Uncoupled Base Rocking and Shear Mechanisms for ...

Uncoupled Base Rocking and Shear Mechanisms for Controlling Higher-mode Effects in High-rise Buildings

by

Fei Tong

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Civil and Mineral Engineering University of Toronto

© Copyright by Fei Tong (2020)

ii

Uncoupled Base Rocking and Shear Mechanisms for Controlling Higher-mode Effects in High-rise Buildings

Fei Tong

Doctor of Philosophy

Department of Civil & Mineral Engineering

University of Toronto

2020

Abstract

While modern seismic design philosophies prioritize life safety under major earthquakes,

structural damage is not precluded. As for high-rise buildings, earthquake-induced damage

can be rather extensive and exacerbated as a result of higher-mode effects. This may lead

to repair and replacement being infeasible or uneconomical.

A large body of research has been conducted on low-damage systems for high-rise

buildings. Concepts involved in these systems essentially fall into three categories

including rocking mechanisms, seismic isolation, and the combination thereof. While

rocking systems have limited efficiency in limiting higher-mode effects, , base isolation

can also be challenging for high-rise buildings due to base isolators being overloaded

axially while undergoing significant lateral deformations. Combining both concepts by

allowing base-isolated structures to rock at the base as well cannot fully resolve all these

problems, leading to design and implementation challenges.

Abstract

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This dissertation proposes a novel system consisting of uncoupled rocking and shear

mechanisms incorporated at the base of RC core-wall buildings. Acting in parallel, the dual

mechanism allows for an independent control of the flexural and shear responses of

structures and an effective mitigation of the higher-mode response. A physical

implementation is developed for the proposed system after the fundamental kinematics

defining the system are understood. This physical embodiment is then designed, detailed,

and numerically validated using a reference 42-storey benchmark building. Results of

extensive nonlinear dynamic analyses indicate that the proposed system is efficient in

mitigating higher-mode effects and minimizing damage to RC core-wall high-rise

buildings. To generalize the design of the proposed system, closed-form analytical studies

and parametric nonlinear time history analyses are conducted. Based on these analyses,

general procedures are developed for the preliminary design of the proposed system.

Acknowledgements

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Acknowledgements

After seven years of experience working as an engineer, I chose to return to academia. This deliberate choice introduced profound changes to my career, life, and soul, which makes my PhD study mean to me a lot more than an educational accomplishment. Along this journey, I received support and help from numerous people whom I can never thank enough.

My sincerest gratitude first goes to my supervisor, Professor Constantin Christopoulos. I am grateful for the opportunity he gave me to work on this topic that has been challenging the field for decades, and his inspiring guidance and insightful criticism. His unshakable faith and inexhaustible optimism are contagious, stimulating people especially myself to pursue excellence unceasingly.

I want to extend my acknowledgement to Professors Evan Bentz, Oh-Sung Kwon, Oya Mercan, and Sanda Koboević for spending valuable time reviewing my dissertation. Their excellent comments brought this work to a higher level of quality and sparked new thinking for future research.

Wholeheartedly, I would like to thank Professor Michael Collins who offered me priceless mentorship on teaching. He, to me, is a role model of being a great engineer, researcher, and educator at the same time. I would also send special thanks to Professors Peter Wright and Satu Repo for their thoughtful care about my career, family, and personal life.

I will always remember those pleasant moments that I had with my colleagues including Michael Montgomery, Lydell Wiebe, Jack Guo, Jeffrey Erochko, Min Sun, Xu Huang, Renée MacKay-Lyons, Michael Gray, Farbod Pakpour, Jeffrey Salmon, Luis Ardila-Bothia, Myron Zhong, Pedram Mortazavi, and many others.

I am speechless when I am trying to thank my family who unreservedly supported me in many ways. As a historian, my wife, Dr. Xin Chen, might be the most knowledgeable among her colleagues about seismic engineering, for she is always my first audience and the most patient one listening about my new ideas, difficulties, breakthrough, and success in work. Over the past years, we hand in hand walked through countless challenging times during which her trust, tolerance, and resilience were essential for what we achieved today. Among these achievements, our son, Daniel, is the most cherished one. His laughters, hugs, and curiosity about everything are the best anodyne and stimulus that made me stronger. I owe to my parents who in their seventies relentlessly helped take care of daily life of the family while both Xin and I were working around the clock for dissertations. They and my parents-in-law also generously provided us financial supports which bought us time and concentration on research. I thank them from the bottom of my heart. I wish my father-in-law see our accomplishments in heaven and his soul rest in peace. I dedicate my dissertation to my beloved family members.

Table of Contents

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Table of Contents

Abstract ............................................................................................................................... ii

Acknowledgements ............................................................................................................ iv

Table of Contents ................................................................................................................ v

List of Tables ..................................................................................................................... xi

List of Figures ................................................................................................................... xii

List of Symbols ................................................................................................................ xvi

Chapter 1 Introduction ................................................................................................... 1

1.1 Urbanization and Development of High-rise Buildings ....................................... 1

1.2 Modern Seismic Design and its Limitations ........................................................ 2

1.3 Low-damage Design towards Seismic Resilience ............................................... 4

1.4 Research Objectives ............................................................................................. 5

1.5 Research Methodologies ...................................................................................... 5

1.6 Organization ......................................................................................................... 7

Chapter 2 Background and Literature Review .............................................................. 9

2.1 Introduction .......................................................................................................... 9

2.2 Seismic Demands on RC Wall Buildings .......................................................... 10

2.2.1 Higher-mode effects on elastic structures ................................................... 10

2.2.2 Numerical studies on higher-mode effects in inelastic structures .............. 11

2.2.3 Experimental studies on inelastic higher-mode effects .............................. 16

2.2.4 Evaluation of higher-mode effects .............................................................. 18

2.3 Damage to RC Wall Structures .......................................................................... 25

2.3.1 Flexural hinges at the base of RC walls ...................................................... 25

2.3.2 Distributed plasticity due to higher-mode effects ....................................... 27

2.4 Low-damage Systems for High-rise Buildings .................................................. 28

2.4.1 Dynamics of rocking ................................................................................... 28

2.4.2 Rocking wall systems ................................................................................. 30

2.4.3 Seismic isolation ......................................................................................... 32

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2.4.4 Systems with dual seismic protections ....................................................... 39

2.5 Summary ............................................................................................................ 40

Chapter 3 Considered Resilient Concepts for High-Rise Buildings ............................ 41

3.1 Introduction ........................................................................................................ 41

3.2 Base Rocking Storey and Self-centering Slider ................................................. 42

3.2.1 Self-centering energy dissipating braces .................................................... 42

3.2.2 Base rocking storey ..................................................................................... 43

3.2.3 Lateral-force-limiting self-centering slider ................................................. 45

3.2.4 Remarks ...................................................................................................... 46

3.3 Three-dimensional Decoupled Rocking and Shear Mechanisms ....................... 49

3.3.1 Rocking pyramid and additional rocking toes ............................................ 49

3.3.2 Decoupled shear mechanism ....................................................................... 51

3.3.3 Remarks ...................................................................................................... 52

3.4 Torsion-resistant Shear Mechanism and Cubic Rocking Block ......................... 54

3.4.1 Torsion-resistant shear mechanism ............................................................. 54

3.4.2 Improved structural arrangements for shear mechanism ............................ 55

3.4.3 Cubic rocking block with rocking toes at corners ...................................... 55

3.4.4 Remarks ...................................................................................................... 57

3.5 From Rocking to Wobbling ................................................................................ 58

3.5.1 Multi-phased rocking mechanism ............................................................... 58

3.5.2 Base wobbling mechanism ......................................................................... 60

3.5.3 Remarks ...................................................................................................... 61

3.6 Summary ............................................................................................................ 62

Chapter 4 Uncoupled Base Rocking and Shear Mechanisms: MechRV3D ................. 65

4.1 Introduction ........................................................................................................ 65

4.2 Uncoupled Base Rocking and Shear Mechanism System: Concept .................. 66

4.3 Idealized Configuration of the Proposed System ............................................... 68

4.3.1 Rocking mechanism .................................................................................... 69

4.3.2 Shear mechanism ........................................................................................ 70

4.3.3 Comparison with conventional systems ...................................................... 72

4.4 Numerical Modelling of the Idealized Configuration ........................................ 74

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4.4.1 PEER benchmark building and stick model ............................................... 74

4.4.2 Rocking mechanism .................................................................................... 75

4.4.3 Shear mechanism ........................................................................................ 77

4.4.4 P-𝛥 effects ................................................................................................... 78

4.5 Mechanics of the MechRV3D System ............................................................... 79

4.6 Dynamic Response of the MechRV3D System ................................................. 85

4.7 Physical Embodiment of the MechRV3D System: Shear Transmitters ............. 87

4.7.1 Hinged plates .............................................................................................. 87

4.7.2 Gear connections ......................................................................................... 88

4.7.3 Remarks ...................................................................................................... 90

4.8 Physical Embodiment of the MechRV3D System: Shear Mechanism .............. 92

4.8.1 Steel plate shear wall panels ....................................................................... 92

4.8.2 Buckling-restrained steel plate shear walls ................................................. 95

4.8.3 Steel slit shear wall panels .......................................................................... 97

4.8.4 Unbonded buckling-restrained brace frames ............................................ 100

4.9 Physical Embodiment of the MechRV3D System: Rocking Mechanism ........ 102

4.9.1 Docker trusses ........................................................................................... 102

4.9.2 Rocker ....................................................................................................... 103

4.9.3 Potential problems with ball rollers .......................................................... 105

4.9.4 Moment-free mega-columns ..................................................................... 106

4.9.5 Hinged buckling-restrained Braces ........................................................... 106

4.9.6 Concrete hinges ......................................................................................... 108

4.9.7 Ball pin joints ............................................................................................ 109

4.9.8 Telescopic pipe-pin hinges, rocking columns and kinematic isolation .... 111

4.9.9 Composite mega-columns with pipe-pin rolling joints ............................. 114

4.10 Physical Embodiment of the MechRV3D system: Overall System ................. 118

4.10.1 Integrated dual-mechanism system ........................................................... 118

4.10.2 MechRV3D-incorporated benchmark building ........................................ 118

4.10.3 Lateral equilibrium of the MechRV3D system ......................................... 124

4.11 Summary .......................................................................................................... 126

Chapter 5 Numerical Validation of the Proposed MechRV3D System .................... 127

5.1 Introduction ...................................................................................................... 127

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5.2 PEER Benchmark Building .............................................................................. 128

5.2.1 PEER Tall Buildings Initiative ................................................................. 128

5.2.2 Building geometry .................................................................................... 130

5.2.3 Seismic hazard and ground motions ......................................................... 130

5.2.4 Gravity loading allowance ........................................................................ 133

5.2.5 Load combination ..................................................................................... 133

5.2.6 Material properties .................................................................................... 134

5.2.7 Acceptance criteria .................................................................................... 134

5.2.8 Structural sizes .......................................................................................... 135

5.3 Design of the MechRV3D System ................................................................... 137

5.3.1 Design of the rocking mechanism ............................................................. 137

5.3.2 Design of the rolling mega-columns ......................................................... 139

5.3.3 Design of the Shear Mechanism ............................................................... 143

5.4 Advanced Nonlinear Modelling of the Benchmark Building .......................... 148

5.4.1 Modelling strategies .................................................................................. 148

5.4.2 Modelling techniques for RC shear walls ................................................. 149

5.4.3 Modelling of the RC core of the benchmark building .............................. 153

5.4.4 Modelling of coupling beams ................................................................... 157

5.4.5 P-𝛥 effects and gravity loads .................................................................... 159

5.4.6 Structural damping .................................................................................... 160

5.4.7 Ground motions used in the NLRHAs ...................................................... 162

5.4.8 Validation of the WCFA Model ............................................................... 165

5.5 Benchmark Building with a Rocking-only Base-mechanism .......................... 167

5.5.1 MCE responses of the 1M0V-based benchmark building ........................ 167

5.5.2 Incremental Dynamic Analyses ................................................................ 169

5.5.3 Remarks .................................................................................................... 171

5.6 Advanced Nonlinear Modelling of the MechRV3D System ........................... 172

5.6.1 Proposed model for rocking and rolling cylinders .................................... 172

5.6.2 Modelling of the BRBFs ........................................................................... 178

5.7 Numerical Validation of the MechRV3D System ............................................ 179

5.7.1 Responses of the shear mechanism ........................................................... 179

5.7.2 Seismic performance of the 1M1V-based PEER benchmark building .... 181

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5.8 Summary .......................................................................................................... 185

Chapter 6 Modal Analysis on Generally Supported Cantilever Systems .................. 187

6.1 Introduction ...................................................................................................... 187

6.2 Continuum Beam Analogy ............................................................................... 188

6.3 Equation of Motion governing Distributed Systems ........................................ 190

6.4 Eigenvalue Analysis on Elastically Supported Beams ..................................... 191

6.5 Cantilever Beams with Special Base Constraints ............................................ 196

6.5.1 Fully fixed: R → ∞, T → ∞ ..................................................................... 196

6.5.2 Rotationally flexible and translationally fixed: R ≠ 0, T → ∞ ................ 196

6.5.3 Rotationally free and translationally fixed: R → 0, T → ∞ ..................... 197

6.5.4 Rotationally fixed and translationally free: R → ∞, T → 0 ...................... 198

6.6 Modal Analysis on Generally Supported Beams ............................................. 199

6.6.1 Effective modal mass ................................................................................ 200

6.6.2 Displacements ........................................................................................... 202

6.6.3 Rotation angles (inter-storey drift ratios) .................................................. 208

6.6.4 Overturning moments ............................................................................... 212

6.6.5 Shear forces ............................................................................................... 216

6.7 Summary .......................................................................................................... 220

Chapter 7 Parametric Analyses and Design Recommendations ................................ 221

7.1 Introduction ...................................................................................................... 221

7.2 Governing Response Quantities and Approaches for Evaluation .................... 222

7.3 Nonlinear Parametric Analyses ........................................................................ 224

7.3.1 Generic buildings and fundamental period, T1 ......................................... 224

7.3.2 Rocking mechanism: Moment reduction factor, RM ................................. 227

7.3.3 Shear mechanism: Shear reduction factor, μV ........................................... 230

7.3.4 Shear mechanism: Initial stiffness, Kb1 ..................................................... 231

7.3.5 Shear mechanism: Post-yielding stiffness, Kb2 ......................................... 232

7.3.6 Summary of the control parameters .......................................................... 232

7.3.7 Rolling mega-columns .............................................................................. 233

7.3.8 P-𝛥 effects and gravity loads .................................................................... 233

7.3.9 Damping model ......................................................................................... 234

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7.3.10 Seismic hazard .......................................................................................... 234

7.3.11 Ground motion selection and scaling ........................................................ 235

7.3.12 Procedures for the parametric NLRHAs ................................................... 238

7.3.13 Base displacements, 𝛥b ............................................................................. 238

7.3.14 Rocking rotations at the base of the rocker, 𝜃rock ...................................... 241

7.3.15 IDRs, 𝛿s ..................................................................................................... 243

7.4 Design Charts and Design Recommendations ................................................. 245

7.4.1 Design charts ............................................................................................. 245

7.4.2 Recommended design framework ............................................................. 247

7.4.3 Other design considerations ...................................................................... 252

7.4.4 Validation of the preliminary design ........................................................ 253

7.4.5 Lock-up devices ........................................................................................ 253

7.5 Summary .......................................................................................................... 254

Chapter 8 Conclusion ................................................................................................ 256

8.1 Introduction ...................................................................................................... 256

8.2 Summary .......................................................................................................... 256

8.3 Conclusion ........................................................................................................ 259

8.3.1 Uncoupled flexural and shear responses ................................................... 259

8.3.2 Practice-oriented design of the proposed system ...................................... 259

8.3.3 Validation of the MechRV3D system ....................................................... 260

8.3.4 Theoretical study on generally supported cantilever systems ................... 261

8.3.5 Design framework ..................................................................................... 262

8.3.6 Original contributions .............................................................................. 262

8.4 Limitations and Recommendations for Future Research ................................. 264

8.4.1 Numerical analysis .................................................................................... 264

8.4.2 Experimental validation ............................................................................ 265

8.4.3 Development of design procedures ........................................................... 265

8.4.4 Further development of the MechRV3D system ...................................... 266

References ....................................................................................................................... 267

List of Tables

xi

List of Tables

Table 2.1 Dynamic shear amplification factors proposed by Boivin and Paultre [2012b] ......... 20

Table 5.1 Ground motions selected for the seismic design (from [Moehle et al. 2011]) ......... 132

Table 5.2 Superimposed dead loads and live loads (from [Moehle et al. 2011]) ..................... 133

Table 5.3 Concrete strength and modulus of elasticity ............................................................. 134

Table 5.4 Reinforcement steel strength and modulus of elasticity ........................................... 134

Table 5.5 Acceptance criteria under MCE level earthquakes (from [Moehle et al. 2011]) ...... 135

Table 5.6 Thickness and gravity weight of the floor slabs (from [Moehle et al. 2011]) .......... 135

Table 5.7 Sizes of gravity columns in [mm] (from [Moehle et al. 2011]) ................................ 136

Table 5.8 Thickness of RC core walls (from [Moehle et al. 2011]) ......................................... 136

Table 5.9 Design rocking moments and dimensions of the rocking mechanism ..................... 138

Table 5.10 Design parameters of the shear mechanism ............................................................. 147

Table 5.11 Ground motions used for verifying the design of the benchmark building .............. 163

Table 5.12 Ground motions used in NLRHAs conducted by MacKay-Lyon [2013] ................. 163

Table 6.1 βnH values for special base constraint conditions..................................................... 192

Table 7.1 Fundamental periods and structural heights (from [Xu et al. 2014]) ....................... 224

Table 7.2 Properties of the generic high-rise buildings ............................................................ 225

Table 7.3 Summary of the design parameters .......................................................................... 233

Table 7.4 Seismic design parameters considered for the parametric analyses ......................... 235

Table 7.5 Parameters for the selection of ground motion records ............................................ 236

Table 7.6 Selected ground motions .......................................................................................... 237

List of Figures

xii

List of Figures

Figure 2.1 Design envelope for storey shears ............................................................................. 19

Figure 2.2 Effective modal substitute structures (from [Pennucci et al. 2015]) ......................... 24

Figure 2.3 RC wall damage observed in the 2010 Chile Earthquake ......................................... 26

Figure 2.4 Aseismic joints proposed by Stevenson [1868] ........................................................ 32

Figure 2.5 Uplift-restrained elastomeric bearings (from [Griffith et al. 1990]) ......................... 34

Figure 2.6 Tension-resistant rubber bearings (from [Lu et al. 2016]) ........................................ 35

Figure 2.7 Tension-resistant base isolators (from [Hu et al. 2017]) ........................................... 35

Figure 2.8 Uplift-restrained Teflon disc bearing (from [Nagarajaiah et al. 1992]) .................... 36

Figure 2.9 XY-FP isolators (from [Roussis and Constantinou 2006]) ....................................... 36

Figure 2.10 Cross linear bearing (from [Arima et al. 2000]) ........................................................ 36

Figure 2.11 Rubber bearing with a loose detail. (from [Kikuchi et al. 2005]) ............................. 37

Figure 2.12 Kajima winkler method (from [Kajima Corporation 2006]) ..................................... 37

Figure 2.13 Base isolation arranged in-series with base rocking mechanism .............................. 38

Figure 3.1 SCED braces (from [Christopoulos et al. 2008]) ...................................................... 42

Figure 3.2 Base rocking storey ................................................................................................... 44

Figure 3.3 Self-centering slider system ...................................................................................... 45

Figure 3.4 Rocking pyramid with SCED braces as rocking toes ............................................... 49

Figure 3.5 Shift of rocking toes .................................................................................................. 50

Figure 3.6 Decoupled rocking and shear mechanisms ............................................................... 51

Figure 3.7 Shear mechanism with torsional resistance ............................................................... 54

Figure 3.8 Alternative arrangements of the lateral bracing components .................................... 55

Figure 3.9 Cubic rocking block with rocking toes at corners ..................................................... 56

Figure 3.10 Rocking block with distributed rocking toes............................................................. 58

Figure 3.11 Multi-phased rocking action ..................................................................................... 59

Figure 3.12 Base wobbling mechanism ........................................................................................ 60

Figure 3.13 Evolution of concepts ................................................................................................ 63

Figure 4.1 Concept of the uncoupled base rocking and shear mechanisms ............................... 66

Figure 4.2 Idealized configuration of the MechRV3D system ................................................... 68

Figure 4.3 Rocking states of the rocker ...................................................................................... 69

List of Figures

xiii

Figure 4.4 Transfer of lateral shears ........................................................................................... 71

Figure 4.5 Transfer of torsional moments .................................................................................. 72

Figure 4.6 MechRV3D system and a conventional base rocking system ................................... 73

Figure 4.7 PEER benchmark building (adapted from [Moehle et al. 2011]).............................. 74

Figure 4.8 Schematic model of the idealized configuration ....................................................... 76

Figure 4.9 Static pushover analysis scheme ............................................................................... 79

Figure 4.10 Lateral capacity curves and base rocking rotations ................................................... 80

Figure 4.11 Base overturning moments governed by Mrock .......................................................... 81

Figure 4.12 Rigid rocking block and flexible rocking structure ................................................... 82

Figure 4.13 Response of the shear mechanism in the first-mode pushover ................................. 83

Figure 4.14 Shear mechanism engaged under a different loading profile .................................... 84

Figure 4.15 Redistribution of gravity loads on rocking toes ........................................................ 84

Figure 4.16 Hystereses of the rocking mechanism ....................................................................... 85

Figure 4.17 Variation of gravity loads carried by the rocking toes .............................................. 85

Figure 4.18 Time histories of the lateral resistance ...................................................................... 86

Figure 4.19 V-𝛥skirt hystereses of the shear mechanism ................................................................ 86

Figure 4.20 Hinged plates as the shear transmitters ..................................................................... 87

Figure 4.21 Gear connections as the shear transmitters ............................................................... 88

Figure 4.22 Moment resistance contributed by leeward gear connections ................................... 89

Figure 4.23 Moment resistance contributed by leeward and side gear connections ..................... 90

Figure 4.24 Relation between Ωκ and κ (from [Qu and Bruneau 2009]) ...................................... 93

Figure 4.25 Schematic strip model for the SPSW panel .............................................................. 94

Figure 4.26 Ring-shaped steel plate shear wall panel................................................................... 96

Figure 4.27 Slotted steel plate shear wall panel (from [Jin et al. 2016]) ...................................... 96

Figure 4.28 Steel slit shear wall panel (from [He et al. 2016]) .................................................... 98

Figure 4.29 Schematic model of SSSW panels ............................................................................ 99

Figure 4.30 Shear mechanism consisting of BRBFs .................................................................. 100

Figure 4.31 Practical applications of mega-BRBs ...................................................................... 101

Figure 4.32 Docker trusses across the core-to-rocker joint ........................................................ 102

Figure 4.33 Skeleton truss in the rocker ..................................................................................... 103

Figure 4.34 Moment-free mega-columns ................................................................................... 106

Figure 4.35 Hinged BRBs used as mega-columns ..................................................................... 107

Figure 4.36 BRB mega-columns with two-way hinges .............................................................. 108

Figure 4.37 Concrete hinges (from [Schacht and Marx 2015]) .................................................. 109

List of Figures

xiv

Figure 4.38 Mega-columns with ball pin joints .......................................................................... 110

Figure 4.39 Telescopic pipe-pin hinge (from [Zaghi and Saiidi 2010]) ..................................... 111

Figure 4.40 Kinematic isolation by rocking columns ................................................................. 113

Figure 4.41 Composite mega-columns with a tube-in-tube built-up .......................................... 115

Figure 4.42 Proposed pipe-pin rolling joint ................................................................................ 116

Figure 4.43 Pipe-pin rolling joints at the foundation level ......................................................... 117

Figure 4.44 Physical embodiment of the MechRV3D system ................................................... 118

Figure 4.45 Indicative construction sequence for the MechRV3D system ................................ 119

Figure 4.46 MechRV3D system incorporated in the benchmark building ................................. 121

Figure 4.47 Details allowing for the slide at skirt-to-retaining wall joints ................................. 122

Figure 4.48 P-𝛥 effects of gravity columns in the basement ...................................................... 123

Figure 4.49 Lateral equilibrium of the MechRV3D system ....................................................... 124

Figure 5.1 Structural layout of the benchmark building (from [Moehle et al. 2011]) .............. 129

Figure 5.2 Seismic fault map of Los Angeles (from [USGS 2019]) ........................................ 130

Figure 5.3 Hazard disaggregation at the MCE level (from [Moehle et al. 2011]) .................... 131

Figure 5.4 Spectra of scaled ground motions at SLE43 (from [Moehle et al. 2011]) .............. 132

Figure 5.5 Composite mega-column section ............................................................................ 140

Figure 5.6 Spherical contact between the pad and cap ............................................................. 143

Figure 5.7 MVLEM and the modified models ......................................................................... 151

Figure 5.8 Equivalent truss models for RC walls ..................................................................... 152

Figure 5.9 Layout of the beam-column elements representing wall piers ................................ 153

Figure 5.10 Fibre sections of the wall piers ................................................................................ 154

Figure 5.11 WCFA model of the benchmark building ............................................................... 156

Figure 5.12 Schematic models of coupling beams (from [Naish et al. 2013b]) ......................... 158

Figure 5.13 Validation of the Vn-hinge model against the test results ....................................... 159

Figure 5.14 Equivalent viscous damping versus building height [LATBSDC 2020]. ............... 162

Figure 5.15 MCE response spectra of the ground motions ........................................................ 164

Figure 5.16 Periods and peak responses of the benchmark building at the MCE level .............. 165

Figure 5.17 Validation of the WCFA model against the reference analyses .............................. 166

Figure 5.18 MCE responses of the 1M0V-based benchmark building ...................................... 168

Figure 5.19 Incremental dynamic responses of the 1M0V-based benchmark building ............. 170

Figure 5.20 Planar rocking model (redrawn from [Vassiliou et al. 2017a]) ............................... 172

Figure 5.21 Proposed model for a three-dimensional rocking cylinder ..................................... 174

Figure 5.22 Validation of the proposed three-dimensional rocking model ................................ 176

List of Figures

xv

Figure 5.23 Schematic model of the fibre-based rolling section ................................................ 177

Figure 5.24 Validation of the BRB model .................................................................................. 178

Figure 5.25 Lateral response of the MechRV3D system ............................................................ 179

Figure 5.26 Ultimate lateral response of the MechRV3D system .............................................. 181

Figure 5.27 Mean strains of longitudinal reinforcement and concrete ....................................... 182

Figure 5.28 Chord rotations of the coupling beams ................................................................... 183

Figure 5.29 MCE IDRs and PFAs of the 1M1V-based building ................................................ 184

Figure 6.1 Cantilever beam analogy for RC core-wall high-rise buildings .............................. 189

Figure 6.2 Domain of the nondimensional stiffnesses R and T ................................................ 192

Figure 6.3 Variation of βnH with R and T (to be continued) .................................................... 193

Figure 6.4 Variation of the modal participation mass ratios, 𝑀∗ , with R and T ....................... 201

Figure 6.5 Variation of 𝛤 𝜙 with R and T .............................................................................. 205

Figure 6.6 Variation of 𝑢 , with R and T .......................................................................... 207


Figure 6.8 Variation of δ , with R and T ......................................................................... 211


Figure 6.10 Variation of 𝑀 , with R and T ......................................................................... 215

Figure 6.11 Variation of 𝛤 𝜙 with R and T ............................................................................. 217

Figure 6.12 Variation of 𝑉 , with R and T ........................................................................... 219

Figure 7.1 Relations between T1 and H (redrawn from [Xu et al. 2014]) ................................ 224

Figure 7.2 Geometry of the generic RC core-wall buildings .................................................... 227

Figure 7.3 Numerical model for the nonlinear parametric analyses ......................................... 228

Figure 7.4 MCER response spectrum for the parametric analyses ............................................ 235

Figure 7.5 Response spectra of the scaled ground motions ...................................................... 237

Figure 7.6 Displacement demands at the base of generic buildings ......................................... 239

Figure 7.7 Displacement and strength relation of base-isolated structures .............................. 240

Figure 7.8 Rocking rotations at the base of the rocker ............................................................. 242

Figure 7.9 Inter-storey drift ratios of the generic buildings ...................................................... 244

Figure 7.10 Preliminary design charts based on IDRs and base displacements ......................... 246

Figure 7.11 Acceptable design area ............................................................................................ 249

Figure 7.12 V-𝛥 relations of the overall lateral response and the BRBFs .................................. 251

List of Symbols

xvi

List of Symbols

Chapter 2

Cg normalized peak ground acceleration

Dm dynamic amplification factor

g gravitational acceleration

H building height

Hf coefficient applied to elastic higher-mode shears

Mb base overturning moment

MEd design bending moment

MEQ total seismic mass 𝑀∗ effective modal mass of the ith mode

Mo expected flexural strength of RC walls

MRd design flexural resistance

n number of storeys

PGA peak ground acceleration

QDU ultimate base shear

QFU fluctuating component of QDU

QSU static component of QDU

R force reduction factor

Rd force reduction factor related to ductility

Rh force reduction factor applied to the elastic higher-mode response

Ro force reduction factor related to overstrength

Rpf ratio of the elastic higher-mode response to the first-mode response

Sa(0.2) pseudo-acceleration spectrum ordinate for period equal to 0.2 sec

Sa(2.0) pseudo-acceleration spectrum ordinate for period equal to 2.0 sec

Sa(Ti) pseudo-acceleration spectrum ordinate for Ti

Sa(Tc) pseudo-acceleration spectrum ordinate for Tc

List of Symbols

xvii

SMS MCE pseudo-acceleration spectrum ordinate for short period

T1 fundamental period

Ti natural period for the ith mode

Va base shear demand accounting for dynamic amplification

Vb base shear demand

Vd design base shear

VE1 elastic shear demand contributed by the first mode

VEi,j elastic shear demand at the ith storey contributed by the jth mode

Vi shear demand at the ith storey

W total seismic weight 𝛾Rd overstrength factor due to strain-hardening 𝛾w base flexural overstrength factor 𝛥r roof displacement 𝜃 inter-storey drift ratio 𝜇 ductility demand 𝜇𝜙 flexural ductility of base plastic hinges 𝜉 height ratio to define storey shear envelope

ρ beam-to-column stiffness ratio 𝜙o flexural overstrength factor

ωV dynamic shear amplification factor

Chapter 4

b half of the width of rectangular rocking block

dc2c centre-to-centre distance between ball rollers

F resultant of lateral inertial forces

Fi inertial force at the ith floor

H height of the benchmark building above the ground level

h half of the height of rectangular rocking block

hi height of the ith floor level above the ground

hrocker depth of the rocker

jdF lever arm for F

List of Symbols

xviii

jdsc lever arm for Wsc

Mb base overturning moment

Mb base overturning moment of RC core

Mb,F base overturning moment created by the applied lateral forces

Mb,P-𝛥 base overturning moment due to P-𝛥 effects

Mrock rocking moment

mi seismic mass at the ith floor

Ptoe gravity load carried by rocking toes

R vertical base reaction at rocking toes

r dimension of rectangular rocking block

SMS MCER, 5% damped, short-period spectral response acceleration parameter

V lateral resistance at the base of structures

W weight of rectangular rocking block

Wcore gravity loads tributary to the RC core

WEQ total seismic weight

Wrckr self-weight of the rocker

Wsc total self-centering gravity load 𝛼 slenderness of rectangular rocking block 𝜃rock base rocking rotation angle 𝜃roof roof drift ratio 𝛥rckr lateral displacement of the rocker 𝛥roof lateral displacement at the roof level 𝛥skirt lateral displacement of the skirt diaphragm

Chapter 5

Ay1 cross-sectional area of the yielding segment of one single BRB

dc2c,EW centre-to-centre distance of the mega-columns in the east-west direction

dc2c,NS centre-to-centre distance of the mega-columns in the north-south direction

dimrckr,EW dimension of the rocker on plan in the east-west direction

dimrckr,EW dimension of the rocker on plan in the north-south direction

Ec modulus of elasticity of concrete

List of Symbols

xix

EIeff effective flexural rigidity of the rolling mega-columns

Fy specified yielding strength of structural steel used in BRBs; or specified

yielding strength of diagonal reinforcement used in coupling beams

f height of the spherical cap 𝑓 compressive strength of concrete

fu ultimate tensile strength of steel reinforcement

fy yielding strength of steel reinforcement

g gap distance of rolling section fibres

H height of the benchmark building above the ground level

h height of the spherical cap off the centre of the base

hc height of the rolling mega-columns; or half of the height of rigid rocking

cylinder

hf height of the BRBFs

MrEW overturning moments of the benchmark building about the east-west

direction

MrNS overturning moments of the benchmark building about the north-south

direction

Mrock,rEW rocking moment about the east-south direction

Mrock,rNS rocking moment about the north-south direction

My minimum base overturning moment strength of the benchmark building

nf number of BRBFs in one principal direction

R radius of the sphere

Rc dimension of rigid rocking cylinder

Ry material overstrength factor of structural steel used in BRBs; or material

overstrength factor of diagonal reinforcement used in coupling beams

r radius of the base of the spherical cap

rc cross-sectional radius of rigid rocking cylinder

SMS MCER, 5% damped, short-period spectral response acceleration parameter

t time

V1M0V base shear demand obtained from the rocking-only scenario

V1M0V,EW V1M0V in the east-west direction

List of Symbols

xx

V1M0V,NS V1M0V in the north-south direction

VNS storey shears of the benchmark building in the north-south direction

V lateral resistance provided by the MechRV3D system

Vc buttressing force required by the rolling mega-columns

Vf lateral resistance provided by the BRBFs

Vu ultimate lateral resistance provided by the MechRV3D system; or ultimate

shear strength of coupling beams

Vu,c maximum buttressing force required by the rolling mega-columns

Vu,f ultimate lateral resistance provided by the BRBFs

Vu,gravcol maximum buttressing force required by the gravity columns

Vy expected yielding shear strength of coupling beams

Wcore gravity loads tributary to the RC core

Wrckr self-weight of the rocker

Wsc total self-centering gravity load 𝛼BRB inclined angle with respect to the horizontal direction 𝛼c slenderness of rigid rocking cylinder β compression strength adjustment factor of BRBs 𝛥f peak lateral drift of the BRBFs 𝛥rckr peak lateral displacement of the rocker 𝛥skirt peak lateral displacement of the skirt diaphragm 𝛥X horizontal displacement recorded at the mid-height of rigid rocking cylinder

in axis-X 𝛥Z horizontal displacement recorded at the mid-height of rigid rocking cylinder

in axis-Z 𝛿f peak drift ratio of the BRBFs 𝛿s,EW inter-storey drift ratio of the benchmark building in the east-west direction 𝛿s,core inter-storey drift ratio due to deformation of RC core 𝛿s,rock inter-storey drift ratio due to base rocking rotation 𝛿s,tot total inter-storey drift ratio of the benchmark building

εy yielding strain of longitudinal reinforcement used in RC core walls 𝜃 nutation angle of rigid rocking cylinder

List of Symbols

xxi

𝜃rNS base rocking rotation angle about the north-south direction

κV shear reduction factor

ρ radial coordinate of rolling section fibres

φ precession angle of rigid rocking cylinder 𝜒 axial compressive resistance reduction factor due to buckling effect

ω strain-hardening adjustment factor of BRBs

Chapter 6

An(t) pseudo-acceleration of the SDOF system for the nth mode

Asin coefficient of the term sin

Asinh coefficient of the term sinh

CRT coefficient used in the mode shape function

Dn(t) deformation of the SDOF system for the nth mode

EI flexural rigidity of the cantilever beam


H height of the cantilever beam

KR stiffness of the rotational spring at the base of the cantilever beam

KT stiffness of the translational spring at the base of the cantilever beam 𝐿 𝐿 = 𝑚 𝜙 (𝑧)𝑑𝑧

M bending moment of the cantilever beam

m uniformly distributed seismic mass of a continuum cantilever beam

Mn generalized modal mass of the nth mode 𝑀∗ effective modal mass of the nth mode 𝑀∗ normalized effective modal mass of the nth mode

Mn(z,t) bending moment due to the nth mode

Mno peak value of Mn(z,t) 𝑀 normalized Mno 𝑀 , cumulative modal contribution of the nth mode to the total response

n mode number

q(t) time function

List of Symbols

xxii

qn(t) time function of the nth mode

R normalized stiffness of the rotational spring

Sa(Tn) pseudo-acceleration spectrum ordinate corresponding for period Tn

T normalized stiffness of the translational spring

Tn natural period of the nth mode

t time 𝑢 (𝑡) ground acceleration

u(z,t) lateral deflection of the cantilever beam

un(z,t) lateral deflection due to the nth mode

uno peak value of un(t) 𝑢 normalized uno 𝑢 , cumulative modal contribution of the nth mode to the total response

V shear force of the cantilever beam

Vn(z,t) shear force due to the nth mode

Vno peak value of Vn(z,t) 𝑀𝑉 normalized Vno 𝑉 , cumulative modal contribution of the nth mode to the total response

Z normalized position coordinate

z position coordinate along the height of the cantilever beam

β frequency parameter

βn frequency parameter of the nth mode

Γn modal participation factor of the nth mode 𝛿(z,t) first-order derivative of u(z,t) with respect to z 𝛿no peak value of 𝛿(t) 𝛿 normalized 𝛿no 𝛿 , cumulative modal contribution of the nth mode to the total response ζn damping ratio of the nth mode 𝜙 mode shape 𝜙n mode shape of the nth mode

ω natural circular frequency of vibration

List of Symbols

xxiii

ωn natural circular frequency of the nth mode

Chapter 7

Ay,BRB1 cross-sectional area of the yielding segment of one single BRB

B dimension of generic buildings on plan

bBRBF post-yielding stiffness ratio of the BRBFs

dc2c centre-to-centre distance of the mega-columns

EI flexural rigidity of a continuum cantilever beam or of the stick model

Es modulus of elasticity of structural steel


H height of generic buildings

Heff effective height of generic buildings

HBRBF height of BRBFs

hc height of rolling mega-columns

hgravcol height of gravity columns in the basement

K1,BRBF initial lateral stiffness of the BRBFs

K2,BRBF post-yielding stiffness of the BRBFs

Kb1 initial lateral stiffness of the shear mechanism

Kb2 post-yielding stiffness of the shear mechanism

KF axial stiffness modification factor of BRBs

I(Kb1) degree of isolation based on the initial stiffness

I(Kb2) degree of isolation based on the post-yielding stiffness

Lwp work-point length of BRBs

Mact activation moment of the elastic bilinear rotational spring

Mb,MCE base overturning moment demand at the MCE level

Mb,min minimum base overturning moment strength

Mb,SLE base overturning moment demand at the SLE level

Mb,wind base overturning moment demand due to wind loads

Mrock design rocking moment

m uniformly distributed mass of a continuum cantilever beam

mEQ seismic mass lumped at each floor

List of Symbols

xxiv

nBRBF number of BRBFs in one principal direction

ns number of storeys

RM moment reduction factor

RM,EQ moment reduction factor determined for seismic loads

T1 target fundamental period

T1,a fundamental period obtained from analysis

T1,FB fundamental period of fixed-based structures

Tb isolated period

Tb1 isolated period based on the initial stiffness

Vb,1M0V base shear demand obtained from the rocking-only scenario

Vy yielding strength of the shear mechanism

Vy,BRBF yielding strength of the BRBFs

WEQ total seismic weight 𝛼BRB inclined angle with respect to the horizontal direction 𝛼K post-yielding stiffness ratio of the shear mechanism 𝛥b base displacement of generic buildings [𝛥b] maximum acceptable base displacement 𝛥y base displacement at the yielding of the shear mechanism 𝛿s inter-storey drift ratio of generic buildings [𝛿s] maximum acceptable inter-storey drift ratio

εy,BRB yielding strain of BRBs 𝜃rock base rocking rotation angle λBRB yielding length ratio of BRBs 𝜇V higher-mode shear reduction factor

Chapter 8

R normalized stiffness of the rotational spring

RM moment reduction factor

T normalized stiffness of the translational spring

T1 fundamental period

V1M0V base shear demand obtained from the rocking-only scenario

1

Chapter 1 Introduction

1.1 Urbanization and Development of High-rise Buildings

Urbanization has been happening at an incredible pace around the globe. While, in 1950,

just barely one out of three people in the world was living in cities, the urban population

surpassed the rural population in 2010, while the world’s population increased nearly

threefold during the same period. According to the latest statistics [United Nations 2019],

by 2050, it is projected that 6.7 billion people will be urban residents, representing 68% of

the anticipated world’s population.

With this population agglomeration, urban densification also leads to the concentration of

social assets in cities which provide more than 80% of the global GDPs [Weiss 2001;

Dobbs et al. 2011]. In the meanwhile, a large number of these populous and economically

important cities are located in areas that are at higher risk of natural disasters. In 2018,

1087 cities with over 300000 inhabitants were exposed to at least one of the six major

natural disasters (earthquakes, cyclones, floods, droughts, landslides, and volcano

eruptions), representing 58% of the total 1860 cities with a similar population size around

the world [Gu 2019]. These 1087 cities were home to about 1.6 billion people, accounting

for 64% of the total population of the 1860 cities [Gu 2019]. Hence, in terms of socio-

economic losses and casualties, cities are heavily risk-concentrated [Gencer 2013]. In this

context, one of the critical challenges is to accommodate the skyrocketing urban population

in a resilient and sustainable way. This is now a broad consensus in the 2030 Agenda for

Sustainable Development [United Nations 2015].

To address these challenges, high-rise buildings have been adopted as an efficient solution

given the scarcity of land in cities. As a result, high-rise construction has sharply increased

in many countries over the past few decades. As of 2019, over sixteen hundred 200 m-plus

buildings were in operation around the world [CTBUH 2019]. Many earthquake-prone

cities are leading in this global high-rise development.


2

1.2 Modern Seismic Design and its Limitations

Modern seismic design has undergone significant developments over the past 100 years.

Prior to the 1920s, “engineers were used to thinking of only gravity loads that push straight

down, and of constant wind loads”, as Housner recalled [Scott 1997]. After the 1930s,

earthquake loads were specifically considered in structural design as a rule-of-thumb

fraction of the weight of structures.

In the 1970s, Park and Paulay proposed capacity design principles that allow for a

hierarchical allocation of strength in a structure. Following these principles, some elements

in a structure are intended to yield and undergo significant inelastic deformations during

major earthquakes. These designated yielding elements lead to the other components in the

structure being less sensitive to seismic excitations and could be capacity designed to

remain elastic. These design principles were soon adopted in major seismic codes around

the world and profoundly influence even current design practice.

Current design codes follow a force-based approach, requiring a minimum strength of

structures to be ensured at design level earthquakes. However, this single point check may

not be sufficient to ensure that the expected seismic performance can be achieved under

earthquakes at the other intensity levels. In the Vision 2000 report [SEAOC 1995], the

Structural Engineers Association of California (SEAOC) recommended that ordinary

buildings can be designed to meet categorized performance objectives which represent

combinations of performance levels and seismic hazard intensities. This is recognized as

the first document that outlines a framework of performance-based seismic design (PBSD).

Following this framework, structures are expected to remain operational under frequent

events, undergo significant damage during design level earthquakes without jeopardizing

life-safety, and avoid collapse during rare earthquakes with a probability of exceedance of

2% in 50 years. A similar design framework was adopted in FEMA-273 [FEMA 1997] and

its substitute prestandard FEMA-356 [FEMA 2000], both of which were developed for

retrofitting existing buildings and eventually replaced by the standard document ASCE-41

[ASCE 2017]. FEMA-450 [FEMA 2003] provides guidance for the design of new

buildings following PBSD procedures.


3

The concept of PBSD has also been applied to the design of high-rise buildings more

recently. The Tall Buildings Initiative [Moehle et al. 2011] that was launched by the Pacific

Earthquake Engineering Research Centre (PEER) marked a significant contribution in this

direction. This program provided quantitative guidance to all the design steps ranging from

performance objectives to seismic hazard analysis, from nonlinear modelling to acceptance

criteria, and from engineering demands to loss assessment, all of which lead to a better

prediction of the seismic performance of high-rise buildings during major earthquakes and

a probability-based estimation of potential socio-economic impacts associated with these

seismic responses.

While life-safety and collapse-prevention are prioritized during major seismic events,

structural damage is not precluded given the expected inelastic response of structures when

following capacity design principles. As a result, structures that are designed in this way

are expected to sustain extensive damage and therefore repair or replacement of damaged

elements may become either technically infeasible or economically impractical. Structures

with sustained damage may not be adequate in resisting aftershocks and future earthquakes

and have to be demolished despite a low risk of collapsing. This leads to significant

downtime and tremendous indirect losses, as clearly highlighted during the devastating

Christchurch Earthquake on February 22, 2011 [Kam et al. 2011].


4

1.3 Low-damage Design towards Seismic Resilience

With the increasing socio-economic demand on rapid recovery after disasters, seismic

resilience has gradually become an enhanced objective for the design of high-rise buildings.

The seismic design of high-rise structures is evolving from performance-based methods to

resilience-targeted approaches. In line with this trend, some design guidelines [Almufti and

Willford 2013] have been proposed to facilitate the design of the next generation high-rise

buildings.

At the same time, numerous high seismic performance systems have been proposed for

high-rise buildings, aiming at achieving seismically resilient structures. These systems

have different configurations and involve varied nonlinear mechanisms that are expected

to limit seismic demands, absorb earthquake energy, and undergo significant inelastic

deformations or rigid-body motions during major earthquakes. From the strategic

perspective, most of these systems can be related to three basic concepts including base

rocking, base isolation, and combined uses of these two mechanisms. However, when

applied to high-rise buildings, these systems exhibit a number of limitations. While

allowing structures to rock at the base has been proven ineffective in controlling the higher-

mode response, base isolating high-rise buildings is challenging due to isolators that can

be overloaded vertically while undergoing significant lateral deformations. Further,

introducing both rocking and isolating mechanisms at the base of structures to improve the

base isolated structure’s response is also challenging, particularly when these two

mechanisms are arranged in series. Hence, there is an urgent need to seek more efficient

and realistic solutions for controlling the higher-mode response and achieving enhanced

seismic resilience for high-rise buildings.


5

1.4 Research Objectives

To this end, this dissertation focuses on the development of a novel system that provides

dual seismic protective mechanisms that act independently at the base of high-rise

buildings. This system is expected to effectively mitigate higher-mode effects in high-rise

buildings, leading to a low-damage design of these structures. In line with this motivation,

specific research objectives are outlined as follows:

• to propose a structural concept that allows for the uncoupling of the flexural and

shear responses of tall structures;

• to develop a possible implementation of the proposed structural concept;

• to validate the feasibility of the proposed system through numerical analyses;

• to investigate the impact of the proposed system on structures in a general sense

through analytical studies and parametric analyses;

• to recommend a design framework to facilitate the preliminary design of the

proposed system.

1.5 Research Methodologies

To achieve these objectives, the research work began with an extensive conceptual design

during which a variety of possible ideas were developed for the intended base-mechanism

system. Pros and cons were examined for each of these concepts, leading the conceptual

design towards a scheme that in principle met the original design intent of this research.

This scheme was selected as the system that was further studied in this dissertation.

The feasibility of this proposed system was investigated numerically in two stages. In the

first stage, the proposed system was represented using an idealized configuration on which

nonlinear static and dynamic analyses were conducted to understand the fundamental

mechanics that govern the proposed system. In these preliminary analyses, a 42-storey RC

core-wall building was used as a reference structure and represented using a simplified

model.

Based on this idealized configuration, a physical embodiment was proposed, designed and

detailed as a possible way of implementing the proposed system for practical applications.


6

In the second stage of the feasibility study, this embodiment design was numerically

validated using advanced nonlinear models and nonlinear response history analyses. In

these analyses, the same benchmark building was used as the reference structure. In

contrast to the first stage, a three-dimensional inelastic model was built for the benchmark

building and validated as a good representation of a conventional design conducted

following performance-based design approaches.

At the base of the reference structure, the proposed system was incorporated in (1) a

rocking-only system and (2) the full dual-mechanism, respectively. While the former only

allows for nonlinear flexural response at the base of the structure, the latter allows the

nonlinearity to occur in both flexure and shear. These two case scenarios were compared

with the conventional fixed-based design, in terms of the seismic performance of the

superstructure. These comparisons are expected to demonstrate the differential efficacy of

a single flexural mechanism and the proposed dual-mechanism system in controlling the

higher-mode response and minimizing damage to these structures.

After this case study, the research work extended to a more general scope through

analytical studies on continuum cantilever systems that were flexibly supported in both

rotational and translational degrees-of-freedom at the base, simulating high-rise buildings

with the proposed system incorporated at their base. These closed-form studies are

expected to provide insights on the impact of base fixity on higher-mode effects in high-

rise buildings.

