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 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 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 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 ͵Ǥͷͺ ൈ ͳǤͺʹͻ (ͳʹ ൈ ) and is ͷǤʹͺ͵ (ͳ െ 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 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 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 under level II earthquakes. The drift angle limit for conventional frames subjected to this level of earthquake is ͲǤͲͳ. The structure was also required to remain elastic without damage when drift angle is less than ͲǤͲͳ (Kasai et al 2008). ʹǤͳǤ͵ 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 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 ͳǤʹthick and oriented parallel to the Y-direction. Reinforcement for the floor slabs is typically a single layer of ԄͳͲ̷ͳͷͲ 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 The nominal yield strengths of steel are ʹͻͷ and ͵ʹͷ for columns and beams, respectively. Their expected ultimate strengths are ͶͲͲ for columns and ͶͻͲ 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 ʹͳ with the expected compression strength of standard samples in the compression test is ʹͶ. The concrete slabs are reinforced by SD295A grade reinforcement. The 17 Table 2-1 Actual yield and ultimate strengths of steel Member ሺሻ ሺሻ Columns 346 – 398 430 – 470 Beams 331 – 422 510 – 557 Source: Kasai et al 2010 nominal yield stress for this steel grade is ʹͻͷ, 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 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 ǤǤ 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 ͲǤͶܵ௦ ܶ 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 ൌ ͳͺͲȀ to ͵ͲȀ) 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 ͲǤͶͷ ܵଵ () (**) ͳǤͳͺ ͲǤͳ͵ ͲǤ͵͵ (*) (*) MCE = maximum considered earthquake, ʹΨ probability of exceedance in ͷͲ (ʹȀͷͲ) DBE = design basis earthquake, ͳͲΨ probability of exceedance in ͷͲ (ͳͲȀͷͲ) Service level: ͷͲΨ probability of exceedance in ͷͲ (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 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 ܶ ܶ ൏ ͲǤͳݏ ͲǤͳ ݏ ܶ ͲǤͶݏ ܶ ͲǤͶݏ ൢ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 ۓ ͲǤͺܶ ଶ ۖ ۖ ܩെ ܩ௦ଶ ۖܩ௦ଶ ௦ଵ ሺܶ െ ͲǤͺܶଶ ሻ ۖ ͲǤͺሺܶଵ െ ܶଶ ሻ ۔ ܩ௦ଵ ۖ ۖ ۖ ܩ ܩ௦ଵ െ ͳǤͲ ൬ͳ െ ͳ ൰ ۖ ௦ଵ ͳ ܶ ͳǤʹܶଵ ە ͳǤʹܶଵ െ ͲǤͳ ܶ ͲǤͺܶଶ ͲǤͺܶଶ ൏ ܶ ͲǤͺܶଵ ͲǤͺܶଵ ൏ ܶ ͳǤʹܶଵ 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 ۖͲǤͺͶ ܶ ۔ ۖ ͳ ەǤ͵ͷ ܶ ൏ ͲǤͷݏ ͲǤͷ ݏ ܶ ൏ ͲǤͶݏ (3.1-12) ͲǤͶ 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 ܶ ܩ௦ ൌ ͳǤͷ ͲǤͶ ۔ ۖ ݃ ە௩ ܶ ൏ ͲǤͶݏ ͲǤͶ ݏ ܶ ൏ ͲǤͶ ͲǤͶ ݃௩ ݏ ͳǤͷ ݃௩ ݏ ܶ ͳǤͷ ݃௩ ൌ ʹǤͲ͵ for soil type II and ݃௩ ൌ ʹǤͲ 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 ͲǤͷȀ. 