫ ێ‬ଶଵ
Ͳ
‫ێ‬
Ͳ
‫ێ‬
‫Ͳ ێ‬
Ͳ
ൌ‫ێ‬
‫ێ‬െ‫ܭ‬ଵଵ
‫ێ‬െ‫ܭ‬ଶଵ
‫Ͳ ێ‬
‫Ͳ ێ‬
‫Ͳ ێ‬
‫Ͳ ۏ‬
‫ܭ‬ଵଶ
‫ܭ‬ଶଶ
Ͳ
Ͳ
Ͳ
Ͳ
െ‫ܭ‬ଵଶ
െ‫ܭ‬ଶଶ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
‫ܭ‬௩
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െ‫ܭ‬௩
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െ‫ܭ‬ଵଵ
െ‫ܭ‬ଶଵ
Ͳ
Ͳ
Ͳ
Ͳ
‫ܭ‬ଵଵ
‫ܭ‬ଶଵ
Ͳ
Ͳ
Ͳ
Ͳ
െ‫ܭ‬ଵଶ
െ‫ܭ‬ଶଶ
Ͳ
Ͳ
Ͳ
Ͳ
‫ܭ‬ଵଶ
‫ܭ‬ଶଶ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െ‫ܭ‬௩
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
‫ܭ‬௩
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ‫ې‬
‫ۑ‬
Ͳ
‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ۑ‬
Ͳ‫ے‬
Z
z
Element nodes
2
Y
Global coordinate system
y
1
x
Basis coordinate system
X
Figure 6-13
Basis coordinate system of TPB in global coordinate system
(6.5-4)
192
The force vector of the TFP element in global coordinate system:
۴்ி௉ሺଵଶൈଵሻ ൌ ்ܽ ൈ ‫்ܨ‬ி௉ሺଷൈଵሻ
۴்ி௉ሺଵଶൈଵሻ
െ‫ܨ‬௫
‫ۓ‬െ‫ۗ ܨ‬
௬
ۖ
ۖ
ۖെ‫ܨ‬௩ ۖ
ۖ Ͳ ۖ
ۖ Ͳ ۖ
ۖ
ۖ
Ͳ
ൌ
‫ܨ ۔‬௫ ۘ
ۖ ‫ܨ‬௬ ۖ
ۖ ‫ۖ ܨ‬
ۖ ௩ ۖ
ۖ Ͳ ۖ
ۖ Ͳ ۖ
‫ۙ Ͳ ە‬
(6.5-5)
Equation (6.5-4) and Equation (6.5-5) suggest that the developed TPB element
does not have any rotational resistance. To account for the eccentricity of the reaction
forces, the overturning moment and torque are computed and equally distributed to the
element’s nodes. The resistance to these moments and torque is provided by the supports
and/or the element connected to the TPB element’s nodes.
From Figure 6-14:
‫ܯ‬௬ ൌ
ܶൌ
where ݄ is the height of isolator.
Similarly:
‫ܨ‬௩ Ǥ ‫ݑ‬௫ െ ‫ܨ‬௫ Ǥ ݄
ʹ
‫ܨ‬௫ Ǥ ‫ݑ‬௬ െ ‫ܨ‬௬ Ǥ ‫ݑ‬௫
ʹ
(6.5-6)
(6.5-7)
193
‫ܯ‬௫ ൌ
‫ܨ‬௬ Ǥ ݄ െ ‫ܨ‬௭ Ǥ ‫ݑ‬௬
ʹ
(6.5-8)
‫ܨ‬௩
z
y
‫ܯ‬௬
݄
‫ܨ‬௬
‫ܨ‬௫
ʹ
ܶ
‫ݑ‬௬
ͳ
‫ܨ‬௫
‫ܨ‬௫
‫ݑ‬௫
‫ܯ‬௬
ʹ
ͳ
x
ܶ
‫ܨ‬௩
‫ܨ‬௬
(a)
(b)
Figure 6-14
‫ܨ‬௫
‫ݑ‬௫
Force diagram for computing overturning moment and torsion
(a) ‫ ݔ‬െ ‫ ݖ‬elevation view
(b) ‫ ݔ‬െ ‫ ݕ‬plan view
x
Chapter 7
ƒŽ›–‹ ƒŽ‘†‡Ž‹‰‘ˆ–Š‡—‹Ž†‹‰’‡ ‹‡
An analytical model of the building specimen was developed both in SAP2000
v14 and OpenSees v2.2.2. The OpenSees model was the primary model for analysis,
comparison with and calibration of test data, and further investigation. The SAP model
was used for computing nodal gravity load and lumped mass distribution to the OpenSees
model, as well as verification of the comparing static response and modal analysis of the
OpenSees model.
The SAP model (Figure 7-1 (a)) is a 3-dimensional model using linear
constitutive behavior, and neglecting secondary (P-Delta or large deformation) effects. A
center line approach was used, meaning beam and column nodes connect together at their
centerline (or centroid of the sections) and slabs connect to their edge beams at their
centerlines. Distributed gravity load and mass were applied to the slab and frame
elements according to the realistic distribution of these loads. To account for the
composite interaction between slabs and beams, the beam stiffnesses were uniformly
scaled by a common factor of ʹǤʹͲ so that the fundamental period of the model matched
the fundamental period of the specimen obtained from the processed test data, which is
approximately ͲǤ͸ͻ• in Y direction. This suggests that the beam stiffness in this bare
frame centerline model should be scaled by an average of ʹǤʹͲ to account for the
composite interaction of the beams and slabs. The SAP model is rather simple and is not
presented here in detail.
195
(b)
(a)
Figure 7-1
Models of the specimen
(a) SAP model
(b) OpenSees model
The OpenSees model (Figure 7-1 (b)) is a 3-dimensional frame model. Slabs were
not explicitly modeled, but their effect was accounted for through application of
diaphragm constraints and composite beam sections. The beam-column connection
behavior was represented by a panel zone model. The nonlinear material behavior was
considered through nonlinear material models, and geometric nonlinearities were also
included using P-Delta transformation. Mass and gravity load were lumped to nodes.
Rayleigh damping model was employed to represent energy dissipation. Details of the
OpenSees model are described in the following sections.
196
7.1
Material Models
Materials of the specimen were assumed to be homogeneous and isotropic.
Two uniaxial material models were used for modeling the behavior of steel and
concrete materials. Uniaxial materials in OpenSees represent stress-strain or forcedisplacement relationships in a single direction. In general, the stress and strain state at
any point in the structures is a 3-dimensional state. However, in analysis of a structure
containing primarily slender 1-dimensional members, the effect of shear deformation is
usually neglected. The combination of bending moment and axial force in a 1dimensional element results in normal stress acting in the axial direction of the element as
shown in Figure 7-2. Therefore, uniaxial material models are sufficient to capture
behavior of 1-dimensional elements.
The Giuffre-Menegotto-Pinto steel material model (CEB, 1996), which is
implemented in OpenSees as Steel02 material, was used to model behavior of steel
material. This material model can capture both kinematic hardening and isotropic
(a)
(b)
Figure 7-2
(c)
Normal stress in a cross section of 1-dimensional elements
(a) Normal stress caused by bending moment
(b) Normal stress caused by axial force
(c) Normal stress caused by axial force and bending moment
197
hardening. The monotonic and cyclic stress-strain relations for this material with
kinematic hardening are shown in Figure 7-3. The material behavior is controlled by: (1)
yield stress ߪ௬ , (2) initial stiffness ‫ܧ‬, (3) post-yield stiffness ratio ܾ ൌ ‫ܧ‬௣ Ȁ‫ܧ‬, (4)
parameters ܴ଴ ǡ ܴܿଵ ǡ ܴܿଶ that control the transition from elastic to plastic branches, and (5)
optional parameters that control isotropic behavior (which was not used in this model).
Values of ܴ଴ ൌ ͳͲ‫Ͳʹ݋ݐ‬, ܴܿଵ ൌ ͲǤͻʹͷ and ܴܿଶ ൌ ͲǤͳͷ are recommended (OpenSees
manual).
The concrete model proposed by Kent and Park (1971) and modified by Scott et
al. (1982), implemented into OpenSees as Concrete01 material, was used for modeling
concrete behavior in the model. The envelope monotonic and cyclic stress-strain relations
for this model are presented in Figure 7-4. Parameters that determine the response are: (1)
concrete compressive strength ߪ௣௖ , (2) strain at compressive strength ߳௣௖ , (3) crushing
strength ߪ௣௖௨ , and (4) strain at crushing strength ߳௣௖௨ . Any tensile resistance of concrete
Stress, σ (MPa)
400 σ
400
y
200
200
0
0
-200
-200
-400
-0.02
εy
-0.01
0
Strain, ε
0.01
0.02
-400
-0.02
-0.01
0
Strain, ε
(a)
(b)
Figure 7-3
Behavior of steel material model
(a) Backbone curve
(b) Hysteresis loop due to cyclic load
0.01
0.02
Compressive stress, σc (MPa)
198
30
25
30
σpc
25
20
20
15
15
10
10
5
0
-5
-5
σpcu
unloading = reloading path
P
5
0
εpcu
εpc
0
5
10
Compressive strain, ε
15
-5
-5
-3
x 10
c
(a)
R ε
R
εP
0
5
10
Compressive strain, ε
c
15
-3
x 10
(b)
Figure 7-4
Behavior of concrete material model
(a) Backbone curve
(b) Hysteresis loop due to cyclic load
is neglected in this material model. The envelop curve before reaching the strength ߪ௣௖ is
a parabola with tangent slope computed according to:
‫ܧ‬௖ ൌ
ʹߪ௣௖
߳
ቆͳ െ
ቇ
߳௣௖
߳௣௖
(7.1-1)
The unloading path (from ܲ to ܴ) and reloading path (ܴ to ܲ) in Figure 7-4 (b) of
the Kent-Park-Scott model are identical and determined by:
ଶ
‫߳ ۓ‬௉ ൌ ͲǤͳͶͷ ቆ ߳ோ ቇ ൅ ͲǤͳ͵ ቆ ߳ோ ቇ ݂݅ ቆ ߳ோ ቇ ൏ ʹ
ۖ߳௣௖
߳௣௖
߳௣௖
߳௣௖
߳ோ
߳ோ
‫߳ ۔‬௉
ൌ ͲǤ͹Ͳ͹ ቆ
െ ʹቇ ൅ ͲǤͺ͵Ͷ݂݅ ቆ ቇ ൒ ʹ
ۖ
߳௣௖
߳௣௖
‫߳ ە‬௣௖
(7.1-2)
This concrete model can be used for both confined and unconfined concrete (Kent
and Park, 1971). The two models are different in the slope of the softening line of the
backbone curve after concrete reaches to its strength. The unconfined concrete model was
used because the concrete of the slab is not confined.
199
7.2
Modeling Columns
The displacement-based nonlinear elements were used to model the columns.
Force-based elements are known to provide improved accuracy compared to
displacement-base elements without discretization (Neuenhofer and Filippou, 1997).
Force-based elements were tested extensively for the building model; however, multiple
levels of internal iteration caused convergence problems at the element level, preventing
these elements from utilized.
Contrarily to force-based elements, which are based on the interpolation of force
distribution, displacement-based elements are formulated base on the interpolation of
displacement distribution. The interpolation function (of force) in force-based elements is
always exact but the displacement (hence curvature) interpolation function in the
displacement-based element is not accurate in the nonlinear range. This can be illustrated
conceptually in Figure 7-5. The figure shows distribution of moment and curvature of the
two elements after yielding. The distribution of bending moment is still linear but the
curvature distribution is no longer linear throughout the beam. In force-based elements,
the interpolation function is linear on moment distribution so that it can capture the
moment distribution. The interpolation function in displacement-based element produces
linear distribution of curvature along the beam so that cannot capture the behavior of the
beam.
Dividing a frame member into several elements can improve the performance of
displacement-based elements since the nonlinear curvature distribution can be captured
(Neuenhofer and Filippou, 1997). When a column enters the inelastic response range, the
200
Force-based = exact
(c)
Displacement based
(a)
‫ܯ‬
‫ܯ‬௣௟
‫ܯ‬
‫ܯ‬௣௟
‫ܯ‬
‫ܯ‬
(b)
ߢ௣௟
ߢ
ߢ௣௟
Displacement based
Force-based = exact
Figure 7-5
Behavior of force-based elements and displacement-based elements
(a) Bending moment distribution
(b) Curvature distribution
(c) Moment – curvature relationship
distribution of bending moment is approximately linear because the moment induced by
transverse distributed loading from the column inertial force is small compared to the
moment from rotation of the two ends. The linear distribution of bending moment results
in the concentration of plasticity at the two ends, so that these areas in particular can be
discretized to improve the accuracy.
Figure 7-6 compares the simulated moment-rotation relationship of a typical
column member of the specimen developed from force-based elements and displacementbase elements. The member is a steel member with yield strength of ʹͻͷƒ. The
dimensions of the hollow square section are ͵ͷͲ ൈ ͵ͷͲ ൈ ͳͻ. The length
of the member is ͵. The member was subjected to bending moments at its ends as
shown in Figure 7-6 (a). The member was modeled by: (1) one force-based element with
201
1.4
‫ܯ‬
Moment, M (MNm)
1.2
ߠ
1
0.8
0.6
0.4
1 force ele., 7 pts.
3 disp. ele., 5 pts.
1 force ele., 5 pts.
0.2
‫ܯ‬
(a)
Figure 7-6
0
0
0.005
0.01
0.015
Rotation, θ (rad)
0.02
0.025
(b)
Behavior of a bending member simulated by force-based element
and displacement-based elements
(a) Bending member, (b) Moment-rotation relationship
7 integration points along the element, (2) three displacement-based elements with 5
integration points per element, and (3) one force-based element with 5 integration points.
For case (2), the length of the two end elements, where plasticity is expected to occur
first, equals the depth of the section (͵ͷͲሻ. Based on the moment-rotation
relationships developed for the 3 cases (Figure 7-6 (b)), 3 displacement-based elements
with 5 integration points per element gives similar results to 1 force-based element with 7
integration points. All column members in the building model were modeled using the
discretization of case (2).
To capture the interaction between axial force and bending moment in the column
response, fiber sections were used. In the fiber section approach, a section is divided into
small areas called fibers. From the trial section deformations, including centroidal axial
strain and curvature, the axial strain at every fiber is computed, and the axial stress and
tangent modulus are derived from the stress-strain relationship. The axial stress and
tangent modulus are numerically integrated over section fibers to compute the resultant
202
forces (axial force and bending moments) and stiffnesses (axial and bending) of the
section.
7.3
Modeling Beams
Beam members were also modeled by displacement-based elements because of
the convergence difficulties mentioned earlier. Each member was also divided into
several elements for improving the performance of displacement-based elements. The
influence of distributed transverse load, including gravity load and inertial force on the
bending moment distribution is larger in beams than in columns, so the beam moment
distribution may not be approximately linear. Thus, plasticity may occur anywhere along
the beam length. As such, each beam member was divided into nearly equal length
elements. This disceretization approach also improves the distribution of mass – which is
lumped to nodes.
͹Ǥ͵Ǥͳ ”‹ƒ”›‡ƒ•
In the OpenSees model, each primary beam member, which is supported by
columns, was divided into at least 8 displacement-based elements depending on how it
connects with other members. Figure 7-7 schematically shows the discretization of
typical ͷ and ͹ primary beam members. Primary beams were connected to columns
through a panel zones model described later (Section 7.4).
Primary beams were modeled with composite secions to account for the effect of
slab contribution. The slab effective width in the composite section was selected
according to the recommendation of AISC (2005), which states that the width of concrete
203
Secondary beam
(a)
Big section
Small section
2 elements
2 x 4 elements
Secondary beam
(b)
Big section
Small section
2 elements
2 x 4 elements
Figure 7-7
Discretization of typical primary beams
(a) 5 m beams
(b) 7 m beams
slab for each side of the section is the minimum value of (1) one-eighth of the beam span,
(2) one-half the distance between the beams and (3) the distance to the edge of the slab.
Reinforcement of slabs parallel to the beam was also included in the section model.
For non-symmetric sections such as these composite sections, when the material
behavior becomes nonlinear, the neutral plane of the section moves and the geometric
centerline deforms axially under pure bending loads. In that case, the rigid diaphragm
constraint, which prevents the axial deformation of the centerline, introduces an axial
force to the bent beam. The existence of axial force changes the behavior of beams
significantly, as demonstrated Figure 7-8, which compares the bending behavior of
204
ܲ
ȟ
Concrete
Steel
(b)
4
1
2
0.5
Force, P (MN)
Displacement, Δ (cm)
(a)
0
0
-0.5
-2
Axially restrained
Axially unrestrained
-1
-4
-4
(c)
-2
0
2
Displacement, Δ (cm)
4
(d)
Figure 7-8
Behavior of a composite fiber section beam with and without axial restraint
(a) Beam configuration, (b) Composite section of the beam
(c) Displacement protocol, (d) Relationship between force and displacement
composite section with and without axial deformation restraint. In this numerical
investigation, a pseudo-static cyclic transverse displacement profile ȟ (Figure 7-8 (c))
was applied to the midpoint of a ͷ long beam (Figure 7-8 (a)). The beam section, a
representative section from the model, (Figure 7-8 (b)) is the combination of a steel Isection and a concrete slab section where the I-section is ͶͲͲ ൈ ʹͲͲ ൈ ͻ ൈ ͳͻ and
the concrete slab portion is ͳʹͷ ൈ ͳͳǤ͹ͷ . Because of the corrugated deck of the slab,
a gap between the slab and the I-section of ͵Ǥ͹ͷ  is modeled. Strength of steel and
compressive strength of concrete are ͵ʹͷƒ and ʹͶƒ, respectively. The
relationship between the driven force ܲ and the displacement ȟ is shown in Figure 7-8
(d). It is clear that the relationship between ܲ and ȟ when the axial deformation of the
205
beam is restrain is significantly different from that relationship when the axial
deformation of the beam is not restrained. The hysteresis loop of the axial restrained
beam is almost symmetric while the loop of the axial unrestrained beam is nonsymmetric with the strength and stiffness in the positive displacement side is much
smaller than that in the negative displacement side. The area of the loop at each cycle is
smaller for the axial unrestrained beam than for the axial restrained beam. In short, the
strength, stiffness and dissipated energy are totally different between the two models.
To avoid the inadvertent effect of axial force on bending behavior of the
composite section of beams, the axial behavior and bending behavior were decoupled
through the use of resultant sections for moment-curvature and axial force-strain.
Moreover, in resultant section, neutral planes always contain the geometric centerline so
that the centerline never deforms under bending.
The resultant section behavior of the beams was determined from a section
analysis, as illustrated in Figure 7-9. The composite section of a beam was modeled as a
fiber section (Figure 7-9 (a)) and its pure bending cyclic behavior determined by section
analysis is plotted in Figure 7-9 (b). This cyclic behavior was approximately captured by
combining the OpenSees Steel02 material model (Figure 7-9 (c)) with a hysteresis model
(Figure 7-9 (d)) in parallel.
͹Ǥ͵Ǥʹ ‡ ‘†ƒ”›‡ƒ•
Secondary beams, which are supported by primary beams, were modeled as
elastic beam column elements with assumed elastic composite sections. The width of the
206
1.5
Reinforcement
Moment, M (MNm)
1
Concrete slab
Steel I-section
0.5
0
-0.5
-1
Fiber Section
Resultant Section
-1.5
-2
-0.02
0.02
(b)
1.5
1.5
1
1
Moment, M (MNm)
Moment, M (MNm)
(a)
-0.01
0
0.01
Curvature, κ (rad/m)
0.5
0
-0.5
-1
0.5
0
-0.5
-1
-1.5
-1.5
-2
-0.02
-0.01
0
0.01
Curvature, κ (rad/m)
(c)
0.02
-2
-0.02
-0.01
0
0.01
Curvature, κ (rad/m)
0.02
(d)
Figure 7-9
Composite section behavior
(a) Fiber section for section analysis
(b) Moment-curvature relationship of the section
(c) Component 1 of resultant section modeled by steel material model
(d) Component 2 of resultant section modeled by pinching material model
concrete slab in these composite sections was determined similarly to that of primary
beams. The secondary beams in the model were also divided into 8 elements for the
purpose of distributing mass evenly.
7.4
Modeling Panel Zones
Krawinkler panel zone model (Krawinkler, 1978) described in detail by Charney
and Downs (2004), was used to model the connection between beams and columns.
207
According to this model, the panel zone is modeled by 8 rigid elements and 2 elastic
perfectly plastic rotational springs, one representing the shear behavior of the panel zone
(or the web, lying in the working plane) and, one representing the bending behavior of
flanges (perpendicular to the working plane) as shown in Figure 7-10 (b).
The initial stiffness and yield strength of the spring representing the shear
behavior of panel zone are computed as:
ܵ௉ ൌ ‫ܸܩ‬௉
(7.4-1)
‫ܯ‬௒௉ ൌ ͲǤͷͺ‫ܨ‬௒ ܸ௉
(7.4-2)
where: ܵ௉ ൌ initial stiffness of the spring, ‫ ܩ‬ൌ shear modulus of steel, ܸ௉ ൌ volume of
the panel zone, ‫ܯ‬௒௉ ൌ yield strength of the spring, and ‫ܨ‬௒ ൌ yield strength of steel
material.
Panel web
Spring
representing
panel web
Rigid element
Hinge
Beam
Beam
Column
Column
(a)
Figure 7-10
Spring
representing
column flanges
(b)
Panel zone model for beam to column connection
(a) Beam to column connection
(b) Analytical model of panel zone
208
The initial stiffness and yield strength of the spring representing bending behavior
of flanges are:
ଶ
ܵி ൌ ͲǤ͹ͷ‫ܾܩ‬஼௙ ‫ݐ‬஼௙
(7.4-3)
ଶ
‫ܯ‬௒ி ൌ ͳǤͺͲ‫ܨ‬௒ ܾ஼௙ ‫ݐ‬஼௙
(7.4-4)
where: ܵி ൌ initial stiffness of the spring, ܾ஼௙ ൌ flange width of column, ‫ݐ‬஼௙ ൌ flange
thickness of column, ‫ܯ‬௒ி ൌ yield strength of the spring.
The panel zone can also be modeled using “Scissor model” with proper
parameters so that it is equivalent to Krawinkler model. However, as noted by Charney
and Downs (2004), the scissor model is not recommended when the lengths of the beams
on the 2 sides of the panel zone or the heights of the columns below and above the panel
are unequal. Both conditions exist in the building specimen so that the Krawinkler was
used despite its computational expense. In this building model, where columns are fully
welded to primary beams in both directions, the panel zones in two directions were
independently modeled by two Krawinkler panel zone models
Elastic truss elements equivalent to Krawinkler model were also used to model
gusset plates. The gusset plates were an integral part of the specimen for attaching
various dampers in the March 2009 test (Kasai et al., 2010). The dampers were removed
in the test reported in this dissertation. Figure 7-11 shows a typical gusset plate and its
finite element model. Finite element analysis of the model subjected to gravity load
suggests that the gusset resistance is in the diagonal direction, and it can be modeled as a
diagonal truss.
209
(b)
(a)
Figure 7-11
Gusset plate and its finite element model
(a) Gusset plate
(b) Von-Mises stress due to gravity load
To derive the stiffness of an equivalent truss, consider the panel zone diagram of a
gusset plate in Figure 7-12. In this diagram, ‫ ܨ‬is the force applied to the equivalent truss
and ‫ ܯ‬is the internal moment in the spring representing the gusset plate. The column
flange spring has been eliminated.
From moment equilibrium at the spring (Figure 7-12 (a)), the relationship
between ‫ ܨ‬and ‫ ܯ‬can be written as:
‫ ܯ‬ൌ ‫ܨ‬Ǥ
‫ݔ‬Ǥ ‫ݕ‬
ඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ
(7.4-5)
In the elastic range:
ܵ௉ ߠ ൌ ‫ܨ‬Ǥ
‫ݔ‬Ǥ ‫ݕ‬
ඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ
(7.4-6)
210
‫ݔ‬
‫ܯ‬
‫ܨ‬
‫ܯ‬
‫ܨ‬
‫ܯ‬
Equivalent truss
Ͳ
‫ݕ‬
Ͳ
Panel zone
Ͳ
Ͳ
‫ܨ‬
‫ܨ‬
(a)
‫ݔ‬
ߙଵ ߙଶ ‫ݕ‬
‫ܮ‬
(b)
Figure 7-12
Panel zone model and equivalent truss of the gusset plate
(a) Model and force diagram
(b) Geometry of the model
where ܵ௉ is elastic stiffness of the gusset plate computed from Equation (7.4-1), ߠ is
deformed angle of the spring. From the geometry relationship in Figure 7-12 (b):
‫ ܮ‬ൌ ‫ݔ‬Ǥ •‹ ߙଵ ൅ ‫ݕ‬Ǥ •‹ ߙଶ
(7.4-7)
Differentiate Equation (7.4-7) with respect to ߙଵ and ߙଶ and neglect the higher terms:
ȟ‫ ܮ‬ൌ ‫ݔ‬Ǥ ‘• ߙଵ Ǥ ȟߙଵ ൅ ‫ݕ‬Ǥ ‘• ߙଶ Ǥ ȟߙଶ
(7.4-8)
Noting that ‘• ߙଵ ൌ ‫ݕ‬Ȁඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ and ‘• ߙଶ ൌ ‫ݔ‬Ȁඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ , Equation (7.4-8) becomes:
211
ȟ‫ ܮ‬ൌ
‫ݔ‬Ǥ ‫ݕ‬
ඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ
ሺȟߙଵ ൅ ȟߙଶ ሻ ൌ ‫ݔ‬Ǥ ‫ݕ‬
ඥ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ
ߠ
(7.4-9)
Substitute Equation (7.4-9) into Equation (7.4-6):
‫ݔ‬ଶ ൅ ‫ݕ‬ଶ
‫ ܨ‬ൌ ܵ௉ ଶ ଶ Ǥ ȟ‫ ܮ‬ൌ ‫ܭ‬௧௥௨௦௦ Ǥ ȟ‫ܮ‬
‫ݕ ݔ‬
(7.4-10)
where ‫ܭ‬௧௥௨௦௦ is stiffness of the equivalent elastic truss.