Further to these theoretical studies, parametric nonlinear analyses were also conducted,

considering multiple design parameters that govern the responses of high-rise buildings

and the proposed base-mechanism system. Based on these analyses, general procedures

were developed to facilitate the preliminary design of the proposed system.


7

1.6 Organization

This dissertation is organized as follows.

Chapter 2 provides a detailed review on higher-mode effects that are a major challenge for

the design of high-rise buildings. This includes numerical studies and experimental tests

that were conducted previously on this subject, and approaches that were proposed for

evaluating the higher-mode-induced dynamic amplification of a building’s seismic

response. After examining previously observed damage to RC wall structures, a variety of

low-damage systems are reviewed, which defines the main directions of this study.

A series of novel structural systems that were developed at the conceptual design stage of

this thesis are presented in Chapter 3. While improvements of each generation of systems

are highlighted, critical limitations are also identified, converging to a more promising

design scheme that is investigated in depth in the following chapters.

In Chapter 4, the proposed system is investigated numerically through an idealized

configuration. After presenting the fundamental mechanics of the proposed system, a

physical implementation is proposed. A few design options that were considered during

the schematic design stage of this study are discussed before a detailed design of the novel

system is proposed.

Chapter 5 describes the numerical validation that was conducted to verify the feasibility of

the proposed system. Advanced nonlinear modelling is at the heart of this chapter,

including a novel modelling approach that was developed for the proposed system. Seismic

performance of a reference high-rise building is investigated with the proposed system

incorporated at the base of the structure.

Chapter 6 demonstrates an analytical study that sheds light on the dynamic response of

high-rise buildings under generalized base conditions that are analogous to the constraints

provided by the proposed system. Results of this study provides insights on higher-mode

effects of high-rise structures and more specifically with uncoupled shear and flexural

mechanisms at their base.


8

Chapter 7 discusses parametric studies that were conducted to generalize the design of the

proposed systems. Based on these studies, a design framework is outlined to facilitate the

preliminary design of the proposed system.

Chapter 8 summarizes the major findings and contributions of this thesis and provides

recommendations for future research.

9

Chapter 2 Background and Literature Review

2.1 Introduction

This chapter provides a background upon which the present research was initiated. Section

2.2 provides detailed discussions regarding the significance of higher modes on the seismic

response of high-rise buildings in elastic and more importantly inelastic ranges. These

discussions include numerical and experimental studies that have been conducted

previously on higher-mode effects, and proposed approaches for evaluating the dynamic

shear amplification. Section 2.3 presents the different types of damage that have been

observed in RC walls in past earthquakes. To achieve low-damage design, a few high-

performance systems have been proposed, including rocking systems, base isolation

systems and dual protection systems, which are discussed in Section 2.4. These damage-

resistant systems are carefully reviewed with their limitations identified. Section 2.5

concludes this chapter with a brief summary.


10

2.2 Seismic Demands on RC Wall Buildings

2.2.1 Higher-mode effects on elastic structures

High-rise buildings are characterized by high lateral flexibility which makes these

structures more sensitive to dynamic loads in terms of the response to high-frequency

vibration modes. During seismic loading, these higher modes can be more significantly

excited, leading to the seismic response of high-rise structures being more complex than

lower-rise buildings whose response is primarily governed by the fundamental modes of

the structure. This difference can be intuitively perceived by interpreting modal responses

from a pseudo-acceleration response spectrum. Unlike low-rise buildings, slender

structures usually have a longer fundamental period that likely exceeds the predominant

period of earthquake ground motions, which leads to an attenuated first-mode response. In

addition, the periods of higher modes (particularly the second and third ones), which are

typically a fraction of the fundamental one, become longer as well and more likely to fall

in the high-acceleration region of the spectrum, and consequently making these modes

more significantly excited under seismic ground motions. This trend can be further

intensified if the ground motion displays greater high-frequency contents. As a result,

higher-mode contributions to the total response cannot be ignored for high-rise buildings

as they typically are for low-rise structures.

Studies on higher-mode effects can be traced back to the 1940s. Biot [1943] compared the

relative modal significance using an “effectiveness factor”, Cn, which is equivalent to the

modal contribution factor, 𝑟 , as defined in [Chopra, 2000]. Based on these comparisons,

Biot [1943] stated that higher modes were generally less important than the fundamental

one. This is believed to be an incomplete conclusion because the Cn-factors only indicated

the static component of modal responses without accounting for the dynamic amplification.

Clough [1955] bridged this gap by using a modal superposition method by which dynamic

shear demands resulting from the first three modes were added to compute the seismic

response of the structure. In the most critical combination, the base shear contributed by

the second and third modes was found to be 15% of that due to the first-mode, leading to

the total base shear demand (the minimum among the varied calculations) being 2.4 times


11

the value prescribed in the 1952 edition of Uniform Building Code. Nevertheless, Clough

[1955] did not see this exceedance as a concern with one of the arguments being that

material yielding may prevent dynamic resonance, which leads to the elastic shear

amplification, from happening, such that shear demands would be somewhat limited at the

yielding level. However, this shear-limiting effect was not justified by Clough [1955] for

all the modes especially the high-frequency ones.

Using a 5-storey frame structure as an example, Chopra [2000] conducted modal response

spectrum analyses (MRSA) in which the fundamental period, T1, and the beam-to-column

stiffness ratio, ρ, were chosen as parameters. Results of these analyses indicated that the

percentage of higher-mode contribution increases with increasing T1 and decreasing ρ,

particularly when T1 was in the velocity- and displacement-sensitive regions of the

spectrum. This implies that slender structures, such as RC wall buildings, that are

characterized by longer natural periods and flexure-dominated behaviour, have innate

susceptibility to more pronounced higher-mode effects.

2.2.2 Numerical studies on higher-mode effects in inelastic structures

RC frame structures

Most of the early studies on higher-mode effects were focused on elastic systems. However,

given the principles of ductile capacity design, modern earthquake-resistant structures are

expected to respond inelastically under strong ground motions, developing yielding

mechanisms in a predetermined way to make structures more robust to a wider range of

extreme loading conditions and the design forces less dependent on the amplitude of the

applied ground motion. However, this intended effect may not be fully achieved in MDOF

structures. This was first pointed out by Park and Paulay [1975] for multi-storey RC frame

structures where, despite the formation of beam-mechanisms, higher-mode effects induced

unexpected moment re-distributions to columns, leading to these elements being yielded

even though they were designed to remain elastic.

As for cantilever shear walls, the designated yielding mechanism is intended to occur only

at the base of the wall in the form of flexural hinges and is not distributed throughout the

structure as in frame structures. Given this concentrated plasticity, whether higher-mode


12

effects can cause unexpected failure modes to wall structures has sparked a considerable

amount of research over the past decades.

RC wall structures

Blakeley et al. [1975] first investigated higher-mode effects of cantilever walls in the

inelastic response range. From response spectrum analyses where a critical combination of

the first three modes was assumed, they found that the ratio of the maximum base shear

demand at the expected flexural capacity to that derived from the code-defined lateral load

distribution ranged from 1.4 to 4.0 for cantilever walls of different heights and design

parameters. In comparison, from inelastic dynamic analyses that were conducted on 6- to

20-storey cantilever walls under varied ground motions, the dynamic shear amplification

ratio was found to vary from about 1.0 to 3.4, which is slightly lower than that obtained

from the response spectrum analyses. This slight reduction may result from that in the

inelastic dynamic analyses, beam-column elements with lumped plastic hinges at both ends

were used for all storeys of the walls, allowing for the inelastic response to occur

throughout the height of the structures. This dynamic amplification was found to be more

significant for storey shears than for base overturning moments. As a result, a base

moment-to-shear ratio as low as 23% of the wall height, H, was reported by Blakeley et al.

[1975]. This is much lower than the effective height that is calculated from a first-mode

lateral force distribution. Blakeley et al. [1975] also drew attention to the possibility of

flexural yielding along the height of walls which could result in unintended plastic hinges

above the base.

Derecho et al. [1978a] carried out similar studies in which inelastic dynamic analyses were

conducted parametrically on cantilever walls of different heights. Based on the analysis

results, Derecho et al. [1978a] concluded that the dynamic shear amplification increased

with increasing fundamental period, T1, and increasing flexural ductility of the base plastic

hinge, 𝜇𝜙. These are similar to the findings of Blakeley et al. [1975]. The only difference

is that Blakeley et al. [1975] used the flexural overstrength, Mo, rather than ductility to

account for the impact of inelasticity, such that they found the dynamic amplification to be

inversely correlated with Mo.


13

These pioneering findings were followed by a large number of follow-up studies.

Kabeyasawa and Ogata [1984] conducted numerical analyses on wall-frame structures to

evaluate the dynamic magnification of shear demands for use in the ultimate state design.

Keintzel [1984] and Eibl and Keintzel [1988] investigated the impact of higher modes on

base shears of cantilever walls in the context of German seismic codes, which provided a

basis for the draft Eurocode 8. Amaris [2002] studied the higher-mode-induced dynamic

amplification in both shears and moments for cantilever walls that were designed using the

displacement-based design approach. Rutenberg and Nsieri [2006] re-examined the

European code provisions on the seismic shear design of cantilever walls, considering the

influence of non-simultaneous yielding in multiple walls. Rejec et al. [2012] corrected the

misconceptions in the use of the Eibl and Keintzel model [1988] for predicting nonlinear

dynamic shear amplification and recommended to extend the use of the Eurocode 8

approach that is formulated for evaluating shears in highly ductile walls to RC walls with

moderate ductility in order to achieve a conservative design.

In most of these studies, nonlinear response history analyses (NLRHAs) were conducted

under strong ground motions. While inelastic models were built for RC walls, plasticity

was only allowed for at the base (representing the expected flexural hinge) with the rest of

the structure being assumed to remain elastic. The inelastic seismic response was then

compared against design demands that were determined using code-prescribed equivalent

lateral force (ELF) methods and MRSAs. Across these comparisons made in varied code

contexts, exceedance of code prescribed values was consistently reported and attributed to

an inadequate consideration of higher-mode effects. This inadequacy resulted in

unconservative design demands, in terms of magnitude and the height-wise distribution of

these demands, which implied potential unintended failure modes of structures under major

earthquakes. While this problem is inherent with the ELF method which is based on the

first mode only, it still exists when using the MRSA approach even though multiple

vibration modes are considered.

Modal sensitivity to base rotational constraint

In fact, the MRSAs’ inclusion of multiple modes is negated by the use of an identical

reduction factor for all modes in the design of ductile structures. Paulay and Priestley [1992]


14

pointed out that vibration shapes of the second and third modes of an elastic cantilever are

similar irrespective of a fixed or hinged base condition. This independence of hyperbolic

higher modes to the rotational boundary condition explains Keintzel’s earlier statement

[1990] that the flexural hinging at the base of cantilever walls may limit the first-mode

response but barely affects those associated with the higher modes. Sangarayakul and

Warnitchai [2004] looked into this phenomenon by using an approximate modal

decomposition of the inelastic dynamic responses of wall buildings and found that, as the

intensity of ground shaking increased, the first mode was yielded at a lower intensity level,

and, due to an increased hysteretic damping, the modal response soon approached a

maximum bound, which was referred to as the modal saturation. In contrast, the higher

modes were saturated at higher intensity levels, which led to these modes dominating the

total inelastic response. Munir et al. [2012] applied this nonlinear modal decomposition

approach to a 40-storey core-wall building and revealed that while a significant hysteresis

was developed in the first mode as a result of the base flexural plastic hinge, a yielding

mechanism was either just slightly engaged or not at all in the higher modes.

The differential modal sensitivity to base rotational constraint was analytically verified by

Wiebe and Christopoulos [2009] who studied modal responses of a continuum cantilever

beam with the base constraint in a pinned condition. Wiebe and Christopoulos [2009] found

that, while the fundamental sway mode vanishes as a result of this rotational release, the

first and second pinned-base modes exhibited similar vertical distribution profiles of

displacements, lateral forces, shears and moments as those in the fixed-base second and

third modes, except a zero-moment at the base due to the idealized pinned condition. This

suggested that these non-sway sinusoidal modes were insignificantly affected by the pinned

condition at the base of the cantilever, allowing seismic demands to continually increase in

these modes. Wiebe and Christopoulos [2015a] extended this closed-form study and

demonstrated an evolution of modal responses as the rotational fixity was incrementally

reduced at the base of cantilevers. A similar theoretical study was done by Pennucci et al.

[2015].

Assuming inelasticity in the first mode only, Priestley [2003a] proposed a modified modal

superposition (MMS) method by which the ductile first mode response is combined with


15

elastic higher-mode responses. When this approach was applied to cantilever walls, the

calculated shears and moments closely matched the NLRHA results obtained by Priestley

and Amaris [2002]. However, when using the MMS for RC frames, Priestley [2003b]

obtained highly conservative predictions which were attributed to two traits of frame

structures: narrowly separated periods and more distributed plasticity. The former may lead

to higher-mode periods being shifted off the resonant response in an acceleration spectrum.

This tendency can be further enhanced by the latter which may lead to higher mode periods

being elongated to a greater extent than in cantilever walls.

Realistic estimates of higher-mode effects in inelastic structures

Nevertheless, higher-mode effects can be unduly overestimated for wall structures as well,

particularly when softening effects throughout the height of walls are possible in reality

but not realistically allowed for in the nonlinear modelling as the cases in the studies

reviewed previously. To this end, more sophisticated modelling techniques were used for

walls to achieve an accurate evaluation of inelastic higher-mode effects. When examining

the adequacy of the Canadian code provisions in allowing for higher-mode effects, Boivin

and Paultre [2012a] accounted for distributed flexural plasticity, inelastic shear

deformation, and shear-flexure interaction of cantilever walls, which they stressed, if not

considered, could produce overrated predictions and incorrect trends. Similar conclusions

were drawn by Mehmood et al. [2017] after investigating the seismic response of a 20-

storey core-wall buildings. Rad and Adebar [2008] allowed for distributed plastic hinging

along the height of cantilever walls and decreasing shear stiffness due to the diagonal

cracking, which reduced dynamic shear amplification by 40%. Pennucci et al. [2015]

included the degree of coupling (DOC) as a third parameter (in addition to the fundamental

period and ductility demand or flexural overstrength) that affects the inelastic higher-mode

response, and showed that coupled shear walls with a higher DOC may have greater loss

of stiffness over the height of the structure than lightly coupled walls, resulting in a larger

reduction of seismic demands. The influence of coupling beams was also highlighted by

Panagiotou [2017] who compared the seismic response of core-wall buildings with and

without coupling beams, and found that the presence of coupling beams could lead to a


16

reduction of 28% and 20% in base shears and peak accelerations respectively, both of

which are highly affected by higher-mode effects.

2.2.3 Experimental studies on inelastic higher-mode effects

Early dynamic tests at University of Illinois

Experimental evidence of higher-mode effects is limited since most of the tests that have

been conducted on RC walls were under static or pseudo-dynamic loading conditions. In

the 1970s and 1980s, a series of dynamic experiments were carried out at the University of

Illinois, including work done by Aristizabal-Ochoa [1976], Abrams [1979], Moehle [1980],

and Eberhard [1989]. In total, eleven small-scale nine/ten-storey coupled shear walls and

frame-wall structures were tested under the strong ground motions recorded from the 1940

Imperial Valley Earthquake and 1952 Tehachapi Earthquake. Results of these experiments

supported the necessity to consider the varying distribution of lateral inertial forces due to

the influence of higher modes for the design of RC wall buildings.

Full-scale shake-table tests at University of Berkeley, San Diego

Panagiotou et al. [2007a, 2007b] conducted seminal full-scale shake-table tests on two 7-

storey RC wall slices with rectangular and T-shaped sections respectively. Both specimens

were designed using a displacement-based method that was proposed by Panagiotou et al.

[2011a] to allow for the higher-mode influence on expected peak forces.

While the specimens responded with the expected inelasticity under strong ground shaking,

significant higher-mode effects were observed [Panagiotou et al. 2011b]. Firstly, the base

shear of both specimens reached a peak value at a different times than when the maximum

base overturning moments were reached. At the instant of the peak base shear, floor

accelerations measured at lower storeys were larger than those in upper storeys, leading to

the resultant lateral force located at 46% of the height of the structure. This evidently

marked the dominance of higher modes on lateral inertial forces and, in turn, on storey

shears. This led to a dynamic amplification ratio of 4.2 for the base shear at the DBE level,

compared with a ratio of only 2.7 for the base overturning moment. As a result, the effective

height of lateral forces was largely reduced to about half the height of the walls.


17

As the intensity of table shaking increased, greater dynamic shear amplification was

observed in lower storeys. A bulged shape was not seen in the shear envelope at top levels

of the specimens. This was likely a result of the small number of storeys which was selected

to ensure the validity of the design assumption that the lateral deformation of the specimens

would be governed by the first mode only. This limitation may also explain the observation

that lateral drifts of the specimens were concentrated in the bottom three storeys, resulting

in a straight-line distribution of inter-storey drift ratios (IDRs) in upper storeys.

Large-scale shake-table tests at Ecole Polytechnique, Montreal

Ghorbanirenani et al. [2012] carried out shake-table tests on two half-scale 8-storey

cantilever walls that were designed in compliance with the Canadian codes provisions and

subjected to an artificial accelerogram that represented high-frequency earthquake ground

motions in eastern North America.

Dynamic responses measured at the DBE level (2% probability of exceedance in 50 years)

were investigated in both time and frequency domains. It was observed that base shears,

Vb, and IDRs of the sixth storey, 𝜃6, were dominated by the second mode and considerably

influenced by the third one as well. In contrast, roof displacements, 𝛥r, base overturning

moments, Mb, and IDRs of the first storey, 𝜃b, were basically first-mode responses.

Affected by higher modes to varied extents, some responses were not found to be strongly

correlated to each other as they would be in a structure whose response is governed by the

first mode. A weakened correlation was observed between 𝛥r and 𝜃b whose peak values

occurred at different times. This was caused largely by significant inelastic drifts that

occurred in upper storeys due to higher-mode effects. The magnitude of these rotations was

comparable with (at DBE) or even greater (at 200%DBE) than that of the base plastic hinge.

Distributed plasticity also led to the IDR envelope following a highly curved profile which

significantly differed from that obtained in [Panagiotou et al. 2011b] where no plastic

hinges were observed above the base of the walls.


18

Hybrid tests at University of Sherbrooke

Fatemi et al. [2020] conducted hybrid tests on a ductile RC wall that is the lateral-force-

resisting system of an 8-storey building located in Rivière-du-Loup, Québec, Canada.

While only the base hinge zone was physically modelled and tested, responses of the upper

part of the wall were represented using a nonlinear computer model. The wall specimen

was designed to the latest Canadian code provisions where new dynamic amplification

factors were recommended to allow for inelastic higher-mode effects. However, in order

to emphasize the higher mode effects, these factors were not adopted in the design of the

specimen. As a result, maximum base shears were measured to be 2.15 times the design

value at the DBE level (2% probability of exceedance in 50 years) and 3.01 times at the

200%DBE level. The new code-recommended factor would be defined as 1.5 for this

structure which still appears as an unconservative estimate.

2.2.4 Evaluation of higher-mode effects

Quite a few formulae have been proposed over the past decades for a reasonable evaluation

of higher-mode effects, on which Rutenberg [2013] provided a comprehensive overview.

These quantitative predictions are presented either using a lumped dynamic amplification

factor; or, more directly, through a modified modal superposition approach; or, with a

slightly increased complexity, based on a concept of substitute structures. In this section,

the formulations defining these three approaches are presented.

Dynamic amplification factors

Blakeley et al. [1975] first recommended dynamic amplification factors, 𝜔V, that were used

to rectify code-specified shear demands. Paulay and Priestley [1992] formulated these

factors into Equation (2.1), where only the number of storeys, n, is involved as the variable,

without reflecting the identified influence of periods and flexural overstrength. This

formula was adopted in the New Zealand Standard up to the current edition, which affected

many other seismic codes.

𝜔 = 0.9 + 𝑛 10⁄ 𝑛 ≤ 61.3 + 𝑛 30⁄ ≤ 1.8 𝑛 > 6 (2.1)


19

Realizing the importance of periods to this phenomenon, Priestley [2003a] updated

Blakeley [1975]’s formula by replacing the factor, n, with BT1 which is a function of the

fundamental period, T1, as shown in Equation (2.2). In these expressions, displacement

ductility, 𝜇, is included as well.

𝜔 = 1 + 𝜇𝜙 𝐵 0.067 ≤ 𝐵 = 0.067 + 0.4(𝑇 − 0.5) ≤ 1.15

(2.2)

In the context of Eurocode 8, Rutenberg and Nsieri [2006] proposed a formula where the

dynamic shear amplification is linearly proportional to the fundamental period, T1, and

ductility demand, 𝜇, as expressed in Equation (2.3). They restricted the validity of this

expression to a single cantilever wall or multiple walls with an equal length.

𝜔 = 0.75 + 0.22(𝑇 + 𝜇 + 𝑇 𝜇) (2.3)

In addition to Equation (2.3), Rutenberg and Nsieri [2006] also proposed a design envelope

for storey shears, as shown in Figure 2.1. The higher-mode influence on the distribution of

storey shears was accounted for through a height ratio, 𝜉, that was calculated as 𝜉 = 1.0-

0.3T1 and 𝜉 ≥ 0.5.

Figure 2.1 Design envelope for storey shears

To keep a consistent format to the one used in the Canadian codes, Boivin and Paultre

[2012b] proposed the dynamic shear amplification factors that are listed in Table 2.1. They

adopted the approach used by Rutenberg and Nsieri [2006] to define the shear envelope,


20

but slightly adjusted the expression of 𝜉 to be 1.5-T1 and bounded them between 0.5 and

1.0. In Table 2.1, Rd and Ro are the reduction factors related to ductility and overstrength

respectively, and γw is the base flexural overstrength factor.

Table 2.1 Dynamic shear amplification factors proposed by Boivin and Paultre [2012b]

RdRo/γw T1 ≤ 0.5 sec T1 ≥ 1.0 sec

2.80 1.0 2.0 1.87 1.0 1.5

≤1.40 1.0 1.0 Modified modal superposition

Kabeyasawa [1988] suggested to estimate ultimate base shears, QDU, as an arithmetic sum

of a static component, QSU, representing the maximum possible first-mode base shear, and

a fluctuating component, BFU, allowing for the higher-mode contribution using Equation

(2.4) as follows,

𝑄 = 𝑄 + 𝐵 = 𝑄 + 𝐷 × 𝑊 × 𝐶 (2.4)

where W is the total seismic weight, and Cg is the normalized peak ground acceleration. Dm

in Equation (2.4) denotes a cumulative dynamic amplification factor that covers all the

higher modes, and is evaluated with respect to the ground acceleration using simplified

higher-mode shapes. Equation (2.4) is the first proposal that is based on a concept of modal

decomposition, even though the response of higher modes is computed in a lumped manner

rather than individually.

Eibl and Keintzel [1988] proposed a SRSS modal superposition method for calculating the

base shear demand, including the first two modes. Following this method, the dynamic

amplification factor was calculated as follows,

𝜔 = 𝛾 𝑀𝑀 + √0.1 𝑆 (𝑇 )𝑆 (𝑇 ) 𝑅 (2.5)


21

where γRd is the overstrength factor due to strain-hardening, MRd and MEd are respectively

the design flexural resistance and the design bending moment, and R is the force reduction

factor. In Equation (2.5), the first term under the square root represents the ductile first-

mode base shear considering overstrength, while the second term represents the elastic

second mode contribution which is √0.1 𝑆 (𝑇 ) 𝑆 (𝑇 )⁄ times the first-mode response,

where Sa(Tc) and Sa(T1) are the spectral accelerations at the corner period, Tc, and the

fundamental period, T1.

In a similar format, more higher modes are included in the MMS procedure proposed by

Priestley [2003a], and superimposed with the ductile first-mode response following

Equation (2.6) as follows,

𝑉 = 𝜙 𝑉 ,𝑅 + 𝑉 , + 𝑉 , + ⋯ (2.6)

where Vi is the shear demand at the ith storey, VEi,j is the elastic shear demand at the ith

storey contributed by the jth mode (j = 1, 2, 3, …), and R is the force reduction factor.

Assuming that R is equal to the ductility factor, 𝜇, this expression can be rewritten as

𝑉 = 𝑉 , + 𝜇 𝑉 , + 𝑉 , + ⋯ (2.7)

where Vi,j stands for modal shears obtained from an inelastic response spectrum. This

indicates that the dynamic amplification increases with increasing intensity.

Panagiotou [2017] proposed the enhanced response spectrum analysis (ERSA) method for

calculating the actual base shear demand, Vb. In the ERSA, inelastic higher-mode responses

were accounted for by using different force reduction factors, Rh, rather than R which was

used for the first mode only.


22

𝑉 = 𝑀∗ 𝑆 (𝑇 )𝑅 𝜙 + 𝑀∗ 𝑆 (𝑇 )𝑅 + 𝑀 − 𝑀∗ − 𝑀∗ 𝑃𝐺𝐴𝑅 (2.8)

where 𝑀∗ and Ti are respectively the effective mass and period of the ith mode, Sa(Ti) is

the spectral ordinate of the design response spectrum at period Ti, MEQ is the total seismic

mass of the structure, and PGA is the peak ground acceleration. In the calculation of the

higher-mode reduction factors, the influence of coupling beams was accounted as follows,

without coupling beams 𝑅 = 2.2(1 + 0.05𝑇 )[0.8 + 0.26(𝑆 𝑔⁄ − 1)] (2.9)

with coupling beams 𝑅 = 2.8(1 + 0.05𝑇 )[0.8 + 0.26(𝑆 𝑔⁄ − 1)] (2.10)

In Equations (1.9) and (1.10), SMS is the short-period spectral acceleration at the MCE level,

and 𝑔 is the gravitational acceleration.

Compared with a simple shear amplification factor, these modified modal superposition

approaches require more computational effort. However, these multi-mode methods can

provide a relatively more rational estimate on higher-mode contributions by evaluating

individual modal responses, especially, in some of these approaches, where the inelastic

response of higher modes is accounted for using a different force reduction factor for higher

modes than for the first-mode response.

Substitute structures

Yathon [2011] suggested to predict actual shear demands of cantilever walls by conducting

MRSAs on the structure with a fixed and pinned base respectively. From the fixed-based

analysis, elastic first-mode shears, VE1, are extracted and applied with the force reduction

factor, R. The pinned-based system serves as a substitute structure representing the inelastic

response of walls. From this pinned-based analysis, elastic higher-mode shears are obtained

but only a fraction of them is added to the inelastic first-mode response which has been


23

calculated as VE1/R. This fractional factor, Hf, increases with increasing R as calculated

using the following equation,

𝐻 = 1.1 − 𝑒 . ( ) 𝑆 (0.2) 𝑆 (2.0) ≥ 8.0⁄1.0 − 𝑒 . ( ) 𝑆 (0.2) 𝑆 (2.0) < 8.0⁄ (2.11)

where Sa(0.2) and Sa(2.0) are the spectral accelerations at periods of 0.2 and 2.0 sec

respectively.

To avoid running MRSAs twice in practical design situations, Yathon [2011] made a

simplification that the elastic higher-mode response of the pinned-based substitute system

can be approximated by multiplying the elastic first-mode response of the fixed-based

system with a factor, Rpf, which is calculated based on the period of the second mode, T2,

as follows

𝑅 = 2𝑇 𝑇 < 0.25𝑠0.5 𝑇 ≥ 0.25𝑠 (2.12)

As such, the dynamically amplified shear demands, Va, can be calculated based on the

elastic first-mode shear, VE1, as follows,

𝑉 = 𝑉𝑅 + 𝐻 𝑅 𝑉 (2.13)

The substitute system used in Yathon’s [2011] approach is an idealized equivalence since

the flexural stiffness and strength are both equal to zero at the pinned base, which is not

the case for actual flexural hinges at the base of RC walls. However, this should not

essentially affect the validity of this method given the fact that the higher vibration mode

shapes are not significantly sensitive to the base rotational fixity.


24

Pennucci et al. [2015] developed a more strict method for coupled shear walls, where

inelastic higher-mode responses were obtained from a series of equivalent elastic substitute

structures, as illustrated in Figure 2.2. This equivalency was achieved through using the

effective stiffness of the plastic hinge at the base of walls and both ends of coupling beams.

This implies that dynamic properties of these mode-specific substitute structures would

vary as the overall ductility demand increases in the original structure. However, Pennucci

et al. [2015] made a simplification on damping and assigned all higher modes with a

constant damping ratio equal to the elastic damping of the original structure. Responses

contributed by these effective higher modes were then superimposed with the ductile first-

mode response.

Figure 2.2 Effective modal substitute structures (from [Pennucci et al. 2015])


25

2.3 Damage to RC Wall Structures

Modern seismic design philosophies target life-safety and collapse-prevention under major

earthquakes. However, given capacity design principles, structural damage is not precluded

as a result of the intended ductile response of designated elements. As such, structures that

are designed to current seismic codes are expected to sustain damage which can be more

extensive for high-rise buildings due to pronounced higher-mode effects. This damage may

cause interruptions to the immediate re-occupancy and continued operations, which results

in significant losses. In the following section, the different types of damage that are

expected in ductile wall structures is presented.

2.3.1 Flexural hinges at the base of RC walls

While base flexural hinges help limit forces that are applied to cantilever walls, inelastic

curvature demands can result in substantial damage at the base of the structure, including

concrete crushing, yielding of steel reinforcement, and unintended buckling or even

fracture of longitudinal reinforcement. In addition, large diagonal cracking may also occur

as a result of the interaction between the flexural demand and shear forces which can be

significantly amplified due to the higher-mode response. Given the flexural overstrength

and the shear-flexure interaction, the plastic hinging zone can extend over multiple storeys

in the height of RC walls. This has been numerically demonstrated in [Bohl and Adebar,

2011] in which a formula was proposed to provide a lower-bound estimate on the

equivalent plastic hinge length, 𝑙 , over which the maximum inelastic curvature is uniform

at the base of RC walls. Assuming a linear distribution of inelastic curvatures at the base

of walls, 𝑙 is half of the length, 𝑙∗ , over which curvatures exceed the yielding limit. Based

on the proposed lower-bound formula [Bohl and Adebar, 2011], for 55 m high isolated

shear walls, 𝑙 can reach 4.59 m, implying an 9.2 m-high plastic hinge zone at the base of

the wall. This was compared with 𝑙∗ = 11.2 m which was obtained by Bohl and Adebar

[2011] from a more sophisticated nonlinear finite element analysis.

This type of damage has also been observed during the shaking-table tests conducted by

Panagiotou et al. [2007a, 2007b] and Ghorbanirenani et al. [2012] respectively, and in past

major earthquakes, as shown in Figure 2.3. Birely [2012] presented a comprehensive


26

overview of RC wall damage that was observed in over 20 major earthquakes, and failure

modes of walls that were witnessed from a large number of experimental studies.

Figure 2.3 RC wall damage observed in the 2010 Chile Earthquake


27

2.3.2 Distributed plasticity due to higher-mode effects

If not properly accounted for in the design, higher-mode effects may also lead to flexural

plasticity spreading over the height of buildings. Blakeley et al. [1975] predicted

unintended plastic hinging above the base of cantilevered walls where the design moment

envelope did not adequately account for higher-mode amplification. This phenomenon was

also revealed in other analytical studies [Boivin and Paultre 2012a; Luu et al. 2013] and

observed in shaking-table tests that Ghorbanirenani et al. [2012] conducted on scaled 8-

storey walls.

As for coupled shear walls, structural damage is expected in coupling beams whose flexural

and flexural-shear yielding, along with the flexural hinging at the base of wall piers, form

a mechanism that limits base overturning moments. Such damage was observed in past

earthquakes as shown in Figure 2.3. It was also reported by Harries and McNeice [2006]

and MacKay-Lyons [2013] who respectively conducted nonlinear dynamic analyses on 30-

and 42-story RC core-wall buildings.

Hence, RC wall buildings designed to current codes are expected to sustain considerable

damage such that repair or replacement may be either financially prohibitive or technically

infeasible. As an example of this, after the Mw6.2 aftershock that struck Christchurch on

February 22, 2011, 48% of the post-1970’s RC wall buildings were tagged as “Restricted

Use” or “Unsafe” and a large number of them had to be demolished even though they did

not collapse [Kam et al. 2011].


28

2.4 Low-damage Systems for High-rise Buildings

To minimize earthquake-induced damage to structures, a few high-performance systems

have been proposed to prevent structures from being significantly affected by seismic

excitations. In terms of strategies, these system can be grouped into three categories: (1)

rocking systems, (2) base isolation, and (3) the combination thereof. The following sections

discuss about the dynamics and applications of these low-damage systems.

2.4.1 Dynamics of rocking

The idea of allowing structures to rock at the base is not a new concept but can be observed

in ancient Greek and Roman temples that have survived devastating earthquakes over the

past 2500 years. The apparent earthquake resistance of these historical structures is

achieved by their freestanding columns that rock under strong ground shaking while

supporting epistyles and friezes atop. These massive elements form articulated frames that

display negative stiffness and limited damping which sources from the impact action only.

These characteristics make the primitive rocking concept unconventional from the

perspective of modern seismic design where structural redundancy, ductility and energy

dissipation are highly emphasized [Makris 2014]. Whereas the negative stiffness sounds

undesirable for it potentially results in instability, once the rocking action begins, it is not

the case under dynamic conditions where major seismic resistance is provided by rotational

moments of inertia. In fact, as Makris [2014] emphasized, it is the negative stiffness of

rocking systems that isolates structures from seismic resonance, and the gravity-recentering

capacity that eliminates permanent displacements.

Scientific studies on dynamics of rocking was initiated from the efforts made in very early

studies to determine the critical horizontal acceleration that can overturn a freestanding

column. Based on extensive experiments, Milne [1885a] suggested that this overturning

acceleration could be estimated as g(b/h), where g is the acceleration of gravity and b/h is

the width-to-height ratio, known as the slenderness of the column. Kirkpatrick [1927]

provided a remarkably improved prediction where the rotational inertia of a rocking

column and the period of ground accelerations were included, both of which are key

parameters that govern rocking dynamics.


29

Inspired by golf-ball-on-a-tee type structures that survived during the Chilean Earthquake

in 1960, Housner [1963] systematically studied the dynamic response and stability of

rocking structures, and advanced Kirkpatrick’s finding on the size-frequency scale effect

giving analytical elucidations on two counterintuitive phenomena: (1) among two blocks

that are geometrically similar, the larger one has greater stability than the smaller; and (2)

tall slender structures that can be toppled by long-period accelerations have much greater

chance to survive earthquake ground motions.

Following Housner’s seminal paper, a large number of researchers further investigated

rocking dynamics. Yim, Chopra, and Penzin [1980] investigated the rocking response of

rigid blocks subjected to earthquake ground motions. This study revealed that the response

of the block is very sensitive to small changes in its size and slenderness ratio and to the

details of ground motion. The steady state rocking response of rigid blocks under sinusoidal

base motion was studied analytically [Tso and Wong 1989] and experimentally [Wong

and Tso 1989]. Makris and Vassiliou [2012] suggested an analytical method that provides

a minimum slenderness that is required to ensure a free-standing column withstand a pulse-

like motion, given the acceleration amplitude and duration of this excitation. Makris and

Vassiliou [2013] further investigated planar rocking frames that consist of a series of free-

standing columns capped with a freely supported rigid beam, and concluded that a heavier

cap beam leads to increased stability of the rocking frame despite the raised centroid of

gravity of the cap. This well explained the excellent earthquake-resistance of ancient Greek

temples with free-standing columns supporting massive epistyles and the frieze atop.

Makris and Konstantinidis [2003] also proposed the concept of rocking spectra which focus

on the rotation-related responses of inverted pendulum oscillators, and highlighted the

difference of rocking spectra from conventional response spectra which reflect

translational motions of regular SDOF oscillators. Vassiliou et al. [2017] extended the

study on planar rocking dynamics into three-dimensional rocking behaviour, and provided

theoretical solutions to equations that govern the rocking motion of an inverted pendulum

cylinder. Vassiliou et al. [2017]’s solutions will be used as a reference in Chapter 5 to

validate an innovative modelling technique that is developed in this dissertation for

modelling an unconventional rocking system that is proposed in this study.


30

2.4.2 Rocking wall systems

The feasibility of using rocking systems for building structures was first studied in the

Precast Seismic Structural Systems (PRESSS) program [Priestley 1991, 1996; Priestley et

al. 1999]. In this program, controlled rocking systems were developed for precast concrete

wall panels that are allowed to uplift at the base, and for concrete moment frames in which

gaps are allowed to open at beam-to-column connections. These articulated joints are

clamped tight using unbonded post-tensioned strands until decompression occurs as a result

of a threshold lateral force that activates the rocking action. The decompressed joints open

angles as the rotation increases, inducing a softening effect that prevents the structure from

carrying additional lateral loads. Stretched due to this gap opening, the prestressed strands

develop elastic clamping forces that tend to close the opened angle, and, at a global level,

have the whole structure re-centered. This force-limiting effect and self-centering capacity

provide rocking structures with low-damage responses and minimal residual deformations,

which have been verified numerically [Kurama 1999; Perez et al. 2004] and experimentally

[Priestley et al. 1999].

In absence of cast-in-situ ductile joints, energy dissipation in the PRESSS wall system is

achieved using supplemental damping devices that connect adjacent wall panels along their

vertical joints. These energy dissipating connectors are implemented using U-shaped steel

plates [Priestley et al. 1999] or friction dampers [Kurama, 2001], whose hysteretic response

is engaged when wall panels rock individually inducing large vertical movements at their

joints. As such, the bilinear backbone curve of rocking walls is expanded into a flag-shaped

hysteresis under seismic loads. Sritharan et al. [2008] and Sritharan et al. [2015] proposed

an alternative configuration, referred to as the PreWEC (Precast Wall with End Column)

system, which consists of one single post-tensioned wall sided by two end columns that are

post-tensioned separately and connected to the central wall through a set of oval-shaped

metallic dampers. Guo et al. [2014] proposed a similar system but used distributed friction

dampers instead of metallic ones.

The single-wall arrangement is also used in another type of rocking wall system in which,

however, damping devices are attached to both rocking toes of the wall. These systems

share a similar strategy of exploiting the uplift-induced large displacements expected at


31

rocking toes, but are implemented using varied damping mechanisms, including viscous

dampers [Kurama 2000], and hybrid use of different types of dampers [Kam et al. 2006;

Marriott et al. 2007, 2008].

Kurama [2002] proposed a damping scheme for rocking walls where internally grouted

mild steel rebars are used along with unbonded post-tensioned strands. These rebars are

placed near the rocking toes with a wrapped length across the rocking joint where the steel

and concrete bond is prevented. When uplifting is initiated, yielding occurs to these bars in

this ungrouted length, dissipating earthquake energy. This damping solution was inspired

by the detailing originally proposed for the PRESSS rocking frames [Priestley et al. 1999].

Similar hybrid rocking wall systems were also investigated by Holden et al. [2003],

Restrepo and Rahman [2007], Hamid and Mander [2010], and Smith et al. [2013].

The concept of rocking walls has been used in practice for new constructions of buildings.

Stevenson and Panian [2009] used reinforced and post-tensioned rocking core walls in the

new development of David Brower Centre whose site is just 1 km away from the Hayward

Fault. A similar rocking core-wall solution is seen in San Francisco Public Utilities

Commission’s New Headquarters for which Panian and Bucci [2013] used a U-shaped

profile for the post-tensioned tendons such that anchorage is all made at the top of the walls.

Muthukumar and Sabelli [2013] used the controlled rocking technology to the Air Traffic

Control Tower at the San Francesco International Airport which is a 67 m tall cast-in-site

reinforced concrete cylinder. The self-centering mechanism is provided by post-tensioned

tendons that are in a J-shaped profile and anchored in the foundation. Engineers at Arup

and the University of California, Berkeley [Nielsen 2009] studied a 50-storey RC core that

rocks at the base under gravity loads. Lu and Panagiotou [2015] studied the seismic

responses of a 20-storey rocking core-wall building in three dimensions.

However, base rocking joints are not effective in controlling higher-mode effects. This is

confirmed numerically by Nielsen [2009] and Kurama et al. [1999] and experimentally by

Priestley et al. [1999]. The concept of distributed moment mechanisms was then proposed

to address this problem. Panagiotou and Restrepo [2009] proposed a concept of allowing

for dual plastic hinges to form over the height of RC walls. Munir and Warnitchai [2013]


32

proposed to introduce multiple flexural hinges into walls. These systems still imply

material yielding in structures. Following a similar idea, Wiebe and Christopoulos [2009]

studied multiple rocking joints for wall systems and later extended it to base rocking steel

frames [Wiebe et al. 2013]. However, these systems with distributed flexural mechanisms

were found less efficient in reducing shear demands than overturning moments [Panagiotou

and Restrepo 2009; Wiebe et al. 2013].

2.4.3 Seismic isolation

Given the persistent dominance of higher-modes in both base-hinged and rocking walls, it

is evident that a flexural mechanism is not an ideal mechanism for limiting the forces

induced in tall structures during seismic loading. Stevenson [1868] was first to observe that

it is crucial to break the shear continuity of structures in order to prevent the propagation

of earthquake shocks. Based on this idea, Stevenson [1868] proposed a concept of

“aseismic joints”, as shown in Figure 2.4, that horizontally cut through buildings at the

base and separate the superstructure from ground shaking. This separation was physically

achieved using assemblies of spherical balls and concave cups that permit significant

translational movements to occur at this joint level. This strategy is highly similar to

modern friction pendulum isolators. Following Stevenson’s invention, alternative base

isolation schemes were proposed by Touaillon [1870], Milne [1885b], and Calantarients

[1907], most of which are variations of Stevenson’s earlier idea.

Figure 2.4 Aseismic joints proposed by Stevenson [1868]


33

Modern applications of base isolation commenced when laminated rubber bearings (LBRs)

became commercially available. Reinforced by steel plates in a sandwiched built-up, LRBs

have high flexibility in the horizontal direction to accommodate large shear deformations

while carrying high gravity loads with sufficient vertical stiffness. As a result of the

increased lateral flexibility at the base level, the fundamental period of isolated buildings

is elongated such that the superstructure is decoupled from the damaging effects of ground

motion. A side-effect of the period-elongation is the significantly enlarged displacement

which has to be accommodated by the isolators while maintaining the gravity load-bearing

capacity and stability. External dampers are often used to dissipate energy so as to mitigate

the high deformation demand. Supplemental damping may also be obtained by using lead-

plugged rubber bearings or high-damping rubber bearings. As Naeim and Kelly [1999]

summarized, this is the only practical way by which storey drifts and floor accelerations

can be simultaneously reduced for building structures.

Sliding systems represent an alternative strategy to achieve seismic isolation. Among

varied sliding isolators, friction pendulum systems have been widely used. In these systems,

while the low-friction interface acts as a shear fuse, gravity loads from the superstructure

provide the restoring force. In addition to single concave friction pendulum systems,

derivative systems were also developed including multiple spherical sliding faces [Fenz

and Constantinou 2006, 2008].

It has been widely accepted that base isolation is more efficient for low-rise buildings since

the seismic response of these structures is primarily governed by the first mode and the

base isolating effect can effectively elongate the fundamental period and thereby control

the total dynamic response. In Japan however, the application of this technology has been

extended to high-rise buildings. Completed in 1999, the Sendai MT Building became the

first base-isolated building in Japan with a height exceeding 60 m [Hara et al. 1997].

Despite the reduced effectiveness in period-shifting, inclusion of base isolators still leads

to large reductions in story drifts and floor accelerations, both of which are primary causes

of damage and highly influenced by higher modes. These benefits of base-isolating high-

rise buildings were investigated by Terashima et al. [1997] and Ogura et al. [1997].

Calugaru and Panagiotou [2014] evaluated the performance of a base-isolated 20-story RC


34

core-wall building numerically. Becker et al. [2015], through a comparative study,

highlighted the different extents to which base-isolated high-rise buildings are accepted in

Japan and the United States.

Most of these studies drew attention to the fact that significant overturning moments in

slender structures can lead to base isolators being loaded in tension or overstressed in

compression beyond their acceptable range. Ishii et al. [2012] investigated this issue

numerically and reported severe damage in elastomeric bearings due to buckling under

ultimate seismic excitations. Takaoka et al. [2011] observed both tensile and buckling

fractures in laminated rubber bearings during shaking-table tests conducted on slender

isolated structures. Pan et al. [2008] pointed out that tension in conventional bearings

should be avoided in practical design. The Canterbury Earthquake Royal Commission

[2012] suggested that tall and slender structures may not be appropriate for base isolation.

These concerns and limitations, as Becker et al. [2015] concluded, explain why base

isolation is of limited use for high-rise buildings outside of Japan.

To resolve the problem, varied tension-resistant isolation devices were developed. Griffith

et al. [1990] proposed a two-bolt lockup device, as shown in Figure 2.5, that provides

uplift-restraint and displacement control for multilayer elastomeric bearings. Lu et al.

[2016] developed tension-resistant rubber bearings relying on preloaded strands, as shown

in Figure 2.6. Similar tension-resistant mechanisms were developed by Hu et al. [2017]

who used steel arms at the perimeter of rubber bearings to carry tensile forces while

allowing for bi-directional deformations of the bearing, as shown in Figure 2.7.

Figure 2.5 Uplift-restrained elastomeric bearings (from [Griffith et al. 1990])


35

Figure 2.6 Tension-resistant rubber bearings (from [Lu et al. 2016])

Figure 2.7 Tension-resistant base isolators (from [Hu et al. 2017])

As for sliding isolation bearings, similar tension-resistant design have been proposed.

Nagarajaiah et al. [1992] used an uplift-resistant device to prevent Teflon-disc sliding

bearings from being subjected to tension, as shown in Figure 2.8. Roussis and Constantinou

[2006] proposed an uplift-restrained friction pendulum isolation system, named as the XY-

FP isolators as shown in Figure 2.9, in which orthogonal sliding is allowed between two

concave steel beams that are locked up vertically to prevent uplifting from the sliding

bearing. In Japan, Cross Linear Bearings were used as a seismic isolation system in which

low-friction sliding is achieved using steel balls. These balls recirculate within orthogonal

grooved linear guides which provide tension-restraint simultaneously, as shown in Figure

2.10 [Arima et al. 2000]. However, when used for highly slender structures or under

extreme seismic events, these rubber isolators and sliding bearings with high tensile

resistance are still challenging to design and implement.


36

Figure 2.8 Uplift-restrained Teflon disc bearing (from [Nagarajaiah et al. 1992])

Figure 2.9 XY-FP isolators (from [Roussis and Constantinou 2006])

Figure 2.10 Cross linear bearing (from [Arima et al. 2000])


37

In addition to the above concepts of resisting the tension, another strategy to alleviate the

tension in isolation bearings is to release the restrained tendency overturning. This can be

achieved by allowing base isolators to uplift, as Kikuchi et al. [2005] proposed for a 90 m

tall building that is located in Tokyo and base isolated using rubber bearings. For the corner

bearings, a loose detail was applied to the anchor bolts, allowing these bearings to step up

to a certain extent during strong earthquakes, as shown in Figure 2.11. For another 47-

storey building in Tokyo, uplifting of bearings was realized through a series of steel

elements, termed winkler plates, that loosely attached bearings to the foundation, allowing

for free uplifting while refraining lateral movements, as shown in Figure 2.12.

Figure 2.11 Rubber bearing with a loose detail. (from [Kikuchi et al. 2005])

Figure 2.12 Kajima winkler method (from [Kajima Corporation 2006])


38

Calugaru [2013] further extended the partially stepping base isolation scheme and proposed

a base-isolated rocking system for a 20-storey RC core-wall building. As shown in Figure

2.13, while the whole building is seismically isolated using friction pendulum bearings at

the bottom level of the basement, post-tensioned RC core walls are allowed to rock at the

base, which results in a dual-mechanism protection for the superstructure.

Figure 2.13 Base isolation arranged in-series with base rocking mechanism (from [Calugaru 2013])

However, since the two mechanisms are arranged in series, isolating bearings have to carry

axial tension forces prior to the activation of the rocking action. Calugaru [2013] reported

tensile forces being 2.37 MN (mean) and 4.94 MN (mean + std dev) while they were

undergoing shear deformations of 830 mm (mean) and 1100 mm (mean + std dev) at the

MCE level.

These systems with a tension-releasing mechanism can control tensile forces in bearings.

However, bearings still have to carry significant forces until either uplifting occurs at the

bottom of the isolators or the superstructure steps up. In addition, since the flexural and

shear responses are fully coupled, it is still challenging to design against compressive

overloading.


39

2.4.4 Systems with dual seismic protections

Wiebe et al. [2013] proposed a dual seismic protection concept for multi-storey steel frames.

In this system, a self-centering brace was used in the first storey of a base rocking frames

that are recentered by using post-tensioned tendons, which leads to a well integrated dual-

mechanism system. However, in this system, the shear fuse remains in-series with the

rocking mechanism, leading to the first storey – which acts as a soft storey that undergoes

inelastic lateral drifts – carrying the gravity load plus the clamping force that the post-

tensioned tendons impose for recentering the rocking frame.