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 ͲȀ 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 ͺͺ and ͶͷͲ while the average static load on bearings was ͷͻͷȀ, 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 Ǧ Consider a bearing with characteristic dimensions and friction coefficients shown in Figure 3-15. Let the effective radii of spherical surfaces be: ଵ ൌ ଵ െ ଵ Ǣଶ ൌ ଶ െ ଶ Ǣ ଷ ൌ ଷ െ ଷ 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 כݑൌ ݑ௧ 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: כݑൌ 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: כݑൌ 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͵ 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 ͲǤʹͷ. Table 3-8 Design parameters of triple pendulum bearings ߤଵ ൌ ͲǤͲʹ ܴଵ ൌ Ͷͷ ݄ଵ ൌ ͵ͺ ݀ଵ ൌ ͷͳ ܮଵ ൌ Ͷͳͻ ߤଶ ൌ ͲǤͲͺ ߤଷ ൌ ͲǤͲͺ ܴଶ ൌ ͵ͻʹ ܴଷ ൌ ͵ͻʹ ݄ଶ ൌ ͳͳͶ ݄ଷ ൌ ͳͳͶ ݀ଶ ൌ ͷͳͶ ݀ଷ ൌ ͷͳͶ ܮଶ ൌ ͵ͺͶͺ ܮଷ ൌ ͵ͺͶͺ Length is in 58 ݂ ͲǤʹͷ ܶ ൌ 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 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 ܦ௫ , 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 ܦ௫ 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 ܦ௫ . Step 5: Compute the effective period and damping ratio: ܯ ܶ ൌ ʹߨඨ ܭ ߞ ൌ ܣ ଶ ʹߨܭ ܦ௫ (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 ܦ௫ ሺ ሻ ܶ ሺሻ 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 . 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 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 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 ͷͳ݇ܰ 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 ͺ݉݉. The thickness of ͻͷ݉݉ was chosen for conservatism. The connecting plates were produced by milling plate steel having thickness of ͳͲͳǤ݉݉ (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 ͲǤͷͷ݉ 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 ͲǤ݉݉ȀͳͲͲͲ݉݉. The vertical displacement of the acting load is ͲǤͺͺ݉݉, 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 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͵ 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 Ͷݏ݊ݐ, 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 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 ͲǤݏ, 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 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): ሺܺᇲ െ ܺ ሻଶ ሺܻᇲ െ ܻ ሻଶ ൌ ܮଶ ሺܺᇲ െ ܺ ሻଶ ሺܻᇱ െ ܻ ሻଶ ൌ ܮଶ ሺܺᇲ െ ܺௗ ሻଶ ሺܻᇲ െ ܻௗ 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 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 ܴǡ௧ 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 ܴǡ௧ 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͵ 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 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 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 ͵ 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 ሺݑ௫ െ ݑ ሻ (5.1-1) Or: ߤଵ ൌ ܣ ʹሺݑ௫ െ ݑ ሻ (5.1-2) 112 ݂ (a) ߤଵ ݑ ݑ௫ െߤଵ ݑ ݇ଵ ݑ ݑ௫ 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 ܣ௨௧ ൌ ʹߤௗ ሺݑ௫ െ 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 ൌ ܣ௨௧ െ Ͷݑଶ ߤଵ ʹሺݑ௫ െ 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 ൌ ܣ௨௧ െ Ͷݑଶ ߤଵ ݇ ݑଶ ʹሺݑ௫ െ ݑ െ ʹݑଶ ሻ (5.1-6) Given the experimental areas ܣ and ܣ௨௧ of the inner and outer normalized hysteresis loops, the maximum and minimum displacements ݑ௫ 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 ൌ ߤ௫ െ ሺߤ௫ െ ߤ ሻ ሺെܽݑሶ ሻ (5.1-9) where ߤ௫ 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 ൌ ܹܣିଵ (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 ͲǤͲʹȀ was characterized as low while velocity greater than ͲǤʹͷȀ 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 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 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 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͵ 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 ͻǤΨ 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 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 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 ͲǤʹͷ. 