7.5
Modeling Gravity Load and Mass
As mentioned earlier, the OpenSees model is a bare frame model without slabs so
that the gravity load and mass of the specimen must be applied directly to beams and
columns of the model. Static analysis of the SAP model, which calculated the transfer of
gravity load from slabs to beams, was used to compute distributed loads to the frame
elements in the OpenSees model. Specifically, static analysis of the SAP model subjected
to gravity load was carried out to determine the internal forces in beams. From the shear
forces at the two ends of a beam element, the equivalent uniform load ‫ ݒ‬on the beam
element was computed according to:
‫ݒ‬ൌ
ܸ௜ െ ܸ௝
‫ܮ‬
(7.5-1)
where ܸ௜ and ܸ௝ are shear forces at the two ends of the element, and ‫ ܮ‬is the length of the
element. Because of the bending stiffness of the slab, the slab gravity load does not
transfer entirely to beams but some transfers to the corner slab nodes. In the OpenSees
model, these loads were applied as concentrated loads to the corresponding corners. The
212
mass of the OpenSees model was directly derived from gravity load and was lumped to
every node of the model.
Table 7-1 summarizes the weight and the eccentricity of gravity center from the
geometric center of the SAP and OpenSees models. The weight of all floors in SAP
model, estimated directly from the nominal weight and dimension of all components of
the specimen, is shown in column 2. By this approach, the total weight of the specimen is
ͷͳʹʹ. However, the measured weight of the specimen from the test was actually
ͷʹʹͲ (Section 4.5.2). To match the measured weight, the weight in the analytical
model was uniformly increased by a factor of ͷʹʹͲȀͷͳʹʹ ൌ ͳǤͲͳͻ. Column 3 shows the
factored weight at all floors, which was used in the OpenSees model. The last 2 columns
show eccentricity of the weight at every floor from its geometric center. In general, the
gravity centers of all floors shift to Northeastern direction. At base, the gravity center
shifts to the West side because the column bases concentrate at that side and there is a
half of the weight of staircase in the Southwestern side. Eccentricity in Y direction at
213
Table 7-1
Weight of analytical models
Floor
Weight from SAP
(݇ܰ)
Modified weight
(݇ܰ)
Roof
1153.218
5
Eccentricity
X ሺ݉ሻ
Y ሺ݉ሻ
1175.378
0.093
-0.848
770.853
785.666
0.197
-0.397
4
780.666
795.667
0.207
-0.243
3
782.228
797.259
0.272
-0.215
2
792.267
807.491
0.216
-0.235
Base
842.352
858.539
-0.004
0.307
Sum
5121.584
5220.000
0.156
-0.318
floor 5 is larger than other floors. The absence of the staircase in this story contributes to
this larger eccentricity. Eccentricity at roof is largest because of the eccentricity steel
weights.
In the analysis process, the static analysis of the model subjected to gravity load
had been analyzed before a ground motion excitation was applied. This gravity load is
acting during the excitation of ground.
7.6
Support Conditions
In the isolated base configuration, the TPBs were modeled by the TPB elements
described in Chapter 6. The geometric parameters of bearings, i.e. ‫ܮ‬௜ and ‫ݑ‬ഥప , were taken
from the designed values (Tables 3-8, 6-1). The general friction coefficient model of
bearings estimated from the sine-excitation test (Section 5.1) was used in the model.
214
The compression stiffness of the bearings was selected to obtain the overall best
agreement between vertical reactions of the model and the test data. Response to the
70LGP motion was selected for tuning the compression stiffness. The vertical component
of this record was strong, but did not induce significant uplift. The analytical model was
analyzed at each of several values of compressive stiffness, ‫ܭ‬௭ , varied over a wide range,
wherein the difference in total vertical reaction between the analysis and test data was
evaluated by:
௡
ߪோ௭ ൌ ඩ෍൫ܴ௭ǡ௔௡௔௟ǡ௜ െ ܴ௭ǡ௧௘௦௧ǡ௜ ൯
ଶ
(7.6-1)
௜ୀଵ
where ܴ௭ǡ௔௡௔௟ǡ௜ and ܴ௭ǡ௧௘௦௧ǡ௜ are the total vertical reactions at time step ݅ computed from
analysis and test data, respectively.
Figure 7-13 shows the relationship between the total deviation of vertical reaction,
ߪோ௭ , and compressive stiffness of bearings, ‫ܭ‬௭ , determined by the procedure described in
the previous paragraph. The relationship suggests that a compressive stiffness of
ͳǤʹͲȀ is a best fit to the test data. This value is slightly lower than the expected
vertical stiffness of the TPB in series with the connection compound, including the steel
plates and load cells that measured the reaction of the bearings (Section 4.1.1). The
analysis of the finite element model suggested the vertical stiffness of the connection
compound to be ʹǤͶͲȀ (Section 4.1.1). This vertical stiffness in series with
vertical stiffness of the TPB, ͷǤ͵Ȁ as mentioned earlier, produces the vertical
215
Total deviation, σRz (MN)
13
12.5
12
11.5
0.6
0.8
1
1.2
1.4
1.6
Compressive stiffness,K (MN/mm)
z
Figure 7-13
Total deviation of vertical reaction for tuning vertical stiffness of isolators
stiffness of ͳǤ͸ͷȀ. The increased flexibility is likely due to the flexibility of the
contact surfaces between these elements and/or the modeling approximation.
Vertical tension stiffness of ͳͲͲȀ was selected for the bearings.
The total vertical reaction history from analytical model with compression
stiffness of bearing of ͳǤʹȀ is plotted against the total vertical stiffness from the
test in Figure 7-14. The comparison shows the agreement between analytical and
experimental data.
In the fixed base configuration, the model was fixed at the bottom surface of the
column bases.
216
Figure 7-14
7.7
Total vertical reaction of the isolated base structure
subjected to 70LGP excitation
Modeling Damping
Rayleigh damping, a convenience damping model for applying classical damping,
was used to represent energy dissipation in the structure. This damping model is the
combination of mass proportional and stiffness proportional components and is a form of
classical damping, meaning that the damping matrix in modal coordinates is diagonal. As
showed in details by Chopra (2007), the damping matrix for Rayleigh damping can be
determined as:
ሾ‫ܥ‬ሿ ൌ ܽெ ሾ‫ܯ‬ሿ ൅ ܽ௄ ሾ‫ܭ‬ሿ
(7.7-1)
where ሾ‫ܥ‬ሿǡ ሾ‫ܯ‬ሿ and ሾ‫ܭ‬ሿ are the global damping matrix, mass matrix and stiffness matrix,
and ܽெ and ܽ௄ are mass proportional and stiffness proportional constants. The damping
ratio ߞ௡ for the nth mode is:
ߞ௡ ൌ
ܽெ ͳ
ܽ௄
൅ ߱௡
ʹ ߱௡
ʹ
(7.7-2)
217
where ߱௡ is angular frequency of nth mode. The constants ܽெ and ܽ௄ can be determined
by prescribing the damping ratios ߞ௜ , ߞ௝ of 2 different modes, according to:
ͳ ͳȀ߱௜
൤
ʹ ͳȀ߱௝
߱ ௜ ܽெ
ߞ௜
൨ ቄ ܽ ቅ ൌ ൜ߞ ൠ
߱௝
௄
௝
(7.7-3)
The response of the fixed base building to white nose excitations was analyzed
(Sasaki et al., 2012) to find the periods and damping ratios of natural modes of the
structure. Table 7-2 summarizes the periods and damping ratios of first 3 modes in both
directions determined from this process. By comparing the analytical results and test
results, the Rayleigh damping curve passing through damping ratios of ʹǤʹΨ at periods
of ͲǤ͹• (ͳǤͶʹͻ œሻ and ͲǤͳͷ• (͸Ǥ͸͹ œ) was found to give a good match. This damping
model was used throughout the analysis of the fixed base model. Figure 7-15 shows the
Table 7-2
Natural periods and damping ratios of the fixed base configuration
White noise X
White noise Y
White noise 3D
Period
(s)
Damping
ratio (%)
Period (s)
Damping
ratio (%)
Period (s)
Damping
ratio (%)
Mode 1 X
0.652
3.30
n/a
n/a
0.677
4.09
Mode 2 X
0.204
1.62
n/a
n/a
0.205
1.95
Mode 3 X
0.112
3.31
n/a
n/a
0.112
3.74
Mode 1 Y
n/a
n/a
0.677
2.54
0.686
3.49
Mode 2 Y
n/a
n/a
0.211
1.65
0.212
1.93
Mode 3 Y
n/a
n/a
0.113
2.64
0.113
3.61
218
Damping ratio, ζ (%)
10
Fixed-base model
Isolated-base model
Fixed-base test
8
6
4
2
0
0
2
Figure 7-15
4
6
Frequency, f (Hz)
8
10
Rayleigh damping models
selected Rayleigh damping model for the fixed base analytical model compared to the
damping values computed from test data.
Frequency analysis of test data showed that the isolation system was mainly
working at a frequency around ͲǤͷ œ and the peak amplitude in the responses of
superstructure occurred at a frequency as high as around ͹ œ, which was the major
component of the response to some excitations (see Figures 5-37 to 5-40). A damping
model was needed that provides low damping at these frequencies, so that it does not
overdamp the isolation system or higher frequency structural modes. The Rayleigh
damping model satisfying this criterion, which was calibrated to ͳǤͷΨ and ʹǤͷΨ
damping ratio at ͲǤͷ œ and ͸Ǥ͸͹ œ, respectively (Figure 7-15), gives unacceptably low
damping at intermediate frequencies such as ͵ œ, which is the 2nd mode or 1st structural
mode in both directions. This causes these frequency components to be blown up,
especially when the model is subjected to motions rich in this frequency component. To
solve this difficulty, additional dampers were added to apply extra damping to these
219
modes. From modal analysis, the relative displacement between base and roof in the 2nd
mode was observed to be much larger than in any other mode, which is valid for both
directions. Thus the additional damper with damping coefficient ܿ was connected
between the center node of the base and the center node of the roof as shown in Figure 716. At these center nodes, the displacements in the torsional mode shape are zero, thus
the damper is inactive in the torsional mode.
ܿ
ܼ
ܺ
Figure 7-16
Additional damper for adjusting damping of the 2
nd
mode in the ܺ direction
The following approach was used to verify that the additional damper has the
intended effect, and compute the damping constant ܿ. Let the degree of freedoms in ܺ
direction be ‫ ݌‬and ‫ ݍ‬at the base and the roof, respectively. The global damping matrix
contributed by the damper then can be written as:
220
ǤǤ
‫ۍ‬Ǥ Ǥ
‫ێ ݌‬Ǥ Ǥ
‫ێ‬Ǥ Ǥ
ሾ‫ܥ‬ሿ ൌ ‫ێ‬
‫ێ‬ǣ
‫ێ‬Ǥ Ǥ
‫ێ ݍ‬Ǥ Ǥ
‫ێ‬Ǥ Ǥ
‫ۏ‬Ǥ Ǥ
‫݌‬
ǣ
ǣ
Ͳ Ͳ
Ͳ ܿ
Ͳ Ͳ
ǣ
‫ڭ‬
Ͳ Ͳ
Ͳ െܿ
Ͳ Ͳ
ǣ
ǣ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
ǤǤ
ǤǤ
‫ڮ‬
ǤǤ
‫ڰ‬
ǤǤ
‫ڮ‬
ǤǤ
ǤǤ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
‫ݍ‬
ǣ
Ͳ
െܿ
Ͳ
‫ڭ‬
Ͳ
ܿ
Ͳ
ǣ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
Ͳ
Ͳ
Ͳ
ǣ
ǤǤ
Ǥ Ǥ‫ې‬
Ǥ Ǥ‫ۑ‬
Ǥ Ǥ‫ۑ‬
‫ۑ‬
ǣ‫ۑ‬
Ǥ Ǥ‫ۑ‬
Ǥ Ǥ‫ۑ‬
Ǥ Ǥ‫ۑ‬
Ǥ Ǥ‫ے‬
(7.7-4)
When converted to modal coordinates, this damping matrix becomes:
‫כ‬
ሾ‫ כ ܥ‬ሿ ൌ ሾȰሿ் ሾ‫ܥ‬ሿሾȰሿ ൌ ൣܿ௜௝
൧
where ሾȰሿ ൌ ሾሼ߶ሽଵ
(7.7-5)
ǥ ሼ߶ሽ௡ ሿ is the matrix of mode shapes, and ሼ߶ሽ௜ is the ݅ ௧௛ mode
shape.
‫כ‬
The term ܿ௜௝
in Equation (7.7-5) can be written as:
‫כ‬
ܿ௜௝
ൌ ሼ߶ሽ்௜ ሾ‫ܥ‬ሿሼ߶ሽ௝
(7.7-6)
Expanding Equation (7.7-6):
‫כ‬
ܿ௜௝
ൌ ܿ൫߶௣௜ Ǥ ߶௣௝ ൅ ߶௤௜ Ǥ ߶௤௝ െ ߶௣௜ Ǥ ߶௤௝ െ ߶௣௝ Ǥ ߶௤௜ ൯ ൌ ܿǤ ܿҧ௜௝
(7.7-7)
Thus, Equation (7.7-5) becomes:
ሾ‫ כ ܥ‬ሿ ൌ ܿൣܿҧ௜௝ ൧
(7.7-8)
Table 7-3 shows ൣܿҧ௜௝ ൧ computed for the first 14 modes, with frequencies ranging
from ͲǤͷ œ to ͳͲ œ. Observe that ܿହǡହ is much larger than any other ܿ௜௝ and the matrix
is approximately diagonal. This means that the damping contributed by the damper is
221
approximately classical damping and the damper effect mainly mode 5, which is the 2nd
mode in ܺ direction with frequency of ʹǤͻͳ œ.
With this result, the damping ratio of the 2nd mode can be controlled.
Using classical damping theory, the damping ratio of the nth mode is computed as:
‫כ‬
ܿ௡௡
ߞ௡ ൌ
‫כ‬
ʹ߱௡ Ǥ ݉௡௡
(7.7-9)
‫כ‬
where ݉௡௡
ൌ ሼ߶ሽ்௡ ሾ‫ܯ‬ሿሼ߶ሽ௡ is modal mass of mode n.
From Equations (7.7-7) and (7.7-9):
ܿൌ
‫כ‬
ʹǤ ߞ௡ Ǥ ߱௡ Ǥ ݉௡௡
ܿҧ௡௡
(7.7-10)
Using Equation (7.7-10) the damping constant ܿ of the additional damper was
computed to increase the damping ratio of the 2nd mode in ܺ direction by ߞ௡ . A similar
approach was used to increase the damping ratio of the 2nd mode in ܻ direction.
0.0061
0.0094
0.001
0.0023
0.0035
-0.0043
0.0007
0.0015
-0.0001
-0.002
0.0003
0.0309
0.0009
0.0125
-0.0003
0.0021
0.0001
-0.0024
-0.0102
0.0764
0.0003
0.0014
0.0001
0.3284
-0.0001
0.009
-0.0041
-0.001
0.1328
-0.011
-0.0049
0.0007
0.0002
-0.0209
0.0017
-0.0003
-0.0106
0.0014
0.0003
-0.0454
0.0038
-0.0033
-0.0066
-0.0004
-0.0163
0.0022
0.0005
-0.0702
0.0058
-0.005
0.0255
0.0028
-0.0178
-0.0012
-0.0441
0.0059
0.0014
-0.1896
0.0157
-0.0136
-0.0074
-0.044
-0.0007
-0.0015
-0.0027
-0.0105
-0.0272
0.0096
-0.0018
-0.0022
1.4116
0.0036
0.0007
-0.0008
-0.0008
0.0009
0.0236
0.0052
-0.0005
-0.1168
-0.0032
0.0004
-0.0002
0.0097
-0.0008
0.0128
0.0015
0.1015
-0.0017
0.0001
-0.0084
-0.0004
0.0038
0.0073
0.0549
-0.0005
-0.0045
-0.0001
0.0039
0.0162
0.0021
-0.0013
0.0012
0.0006
Global damping matrix in modal coordinates system contributed by unit damping coefficient damper in the X direction
Notes: The matrix was computed from first 14 modes, whose frequencies range from ͲǤͷ‫ ݖܪ‬to ͳͲ‫ݖܪ‬
0.0002
Table 7-3
222
223
7.8
Adjusting Vertical Reaction
As presented in Section 4.2, the base of the specimen was warped so that the
distribution of vertical load on bearings was different from the expected vertical load
estimated from the analytical model, where the base is assumed plane. Figure 7-17 shows
the expected distribution of load on bearings based on the analysis of the calibrated
model and the actual distribution of load on the bearings after installation presented in
Table 4-10. Obviously, the discrepancy between the two distributions is large.
Because the response of friction bearings depends on axial load acting on them,
the initial vertical reactions of the analytical model should be matched to the test data to
get good analytical results. The following process of load redistribution was applied to
adjust the load on bearings in analytical model. Let the actual vertical reaction at bearing
݅ at the beginning of a test simulation be ܴ௜ǡ௧௘௦௧ and the vertical reaction at bearing ݅ in
Actual
Expected
N
Figure 7-17
Distribution of the initial static vertical reaction at bearings
224
the analytical model subjected to gravity load be ܴ௜ǡ଴ . The additional reaction ȟܴ௜ needed
at bearing ݅ so that the initial analytical reaction matches the test data is:
ȟܴ௜ ൌ ܴ௜ǡ௧௘௦௧ െ ܴ௜ǡ଴
(7.8-1)
Additional forces were applied to the top of the bearings to increase the reaction
at bearing ݅ in the analytical model by ȟܴ௜ . The value of these additional force was
determined as follows.
The reaction ‫ݎ‬௜௝ was measured from the analytical model, where ‫ݎ‬௜௝ ൌ reaction at
തതതതത
bearing ݅ (݅ ൌ ͳǡ
ͻ) due to a unit vertical load applied at the top of bearing ݆ (݆ ൌ തതതതത
ͳǡ ͻሻ. It
should be noted that:
ଽ
෍ ‫ݎ‬௜௝ ൌ ͳ
(7.8-2)
௜ୀଵ
If the behavior of the system remains linear, the vertical reaction at bearing ݅
caused by a vertical load ܲ௝ applied at the top of bearing ݆ is:
ܴ௜௝ ൌ ‫ݎ‬௜௝ Ǥ ܲ௝
(7.8-3)
The total vertical reaction ܴ௜ at bearing ݅ when each bearing is subjected to a
vertical load ܲ௝ is:
ଽ
ܴ௜ ൌ ෍ ‫ݎ‬௜௝ Ǥ ܲ௝
௝ୀଵ
(7.8-4)
225
Based on Equations (7.8-1) and (7.8-4), the additional vertical loads ܲ௝ s needed
for adjusting the initial vertical reactions in the analytical model such that they match the
initial reactions measured from test can be obtained by solving the following system of
linear equations:
ଽ
݅ ൌ തതതത
ͳǡͻ
෍ ‫ݎ‬௜௝ Ǥ ܲ௝ ൌ ȟܴ௜ ǡ
௝ୀଵ
(7.8-5)
From Equation (7.8-1):
ଽ
ଽ
ଽ
෍ ȟܴ௜ ൌ ෍ ܴ௜ǡ௧௘௦௧ െ ෍ ܴ௜ǡ଴ ൌ ܹ௧௘௦௧ െ ܹ௠௢ௗ௘௟
௜ୀଵ
௜ୀଵ
(7.8-6)
௜ୀଵ
where ܹ௧௘௦௧ and ܹ௠௢ௗ௘௟ are the weight of the specimen and the weight of the model,
respectively.
If the weight of the model is identical to the weight of the specimen, then:
ଽ
෍ ȟܴ௜ ൌ Ͳ
(7.8-7)
௜ୀଵ
From Equations (7.8-5):
ଽ
ଽ
ଽ
෍ ෍ ‫ݎ‬௜௝ Ǥ ܲ௝ ൌ ෍ ȟܴ௜ ൌ Ͳ
௜ୀଵ ௝ୀଵ
Or:
௜ୀଵ
(7.8-8)
226
ଽ
ଽ
෍ ܲ௝ ෍ ‫ݎ‬௜௝ ൌ Ͳ
௝ୀଵ
௜ୀଵ
(7.8-9)
Introducing Equation (7.8-2) into Equation (7.8-9):
ଽ
෍ ܲ௝ ൌ Ͳ
௝ୀଵ
(7.8-10)
This means that when the weight of the analytical model equals the weight of the
specimen, the additional set of loads computed from Equation (7.8-5) does not change the
total vertical load on the structure.
Because the initial reactions changed from simulation to simulation (Figure 4-28),
the analytical reactions were modified independently at the beginning of every
simulation.
7.9
Effect of Modeling Assumption on Response of the Analytical Models
͹ǤͻǤͳ ˆˆ‡ –‘ˆ ”ƒ‡‡ –‹‘ƒ†‘‡ –‹‘••—’–‹‘•
To investigate the effect of the frame section model and the beam-to-column
connection model on the response of the structure model, responses of 4 different models
to selected motions were compared. The Resultant Section model uses resultant beam
sections as described. The Fiber Section model uses the fiber section for the composite
beams. The difference in behavior between resultant beam section and fiber beam section
in a rigid diaphragm floor is presented in Figure 7-8 (d) where the resultant section and
fiber section are represented by the Axially unrestrained and Axially restrained curves,
227
respectively. In the Bare Section model, the composite effect is neglected and only steel
sections are accounted. All these 3 models use the panel zone model for beam-to-column
connection. The No Panel Zone model is similar to the Resultant Section model, except
that the beam-to-column connection is a centerline rigid connection (i.e. a beam and a
column are connected at their center lines and no angle deformation at the connection
exists).
Figures 7-18 and 7-19 show roof acceleration and roof drift histories at geometric
center of the fixed base models to 35RRS motion along with the test data. The 35RRS
motion is a 3D motion with scale factor of 35% applied to all 3 components. The 35RRS
Acc. X, aX (g)
1
0.5
0
-0.5
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
-1
Acc. Y, aY (g)
1
0.5
0
-0.5
-1
0
1
Figure 7-18
2
3
4
5
Time, t (s)
6
7
8
9
Roof acceleration of the fixed base structure subjected to 35RRS:
test vs. analysis with different frame models
10
228
Drift, δX (%)
0.5
0
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
-0.5
1
Drift, δY (%)
0.5
0
-0.5
-1
0
1
Figure 7-19
2
3
4
5
Time, t (s)
6
7
8
9
10
Roof drift of the fixed base structure subjected to 35RRS:
test vs. analysis with different frame models
motion generated maximum response to the specimen during the test series. In general,
the Bare Section model amplified the response of the structure during the time histories.
The Fiber Section model also tends to exaggerate the response amplitude along the
history but reduces the peak response, which occurs near the beginning of the response
histories. The panel zone model also has significant influence on the time-history
response of the analytical model since the No Panel Zone model amplifies the response
compared to the response of the model with panel zones (the Resultant Section model).
The peak floor acceleration and peak story drift of these models subjected to 35RRS are
presented in Figures 7-20 and 7-21. The difference in the peak floor acceleration between
these models is not quite obvious, but the difference in the peak story drift is evident. The
229
6
5
Floor
Bare Section
No Panel Zone
4
Fiber Section
Resultant Section
3
Test
2
1
0
0.2
0.4
Peak floor acc., A
Xmax
Figure 7-20
0.6
(g)
0
0.4
0.6
0.8
1
Peak floor acc., A
(g)
1.2
Ymax
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
4
5
4
3
3
2
2
1
0
0.2
Peak floor acceleration of the fixed base structure subjected to 35RRS:
test vs. analysis with different frame models
5
Story
0.8
0.2
0.4
Peak drift, δ
Xmax
Figure 7-21
0.6
(%)
0.8
1
0
0.2
0.4
0.6
Peak drift, δ
0.8
(%)
1
1.2
Ymax
Peak story drift of the fixed base structure subjected to 35RRS:
test vs. analysis with different frame models
Bare Section model overestimates the peak story drift while the Fiber Section
underestimates the response. The Resultant Section model gives the best match to the test
data, compared to other models. The comparison of floor spectra at roof in Figure 7-22
further confirms this observation. Recall that the moment-curvature relationship of a fiber
section in a rigid diaphragm constraint floor is symmetric (Figure 7-8 (d)) and the initial
stiffness equals the elastic stiffness of the section. This means that the Fiber Section
model can represent the elastic model in the elastic range. The comparison of the roof
230
8
5
Spectral acc., SAY (g)
Spectral acc., SAX (g)
6
4
3
2
1
0 -2
10
-1
10
Figure 7-22
0
1
10
Period, T (s)
10
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
6
4
2
0 -2
10
-1
10
0
1
10
Period, T (s)
10
Floor spectra at roof of the fixed base structure subjected to 35RRS:
test vs. analysis with different frame models
drift history of these 2 models subjected to 35RRS in Figure 7-23 supports this
conclusion.
The comparison between test data and analytical response of the isolated base
models with different frame section models subjected to 70LGP is presented in Figures 724 to 7-28. The 70LGP motion was the second largest table motion in vertical
Drift, δX (%)
0.2
0.1
0
-0.1
Elastic Section
Fiber Section
-0.2
Drift, δY (%)
0.5
0
-0.5
0
1
2
Figure 7-23
3
4
5
Time, t (s)
6
7
8
Roof drift of the fixed base model subjected to 35RRS:
Elastic Section vs. Fiber Section
9
10
231
0.2
aX (g)
0.1
0
-0.1
-0.2
aY (g)
0.5
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
0
-0.5
4
Figure 7-24
5
6
7
t (s)
8
9
10
Roof acceleration of the isolated base structure subjected to 70LGP:
test vs. analysis with different frame models
acceleration with peak vertical acceleration of ͲǤ͸ͺ͹‰ (the largest one was 88RRS, which
caused the uplift of the entire building). Period of high frequency components in
acceleration response of these models is significantly different (Figures 7-24 and 7-28),
but low frequency components, corresponding to the response of the isolation system, in
the response of these models are almost identical. Similar to the fixed base configuration,
in the isolated base configuration, the Bare Section model overestimates the story drift
and the Fiber Section model underestimates it (Figures 7-25 and 7-27). The difference in
peak floor acceleration between these models is not very clear, except that the peak floor
acceleration in the Bare Section model is too low in the X direction (Figure 7-26). The
effect of panel zone model in the response to this motion is small. In general, the
232
Resultant Section model gives the best match to the test data. These observations are also
valid for the response to the 80TCU motion (Figures 7-29 to 7-33), which is a horizontal
motion.