This in-series dual-mechanism system works reasonably for low- to medium-rise buildings

for which the inelastic drift demand in the soft storey is low (with approximate magnitudes

of 0.82% (mean) and 1.81% (mean + std dev) under MCE earthquakes as observed for a

12-storey steel frame that was designed with the two mechanisms incorporated [Wiebe and

Christopoulos 2015c]).

However, if this system were applied to high-rise buildings, MCE drifts would be largely

increased in the soft storey which at the same time has to carry high gravity loads from the

superstructure plus the clamping force provided by tendons. As a result, significant P-𝛥 effects can be induced causing concerns to the overall stability of the structure.

From a practical perspective, while it is straightforward to incorporate a shear fuse using

nonlinear braces into steel frames, it is rather challenging to implement a similar lateral-

force-limiting mechanism within three-dimensional RC core-wall structures which are the

focus of this dissertation.


40

2.5 Summary

This chapter reviewed causes and consequences of higher-mode effects that are often

pronounced for high-rise buildings and challenge the seismic design of these structures.

Through high-frequency modes, seismic demands, especially shear forces, can be

significantly magnified, leading to code-based design being likely on the unsafe side. This

higher mode-induced dynamic amplification has been confirmed numerically and verified

experimentally. The consequential impact of this phenomenon has also been observed in

major earthquakes during which varied types of damage happen to RC wall structures,

including distributed damage that is related to higher-mode response.

This chapter also went through a wide range of high seismic performance systems that have

been developed to achieve low-damage design of high-rise buildings. While rocking

systems are proven efficient in limiting the first-mode response, higher modes are barely

affected despite the activated rocking action at the base of structures. At this point, base

isolation systems provide an enhanced mitigation to higher-mode effects. However,

bearings under base-isolated high-rise buildings can be subjected to large axial loads while

undergoing significant shear deformations, which put these isolators at risk of fracture in

tension or buckling in compression. Varied schemes have been proposed to resolve this

bearing-in-tension problem. However, all these approaches have limitations, leading to the

design and implementation remaining challenging under extreme seismic events.

Hence, new high-performance systems are highly needed for high-rise buildings to mitigate

higher-mode effects and achieve low-damage design in a more practical and robust way.

To address this need, this dissertation aims at developing a solution at a system level to

achieve minimal damage for RC core-wall high-rise buildings. This work begins with an

extensive exploration of possible concepts as will be presented in the following chapter.

41

Chapter 3 Considered Resilient Concepts for High-Rise Buildings

3.1 Introduction

As reviewed in the previous chapter, mitigating higher-mode effects is a crucial step

towards a low-damage design of high-rise buildings. To achieve this goal, an additional

mechanism that can effectively limit high-frequency response over the height of the

structure is needed, since a single flexural mechanism at the base of these structures is

ineffective at controlling higher mode effects. As reviewed in Chapter 2, it is not ideal to

introduce a second flexural mechanism, since previous studies have confirmed that

multiple flexural fuses are not as efficient in reducing the dynamic shear amplification over

the height of the building. Hence, a shear mechanism becomes essential to provide dual

seismic protection along with the rocking action at the base of structures.

With the goal of defining a dual seismic protection system, a series of structural systems

were developed during the conceptual design stage in this study, as will be discussed in

this chapter in an evolving manner. Section 3.2 describes the first system that consists of a

base rocking storey in series with a planar shear fuse component. This initial concept is

significantly improved in Section 3.3 to allow for three-dimensional rocking action and,

more importantly, a decoupled shear mechanism. Using this system as a basis, more

variations are proposed in Sections 3.4 and 3.5 for both mechanisms, with their advantages

and limitations discussed. Section 3.6 makes an illustrative summary of the evolution of

the different concepts that were considered in this thesis.

During the concept development stage, RC core-wall structures were considered as a target

system since it is one of the most widely used lateral-force-resisting systems for high-rise

buildings. At the same time, applicability to general structural forms was considered when

each generation of concepts was defined and studied.


42

3.2 Base Rocking Storey and Self-centering Slider

3.2.1 Self-centering energy dissipating braces

In the first-generation of the system, the self-centering idea was used for the development

of both flexural and shear mechanisms. This recentering behaviour is achieved through

self-centering energy dissipating (SCED) braces that were proposed by Christopoulos et al.

[2008]. A typical SCED brace consists of two structural members, a tension system, and

an energy dissipating mechanism, as shown in Figure 3.1. At both ends of the SCED brace,

axial loads are exerted on different structural members respectively, making these members

tend to move relative to each other. However, this relative movement will not be initiated

until the applied axial load overcomes the clamping force that the pretensioned tendons

impose on the structural members via abutting elements. Up to this critical state, the whole

brace displays a high axial stiffness that is primarily provided by the structural members.

Figure 3.1 SCED braces (from [Christopoulos et al. 2008])


43

Once the clamping force is overcome, the two structural members start undergoing large

relative displacements, engaging the energy dissipating mechanism that is attached

between these two moving parts. In this stage, the stiffness of the overall brace is governed

by the prestressed tendons, and therefore much smaller than the initial stiffness prior to the

decompression. This reduced stiffness allows the SCED brace to accommodate significant

axial deformation and display a bilinear force-deformation relationship. However, this

nonlinearity is not caused by material plasticity, since all components of the brace, except

the damping device, are designed to work in the elastic range. Instead, the softening effect

results from the loading shift between the structural members and the tendons.

Being further elongated, the pretensioned tendons develop increased tensile forces that tend

to pull the structural members back to their initial aligned position. This restoring force

makes the whole brace self-centering. Once realigned, the structural members act as a

whole and take over the axial load again until they are separated upon a decompression that

is activated under a reversed load. This load reversal leads to a symmetric self-centering

force-deformation relationship in the opposite direction. Given the supplemental energy

dissipation, the whole SCED brace displays a flag-shaped hysteresis.

3.2.2 Base rocking storey

To prevent flexural plastic hinges at the base of RC walls, the idea of base rocking was

adopted in place of cast-in-situ construction. This intended rocking action is expected to

occur in a storey high space where a series of steel braces are arranged under the footprint

of the RC core, forming a truss tube, as shown in Figure 3.2. Each side of this tube consists

of two diagonal elements that are arranged in an inverted chevron pattern in the middle,

and two vertical elements that are located at corners. The diagonals are made of

conventional steel braces with adequate load-bearing capacity to carry the entire gravity

load from the RC core without yielding. The intersecting point of the diagonals is

connected to the foundation through a ball-pin joint about which the diagonals can pivot in

and out of the plane. The corner verticals are made of SCED braces described in the

previous section. To avoid initial compression forces, these SCED braces are installed after

the gravity-induced settlement of the superstructure has occurred.


44

Figure 3.2 Base rocking storey

Given this configuration, overturning moments that are developed at the base of the RC

core cause axial forces to the verticals, tension on one side and compression on the other.

Once the base moment reaches a level at which the induced axial forces are high enough

to activate the SCED braces, the RC core starts pivoting about the intersecting point of the

central chevron braces. At the same time, the activated SCED verticals provide restoring

forces that, along with the superstructure’s weight, bring the tilted RC core back to the at-

rest position. As such, the intended rocking action is achieved at the base of the

superstructure. Activation moments of this rocking mechanism can be obtained by

choosing a proper pretension force for the SCED braces. It is also noteworthy that instead

of an articulated rocking joint where angles open, the rocking action achieved in this system

relies on the elongation and shortening of the SCED braces that happens over a storey

height. In this sense, the truss tube acts as a base rocking storey.

In this rocking storey, the chevron braces are designed to remain elastic during earthquakes.

While carrying gravity loads from the superstructure, these elastic elements transfer base

shears of the RC core to the foundation as well without capping off the shear demands.

Hence, it is necessary to have a separate lateral-force-limiting mechanism that works in the

transverse direction, as will be discussed in the following section.


45

3.2.3 Lateral-force-limiting self-centering slider

This lateral-force-limiting mechanism is located between the RC core and the base rocking

storey, as shown in Figure 3.3. This system consists of a thick concrete transfer plate that

is located on the roof of the truss tube. This transfer plate serves as a rigid diaphragm that

engages all the members in the truss tube to undergo identical horizontal movements.

Figure 3.3 Self-centering slider system

On the top face of this transfer plate, there is a circular recess in which a series of SCED

braces are polar arrayed between the wall of the recess and a ring-shaped component that

is located at the centre. Fixed to the base of the RC core via an inverted cone, this ring is

allowed to move in any horizontal direction within the recess, leading to the lateral mobility

of the superstructure relative to the transfer plate. This makes the ring act as a slider that is

a key component used in sliding base isolation systems. The horizontal motion of this slider

is self-centered given the SCED braces surrounding the ring. These braces can be designed

such that their flag-shaped hysteretic response is activated once the lateral force reaches a

level under which the base shear of the superstructure is expected to be limited. Upon this

activation, those braces whose orientation more closely aligns with the loaded direction

will undergo larger axial deformations, which meanwhile generates greater recentering

forces that bring the slider, in conjunction with the RC core, back to the central position.

As such, a self-centering slider system is formed, providing the needed shear mechanism.

The configuration shown in Figure 3.3 leads to the slider system having no capacity in

transferring overturning moments from the superstructure. This limitation and possible

solutions are discussed in Section 3.2.4.


46

It is noteworthy that this multi-directional self-centering shear mechanism has a striking

resemblance with bike wheels. While the slider ring is equivalent to the hub of a wheel, the

SCED braces are analogous to the spokes. Both the SCED braces and the spokes are

prestressed components that can accommodate large deformations, and, more importantly,

are capable of having the deformed system restore its original shape. This proposed sliding

system alone can be further developed to be a novel seismic isolation system for which no

additional recentering devices or supplemental dampers would be needed.

3.2.4 Remarks

As individual mechanisms, the rocking storey and the slider system are capable of limiting

flexural and shear responses respectively. However, at a system level, a few limitations

were identified for this first-generation concept. Firstly, the rocking storey and the slider

system act in series. This means that the shear mechanism has to transfer overturning

moments from the RC core to the base rocking mechanism. This requirement cannot be

met by the slider system as aforementioned, because the horizontal SCED braces are

incapable of resisting out-of-plane actions. This problem may be resolved by adding a few

vertical rods between the core walls and the wall of the circular recess for moment transfer.

However, these elements may restrict the relative horizontal movement between the core

and the transfer plate, which is intended to be solely controlled by the SCED braces. This

redundancy leads to an unclear load path and additional complexity to the overall system.

Assuming that the vertical rods were in place, transferring overturning moments, and that

the increased complexity could be accepted somehow, this first-generation system still

displays behaviours that are different from the intended response. For example, the rocking

storey only allows the RC core to pivot about a single principal axis of the structure at a

time with limited ability to accommodate bi-directional rocking motions. Even during

unidirectional pivoting, the rotation of diagonal braces can be impeded by the other braces

that are in the orthogonal elevations given that these elastic braces are not expected to

elongate or shorten significantly. The limited mobility in both unidirectional and bi-

directional pivoting was identified as another limitation of the rocking storey.


47

In addition, the rocking storey pivots about central axes of the structure with the vertical

SCED braces at corners providing restoring forces only rather than acting as rocking toes.

Once the pivoting body leaves its plumb position, gravity loads of the superstructure do

not recentre the structure but tend to overturn it, causing overall stability concerns.

Despite these shortcomings, this early concept set up some fundamental rules for the

ensuing concept development. Firstly, the anticipated dual-mechanism system shall be

located at the base of these structures. This arrangement can avoid discrete gap opening (as

in the case of using multiple rocking systems) or significant lateral displacements (as in the

case of using mid-storey seismic isolation) over the height of the superstructure, both of

which may require complicated detailing and cause concerns to occupants. Locating

mechanisms at the base also allows for a flexibility in construction sequencing, since the

superstructure and the base mechanisms can be built independently. As such, conventional

technologies and equipment that are efficient for high-rise construction remain applicable.

In addition, the proposed base rocking storey makes it possible to arrange the pretensioned

tendons in a shorter length. This will considerably reduce maintenance requirements and

interruption to occupancy, both of which are typical concerns for the previously proposed

rocking systems in which post-tensioned strands are extended throughout the height of

structures.

Located at the base, the proposed mechanism system serves as a supplemental foundation

to the superstructure. This foundation provides a platform on top of which the RC core

rests, moves, and rotates as a whole rather than as individual wall piers. This is realized

relying on the transfer plate that collects superincumbent gravity loads, lateral forces and

overturning moments and convey these loads to the base mechanisms as resultant actions.

This strategy makes the proposed base mechanism system adaptive to various high-rise

buildings with different structural forms.

During the conceptual design, the focus was put on achieving the intended global

behaviours. Detailing issues are addressed in Chapter 4 for the proposed system. These

discussions about the detailed design will demonstrate how components of the proposed


48

system are connected to each other and how the whole system is incorporated into a high-

rise building and connected to the foundation.


49

3.3 Three-dimensional Decoupled Rocking and Shear Mechanisms

To address the limitations pointed out in the previous section, significant modifications

were made to the first-generation system to account for the three-dimensional response of

core structures. In addition, the shear mechanism is relocated so as to not to interact with

the rocking mechanism. These design developments define the second-generation of the

concept that was developed.

3.3.1 Rocking pyramid and additional rocking toes

In general cases, seismic excitations cause critical responses along both principal axes of

buildings simultaneously. For core-type structures, these seismic effects typically subject

the core to bi-directional overturning moments. If this core has a rocking mechanism

incorporated at the base, rocking actions can be activated about both orthogonal directions

at the same time, provided that the rocking moment is reached in each direction. As a result,

the core rotates within a skewed vertical plane that does not pass through any of the

principal axes. To provide a supporting point for this skewed pivoting motion, an inverted

pyramid-shaped block is introduced in the rocking storey, as shown in Figure 3.4, replacing

the truss tube used in the first-generation system.

Figure 3.4 Rocking pyramid with SCED braces as rocking toes


50

This pyramid is a solid concrete block with the transfer plate at the top face, forming a

monolithic rigid body, referred to as the rocking pyramid. Regardless of in which vertical

plane the pivoting motion takes place, this rocking pyramid carries gravity loads to the

foundation through its apex at the bottom. At this location, a two-way dowel connection is

provided to restrict horizontal sliding. This dowel includes two parts: one is attached to the

apex, and the other is fixed to the foundation. Each part consists of protruded cubes and

cavities that are arranged on a common 3-by-3 grid but in different jigsaw patterns, such

that both parts can be plugged into each other, forming a set of shear keys.

Vertical SCED braces are retained at corners of the pyramid for their recentering capacity.

In these braces a gap element is introduced and has a large compressive rigidity once the

gap is closed up. As the pyramid starts pivoting, restoring forces and flag-shaped hystereses

can be fully developed in SCED braces that are in tension. However, those in compression

will soon be locked up and take on gravity loads along with the pyramid. If the pyramid

tends to rotate further, slight uplifting may occur at the apex where the dowel connection

allows for this vertical detachment. At this instant, the locked-up SCED braces support the

entire gravity load and act as a new pivoting point until the pyramid settles back in contact

with the foundation again at the apex. This shift of pivoting points between the central apex

of the pyramid to peripheral SCED braces allows for rocking motions in any of the vertical

planes, as shown in Figure 3.5. For clarity, this figure does not show the shear mechanism

which will be further developed and included in the next section.

Figure 3.5 Shift of rocking toes


51

3.3.2 Decoupled shear mechanism

In the second-generation concept, the shear mechanism is achieved using four nonlinear

lateral bracing components that are located at the periphery of the rocking pyramid, one

on each side, as shown in Figure 3.6. These components are orientated horizontally and

perpendicular to the pyramid at the mid-point of each side. With the far end fixed to the

foundation, each of these bracing components is connected to the pyramid via a special

joint consisting of ball rollers, as shown in Figure 3.6. These conceptual ball-roller joints

transfer lateral shears from the pyramid to the bracing components, and are therefore

referred to as the shear transmitters. With a low rolling friction assumed, these shear

transmitters induce a minimal impedance to the relative moments between the rocking

pyramid and lateral bracing components in the vertical direction. As such, the shear

mechanism is decoupled from the rocking mechanism, carrying no overturning moments

or vertical forces.

Figure 3.6 Decoupled rocking and shear mechanisms


52

In the transverse direction, the lateral bracing components are interlinked using a rigid ring

beam, which runs around the rocking pyramid. When the pyramid pivots about the apex at

the bottom, lateral movements arise at the top level. The rigid ring beam ensures that this

horizontal displacement can be identically transmitted to all the bracing components. This

synchronization is important to guarantee a simultaneous engagement of the bracing

components that are oriented in the same direction.

The inner side of the ring beam is made of a curved surface that is concave towards the

rocking pyramid. This concave surface provides a track that holds the ball rollers in

position while accommodating their rolling motions. Low-friction coating can be applied

to this track to ensure smooth motions of the ball rollers along the ring beam and in the

vertical direction. This detail helps enhance the decoupled mechanics that are achieved in

the second-generation system.

3.3.3 Remarks

This second-generation system marks an important step in the overall evolution of the

concepts in the sense that it decouples the flexural and shear responses into two

independent components. This separation in structural behaviour makes it possible to

independently control these two critical responses using different nonlinear mechanisms at

a system level. Each of the mechanisms can be realized in varied ways using different

materials and components. When these design choices are made, engineers do not need to

consider whether a system that is efficient in resisting shear demands is also adequate in

resisting flexural demands and vice versa. This design convenience is not available in

conventional RC structures where shear-flexure interaction in flexural hinge regions is

inherent given the innate steel-and-concrete bond at the material level; this design

flexibility may also be compromised in some existing high-performance systems in which

the dual mechanism remains in-series.

From the flexural perspective, it is important to introduce the rocking pyramid that provides

a pivoting point at the centre of the core’s footprint. This allows the superstructure to pivot

about any horizontal axes, which more realistically represents the three-dimensional

motion of a rocking body. In addition, this all-direction pivoting motion is further enhanced


53

to be a rocking action in which the superstructure can lean upon peripheral SCED braces

that are rigidly locked up in compression, acting as additional supports. As long as the

centroid of gravity of the core remains within the scope of these peripheral supports, gravity

loads contribute to a recentering force that helps stabilize the structure. This is a significant

improvement compared with the first-generation system in which the gravity load causes

destabilizing overturning moments only.

Alongside these advantages, some concerns arise for the second-generation system. One of

the issues to be highlighted is the response in torsion. Whereas the lateral bracing

components provide shear resistance, they do not develop resistance to react in plane

torsion that arise from the superstructure. This is because these elements are located at the

midpoint of each side of the pyramid, where a limited lever arm can be obtained. Since

they are not resisted at the ground level, the torsional moment travels down to the apex of

the pyramid. However, the jigsawed dowel connection at the apex neither allows for

rotations in the horizontal plane so as to release the torque, nor acts as an efficient torsion-

resisting element given the small size of this connection and its limited lever arms. Hence,

an alternate solution is needed to achieve torsional equilibrium of the overall structure.

The rocking mechanism is also reviewed in terms of whether it carries lateral forces. At

this location, while the dowel key refrains the pyramid from sliding, the interlocked cubes

contributes considerable lateral resistance. A similar situation occurs for the SCED braces

as well. When the pyramid pivots, the SCED braces are in an inclined position. Then the

horizontal component of the axial forces become a portion of lateral resistance in addition

to what is provided by the lateral bracing components. Under these circumstances, the

separated force flow that is achieved at the ground level is compromised. This means that

while the rocking mechanism is engaged to limit base overturning moments, it has to carry

shear forces simultaneously. This is another problem to be addressed in the ensuing concept

developments.


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3.4 Torsion-resistant Shear Mechanism and Cubic Rocking Block

In the third phase of the concept development, a feasible way to achieve torsional resistance

was prioritized. Rather than introducing a new separate torsion mechanism, it was preferred

to rationalize the current shear mechanism to meet the equilibrium requirement. Meanwhile,

significant adjustments were made to the rocking mechanism, aiming at minimal lateral

resistance and a better integrated configuration.

3.4.1 Torsion-resistant shear mechanism

To make the shear mechanism torsion-resistant, it is necessary to create a lever arm upon

which the lateral bracing components can develop a resisting torque. For this purpose, two

lateral bracing components are used on each side of the rocking block, being apart from

each other as much as possible, as shown in Figure 3.7. When a pure torque arises at the

base of the RC core, the rocking block tends to rotate in the horizontal plane. This rotational

tendency, either counterclockwise as shown in Figure 3.7 (a) or clockwise as shown in

Figure 3.7 (b), leads to a tight contact in shear transmitters at diagonal positions. Through

these contact points, the torque is transferred to the rigid ring beam which in turn engage

all the lateral bracing components. These bracing components will be in compression if

they align with the shear transmitters that are in contact, or in tension if they are in a loose

state. More than two lateral bracing components can be used on each side of the rocking

block. This can reduce the design demand on individual elements.

Figure 3.7 Shear mechanism with torsional resistance


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3.4.2 Improved structural arrangements for shear mechanism

For illustration, the lateral bracing components are shown as one-dimensional elements

oriented horizontally at the ground level. However, it is inconvenient or unrealistic to

follow this layout in practice. It is also difficult to hold the rigid ring beam in position

vertically while allowing it move horizontally. In addition, special details may be required

to provide lateral reactions to the bracing components, if horizontally arranged, which will

complicate the construction of the ground floor system.

In fact, the lateral bracing components can be placed in the basement at the periphery of

the rocking block. On plan, they can be arranged orthogonally in both principal axes of the

structure, as shown in Figure 3.8 (a), or following a circumferential layout as shown in

Figure 3.8 (b). In both schemes, the lateral bracing components are connected to the ground

floor slab at the top and to the foundation at the bottom. In response to the lateral

displacement of the rocking block, the ground floor moves horizontally, inducing shear

deformations into these nonlinear lateral bracing components which then act as shear fuses.

Figure 3.8 Alternative arrangements of the lateral bracing components

3.4.3 Cubic rocking block with rocking toes at corners

In order to minimize the shear resistance contributed by the rocking mechanism, the dowel

connection at the apex of the rocking pyramid is removed. At the same time, the inverted

pyramid is transformed into a cubic block that is made of concrete as well, as shown in

Figure 3.10. Four concrete toes are extended from the bottom face of this block, each at a

corner. These toes provide supports when the superstructure undertakes three-dimensional

rocking motions.


56

Figure 3.9 Cubic rocking block with rocking toes at corners

The concrete rocking toes are assigned a hollow tubular section inside which a SCED brace

is enclosed, forming a capsule-like configuration, as shown in Figure 3.9. The SCED brace

is connected to the ceiling of the hollow at the top through a three-dimensional pinned joint.

At the bottom, the brace is pin-connected to a steel element that is contained in the

foundation and allowed to move as a slider in any horizontal direction within the cavity.

However, the vertical movement of this steel slider is fully restricted to provide reliable

anchorage to the post-tensioned tendons.

As shown in Figure 3.9, the SCED brace consists of multiple conceptual components: (1)

a top member (in green) that is connected to the rocking block; (2) a bottom member (in

blue) that is pinned to the steel slider; and (3) an intermediate member (in red) that is

clamped tight with the blue member through post-tensioned tendons. The green and red

members form a hook mechanism that transfers forces only when the brace is elongated in

a toe at which the rocking block lifts up. Once the hook is engaged, the green member pulls

apart the red and blue, leading to the post-tensioned tendons providing restoring forces and

the damping devices dissipating energy, as shown in Figure 3.9. These encapsulated SCED

braces are intended to be activated in tension only, carrying no compression loads.


57

3.4.4 Remarks

Torsional stability is achieved in this third-generation system. The system also becomes

more realistic given the alternative layouts of the lateral bracing components. Located in

the basement, these bracing components can be made using both diagonal braces and two-

dimensional shear panels. This provides considerable flexibility for the design of the shear

mechanism. As a result of this arrangement, horizontal displacements are expected at the

ground floor slab. Special detailing is then required at the basement wall to accommodate

this movements, which will be addressed in the detailed design.

The current rocking system has its loading-bearing units and restoring mechanism better

integrated through the capsule configuration. This makes the rocking toes multi-functional

in the sense that the concrete part carries compression only, while the tension-only SCED

brace is dedicated to providing restoring forces. Which action is in effect depends on if a

rocking toe is in contact with the foundation or is lifting off it.

When engaged in tension, the SCED braces provide minimal lateral resistance since they

essentially remain vertical even if the rocking block is in a tilted position. This verticality

is ensured by relying on the steel slider that allows the bottom of the braces to align with

the top vertically at all times. As a result, these braces have no force components projected

in the horizontal directions and hence carry no lateral forces.

However, the rocking mechanism may still attract lateral shears on areas where the rocking

toes are in contact with the foundation. Despite that the friction at these contact areas could

be reduced by applying lubricants (e.g. PTFE layers), high normal forces caused by the

superstructure’s weight may still lead to significant frictional forces. While contact is

unavoidable between the rocking block and the foundation in order to carry gravity loads,

frictional forces arising due to this contact makes the rocking mechanism part of lateral-

force-resisting system, which may compromise the shear-force-limiting effect obtained

through the shear mechanism. This problem becomes one of the significant challenges to

be addressed in this study.


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3.5 From Rocking to Wobbling

With four distinct supports at the base, the rocking block may pivot on one toe at an instant,

and then lean towards an adjacent one, or even settle back and jump onto the diagonal

corner. This discrete shift of contact points with the foundation may lead to abrupt impacts.

To achieve a smoother rocking action, a multi-phased rocking system is explored, and then

further extended to a concept of a wobbling mechanism.

3.5.1 Multi-phased rocking mechanism

In addition to the existing supports, more concrete toes are arranged across the bottom face

of the rocking block, as shown in Figure 3.10. These toes are distributed on concentric

squares with one being located at the centre. While the central toe is the longest reaching

the top face of the foundation, those on the intermediate and perimeter squares gradually

get shorter, such that soffits of all the toes follow an imaginary curved surface that concaves

upwards. This leaves a non-zero gap distance under the intermediate toes, and a larger gap

under the perimeter ones, both measured from the foundation level.

Figure 3.10 Rocking block with distributed rocking toes


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When the rocking block starts pivoting, it firstly rotates on the central toe which initially

rests atop the foundation, as shown Figure 3.11 (a). As the rotation amplifies, gaps can be

narrowed and closed under the intermediate toes which the block is leaning towards. These

toes, once in contact with the foundation, are engaged to take over gravity loads from the

central toe. After this load redistribution is completed, the rocking block may uplift at the

central toe and pivot completely on the newly engaged ones, which marks another phase

of the rocking action, as shown in Figure 3.11 (b). If the rocking block rotates further, a

similar gravity load transfer and shift of rocking toes can happen between the intermediate

and perimeter toes, leading to a third rocking phase, as shown in Figure 3.11 (c). In these

ways, a multi-phased rocking action is achieved and anticipated to be more continuous and

smoother than the discrete rocking observed in the third-generation system.

Figure 3.11 Multi-phased rocking action

In line with the concept of multi-phased rocking, it can be imagined that if more toes are

added at the base of the rocking block fitting the concave envelope, the smoothness of the

rocking action can be further enhanced. When the number of toes is large enough such that

they physically form a spherical segment, the superstructure is then allowed to roll atop the

foundation through a continuous curved surface at the base rather than rocking on discrete

toes. This implies a wobbling action that is studied as a fifth option for limiting overturning

moments at the base of structures.


60

3.5.2 Base wobbling mechanism

The idea of using a base wobbling mechanism was initially inspired by roly-poly toys. A

roly-poly toy, also called a round-bottomed toll, has a hollow hemispherical base in which

weights are placed to bring the centre of gravity below the centre of the hemisphere. When

the toll is pushed off the vertical orientation, the centre of gravity is raised up but never

goes beyond the contact point with the ground even if the toll rotates through a large angle.

Once the push is removed, the added weight uprights the toll after cycles of wobbling, and

brings it back to a stable equilibrium with minimum gravitational potential energy.

As for building structures, to allow for wobbling motions at the base of the central core,

the cubic block that was used in the previous rocking systems need to be fattened at the

bottom to form a spherical cap, as shown in Figure 3.12. This spherical cap has an enlarged

dimension such that the centre of gravity of the building – which is located at an elevated

level where large lateral displacements are expected – will be kept within the scope where

supports can be obtained at the base. As the superstructure is subjected to overturning

moments, the wobbling block starts rolling against the top face of the foundation, such that

the moment demand at the base of the structure is limited.

Figure 3.12 Base wobbling mechanism


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Different from the previous rocking systems, SCED braces are not included in the spherical

wobbling block. This is to ease the construction of this concrete block with a curved profile.

As for the restoring force that is required to bring the superstructure back to the vertical

orientation, gravity loads that are tributary to the RC core can be relied on since these loads

are usually large in magnitude and typically invariant during earthquakes.

Similar as in the previous rocking mechanisms, shear transmitters are arranged near the top

level of the wobbling block, transferring lateral forces and torques to the shear mechanism.

Since the lateral bracing components are torsional resistant, the wobbling block is refrained

from rotating within the horizontal plane.

3.5.3 Remarks

Whereas the rolling motion of the wobbling block is smooth, resulting from a continuous

shift of contact points, some additional concerns arise. Firstly, given the spherical base, the

wobbling mechanism can be easily activated under a very low overturning moment. This

can lead to the whole structure swaying under service level wind loads and frequent

earthquakes, which is typically not acceptable. This problem might be resolved by reducing

the curvature of the spherical base. Nevertheless, a much flatter base would approach the

case scenario where a cubic block discretely rocks on four corners.

Regardless of whatever curvatures, contact areas between the wobbling block and the

foundation cannot be avoided. Friction forces that are developed at this interface can still

be large in magnitude, making the wobbling block carry substantial shears. This repeats

the design dilemma that was pointed out previously. And this issue will be resolved in the

proposed detailed design of the system as discussed in Chapter 4.


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3.6 Summary

This chapter discussed a wide range of concepts that were developed to achieve uncoupled

flexural and shear responses at the base of tall structures. The conceptual design went

through an evolving process in which five generations of systems were proposed for each

of the rocking and shear mechanisms. The design is initiated from an alpha configuration

that had in-series mechanics and only allowed for planar rocking motions. By introducing

a pyramid-shaped block, a three-dimensional rocking action was achieved. More

importantly, through a novel structural arrangement, the shear force flow was completely

separated from the flexural loading path. These improvements established a basic

framework for further developing the desired uncoupled dual mechanism. Varied options

are explored for the rocking mechanism, including the four-toe rocking system and the

multi-phased rocking scheme which was extended to be a base wobbling mechanism. In

parallel, structural arrangements were adjusted for the shear mechanism to be torsionally

resistant, as well as more realistic and adaptive to be incorporated into real building

structures. These concept evolutions are schematically summarized in Figure 3.13.

Each generation of these systems provided valuable insights towards achieving the desired

structural system. Meanwhile, limitations were also identified and are also considered as

the final selected system is defined. Based on this concept development work, the main

guiding principles for the final definition of the system as presented in the following

chapters are :

• The dual-mechanism system shall be located at the base of high-rise buildings for

minimal impact on the superstructure during construction and during service life.

• The RC core walls shall rest on top of the rocking block as a whole rather than as

individual wall piers.

• The self-centering effect is achieved using the structure’s weight that is attributed

to the RC core of high-rise buildings. No prestressed components are involved.


63

Figure 3.13 Evolution of concepts


64

• The scheme of using a cubic rocking block with four rocking toes is adopted. The

concept of wobbling is not preferred due to its vulnerability under wind loads and

frequent earthquakes.

• As part of the shear mechanism, lateral bracing components are arranged in

basement stories. While the circumferential layout is structurally feasible, the radial

arrangement is recommended for its better architectural compatibility.

The rocking mechanism may still carry lateral forces at contact points where gravity loads

are supported. This problem is to be resolved in the stage of detailed design as will be

discussed in Chapter 4.

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Chapter 4 Uncoupled Base Rocking and Shear Mechanisms: MechRV3D

4.1 Introduction

The ideas that were developed in the previous chapter converged to a novel concept that

allows for an independent control of the flexural and lateral responses of structures using

distinct nonlinear mechanisms. This concept is proposed in Section 4.2 of this chapter

through a two-dimensional illustration, and then, in Section 4.3, developed into a three-

dimensional structural system with an idealized configuration. To understand the

governing mechanics of the proposed system, this idealized configuration is numerically

modelled in Section 4.4 and analyzed using nonlinear procedures under both static and

seismic loading conditions, as will be discussed in Sections 4.5 and 4.6 respectively.

In this chapter, a physical embodiment is also proposed as a possible way of implementing

the proposed system in practice. The development of this physical system is discussed for

the shear transmitters in 4.7, the shear mechanism in Section 4.8, and for the rocking

mechanism in Section 4.9. In each of these sections, a few possible design options are

reviewed before the final design scheme is introduced in detail. The proposed mechanism

components are assembled in Section 4.10. In this section, the relation of the integrated

system with the entire building is discussed. In addition, the characteristic lateral

equilibrium of the MechRV3D system is also reviewed. Section 4.11 summarizes the main

findings of the chapter.


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4.2 Uncoupled Base Rocking and Shear Mechanism System: Concept

As schematically shown in Figure 4.1 (a), this proposed system, is located at the base of

structures, consisting of a flexural mechanism at the centre and a shear mechanism at the

periphery.

Figure 4.1 Concept of the uncoupled base rocking and shear mechanisms

The flexural mechanism relies on a rocking action to control base overturning moment

demands. This rocking action is intended to occur at the bottom level of a monolithic block

that is introduced at the base of the structure and allowed for stepping atop the foundation.

Having large rigidity, this stepping block acts as a load-path-decoupler that carries vertical

forces induced by flexure and gravity loads down to the foundation, while directing lateral

forces from the superstructure to the peripheral shear mechanism, as shown in Figure 4.1

(b).


67

The shear mechanism consists of nonlinear elements that limit lateral force demands that

can be developed at the base of the superstructure. Both mechanisms are specially detailed

at their intersections and joints to the foundation, such that the rocking mechanism carries

minimal lateral force at the rocking toes, while the shear mechanism provides minimal

moment resistance at the shear transmitters, as illustrated in Figure 4.1 (b). By these means,

these two mechanisms are physically separated and behaviourally uncoupled, which makes

it possible to control the flexural and lateral responses independently.


68

4.3 Idealized Configuration of the Proposed System

The proposed dual-mechanism system acts in three-dimension and is therefore referred to

as MechRV3D, which indicates the three-dimensional action (3D) of the rocking (R) and

shear (V) mechanisms (Mech). Figure 4.2 shows a simplified configuration of the

MechRV3D system, in which behaviours of the mechanism components are idealized. This

idealized system is first studied to understand governing mechanics of the proposed system.

Figure 4.2 Idealized configuration of the MechRV3D system


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4.3.1 Rocking mechanism

The rocking mechanism is located in the basement of a building under the RC core walls

as shown in Figure 4.2. At the ground level, the RC walls are cast monolithically with the

aforementioned near-rigid block which is referred to as the rocker hereafter. The rocker

rests on top of the foundation via four toes idealized as ball rollers that allow the rocker to

uplift freely while being able to move horizontally with negligible shear resistance. Once

base overturning moments, Mb, overcome the resisting moment created by the total gravity

load, Wsc, upon the lever arm, jdsc – where Wsc is the sum of the gravity load tributary to

the RC core, Wcore, and the self-weight of the rocker, Wrckr – the intended rocking action is

activated. This activation moment is referred to as the rocking moment, Mrock, and

calculated as Mrock = Wsc×jdsc, as illustrated in Figure 4.1 (b).

When Mrock is reached about a single principal direction, the rocker tends to leave its at-

rest position as shown in Figure 4.3 (a) and pivots about two ball rollers that line up along

this direction, forming a line-pivoting state shown in Figure 4.3 (b). If Mrock is overcome

about both principal directions, the rocker pivots about a single roller as shown in Figure

4.3 (c), which leads to the most critical scenario where this roller carries the entire gravity

load from the superstructure. Point- and line-pivoting are basic rocking states that depict

seismic motions of the rocker under bi-directional moments.

Figure 4.3 Rocking states of the rocker

In the rocking mechanism, no prestressed components are used. Instead, the self-centering

effect is totally achieved through the gravity load carried by the RC core, which is typically

about 50% of the expected gravity load of the entire building. This portion of recentering

force is basically invariant given its magnitude being primarily governed by structural


70

weights and superimposed dead loads that are relatively well defined in a structure. This

makes it convenient to achieve a required rocking moment by simply adjusting the centre-

to-centre distance between rollers, and also ensures that this flexural capacity stays

essentially constant throughout seismic excitations.

Vertical ground accelerations may cause fluctuations to the recentering force and thereby

to the rocking moment, Mrock. Typically the effects of vertical accelerations are not

considered in the lateral analysis for high-rise buildings without significant discontinuities

in gravity-load-resisting elements, which is the case for the reference building that was

used in this study, as discussed in [LATBSDC 2020] and [PEER 2017]. This is primarily

due to the fact that the vertical accelerations are of higher frequency and axial effects

caused by vertical accelerations fluctuate multiple times within one lateral deformation

cycle.

As such, vertical seismic excitations are not applied along with the horizontal components

in nonlinear seismic analyses that will be discussed in this dissertation. However, these

effects, which can be evaluated when required as 0.12SMSWsc according to [LATBSDC

2020] and [PEER 2017], where SMS is the MCER short-period spectral acceleration, and

Wsc is the total recentering gravity load as defined previously, are accounted for when

considering the design of the base rocking mechanism.

4.3.2 Shear mechanism

The shear mechanism is positioned on the periphery of the rocker, as shown in Figure 4.2.

It consists of a skirt diaphragm, or skirt in short, and a series of lateral bracing components

that connect the skirt to the foundation. The skirt consists of the ground floor slab outside

of the central core area, which has sufficient in-plane stiffness to act as a rigid diaphragm.

The lateral bracing components are made of nonlinear elements and arranged in parallel on

each side of the rocker. Base shears of the RC core arising at the top level of the rocker are

transferred to the skirt which then engages the lateral bracing components that provide

shear strength, stiffness and ductility in both principal directions. Gravity loads imposed

on the skirt (the ground floor slab) are carried by vertical boundary elements of the lateral

bracing components and gravity columns in the basement of the building.


71

The rocker and the skirt are not cast monolithically but have a gap between them, as shown

in Figure 4.2. Along this gap, a series of shear transmitters are arranged in alignment with

the lateral bracing components. These transmitters are embedded into the side face of the

rocker on one end and into the inner edge of the skirt on the other end, linking these two

components at the ground level. Each transmitter is assumed to be rigid in the longitudinal

axis normal to the rocker’s side face, transferring lateral forces and movements. The shear

transmitters can carry axial forces in both tension and compression or in compression only,

depending on how these transmitters are physically implemented. In Section 4.7, a

structural connection and a mechanical joint will be proposed as possible implementations

for both types of actions. Regardless of axial loading modes, all the shear transmitters are

not expected to carry forces in the vertical direction, such that the shear mechanism

imposes minimal impedance to the rocking motions.

When the base shear of the RC core is imposed, the rocker tends to move in the loaded

direction. This lateral force and movement are then transferred via the shear transmitters to

the skirt which in turn engages the lateral bracing components that are oriented in the same

line of action, as shown in Figure 4.4. These engaged bracing components, regardless of

on which side of the rocker they are, share a common deformation that is essentially equal

to the horizontal displacement of the rocker, given the negligible axial deformability of the

shear transmitters and the diaphragm effect of the skirt. However, as noted in Figure 4.4,

lateral bracing components that are oriented in the orthogonal direction will not be engaged

due to a negligible out-of-plane stiffness.

Figure 4.4 Transfer of lateral shears


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In reality, accidental eccentricity may induce torsional effects at elevated floors of the

superstructure which are cumulated over the height and form a torsional moment at the top

level of the rocker. This torque is equilibrated by a torsional resistance that is developed

by all the lateral bracing components. In transferring torsional moments, shear transmitters

are engaged in varied ways depending on if they can carry load in one way or both ways,

as illustrated in Figure 4.5.

Figure 4.5 Transfer of torsional moments

4.3.3 Comparison with conventional systems

Characterized by the uncoupled mechanics, the proposed system differs from conventional

RC structures where the bond between steel and concrete materials leads to an inherent

shear-flexure interaction. In addition, the dual-mechanism of the MechRV3D system is not

present in RC wall systems where seismic effects are limited replying on a single

mechanism in the form of base flexural hinges.

The proposed system is conceptually compared with a conventional rocking system. In the

latter case as shown in Figure 4.6 (b), horizontal displacements are restricted at the base of

the structure where rocking action takes place. As for the MechRV3D system shown in

Figure 4.6 (a), while the lateral resistance is still provided at the base of the core, the

rocking section is lowered down to the bottom level of the rocker. This change does not

essentially affect the flexural response, since the overturning moment, F×jdF, is governed

by the self-centering moment, Wsc×jdsc, regardless of whether it is evaluated with respect


73

to the actual rocking toe in the both cases or, in case (a), the node VR where the forces V

and R intersect. However, the proposed system separates lateral forces and vertical loads

at the ground level. This forms the basis of the uncoupled mechanics which are not present

in conventional base rocking systems.

Figure 4.6 MechRV3D system and a conventional base rocking system

The MechRV3D system is also differentiated from the base-isolated rocking system that

was proposed by Calugaru [2013] and reviewed in Chapter 2. The latter system represents

a category of existing dual-mechanism solutions where the flexural and shear mechanisms

act in series. This in-series arrangement leads to shear fuse elements undergoing significant

deformations while carrying significant gravity loads and large flexure-induced tensile

forces until the base overturning moment is limited upon the activation of the rocking

action. In contrast, shear-flexure interaction of this kind is not present in the proposed

MechRV3D system.


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4.4 Numerical Modelling of the Idealized Configuration

To understand the governing mechanics of the proposed system, the idealized configuration

described in the previous section was first investigated numerically. For this preliminary

study, advanced nonlinear models were built for the MechRV3D system and used for

nonlinear static and dynamic analyses that were conducted using the OpenSees (Open

System for Earthquake Engineering Simulation) [McKenna et al. 2010]. In these analyses,

a benchmark building [Moehle et al. 2011] that has been studied by the Pacific Earthquake

Engineering Research Centre (PEER) at the University of California, Berkeley, was used

as a reference structure.

4.4.1 PEER benchmark building and stick model

This benchmark building was studied by the PEER as part of the Tall Buildings Initiative

project [Moehle et al. 2011]. The building is a residential development including 42 storeys

above the ground and 4 storeys in the basement. The lateral-force-resisting system consists

of RC walls that are located at the centre of the building, forming a core that is about 10

m-by-15 m on plan and 125 m tall above the ground level, as shown in Figure 4.7. Given

the aspect ratio over 10 and the fundamental period of about 4 sec, this building is expected

to be susceptible to pronounced higher-mode effects.

Figure 4.7 PEER benchmark building (adapted from [Moehle et al. 2011])


75

In this chapter, the PEER benchmark building is first used for investigating the idealized

configuration of the proposed system. This study focuses on fundamental mechanics of the

base mechanisms. For this purpose, the benchmark building is represented using a

simplified model including only the central core that is modelled as an elastic stick with

lumped mass, providing a MDOF system whose lateral seismic response is contributed by

multiple vibration modes. This is needed to preliminarily verify if the MechRV3D system

can limit higher-mode effects as expected. In Chapter 5, the same benchmark building will

be further used as a reference structure for nonlinear numerical analyses conducted to

validate a detailed design of the MechRV3D system. In that chapter, nonlinear properties

of the benchmark building will be considered in detail.

Figure 4.8 schematically shows the nonlinear model that was built for the preliminary study.

The stick model of the RC core consists of beam-column elements, one per storey, as

shown in Figure 4.8 (a) . These elements were assigned elastic material properties and gross

sectional properties of the core about both strong and weak axes. Seismic inertias were

lumped at floor levels, including horizontal masses and rotational moments of inertia

within the horizontal plane. For simplicity, basement storeys were not included. The core

stick was fixed at the ground level to the centre of the top face of the rocker as can be seen

from in Figure 4.8.

4.4.2 Rocking mechanism

Despite a solid block, the rocker of the MechRV3D system was modelled using frame

elements for the sake of computational efficiency. These frame elements were assigned

large axial and flexural rigidities, and then interconnected to establish a cage-like enclosure

simulating the negligible deformability and the encasement that the rocker provides to the

core at the base. As shown in Figure 4.8 (c), at the top face of the rocker, rigid elements

connect to a node at the centre where the core stick is encased, and to nodes on the perimeter

where lateral bracing components are framed in. At the bottom face of the rocker, four

nodes that represent rocking toes are interconnected using rigid elements. The top and

bottom faces are then connected at the perimeter through interweaved elements that make

the skeleton model take on an enclosed volume. Rigid truss elements are also placed inside

this enclosed cage, carrying gravity loads from the core stick down to the ball rollers.


76

Figure 4.8 Schematic model of the idealized configuration


77

At the location of each ball roller, a pair of nodes with identical coordinates were created

to respectively represent the rocking toe in the soffit of the rocker and its potential contact

area on the foundation, as can be seen in Figure 4.8 (d). Each pair of nodes was connected

using a zero-length spring that was oriented vertically and assigned no tensile resistance

but a large rigidity in compression. This compression-only response was implemented

using the elastic-no-tension (ENT) uniaxial-material model formulated in OpenSees

[McKenna et al. 2010] to capture the free uplifting action of ball rollers. In parallel with

the ENT spring, another linear elastic spring with very small stiffness was used to ensure

the numerical stability. The foundation nodes are fixed vertically. However, the horizontal

degree-of-freedom of these nodes were slaved to that of corresponding rocking-toe nodes

such that these foundation nodes acted as potential landing points for the rocker.

4.4.3 Shear mechanism

Similarly, components of the shear mechanism were represented using nonlinear springs

as well. As shown in Figure 4.8 (e), lateral bracing components on each side of the rocker

were modelled using zero-length elements that were horizontally oriented and assigned an

elasto-perfectly plastic (EPP) force-deformation relation. With one end being fixed to the

foundation, each of these EPP springs was linked in series to a gap element at the other

end. These gap elements with a near-zero gap distance were modelled using ENT springs

with large compression rigidity, simulating the shear transmitters that link the rocker and

the skirt. In this idealized simulation, the skirt diaphragm was not explicitly modelled as a

floor slab as it is in reality, but represented by the intersecting nodes, or skirt nodes,

between the gap elements and the EPP springs. These skirt nodes were constrained by a

horizontal rigid diaphragm constraint such that they would, as the physical skirt diaphragm,

undergo identical displacements and rotations within the horizontal plane. Given the

negligible deformability of the shear transmitters, horizontal movements of the rocker are

transmitted to the skirt which then turns these rigid-body movements into deformations of

the lateral bracing components (EPP springs).


78

4.4.4 P-𝛥 effects

Gravity columns outside of the central core area were not explicitly modelled. However,

P-𝛥 effects were accounted for using leaning columns that were modelled using the

corotational truss element in OpenSees [McKenna et al. 2010], as shown in Figure 4.8 (a).

The cross-sectional area of these truss elements was set to be equal to the total area of the

physical gravity columns. While the leaning column was pinned at the foundation level, it

was constrained by the rigid floor diaphragms that were defined at each level of the building

above the ground. At the ground level, the leaning column was horizontally slaved to the

skirt-node diaphragm (defined in Section 4.4.3) to capture the effect that gravity columns

sway at this level with the moving skirt. The core stick was modelled including P-𝛥 effects

as well. Therefore, gravity loads were imposed separately to the leaning column and the

core stick at each floor level according to their tributary areas.


79

4.5 Mechanics of the MechRV3D System

The static response of the MechRV3D system was first studied using the OpenSees model

built in Section 4.4. For this purpose, pushover analyses were conducted on the idealized

system, as schematically shown in Figure 4.9. These static nonlinear procedures were

carried out in a displacement-controlled manner with the lateral drift at the roof level, 𝛥roof,

taken as the reference. Lateral forces, Fi, that were applied at floor levels, were calculated

using Equation (4.1) as follows,

𝐹 = 𝑚 ℎ∑ 𝑚 ℎ (4.1)

where mi is the seismic mass assigned to the ith floor, and hi is the floor level above the

ground. This loading profile takes a shape of an inverted triangle which basically follows

the first-mode lateral force distribution. Starting from the at-rest position, a maximum drift

of 1.5%H, where H is the total height of the building above the ground, was targeted at the

roof level. This peak drift ratio was believed to be sufficient to effectively engage the base

mechanisms. The target peak drift was reached with a displacement incremental of 0.1 mm

which led to good numerical stability.

Figure 4.9 Static pushover analysis scheme


80

For the preliminary analyses, the centre-to-centre distances between the ball rollers, dc2c,

were set to be 9900 mm along the east-west direction and 14855 mm along the north-west

direction respectively. Given the total self-centering weight, Wsc of 242 MN, these rocking-

toe distances result in rocking moments of 1200 MN-m about the north-west direction and

1800 MN-m about the east-west direction correspondingly. The total lateral resistance was

set to be 84.7 MN (EW) and 86.1 MN (NS) which are equivalent to 19.2% and 19.5% of

the total seismic weight, WEQ, of 441 MN. These resisting forces are provided by eight

lateral bracing components (EPPs) in each principal direction.