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 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 . 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 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 ͲǤͶͺΨ 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 ͲǤͻͲΨ 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 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 , 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 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 ͳͲͲȀͲ, 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 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 ܮଵ ݂௬ଵ ൌ ߤଵ ݇ଶ ൌ ͳ ܮଶ ݂௬ଶ ൌ ߤଶ ݇ଷ ൌ ͳ ܮଷ ܨ ܹ ݂௬ଷ ൌ ߤଷ ீݎଵ ൌ ݑതଵ ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ ீݎଶ ൌ ݑതଶ ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ ீݎଷ ൌ ݑതଷ ᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ Figure 6-1 Theoretical series model for multi-stage behavior of TFP Table 6-1 Parameters of theoretical series model Elastic model Friction model ܮଵ ൌ ʹܮଵ ݂௬ଵ ൌ ߤଵ ܮଶ ൌ ܮଶ െ ܮଵ ݂௬ଵ ൌ ߤଶ ݑതଶ ൌ ሺͳ െ ܮതଵ Ȁܮതଶ ሻ݀ଶ ܮଷ ൌ ܮଷ െ ܮଵ ݂௬ଵ ൌ ߤଷ ݑതଷ ൌ ሺͳ െ ܮതଵ Ȁܮതଷ ሻ݀ଷ ሺכሻ Gap model ݑതଵ ൌ ݑ௧ െ ݑതଶ െ ݑതଷ ሺכሻ ݑ௧ ൌ 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 ݇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 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 ሽ்ାଵ at step ݊ ͳ, compute a trial stress ௧ assuming elastic response and evaluate the yield function ܨାଵ : 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 ܨାଵ ii. ߦ௫ ௧ ൌ ብ൜ ൠ ብ െ ݂ ߦ௬ ାଵ (6.2-8) (6.2-9) ௧ Elastic condition: If ܨାଵ Ͳ Update state: ሺǤ ሻାଵ ൌ 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 ܨାଵ Ͳ Compute normal vector: ݊௫ ͳ ߦ௫ ௧ ቄ݊ ቅ ൌ Ǥ൜ ൠ ௬ ԡߦԡ௧ ߦ௬ ାଵ ାଵ (6.2-12) Compute consistency parameter: ௧ ܨାଵ 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 ൫ߦ௫ǡାଵ 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 ൫ߦ௫ǡାଵ ൯ ܨାଵ Ǥ ൭ͳ െ ଶ ൱ൡ ௧ ԡߦԡାଵ 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 ൫ߦ௫ǡାଵ ܨାଵ Ǥ ൭ͳ െ ଶ ൱ൡ ௧ ԡߦԡାଵ ൫ԡߦԡ௧ ൯ (6.2-24) ାଵ Similarly, ௧ ଶ ݇௬௬ ߦ௬ ݇ଵଶ ൌ ݇ଵ െ Ǥ ቐ൭ቆ ቇ ൱ ԡߦԡ ݇ଵ ܪ ାଵ ଶ ௧ ௧ ൫ߦ௬ǡାଵ ൯ ܨାଵ Ǥ ൭ͳ െ ൱ൡ ௧ ଶ ԡߦԡ௧ ൫ԡߦԡ ൯ ାଵ (6.2-25) ାଵ ௧ ݇௫௬ ൌ ݇௬௫ ௧ ൫ߦ௫ Ǥ ߦ௬ ൯ାଵ ݇ଵଶ ܨାଵ ൌെ 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 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 ௧௧௦௧௦௦௧௫ 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) ௧௧௦௧௦௦௧௫ 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 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 Figure 6-7 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 ൌ ሺଶൈ଼ሻ 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 ݇Ǥሺଶൈଶሻ Ǥ ܽሺଶൈ଼ሻ (6.4-8) ୀଵ where ݇Ǥሺଶൈଶሻ 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 ሺͳǡሻ ൌ െ݇Ǥଵ ሺͳǡʹሻ െ ݇Ǥଶ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͳǡሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺͳǡͺሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺʹǡʹሻ ൌ ݇Ǥଵ ሺʹǡʹሻ ݇Ǥଶ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺʹǡ͵ሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺʹǡͶሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺʹǡͷሻ ൌ െ݇Ǥଵ ሺͳǡʹሻ െ ݇Ǥଶ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺʹǡሻ ൌ െ݇Ǥଵ ሺʹǡʹሻ െ ݇Ǥଶ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺʹǡሻ ൌ 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͵ǡሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺ͵ǡሻ ൌ െ݇Ǥହ ሺͳǡͳሻ െ ݇Ǥ ሺͳǡͳሻ ்݇ிሺ଼ൈ଼ሻ ሺ͵ǡͺሻ ൌ െ݇Ǥହ ሺͳǡʹሻ െ ݇Ǥ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͶǡͶሻ ൌ ݇Ǥହ ሺʹǡʹሻ ݇Ǥ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͶǡͷሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺͶǡሻ ൌ Ͳ ்݇ிሺ଼ൈ଼ሻ ሺͶǡሻ ൌ െ݇Ǥହ ሺͳǡʹሻ െ ݇Ǥ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͶǡͺሻ ൌ െ݇Ǥହ ሺʹǡʹሻ െ ݇Ǥ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͷǡͷሻ ൌ ݇Ǥଵ ሺͳǡͳሻ ݇Ǥଶ ሺͳǡͳሻ ݇Ǥଷ ሺͳǡͳሻ ݇Ǥସ ሺͳǡͳሻ 181 ்݇ிሺ଼ൈ଼ሻ ሺͷǡሻ ൌ ݇Ǥଵ ሺͳǡʹሻ ݇Ǥଶ ሺͳǡʹሻ ݇Ǥଷ ሺͳǡʹሻ ݇Ǥସ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺͷǡሻ ൌ െ݇Ǥଷ ሺͳǡͳሻ െ ݇Ǥସ ሺͳǡͳሻ ்݇ிሺ଼ൈ଼ሻ ሺͷǡͺሻ ൌ െ݇Ǥଷ ሺͳǡʹሻ െ ݇Ǥସ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺǡሻ ൌ ݇Ǥଵ ሺʹǡʹሻ ݇Ǥଶ ሺʹǡʹሻ ݇Ǥଷ ሺʹǡʹሻ ݇Ǥସ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺǡሻ ൌ െ݇Ǥଷ ሺͳǡʹሻ െ ݇Ǥସ ሺͳǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺǡͺሻ ൌ െ݇Ǥଷ ሺʹǡʹሻ െ ݇Ǥସ ሺʹǡʹሻ ்݇ிሺ଼ൈ଼ሻ ሺǡሻ ൌ ݇Ǥଷ ሺͳǡͳሻ ݇Ǥସ ሺͳǡͳሻ ݇Ǥହ ሺͳǡͳሻ ݇Ǥ ሺͳǡͳሻ ்݇ிሺ଼ൈ଼ሻ ሺǡͺሻ ൌ ݇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): ൨൜ ൠ ൌ ቄ ቅ (6.4-10) From the second equation of Equation (6.4-10): ൌ െି Ǥ Ǥ Substitute Equation (6.4-11) into the first equation of Equation (6.4-10): (6.4-11) 183 ൫ െ Ǥ ି Ǥ ൯ ൌ (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 ൌ ்ܭሺଶൈଶሻ ܨ௩ (6.4-15) where ்ܭሺଶൈଶሻ is the actual horizontal tangent stiffness of the TPB element in its basis coordinate system. From Equation (6.4-15), ்ܭሺଶൈଶሻ can be computed from ்݇ሺଶൈଶሻ as follows: ்ܭிሺଶൈଶሻ ൌ 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 ்ܨሺଶൈଶሻ can be computed from normalized horizontal force vector ்݂ሺଶൈଶሻ as: ݂௫ ܨ௫ ்ܨிሺଶൈଵሻ ൌ ܨ௩ ൈ ்݂ሺଶൈଶሻ ൌ ܨ௩ ൈ ൜ ൠ ൌ ൜ ܨൠ ݂௬ ௬ (6.4-17) ǤͶǤʹ 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 and force vector given the trial deformation ௧ and the converged state (ܿ )ݒ݊of the last time step. Knowledge of the converged state includes normalized tangent stiffness matrix ௩ , normalized force vector 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 ǡ௩ǡ 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 ݑ௩ ǻݑଵ ݑଶ ݑ௧ ݑଷ ǻݑଷ ǻݑଶ Figure 6-8 Inverse Newton – Raphson iteration ݑ 186 Begin TPB element ௧ ǡ ௩ ௩ ǡ ௩ ௩ǡଵ ǡ ௩ǡଶ ǡ ௩ǡଷ ௩ǡଵ ǡ ௩ǡଶ ǡ ௩ǡଷ ǡ௩ǡǤଵ ǡ ǡ௩ǡǤଷ ǡ ǡ௩ǡǤହ ௩ǡǤଵ ǡ ௩ǡǤଷ ǡ ௩ǡǤହ ݅ ൌ Ͳǡ ܹ ൌ ݇௩ ݑ௩௧ ଵ ൌ ௩ ǡ ଵ ൌ ௩ ǡ ଵ ൌ ௩ ݅ ൌ݅ͳ ઢ ൌ ௧ െ ǡઢ ൌ ઢ Call Element Group 2, get: Call Element Group 1, get: ሺǤଷ ሻ ǡ ൫ǡǤଷ ൯ ǡ ሺǤଷ ሻ ሺǤଵ ሻ ǡ ൫ǡǤଵ ൯ ǡ ሺǤଵ ሻ ሺǤଶ ሻ ǡ ൫ଵ ൯ ǡ ൫ଵ ൯ ାଵ ൌ ሺ ሻ ǡ ାଵ ൌ ൫ ൯ ାଵ ൌ ઢ ݎݎܧൌ ԡ௧ െ ାଵ ԡ N ሺǤସ ሻ ǡ ൫ଶ ൯ ǡ ൫ଶ ൯ Call Element Group 3, get: ሺǤହ ሻ ǡ ൫ǡǤହ ൯ ǡ ሺǤହ ሻ ሺǤ ሻ ǡ ൫ଷ ൯ ǡ ൫ଷ ൯ Y ݎݎܧ ݈ܶ ൌ ܹǤ ሺሻାଵ ǡ ൌ ܹǤ ሺሻାଵ Return ǡ Update: ௩ ൌ ାଵ ǡ ௩ ൌ ାଵ ǡ ௩ ൌ ௧ ௩ǡଵ ൌ ൫ଵ ൯ ǡ ௩ǡଶ ൌ ൫ଵ ൯ ǡ ௩ǡଷ ൌ ൫ଷ ൯ ௩ǡଵ ൌ ൫ଵ ൯ ǡ ௩ǡଶ ൌ ൫ଶ ൯ ǡ ௩ǡଷ ൌ ൫ଷ ൯ ǡ௩ǡǤଵ ൌ ൫ǡǤଵ ൯ ǡ ǡ௩ǡǤଷ ൌ ൫ǡǤଷ ൯ ǡ ǡ௩ǡǤହ ൌ ൫ǡǤହ ൯ ௩ǡǤଵ ൌ ሺǤଵ ሻ ǡ ௩ǡǤଷ ൌ ሺǤଷ ሻ ǡ ௩ǡǤହ ൌ ሺǤହ 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͵ 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 