Drift X, δX (%)
0.1
0.05
0
-0.05
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
-0.1
Drift Y, δY (%)
0.2
0.1
0
-0.1
-0.2
0
2
Figure 7-25
4
6
8
10
Time, t (s)
12
14
16
18
Roof drift of the isolated base structure subjected to 70LGP:
test vs. analysis with different frame models
20
233
6
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
5
Floor
4
3
2
1
Table
0
Figure 7-26
0.1
0.2
0.3
0.4
Peak acc. X, AXmax (g)
0
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
4
0.8
5
4
3
3
2
2
1
0
0.2
0.4
0.6
Peak acc. Y, AYmax (g)
Peak floor acceleration of the isolated base structure subjected to 70LGP:
test vs. analysis with different frame models
5
Story
0.5
0.05
0.1
Peak drift X, δ
Xmax
Figure 7-27
0.15
(%)
0.2
1
0
0.1
Peak drift Y, δ
0.2
(%)
Ymax
Peak story drift of the isolated base structure subjected to 70LGP:
test vs. analysis with different frame models
0.3
234
2
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
1
Spectral acc., SAY (g)
Spectral acc., SAX (g)
1.5
0.5
0 -2
10
-1
10
Figure 7-28
0
1
10
Period, T (s)
10
1.5
1
0.5
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Floor spectra at roof of the isolated base structure subjected to 70LGP:
test vs. analysis with different frame models
Acc. X, aX (g)
0.2
0.1
0
-0.1
-0.2
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
Acc. Y, aY (g)
0.2
0.1
0
-0.1
-0.2
0
Figure 7-29
5
10
15
20
Time, t (s)
25
30
35
Roof acceleration of the isolated base structure subjected to 80TCU:
test vs. analysis with different frame models
40
235
Drift X, δX (%)
0.2
0.1
0
-0.1
Test
Bare Section
No Panel Zone
Fiber Section
Resultant Section
-0.2
Drift Y, δY (%)
0.2
0.1
0
-0.1
-0.2
0
5
Figure 7-30
6
5
Floor
4
10
15
20
Time, t (s)
25
30
35
40
Roof drift of the isolated base structure subjected to 80TCU:
test vs. analysis with different frame models
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
3
2
1
Table
0
Figure 7-31
0.2
0.4
0.6
Peak acc. X, AXmax (g)
0.8
0
0.1
0.2
0.3
0.4
Peak acc. Y, AYmax (g)
Peak floor acceleration of the isolated base structure subjected to 80TCU:
test vs. analysis with different frame models
0.5
236
5
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
Story
4
5
4
3
3
2
2
1
0
0.1
Peak drift, δ
0.2
(%)
1
0
0.3
Xmax
Figure 7-32
0.8
0.4
0.2
0 -2
10
Figure 7-33
0.3
Peak story drift of the isolated base structure subjected to 80TCU:
test vs. analysis with different frame models
Bare Section
No Panel Zone
Fiber Section
Resultant Section
Test
0.6
0.2
(%)
Ymax
Spectral acc., SAY (g)
Spectral acc., SAX (g)
0.8
0.1
Peak drift, δ
-1
10
0
10
Period, T (s)
1
10
0.6
0.4
0.2
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Floor spectra at roof of the isolated base structure subjected to 80TCU:
test vs. analysis with different frame models
͹ǤͻǤʹ ˆˆ‡ –‘ˆƒ’‹‰‘†‡Ž
The effect of damping model on the response of the fixed base configuration
model is investigated by comparing the responses of the two analytical models with
different damping and the test data. The Rayleigh Damping model uses Rayleigh
damping calibrated to a damping ratio of
ʹǤʹΨ at periods of ͲǤ͹• and ͲǤͳͷ• as
described in Section 7.7. The Stiffness Damping model uses stiffness proportional
237
damping calibrated to ʹǤʹΨ damping ratio at period of ͲǤ͹•, which is approximately the
period of the 1st mode of the analytical model.
The comparison shows slightly difference in acceleration (Figures 7-34 and 7-35)
and drift (Figures 7-36 and 7-37) between the 2 models. The Stiffness Damping model
predicts smaller response than that of the Rayleigh Damping model. As expected,
because the stiffness damping adds higher damping to higher frequency mode, the high
frequency (or short period) spectral acceleration is smaller in the Stiffness Damping
model than in the test data and Rayleigh Damping model (Figure 7-38). In general, the
Rayleigh Damping model gives better predicted response than the Stiffness Damping
model.
Acc. X, aX (g)
1
0.5
0
-0.5
Stiffness Damping
Rayleigh Damping
Test
-1
Acc. Y, aY (g)
1
0.5
0
-0.5
-1
0
1
Figure 7-34
2
3
4
5
Time, t (s)
6
7
8
9
Roof acceleration of the fixed base structure subjected to 35RRS:
test vs. analysis with different damping models
10
238
6
Rayleigh Damping
Stiffness Damping
Test
Floor
5
4
3
2
1
0
0.2
0.4
Peak floor acc., A
Xmax
Figure 7-35
0.6
(g)
0.8
0
0.2
0.4
0.6
Peak floor acc. A
Ymax
0.8
(g)
1
Peak floor acceleration of the fixed base structure subjected to 35RRS:
test vs. analysis with different damping models
Drift, δX (%)
0.5
0
Stiffness Damping
Rayleigh Damping
Test
-0.5
Drift, δY (%)
1
0.5
0
-0.5
-1
0
1
Figure 7-36
2
3
4
5
Time, t (s)
6
7
8
Roof drift of the fixed base structure subjected to 35RRS:
test vs. analysis with different damping models
9
10
Story
239
5
5
4
4
3
3
Rayleigh Damping
Stiffness Damping
Test
2
1
0
0.1
2
0.2
0.3
0.4
Peak drift X, δ
(%)
1
0
0.5
Xmax
Figure 7-37
5
1.5
1
0.5
0 -2
10
Figure 7-38
1
Peak story drift of the fixed base structure subjected to 35RRS:
test vs. analysis with different damping models
Rayleigh Damping
Stiffness Damping
Test
2
0.4
0.6
0.8
Peak drift Y, δ
(%)
Ymax
Spectral acc., SAY (g)
Spectral acc., SAX (g)
2.5
0.2
-1
10
0
10
Period, T (s)
1
10
4
3
2
1
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Floor spectra at roof of the fixed base structure subjected to 35RRS:
test vs. analysis with different damping models
Four analytical models were used to investigate the effect of damping model on
response of the isolated base configuration model. The Rayleigh ͳǤͷΨ, Rayleigh ʹǤͷΨ
and Rayleigh ͵ǤͷΨ models use Rayleigh damping calibrated to ͳǤͷΨ damping ratio at
period of ʹ• (the main period component of response of the isolation system) and ͳǤͷΨ,
ʹǤͷΨ and ͵ǤͷΨ damping ratio, respectively, at period of ͲǤͳͷ•. The Stiffness Damping
model uses stiffness proportional damping with ͳǤͷΨ damping ratio at period of ʹ•. The
240
comparison of the responses of these models subjected to 70LGP and the test data
(Figures 7-39 to 7-43) shows that the Stiffness Damping model significantly damps out
high frequency components of the response and underestimates the peak responses. Since
the high frequency components dominate the response of the isolated base structure
subjected to this 3D motion, stiffness proportional damping is not appropriate for
predicting the response. The analytical response is also sensitive to the value of calibrated
damping ratio in the Rayleigh damping model (i.e. Rayleigh ͳǤͷΨ, Rayleigh ʹǤͷΨ and
Rayleigh ͵ǤͷΨ). However, it is difficult to tell what Rayleigh damping model is correct.
In the range of the investigated damping ratio, the discrepancy between analytical
Acc. X, aX (g)
0.2
0.1
0
-0.1
-0.2
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Acc. Y, aY (g)
0.5
0
-0.5
4
Figure 7-39
5
6
7
8
Time, t (s)
9
10
11
Roof acceleration of the isolated base structure subjected to 70LGP:
test vs. analysis with different damping models
12
241
response and test data due to different damping may be smaller than the discrepancy due
to other assumptions.
Contrary to the analytical response to 70LGP, the analytical response to 80TCU is
not very sensitive to damping model (Figures 7-44 to 7-47), except that the high
frequency spectral acceleration is damped out stronger in the Stiffness Damping model
than in other models (Figure 7-48). However, because the high frequency (short period)
components do not dominate the response to the 80TCU motion, which is a horizontal
motion, the difference between responses of Stiffness Damping model and Rayleigh
damping models is minor and may be neglected compared to the inaccuracy from other
sources.
242
Drift X, δX (%)
0.1
0.05
0
-0.05
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
-0.1
Drift Y, δY (%)
0.2
0.1
0
-0.1
-0.2
4
5
Figure 7-40
6
7
8
Time, t (s)
9
10
11
12
Roof drift of the isolated base structure subjected to 70LGP:
test vs. analysis with different damping models
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Test
5
4
Floor
6
3
2
1
Table
0
Figure 7-41
0.1
0.2
0.3
0.4
Peak acc. X, AXmax (g)
0.5
0
0.2
0.4
0.6
Peak acc. Y, AYmax (g)
Peak floor acceleration of the isolated base structure subjected to 70LGP:
test vs. analysis with different damping models
0.8
243
5
5
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Test
Story
4
3
2
3
2
0.05
0.1
Peak drift, δ
Xmax
Figure 7-42
0.15
(%)
1
0
0.2
0.3
2
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
1
0.5
Figure 7-43
0.2
(%)
Peak story drift of the isolated base structure subjected to 70LGP:
test vs. analysis with different damping models
1.5
0 -2
10
0.1
Peak drift, δ
Ymax
Spectral acc., SAY (g)
1
0
Spectral acc., SAX (g)
4
-1
10
0
10
Period, T (s)
1
10
1.5
1
0.5
0 -2
10
-1
10
0
10
Period, T (s)
Floor spectra at roof of the isolated base structure subjected to 70LGP:
test vs. analysis with different damping models
1
10
244
Acc. X, aX (g)
0.2
0.1
0
-0.1
-0.2
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Acc. Y, aY (g)
0.2
0.1
0
-0.1
-0.2
0
Figure 7-44
5
10
15
20
Time, t (s)
25
30
35
Roof acceleration of the isolated base structure subjected to 80TCU:
test vs. analysis with different damping models
40
245
Drift X, δX (%)
0.2
0.1
0
-0.1
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
-0.2
Drift Y, δY (%)
0.2
0.1
0
-0.1
-0.2
0
5
Figure 7-45
10
15
20
Time, t (s)
25
30
35
40
Roof drift of the isolated base structure subjected to 80TCU:
test vs. analysis with different damping models
6
5
Floor
4
3
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Test
2
1
Table
0
Figure 7-46
0.2
0.4
0.6
Peak acc. X, AXmax (g)
0.8
0
0.1
0.2
0.3
0.4
Peak acc. Y, AYmax (g)
Peak floor acceleration of the isolated base structure subjected to 80TCU:
test vs. analysis with different damping models
0.5
5
5
4
4
3
2
1
0
0.05
0.1
Peak drift, δ
Xmax
Figure 7-47
0.8
Spectral acc., SAX (g)
3
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
Test
2
0.15
(%)
1
0
0.2
0.2
Figure 7-48
0.2
0.8
0.4
0 -2
10
0.15
(%)
Peak story drift of the isolated base structure subjected to 80TCU:
test vs. analysis with different damping models
Test
Rayleigh 1.5%
Rayleigh 2.5%
Rayleigh 3.5%
Stiffness Damping
0.6
0.05
0.1
Peak drift, δ
Ymax
Spectral acc., SAY (g)
Story
246
-1
10
0
10
Period, T (s)
1
10
0.6
0.4
0.2
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Floor spectra at roof of the isolated base structure subjected to 80TCU:
test vs. analysis with different damping models
Effect of additional interstory dampers on response of the isolated base model is
investigated by comparing floor spectra at roof of 2 models with and without these
dampers. The Damper model uses the interstory dampers and the No Damper model do
not use them. Both these models use resultant section and panel zone connection. The
comparison in Figures 7-49 and 7-50 shows that the No Damper model significantly
amplifies the frequency component with period between ͲǤ͵• and ͲǤͶ• (frequency
247
between ʹǤͷ œ and ͵Ǥ͵ œ), which is the period of the first structural mode in both X and
Y directions.
2
2
Damper
No Damper
Test
1.5
SAY (g)
SAX (g)
1.5
1
0.5
0 -2
10
1
0.5
-1
0
10
10
1
10
0 -2
10
-1
T (s)
Figure 7-49
0.8
0.6
SAY (g)
SAX (g)
1
10
Floor spectra at roof of the isolated base structure subjected to 70LGP:
test vs. analysis with and without interstory damper models
0.6
0.4
0.2
Damper
No Damper
Test
0.4
0.2
-1
0
10
10
T (s)
Figure 7-50
10
T (s)
0.8
0 -2
10
0
10
1
10
0 -2
10
-1
0
10
10
T (s)
Floor spectra at roof of the isolated base structure subjected to 80TCU:
test vs. analysis with and without interstory damper models
1
10
Chapter 8
‡•’‘•‡•‘ˆ–Š‡ƒŽ›–‹ ƒŽ‘†‡Ž•ƒ†
˜ƒŽ—ƒ–‹‘‘ˆ‡”ˆ‘”ƒ ‡„Œ‡ –‹˜‡•
In this chapter, analytical response of the OpenSees models described in Chapter
7 is presented. The model is validated by comparing its response to the experimental data.
Only selected response to some motions is presented here; the full comparison for all
simulations can be found online at NEEShub website. The response of the verified
analytical model subjected to the selected motions representing different earthquake
scenarios is also analyzed for checking the performance objectives of the isolated
structure.
8.1
Fundamental Properties of the Model
ͺǤͳǤͳ ‘†ƒŽ ˆ‘”ƒ–‹‘
Modal analysis of the fixed base model was carried out to find the mode shapes
and corresponding natural periods. Figure 8-1 shows the first 3 modes and the first
vertical mode of the model. The first mode of the model is a torsional-translational mode
in Y direction with period of ͲǤ͸ͺ͹•. The second mode has period of ͲǤ͸͸͸• and is a
torsional-translational mode in X direction. The third mode is a torsional mode with
period of ͲǤͶ͹ʹ•. The first vertical mode has period of ͲǤͳͶʹ• and is dominated by
vibration at roof where the additional steel weights were installed (Figure 2-8, Chapter 2).
This mode is not a pure vertical mode but a lateral-vertical coupling mode at which the
horizontal deformation is accompanied with the vertical deformation. The lateral-vertical
249
coupling modes cause horizontal response even if the structure is only subjected to
vertical excitations. This type of mode should exist in any asymmetric structure.
Z
Z
X
T=0.687 s
X
T=0.666 s
(b)
(a)
Z
Z
X
T=0.472 s
(c)
Figure 8-1
X
T=0.142 s
(d)
The first 3 modes and the first vertical mode of the fixed base model
(a) The first mode (Y-direction) (b) The second mode (X-direction)
(b) The third mode (torsion) (d) The first vertical mode
250
Table 8-1 compares the periods of the first 3 modes in each direction between
analytical model and test data provided by collaborative E-defense team. The
experimental modes were determined from responses of the structure to one-dimensional
white noise excitations. These mode shapes at vertical center line passing through the
geometric centers of floors are compared in Figure 8-2. The modes determined by
analytical model and test data are in close agreement.
Table 8-1
Natural periods of the fixed base configuration
Period (s)
Mode
Test
Analysis
1X
0.652
0.666
2X
0.204
3X
0.112
Period (s)
Mode
Test
Analysis
1Y
0.677
0.687
0.213
2Y
0.211
0.219
0.108
3Y
0.113
0.111
Note: Test periods were computed from 1D white noise excitation test.
Values from 3D test are slightly different (see Table 7-2)
Test
Analysis
Mode 1X
Mode 1Y
Figure 8-2
Mode 2X
Mode 3X
Mode 2Y
Mode 3Y
Shapes of first 3 modes in X and Y directions
251
ͺǤͳǤʹ —•Š‘˜‡”—”˜‡ƒ†–”‡‰–Š‘ˆ–Š‡‘†‡Ž
Pushover analysis of the analytical model was performed to develop the pushover
curves and estimate the strength of the specimen. In pushover analysis, static loads
following a specified load pattern are applied at control points in each floor. In
earthquake engineering applications, the load pattern is determined from expected
distribution of equivalent static loads caused by earthquakes, which is usually assumed to
be “linear” for low rise buildings, because the vibration of the first mode is assumed to
dominate. These loads are determined by:
ܲ௜ ̱
‫ݖ‬௜ ‫ݓ‬௜
σ ‫ݖ‬௜ ‫ݓ‬௜
(8.1-1)
where ܲ௜ is horizontal load acting at floor ݅, ‫ݖ‬௜ is the height at floor ݅ computed from the
base, and ‫ݓ‬௜ is the weight of floor ݅. The pushover loads were applied horizontally at
mass center of each floor, where earthquake load is expected to act. Locations of these
centers are shown in Table 7-1 of Chapter 7. In displacement-control pushover analysis,
the displacement of a control point, which is usually the displacement at roof, is increased
monotonically to create a monotonic pushover curve. At each incremental displacement,
the load increment and base shear are determined by analyzing the change in state of the
structure from previous load state. The pushover analysis follows gravity analysis and the
gravity loads are active during the pushover process.
Figure 8-3 shows global pushover curves and story pushover curves of the model
in X and Y direction. The global pushover curves indicate that the model is slightly stiffer
in X direction than in Y direction. This confirms the result from modal analysis that the
252
first mode period in Y direction is slightly longer than the first mode period in X
direction. The first yield of the model occurs at a roof drift of about ͲǤ͹ͷΨand base
shear coefficient of ͲǤͺ. Story 5 is weakest in both directions, and starts yielding at about
ͲǤͷͷΨ drift. All other stories yield at about ͲǤ͹ͷΨ drift.
2
Base shear coefficient, S
B
X direction
Y direction
1.5
1
0.5
0
0
1
2
Roof drift, δr (%)
3
(a)
2
Normalized story shear, S
st
2
1.5
St.1
St.2
St.3
St.4
1
St.5
0.5
0
0
1
2
Story drift, δ (%)
(b)
Figure 8-3
3
1.5
St.1
St.2
St.3
St.4
1
St.5
0.5
0
0
1
2
Story drift, δ (%)
(c)
Pushover curves of the fixed base model
(a) Global pushover curves
(b) Pushover curves for each story in X direction
(c) Pushover curves for each story in Y direction
3
253
8.2
Responses of the Fixed Base Model
ͺǤʹǤͳ ‡Ž‡”ƒ–‹‘‡•’‘•‡–‘ͺͲƒ†͵ͷ
Time history accelerations of the fixed base model subjected to 80WSM motion
and 35RRS motion were selected to compare with the test data. These input motions are
3D motions and their 3 components were uniformly scaled. Figures 8-4 and 8-5 compare
horizontal acceleration response of analytical model and test data at geometric center of
all floors above base level caused by 70WSM excitation. The results shows that the
model cannot capture every detail of the response, but it can capture the trend of response
very well. Specially, the frequency is well captured but the amplitude is not quite
accurate.
254
0.5
Roof
0
Test
Analysis
-0.5
0.5
Floor 5
-0.5
0.5
X
Acceleration, a (g)
0
Floor 4
0
-0.5
0.5
Floor 3
0
-0.5
0.5
Floor 2
0
-0.5
0
Figure 8-4
5
10
15
Time, t (s)
20
25
Acceleration response in X direction of the fixed base structure
subjected to 80WSM: analytical model vs. test data
30
255
0.5
Roof
0
Test
Analysis
-0.5
0.5
Floor 5
-0.5
0.5
Y
Acceleration, a (g)
0
Floor 4
0
-0.5
0.5
Floor 3
0
-0.5
0.5
Floor 2
0
-0.5
0
Figure 8-5
5
10
15
Time, t (s)
20
25
30
Acceleration response in Y direction of the fixed base structure
subjected to 80WSM: analytical model vs. test data
Time histories of horizontal acceleration response at geometric center of all floors
above base level due to 35RRS are compared for analysis and test in Figures 8-6 and 8-7.
These results again confirm that the model can capture the trend of the response. Overall
256
peak acceleration response during 35RRS was observed in Y direction and was much
larger than the peak response from 80WSM. However, the large response just spans a
short duration, because 35RRS is a pulse-like motion in Y direction (see Figure 4-14 for
the time history of the RRS motion generated by the shake table).
257
1
Test
Analysis
Roof
0
-1
1
Floor 5
-1
1
X
Acceleration, a (g)
0
Floor 4
0
-1
1
Floor 3
0
-1
1
Floor 2
0
-1
0
Figure 8-6
5
10
Time, t (s)
15
Acceleration response in X direction of the fixed base structure
subjected to 35RRS: analytical model vs. test data
20
258
1
Roof
0
Test
Analysis
-1
1
Floor 5
-1
1
Y
Acceleration, a (g)
0
Floor 4
0
-1
1
Floor 3
0
-1
1
Floor 2
0
-1
0
Figure 8-7
5
10
Time, t (s)
15
20
Acceleration response in Y direction of the fixed base structure
subjected to 35RRS: analytical model vs. test data
At the end of the response to 35RRS (Figure 8-6 and 8-7), acceleration response
of the analytical model are observed to damped out more slowly in X direction than in Y
direction. This happens because the damping ratios of the first mode in the analytical
259
model are about the same in both directions (In fact the damping ratio of the first mode of
the model is higher in Y direction than in X direction since the first mode frequency is
smaller in Y direction than in X direction. See the Rayleigh damping model on Figure 715 for clarification), while analysis of the test data shown in Table 7-2 indicates that
damping ratio of the first mode of the specimen is higher in X direction than in Y
direction. The first mode response dominates the response of the structure at the end of
the record, where excitation is very small and the structure can be considered to vibrate
freely.
This trend of response due to the effect of damping can also be observed from
other types of responses such as story drift and internal forces of columns, which are
represented later in this section.
Peak acceleration profiles (peak acceleration vs. floor level) from these
simulations are shown in Figure 8-8. The distribution of peak floor acceleration in both
directions does not strictly follow the shape of the 1st mode. This may come from the
contribution of higher modes in the response and is captured well by the analytical
model. The existence of high frequency components can be seen in the time history
response of acceleration. The response spectra of these accelerations presented later show
this more obviously.
260
Roof
Roof
5
Y Test
X Test
4
Y Analysis
X Analysis
3
X Test
2
Table
0
0.2
0.4
Peak acceleration, A (g)
Floor
Floor
5
4
Y Analysis
3
Y Test
2
0.6
Table
0
(a)
Figure 8-8
X Analysis
0.2
0.4
0.6
0.8
Peak acceleration, A (g)
1
(b)
Peak floor acceleration of fixed base structure: analytical model vs. test data
(a) 80WSM excitation
(b) 35RRS excitation
Besides the acceleration history and their peaks, the response spectra of these
accelerations were also developed and compared. These floor spectra are plotted in
Figures 8-9 and 8-10. The comparison from these figures shows that the analytical model
can capture the period of peaks well, but discrepancies in amplitude result from
differences in damping ratio. The result also shows that the participating of higher modes
to acceleration response is significant. Note that the free vibration at the end of record in
analytical model does not affect the spectral peaks.
261
3
3
Test
Y-Roof
Analysis
2
X-Roof
2
1
1
A
Spectral acceleration, S (g)
0 -2
-1
10
10
3
X-Floor 5
2
0 -2
-1
10
10
3
Y-Floor 5
2
1
1
0 -2
-1
10
10
3
X-Floor 4
2
0
10
0
10
10
10
1
1
0 -2
-1
10
10
2
Y-Floor 4
0
10
0
10
10
10
1
1
1
1
0 -2
-1
10
10
2
X-Floor 3
0
10
10
1
0 -2
-1
10
10
1.5
Y-Floor 3
1
0
10
10
1
1
0.5
0 -2
-1
10
10
1
X-Floor 2
0
10
10
1
0 -2
-1
10
10
1.5
Y-Floor 2
1
0
10
10
1
0.5
0.5
0 -2
10
Figure 8-9
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
Floor spectra of the fixed base structure subjected to 80WSM:
analytical model vs. test data
10
1
262
3
6
X-Roof
Test
Analysis
2
1
0
10
10
1
1
A
4
2
0 -2
-1
10
10
2
X-Floor 5
Spectral acceleration, S (g)
Y-Roof
0 -2
-1
10
10
4
Y-Floor 5
0
10
10
1
2
0 -2
-1
10
10
2
X-Floor 4
0
10
10
1
1
0 -2
-1
10
10
4
Y-Floor 4
0
10
10
1
2
0 -2
-1
10
10
1.5
X-Floor 3
1
0
10
10
1
0.5
0 -2
-1
10
10
3
Y-Floor 3
2
0
10
10
1
1
0 -2
-1
10
10
1.5
X-Floor 2
1
0
10
10
1
0.5
0 -2
-1
10
10
1.5
Y-Floor 2
1
0
10
10
1
0.5
0 -2
10
-1
0
10
10
Period, T (s)
Figure 8-10
10
1
0 -2
10
-1
0
10
10
Period, T (s)
Floor spectra of the fixed base structure subjected to 35RRS:
analytical model vs. test data
10
1
263
ͺǤʹǤʹ –‘”›”‹ˆ–‡•’‘•‡–‘ͺͲƒ†͵ͷ
Story drift response of the analytical model subjected to 80WSM and 35RRS was
also compared with the test data. The time history responses of story drift are plotted in
Figures 8-11 to 8-14, and peak drift profiles are shown in Figure 8-15. These drifts were
computed at geometric center of slabs.
The comparison shows that the model can capture responses recorded from the
test. The comparison of drift response to 35RRS (Figure 8-13) also confirms the
comment that the damping of the model is lower than the damping of the specimen in X
direction since the analytical response damps out slower than test response at the end of
the record.
The peak story drift trends to be moderate at first story, increase to a peak at
second story, and decrease in each of the higher stories, which is observed both directions
in both analytical and test data. The drift responses to all other motions of the fixed base
configuration also follow this trend.