Capacity curves that were obtained from the first-mode pushover analyses are plotted in

black lines in Figure 4.10, indicating the relationship between the total applied lateral force,

F, and the roof drift ratio (RDR), 𝜃roof, which is calculated as 𝛥roof /H. In the same figure,

rotation angles at the base of the rocker due to the rocking action, 𝜃rock, are plotted in blue

lines, showing their variation with 𝜃roof. From these diagrams, it can be seen that, in the

initial low-RDR stage, no uplifting occurred at the base of the rocker such that the roof

deflection resulted completely from the flexural deformation of the core. Given no rotation

at the base of the rocker and the assumed elasticity of the core, the influence of P-𝛥 effects

was trivial in this linear elastic stage. This explains why the solid line (with P-𝛥 effects)

and dashed line (without P-𝛥 effects) in Figure 4.10 are very close to each other prior to

the activation of rocking action.

Figure 4.10 Lateral capacity curves and base rocking rotations

0.0

0.5

1.0

1.5

0

1

2

3

4

0.0 0.5 1.0 1.5

𝜃 rock

,rNS[

%]

F EW/W

EQ[%

]

𝜃roof,EW [%]

applied lateral force limited through rocking action P-𝛥 effects

included

P-𝛥 effects neglected

rocking actionactivated


81

As the control displacement was further increased, the applied lateral forces scaled up

proportionally. When the RDR reached a threshold level, which, in this particular case, was

0.24%, the rotation of the rocker started taking finite angles, which marked the activation

of the intended rocking action. At this instant, the resultant applied force, F, reached its

peak value of 3.1%WEQ. As the core was pushed further, a major portion of the roof drift

was contributed by the rigid-body rotation of the rocker at the base, which can be seen from

the close-to-unity slope of the 𝜃rock-𝜃roof diagram plotted in blue in Figure 4.10. This

increasing base rocking rotation led to increased lateral displacements in the superstructure

upon which gravity loads created larger P-𝛥 effects, resulting in a continuous reduction in

the applied lateral force. This can be seen in Figure 4.10 from the post-rocking portion of

the F-𝜃roof curve that declines being compared to the dashed line indicating no P-𝛥 effects.

The overturning moment, Mb,F, that is created by the applied force F at the base of the core

over a constant effective height, followed a similar trend in both prior- and post-rocking

phases, as indicated by the dashed line in Figure 4.11. Plotted in the same diagram using

solid lines is the total base overturning moment, Mb, that includes both the applied moment,

Mb,F, and that caused by the P-𝛥 effect of the superstructure, Mb,P-𝛥. Once the rocking action

is activated, Mb is effectively capped off and anchored at the level of the rocking moment,

Mrock. This means that the maximum possible moment demand at the base of the RC core

is entirely governed by the rocking mechanism, since the moment capacity Mrock is solely

determined by the recentering gravity load, Wsc, without any contributions from the shear

mechanism.

Figure 4.11 Base overturning moments governed by Mrock

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mb,

rNS

/ Mro

ck,rN

S

𝜃rock,rNS [%]

Mb,P-𝛥base overturning moment

due to P-𝛥 effects Mb,Fbase overturning moment created by applied lateral forces

total base overturning moment Mb = Mb,F + Mb,P-𝛥

rocking actionactivated


82

In Figure 4.11, the post-rocking portion of the Mb-curve keeps constant even after the

rocking action is activated, which implies that the resisting moment Mrock = Wsc×jdsc does

not decrease as it typically does for a rigid rocking block. This counterintuitive comparison

can be explained using Figure 4.12 as an illustration. For a rigid block that undergoes planar

rocking as shown in Figure 4.12 (a), once a gap opens at the base of the block, the centroid

of gravity moves closer to the rocking toe in the horizontal direction, leading to the lever

arm of the recentering force, W, decreasing. Given a constant W, the shortening lever arm

results in a declining moment resistance that can be provided by the rocking body, which

is reflected as a declining line in a typical moment-rotation curve of a rigid rocking block.

Once the rocking rotation, 𝜃rock, reaches the rocking block’s slenderness, defined as tan𝛼 =

b/h, the moment-carrying capacity becomes zero. If the block further rotates, it will topple

over under the gravity load, W.

Figure 4.12 Rigid rocking block and flexible rocking structure


83

In contrast, real high-rise buildings, at the base of which the proposed system is to be

incorporated, are flexible structures. Under lateral seismic forces, the centroid of gravity of

the superstructure (which can be roughly assumed to be located at the mid-height of the

RC core) displaces horizontally as a result of both structural deformation and the base

rocking rotation, as shown in Figure 4.12 (b). Despite the magnitude of this deflection, the

entire gravity load, Wcore, that is carried by the core, is definitely transferred to the top

centre of the rocker. This suppressing force, denoted as Wcore@toprckr to highlight its location,

is vertically away from the rocking toe by just the depth of the rocker. This Wcore@toprckr,

along with the rocker’s weight, Wrckr, create the moment resistance, Mrock, upon the lever

arm, jdsc. As will be revealed in Chapter 5, under MCE level earthquakes, the maximum

base rotation of the rocker can be at a magnitude of 1% - 2%, which leads to the lever arm

being reduced by 2% - 3% for Wcore@toprckr and even less for Wrckr. Given the constant

recentering forces, these reductions in the lever arm are too trivial to cause a declining

moment resistance, resulting in a constant Mrock, as seen in the static response in Figure

4.11 and will be seen in the dynamic response as to be discussed in the next section.

In the first-mode pushover, the shear mechanism was not actually engaged, resulting in an

elastic V-𝛥skirt relationship, as shown in Figure 4.13. This is expected when the proposed

system is subjected to a static lateral force distribution of which the moment-to-shear ratio

is invariant throughout the loading process, and when the shear mechanism is assigned a

yielding strength that is greater than the force that is needed to activate the rocking action.

Figure 4.13 Response of the shear mechanism in the first-mode pushover

0 200 400 600

0

1

2

3

4

0

1

2

3

4

0 2 4 6

𝛥roof,EW [mm]

F EW/ W

EQ[%

]

V EW/ W

EQ[%

]

𝛥skirt,EW [mm]

shear mechanismremaining elastic

lateral-load-carrying capacitylimited by rocking action


84

Another static pushover analysis was conducted using a different load distribution whose

resultant force was applied laterally at a level of 0.1H above the ground. This led to a lever

arm that is much smaller than the effective height, estimated as 2H/3, in the previous first-

mode pushover. In this analysis, the shear mechanism was effectively engaged and yielded

when F reached 19.2%WEQ, which was identical to the total strength assigned to the EPP

springs, as seen in Figure 4.14. This also confirms that the shear mechanism is the only

provider of the lateral resistance and the rocking mechanism makes no contribution.

Figure 4.14 Shear mechanism engaged under a different loading profile

Given the short lever arm, the applied force, F, even at the peak level, did not create a base

overturning moment that overcomes Mrock. This made the rocking toes on the windward

side keep carrying gravity loads even though a majority of these loads had been transferred

to the leeward toes. This redistribution of the gravity loads is shown in Figure 4.15. Since

no rocking occurred, the impact of P-𝛥 effects to the overall lateral response was small, as

can be seen in Figure 4.14. Whereas it is certainly possible to find a static loading pattern

that engages the dual mechanism simultaneously, it is more natural to do so under a

dynamic loading condition as higher modes will be included automatically.

Figure 4.15 Redistribution of gravity loads on rocking toes

0 20 40 60 80 100 120

0

5

10

15

20

25

0

5

10

15

20

25

0 20 40 60 80 100 120

F EW/ W

EQ[%

]

V EW/ W

EQ[%

]

inelastic shear mechanism

lateral-load-carrying capacity limited by inelastic shear mechanism

𝛥roof,EW [mm]

𝛥skirt,EW [mm]

0

25

50

75

100

0 50 100 150

P - gravity loads carried byleeward and windward rocking toes

𝛥roof,EW [mm]

P / Wsc [%]

leeward rocking toes

windward rocking toes

rocking toesNW

SW

NE

SE


85

4.6 Dynamic Response of the MechRV3D System

Nonlinear response history analyses (NLRHAs) were conducted using the idealized model

to investigate the seismic response of the MechRV3D system. The ground motion recorded

at the Saratoga Aloha Ave Station during the Mw6.9 1989 Loma Prieta Earthquake was

used and scaled to the MCE level. Subjected to this bi-directional ground shaking, the

intended rocking action was effectively achieved, displaying a self-centering moment-

rotation hysteresis about both principal directions, as shown in Figure 4.16. Similar as in

Figure 4.11, the base overturning moment was governed by Mrock that is constant.

(a) about the north-south direction (b) about the east-west direction

Figure 4.16 Hystereses of the rocking mechanism

Bi-directional rocking led to the line-pivoting and point-pivoting states as illustrated in

Figure 4.3. As a result, gravity loads were either shared by all the rocking toes or alternately

carried by some of them, or in the most critical case by a single roller. This dynamic

redistribution of gravity loads is demonstrated in Figure 4.17.

Figure 4.17 Variation of gravity loads carried by the rocking toes

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Mb,

rNS

/ Mro

ck,rN

S

𝜃rock,rNS [%]

rocking action activated


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Mb,

rEW

/ Mro

ckrE

W

𝜃rock,rEW [%]



0 5 10 15 20 25 30 35 40 45 500

255075

100

Time [sec]

P toe

/ Wsc

[%]

NE

SE

NW

SW

rockingtoes

(a) full time history

0 1 2 3 4 5 6 7 8 9 100

25

50

75

100

Time [sec]

P toe

/ Wsc

[%]

NESE

NWSW

rockingtoes

(b)truncatedtime history(0 - 10 sec)


86

The shear mechanism was also engaged in this seismic analysis. As shown in Figure 4.18,

the total lateral resistance, V, was effectively limited to 19.2%WEQ, compared to a reference

analysis where the lateral bracing springs were intentionally set as elastic. This cutoff force

was identical to the total strength assigned to the EPP springs, which confirmed that only

the shear mechanism provided lateral resistance. Energy dissipation was achieved through

the V-𝛥skirt hysteresis shown in Figure 4.19.

Figure 4.18 Time histories of the lateral resistance

Figure 4.19 V-𝛥skirt hystereses of the shear mechanism

-50

0

50

0 5 10 15 20 25 30 35 40 45 50 55

V EW/ W

EQ[%

]

Time [sec]

(a) full response in the east-west direction see truncated response in (c)

-50

0

50

0 5 10 15 20 25 30 35 40 45 50 55

V NS

/ WEQ

[%]

Time [sec]

(b) full response in the north-south directionsee truncated response in (d)

-50

0

50

5 6 7 8 9 10

engaged shear mechanism

non-yieldingshear mechanism

V EW/ W

EQ[%

]

Time [sec]lateral shear

capped off

(c) truncated response in the east-west direction

-50

0

50

5 6 7 8 9 10

V NS

/ WEQ

[%]

Time [sec]

(d) truncated response in the north-south direction

-30

30

-50 50

𝛥skirt,EW [mm]

VEW / WEQ [%]

(a) east-west -30

30

-50 50

𝛥skirt,NS [mm]

VNS / WEQ [%]

(b) north-south


87

4.7 Physical Embodiment of the MechRV3D System: Shear Transmitters

The previously presented study on the idealized configuration demonstrated that the

uncoupled mechanics of the proposed system do allow for independent flexural and lateral

responses as intended. To achieve a practical and buildable embodiment of this idealized

configuration, a few design options were considered and preliminarily investigated for key

mechanism components, as will be discussed in the subsequent sections. This discussion

starts from design options of the shear transmitters, for which possible implementations

were proposed from the perspectives of structural and mechanical engineering respectively.

4.7.1 Hinged plates

As idealized in Section 4.3.2, the shear transmitters are expected to be effective axial load-

bearers while restricting to a minimal extent the relative movements between the rocker

and the skirt in the vertical direction. To achieve these differential load-carrying capacities,

hinged plates were considered as an option, as illustrated in Figure 4.20. In this scheme, a

series of short steel plates are used to bridge the gap between the rocker and the skirt.

Relying on the large membrane stiffness and in-plane force resistance, these elements are

capable of transferring lateral shears and movements as intended. At the same time, these

steel plates are designed to form stable out-of-plane flexural hinges near both ends where

they are encased to the rocker and the skirt. These plastic hinges can be developed under a

small transverse deflection induced by the uplifting of the rocker. Having a low weak-axis

flexural capacity, these hinges limit vertical forces that can be induced by the plates at sides

of the rocker and thereby minimize the moment resistance to the rocking action.

Figure 4.20 Hinged plates as the shear transmitters


88

The hinged plates carry axial loads in both tension and compression. Therefore the design

of the axial loading capacity shall be based on the compression load case which is more

critical. Taking the benchmark building (as introduced in Section 4.4.1 and will be

discussed in detail in Chapter 5) as an example, if four metre-long hinged plates are used

on each side of the rocker, a cross-section that is 400 – 700 mm wide and 40 – 60 mm thick

can be adequate to provide horizontal force transfer while limiting the additional moment

resistance to be less than 5% of the intended rocking moment. As such, it is reasonable to

assume that the shear mechanism makes a minimal contribution to the moment resistance

at the base of the superstructure. In modelling the rocking action, the impact of the hinged

plates can be neglected.

4.7.2 Gear connections

In addition to the hinged-plate option, an articulated joint that is usually used in mechanical

engineering was proposed as a second way of implementing the shear transmitters. This

joint consists of a protruding tooth part and a recessed groove part, resembling a

mechanical gear connection as shown in Figure 4.21. While the tooth is encased into the

rocker, the groove is attached to the skirt. Both parts can be made of high-strength cast

steel and are assumed to be undeformable. Unlike the hinged plates that can be axially

engaged in both tension and compression, this gear connection is a compression-only joint,

as defined in Section 4.3.2, that works in contact with a near-zero gap distance between the

tooth and the groove.

Figure 4.21 Gear connections as the shear transmitters


89

When the rocker is loaded laterally and leans towards the skirt on the leeward side, as

illustrated in Figure 4.22, gear connections on this side of the rocker immediately come

into a firm contact, transferring lateral forces and movements. On the windward (opposite)

side of the rocker, the gear connections are in a loose contact if no rocking occurs or in a

detached state if the rocker steps up. In both cases, these windward gear connections

transfer no horizontal forces. As for the gear connections on the two orthogonal sides of

the rocker, as indicated in Figure 4.22, they are not expected to be engaged transversely to

carry any lateral shears, because the cantilever arms of the grooves will be bending out-of-

plane. This can be achieved by carefully machining the gear teeth and grooves, such that

longitudinal engagement always occurs first in the leeward gear connections, preventing

transverse contact from happening in the side gear connections.

Figure 4.22 Moment resistance contributed by leeward gear connections

Under this pattern of engagement shown in Figure 4.22, the engaged leeward gear

connections may induce vertical forces at the side of the rocker due to friction, and thereby

create unintended resisting moment to the rocking action. However, this resistance is

anticipated to be small given a very short lever arm between these frictional forces and the

rocking toe. To further reduce this moment resistance, PTFE coating can be applied to gear

teeth and grooves to minimize the friction. Still taking the PEER benchmark building an

example, when the ultimate lateral resistance is developed in the shear mechanism, the

gear-connection-induced moment is less than 1.5% of the designated rocking moment.


90

Even if, in an unanticipated case, contact occurs at both the leeward and side gear

connections in the longitudinal and transverse directions respectively, as illustrated in

Figure 4.23, the moment resistance due to the frictional forces is slightly increased but still

under 2.0% of the rocking moment based on a conservative calculation.

Figure 4.23 Moment resistance contributed by leeward and side gear connections

Hence, it is rational to ignore the moment resistance that may be created by the frictional

forces in the gear connections, regardless of whether these connections are engaged on the

leeward side only or on the orthogonal sides as well. Due to the reason explained previously,

it is not recommended to have the lateral shear entirely transferred through side gear

connections that are transversely engaged. Compared with the out-of-plane stiffness of the

grooves, it is more efficient to rely on the axial contact that exhibits a higher rigidity.

The grooves can be made deeper to guide the vertical motion of the teeth during uplifting.

In addition, the width of the groove can be larger at the top and bottom faces to ease the re-

entry of the teeth. By these means, the overall rocking action is prevented from being

impeded by jamming of the gear connections.

4.7.3 Remarks

In sum, both hinged plates and gear connections are feasible options to achieve the intended

horizontal transmission of forces and motions while imposing negligible impedance to free

rocking actions. Despite the difference in axial engagement (two-way or one-way), these

two types of shear transmitters play equivalent roles in getting the shear mechanism

activated. For simplicity, gear connections will be displayed in illustrations of the physical

embodiment design that is proposed in this study. At the same time, numerical modelling


91

will also be based on the gear-connection option. However, in terms of the resultant global

behaviour of the MechRV3D system, this modelling is assumed to be applicable to the

hinged-plate option as well.


92

4.8 Physical Embodiment of the MechRV3D System: Shear Mechanism

4.8.1 Steel plate shear wall panels

The shear response of the proposed system is intended to be ductile and to have stable

energy dissipation. For this purpose, steel plate shear wall (SPSW) panels were first

considered to be used as lateral bracing components. SPSWs have been widely used as a

lateral-force-resisting system in seismic design. These panels are filled with a steel web

plate in a thickness that is much smaller than dimensions of the panel. This slender infill

plate is not stiffened across its area but only restrained on boundaries using beams and

columns. These horizontal boundary elements (HBEs) and vertical boundary elements

(VBEs) are usually assigned a high rigidity, such that when the panel is subjected to lateral

forces, diagonal tensile and compressive principal stresses can be effectively developed

across the web plate which is in a pure shear stress state. Due to the high slenderness, the

unstiffened web plate elastically buckles in the compressive direction at a low level of

lateral forces. Nevertheless, the panel can continue carrying lateral loads through a tension

field action that develops in the tensile direction, providing lateral strength, stiffness,

ductility and energy dissipation.

For the proposed MechRV3D system, one-storey and single-bay SPSW panels are placed

at locations that are held for the lateral bracing components. They are connected to the skirt

diaphragm and the foundation through top and bottom HBEs respectively. These panels

are expected to carry in-plane forces only, given negligible out-of-plane stiffness. This can

be practically achieved by providing a pinned connection at the base of the columns to

release the restraint to out-of-plane rotations. The thickness of the web plate is determined

based on an overall lateral resistance that is required of the shear mechanism for achieving

an optimal control of the higher-mode response. As the intended tension field mechanism

is developed in the web plate, HBEs and VBEs are capacity designed to remain essentially

elastic with only plastic hinges allowed at both ends of the HBEs. As such, the boundary

frame action must be properly accounted for in the design, otherwise it may lead to lateral

overstrength that unfavourably allows for excessive shear demands being developed at the

base of the superstructure, resulting in pronounced higher-mode effects. There is a balanced

design where the boundary elements are adequate to allow for the formation of the tension


93

field mechanism, while the associated frame action tops up the lateral load-bearing capacity

of this mechanism to the desired strength level. To this regard, Qu and Bruneau [2009]

developed a design chart, as shown in Figure 4.24, from which balanced strength ratios

between the web plate and boundary frame can be estimated for panels with varied aspect

ratios. In this chart, κ denotes the portion of the design lateral resistance that is allocated to

the web plate, and Ωκ is the overstrength factor that indicates the exceedance of the total

expected capacity over the design strength.

Figure 4.24 Relation between Ωκ and κ (from [Qu and Bruneau 2009])

Following these considerations, SPSW panels were designed for the MechRV3D system.

In each principal direction, eight panels were incorporated, four on each side of the rocker.

Various aspect ratios were considered in the design. Among these options, panels that are

4 m tall and 7 m wide were chosen for a preliminary numerical investigation. Given the

aspect ratio of 1.75, it was determined to have the web plate provide 55% of the total lateral

resistance according to the chart shown in Figure 4.24. This resulted in a 2.7 mm thick

plate, a W-shaped section of W760×484 for VBEs, and a 700 mm deep built-up section for

HBEs. In this design, yielding stresses were assumed to be 250 MPa for the web plate and

345 MPa for all the boundary elements.

For validation purposes, this design option was numerically assessed through NLRHAs in

which a strip model was used to simulate the tension field action of SPSW panels. This

modelling approach was developed by Thorburn et al. [1983] who was inspired by a similar

investigation that Wagner [1931] conducted on thin aluminium webs of girders used in

aircrafts. In this model, as schematically shown in Figure 4.25, the continuum of the web


94

plate was discretized into two groups of strips leaning leftward and rightward respectively.

All the strips share a common angle of 45° to the vertical direction. These strips were

modelled using force-based beam-column elements that consist of non-linear fibre sections.

Pinned to the surrounding HBEs and VBEs at both ends, these fibre-based strips would

only be engaged axially, acting as two-force members. At the midway of the strips, a hook

element was arranged, such that these strips carry tensile forces only. Having no resistance

in compression, the left- and right-leaning strips would be engaged alternately when the

panel is subjected to reversed cyclic lateral loads. Eleven strips were used in each

directional group, which is more than a minimum required number of ten as recommended

in [Bruneau et al. 2011].

Figure 4.25 Schematic strip model for the SPSW panel

SPSW panels modelled in this way were incorporated into an overall model that includes

the rocking mechanism and the PEER benchmark building as introduced in Section 4.4.1.

At this development stage, the rocking mechanism was still a single block as idealized in

Section 4.3.1 and modelled in Section 4.4.2. More realistic design of the rocking

mechanism will be introduced in Section 4.9, followed by more sophisticated modelling

and numerical validations in Chapter 5. The PEER benchmark building was represented

explicitly using a three-dimensional nonlinear model allowing for distributed plasticity.

Since this section focuses on the feasibility of using SPSW panels, the modelling of the

benchmark building will be discussed in Chapter 5 along with numerical validations of a

physical embodiment that is proposed for the MechRV3D system.


95

NLRHAs were conducted on this overall model using a suite of seven ground motions that

were scaled to the MCE level (seismic hazards and ground motion scaling will be discussed

in detail in Chapter 5 as well). Results of these analyses indicated that excessive lateral

displacements were anticipated at the base of the superstructure, resulting in drift ratios

about 6% in the SPSW panels. This excessive drift demand was attributed to the pinched

hystereses that largely reduced the energy dissipation capacity. And this pinching effect is

an inherent characteristic of SPSW panels whose lateral load-bearing capacity is limited as

the tension field action fades out in one direction but has yet developed in the other diagonal

direction until the panel distortion exceeds the peak lateral drift that is reached in a previous

cycle. This slack in the load reversal results from buckled zones of the web plate that are

loosened and cannot carry tensile forces [Bruneau et al. 2011]. Hence, other types of steel

shear panels where buckling is restrained by some means were also studied.

4.8.2 Buckling-restrained steel plate shear walls

Varied concepts have been developed to prevent out-of-plane buckling of steel plate shear

panels. Maurya et al. [2013] proposed ring-shaped steel plate shear walls (RS-SPSWs), as

shown in Figure 4.26 (a), that are characterized by a perforated pattern in which a solid

steel panel is cut, leaving rows of steel rings that are interconnected by links in both

diagonal directions. As a result of these circular cutouts, excessive materials are removed

from zones where buckling would otherwise occur in compression. Instead of a tension

field action, the yielding mechanism of the RS-SPSW panels is formed by flexural hinges

in the rings as shown in Figure 4.26 (b). Efficiency of these panels was verified by Egorova

et al. [2014] through experimental tests. The test results indicated that these buckling-

restrained shear panels display fuller hystereses and greater energy dissipation capacity

than specimens made of conventional SPSWs.


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Figure 4.26 Ring-shaped steel plate shear wall panel

Jin et al. [2016] proposed a different buckling-restraining scheme in which a concrete panel

is attached to each side of the infill steel plate, forming a sandwiched configuration as

shown in Figure 4.27. In the steel plate, inclined slots are cut, transforming the solid plate

into a series of diagonal steel strips, as shown in Figure 4.27 (c). Along lines projected

from these slots, holes are drilled in both concrete panels such that all the three panels can

be bolted together. When the infill steel plate is subjected to lateral loads, the steel strips

are engaged developing yielding mechanisms in both tension and compression without

slack, since the out-of-plane buckling of these strips has been restrained by the concrete

panels. This makes these strips act as a series of buckling-restrained braces, displaying a

stable hysteresis and providing increased energy dissipation.

(a) front view (b) side view (c) slotted steel plate

Figure 4.27 Slotted steel plate shear wall panel (from [Jin et al. 2016])


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Despite the improved hysteretic behaviour, some limitations were also identified that

questioned the practicality of using these variants of SPSWs in the MechRV3D system.

The first concern was special steel fabrication that would be required to cut a solid plate

into the ring-shaped pattern of the RS-SPSWs or paralleling slots in the slotted SPSWs,

which may also lead to an increased cost. Additionally, in the latter option, extra concrete

panels are needed for the buckling-restraining purpose, which adds more complexity into

construction. Furthermore, developments of these buckling-restrained shear panels were in

the early stages in terms of design methods, manufacturing standards, and product

prequalification. For these reasons, these two panels were not adopted as the lateral bracing

components in the MechRV3D system.

4.8.3 Steel slit shear wall panels

Steel slit shear wall (SSSW) panels were also considered as another possible option for the

shear mechanism of the MechRV3D system. Proposed by Hitaka and Matsui [2003], this

type of panels consists of multiple rows of steel segments, referred to as links, that are

formed by cutting vertical slits in a solid plate, as shown in Figure 4.28. Between adjacent

rows of links, a belt area is not slit forming a stiffer band zone. This slit plate is constrained

at the top and bottom through HBEs, but unrestrained at both vertical edges. Given these

boundary conditions, the links act as flexural members that bend in plane when the panel

undergoes lateral drifts. As the drift increases, plastic hinges can be developed at both ends

of the links where they are connected to HBEs and band zones. These flexural hinges form

a yielding mechanism that determines the lateral strength of the panel and provides ductility

and energy dissipation. In contrast to the tension field action in SPSW panels, this flexural

yielding mechanism, once developed, has no buckling involved, and therefore can provide

increased energy dissipation through fuller hystereses with minimal pinching effects.


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Figure 4.28 Steel slit shear wall panel (from [He et al. 2016])

This being said, the formation of the preferred flexural mechanism may be hindered if the

flexural links are not assigned proper length-to-width ratio (l/b) or width-to-thickness ratio

(b/tw), which may lead to local buckling or even fracture. To prevent these premature failure

modes, He et al. [2016] suggested to build SSSW panels using low-yield-point (LYP) steel

whose yielding stress is just a fraction of normal strength materials. As such, to achieve an

equal load-bearing capacity, LYP SSSWs are typically thicker than panels that are made

of normal strength steel. The increased thickness makes the links unsusceptible to lateral-

torsional buckling. Additionally, as experimentally validated by He et al. [2016], high

ductility and significant strain hardening of the LYP steel make it possible to distribute the

shear deformation demand more evenly among rows of links, such that the intended

flexural mechanism can be developed throughout of the panel. This leads to fatter and more

stable hystereses and further enhances energy dissipation.

In this study, the feasibility of using SSSW panels was investigated numerically. SSSW

panels were arranged around the rocker following the same layout used in the SPSW option.

These panels are 3.25 m high and 5 m wide, both of which are centreline distances between

the HBEs and VBEs respectively. Given the depth of the HBEs being 700 mm, the net

height of the slit plate is H = 2550 mm. A plate thickness, tw, equal to 30 mm, was chosen

to provide the lateral strength that is required for an optimal control of higher-mode effects.

The web plate was cut to form two rows (m = 2) of 14 (n = 14) flexural links. Each of the

links (n = 14) is 765 mm long (l = 765 mm) and 255 mm wide (b = 255 mm). This leads to

a length-to-width ratio, l/b, equal to 3, that is in the range recommended by He et al. [2016].


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The width-to-thickness ratio, b/tw, is 8.5 greater than a threshold suggested by He et al.

[2016]. This can be optimized in detailed design by increasing the thickness of the plate.

Given the layout of the links, a slit fraction, β = ml/H, equal to 0.6, is obtained. This is close

to the lower bound of 0.65 as suggested in [He et al. 2016]. The panel height-to-thickness

ratio, H/tw, is equal to 85, which is adequately low to prevent global out-of-plane buckling

of the slit plate according to the limit of 100 as recommended by Cortés and Liu [2011].

This design option was numerically modelled as illustrated in Figure 4.29. In this model,

links were represented using nonlinear fibre-based beam-column elements allowing for

distributed plasticity. Band zones and boundary elements were modelled using elastic

beam-column elements. Rigid elements were used to simulate the depth of the HBEs. This

modelling technique was validated against Specimen 1 that was tested by He et al. [2016]

and a close match was obtained. SSSW panels modelled in this way were incorporated into

an overall model where the superstructure of the reference building was represented using

the three-dimensional nonlinear model of the RC core that will be discussed in Chapter 5.

Figure 4.29 Schematic model of SSSW panels

Using these models, NLRHAs were conducted at the MCE level. Compared to the case of

using SPSW panels, the lateral displacements at the base of the structure was reduced by

45% (peak) and 35% (mean), which benefits from the increased energy dissipating capacity


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of the SSSW panels. However, these SSSW panels still have to undergo a mean drift ratio

of 4.9%, which is still higher than their deformation capacity. Further parametric analyses

were then conducted to seek a better design of SSSW panels. These analyses indicated that

a panel height of 7.2 m is required for a realistic design. Associated with this height, 76

mm thick plates would be required to prevent out-of-plane buckling and maintain the in-

plane flexural mechanism. However, LYP steel plates in this thickness are not readily

available and may require special procurements and extra cost. Given this limitation, using

LYP SSSW was not viewed as the best choice for the MechRV3D system.

4.8.4 Unbonded buckling-restrained brace frames

Given all the previous design considerations, buckling-restrained brace frames (BRBFs)

are adopted to serve as the lateral bracing components in the MechRV3D system, as shown

in Figure 4.30. This is firstly because that BRBs provide high ductility and stable hystereses

in both tension and compression. And these highly desired properties are achievable even

if the brace length is increased, given that the yielding segment is restrained from buckling.

This allows for the flexibility of using BRBFs with greater height to accommodate high

drift demands without excessive strains in the steel.

Figure 4.30 Shear mechanism consisting of BRBFs

As shown in Figure 4.30, each BRBF has a diagonally arranged brace that minimizes its

impact on the basement space. All the BRBs are inclined towards the rocker such that the

numbers of braces in tension and compression are always identical when the frames sway


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back and forth in each of the principal directions. These BRBs, as indicated in Chapter 5,

are expected to be around 10 m long and carry about 8 MN of force. This size and capacity

have previously been used in practice, for example the 10 m-long and 9.8 MN BRBs used

in the L.A. Live Hotel [CVSIC 2018] as shown in Figure 4.31 (a) and the 20 m-long and

8.3 MN BRBs used in Osaka International Conference Centre [Hikone et al. 2001] as

shown in Figure 4.31 (b) .

(a) L.A. Live Hotel (from [CVSIC 2018]) (b) Osaka International Conference Centre (Photo courtesy of Fei Tong)

Figure 4.31 Practical applications of mega-BRBs

BRBs are prone to sustain permanent plastic deformations after earthquakes at the MCE

level. However, MCE earthquakes are rare events. During these events, the number of

extreme deformation cycles is also limited and unlikely exceeds the low-cycle fatigue limit

of typical BRBs. Hence, despite the irreversible plasticity, BRBs would still be capable of

resisting seismic loads during aftershocks that follow shortly.

As for the residual deformation, they can be repaired by replacing the deformed BRBs with

new elements and pushing the whole system back to its original position. Alternatively,

self-centering braces or self-centering BRBs can be used instead as the lateral bracing

components in the MechRV3D system to achieve a recentering lateral response.


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4.9 Physical Embodiment of the MechRV3D System: Rocking Mechanism

4.9.1 Docker trusses

Given the intended rocking action, RC walls of the superstructure can be capacity designed

to remain elastic under the base overturning moment that is governed by the designated

rocking moment. To deal with this moment demand, along with gravity loads and lateral

forces, cast-in-situ construction is adopted at the conjunction between the RC core and the

rocker. Across this joint, longitudinal reinforcing bars in the RC walls are extended and

anchored in the rocker.

In addition to this conventional anchorage, steel trusses are introduced across the core-to-

rocker joint, as shown in Figure 4.32. These trusses are encased in RC walls on each

elevation of the core, and are connected to each other at corners, forming a spatial tube.

This truss tube extends over the bottom two storeys of the building and is anchored in the

rocker, as will be further discussed in the following section. As such, the truss tube serves

as a docker that enhances the structural integrity at the core-to-rocker joint. Additionally,

this docker efficiently collects superincumbent loads and carry them to the rocker which

then directs these loads to the rocking and shear mechanisms respectively as intended.

Figure 4.32 Docker trusses across the core-to-rocker joint

The docker is built using Vierendeel trusses such that no obstruction will be induced to

block door openings underneath coupling beams of the RC core. Shear studs are distributed

along truss elements to tightly bond these steel members to the surrounding concrete. Being

fully laterally restrained, all the truss elements will not be prone to buckling problems.


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4.9.2 Rocker

High rigidity is crucial for the rocker to effectively decouple and transfer vertical and lateral

forces while this block undergoes three-dimensional rotations and translational movements.

For this purpose, the rocker is made of solid concrete whose deformation can be neglected.

Hence, massive concrete pouring will be needed when the rocker is built. This is however

not a challenging construction procedure since sequential casting and thermal crack control

are well developed techniques that have been commonly used in practice. One example of

this was seen in the building for the China Central Television New Headquarters (CCTV)

where the foundation raft is 7.5 m deep [Carroll et al. 2005].

Figure 4.33 Skeleton truss in the rocker


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Inside the rocker, a three-dimensional steel truss is embedded, acting as a skeleton as shown

in Figure 4.33. Top chords of this skeleton truss are at a level that is close to the top face

of the rocker, providing anchorage to the docker trusses extended from the RC core walls.

Diagonal braces are arranged within the area enclosed by the top chords, forming a rigid

top-chord plane. From the perimeter of this top-chord plane, a series of short arms are

extended horizontally reaching side faces of the rocker where these arms provide

embedment to the gear teeth. In this way, lateral forces from the superstructure are carried

to the shear mechanism through the gear connections.

The bottom face of the skeleton truss has a similar interconnected topology to ensure a high

in-plane rigidity. This bottom-chord plane is near the soffit of the rocker where four rocking

toes, made of cast steel, are attached to the skeleton truss, each at a corner, as shown in

Figure 4.33. The top- and bottom-chord planes are connected using interlaced diagonal

members, forming a three-dimensional enclosure with a high spatial rigidity. By these

means, the skeleton truss provides a definite loading path through which vertical forces

that are carried in by the docker truss can be efficiently directed to the rocking toes.

Given the docker and the skeleton, bending moments developed at the base of individual

wall piers plus couple moments resulting from the axial forces of the walls are transferred

from the core to the rocker in a well distributed manner, leading to these moments being

exerted onto the rocker as a lumped global overturning moment rather than local loading

effects. A similar load dispersion also happens with the transfer of gravity loads and lateral

forces. These being said, longitudinal reinforcing bars that are extended from the wall piers

into the rocker may still causes local stresses near the top face of the rocker where they are

anchored. This can be addressed by providing adequate anchorage and proper confinement.

In fact, this local effect is limited since the steel trusses are expected to carry the major

portion of these actions across the core-to-rocker joint.

At the same time, the skeleton truss also serves as reinforcement that prevents the rocker

from working in a cracked condition. For this purpose, shear studs are distributed along

members of the truss, increasing the bond with the surrounding concrete. In addition to this

internal reinforcement, control of cracking may be further enhanced by using post-


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tensioned tendons (not shown in Figure 4.33). Since the rocker is primarily loaded in the

vertical direction transferring gravity loads, the post-tensioned tendons are arranged

horizontally along both principal directions and distributed over the height of the rocker.

These tendons are threaded through the rocker and anchored to side faces. Clamping forces

provided by these tendons are self-equilibrated and hence do not affect the intended

mechanics of the MechRV3D system.

4.9.3 Potential problems with ball rollers

In the idealized configuration, ball rollers are assumed not to carry any lateral forces when

they are rolling on top of the foundation. However, this assumption may be of doubtful

validity when real contact conditions are taken into account. First of all, frictional forces

will develop during the rolling motion. It may be arguable that this rolling friction is usually

small in magnitude and therefore can be ignored. Nevertheless, even if this argument is

deemed reasonable and accepted, cumulative deposit of debris can still significantly

increase the resistance to the rolling motion and even block the rotation of the rollers. While

this problem can hardly be avoided using debris shields because there is little chance to

attach any protective boards around the rollers that spin in full cycles, it is also impractical

to clean up the cumulated debris given the limited space between the rocker and the

foundation.

When ball rollers are restrained from a rolling motion, horizontal sliding may be engaged

with a friction coefficient greater than that of rolling friction. Although this sliding friction

coefficient decreases with increasing normal force, under the enormous gravity load from

the superstructure, the sliding friction force can still provide a lateral resistance that is

considerable or even comparable with what the shear mechanism is expected to provide.

This would obviously negate the intended uncoupled mechanics. Moreover, the sliding

friction coefficient can be affected by many factors, including the magnitude of normal

forces, deformability of bodies in contact, sliding velocity, changing roughness of the

surface, and, if some lubricant applied, aging problems and maintenance. All these factors

induce uncertainty to the response of the proposed system. Hence, the intended shear-free

condition at the base of the rocking mechanism must be achieved in a more practical and

reliable way.


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4.9.4 Moment-free mega-columns

The idealized ball rollers could also be realized by using a group of columns that stand

between the rocker and the foundation, as schematically shown in Figure 4.34. These

columns are large in size and designed for high load-bearing capacity to support gravity

loads from the superstructure. Being allowed to uplift at the bottom, these mega-columns

are also specially detailed such that a pinned condition can be achieved at both ends,

leading to minimal moment resistance when rotations occur. Having both ends moment-

free, these mega-columns carry no lateral forces when they sway horizontally in response

to the lateral movement of the rocker. To achieve the pinned condition, different possible

detailing options were explored ranging from structural joints to mechanical connections,

as will be discussed subsequently.

Figure 4.34 Moment-free mega-columns

4.9.5 Hinged buckling-restrained Braces

BRBs were first considered as a possible way to build the mega-columns. With a steel core

plate that is laterally restrained from buckling, BRB mega-columns can be designed to

carry high compression loads. Where necessary, the load-bearing capacity can be increased

using multiple plates in parallel as the steel core. In addition, an external steel case can be

provided as an additional confinement to the concrete that confines the steel core plates.

Unlike in their conventional use, these BRBs are designed not to yield at the encased

segment of the steel core. Instead, steel plates are expected to yield flexurally and form a


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plastic hinge near both ends of the BRB. This can be achieved by having the steel core

plates exposed in a small notch where the concrete confinement is removed, as shown in

Figure 4.35. These notches are of a small gap distance such that bare steel plates within

this unrestrained length will not buckle prior to the formation of flexural hinges. The

intended flexural yielding occurs about the weak axis of the steel plates resulting in a plastic

moment that is greater than zero. Whereas this non-zero flexural resistance does not

rigorously render a pin condition, it may be reasonably deemed so since the out-of-plane

moment capacity of steel plates is typically small.

Figure 4.35 Hinged BRBs used as mega-columns

Nevertheless, a similar near-pin condition cannot be achieved in the orthogonal direction

where the in-plane plastic moment of steel plates is significant. This made the viability of

using hinged BRBs questionable since the mega-columns may sway in any horizontal

directions, requiring a universal pin connection at both ends. To resolve this problem, it

was attempted to form a two-way hinge by orientating the steel plates orthogonally in two

separate notches near each end of the BRBs, as shown in Figure 4.36. However, this

arrangement requires twisting the steel plates by 90 degrees within a short length between

two adjacent notches, which, if not impractical technically, will cause considerable

fabrication cost. Hence, this option of using hinged BRB mega-columns was not adopted.


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Figure 4.36 BRB mega-columns with two-way hinges

4.9.6 Concrete hinges

To address the demand of multi-directional pin joints, the feasibility of using concrete

hinges was examined. Concrete hinges have been successfully used in bridge engineering

for over 120 years. This type of joints is known for the high load-bearing capacity and

moderate rotational flexibility [Schacht and Marx 2015]. Since, in 1880, Claus Köpcke

developed a saddle bearing, varied types of concrete hinges were proposed. Noteworthy

examples of these include Mesnager hinges, as shown in Figure 4.37 (a), that transfer loads

through reinforcing bars that intersect at a narrow throat between the connected elements.

However, this type of hinge is prone to buckling of the reinforcing bars if these bars are

left unrestrained at the throat, or, if the bars are restrained, tension cracks in connected

members due to lack of transverse confinement. Armand Considère proposed a different

hinge, as shown Figure 4.37 (b), where spiral reinforcement is introduced to provide

confinement to concrete at the throat, leading to a tri-axial stress state and therefore

significantly increased load-bearing capacity. However, Considère hinges may not render

a joint with negligible moment resistance as required for the proposed system in this study,

because the reinforcement passing through the throat and those longitudinal bars linking

connected members at the perimeter may largely increase the moment capacity of the hinge.


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(a) Mesnager hinge (b) Considère hinge (c) Freyssinet hinge

Figure 4.37 Concrete hinges (from [Schacht and Marx 2015])

Under such circumstances, a plain concrete hinge that was developed by Eugène Freyssinet

was considered. In a Freyssinet hinge, as shown in Figure 4.37 (c), the favourable tri-axial

stress state of concrete is established at the throat, which does not rely on the presence of

any reinforcement, but requires of a limited length of the throat. Preliminary calculations

were done to examine the viability of using Freyssinet hinges at both ends of the mega-

columns. Results of these calculations indicated that a large cross-sectional area is needed

at the throat to achieve adequate load-bearing capacity. This unfavourably resulted in a

considerable moment capacity which, if applied at both ends of the mega-columns, would

lead to a large lateral force resistance that can reach nearly 40% of what is expected to be

provided by the shear mechanism. This means that the rocking mechanism will act as part

of the lateral-force-resisting system while controlling the flexural response. This is not

acceptable given the intended uncoupled mechanics. Hence, Freyssinet hinges were not

adopted in the proposed system.

4.9.7 Ball pin joints

Given the limitation of structural joints, possibility of using mechanical connections was

considered to achieve a pin-pin boundary condition for the mega-columns. One option of

this kind is shown in Figure 4.38 where mega-columns are connected to the rocker and the

foundation through ball pin joints which have been widely used in mechanical engineering.

At the top of the mega-columns, a ball pin retainer is fixed to the bottom face of the rocker,

accommodating a ball pin that is connected to the mega-column. The ball pin joint allows

for rotations about any horizontal axes developing insignificant rotational restraint at the

joint. And this limited restraint can be further minimized by applying some lubricant


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materials between the pin and the retainer. A multi-directional pin joint is configured in a

similar way at the bottom end of the mega-columns. The only difference is that the mega-

columns are allowed to uplift at the bottom using the same stepping mechanism as used for

the hinged BRBs. By these means, the expected non-lateral-force-resisting mega-columns

are achieved.

Figure 4.38 Mega-columns with ball pin joints

In fact, ball-pinned mega-columns can be viewed as a transformation of the ball rollers that

were idealized in Section 4.3. Whereas these two configurations are equivalent in terms of

a near-zero shear resistance, ball-pinned mega-columns follow the lateral movement of the

rocker through chord rotations over their height rather than relying on full-cycle spinning

as in the case of the idealized ball rollers. This makes it feasible to apply some dust seal to

prevent the ball pin joints from being stuck by debris, and also, if debris still deposits,

allows for space where cleaning work can be done. This facilitates to maintain a condition

of trivial rotational restraint at both ends of the mega-columns.

Nevertheless, ball-pinned mega-columns were not adopted due to concerns of availability

and buildability. Used as typical products in mechanical engineering, ball pin joints may

not be readily available for sizes and load-bearing capacities that are required for building

structures, especially for the proposed MechRV3D system where the rocking mechanism

has to carry large gravity loads from the superstructure. As a result, special fabrication may

be requested to meet different industrial standards. This may cause an increased building

cost, slow down construction speed, and require substantial investments in qualifying new

products. These impacts compromise the practicality of using ball pin joints. This being


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said, the idea of achieving moment-free connections through mechanical articulations is on

the right track, as long as the chosen mechanical joint is realistic to be built.

4.9.8 Telescopic pipe-pin hinges, rocking columns and kinematic isolation

Following this direction, consideration was given to a telescopic pipe-pin hinge that was

developed by bridge engineers at the California Department of Transportation (Caltrans),

and later validated by Zaghi and Saiidi [2010] numerically and experimentally. This type

of hinge is intended to block moments from being transferred between RC bridge bent caps

and piers. For this purpose, the hinge was designed to have an articulated configuration,

consisting of a steel pipe that is centrally embedded in the pier and an inverted can that is

embedded in the cap, as shown in Figure 4.39. The pipe and the can form a shear key that

transfers lateral forces but carries no vertical loads. Gravity loads from the superstructure

are transferred through a hinge throat that is a circular bearing area around the shear key.

In this area, a thin layer of concrete extrudes from the cap and comes into contact with the

flat surface of the pier. Whereas friction developed at this interface somewhat contributes

to the shear transfer, the major portion of the lateral force is carried through the shear key.

Figure 4.39 Telescopic pipe-pin hinge (from [Zaghi and Saiidi 2010])

For bridges, this pipe-pin hinge is installed at pier-to-cap joints only. Piers are still fully

fixed at the bottom, serving as the lateral-force-resisting system of the structure. However,

columns with a single pin at the top do not fit the intended role of the mega-columns in the


112

proposed MechRV3D system, because the latter ones are expected to be moment-free at

both ends and therefore provide no lateral resistance at all.

This being said, it seems possible to remove this limitation by having the top-pinned mega-

columns rotationally released at the bottom as well as using the telescopic pipe-pin joint,

as shown in Figure 4.40 (a). In this configuration, mega-columns are allowed to rock

against the soffit of the rocker and the top face of the foundation, for which the flat bearing

area of the hinge throat serves as the rocking section, as shown in Figure 4.40 (c). In other

words, the mega-columns act equivalently as flat-ended rocking columns, as shown in

Figure 4.40 (b). When the rocker moves horizontally, these rocking mega-columns tend to

tilt back and forth between the rocker and the foundation, being recentered by

superimposed gravity loads as illustrated in Figure 4.40 (d). Whereas these flat-ended

rocking columns display a decreasing load-bearing capacity once the rocking action is

activated, lateral resistance can still be developed prior to the activation with a magnitude

that is linearly proportional to the weight imposed atop these columns. As for the

MechRV3D system, this weight would be the total gravity load from the RC core, which

is typically large for high-rise structures. As a result, the pre-rocking lateral resistance can

be significant such that the mega-columns, as part of the rocking mechanism, can play a

role of lateral-force-resisting members along with the shear mechanism. This is

incompatible with the intended uncoupled mechanics.


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Figure 4.40 Kinematic isolation by rocking columns

In fact, rocking columns as shown in Figure 4.40 form a kinematic isolation system which

was already used in ancient temples and provided excellent seismic resistance as reviewed

in Chapter 2. Modern applications of this technology have been seen in Sochi, a highly

seismic area in Russia, where low-rise buildings are base isolated using flat-ended rocking

columns that freestand in the first storey. However, to the author’s knowledge, these

applications were not well documented in written publications, except some videos that are

narrated in Russian.

Given the properties of kinematic isolation, rocking columns can be used to base isolate

structures. This can be feasible for low-rise buildings whose dynamic response is primarily

governed by the first mode. However, for high-rise structures, significantly increased

gravity loads require a larger section size for the rocking columns. When the gravity load

is imposed on these columns, the rocking moment is increased accordingly. As a result, the


114

lateral resistance provided by these rocking columns could be largely increased in such

cases, and consequently the shear demand that can be developed at the base of the structure

would also increase, leading to a shear resistance and potentially greater higher-mode

effects. In the proposed MechRV3D system, the separate and dedicated shear mechanism

allows for a flexibility in defining a base shear that is optimal for higher-mode control.

Hence, the scheme of using rocking columns alone at the base of high-rise structures was

not adopted in this study. The research effort is still focused on the proposed dual-

mechanism system with the uncoupled mechanics.

4.9.9 Composite mega-columns with pipe-pin rolling joints

Inspired by the telescopic pipe-pin hinge shown in Figure 4.39, this study proposed a

special detailing, referred to as a pipe-pin rolling joint, which is used at both ends of the

mega-columns. To be compatible with this joint, a tube-in-tube built-up and a composite

section are proposed for the mega-columns, as shown in Figure 4.41. The inner steel tube

confines a concrete section that is reinforced using multiple steel circular hollow sections

(CHSs) in addition to mild reinforcing bars. As will be verified in Chapter 5, this CHS-

reinforced section alone can provide adequate load-bearing capacity such that a single

mega-column can support the entire gravity load of the superstructure in the elastic range.

Surrounding the inner section is a ring-shaped concrete section that is confined by the outer

steel tube. A series of steel pipes are encased in this section following a polar array. These

pipes, filled with concrete and referred to as the CFT (concrete-filled tube) pipes, are

extended from the mega-column and become part of the proposed rolling joint as will be

discussed subsequently.


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Figure 4.41 Composite mega-columns with a tube-in-tube built-up

Rather than being cast-in-situ, these mega-columns are prefabricated and then dry plugged

into a few indentations, referred to as sockets, that are preformed in the soffit of the rocker,

as shown in Figure 4.42. At the perimeter of each socket, a series of inverted steel cans are

embedded at positions projected from the CFT pipes that extend from the corresponding

mega-column. When the mega-column is inserted, these cans accommodate the CFT pipes,

forming pipe-and-can pairs. Vertically, the pipes and the cans are free to move relative to

each other, which allows the rocker to uplift atop each mega-column. As such, in the

proposed physical embodiment, the intended rocking action occurs at the top level of the

mega-columns instead of the foundation level. The length of the CFT pipes is carefully

determined such that the portion within the cans is sufficiently longer than the expected

uplifting distance and will never be pulled out. This allows the pipe-and-can pairs to act as

dowels laterally, preventing the mega-columns from sliding off even if the rocker steps up.

While pull-out is precluded, the CFT pipes are neither expected to touch the base of the

cans, nor carry any gravity loads. This detachment can be ensured by limiting the length of

the pipes and increasing the depth of the cans.


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Figure 4.42 Proposed pipe-pin rolling joint

Provided no vertical contact in the pipe-and-can pairs, gravity loads from the superstructure

are transferred to the mega-columns through a spherical cap that is concentrically anchored

to the inner section of the mega-columns, as shown in Figure 4.42. These spherical caps

are made of high-strength machined or cast steel for high load-bearing capacity. They are

in direct contact with the central area of the sockets where a thick high-strength steel plate,

referred to as a load-dispersing pad, is attached as shown in Figure 4.42. When the rocker

moves horizontally, the spherical cap rolls against the bottom face of the rocker within the

socket unless they are separated as a result of the uplifting of the rocker. This rolling motion

of a spherical surface against a plane causes little rotational restraint at the proposed joint.

In addition, when the joint undergoes rotations, no moment resistance is developed either

by the pipe-and-can pairs given the unrestrained vertical mobility. This moment-free design

is further enhanced by enlarging the cans near the free end of the pipes in order to allow

the pipes to rotate without bending. By all these means, the intended pin joints are achieved

at column-to-rocker connections, allowing for free uplifting of the rocker at the same time.

More importantly, this pinned condition remains valid when the proposed joint rotates

about any horizontal axes. This benefits from the polar symmetry of the joint in terms of

both geometry and rolling mobility.


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The proposed all-direction rolling joint, with its configuration flipped upside down, is also

used at the bottom of the mega-columns, allowing these columns to roll free of moments

in the sockets at the foundation level. With moments released at both ends, these spherically

ended mega-columns carry no lateral forces and referred to as the rolling mega-columns,

differentiating from flat-ended rocking columns that are lateral-force-resisting as pointed

out in Section 4.9.8. These rolling mega-columns are loaded in compression at all times

and therefore never lift up off the foundation. At both column-to-rocker and column-to-

foundation joints, the rolling motion is restricted within the central area of the sockets that

is enclosed by the pipe-and-can dowels. In this area, while the aforementioned load-

dispersing pads help reduce stress concentration, the rocker and the foundation are also

locally reinforced using layers of orthogonal rebars for an increased bearing capacity.

Figure 4.43 Pipe-pin rolling joints at the foundation level


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4.10 Physical Embodiment of the MechRV3D system: Overall System

4.10.1 Integrated dual-mechanism system

Given all these proposed configurations for the shear transmitters, and the rocking and

shear mechanisms, the MechRV3D system can be integrated as shown in Figure 4.44. As

discussed in Section 4.7.3, gear connections are used in this embodiment as one of the

feasible options developed for the shear transmitters. These articulated connections can be

replaced using hinged plates without compromising the intended functionality of the shear

transmitters.

Figure 4.44 Physical embodiment of the MechRV3D system

4.10.2 MechRV3D-incorporated benchmark building

Construction sequence

The integrated MechRV3D system is then incorporated into the basement of the benchmark

building. Figure 4.45 indicatively shows a possible construction sequence following which

the proposed system can be built in coordination with the construction of the basement.

The first two steps are typical construction procedures for conventional RC raft and pile (if

necessary) foundations. At the locations where the BRBFs will be installed, the raft is

thickened to provide a foundation to these frames. At the locations where the rolling mega-

columns will be erected, recesses are preset to hold the space for sockets that accommodate

these columns.


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Figure 4.45 Indicative construction sequence for the MechRV3D system


120

The erection of rolling mega-columns can be conducted in Step 3. These elements are

prefabricated in fabrication shop and transported on site. Given the dry plug-in design of

the pipe-pin joints, these columns can be inserted into the sockets that were constructed in

the previous step. Temporary propping is needed to ensure these rolling columns stand

upright before they are laterally braced by the BRBFs via the rocker.

At the same time, local frame structures can be built for the basement storeys independently,

following conventional concrete construction procedures. Gravity columns that will be

extended into the superstructure are detached from underground floor slabs, which ensures

these columns sway between the ground and foundation levels following the lateral

movement of the skirt. Before the ground slab is cast, temporary supports are needed to

brace these columns.

At levels where the underground slabs meet the BRBFs, recesses will be preset in the the

slabs to accommodate these frames and their expected lateral deformation. At levels where

the underground slabs meet the foundation to the BRBFs, the slabs can be cast with the

foundation to obtain vertical supports. These arrangements are demonstrated in the sketch

beside Step 3 in Figure 4.45.

After these gravity-load-bearing elements are built, the BRBFs can be erected on top of the

raised foundation. Once installed, these frames are ready to provide lateral bracing that will

be needed subsequently. Firstly, a box-shaped formwork, which is used for casting the

concrete rocker, is erected atop the mega-columns that are being held in position. Laterally,

this formwork is braced by the BRBFs through temporary propping trusses. The skeleton

truss of the rocker is then placed into the formwork with the rocking toes in contact with

the mega-columns. On top of the skeleton truss, the docker truss, which will be cast into

the RC core walls, is welded as well. The assemblage of these two trusses are held upright

using temporary supports. At the ground level, formwork is also installed for cast the

concrete slab (the skirt). This formwork can be supported by temporary propping structures

and the BRBFs. Parts of the shear transmitters should be embedded at the inner edge of the

skirt formwork, meeting their corresponding parts that extend out from the top chord of the

skeleton truss.


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In Step 5, concrete is cast into the formwork, leading to the rocker and the skirt taking their

shapes. As will be described in a later section, the outer edge of the skirt needs to be

extended beyond the retaining wall and allowed to move horizontally with minimal

impedance. At the skirt-to-retaining wall joint, waterproof seals need to be applied. After

concrete elements obtain adequate strength, the temporary props can be dismantled. Then

the construction can proceed with the superstructure. This follows typical construction

techniques used for conventional RC core-wall buildings, and therefore is not discussed in

detail here.

Figure 4.46 shows the MechRV3D system that has been incorporated into the basement of

the benchmark building. Relations between this system and conventional elements in the

benchmark building are discussed subsequently.

Figure 4.46 MechRV3D system incorporated in the benchmark building

Slab-to-core wall joints

In response to the rocking action at the base, the RC core walls may undergo vertical

displacements over the height of the structure, leading to concrete floor slabs that are cast

monolithically with the walls being subjected to out-of-plane bending moments. The

reactive moments that the slabs impose back on the walls provide unintended hindrance to

the gravity-recentered rocking action of the core and thereby of the rocker at the base. To


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minimize this effect, slab-to-core wall joints can be carefully detailed to form a near-pinned

connection without compromising the capacity of transferring gravity loads and horizontal

forces that are collected by the slab diaphragms. For example, Freyssinet hinges can be

applied at these junctions. Alternatively, similar hinged steel plates as used for the shear

transmitters can also be used at the slab-to-core wall joints. These plates can be easily

replaced after a major seismic event if they are designed with bolted connections.

In fact, slab-and-wall interactions of this kind are not only induced due to the base rocking

action, but commonly exist in conventional RC wall structures. When flexural hinges are

formed at the base of conventional walls, inelastic rotations cumulated within the base

hinging zone will also cause elongation of the walls and vertical uplift, which can result in

a slab-to-wall interaction that is similar to what is anticipated for the MechRV3D system.

And this interaction has been well addressed in conventional constructions.

Skirt-to-retaining wall joints

At the ground level, the skirt diaphragm (ground floor slab) is expected to undergo lateral

displacements due to the inelastic shear deformation of the BRBFs. As a result, the ground

floor slab may slide horizontally atop retaining walls surrounding the basement. This

relative movement is not present in fixed-based buildings, but commonly seen in

conventional base-isolated structures.

Figure 4.47 Details allowing for the slide at skirt-to-retaining wall joints


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To accommodate this movement, the concept of a seismic moat cover that is widely used

for base-isolated buildings can be used. Figure 4.47 shows two possible details that allow

for sliding movements at skirt-to-retaining wall junctions. In both detailed designs, the

ground floor slab is extended beyond the retaining wall and allowed to move in any

horizontal directions. Deformable caulks can be filled between the extended slab and the

retaining wall for waterproof. The caulks may be damaged due to seismic movements, but

can be readily replaced after the event. Compared to conventional base isolation, the base

displacement demand required by the MechRV3D system is largely reduced to be around

250 mm (as will be revealed in Chapter 5). This makes it practical to use commercial

products of seismic moat covers that can typically accommodate horizontal displacements

up to 1 m [FEMA 2012].

P-𝛥 effects induced by gravity columns in basement

As can be interpreted from Figure 4.46, gravity columns in the basement are expected to

sway in response to the lateral movement of the skirt diaphragm. This leads to local P-𝛥

effects at the ground level, which will be resisted by the BRBFs. To minimize this effect,

it is recommended to allow the gravity columns to sway between the ground level and the

foundation without being restrained by the intermediate floor slabs. Mechanics that support

this arrangement are illustrated in Figure 4.48. Given the gravity load Wgravcol, that is carried

by the gravity column at the ground floor, and the lateral displacement of the skirt, 𝛥skirt,

the buttressing force Vgravcol, that is required to stabilize the column, is inversely

proportional the column height hgravcol. This explains why using a column height equal to

the full depth of the basement is preferred.

Figure 4.48 P-𝛥 effects of gravity columns in the basement


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However, this arrangement leads to intermediate basement floor slabs being detached from

and not being supported by the basement-high gravity columns. To address this problem,

short columns can be locally added where the detachment occurs. And these local gravity

columns are only needed between the second basement storey and the foundation. This

zone is not affected by the lateral movement of the skirt diaphragm.

4.10.3 Lateral equilibrium of the MechRV3D system

In the physical embodiment design (shown in Figure 4.44), the rolling mega-columns and

the rocker form a three-dimensional rolling frame that can sway in any horizontal

directions, providing minimal lateral resistance at the base of the superstructure. In this

rolling frame, the rolling mega-columns act as leaning columns whose stability is ensured

by buttressing forces that are provided by the BRBFs via the rocker. As such, P-𝛥 effects

are induced, generating a negative lateral stiffness. This P-𝛥 effect is intended in the design

since the induced negative stiffness, as the rolling frame sways, leads to an offset to the

post-yielding overstrength of the BRBFs, which allows for a better limit on the base shear

and consequently of the higher-mode effects.

Figure 4.49 Lateral equilibrium of the MechRV3D system


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The BRBFs provide buttressing forces to the rolling mega-columns via the rocker. When

calculating the total buttressing force, Vc, to these columns, the recentering gravity load,

Wsc, is accounted for, regardless of the number of mega-columns that are engaged in

carrying this weight. The number of vertically engaged mega-columns can be four if no

rocking action is activated, two if the rocking occurs in the line-pivoting state, or one in

the point-pivoting state. For a mega-column that is vertically disengaged at an instant, a

small buttressing force, vc, is still required to resist the overturning moment that is caused

by the self-weight of the column, w, and its reaction at the foundation, as illustrated in

Figure 4.49. Since w is negligibly small compared to Wsc, vc is negligible compared to Vc.

The BRBFs also stabilize gravity columns in the basement by providing a buttressing force

Vgravcol, as discussed in Section 4.10.2 and illustrated in Figure 4.49. This buttressing force,

as evaluated in Figure 4.48, is transferred through the skirt diaphragm, as shown in Figure

4.46. As a result, the shear demand, V, that can be developed at the base of the RC core, is

limited by the shear strength of the BRBFs, Vf, minus the buttressing forces, Vc and Vgravcol ,

that are respectively required to ensure the stability of the rolling mega-columns and the

gravity columns in the basement.


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4.11 Summary

This chapter proposed the MechRV3D system, that consists of independently acting

rocking and shear mechanisms at the base of high-rise buildings. Relying on the distinct

and dedicated nonlinear mechanisms, this proposed system limits flexural and shear

demands that are induced by earthquakes. These characteristic uncoupled mechanics were

verified through numerical analyses that were conducted on an idealized configuration of

the MechRV3D system.

A physical embodiment was also developed to implement the proposed MechRV3D system

in practice. In this embodiment design, BRBFs are used as the lateral bracing components

of the shear mechanism. Innovative composite mega-columns and pipe-pin rolling joints

were proposed to ensure that the rocking mechanism provides minimal lateral resistance.

In addition, hinged plates and gear connections were proposed as the shear transmitters to

essentially eliminate the interaction between the rocking and shear mechanisms. All these

design schemes evolved from a series of design options that were discussed in detail in this

chapter. These design efforts were made in order to enhance the practicality of the proposed

system. As a result, most of the components in the MechRV3D system are elements that

have been widely used in practice. However, these components are integrated and engaged

in an unconventional way, which marked the originality of this research.

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Chapter 5 Numerical Validation of the Proposed MechRV3D System

5.1 Introduction

The numerical analyses that were conducted in Chapter 4 demonstrated the fundamental

kinematics of the proposed system. It was also confirmed that the uncoupled flexural and

shear response could be achieved as intended. However, these findings were limited to the

conceptual level, given the idealized configuration and properties that were assumed in the

preliminary studies.

In this chapter, the MechRV3D system is studied more in depth through the physical

embodiment proposed in Chapter 4. For this purpose, the PEER benchmark building is

used as the reference structure for which more details are introduced in Section 5.2

regarding the conventional design. With respect to this building, the MechRV3D system is

designed in Section 5.3 with varied properties considered for both mechanisms.

Based on the parametric design of the base mechanisms, nonlinear numerical studies are

conducted to validate the feasibility of the MechRV3D system in a real application. This

begins with the introduction of an advanced nonlinear model that was built for the original

benchmark building as described in Section 5.2. After being validated, this superstructure

model is used in Section 5.5 for investigating a special case where only the rocking

component of the proposed system is activated. This rocking-only scenario provides an

additional reference point upon which the efficiency of the dual-mechanism system can be

fully assessed against.

Prior to the full assessment, advanced nonlinear models were built, as described in Section

5.6, for both the rocking and shear mechanisms as they were physically implemented in

Chapter 4. These mechanism models are incorporated at the base of the benchmark building

in Section 5.7 where extensive nonlinear dynamic analyses are carried out on the integrated

system. Results of these analyses are compared to the conventional design and the rocking-

only scenario. Major conclusions are then summarized in Section 5.8.


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5.2 PEER Benchmark Building

5.2.1 PEER Tall Buildings Initiative

In 2006, the Pacific Earthquake Engineering Research Centre (PEER) launched a Tall

Buildings Initiative (TBI) program [Moehle et al. 2011] aiming at the advancement of

performance-based seismic design (PBSD) procedures and the development of design

guidelines for high-rise buildings. This research project looked into high-rise buildings that

have fundamental periods longer than 1.0 sec where seismic responses are often highly

influenced by higher modes of vibration. Three different lateral-force-resisting systems

were studied, including a concrete core-wall system, a dual system consisting of concrete

core and special moment frame, and a steel buckling-restrained braced frame system. These

systems were designed for a hypothetical yet realistic building following the PBSD

procedures recommended by the Los Angeles Tall Buildings Structural Design Council

(LATBSDC) [LATBSDC 2008] and the guidelines proposed by the PEER. The two PBSDs

were compared to a prescriptive design for which the International Building Code (IBC)

[ICC 2006] was followed.

In this dissertation, the RC core-wall building is selected as the reference structure, largely

because this structural system is most commonly used for high-rise construction around

the world. The design is based on the LATBSDC guidelines [LATBSDC 2008] and was

conducted by Magnusson Klemencic Associates (MKA). MKA also verified the design by

carrying out nonlinear response history analyses (NLRHAs) at varied hazard levels. The

structural layout of this conventional design is shown in Figure 5.1.

Being representative structurally and architecturally, the PEER benchmark buildings have

been referenced in a number of other research investigations. As an example, MacKay-

Lyons [2013] used the RC core-wall system in a numerical study on a novel viscoelastic

coupling damper that was proposed by Christopoulos and Montgomery [2013] for

enhanced wind and seismic performance of high-rise buildings. In this study, MacKay-

Lyons [2013] carried out nonlinear dynamic analyses on the building that was designed to

the LATBSDC guidelines [LATBSDC 2008]. These analyses, along with those done by


129

MKA, are used as reference analyses for validation purposes as is presented in Section

5.4.8.

Figure 5.1 Structural layout of the benchmark building (from [Moehle et al. 2011])


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5.2.2 Building geometry

The benchmark building is a residential development consisting of 42 storeys above the

ground and 4 storeys in the basement, as shown in Figure 5.1. With a typical storey height

of 2.95 m, the building reaches a height of 125 m at the roof level. On plan, typical floors

of the tower cover a 33 m-by-33 m area, at the centre of which coupled concrete walls are

located, forming a closed core that is approximately 10.5 m-by-15.5 m. This central core

extends throughout the height of the building, being encased at the foundation level which

is 12.2 m below the ground.

5.2.3 Seismic hazard and ground motions

The PEER benchmark building is located in Los Angeles, California, where high seismicity

is expected. The site of the building is anticipated to be affected by multiple active faults

from different distances, including Puente Hills fault (1.5 km), Hollywood fault (7.3 km) ,

Raymond fault (8.8 km), Santa Monica fault (11.5 km), Elsinore fault (24.5 km), Sierra

Madre fault system (40 km), and San Andres fault (56 km), as shown in Figure 5.2.

Figure 5.2 Seismic fault map of Los Angeles (from [USGS 2019])


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In conventional seismic design, probabilistic hazard analyses were conducted to identify

dominant seismic scenarios. From these analyses, it was concluded that for the maximum

considered earthquake (MCE) with a return period of 2475 years, the seismic hazard of the

site is dominated by relatively large magnitude-small distance events or extremely large

magnitude-long distance events for long periods, and by large magnitude-small distance

events for shorter periods. These trends for the dominant magnitude-distance scenarios can

be seen from the hazard disaggregation charts shown in Figure 5.3. Details on the seismic

hazard analysis can be found in [Moehle et al. 2011].

(a) T = 5.0 sec

(b) T = 1.0 sec

Figure 5.3 Hazard disaggregation at the MCE level (from [Moehle et al. 2011])


132

Based on the seismic hazard analysis, a set of seven seed ground motions were selected for

the design, as listed in Table 5.1. These records were spectrally modified and amplitude

scaled to match the target design spectrum (43-year return period, 2.5% damped) [Moehle

et al. 2011]. Figure 5.4 shows the response spectra of the scaled ground motions and the

mean spectrum that matches the design spectrum in the medium- and long-period ranges.

Table 5.1 Ground motions selected for the seismic design (from [Moehle et al. 2011])

Earthquake Year Mw Station Rm [km]Denali, Alaska 2002 7.90 TAPS Pump Station #9 54.78 Loma Prieta, California 1989 6.93 Saratoga – Aloha Ave 8.50 Northridge, California 1994 6.69 Sylmar – Converter Station 5.35 Denali, Alaska 2002 7.90 Carlo 50.94 Chi-Chi, Taiwan 1999 7.62 CHY109 50.53 Denali, Alaska 2002 7.90 TAPS Pump Station #8 104.9 Landers, California 1992 7.28 Yermo Fire Station 23.62

Figure 5.4 Spectra of scaled ground motions at SLE43 (from [Moehle et al. 2011])


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5.2.4 Gravity loading allowance

In the conventional design, superimposed dead loads and live loads were specified as listed

in Table 5.2. These loading allowances did not include the weight of floor slabs, gravity

columns, and structural walls. These structural weights were computed from the actual

member sizes as summarized in Section 5.2.8.

Table 5.2 Superimposed dead loads and live loads (from [Moehle et al. 2011])

Usage Superimposed Dead Loads [kPa]

Live Loads [kPa]

roof 1.3 1.2 residential (elevated floors outside of core walls) 1.3 1.9 retail (ground level, under tower footprint) 5.3 4.8 exit area (inside core walls) 1.3 4.8 parking 0.1 1.9 facade (on elevation) 0.7 -

5.2.5 Load combination

In the conventional design [Moehle et al. 2011], the seismic performance was assessed at

the SLE and MCE levels respectively. For the both checks, one single load combination

case was considered as shown in Equation (5.1),

1.0 D + 0.25 L + 1.0 E (5.1)

where D is the expected dead load, L is the unreduced live load, and E is the seismic effect.

While E was computed through response spectrum analyses for the serviceability check, it

was evaluated through NLRHAs at the MCE level using the selected suite of ground motion

records. In these analyses, the two-component records were not rotated to match the

building axes.


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5.2.6 Material properties

In Table 5.3 and Table 5.4, material properties are listed for concrete and reinforcing steel

that were used in the conventional design [Moehle et al. 2011]. The specified and expected

values were used for the initial design and the MCE-level checks respectively.

Table 5.3 Concrete strength and modulus of elasticity

Structural Members Nominal fcʹ [MPa]

Expected fcʹ [MPa]

Nominal Ec [MPa]

Expected Ec [MPa]

shear walls and columns 55.2 71.7 31600 35000beams and floor slabs 37.9 49.6 27400 30300

basement walls 34.5 44.8 26400 29100foundation mats 41.4 49.6 28300 30300

Table 5.4 Reinforcement steel strength and modulus of elasticity

Structural Members Steel Grade Nominal fy [MPa]

Expected fy [MPa]

Expected fu [MPa]

shear walls ASTM A706 Grade 60 414 483 724coupling beams ASTM A615 Grade 75 517 586 896

5.2.7 Acceptance criteria

While the LATBSDC procedures [LATBSDC 2008] were followed in the conventional

design [Moehle et al. 2011], some exceptions were applied:

• The minimum base shear of 0.03W required in [LATBSDC 2008] was waived,

provided that the minimum strength was established using the serviceability

earthquake in conjunction with design for wind forces.

• The serviceability check was conducted under frequent earthquakes with a 25-year

return period instead of a 43-year return period as specified in [LATBSDC 2008].

The seismic performance was also checked at the MCE level in the conventional design.

To ensure the performance objectives being achieved at this intensity level, acceptance

criteria were adopted in the conventional design as listed in Table 5.5. In the evaluation of

seismic responses, mean values of the seven ground motions were used.


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Table 5.5 Acceptance criteria under MCE level earthquakes (from [Moehle et al. 2011])

Seismic Responses Acceptance Criteriainter-storey drift ratios 3%coupling beam rotations 6%strains of steel reinforcement in core walls 50×10-3 (tension) 20×10-3 (compression)strains of confined concrete in core walls 15×10-3 (compression) shear component of core walls elastic response

5.2.8 Structural sizes

Based on these design assumptions and criteria, sizes were determined in the conventional

design for structural members as summarized below. These structural sizes will be used in

Section 5.4 where an independent nonlinear model is built for the benchmark building for

validating the proposed system in this dissertation. For more details of the conventional

design outcome, references shall be made to [Moehle et al. 2011].

Floor slabs

In the conventional design [Moehle et al. 2011], post-tensioned concrete slabs were used

for elevated floors, while reinforced concrete slabs were used for the roof, ground, and

basement levels. Slab thicknesses and gravity loads are summarized in Table 5.6.

Table 5.6 Thickness and gravity weight of the floor slabs (from [Moehle et al. 2011])

Levels Roof Residential/Hotel Ground Floor Basement LevelsThickness [mm] 254 203 305 254

Weight [kPa] 6.1 4.9 7.3 6.1

Gravity columns

Gravity columns were assigned squared sections with dimensions varying with the floor

levels as summarized in Table 5.7. Figure 5.1 shows the specific locations of these columns

on the plan of the benchmark building. At the ground level, the total area of these column

sections is approximately 14.2 m2. This will be used in Section 5.4 where numerical

modelling of this benchmark building is discussed.


136

Table 5.7 Sizes of gravity columns in [mm] (from [Moehle et al. 2011])

Gravity Columns D3.5 D5 D6 E5 E6 F4 F5

Level 34 - Roof 457 457 457 457 457 457 457Level 22 - 34 533 483 457 559 457 533 457Level 12 - 34 635 610 457 660 483 660 533

Fndn – Level 12 762 737 610 838 660 864 711RC core walls and coupling beams

The conventional design of the core is summarized in Table 5.8. The design report [Moehle

et al. 2011] indicates ratios of the steel reinforcement in the RC walls. Coupling beams

were assigned a constant depth 762 mm (30 in) throughout the building with a width equal

to the thickness of the walls to which they are connected. All the coupling beams were

diagonally reinforced, for which details can be found in Appendix A in the design report

[Moehle et al. 2011].

Table 5.8 Thickness of RC core walls (from [Moehle et al. 2011])

Floor Levels Walls on the North and South Elevations (mm)

Walls on the East and West Elevations (mm)

Level B4 – Level 13 813 711 Level 13 – Level 31 610 610

Leve 31 – Roof Level 533 533


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5.3 Design of the MechRV3D System

With respect to the benchmark building, the proposed MechRV3D system was designed

such that the overall structure is close to the conventional design in terms of minimum

flexural strength, while, laterally, the MechRV3D system reduces base shear demands to a

level where higher-mode responses can be better controlled.

5.3.1 Design of the rocking mechanism

In the conventional design, RC core walls and coupling beams were designed to remain

essentially elastic under service level earthquakes [Moehle et al. 2011]. At this hazard level,

overturning moment demands at the base of the structure, denoted as My, were determined

to be My,rNS = 801 MN-m and My,rEW = 1249 MN-m, where the subscripts rNS and rEW

indicate moments about the north-south (NS) direction and east-west (EW) directions

respectively. These minimum flexural strengths were taken as threshold moments at which

the intended rocking action tends to be activated about the corresponding axes. These

activation moments are referred to as the rocking moments and denoted as Mrock,rNS and

Mrock,rEW respectively. As such, the rocking mechanism is deemed to have a comparable

strength to the fixed-based design. However, the mechanical rocking action, once activated,

leads to a sharp cut-off of base overturning moments, while conventional RC cores can

further develop an increased post-yielding flexural capacity due to the overstrength at the

material and system levels. In addition, as discussed in Section 4.3.1, vertical ground

acceleration may lead to Mrock fluctuating by 25% to 30% around the constant component

created by the gravity loads. To account for impacts of the difference in terms of

overstrength and the vertical seismic response, varied rocking moments were considered

in the design of the rocking mechanism, ranging from 0.75 to 2.0 times My, which are listed

in Table 5.9 along with the basic design.

The designated rocking moments are achieved by relying on gravity loads that are tributary

to the central core of the building, including structural weights, superimposed dead loads,

and 25% of the specified live loads. The total amount of these loads, denoted as Wcore, is

equal to 206 MN. It is noted that nearly 95% of Wcore originates from the dead weight,

which leads to the rocking moment be essentially constant during earthquakes.


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Table 5.9 Design rocking moments and dimensions of the rocking mechanism

Design Rocking Moments 0.75×My 1.0×My (basic design) 1.5×My 2.0×My

Mrock,rNS (MN-m) 600 800 1200 1600Mrock,rEW (MN-m) 900 1200 1800 2400dc2c,EW (m) 5.4 7.2 10.2 12.8dc2c,NS (m) 8.1 10.7 15.2 19.1

dimrckr,EW (m) 10.5 11.0 14.0 17.0dimrckr,NS (m) 15.5 15.5 19.0 23.0

Wrckr/Wcore - 10% 10% 15% 23%Wsc (MN) 227 227 237 253

In addition, the self-weight of the rocker also contributes to resist potential rocking motions

at the base of the structure and therefore was counted in for evaluating Mrock. This weight,

denoted as Wrckr, along with Wcore, leads to a total self-centering weight, Wsc, that tends to

bring the entire system back to the upright position during earthquakes. About each

principal direction, this re-centering effect is achieved with a lever arm that is equal to one

half of the centre-to-centre distance, dc2c, between the two rolling mega-columns that serve

as rocking toes. Given the designated rocking moments, these column distances, denoted

as dc2c,EW and dc2c,NS, were back calculated using Equation (5.2) and are listed in Table 5.9.

dc2c,EW =2Mrock,rNS

Wsc=

2Mrock,rNS

Wcore + Wrckr

(5.2)

dc2c,NS =2Mrock,rEW

Wsc=

2Mrock,rEW

Wcore + Wrckr

According to Equation (5.2), dc2c increases as Mrock becomes larger. This requires a greater

overall dimension, dimrckr, for the rocker on plan such that the rolling mega-columns and

their corresponding sockets will not hit side faces of the rocker. The larger the rocker, the

heavier it will be. This explains why, in Table 5.9, Wrckr increases as Mrock rises, ranging

from 10% to 23% of Wcore. In calculating Wrckr, a constant depth of 5.0 m was assumed for

the rocker to achieve the intended high rigidity and to ensure that an effective strut-and-tie

model can be developed, carrying gravity loads from the RC core to the mega-columns.


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5.3.2 Design of the rolling mega-columns

While providing vertical support to the rocking superstructure, the rolling mega-columns

accommodate horizontal movements of the rocker by swaying laterally and rolling via the

pipe-pin joints. In a displaced position, these spherically capped columns also contribute

negative stiffness to the system because of geometric effects, adjusting the post-yield

response of the shear mechanism as discussed in Chapter 4. Demands arising from these

concurrent actions were addressed in the design of the rolling columns as discussed

subsequently.

Column height

Before sizing the cross section, the height of the rolling mega-columns, hc, was firstly

determined, since it affects its compression load-carrying capacity. Apart from the

influence on strength, the column height also affects the rigid-body rotation that the pipe-

pin joints may undergo in response to the sway of the rocker, and determines the negative

stiffness that the rolling columns can induce to the overall lateral response of the

MechRV3D system.

At this stage of the study, a height of 8 m was chosen for all the four mega-columns, as

shown in Figure 4.40. This design choice was made after a number of trial-and-error

analyses whose results indicated that the P-𝛥 effect caused at the chosen height offsets the

overstrength of the BRBFs without jeopardizing the overall stability of the structure, as

will be demonstrated in Section 5.7. This semi-empirical design will be further rationalized

in Chapter 7 where an optimal column height can be selected from general design charts

that are generated based on parametric analyses.

The column height equal to 8 m leads to the pipe-pin joints undergoing rigid-body rotation

of around 3.1% (3.1% = 250 mm/8000 mm), given the expected horizontal displacement

of the rocker equal to 250 mm (mean MCE response as will be revealed in Section 5.7).

Deformations in this magnitude can be challenging for conventional cast-in-situ joints, but

are addressable for the proposed pipe-pin articulation. While there is minimum impedance

to rotations at the interface between the load-dispersing pad and the spherical cap, the

expected rotation can also be allowed by the pipe-and-can pairs as long as the free end of


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the pipes is impeded from touching the bottom of the cans. In the proposed detailing as

shown in Figure 4.40, a gap of 300 mm is left between the pipes and cans in the vertical

direction. In the same direction, these two parts can move towards each other by 93 mm

(93 mm = 3.1% × 3000 mm, where 3000 mm is the diameter of the circular grid along

which the CFT pipes are arranged) as a result of the 3.1% rotation. This movement demand

is less than one-third of the preset 300 mm of available space.

In a general sense, if a greater rotation is anticipated for the pipe-pin joints, the rotational

mobility of the joint can be readily enhanced by simply increasing the depth of the steel

cans.

Axial resistance in compression

During the bidirectional rocking motion, the point-pivoting state as illustrated in Figure 4.3

(c), is deemed the most critical loading case in which one single mega-column carries the

entire self-centering weight, Wsc. Following the basic design of the rocking mechanism,

Wsc is expected to be 227 MN, including Wcore = 206 MN and Wrckr = 10%Wcore = 22.6 MN.

Subject to this load, the mega-column is in pure axial compression free of moments at both

ends. Accordingly, the steel reinforced concrete (SRC) section, as proposed in Figure 4.36,

was sized as shown in Figure 5.5, such that the rolling mega-columns remain elastic.

Figure 5.5 Composite mega-column section


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Different from conventional SRC columns, multiple steel sections that are separated from

each other are used to reinforce the inner section of the mega-columns, which is the area

directly loaded via the spherical cap. Code provisions provide procedures for the design of

composite sections, their applicability is however limited to sections with only one steel

profile encased. ArcelorMittal [Bogden et al. 2017] extended the traditional composite

design specified in Eurocode 4 and made it applicable to multiple steel sections. Deng et

al. [2016] validated these proposed methods experimentally through static and quasi-static

tests and numerically using finite element analyses. The extended approach in the Eurocode

4 formation were followed in this dissertation for the design of the rolling mega-columns.

In the design shown in Figure 5.5, the total area of the CHSs represents 3% of the gross

area of the inner section, while the ratio of the steel reinforcement is 2%. When calculating

capacities contributed by different materials, yielding stresses of 450 MPa and 500 MPa

were assumed for the structural steel and the longitudinal bars respectively, and a specified

compressive strength of 60 MPa was assumed for the concrete. Material strength reduction

factors of 0.90, 0.85, and 0.65 were applied correspondingly. Based on these properties,

the total factored sectional capacity was calculated as 256 MN. This sectional resistance

does not have to be reduced due to the M-N interaction because the mega-columns are

purely axially-loaded members. Nevertheless, the impact due to potential buckling effect

was accounted for by using a reduction factor, χ = 0.989, which is very close to unity

because of the very large flexural rigidity (EIeff = 43.0×1015mm4) of the composite section,

and the relatively small unbraced length of the mega-columns, hc = 8000 mm, which was

determined previously. In calculating EIeff, long-term creep effects were allowed for by

applying a reduction factor of 0.5 to the elasticity modulus of the concrete, considering the

fact that nearly 95% of Wsc is permanent. As a result, calculations led to a factored axial

compression resistance equal to 253 MN, which surpasses the demand by more than 10%.

It is noteworthy that, in all these calculations, confinement effects provided by the CHSs

and the inner steel tube were not included as a strength reservation. In addition, the outer

ring-shaped section of the mega-columns were not accounted for in calculating the strength

or stiffness. This exterior layer primarily provides encasement to the CFT pipes.


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The design compression load of 227 MN does not allow for an additional demand that can

be induced by vertical seismic excitations. While vertical response spectrum analyses can

be conducted to evaluate the vertical earthquake response accurately, [LATBSDC 2020]

and [PEER 2017] recommend an estimation which, if applied to the benchmark building,

is 0.12SMSWsc, where SMS = 2.369 is the MCER short-period spectral ordinate for Los

Angeles, and Wsc = 227 MN as calculated previously. Following this expression, the

compression load resulting from the vertical seismic can be 64.5 MN, leading to the total

design demand being increased to be 292 MN. Despite this increase by 28.4%, the

composite column section shown in Figure 5.5 is adequate if the capacity margin resulting

from the confinement effect is counted in. Giakoumelis and Lam [2004] recommended to

bump up the load capacity of confined concrete by 30%. Baig et al. [2006] observed 60%

increase in compression strength during the experimental tests on concrete-filled steel tubes.

To be conservative, in this study, a 30% increase was assumed for the confined concrete

section of the mega-columns. As a result, the total compression load resistance of a single

mega-column was updated to be 300 MN which is still greater than the increased demand.

Stress concentration at pad-to-cap contacts

After the section design ensures an adequate axial resistance, the high bearing stress that is

anticipated at the pad-and-cap contact is another critical effect governing the design of the

mega-columns. Following the detailing shown in Figure 5.5, this contact stress can reach

1084 MPa in the most critical case scenario where the entire gravity load of 227 MN from

the superstructure and the rocker is carried by a single mega-column. Although steel

castings with a yielding stress of 1000 MPa, 1240 MPa, and up to 1450 MPa have been

specified in industrial standards for example in ASTM A148 [ASTM 2019], as Lynch

[2011] pointed out, most steel foundries produce cast steels with a yielding limit no greater

than 900 MPa.

To alleviate the stress concentration, a possible way is to design the load-dispersing pad,

which is currently flat, as a spherical shape and make it concave towards the cap, as shown

in Figure 5.6. By this means, a spherical contact is formed between the two parts, providing

an enlarged bearing area equal to 0.440×106 mm2 which was calculated following the

Hertzian contact theory [Budynas and Nisbett 2015; Boresi and Schmidt 2003]. In this


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circular bearing area, the maximum contact stress, or Hertzian stress, was reduced from

over 1000 MPa to 773 MPa, which is less than the yielding strength of 840 MPa that is

specified for Grade 840-1030 low-alloy cast steel in ISO 9477 - High Strength Cast Steels

for General Engineering and Structural Purposes [ISO 2015]. Even if the expected gravity

load is amplified by a factor of 1.284 to be 292 MN to allow for the vertical seismic effect,

the peak bearing stress reaches 841 MPa in a contact area of 0.521×106 mm2, which is

basically at the yielding limit of 840 MPa. Low-alloy cast steels at this specified strength

level (Grade 840-1030) are commercially supplied by numerous manufacturers around the

world. To the author’s knowledge, some Canadian foundries provide alloy steel castings

for all the grades (up to 1450 MPa) that are specified in ASTM A148 [ASTM 2019].

Figure 5.6 Spherical contact between the pad and cap

In this proposed detail, the spherical cap rolls against the spherical pad, but the contact area

is much smaller than the hemisphere, leading to little rotational constraint being generated

at the joint. As a result, the rolling mega-columns remain free of moments at both ends and

induce negative stiffness as intended. While additional research is needed to investigate

this spherical contact design more in-depth, numerical studies in this dissertation are based

on the spherical cap-to-flat pad model without loss of accuracy in capturing the global

response of the overall system.

5.3.3 Design of the Shear Mechanism

The design of the shear mechanism was initiated by first determining the minimum lateral

resistance that the MechRV3D system is expected to provide at the base of the RC core.

This lower-bound shear strength was set to be just greater than the base shear demand that


144

is expected at the onset of the rocking action. This is to ensure that the intended rocking

action can be effectively engaged during the seismic response of the overall system. Taking

the basic design of the rocking mechanism as an example, these lower-bound shear limits

were evaluated as 9.60 MN for the east-west direction and 14.4 MN for the north-south

direction, given the corresponding design rocking moments Mrock,rNS = 800 MN-m and

Mrock,rEW = 1200 MN-m and an effective height of 2H/3 assumed for lateral seismic forces,

where H is the total height of the building above the ground. If a lateral yielding strength

lower than these shear limits is used, there is a possibility that the shear mechanism would

be engaged prior to the activation of the rocking action. Once the base shear is capped off,

there will be no significant increase in the base overturning moment, which can lead to the

rocking mechanism not being activated. As a result, the intended dual-mechanism system

becomes a conventional base isolation undergoing large lateral deformations.

Upper-bound strength limits were also explored for the shear mechanism through a special

case scenario in which the rocking action was designed to be activated as intended, but the

shear mechanism was intentionally set to be elastic under all loading conditions. In this

scenario, the proposed dual-mechanism system becomes a rocking-only system which, for

ease of reference, is labelled as 1M0V, where the numbers 1 and 0 flag the activation of the

rocking (M) and shear (V) mechanisms – 1 meaning activated while 0 referred to an inactive

mechanism (remaining elastic). In this 1M0V scenario, shear reactions at the base of the

RC core can reach maximum possible values, including the first-mode response which is

limited due to the activated rocking action, and higher-mode contributions which are

excited at any given hazard level but not significantly affected by the activated rocking

mechanism. These base shears, denoted as V1M0V, define the upper-bound limits to the

lateral resistance of the MechRV3D system. In Section 5.5, the 1M0V-based benchmark

building will be investigated through NLRHAs that are conducted at the MCE level. From

these analyses, mean values of V1M0V are predicted to be V1M0V,EW = 67.0 MN and V1M0V,NS

= 64.7 MN.

These lower- and upper-bound shear limits define the range from which a design value for

the lateral resistance can be selected for the MechRV3D system. While a higher value

better controls the horizontal displacement at the base of the structure, a lower resistance


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is advantageous since it reduces the shear transmitted to the superstructure and thereby the

higher-mode effects on the inelastic response. To find a balanced design that is efficient in

mitigating the higher-mode effect as well as controlling the base displacement, a series of

values were considered for the lateral resistance, Vu, which is calculated as Vu = κVV1M0V,

where κV ranges from 0.5 to 0.8, and the subscript u indicates that this is the ultimate lateral

resistance that the MechRV3D system provides at the base of the RC core. This maximum

base shear reaction occurs when the BRBFs, as a whole, develop the ultimate shear strength,

Vu,f. At the same time, the BRBFs also provide buttressing forces Vu,c to the rolling mega-

columns and Vu,gravcol to the gravity columns in the basement, as discussed in Section 4.10.3.

As such, Vu can be calculated as follows,

Vu = Vu,f - Vu,c - Vu,gravcol (5.3)

Then the required ultimate strength of the BRBFs is

Vu,f = Vu + Vu,c + Vu,gravcol (5.4)

This Vu,f can be used to design the BRBFs. To achieve a conservative design at this proof-

of-concept stage, the term Vu,gravcol in Equation (5.4) was ignored. As will be revealed in

later sections, Vu,gravcol is approximately 10% of Vu,f. As such, the BRBFs that are sized

according to this reduced Vu,f will lead to an overestimated displacement at the base of the

RC core, which is conservative for the design of the MechRV3D system. Hence, the design

shear strength that was adopted for the BRBFs becomes,

Vu,f = Vu + Vu,c (5.5)

In Equation (5.5), Vu,f is provided by eight 7.5 m-tall single-span BRBFs in each principal

direction. These BRBFs are intended to carry in-plane forces only. This can be practically

achieved by releasing the constraint to the out-of-plane rotation at the column bases, which

leads to a negligible translational stiffness in this direction. In the plane of the BRBFs, a

pinned condition is applied at the column bases as well as at the beam-to-column joints. As

such, the lateral force that is distributed to each frame will be entirely carried by the


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diagonal brace. All the BRBs are inclined at an angle of αBRB = 45° and oriented towards

the rocker. As the skirt, along with the rocker, moves horizontally along each principal axis

in both positive and negative directions, four BRBs will be in tension while the other four

braces will be in compression, undergoing drifts of similar magnitude. As such, Vu,f can be

calculated as follows:

Vu,f = nf

2Ay1 RyFy (ω)(1+β)cos αBRB (5.6)

where nf the number of the BRBFs in each principal direction, Ay1 is the cross-sectional

area of the yielding segment of a single BRB, RyFy is the expected yielding strength that is

calculated as a product of the specified yielding strength, Fy, and the material overstrength

factor, Ry. In this design, low-yield point (LYP) steel with strength properties of Fy = 100

MPa and Ry = 1.2 was used to achieve high initial lateral stiffness given the same strength

demand. This type of BRBs have been used in practice, for example in the Sankyo Tokyo

Head Office [Hayashi et al. 1998]. Factors 𝜔 and 𝛽 in Equation (5.6) account for the

asymmetric overstrength of BRBs, with 𝜔 representing the ratio of the ultimate axial tensile

strength to the expected yielding strength, while 𝛽 indicates the further amplification of the

strength in compression due to friction of the core. Their values, as listed in Table 5.10,

were assumed based on the peak drift ratios 𝛿f of the BRBFs corresponding to varied design

strengths, Vu, that were considered in the design.

The buttressing force, Vu,c, in Equation (5.3) equilibrates the overturning moment that the

total gravity load, Wsc, creates upon the lateral deflection of the rolling mega-columns,

which is determined by the horizontal displacement of the rocker, 𝛥rckr. Accordingly, Vu,c

can be calculated as follows:

Vu,c = WscΔrckr

hc (5.7)

Given the negligible deformability of the gear teeth and the large in-plane rigidity of the

skirt diaphragm, the horizontal displacement of the rocker, 𝛥rckr, is essentially equal to that


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of the skirt, 𝛥rckr, which imposes to the BRBFs similar lateral drifts, 𝛥f. Hence, the Equation

can be rewritten as follows:

Vu,c = WscΔskirt

hc= WscΔf

hc (5.8)

where 𝛥f can be substituted by 𝛿f × hf, leading to Vu,c being evaluated as follows:

Vu,c = Wsc δfhf

hc (5.9)

Re-arranging Equations (5.5) to (5.9), Ay1 can be calculated using Equation (5.10). The

calculated areas are listed in Table 5.10.

Ay1 = Vu +Wsc δfhf

hcnf2 RyFy (ω)(1+β)cos αBRB

(5.10)

Table 5.10 Design parameters of the shear mechanism

V1M0V κV = Vu / V1M0V Vu 𝛿f ω β Ay1

67.0 MN (EW) 64.7 MN (NS)

0.8 53.6 MN (EW)

1.5% 1.3 1.2 585 cm2 (EW)

51.8 MN (NS) 566 cm2 (NS)

0.7 46.9 MN (EW)

2.0% 1.5 1.2 434 cm2 (EW)

45.3 MN (NS) 420 cm2 (NS)

0.6 40.2 MN (EW)

3.0% 1.8 1.2 347 cm2 (EW)

38.8 MN (NS) 336 cm2 (NS)

0.5 33.5 MN (EW)

4.0% 2.0 1.2 281 cm2 (EW)

32.4 MN (NS) 274 cm2 (NS)


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5.4 Advanced Nonlinear Modelling of the Benchmark Building

To validate the feasibility the MechRV3D system, the seismic performance of the

benchmark building was studied numerically with the base mechanisms incorporated as

they were designed in Section 5.3. An advanced nonlinear model was first built for the

benchmark building and was then used throughout the nonlinear dynamic analyses that

were carried out for the validation of the proposed system.

5.4.1 Modelling strategies

Extensive NLRHAs on this 42-storey RC wall structure were anticipated to be very

computationally expensive. To increase the efficiency of these analyses, some

simplifications were made. Firstly, only the RC core of the benchmark building was

modelled explicitly with wall piers and coupling beams represented using nonlinear

elements. The core was assumed to be fixed at the ground level, excluding the basement

and retaining walls. Considering the large lateral stiffness that these underground structures

provide, the ground-level fixity was deemed reasonable, causing no essential change to the

global response of the superstructure. It was noted though that removing the underground

portion of the core walls makes it impossible to see shear reversals due to backstay effects.

This is however inconsequential in this study that focuses on the impact of the base-

mechanism system on the superstructure rather than the design of RC walls.

Without significant openings, 200 mm-thick concrete slabs display large in-plane stiffness.

This allows for these slabs to be simulated through rigid diaphragm constraints at elevated

floors instead of using plane-stress finite elements, which dramatically reduced the number

of degrees-of-freedom and thereby the time for analyses. In absence of physical slabs,

distributed gravity loads and seismic inertia were assigned in equivalent ways as discussed

in later sections. Gravity load-bearing columns were not modelled explicitly but were

represented using leaning columns that contributed no lateral resistance. With these gravity

columns and the floor slabs excluded, potential slab-column outrigger effects were not

considered. This is reasonable in this study where the seismic response at the MCE level is

emphasized. At this hazard level, significant cracking is expected at junctions of the

concrete slabs with the gravity columns and the RC core.


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5.4.2 Modelling techniques for RC shear walls

Modelling RC shear walls is a challenging task, requiring adequate representation of the

axial, flexural, and shear responses of walls as well as their interactions both at the local

and global levels. This is compounded in the context of modelling of high-rise walls for

which extended nonlinearity requires a balance between computational accuracy,

efficiency, and numerical stability. Over the past 50 years, a wide range of methods have

been proposed for modelling RC shear walls. These approaches are briefly overviewed

before the modelling of the benchmark building is discussed.

Lumped plasticity models

Early modelling techniques [Clough et al. 1965; Giberson 1967] represent RC walls using

beam-column elements with flexural hinges at both ends. These lumped plasticity models

are computationally efficient and numerically stable, but the associated analysis results can

be unreliable since the prescribed moment-rotation relations of the plastic hinges do not

account for the impact of fluctuating axial forces. In addition, these hysteretic rules are

often constructed phenomenologically rather than being derived from material properties.

This made these models less adaptable to changes in wall geometries, loading histories,

and other conditions for which the flexural hysteretic responses were initially formulated.