ǡ௩ and the displacement ǡ௩ of the element group; the plastic deformation ǡ௩ǡ 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 and ઢ passed from the TPB element level. The new values of plastic deformation ݑǡ 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 ݑ௩ǡ ǡ ݇௩ǡ ݑǡ௩ǡ ǡ ݍ௩ǡ ݂௧ ݆ൌͲ ൌ ݂ ȟ݂ ǡ ൫݂ ൯ ൌ ݂ ଵ ൫݇ ൯ ൌ ݇௩ǡ ǡ ൫ݑ ൯ ൌ ݑ௩ǡ ଵ ଵ ݆ ൌ݆ͳ ିଵ ൫ȟ݂ ൯ ൌ ݂௧ െ ൫݂ ൯ ǡ ൫ȟݑ ൯ ൌ ൫݇ ൯ ൫ȟ݂ ൯ Solving circular elastic gap element, get ൫݇ǡ ൯ ǡ ൫݂ǡ ൯ Solving bilinear plasticity element, get: ൫݇ǡ ൯ ǡ ൫݂ǡ ൯ ǡ ൫ݑǡ ൯ ǡ ሺݍ ሻ ൫݇ ൯ ାଵ N ݎݎܧ ݈ܶ Y ൌ ൫݇ǡ ൯ ൫݇ǡ ൯ ൫ݑ ൯ ൫݂ ൯ ାଵ ାଵ ൌ ൫ݑ ൯ ൫ȟݑ ൯ ൌ ൫݂ǡ ൯ ൫݂ǡ ൯ ݎݎܧൌ ቛ݂௧ െ ൫݂ ൯ ାଵ Return ൫݇ǡ ൯ ǡ ൫ݑǡ ൯ ǡ ሺݍ ሻ ǡ ൫݇ǡ 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 ݂௧௧ 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: ்ܭிሺଷൈଷሻ ܭଵଵ ܭ ൌ ଶଵ Ͳ ்ܨிሺଷൈଵሻ 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 ൌ ்ܽ ൈ ்ܭሺଷൈଷሻ ൈ ܽ 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: ۴்ிሺଵଶൈଵሻ ൌ ்ܽ ൈ ்ܨிሺଷൈଵሻ ۴்ிሺଵଶൈଵሻ െ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 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 ͲǤͻ 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 ܴ ൌ ͳͲͲʹݐ, ܴܿ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 ߪ௨ , 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 ۖ߳ ߳ ߳ ߳ ߳ோ ߳ோ ߳ ۔ ൌ ͲǤͲ 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 ʹͻͷ. 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 (͵ͷͲሻ. 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 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 ൈ ͳͳǤͷ . Because of the corrugated deck of the slab, a gap between the slab and the I-section of ͵Ǥͷ is modeled. Strength of steel and compressive strength of concrete are ͵ʹͷ and ʹͶ, 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 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 ܵி ൌ ͲǤͷܾܩ ݐ (7.4-3) ଶ ܯி ൌ ͳǤͺͲܨ ܾ ݐ (7.4-4) where: ܵி ൌ initial stiffness of the spring, ܾ ൌ flange width of column, ݐ ൌ flange thickness of column, ܯி ൌ 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 ܭ௧௨௦௦ Ǥ ȟܮ ݕ ݔ (7.4-10) where ܭ௧௨௦௦ 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 ൌ ඩ൫ܴ௭ǡǡ െ ܴ௭ǡ௧௦௧ǡ ൯ ଶ (7.6-1) ୀଵ where ܴ௭ǡǡ and ܴ௭ǡ௧௦௧ǡ 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 ͳǤͷȀ. 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 ሻ and ͲǤͳͷ (Ǥ ) 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 and the peak amplitude in the responses of superstructure occurred at a frequency as high as around , 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 and Ǥ , respectively (Figure 7-15), gives unacceptably low damping at intermediate frequencies such as ͵ , 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 to ͳͲ . 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 . 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 ܴǡ௧௦௧ 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 ൌ ܴǡ௧௦௧ െ ܴ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 ൌ ܴǡ௧௦௧ െ ܴǡ ൌ ܹ௧௦௧ െ ܹௗ ୀଵ ୀଵ (7.8-6) ୀଵ where ܹ௧௦௧ 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 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 ͲǤͺ (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 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 and ͵Ǥ͵ ), 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 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 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 ͲǤͺ. 