264
0.5
Test
Analysis
Story 5
0
-0.5
0.5
Story 4
0
X
Drift, δ (%)
-0.5
0.5
Story 3
0
-0.5
0.5
Story 2
0
-0.5
0.5
Story 1
0
-0.5
0
Figure 8-11
5
10
15
Time, t (s)
20
25
30
Drift response in X direction of the fixed base structure subjected to 80WSM:
analytical model vs. test data
265
0.5
Test
Analysis
Story 5
0
-0.5
0.5
Story 4
0
Y
Drift, δ (%)
-0.5
0.5
Story 3
0
-0.5
0.5
Story 2
0
-0.5
0.5
Story 1
0
-0.5
0
Figure 8-12
5
10
15
Time, t (s)
20
25
30
Drift response in Y direction of the fixed base structure subjected to 88WSM:
analytical model vs. test data
266
1
Test
Analysis
Story 5
0
-1
1
Story 4
0
X
Drift, δ (%)
-1
1
Story 3
0
-1
1
Story 2
0
-1
1
Story 1
0
-1
0
Figure 8-13
5
10
Time, t (s)
15
20
Drift response in X direction of the fixed base structure subjected to 35RRS:
analytical model vs. test data
267
1
Test
Analysis
Story 5
0
-1
1
Story 4
0
Y
Drift, δ (%)
-1
1
Story 3
0
-1
1
Story 2
0
-1
1
Story 1
0
-1
0
Figure 8-14
5
10
Time, t (s)
15
20
Drift response in Y direction of the fixed base structure subjected to 35RRS:
analytical model vs. test data
268
5
5
4
X Test
Y Test
X Analysis
Y Analysis
3
Story
Story
4
3
2
2
1
0
0.1
0.2
0.3
Peak drift, δmax (%)
0.4
1
0
0.2
0.4
0.6
Peak drift, δ
(%)
(a)
Figure 8-15
0.8
1
max
(b)
Peak story drift of fixed base structure: analytical model vs. test data
(a) 80WSM excitation
(b) 35RRS excitation
ͺǤʹǤ͵Ǥ ‘Ž— ‘” ‡•‡•’‘•‡–‘ͺͲƒ†͵ͷ
Internal forces in the column at the Northeast corner are compared for the
analytical model and test. The observed forces represent only dynamic variation because
this is the only component measured during the test. Figure 8-16 to 8-21 shows time
history responses of bending moments about X- and Y-axis and the axial force at the
bottom section of the column at every story. Sign conventions of these internal force
components are: (1) axial force causing tension in the column is positive, (2) bending
moment causing tension on the fiber in the positive side of the global coordinate is
positive. Figure 8-22 shows peak internal forces at every section in the column. The peak
internal forces at a section were enveloped from the history response of that section,
which was linearly extrapolated from the response at the measured sections at every time
step. The comparison shows the good agreement between analytical and experimental
data.
269
Observation of the time history responses (Figure 8-16 to 8-21) shows that the
responses at story 5 contains high frequency components with stronger amplitude
compare to responses at lower stories. This may come from the effect of vertical
vibration at roof whose frequency is about ͹ œ or period of ͲǤͳͶʹ•. The high frequency
component is stronger in the bending moment about X-axis than about Y-axis, which is
accordant with the asymmetry in Y-direction.
In each column segment bounded by the two adjacent floors, the peak bending
moments are large at the two ends and small at middle and the distribution of peak
bending moment from the middle section to the two ends is linear. This caused by the
reverse bending of linear distribution moment in columns when structures subjected to
lateral load.
Even though the maximum peak story drift occurs at story 2, the peak bending
moment at this story is not necessarily largest. The bending moment depends not only on
story drift, which is the difference in displacement between the two column ends, but also
on rotation of the ends, which can be affected by the beams connected to these ends.
Moreover, the section of the column is larger at story 1 than at story 2 so that the internal
force could be larger at story 1 than story 2 even if the deformation is larger at story 2.
The contribution of end rotations on bending moment can also explain why the peak
bending moments about X-axis in the column subjected to 35RRS (Figure 8-22 (b)) are
about the same in both analytical and experimental data, while the discrepancy in peak
story drift in Y direction is much larger between the two (Figure 8-15 (b)).
270
50
Story 5
0
Test
Analysis
-50
-100
100
Story 4
-100
100
Story 3
X
Bending moment, M (kNm)
0
0
-100
-200
100
Story 2
0
-100
-200
400
Story 1
200
0
-200
-400
0
Figure 8-16
5
10
15
Time, t (s)
20
25
Dynamic bending moment about X-axis in the NE column of the fixed base
structure subjected to 80WSM: analytical model vs. test data
30
271
100
Story 5
0
Test
Analysis
-100
200
Story 4
100
Bending moment, MY (kNm)
0
-100
200
Story 3
0
-200
200
Story 2
0
-200
400
Story 1
200
0
-200
-400
0
Figure 8-17
5
10
15
Time, t (s)
20
25
Dynamic bending moment about Y-axis in the NE column of the fixed base
structure subjected to 80WSM: analytical model vs. test data
30
272
100
Test
Analysis
Story 5
0
-100
200
Story 4
Axial force, P (kN)
0
-200
400
Story 3
200
0
-200
-400
400
Story 2
200
0
-200
-400
500
Story 1
0
-500
0
Figure 8-18
5
10
15
Time, t (s)
20
25
Dynamic axial force response in the NE column of the fixed base structure
subjected to 80WSM: analytical model vs. test data
30
273
100
Story 5
0
Test
Analysis
-100
-200
200
Story 4
Bending moment, MX (kNm)
0
-200
200
Story 3
0
-200
200
Story 2
0
-200
1000
Story 1
0
-1000
0
Figure 8-19
5
10
Time, t (s)
15
Dynamic bending moment about X-axis in the NE column of the fixed base
structure subjected to 35RRS: analytical model vs. test data
20
274
100
Story 5
Test
Analysis
0
-100
200
Story 4
Bending moment, MY (kNm)
0
-200
200
Story 3
0
-200
200
Story 2
0
-200
400
Story 1
200
0
-200
-400
0
Figure 8-20
5
10
Time, t (s)
15
Dynamic bending moment about Y-axis in the NE column of the fixed base
structure subjected to 35RRS: analytical model vs. test data
20
275
200
Test
Analysis
Story 5
0
-200
400
Story 4
200
0
Axial force, P (kN)
-200
-400
500
Story 3
0
-500
-1000
1000
Story 2
0
-1000
1000
Story 1
0
-1000
-2000
0
Figure 8-21
5
10
Time, t (s)
15
Dynamic axial force response in the NE column of the fixed base structure
subjected to 35RRS: analytical model vs. test data
20
276
6
6
6
5
5
4
4
4
3
3
3
2
2
2
Test
Analysis
Floor
5
1
0
200
Peak moment, M
x,peak
400
(kNm)
1
0
200
Peak moment, M
400
(kNm)
1
0
y,peak
200
400
Peak axial foce, P
(kN)
peak
(a)
6
6
6
5
5
4
4
4
3
3
3
2
2
2
Test
Analysis
Floor
5
1
0
500
Peak moment, M
x,peak
1000
(kNm)
1
0
200
Peak moment, M
400
(kNm)
y,peak
1
0
1000
Peak axial foce, P
peak
2000
(kN)
(b)
Figure 8-22
Peak dynamic forces at every section of column NE of the fixed base structure:
analytical model vs. test data
(a) Subjected to 80WSM, (b) Subjected to 35RRS
277
8.3.
Responses of the Isolated Base Model
Analysis of the isolated base model to 100TAK, 80TCU and 100TAB were
selected to compare to the test data. The 100TAK motion was a full 3D motion applied at
100% scale factor while the 80TCU motion was a horizontal XY motion applied at a
scale factor of 80% the original record. The 100TAB motion, which caused the largest
peak displacement to the isolation system of approximately ͹Ͳ , was a nominally
horizontal XY input motion applied at 100% scale factor. However, the recorded table
motion for 100TAB contained a non-negligible vertical component, which affected the
responses of the superstructure dramatically (see Chapter 9). Accordingly, 80TCU
motion was chosen to represent response of the isolated structure to a strong horizontal
motion, while 100TAK was chosen to represent the response of the isolated structure to a
strong 3D motion. The influence of friction model on the isolation system response was
investigated for 100TAK and 100TAB, since the response was sensitive to the friction
model for these motions.
ͺǤ͵Ǥͳ ˆˆ‡ –‘ˆ ”‹ –‹‘‘†‡Ž‘–Š‡‡•’‘•‡‘ˆ–Š‡ •‘Žƒ–‹‘›•–‡
Response of the isolation system to 100TAK and 100TAB motions using different
analytical models with different friction coefficient models was investigated. These
friction coefficient models were introduced in Section 5.1. In the “Const. ߤ௔௩௚ ” model,
the friction coefficients are constant (invariant with respect to axial force and velocity)
and set equal to the average values from all loops of all bearings computed from the
equivalent area approach (Column (5) and Column (10) of Table 5.1). The “Gen. ߤ଴ ”
model includes dynamic variation of friction coefficients with axial load and velocity
278
(Equation 5.1-9), where parameters were computed according to Figures 5-10 and 5-12.
The “Const. ߤ଴ ” is similar to “Const. ߤ௔௩௚ ”, but friction coefficients computed from the
area of the first cycle of the hysteresis loop (Column (2) and Column (6) of Table 5.1).
First cycle parameters were used because during a single simulation as well as during real
earthquakes, the displacement trace of one bearing is not expected to repeat on the same
route, except at small displacement, where sliding is limited to the inner slider.
Figures 8-23 and 8-24 show displacement histories of the center bearing subjected
to the 100TAK and 100TAB motions. The comparison shows good agreement between
analytical results and experimental results. The difference in responses of the 3 models is
not obvious at small displacement, but becomes significant at large displacement. This
may cause difficulty in predicting of displacement of large-scale isolation systems
subjected to strong motions. Contrarily to common intuition, although the friction
coefficient is higher in “Const. ߤ଴ ” model than in “Const. ߤ௔௩௚ ” model, the peak
displacement in the “Const. ߤ௔௩௚ ” model is smaller when subjected to 100TAK motion.
This does not occur to 100TAB motion.
The “Gen. ߤ଴ ” model gives the best match to analytical results in both excitations,
though the “Const. ߤ௔௩௚ ” model produces similar displacement history to that of the
“Gen. ߤ଴ ” model in response to 100TAK. The peak displacement in response to 100TAB
motion is very sensitive to friction coefficients: when friction coefficient ߤଶ (=ߤଷ )
reduces from ͳͳǤͲ͵Ψ in “Const. ߤ଴ ” model to ͻǤͻʹΨ in “Const. ߤ௔௩௚ ” model (reduces
ͳͲǤͲ͸Ψ), the peak displacement increases from ͸ʹǤͶ͵  to ͹ͶǤͲͺ  (increases
279
ͳͺǤ͸͸Ψ). The difference in ߤଵ between these models is small (ͳǤ͸ͻΨ compare to
ͳǤ͸ʹΨ). This peak displacement changing is െͷǤͶͳΨ in response to 100TAK motion.
Normalized hysteresis loops of the isolation system (all bearings) subjected to
these motions are shown in Figures 8-25 and 8-26. The analytical loop of the “Gen. ߤ଴ ”
model matches well with the experimental loop in response to 100TAB. In response to
100TAK, the analytical loop looks much fatter than the experimental loop in X direction
on the negative displacement side. However, this is limited to a duration of around ͳ‫ݏ‬
(from ͷ െ ͸•, Figure 8-23). Afterwards, when the isolation system continues moving in
the positive direction, the analytical loop becomes smaller than the experimental loops.
Thus the total dissipated energy from the analytical model and experimental data are
balanced and converge to the same value after this cycle. This energy balance is
confirmed from the energy dissipated by the isolation system presented in Figure 8-27.
After about ͺ•, the amount of energy dissipated from both data are equal. This plot also
confirms that the “Gen ߤ଴ ” model better matches the experimental data than other 2
models, and was thus used for all final analyses.
Vector-sum disp., u t (cm)
Disp. Y, u Y (cm)
Disp. X, u X (cm)
280
50
0
Test
-50
Gen. μ0
Const. μ0
Const. μavg
50
0
-50
60
40
20
0
0
2
Figure 8-23
4
6
8
10
Time, t (s)
12
14
16
18
Displacement history of the center isolator subjected to 100TAK:
analytical model vs. test data
20
Disp. X, u X (cm)
281
50
0
-50
Gen. μ0
Vector-sum disp., u t (cm)
Disp. Y, u Y (cm)
Test
Const. μ0
Const. μavg
50
0
-50
60
40
20
0
0
5
10
15
20
25
Time, t (s)
Figure 8-24
Displacement history of the center isolator subjected to 100TAB:
analytical model vs. test
Test
Gen. μ0
Const. μ0
0.2
Norm. force Y, f Y
Norm. force X, f X
0.2
0.1
0
-0.1
-0.2
Const. μavg
-50
Figure 8-25
0
Disp. X, uX (cm)
50
0.1
0
-0.1
-0.2
-50
0
Disp. Y, uY (cm)
50
Global normalized loop of the isolation system subjected to 100TAK:
analytical model vs. test data
282
Gen. μ0
Test
Const. μ0
0.2
Norm. force Y, f Y
Norm. force X, f X
0.2
Const. μavg
0.1
0
-0.1
-0.2
-50
0
Disp. X, u (cm)
0.1
0
-0.1
-0.2
50
-50
X
Figure 8-26
0
Disp. Y, u (cm)
50
Y
Global normalized loop of the isolation system subjected to 100TAB:
analytical model vs. test
Dissipated energy, E p (MNm)
3
Test
Gen. μ
0
2.5
Const. μ0
Const. μ
avg
2
1.5
1
0.5
0
0
2
Figure 8-27
4
6
8
10
Time, t (s)
12
14
16
18
Energy dissipated by the isolation system during 100TAK:
analytical model vs. test data
ͺǤ͵ǤʹǤ ‡•’‘•‡‘ˆ–Š‡ •‘Žƒ–‹‘›•–‡–‘ͺͲ
Displacement of the center isolator subjected to 80TCU motion is presented in
Figure 8-28. Isolator displacement from the test did not start from zero due to a residual
displacement from the previous simulation, which causes an initial offset between
20
283
analytical and experimental displacement. Centering the test data to have zero initial
displacement is not advantageous, since any discrepancy due to initial displacement tends
to be minimized after a few large cycles. This can be easily seen in the displacement
history in the X direction. Despite of the difference in initial displacement, the
displacement history from analysis agrees well with the displacement recorded from the
test.
Three components of isolator reaction of the center bearing subjected to the
80TCU motion are plotted in Figure 8-29. The comparison shows that the analytical
model cannot represent the isolator reaction as well as the displacement, especially in
vertical direction. This may result from two main causes. First, as mentioned before,
because the base of the specimen was not perfectly plane, shimming plates were applied
to redistribute forces to the bearings. This process applied “pre-stress” to the specimen,
which may affect the distribution of vertical reaction on bearings during their dynamics
response, and was not represented in the analytical model. Second, in the isolation system
using TPBs, if the isolation system is limited to translation, the vertical displacements at
all bearings are identical. However, under torsional displacement of the isolation system,
the vertical displacement of individual bearings will be different due to the curvature of
the TPBs. This difference in vertical displacement will redistribute the vertical force to
bearings in a way that cannot be captured in the current TPB element. A difference in
vertical reaction leads to a difference in lateral reaction since the lateral and vertical
behaviors of TPB are coupled.
284
Disp. X, u X (cm)
60
Test
Analysis
40
20
0
-20
Disp. Y, u Y (cm)
-40
40
20
0
-20
-40
0
5
10
15
20
25
Time, t (s)
30
35
40
45
(a)
Test
Analysis
40
30
Disp. Y, u Y (cm)
20
10
0
-10
-20
-30
-40
-60
-40
-20
0
Disp. X, uX (cm)
20
40
(b)
Figure 8-28
Displacement of the center isolator when the isolated base structure
subjected to 80TCU: analytical model vs. test
(a) Displacement history, (b) Displacement trace
60
285
Test
Analysis
RX (kN)
100
0
-100
RY (kN)
100
0
-100
RZ (kN)
1000
500
0
0
Figure 8-29
5
10
15
20
25
Time, t (s)
30
35
40
45
Reactions at center bearing of the isolated base structure subjected to 80TCU:
analytical model vs. test
Hysteresis loops, both absolute and normalized, of the center bearing and of the
whole isolation system are plotted in Figure 8-30. The global loop of the whole isolation
system was developed using displacement of the center bearing and the total base shear,
which is the sum of the forces in all 9 TPBs. The comparison indicates that the analytical
loops match the experimental loops better at the global level than at the center bearing.
This means that the model can capture global behavior of isolation system better than the
behavior of individual bearing. This comes from the difficulty in predicting the vertical
response and reaction in individual bearings as explained before. The global behaivor of
the isolation system, rather than the behavior of individual bearing, is significant to the
response of the isolated structure.
286
50
100
Test
Analysis
Force Y, F Y (kN)
Force X, F X (kN)
100
0
-50
-100
-60
-40
-20
0
20
Disp. X, u (cm)
40
-50
-100
-60
0.2
0.1
0
-0.1
-40
-20
0
20
Disp. X, u (cm)
-40
40
-20
0
20
Disp. Y, u (cm)
40
60
-20
0
20
Disp. Y, u (cm)
40
60
-20
0
20
Disp. Y, u (cm)
40
60
-20
0
20
Disp. Y, u (cm)
40
60
Y
(a)
Norm. force Y, f Y (kN)
Norm. force X, f X (kN)
0
60
X
-0.2
-60
50
60
0.2
0.1
0
-0.1
-0.2
-60
-40
X
Y
1000
Force Y, F Y (kN)
Force X, F X (kN)
(b)
0
-1000
-60
-40
-20
0
20
Disp. X, u (cm)
40
1000
0
-1000
60
-60
-40
X
Y
0.2
Norm. force Y, f Y (kN)
Norm. force X, f X (kN)
(c)
0.1
0
-0.1
-0.2
-60
-40
-20
0
20
Disp. X, u (cm)
40
60
X
0.2
0.1
0
-0.1
-0.2
-60
-40
Y
(d)
Figure 8-30
Hysteresis loops of the center bearing and of the isolation system when the
isolated base structure subjected to 80TCU: analytical model vs. test
(a) Hysteresis loop of the center bearing; (b) Normalized loop of the center bearing
(c) Global hysteresis loop; (d) Global normalized loop
287
ͺǤ͵Ǥ͵Ǥ ‡Ž‡”ƒ–‹‘‡•’‘•‡–‘ͳͲͲƒ†ͺͲ
Horizontal acceleration at geometric center of slabs of the isolated base structure
subjected to 100TAK and 80TCU were investigated. Time history responses of these
accelerations are plotted in Figures 8-31 to 8-34. Their peak values are plotted against
height (peak acceleration profiles) in Figure 8-35.
As expected, the isolation system reduces floor acceleration compared to the input
acceleration, as seen from the acceleration profile plots in Figure 8-35. However, the
reduction is much smaller in acceleration for 100TAK than in acceleration for 80TCU.
The distribution of peak floor acceleration is also different in nature for these two
motions. Under 80TCU, which is an XY horizontal motion, the peak floor acceleration is
almost uniform throughout the height (Figure 8-35 (b)), as expected for isolated
structures when the participation of higher modes is eliminated. Under 100TAK, which is
a 3D motion, the peak floor acceleration does not follow the uniform distribution (Figure
8-35 (a)), especially in X direction where isolator displacement, and hence base shear
coefficient, is larger (Figure 8-23). Participation of higher frequencies in floor
acceleration is also stronger in response to 100TAK than in response to 80TCU (Figures
8-31 to 8-34). To corroborate this observation, significant peaks are observed at several
lower periods (higher frequencies) in floor response spectra for 100TAK (Figure 8-36)
compare to 80TCU (Figure 8-37), which indicates greater participation of higher modes
in 3D excitation. All these phenomena can easily be explained by vertical-horizontal
coupling theory presented in detail in the next chapter. The peak X direction spectral
acceleration during 80TCU is observed at period as long as about 4.5 s. This long period
288
component is not necessarily from the isolation system since the input excitation itself is
strong at this period (see Figure 3-10). The comparison shows good agreement between
analytical response and experimental response.
289
Test
Analysis
0.5 Roof
0
-0.5
0.5 Floor 5
0
X
Acceleration, a (g)
-0.5
0.5
Floor 4
0
-0.5
0.5
Floor 3
0
-0.5
0.5
Floor 2
0
-0.5
0.5 Base
0
-0.5
0
Figure 8-31
5
10
Time, t (s)
15
Acceleration response in X direction of the isolated base structure
subjected to 100TAK: analytical model vs. test
20
290
0.5
Test
Analysis
Roof
0
-0.5
0.5
Floor 5
0
Y
Acceleration, a (g)
-0.5
0.5
Floor 4
0
-0.5
0.5
Floor 3
0
-0.5
0.5
Floor 2
0
-0.5
0.5
Base
0
-0.5
0
Figure 8-32
5
10
Time, t (s)
15
Acceleration response in Y direction of the isolated base structure
subjected to 100TAK: analytical model vs. test
20
291
0.2
Test
Analysis
Roof
0
-0.2
0.2
Floor 5
0
X
Acceleration, a (g)
-0.2
0.2
Floor 4
0
-0.2
0.2
Floor 3
0
-0.2
0.2
Floor 2
0
-0.2
0.2
Base
0
-0.2
0
5
Figure 8-33
10
15
20
25
Time, t (s)
30
35
40
Acceleration response in X direction of the isolated base structure
subjected to 80TCU: analytical model vs. test
45
292
0.2
Test
Analysis
Roof
0
-0.2
0.2
Floor 5
0
Y
Acceleration, a (g)
-0.2
0.2
Floor 4
0
-0.2
0.2
Floor 3
0
-0.2
0.2
Floor 2
0
-0.2
0.2
Base
0
-0.2
0
5
Figure 8-34
10
15
20
25
Time, t (s)
30
35
40
Acceleration response in Y direction of the isolated base structure
subjected to 80TCU: analytical model vs. test
45
293
Roof
Roof
X Test
Y Test
X Analysis
Y Analysis
5
Floor
Floor
4
3
4
3
2
2
1
Table
0
5
0.2
0.4
0.6
0.8
Peak acceleration, Amax
(g)(g)
(a)
Figure 8-35
1
Table
0
1
0.2
0.4
0.6
Peak acceleration, Amax
(g) (g)
(b)
Peak floor acceleration of fixed base structure: analytical model vs. test
(a) Subjected to 100TAK
(b) Subjected to 80TCU
0.8
294
4
2
X-Roof
Y-Roof
Test
Analysis
2
1
0 -2
-1
10
10
2
X-Floor 5
0
10
10
1
A
Spectral acceleration, S (g)
1
0
10
10
1
2
10
1
0 -2
-1
10
10
2
Y-Floor 4
0
10
10
1
1
0 -2
-1
10
10
4
X-Floor 3
0
10
10
1
2
0 -2
-1
10
10
2
Y-Floor 3
0
10
10
1
1
0 -2
-1
10
10
1
X-Floor 2
0
10
10
1
0.5
0 -2
-1
10
10
2
Y-Floor 2
0
10
10
1
1
0 -2
-1
10
10
4
X-Base
0
10
10
1
2
Figure 8-36
0
10
1
0 -2
-1
10
10
4
X-Floor 4
0 -2
10
0 -2
-1
10
10
2
Y-Floor 5
0 -2
10
2
Y-Base
10
-1
0
10
10
1
1
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
Floor spectra of the isolated base structure subjected to 100TAK:
analytical model vs. test
10
1
295
1
Test
Analysis
X-Roof
0.5
0
10
10
1
A
Spectral acceleration, S (g)
0.5
0 -2
-1
10
10
1
Y-Floor 5
0
10
10
1
0.5
0 -2
-1
10
10
1
X-Floor 4
0
10
10
1
0.5
0 -2
-1
10
10
1
Y-Floor 4
0
10
10
1
0.5
0 -2
-1
10
10
1
X-Floor 3
0
10
10
1
0.5
0 -2
-1
10
10
1
Y-Floor 3
0
10
10
1
0.5
0 -2
-1
10
10
1
X-Floor 2
0
10
10
1
0.5
0 -2
-1
10
10
1
Y-Floor 2
0
10
10
1
0.5
10
-1
0
10
10
1
0.5
0 -2
10
Y-Roof
0.5
0 -2
-1
10
10
1
X-Floor 5
0 -2
10
1
X-Base
1
0 -2
10
1
Y-Base
10
-1
0
10
10
1
0.5
-1
0
10
10
Period, T (s)
Figure 8-37
10
1
0 -2
10
-1
0
10
10
Period, T (s)
Floor spectra of the isolated base structure subjected to 80TCU:
analytical model vs. test
10
1
296
ͺǤ͵ǤͶǤ –‘”›”‹ˆ–‡•’‘•‡–‘ͳͲͲƒ†ͺͲ
Analytical and experimental time histories of story drift computed at geometric
center of slab are shown in Figures 8-38 to 8-41 for 100TAK and 80TCU, and peak
profiles are plotted in Figure 8-42. Both the time history drift plots and peak story drift
plots show the good match between analytical and experimental responses.
Distribution of peak story drift subjected to 80TCU follows the trend described
for the fixed base structure. But the distribution is more complicated in response to
100TAK. In this 3D excitation, the distribution of peak story drift is different in X
direction from Y direction. The common distribution trend of the peak story drift
subjected to motions with small vertical excitation is different from the distribution
subjected to motions with strong vertical excitation. These trends are addressed in
Section 8.4.
The initial conclusion from observing acceleration and story drift responses of the
isolated base structure to 3D excitations is that the existence of vertical excitation
complicates the response of the isolated base structure, but it can still be predicted with
high accuracy by a well-calibrated analytical model.