Hence, these macroscopic models may be useful for a glimpse at the global behaviour of

structures, but are lacking precision when a close look into the inelastic response is required.

Fibre section models

Fibre section models were developed to fill this gap. Following this method, RC walls are

represented using beam-column elements whose cross-section is discretized into a number

of fibres. Inelastic stress-strain relations are assigned to these fibres, reflecting the

constitutive laws of concrete and steel reinforcement. Fibre sections follow the plane

sections remain plane assumption and capture the axial-flexure interaction. While the

sectional response is derived from the integration over the fibres, the element behaviour is

monitored at selected integration points allowing for distributed plasticity over the length.


150

Fibre beam-column elements can be formulated using a displacement-based method where

element deformations are approximated using presumed shape functions, or a force-based

method [Ciampi and Carlesimo 1986; Spacone et al. 1992; Taucer et al. 1991] where forces

are accurately defined through linear interpolations. Neuenhofer and Filippou [1997, 1998]

compared these two formulations numerically and stated that only one force-based element

is needed for a wall to achieve an acceptable accuracy that, if using the displacement-based

method, can only be obtained at a finer meshing. This advantage largely stems from the

exact force functions that lead to equilibrium conditions unconditionally satisfied over the

length of the elements regardless of the material inelasticity or geometric nonlinearity.

Despite their merits, fibre sections have no inherent ability to account for the shear response

of walls. However, inelastic shear deformations were observed for RC walls in numerous

experiments [Oesterle et al. 1979; Hiraishi 1984; Thomsen and Wallace 1995; Tran and

Wallace 2012]. The significance of the shear-flexure interaction was also pointed out by

Massone and Wallace [2006], Wallace [2007], and Beyer et al. [2011]. The inability of

capturing the shear response is a limitation of fibre section models that needs to be

addressed.

Multiple-vertical-line-element models

To account for shear flexibility, Vulcano et al. [1988] developed a multiple vertical line

element model (MVLEM) as an extension to the original model proposed by Kabeyasawa

et al. [1983]. In this model, uniaxial fibres are arranged vertically between two rigid beams,

representing the axial and flexural responses of walls, while a horizontal spring is used for

allowing for shear deformations, as shown in Figure 5.7. Although this shear spring can be

assigned inelastic force-deformation properties, Colotti [1993] pointed out that the inelastic

shear was not adequately described by the MVLEM especially under high shear stresses,

and that the shear and flexural responses were still independent in this model.

Colotti [1993], Milev [1996], and Chen and Kabeyasawa [2000] proposed to replace the

shear spring in the MVLEM with a shear panel whose biaxial stress-strain relations capture

the M-N-V interaction the web of walls. These modified MVELM introduced plane-stress

finite elements that significantly increase the computation cost.


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(a) Kabeyasawa et al. [1983] (b) Vulcano et al. [1983]

(c) Milev [1996] (d) Kolozvari et al. [2015]

(e) Fischinger et al. [2012]

Figure 5.7 MVLEM and the modified models

Petrangeli et al. [1999] proposed a concept to superimpose the strain fields that result from

axial, flexure, and shear responses at the fibre level. Based on this framework, Massone et

al. [2006, 2009] and Kolozvari et al. [2015] developed modified MVLEMs where a shear

spring was assigned to each vertical fibre forming a strip that reflects the RC panel

behaviour. Fischinger et al. [2012] proposed a similar MVLEM where the shear springs

that were coupled with vertical fibres accounted for aggregate interlock, dowel action, and

horizontal reinforcement resistance.


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Truss models

Based on the strut-and-tie concept, Panagiotou et al. [2012] proposed an equivalent truss

model by which RC walls are represented using longitudinal, transverse, and diagonal truss

elements that are connected at nodes and assigned with nonlinear uniaxial constitutive laws.

Given this diagrid topology, the truss model represents the shear-flexure interaction by

evaluating normal stress and strain considering biaxial stress state of the concrete diagonals.

Lu and Panagiotou [2014] extended the in-plane truss model to three-dimensional beam-

truss version which was used for modelling non-planar RC walls. Lu et al. [2016] and Lu

and Panagiotou [2016] and Alvarez et al. [2019] also used the truss model [Panagiotou et

al. 2012] to model coupled shear walls.

(a) in-plane truss model [Panagiotou et al. 2012]

(b) three-dimensional beam-truss model [Lu and Panagiotou 2014]

Figure 5.8 Equivalent truss models for RC walls


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5.4.3 Modelling of the RC core of the benchmark building

Among these numerical models, those allowing for biaxial stress and strain outperform

other models in providing a reliable representation of the inelastic response of RC walls,

particularly squat walls that are shear-dominated. However, these approaches are

computationally expensive due to the substantial nonlinearity that occurs under severe

seismic loading. In this study, extensive NLRHAs would be conducted on a complex high-

rise core structure for which distributed plasticity was expected. These analyses needed to

be repeated under numerous ground motions, and were then extended parametrically to

cover varied design scenarios of the MechRV3D system which is highly nonlinear as well.

Given these computational challenges, modelling with fibre-based beam-column elements

was adopted as a practical mean for representing the RC core walls of the benchmark

building. In fact, this modelling choice has been a common practice in the design of RC

wall-type high-rise structures. It is also recommended in a few major consensus documents

[PEER 2017; LATBSDC 2020; NIST 2017; CTBUH 2017] that guide nonlinear modelling

and performance-based seismic design.

The fibre elements were located at the centroid of wall piers as shown in Figure 5.9 (a).

Each wall pier was represented using single force-based beam-column element per storey.

This mesh size is sufficiently fine since the force-based formulation allows for a nonlinear

distribution of curvature, as pointed out by Neuenhofer and Filippou [1997; 1998] and

confirmed by Beyer et al. [2008]. Over the height of the structure, these vertical frame

elements remain aligned to each other without shifting off the centroid location even at

levels where the wall thickness changes, as indicated in Figure 5.9 (b).

Figure 5.9 Layout of the beam-column elements representing wall piers


154

Over the length of the beam-column elements, five integration points were used to derive

the element response following the Gauss-Lobatto algorithm. The cross-section of wall

piers was discretized into fibres, as shown in Figure 5.10, representing steel reinforcement

and concrete. Only the confined concrete was included with the 25 mm thick concrete

covers neglected since spalling of these unconfined layers was anticipated at the early stage

of major earthquakes. The steel fibres were located at the perimeter of the confined concrete,

which closely matches the physical arrangement of the steel reinforcement. Areas of these

steel fibres were calculated according to the reinforcement ratios that were determined for

each of the wall piers in the conventional design of the benchmark building.

Figure 5.10 Fibre sections of the wall piers

For the concrete and steel fibres, expected material properties were used following the

values listed in Table 5.3 and Table 5.4. For the concrete fibres, the modulus of elasticity

was equal to 35000 MPa, while the compressive strength was 93.2 MPa, which is 1.3 times

the expected strength allowing for the confinement effect. This strength was assumed to

occur at the compressive strain of 5.0×10-3. The uniaxial material model, Concrete04,

formulated in OpenSees [McKenna et al. 2010] was used to define the stress-strain relation

of the concrete fibres. The tensile strength of the concrete was neglected. The stress-strain

hysteresis of the steel fibres was defined using the material model, Steel02, in the OpenSees

[McKenna et al. 2010]. The expected yielding strength was equal to 483 MPa and the

Young’s Modulus was 200 GPa according to Table 5.4.


155

For the shear response, the guidelines [PEER 2017; LATBSDC 2020; ATC 2017; CTBUH

2017] suggest to model it as an elastic action that is decoupled from the bending behaviour.

This superimposed shear component is usually assigned an equivalent stiffness accounting

for cracked wall properties. These simplifications were believed relevant to this study for

several reasons: (1) the shear response of the RC core was designed to remain elastic in the

conventional design; (2) the high slenderness of the RC core leads to the seismic response

flexurally dominated; and (3) upon the activation of the base mechanisms, shear demands

were expected to be further limited in the superstructure. Hence, this decoupled shear

modelling was adopted in this study and used with the fibre elements.

The elastic shear component is aggregated into the fibre sections as an independent

sectional property. An effective shear stiffness was taken as a fraction of the gross sectional

stiffness. The PEER guidelines [PEER 2017] and the LATBSDC guidelines [LATBSDC

2020] both recommend to use 0.2EcAw or 0.5GcAw for RC walls at the MCER level, when

the axial and flexural responses are modelled using fibre elements. The same reduction in

the shear stiffness was adopted by MacKayLyons [2013] who numerically investigated the

seismic performance of the same benchmark building using a separate model built in

Perform3D. Following these recommendations, 0.2EcAw was used in this study.

The torsional component of wall sections cannot be accounted for through fibre sections.

In this study, the torsional action was also modelled as an independent sectional property

that was aggregated with the fibre section. However, a very small torsional stiffness was

assigned, as recommended by Xenidis et al. [1993]. This complies with the common

practice of having torsional moments resisted by in-plane shear forces of the core walls.

The fibre beam-column elements are a centreline skeleton model of wall piers and cannot

reflect the physical width of these walls or connections with coupling beams. To account

for this effect, the wide-column frame analogy (WCFA) was used. This approach was

initially developed by Clough et al. [1964] and MacLeod [1973] for modelling the in-plane

response of RC walls and later extended by MacLeod and Hosny [1977], Stafford Smith

and Abate [1981] for non-planar walls. Beyer et al. [2008] validated this modelling

technique against experimental test results.


156

For the benchmark building, the WCFA method was applied to the fibre-section elements,

forming a three-dimensional full model of the RC core, as shown in Figure 5.11. From the

vertical beam-column elements, horizontal links extended out and reached the physical

edges of the wall piers to realize the wide-column effect. These links were assigned very

large axial and flexural rigidities to simulate the width of wall piers, maintain the plane-

section-remains-plane assumption, and ensure that the displacements and rotations at wall

edges would be compatible with the flexure of the wall piers [Stafford Smith and Coull

[1991]. Torsional flexibility was considered for the horizontal links to account for warping

actions of the core. As recommended by Beyer et al [2008], the torsional stiffness was set

to be 0.25×(Gchstw3/3), where hs is the storey height, tw is the wall thickness, and the factor

of 0.25 allowed for a reduced torsional rigidity due to cracking. The free ends of the

horizontal links were connected to the adjacent coupling beams.

Figure 5.11 WCFA model of the benchmark building


157

5.4.4 Modelling of coupling beams

Coupling beams in the benchmark building were designed to have aspect ratios around 2.0,

leading to the transverse response being dominated by shear. In the conventional design

[Moehle et al. 2011], these short beams were diagonally reinforced. The intersecting

bundles of reinforcement act as braces in a truss, carrying shear forces in a ductile manner

and providing enhanced energy dissipation. To prevent the diagonal reinforcement from

buckling, confinement was provided with stirrups that go around the entire section of the

beam rather than being immediately attached to the diagonal bar-bundles. This full-section

confinement was introduced in ACI 318-08 [ACI 2008] as an alternative to the traditional

diagonal confinement which may cause construction difficulty.

Naish et al. [2013a] investigated the feasibility of the full-section confinement for coupling

beams. Static cyclic tests were conducted on 1/2-scale specimens in two different aspect

ratios equal to 2.4 and 3.33. All the specimens were diagonally reinforced but confined in

different ways. Comparing the cyclic response, the test results confirmed that the fully

confined coupling beams are equivalent to the diagonally confined counterparts. In these

tests, constraints to the axial deformation were not considered but the impact of floor slabs

were accounted for.

Based on the experimental tests, Naish et al. [2013b] developed a numerical model for

coupling beams. In this model, the nonlinear force-deformation relation of coupling beams

is idealized as a linearized backbone curve. Characteristic points on this curve are defined

using parameters including effective stiffness, rotation that leads to significant degradation

in strength, and residual strength. These parameters were calibrated against the measured

response and modified to reflect scale effects.

Naish et al. [2013b] suggested two ways to implement the proposed model: (1) Mn-hinge

model consisting of an elastic beam element and flexural hinges at both ends; and (2) Vn-

hinge model consisting of an elastic beam element with a displacement-based shear hinge

at the midspan. To account for the softening effect due to the slip/extension at the beam-

to-wall joints, elastic rotational springs can be incorporated. These two compound models

are schematically shown in Figure 5.12.


158

Figure 5.12 Schematic models of coupling beams (from [Naish et al. 2013b])

In the WCFA model of the benchmark building, coupling beams were modelled using the

Vn-hinge model. For simplicity, the flexibility due to the slip/extension was not reflected

using additional rotational springs but accounted for by assigning the effective stiffness of

0.15EcIg to the elastic beam segments, as shown in Figure 5.11. This alternative formation

of the Vn-hinge model was used by Naish [2010] in inelastic time history analyses that

were conducted on a different PEER TBI benchmark building.

The shear hinge in the Vn-hinge model was implemented using a zero-length nonlinear

spring that was oriented vertically. The force-rotation relation of this spring was defined

using the uniaxial-material model, hysteretic, in OpenSees. The expected yielding force

was calculated as Vy = 2(RyFy)(As)sin(α), where RyFy is the expected steel yielding strength,

As and α are the area and inclined angle of the diagonal reinforcement. The ultimate shear

strength, Vu, was assumed to be 1.3Vy, while the residual strength was taken as 0.3Vu. Chord

rotations corresponding to the ultimate and residual strengths were set to 6.0% and 9.0%,

respectively. These modeling parameters were recommended by Naish [2013b]. Hysteretic

parameters of the Vn-hinge model were calibrated against the specimen CB24F tested by

Naish [2013a], following the same cyclic loading protocol. Compared with the test results

(plotted in red in Figure 5.13), the shear response predicted using the Vn-hinge model is

reasonably accurate. Hence, this validated component model was then incorporated into

the overall model as shown in Figure 5.11.


159

Figure 5.13 Validation of the Vn-hinge model against the test results

5.4.5 P-𝛥 effects and gravity loads

As mentioned earlier, the gravity columns in the benchmark building were not physically

included in the WCFA model for simplicity. Instead, these elements were represented using

a cluster of leaning columns, one per storey. These leaning columns were modelled using

corotational truss elements that account for the geometric nonlinearity but provide no

lateral resistance. The cross-sectional area of these leaning columns was set to be equal to

the total area of the physical gravity columns in the benchmark building, which is about 15

m2. At floors above the ground, the horizontal displacements were enforced to follow the

movement of the central core, relying on rigid floor diaphragm.

Gravity loads were separately imposed on the RC core and the gravity columns according

to their tributary areas. Following the load combination case specified in the conventional

design, these gravity loads included the entire structural weight, superimposed dead loads,

and 25% of the specified live loads. The gravity loads that tributary to the core is equal to

206 MN, while that tributary to the gravity columns is equal to 210 MN. The gravity loads

were applied at each floor level before the entire model was subjected to seismic excitations.

-800

-600

-400

-200

0

200

400

600

800

-15 -10 -5 0 5 10 15

V [k

N]

Chord Rotation [%]

Test [Naish 2013a]

Vn-Hinge Model


160

5.4.6 Structural damping

Under seismic loads, structural damping is provided from material nonlinearity, which is

referred to as hysteretic damping, and from non-yielding mechanisms, which is referred to

as inherent damping. In the WCFA model of the benchmark building, the hysteretic

damping was directly captured through the inelastic response of the yielding elements.

Since the hystereses that were assigned to these elements were derived from actual member

sizes and material constitutive laws, they were expected to represent the energy dissipation

reliably.

In s contrast, modelling the inherent damping is a controversial topic that lacks consensus

[Charney et al. 2017]. Current practice is to model the inherent damping as viscous

damping, being mass- or stiffness-proportional, or the combination thereof (Rayleigh

damping). These classical damping models lack physical evidence since they lead to

frequency-dependent damping forces, as opposed to amplitude-dependent inherent

damping. However, these models are mathematically convenient such that a MDOF system

can be readily decoupled, relying on the orthogonality of elastic modes. Therefore, using

classical damping models for elastic systems may be inaccurate but the discrepancy is

assumed to be inconsequential [Charney et al. 2006].

However, classical damping models can cause significant uncertainty if used for nonlinear

analyses. This stems from the dependence of proportional damping matrices on the initial

structural stiffness which can drastically changes in the inelastic range. If the damping

matrix is not updated as the structure softens, significant artificial damping forces may

arise. Unrealistic damping forces may be also provoked where plastic hinges are used or

mechanical motions are introduced due to the abrupt change in stiffness. These issues of

spurious damping have been pointed out by a few researchers [Chrisp 1980; Ledger and

Dussault 1992; Bernal 1994; Hall 2006; Charney 2006]. Apart from the numerical

unreliability, the mathematical convenience of using classical damping is jeopardized in

the inelastic range where vibration modes are no longer decouplable and solving nonlinear

equations relies on direct integration instead of modal decomposition.


161

A wide range of approaches have been proposed to remediate the spurious damping issue.

Charney [2006, 2008] recommended to compute damping forces based on tangent stiffness

so as to reflect the ongoing softening of the structure. Bernal [1994] suggested to condense

massless degrees-of-freedom before damping matrices are assembled, believing that

coordinates with small inertias are sources of unrealistic damping forces. Chopra and

McKenna [2016] revealed that a damping matrix constructed by superimposing modal

damping matrices can eliminate the spurious damping forces. Puthanpurayil et al. [2016]

developed approaches to derive damping matrices at the elemental level and then assemble

them to form the global damping matrix in a similar way mass and stiffness matrices are

constructed. Grammatikou et al. [2019] modified the hysteresis of the inelastic elements to

allow for energy dissipation prior to yielding. Luco and Lanzi [2017] proposed to calculate

damping forces based on the first derivative of restoring forces with respect to the time,

which leads to the damping forces being proportional to the elastic component of velocities.

While numerous new methods are emerging, Rayleigh damping models are still universally

used in practical design and nonlinear analysis of high-rise buildings. Ledger and Dussault

[1992] stated that for multistorey buildings with periods greater than 1.5 sec, the seismic

response is not sensitive to the type of Rayleigh damping models that are used. Chopra and

McKenna [2016] drew a similar conclusion that as long as the nonlinear behaviour of

structures is properly modelled allowing for distributed plasticity, acceptable predictions

on the inelastic response can be obtained even when Rayleigh damping models are used.

In the latest design codes and guidelines [ASCE 2016; ATC 2010; PEER 2017; LATBSDC

2020; NIST 2017], Rayleigh damping models are still recommended for use.

In this study, the inherent damping was modeled as Rayleigh damping. A damping ratio of

2.5% was assumed, following the range that is recommended in [PEER 2010], [ASCE

2016], [NIST 2017], [PEER 2017], and [LATBSDC 2020], as illustrated in Figure 5.14.

This damping ratio was assigned at periods of 1 sec and 5 sec. These anchored periods

covered the elongated fundamental period (about 4.0 sec) and significant higher modes of

the benchmark building. These assumptions are also consistent with the damping model

that was adopted by MacKay-Lyons [2013] which will be taken as a reference analysis for

validating the WCFA model of the benchmark building in Section 5.4.8.


162

Figure 5.14 Equivalent viscous damping versus building height [LATBSDC 2020].

To minimize spurious damping forces, tangent stiffness matrices that are determined at the

end of each last converged time step were used in computing the damping matrix in order

to account for the gradual reduction in stiffness due to material plasticity. To further avoid

unintended damping, no Rayleigh damping was included on the zero-length elements and

the plastic hinges in the WCFA model. These measures follow good practices that are

suggested in [NIST 2017].

5.4.7 Ground motions used in the NLRHAs

In the conventional design, MKA used a suite of seven ground motions as listed in Table

5.1 and repeated here in Table 5.11. For verifying the design, nonlinear time history

analyses were conducted using a slightly different set of ground motions as listed in Table

5.11 as well. For both sets of records, limited detail was provided about how these ground

motions were scaled and modified in the design report [Moehle et al. 2011]. This made it

difficult to duplicate these ground motion records for carrying out NLRHAs on the WCFA

model of the benchmark building in this study.


163

Table 5.11 Ground motions used for verifying the design of the benchmark building

Earthquake Year Mw Station Design VerificationSuperstition Hills, California 1987 6.54 Parachute Site Test ×Denali, Alaska 2002 7.90 TAPS Pump Station #9 × ×Northridge, California 1994 6.69 Sylmar – Converter Station × ×Loma Prieta, California 1989 6.93 Saratoga – Aloha Ave × ×Northridge, California 1994 7.14 Sylmar Hospital ×Landers, California 1992 7.28 Yermo Fire Station × ×Kocaeli, Turkey 1999 7.51 Izmit × ×Denali, Alaska 2002 7.90 Carlo × Chi-Chi, Taiwan 1999 7.62 CHY109 ×

MacKay-Lyon [2013] conducted NLRHAs on the same benchmark building. In these

analyses, the suite of records that was used by MKA for the design verification was used

except the Northridge record obtained at Sylmar Hospital station being replaced by a record

obtained at Izmit station during the 1999 Duzce Earthquake in Turkey. This replacement

was probably to avoid using more than one record from the same event. This modified suite

of ground motions as well as the scale factors, as listed in Table 5.12, were used in this

study in order to ensure the consistency when validating the WCFA model against the

reference analyses conducted by MKA [Moehle et al. 2011] and MacKay-Lyon [2013], as

discussed in Section 5.4.8.

Table 5.12 Ground motions used in NLRHAs conducted by MacKay-Lyon [2013]

Earthquake Mw Station SF (MCE) Pulse Tp [sec]

Superstition Hills, CA 6.54 Parachute Site Test 1.24 FN 2.3Denali, AK 7.90 TAPS Pump Station #9 3.09 - -Northridge, CA 6.69 Sylmar – Converter Station 1.24 FN 3.5Loma Prieta, CA 6.93 Saratoga – Aloha Ave 2.99 FN 4.5Duzce, Turkey 7.14 Duzce 0.89 FP 5.6Landers, CA 7.28 Yermo Fire Station 1.68 FN 7.5Kocaeli, Turkey 7.51 Izmit 2.41 - -


164

These records were only amplitude-scaled in [MacKay-Lyon 2013] to ensure their mean

SSRS spectrum matched the target design spectrum. The scale factors listed in Table 5.12

are all less than 4, being within the range recommended in [ASCE 2016] and [NIST 2011]

for scaling factors. The matching was ensured in the period range of 0.2TNS to 1.5TEW

(where TNS and TEW were the fundamental periods in the north-south and east-west

directions respectively, TNS < TEW), to cover the elongated periods in the inelastic range and

higher modes. Spectra of the scaled records were plotted in Figure 5.15.

(a) pseudo-acceleration spectra

(b) relative displacement spectra

Figure 5.15 MCE response spectra of the ground motions

Each of the ground motion record contains two orthogonal components. They were input

into the structure for one analysis and then rotated by 90 degrees for another analysis. In

total, 14 analyses were run for the conventional design under the MCE level hazard. In

each of the analysis, the accelerations were fully used without any down sampling.

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8

S a[g

]

T [sec]

Superstition 1987 at Parachute Site TestNorthridge 1994 at Sylmar Converter StationLoma Prieta 1989 at Saratoga-Aloha AveDuzce 1999 at DuzceLanders 1992 at Yermo Fire StationKocaeli 1999 at IzmitDenali 2002 at TAPS Pump Station #95% MCE SpectrumMEAN

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8

S d[m

]

T [sec]


165

5.4.8 Validation of the WCFA Model

The fidelity of the WCFA model was validated against the NLRHAs that MKA [Moehle

et al. 2011] and MacKay-Lyons [2013] conducted separately using Perform3D. Periods

and mean values of the MCE responses are listed in Figure 5.16. while peak response

envelopes over the height of the building are shown in Figure 5.17. The results of the

WCFA model are reasonably close to those obtained in the reference analyses. Hence, this

WCFA model was considered to be a reasonable representation of the benchmark building

and is used consistently throughout the comparative studies on the MechRV3D system that

are presented in the following sections.

(a) natural periods

(b) base shears (c) base overturning moments

(d) peak IDRs (e) peak floor accelerations

Figure 5.16 Periods and peak responses of the benchmark building at the MCE level

4.1 4.2 4.4

3.2 3.4

3.3

2.6

2.3

2.2

012345

WCFA MKA MacKay-Lyons

Perio

ds [s

ec]

Mode 1 Mode 2 Mode 3

58.3

61.0 66

.1

54.7 61

.8

62.0

0

20

40

60

80


Base

She

ars [

MN]

East-West North-South

1716

1664

2323

2209

0500

1000150020002500


about North-South about East-West

Base

Ove

rtur

ning

M

omen

ts [M

N-m

]

n.a.

2.5

2.0 2.

5

1.6

1.3 1.

8

0.0

1.0

2.0

3.0



IDR

[%]

0.74

0.75

1.04

0.70

0.70 0.

81

0.0

0.5

1.0

1.5



PFA

[g]


166


(a) storey shears.

n.a.

(b) storey overturning moments.

(c) IDRs.

Figure 5.17 Validation of the WCFA model against the reference analyses


167

5.5 Benchmark Building with a Rocking-only Base-mechanism

The performance of the benchmark building was first studied numerically with a rocking-

only system incorporated at the base and compared to the fixed-based reference building.

The rocking moments listed in Table 5.9 were considered. The WCFA model built in

Section 5.4 was used to capture the inelastic response of the RC core. The ground motions

shown in Figure 5.15 were used for conducting NLRHAs at the MCE level. In these

analyses, the shear mechanism was intentionally set to remain elastic. As a result, the

rocker and the mega-columns would not sway significantly. For computational efficiency,

for the 1M0V-based analyses, the rocking and shear mechanisms were modelled using

zero-length springs as described in Section 4 of Chapter 4.

5.5.1 MCE responses of the 1M0V-based benchmark building

Storey overturning moments for the 1M0V- and fixed-based buildings are compared in

Figure 5.18 (a). It can be seen that the rocking mechanism was activated, reducing the

overturning moments in the lower half of the building to varied extents. While this largely

reduced inelastic strains of the longitudinal reinforcement when Mrock did not exceed My,

as shown in Figure 5.18 (b), the strain reduction was just noticeable in the high-Mrock cases

where the storey overturning moments were of similar magnitudes to the fixed-based

building, as shown in Figure 5.18 (a). Figure 5.18 (d) shows the impact of rocking moments

on the rocker’s rotation. While the smaller activation moments allowed the rocker to tilt by

0.6% and 0.8%, the rocking motion was largely reduced as Mrock increased and even barely

occurred when Mrock was set to 2.0×My.

The base rocking action influenced the response of the upper storeys of the building

significantly less. This can be seen from Figure 5.18 (b) where steel strains in the 1M0V-

based building remained nearly unchanged in the upper half of the structure regardless of

the base rocking moments. A similar phenomenon was observed for the deformations of

the coupling beams, as shown in Figure 5.18 (c). When Mrock was less or equal to My, chord

rotations of the coupling beam, CB02, were reduced at most floors in the 1M0V-based

building. However, the reduction at the upper levels was not as large as that in the lower

floors, with those beams reaching deformations beyond the limit of 2%, which, according


168

to Naish [2010] leads to a major damaged state and requires substantial repair. As for the

cases where Mrock was greater than My, inelastic rotations of the coupling beams were

minimally affected or even increased. These observations reaffirmed that a base flexural

mechanism alone is insufficient to prevent unintended plastic hinging or damage to

coupling beams, both of which were greatly influenced by higher-mode effects.

(a) mean storey overturning moments

(b) tensile steel strains at the north-west corner of the core

Figure 5.18 MCE responses of the 1M0V-based benchmark building


169

(c) mean chord rotations of the coupling beam CB02

(d) time history of the base rocking rotation (Superstition, truncated)

Figure 5.18 MCE responses of the 1M0V-based benchmark building (continued).

5.5.2 Incremental Dynamic Analyses

Incremental dynamic analyses were conducted on the 1M0V-based building where the

basic rocking mechanism (Mrock = My) was considered. The Superstition record was used

in these analyses with scaling from 25% to 150% of MCE. As shown in Figure 5.19 (a),

the envelope of story shears basically follows a straight line at 25% MCE and starts curving

out at 50% MCE. It then gradually displays zig-zag profiles as the intensity increases,


170

which highlights the increasing dominance of higher modes in the total response. The

envelopes of storey overturning moments, as shown in Figure 5.19 (b), significantly

ballooned out in lower stories at 50% MCE and for higher intensities, but were all anchored

at the base of the core, given the designated rocking moment. Whereas dynamic

amplification in shears and moments was not directly proportional to the ground shaking

intensity, inelastic deformations clearly increased as the ground shaking intensified, as

shown in Figure 5.19 (c) and (d).

(a) storey shears

(b) storey overturning moments

Figure 5.19 Incremental dynamic responses of the 1M0V-based benchmark building


171

(c) chord rotations of the coupling beam, CB31

(d) tensile strains of the longitudinal reinforcement

Figure 5.19 Incremental dynamic responses of the 1M0V-based benchmark building

5.5.3 Remarks

In sum, introducing a rocking mechanism alone at the base of the RC core may effectively

reduce inelastic deformations near the base of the structure, but barely affects the nonlinear

response in the upper levels where higher-mode contributions are significant. While these

inelastic responses cannot be eliminated throughout the height of the structure, their

magnitudes increase as the ground shaking intensifies. These observations confirmed that

the rocking-only scheme is not efficient enough in mitigating higher-mode effects or

achieving minimal damage to structures.


172

5.6 Advanced Nonlinear Modelling of the MechRV3D System

5.6.1 Proposed model for rocking and rolling cylinders

Given the articulated joints, the proposed rolling mega-columns are unconventional

compared with cast-in-situ columns or rocking columns that are flat-ended, which makes

it challenging to model these elements. Whereas using solid finite elements would be rather

accurate, it is computationally expensive for the overall model in which distributed

plasticity of the superstructure must be considered. To this end, a simplified model that

only involves frame elements and fibre sections is proposed in this dissertation. The

development of this approach starts from a two-dimensional rocking model that Vassiliou

et al. [2017a] proposed and then goes through a two-step procedure as discussed

subsequently.

In-plane Rocking Model

Vassiliou et al. [2017] proposed a model to capture the response of in-plane rocking

systems using OpenSees [McKenna et al. 2010], as shown in Figure 5.20. Following this

approach, the rocking body is represented using beam-column elements that can be elastic

or nonlinear. At the base of the rocking body are two nodes that have identical coordinates.

One node is connected to the beam-column element, and the other is fixed to the foundation.

These two nodes are linked using a zero-length section element that simulates the rocking

surface. This zero-length section is built using a row of nonlinear fibres that are assigned

no resistance in tension and an elastic response in compression.

Figure 5.20 Planar rocking model (redrawn from [Vassiliou et al. 2017a])


173

Vassiliou et al. [2017a] numerically validated this model through simulating the free

rocking motion of a rigid block that is released from an inclined position. The analysis

results were compared with the theoretical solution derived with the Housner [1963] model,

from which a close match was obtained. In these analyses, sensitivity studies were

conducted accounting for the influence of the number of the fibres, mesh size of the rocking

body, and integration time step. It was concluded by Vassiliou et al. [2017a] that two fibres

are adequate to capture the rocking action, and responses are essentially identical when five

or more beam-column elements are used for the rocking body. Vassiliou et al. [2017a] also

pointed out that the model is insensitive to the integration time step as long as it is

sufficiently shorter than the periods of dominant motion components. Vassiliou et al.

[2017a] recommended to use the Hilber-Hughes-Taylor (HHT) integration algorithm

formulated in OpenSees [McKenna et al. 2010] and indicated that the induced numerical

damping dissipates the kinetic energy contained in high-frequency impact waves in the

rocking body but does not affect the rocking motion whose frequency is relatively low.

This is confirmed since the predicted responses matched the Housner model closely as long

as the dissipation factor 𝛼b is smaller than 1.

Three-dimensional Rocking Model

The two-fibre rocking section only allows a rectangular block to pivot in a single vertical

plane about edge toes at the base. However, it cannot account for rocking modes of a

freestanding cylinder whose motion can be viewed as a composition of three elemental

rotations including nutation, precession, and spinning that can occur simultaneously. To

capture these complex motions, the planar rocking model is extended into a three-

dimensional one in this dissertation, as shown in Figure 5.21.


174

Figure 5.21 Proposed model for a three-dimensional rocking cylinder

In this extended model, the cylinder is represented using one single elastic beam-column

element. This is less than five as recommended in [Vassiliou et al. 2017a], but does not

significantly affect the accuracy as can be seen in the subsequent numerical validation.

Uniformly distributed mass is applied along the height of this beam-column element.

However, the moment of inertia of this element about its base point is different from that

of the physical cylinder pivoting about edge toes. This is compensated by assigning a

moment of inertia at the top and bottom nodes of the beam-column element.

The rocking surface at the base is still modelled using a zero-length section element. This

section is of circular shape that is identical to the cross-section of the cylinder. Eight fibres

are evenly distributed on the perimeter. Each fibre is assigned an area equal to 1/8 of the

cross-sectional area of the cylinder. All the fibres are tension free and elastic in

compression. The compressive stiffness of each fibre is set to be ten times the axial stiffness

of the cylinder. Rigid-elastic shear and torsional components are aggregated with the fibre

section such that sliding and spinning are restrained at the rocking surface. No Rayleigh

damping is included in the nonlinear fibres that are expected to undertake abrupt stiffness

changes.


175

For validation purposes, this three-dimensional rocking model was used to predict the free

vibration of a cylinder. As shown in Fig. 4, this cylinder has a dimension Rc = 6 m and a

slenderness 𝛼c = 0.2. It was released from an initial nutation angle 𝜃0 = 0.1 rad and an initial

precession angular velocity (d𝜑/dt)0 = 0.5 rad/sec. The loading pattern of multiple support

excitations formulated in OpenSees [McKenna et al. 2010] was used to apply these initial

conditions as imposed motions on the beam-column element that represents the cylinder.

If more than one element was used, these imposed motions could not be reflected

accordingly over the height of the cylinder; this justifies the use of a single element.

The free vibration response is shown in Figure 5.22 and compared with the results that

Vassiliou et al. [2017b] obtained using a theoretical model. In the latter study, closed-form

equations of motions and analytical solutions were derived for a rigid cylinder that is

allowed to rock and wobble in three dimensions on a rigid surface without any damping

mechanism involved. As can be seen from Figure 5.22, the numerical results reasonably

match the theoretical solutions in terms of movement trajectories, angles and angular

velocities of both nutation and precession. However, some discrepancy is seen in time

histories of the nutation angle. While peak amplitudes are constant in the theoretical

solution, the numerical result displays some decay. This is expected since the analytical

model accounts for no energy dissipation, but, as pointed out by Vassiliou et al. [2017a],

in the numerical analysis, kinetic energy is transformed from the rocking motion to the

form of impact waves in the rocking body. The impact waves are high-frequency and

dissipated by the numerical damping induced by the HTT algorithm. As a result of the

decay in nutation angle, the orbit recorded at the mid-height of the cylinder is not as polar

symmetrical in the numerical analysis as in the theoretical solution.


176

(a) orbits recorded at the mid-height of the cylinder

(b) time histories of the nutation angle (c) time histories of the nutation velocity

(d) time histories of the precession angle (e) time histories of the precession velocity

Figure 5.22 Validation of the proposed three-dimensional rocking model

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

𝛥X [m]

𝛥Z [m] Numerical results fromthe proposed model

0

0.05

0.1

0 5 10 15 20 25Time [sec]

𝜃[rad]

numerical results -0.5

0

0.5

0 5 10 15 20 25d𝜃/dt [ra

d/se

c]

Time [sec]numerical results

0

20

40

0 5 10 15 20 25

𝜑[rad]

Time [sec]

numerical results

0

5

0 5 10 15 20 25d𝜑/dt [ra

d/se

c]

Time [sec]

numerical results


177

The rocking model developed in the previous section is further modified to capture three-

dimensional rolling motions. In the rocking surface, multiple layers of fibres are introduced

and arranged on a polar grid, as shown in Figure 5.23. All the fibres remain compression-

only but have to undergo varied gap distances before they can be engaged to carry loads.

These gap distances, denoted as g, are determined based on the geometry of the spherical

cap, as illustrated in Figure 5.23. By these means, the modified fibre section simulates a

smooth transition of the contact point when the spherical cap rolls against the rocker or the

foundation, and is referred to as the rolling section. Sliding and spinning are also restricted

by aggregating rigid elastic shear and torsional components to the fibre section. These

assumptions are reasonable given the physical dowel action provided by the pipe-and-can

pairs.

Figure 5.23 Schematic model of the fibre-based rolling section


178

5.6.2 Modelling of the BRBFs

The BRBFs were modeled as pinned at the column bases and beam-to-column joints. Their

out-of-plane stiffness was neglected such that lateral forces in one direction would be

entirely carried by the frames in the same direction. The BRBFs in both principal directions

were interconnected at the beam-to-column joints through rigid truss elements to achieve

the skirt diaphragm action. These frames were also connected to the gap elements that were

used in Section 4 in Chapter 4 to model the gear teeth. BRBs were modeled using fiber-

based truss elements. The stress-strain relation of the fibers was defined using the uniaxial

material, Steel4, which Zsarnóczay [2013] implemented in OpenSees [McKenna et al.

2010]. Asymmetrical hardening in tension and compression was accounted for using the

parameters recommended by Zsarnóczay [2013]. The BRB model was calibrated against

the specimen C500W-II tested by Zsarnóczay [2013].

Figure 5.24 Validation of the BRB model


179

5.7 Numerical Validation of the MechRV3D System

5.7.1 Responses of the shear mechanism

The proposed MechRV3D system is referred to as 1M1V since both mechanisms were

activated as designed in Section 5. NLRHAs were conducted at the MCE level on the

1M1V-based benchmark building, and compared to the fixed- and 1M0V-based results. In

these analyses, the component models of the rolling mega-columns and BRBFs were

incorporated as described in Sections 5.6.1 and 5.6.2. While only the basic design (Mrock =

My) was considered for the rocking mechanism, design variants of the shear mechanism

were included as listed in Section 5.3.3. The 1M1V-based building was subjected to the

same suite of ground motions as the ones shown in Figure 5.15. Numerical convergence

was reached for all analyses except for the case of the Northridge Earthquake for a 𝜅V=0.5,

where numerical instability was caused by large deformations of the BRBFs due to the low

activation strength.

Figure 5.25 Lateral response of the MechRV3D system

-80

80

-305 305

BRBFs

rolling mega-columns 𝛥skirt[mm]

MechRV3D

V[MN]

V=Vf -Vc

V = Vf + Vc

𝜅V = 0.5

Superstition, MCEDirection: east-west -80

80

-300 300

𝛥skirt [mm]

V[MN]𝜅V = 0.6

-80

80

-300 300

𝛥skirt[mm]

V[MN]𝜅V = 0.7

-80

80

-300 300

𝛥skirt[mm]

V[MN]𝜅V = 0.8


180

The overall lateral response of the MechRV3D system is depicted using the V-𝛥skirt

hystereses as shown in Figure 5.25. As 𝜅V increased from 0.5 to 0.8, the horizontal

movement of the skirt, 𝛥skirt, decreased, resulting in fuller hystereses at lower design

strengths and thinner ones for the higher ones. At all the lateral strength levels, inelastic

excursions of the BRBFs were achieved as intended along both principal directions. In each

cycle of the hystereses, when the skirt approached peak displacements, the lateral resisting

force provided by the BRBFs (including the hardening effects) was reduced by the reverse

forces that the rolling mega-columns created at the same displacement. When forces in the

BRBFs reversed, the force created by the rolling columns did not and thereby became

temporarily additive to the former. This led to a further curving-out around the yielding arc

on the unloading and reloading paths, as highlighted in Figure 5.25. Eventually, the overall

hystereses followed a shape that is less skewed than they would otherwise be without the

softening effect of the rolling columns. This helped impose a more rigorous limit on shear

forces that developed at the base of the core, which is desirable for controlling higher-mode

responses.

Figure 5.26 (a) summarizes the actual lateral resistance that the MechRV3D system

provided to the base of the core. Peak responses under the considered ground motions and

their mean values are plotted. Comparing the 1M1V-based responses to those of the 1M0V-

based scenario, consistent reductions are observed. Whereas these reductions varied for the

different ground motions, the mean values closely matched the design lateral strengths. In

addition, these ultimate lateral forces were achieved with the BRBFs undergoing the drift

ratios that are approximately equal to the assumed values, as can be seen in Figure 5.26 (b).

The highest BRBF drift ratios were 2.9% (EW) and 3.3% (NS, not shown), which were

obtained when 𝜅V = 0.6. This resulted in strains of 2.8% (EW) and 3.2% (NS) in the

yielding segment of the BRBs. The mean-plus-one-standard-deviation values of these

strains reached 5.1% (EW) and 6.2% (NS), which, according to Tremblay et al. [2004] is

realistically achievable using BRBs that are made of high-ductility Japanese steel grades.

Corresponding to the BRBF drifts, the lateral displacement at the base of the core did not

exceed 250 mm (mean) or 465 mm (mean + std) in both principal directions.


181

(a)

(b)

Figure 5.26 Ultimate lateral response of the MechRV3D system

5.7.2 Seismic performance of the 1M1V-based PEER benchmark building

Figure 5.27 shows mean concrete and longitudinal reinforcement strains at critical

locations of the core. At the north-west corner, the activation of the shear mechanism led

to reductions in steel strains around the mid-height of the building and stories above.

Smaller strains were obtained as the design lateral resistance was reduced. In the case of 𝜅V = 0.6, rebars around the mid-height remained totally in the elastic range. This was

achieved in upper stories as well despite a slight exceedance of just 5%𝜀y at the 31st floor.

At the south-west corner, yielding was not completely avoided in most of the upper stories,

inelastic strains were cut down by over 50% however in the case of 𝜅V = 0.6, when

compared to those in the fixed-based reference building. High strains at this location appear

to be an isolated issue. It may be the result of the 914 mm-long wall pier at this corner

which behaves more like a column than a wall. In the lower half of the 1M1V-based

building, while inelastic steel strains were further reduced at the SW corner, yielding of

rebars was totally avoided at the NW corner. Concrete strains were well below the

acceptable limit of 1.5% in the conventional design. They were nearly halved after the

1M1V configuration was used, as shown in Figure 5.27.

1.0

0.9

0.8

0.7

0.6

1.0

0.8

0.8

0.7

DIV

ERG

ED

1.0

0.5

0.5

0.5

0.4

1.0

0.9

0.8

0.6

0.6

1.0

0.8

0.7

0.6

0.6

1.0

0.7

0.6

0.5

0.5

1.0

1.1

1.0

0.9

0.8

V1M0V67.0MN

52.2MN 46.8MN 40.6MN 34.6MN

0

50

100

150

Superstition Northridge Loma Prieta Duzce Landers Kocaeli Denali MEAN

Vu[MN]

1M1V Design Lateral Resistance0.8V1M0V53.6MN

1M0V0.7V1M0V46.9MN

0.6V1M0V40.2MN

0.5V1M0V33.5MN

(a)

1.5%2.1%

2.9%2.5%

0%

1%

2%

3%

4%

5% 1M1V DesignLateral Resistance 0.8V1M0V 0.7V1M0V 0.6V1M0V 0.5V1M0V

1.5% 2.0% 3.0% 4.0%

Actual BRBF Drift Ratios

Assumed BRBF Drift Ratios

(b)


182

(a) north-west corner

(b) south-west corner

Figure 5.27 Mean strains of longitudinal reinforcement and concrete

No coupling beams in the fixed-based building exceeded the limit of 6% that was adopted

as the maximum acceptable deformation in the design of the benchmark building. However,

as shown in Figure 5.28, a great number of them sustained rotations over 2% which is

suggested by Naish [2010] as the repair limit. This issue was just partially addressed in the

1M0V-based scenario, leaving the beams located within the top 1/3 to 2/3 of the building

still in exceedance of the repair limit. In the 1M1V-based building, the number of beams

that required substantial repair was largely reduced. In the case of 𝜅V=0.6, the repair limit

was surpassed only at 6 to 9 floors which were about 80% less than the numbers obtained

in the fixed-based building. Even for these limit-exceeding beams, for example, CB02 on

the south elevation, chord rotations were just marginally above 2%, compared to rotations

over 5% in the conventional design.


183

(a) CB02

(b) CB21

(c) CB31

Figure 5.28 Chord rotations of the coupling beams


184

IDRs were also checked to ensure reduced deformation demands on non-structural

elements. In this paper, total IDRs, 𝛿s,tot, include lateral deformations of the RC core, 𝛿s,core,

and base rocking rotations, 𝛿s,rock. As shown in Figure 5.29 (a), 𝛿s,tot in the 1M1V-based

building kept declining as 𝜅V decreased. When 𝜅V = 0.6, the peak value of 𝛿s,tot dropped

from 2.5% to 2.0%, which has been experimentally proven to be a drift level that can be

accommodated using innovative partition and cladding systems with just minor damage.

In the lower storeys, 𝛿s,tot in the 1M0V- and 1M1V-based buildings were higher than those

in the fixed-based building, since IDRs in this zone are highly influenced by the rocking

rotations. The real deformation of the core, 𝛿s,core, was considerably reduced, with the peak

value of 1.6% in the case of 𝜅V = 0.6 which was about 2/3 of that in the fixed-based building

and a half of the acceptance criteria adopted in the conventional design. Floor accelerations

were significantly reduced in the 1M1V-based building as shown in Figure 5.29 (b). Again, 𝜅V = 0.6 was found to be the optimal design choice as it achieved nearly 50% reduction in

PFAs.

(a) IDRs

(b) PFAs

Figure 5.29 MCE IDRs and PFAs of the 1M1V-based building


185

5.8 Summary

In this chapter, the feasibility of the proposed MechRV3D system was validated. This was

conducted numerically by carrying out extensive NLRHAs at the MCE level. For this

validation purpose, an advanced nonlinear model was built for the benchmark building that

was used as the reference structure. This model allows for distributed plasticity of the RC

core walls using fibre-based elements and three-dimensional actions relying on the wide-

column frame analogy. NLRHAs were first conducted on this WCFA model under a suite

of ground motions that were scaled to match the MCE design spectrum. The analysis results

indicated the fidelity of this reference model which was then used throughout in the ensuing

validating analyses.

With respect to the benchmark building, the physical embodiment of the MechRV3D

system was designed covering a range of activation strengths for both intended base

mechanisms. Based on this design, the MechRV3D system was simulated using skeleton

models representing highly nonlinear behaviours of the mechanism components. An

innovative modelling technique was proposed for modelling three-dimensional rocking and

rolling actions of a cylindrical column. These base-mechanism models were then

incorporated into the WCFA model of the benchmark building for further investigations.

Before validating the proposed system in its dual-mechanism formation, a rocking-only

system was first investigated through NLRHAs and IDAs. The analysis results clearly

demonstrated that the base rocking mechanism alone is instrumental in limiting the seismic

response in the lower portions of the structure and even eliminate the flexural hinges in

bottom storeys, but is inefficient in reducing inelastic deformations that are distributed over

the height of the structure as a result, in great part, of higher-mode effects. These

deformations can still induce structural damage to the extent that exceeds the repair limit.

This issue was resolved once the shear mechanism was effectively engaged. This was

observed from the NLRHAs that were conducted on the 1M1V-based benchmark building.

The analysis results confirmed that the proposed system eliminated the base plastic hinges,

prevented unintended flexural hinges in the upper stories of the building, and minimized

the damage to the coupling beams. The MCE deformation demand on non-structural


186

elements were also largely reduced to the level that can be safely accommodated using

innovative low-damage partition and cladding systems [Tasligedik et al. 2015; Araya-

Letelier et al. 2019; Okazaki et al. 2007]. Varied design options were examined

parametrically for the dual-mechanism. While Mrock equal to My was chosen to achieve a

strength similar to the conventional design, a Vu of 0.6V1M0V was found in this study to be

an optimal design for the shear activation that was efficient in both higher-mode mitigation

and base movement control. Additional theoretical studies is presented in subsequent

chapters to develop a general design procedure that achieves an optimal balance of

superstructure drift reduction and acceptable lateral deformation of the base shear

mechanism.

In contrast to conventional fixed-base design approaches, the MechRV3D system allows

the structure that rests on top of it to be more directly capacity designed for the rocking

moment and the lateral resistance assigned to the base mechanisms. While the proposed

system was investigated with respect to a specific RC core-wall building in this chapter,

the concept can be extended in future research to determine optimal design parameters of

the MechRV3D system for various other high-rise building lateral-force-resisting systems.

A wider range of ground motions will also be included in these studies to affirm the

feasibility of the proposed system under varied seismic loading conditions.

187

Chapter 6 Modal Analysis on Generally Supported Cantilever Systems

6.1 Introduction

After confirming the efficiency of the proposed system for a representative RC core-wall

building, the study on the MechRV3D system was further extended to more general

building cases. As a first step of this generalization, this chapter discusses an analytical

model that was formulated to evaluate modal contributions to the seismic response of high-

rise structures and how these could be used to establish the properties of structures with the

MechRV3D system. This analytical study is focused on defining the varying patterns of

modal contributions as both shear and flexural constraint conditions are changed at the base

of structures. While helping better understand higher-mode effects, these analytical studies

also provide guidance for the parametric analyses that are presented in Chapter 7.