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 ͲǤͶʹ. 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 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 ͲǤͷΨand 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 ͲǤͷΨ 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 ͺͲ͵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 ͺͲ͵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 ͺͲ͵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 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 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. ߤ௩ ” 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. ߤ௩ ”, 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. ߤ௩ ” model, the peak displacement in the “Const. ߤ௩ ” 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. ߤ௩ ” 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. ߤ௩ ” model (reduces ͳͲǤͲΨ), the peak displacement increases from ʹǤͶ͵ to ͶǤͲͺ (increases 279 ͳͺǤΨ). The difference in ߤଵ between these models is small (ͳǤͻΨ compare to ͳǤʹΨ). 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 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 ͳͲͲͺͲ 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 ͳͲͲͺͲ 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 ͳͲͲͺͲ 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 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 ͲǤͺ 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 ǣ Ǧ 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 (ൌ ܹȀሺͲ 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 ǣ Ȁ 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 ൌ ͺǤͷ ݇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͵ , 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 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 σଷୀଵ 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 ൌ ͺǤͷݐ ݂ ݇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 ͲǤͺͲ, 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͵ ͵ 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 ቀܽǡଷ ሺܺǡ ܻǡ ܼǡ ݐሻ െ ܽǡ ሺܺǡ ܻǡ ܼǡ ݐሻቁ Ǥ ݀݉ (9.2-28) ൌ ݂ሺݐሻǤ න ቀܽǡଷ 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: ௨௦ 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 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 ൌ ݐݏ݊ܥ 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 σୀଵ ݉ 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 σୀଵ ݉ 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 (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 Ȁܵǡ 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 ͲǤͳʹ 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 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 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 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͵ Ǧ Ǧ Ǧ 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. 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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 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 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

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