297
0.2
Test
Analysis
Story 5
0
-0.2
0.2
Story 4
0
-0.2
X
Drift, δ (%)
0.2
Story 3
0
-0.2
0.2
Story 2
0
-0.2
0.2
Story 1
0
-0.2
0
5
Figure 8-38
10
Time, t (s)
15
Drift response in X direction of the isolated base structure
subjected to 100TAK: analytical model vs. test
20
298
0.2
Test
Analysis
Story 5
0
-0.2
0.2
Story 4
0
-0.2
Y
Drift, δ (%)
0.2
Story 3
0
-0.2
0.2
Story 2
0
-0.2
0.2
Story 1
0
-0.2
0
Figure 8-39
5
10
Time, t (s)
15
20
Drift response in Y direction of the isolated base structure subjected to 100TAK:
analytical model vs. test
299
0.2
Test
Analysis
Story 5
0
-0.2
0.2
Story 4
0
X
Drift, δ (%)
-0.2
0.2
Story 3
0
-0.2
0.2
Story 2
0
-0.2
0.2
Story 1
0
-0.2
0
5
Figure 8-40
10
15
20
25
Time, t (s)
30
Drift response in X direction of the isolated base structure
subjected to 80TCU: analytical model vs. test
300
0.2
Test
Analysis
Story 5
0
-0.2
0.2
Story 4
0
Y
Drift, δ (%)
-0.2
0.2
Story 3
0
-0.2
0.2
Story 2
0
-0.2
0.2
Story 1
0
-0.2
0
5
Figure 8-41
10
15
20
25
Time, t (s)
30
Drift response in Y direction of the isolated base structure
subjected to 80TCU: analytical model vs. test
301
5
5
X Test
Y Test
X Analysis
Y Analysis
4
Story
Story
4
3
2
2
1
0
3
0.05
0.1
0.15
Peak drift, δmax (%)
0.2
0.25
1
0
0.05
0.1
0.15
Peak drift, δmax (%)
(a)
Figure 8-42
0.2
(b)
Peak story drift of the isolated base structure: analytical model vs. test
(a) Subjected to 100TAK
(b) Subjected to 80TCU
ͺǤ͵ǤͷǤ ‘Ž— ‘” ‡•‡•’‘•‡–‘ͳͲͲƒ†ͺͲ
Figures 8-43 to 8-48 show the time history of dynamic components of the internal
forces in the Northeast column, computed at the bottom section of each column segment
at every story as described for the fixed base structure. The sign convention for these
internal forces is as same as the sign convention for them in the fixed base structure.
Furthermore, Figure 8-49 plots the envelope of the peak internal forces over the height of
the column.
Similar to the internal force in the column of the fixed base structure, a high
frequency component is present and becomes stronger in the internal force histories at
upper stories compared to lower stories. This high frequency component is much stronger
in the internal force to 100TAK than in the internal force to 80TCU, again because of the
vertical excitation component. In the axial force response to 100TAK, the strong high
302
frequency component appears throughout the column, which differs from the fixed base
structure, where the high frequency component was only strong near the roof level.
50
Story 5
0
-50
50
Test
Analysis
Story 4
0
-100
50
X
Bending moment, M (kNm)
-50
Story 3
0
-50
-100
50
Story 2
0
-50
-100
200
Story 1
0
-200
0
Figure 8-43
5
10
Time, t (s)
15
20
Dynamic bending moment about X-axis in the NE column of the isolated base
structure subjected to 100TAK: analytical model vs. test
303
100
Story 5
0
Test
Analysis
-100
100
Story 4
-100
100
Story 3
Y
Bending moment, M (kNm)
0
0
-100
100
Story 2
0
-100
200
Story 1
0
-200
0
Figure 8-44
5
10
Time, t (s)
15
20
Dynamic bending moment about Y-axis in the NE column of the isolated base
structure subjected to 100TAK: analytical model vs. test
304
200
Test
Analysis
Story 5
0
-200
200 Story 4
0
Axial force, P (kN)
-200
400
Story 3
200
0
-200
-400
400
Story 2
200
0
-200
-400
500
Story 1
0
-500
0
Figure 8-45
5
10
Time, t (s)
15
20
Dynamic axial force in the NE column of the isolated base structure subjected
to 100TAK: analytical model vs. test
305
20
Story 5
0
-20
40
Test
Analysis
Story 4
20
0
Bending moment, MX (kNm)
-20
-40
50
Story 3
0
-50
50
Story 2
0
-50
200
Story 1
0
-200
0
Figure 8-46
5
10
15
20
25
Time, t (s)
30
35
40
45
Dynamic bending moment about X-axis in the NE column of the isolated base
structure subjected to 80TCU: analytical model vs. test
306
40
Story 5
20
0
-20
Test
Analysis
-40
100
Story 4
50
-50
100
Story 3
Y
Bending moment, M (kNm)
0
50
0
-50
100
Story 2
50
0
-50
200
Story 1
0
-200
0
Figure 8-47
5
10
15
20
25
Time, t (s)
30
35
40
45
Dynamic bending moment about Y-axis in the NE column of the isolated base
structure subjected to 80TCU: analytical model vs. test
307
40
Story 5
20
0
-20
Test
Analysis
-40
100
Story 4
Axial force, P (kN)
0
-100
200
Story 3
100
0
-100
200
Story 2
0
-200
400
Story 1
200
0
-200
0
Figure 8-48
5
10
15
20
25
Time, t (s)
30
35
40
45
Dynamic axial force in the NE column of the isolated base structure subjected
to 80TCU: analytical model vs. test
308
6
6
6
5
5
4
4
4
3
3
3
2
2
2
Test
Analysis
Floor
5
1
0
100
Peak moment, M
x,peak
200
(kNm)
1
0
200
Peak moment, M
y,peak
400
(kNm)
1
0
200
400
Peak axial foce, P
(kN)
peak
(a)
6
6
6
5
5
4
4
4
3
3
3
2
2
2
Test
Analysis
Floor
5
1
0
100
Peak moment, M
x,peak
200
(kNm)
1
0
100
Peak moment, M
y,peak
200
(kNm)
1
0
200
Peak axial foce, P
peak
400
(kN)
(b)
Figure 8-49
8.4
Peak dynamic forces at every section of column NE of the isolated base
structure: analytical model vs. test
(b) Subjected to 100TAK, (b) Subjected to 80TCU
Checking Performance Objectives
The calibrated model is considered to be reliable for analyzing other earthquake
simulations than the ones selected for the test program. Response of the analytical model
subjected to the target ground motions (not table motions) representing different
earthquake scenarios was analyzed for checking the performance objectives. These target
motions with scale factors given in Tables 3-4, 3-6 and 3-7. All motions representing the
309
U.S code and Japan code were applied as 3D motions, while the motions representing
long duration, long period, subduction motions were applied as horizontal XY motions.
The peak displacement of the isolation system, the peak floor acceleration of all floors,
and the peak story drift of all stories subjected to every ground motion were recorded and
plotted in Figures 8-50 to 8-52. The target displacements of the isolation system at
different earthquake scenarios and the design limit of floor acceleration and story drift are
also presented in these figures for reference. Recall that the target displacements of the
isolation system (Section 3.5.4) were estimated based on the designed values of friction
coefficients, which is smaller than the actual values.
Despite the change in friction coefficient, the peak displacements of the isolation
system in the calibrated model subjected to 80WSM, 88RRS, 130ELC, 100KJM,
100SCT, 100IWA and 100SAN are similar to these of the pre-test model (Figure 3-25).
All these peak displacements are smaller than ͲǤͷ. For the motions that induce the peak
displacement larger than ͲǤͷ, the peak displacements of the calibrated model vary
significantly from the peak displacements of the pre-test model. The 100TAK motion
induces larger displacement in the calibrated model than in the pre-test model. The peak
displacement is most sensitive to modeling assumptions for the MCE motions. The peak
isolator displacements subjected to 89TCU motion and 103TAB motion are evidently
smaller in the calibrated model than in the pre-test model, even though the scale factors
for these motions were smaller in the pre-test analysis, equal to 0.8 and 1.0, respectively.
Calibrated and pre-test peak displacements of LGP and SYL motions are incomparable
since they have different scale factors (0.7 and 1.09 for LGP motion, and 1.0 and 1.22 for
310
SYL motion for pre-test and calibrated models, respectively). Recall that the final pre-test
analysis was focused on a safety check, and thus utilized the final scale factors of the
input motions as planned for the test.
Peak story drift of all stories over all excitations is well below the design limit,
which represents the performance objective for the isolated structure. However, the
acceleration objective was not met. As shown in Figure 8-52, peak horizontal floor
accelerations of the model subjected to motions with large vertical excitation all exceed
the design limit, even though the vertical ground acceleration was not amplified in this
analysis like it was in the table motions. This result is much different from the result of
the pre-test analysis (Figure 3-27). The difference comes from the vertical-lateral
coupling effect, which was suppressed in the pre-test model by uncoupling bearing and
excessive damping of higher modes.
The distribution of the peak floor acceleration and peak story drift are plotted in
Figures 8-53 and 8-54. The distribution of small peak acceleration is almost constant
throughout the height. At larger peak distribution, the acceleration is small at Floor 2 and
Floor 5, which are nodes of the higher mode excited by the lateral-vertical coupling
effect.
Similar to the distribution of peak floor acceleration, the distributions of peak
story drift also differ from small to large drift levels. At small drift level, the peak story
drift is largest at the 1st or 2nd stories and decrease with height, which matches the
distribution of the peak story drift of the fixed base configuration. At larger drift levels,
the story drift also follow this trend, except that the story drift is smallest at the 3rd story,
311
where the slope of the excited higher mode is minimized (e.g. difference in acceleration
of 3rd and 4th floor, Figure 8-54). Since the participation of higher modes due to vertical
excitation even affects the drift distribution, the change to the behavior of the isolated
structure is significant.
Bearing limit
1
MCE
0.8
Service
DBE
MCE
Japan code
100SAN
100IWA
100SCT
100TAK
100KJM
130ELC
0
89TCU
Service
103TAB
0.2
122SYL
DBE
109LGP
0.4
88RRS
0.6
80WSM
Peak iso. disp., d max (m)
1.2
Long period,
long duration,
subduction
Ground motion
Figure 8-50
Peak isolator displacement of the analytical model subjected to
selected motions representing different earthquake scenarios
312
0.6
Design limit
Peak drift, δmax (%)
0.5
0.4
0.3
0.2
Service
DBE
Japan code
MCE
100SAN
100IWA
100SCT
100TAK
100KJM
130ELC
89TCU
103TAB
122SYL
109LGP
80WSM
0
88RRS
0.1
Long period,
long duration,
subduction
Ground motion
Figure 8-51
Peak story drift of the analytical model subjected to selected motions
representing different earthquake scenarios
2.5
2
1.5
1
Service
DBE
MCE
Japan code
Ground motion
Figure 8-52
100SAN
100SCT
100TAK
100KJM
130ELC
89TCU
103TAB
122SYL
109LGP
88RRS
0
100IWA
Design limit
0.5
80WSM
Peak floor acc., A max (g)
3
Long period,
long duration,
subduction
Peak floor acceleration of the analytical model subjected to selected motions
representing different earthquake scenarios
313
5
Design limit
Story
4
3
2
1
0
Figure 8-53
0.1
0.2
0.3
Peak drift, δmax (%)
0.4
0.5
0.6
Distribution of peak story drift of the analytical model subjected to selected
motions representing different earthquake scenarios
Roof
Floor
4
3
Design limit
5
2
Base
Ground
0
Figure 8-54
0.5
1
1.5
Peak acc., Amax (g)
2
2.5
Distribution of peak floor acceleration of the analytical model subjected to
selected motions representing different earthquake scenarios
3
Chapter 9
ˆŽ—‡ ‡‘ˆ‡”–‹ ƒŽš ‹–ƒ–‹‘
‘–Š‡‡•’‘•‡‘ˆ–Š‡–”— –—”‡
As shown in Chapter 5, the existence of vertical excitation amplifies the
horizontal responses of the structure. The amplification occurs in both the isolated base
structure and the fixed base structure, though small in the fixed base structure. This
phenomenon is hereafter referred to as the “vertical-horizontal coupling effect”. The
amplification comes from the participation of high frequency components, which are the
major components in some responses of the isolated structure. The sources of this
horizontal-vertical coupling effect are investigated in detail in this chapter. The analytical
model developed in Chapter 7 and validated in Chapter 8 is used for the investigation.
9.1
Identifying the Sources of the Vertical- Horizontal Coupling Effect
The sources of the vertical-horizontal coupling effect are identified based on the
investigation of the responses of the analytical models subjected to selected motions. The
88RRS table motion of the fixed base test, which has the strongest vertical component, is
selected for investigating the fixed base structure model. The 70LGP table motion of the
isolated base test is used for the isolated base structure model. This motion has strong
vertical acceleration with the peak vertical acceleration of ͲǤ͸ͺ͹‰ but does not lead to
significant uplift of the isolated structure. Comparison of the analytical data and test data
(Figures 9-1 to 9-4) shows that the analytical models capture the responses to the selected
motion well and they can be used for the investigation.
Acceleration Y, aY (g)
Acceleration X, aX (g)
315
0.5
0
-0.5
Test
Analysis
-1
2
1
0
-1
-2
0
Figure 9-1
1
2
3
4
5
Time, t (s)
6
7
8
9
10
Horizontal acceleration at roof of the fixed base structure subjected to the
88RRS motion
6
Story
5
4
3
X - Test
Y - Test
X - Analysis
Y - Analysis
2
1
0
0.5
1
Peak floor acc., A
(g)
1.5
max
Figure 9-2
Peak horizontal floor acceleration of the fixed base structure subjected to the
88RRS motion
Acceleration Y, aY (g)
Acceleration X, aX (g)
316
0.5
Test
Analysis
0
-0.5
0.5
0
-0.5
0
5
10
15
Time, t (s)
Figure 9-3
Horizontal acceleration at roof of the isolated base structure subjected to the
70LGP motion
6
Story
5
4
X - Test
Y - Test
X - Analysis
Y - Analysis
3
2
1
0
0.2
0.4
Peak floor acc., A
max
Figure 9-4
0.6
(g)
0.8
Peak horizontal floor acceleration of the isolated base structure subjected to
the 70LGP motion
ͻǤͳǤͳ ‘—” ‡•ˆ”‘—’‡”•–”— –—”‡ǣ‡”–‹ ƒŽǦ ‘”‹œ‘–ƒŽ‘—’Ž‹‰‘†‡•
As mentioned earlier in Chapter 8, the first vertical mode of the fixed base
structure is not a pure vertical mode but is a vertical- horizontal coupling mode where the
317
horizontal deformation is accompanied with the vertical deformation. This verticalhorizontal coupling mode also exists in the isolated base structure as well. The modal
analysis indicated that most vertical modes and many horizontal modes of the analytical
models are vertical- horizontal coupling modes. These vertical-horizontal coupling modes
excite the horizontal responses even if the structures are only subjected to vertical
excitation. Figures 9-5 and 9-6 show the mode shapes of the first vertical modes of the
fixed base structure model and the isolated base structure model, respectively. These
modes in the figures are for demonstrating the coupling modes, and are not necessarily
the main modes contributing to the coupling behavior. The horizontal stiffness of the
isolators used in modal analysis of the isolated base model was the first stiffness of the
backbone curve of the isolation system (ൌ ܹȀሺͺͶ ሻ, Figure 3-23), which corresponds
to the period of ͳǤͺͶ•, the major response component of the isolation system. The mode
shapes of these two modes show that the horizontal deformation is larger in the Ydirection than in the X-direction, where the structure is almost symmetric.
(a)
Figure 9-5
(b)
A horizontal-vertical coupling mode of the fixed base structure model
(a) in the X-direction, (b) in the Y-direction
318
(a)
Figure 9-6
(b)
A horizontal-vertical coupling mode of the isolated base structure model
(a) in the X-direction, (b) in the Y-direction
Figure 9-7 shows the time-history of the horizontal acceleration response at the
geometric center of the roof floor of the fixed base model subjected to only the vertical
component of the 88RRS motion. The peak horizontal floor acceleration at all floors of
the fixed base model is presented in Figure 9-8. The horizontal acceleration due to the
pure vertical excitation is strong.
Horizontal acceleration at roof of the isolated base model subjected to the vertical
components of 70LGP motions is plotted in Figures 9-9. To investigate the effect of the
effective horizontal stiffness of the isolation system on the horizontal response of the
isolated base model subjected to vertical excitation, 2 models with different horizontal
stiffness were used. The “1st slope” model used the first stiffness of the backbone curve
of the isolation system, which was used in the modal analysis of the isolated base model
as mentioned earlier. The “2nd slope” model used the second stiffness of the backbone
curve of the isolation system (ൌ ܹȀሺ͹͹Ͳ ሻ, Figure 3-23), which corresponds to the
period of ͷǤͷ͹•. The comparison shows little difference between the 2 models. Peak
319
floor acceleration of all floors of the 2 models in Figure 9-10 also shows insignificant
difference between these models. The peak horizontal acceleration response of the
isolated base models subjected to vertical excitation is also larger in the Y-direction than
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0
1
Figure 9-7
2
3
4
5
Time, t (s)
6
7
8
9
10
Horizontal acceleration at roof of the fixed base model subjected to the
vertical component of 88RRS motion
6
5
Floor
Acceleration Y, aY (g)
Acceleration X, aX (g)
in the X-direction, similar to the response of the fixed base model.
4
X-direction
Y-direction
3
2
Base1
0
Figure 9-8
0.1
0.2
0.3
Peak floor acc., Amax (g)
0.4
Peak horizontal floor acceleration of the fixed base model subjected to the
vertical component of 88RRS motion
Acceleration X, aX (g)
320
0.1
1st slope
2nd slope
0.05
0
Acceleration Y, aY (g)
-0.05
-0.1
0.1
0.05
0
-0.05
-0.1
0
5
10
15
Time, t (s)
Figure 9-9
Horizontal acceleration at roof of the isolated base model subjected to the
vertical component of 70LGP motion
6
Floor
5
4
X-1st slope
Y-1st slope
X-2nd slope
3
Y-2nd slope
2
Base1
0
Figure 9-10
0.05
0.1
0.15
Peak floor acc., Amax (g)
0.2
Peak horizontal floor acceleration of the isolated base model subjected to the
vertical component of 70LGP motions
Next, it will be shown that the coupling in the fixed base structure is limited to
modal coupling effects, and thus can be predicted by linear superposition of response to
horizontal excitation and response to vertical excitation. For the isolated base structure,
however, superposition of horizontal and vertical response significantly underestimates
321
the response from full 3D analysis, indicating that additional nonlinear sources of
coupling must be present.
Figure 9-11 shows the horizontal acceleration history at the geometric center at
the roof of the fixed base structure model subjected to 88RRS motion. The responses
from 2 cases are plotted in this figure. In the “3D” case, the response is from the model
subjected to 3 components of the 88RRS motion simultaneously. In the “XY+Z” case, the
response is the linear combination of the response of the model to the horizontal
components of the motion and the response of the model to the vertical component of the
motion (the horizontal components and the vertical component were applied to the model
separately). The peak horizontal floor acceleration at every floor is presented for the 2
cases in Figure 9-12. The comparison shows that the responses from the 2 cases are
almost identical. The small discrepancy between the 2 cases likely comes from the
Acceleration Y, aY (g)
Acceleration X, aX (g)
nonlinear behavior of the model. This comparison means that the horizontal response to
0.5
0
3D
XY+Z
-0.5
-1
1
0.5
0
-0.5
-1
0
1
Figure 9-11
2
3
4
5
Time, t (s)
6
7
8
9
Horizontal acceleration at roof of the fixed base model subjected to
88RRS motion: 3D vs. XY+Z
10
322
6
Floor
5
4
3
X-3D
Y-3D
X-XY+Z
Y-XY+Z
2
Base1
0.2
Figure 9-12
0.4
0.6
Peak floor acc., A
max
0.8
(g)
1
Peak horizontal floor acceleration of the fixed base model subjected to
88RRS motion: 3D vs. XY+Z
the 3D excitation equals the horizontal response to the horizontal components of
excitation plus the horizontal response to the vertical excitation. The applicability of the
superposition principle implies that the vertical-horizontal coupling effect is purely from
the coupling behavior of the structure represented by the vertical-horizontal coupling
modes.
Figure 9-13 shows the horizontal floor spectra of the fixed base model subjected
to both 3D and XY excitations of the 88RRS motion. This figure shows that the
amplification of the response to the 3D excitation compared to the response to the XY
excitation mainly comes from the ͲǤͳͳ• period component. The modal information of
the first 20 modes of the fixed base model (Table 9-1) indicates that mode 10 could cause
the amplification. Figure 9-14 shows that this mode is controlled by the vertical vibration
at the 5th floor accompanied with the vertical vibration at lower floors.
323
4
4
3D
XY
X-Roof
Y-Roof
2
2
0
0
4
4
Spectral acceleration, S A (g)
X-Floor 5
Y-Floor 5
2
2
0
0
4
4
X-Floor 4
Y-Floor 4
2
2
0
0
4
4
X-Floor 3
Y-Floor 3
2
2
0
0
4
4
X-Floor 2
Y-Floor 2
2
0
-2
10
Figure 9-13
2
10
-1
10
Period, T (s)
0
10
1
0
-2
10
10
-1
10
Period, T (s)
0
10
Horizontal floor spectra of the fixed base model subjected to 88RRS motion:
3D vs. XY excitations
1
324
Table 9-1
Modal information of the first 20 modes of the fixed base model
Contribution Contribution
factor to Xfactor to Ydisplacement displacement
at roof due
at roof due
to vertical
to vertical
excitation
excitation
Mode
Period (s)
Frequency
(Hz)
Effective
mass ratio
in vertical
direction
(%)
1
0.69
1.46
0.000
0.011
0.094
2
0.67
1.50
0.000
0.084
-0.035
3
0.47
2.12
0.000
0.002
0.001
4
0.22
4.56
0.003
0.005
0.039
5
0.21
4.69
0.000
0.003
-0.002
6
0.16
6.40
0.004
0.005
0.002
7
0.14
7.06
20.943
-0.005
0.057
8
0.11
9.03
0.055
-0.003
-0.027
9
0.11
9.27
0.153
-0.023
0.009
10
0.11
9.48
16.284
0.034
-0.084
11
0.10
9.69
0.153
-0.003
-0.001
12
0.10
10.19
5.400
-0.002
0.018
13
0.09
11.00
0.552
-0.001
0.007
14
0.09
11.19
10.921
-0.001
0.019
15
0.09
11.74
0.166
-0.001
0.000
16
0.08
12.04
1.431
0.000
-0.001
17
0.08
12.22
0.104
0.002
0.001
18
0.08
12.45
0.057
-0.003
-0.002
19
0.08
12.59
6.935
0.002
0.012
20
0.08
12.88
1.849
-0.002
-0.008
325
YZ view
XZ view
Figure 9-14
th
The 10 mode shape of the fixed base model
In contrast to the fixed base model, the response of the isolated base model is
much larger in the 3D case than in the XY+Z case (Figures 9-15 and 9-16). In these
results, the response to the vertical (Z) component of excitation was taken from the
isolated base model with the first stiffness of the isolation system. As asserted earlier, the
obvious difference between the 2 cases implies that besides the vertical-horizontal
coupling modes of the superstructure, other sources contribute to the vertical-horizontal
coupling effect. These sources come from the isolation system and are investigated next.
Acceleration Y, aY (g)
Acceleration X, aX (g)
326
0.4
3D
XY+Z
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
0
5
10
15
Time, t (s)
Figure 9-15
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: 3D vs. XY+Z
6
Floor
5
4
3
2
1
0
Figure 9-16
X-3D
Y-3D
X-XY+Z
Y-XY+Z
0.2
0.4
0.6
Peak floor acc., Amax (g)
Peak horizontal floor acceleration of the isolated base model subjected to
70LGP motion: 3D vs. XY+Z
ͻǤͳǤʹ ‘—” ‡•ˆ”‘ •‘Žƒ–‹‘›•–‡ǣ‡”–‹ ƒŽ‡ˆ‘”ƒ–‹‘Ȁ’Ž‹ˆ–‘ˆ–Š‡
•‘Žƒ–‹‘›•–‡ƒ†‘—’Ž‹‰‡Šƒ˜‹‘”‘ˆ–Š‡•
To verify that the isolation system has a contribution to the vertical-horizontal
coupling effect, the responses of the structure with two different isolator models are
compared. In the “Full 3D” model, the 3D TPB elements with the general friction model
327
are used. In the “Z Restrained” model, the vertical deformation of the TPB elements is
restrained so that any vertical-horizontal coupling behavior of the isolator is eliminated.
The static vertical reaction of the bearing for the model subjected to gravity load is taken
as the vertical load in the restrained bearing. The horizontal roof acceleration of these 2
isolated base structure models are compared in Figures 9-17 and 9-18. The acceleration
of the Z Restrained model is much smaller than the acceleration of the Full 3D model.
According to the characteristic of the 2 bearing models, the difference in the response
comes from 2 sources: (1) the vertical flexibility of the isolation system, including
capacity for uplift, and (2) the vertical-horizontal coupling behavior of the TPBs. The
effect of these 2 sources on the vertical-horizontal coupling effect is investigated in detail
Acceleration Y, aY (g)
Acceleration X, aX (g)
next.
0.4
Full 3D
Z Restrained
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
0
5
10
15
Time, t (s)
Figure 9-17
Horizontal acceleration at roof of the isolated base model subjected to 70LGP
motion: 3D vs. Z Restrained
328
X - Full 3D
Y - Full 3D
X - Z Restrained
Y - Z Restrained
6
Floor
5
4
3
2
1
0
Figure 9-18
0.2
0.4
0.6
Peak floor acc., Amax (g)
0.8
Peak horizontal floor acceleration of the isolated base model subjected to
70LGP motion: 3D vs. Z Restrained
a) Vertical Deformation/Uplift of the Isolation
The rocking of the structure on the isolation system during the dynamic response
of the system, which is caused by the overturning, uneven distribution of mass, gravity
load and stiffness of the system, may amplify the horizontal response of the
superstructure. This phenomenon is conceptually demonstrated in Figure 9-19. Since the
isolated structure is not rigid so that the uneven vertical deformation may be amplified or
reduced when transferring to the horizontal response, depending on its frequency content.
If the frequency content of the uneven deformation resonates with a natural frequency of
the isolated structure, it is amplified. If the frequency of the uneven deformation is too
high, it is reduced and very little effect on horizontal response occurs. The uneven
vertical deformation in the isolation system is caused by the overturning and vertical
dynamic response of the system so that the dominant frequency is not very high and can
be amplified when transferring to the horizontal response. The comparison of the
responses of the 2 isolated base structure models with different TPB models in Figures 9-
329
20 and 9-21 shows the effect of the vertical deformation of the isolation system on the
horizontal response during 70LGP. The “Z Restrained” model mentioned in these figures
has the same meaning as before. The “Uncoupled” model uses TPB elements where the
horizontal response of the elements is independent of the vertical force. The Uncoupled
model is similar to the Z Restrained model except that the vertical stiffness of the TPB
elements in the Uncoupled model is as same as the vertical stiffness of the TPB elements
in the Full 3D model, where the compression stiffness is ͳʹ ൈ ͳͲ଼ Ȁ and tension
stiffness is very small (ͳͲͲȀ) (Section 7.6). Note that the TPB elements in the
Uncoupled model and Z Restrained model are horizontally-vertically uncoupled so that
the difference in responses between these two models are purely from the vertical
deformation of the isolation system, including uplift. The difference in the 2 responses is
small in this case.