The analytical study was initiated from the formulation of a beam analogy to high-rise

buildings which in many instances can be idealized as a cantilever. With distributed mass

and elasticity, this equivalent cantilever beam is elastically supported at the base in both

rotational and translational directions, simulating the boundary conditions that the

proposed MechRV3D system provides to high-rise buildings. These are discussed in

Section 6.2 as well as a review of previous studies.

In Sections 6.3 to 6.5, frequency equations and closed-form solutions are derived for a

generally supported cantilever system and for some special boundary condition cases.

These formulations facilitate the study of the impact of varying base constraints on the

effective modal mass and modal contributions to varied seismic responses as will be

discussed in Section 6.6. This chapter concludes in Section 6.7 with a summary of the main

findings.


188

6.2 Continuum Beam Analogy

For conventional fixed-based buildings, the lateral response is governed by overall bending

due to the axial elongation and shortening of structural elements and/or transverse racking

due to differential deflections between adjacent building floors. To capture these dominant

deformations, Khan and Sbarounis [1964] proposed an analogous model that consists of

two continuum cantilever beams, one representing the flexural response and the other

representing the shear response. This continuum analogy makes it feasible to present

structural responses through analytical expressions. Coull and Choudhury [1967] derived

differential equations and closed-form solutions for the static response of coupled shear

walls. In these derivations, discretely distributed coupling beams were also smeared as a

continuum medium that connects two cantilever beams, each accounting for both flexural

and shear responses of RC walls.

Pennucci et al. [2015] used a cantilever beam analogy in an analytical study on higher-

mode effects for isolated and coupled shear walls. In this equivalent system, the cantilever

beam was restrained translationally at the base but released rotationally, simulating base-

hinged RC walls. For the flexible rotational constraint, a pinned condition was applied

without considering the variation in flexural stiffness at the base. This pinned cantilever

was compared to a totally encased counterpart to highlight the significance of higher-mode

contributions.

Wiebe and Christopoulos [2015a] extended the use of the continuum beam analogy for

conducting modal analysis on controlled rocking systems. In this analysis, the base rocking

joint was idealized as an elastic rotational spring at the base of a cantilever beam. In this

study, a range of rotational stiffnesses were considered, leading to the base constraint

varying from a fixed to a pinned condition. During this process, no translational movements

were allowed at the base of the cantilever, because the horizontal displacement is impeded

at the base of rocking systems.

In this study, a single cantilever continuum beam was used to represent the overall lateral

response of RC core walls, as shown in Figure 6.1. This beam was assigned distributed

mass, m, and flexural rigidity, EI, constant over the height of the cantilever, H. The constant


189

elasticity leads to an elastic response of the beam, which matches the expectation that

minimal yielding will occur in the superstructure when the MechRV3D system is

incorporated at the base, as has been verified in Chapter 5. At the current stage, only the

flexural behaviour is considered for the beam, which is reasonable for slender wall

structures that are primarily flexure-dominated. The influence of the shear flexibility will

be included in future research.

Figure 6.1 Cantilever beam analogy for RC core-wall high-rise buildings

This formulation is an extension of the models used by Pennucci et al. [2015] and Wiebe

and Christopoulos [2015a] (where only the rotational flexibility at the base was considered)

with both shear and flexural constraints softened at the base in order to simulate the

nonlinear boundary conditions that are imposed as a result of the concurrent engagement

of the rocking and shear mechanisms. This is implemented by using a translational spring

and a rotational spring at the base of the cantilever, as shown in Figure 6.1. Both springs

are elastic, being assigned an axial stiffness, KT, and a rotational stiffness, KR, respectively.

KT and KR decrease from a very large value to a very small one, simulating the

corresponding constraints varying from a fully fixed to a completely released condition.

This variation in stiffnesses leads to a series of elastic systems, each consisting of a

cantilever that is elastically supported in both translational and rotational directions. Given

these two elastic supports, conducting analytical studies on these continuum beams become

more challenging. At the same time, interesting findings are expected from closed-form

analyses regarding the impact of the dual flexible shear and flexural supports on higher-

mode contributions.


190

6.3 Equation of Motion governing Distributed Systems

With the uniformly distributed mass m and a constant elasticity EI over the height, the

cantilever beam forms a distributed system. Using a differential segment shown in Figure

6.1, Chopra [2000] defines the equation that governs the lateral vibration of the cantilever,

𝑚 𝜕 𝑢𝜕𝑡 + 𝜕𝜕𝑧 𝐸𝐼 𝜕 𝑢𝜕𝑧 = 0 (6.1)

In this partial differential equation, u represents the lateral deflection of the cantilever, and

varies with position z and time t. To separate the position-dependent component and the

time-variant component, this deflection function u(z, t) is rewritten as

𝑢(𝑧, 𝑡) = 𝜙(𝑧)𝑞(𝑡) (6.2)

With this variable separation, Equation (6.1) is transformed into two equations of motion,

𝑞(𝑡) + 𝜔 𝑞(𝑡) = 0 (6.3)

𝜙 (𝑧) = 𝛽 𝜙(𝑧) (6.4)

that respectively govern the time function q(t) and the spatial function 𝜙(z) which is the

mode shape. The variable β in Equation (6.4) is a parameter that is related to the natural

frequency, ω, of the distributed system, and is defined as

𝛽 = 𝑚𝜔𝐸𝐼 (6.5)

Equation (6.4) defines an eigenvalue problem to which the general solution is,

𝜙(𝑧) = 𝐶 (cos 𝛽𝑧 + cosh 𝛽𝑧) + 𝐶 (cos 𝛽𝑧 − cosh 𝛽𝑧)+ 𝑆 (sin 𝛽𝑧 + sinh 𝛽𝑧) + 𝑆 (sin 𝛽𝑧 − sinh 𝛽𝑧)

(6.6)

where constants Cp, Cm, Sp, and Sm and the frequency parameter β can be solved based on

the boundary conditions of the cantilever beam.


191

6.4 Eigenvalue Analysis on Elastically Supported Beams

As shown in Figure 6.1, the cantilever beam is supported using one elastic translational and

one rotational spring at the base. These boundary conditions are described as below,

𝑧 = 0 elastic translational spring 𝐸𝐼𝜙 (0) = −𝐾 𝜙(0) (6.7)

elastic rotational spring 𝐸𝐼𝜙 (0) = 𝐾 𝜙 (0) (6.8) 𝑧 = 𝐻 no flexural constraint 𝜙 (𝐻) = 0 (6.9)

no shear constraint 𝜙 (𝐻) = 0 (6.10)

Applying these boundary conditions to the beam, a frequency equation is obtained as

1 + 1cos 𝛽𝐻 cosh 𝛽𝐻 = 𝛽𝐻𝑅 (tan 𝛽𝐻 − tanh 𝛽𝐻)+ (𝛽𝐻)𝑇 (tan 𝛽𝐻 + tanh 𝛽𝐻) + (𝛽𝐻)𝑅𝑇 1 − 1cos 𝛽𝐻 cosh 𝛽𝐻

(6.11)

where R and T are nondimensional stiffnesses of the rotational and translational springs

respectively, being defined as follows,

𝑅 = 𝐾 𝐻𝐸𝐼 (6.12)

𝑇 = 𝐾 𝐻𝐸𝐼 (6.13)

As aforementioned, KR and KT are expected to vary over a wide range to allow for the

rotational and translational constraints at the base of the cantilever to vary from full fixity

to a released condition. For this purpose, R and T are assigned values as listed in Figure

6.2. These values of R and T form a domain as shown in Figure 6.2. Special combinations

of R and T lead to special constraint conditions as marked in Figure 6.2. The cantilever

systems studied in this chapter represent the generally supported scenarios.


192

log 𝑅 = 6, 4, 2, 1, 0, −1, −2, −4, −6 log 𝑇 = 6, 4, 2, 1, 0, −1, −2, −4, −6

Figure 6.2 Domain of the nondimensional stiffnesses R and T

Equation (6.11) is a transcendental equation to which closed-form solutions are not

available. Numerical roots were solved for βnH, where n is the mode number, as plotted in

Figure 6.3 and listed in Table 6.1 for selected R and T.

Table 6.1 βnH values for special base constraint conditions

R = 10-6 R = 10-2 R = 101 R = 106

T = 106

β1H = 0.0416 β2H = 3.9266 β3H = 7.0684 β4H = 10.2096 β5H = 13.3506

0.41593.92787.0691

10.210113.3510

1.72274.39947.4507

10.520913.6123

1.87514.69407.8543

10.994214.1343

T = 101

0.0416 2.4548 4.8272 7.8741

11.0032

0.41572.45544.82907.8754

11.0041

1.50782.60645.34118.3657

11.4376

1.57712.64825.52868.6472

11.7840

T = 10-2

0.0416 0.4472 4.7301 7.8532

10.9956

0.29800.62404.73227.8545

10.9965

0.31622.23535.29448.3527

11.4322

0.31622.36535.49788.6394

11.7810

T = 10-6

0.0298 0.0625 4.7300 7.8532

10.9956

0.03160.58804.73217.8545

10.9965

0.03162.23495.29448.3527

11.4321

0.03162.36505.49788.6394

11.7810


193

(a) Mode 1

(b) Mode 2

(c) Mode 3

Figure 6.3 Variation of βnH with R and T (to be continued)


194

(d) Mode 4

(e) Mode 5

Figure 6.3 Variation of βnH with R and T (continued)

Given these solutions to the frequency parameter, mode shapes are derived and presented

in closed-form expression as follows:

𝜙 (𝑍) = 𝐴 (cos 𝛽 𝐻𝑍 − cosh 𝛽 𝐻𝑍) + 𝐶 (𝛽 𝐻)𝑇 (cos 𝛽 𝐻 + cosh 𝛽 𝐻𝑍)+ 𝐶 (sin 𝛽 𝐻𝑍 − sinh 𝛽 𝐻𝑍) − 𝛽 𝐻𝑅 (sin 𝛽 𝐻𝑍 + sinh 𝛽 𝐻𝑍) (6.14)

where 𝑍 = 𝑧/𝐻 is the position coordinate normalized to the height of the cantilever beam, 𝐴 is an arbitary constant that scales mode shapes. CRT, a coefficient that depends on the


195

nondimensional stiffnesses, R and T, and the frequency parameter, βnH, is calculated as

follows,

𝐶 = 𝛽 𝐻𝑅 (sinh 𝛽 𝐻 − sin 𝛽 𝐻) + (cosh 𝛽 𝐻 + cos 𝛽 𝐻)(𝛽 𝐻)𝑇 (cosh 𝛽 𝐻 − cos 𝛽 𝐻) − (sinh 𝛽 𝐻 + sin 𝛽 𝐻) (6.15)

The above eigenvalue analysis is valid in a general sense for elastically supported

cantilever beams. When R and T take particular values that indicate special constraint

conditions, these analytical results reduce to simpler formats as briefly summarized in the

next section.


196

6.5 Cantilever Beams with Special Base Constraints

6.5.1 Fully fixed: 𝑅 → ∞, 𝑇 → ∞

When both R and T approach an infinitely large value, the rotational and translational

springs provide as large reactions as needed to refrain the cantilever from deforming at the

base. This scenario resembles conventional structures that are fixed at the grade.

With 𝑅 → ∞, 𝑇 → ∞, all the three terms on the right side of Equation (6.11) approach to

zero. The frequency equation reduces to be

1 + cos 𝛽𝐻 cosh 𝛽𝐻 = 0 (6.16)

𝛽 𝐻 that satisfy this equation are close to the values listed in Table 6.1 for (R, T) = (106,

106). For n > 4, 𝛽 𝐻 is approximately equal to (2n-1)π/2.

Accordingly, the mode shape Equation (6.14) can be reduced as

𝜙 (𝑍) = 𝐴 (cosh 𝛽 𝐻𝑍 − cos 𝛽 𝐻𝑍)− cosh 𝛽 𝐻 + cos 𝛽 𝐻sinh 𝛽 𝐻 + sin 𝛽 𝐻 (sinh 𝛽 𝐻𝑍 − sin 𝛽 𝐻𝑍)

(6.17)

which is identical to the expression in Chopra [2000].

6.5.2 Rotationally flexible and translationally fixed: 𝑅 ≠ 0, 𝑇 → ∞

In Chapter 5, a rocking-only configuration of the MechRV3D system was investigated. The

cantilever beam analogy to this scenario can be obtained by assigning R a finite value while

keeping T very large. As such, the frequency equation becomes

1 + 1cos 𝛽𝐻 cosh 𝛽𝐻 = 𝛽𝐻𝑅 (tan 𝛽𝐻 − tanh 𝛽𝐻) (6.18)

Multiplying both sides of this equation by (cos 𝛽𝐻 cosh 𝛽𝐻), it gives

1 + cos 𝛽𝐻 cosh 𝛽𝐻 = 𝛽𝐻𝑅 (sin 𝛽𝐻 cosh 𝛽𝐻 − cos 𝛽𝐻 cosh 𝛽𝐻) (6.19)


197

which is identical to the frequency equation in Wiebe and Christopoulos [2015a] for base

rocking systems. 𝛽 𝐻 satisfying Equation (6.18) are provided in Table 6.1 for (R, T) = (101,

106) and (R, T) = (10-2, 106) as examples. Solutions for other R values can be obtained from

Table 6.1.

Regarding mode shapes, any term in Equations (6.14) and (6.15) that contains T in the

denominator will vanish as T approaches infinity, leading to

𝜙 (𝑍) = 𝐴 (cos 𝛽 𝐻𝑍 − cosh 𝛽 𝐻𝑍) + 𝐶 (sin 𝛽 𝐻𝑍 − sinh 𝛽 𝐻𝑍)− 𝛽 𝐻𝑅 (sin 𝛽 𝐻𝑍 + sinh 𝛽 𝐻𝑍) (6.20)

𝐶 = 𝛽 𝐻𝑅 (sin 𝛽 𝐻 − sinh 𝛽 𝐻) − (cosh 𝛽 𝐻 + cos 𝛽 𝐻)(sinh 𝛽 𝐻 + sin 𝛽 𝐻) (6.21)

After combining similar terms for (sin 𝛽 𝐻𝑍) and (sinh 𝛽 𝐻𝑍), the mode shape function

can be rewritten as

𝜙 (𝑍) = 𝐴 [cosh 𝛽 𝐻𝑍 − cos 𝛽 𝐻𝑍 + 𝐴 sin 𝛽 𝐻𝑍 − 𝐴 sinh 𝛽 𝐻𝑍] (6.22)

𝐴 = cos 𝛽 𝐻 + cosh 𝛽 𝐻 + 2𝛽 𝐻𝑅 sinh 𝛽 𝐻sin 𝛽 𝐻 + sinh 𝛽 𝐻 (6.23)

𝐴 = cos 𝛽 𝐻 + cosh 𝛽 𝐻 − 2𝛽 𝐻𝑅 sin 𝛽 𝐻sin 𝛽 𝐻 + sinh 𝛽 𝐻 (6.24)

which is the format used by Wiebe and Christopoulos [2015a] to present the mode shape

of a cantilever beam supported by a rotational spring at the base.

6.5.3 Rotationally free and translationally fixed: 𝑅 → 0, 𝑇 → ∞

If the rotational constraint is further relaxed, a pinned condition is formed at the base of the

cantilever. With R approaching nil while T remaining infinitely large, the frequency

equation becomes as simple as follows,

tan 𝛽𝐻 − tanh 𝛽𝐻 = 0 (6.25)


198

which governs the free vibration of the pin-supported cantilever beam used by Pennucci et

al. [2015] as an analogy to wall structures. 𝛽 𝐻 that satisfy Equation (6.25) are listed in

Table 6.1 for (R, T) = (10-6, 106).

With 𝑅 → 0, 𝑇 → ∞, the mode shape function is simplified as

𝜙 (𝑍) = 𝐴 (𝛽 𝐻) sinh 𝛽 𝐻 sin 𝛽 𝐻𝑍 + sin 𝛽 𝐻 sinh 𝛽 𝐻𝑍sinh 𝛽 𝐻 + sin 𝛽 𝐻 (6.26)

6.5.4 Rotationally fixed and translationally free: 𝑅 → ∞, 𝑇 → 0

In contrast to the previous two scenarios, when the cantilever beam is fully rotationally

restrained at the base ( 𝑅 → ∞ ) but allowed to move laterally, the system becomes

analogous to base-isolated structures. Considering the fact that base isolators are typically

engaged at a low levels of lateral strength, it can be assumed that T approaches zero.

Having 𝑅 → ∞, 𝑇 → 0, the frequency equation becomes,

tan 𝛽𝐻 + tanh 𝛽𝐻 = 0 (6.27)

And the mode shape function is as follows,

𝜙 (𝑍) = 𝐴 cos 𝛽 𝐻𝑍 + cosh 𝛽 𝐻𝑍cosh 𝛽 𝐻 − cos 𝛽 𝐻 (6.28)


199

6.6 Modal Analysis on Generally Supported Beams

Having obtained natural frequencies and mode shapes, it is feasible to solve modal

responses of the cantilever beam under earthquake excitation. This modal analysis was

conducted for the general case in which the cantilever is elastically supported at the base

as investigated in Section 6.4.

In terms of the mode shapes that are orthogonal to each other, the lateral deflection 𝑢(𝑧, 𝑡)

can be presented in a modal combination format as follows,

𝑢(𝑧, 𝑡) = 𝑢 (𝑧, 𝑡) = 𝜙 (𝑧)𝑞 (𝑡) (6.29)

As such, the equation of motion, that governs the elastically supported cantilever beam

subjected to seismic excitation, 𝑢 (𝑡), can be decoupled into a series of modal equations,

as outlined by Chopra [2000] for classically damped systems,

𝑞 (𝑡) + 2𝜁 𝜔 𝑞 (𝑡) + 𝜔 𝑞 (𝑡) = −𝛤 𝑢 (𝑡) (6.30)

In this equation, 𝜁 is the damping ratio of the nth mode, and 𝛤 is the modal participation

factor defined as,

𝛤 = 𝐿𝑀 = 𝑚 𝜙 (𝑧)𝑑𝑧𝑚 [𝜙 (𝑧)] 𝑑𝑧 = 𝑚𝐻 𝜙 (𝑍)𝑑𝑍𝑚𝐻 [𝜙 (𝑍)] 𝑑𝑍 = 𝜙 (𝑍)𝑑𝑍[𝜙 (𝑍)] 𝑑𝑍

(6.31)

Defining 𝑞 (𝑡) = 𝛤 𝐷 (𝑡), Equation (6.30) can be rewritten as

𝐷 + 2𝜁 𝜔 𝐷 + 𝜔 𝐷 = −𝑢 (6.32)

which governs the deformation, 𝐷 , of the nth SDOF system subjected to the same seismic

load, 𝑢 .


200

6.6.1 Effective modal mass

The effective modal mass of the nth SDOF system can be calculated as

𝑀∗ = 𝛤 𝐿 = 𝑚𝐻 𝜙 (𝑍)𝑑𝑍[𝜙 (𝑍)] 𝑑𝑍 (6.33)

which, if normalized to the total seismic mass 𝑚𝐻, leads to a nondimensional ratio of

modal participation mass, 𝑀∗ , calculated as follows,

𝑀∗ = 𝑀∗𝑚𝐻 = 𝜙 (𝑍)𝑑𝑍[𝜙 (𝑍)] 𝑑𝑍 (6.34)

Independent of the normalization of mode shapes, 𝑀∗ varies when the rotational and

translational constraints are changed at the base of the cantilever beam, as shown in Figure

6.4. It can be seen that as R and T decrease either individually or simultaneously, fewer

modes are needed to satisfy a required cumulative participation mass ratio, for example,

90% as a limit that is typically targeted in practice.

For a given R, as T decreases, the first mode gradually dominates over higher modes in

terms of mass contribution. This is actually the way base-isolated structures work by having

a major portion of the total mass concentrated in the fundamental mode which is basically

a rigid-body sliding motion. With T decreasing, this tendency of the first-mode dominance

ramps up faster under higher R values, but slows down as the rotational constraint is

gradually relaxed as well. This leads to a comparison that, for example, when R = 106, the

first mode contributes over 99% of the total mass when T reaches 100, however, a

dominance to the same extent is not seen until T drops to the level of 10-2 when R = 10-2 or

the level of 10-6 when R = 10-6. In scenarios of this kind where R and T are both low, a

significant amount of mass is contributed by the first two modes that allow for rigid-body

rocking and sliding motions as a result of ineffective rotational and translational constraints

at the base of the cantilever beam.


201

Figure 6.4 Variation of the modal participation mass ratios, 𝑀∗ , with R and T


202

Higher mode mass participation is considerable when T is no lower than 102, corresponding

to the top three charts in Figure 6.4. In these cases, whereas decreasing R leads to 𝑀∗

declining for all the higher modes, this reduction is however limited, if not trivial, and much

less efficient than that obtained by reducing T. This to some extent validates that a flexural

mechanism alone at the base of structures is insufficient in limiting the higher-mode

response for which a shear mechanism is indispensable.

6.6.2 Displacements

The SDOF deformation, 𝐷 (𝑡), as expressed by Equation (6.32) is correlated with the

pseudo-acceleration, 𝐴 (𝑡), as follows,

𝐴 (𝑡) = 𝜔 𝐷 (𝑡) (6.35)

Recalling 𝛽 = 𝑚𝜔 𝐸𝐼⁄ defined in Equation (6.5), 𝐷 (𝑡) can be presented as

𝐷 (𝑡) = 𝑚𝐴 (𝑡)𝛽 𝐸𝐼 = 𝑚𝐻 𝐴 (𝑡)𝐸𝐼(𝛽 𝐻) (6.36)

𝐷 (𝑡) is also involved in the expression of the modal displacement of the cantilever beam,

if the definition 𝑞 = 𝛤 𝐷 (𝑡) is substituted into Equation (6.29). This gives

𝑢 (𝑧, 𝑡) = 𝜙 (𝑧)𝑞 (𝑡) = 𝛤 𝜙 (𝑧)𝐷 (𝑡) (6.37)

Replacing 𝐷 (𝑡) using Equation (6.36), 𝑢 (𝑧, 𝑡) can be rewritten as follows,

𝑢 (𝑍, 𝑡) = 𝑚𝐻𝐸𝐼 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝐴 (𝑡) (6.38)

Then the peak absolute value of 𝑢 is

𝑢 = 𝑚𝑔𝐻𝐸𝐼 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.39)


203

where 𝑆 (𝑇 ) is the spectral acceleration normalized by the gravitational acceleration, 𝑔,

corresponding to natural period of 𝑇 . Then 𝑢 can be normalized as follows by dividing

the term 𝑚𝑔𝐻 𝐸𝐼⁄ ,

𝑢 = 𝑢𝑚𝑔𝐻 𝐸𝐼⁄ = 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.40)

In Equation (6.40), (𝛽 𝐻) is a unitless quantity based on a dimensional analysis as below,

𝑑𝑖𝑚(𝛽 𝐻) = 𝑑𝑖𝑚 𝑚𝜔 𝐻𝐸𝐼= [𝐹] [𝑇] [𝐿]⁄[𝐿] 1[𝑇] [𝐿][𝐹][𝐿] [𝐿] = 1

(6.41)

where [𝐹], [𝐿], and [𝑇] are units of force, length, and time respectively.

Using Equation (6.31), 𝛤 𝜙 in Equation (6.40) can be rewritten as

𝛤 𝜙 = 𝜙 (𝑍) 𝜙 (𝑍)𝑑𝑍[𝜙 (𝑍)] 𝑑𝑍 (6.42)

In this equation, 𝜙 (𝑍) and the two intergrals are unitless. In addition, any scaling factor

applied to 𝜙 (𝑍) will be cancelled out from the numerator and denominator. Hence, 𝛤 𝜙

is a nondimensional quantity and invariable regardless of how mode shapes are scaled.

As such, 𝑢 in Equation (6.40) is dimensionless and independent of the normalization of

mode shapes. It includes the static modal displacement pattern regulated by 𝛤 𝜙 and the

dynamic amplification associated with 𝑆 . The impact of the varying base constraints on 𝛤 𝜙 and 𝑢 is demonstrated in Figure 6.5 to Figure 6.6 respectively.


204

Static modal response, 𝛤 𝜙

As can be seen from Figure 6.5, when the cantilever is fixed translationally (T = 106) at the

base, all the modal displacement profiles are not sensitive to R, except, for the first mode,

the deflection displays a flexural type response if the cantilever is rotationally restrained

(R = 106) as well, and gradually straightens up as the rotational constraint is relaxed (R

decreasing) allowing for rigid-body rotations about the base. For T < 106, the displacement

profiles 𝛤 𝜙 (n ≥ 3) remain insensitive to the variation of R. These higher-mode

deflections rapidly vanish once T drops to the level of 101 and lower, regardless of R values.

Comparatively, the deflection patterns of the first two modes are more complex. Factors

that influence the variation of 𝛤 𝜙 (n ≤ 2) include not only the values of R and T but also

their relative magnitudes. For the first mode, when the translational constraint remains

fairly effective (T = 104 and 102), the deflection profile follows a similar flexure-to-rotation

transformation with decreasing R as observed for T = 106. However, when T drops to 101,

this trend is only partially reproduced in the sense that the both deflection profiles still exist

but tend to develop at a displaced position for R = 106 which is much greater than T or R =

101 and 100 which are comparable with T. Obviously, the reduced translational fixity is

attributed to this displaced first-mode shape. However, if the rotational constraint is even

weaker (R = 10-6 << T = 101), the displacement disappears, leading to the first-mode, which

is now a rigid-body rotation, developing at the original base of the cantilever.

As T further decreases to the level of 100 and under, the first-mode no longer develops in

flexure but in different rigid-body motions. The deflection profile, 𝛤 𝜙 , displays a rigid-

body slide off the original position when the base constraint is relatively higher in rotation

than in translation (R > T); a rigid-body pivoting about the original base when the rotational

constraint is weaker; or a combination of these two motions when R and T are comparable.

The second mode can also be displaced due to a reduction in the translational fixity, as can

be seen for T = 102 in Figure 6.5. With T further decreasing, 𝛤 𝜙 tends to be utterly

isolated when R is relatively higher. However, when R is comparable with or lower than T,

the second mode displays a rigid-body pivoting motion from a displaced position, implying

a significant modal contribution to the base displacement as revealed in later sections.


205

Figure 6.5 Variation of 𝛤 𝜙 with R and T


206

Dynamic modal response, 𝑢

Dynamic modal contributions to the displacement are calculated following Equation (6.40).

The total dynamic response of the displacement is defined as the square root of the summed

squares (SRSS) of the contributions from the first five modes. Following this SRSS rule, a

cumulative contribution is calculated for each mode as follows,

𝑢 , = 𝑢 (6.43)

Cumulative modal contributions, 𝑢 , , are plotted for varied combinations (R, T) in

Figure 6.6 in which color patches indicate the contribution from each individual mode.

As can be seen from Figure 6.6, the displacement of the cantilever is primarily dominated

by the first mode in most of the (R, T) scenarios. This dominance is maintained irrespective

of whether the deflection results from the flexural deformation of the beam or the rigid-

body motions (rotation and/or sliding). This is why the curves that represent cumulative

higher-mode contributions overlap with that of the first mode response in many charts in

Figure 6.6. The only exceptions are seen in the charts where T is 100, 10-2, and 10-4, while

R is rather smaller than T. In these scenarios, the deflection near the base of the cantilever

is largely attributed to the second mode in which significant displacements are developed

in the same region, as observed from the static modal response shown in Figure 6.5. The

first mode resumes its dominance on deflections in the upper part of the cantilever.


207

Figure 6.6 Variation of 𝑢 , with R and T


208

6.6.3 Rotation angles (inter-storey drift ratios)

Inter-storey drift ratios are relevant to building structures where seismic inertias are lumped

at discrete floors. For the continuum cantilever beam with distributed mass and elasticity,

an equivalent response quantity is the rotation angle which represents the deflection change

per unit length in the height of the cantilever. This quantity, denoted as 𝛿(z,t), can be

calculated by taking the first order of derivative of the displacement about the position

variable z. The peak modal response 𝛿 and its nondimensional format, 𝛿 , are expressed

as follows,

𝛿 = 𝑚𝑔𝐻 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.44)

𝛿 = 𝛿𝑚𝑔𝐻 = 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.45)


The term 𝛤 𝜙 in Equation (6.45) represents the static modal contribution to the rotation.

The variation of 𝛤 𝜙 with R and T is plotted in Figure 6.7. For the first mode, For T ≥

101, the vertical distribution of the rotation angle displays a curved profile when R = 106.

At this moment, no rotation occurs at the base of the cantilever. As R decreases, the 𝛤 𝜙

curves get less curvy, and have a non-zero rotation at z = 0 and decreasing rotations in the

upper part of the cantilever. When R reaches 10-6, the 𝛤 𝜙 profiles approach a vertical line,

indicating a constant rotation throughout the height of the cantilever. This is consistent with

the modal deflection which is in a rigid-body rotation pattern in these (R, T) scenarios, as

discussed in the previous section. As T is reduced to the level of 100 or under, the rotation

angle is essentially constant and small (near zero) over the height of the cantilever if R is

no smaller than T. This is because under these base constraint conditions, the first-mode

deflection is basically a rigid-body slide as observed in Figure 6.5. In the cases where R is

smaller than T, the rotation angle is still constant over the height of the cantilever but in a

higher magnitude.


209



210

As for the second mode, for T ≥ 101, non-zero rotations are also seen at the base of the

cantilever as R decreases, but the rotation always varies in the height. When T drops to the

level of 100 and under, the variation of the second-mode response follows a similar trend

as seen in the first mode. As T decreases from 106 to 10-6, rotations developed in the other

higher modes (n ≥ 3) rapidly vanish especially after T reaches 101.

Dynamic modal response, 𝛿

Cumulative modal responses to the rotation angle are plotted in Figure 6.8. It can be seen

that the first mode completely dominates the response when an effective translational

constraint is in place (T ≥ 102). Keeping T in this range, the rotation angle varies at different

levels in the height of the cantilever when R takes a higher value. As R decreases to 10-2 or

smaller, rotations are basically constant and distribute in a straight line profile.

As T further decreases, the first-mode contribution still represents a significant percentage

of the total response. However, for those scenarios where R is comparable or smaller than

T, the second-mode contribution is considerably increased.


211

Figure 6.8 Variation of δ , with R and T


212

6.6.4 Overturning moments

The modal contribution to overturning moments is presented as follows,

𝑀 (𝑧, 𝑡) = 𝐸𝐼𝑢 (𝑧, 𝑡) = 𝐸𝐼𝛤 𝜙 (𝑧)𝐷 (𝑡) (6.46)

Replacing 𝐷 (𝑡) using Equation (6.36), and considering 𝜙 (𝑧) = 𝜙 (𝑍) 𝐻⁄ , Equation

(6.46) is rewritten as

𝑀 (𝑍, 𝑡) = 𝑚𝑔𝐻 𝛤 [𝜙 (𝑍) 𝐻⁄ ](𝛽 𝐻) 𝐴 (𝑡) = 𝑚𝑔𝐻 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝐴 (𝑡) (6.47)

Then the peak modal moment and its nondimensional formation are as follows respectively,

𝑀 = 𝑚𝑔𝐻 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.48)

𝑀 = 𝑀𝑚𝑔𝐻 = 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.49)


In Equation (6.49), 𝛤 𝜙 represents the static modal response of overturning moments.

This quantity is nondimensional and independent of the normalization of mode shapes.

Charts in Figure 6.9 demonstrate how 𝛤 𝜙 are affected as the base fixity varies

rotationally and translationally. For the first mode, rotational relaxation at the base of the

cantilever leads to a consistent reduction in overturning moments throughout the height of

the beam. When R = 10-6, which indicates a near-pinned condition, no moments can be

developed in the beam through the first mode which, in this situation, becomes a rigid-

body rotation. However, a similar reduction on higher-mode moments is only visible near

the base of the cantilever but insignificant over the height. Even if R takes an extremely

low value of 10-6, the static higher-mode moments can hardly be eliminated, unless T

decreases at the same time to a magnitude below 102. This suggests the higher effectiveness

of softening the translational constraint at the base of the cantilever beam than relaxing the

rotational constraint in limiting higher-mode contributions to moments.


213



214

Dynamic modal response, 𝑀

Based on Equation (6.49), cumulative modal contributions, 𝑀 , , are calculated and

plotted in Figure 6.10 for the first five modes. As can be seen from these charts, releasing

the rotational constraint consistently leads to a reduced total overturning moment near the

base of the cantilever. This reduction is primarily achieved by limiting the first-mode

contribution that dominates regardless of T values.

When T > 101, decreasing R cannot smoothen the ballooned profile of the total overturning

moment at the upper levels of the cantilever, despite an effective reduction in the first mode

contribution. This leads to an increased percentage of the second-mode contribution in the

total response. This second-mode dominance in overturning moments can be readily

mitigated by releasing the translational constraint. It can be clearly seen in Figure 6.10 that,

with decreasing T, the ballooning in the total moment profile is gradually flattened, leading

to a smooth curve that is very close to the pattern of the first-mode moment distribution. In

addition, the magnitude of the total moments is dramatically reduced as T drops down.


215

Figure 6.10 Variation of 𝑀 , with R and T


216

6.6.5 Shear forces

Following similar procedures, the peak modal contribution to shear forces and the

normalized format are derived as follows,

𝑉 = 𝑚𝑔𝐻 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.50)

𝑉 = 𝑉𝑚𝑔𝐻 = 𝛤 𝜙 (𝑍)(𝛽 𝐻) 𝑆 (𝑇 ) (6.51)

Static modal response, 𝛤 𝜙 𝛤 𝜙 in Equation (6.51) represents the static modal shear. This quantity is nondimensional

and independent of the normalization of mode shapes, and plotted in Figure 6.11. As can

be seen from Figure 6.11, a rotational release at the base of the cantilever effectively

reduces shear forces that are contributed by the first mode. Once the rotational constraint

essentially disappears (R = 10-6), no shear forces can be developed through the first mode

at any level of the cantilever. This is consistent with the zero-moment distribution achieved

in the first mode when R = 10-6, as observed in Figure 6.9.

To a very limited extent, decreasing R may reduce shear forces contributed by the higher

modes over the height and at the base of the cantilever. However, as long as an effective

translational constraint is in place (T > 101), the higher-mode shears cannot reach zero even

though a zero base moment is achieved in these modes when R = 10-6, as seen in Figure

6.9. Nevertheless, the equilibrium is still maintained between the non-zero shears and the

zero base moment owing to the sinusoidal profile of higher-mode shears.

Similar to what was concluded for moments, higher-mode contributions to shear forces can

only be efficiently limited by releasing the translational constraint. As can be seen in Figure

6.11, static higher-mode shears can be largely reduced when T = 102 and nearly eliminated

if T is further reduced.


217



218

Dynamic modal response, 𝑉

Figure 6.12 shows the cumulative modal contributions to shear forces for the first five

modes. When T is no lower than 102, the second mode contributes a significant portion of

the shear forces of the cantilever except around the mid-height where the third-mode

contribution becomes considerable. As the base rotational constraint approaches a pinned

condition, these higher-mode shears represent an increased percentage of the total response

while the first-mode contribution is essentially eliminated. This higher-mode dominance

leads to the shear envelop ballooning out in the two aforementioned zones. As T decreases,

these highly curvy shear distributions gradually straighten up approaching a triangular

profile for the first mode.


219

Figure 6.12 Variation of 𝑉 , with R and T


220

6.7 Summary

This chapter investigated the dynamic response of cantilevers representing high-rise

buildings that are not fully fixed at the base. For the sake of generalization, a cantilever

beam with distributed mass and elasticity was used as an equivalent system. To achieve an

analogy to the proposed MechRV3D system that consists of both rocking and shear

mechanisms at the base, this cantilever beam is elastically supported using rotational and

translational springs. A broad range of stiffness values were considered for these springs,

simulating the variation of the corresponding constraints from a fixed to released condition.

Based on this beam analogy, analytical studies were conducted, providing solutions to

natural frequencies and mode shapes for the elastically supported beams. In addition, modal

properties and responses were derived and presented in closed-form expressions. These

results facilitated the investigation on the sensitivity of both modal contributions and total

responses to the varying amounts of fixity in both rotational and translational constraints.

These correlations were illustrated graphically and discussed in detail.

In sum, based on this analytical work, it was found that a translational relaxation at the base

of the cantilever beam is more efficient than a rotational release in reducing higher-mode

contributions particularly to shear forces and overturning moments, while displacements

are primarily dominated by the fundamental mode.

It is also interestingly found that higher-mode (especially the second-mode) contributions

can also dominate the displacement at the base of the cantilever when the translational

constraint is somewhat relaxed but the rotational constraint is even weaker. The concurrent

sliding and rotation well match the effective conditions that the MechRV3D system

provides at the base of high-rise buildings after both mechanisms are activated. Hence, for

design purposes, it is necessary to predict the base displacement demand by conducting

nonlinear time history analyses as will be discussed in Chapter 7. Empirical equations used

in conventional design of base rocking systems and base-isolated structures may not be

relevant because the base displacement is largely determined by the first mode for these

structures.

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Chapter 7 Parametric Analyses and Design Recommendations

7.1 Introduction

The analytical study presented in Chapter 6 provided insight on the impact of base fixity

on the dynamic response of high-rise cantilevered structures. In this chapter, parametric

analyses are carried out to provide a basis upon which general procedures can be developed

to facilitate the preliminary design of the proposed MechRV3D system.

The parametric study is conducted through a series of NLRHAs. In these analyses, the

superstructure is represented using a stick model, while the rocking and shear mechanisms

are represented using nonlinear springs. The characteristics of the superstructure and base

mechanisms are defined using a group of control parameters, as discussed in Section 7.2.

For each of these parameters, a range of values are selected to keep the results relevant to

design practice and practical in terms of computational efforts. The parametric analyses are

conducted at the MCE level. Analysis results are summarized for governing responses of

the superstructure and the MechRV3D system, as discussed in Section 7.3.

Based on findings obtained from the parametric analyses, diagram charts are developed in

Section 7.4 for critical response quantities including the lateral displacement at the base of

the superstructure and the inter-storey drift ratio. These charts reflect the influence of the

considered design parameters, and therefore can be used as preliminary design aids. This

section also suggests procedures following which these charts can be used to reach a

reasonable preliminary design of the MechRV3D system which can then be verified with

NLRHAs.

After summarizing the findings from this study, Section 7.4.4 discusses main elements that

form the basis for a design methodology in a performance-based framework.


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7.2 Governing Response Quantities and Approaches for Evaluation

For general high-rise buildings, a reasonable design of the MechRV3D system is expected

to lead to minimal damage in the superstructure. To reduce displacement-sensitive damage,

peak inter-storey drift ratios (IDRs), 𝛿s, are taken as a measure of the seismic response of

high-rise buildings with the proposed system incorporated at the base. For these structures

that are allowed to uplift at the base, IDRs include the shear distortion of RC core walls as

well as the rigid-body rotation at the base of the core as a result of the rocking action.

At the same time, deformations in the base mechanisms need to be controlled under a level

that is practical to achieve. This is particularly critical for the shear mechanism where large

drift ratios may result in unrealistic strains and even fractures in the BRBs. The drift ratio

of the BRBFs is highly affected by the peak displacement at the base of the superstructure, 𝛥b, which is also considered for choosing a reasonable design of the proposed system. 𝛥b and 𝛿s depict how the total earthquake-induced deformation demand will be shared

between the base mechanisms and the superstructure respectively. The trade-off between

these two responses and the challenge of reaching a balanced design have been discussed

in Chapter 5. There are certainly other responses that are important for design

considerations, including peak floor accelerations and rocking rotations at the base of the

rocker. In this chapter, these responses are verified after preliminary sizes have be obtained

for the MechRV3D system, meeting the acceptance criteria set for 𝛥b and 𝛿s.

Approaches for a quick estimate on storey drifts and base displacements are available in

previous studies where rocking actions or base isolation has been considered. In the design

of controlled rocking systems, Wiebe and Christopoulos [2015b] represented multi-storey

steel frames using an equivalent SDOF system as proposed by Priestley et al. [2007].

Instead of directly calculating storey drifts of the frame, Wiebe and Christopoulos [2015b]

estimated the displacement demand of the SDOF substitute system, assuming that the

frame drifts were entirely attributed to the base rocking rotation. This correlation was

sufficiently accurate for short structures that are dominated by the first-mode response, but

may not adequately capture higher-mode contributions for slender structures as pointed out

by Priestley et al. [2007]. This is true for high-rise buildings especially when the base fixity


223

is softened rotationally and translationally. As demonstrated in Chapter 6, when these two

constraints are softened and the rotational one is proportionally softer, the second mode

considerably contributes to the inter-storey drift ratios.

As for the base displacement demand, a similar SDOF equivalency is usually used in the

conventional design of base-isolated structures. Typically short and stiff, these structures

are intended to move laterally as a rigid body atop the isolation layer, attracting nearly the

entire modal participation seismic mass in this sliding mode. This makes it feasible to

construct a SDOF substitute system using the structural mass and properties of the base

isolators. As Skinner et al. [1993] suggested, the peak displacement of this SDOF system

can be readily computed and empirically modified, providing an estimated demand at the

base of the original structure. Nevertheless, this first mode-based approach may not be

valid for high-rise buildings at the base of which rotations and horizontal movements are

allowed simultaneously. Given these unconventional constraints, the second-mode

contribution to the displacements near the base of the superstructure can be significant or

even dominant, as has also been revealed in the analytical study that was presented in the

previous chapter.

Based on these considerations, approaches involving SDOF equivalence of MDOF systems

were not used in this study. Instead, nonlinear time history analyses were conducted to

account for the higher-mode contribution to the chosen governing responses. In these

analyses, a number of design parameters that describe the characteristics of the

superstructure and the base mechanisms were considered. For each parameter, a series of

values were assigned to cover a range that is relevant to design practice. Relying on the

analysis results, correlations between the design parameters and the governing responses

were identified, aiding the preliminary design of the MechRV3D system.


224

7.3 Nonlinear Parametric Analyses

7.3.1 Generic buildings and fundamental period, T1

The parametric analyses were conducted on generic high-rise buildings in which a RC core

is the lateral-force-resisting system. These buildings range from 45 m to 375 m of height,

or 15 to 125 storeys with an assumed storey height equal to 3.0 m. Xu et al. [2014]

suggested regression relations between natural periods, T1, and the structural height, H, for

tall buildings based on 414 completed high-rise developments with heights ranging from

50 m to 600 m. These relations are listed in Table 7.1 and redrawn in Figure 7.1.

Table 7.1 Fundamental periods and structural heights (from [Xu et al. 2014])

H T1 = Lower ~ Upper Bound T1 = Average

H ≥ 250 m T1 = 0.30√H~0.40√H T1 = 0.35√H

150 m ≤ H < 250 m T1 = 0.25√H~0.40√H T1 = 0.325√H

100 m ≤ H < 150 m T1 = 0.20√H~0.35√H T1 = 0.275√H

50 m ≤ H < 100 m T1 = 0.15√H~0.30√H T1 = 0.225√H

H < 50 m T1 = 0.08√H~0.15√H T1 = 0.115√H

Figure 7.1 Relations between T1 and H (redrawn from [Xu et al. 2014])

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

Stru

ctur

al H

eigh

ts, H

[m]

Fundamental Periods, T1 [sec]

600m

250m

150m

100m

50m

0.35H0.5

0.325H0.5

0.275H0.5

0.225H0.5

0.115H0.5


225

The range of building heights considered in this parametric study is included in the height

range that was covered by the referenced buildings in [Xu et al. 2014]. Taking the average

of the lower and upper bounds of periods listed in Table 7.1, the fundamental periods, T1,

of the generic buildings were calculated, ranging from 1 to 7 sec. Within this range, specific

values were chosen for T1 which represents one of the design parameters in this the

parametric study. These representative periods are listed in Table 7.2 along with the

associated building heights and number of storeys, ns.

Table 7.2 Properties of the generic high-rise buildings

T1 H Number of Storeys H/B B mEQ EI T1,a

[sec] [m] ns - [m] [ton] [MN-m2] [sec]

1 45 15 3 15 229 1.068×106 1.03

2 90 30 4 23 516 9.309×106 2.03

4 150 50 5 30 917 31.52×106 4.04

6 300 100 7 43 1872 452.8×106 6.03

7 375 125 8 47 2240 970.1×106 7.03

For all these generic buildings, the RC core is located at the centre of the floor plan. Both

the core and the floor plan are square in shape, as schematically shown in Figure 7.2. Floor

dimensions, B, were calculated based on varied aspect ratios, H/B, that were assumed for

the generic buildings in different heights. These aspect ratios and floor sizes, as listed in

Table 7.2, are realistic from a practical design perspective.

For each floor in the generic buildings, a uniformly distributed area load w = 10 kPa was

assumed, allowing for dead and live loads that are typically specified in practical designs.

Along with the floor geometry, a seismic mass, mEQ, that is lumped at each floor level, is

computed for the five generic buildings, as listed in Table 7.2.

For the parametric analyses, all the prototype RC core buildings were represented using an

elastic stick model. This simplification is justifiable since the superstructure is expected to

remain essentially elastic when the MechRV3D system is incorporated at the base, as has


226

been verified in Chapter 5. At floor levels of the stick model, the seismic mass, mEQ, that

is calculated in Table 7.2, was assigned to the degree of freedom in the horizontal direction.

No moment of inertias about the vertical axis were applied, since, in this parametric study,

only the lateral response in one of the principal directions was considered. Torsional effects

can be included in more detailed design stage. In addition, rotational inertias about any

horizontal axes were not considered either, because building floors are expected to move

primarily horizontally even if rocking action is activated at the base of the structure. The

stick model consists of elastic beam-column elements to which a constant flexural rigidity,

EI, was assigned. To determine the value of EI, a fixed-based continuum cantilever beam

was used as an analogy for the stick model. This continuum cantilever beam has a

uniformly distributed mass m that is equal to (ns×mEQ)/H, and a flexural rigidity equal to

EI which is constant over the height of the cantilever. The fundamental period of this

cantilever beam can be calculated as follows,

𝑇 = 2𝜋𝜔 = 2𝜋(𝛽 𝐻)𝐻 𝐸𝐼𝑚 = 2𝜋𝐻(𝛽 𝐻) 𝑚𝐸𝐼 (7.1)

where ω1 is the natural circular frequency of the first mode, while β1 is the first-mode

frequency parameter as defined in Equation (6.5). Given the fixed-based condition, β1H is

equal to 1.8751 as listed in Table 6.1. Rearranging Equation (7.1), the value of EI can be

calculated as follows,

𝐸𝐼 = 4𝜋 𝐻 𝑚[(𝛽 𝐻) 𝑇 ] (7.2)

where T1 can be substituted using the target periods listed in the first column of Table 7.2.

The calculated EI values are listed in Table 7.2 as well.

Given mEQ and EI, eigenvalue analysis was conducted using OpenSees [McKenna et al.

2010] for each of the generic buildings using the corresponding stick model that is fixed at

the base. These analyses provided actual fundamental periods, T1,a, as listed in Table 7.2,

which well match the target periods.


227

Figure 7.2 Geometry of the generic RC core-wall buildings

7.3.2 Rocking mechanism: Moment reduction factor, RM

The rocking mechanism of the MechRV3D system was simulated using a zero-length

rotational spring at the base of the core stick. This spring was assigned an elastic bilinear

moment-rotation relation with the initial stiffness set to a very large value. The strength of

this rotational spring, Mact, defines the moment capacity of the overall system which is

equal to the recentering moment, Mrock, provided by the physical rocking mechanism. As

discussed in Section 4.5 in Chapter 4, Mrock stays essentially constant given that the

expected rocking rotation is very small ( 1% to 2% as discussed in Section 7.3.14). As such,

the nonlinear curve in the bilinear relation of the rotational spring is set to be flat.


228

Figure 7.3 Numerical model for the nonlinear parametric analyses

In practical design of high-rise buildings, a minimum flexural strength is usually ensured

such that structures remain essentially elastic under earthquakes at the SLE hazard level.

In the design of the MechRV3D system, this conventional design practice can be retained

by having the rocking mechanism not engaged under base overturning moments that are

lower than the SLE demand. This can be ensured by assigning an activation moment, Mrock,

of the rocking mechanism that satisfies the following requirement,

𝑀 = 𝑀 , (7.3)

where Mb,SLE is the elastic base overturning moment demand expected at the SLE level.

Whereas a specific probability of exceedance has been defined for SLE and MCE events

respectively, the ratio between the spectral responses at these two hazard levels may vary

geographically due to the difference in seismicity. The CTBUH working group [Golesokhi

et al. 2017] indicated that, in terms of peak ground acceleration (PGA), the hazard intensity

at the SLE level can be 1/8 to 1/4 of that at the MCER level. Considering that the maximum

spectral acceleration is typically around 2.25 times the PGA (the spectral acceleration for

a zero period), it is reasonable to assume that elastic seismic responses at the MCE level

can be 4 to 8 times the SLE responses. Applied to the base overturning moment demand,

this assumption leads to


229

𝑀 , = (4~8) × 𝑀 , (7.4)

where Mb,MCE is the elastic base overturning moment demand obtained at the MCE level.