Figure 9-19
Influence of rocking on the isolation system to the horizontal response of the
isolated structure
Acceleration Y, aY (g)
Acceleration X, aX (g)
330
0.2
Uncoupled
Z Restrained
0.1
0
-0.1
-0.2
0.4
0.2
0
-0.2
-0.4
0
5
10
15
Time, t (s)
Figure 9-20
Horizontal acceleration at roof of the isolated base model subjected to
70LGP motion: Uncoupled vs. Z Restrained
6
5
Floor
X - Uncoupled
4
Y - Uncoupled
X - Z Restrained
3
Y - Z Restrained
2
1
0.1
0.2
0.3
0.4
Peak floor acc., Amax (g)
0.5
Figure 9-21. Peak horizontal floor acceleration of the isolated base model subjected to
70LGP motion: Uncoupled vs. Z Restrained
The vertical deformation of the TPB can be divided into 2 parts. One part is from
the vertically flexibility of the bearings and their supports and the other part is from the
uplift of the bearings. Next, it is shown that uplift of the bearings can be a significant
source of amplification in some motions. The effect of uplift is investigated by comparing
331
the response of the Uncoupled model and the Const Kz model as shown in Figures 9-22
and 9-23. The Const Kz model is similar to the Uncoupled model but the tension stiffness
and compression stiffness of the bearings are the same so that the bearing uplift in this
model is eliminated. As expected, the difference between responses of these 2 models to
70LGP motion is very small since the uplift is small. For motions with very strong
vertical component such as 88RRS, the uplift is large so that its effect on the horizontal
response becomes significant, which is observed from the responses of the 2 models to
Acceleration Y, aY (g)
Acceleration X, aX (g)
88RRS motion presented in Figures 9-24 and 9-25.
0.2
Uncoupled
ConstKz
0.1
0
-0.1
0.4
0.2
0
-0.2
-0.4
0
5
10
Time, t (s)
Figure 9-22
Horizontal acceleration at roof of the isolated base model subjected to
70LGP motion: Uncoupled vs. Const Kz
15
332
6
Floor
5
4
X - Uncoupled
Y - Uncoupled
X - ConstKz
Y - ConstKz
3
2
1
0
Acceleration Y, aY (g)
Acceleration X, aX (g)
Figure 9-23
0.1
0.2
0.3
0.4
Peak floor acc., Amax (g)
0.5
Peak horizontal floor acceleration of the isolated base model subjected to
70LGP motion: Uncoupled vs. Const Kz
0.4
Uncoupled
ConstKz
0.2
0
-0.2
-0.4
0.5
0
-0.5
-1
0
1
Figure 9-24
2
3
4
5
Time, t (s)
6
7
8
9
Horizontal acceleration at roof of the isolated base model subjected to
88RRS motion: Uncoupled vs. Const Kz
10
333
6
Floor
5
4
X - Uncoupled
Y - Uncoupled
X - ConstKz
Y - ConstKz
3
2
1
0
Figure 9-25
0.5
1
Peak floor acc., Amax (g)
1.5
Peak horizontal floor acceleration of the isolated base model subjected to
88RRS motion: Uncoupled vs. Const Kz
b) Vertical-Horizontal Coupling Behavior of the TPB
Next, it is shown that the vertical-horizontal coupling of the bearing is a
significant source of amplification of horizontal accelerations beyond the vertical
flexibility of the bearings. Figure 9-26 and 9-27 compares the responses of the Full 3D
model, which includes the coupling behavior of TPBs, to the Uncoupled model subjected
to 70LGP motion. The horizontal accelerations are much larger for the Full 3D model
than for the Uncoupled model. The mechanism of the effect of the vertical-horizontal
coupling behavior of the TPBs to the horizontal responses of the isolated structure is
analytically explained in Section 9-2.
0.4
Full 3D
Uncoupled
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
0
5
10
Time, t (s)
Figure 9-26
Horizontal acceleration at roof of the isolated base model subjected to
70LGP motion: Full 3D vs. Uncoupled
6
5
Floor
Acceleration Y, aY (g)
Acceleration X, aX (g)
334
X - Full 3D
Y - Full 3D
X - Uncoupled
Y - Uncoupled
4
3
2
1
0
Figure 9-27
0.2
0.4
0.6
Peak floor acc., Amax (g)
Peak horizontal floor acceleration of the isolated base model subjected to
70LGP motion: Full 3D vs. Uncoupled
15
335
9.2
Effect of the Vertical-Horizontal Coupling Behavior of Friction Bearings on
Responses of the Isolated Structures
ͻǤʹǤͳ ‹‰‹†–”— –—”‡•
To understand how the vertical-horizontal coupling behavior of friction bearings
affects the horizontal acceleration under the existence of vertical acceleration, consider a
simple system shown in Figure 9-28 (a). In this system, the mass ݉ is supported by a
friction bearing whose horizontal normalized hysteresis loop is illustrated schematically
in Figure 9-28 (b). The system is subjected to 2 cases of excitation as shown in Figures 929 (a) and 9-29 (b). Only horizontal acceleration ܽ௚௑ ሺ‫ݐ‬ሻ is applied in the first case while
both horizontal acceleration ܽ௚௑ ሺ‫ݐ‬ሻ and vertical acceleration ܽ௚௓ ሺ‫ݐ‬ሻ are applied in the
second case. The horizontal acceleration responses of the mass ݉ in these two cases are
ܽ௑ǡ௑ ሺ‫ݐ‬ሻ and ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻ, which are refer to as “X excitation” and “X + Z excitation”,
respectively.
݂
݉
superstructure
ܼ
‫ݑ‬
isolator
ܺ
(a)
Figure 9-28
(b)
Single mass isolated system with friction bearing
(c) System configuration
(d) Normalized hysteresis loop of the isolator
Z component
336
݉
݉
X component
X component
(b)
(a)
ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻ
ܽ௑ǡ௑ ሺ‫ݐ‬ሻ
݉Ǥ ܽ௑ǡ௑ ሺ‫ݐ‬ሻ
݉Ǥ ݃
݉Ǥ ݃
ܴ௑ǡ௑ ሺ‫ݐ‬ሻ
ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ
݉Ǥ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻ
ܴ௓ǡ௑ ሺ‫ݐ‬ሻ
ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻ
(d)
(c)
Figure 9-29
ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻ
݉Ǥ ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻ
Two cases of excitation on the isolated single mass system
(a) X excitation
(b) X + Z excitation
(c) Free body diagram of ݉ in X excitation
(d) Free body diagram of ݉ in X + Z excitation
Many analytical and experimental studies concluded that the effect of vertical
excitation on horizontal displacement of isolators is small (see Section 1.1). In this
theoretical derivation, assume that the horizontal displacement of the bearing ‫ݑ‬ሺ‫ݐ‬ሻ is
identical in the 2 cases. Because of this assumption, the normalized force of the bearing,
݂ሺ‫ݐ‬ሻ, is also identical in the 2 cases.
At any time ‫ݐ‬, the vertical reactions in X excitation is:
337
ܴ௓ǡ௑ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ݃
(9.2-1)
ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ቀ݃ ൅ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻቁ
(9.2-2)
and X + Z excitation is
where ݃ is the gravity acceleration and ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻ is the vertical acceleration in X + Z
excitation. These reactions can easily be derived by applying D’Alembert’s principle to
the free body diagrams of the two systems in Figures 9-29 (c) and 9-29 (d) while
observing that the vertical acceleration of the mass is zero during X excitation. From
these vertical reactions and definition of the normalized force, the horizontal reaction of
the 2 cases can be computed:
ܴ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ݃Ǥ ݂ሺ‫ݐ‬ሻ
(9.2-3)
ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ቀ݃ ൅ ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻቁ Ǥ ݂ሺ‫ݐ‬ሻ
(9.2-4)
Applying D’Alembert’s principle again to the free body diagrams in the
horizontal direction:
݉Ǥ ܽ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ܴ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ݃Ǥ ݂ሺ‫ݐ‬ሻ
(9.2-5)
݉Ǥ ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ቀ݃ ൅ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻቁ Ǥ ݂ሺ‫ݐ‬ሻ
(9.2-6)
Subtracting Equation (9.2-5) from Equation (9.2-6) and dividing both sides by ݉
give:
338
ȟܽ௑ ሺ‫ݐ‬ሻ ൌ ܽ௑ǡ௑௓ ሺ‫ݐ‬ሻ െ ܽ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻǤ ݂ሺ‫ݐ‬ሻ
(9.2-7)
Equation (9.2-7) shows that during dynamic response, there is a change or
amplification in horizontal acceleration due to the resultant to vertical acceleration of the
system. This explains why the coupling behavior of friction bearings contributes to
horizontal – vertical coupling effect. The equation also suggests that the amount of
horizontal acceleration transferred from vertical acceleration equals the product of
vertical acceleration and normalized force. This is exact for single mass systems or for
systems with rigid superstructures. The influence of overturning on vertical acceleration
has been neglected, because the loads from overturning are balanced over the isolation
system.
Numerical Example 9.1
Consider a rigid cantilever supported by a TPB as shown in Figure 9-30. The
masses and dimensions of the cantilever and the normalized backbone curve of the
bearings are also shown in this figure. In fact, the specific values of dimensions and
masses do not influence coupling, according to Equation (9.2-7). The dimensions (story
heights) and masses, corresponding to the height and floor mass of the 5-story specimen,
will be used in a follow-up examples. The backbone curve of the TPB is identical to the
designed backbone curve of the bearing used in the test. The vertical stiffness of the
bearing is selected to be very large such that the vertical acceleration in the superstructure
equals the input vertical acceleration, and is easily controlled.
339
݉଺ ൌ ͳʹͲ–
݄ ൌ ͷ ൈ ͵݉
݉ହ ൌ ͺͲǤͳ–
݂
݉ସ ൌ ͺͳǤͳ–
݉ଷ ൌ ͺͳǤ͵–
Rigid
structure
݉ଶ ൌ ͺʹǤ͵–
ͲǤʹ͹
ͲǤʹͳ
ͲǤͲͺ
ͲǤͲʹ ͳ
݉ଵ ൌ ͺ͹Ǥͷ–
݇ଵ ൌ
ͳ
ͳ
ǡ ݇ ൌ
ͺͶ  ଶ ͹͹Ͳ 
݇ଵ
݇ଵ
ͳ
݇ଶ
ͷ
ͳͲͺ
ͳ
ͳͳ͵
‫ݑ‬ሺ ሻ
(b)
(a)
Figure 9-30
System for Numerical Example 9.1
(a) System configuration
(b) Normalized hysteresis loop of the isolators
The system is subjected to X excitation and X+Z excitation as defined earlier. The
two components of input acceleration are shown in Figure 9-31. The horizontal
component of excitation is identical to the one applied in the sine-excitation test to
determine the characteristics of the bearings. The frequency of this excitation, which is
ͳȀ͵ œ, was selected such that it does not resonate with any frequency of the isolation
system nor the isolated structure. The amplitude of ͲǤͳ͵‰ was selected so that it can
drive the isolation system to a peak displacement of about half of displacement limit. The
frequency of the vertical component (ʹ ‫ )ݖ‬was selected so that there are several vertical
cycles during one horizontal cycle, which is consistent with earthquake excitation. The
amplitude of the vertical component (ͲǤͺ‰) was high enough to easily observe the
vertical-horizontal coupling effect but not high enough to induce uplift.
Acc., a z (g)
Acc., a x (g)
340
0.2 (a)
0
-0.2
1
(b)
0
-1
0
2
Figure 9-31
4
6
8
10
12
Time, t (s)
14
16
18
20
22
Input acceleration components for Numerical Example 9.1
(a) Horizontal component
(b) Vertical component
Responses of the TPB for the two cases of excitation are presented in Figure 9-32.
Both absolute and normalized hysteresis loops are plotted. The figure shows that the
hysteresis loops from the two cases of excitation are significantly different although the
normalized loops are similar. This means that the horizontal shear force from the two
cases are significantly different but the ratios between horizontal shear and vertical load
(or normalized force) are almost the same. The displacements of isolator in the two cases
are slightly different.
Figure 9-33 shows horizontal acceleration at roof to evaluate the amplification of
acceleration due to the vertical-horizontal coupling effect. Presented in Figure 9-33 (a)
are time histories of the horizontal acceleration at the roof from the two cases. The
difference between these acceleration histories is evident. The peak acceleration during X
+ Z excitation, which is ͲǤ͵ʹ͹‰, is 1.85 times larger than the peak acceleration during X
excitation. The peak accelerations in the two cases may happen at different times, so that
341
it is generally inconsistent to evaluate the amplification due to vertical-horizontal
coupling effect based only on the peaks.
The change in acceleration at the roof during X + Z excitation relative to X
excitation, which is defined by equation (9.2-7), is plotted in Figure 9-33 (b). The
“Computed” value is computed directly by subtracting the acceleration from X + Z
excitation by the acceleration from X excitation (the first part of Equation 9.2-7). The
Estimated value is obtained by multiplying the vertical acceleration by the global
normalized force (the second part of Equation 9.2-7). The Estimated and Computed
curves differ slightly only because Equation 9.2-7 assumes that the isolator displacements
2000
0.2
1000
0.1
0
-1000
X and
+ Z excitation
X excitation only
-2000
-0.75
-0.5
-0.25
0
0.25 0.5
Displacement, u (m)
X
(a)
Figure 9-32
0.75
Norm. force, fx
Force, Fx (kN)
are the same in X + Z excitation and X excitation cases, which is not strictly correct.
0
-0.1
-0.2
-0.75
-0.5
-0.25
0
0.25 0.5
Displacement, u (m)
(b)
X
Hysteresis loops of isolation system in Numerical Example 9.1
(a) Actual hysteresis loop
(b) Normalized hysteresis loop
0.75
342
Acc., ax (g)
0.4
X and
+ Z excitation
X excitation only
0.327
(a)
0.2
0.177
0
-0.2
Diff. acc., Δ ax (g)
-0.4
0.2 (b)
Computed
Estimated
0.1
0
-0.1
-0.2
0
Figure 9-33
2
4
6
8
10
12
Time, t (s)
14
16
18
20
22
Horizontal acceleration of the isolated structure in Numerical Example 9.1
(a) Horizontal acceleration
(b) Difference in horizontal acceleration
ͻǤʹǤʹ ƒ–‹Ž‡˜‡”–”— –—”‡•
Consider the cantilever structure including 3 masses subjected to the 2 cases of
excitation as shown in Figure 9-34. The cantilever is flexible in the horizontal direction
but assumed rigid in the vertical direction. The isolator is also assumed rigid in the
vertical direction as well (similar to the assumption in Numerical Example 9.1). The
number of masses and the equality of masses are selected for convenience of deriving
equations but do not alter the generality of the results. The horizontal acceleration
response of masses in the 2 cases is illustrated to the right side of the cantilever (Figure 934). Let the horizontal acceleration of the mass ݉௜ under X excitation be ܽ௜ǡ௑ ሺ‫ݐ‬ሻ. The
horizontal acceleration ܽ௜ǡ௑௓ ሺ‫ݐ‬ሻ of mass ݉௜ in X + Z excitation can be separated into 2
parts:
343
݉Ǥ ܽଷǡ௑ ሺ‫ݐ‬ሻ
݉ଷ ܽଷǡ௑ ሺ‫ݐ‬ሻ
݉Ǥ ȟܽଷ ሺ‫ݐ‬ሻ ݉Ǥ ܽଷǡ௑ ሺ‫ݐ‬ሻ ݉ଷ ܽଷǡ௑ ሺ‫ݐ‬ሻ
ȟܽଷ ሺ‫ݐ‬ሻ
݉Ǥ ܽଶǡ௑ ሺ‫ݐ‬ሻ
݉ଶ ܽଶǡ௑ ሺ‫ݐ‬ሻ
݉Ǥ ȟܽଶ ሺ‫ݐ‬ሻ
݉Ǥ ܽଶǡ௑ ሺ‫ݐ‬ሻ ݉ଶ
ܽଶǡ௑
ȟܽଶ ሺ‫ݐ‬ሻ
݉Ǥ ܽଵǡ௑ ሺ‫ݐ‬ሻ
݉ଵ ܽଵǡ௑ ሺ‫ݐ‬ሻ
݉Ǥ ȟܽଵ ሺ‫ݐ‬ሻ
݉Ǥ ܽଵǡ௑ ሺ‫ݐ‬ሻ
ܽଵǡ௑
ȟܽଵ ሺ‫ݐ‬ሻ
Z component
݉ଵ
݉ଵ ൌ ݉ଶ ൌ ݉ଷ ൌ ݉
ଷ
ଷ
݉ ෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ
௜ୀଵ
X component
ଷ
݉ ෍ ȟܽ௜ ሺ‫ݐ‬ሻ ݉ ෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ
௜ୀଵ
௜ୀଵ
(a)
Figure 9-34
X component
(b)
Two cases of excitation on the isolated cantilever structure
(a) X excitation
(b) X + Z excitation
ܽ௜ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ ൅ ȟܽ௜ ሺ‫ݐ‬ሻ
(9.2-8)
where ȟܽ௜ ሺ‫ݐ‬ሻ is the difference of horizontal acceleration of mass ݉௜ during X + Z
excitation compared to X excitation. ȟܽ௜ ሺ‫ݐ‬ሻ can be positive or negative over the time
history response.
The horizontal inertia force at all masses due to the horizontal acceleration is
presented to the left side of the cantilever in Figure 9-34. Applying D’Alembert’s
principle to the horizontal direction, the horizontal base reaction from the 2 cases can be
computed from the horizontal inertia forces:
344
ଷ
ܴ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ݉ ෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ
(9.2-9)
௜ୀଵ
ଷ
ଷ
ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݉ ෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ ൅ ݉ ෍ ȟܽ௜ ሺ‫ݐ‬ሻ
௜ୀଵ
(9.2-10)
௜ୀଵ
where ܴ௑ǡ௑ ሺ‫ݐ‬ሻ and ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ are horizontal base reactions during X excitation and X + Z
excitation, respectively. These reactions can also be computed from the vertical reaction
(ܴ௓ǡ௑ ሺ‫ݐ‬ሻǡ ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻሻand the normalized horizontal force (݂௑ ሺ‫ݐ‬ሻǡ ݂௑௓ ሺ‫ݐ‬ሻ) as follows:
ܴ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ݂௑ ሺ‫ݐ‬ሻǤ ܴ௓ǡ௑ ሺ‫ݐ‬ሻ
(9.2-11)
ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݂௑௓ ሺ‫ݐ‬ሻǤ ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻ
(9.2-12)
The vertical reactions ܴ௓ǡ௑ ሺ‫ݐ‬ሻ and ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻ can be computed based on the
equilibrium in the vertical direction of the 2 cases:
ܴ௓ǡ௑ ሺ‫ݐ‬ሻ ൌ ͵݉݃
(9.2-13)
ܴ௓ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ͵݉൫݃ ൅ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻ൯
(9.2-14)
where ݃ is the acceleration due to gravity and ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻ is the vertical acceleration of all
masses ݉௜ during X + Z excitation, which equals the vertical ground acceleration since
the isolator and the cantilever are rigid in the vertical direction.
Substituting Equations (9.2-13) and (9.2-14) into Equation (9.2-11) and (9.2-12)
and applying the simplifying assumption that the displacements, hence the normalized
forces, between the 2 cases are the same:
345
ܴ௑ǡ௑ ሺ‫ݐ‬ሻ ൌ ݂ሺ‫ݐ‬ሻǤ͵݉݃
(9.2-15)
ܴ௑ǡ௑௓ ሺ‫ݐ‬ሻ ൌ ݂ሺ‫ݐ‬ሻǤ͵݉ ቀ݃ ൅ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻቁ
(9.2-16)
where ݂ሺ‫ݐ‬ሻ is the normalized force of the 2 cases.
Combining Equation (9.2-9) and Equation (9.2-15):
ଷ
෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ ൌ ͵݃Ǥ ݂ሺ‫ݐ‬ሻ
(9.2-17)
௜ୀଵ
Combining Equation (9.2-10) and Equation (9.2-16):
ଷ
ଷ
෍ ܽ௜ǡ௑ ሺ‫ݐ‬ሻ ൅ ෍ ȟܽ௜ ሺ‫ݐ‬ሻ ൌ ͵ ቀ݃ ൅ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻቁ Ǥ ݂ሺ‫ݐ‬ሻ
௜ୀଵ
(9.2-18)
௜ୀଵ
Subtracting Equation (9.2-17) from Equation (9.2-18):
ଷ
෍ ȟܽ௜ ሺ‫ݐ‬ሻ ൌ ͵ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻǤ ݂ሺ‫ݐ‬ሻ
(9.2-19)
௜ୀଵ
or:
തതതതത
ȟܽ௑ ሺ‫ݐ‬ሻ ൌ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻǤ ݂ሺ‫ݐ‬ሻ
(9.2-20)
ଵ
where തതതതത
ȟܽ௑ ሺ‫ݐ‬ሻ ൌ ଷ σଷ௜ୀଵ ȟܽ௜ ሺ‫ݐ‬ሻ is the average increase of the horizontal acceleration of all
masses of the cantilever structure in X + Z excitation compare to X excitation. Equation
(9.2-20) is similar to Equation (9.2-7) and implies that the average horizontal acceleration
amplification transferred from vertical acceleration equals the product of vertical
346
acceleration and normalized force. Suppose, at any time ‫ݐ‬଴ , the increasing horizontal
acceleration differs from mass to mass, say ȟܽଷ ሺ‫ݐ‬଴ ሻ ൌ ʹȟܽଶ ሺ‫ݐ‬଴ ሻ ൌ ͵ȟܽଵ ሺ‫ݐ‬଴ ሻ, Equation
(9.2-19) becomes:
ͳ ͳ
൬ ൅ ൅ ͳ൰ ȟܽଷ ሺ‫ݐ‬଴ ሻ ൌ ͵ܽ௓ǡ௑௓ ሺ‫ݐ‬଴ ሻǤ ݂ሺ‫ݐ‬଴ ሻ
͵ ʹ
(9.2-19)
or:
ȟܽଷ ሺ‫ݐ‬଴ ሻ ൌ
ͳͺ
ሺ‫ ݐ‬ሻǤ ݂ሺ‫ݐ‬଴ ሻ ൐ ܽ௓ǡ௑௓ ሺ‫ݐ‬଴ ሻǤ ݂ሺ‫ݐ‬଴ ሻ
ܽ
ͳͳ ௓ǡ௑௓ ଴
(9.2-20)
In other words, the amplification of horizontal acceleration is minimum in the rigid
superstructure, where the horizontal acceleration is identical everywhere. However, in a
flexible structure, the peak amplification of horizontal acceleration is always larger than
the average value estimated by Equation (9.2-20).
The general form of Equation (9.2-19) when ݉ଵ ് ݉ଶ ് ݉ଷ is:
ଷ
ଷ
෍ ݉௜ Ǥ ȟܽ௜ ሺ‫ݐ‬ሻ ൌ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻǤ ݂ሺ‫ݐ‬ሻ ෍ ݉௜
௜ୀଵ
(9.2-21)
௜ୀଵ
Numerical Example 9.2
This example demonstrates that the theoretical amplification of horizontal
acceleration assuming a rigid structure is a lower bound to the actual amplification in a
flexible structure. The model used in this numerical example (Figure 9-35) is similar to
the one in Numerical Example 9.1 except that the superstructure is now a cantilever shear
structure (rotation of masses is restrained) instead of a rigid structure. The shear stiffness
݄ ൌ ͷ ൈ ͵݉
347
݉଺ ൌ ͳʹͲ‫ݐ‬
݇ହ ൌ ͸ͳǤͳ‫ܰܯ‬Ȁ݉
݉ହ ൌ ͺͲǤͳ‫ݐ‬
݇ସ ൌ ͹͸Ǥʹ‫ܰܯ‬Ȁ݉
݉ସ ൌ ͺͳǤͳ‫ݐ‬
݇ଷ ൌ ͻ͵Ǥ͵‫ܰܯ‬Ȁ݉
݉ଷ ൌ ͺͳǤ͵‫ݐ‬
݇ଶ ൌ ͳͲͷ‫ܰܯ‬Ȁ݉
݉ଶ ൌ ͺʹǤ͵‫ݐ‬
݇ଵ ൌ ͳ͵ͳ‫ܰܯ‬Ȁ݉
݉ଵ ൌ ͺ͹Ǥͷ‫ݐ‬
݂
݇ଵ ൌ
ͲǤʹ͹
ͲǤʹͳ
ͲǤͲͺ
ͲǤͲʹ ͳ
ͳ
ͳ
ǡ ݇ ൌ
ͺͶܿ݉ ଶ ͹͹Ͳܿ݉
݇ଵ
݇ଵ
ͳ
݇ଶ
ͷ
ͳͲͺ
ͳ
ͳͳ͵
‫ݑ‬ሺܿ݉ሻ
(b)
(a)
Figure 9-35
System for Numerical Example 9.2
(a) System configuration
(b) Normalized hysteresis loop of the isolators
of the cantilever is taken from the story stiffness in the X direction of the 5-story
specimen model based on pushover analysis. Eigenvalue analysis of this cantilever model
when the base is fixed predicts that the natural periods of the superstructure are ͲǤ͸ͺͲ•,
ͲǤʹͶͻ•, ͲǤͳͷ͵•, ͲǤͳͳ͸• and ͲǤͲͻ͸•. The model was subjected to the two cases of
excitation as described in the Numerical Example 9.1.
The TPB responses in this example are plotted in Figure 9-36 and the horizontal
acceleration at the roof for the 2 excitations are plotted in Figure 9-37. In general, the
hysteresis shear responses of the bearing are similar to those from the previous example.
Participation of higher modes in responses can be seen in the horizontal acceleration
response at the roof for X excitation (Figure 9-37 (a)). The horizontal acceleration in X +
Z excitation is observed to be 3.9 times the horizontal acceleration in X excitation (it was
1.85 in the previous example). However, as mentioned before, this amplification factor
348
reflects the difference in the peaks and not the actual amplification as a function of time
due to vertical-lateral coupling effect. In Figure 9-37 (b), the estimated average horizontal
acceleration difference (the Estimated line) computed from Equation (9.2-20) is far below
the actual acceleration “transferred” from the vertical acceleration (the Computed line),
hence demonstrating significant additional amplification due to structural flexibility.
X and
+ Z excitation
X excitation only
0.2
1000
Norm. force, fx
Force, Fx (kN)
2000
0
-1000
-2000
-0.75 -0.5
-0.25
0
0.25 0.5
Displacement, u (m)
0.75
0.1
0
-0.1
-0.2
-0.75
-0.5
X
-0.25
0
0.25 0.5
Displacement, u (m)
Figure 9-36
Hysteresis loops of isolation system in Numerical Example 9.2
(a) Actual hysteresis loop, (b) Normalized hysteresis loop
1 (a)
Acc., ax (g)
X
(b)
(a)
0.75
0.261
0
-1
X and
+ Z excitation
X excitation only
-1.02
Diff. acc., Δ ax (g)
-2
Computed
Estimated
1 (b)
0.5
0
-0.5
-1
0
Figure 9-37
2
4
6
8
10
12
Time, t (s)
14
16
18
20
22
Horizontal acceleration of the isolated structure in Numerical Example 9.2
(a) Horizontal acceleration, (b) Difference in horizontal acceleration
349
ͻǤʹǤ͵ ‡‡”ƒŽ͵ Ž‡š‹„Ž‡–”— –—”‡•
The approach used for developing equations to estimate the horizontal
acceleration response due to the vertical acceleration of the superstructure in the rigid
superstructure and cantilever superstructure is extended to derive a general equation for
the 3D flexible structures. The equation for increasing acceleration in a specific
horizontal direction (X) is derived as follows.
The total vertical reaction in the horizontal excitation case is
ܴ௓ǡ௑௒ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ݃ ൅ න ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉
(9.2-22)
௠
The total vertical reaction in the 3D excitation case is:
ܴ௓ǡଷ஽ ሺ‫ݐ‬ሻ ൌ ݉Ǥ ݃ ൅ න ܽ௓ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉
(9.2-23)
௠
In Equation (9.2-22), ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ is the vertical inertia force of an
infinitesimal mass ݀݉ with vertical acceleration ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ in the horizontal
excitation case. The acceleration ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ varies through space and time. The
term ܽ௓ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ in Equation (9.2-23) has the same meaning in the 3D excitation
case.
The total horizontal reaction in the 2 cases:
350
ܴ௑ǡ௑௒ ሺ‫ݐ‬ሻ ൌ ቌ݉Ǥ ݃ ൅ න ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ቍ Ǥ ݂௑ ሺ‫ݐ‬ሻ
(9.2-24)
௠
ܴ௑ǡଷ஽ ሺ‫ݐ‬ሻ ൌ ቌ݉Ǥ ݃ ൅ න ܽ௓ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ቍ Ǥ ݂௑ ሺ‫ݐ‬ሻ
(9.2-25)
௠
Applying D'Alembert's principle to the X-direction for the 2 cases:
න ܽ௑ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻǤ ݀݉ ൌ ቌ݉Ǥ ݃ ൅ න ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ቍ Ǥ ݂௑ ሺ‫ݐ‬ሻ
௠
(9.2-26)
௠
න ܽ௑ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻǤ ݀݉ ൌ ቌ݉Ǥ ݃ ൅ න ܽ௓ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉ቍ Ǥ ݂௑ ሺ‫ݐ‬ሻ (9.2-27)
௠
௠
In Equation (9.2-26), ܽ௑ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻǤ ݀݉ is the inertia force in the X-direction of
an infinitesimal mass ݀݉ with horizontal acceleration ܽ௑ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ. The acceleration
ܽ௑ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ varies through space and time. The term ܽ௑ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻǤ ݀݉ in
Equation (9.2-27) has the same meaning.
Subtracting Equation (9.2-27) and Equation (9.2-26):
න ቀܽ௑ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ െ ܽ௑ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻቁ Ǥ ݀݉
௠
(9.2-28)
ൌ ݂ሺ‫ݐ‬ሻǤ න ቀܽ௓ǡଷ஽ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ െ ܽ௓ǡ௑௒ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻቁ ݀݉
௠
Or:
351
න ȟܽ௑ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻǤ ݀݉ ൌ ݂ሺ‫ݐ‬ሻǤ න ȟܽ௓ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉
௠
(9.2-29)
௠
In case the superstructure is a multistory building with mass lumped to floors and
each floor is a rigid diaphragm, Equation (9.2-29) can be rewritten as:
௡௨௠௕௘௥௢௙௙௟௢௢௥௦
෍
ȟܽ௑௜ ሺ‫ݐ‬ሻǤ ݉௜ ൌ ݂ሺ‫ݐ‬ሻǤ න ȟܽ௓ ሺܺǡ ܻǡ ܼǡ ‫ݐ‬ሻ݀݉
௜ୀଵ
௠
(9.2-30)
where ȟܽ௑௜ is the difference in horizontal acceleration at floor ݅ between the 3D
excitation case and the horizontal case; ݉௜ is the total mass of floor ݅.
The effect of the vertical excitation on the horizontal response of the isolated
structures using friction bearings can be qualitatively explained as follows:
o The vertical excitation excites the vertical response of the isolated
structure.
o The vertical response causes the fluctuation in the vertical reaction.
o The fluctuation in the vertical reaction changes the horizontal reaction and
stiffness due to the coupling behavior of friction bearings.
o The changing of the horizontal reaction and stiffness in turn excites the
horizontal response of the isolated structure.
This qualitative explanation suggests that the frequency component of the vertical
response may have significant influence on the horizontal response of the isolated
structure. This effect is investigated in the next section.
352
ͻǤʹǤͶ ˆˆ‡ –‘ˆ ”‡“—‡ ›‘ˆ–Š‡‡”–‹ ƒŽ‡•’‘•‡‘–Š‡ ‘”‹œ‘–ƒŽ‡•’‘•‡
‘ˆ–Š‡ •‘Žƒ–‡†–”— –—”‡•
The isolated cantilever structure used in Numerical Example 9.2 (Figure 9-33) is
used for investigating the effect of the frequency of the vertical response on the
horizontal response of the isolated structure. The horizontal excitation is identical to the
excitation in Numerical Examples 9.1 and 9.2. The vertical excitation is a sine wave with
peak acceleration of ͲǤͺ‰, and the period ranging from ͲǤͲͷ• to ͳǤͲ•. As mentioned
earlier, the vertical-horizontal coupling in friction bearings is driven by the fluctuation in
the vertical force. In this sense, varying of the period of the vertical excitation
demonstrates the influence of the vertical force fluctuation period in isolated structures
with different vertical periods.
For further investigation, the eigenvalue analysis of the isolated structure was
carried out. The first slope stiffness of the isolator (equivalent to a period of ͳǤͺͶ•) was
used for the isolator in the eigenvalue analysis. The frequencies of the higher modes
(above the first mode) are not much affected by the selected value of the isolator stiffness
for modal analysis. The mode shapes and natural periods of the 6 modes of the isolated
structure are shown in Figure 9-38. The envelope shapes of these modes are also
developed (Figure 9-39) for future use.
Peak horizontal acceleration at every mass of the cantilever is plotted against the
period of the vertical excitation in Figure 9-40. All masses are named Mass 1, 2, .., 6
from the bottom to the top of the cantilever. In general, local peaks are observed where
the period of the vertical excitation is close to the natural period of a structural mode,
353
Mode 1, T = 1.926 s
Mode 2, T = 0.392 s
Mode 3, T = 0.203 s
Mode 4, T = 0.139 s
Mode 5, T = 0.11 s
Mode 6, T = 0.092 s
Figure 9-38
Mode shapes of the isolated cantilever
Mode 1, T = 1.926 s
Mode 2, T = 0.392 s
Mode 3, T = 0.203 s
Mode 4, T = 0.139 s
Mode 5, T = 0.11 s
Mode 6, T = 0.092 s
Figure 9-39
Envelope mode shapes of the isolated cantilever
except at the first natural mode, or the second mode of the isolated structure, which is
partly explained later. In other words, when the period of the vertical excitation matches
the period of a horizontal natural mode, it excites that mode and amplifies the horizontal
acceleration. This conclusion is strengthened by observing the distribution of peak
horizontal acceleration at different vertical excitation period (Figure 9-41). This figure
Mode 2
Mode 3
Mass 1
Mass 2
Mass 3
Mass 4
Mass 5
Mass 6
1.5
1
0.5
0.1
0.3
0.4
0.5
0.6
0.7
Period of vertical excitation, T (s)
0.8
0.9
1
Peak horizontal acceleration at different periods of the vertical excitation
Mode 6
Mode 5
Mode 4
Figure 9-40
0.2
Mode 2
0
0
Mode 3
Peak horizontal acc., Amax (g)
2
Mode 6
Mode 5
Mode 4
354
6
Mass
5
4
3
2
1
0
0.1
Figure 9-41
0.2
0.3
0.4
0.5
0.6
0.7
Period of vertical excitation, T (s)
0.8
0.9
1
Peak acceleration distribution at different periods of vertical excitation
shows that when the period of vertical excitation is close to the period of a natural mode,
the distribution of peak horizontal acceleration is proportional to the envelope mode
shape of that mode (Figure 9-39). As the period of the vertical excitation becomes
significantly larger than the period of the first structural mode, the distribution of the
355
peak horizontal acceleration matches with the shape of the first mode (the isolator mode),
and the peak horizontal acceleration becomes small (Figure 9-40). This observation
suggests that the very large vertical flexibility of the isolated structure reduces the
vertical-horizontal coupling effect. This case is likely more applicable to bridges than to
buildings. Vertical-horizontal coupling effect also becomes small when the period of the
vertical excitation is very small, except for the horizontal acceleration response at base,
where the structure is in contact with the isolator. At the short period (or high frequency)
vertical excitation, the change in horizontal reaction due to fluctuation of the vertical
reaction transfers directly to the mass right above the isolator.
As mentioned earlier, the local peak acceleration does not occur at the period of
the first structural mode but at a neighboring period (Figure 9-40). The physical meaning
behind this phenomenon was not known, but it can be understood mathematically.
Consider Equation (9.2-21), which is extended to 6 masses as follows:
଺
଺
෍ ݉௜ Ǥ ȟܽ௜ ሺ‫ݐ‬ሻ ൌ ܽ௓ǡ௑௓ ሺ‫ݐ‬ሻǤ ݂ሺ‫ݐ‬ሻ ෍ ݉௜
(9.2-31)
௜ୀଵ
௜ୀଵ
At a certain time step ‫ ݐ‬ൌ ‫ݐ‬଴ , the right hand side of the equation is a constant and the
amplification accelerations ȟܽ௜ ሺ‫ݐ‬଴ ሻ depend on the distribution of ȟܽ௜ ሺ‫ݐ‬଴ ሻ throughout the
structure. Let ȟܽ௜ ሺ‫ݐ‬଴ ሻ be normalized by ȟܽ଺ ሺ‫ݐ‬଴ ሻ, so that Equation (9.2-31) becomes:
଺
ȟܽ଺ ሺ‫ݐ‬଴ ሻ ෍ ݉௜ Ǥ
௜ୀଵ
ȟܽ௜ ሺ‫ݐ‬଴ ሻ
ൌ ‫ݐݏ݊݋ܥ‬
ȟܽ଺ ሺ‫ݐ‬଴ ሻ
(9.2-32)
356
From this equation, ȟܽ଺ ሺ‫ݐ‬଴ ሻ, hence ȟܽ௜ ሺ‫ݐ‬଴ ሻ, is largest when the distribution of ȟܽ௜ ሺ‫ݐ‬ሻ
minimizes the absolute value of the summation σ଺௜ୀଵ ݉௜ Ǥ
୼௔೔ ሺ௧బ ሻ
.
୼௔ల ሺ௧బ ሻ
When the period of the vertical excitation is close to the period of a natural mode,
it mainly excites this mode. However, the participation of other modes still exists. If the
combination of these modes minimizes the absolute value of the summation
σ଺௜ୀଵ ݉௜ Ǥ
୼௔೔ ሺ௧బ ሻ
, the horizontal acceleration response is maximized. This means that the
୼௔ల ሺ௧బ ሻ
local peak of acceleration does not necessarily occur right at a modal period, but can
occur at a neighboring period. This phenomenon affects the peaks near all modal periods,
but is the most obvious near the 1st structural mode because of the period gap between
modes. Understanding of the dynamic response of multi degree of freedom systems with
varying stiffness (at base) is required to fully understand the physical meaning of this
phenomenon.
The influence of vertical excitation on the horizontal response explains the
existence of the high frequency (or short period) components in the horizontal
acceleration response of the test specimen subjected to 3D excitation (see Section 5.5.1
and Section 8.3). It also explains the distribution patterns of the peak floor acceleration
(Figure 5-28) and peak story drift (Figure 5-30). As mentioned before, the first vertical
mode of the isolated structure has a frequency of ͸Ǥͻ œ (period of ͲǤͳͶͷ•). This mode is
dominated by the vertical roof vibration, where the additional steel weights were
installed. Beside this mode, other vertical modes dominated by the vertical vibration at
other floors (at around ͲǤͳ•) also exists, as can be observed from the frequency
357
component of the total vertical reaction shown later, but the vertical mode at ͲǤͳͶͷ• is
seen to be the strongest among them. When the isolated structure is subjected to a vertical
excitation, the first vertical mode is excited and causes oscillation of the total vertical
isolator force dominated by a period component of about ͲǤͳͶͷ• (Figure 9-42 and 9-43).
This fluctuation in the total vertical reaction causes the horizontal stiffness of the
isolation system and base shear to oscillate. This short period component in the base
shear and horizontal stiffness excites or amplifies horizontal modes with period close to
ͲǤͳͶͷ•. The mode shapes and periods of horizontal modes having periods nearest to
ͲǤͳͶͷ• are plotted in Figure 9-44. The distribution of the peak acceleration and the peak
story drift caused by these modes are in Figure 9-45. These distributions match the peak
floor acceleration profiles (Figure 5-28) and the peak drift profiles (Figure 5-30) from the
test data. The floor spectra of the isolated base structure subjected to 70LGP (Figure 946) confirms this explanation. The floor spectra in both directions at floors 1, 3, 4 and
roof have a peak at a period around ͲǤͳ͹•. At floors 2 and 5, the spectral acceleration at
this period is small. Besides the peak at period around ͲǤͳ͹•, small local peak at around
ͲǤͳͳ• is also observed in both directions. This local peak corresponds with the local peak
at around ͲǤͳͳ• in the Fourier spectrum of the vertical reaction (Figure 9-43) caused by
the vertical mode controlled by vibration at lower floors.
358
8000
6000
4000
2000
0
0
2
Figure 9-42
4
6
8
10
Time, t (s)
12
14
16
18
Total vertical reaction of the isolated structure subjected to 70LGP
200
Spectral amplitude
Vertical reaction, RZ (kN)
10000
T=0.145 s
150
100
50
0 -2
10
Figure 9-43
-1
10
0
10
Period, T (s)
1
10
Fourier spectrum of the dynamic vertical reaction of the isolated structure
subjected to 70LGP
20
359
Mode 7: fT==5.63
Hzs
0.178
Y-direction
Mode 8: T
f ==5.87
0.170Hzs
X-direction
Figure 9-44
Modes of the isolated base structure model with periods close to the period
of the first vertical mode
Figure 9-45
Distribution of the peak acceleration and the peak story drift of modes with
period close to the frequency of the first vertical mode
(a) Peak acceleration (b) Peak story drift
360
2
2
X-Roof
Y-Roof
1
1
0
0
T=0.17s
T=0.17s
2
2
Spectral acceleration, S A (g)
X-Floor 5
Y-Floor 5
1
1
0
0
2
2
Y-Floor 4
X-Floor 4
1
1
0
0
2
2
Y-Floor 3
X-Floor 3
1
1
0
0
2
2
X-Floor 2
Y-Floor 2
1
1
0
0
2
4
X-Base
Y-Base
1
0
-2
10
Figure 9-46
2
10
-1
10
Period, T (s)
0
10
1
0
-2
10
10
-1
10
Period, T (s)
0
10
1
5% damped floor spectra of the isolated structure subjected to 70LGP
Figures 9-47 and 9-48 respectively show total vertical reaction and its Fourier
spectrum of the isolated base structure subjected to 88RRS. The dominating period of the
vertical reaction is slightly longer than the period of the first vertical mode (ͲǤͳͶͷ•),
361
which may be resulted from the nonlinear behavior of the slabs due to strong vertical
excitation. The floor spectra in both directions in 88RRS (Figure 9-49) also have a peak
at a period around ͲǤͳ͹•, similar to the floor spectra in 70LGP (Figure 9-46). However,
the peak at period around ͲǤͳͳ• is much stronger in 88RRS than in 70LGP. Local peaks
at shorter period (around ͲǤͲͷͷ•) are also observed in the floor spectra subjected to
88RRS. The vertical impact and rocking of the specimen on the isolation system during
88RRS could trigger/amplify these short period components.
4
x 10
1.5
1
0.5
0
0
1
Figure 9-47
2
3
4
5
Time, t (s)
6
7
8
9
Total vertical reaction of the isolated structure subjected to 88RRS
350
300
Spectral amplitude
Vertical reaction, RZ (kN)
2
T=0.145 s
250
200
150
100
50
0 -2
10
Figure 9-48
-1
10
0
10
Period, T (s)
1
10
Fourier spectrum of the dynamic vertical reaction of the isolated structure
subjected to 88RRS
10
362
4
4
3D
88RRS
2D
88RRSXY
X-Roof
2
0
Y-Roof
2
0
T=0.17 s
T=0.17 s
4
4
Spectral acceleration, S A (g)
X-Floor 5
Y-Floor 5
2
2
0
0
4
4
X-Floor 4
Y-Floor 4
2
2
0
0
4
4
Y-Floor 3
X-Floor 3
2
2
0
-2
-1
10
10
4
X-Floor 2
10
0
10
1
0
4
Y-Floor 2
2
2
0
0
4
4
X-Base
Y-Base
2
0
-2
10
Figure 9-49
2
10
-1
10
Period, T (s)
0
10
1
0
-2
10
10
-1
10
Period, T (s)
0
10
5% damped floor spectra of the isolated structure subjected to 88RRS and
88RRSXY
1
363
9.3
Effect of the Roof Steel Weights on the Horizontal Response of the Tested
Specimen
The previous section indicates that the frequency content of the vertical vibration
of isolated structures influences their lateral response. This means that the steel weights at
roof simulating additional weights (Section 2.3) may affect the horizontal response of the
tested specimen in the isolated base configuration. This effect is investigated here using
the calibrated analytical model.
Modal analysis of the isolated base model shows that when the steel weights at
roof is removed, the first vertical mode has period of ͲǤͳͳ• (Figure 9-50 (a)), instead of
ͲǤͳͶͷ•, when the steel weight was included. The periods and mode shapes of the 2nd and
3rd horizontal structural modes in the Y-direction are also plotted in Figure 9-50 for later
use. The change in the period of the first vertical mode changes the dominant period
content of the total vertical reaction. Figure 9-51 shows total vertical reaction of the
isolated base models with and without steel weights at roof subjected to
Mode 10: T = 0.110 s
Mode 7: T = 0.167 s
Mode 12: T= 0.100 s
(a)
(b)
(c)
Figure 9-50
Selected mode shapes of the isolated base model without the roof weights
(a) The first vertical mode
(b) The second horizontal structure mode in the Y-direction
(c) The third horizontal structure mode in the Y-direction
364
Vertical reaction (kN)
10000
With Roof Weight
No Roof Weight
8000
6000
4000
2000
0
0
2
Figure 9-51
4
6
Time, t (s)
8
10
12
Total vertical reactions of the isolated base models with and without roof
weights subjected to 70LGP
70LGP. The amplitudes of the oscillation of these reactions are close, but the spectral
amplitudes of Fourier spectra of these reactions (Figure 9-52) shows that the dominant
period of the vertical reaction shifts from ͲǤͳͷͲ• in the With Roof Weight case to
ͲǤͳʹͲ• in the No Roof Weight case. These dominant periods are slightly longer than the
period of the 1st vertical modes of both cases, which could result from the nonlinear
5
2.5
x 10
No Roof Weight
With Roof Weight
T = 0.120 s
Spectral amplitude
2
T = 0.150 s
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
Period, T (s)
Figure 9-52
Fourier spectra of the dynamic vertical reaction of the isolated base models
with and without roof weights subjected to 70LGP
365
behavior of the composite beams. The period of ͲǤͳʹ• in the No Roof Weight model is
between the periods of the 2nd and 3rd horizontal structural modes (Figure 9-50), so that
both of them can be excited. Note that when the dominant vertical period falls between
the periods of the two horizontal modes, the horizontal floor acceleration may
significantly reduce (Figure 9-40). This reduction occurs to the tested specimen. As can
be seen from the 5% damped floor spectra of the 2 models subjected to 70LGP in Figure
9-53, the peak floor acceleration (the spectral acceleration at very small period, ͳͲିଶ •) is
smaller in No Roof Weight model than in With Roof Weight model. Especially, the
spectral acceleration at the period of the 2nd horizontal structural mode (around ͲǤͳ͹•) is
much smaller in No Roof Weight model than in With Roof Weight model, which
confirms that the shifting of the dominant period of the vertical reaction from ͲǤͳͷ• to
ͲǤͳʹ• significantly reduces the participation of the 2nd horizontal structural mode. To
compare the acceleration amplification due to 3D excitation between the two models, the
ratio ܵ஺ǡଷ஽ Ȁܵ஺ǡ௑௒ between the spectral floor accelerations subjected to 3D and XY
excitations from the two models were computed and plotted in Figure 9-54. The
comparison shows that the amplification in With Roof Weight model is generally larger
than the amplification in No Roof Weight model. In other words, the horizontal-vertical
coupling effect subjected to 70LGP is stronger in the model with steel weight at roof.
366
2
2
With Roof Weight
X-Roof
Y-Roof
No Roof Weight
1
1
0
0
2
X-Floor 5
Y-Floor 5
1
1
0
0
A
Spectral acceleration, S (g)
2
2
X-Floor 4
2 Y-Floor 4
1
1
0
0
2
2
X-Floor 3
Y-Floor 3
1
1
0
0
2
2
X-Floor 2
Y-Floor 2
1
1
0
0
2
3
Y-Base
X-Base
2
1
1
0 -2
10
Figure 9-53
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
10
5% damped floor spectra of the isolated base model subjected to 70LGP:
with vs. without roof weights
1
367
4
With Roof Weight
No Roof Weight
X-Roof
4
Y-Roof
2
2
0
0
4
4
Spectral acceleration ratio, S
A,3D
/S
A,XY
X-Floor 5
Y-Floor 5
2
2
0
0
4
4 Y-Floor 4
X-Floor 4
2
2
0
0
4
4
X-Floor 3
Y-Floor 3
2
2
0
0
4
4 Y-Floor 2
X-Floor 2
2
2
0
0
4
X-Base
4 Y-Base
2
0 -2
10
Figure 9-54
2
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
10
1
5% damped floor spectra ratio of the isolated base model subjected to 70LGP:
with vs. without roof weights
368
The effect of the steel weights at roof on the response of the isolated base model
to 88RRS was also investigated. Similar to the total vertical reaction subjected to 70LGP,
the total vertical reaction subjected to 88RRS of the two models with and without roof
weights is similar in the amplitude (Figure 9-55) but the dominant period shifts from
ͲǤͳ͸ʹ• in the With Roof Weight model to ͲǤͳʹͷ• in the No Roof Weight model (Figure
9-56). However, in contrast to the response to 70LGP, in the response to 88RRS, the steel
weights at roof do not necessarily increase the response of the isolated specimen since the
peak horizontal floor acceleration (spectral acceleration at very small period, ͳͲିଶ •) at
some floors is larger in the No Roof Weight model than in the With Roof Weight model
(Figure 9-57). The acceleration amplification due to 3D excitation, compared to 2D
excitation, is also larger in the No Roof Weight model than in the With Roof Weight
model at some floors (Figure 9-58). The floor spectra trends for With Roof Weight model
compared to No Roof Weight model described earlier for the response to 70LGP also do
not apply to the response to 88RRS (Figure 9-57), i.e. the local peak of the spectral
acceleration
corresponding
to
2nd
the
horizontal
structural
Vertical reaction, RZ (kN)
15000
mode
is
not
With Roof Weight
No Roof Weight
10000
5000
0
0
Figure 9-55
1
2
3
4
5
Time, t (s)
6
7
8
9
10
Total vertical reactions of the isolated base structures with and without roof
weights subjected to 88RRS
369
5
4
x 10
With Roof Weight
No Roof Weight
3.5
T = 0.125 s
Spectral amplitude
3
T = 0.162 s
2.5
2
1.5
1
0.5
0 -2
10
-1
0
10
10
1
10
Period, T (s)
Figure 9-56
Fourier spectrum of the dynamic vertical reaction of the isolated base
structures with and without roof weights subjected to 88RRS
necessarily smaller in No Roof Weight model than in With Roof Weight model,
especially in the X-direction. The period of this local peak is also shorter in No Roof
Weight model than in With Roof Weight model. The inconsistency in the response to
88RRS compared to the response to 70LGP may come from the rocking of the isolated
structure on the isolation system (in response to 88RRS), the nonlinear behavior of the
composite beams in response to 88RRS (with very strong vertical component), or the
sensitivity of the response to the period of the vertical reaction when this period falls
between the periods of the 2 horizontal structural modes (Figure 9-40). The inconsistency
may also come from an unknown source, due to very strong vertical excitation.
370
With Roof Weight
4 Y-Roof
No Roof Weight
2
2
0
0
4 X-Floor 5
4 Y-Floor 5
2
2
0
0
4 X-Floor 4
4 Y-Floor 4
2
2
0
0
4 X-Floor 3
4 Y-Floor 3
2
2
0
0
4 X-Floor 2
4 Y-Floor 2
2
2
0
0
4 X-Base
4 Y-Base
2
2
A
Spectral acceleration, S (g)
4 X-Roof
0 -2
10
Figure 9-57
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
10
5% damped floor spectra of the isolated base model subjected to 88RRS:
with vs. without roof weights
1
371
20
20
10
10
0
0
20
20
Spectral acceleration ratio, SA,3D/SA,XY
X-Floor 5
Y-Floor 5
10
10
0
0
20
20
X-Floor 4
Y-Floor 4
10
10
0
0
20
With Roof Weight
No Roof Weight
Y-Roof
X-Roof
20
X-Floor 3
Y-Floor 3
10
10
0
0
20
20
X-Floor 2
Y-Floor 2
10
10
0
0
20
20
X-Base
Y-Base
10
0 -2
10
Figure 9-58
10
-1
0
10
10
Period, T (s)
10
1
0 -2
10
-1
0
10
10
Period, T (s)
10
1
5% damped floor spectra ratio of the isolated base model subjected to 88RRs:
with vs. without roof weights
Chapter 10
‘ Ž—•‹‘•ƒ†‡ ‘‡†ƒ–‹‘•
10.1
Conclusions
Experimental and analytical seismic response of a full-scale 5-story steel moment
frame, both in isolated base with triple pendulum bearings (TPB) and fixed base
configurations, subjected to 3D earthquake excitations was studied in this dissertation.