As such, the moment strength requirement outlined in Equation (7.3) can be rewritten as

𝑀 = 𝑀 ,𝑅 , (7.5)

where RM,EQ is the moment reduction factor of the rocking mechanism for seismic loads.

In this parametric study, the following values were considered for RM,EQ to account for the

fluctuating ratios between the SLE and MCE responses,

𝑅 , = 4, 6, 8 (7.6)

In addition to a minimum seismic resistance, it is also essential to ensure no gap opening

at the base of the structure under ultimate wind loads. This requires Mrock ≥ Mb,wind, where

Mb,wind is the wind-induced moment at the base of the structure. Hence, Equation (7.5) shall

be extended to be as follows,

𝑀 = max 𝑀 , 𝑅 ,⁄ , 𝑀 , (7.7)

As a result, in regions where the strength design of high-rise buildings is likely to be

governed by wind load, a moment reduction factor that is smaller than RM,EQ may be needed

for the design of the rocking mechanism.

This study currently focuses on high-seismicity regions where seismic loads dominate the

design over wind effects. For the parametric analyses, the values of RM,EQ included in

Equation (7.7) were considered as the moment reduction factor, RM, that would be used for

determining the strength of the rocking mechanism. Then, Equation (7.5) is rewritten as

𝑀 = 𝑀 ,𝑅 (7.8)


230

7.3.3 Shear mechanism: Shear reduction factor, μV

The shear mechanism was represented using a zero-length translational spring, as shown

in Figure 7.3. A hysteretic bilinear force-deformation relation was assigned to this spring,

accounting for the inelastic hysteresis of the BRBFs. The yielding strength of the shear

mechanism, Vy, is deemed to consist of two portions, as defined below,

𝑉 = 𝑀𝐻 + 𝜇 𝑉 , − 𝑀𝐻 (7.9)

In this equation, Mrock is determined using Equation (7.8), and Heff is the effective height

for the first mode of a flexural type cantilever, being assumed to be 0.726H. The ratio of

these two quantities gives the base shear demand that is expected at the onset of the rocking

action, and sets the lower bound of Vy that prevents a premature engagement of the shear

mechanism, as discussed in Section 5.3.3 in Chapter 5. Vb,1M0V in Equation (7.9) denotes

the base shear demand that is obtained at the MCE level from the system shown in Figure

7.3 but in a rocking-only format where only the rocking action is activated while the shear

spring remains non-yielding. Having the rocking-associated shear, Mrock/Heff deducted from

Vb,1M0V, the parenthesized term in Equation (7.9) is an indication of the elastic MCE shear

demand developed through the higher modes, assuming that the rocking at the base of the

structure barely affects these high-frequency modes. A coefficient, μV, ranging from zero

to unity, is then applied to Vb,1M0V, leading to varied levels of the higher-mode shear demand.

This higher-mode component, along with the rocking-related term, define the yielding

strength of the shear mechanism.

The coefficient μV is hereafter referred to as the shear reduction factor and is considered as

one of the control parameters in the parametric analyses. Within the range from 0 to 1, the

following values were considered for μV,

𝜇 = 0, 0.2, 0.5, 0.7, 1 (7.10)


231

When μV = 0, the shear mechanism is expected to yield once the rocking action is activated.

When μV = 1, the proposed dual-mechanism system reduces to its rocking-only version,

allowing for the elastic higher-mode response to develop as much as possible.

It is worth pointing out that the shear strength definition used in this parametric study is

slightly different from that used in Chapter 5 where the PEER benchmark building was

investigated. While previously the strength was defined by the ultimate lateral resistance,

Vu, in this parametric study, the yielding strength, Vy, is used instead as a measure of the

shear resistance. In both studies, a shear reduction factor is involved to reduce Vb,1M0V in

order to achieve the shear strength. However, in Chapter 5, this factor, denoted as κV, was

directly applied to Vb,1M0V without distinguishing between the contributions from the

rocking mode and the higher modes. This compares with Equation (7.9) where the factor,

μV, is only used to reduce the elastic higher-mode shear (the term in the parentheses) at the

MCE level. The latter scheme is deemed more rational, being better compatible with

capacity design principles.

7.3.4 Shear mechanism: Initial stiffness, Kb1

The initial lateral stiffness of the shear mechanism, denoted as Kb1 in Figure 7.3, is

represented using a unitless factor, I(Kb1), which is defined as follows,

𝐼(𝐾 ) = 𝑇𝑇 (7.11)

where T1 is the fundamental period of the stick model shown in Figure 7.3 when the

structure is translationally fixed at the base, while Tb1 is the fundamental period of an

equivalent linear elastic SDOF system that has a mass equal to the total mass of the original

stick model and a lateral stiffness equal to Kb1. The ratio of these two periods indicates the

elastic stiffness of the shear mechanism relative to the lateral rigidity of the superstructure.

The representation in Equation (7.11) was inspired by Skinner et al. [1993] who used a

similar ratio, I(Kb) = Tb/T1,FB, to measure the period shift induced by base isolation systems,

where Kb is the shear stiffness of a linear isolator, Tb is the isolated period, and T1,FB is the

fixed-based period. The authors referred to I(Kb) as the degree of isolation (or isolation


232

ratio) and presented it as, I(Kb1), and I(Kb2), which are respectively evaluated based on the

initial stiffness, Kb1, and nonlinear stiffness, Kb2, of base isolators. Skinner et al. [1993]

stated that a structure can be well isolated when I > 2 and investigated some representative

base-isolated low-rise structures for which I(Kb2) ranged from 3 to 24 and even for cases

where I(Kb1) was above 3.

For high-rise buildings, isolation ratios at these magnitudes are difficult, if not impractical,

and unnecessary to achieve since the fixed-based period of these structures is already long.

Hence, in this parametric study, I(Kb1) is primarily taken as a relative lateral stiffness of the

shear mechanism, and therefore assigned low values as follows,

𝐼(𝐾 ) = 0.1, 0.2, 0.3 (7.12)

7.3.5 Shear mechanism: Post-yielding stiffness, Kb2

The post-yielding stiffness of the shear mechanism is denoted as Kb2 in Figure 7.3. This

quantity can certainly be represented using I(Kb2) following the concept of the degree of

isolation. However, this would not be as relevant for base-isolated short structures as for

high-rise buildings as explained above. Hence, in this parametric study, a post-yielding

stiffness ratio, 𝛼K = Kb2/Kb1, was used instead and assigned a single value of 𝛼K = 0.02.

7.3.6 Summary of the control parameters

Based on the numerical studies conducted on the PEER benchmark building in Chapter 5,

the five variables that are defined in the previous sections were identified as the governing

design parameters that determine the global seismic behaviour of high-rise buildings with

the MechRV3D system incorporated at the base of the structure. For the purpose of guiding

a preliminary design, this study focused on these parameters at this stage. Other more

detailed properties, such as the degree of coupling of RC core walls, may also be influential

and will be considered in future research.

For clarity, all the considered parameters are summarized in Table 7.3. Permutations of

these parameters and their values generated 225 case scenarios representing varied

properties of the generic buildings and the MechRV3D systems.


233

Table 7.3 Summary of the design parameters

Parameters ValuesT1 1 2 4 6 7 secRM 4 6 8μV 0.0 0.2 0.5 0.7 1.0

I(Kb1) 0.1 0.2 0.3αK 0.02

7.3.7 Rolling mega-columns

In the numerical model shown in Figure 7.3, the rolling mega-columns in the physical

MechRV3D system are not included. This leads to the exclusion of the negative lateral

stiffness that will be induced as these columns sway. As a result, the parametric NLRHAs

based on this model are anticipated to provide a slight underestimate of displacement

demand at the base of the superstructure. Taking the case of the PEER benchmark building

as an example, this underestimation can be around 10%-15% of the base displacements

obtained from the NLRHAs conducted in Chapter 5.

7.3.8 P-𝛥 effects and gravity loads

Physical gravity columns in the generic buildings were represented using leaning columns

that account for the P-𝛥 effect, as shown in Figure 7.3. Meanwhile, the core stick elements

were also modelled as P-𝛥 sensitive, reflecting the second-order geometric nonlinearity

that is developed at the central core. Based on these settings, gravity loads were imposed

on the leaning columns and the core stick separately, as shown in Figure 7.3, according to

their tributary areas. It was assumed that 50% of the total gravity load is carried by the core

while the rest portion carried by the gravity columns. This gravity load distribution is a

realistic assumption in the practical design of high-rise buildings.

As recommended in Chapter 4, the gravity columns under the ground level are allowed to

sway between the ground and foundation levels without being restrained at the intermediate

basement floors. This is to minimize the P-𝛥 effect that is caused at the ground level and

transferred to the BRBFs through the skirt diaphragm. Accordingly, the leaning column


234

that represents these underground gravity columns was assigned a length that is equal to

the full height of the basement, as indicated in Figure 7.3.

In practical design, the heights of the basement and superstructure are weakly correlated.

Two buildings of the same height may have different numbers of storeys underground. For

the sake of simplicity and practicality, it was assumed that all the generic buildings that are

considered in this parametric study contain 4 storeys in the basement, each storey being 3

m high. Hence, the height of the underground leaning column in the numerical model is set

to be 12 m, as indicated in Figure 7.3.

7.3.9 Damping model

A Rayleigh damping model was adopted for the model. A damping ratio of 2.5% was

assigned to Modes 1 and 5 for each of the generic building to ensure all the significant

higher modes will not be overdamped. To reflect the structural stiffness that constantly

varies especially in the inelastic range, the last-committed stiffness was used for updating

the damping matrix accordingly.

In the model, Rayleigh damping was not applied to elements that have large rigidities or

are expected to undergo abrupt changes in stiffness. These elements include the rotational

and translational springs, the leaning columns and the rigid links that represent the floor

diaphragms. This was necessary in order to ensure minimal spurious damping in the

analyses.

7.3.10 Seismic hazard

As aforementioned, high seismicity areas were considered in this parametric study. For this

purpose, the seismic hazard in the Los Angeles area was adopted. The seismic parameters

were obtained based on ASCE 7-16 [ASCE 2016] and are listed in Table 7.4. Based on

these parameters, a target MCER response spectrum was developed as shown in Figure 7.4.


235

Table 7.4 Seismic design parameters considered for the parametric analyses

Seismic Design Parameters ValuesLocation Los Angeles (34.05, -118.24)

Risk Category IISite Class C

SS 1.974S1 0.703

SMS 2.369SM1 0.985SDS 1.579SD1 0.657

Figure 7.4 MCER response spectrum for the parametric analyses

7.3.11 Ground motion selection and scaling

To achieve unbiased analyses, a suite of eleven ground motion records were selected,

which meets the requirement specified in ASCE 7-16 [ASCE 2016], and other PBSD

design guidelines [PEER 2017; LATBSDC 2020]. These records are different from those

used in Chapter 5 for the numerical validation of the MechRV3D system. In this suite,

pulse-like ground motions were included to account for near-fault effects. The ground

motion records were selected from the PEER NGA-West 2 [Ancheta et al. 2013], following

the parameters as listed in Table 7.5.

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10

Sa [g

]

T [sec]


236

Table 7.5 Parameters for the selection of ground motion records

Parameters ValuesMagnitude 6-12

Rrup 0 – 120 kmRjb 0 – 120 kmV530 180 – 1200 m/sec D9-95 15 – 60 sec

Fault types All typesPeriod range for scaling 0.1 – 10 sec

Weights 0.2 for [0.5, 3] sec; 0.6 for [3, 8] sec; 0.2 for [8, 10] sec.

Scaling method Minimize MSEDamping ratio 5%

Component RotD100 (maximum direction)

The selected ground motions are listed in Table 7.6. These records were amplitude-scaled

to match the target MCER spectrum plotted in Figure 7.4, using the scale factors that were

calculated by the PEER NGA-West 2 [Ancheta et al. 2013]. Some of these factors exceed

4, which is the limit recommended in [ASCE 2016] and [NIST 2011], as a result of the

need to select a sufficient number of ground motions from as different earthquake events

as possible. This exceedance might induce some bias in terms of the frequency content of

the ground motions. However, this effect is anticipated to be limited since most of these

factors are close to 4 with only two reaching 6.10 and 7.49 respectively. More detailed

studies will need to be conducted on a wider range of ground motions when these

preliminary design procedures are extended into a full performance-based design

framework in future studies.

Figure 7.5 shows the pseudo-acceleration spectra and displacement spectra of the scaled

records and their mean spectra, compared with the target spectra.


237

Table 7.6 Selected ground motions

RSN Earthquake Year Station Mw Tp Scale Factor

15 Kern County 1952 Taft Lincoln School 7.36 - 5.3220 Northern Calif-03 1954 Ferndale City Hall 6.5 - 3.5236 Borrego Mtn 1968 El Centro Array #9 6.63 - 4.3993 San Fernando 1971 Whittier Narrows Dam 6.61 - 7.49143 Tabas_ Iran 1978 Tabas 7.35 6.2 sec 0.70176 Imperial Valley-06 1979 El Centro Array #13 6.53 - 5.35266 Victoria_ Mexico 1980 Chihuahua 6.33 - 3.08285 Irpinia_ Italy-01 1980 Bagnoli Irpinio 6.9 1.7 sec 3.95292 Irpinia_ Italy-01 1980 Sturno (STN) 6.9 3.3 sec 1.74313 Corinth_ Greece 1981 Corinth 6.6 - 6.10341 Coalinga-01 1983 Parkfield - Fault Zone 2 6.36 - 5.80

(a) pseudo-acceleration response spectra.

(b) displacement response spectra.

Figure 7.5 Response spectra of the scaled ground motions

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

S a[g

]

T [sec]

RSN-15 RotD100 RSN-20 RotD100RSN-36 RotD100 RSN-93 RotD100RSN-143 RotD100 RSN-176 RotD100RSN-266 RotD100 RSN-285 RotD100RSN-292 RotD100 RSN-313 RotD100RSN-341 RotD100 Suite Mean - STD DEVSuite Mean + STD DEV Target SpectrumSuite Mean

0

1

2

3

0 1 2 3 4 5 6 7 8 9 10

S d[m

]

T [sec]

RSN-15 RotD100 RSN-20 RotD100 RSN-36 RotD100RSN-93 RotD100 RSN-143 RotD100 RSN-176 RotD100RSN-266 RotD100 RSN-285 RotD100 RSN-292 RotD100RSN-313 RotD100 RSN-341 RotD100 Suite Mean - STD DEVSuite Mean + STD DEV Target Spectrum Suite Mean


238

7.3.12 Procedures for the parametric NLRHAs

Using these scaled ground motions, NLRHAs were conducted on each of the parameter

scenarios listed in Table 7.3 following steps as summarized below.

Step 1 – T1

For each value of T1, NLRHAs were first conducted on the fixed-based structure. From

these analyses, the mean response of the elastic base overturning moment at the MCE level,

Mb,MCE, was obtained. This step required 55 (= 5 × 11) analysis cases.

Step 2 - RM

For each value of RM, a rocking moment, Mrock, was calculated using Mb,MCE that was

obtained from Step 1 and Equation (7.8). This Mrock was assigned to the rotational spring

at the base of the core stick, while the translational spring was set to be elastic, leading to

a rocking-only response of the MechRV3D system. NLRHAs were then carried out on this

rocking-only system using the same suite of ground motions. From these analyses, the

mean response of the base shear, Vb,1M0V, was obtained. This step required 165 (= 3 × 5 ×

11) analysis cases.

Step 3 - μV, I(Kb1), and 𝛼K

Based on Mrock and Vb,1M0V, Vy was calculated following Equation (7.9) for each value of

μV. This was combined with the values of I(Kb1) and 𝛼K, generating 225 different scenarios.

NLRHAs were then conducted on each of these scenarios using the eleven ground motions.

This step required 2475 (= 1 × 3 × 5 × 3 × 5 × 11) analysis cases.

In total, 2695 NLRHAs were conducted. Analysis results are discussed subsequently.

7.3.13 Base displacements, 𝛥b

Mean values of the displacement at the base of the generic buildings are plotted in Figure

7.6, demonstrating the influence of the moment reduction factor, RM, and the fundamental

period, T1. It can be seen that, as T1 gets longer, the superstructure is expected to undergo

larger displacements at the base. In contrast, the base displacement is not sensitive to the

strength of the rocking mechanism.


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Figure 7.6 Displacement demands at the base of generic buildings


240

For given RM and T1, increasing μV leads to reduced displacement demands. While this

trend is just noticeable for buildings with shorter periods, especially when μV reaches no

lower than 0.2, controlling displacement by increasing the shear strength becomes more

efficient for taller structures. At the same time, the relative initial stiffness of the shear

mechanism, which is indicated by I(Kb1), also considerably affects 𝛥b in a way that a higher

stiffness, Kb1, which means a smaller I(Kb1), leads to a more stringent control of the

displacement. This influence of I(Kb1) becomes more conspicuous for long-period

structures.

It is interesting to note that when the period reaches 6 and 7 sec, there is a local hump in

the 𝛥b curves that are generally descending with decreasing μV . This hump increases for a

larger I(Kb1) which results from a softer shear mechanism, and for a smaller RM which leads

to a stronger rocking mechanism. A similar phenomenon was also observed in the

displacement-strength relation that Skinner et al. [1993] developed for base-isolated

structures through an analytical study. This diagram is redrawn in Figure 7.7. It can be seen

that the curves with larger Tb1 have a humped shape at higher shear strengths.

Figure 7.7 Displacement and strength relation of base-isolated structures

In Figure 7.6, mean values of the base displacement are not available for buildings with

fundamental period T1 = 7 sec when μV = 0 which means the shear mechanism may yield

at the onset of the rocking action. The NLRHAs for these scenarios did not converge due

to the large base displacement demand that results from the long period and especially the

low shear resistance.


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7.3.14 Rocking rotations at the base of the rocker, 𝜃rock

Figure 7.8 shows the mean values of the maximum rotations of the rocker due to the rocking

action. Firstly, these diagrams reaffirm that the rocking rotation that happens at the base of

high-rise buildings is typically as small as 0.5% to 2%. Tilting of the typically squat rocker

at this amplitude will barely cause a reduction to the recentering moment during the rocking

motion. This justifies the constant activation moment that was assumed for the rotational

spring in Section 7.3.2.

Within this range of the rocking amplitude, the peak rotation decreases as the fundamental

period gets longer and increases as the moment reduction factor RM increases. However,

both these two trends become less significant when T1 reaches the higher values in the

range of periods. As a result, for any T1 ≥ 4 sec, the rocking rotation is insensitive to the

variation of RM; and, for any given RM, the rocking rotation basically does not vary when

T1 is as long as 6 or even 7 sec.

In any given scenario of (T1, RM), the rocking rotation reaches its lowest amplitude at μV =

0 which leads to the shear mechanism attracting most of the lateral deformation demand of

the overall system. When μV is increased just up to 0.2, a jump can be observed in most of

the rocking rotation curves, which indicates that an effective engagement of the rocking

action alleviates the deformation demand on the shear mechanism. As μV continues to rise,

the rocking rotation keeps increasing as well but at a much reduced rate. The sensitivity of 𝜃rock to the variation of μV is greater for buildings with relatively shorter periods. The

rocking rotation is essentially unaffected by the initial stiffness of the shear mechanism

except when T1 = 1 and 2 sec, the impact of I(Kb1) on 𝜃rock is just noticeable.


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Figure 7.8 Rocking rotations at the base of the rocker


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7.3.15 IDRs, 𝛿s

As aforementioned, IDRs of rocking structures are partially contributed by the rocking

rotation at the base of the structure. For comparison purposes, 𝛿s and 𝜃rock are plotted

together for each given T1 and RM, as shown in Figure 7.9. It is clearly seen that the variation

of IDRs with the control parameters basically follow similar patterns that are observed for

the rocking rotations. However, IDRs are greater, leading to a gap between the solid and

dashed lines, as can be seen in Figure 7.9. This difference is due to the deformation of the

superstructure. As can be seen, for any given RM, IDRs due to structural deformation are

the greatest when T1 = 4 sec and become smaller under shorter and longer periods. For any

given T1, IDRs due to structural deformation decrease with the decreasing RM. In each

scenario of (T1, RM), increasing μV leads to greater IDRs that are due to the structural

deformation.


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Figure 7.9 Inter-storey drift ratios of the generic buildings


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7.4 Design Charts and Design Recommendations

7.4.1 Design charts

Based on the results of the parametric analyses, design charts are produced as shown in

Figure 7.10. In this figure, base displacement demands and IDR demands are plotted

together for each varied combinations of the fundamental period T1 and the moment

reduction factor RM. In each chart, the ascending curves are for IDRs while the descending

curves are for base displacements. While these two response quantities share the same

horizontal axis, which is the shear reduction factor μV, the left vertical axis is read for the

base displacements and the right axis for the IDRs. In each chart, the impact of the initial

stiffness of the shear mechanism is also presented using different line styles as indicated in

Figure 7.10. Based on these design charts, design procedures are recommended in the

following section.


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Figure 7.10 Preliminary design charts based on IDRs and base displacements


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7.4.2 Recommended design framework

Step 1 – Determine T1

At the beginning of a new design project, limited information is known about the potential

development. At this moment, an approximate fundamental period T1 can be estimated

using empirical relations based on the structural height or the number of storeys. These

basic building properties can usually be provided by the client or the architect. If an initial

numerical model of the structure becomes available as the design proceeds, T1 can also be

obtained by conducting a modal analysis using commercial software.

Step 2 – Determine RM

For conventional fixed-based structures, a seismic force reduction factor needs to be chosen

based on the deformation capacity the structural system and the overstrength at the material

and system levels. However, for the rocking mechanism in the MechRV3D system, the

selection of the moment reduction factor RM is not restricted by the ductility or material

properties. As found in Sections 7.3.14 to 7.3.15, for a given T1, the peak response of base

displacements, rocking rotations and IDRs do not vary significantly with different RM.

This being said, it is usually preferred to maintain a minimum flexural strength under

frequent earthquakes especially in the design of high-rise buildings. For this purpose, RM

can be set to be equal to the ratio of the elastic base overturning moments, Mb,MCE and

Mb,SLE, that are obtained at the MCE and SLE levels respectively. These two moment

demands can be obtained either using an equivalent static method or through elastic modal

response spectrum analysis if this is practical. For both technical routes, seismic inertias

would be required. This can be estimated from the architectural specification on building

usage and gravity load allowance.

In case the design is anticipated to be governed by wind loads, both the wind-induced base

overturning moment demand, Mb,wind, and Mb,SLE shall be considered to calculate RM as

𝑅 = 𝑀 , 𝑀 ,⁄ (7.13)

where Mb,min = max [Mb,SLE, Mb,wind].


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Step 3 – Determine the required rocking moment, Mrock

Given the chosen RM, the required rocking moment is

𝑀 = 𝑀 , 𝑅⁄ (7.14)

Step 4 – Specify acceptance design criteria, [𝛿s] and [𝛥b]

A maximum acceptable IDR at the MCE level, [𝛿s], can be set according to code provisions

and alternative design guidelines.

At the same time, a limit of the base displacement, [𝛥b], needs to be specified at the same

hazard level. This will depend on the geometry and configuration of the BRBFs. If the

space is anticipated to be sufficient in the basement to accommodate taller BRBFs, a higher

[𝛥b] can be accommodated.

Step 5 – Choose a design chart

An appropriate design chart can be chosen from Figure 7.10 based on the preliminarily

determined T1 and RM.

Step 6 – Identify acceptable design zone

Mark the specified acceptance criteria [𝛿s] and [𝛥b] on the chosen chart by drawing

horizontal lines, as shown in Figure 7.11. Find the interval μV |[𝛥b] that leads to 𝛥b being

no greater than [𝛥b] and the interval μV|[𝛿s]that leads to 𝛿s being no greater than [𝛿s].

Identify the common area of these two μV intervals. Any μV within this range provides an

acceptable design, as illustrated in Figure 7.11 (a) . In this selection process, the 𝛥b-curve

and the 𝛿s-curve must be based on the same I(Kb1), meaning that they are plotted in the

same line type (solid, dashed, or dotted).


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Figure 7.11 Acceptable design area

Step 7 – Choose μV and I(Kb1)

Within the identified acceptable design zone, designers have the flexibility in choosing

different values for μV and I(Kb1) considering the trade-off between the IDRs and the base

displacements and other design constraints.

Step 8 - Calculate the initial stiffness of the shear mechanism, Kb1

Based on the chosen I(Kb1), the required initial stiffness of the shear mechanism can be


𝐾 = (2𝜋)𝐼(𝐾 )𝑇 𝑊𝑔 (7.15)

where WEQ is the total seismic weight of the structure, 𝑔 is the gravitational acceleration.

Step 9 - Calculate Vy and 𝛥y of the shear mechanism

Based on the chosen μV, the required yielding strength of the shear mechanism can be


𝑉 = 𝑀𝐻 + 𝜇 𝑉 , − 𝑀𝐻 (7.16)


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where, Mrock = Mb,MCE/RM. Vb,1M0V in this equation is the base shear demand that is expected

at the onset of the rocking action in the rocking-only scenario and can be estimated using

the chart for (R, T) = (10-6, 106) in Figure 6.12 in Chapter 6.

Accordingly, the yielding displacement of the shear mechanism can be calculated as

follows,

Δ = 𝑉𝐾 (7.17)

Step 10 – Determine Vy,BRBF and hc

In order to determine the required shear strength for the BRBFs, Vy,BRBF, it is necessary to

modify Vy and Kb1 that are obtained above to allow for the P-𝛥 effect induced by the rolling

mega-columns which are not accounted for in the current parametric analyses. For this

purpose, the following equations need to be solved,

𝑉 = 𝑉 , − 0.5𝑊 ℎ⁄ Δ (7.18)

𝛼 𝐾 = 𝐾 , − 0.5𝑊 ℎ⁄= 𝑏 𝐾 , − 0.5𝑊 ℎ⁄= 𝑏 𝑉 , Δ⁄ − 0.5𝑊 ℎ⁄

(7.19)

In these equations, K1,BRBF and K2,BRBF are respectively the initial and post-yielding stiffness

of the BRBFs. bBRBF, equal to K2,BRBF/K1,BRBF, is the post-yielding stiffness ratio of the

BRBFs and can be set based on experimental tests or empirically assumed in the

preliminary design stage. hc is the height of the rolling columns. The term 0.5WEQ/hc in

Equations (7.18) and (7.19) gives the negative stiffness induced by the rolling columns

under the tributary gravity load equal to 50%WEQ. Vy,BRBF and hc are the unknown variables

in the above two equations and can be solved using trial-and-error. These equations and all

the variables involved are illustrated in Figure 7.12.


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Figure 7.12 V-𝛥 relations of the overall lateral response and the BRBFs

Step 11 – Design the BRBFs

Choose hBRBF, the height of the BRBFs, and 𝛼BRB, the inclined angle of the BRB, such that

Δℎ = 2𝜆 𝜀 ,sin 2𝛼 (7.20)

where λBRB is the yielding length ratio of the BRBs, εy,BRB is the yielding strain of the steel

core of the BRBs. These two quantities needs to be determined based on the BRB products

and can be empirically assumed in the preliminary design.

The cross-section area of the yielding segment of one single BRBs can be calculated as,

𝐴 , = 𝐾 , 𝐿𝑛 (𝐾𝐹)(𝐸 )(cos 𝛼 ) (7.21)

Where nBRBF is the number of BRBFs arranged in each principal direction of the structure

and can be determined according to the space that is available in the basement to

accommodate the BRBFs. Lwp is the work-point length of the BRBs. KF is the axial

stiffness modification factor. Es is the Young’s Modulus of the steel used for the yielding

segment of the BRBs. The initial stiffness of the BRBFs can be calculated as follows,

𝐾 , = 𝑉 , Δ⁄ (7.22)

At this point, a preliminary design of the shear mechanism is completed.


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Step 12 – Determine the distance between the rolling mega-columns

Based on the Mrock determined in Step 3, the centre-to-centre distance between the rolling

mega-columns can be calculated using Equation (5.2) in Chapter 5 which is rewritten here,

𝑑 = 2𝑀 0.5𝑊⁄ (7.23)

Step 13 – Design the axial compression load resistance of the rolling mega-columns

Based on the column height, hc, which has been determined in Step 10, the cross-section

of the rolling mega-columns can be designed to achieve the required axial compression

load resistance. This resistance should be no lower than the gravity load that is tributary to

the RC core, since this load demand is expected to occur in the most critical case in which

the superstructure pivots atop one single column. In the design of the mega-columns, a

safety margin should be reserved to allow for an increased demand due to the vertical

earthquake response. This vertical dynamic amplification can be either estimated using the

empirical equation recommended in [LATBSDC 2020] and [PEER 2017] or by conducting

vertical response spectrum analyses. The axial strength design of the mega-columns

follows a conventional procedure and is not expanded upon here.

7.4.3 Other design considerations

Shear transmitters

Based on the design of the BRBFs, the shear transmitters between the skirt diaphragm and

the rocker can be designed such that these connections remain elastic when the maximum

possible shear strength is developed in the BRBFs. This expected strength shall be

determined based on the cross-sectional area that is actually provided to the BRBs.

Beams and columns in the BRBFs

Based on the expected strength of the BRBs, the beam and columns in each of the BRBFs

shall be capacity designed accordingly. As for the columns, the design axial force shall

include the ground floor load that is tributary to these elements and limited vertical forces

that may be induced by the shear transmitters from the inner edge of the skirt. These two

portions of demand shall be combined with the vertical component of the BRB capacity.


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7.4.4 Validation of the preliminary design

After having obtained the preliminary sizes and properties of the MechRV3D system, the

superstructure can be capacity designed such that both the flexural and shear responses of

the RC core remain elastic when the expected rocking moment and lateral strength are

developed in the rocking and shear mechanisms at the base of the structure.

Numerical models can then be built based on the design of the superstructure and the

preliminary design of the base mechanisms. NLRHAs can be conducted on the integrated

model of the overall system in order to verify the preliminary design. In these analyses,

vertical ground motions could be included along with the bidirectional horizontal

excitations. Results of these analyses provide more accurate predictions on the governing

seismic responses especially the base displacements and inter-storey drift ratios. These

results can be compared with the estimated values that are obtained from the design charts.

Adjustments to the design of the MechRV3D system can be made accordingly.

7.4.5 Lock-up devices

Following the recommended design procedures, the expected seismic performance can be

essentially achieved under the MCE level earthquakes. This being said, for very rare

seismic event that exceeds this level of ground shaking, lock-up mechanisms are

recommended to be integrated with the proposed system so as to allow for a last resort of

protection. For the shear mechanism, bumper walls can be provided a distance away from

the outer edge of the skirt diaphragm. During extreme earthquakes, excessive lateral

movements of the overall structure can be prevented once the sliding of the skirt diaphragm

is stopped by these bumpers at the ground level.

As for the rocking mechanism, pipe-pin rolling joints can be carefully detailed such that

the dowel pipes cannot be pulled out from the soffit of the rocker. However, it is still

beneficial to introduce tension-resistant devices at the rocking surface. These devices can

be designed not to be engaged until an unexpected uplifting occurs. As such, prior to the

engagement of these lock-up devices, the rocking mechanism act as originally intended.


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7.5 Summary

This chapter further generalized the possible application of the proposed MechRV3D

system to generic RC core-wall buildings of different heights. Nonlinear parametric

analyses were conducted to investigate the impact of a key group of design parameters on

the governing response of the superstructure and critical deformations in the base

mechanisms. NLRHAs were substantially conducted on representative high-rise buildings

considering the selected design parameters. Findings obtained from these analyses were

discussed in detail.

Based on the results of the NLRHAs, preliminary design charts were developed in order to

facilitate the preliminary design of the MechRV3D system. Based on these design charts,

a preliminary design methodology was defined with steps by which initial sizes can be

quickly selected for the proposed system as a reasonable starting point for the design.

In these parametric analyses, the rolling mega-columns in the MechRV3D system were not

explicitly included in the numerical model that was used for the parametric analyses. This

leads to a displacement demand at the base of the superstructure that may be

underestimated by as much as 10% - 15%. In addition, complex modifications are required

in order to allow for the impact of the negative stiffness introduced by the rolling columns

such that properties of the BRBFs can be determined properly.

The height of the gravity columns in the basement was assumed constant and equal to 12

m over the height of the basement storeys. However, in practical design, the total height of

the basement will definitely vary. Shorter basements, would lead to larger the P-𝛥 effects

produced by the gravity columns.

Regarding the moment reduction factor RM, the current chosen values may not present a

range that can cover the minimal strength requirement determined by wind loads,

especially when wind loads likely govern the design of high-rise structures in some regions.

In addition, only a single value of the post-yielding stiffness ratio that was deemed

representative was considered while this parameter may also vary as a result of a strain-

hardening and other factors affecting the shear yielding mechanism’s response.


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It is assumed that the above simplifications, which enhanced the computational efficiency

of the parametric study, are acceptable for preliminary design. Further studies into their

effects are needed in the future. In addition, when these preliminary design procedures are

further developed into a full performance-based design methodologies, the validity of this

preliminary design approach need to be verified more thoroughly.

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Chapter 8 Conclusion

8.1 Introduction

This chapter provides an overall review of the work that has been discussed in the previous

chapters. Section 8.2 summarizes the research motivation and background as well as the

methodologies and findings of each chapter. Section 8.3 reviews the major conclusions and

original contributions of this dissertation. Section 8.4 summarizes recommendations for

future research.

8.2 Summary

Accelerating urbanization has created a critical challenge of housing a soaring number of

residents in city areas where land scarcity is a common problem. This leads to the record

high-rise building development that is being observed worldwide in terms of increased

numbers and height of buildings. Many of these new constructions are located in seismic

regions. In this context, an urgent demand arises for disaster resilient structures given a

broad awareness about the shortfalls of current seismic design philosophies in preventing

damage.

Aiming at low-damage design of high-rise buildings, numerous high-performance systems

have been developed as reviewed in Chapter 2. Different in configurations, these systems

aim to desensitize structures to the effects of seismic excitations. In some systems, this is

achieved by allowing structures to rock at the base, while in others, a shear fuse is

introduced at the base using isolators. In terms of efficiency in mitigating higher-mode

response, base isolation outperforms rocking systems due to the former’s versatility in

limiting all modes particularly the high-frequency ones. This advantage provides a

justification for base-isolating high-rise structures despite an inconspicuous effect on the

period shift of the isolated structure. However, significant overturning moments at the base

of high-rise buildings create significant challenges in the design of these isolators.

Modifications were proposed to address this problem by allowing base-isolated high-rise


257

buildings to uplift either at the base of the superstructure or at the bottom of isolators.

However, the isolators still have to carry significant tensile forces prior to the rocking

action and compression loads that are unreduced whatsoever or even increased as a result

of the redistribution of gravity loads from the lifted isolators to those remaining in contact.

This situation results from the in-series arrangement of the rocking and shear mechanisms.

This thesis focused on the development of a novel dual-mechanism system that allows for

the uncoupling of the flexural and lateral responses at the base of high-rise buildings. The

proposal of this uncoupled system came after an intensive exploration during which a series

of innovative concepts were devised and assessed critically as discussed in Chapter 3. This

concept evolved from configurations where the two mechanisms are still interacted to

systems where the intended uncoupled flexure and shear mechanics are achieved. The

system was then further defined to ensure full equilibrium and allow for three-dimensional

motions. Concepts of multi-phased rocking and wobbling were also explored.

An idealized configuration of the MechRV3D system was first investigated numerically

through nonlinear static and time-history analyses, as demonstrated in Chapter 4. Then this

prototype system was further developed into a possible physical embodiment where the

intended mechanism components are implemented using structural elements that have been

widely used in practice. During this schematic design stage, a variety of structural options

were proposed and examined conceptually and numerically for both the rocking and shear

mechanisms. Special detailing was also developed for key connections.

The feasibility of the proposed implementation was numerically validated in Chapter 5

through substantial NLRHAs. In this validation, a 42-storey benchmark RC core-wall

building that has been studied in the PEER TBI project was used as a reference structure.

Advanced nonlinear models were built for this benchmark building, and innovative

modelling techniques were proposed for simulating the newly proposed base mechanism.

Seismic responses of the benchmark building were investigated for varied scenarios in

which the MechRV3D system was introduced at the base of the structure in a rocking-only

format or with the dual mechanism fully engaged.


258

To generalize the findings of the benchmark numerical study, an analytical study was

carried out in Chapter 6 using a continuum cantilever beam analogy. This cantilever beam

was elastically supported in both the rotational and translational directions at the base, with

the fixity in these two constraints varying from a fixed to a fully released condition. This

simulates the range of possible boundary conditions that the MechRV3D system provides

at the base of structures during a nonlinear seismic response. Modal analyses were

conducted on a group of generally supported cantilever systems, providing results on modal

properties and varied response quantities.

Guided by the theoretical framework established in Chapter 6, nonlinear parametric

analyses were conducted in Chapter 7 on a set of generic RC core-wall buildings with

fundamental periods ranging from 1 to 7 sec. Control parameters were chosen for this

parametric study, depicting the characteristic properties of the MechRV3D system. Based

on these parametric analyses, design charts and procedures to facilitate the preliminary

design of the proposed system were proposed.


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8.3 Conclusion

8.3.1 Uncoupled flexural and shear responses

Compared with fixed-based systems, the concepts of base rocking and shear fuses are

unconventional but not new, since they have been widely used in historical structures and

modern buildings. Hence, the primary novelty of the MechRV3D system is not the use of

these concepts but the achievement of uncoupled mechanics through which the flexural

and shear responses are separated behaviourally at the base of structures. This makes it

feasible to control these critical responses independently at a component level through

varied structural elements or mechanical actions. This is in contrast with conventional RC

structures in which flexure and shear are innately intertwined at the material level.

The MechRV3D plays the role of the designated softening system, allowing for a strength

hierarchy in flexure and shear at the base of high-rise buildings and thereby limiting seismic

demands that will be otherwise imparted into the superstructure particularly through higher

vibration modes. The flexural and shear mechanisms of this system can be implemented in

varied ways as long as the desired strengths and movement/deformation capacities are

ensured. Based on expected flexural and lateral strengths, the superstructure is capacity

designed following regular principles. However, what differs from conventional concrete

design is that, having the MechRV3D system incorporated, both the flexural and shear

responses of the superstructure are designed to remain nearly elastic under major

earthquakes. As such, this modularized base softening system can be applied to high-rise

buildings with varied structural systems.

8.3.2 Practice-oriented design of the proposed system

Varied design possibilities were extensively investigated in Chapter 4 to achieve a realistic

implementation of the rocking and shear mechanisms. When a design scheme was turned

down or adopted, the criteria were not only related to the structural efficiency, but also

included considerations about the buildability, design convenience, and commercial

availability of the components comprising the system. This practice-oriented design led to

the proposed rocking and shear mechanisms that are primarily built using conventional

structural elements but integrated and engaged in an unconventional way.


260

The importance of the practicality was also a driving consideration during the detailed

design stage. Following this principle, the rolling pipe-pin joints were proposed to allow

for and regulate the concurrent mechanical motions (uplifting, rolling, and sway) at the

same time at both ends of the mega-columns. As for the shear transmitters, the transfer of

forces and motions were ensured using a typical mechanical connection (gear connections)

and also by a structural solution (hinged steel plates) proposed as an alternative.

8.3.3 Validation of the MechRV3D system

The feasibility of the MechRV3D system was validated by comparing the seismic

performance of the benchmark building with and without the proposed system incorporated

at the base. In the fixed-based scenario, the specified performance objectives are all met

and no responses exceed the acceptance criteria of the conventional design. However,

unintended plastic hinges in upper storeys and significant inelastic rotations in coupling

beams are observed. This indicated conspicuous higher-mode effects despite the formation

of the designated yielding mechanisms in coupling beams and at the base of RC walls.

When the rocking mechanism is introduced alone, the inelastic responses that are

dominated by the first flexural mode are dramatically reduced. Nevertheless, the higher-

mode effects are nearly unaffected. Furthermore, as the seismic excitation intensifies, these

higher-mode dominated inelastic deformations increase significantly even though some

saturation is observed in the force-related responses. These phenomena reaffirmed the

inadequacy of a base flexural mechanism, regardless if it is a plastic hinge or a rocking

joint, in limiting higher-mode responses, particularly in the inelastic range.

In comparison, when the shear mechanism is engaged at a proper strength level (which was

determined as 0.6V1M0V), plastic hinges are essentially eliminated from the RC walls, chord

rotations of the coupling beams are significantly reduced under a repairable level. In

addition, inter-storey drift ratios and floor accelerations are also largely reduced. At the

same time, the deformation demands in the base mechanisms are acceptable. This shear

strength level is deemed as an optimal design specifically for the benchmark building.


261

During the numerical validation, it was challenging to capture the three-dimensional rolling

motions of the spherically capped mega-columns. In this study, an advanced modelling

technique was proposed to simulate the rolling surface using zero-length fibre sections.

This modelling approach was verified against theoretical solutions before it was

incorporated into the overall mechanism model.

8.3.4 Theoretical study on generally supported cantilever systems

The analytical study discussed in Chapter 6 provided extended insights that helped better

understand the sensitivity of modal properties and responses to the variation of the base

fixity in both the rotational and translational degrees-of-freedom. While details were

discussed in Chapter 6, a few general conclusions are drawn here.

Firstly, as the base rotational and translational constraints are gradually relaxed, the seismic

mass of the structure tends to rapidly concentrate into the first or the first two modes,

leading to the desired cumulative participation mass ratio (90%) being satisfied by fewer

modes.

Softening the translational constraint is more effective in limiting the response in all the

vibration modes but particularly the higher-mode contributions. In contrast, relaxing the

rotational fixity only affects those responses that are dominated by the first mode, for

example the top deflection and the base overturning moment of the cantilever.

It is also found that modal contributions to varied response quantities (deflections, rotation

angles, overturning moments, and shear forces) are not only affected by the stiffnesses of

the rotational and translational constraints, but also sensitive to their relative magnitudes.

Particularly when the non-dimensional rotational stiffness R is comparable with or smaller

than the translational counterpart T, the second-mode contribution can be considerable for

rotation angles and dominant for the base displacement. For the overturning moments and

shear forces, some counterintuitive phenomena were also observed in these combinations

of (R, T) as explained in Chapter 6 in detail.


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8.3.5 Design framework

Nonlinear parametric analyses presented in Chapter 7 demonstrated the variation of

governing response quantities with selected design parameters. According to the analysis

results, the trade-off between the base displacement demand and the inter-storey drift ratios

of the superstructure are demonstrated. For any given combination (T1, RM), allocating

more deformation demand into the shear mechanism will lead to a reduced deformation in

the superstructure, and vice versa. Based on this correlation, design charts were developed

based on these two response quantities. Design procedures were also recommended to

reach a preliminary design of the proposed system.

8.3.6 Original contributions

In sum, original contributions have been presented on a number of different aspects in this

dissertation. Firstly, this study proposed an unconventional system that is characterized by

the uncoupled mechanics at the base of high-rise buildings. This uncoupled system

provided a modularized framework to achieve an independent control of seismic flexural

and shear responses and an efficient mitigation of higher-mode effects of slender structures.

As a result, a new direction is created for the seismic design of high-rise buildings towards

achieving low-damage and resilient structures.

The physical embodiment of the MechRV3D displays novelty from a practical perspective

as well, especially the innovative rolling pipe-pin joint that was proposed in this study.

Prioritizing practical implementation, multiple innovative concepts were developed to

achieve the desired seismic protection by using existing elements and technologies in a

new way that leads to enhanced structural efficacy.

On the analysis front, novel work in this research included the analytical study that was

conducted on generally supported cantilever systems. Previous studies of this kind only

considered release in the rotational constraint only. By accounting for both rotational and

translational flexibility, this dissertation provided a more general understanding and further

insights on higher-mode effects in slender structures that respond in the nonlinear stage. In

addition, the modelling technique proposed for simulating three-dimensional rocking and


263

rolling motions provided a numerical solution that balances accuracy and computational

efficiency.


264

8.4 Limitations and Recommendations for Future Research

There are limitations in this study, as have been highlighted in the previous chapters. These

aspects are summarized here, with additional suggestions for future research.

8.4.1 Numerical analysis

In the numerical analyses used in this dissertation, components of the MechRV3D system

were simulated using skeleton models; for example the rocker and the rolling mega-

columns. In addition, contacting joints were represented using nonlinear springs rather than

solid elements that reflect actual physical shapes. While these models reliably

demonstrated the intended mechanics of the proposed system and the global behaviour of

the overall system, more sophisticated modelling is needed in future research to further

verify the local response of the proposed system.

This modelling need can be satisfied by using finite element analyses that involve solid

elements and dedicated contact elements. By these means, higher accuracy can be achieved

when investigating stress distributions that are anticipated as being critical at the rolling

pipe-pin joints and the shear transmitters (the gear connections or hinged plates). Following

a similar approach, the rocker can be modelled in more detail to account for the

deformability of the concrete block, the reinforcing effect of the embedded steel truss, and

the impact of post-tensioned strands. Results of these analyses can facilitate the detailed

design of the aforementioned components.

In future numerical analyses, the impact of vertical ground excitations needs to be

investigated in more detail. This can be achieved by conducting response spectrum

analyses using a code-specified vertical spectrum, or, more accurately, by carrying out full

dynamic time history analyses in which ground motions are input in both horizontal

directions and in the vertical direction. In the latter case, pulse-like ground motion records

shall be involved to reflect near-fault effects which may induce more significant vertical

amplification.

In addition, in extended analyses, ground motions representing varied seismicity should be

considered to investigate the sensitivity of MechRV3D-incorporated high-rise buildings to


265

seismic excitations enriched by different frequency contents such as higher frequency

content that characterizes Eastern North American seismicity.

8.4.2 Experimental validation

At the current stage, the feasibility of the MechRV3D system has been validated only

numerically through nonlinear static and time history analyses that were conducted on both

the idealized configuration and the physical embodiment of the system. In the next stage

of this system’s development, large-scale experimental tests are needed to further verify

the numerical modelling, confirm the seismic response of the proposed system, and

investigate the three-dimensional kinematics of the dual mechanism in depth.

For these purposes, shaking-table tests would be advantageous over static and pseudo-

dynamic experiments, since the higher-mode response of the superstructure can be

accounted for under dynamic loading conditions. In fact, shaking-table tests can be more

reliable for the MechRV3D system in which most of the mechanism components are

expected to act in the elastic range as rigid bodies.

Considering space limitations in laboratories, hybrid simulation techniques will be

indispensable for testing the proposed system which is intended for high-rise structures. As

such, while models of the base mechanisms must be physically built, the dynamic response

of the superstructure can be captured numerically. To facilitate a smooth communication

between these two modules, a reliable interface protocol is needed. It can be anticipated

that hybrid shaking-table tests can be particularly efficient for the proposed system since

the superstructure is expected to remain essentially elastic provided that it is capacity

designed properly according to the expected strengths of the base mechanisms. As a result,

the required computational efforts may be more reasonable for completing the nonlinear

iterations in the integration module.

8.4.3 Development of design procedures

The design procedures recommended in Chapter 7 are essential steps that allow for a quick

selection of properties for the MechRV3D system as a reasonable starting point in the

preliminary design. These procedures can be further extended to further aid the design by


266

replacing the simplifications that were made in the parametric study with more realistic

considerations. For example, the rolling mega-columns in the MechRV3D system can be

explicitly included in the numerical model that was used for the nonlinear parametric

analyses. As such, the P-𝛥 effect induced by these columns will be directly accounted for,

leading to enhanced accuracy of the design charts.

New parameters may be introduced so as to cover more design possibilities that are relevant

in practice. For instance, the height of the basement that was assumed constant in the

parametric analyses may be taken as a variable, such that P-𝛥 effects induced by

underground gravity columns can be captured over a wider range of possible geometries.

In addition, a wider range of values can be considered for the post-yielding lateral stiffness,

which allows for some flexibility in implementing the shear mechanism using BRBFs with

different geometries, materials, or even other types of shear panels.

The current design recommendations focused on the seismic response at the MCE hazard

level, which usually governs the design. Further studies are needed to formulate the design

procedures in a performance-based design framework in which the seismic performance is

verified at multiple hazard levels. While multi-level performance objectives can be readily

determined for the superstructure based on available guidelines for the design of high-rise

buildings, specifications on the limit states that govern the MechRV3D system require

additional research including experimental verifications.

8.4.4 Further development of the MechRV3D system

As part of the longer-term plan, the application of the MechRV3D system can be extended

to other structural systems of high-rise buildings. This is theoretically viable since the

superstructure, regardless of its lateral-force-resisting system, can always be capacity

designed for the expected flexural and lateral strengths of the base mechanisms. This being

said, studies are needed for incorporating these base mechanisms into varied structural

systems in an integrated manner that is practical engineeringly and architecturally.

267

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