Key results and conclusions are summarized as follows.
ͳͲǤͳǤͳ
‡•–‡•—Ž–•
While the effectiveness of base isolation has been verified many times at reduced
scale, limited verification of realistic full scale structures under very large ground
motions has been performed to date. The full scale test result in this research program
served as a full scale proof of concept that the demand in the isolated base structure
(including floor acceleration, story drift and torsion drift) is significantly reduced
compared to the demand in the fixed base structure. Even though the structure was
asymmetric due to irregular stiffness and mass configurations, the torsion at base of the
isolated structure during the test was small and did not increase proportional to the peak
displacement. The peak story drift ratio of the isolated base structure easily met the
performance objective of ͲǤͷΨ for the maximum considered earthquake (MCE) for all
excitations. The drift objective was also met in the analysis of the revised analytical
model subjected to the selected earthquake motions scaled to represent different
earthquake levels (Service, DBE, MCE). However, the peak floor acceleration observed
373
during the test was larger than expected values and did not meet the design objective, of
ͲǤ͵ͷ‰ for the MCE. Further investigation of the test data revealed that the verticalhorizontal coupling effect caused the increase in acceleration.
From the sine wave characterization test data, a friction coefficient model that
accounts for the dependence of friction coefficient on velocity and vertical force was
derived and calibrated. In the calibrated data, the variation of friction coefficient with
vertical force was more significant than the variation of the friction coefficient with
velocity. This observation is valid for the range of vertical force of the bearings observed
in this full-scale test. The bearings were found to respond with a higher friction
coefficient than the design value, which caused the peak displacement of the isolation
system to be smaller than expected.
Since the bearing concave plates were large in curvature, corresponding to a
tangent isolation period of ͷǤͷ͹•, the residual displacement of the isolation system was
large following some motions. The maximum residual displacement during the test was
ͳͲǤͺ , which is about ͳͲΨ of the displacement limit and ͳͷΨ of the peak
displacement of the isolation system. All bearings uplifted at least once during the test.
However, there was no evidence that the local uplift altered the global response of the
isolation system and the isolated structure.
ͳͲǤͳǤʹ
ƒŽ›–‹ ƒŽ‘†‡Ž‹‰ƒ†‡”‹ˆ‹ ƒ–‹‘
A 3D TPB element was developed as part of this research. The horizontal
behavior of the element is based on the series combination of the bidirectional
374
elastic/perfectly plastic model and the circular gap model, which is an extension of the
1D series model developed by previous researchers. The vertical-horizontal coupling of
TPBs is accounted for by calculating the actual horizontal force and tangent stiffness to
be proportional to the instantaneous vertical force on the bearing. The friction coefficient
model for the element is a general model that accounts for the influence of velocity and
axial force on the friction coefficient. A numerical procedure for implementing the
element in a finite element program was described in detail and demonstrated by
implementation in OpenSees.
Besides the TPB element, an analytical model of the building specimen was also
developed and calibrated in OpenSees. In this 3D frame model, floor slabs were
represented through the composite beam sections and rigid diaphragm constraints.
Material nonlinearity and P-Delta effects were considered. Flexibility of beam to column
connections was also accounted for through a panel zone model. A combination of
Rayleigh damping to the superstructure and additional interstory viscous dampers was
used to simulate the energy dissipation. Mass and gravity loads were lumped to every
node of the model.
The significance of various modeling assumptions on the accuracy of analytically
predicted dynamic response was investigated for both fixed base and isolated base
models. The analytical model with bare beam sections (i.e. the composite effect from
slabs was not included) significantly overestimated the peak response while the analytical
model with composite effects modeled by fiber sections underestimated the peak
response compared to the test data. The analytical model with composite effects modeled
375
by resultant section moment curvature relations led to the best match to the test data. The
time history responses of the fixed base building were overestimated when the panel zone
connection behavior was not included, but the effect of panel zones on the response of the
isolated base model was small. The effect of the damping model on response of the
analytical models was also investigated. The result showed that stiffness proportional
damping is inappropriate to predict the response of the isolated base model subjected to
3D motions since the high frequency of response components, which were dominant in
some motions, were damped out.
Comparison indicated good agreement between the analytical response of both
isolated base and fixed base models and the full-scale test data, which verified the newly
developed 3D TPB element and the modeling process. The analytical model was used for
further investigations.
ͳͲǤͳǤ͵
‡”–‹ ƒŽǦŠ‘”‹œ‘–ƒŽ‘—’Ž‹‰‹‡•’‘•‡‘ˆ ‹š‡†Ǧ„ƒ•‡–”— –—”‡
ƒ† •‘Žƒ–‡†Ǧ„ƒ•‡–”— –—”‡™‹–Š•
The test data showed that the existence of the vertical component amplified the
horizontal response of both the fixed-base and isolated-base structures. The test data and
response of the analytical models were investigated to understand this vertical-horizontal
coupling effect. The sources and mechanism of this effect in the two configurations were
identified.
In the fixed base structure, the vertical horizontal coupling was mostly a result of
the vertically-horizontally coupled modes of the structure. In these modes, the horizontal
displacement is accompanied with the vertical displacement. When the structure was
376
subjected to a pure vertical excitation, these modes were excited and horizontal response
is observed. The contribution of nonlinear behavior to the vertical-horizontal coupling
effect was very small in the fixed base building, such that the coupled behavior could be
predicted by a linear modal analysis.
In the isolated-base structure, three main sources of the vertical-horizontal
coupling were identified: (1) the vertically-horizontally coupled modes (as described
above), (2) rocking of building on the isolators, and (3) vertical-horizontal coupling of
TPBs. The uneven vertical displacement of supports subjected to vertical excitation (due
to the uneven distributions of mass, stiffness and weight of both the isolation system and
isolated structure) caused the rocking of the isolated structure on the isolation system,
which contributed to the horizontal response of the isolated structure. Because of the
vertical-horizontal coupling of TPBs, the variation of the total vertical force when the
system was subjected to vertical excitation introduced a high frequency oscillation into
the base shear and horizontal stiffness of the isolation system. The base shear and
horizontal stiffness with high frequency variation served as horizontal input to the
isolated structure and amplified the horizontal response. The frequency of the total
vertical reaction, and hence the base shear, was dominated by the frequency of the
fundamental vertical mode. The horizontal modes with frequency close to the frequency
of the dominating vertical mode were amplified. The relative contribution of each source
to the vertical-horizontal coupling effect was not determined, but the numerical
investigation of a cantilever structure isolated by a TPB suggested that the vertical-
377
horizontal coupling behavior of the TPB was the most significant source of amplification
of horizontal accelerations.
Since the observed vertical-horizontal coupling effect is very strong in the
isolated base structure, it is recommended that the vertical component of ground motions
be included in analysis and design of the structures isolated by friction bearings.
Neglecting the vertical component of ground motions may lead to significant
underestimation of the superstructure horizontal accelerations.
10.2
Recommendations for Future Studies
This research program has led to meaningful contributions in understanding the
response characteristics of full-scale buildings isolated with TPBs as well as validating
and improving present modeling approaches. Further investigation of the following topics
is needed to make practical recommendation for design.
1. Response of isolation system and isolated structure when the displacement
limit of the isolation system is reached. Friction bearings can be theoretically
designed to accommodate the peak required displacement of an isolation
system subjected to any extreme earthquake motion. However, for a more
economical design, most isolator designs limit the displacement demand to
reduce the bearing size and isolation gap. In extreme earthquakes, the required
displacement may be larger than the displacement limit of the isolation system
and impact occurs. The response of both isolation systems and isolated
structures in this extreme condition has not yet been experimentally studied at
378
full scale level. This objective was initially included in this testing program,
but the test could not be accommodated due to safety concerns.
2. Effectiveness of the stiffening stages of TPBs in reducing peak displacement
or slowing down the movement of isolated structures before reaching the
displacement limit. A TPB can be designed to provide stiffening stages at the
end of its backbone curve. These stiffening stages may help slow down the
movement of isolated structure before reaching the displacement limit and
thus reduce the peak displacement. However, this effect needs to be
understood well to propose the optimum parameters for these stages.
3. Influence of initial displacement on response of isolation systems and isolated
structures. A friction isolation system may possess a residual displacement
after experiencing an earthquake, which may affect the response of the
isolation system in a future earthquake. In comparing the analytical
displacement history with zero initial displacement to the experimental
displacement history with an initial displacement, the two histories were offset
at the beginning but became aligned after a large pulse. On the other hand,
these displacement histories could be offset until the end of the record in a
smaller motion. These initial results suggest further investigation to
thoroughly understand the effect.
4. Effect of temperature on the variation of friction coefficient. The effect of
temperature on friction coefficient was not included in this study. As pointed
out, temperature does not appear to affect the response of the isolation system
at large displacements since the displacement trace is not overlapped. At small
379
displacements, however where sliding is limited to the inner slider, the effect
may be significant. Ignoring this effect may inaccurately predict response of
the isolation system and structure under frequent earthquakes, and lead to
inaccuracy in prediction of residual displacement.
5. Systematically estimating horizontal response due to 3D excitations. The
sources and mechanisms of the horizontal-vertical coupling effect were
investigated in this study. However, the contribution of each source and a
systematic way to account this coupling effect in design has not yet been
developed. Further investigation is needed to make recommendations for the
design process.
380
‡ˆ‡”‡ ‡•
American Institute of Steel Construction (AISC). Steel construction manual, 13rd edition.
2005.
American Society of Civil Engineers (ASCE). Minimum design loads for buildings and
other structures. ASCE 2005.
Becker TC, Mahin SA. Experimental and analytical study of the bi-directional behavior
of the triple friction pendulum isolator. Earthquake Engineering and
Structural Dynamics 2012; 41(3):355-373.
Bowden FP, Tabor D. The friction and lubrication of solids – part II. Oxford University
Press, London, Great Britain, 1964.
Calvi GM, Ceresa P, Casarotti C, Bolognini D, Auricchio F. Effects of axial force
variation in the seismic response of bridges isolated with friction
pendulum systems. Journal of Earthquake Engineering 2004; 8(1): 187224.
Charney FA, Downs WM. Modeling procedures for panel zone deformations in moment
resisting frames. ECCS/AISC workshop: Connections in Steel Structures
V, Amsterdam, Netherlands, 2004.
Chopra AK. Dynamics of structures, 3rd edition. Pearson Prentice Hall, New Jersey,
2007.
Comite Euro-interational du Beton (CEB). RC elements under cyclic loading, state of the
art report. Thomas Telford Publications, London, England, 1996.
Constantinou MC, Mokha A, Reinhorn A. Teflon bearings in base isolation. II:
Modeling. Journal of Structural Engineering (ASCE) 1990; 116(2): 455474.
381
Erduran E, Dao ND, Ryan KL. Comparative response assessment of minimally compliant
low-rise conventional and base-isolated steel frames. Earthquake
Engineering and Structural Dynamics 2011. 40(10): 1123-1141.
Eroz M, DesRoches R. Bridge seismic response as a function of the friction pendulum
system (FPS) modeling assumptions. Engineering Structures 30 (2008):
3204-3212.
Fenz DM, Constantinou MC. Spherical sliding isolation bearings with adaptive behavior:
Theory. Earthquake Engineering and Structural Dynamics 2008;
37(2):163-183.
Fenz DM, Constantinou MC. Modeling triple friction pendulum bearings for responsehistory analysis. Earthquake Spectra 2008; 24(4):1011-1028.
Fenz DM, Constantinou MC. Development, implementation and verification of dynamic
analysis models for multi-spherical sliding bearings. Technical Report
MCEER-08-0018, 2008.
Hyogo Earthquake Engineering Research Center. Design and construction drawing for
testing of value-added damped building. NIED 2008.
Kasai K, Ooki Y, Ishii M, Ozaki H, Ito H, Motoyui S, Hikino T, Sato E. Value-added 5story steel frame and its components: Part 1 – full scale damper tests and
analyses. The 14th World Conference on Earthquake Engineering 2008.
Kasai K, Ito H, Ooki Y, Hikino T, Kajiwara K, Motoyui S, Ozaki H, Ishii M. Full scale
shake table tests of 5-story steel building with various dampers. 7CUEE &
5ICEE Conference Proceedings 2010.
Kasai K. Personal communication. 2012
Kelly JM. Earthquake-Resistant Design with Rubber, 2nd Ed. Springer, London, Great
Britain, 1997.
382
Kent DC, Park R. Flexural members with confined concrete. Journal of the Structural
Division, Proceedings of the ASCE 1971; 97(ST7): 1969-1990.
Krawinkler H. Shear in beam-column joints in seismic design of steel frames.
Engineering Journal (AISC) 1978; 15(3): 82-91.
Lin BC, Tadjbakhsh I. Effect of vertical motion on friction-driven isolation systems.
Earthquake Engineering and Structural Dynamics 1986; 14: 609-622.
Mokha A, Constantinou MC, Reinhorn AM. Teflon bearing in aseismic base isolation:
experimental studies and mathematical modeling. Technical Report
NCEER-88-0038, National Center for Earthquake Engineering Research,
State University of New York at Buffalo, 1988.
Mokha A, Constantinou M, Reinhorn A. Teflon bearings in base isolation. I: Testing.
Journal of Structural Engineering (ASCE) 1990; 116(2): 438-454.
Morgan TA, Mahin SA. The use of innovative base isolation systems to achieve complex
seismic performance objectives. PEER-2011/06 2011.
Mosqueda G, Whittaker AS, Fenves GL. Characterization and modeling of friction
pendulum bearings subjected to multiple components of excitation.
Journal of Structural Engineering (ASCE) 2004; 130(3): 433-442.
Naeim F, Kelly JM. Design of seismic isolated structures: From theory to practice. John
Wiley & Sons, Inc. New York, USA, 1999.
Neuenhofer A, Filippou FC. Evaluation of nonlinear frame finite-element models.
Journal of Structural Engineering (ASCE) 1997; 123(7): 958-966.
OpenSees, http://opensees.berkeley.edu/
Otani S, Hiraishi H, Midorikawa M, Teshigawara M. New seismic provision in Japan.
American Concrete Institute 2002. 197: 87-104.
383
Pan P, Zamfirescu D, Nakashima M, Nakayasu N, Kashiwa H. Base-isolation design
practice in Japan: Introduction to the post-Kobe approach. Journal of
Earthquake Engineering 2005; 9(1): 147-171.
Panchal VR, Jangid RS, Soni DP, Mistry BB. Response of the double variable frequency
pendulum isolator under triaxial ground excitations. Journal of Earthquake
Engineering 2010, 14: 527-558.
Rabiei M, Khoshnoudian F. Response of multistory friction pendulum base-isolated
buildings including the vertical component of earthquakes. Canadian
Journal of Civil Engineering 2011, 38: 1045-1059.
Sasaki T, Sato E, Ryan K, Okazaki T, Mahin S, Kajiwara K. NEES/E-Defense baseisolation tests: Effectiveness of friction pendulum and lead-rubber
bearings systems. World Conference in Earthquake Engineering 2012.
Sato E, Furukawa S, Kakehi A, Nakashima M. Full-scale shaking table test for
examination of safety and functionality of base-isolated medical facilities.
Earthquake Engineering and Structural Dynamics 2011; 40: 1435-1453.
Sato K, Okazaki T. Personal communication. 2011.
Sayani, Prayag J. and Keri L. Ryan. Comparative evaluation of base-isolated and fixedbase buildings using a comprehensive response index. Journal of
Structural Engineering (ASCE) 2009, 135(6):698-707.
Scott BD, Park R, Priestley MJN. Stress-strain behavior of concrete confined by
overlapping hoops at low and high strain rates. ACI Journal Proceedings
1982; 79(1): 13-27.
Shakib H, Fuladgar A. Effect of vertical component of earthquake on the response of
pure-friction base-isolated asymmetric buildings. Engineering Structure 25
(2003): 1841-1850.
384
Shome N, Cornell CA, Bazzurro P, Carballo JE. Earthquakes, records, and nonlinear
responses. Earthquake Spectra 1998; 14(3): 469–500.
Simo J, Hughes T. Computational Inelasticity. Springer, New York, 1998.
Somerville P, Smith N, Punyamurthula S, Sun J. Development of ground motion time
histories for phase 2 of the FEMA/SAC steel project. Report No.
SAC/BD-97/04, 1997.
Stone WC, Yokel FY, Celebi M, Hanks T, Leyendecker EV. Engineering Aspect of the
September 19, 1985 Mexico Earthquake. NBS Building Science Series
165, National Breau of Standards, Washington DC, 1987.
Tsai CS, Lin YC, Su HC. Characterization and modeling of multiple friction pendulum
isolation system with numerous sliding interfaces. Earthquake Engineering
and Structural Dynamics 2010; 39(13):1463-1491.
USGS. www.usgs.gov
Zayas VA, Low SS, Mahin SA. The FPS earthquake resisting system experimental
report. Report No. UCB/EERC-87/01, 1987.
Zayas VA. Personal communication. 2012
385
Appendix A
‡•‹‰”ƒ™‹‰•‘ˆ–Š‡‘‡ –‹‘••‡„Ž‹‡•
Below are design drawings of the connection assemblies, which include load cells
and connecting plates connecting the isolators to the shake table.
386
387
388
389
390
391
392
393
394
395
396
397
398
Appendix B
’‡ –”ƒ‘ˆƒ„Ž‡ƒ†ƒ”‰‡–‘–‹‘•
Below are 5% damped acceleration response spectra of the target motions and the
motions generated by the shake table for both the isolated base and fixed base tests.
399
1
0.8
0.6
Table motion
Target motion
X
X
0.4
0.5
0.2
0
10
Spectral acceleration, S
A
0.8
-1
10
0
1
10
10
A
1.5
Y
0.6
0.4
0.2
0 -2
10
0 -2
10
1
10
(g)
-1
10
-1
10
0
1
10
10
0.8
Spectral acceleration, S
(g)
0 -2
10
Y
1
0.5
0 -2
10
1.5
Z
0.6
-1
10
Z
0
1
10
10
Table motion
Target motion
1
0.4
0.5
0.2
-1
10
0
10
Figure C-1: 80WSM for isolated-base test
1
0
1
10
10
A
(g)
A
Spectral acceleration, S
-1
10
(g)
0.5
Y
2
1
-1
10
0
1
10
10
0 -2
10
Table motion
Target motion
-1
10
0
1
10
10
4
Y
3
2
1
0 -2
10
-1
10
0
1
10
10
2
4
Z
3
1
1
0.5
-1
10
0
10
Period, T (s)
Figure C-3: 88RRS for isolated-base test
Z
1.5
2
0 -2
10
1
10
X
X
3
0 -2
10
0
10
Period, T (s)
1.5
Table motion
Target motion
1
0 -2
10
-1
10
Figure C-2: 130ELC for isolated-base test
3
2
0 -2
10
1
10
Period, T (s)
Spectral acceleration, S
0 -2
10
1
10
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-4: 100SYL for isolated-base test
400
3
1.5
X
Table motion
Target motion
2
1
-1
0
10
1
10
A
10
(g)
0 -2
10
3
Y
2
1
0 -2
10
1.5
-1
10
0
10
1
10
0 -2
10
0
2
1
0 -2
10
0.5
1
-1
0
10
Period, T (s)
0 -2
10
1
10
Figure C-5: 50TAB for isolated-base test
10
Y
2
10
1
10
-1
10
0
1
10
3
Z
0 -2
10
-1
10
3
1
10
Z
-1
10
0
1
10
Period, T (s)
10
Figure C-6: 70LGP for isolated-base test
0.8
0.6
X
0.5
Spectral acceleration, S
Spectral acceleration, S
A
(g)
1
Table motion
Target motion
1.5
X
1
Table motion
Target motion
X
0.4
Table motion
Target motion
-1
10
0
10
Y
0.5
0 -2
10
-1
10
0
10
1
10
0.04
0
1
10
10
1.5
Y
1
0.5
0 -2
10
-1
10
0
1
10
10
0.08
Z
Z
0.06
0.02
0.04
0.01
0.02
0 -2
10
-1
10
A
1
0.03
0 -2
10
1
10
Spectral acceleration, S
Spectral acceleration, S
A
(g)
0 -2
10
0.5
(g)
0.2
-1
10
0
10
Period, T (s)
1
10
Figure C-7: 50TCU for isolated-base test
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-8: 70TCU for isolated-base test
401
2
1.5
0.8
Table motion
Target motion
X
1
0.4
0.5
0.2
0
A
10
3
Y
2
1
0 -2
10
0 -2
10
1
10
(g)
-1
10
-1
10
0
1
10
0.08
10
Spectral acceleration, S
Spectral acceleration, S
A
(g)
0 -2
10
X
0.6
0.06
0
1
10
10
0.8
Y
0.6
0.4
0.2
0 -2
10
0.06
Z
-1
10
-1
10
Z
0
1
10
10
Table motion
Target motion
0.04
0.04
0.02
0.02
0 -2
10
-1
10
0
0 -2
10
1
10
Period, T (s)
10
Figure C-9: 100IWA for isolated-base test
X
2
0
1
10
10
A
(g)
-1
10
Y
3
2
1
-1
10
0
1
10
10
1.5
Spectral acceleration, S
(g)
A
Spectral acceleration, S
Table motion
Target motion
X
1
4
0 -2
10
1
10
3
Table motion
Target motion
1
0 -2
10
0
10
Period, T (s)
Figure C-10: 100SAN for isolated-base test
3
2
-1
10
0 -2
10
-1
10
0
1
10
10
4
Y
3
2
1
0 -2
10
-1
10
0
1
10
10
2
Z
Z
1.5
1
1
0.5
0 -2
10
0.5
-1
10
0
10
Period, T (s)
1
10
Figure C-11: 100TAK for isolated-base test
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-12: 100KJM for isolated-base test
402
3
1.5
Table motion
Target motion
2
X
X
1
0.5
-1
10
0
10
-1
10
-1
10
10
0
10
1
0
10
0
10
1.5
A
3
Table motion
Target motion
0 -2
10
1
10
(g)
0 -2
10
Y
2
1
0 -2
10
-1
10
0
1
10
0.4
10
Spectral acceleration, S
Spectral acceleration, S
A
(g)
1
Y
1
0.5
0 -2
10
0.1
Z
10
1
Z
0.3
0.2
0.05
0.1
0 -2
10
-1
10
0
10
Figure C-13: 88RRSXY for isolated-base test
6
X
0 -2
10
1
10
Period, T (s)
2
0
1
10
10
A
(g)
-1
10
Y
3
2
1
-1
10
0
1
10
10
3
Spectral acceleration, S
(g)
A
Spectral acceleration, S
Table motion
Target motion
X
4
4
0 -2
10
1
6
2
0 -2
10
10
Period, T (s)
Figure C-14: 80TCU for isolated-base test
Table motion
Target motion
4
-1
10
0 -2
10
-1
10
-1
10
10
0
10
1
0
10
0
10
6
Y
4
2
0 -2
10
10
1
0.8
Z
0.6
2
Z
0.4
1
0.2
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-15: 80TAB for isolated-base test
0 -2
10
-1
10
10
Period, T (s)
1
Figure C-16: 90TAB for isolated-base test
403
6
1
X
Table motion
Target motion
Table motion
Target motion
4
X
0.5
-1
0
10
6
Y
4
2
0 -2
10
0.8
0 -2
10
1
10
-1
10
0
1
10
10
0.4
0.2
0 -2
10
0.02
0.2
0.01
0 -2
10
-1
10
0
0 -2
10
1
10
Period, T (s)
10
Figure C-17: 100TAB for isolated-base test
X
X Target motion
0.6
2
0.4
1
0.2
-1
0
10
0 -2
10
1
10
(g)
10
1
10
Z
-1
10
0
1
10
Period, T (s)
10
0.8
Table motion
Table motion
Target motion
-1
10
X
0
1
10
10
0.8
A
6
Y
4
2
0 -2
10
-1
10
0
10
1
10
1.5
Spectral acceleration, S
(g)
0
10
Figure C-18: 100SCT for isolated-base test
4
0 -2
10
-1
10
0.04
Z
0.4
A
1
10
Y
0.6
0.03
Spectral acceleration, S
0
10
0.8
0.6
3
-1
10
A
10
(g)
0 -2
10
Spectral acceleration, S
Spectral acceleration, S
A
(g)
2
Y
0.6
0.4
0.2
0 -2
10
-1
10
0
1
10
10
0.8
Z
Z
0.6
1
0.4
0.5
0.2
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-19: 115TAK for isolated-base test
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-20: 80WSM for fixed base test
404
1
1
Table motion
Target motion
Table motion
Target motion
X
0.5
0.5
0
10
A
1
Y
0.5
0 -2
10
0 -2
10
1
10
(g)
-1
10
-1
10
0
1
10
0.06
10
Spectral acceleration, S
Spectral acceleration, S
A
(g)
0 -2
10
0.5
0 -2
10
0.02
0.5
0
0 -2
10
1
10
Period, T (s)
10
Figure C-21: 35RRSXY for fixed-base test
1
10
Y
1
-1
0
10
-1
10
0
1
10
1.5
Z
10
-1
10
1
0.04
0 -2
10
X
10
Z
-1
10
0
1
10
Period, T (s)
10
Figure C-22: 35RRS for fixed-base test
1
1.5
Table motion
Target motion
X
1
Table motion
Target motion
X
0.5
0.5
0
10
1
-1
10
0
1
10
10
A
1.5
Y
0.5
0 -2
10
0 -2
10
1
10
(g)
-1
10
-1
10
0
1
10
10
4
Spectral acceleration, S
Spectral acceleration, S
A
(g)
0 -2
10
Y
1
0.5
0 -2
10
-1
10
0
1
10
10
0.06
Z
3
Z
0.04
2
0.02
1
0 -2
10
-1
10
0
10
Period, T (s)
1
10
Figure C-23: 88RRS for fixed-base test
0 -2
10
-1
10
0
10
Period, T (s)
Figure C-24: 70IWA for fixed base test
1
10
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement