Seismic performance of new hybrid ductile-rocking braced

Seismic performance of new hybrid ductile-rockingbraced frame system

by

Justin Binder

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied ScienceGraduate Department of Civil Engineering

University of Toronto

© Copyright 2016 by Justin Binder

Seismic performance of new hybrid ductile-rockingbraced frame system

Justin Binder

Master of Applied Science

Graduate Department of Civil Engineering

University of Toronto

2016

A new hybrid ductile-rocking (HDR) seismic-resistant system is proposed which consists of

a code-designed buckling-restrained braced frame (BRBF) that is free to rock on its foundation.

The goal of this system is to reduce the disadvantages associated with BRBFs, such as excessive

drift concentrations and residual deformations, while maintaining their reliable limit on forces and

accelerations. A lockup device ensures the full code-compliant strength at a predetermined column

uplift, and supplemental energy dissipation elements reduce the overall response.

Buildings of 2, 4, and 6-storeys in height were designed for Los Angeles, California, and studies

were performed to investigate how the energy dissipation strength, lockup base rotation, and verti-

cal mass modelling choices affected the system’s performance and dynamic response. An example

detail was developed that included a cast steel energy dissipating device. These studies showed

that the HDR system achieved a significant reduction in brace damage over conventional BRBF

structures.

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Acknowledgements

I would like to thank Professor Constantin Christopoulos for serving as my supervisor, and pro-

viding support, motivation, and an inspiring vision for the future of earthquake engineering. I want

to thank Dr. Michael Gray, my industry supervisor at Cast Connex Corporation, for being an end-

less source of encouragement, motivation, and technical guidance, and for constantly encouraging

me to ”just do it!”. As well, my thanks go to Carlos De Oliveira and the rest of the Cast Connex

team for supporting this research and exposing me to incredible engineering projects. I am grateful

to Tarana Haque for listening to me practice my presentation and giving useful feedback.

I want to thank Professor Oh-Sung Kwon for his thoughtful review of this thesis, and my col-

leagues at the University of Toronto, in particular Deepak Pant, for helping to resolve challenging

technical problems.

My thanks go to Jacob Binder, Raquel Binder, Jake Yanowski, Giselle Hausman, David Gut-

stein, and my band Pudding for always being there for me.

Most importantly, I want to thank my parents, Sari and David Binder, for their love, wisdom,

and support.

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Contents

1 Introduction 1

1.1 Conventional ductile design versus base-rocking . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background and literature review 7

2.1 Buckling-restrained braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Seismic performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Base rocking structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Flag-shaped hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Early studies of rocking structures . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 The PRESSS program and rocking of concrete walls . . . . . . . . . . . . 14

2.2.4 Rocking steel systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.5 Higher mode effects in controlled rocking structures . . . . . . . . . . . . 19

2.2.6 Compatibility between rocking frame and rest of structure . . . . . . . . . 22

2.3 Combining BRBs and base rocking for improved performance . . . . . . . . . . . 23

2.3.1 Examples of combined self-centering and plastic systems . . . . . . . . . . 23

2.3.2 Proposed combined seismic system . . . . . . . . . . . . . . . . . . . . . 24

2.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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3 Mechanics of hybrid ductile-rocking 25

3.1 Mechanics of a rocking joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Modifying conventional structures to become a HDR system . . . . . . . . . . . . 28

3.2.1 Drifts from brace deformation and rocking . . . . . . . . . . . . . . . . . 28

3.2.2 Overview and mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.3 P �� effects in HDR systems . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.4 Residual drifts in HDR systems . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Reference structures 40

4.1 Modeling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Nonlinear time-history analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Selection and scaling of ground motions . . . . . . . . . . . . . . . . . . . 44

4.2.2 Dynamic modeling assumptions . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.3 Reference structure results . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Peak interstorey drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Peak floor displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Peak residual interstorey drifts . . . . . . . . . . . . . . . . . . . . . . . . 49

Peak storey shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Peak storey overturning moment . . . . . . . . . . . . . . . . . . . . . . . 52

Peak column compression . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Peak storey accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Parametric study 56

5.1 Overview of parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1.2 Modeling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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5.2 Response of example structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Pushover response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.2 Push-pull response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.3 Sample record . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Results of parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Peak interstorey drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Peak base rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Peak drift minus base rotation . . . . . . . . . . . . . . . . . . . . . . . . 76

Residual drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Residual base rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Global uplift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Number of records that engaged the lockup device . . . . . . . . . . . . . 82

Energy dissipated by buckling restrained braces . . . . . . . . . . . . . . . 85

Maximum foundation tension . . . . . . . . . . . . . . . . . . . . . . . . 87

Maximum column compressive force . . . . . . . . . . . . . . . . . . . . 89

Maximum base overturning moment . . . . . . . . . . . . . . . . . . . . . 89

Maximum base shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Maximum storey accelerations . . . . . . . . . . . . . . . . . . . . . . . . 94

Summary of parametric study conclusions . . . . . . . . . . . . . . . . . . 94

5.4 Investigation of vertical mass modeling on analysis results . . . . . . . . . . . . . 97

5.4.1 Literature review of rocking studies relevant to vertical mass modeling . . 97

5.4.2 Parametric study on vertical mass modeling . . . . . . . . . . . . . . . . . 101

5.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Example design of 6-storey BRB HDR structure with cast steel energy dissipation el-

ements 110

6.1 Cast steel yielding connector used as rocking fuse element . . . . . . . . . . . . . 111

Yielding brace system properties . . . . . . . . . . . . . . . . . . . . . . . 111

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6.2 Validation of cast steel material in OpenSees . . . . . . . . . . . . . . . . . . . . 112

6.3 Design of 6-storey HDR frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.1 Superstructure design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.2 Fuse and lockup properties . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.3 Modeling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.4 Pushover response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3.5 Push-pull response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3.6 Time-history results under a sample record . . . . . . . . . . . . . . . . . 119

6.3.7 Response of HDR structure to suites of records . . . . . . . . . . . . . . . 123

Peak interstorey drift and drift minus base rotation . . . . . . . . . . . . . 123

Peak floor displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Base rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Residual drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Energy dissipated by buckling restrained braces . . . . . . . . . . . . . . . 127

Maximum foundation tension . . . . . . . . . . . . . . . . . . . . . . . . 128

Maximum column compressive force . . . . . . . . . . . . . . . . . . . . 129

Maximum storey overturning moment . . . . . . . . . . . . . . . . . . . . 130

Maximum storey shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Maximum storey accelerations . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4 Preliminary Detail Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4.1 Overview of detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.2 Design forces and material properties . . . . . . . . . . . . . . . . . . . . 135

6.4.3 Energy dissipation elements . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4.4 Gusset plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.5 Base plate assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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7 Summary and conclusions 143

7.1 Background and literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2 Mechanics of the HDR system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.3 Parametric study on hybrid ductile-rocking BRBFs . . . . . . . . . . . . . . . . . 145

7.4 Detail design of 6-storey BRB HDR frame with cast steel fuse . . . . . . . . . . . 147

7.5 Framework for application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.6 Low-damage, economical seismic design . . . . . . . . . . . . . . . . . . . . . . 150

A Design of reference structures 159

A.1 Description of reference structures and preliminary design . . . . . . . . . . . . . 159

A.2 Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.3 Design of Structural Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B Push-pull analysis overview of building models for parametric study 176

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List of Figures

1.1 Comparison of seismic design philosophies . . . . . . . . . . . . . . . . . . . . . 2

2.1 CBF and BRB hystereses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Anatomy of a buckling restrained brace . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Example of failed BRB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Typical flag shaped hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Concrete bridge pier designed to step . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Moment frame with column uplift and energy absorbing devices . . . . . . . . . . 14

2.7 Yielding base plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Retrofitted bridge steel truss pier using controlled rocking approach . . . . . . . . 18

2.9 Test structure with bumper detail and shear fuse . . . . . . . . . . . . . . . . . . . 20

2.10 Experimental setup of controlled rocking steel frame with higher mode mitigation 21

2.11 Possible details for connecting rocking frame to diaphragm and collectors . . . . . 22

2.12 SCED brace setup for testing and hysteretic behavior . . . . . . . . . . . . . . . . 23

3.1 Rocking structure mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Contribution to system deformations from two ductile mechanisms . . . . . . . . . 28

3.3 Mechanics of hybrid ductile-rocking under smaller amplitude displacements . . . . 31

3.4 Mechanics of hybrid ductile-rocking under larger amplitude displacements . . . . . 32

3.5 P-Delta effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Effect of loading history on residual drifts in conventional and HDR structures . . 38

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4.1 Plan and elevation of reference structures . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Calibration of BRB OpenSees model to test results . . . . . . . . . . . . . . . . . 42

4.3 Schematic of Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Summary of ground motion suite scaled to DBE . . . . . . . . . . . . . . . . . . . 47

4.5 Acceleration and displacement spectra of scaled ground motion suite . . . . . . . . 48

4.6 Peak interstorey drift results for reference structures . . . . . . . . . . . . . . . . . 48

4.7 Peak floor displacement results for reference structures . . . . . . . . . . . . . . . 49

4.8 Residual drift results for reference structures . . . . . . . . . . . . . . . . . . . . . 50

4.9 Storey shear results for reference structures . . . . . . . . . . . . . . . . . . . . . 51

4.10 Overturning moment results for reference structures . . . . . . . . . . . . . . . . . 52

4.11 Peak column compression results for reference structures . . . . . . . . . . . . . . 53

4.12 Acceleration results for reference structures . . . . . . . . . . . . . . . . . . . . . 54

5.1 Photos of Yielding Brace System . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Schematic of rocking joint modeling assumptions . . . . . . . . . . . . . . . . . . 61

5.3 Pushover response of 6-storey fixed base frame and 6-storey frame with ✓lock = 1%

and � = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Push-pull response of 6-storey fixed base frame and 6-storey framewith ✓lock = 1%

and � = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5 Foundation element behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 Roof displacement and system forces of example model compared to fixed base

response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.7 Hysteretic response of 6-storey fixed base and HDR frames . . . . . . . . . . . . . 69

5.8 Median peak interstorey drift results . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.9 Median drift profiles for 6-storey structure, � = 0 . . . . . . . . . . . . . . . . . . 73

5.10 Median peak base rotation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.11 Median peak interstorey drift minus base rotation results from parametric study . . 77

5.12 Median drift minus base rotation profiles for 6-storey structure, � = 0 . . . . . . . 78

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5.13 Median peak residual drift results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.14 Residual base rotation results from parametric study . . . . . . . . . . . . . . . . . 81

5.15 Median peak global uplift results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.16 Number of records that engaged the lockup device . . . . . . . . . . . . . . . . . . 84

5.17 Median total energy dissipated by braces results . . . . . . . . . . . . . . . . . . . 86

5.18 Median peak foundation tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.19 Median peak first storey column compression results . . . . . . . . . . . . . . . . 90

5.20 Median peak base overturning moment results . . . . . . . . . . . . . . . . . . . . 91

5.21 Base moment example results for 2, 4, and 6-storey frames with � = 1.0 and

✓lock = 1.0%, record ID#1 scaled to DBE . . . . . . . . . . . . . . . . . . . . . . 92

5.22 Median peak base shear results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.23 Median peak storey acceleration results . . . . . . . . . . . . . . . . . . . . . . . 95

5.24 Examples of column force spikes in literature . . . . . . . . . . . . . . . . . . . . 99

5.25 Schematic of vertical mass modeling assumptions . . . . . . . . . . . . . . . . . . 102

5.26 2-storey response with different vertical mass modeling assumptions . . . . . . . . 104



5.29 Peak drift profiles for 2, 4, and 6-storey structures with different vertical mass mod-

eling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1 Cast steel yielding fuse numerical model calibration . . . . . . . . . . . . . . . . . 112

6.2 Pushover response of 6-storey fixed base frame and 6-storey frame with cast steel

fuse and ✓lock = 1% and � = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Push-pull response of 6-storey fixed base frame and 6-storey frame with cast steel

fuse and ✓lock = 1% and � = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4 Foundation element behaviour for model with YBS fuse . . . . . . . . . . . . . . 120

6.5 Roof displacement and system forces of HDR example design with cast steel fuse

compared to fixed base response . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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6.6 Hysteretic response of 6-storey fixed base structure and HDR structure with cast

steel fuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.7 Peak interstorey drift results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.8 Peak floor displacement results for fixed base and HDR frames . . . . . . . . . . . 126

6.9 Residual drifts results for fixed base and HDR structure . . . . . . . . . . . . . . . 128

6.10 Peak column compression results for 6-storey fixed base and HDR frames . . . . . 129

6.11 Peak storey overturning moment results for fixed base and HDR frames . . . . . . 130

6.12 Peak storey shear results for fixed base and HDR frames . . . . . . . . . . . . . . 131

6.13 Peak storey acceleration results for fixed base and HDR structures . . . . . . . . . 132

6.14 Overview of HDR column-foundation detail . . . . . . . . . . . . . . . . . . . . . 133

6.15 Overview of proposed construction sequence for HDR column-foundation detail . 134

6.16 Cast steel energy dissipating supplemental fuse for HDR column-foundation con-

nection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.17 Gusset plate and column detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.18 Base plate assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.19 Components of base plate assembly drawings . . . . . . . . . . . . . . . . . . . . 140

A.1 Mode shapes from SAP2000 modal analysis . . . . . . . . . . . . . . . . . . . . . 168

A.2 Unbalanced force on beams in BRBFs . . . . . . . . . . . . . . . . . . . . . . . . 172

B.1 Push-pull response of 2 storey structures for � = 0, � = 0.5, and � = 1.0 . . . . . 177

B.2 Push-pull response of 2 storey structures for � = 1.2, � = 1.4, and � = 1.6 . . . . 178





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List of Tables

4.1 Reference frame elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Material parameters for nonlinear buckling restrained brace elements . . . . . . . . 42

4.3 Earthquake records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Parameters for parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Yielding brace system device characteristics . . . . . . . . . . . . . . . . . . . . . 58

5.3 Value of ED for each building model . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Ratio of rocking moment to yield moment . . . . . . . . . . . . . . . . . . . . . . 60

5.5 Total energy dissipated by braces in fixed base structures . . . . . . . . . . . . . . 85

5.6 Summary of first storey response for vertical mass parametric study . . . . . . . . 107

6.1 YBS fuse geometric parameters for OpenSees calibration . . . . . . . . . . . . . . 113

6.2 YBS fuse material modeling parameters for OpenSees calibration . . . . . . . . . 113

6.3 Example design fuse properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4 Summary of statistical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.5 HDR structure lockup engagement . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.6 Connection design forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.7 Material properties for detail design . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.2 Seismic loading parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.3 Gravity loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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A.4 Live loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.5 Effective seismic weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

A.6 Determination of approximate fundamental period . . . . . . . . . . . . . . . . . . 163

A.7 Calculation of seismic base shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

A.8 Lateral seismic force calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.9 Modal analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.10 Response spectrum analysis parameters . . . . . . . . . . . . . . . . . . . . . . . 169

A.11 Storey deflection and drift calculations . . . . . . . . . . . . . . . . . . . . . . . . 169

A.12 Buckling restrained brace parameters . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.13 BRB design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.14 Column sections, design forces, and resistances . . . . . . . . . . . . . . . . . . . 172

A.15 Forces and resistances of beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A.16 BRB frame forces and beam calculations . . . . . . . . . . . . . . . . . . . . . . . 175

xiv

Chapter 1

Introduction

1.1 Conventional ductile design versus base-rocking

Current conventional earthquake design procedures prescribed in modern building codes (eg.

ASCE (2010a)) intentionally allow for severe damage to carefully designed structural elements

to achieve a stable seismic response that ensures life safety. During an earthquake, the seismic

force resisting system (SFRS) is designed to experience inelastic deformations in fuse elements,

which dissipate the input energy from the ground motion and protect the rest of the elements of

the SFRS from being overloaded as per capacity design principals (eg. Filiatrault et al. (2013)).

Engineers achieve economical designs by reducing system strength and capacity design forces. As

such, SFRSs with large inelastic deformation capacity and low post-yield stiffness can be cost-

effectively designed for lower system forces.

Buckling restrained braced frames (BRBFs) are one of such very efficient ductile systems.

Buckling restrained braces (BRBs), which are steel braces that are restrained from buckling by

a confining material, yield symmetrically in both tension and compression and have a very low

inelastic stiffness. These features allow for the largest allowable strength reduction factor (R=8

in ASCE (2010a)) as well as low capacity design forces. When distributed along the height of

an SFRS in a braced frame configuration, BRB systems have a large energy-dissipation capability

1

2 CHAPTER 1. INTRODUCTION

(a) (b)concentratedducility

seismic Force

structural displacement

no residual displacement

hystereticenergydistributed

ductility fromBRBs

driftconcentration

seismic force

structural displacement

residual displacement

hystereticenergy

Figure 1.1: Comparison of seismic design philosophies: (a) conventional damage-based bucklingrestrained brace frame system; (b) base rocking system

(Sabelli et al., 2003).

But there are issues with the performance of BRB frames. BRBFs tend to have large residual

deformations compared to other SFRSs (Erochko et al., 2011), as has been observed in general

for seismic systems with low or negative post-yield stiffness (MacRae and Kawashima (1997),

Ruiz-Garcia and Miranda (2006), Christopoulos et al. (2003)). BRB frames are also prone to drift

concentrations, meaning that a few of many distributed ductile elements in an SFRS can experience

a larger amount of cyclic and permanent damage than they were designed for (Sabelli et al., 2003).

Such severe concentrations of inelastic demand can jeopardize the overall performance of the SFRS

and greatly increase the cost of repairs, even leading to the condemnation of the building. Figure

1.1(a) shows a schematic of a conventional ductile BRB frame. This figure illustrates the potential

for drift concentrations in distributed ductility systems and shows how residual deformations are

expected in systems exhibiting elastoplastic hystereses.

More generally, engineers have recognized that the economic toll associated with code-based

designs after an earthquake event is unacceptable. For example the series of earthquakes that

struck Christchurch, New Zealand between September 2010 and February 2011 caused consid-

erable building damage even if there were very few collapses. Over 70% of the buildings in the

Central Business District have been or will be demolished, and rebuilding is expected to cost more

than 20 billion New Zealand Dollars (Wiebe, 2013).

Considering the drawbacks of conventional damage-based systems and BRBs in particular, this

thesis builds on a worldwide effort over the past few decades to design buildings that move beyond

life-safety as a minimum performance level. Such resilient systems seek to minimize damage to

1.1. CONVENTIONAL DUCTILE DESIGN VERSUS BASE-ROCKING 3

structural elements, increase reparability, reduce the economic loss associated with post-earthquake

downtime, reduce accelerations perceived by building occupants, and reduce damage to nonstruc-

tural elements.

One such promising system is self-centering base-rocking structures. Rather than using steel

inelastic deformations to achieve a system-wide ductile mechanism, rocking systems use the open-

ing of a gap at the column base in conjunction with a restoring force from self-weight and/or post-

tensioning cables to provide a non-damage-based system ductility. These rocking systems exhibit

a flag-shaped hysteretic response that typically has zero displacement at zero force in contrast to

conventional yielding systems that have a high potential for residual deformations. Supplemen-

tal replaceable energy dissipation elements are often added to base rocking systems in order to

further-reduce seismic response. Figure1.1(b) shows a schematic of a base-rocking system and its

corresponding hysteresis.

While base-rocking systems can be designed to successfully reduce residual deformations and

concentrate cyclic demand in replaceable fuses, they have drawbacks related to their dynamic re-

sponse and practical implementation. Higher mode effects can greatly increase the force demands

on members since base-rocking frames do not limit seismic shear force at the base and overturn-

ing moments along the height of the structure. In fact, in base rocking structures, the shear forces

transmitted through the height of the building are proportional to the intensity of the earthquake

even if a flexural ductile mechanism forms at the base of the structure. These demands mean that

significant member sizes are required to ensure an elastic superstructure response. Even still, the

superstructure members could experience non-ductile damage if the frame is overloaded. Wiebe

(2013) proposed using multiple rocking sections and/or a ductile brace at the first storey to miti-

gate higher modes, and while these techniques were shown to be highly effective, they represent

an increased complexity and detailing cost that reduces the desirability of base-rocking structures.

Even base-rocking structures that do not have higher mode mitigation have costs associated with

their implementation since unique detailing is required for building uplift deformation compatibil-

ity, post-tensioning elements, and supplemental fuses. Additionally, these systems are not codified


and thus are not as easily implementable as damage-based ductile systems.

Conventionally designed BRB frames and base-rocking structures represent two alternate seis-

mic design philosophies. On one hand, BRBFs are code-approved, relatively easy and cheap to

construct, and feature distributed ductility that ensures low system forces over the height of the

building. However implicit in such a damage-based system are drift concentrations, permanent

deformations, severe damage of ductile elements and potentially large repair or even complete de-

molition costs. On the other hand, controlled rocking frames offer little or no structural damage but

tend to be complicated to detail, expensive, and not necessarily very effective in reducing systems

forces from higher mode effects.

This thesis is aimed at examining the possible benefits of combining buckling restrained braced

frames with base rocking in an optimal way so as to take advantage of the positive aspects of each

system while reducing their respective drawbacks. The proposed system is referred as the hybrid

ductile-rocking system (HDR), and consists of a conventionally designed BRB frame that has a

specially designed column-foundation connection that permits vertical uplift. A lockup is provided

to ensure the full code-designed resistance of the ductile frame after a predetermined amount of

rocking, and supplemental energy dissipation is used to dissipate earthquake energy and reduce the

response of the rocking joint.

1.2 Objectives and organization

The following objectives were developed for this thesis:

1. to present a literature review that gives an overview of BRBFs and rocking structures, and

demonstrates how the performance deficiencies related to BRBs are improved in rocking

frames.

2. to explain using first principles how a conventional ductile system can be modified with base

rocking to achieve the proposed hybrid ductile-rocking system.

1.2. OBJECTIVES AND ORGANIZATION 5

3. to apply this concept to six, four, and two-storey BRB frames, and study the effect of varying

different system parameters during nonlinear time-history analysis.

4. to design and analyze a six-storey frame with a cast-steel yielding fuse and detail that accom-

modates the hybrid ductile-rocking system.

These objectives are addressed in the following chapters as follows. Chapter 2 provides back-

ground on BRBFs and base-rocking structures, and highlights important research relevant to their

development and performance.

Chapter 3 overviews the mechanics of the proposed combined seismic system. It is shown

that by incorporating supplemental energy dissipation and a lockup device, base-rocking can be

added to a ductile frame in order to reduce structural damage. The P �Δ effect is highlighted as it

can reduce system strength during monotonic loading when yielding is expected to occur at large

deformations, and the sensitivity of residual drifts to individual earthquake records is explained.

Chapter 4 presents three reference BRBFs designed for Los Angeles, California, and presents

the validation of these designs using nonlinear analysis. Chapter 5 then uses these frames as the

basis of a parametric study that investigates how conventional ductility and base-rocking can be

optimally combined. This study highlights the energy dissipation strength and amount of rock-

ing allowed at the foundation as two important parameters, and shows that while peak drifts are

relatively similar no matter how much rocking is allowed, BRB displacements, cyclic damage,

and residual drifts can be significantly reduced for modest amounts of allowable rocking before

lockup. It was noted, however, that these benefits were diminished for shorter period structures,

and for cases when the energy dissipation strength was very large.

Chapter 6 presents a detailed 6 storey frame design. A cast steel rocking fuse is selected and

numerically modeled to capture its unique hysteretic properties. This design is analyzed numeri-

cally in order to highlight performance benefits compared to the conventional 6 storey frame. A

column-foundation detail is presented to demonstrate an example of how the combined BRB and

rocking system could be implemented.


This thesis concludes in Chapter 7 with an overview of important results and a discussion of

future research.

Chapter 2

Background and literature review

This chapter presents an overview of buckling restrained braced frames and rocking structures.

Important studies that demonstrate the performance of these two distinct systems are presented in

order to justify the focus of the thesis, which is to study the merits of combining these two systems

into a new seismic design approach to improve seismic resilience of framed structures.

2.1 Buckling-restrained braced frames

This thesis includes buckling restrained braced frames (BRBFs) as a prime example of con-

ventionally ductile steel frames. While there are many other common types of damage-based steel

seismic force resisting systems, BRBFs were chosen for the purpose of this study for the following

reasons:

• BRBFs are common systems in areas of high seismicity.

• Their stable, symmetric hysteric response is readily modeled numerically.

• BRBFs are prone to damage concentrations and large residual drifts, damage states that are

greatly improved in rocking structures.

7

8 CHAPTER 2. BACKGROUND AND LITERATURE REVIEW

Force

Deformation

Decreased bucklingresistance

Peak compressive load is greater than peak tensile load

Deformation

Force(b)(a)

Figure 2.1: CBF and BRB hystereses: (a) typical hysteresis for normal concentric brace; (b) hys-teresis for buckling restrained brace (adapted from Gray, 2012)

2.1.1 Overview

BRB frames are a subset of concentric braced frames (CBF). CBFs are structural systems in

which lateral forces are primarily resisted by the axial deformation of diagonal members. The

centerlines of these members intersect at the centerline of beams and columns at every connection.

Seismically, CBFs are designed so that inelastic deformation occurs in the braces, and the rest of

the structural members (beams, columns), remain essentially elastic. Normal CBF braces have an

asymmetric hysteresis since braces undergo cross-sectional yielding in the tensile direction and

inelastic buckling in the compressive direction. When braces buckle, they form flexural plastic

hinges at their ends and middle point, and it is these hinges that dissipate seismic energy. CBFs

have performance issues that decrease their appeal in high seismic regions, such as their pinched

hysteresis, tendency for a soft-storey response, and premature fracture of buckled braces. For these

reasons, normal CBFs have a limited ductility, as evidenced by the maximum R factor of 6 for

special concentric braced frames (ASCE, 2010a). A typical brace hysteresis is shown in Figure

2.1(a).

In order to improve the ductility of concentric braced frames, engineers have developed buck-

ling restrained braces. A general drawing of this type of brace is shown in Figure 2.2. Typically,

these braces are composed of a steel core that is restrained against buckling by a confining material

so that the brace strength in both tension and compression is governed by cross-sectional yield-

ing. These braces consist of the restrained yielding core, an intermediate region, and a connection

2.1. BUCKLING-RESTRAINED BRACED FRAMES 9

Figure 2.2: Anatomy of a buckling restrained brace (from Gray, 2012)

to the beam-column intersections shown here as a bolted end. Under seismic load, the yielding

core dissipates energy and limits the force transmitted to other elements. The intermediate region

allows for the strain to transition from inelastic in the yielding core to elastic in the connection.

The connection is capacity designed to the yielding core, and detailed to connect to the rest of the

structure via a corner gusset plate (Gray, 2012). Notably, buckling restrained braces do not have

the exact same strength in both tension and compression. Tests have shown that due to friction

between the yielding core and the confining tube caused by an increase in core area from the Pois-

son effect, compression forces are typically in the range of 10% greater than tension forces (Gray,

2012). Low-cycle fatigue life of BRBs has been shown to depend on various factors including the

restraining mechanism used, material properties, local detailing, workmanship, loading conditions,

and loading history. The ductility of BRBs is considered very large, with cumulative nonlinear de-

formations often exceeding 300 times the yield displacement before core fracture (Sabelli, 2000).

2.1.2 Seismic performance

While buckling-restrained braced frames (BRBFs) have a full, symmetric hysteresis, and thus

favourable energy-dissipating characteristics, their low post-yield stiffness compounded by the

presence of P-Δ effects, leaves them vulnerable to large residual drifts and excessive peak drifts

concentrated at a few stories. Sabelli et al. (2003) numerically studied a variety of three and six

storey BRBFs at different hazard levels of 50%, 10% (DBE) and 2% (MCE) in 50 years. They

found that while for lower earthquake levels (50% in 50 years), the drift demand was distributed


Figure 2.3: Example of failed BRB (from Tsai et al. (2008))

relatively evenly along the height of the structures, demands tended to concentrate in the lower sto-

ries when the groundmotion intensity was increased. They noted that the peak drifts were generally

the same for BRBFs designed with R = 6 or R = 8. They also observed mean residual drifts greater

than 0.5% at the DBE level, and 2.2% at the MCE levels - values that would necessitate expensive

building repairs or demolition, as explained byWiebe (2013). Uang and Kiggins (2006) performed

more numerical analyses on some of the structures from Sabelli et al.’s study. They compared the

normal BRBFs to those with an added backup moment frame. These dual system frames had an

increased system post-yield stiffness. They found that while the maximum storey drift ratio was

only reduced by about 10% to 12%, the addition of the moment frames significantly reduced the

residual storey drifts. They recommended incentivizing BRB dual systems by allowing a larger

value of R in the code.

This tendency for BRBFs to have concentrations of inelasticity in a few stories can be dangerous

with respect to collapse performance, since at a certain inter-storey drift brace connection failure

may occur even if the yielding core does not fracture due to low-cycle fatigue. Such a connection

failure could be due to the application of in-plane moments, yielding core instability due to extreme

elongation (plastic hinging), instability due to the transition region butting against the grout, or some

other failure. Figure 2.3 shows what an example of a BRB failure due to gusset plate instability.

Gray et al. (2014) reviewed BRB specimen tests conducted by Black et al. (2002), Merritt et al.

(2003), Meritt et al. (2003), Uriz and Mahin (2008), Christopulos (2005), and Palmer (2012), in

order to determine a reasonable estimate at which drift BRB failure would occur, settling on 6%

2.2. BASE ROCKING STRUCTURES 11

interstorey drift. In their subsequent numerical analysis of a twelve storey BRB frame, they found

that drift tended to concentrate in the lower stories, and that the mean peak drift of seven records

scaled to DBE was around 3%. Under MCE, four of the seven records caused collapse due to the

concentration of drift at the lower stories.

While they tend to accumulate excessive residual and cyclic damage, BRBFs can, on the other

hand, have a very beneficial storey acceleration response. Choi et al. (2008) compared the acceler-

ation response of numerical models of BRBFs, self-centering energy-dissipating (SCED) frames,

and moment-resisting frames (MRF) and found that while the MRF and SCED frame structures

tended to have storey accelerations that exceeded the peak ground acceleration (especially the

MRFs), the BRB frames tended to have storey accelerations that were equivalent or lower than

the peak ground accelerations thus confirming the excellent performance of such ductile structures

with respect to controlling forces and accelerations along the height of the structure.

2.1.3 Summary

Buckling restrained braced frames offer the economy and simple design method of damage-

based concentric frames without the poor performance associated with conventional braces that

can buckle. While they have favorable energy dissipation and storey acceleration performance,

they are prone to large residual drifts. As well, even though their yielding cores are highly ductile,

drift concentrations can cause excessive deformation demands at one storey that can lead to failure

modes besides core fracture.

2.2 Base rocking structures

Base rocking structures are structures that offer improved performance over damage-based de-

signs since most or all residual drifts can be precluded. This section presents an explanation of the

basic self-centering force-deformation response, and a history of studies and tests on base rocking

structures with a discussion of their performance benefits and limitations.


1

1ko

αkof

y

Force

Deformation

βfy

Figure 2.4: Typical flag shaped hysteresis (adapted from Wiebe, 2013)

2.2.1 Flag-shaped hysteresis

Self-centering base-rocking structures mitigate earthquake effects with a ductile mechanism

that is not associated with damage to the structural frame. There are multiple types of self-centering

systems besides base rocking that share a similar hysteresis, such as self-centering energy dissi-

pating (SCED) braces (Erochko and Christopoulos (2014)), and friction damped post-tensioned

moment-resisting frames (Kim and Christopoulos (2008)). Figure 2.4 shows a simplified self-

centering hysteresis and the various parameters that typically govern such a system’s response. This

hysteresis is defined by the initial stiffness, ko, the elastic limit, fy, the nonlinear stiffness ratio, α,

and the energy dissipation parameter, β. A flag-shaped hysteresis is only fully self-centering if β<1.

β=1 means that the structure has half the energy dissipation capacity as an equivalent elastoplastic

systems. For β>1, the potential for residual deformations exists and increases as � is increased.

2.2.2 Early studies of rocking structures

Muto et al. (1960) investigated the overturning resistance of slender structures through dynamic

testing of models on an elastic foundation. They noted that the restoring force in a structure that

can lift from its foundation is maximum right after first lift, and zero when the center of gravity is

vertical over the edge of the foundation. They concluded that the slender, multistory, reinforced

concrete apartment buildings in Japan would not likely overturn under an earthquake similar to

what had been observed in past historical earthquakes.


Figure 2.5: Concrete bridge pier designed to step (from Beck and Skinner (1974))

Housner (1963), analyzed the free oscillations of a rocking block, and derived equations for the

rocking period considering energy losses at rocking impact. The overturning of a rocking block

when subjected to constant, half-sinusoidal, and earthquake horizontal accelerations were exam-

ined. It was shown that the stability of tall slender structures is greater than that which is presumed

from studying its resistance to a monotonic horizontal force.

Beck and Skinner (1974) conducted a feasibility study of an A-shaped reinforced concrete

bridge pier that was designed to step, as shown in Figure 2.5. Using nonlinear time-history of sin-

gle degree of freedom systems subjected to the 1940 El Centro earthquake (N-S), they concluded

that the forces induced on the pier could be greatly reduced by the stepping motion. However,

they found that the displacements of the bridge deck were up to three times larger than in the fixed

structure, with viscous damping between 1% and 3%. These displacements were reduced further

with supplemental damping.

In the late 1970’s multiple studies were performed at the University of California Berkeley on

steel moment frames with columns that were free to uplift. Kelly and Tsztoo (1977) developed

yielding steel torsion bars and added them to the base of a three storey single bay moment frame

that was previously tested by Clough and Huckelbridge (1977), shown in Figure 2.6. They found

that the uplift mechanism successfully reduced the frame forces from the fixed base configuration.

The added energy dissipation reduced peak displacements for one ground motion, although for the

other tested earthquake motion the peak displacements were larger than the fixed based structure

and the structure with uplift but without the torsion device.


(a) (c)(b)

Figure 2.6: Moment frame with column uplift and energy absorbing devices: (a) fixed base frame;(b) frame modified to allow uplift; (c) energy absorbing torsion device shown after simulation tests(from Kelly and Tsztoo (1977))

Huckelbridge (1977) tested a 8.5 m nine-storey, three-bay moment frame. The maximum uplift

observed during the tests was 40 mm. The rocking mechanism successfully reduced the peak forces

in the first storey columns. During some tests, rocking increased the peak storey displacements. The

rocking response was successfully modeled numerically considering about 0.7% tangent stiffness

proportional damping in the first mode.

2.2.3 The PRESSS program and rocking of concrete walls

In the 1990s the Precast Seismic Structural Systems (PRESSS) programwas developed with the

goal of offering a precast concrete seismic system that had increased performance over traditional

systems, while ensuring cost effectiveness. The program worked to develop connections between

precast concrete members that allowed for a concentration of ductility and damage outside of the

main structural elements (Priestley, 1991). Many of these connections were developed to include

a self-centering response where a gap was allowed to form between precast beams and columns,

or between adjacent walls and at wall foundations. The 10-year PRESSS program culminated in

the test of a 60 percent scale five-storey structure that included five different structural systems in

the same structure that included different connections with and without post-tensioning as well as

energy dissipation from steel bars. The test was a success as damage to the structure was minimal,


especially when compared to a conventional reinforced concrete frame subjected to similar drift

demands. Relatively small levels of residual drifts and damage were reported, and the structure

confirmed the direct-displacement design method that was used to design the structure. Higher

than expected floor forces that were recorded during the tests were attributed to higher mode effects

(Priestley, 1999).

Holden et al. (2003) performed tests on a conventional code-designed concrete wall and a pre-

cast wall with vertical carbon fiber post-tensioning and yielding bars that was allowed to rock.

They observed that the code-designed wall performed well in terms of displacement capacity and

energy dissipation, although damage was extensive even at a relatively moderate level of 1% drift

with residual cracks of up to 2 mm wide being observed. At a larger drift of 2.5%, the capacity

of the unit was significantly degraded. In contrast, the post-tensioned rocking wall successfully

precluded damage at drifts exceeding 2.5%.

2.2.4 Rocking steel systems

Wada et al. (2001), studied truss structures that included yielding column splices. Specimen

tests were performed that showed that the damper devices provided excellent deformation capacity

and energy absorbing ability. The displacement, bending moment, and column force responses

were all decreased during a nonlinear analysis of a 16 storeywarehousewith these devices subjected

to one earthquake record. A full-scale test confirmed that the devices protected the truss structure

from column compressive buckling.

Midorikawa et al. (2002) numerically studied a five storey moment resisting frame that had

ductile base plates, and compared it to a similar frame that was fixed to the ground, and a rocking

frame without a ductile fuse. They analyzed the structures under the 1940 El Centro NS and 1995

Kobe NS ground motions, with a time scale shortened to 1/p3, using 0.5% initial stiffness propor-

tional rayleigh viscous damping in the first and second modes. They found that the structures with

ductile base plates reduced the storey shears from the fixed base structure and were similar to the

simple rocking structure, and that the roof displacements and axial forces were similar to the fixed


(a) (b) (c)

Figure 2.7: Yielding base plate: (a) plan view showing ductile wings; (b) photo of yielding base plateimplemented at the bottom of a steel frame; (c) force deformation response for 9 mm thick baseplate showing large post-yield stiffness (From Midorikawa et al. (2006))

base structure for lower amplitude ground motions. They concluded that the ductile base plates

were successful in reducing earthquake effects.

Midorikawa et al. (2006) verified their previous work with a shake table test and numerical

study of a three storey, three dimensional 1 X 2 bay steel frame scaled to one half. The yielding

base plates are shown in Figure 2.7. The base shear was successfully reduced in the structures with

yielding base plates, and this reduction was attributed to the fact that the rocking motion dominated

the drift response. Interestingly, the drift response of the rocking and fixed base structures were

similar. The maximum column tensile force was limited to about twice the yield strength of the

ductile base plates, since the base plates had a significant post-yield stiffness due to second order

axial deformations. This large post-yield stiffness is evident in Figure 2.7(c). Notably, the base

plate transmitted the seismic shear even during the uplift motion. The numerical models matched

the experimental models well. It should be noted that because of the large post-yield stiffness of the

ductile base plates, the structure likely had a positive system stiffness after yielding even though

the rocking was not controlled with post-tensioning, although this value is not stated explicitly.

The positive post-uplift stiffness might not be the case if the structure was attached to a significant

P-Δ column, or if the post-yield stiffness of the ductile base plates was not so large.

Midorikawa et al. (2008) added yielding base plates to a numerical model of a full scale six

storey structure that was tested in the early 1980’s. The structure consisted of an eccentrically


braced frame and moment-resisting frames, and the uplift connections were placed at the bottom of

the braced frame bay. They validated their model of the original fixed braced frame by comparing

numerical results to the test results. They showed that the base shears in the model with the ductile

base plates were significantly reduced from the fixed base model. However, the peak drifts for

both the records that were considered were much larger for the uplift models than the fixed based

configuration, and in turn the inelastic deformation in the moment-resisting frames were greater

than for the fixed-based model. This large increase in drifts was attributed to the reduced system

stiffness after uplift.

Azuhata et al. (2006) added the ductile base plates to the numerical model of a three bay, ten

storey moment resisting frame. Bracing was also added to the uplift models in two configurations:

in the middle bay in the first storey, and in the middle bay in every storey. As well, a fixed base

model was analyzed with bracing in the middle bay. As mentioned above in previous studies,

the yielding base plates had a high post-yield stiffness, modeled as twenty percent of the elastic

stiffness of the fuses, which were determined from previous static tests. The peak displacements of

the rocking structures were less than the fixed structure, although greater than the fixed structure

with braces added. The rocking motion successfully reduced damage compared to both fixed based

frames, although a small amount of damage was observed in the beam ends even in the rocking

configurations.

Tremblay et al. (2004) proposed a new braced frame configuration where BRBs were used

as the columns in the first and second stories of a one bay braced frame that had a continuous

column running down its middle. The rest of the columns were not continuous between floors, but

rather were interrupted by floor beams that extended from the columns of the next bay, through

to the central column. Under seismic response, the bracing bay deformed in global flexure, and

the BRBs deformed inelastically. This is similar to the rocking systems described above, although

there was no gap in this system. Rather than from post-tensioning, a restoring force was contributed

by the flexural deformation of the beams at each storey, as well as the strain hardening in the BRBs.

The authors proposed placing additional BRBs along the height of the structure in order to reduce


(a) (b)

Figure 2.8: Retrofitted bridge steel truss pier using controlled rocking approach: (a) modified steeltruss pier; (b) cyclic pushover and buckling restrained brace behavior (from Pollino and Bruneau(2007))

demands caused by higher mode vibrations.

Tremblay et al. (2008) also used viscous dampers to control the response of braced frames

allowed to rock. They performed a numerical and experimental analysis of of a 2-storey half-

scaled chevron braced frame that was allowed to uplift in order to validate numerical models. No

post-tensioning was used, but the tributary weight of the frame was relied on to provide a restoring

force. Their numerical models were verified through dynamic testing.

Controlled rocking without post-tensioning has been applied to bridge structures as well as

buildings. Pollino and Bruneau (2007) investigated a seismic retrofit technique that allowed steel

truss bridge piers to rock on their foundations. The bridge’s self weight supplied the restoring

force, and buckling restrained braces were placed at the uplift joints to control the uplift load and

add energy dissipation. It was noted that while self-weight of the bridge contributed a negative

post-yield stiffness to the response, the strain hardening of the BRB caused the system post-uplift

stiffness to be positive, and so P �� effects were ignored in their analysis. They also outlined that

the demands on the pier legs included a dynamic effect caused by the excitation of vertical modes.

An amplification of column design forces was presented and validated with nonlinear time-history

analysis. Figure 2.8 shows a depiction of their proposed bridge retrofitted to rock using buckling

restrained braces as supplemental energy dissipation devices.

Roke et al. (2006) introduced the self-centering concentrically braced frame (SC-CBF) system


concept, which consisted of beams, columns, and braces arranged in a conventional CBF config-

uration, but with columns permitted to uplift. Post-tensioning tendons and gravity forces were

intended to supply restoring forces after uplift. Four limit states were identified and included (1)

column decompression; (2) yielding of the PT steel; (3) significant yielding of beams, columns,

and braces, and; (4) failure of beams, columns, and braces. Only limit state (1) was associated with

an immediate occupancy performance level since the other limit states include permanent damage.

Pushover analyses confirmed the intended limit states for a few different configurations of PT and

energy dissipation, and nonlinear analysis confirmed that supplemental energy dissipation elements

can have an important effect on the considered response quantities including uplift displacement.

Eatherton et al. (2008) presented a controlled rocking steel-framed system with replaceable

energy-dissipating fuses. While multiple configurations of the proposed system were deemed pos-

sible, a configuration with two side-by-side concentrically braced frames with shear fuses between

the frames was designed, tested, and compared to numerical models. The testing program included

component tests of multiple types of shear fuses including fuses that contained high performance

fiber reinforced cementitious composites, engineered cementitious composites, steel plates with

straight slits, and steel plates with butterfly cut-outs. A prototype three-storey structure was de-

signed assuming an arbitrary site in California and R=8. The test specimen, bumper detail for

shear transfer, and close up of a butterfly-type shear fuse is shown in Figure 2.9 . The system

demonstrated excellent energy dissipation response and successfully mitigated residual drifts with

a maximum value of 0.2%. The fuses attained cyclic shear strains of less than half their anticipated

capacity, which was sufficient even when the system exceeded more than 3% interstorey drift.

Analytical models successfully captured the system response, although further refinement such as

modeling the strain hardening in the fuse elements was recommended.

2.2.5 Higher mode effects in controlled rocking structures

Wiebe et al. (2012a) and Wiebe et al. (2012b) identified higher mode effects as an important

issue in the design of controlled rocking structures, and used numerical analyses and shake-table


Figure 2.9: Test structure with bumper detail and shear fuse (from Eatherton et al. (2008))

testing to validate multiple higher mode mitigation techniques. The authors noted that traditional

capacity design techniques may not adequately predict member forces for systems with concen-

trated ductility, such as reinforced concrete shear walls that are expected to undergo plastic hinging

at their base and controlled rocking structures which allow column uplift. They used an idealized

fixed-based flexural cantilever and an analogous pinned-based cantilever (for which mode shapes

can be computed even though such a structure is unstable) to demonstrate that while allowing base

rotation limits the forces in the fundamental mode, which becomes a rigid body rotation, the higher

modes of the pinned-base cantilever are still excitable since they do not increase the base moment

and there is no limit to how much shear force can be transmitted through the base. It was noted that

the higher modes of the pinned-base cantilever were similar to the higher modes of the fixed base

structure, and thus the addition of a hinge at the base acts to limit the forces of the first mode while

modifying but not eliminating the higher modes.

In order to mitigate higher mode effects in controlled rocking structures, the authors recom-

mended allowing multiple rocking sections along the height of the structure, thus capping the mo-

ment permitted to develop at the location of the hinge and limiting storey shears since shear force

is the slope of the moment diagram. As well, the authors recommended a shear control device such

as a self-centering energy-dissipating (SCED) brace in the first storey to limit the amount of shear


Figure 2.10: Experimental setup of controlled rocking steel frame with higher mode mitigation(from Wiebe et al. (2012a))

force permitted into the structure. It was noted that due to the equal displacement principal, and by

adding supplemental energy-dissipation, peak system deformations were not expected to be greatly

increase by the added ductility associated with these higher mode mitigation techniques.

In order to validate the proposed techniques, shake-table tests were performed on an eight-storey

frame based on a 30% scaled prototype frame designed for Vancouver, BC. Figure 2.10 shows the

experimental setup. The frame was designed to allow for four different configurations, including

(1) rocking at just the base; (2) rocking at the base and at mid-height; (3) rocking at just the base,

with a SCED brace placed in the first storey, and; (4) rocking at the base and mid-height, with the

SCED brace. A numerical model was created to capture the response of the frame. The frame was

able to withstand earthquakes at more than twice the design level without damage. Higher mode

effects were effectively reduced with the techniques described above, although it was noted that the

upper rocking joint and SCED brace increased the maximum roof displacement by 18% and 6%

on average, respectively. The numerical model provided a good estimate of the seismic response,

although it tended to increase in accuracy for configurations that included higher mode mitigation

techniques.


(a) (b) (c) (d)

Figure 2.11: Possible details for connecting rocking frame to diaphragm and collectors (fromEatherton (2010))

2.2.6 Compatibility between rocking frame and rest of structure

Since rocking structures incorporate vertical motion along with lateral drift when undergoing

seismic excitation, researchers have considered different details to allow compatibility of vertical

deformations along with the transfer of the seismic shear force. Eatherton (2010) presented four

possible details for connecting a rocking frame and diaphragm.

1. A typical detail can be used to connect the rocking frame to the collector and diaphragm, as

in Figure 2.11(a). Some localized damage is expected after a seismic event.

2. The collectors can be split around the rocking frame into adjacent beams, and shear plates

can be used to transfer seismic force and permit uplift, as in Figure 2.11(b). The slab can be

blocked out around the SFRS to reduce damage.

3. The collectors can be split as in option (3), and rollers can be used to transfer shear in com-

pression, as in Figure 2.11(c).

4. The collector can be attached directly to the rocking frame, with the slab blocked out to

reduce damage, and an adjacent beam with shear transfer plates used to contribute seismic

force along with the collector, as in Figure 2.11(d).

2.3. COMBINING BRBS AND BASE ROCKING FOR IMPROVED PERFORMANCE 23

Figure 2.12: SCED brace setup for testing and hysteretic behavior (from Kim (2012))

2.3 Combining BRBs and base rocking for improved performance

2.3.1 Examples of combined self-centering and plastic systems

While self-centering systems have typically been designed to fully mitigate structural damage

or concentrate it in supplemental fuses, self-centering components have been combined with plas-

tic mechanisms as a means to protect the systems under large deformation demands. For example,

Kim and Christopoulos (2008) developed a self-centering friction-damped moment frame that em-

ployed post-tensioning tendons and supplemental friction fuses. They tested details that allowed

for the formation of stable beam plastic hinging under large drift demands, which protected the

post-tensioning elements from being overloaded. Similarly, Kim (2012) studied flag-shaped SDOF

systems with plastic fuses in order to gain insight into the hysteretic behavior of such systems as

the self-centering moment frame described above as well as the self-centering energy-dissipative

brace system (Erochko and Christopoulos, 2014), which can incorporate an external friction fuse to

protect the PT elements under large deformations. Figure 2.12 shows an example of a SCED brace

being tested, as well as a typical hystereses that includes an external friction fuse. Kim (2012) found

that the ductility demands on the flag-shaped SDOF systems decreased with increasing structural

period, system strength, post-yield stiffness, and �, although the effect of � diminished as its value

increased. The demands on the external fuses when subjected to an MCE record set were dimin-

ished with increasing post-yield stiffness, system strength, and ductility before fuse activation, and

it was noted that � had a negligible effect. The authors concluded that external seismic fuses might

not be necessary for structural periods greater than 3 seconds, and for very ductile systems. As well,

it was noted that the displacement capacity of these fuses should be increased when considering


near-fault effects.

2.3.2 Proposed combined seismic system

As described in Chapter 1 the aim of this thesis is to allow some rocking in conventional BRB

frames so as to incorporate the best of each system. This proposed systemwill feature the beneficial

characteristics of BRBFs, including significant force-reduction, acceleration control, and ease of

design. In turn, the performance drawbacks of BRBFs such as large peak and residual deformations

will be lessened by the incorporation of limited base rocking since base rocking structures have

favourable drift concentration and residual drift responses.

2.4 Chapter summary

Section 2.1 provided an overview of buckling restrained braced frames and highlighted their

important performance benefits and drawbacks. BRBFs are highly ductile and exhibit a low post-

yield stiffness, and as such are very effective at reducing system forces and capacity design forces.

Aswell, BRBFs tend to have a favourable storey acceleration response as compared to other SFRFs.

Conversely, the low post-yield stiffness of these systems can cause very large residual drifts and

drift concentrations potentially requiring expensive repairs or even demolition.

In Section 2.2 base rocking systems were briefly reviewed. The fundamental flag-shaped hys-

teretic behaviour of self-centering systems was described, and a history of important studies on

structures that were allowed to rock was presented. It was noted that higher mode effects can lead

to greatly increased system forces even if the force reduction factor associated with the rocking

joint is large. As well, slab-frame compatibility was highlighted as an important detailing issue

related to rocking structures.

Finally, Section 2.3 overviewed previous seismic systems that featured a combination of self-

centering and plastic behaviours, and briefly summarized the benefits of incorporating base-rocking

into BRBFs.

Chapter 3

Mechanics of hybrid ductile-rocking

This section presents an overview of how a conventionally ductile frame can be modified to in-

clude a self-centering response to enhance the performance of conventional highly ductile systems.

3.1 Mechanics of a rocking joint

Wiebe (2013) derived the fundamental behavior of rocking structures by assuming that the

rocking body is perfectly rigid. His equations to describe the behaviour of the rocking joint are

summarized here with the post-tensioning terms removed as they represent the basis of the proposed

system. This derivation corresponds to the rigid rocking body in Figure 3.1(a). As shown in Figure

3.1(b), these equations assume that a rigid frame has a weight,Wself , that acts vertically through the

centroid of the rocking frame, a horizontal distance of dw from the rocking toe. Energy dissipation

elements are provided at the base of the structure, a distance dEDfrom the rocking toe. The energy

dissipation elements are assumed to have a rigid-perfectly-plastic hysteresis, and a yield load of

ED. These energy dissipation elements represents an idealization of a steel yielding device or

friction fuse. This rocking body is associated with the rest of a structure that has a tributary weight

Wtrib, and thus the weight contributing to P-Δ effects not acting on the frame itself,Wtrib �Wself ,

acts on a leaning column. This derivation could be extended to other configurations of energy

dissipation and post-tensioning if desired by adjusting the equations below.

25

26 CHAPTER 3. MECHANICS OF HYBRID DUCTILE-ROCKING

(a)

dw

dED

energy dissipationelement with yield load ED

leaning column

rockingbody

Wself

Mb,rock

dMbreverse

krigid,rock

Wtrib

-Wself

Hw

(b)

Wself

ED

Mb

1

θbase

Wtrib

-Wself

(c)

Wself

ED

θbase

θbase

Wtrib

-Wself

(d) (e)

Figure 3.1: Rocking structure mechanics: (a) definition of rigid rocking body; (b) rigid rocking bodyat rest; (c) rigid rocking body with increasing base rotation; (d) rigid rocking body with decreasingbase rotation; (e) force-deformation behaviour of rigid rocking body (adapted fromWiebe (2013))

The discussion is presented in terms of base moment, Mb , and base rotation, θbase, since the

rocking joint acts as a nonlinear moment-rotation spring. Rocking occurs when the uplifting side

of the foundation is decompressed and the energy dissipation is activated in tension. The rocking

base moment, at which point the body begins to rotate, can be found by calculating the moments

on the rocking body about the rocking toe:

Mb,rock = Wself ⇤ dw + ED ⇤ dED (3.1.1)

Figure 3.1(c) shows the structure after uplift where the base moment is:

Mb = Wself ⇤ dw �Wtrib ⇤Hwθbase + ED ⇤ dED (3.1.2)

Hw is the height of the centre of the tributary weight. Differentiating with respect to the base

3.1. MECHANICS OF A ROCKING JOINT 27

rotation gives the rotational stiffness of the uplift joint as:

krigid,rock = �WtribHw (3.1.3)

As can be seen in Figure 3.1(e), the tributary weight of the structure contributes a negative

lateral stiffness, as is to be expected from the P-� effect. Thus, when there is no post-tensioning

acting on the structure the post-rocking lateral stiffness is negative, as shown in Equation 3.1.3.

When the direction of rocking is reversed, the energy dissipation must yield in compression

before the uplift is reduced. The difference in the base moment between the initial load reversal

and the moment at which rotation begins to reduce is given by:

dM reverseb = 2ED ⇤ dED (3.1.4)

There are certain scenarios, as described by Wiebe (2013), when the displacement response of

a rocking system is not fully described by the base rotation. For example, if the energy dissipation

elements are located between the toes of the rocking body, it is possible that it would slip in tension

only. In this case, global uplift of the system could occur. This would occur under static loading

if the energy dissipation activation load is larger than the sum of the gravity and post-tensioning

forces.

Actual rocking bodies are not perfectly rigid but rather have a lateral stiffness, ko, which is de-

termined by the structural elements and the lateral load distribution being considered. Considering

this, the following non-dimensional parameters are defined in order to characterize a typical CRS:

α =krockko

(3.1.5)

β =dM reverse

b

Mb,rock

(3.1.6)

where α is the nonlinear stiffness ratio and β is the energy dissipation ratio.


(a) (b)

Ψ

W

θ

θ

hn

h2

h1

up

δ1

δbrace,1

δbrace,2

δbrace,n

δ2

δn

δ1

δ2

δn

1st Storey

2nd Storey

Nth Storey

Figure 3.2: Contribution to system deformations from two ductile mechanisms: (a) brace defor-mations; (b) rocking of frame

3.2 Modifying conventional structures to become a HDR system

3.2.1 Drifts from brace deformation and rocking

In a typical frame structure, interstorey drifts are defined as the ratio of interstorey displace-

ment to storey height and are caused by both shear-type and moment-type deformations. In a

braced frame, the shear-type deformations come from brace axial deformations induced by the

storey shears, and moment-type deformations come from column elongation and shortening in-

duced by overturning moments. In a typical low-rise braced frame structure it is the shear-type

deformations that contribute most to the drifts. During seismic loading, drifts thus primarily con-

sist of brace plastic deformations. A braced frame structure undergoing shear-type deformations

is shown in Figure 3.2(a), where δbrace,n is the axial deformation of one brace at storey n, Ψ is the

brace angle, δn is the horizontal deformation of storey n, Δn is interstorey drift ratio at storey n,

and hn is the height of storey n. Considering that interstorey drifts are defined as:

Δn =δn � δn�1

hn

(3.2.1)

3.2. MODIFYING CONVENTIONAL STRUCTURES TO BECOME A HDR SYSTEM 29

and interstorey displacement can be determined from brace geometry as:

δn � δn�1 =δbracecos(Ψ)

(3.2.2)

interstorey drifts from braces can then be defined from brace deformation, assuming there is only

minimal contribution from column shortening:

Δn,brace =δbrace

cos(Ψ)hn

(3.2.3)

If a braced frame is allowed to rock, and neglecting elastic frame deformations, then the drift is

the same value as the rotation angle, as seen in 3.2(b). This can be derived by considering that the

interstorey displacement can be defined as:

δn � δn�1 = θhn (3.2.4)

and thus, using Equation 3.2.1, drift from rocking can be defined as:

Δn,rocking = θ (3.2.5)

Column uplift can be defined as:

δup = θW (3.2.6)

For example, for a 10 m wide braced frame bay, a rotation of 1% (corresponding to 10 cm of

vertical uplift) causes a 1% interstorey drift. This derivation assumes the frame deformation is a

pure rigid-body rotation and neglects the frame elastic deformations.

In an HDR structure, some brace yielding is expected along with rocking. The drift minus base

rotation, �DMR, can be calculated in order to quantify the contribution to drifts from the braces,

which is directly related to the damage in the frame. This value can be computed as:


�DMR = �n � ✓ (3.2.7)

This value can be approximated using Equation 3.2.3.

It should be noted that one primary difference between these two sources of system deforma-

tions is that, assuming a rigid-body response, the drifts from rocking are constant along the height

of the structure. This is in contrast to the drifts from brace deformations, which for a given storey

are calculated solely based on the brace deformations at that storey.

3.2.2 Overview and mechanics

The two types of drifts described in Section 3.2.1 can be combined by incorporating a lockup

device to create the HDR system. The mechanics of an HDR structure are explained in Figure

3.3 and Figure 3.4. These figures present a one-storey flexible structure as a simplified example.

The structure consists of one set of chevron braces with a presumed idealized elastoplastic bilinear

hysteretic behavior with a positive post-yield stiffness. P � � effects are not considered in this

explanation, though they can have a non-trivial effect on the HDR system and are examined in

further detail in Section 3.2.3. There is gravity load acting on the columns of the structure, and it

is this load that serves as the restoring force after uplift. The columns and braces are presumed to

meet at a pin-connection that is not fastened to the ground. Rather, there is a lockup device that

allows a gap to formwhen the columns are decompressed. That gap is limited to a value determined

by the designer. As well, an energy dissipation element is located at the base of the columns in the

gap. This element is idealized as rigid-plastic for this derivation, similar to the way a friction fuse

or a stiff yielding damper with negligible strain hardening would behave.

Figure 3.3(a) shows free-body diagrams of different stages in the structure’s response during a

smaller amplitude displacement cycle. In this context, smaller amplitude displacements refers to

system displacements that do not engage the lockup device. The actual value of these displacements

is set by the design engineer. The effect of different amounts of allowable lockup on the seismic


lockup

1. rest:

2. column decompression (uplift) 3. ED yields in tension

6. ED yields in compression

5. column recompression (ED has zero force)

7. Rocking gap closed

8. Rest

4. maximum rocking displacement

SeismicForce

Displacement

(a)

1,8

2

3 4

5

67

(b)

Figure 3.3: Mechanics of hybrid ductile-rocking: (a) free body diagrams of smaller displacementhistory that does engage the lockup; (b) corresponding system force-deformation curve


1. rest: 2. column decompression (uplift)

3. ED yields in tension 4. maximum rocking displacement

SeismicForce

Displacement

(a)

1

2

3 4

56

7

89

10

(b)

5. braces yield 6. max. plastic def. ofbraces after lockup

7. column recompression 8. ED yields in compression

9. rocking gap closed 10. rest, with residual deformation

Figure 3.4: Mechanics of hybrid ductile-rocking: (a) free body diagrams of larger displacementhistory that engages the lockup; (b) corresponding system force-deformation curve


response of HDR structures is examined in detail in Chapter 5. Each free body diagram corresponds

to a force-displacement point on the cyclic response plot in Figure 3.3(b). Figure 3.3(a)(1) shows

the structure at rest before any seismic load is applied. The exploded view shows how the lockup

is represented schematically by two fixed boundary conditions that are separated by a gap. The

column-brace connection is free to uplift until it comes into contact with the lockup device after

which it can transfer tension forces to the foundation. Rigid-plastic energy dissipation is assumed to

be present in the gap although it is not shown in the diagram for simplicity. Figure 3.3(a)(2) shows

the structure at the onset of column decompression on the left side of the frame. If this were a fixed

base frame, further seismic loading would induce tension into the column, and the system forces

would increase proportionally until the braces experienced nonlinear deformations. In this case,

however, the tension force is carried by the energy dissipation until it yields in Figure 3.3(a)(3).

After the ED yields in tension the structure rocks, and system forces are limited by the strength

of the rocking connection. After the structure displaces to the maximum rocking displacement in

Figure 3.3(a)(4) (in this case the maximum displacement is just less than that which would engage

the lockup), the loading is reversed, the tension in the column is reduced to zero (Figure 3.3 (a)(5)),

and the ED yields in compression (Figure 3.3 (a)(6)). Finally, the rocking gap is closed (Figure

3.3(a)(7)) and the brace elastic deformations are recovered. Figure 3.3 (a)(7) shows the at-rest state

of the frame, which is identical to the initial free-body diagram in Figure 3.3(a)(1).

Figure 3.4 shows free body-diagrams of the structure when cycled past a displacement that

engages the lockup. The first four free-body diagrams are identical to those in Figure 3.3, but rather

than reversing the loading direction, the structure is pushed such that the lockup is engaged and the

braces yield (Figure 3.4(a)(5)). Figure 3.4(a)(6) shows the structure at a maximum deformation

before loading is reversed. There is significant plastic deformation in the braces. Figure 3.4(a)(7)

shows the structure when the tension force in the column is reduced to zero. Figure 3.4(a)(8) shows

the energy dissipation yielding in compression before the rocking gap begins to close again. Figure

3.4(a)(9) shows the frame just after the rocking gap closes, and Figure 3.4(a)(10) shows the frame at

rest after the elastic deformations are recovered. Notably, there is significant residual deformation


from the braces which was not present after the smaller loading excursion shown in Figure 3.3. A

schematic of the corresponding seismic load-deformation response is shown in Figure 3.4(b).

3.2.3 P �� effects in HDR systems

The P � Δ effect, or the second-order effect caused by building gravity loads, can reduce a

structure’s effective strength and post-yield stiffness. However as P � Δ effects are proportional

to lateral displacement, they typically do not have a large impact on braced frames as such struc-

tures have high elastic stiffnesses and thus typically yield at relatively small displacements (MRFs,

for example, are more flexible and are more prone to strength reductions from the P � Δ effect).

The same cannot be said for a braced frames with hybrid ductile-rocking, as while the storey shear

strength is the same as a fixed-base frame the braces are expected to yield at a larger system dis-

placement.

P �� effect and Pushover Response

Figure 3.2.3 shows how the P �Δ effect changes the pushover behavior of a conventional and

HDR frame with varying hysteretic properties. In this figure P is the gravity load acting on the

whole structure (not just the SFRS),H is the effective structural height, V is the seismic force, and

� is the system displacement.

The change in stiffness and strength caused by the P �Δ effect for a conventional elastoplastic

structure (like a braced frame) with zero post-yield is shown in 3.2.3(a). Here, the reduction in

yield strength is:

Vreduction =PδyH

(3.2.8)

where δy is the system yield displacement.

As well as strength reduction, the P � Δ effect reduces system stiffness. For the system with

zero post-yield stiffness in 3.2.3(a), the resulting post-yield stiffness is negative. If the system had

a positive post-yield stiffness, the P � Δ effect would reduce that stiffness while not necessarily


P

H

Vy

(a)

(c)

Seismic Force V

Seismic displacement δ

Vreduction

δy

δy

δup P

H

Vy

(b) Seismic Force V

Seismic displacement δ

Vreduction, HDR

Vdifference, HDR

SeismicForce

Displacement

P

H

Vreduction,HDR

Smaller strength reductionafter first excursion

Legend:conventional system (no P-Δ)conventional system (P-Δ)HDR system (no P-Δ)HDR system (P-Δ)P-Δ effect

columndecompression

lockup

Figure 3.5: P-Delta effect (a) conventional structure pushover behavior: (b) HDR structurepushover behaviour; (c) HDR structure cyclic behaviour


making it negative.

Figure 3.2.3(b) shows how allowing hybrid ductile-rocking increases the reduction in system

strength caused by P � Δ effects. That reduction is defined as:

Vreduction,HDR =P (δy + δup,lateral)

H(3.2.9)

where δup,lateral is the lateral displacement caused by uplift, similar to δn from Equation 3.2.4.

The reduction in strength between the conventional structure and the HDR structure can be derived

as:

Vdifference,HDR =Pδup,lateral

H(3.2.10)

P-Delta effect and cyclic response

The strength reduction derived in Equation 3.2.10 is not characteristic of an HDR structure’s

cyclic response. During a first excursion inducing plastic deformations on the braces, the displace-

ment due to rocking occurs in one direction. For example, if 1% drift before lockup is allowed, that

1% will first occur completely in the direction of first loading, and the superstructure will yield at

a displacement corresponding to 1% plus the elastic deformation of the frame. Here, the strength

reduction of the superstructure is governed by Equation 3.2.9. But depending on how large the

plastic displacement demand on the superstructure is after yield the system displacement range

during which rocking occurs once the load is reversed will change. Thus the absolute displacement

at which the superstructure subsequently yields depends on the system’s displacement history. The

susceptibility of superstructure strength reduction to displacement history can be seen in Figure

3.2.3(c), which shows a generic push-pull response of an HDR structure during 1.25 cycles of de-

formation to a displacement that causes significant brace yielding. The effect of P �Δ is reduced

after the superstructure yields at a smaller system displacement than during the first excursion. In

this figure, a rigid-perfectly plastic energy dissipation is assumed.


System strength and higher modes

The reduction in system strength caused by the P�Δ effect described above would suggest that

an HDR structure will have a lower overall system strength than a fixed-based structure depending

on the magnitude of the second-order effects. However, this reduction in system forces is a function

of superstructure yielding at a displacement larger than that governed by HDR lockup and that is

not always the case. Higher mode demands, which are not mitigated by rocking at the base, could

cause the superstructure to yield before the lockup is engaged, and thus system strength reduction

is not guaranteed. Since the superstructure in an HDR system is ductile, higher mode effects are

mitigated in a similar manner as the fixed-based structure.

3.2.4 Residual drifts in HDR systems

In a traditional flag shaped system, residual drifts are not possible if β<1. In an HDR system

residual drifts are possible if the superstructure yields. Compared to a fixed based structure, it is ex-

pected that for the same displacement demand, an HDR structure will have less residual drifts since

a portion of those drifts are self-centering. However, an analysis of the cyclic behavior of HDR

structures shows that while residual drifts will often be less than a comparable fixed-based struc-

ture, the displacement history, especially during the cycles which determine the peak displacement

response of the structure, is an important factor in determining residuals, and in certain circum-

stances residual displacements could be larger in HDR structures. Figure 3.6 illustrates this point

by comparing a conventional elastoplastic system to an HDR system for three different displace-

ment histories. In Figure 3.6(a), each system is displaced to δsmall, which yields the elastoplastic

system, and causes uplift but no lockup in the HDR system, and then unloaded. The elastoplastic

system is left with significant residual displacement, whereas the HDR system has zero residuals

since it is behaving in the fully self-centering range. In Figure 3.6(b), the structures are displaced

to a larger displacement, δlarge, which yields the elastoplastic system as well as the HDR system,

and then unloaded. Here, since a significant portion of the drifts are self-centering in the HDR

structure, it experiences less residual displacement than the elastoplastic system. Finally, in Figure


SeismicForce

Elasto-Plastic:

HDR:

(a) Displace to δsmall

, unload (b) Displace to δlarge

, unload (c) Displace to δlarge

, displace back to negative yield displacement, unload

Displacement

Displacement

SeismicForce

residualdisplacement




SeismicForce

no residualdisplacement

δsmall

δsmall

δlarge

δlarge

δlarge

δlarge

no residualdisplacement

Figure 3.6: Effect of loading history on residual drifts in conventional and HDR structures

3.6(c), the systems are displaced to δlarge, displaced back to the negative yield displacement of the

elastoplastic system, and then unloaded. In this case, the residual displacement of the elastoplastic

system is zero whereas the HDR structure has a significant residual deformation.

Thus, it is expected that HDR structures will have less residual drifts than fixed-based structures

when the lockup is not engaged. However, when the lockup is engaged, residual drifts could be

less or more, depending on the earthquake time-history. As well, it should be noted that higher

mode effects can cause residual deformations along the height of the superstructure regardless of

whether or not the lockup is engaged.

3.3 Chapter summary

This chapter overviewed themechanics of HDR structures. Equations for calculating drifts from

two ductile mechanisms – brace inelastic deformations and rocking – were presented. Combining

these mechanisms by incorporating a column lockup and energy dissipation device gives the HDR

3.3. CHAPTER SUMMARY 39

seismic system. Free-body diagrams were presented that showed the different system behavior

under smaller and larger amplitude displacements.

The P �Δ effect and its influence on the response of HDR systems was examined more closely

as well. These second order effects can reduce structural strength in an HDR systemmore than they

would in a conventional structure, although the magnitude of that strength reduction depends on

the earthquake time history after the first yielding excursion.

Finally, it was shown that loading history has a large effect on residual displacements in HDR

structures.

Chapter 4

Reference structures

This chapter presents the reference buckling restrained braced frames that were used as the basis

for the hybrid ductile-rocking modifications investigated in Chapter 5. This chapter also presents

the validation of these designs using nonlinear time-history analysis. The design of these frames

is outlined in Appendix A. The structures’ plan and elevation views are shown in Figure 4.1 , and

the frame elements for the 2, 4, and 6 frames are given in Table 4.1.

4.1 Modeling assumptions

The frames were modeled in 2D using OpenSees, the Open System for Earthquake Engineering

Simulation (McKenna, 2006). The BRBs were modeled using corotational truss elements. These

5@ 9.14 m

3@ 9

.14

m

BRBF

Plan View: Elevation of Braced Frames:

0 m

3.66 m

7.32 m

10.97 m

14.63 m

18.29 m

22.0 m BRB Column splices1.524 m above floorwhere indicated

6 Storey 4 Storey 2 Storey

N

Figure 4.1: Plan and elevation of reference structures

40

4.1. MODELING ASSUMPTIONS 41

Table 4.1: Reference frame elements

storey Beam Column BRB core area (cm2(in2))

6 storey frame 6 W18X55 W12X35 16.13 (2.50)5 W18X55 W12X35 25.8 (4.00)4 W18X55 W12X96 32.3 (5.00)3 W18X55 W12X96 38.7 (6.00)2 W18X55 W14X132 41.9 (6.50)1 W18X55 W14X132 45.2 (7.00)

4 Storey frame 4 W18X65 W10X45 22.6 (3.50)3 W18X65 W10X45 38.7 (6.00)2 W18X65 W12X96 48.4 (7.50)1 W18X65 W12X96 51.6 (8.00)

2 Storey frame 2 W18X60 W8X40 22.6 (3.50)1 W18X60 W8X40 32.3 (5.00)

elements included pin connections at the brace ends and take into account second order geometric

effects. The effective length of the braces was reduced by 30% to account for the connection

regionswhere the brace connects to the rest of the frame (beams and columns), which are considered

very stiff, and reduced again by 28.5% to account for the end connection regions of the braces

themselves, which are also very stiff, as described by Gray (2012). This decrease in effective length

was captured numerically by using a scaled young’s modulus ofE = 200000MPa/0.700/0.715 =

400000MPa. The braceswere calibrated to Specimen 99-3 fromBlack et al. (2002) subjected to the

SAC loading history. This specimen had a core cross-sectional area of 51.6 cm2 and a yield stress

of 419 MPa. Figure 4.2 shows the agreement between the OpenSees model and test results. Note

that the material model was calibrated assuming a scaled young’s modulus of E = 286000 Mpa

since the brace tests from Black et al. (2002) did not included the stiff connection regions that are

typical of a real braced frame.

Braces were assigned the Steel02 material model. The parameters that were used to capture the

Bauschinger effect and cyclic strain hardening parameters are shown in Table 4.2 . A schematic of

42 CHAPTER 4. REFERENCE STRUCTURES

Force(kN)

Displacement (mm)

Test results from Black et. al.(Specimen 99-3 subjected to SACbasic loading history)

OpenSees Steel02 model

-75 -50 -25 0 25 50 75

-3000

-2000

-1000

0

1000

2000

3000

Figure 4.2: Calibration of BRB OpenSees model to test results from Black et al. (2002)

Table 4.2: Material parameters for nonlinear buckling restrained brace elements

Steel Material Parameters Bauschinger Effect Parameters Cyclic Strain Hardening Parameters

Fy 248 MPa by 0.01 a1 0.035E 400000 MPa Ro 20 a2 1.0

cR1 0.9 a3 0.01cR2 0.15 a4 1.0

4.1. MODELING ASSUMPTIONS 43

2210 kN/g2340 kN

170.1 kN/g180.0 kN

180.2 kN/g205 kN

180.2 kN/g205 kN

180.2 kN/g205 kN

180.2 kN/g205 kN

180.2 kN/g205 kN

2340 kN/g2670 kN

2340 kN/g2670 kN

2340 kN/g2670 kN

2340 kN/g2670 kN

2340 kN/g2670 kN

9.14 m

masses and loadson frame:

masses and loadson leaning column:

note: except the continuouscolumns, all connections arepin connections

BRBs modeledwith corotational trusselement, Steel02 material

Columns and beams modeledwith elasticBeamColumn element,elastic steel material

6 @

3.6

6 m

rigid bar

Figure 4.3: Schematic of Numerical Model

the numerical model is shown in Figure 4.3.

All steel columns and beams were modeled using elasticBeamColumn elements, and thus col-

umn and beam yielding was not considered in these analyses. This assumption is reasonable since

both beams and columns were capacity designed to remain essentially elastic. Rigid offsets were

not considered in these models. Columns were modeled as continuous between floors, with splices

occurring above the third and fifth floors for the 4 and 6 storey structures. The splices were modeled

as fixed moment connections.

Concrete diaphragms were not explicitly modeled. Instead, rigid diaphragms were modeled by

connecting the leaning column to the SFRS with a truss bar with a large cross-sectional area. These

modeling choices are in contrast to conventional diaphragm modeling where all nodes on a given

floor are restrained to have the same horizontal displacement. This modeling assumption can inter-

fere with structures that are allowed to rock, since the vertical rocking motion has a corresponding

horizontal motion that is expected to be different on the side of the frame that uplifts than the side

that stays in contact with the ground. Results not shown here confirmed that the results of the fixed

base structures were negligibly sensitive to these alternative diaphragm modeling choices.

Leaning column elements were modeled to simulate the P-� effect. These elements were mod-


eled as pinned at each floor and therefore do not contribute to the lateral resistance of the structure.

This assumption is commonly used in numerical analysis, although it is noted that some interstorey

resistance is expected from continuous gravity columns in real buildings. The leaning column was

loaded to represent the gravity loads for half of the tributary area of the structure, minus the tribu-

tary area of the columns in the BRB frames. As per ASCE 7-10 (ASCE, 2010a), the loads applied

to the leaning column correspond to 1.0D+0.25L, whereD is the dead load and L is the unreduced

live load. It is understood that the 0.25 applied to the live load considers the statistical unlikelihood

of (1) the full live load occurring throughout the full building at the same time and (2) the full live

load and earthquake load occurring at the same time, as described byMalley et al. (2010). Note that

the seismic mass assigned to each storey did not include the effect of live load, as it was assumed

that live loads do not contribute to the inertial response of the structure. This assumption would not

be valid if the structure were used as a storage facility, as described in ASCE 7-10 section 12.7.2

(ASCE, 2010a).

Masses were modeled by lumping the horizontal mass associated with the tributary area of the

building that does not act on the SFRS on the leaning column. The rest of the mass was specified as

both vertical and horizontal and placed at the beam column joints. This mass modeling assumption

is discussed in more detail in Section 5.4.

4.2 Nonlinear time-history analyses

4.2.1 Selection and scaling of ground motions

Ten records were selected and scaled to match the DBE spectrum for the reference design

using the two-step scaling procedure described in Wiebe (2013) which is based on the procedure

from FEMA (2009). First, the records were normalized by their peak ground velocities. Next, a

lognormal distribution of spectral accelerations was assumed and the median acceleration spectrum

4.2. NONLINEAR TIME-HISTORY ANALYSES 45

Table 4.3: Earthquake records

ID# Earthquake Station Ma Rb

(km)Vs30c(m/s)

NGAIDd

DBEFactore

1 Northridge (1994) Beverly Hills - 14145 6.7 22 356 953 1.2092 Northridge (1994) Canyon Country - W

Lost Canyon6.7 22 309 960 1.702

3 Duzce, Turkey (1999) Bolu 7.1 44 326 1602 1.1914 Hector Mine (1999) SCSN 99999 Hector 7.1 30 685 1787 2.145 Imperial Valley (1979) UNAM/UCSD Station

6605 Delta6.5 20 275 169 2.16

6 Imperial Valley (1979) USGS Station 5058, ElCentro Array 11

6.5 20 196 174 2.62

7 Kobe (1995) Nishi-Akashi 6.9 20 609 1111 1.4108 Kobe (1995) Shin-Osaka 6.9 34 256 1116 2.559 Kocaeli, Turkey (1999) Duzce 7.5 41 276 1158 1.62110 Landers (1992) Yermo Fire Station 7.3 86 354 900 2.00

a. magnitudeb. hypocentral distancec. average shear wave velocity between 0 m and 30 m depthd. identifier for the PEER NGA database (2005)e. DBE scaling factor

was calculated as:

S̃a(T ) = exp

" nX

i=1

lnSa,i(T )

!/n

#(4.2.1)

where S̃a(T ) is the median acceleration spectrum, Sa,i(T ) is the acceleration spectrum of earth-

quake record i, and n is the number of records. Since this record suite is intended to be used to

analyze multiple structures with varying periods, the suite of records was scaled by a constant scal-

ing factor that minimized the sum of the logarithms of the ratios between the DBE design spectrum

and the median acceleration spectrum over a period range of 0.1 s to 2.0 s. The records are de-

scribed in Table 4.3 along with their DBE scaling factors. An MCE record suite was developed by

scaling the records a further 1.5 times. A third record suite was created by scaling the DBE suite


by 50% in order to represent a lower intensity, more frequent earthquake. Figure 4.4 shows the

time history of each ground motion scaled to DBE, along with their acceleration and displacement

spectra compared to the design spectrum. Figure 4.5 shows all ten acceleration and displacement

spectra with their mean and mean plus standard deviation, along with the design spectra.

4.2.2 Dynamic modeling assumptions

Inherent damping was modeled as 3% Rayleigh tangent stiffness damping in modes 1 and 2,

as used in Choi et al. (2008). A Krylov-Newton algorithm was used, as has been recommended

by Wiebe (2013) for dynamic analysis of structures with column uplift, which are presented in

subsequent sections of this thesis. A time step of 0.001 s was used for analysis which ensured

convergence of relevant parameters while allowing for the analyses to be performed in a timely

manner.

4.2.3 Reference structure results

Peak interstorey drifts

Figure 4.6 shows the peak interstorey drift profiles for the three references structures at the three

different earthquake scaling factors. Interstorey drifts were calculated at each analysis timestep as

the difference in adjacent storey displacements divided by the storey height. It is clear from the

results that all three of the structures had drift concentrations at the lower stories, with the upper

storeys contributing less to overall system lateral deformations. For the 2 storey structure, the

median peak drift at the first storey was significantly larger than the second storey at the DBE level,

and it failed the 2.5% drift requirement stipulated in the code. Similarly, in the 4 storey structure,

the drift in the first two storeys was clearly larger than the third and fourth storey and that structure

failed the 2.5% drift requirement. Finally, the six storey structure also had drift concentrations in

the lower stories and failed the code drift limit of 2%.


time (s) period (s) period (s)

Ground Motion ID#

Accelogram (g) AccelerationSpectrum (g)design spectrum in gray

Displacement Spectrum (mm)design spectrum in gray

1

2

3

4

5

6

7

8

9

10

−1

0

1 0.624

0

2

4

0

500

1000

−1

0

1 0.82

0

2

4

0

500

1000

−1

0

1 0.979

0

2

4

0

500

1000

−1

0

1 0.721

0

2

4

0

500

1000

−1

0

1 0.758

0

2

4

0

500

1000

−1

0

1 0.954

0

2

4

0

500

1000

−1

0

1

−0.718

0 1 2 30

2

4

0

500

1000

−1

0

1 0.620

0 1 2 30

2

4

0

500

1000

0 20 40 60 80 100

−1

0

1

−0.580

0 1 2 30

2

4

0 1 2 30

500

1000

0 20 40 60 80 100

−1

0

1

−0.490

0 1 2 30

2

4

0 1 2 30

500

1000

Figure 4.4: Summary of ground motion suite scaled to DBE


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 30

100

200

300

400

500

600

700

800

900

1000

Sa (g)

period (s)

Sd (mm)

DBE median median+ std. dev.

DBEmedianmedian + std. dev

Figure 4.5: Acceleration and displacement spectra of scaled ground motion suite

3

4

3

4

3

4

0 2 4 60

1

2

3

4

5

6

0 2 4 60

1

2

3

4

5

6

0 2 4 60

1

2

3

4

5

6

5

6

5

6

5

6

0 2 4 60

1

2

3

4

0 2 4 60

1

2

3

4

0 2 4 60

1

2

3

4

333

444

333

444

333

444

000 2 4 66055

166

2

0000 2 4 666055

166

2

0000 2 4 666055

166

2

storeymedian of 10 recordsmedian + st. dev.

peak interstorey drift (%)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.6: Peak interstorey drift results for reference structures


3

4

3

4

3

4


0 2 40

1

2

3

4

5

6

0 2 40

1

2

3

4

5

6

0 2 40

1

2

3

4

5

6

5

6

5

6

5

6

0 2 40

1

2

3

4

0 2 40

1

2

3

4

0 2 40

1

2

3

4

333

444

333

444

333

444

000 2 44055

1

2

0000 2 444055

1

2

0000 2 444055

1

2

peak displacement (% structure height)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.7: Peak floor displacement results for reference structures

Peak floor displacements

Figure 4.7 shows the peak displacements of each structure, normalized by the structure height.

As outlined in the description of the interstorey drift profiles, concentrations of drift were evident

in the lower stories of the structures, and became progressively more severe as the earthquake

intensity was increased.

Peak residual interstorey drifts

Figure 4.8 shows the median peak residual drift profiles for the structures. The residual drifts

were calculated for each record by taking the difference between storey displacements at the end

of the record (the records were padded with zeros to allow sufficient time for the transient response

to damp out) and dividing that value by the storey height. In their study of residual deformations

in BRBs, MRFS, and SCED braced frames, Erochko et al. (2011) used 0.5% residual drift as a

criteria for whether or not buildings would require expensive repairs or even demolition after a

seismic event. This value is based on a previous study by McCormick et al. (2008) that included


3

4

3

4

3

4

0 0.5 1 1.50

1

2

3

4

5

6

0 0.5 1 1.50

1

2

3

4

5

6

0 0.5 1 1.50

1

2

3

4

5

6

5

6

5

6

5

6

0 0.5 1 1.50

1

2

3

4

0 0.5 1 1.50

1

2

3

4

0 0.5 1 1.50

1

2

3

44 4 4

333

444

333

444

333

444

000 0.5 1 1.51.5055

1

2

0000 0.5 1 1.51.51.5055

1

2

0000 0.5 1 1.51.51.5055

1

2


peak residual drift (%)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.8: Residual drift results for reference structures

a study of one building at Kyoto University in Japan as well as a review of previous research.

Their study included physiological and psychological effects of residual deformations on building

occupants. They concluded that if residual drifts were greater than 0.5% after an earthquake then

it was likely less expensive to to rebuild than repair the structure.

Considering this criteria of 0.5% residual drifts, it can be seen from Figure 4.8 that under the

DBE suite the 2 storey structure did not perform particularly well, with a peak median peak residual

drift greater than 0.5%, and only two records below this threshold. Under MCE, the peak residual

drift at the first storey was almost 1.0%. For the 4 storey structure, the results were worse. While

the residuals were below the criterion for the 0.5 DBE suite of records, they exceeded it under DBE

and MCE. The 6 storey structure just failed the criterion under 0.5 DBE (although five out of ten

records were below this threshold), and greatly exceeded it under DBE andMCE. In fact the results

for the 6 storey structure were quite severe— under DBE the residual drifts were greater than 0.5%

for the first three stories, and under MCE they were much greater for the first four storeys. The

fact that the residual drift results worsened as structural height was increased is consistent with past

work on the topic where it was reported that residual drifts are highly sensitive to building height


storeymedian of 10 recordsmedian + st. dev.nominal storey shearoverstrength storey shear

0 2000 40000

1

2

3

4

5

6

0 2000 40000

1

2

3

4

5

6

0 2000 40000

1

2

3

4

5

6

5 5

6

5

6

0 40000

1

2

3

4

0 2000 40000

1

2

3

4

0 2000 40000

1

2

3

4

3

4

3

4

3

4

000 2000 40000

2

0000 2000 40000

1

2

0000 2000 40000

1

2

0

2

peak storey shear (kN)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.9: Storey shear results for reference structures

(see Erochko et al. (2011)) in part due to the fact that the P-� effect, which increases with building

height, exacerbates residual deformations.

Peak storey shears

Figure 4.9 shows the peak storey shears for all the structures. The storey shears were calculated

at each analysis timestep as the sum of the column shears and the horizontal component of brace

forces, assuming undeformed geometry. As well as the results from the nonlinear time-history anal-

yses, the nominal and overstrength storey shear profiles are shown for comparison. The nominal

storey shear profiles were calculated from the nominal brace properties, assuming no overstrength.

The overstrength storey shears were calculated by considering the overstrength factors used in the

frame design, although the increased yield stress used for design was not considered here since the

yield force during these analyses was explicitly modeled, and since it is the expected brace forces

and not the capacity design forces that are more of interest when discussing fuse inelastic demands.

At the upper stories the storey shears were not much greater than the nominal storey shears for all


0 2e4 4e4 6e40

2

4

6

0 2e4 4e4 6e40

2

4

6

0 2e4 4e4 6e40

2

4

6

6 6

0 2e4 4e4 6e40

4

0 2e4 4e4 6e40

2

4

0 2e4 4e4 6e40

2

4

4 4 4

00 2e4 4e4 6e46e40

2

000 2e4 4e4 6e46e46e40

2

000 2e4 4e4 6e46e46e40

2

6e4

storeymedian of 10 recordsmedian + st. dev.nominal overturning momentoverstrength overturning moment

peak overturning moment (kN-m)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.10: Overturning moment results for reference structures

earthquake intensities. As earthquake intensity was increased, however, the storey shears at the

lower storeys increased and even exceeded the overstrength storey shears for a few records un-

der the DBE and MCE earthquakes. This result demonstrates how drift concentrations can cause

greater force demands than expected in the design phase at a few stories.

Peak storey overturning moment

Figure 4.10 shows the peak overturning moment for all the structures. The storey overturning

moment was calculated at each analysis timestep along the centerline of the frame at the bottom

of every storey and included the contributions from column the axial forces and column bending

moments as well as from braces when relevant. Like the storey shears in Figure 4.9 and the column

forces in Figure 4.11, the standard deviation of the overturning moments were very small since

forces in these systems were effectively limited by the strength of the BRBs. The nominal and

overstrength overturning moments are shown for reference, and were calculated considering the

forces from Figure 4.9. For all the frames, the moment values did not increase greatly with increas-


3

4

3

4

3

4

0 2000 40000

1

2

3

4

5

6

0 2000 40000

1

2

3

4

5

6

0 2000 40000

1

2

3

4

5

6

5

6

5

6

5

6

0 2000 40000

1

2

3

4

0 2000 40000

1

2

3

4

0 2000 40000

1

2

3

44 4 4

333

444

333

444

333

444

000 2000 4000055

166

2

0000 2000 4000055

166

2

0000 2000 4000055

166

2


peak column compression (kN)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.11: Peak column compression results for reference structures

ing earthquake intensity, although as with the storey shears, the overturning moments increased at

the lower storeys more than the upper storeys with increasing earthquake intensity.

Peak column compression

Figure 4.11 shows the peak column compressive forces for all the structures. Like the storey

shears, the column forces have very low standard deviations since gravity loads were unchanging

throughout the analyses, and brace forces were limited by their yield strength and low post-yield

stiffness. In general, the column force demands from the seismic loads were much less than their

capacities. The first floor column capacities for the 2, 4, and 6 storey frames are 1809 kN, 5350

kN, and 7760 kN respectively (see Table A.14). These values are much larger than the first floor

peak column forces from Figure 4.11. This discrepancy reflects the large conservatism present in

code-based column seismic design. The capacity design procedure for columns assumes that all

braces achieve their full overstrength at the same time and in the same loading direction which is

unlikely during an earthquake event. This conservatism in column design may however be war-

ranted, considering that it ensures the integrity of the gravity bearing system.


3

4

3

4

3

4

0 0.5 1 1.50

1

2

3

4

5

6

0 0.5 1 1.50

1

2

3

4

5

6

0 0.5 1 1.50

1

2

3

4

5

6

5

6

5

6

5

6

0 0.5 1 1.50

1

2

3

4

0 0.5 1 1.50

1

2

3

4

0 0.5 1 1.50

1

2

3

44 4 4

333

444

333

444

333

444

000 0.5 1 1.51.5055

1

2

0000 0.5 1 1.51.51.5055

1

2

0000 0.5 1 1.51.51.5055

1

2


peak storey accelerations (g)

2 Storey

4 Storey

6 Storey

0.5 DBE DBE MCE

Figure 4.12: Acceleration results for reference structures

Peak storey accelerations

Figure 4.12 shows the peak storey accelerations for all of the structures. The acceleration results

for all of the structures under the 0.5 DBE suite of records clearly show that the median peak

accelerations were of similar magnitude as the peak ground accelerations (PGA). The BRB frame

on average did not magnify or diminish the ground acceleration for the lower level earthquakes.

Under DBE and MCE, however, the peak storey accelerations were, on average, lower than the

PGAs. This result is consistent with Choi et al. (2008)’s study on the peak acceleration response

of BRB frames. This is a very attractive feature of BRB frames that is discussed in more detail in

the following chapter where the proposed hybrid ductile-rocking system is discussed.

4.3 Chapter summary

This chapter presented the numerical modeling assumptions used to generate an OpenSees

model of the three reference frames. The selection and scaling of ground motions was presented,

and the time-history analysis scheme and dynamic modeling assumptions were explained.


The results of the dynamic time-history analyses were presented. The displacement results

demonstrated that concentrations of inelasticity in the lower stories of the reference frames was

an important issue influencing the overall performance of these frames, especially for the 6 storey

structure. As well, the structures fared poorly with regard to residual drifts when compared to

the suggested 0.5% criteria for repairs/demolition from the literature. The system force results

confirmed that the buckling restrained braces effectively reduced the force demand on the system,

although some increased force demands at the lower storeys, where drift concentrations occurred,

were noted.

Chapter 5

Parametric study

This chapter presents the results of a parametric study investigating the seismic response of

BRB frames with the hybrid ductile-rocking system compared to the reference structures that were

presented in Chapter 4. The study explores how the energy dissipation parameter and the amount

of allowable uplift before lockup affect the three reference structures’ seismic performance.

5.1 Overview of parametric study

5.1.1 Parameters

The three parameters considered in this study were the energy dissipation parameter, β, the base

rotation angle before lockup, θlock, and the structure height. Table 5.1 shows the chosen parameters.

β is directly related to the energy dissipation element strength, ED, as a function of the self-weight

acting as a restoring force, Wself , described in Section 3.1. The required ED strength in kN for a

given β can be determined by substituting Equations 3.1.1 and 3.1.4 into Equation 3.1.6 and solving

for ED:

ED =�WselfdwdED(2� β)

(5.1.1)

Six values of � were chosen. � of 0 corresponds to structures with no supplemental energy

56

5.1. OVERVIEW OF PARAMETRIC STUDY 57

Table 5.1: Parameters for parametric study

Storey β θlock(δup)

6 (T1=1.030 s) 0 0.125% (11.43 mm)4 (T1=0.624 s) 0.5 0.25% (22.9 mm)2 (T1=0.396 s) 1 0.5% (45.7 mm)

1.2 1% (91.4 mm)1.4 2% (182.9 mm)1.6 4% (366 mm)

dissipation. In these cases, uplift occurred as soon as the columns were decompressed. � of 0.5

and 1 represent ED values for which residual base rotations were not possible. � of 1.2, 1.4, and

1.6 correspond to systems where residual base rotations were possible.

While the energy dissipation elements modeled in this study could represent a variety of differ-

ent hysteretic fuse options (eg. friction devices or steel yielding devices), a cast steel yielding con-

nection based on the Scorpion Yielding Brace System (YBS) (Gray (2012)) was used as a reference

for this study as well as the basis of the detail design in Chapter 6. The YBS was developed as an

alternative to BRBFs that featured improved ductility and the potential for increased collapse per-

formance because of a post-yield stiffening and strengthening effect at large displacements (Gray

et al. (2014)). The YBS consists of a cast steel connection that is specially designed to concentrate

nonlinear demands in triangular yielding fingers. Figure 5.1 shows two varieties of this device

during laboratory testing and in a steel building under construction. The nominal properties of the

available YBS devices are shown in Table 5.2 .

The elastic stiffnesses of the energy-dissipation elements were determined by comparing the

calculated ED to the available YBS devices from Table 5.2. The stiffness of the YBS device with

the closest yield strength was scaled by the ratio of the target ED to the YBS device strength. For

example, the ED value for the 6-storey structure with � = 0.5 is 402 kN. The closest YBS device

available is the SYC-100 device which has a yield strength of 400 kN and a stiffness of 156 kN/mm.

Thus the stiffness of the ED used in this example is 402/400 ⇤ 156 = 156.7 kN/mm. The chosen

58 CHAPTER 5. PARAMETRIC STUDY

(a) (b)

Figure 5.1: Photos of Yielding Brace System: (a) being tested at the University of Toronto (photoby Justin Binder); (b) implemented as the braced frame of a building (photo by Dr. Michael Gray)

Table 5.2: Yielding brace system device characteristics

Device NominalForce (kN)

kdevice(kN/mm)

�design�device(mm)

SYC-33 149 62 38.0SYC-50 223 92 38.0SYC-75 334 128 39.5SYC-90 400 156 39.8SYC-100 445 170 39.5SYC-120 534 208 39.8SYC-129 574 211 39.5SYC-150 667 261 39.8SYC-172 765 281 39.5SYC-186 827 319 39.6SYC-215 956 351 39.5SYC-248 1103 425 39.6SYC-310 1379 532 39.6

5.1. OVERVIEW OF PARAMETRIC STUDY 59

Table 5.3: Value of ED for each building model

β 6-storey ED (kN)(ED stiffness(kN/mm))

Wself=2410 kN

4-storey ED (kN)(ED stiffness(kN/mm))

Wself=1590 kN

2-storey ED (kN) (EDstiffness (kN/mm))Wself=770 kN

0.5 402 (156.7) 265 (109.3) 128 (53.4)1.0 1205 (464) 795 (292) 385 (150.2)1.2 1808 (697) 1193 (459) 578 (212)1.4 2812 (1085) 1855 (716) 898 (330)1.6 4820 (1860) 3180 (1227) 1540 (594)

ED strengths and stiffnesses are shown in Table 5.3 . Fuse displacement limits and low-cycle

fatigue life were not modeled during this study. Additionally, the post-yield stiffness of the energy

dissipation, bed, was modeled as zero even though the reference YBS fuses have a non-negligible

post-yield stiffening and strengthening effect. The reason for not modeling the post-yield stiffness

in this parametric studywas so that the energy dissipation element strength and lockup displacement

could be studied independently, since a post-yield increase in energy dissipation element strength

could cause the BRBs to engage. The post-yield stiffness of the energy dissipation elements was

explicitly modeled in Chapter 6.

Another way to consider the effect of adding energy dissipation is in the form of the ratio

between the rocking strength,Mb,rock, to the BRB frame fixed base yield moment,My, which can

be determined by assuming all the braces yield under nominal properties, finding the corresponding

storey forces, and summing the moments about the base of the frame. Table 5.4 shows this ratio. It

can be seen that as you increase �, you also approach the yieldingmoment until eventually you have

a structure where the rocking joint engages only after some or all of the braces have yielded. This

scenario could be beneficial in reducing the system forces caused by brace overstrength, although

significant energy dissipation strength is required.

It should be noted that the fundamental periods given in Table 5.1 are slightly different from

the values determined during design (see Figure A.1) since the OpenSees model included features


Table 5.4: Ratio of rocking moment to yield moment

6-storey 4-storey 2-storey

β ED (kN) Mb,rock

My

ED (kN) Mb,rock

My

ED (kN) Mb,rock

My

0 0 0.389 0 0.318 0 0.4530.5 402 0.518 265 0.424 128 0.6041 1205 0.778 795 0.636 385 0.9061.2 1808 0.972 1193 0.795 578 1.1331.4 2812 1.296 1855 1.060 898 1.5101.6 4820 1.944 3180 1.591 1540 2.266

that the SAP2000 models used in the design phase, did not. In particular, the SAP2000 models did

not include P-Δ columns, as second order effects were checked using code equations during the

design phase.

Six values of ✓lock were chosen ranging from 0.125%, which is just over 1 cm of allowable

uplift, to 4%, where the lockup device was not likely to be engaged except potentially under large

rocking demands. The parametric study included a total of 111 different structures including the

reference frames. Push-pull responses of each building model are shown in Appendix B. These

push-pull analyses were performed in the samemanner as will be further described in Section 5.2.2,

and serve as an overview of all the building models analyzed in this study.

5.1.2 Modeling assumptions

The structures were modeled by modifying the OpenSees models of the reference structures

described in Chapter 4. The pinned boundary condition at the base of the columns was replaced

with two nodes in the same location. One node was fully fixed, and the other was free to uplift but

restrained in the horizontal direction in order to transfer base shear. Three zero-length elements

were placed in parallel in order to model the gap, lockup, and energy dissipation. A compression-

only element with a compressive stiffness of 1000 kN/mm was used to allow vertical deformation

and model the compressive stiffness of a nearly rigid foundation, and is referred to as the ’con-

5.2. RESPONSE OF EXAMPLE STRUCTURE 61

restrainedhorizontally

(a) (c) (d)(b)

zero

leng

th

zero length

δup

force force

deformation deformation

(d) force

deformation

9.14 m

6 @

3.6

6 m

Figure 5.2: Schematic of rocking joint modeling assumptions: (a) 6-storey frame; (b) close-up ofcolumn-foundation connection; (c) energy dissipation element; (d) compression-only element; (e)tension-only element with gap

tact’ element since it provided compressive contact between the column and foundation. A yield-

ing element using the Steel01 material in OpenSees was used to model the energy dissipation. A

tension-only element with an initial gap was used to model the lockup. The tensile stiffness of the

lockup element was set at 1000 kN/mm to represent a rigid condition after the lockup was engaged.

The initial gap was set to correspond to the values of θlock that are given in Table 5.1. A mass of 5

kN/g was added to the uplift nodes in order to avoid issues related to acceleration spikes that can

occur in flag-shaped systems with large mass differentials in uplifting structures (see Wiebe and

Christopoulos (2010)). Figure 5.2 shows a schematic of the modeling choices used in this study.

5.2 Response of example structure

In order to better illustrate the behaviour of the proposed HDR system, the response of the

6-storey frame is examined when subjected to a pushover analysis, push-pull analysis, and one

time-history record. For this example, ✓lock = 1%, and � = 1.0.

5.2.1 Pushover response

Figure 5.3 shows the pushover response of the fixed base frame and the corresponding HDR


Percent Roof Drift (%)

Bas

e Sh

ear

(kN

)

Pushover Response

Fuse Force-Deformation

Def. (mm)

Forc

e (k

N)

Contact Force-Deformation

Def. (mm)

Lockup Force-Deformation

Def. (mm)

First Storey Left Brace Force-Deformation

Def. (mm)

Forc

e (k

N)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

2500

3000

Fixed base structure

HDR structure1% lockup, ED=1205 kN

−100 0 100

−1000

0

1000

−100 0 100

−1000

0

1000

−100 0 100

−2000

0

2000

0 50 100 1500

500

1000

1500

2000

column Decompression

fuse yield

post-uplift stiffnesscorresponds to P-Deltaeffect

(a)

(b)

(e)

(d)(c)

Figure 5.3: Pushover response of 6-storey fixed base frame and 6-storey HDR frame with ✓lock =1% and � = 1.0: (a) base shear versus roof drift; (b) fuse force-deformation; (c) contact elementforce-deformation; (d) lockup force-deformation; (e) first storey left brace force-deformation


frame along with force-deformation plots for the zero-length elements of the HDR frame. The

pushover was performed assuming an inverted triangular force distribution. This distribution was

chosen since fixed and rocking structures have slightly different mode shapes (see Wiebe (2013)),

and these analyses are meant to show fundamental behaviour rather than predict performance or

aid in design. Figure 5.3(a) shows the base shear versus roof drift for the two frames. Note that the

difference in elastic stiffness between the two frames was due to the fact that the fixed base frame

was modeled by assuming pinned base conditions at the column-foundation connections, while the

HDR frame was modeled with a compression gap stiffness of 1000 kN/mmwhich contributed some

elastic flexibility to the system. After column decompression the stiffness of the HDR system was

defined by the fuse elastic stiffness up until the fuse yielded, at which point the system stiffness

was solely affected by the P-� effect (since the fuses were modeled with no post-yield stiffness).

In Section 3.2.3, it was explained that the P-� effect can cause a reduction in system strength

during a pushover analysis since the rocking and lockup cause the superstructure to yield at a signif-

icantly larger deformation than under fixed base conditions. The strength reduction is determined

by the P-� effect, which is noted in the Figure 5.3(a) by drawing a line between the yield points of

the two systems, parallel to the post-rocking stiffness of the HDR system.

Figure 5.3(b) shows the the fuse force-deformation plot which started in compression due to

the gravity loads, and deformed elastically until it yielded in tension at 1205 kN. Figure 5.3(c)

shows the elastic-no-tension contact element, which had a large elastic stiffness in compression

and zero stiffness in tension. Figure 5.3(d) shows the lockup force which had no elastic stiffness

in compression and tension until the deformation exceeded the gap corresponding to 1% rotation

(91.4 mm vertical deformation) at which point it had a high tensile stiffness. It should be noted that

the sum of gravity loads acting on the column at the beginning of the analysis was 1205 kN, which

corresponded to the sum of the initial compressive forces in the contact and fuse elements. Finally,

Figure 5.3(e) shows the force-deformation for the first storey left braces, clearly showing how the

deformation demand was smaller for the HDR structure, even though both structures were pushed

to 2% roof drift.


5.2.2 Push-pull response

A push-pull response was performed to further clarify the behaviour of the HDR system. An

inverted triangular force distribution was used, and the structure was cycled at 0.5%, 1%, 2% and

4% roof drift. Figure 5.4 shows the results of the push-pull analysis and accompanying component

hystereses. The base shear versus roof drift can be seen in Figure 5.4(a). The difference between

the fixed base andHDR structures is clear during the lower amplitude displacements where the flag-

shaped hysteretic profile is visible. Under larger amplitude displacements, the lockup engaged the

superstructure, although the base shear experienced by the HDR structure was ultimately lower than

that which was experienced by the fixed base structure due to the P-� effect (as described in Section

3.2.3), and due to the fact that the lower deformation demand on the braces limited the amount of

overstrength developed. Figure 5.4(b) shows the hysteresis for the left fuse. Once activated, the

fuse alternated between its tensile and compressive yield forces while experiencing only tensile

deformations (small compressive deformations were due to compatibility with the compressive

contact element). Figure 5.4(c) shows the elastic-no-tension contact element, which cycled between

high compressive stiffness and zero tensile stiffness. Figure 5.4(d) shows the lockup element, which

cycled between zero stiffness and a high tensile stiffness. Finally, Figure 5.4(e) shows the brace

hysteresis for the first storey left braces. For the same maximum roof displacement, the HDR

structure’s braces experienced less deformation demand. Notably, during the smaller amplitude

displacements (less than those which engaged the lockup), the HDR structure’s braces experienced

zero plastic deformations.

5.2.3 Sample record

The response of this example structure to record ID#1 scaled to DBE (see Figure 4.4) is pre-

sented here to further illustrate the seismic response of the proposed HDR system.

Figure 5.5 shows the behaviour of the foundation elements and the applied ground motion.

Figure 5.5(a) shows the contact deformation, which was equal to the foundation node displacement

and the displacement of all the foundation elements since they acted in parallel. The limit on uplift



Bas

e Sh

ear

(kN

)

Pushover Response

Fuse Hysteresis

Def. (mm)

Fo

rce

(kN

)

Contact Hysteresis

Def. (mm)

Lockup Hysteresis

Def. (mm)

First Storey Left Brace Hystereses

Def. (mm)

Forc

e (k

N)

−5 −4 −3 −2 −1 0 1 2 3 4 5

−3000

−2000

−1000

0

1000

2000

3000

−100 0 100

−1000

0

1000

−100 0 100

−5000

0

5000

−100 0 100−5000

0

5000

−200 −100 0 100 200−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

Fixed base structure

HDR structure1% lockup, ED=1205 kN

(a)

(b)

(e)

(d)(c)

flag shaped duringsmaller amplitudedisplacements

Figure 5.4: Push-pull response of 6-storey fixed base frame and 6-storey frame with ✓lock = 1%and � = 1.0: (a) base shear versus roof drift; (b) fuse hysteresis; (c) contact element hysteresis;(d) lockup hysteresis; (e) first storey left brace hysteresis


0 2 4 6 8 10 12 14 16 18 20−50

0

50

100

0 2 4 6 8 10 12 14 16 18 20−4000

−3000

−2000

−1000

0

1000

0 2 4 6 8 10 12 14 16 18 200

500

1000

1500

2000

2500

time (s)

0 2 4 6 8 10 12 14 16 18 20−3000

−2000

−1000

0

1000

2000

3000

time (s)

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

0.624

Contact Deformation

(mm)

Contact Force(kN)

Lockup Force(kN)

Fuse Force(kN)

(a)

(b)

(c)

(d)

Groundacceleration

(g)

(e)

Left side

Right side

time (s)

Figure 5.5: Foundation element behaviour: (a) elastic-no-tension contact element deformation;(b) contact force; (c) lockup force; (d) fuse force; (e) ground acceleration (ID#1)


from the lockup in this model was set to 91.4 mm, and the lockup was only engaged once on the left

side during the time-history. Figure 5.5(b) shows the elastic-no-tension element force for the left

and right sides of the frame. The elements began in compression due to the initial gravity loading

on the building, and subsequently went into more and less compression as the frame swayed back

and forth, with tension forces prohibited by the uplift. Figure 5.5(c) shows the lockup force, which

was only engaged on the left side once and corresponded to when the left uplift deformation reached

91.4 mm in Figure 5.5(a). Finally, Figure 5.5(d) shows the fuse force on the left and right sides of

the frame. The fuse forces were limited by the yield force of 1205 kN. The compression in the fuses

at the beginning of the time-history was due to the compatibility of deformations between the fuse

and the elastic-no-tension contact element under gravity loads. The sum of the initial compression

in the contact element and the fuse force was equal to the total gravity load acting on the column,

in this case 1205 kN (coincidentally the same value as the fuse yield force).

The roof displacement response and corresponding system forces are shown in Figure 5.6, as

compared to the fixed base structure. The peak roof displacement of the HDR structure was greater

than that of the fixed structure, as observed in Figure 5.6(a). That being said, the rigid body rotation

of the structure shown in gray (calculated as the difference in base uplift between the left and right

side of the frame divided by the frame width) accounts for most of the lateral displacement of the

frame. The difference between the roof displacement and rigid body rotation is shown as a dashed

line and further demonstrates how most of the system displacement is caused by the base rotation.

Figure 5.6(b) compares the base shear response between the fixed and HDR structures. The simi-

larity between the magnitude of the responses is reasonable since the strength of the rocking joint

was close to that of the superstructure (see Figure 5.3), and forces due to higher mode effects were

capped off by the BRBs in both frames. The overturning moment response, shown in Figure 5.6(c),

demonstrates how the rocking joint successfully limited the response, except when the lockup was

engaged and a spike in the base moment was observed (which was subsequently limited by the

braces, as in the fixed structure).

The response of the hysteretic elements in the frames is shown in Figure 5.7. Figure 5.7(a)


0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time (s)

0 2 4 6 8 10 12 14 16 18 20−4000

−2000

0

2000

4000

0 2 4 6 8 10 12 14 16 18 20-6e4

-4e4

-2e4

0

2e4

4e4

6e4

time (s)

RoofDisplacement

(%)

Base Shear(kN)

OverturningMoment(kN-m)

(a)

(b)

(c)

lockupengaged

lockupengaged

Fixed base roof displacementHDR roof displacementHDR rigid body rotationHDR roof disp. minus rigid body rotation

Fixed base structureHDR structure

Rixed base structureHDR structure

Figure 5.6: Roof displacement and system forces of example model compared to fixed base re-sponse: (a) roof displacement and base rotation of fixed and HDR structures; (b) base sheartime-history for fixed base and HDR structures; (c) overturning moment time-history for fixedbase and HDR structures


−100 −50 0 50 100−2000

0

2000

−100 −50 0 50 100−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−100 −50 0 50 100−2000

0

2000

−100 −50 0 50 100−2000

0

2000


6th StoreyBrace Force

(kN)


(kN)


(kN)

3rd StoreyBrace Force

(kN)

2nd StoreyBrace Force

(kN)

1st StoreyBrace Force

(kN)

Deformation (mm)

Fuse Force(kN)

Deformation (mm)

(a) Left Side of Frame Right Side of Frame

(b)

Figure 5.7: Hysteretic response of 6-storey fixed base and HDR frames: (a) BRB hystereses; (b)fuse hystereses


shows the brace hysteresis. From this example, it is clear that the addition of limited base rocking

has reduced the displacement demand relative to the fixed base structure, most evidently in the

lower stories. As well, the residual drifts were visibly reduced, again especially in the lower stories.

Figure 5.7 (b) shows the fuses hystereses for both sides of the frame. The left side fuse was limited

to 91.4 mmwhich was as much as was allowed by the lockup. The right side fuse did not experience

a displacement large enough to engage the lockup on that side.

5.3 Results of parametric study

This section presents the statistical results of the entire parametric study. The response pa-

rameters that were studied were the peak interstorey drifts, peak base rotations, peak drifts minus

base rotations (drift demand on the frame), residual drifts, residual base rotations, global uplift,

number of records that engaged the lockup, energy dissipated by the braces, peak foundation ten-

sion, peak column compression, peak base overturning moment, peak base shear, and peak storey

accelerations.

Peak interstorey drifts

Figure 5.8 shows the median and median plus standard deviation peak interstorey drift results.

The results are organized in groups corresponding to a given value of �, with each group showing

the effect of increasing ✓lock with each bar that is plotted within the group. The black bar always

refers to the corresponding fixed base frame, and is repeated in each group for reference.

A preliminary understanding of these results can be attempted in the context of the equal dis-

placement observation. Considering that the displacements associated with self-centering struc-

tures tend to be similar to those of plastic systems (as was observed in a study of SDOF flag-shaped

and elastoplastic systems by Christopoulos et al. (2002)), a combination of the two systems using

a lockup might mean that the total drifts from the two mechanisms are expected to be similar. That

being said, the drifts from self-centering systems are dependent on various system parameters (�,

5.3. RESULTS OF PARAMETRIC STUDY 71

0

5

0

5

0

5

0

5

0

5

0

5

0

5

0

5

0

5

0.5 DBEmedian peakinterstorey

drift (%)

DBEmedian peakinterstorey

drift (%)

MCEmedian peakinterstorey

drift (%)


drift (%)


drift (%)


drift (%)


drift (%)


drift (%)


drift (%)

(a) 2 storey

(b) 4 storey

(c) 6 storey

fixed

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

code drift limit = 2.5%



Figure 5.8: Median peak interstorey drift results: (a) 2-storey frame; (b) 4-storey frame; (c) 6-storey frame


↵, period, strength), and thus the combination of deformations from the two different ductile mech-

anisms may not be the same, depending on the system being considered. Wiebe and Christopoulos

(2014) studied SDOF flag-shaped systems and demonstrated how in these systems, peak displace-

ments are highly susceptible to changes in period and system strength. This makes sense in the

context of Figure 5.8(a) which shows the peak drift parametric study results for the 2-storey frame.

For the first three � values, the peak drifts clearly increased with increasing ✓lock. This tendency

means that while the structure’s median peak interstorey drift was only slightly greater than the

2.5% peak drift limit from the code, adding column uplift caused the structures to fail that criteria

by a larger margin for most of the ✓lock values under the DBE record set. That being said, the drift

values were still quite similar to the fixed structure, and demonstrate that the peak drifts remained

relatively reasonable even when no supplemental energy dissipation was provided (� = 0). As

� was increased, the rocking joint strength became larger than the BRB system strength, and thus

minimal change in peak drifts was observed between the fixed structure and rocking structures. It

should be noted that the high � models had similar or identical results to the fixed base models for

all of the response parameters.

For the 4-storey structure (Figure 5.8(b)), the results were similar. While the results demonstrate

a trend of increasing drifts with increased ✓lock under the 0.5DBE record suite, the peak drifts were

reduced with increased � (once again, for larger � the results are very similar to the fixed base

results). Under the DBE and MCE suites, however, the peak drifts results were very close to the

fixed value, with some cases of reduced peak drift for higher ✓lock values. For example, while the

fixed base 4-storey structure just failed the 2.5% code criteria, it actually passed that criteria with

✓lock = 1.0% and � = 0.

For the 6-storey structure (Figure 5.8(c)), the results were very similar for the fixed and HDR

structures under 0.5 DBE. Under DBE, the results illustrate the different nature of drifts from the

two ductile mechanisms of rocking and brace yielding. For example, consider the 6-storey models

with � = 0. Figure 5.9 shows the median drift profiles for the ✓lock =0%, 0.125%, 1%, and 4% of

for all three record suites. Under the DBE suite, the fixed base profile has drift concentrations in


0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0.5 DBE Drift (%) 1.0 DBE Drift (%) MCE Drift (%)

storey

fixedθ

lock=0.125%

θlock

=1.0%

θlock

=4.0%

Figure 5.9: Median drift profiles for 6-storey structure, � = 0

the lower stories as was previously described in Chapter 4. With just a small amount of allowable

rocking (✓lock =0.125%), the median peak drift in the first storey was reduced by an amount larger

than 0.125%. This result highlights the second-order nature of drift concentrations since a small

amount of rocking yielded a disproportionate benefit in reduction of drift concentrations. As ✓lock

was increased, the drift concentrations were further reduced, but the drifts in the upper stories were

increased due to the increased amount of rocking motion. For ✓lock = 4.0%, the drifts were pri-

marily contributed from the rocking motion, and the median peak drift profile was much straighter

since rocking motion was not associated with drift concentrations. The MCE results are similar

to the DBE results, although the ✓lock = 4.0% median peak drift profile has a slight concentration

in drifts in the first storey. This results suggests that a large earthquake intensity can cause some

concentrations of drift in HDR structures even if a large amount of rocking is allowed since the

BRBs can still concentrate the demand from higher lateral modes. That being said, the MCE re-

sults shown in Figure 5.9 demonstrate that even with these slight drift concentrations, the frames

that were allowed to rock represent a significant improvement over the fixed base frame.

Peak base rotation

Figure 5.10 shows the mean peak absolute base rotations for each frame. The base rotation was

calculated at each analysis time-step by taking the difference between the column uplift at each

side of the frame and dividing that difference by the frame width. Since many of the peak base

rotations were essentially equal to ✓lock, the mean and mean absolute deviation were used. The


0

2

4

0

2

4

0

2

4

0

2

4

0

2

4

0

2

4

0

2

4

0

2

4

0

2

4

0.5 DBEmean peak base

rotation (%)

DBEmean peak base

rotation (%)

MCEmean peak base

rotation (%)


rotation (%)

DBEmean peak base

rotation (%)

MCEmean peak base

rotation (%)


rotation (%)

DBEmean peak base

rotation (%)

MCEmean peak base

rotation (%)

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

(a) 2 storey

(b) 4 storey

(c) 6 storey

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

meanabsolutedeviation

fixed

θlock

Figure 5.10: Median peak base rotation results: (a) 2-storey structure; (b) 4-storey structure; (c)6-storey structure


results for the 2-storey frame are shown in Figure 5.10(a). The peak base rotations were limited

by the lockup device for lower levels of ✓lock. For example, under DBE, the peak rotations for the

� = 0 models were essentially equal to the ✓lock values (the peak rotations were slightly greater

than ✓lock because of the elastic deformations of the elastic-no-tension and lockup elements) except

for ✓lock = 4.0%, when the peak rotation was less than ✓lock. This is explained by the fact that the

lockups restrained the base rotation only if they were engaged, and the large amount of allowable

rotation for the ✓lock = 4.0% models meant that the lockups were not necessarily engaged. In

general, the peak rotations decreased with increased � and for large � values there was little or no

base rotation at all. Interestingly, for the � = 1.2 models, the base joint was barely engaged under

0.5 DBE, slightly engaged under DBE, and moderately engaged under MCE. This trend can be

understood by looking at the push-pull behaviour of the models in Appendix B. For the 2-storey

� = 1.2 model, the rocking joint engaged after the BRBs yielded, and so the amount of demand

on the rocking joint depended on how much overstrength developed in the BRBs. This effect was

diminished in the larger values of �, (�=1.4 and 1.6), since the rocking joint strength was too strong

to be engaged. These models with high values of � performed similarly to the fixed base structure.

Figure 5.10(b) shows the base rotation results for the 4-storey frame. The results are similar to

the 2-storey frame, although the � = 1.2models engaged the rocking joint before the BRBs yielded

(see Figure B). This difference between the hysteretic response of the 2 and 4-storey structures

was due to the different superstructure strengths (the results are organized by � and not Mb,rock

My

from

Table 5.4, and so a given value of � will have a different Mb,rock

My

ratio for two different frame

designs).

The base rotation results were similar for the 6-storey frame (Figure 5.10(c)). The rotations

were limited by ✓lock for lower ✓lock values, and are less than ✓lock for larger values. As well, the

rotations were reduced with increased �. Notably, the median peak rotation for ✓lock = 4.0% is

less than 4.0% even under MCE for all � values.


Peak drift minus base rotation

Figure 5.11 shows the median and median plus standard deviation peak interstorey drifts minus

base rotations (referred to here as DMR). These values were calculated at each analysis timestep by

subtracting the base rotation from the interstorey drift at each level of the structures. This parameter

is important as it reflects the displacements that are being contributed by the frame itself and not

the rigid body rotation coming from the rocking joint. Alternatively, this value could have been

calculated as a function of the brace deformations, although that would not have taken into account

frame deformations due to column axial deformations. The values presented for the fixed base

frame are equivalent to the peak interstorey drift values shown in Figure 5.8.

Figure 5.11(a) shows the median peak DMR values for the 2-storey frame. Since this shorter

period structure had a larger amount of rotation associated with the rocking joint, allowing uplift

did not immediately reduce the displacement demands on the frame. For example, under DBE for

the models with � =0, 0.5, and 1.0, the DMR values were very similar to the fixed base structure

except for ✓lock =2.0% and 4.0%, where there was a notable reduction. It should be noted that

even though the lockup was not necessarily engaged at all for ✓lock = 4.0%, the DMR value was

not zero. This is because there were still elastic deformations in the frames and some nonlinear

deformations due to higher lateral mode vibrations.

Figure 5.11(b) shows the median peak DMR values for the 4-storey frame. For this frame,

the benefit of increasing ✓lock occurred earlier than it did for the 2-storey frame. For example, at

the DBE level, the fixed base frame did not pass the code criteria of 2.5% drift. However, for

�=0, 0.5, 1.0, the median DMR values were less than the code limit for ✓lock � 1.0%, and for

� = 1.2 the median DMR value was less than the code limit for ✓lock � 0.5%. While the code limit

referred to peak interstorey drifts it could be justified that the DMR values are what really matter

as they reflect demands on the structural members themselves (although peak interstorey drifts are

still important when considering nonstructural elements and compatibility between the SFRS and

the rest of the structural frame). For �=1.4 and 1.6 the DMR values were almost the same as the

fixed values, although for �=1.4 there was a notable drop in DMR values under the MCE suite.


0

5

0

5

0

5

0

5

0

5

0

5

0

5

0

5

0

5


drift minus baserotation (%)

















(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock




Figure 5.11: Median peak interstorey drift minus base rotation results: (a) 2-storey structure; (b)4-storey structure; (c) 6-storey structure


0.5 DBE Drift minus baserotation (%)

1.0 DBE Drift minus base rotation (%)

MCE Drift minus baserotation (%)

storey

fixedθ

lock=0.125%

θlock

=1.0%

θlock

=4.0%

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

Figure 5.12: Median drift minus base rotation profiles for 6-storey structure, � = 0

This implies that even if the rocking joint had a larger strength than the superstructure it could still

reduce displacement demands on the frame if it was engaged by a large earthquake. However this

trend essentially disappeared for �=1.6.

Figure 5.11(c) shows the median peak DMR values for the 6-storey frame. Under the 0.5 DBE

suite of records, there was a notable drop in DMR values for small amounts of ✓lock for � 1.0.

Under the DBE suite the DMR values dropped significantly even with a small amount of ✓lock. For

example, for � = 0.5 the DMR value was less than the code drift limit of 2%, while the fixed base

structure median peak drift was greater than 3%. Under the MCE suite the DMR values reduced

less at smaller values of ✓lock since the larger earthquakes engaged the smaller ✓lock lockups. Once

again the � � 1.2 models behaved like the fixed base structures although � = 1.2 still showed

some reduction in DMR values under DBE and MCE.

DMR drift profiles are shown in Figure 5.12, similar to Figure 5.9. These profiles help to

understand how the deformation demands on the SFRS were reduced by allowing column uplift.

Under 0.5 DBE, allowing ✓lock = 0.125% yielded an almost identical profile to the fixed result

although there was a slight reduction in the first storey value, implying that the concentration of

drift at this storey was slightly reduced. Allowing ✓lock = 1.0% and 4.0% greatly reduced the

demands at this lower earthquake level. Under the DBE suite of records it is clear that allowing

more base rotation reduced the demands on the superstructure beginning with the lower stories. The

✓lock = 4.0% model had almost no demand at the middle stories, although there was still demand

at the lower and upper floors — likely due to higher mode effects. Finally, the MCE results were


similar to the DBE results, with increased ✓lock greatly reducing the DMR values at the mid-height

of the structure.

Residual drifts

Figure 5.13 shows the median and median plus standard deviation peak residual drift results.

The 2-storey results (Figure 5.13(a)) demonstrate that the residual drifts were not necessarily re-

duced even when the rocking joints were fully self-centering. This phenomena was explained in

Section 3.2.4 and is related to the sensitivity of residual drifts to the individual characteristic of

each earthquake record.

Figure 5.13(b) shows the residual drift results for the 4-storey structure. Under the 0.5 DBE,

DBE, and MCE suites of records the residual drifts increased for smaller values of ✓lock and de-

creased to almost zero for larger values, except for when � � 1.2, when the structure behaved like

the fixed base frame. From this result it becomes apparent that combining the rocking and plastic

ductile mechanisms could have a negative effect on residual drifts. As well, the standard deviations

for most of the models were quite large, reflecting how residual drift results were highly sensitive

to the earthquake record.

The results were more promising for the 6-storey frame (5.13 (c)). While allowing just a small

amount of rocking (✓lock = 0.125%) increased the residuals under the 0.5 DBE suite for � = 1.0,

all other values of ✓lock reduced the residual drifts, even when � � 1.2. Under DBE and MCE the

results were similarly beneficial: increasing ✓lock reduced residual drifts.

Residual base rotations

Even though the drifts contributed from column uplift were intended to be self-centering, they

were not necessarily fully recoverable. Figure 5.14 shows the mean and mean plus absolute devi-

ation residual base rotations results. All of the 2-storey frames (Figure 5.14(a)) experienced negli-

gible residual base rotations. The 4-storey frames (Figure 5.14(b)) experienced some residual base

rotations under the MCE suite for a few of the models. For the 6-storey frames (Figure 5.14(c)),


0

1

2

0

1

2

0

5

0

1

2

0

1

2

0

5

0

1

2

0

1

2

0

5

0.5 DBEmedian peak

residual drift (%)

DBEmedian peak

residual drift (%)

MCEmedian peak

residual drift (%)

0.5 DBEmedian peak

residual drift (%)

DBEmedian peak

residual drift (%)

MCEmedian peak

residual drift (%)

0.5 DBEmedian peak

residual drift (%)

DBEmedian peak

residual drift (%)

MCEmedian peak

residual drift (%)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

Figure 5.13: Median peak residual drift results: (a) 2-storey structure; (b) 4-storey structure; (c)6-storey structure


0.5 DBEmean

residual base rotation (%)

DBEmean


MCEmean


0.5 DBEmean


DBEmean


MCEmean


0.5 DBEmean


DBEmean


MCEmean


β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

(a) 2 storey

(b) 4 storey

(c) 6 storey

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

meanabsolutedeviation

fixed

θlock

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

Figure 5.14: Residual base rotation results from parametric study (a) 2-storey structure; (b) 4-storey structure; (c) 6-storey structure


while the median residual base rotations were all near zero for all the models, there were a few

records — especially under the MCE suite — that experienced significant residual base rotations.

During these instances the residual base rotations were equal to the ✓lock value for that model. This

means that the frame essentially became stuck against the lockup on one side. Such a phenom-

ena is a product of the negative stiffness of the systems’ post-rocking response, and implies that

if the lockup were not present a global collapse might have occurred. This phenomena might be

avoidable by incorporating energy dissipation elements with a positive post-yield stiffness.

Global uplift

Figure 5.15 shows the median peak global uplift results for all the structures. Global uplift

was calculated at each analysis timestep as the lesser of the left and right rocking joint vertical

deformations. In general global uplift was negligible except for the lower � values for the 2-storey

frame. The larger global uplift values for this frame were due to the excitation of vertical mass that

is described in more detail in Section 5.4.

Number of records that engaged the lockup device

Figure 5.11 showed how the drift demands experienced by a frame’s ductile elements can be

reduced when a portion of the total drifts are contributed from rigid body rotation associated with

column uplift. In order to fully understand these results, it is important to realized that for a given

✓lock, not every record engaged the lockup. Figure 5.16 shows the number of records that engaged

the lockup device, Nlockup, for each building model and record suite. For small ✓lock values, all or

most of the 10 records engaged the lockup. As ✓lock was increased, less records engaged the lockup.

More records engaged the lockup with increasing earthquake intensity, and less records engaged it

with increased �. For example, for the 4-storey structure (Figure 5.16 (b)), the ✓lock = 1.0% and

� = 0 model engaged the lockup during 9 records under the 0.5 DBE suite, and 10 records under

the DBE and MCE suites. For the 0.5 DBE suite, as � was increasedNlockup dropped from 9 to 7 to

1, and finally to 0 for � > 1.2. In general, the larger � models (�=1.2, 1.4, and 1.6) did not engage


0.5 DBEmedian peak

global uplift (mm)

DBEmedian peak

global uplift (mm)

MCEmedian peak

global uplift (mm)

0.5 DBEmedian peak

global uplift (mm)

DBEmedian peak

global uplift (mm)

MCEmedian peak

global uplift (mm)

0.5 DBEmedian peak

global uplift (mm)

DBEmedian peak

global uplift (mm)

MCEmedian peak

global uplift (mm)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

Figure 5.15: Median peak global uplift results: (a) 2-storey structure; (b) 4-storey structure; (c)6-storey structure


0.5 DBE Nlockup

DBE Nlockup

MCE Nlockup

0.5 DBE Nlockup

DBE Nlockup

MCE Nlockup

0.5 DBE Nlockup

DBE Nlockup

MCE Nlockup

(a) 2 storey

(b) 4 storey

(c) 6 storey

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

θlock

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

fixed

Figure 5.16: Number of records that engaged the lockup device: (a) 2-storey structure; (b) 4-storey structure; (c) 6-storey structure


Table 5.5: Total energy dissipated by braces in fixed base structures

0.5 DBE median(med. + st. dev)dissipated energy

(kN-m)

DBE median(med. + st. dev)dissipated energy

(kN-m)

MCE median(med. + st. dev)dissipated energy

(kN-m)

2-storey 256 (377) 1030 (1511) 2020 (3060)4-storey 477 (740) 1717 (2700) 3850 (6160)6-storey 707 (1170) 2390 (4220) 4580 (8370)

the lockup except under MCE (and DBE for the 4-storey structure).

Energy dissipated by buckling restrained braces

While peak and residual drifts are important response parameters for gauging system perfor-

mance, they do not reflect the cumulative damage experienced by the hysteretic elements in a

seismic force resisting system. Accordingly, the total hysteretic energy dissipated by the super-

structure of each building model was calculated by summing the total dissipated energy of each

brace. The energy dissipated by each brace was calculated by integrating the force-deformation re-

sponse of each brace from each record. Table 5.5 shows the dissipated energy values for the fixed

base structures. Figure 5.17 shows the median results for all the building models as a percentage

of the median DBE fixed base results for each building size. For the 2-storey structure (Figure

5.17(a)), the addition of column uplift reduced the energy dissipated by the braces considerably for

the � = 0, 0.5 and 1.0 models, somewhat for the � = 1.2 models under MCE, and essentially not

at all for the � = 1.4 and 1.6 models. The results were similar for the 4-storey (Figure 5.17(b))

and 6-storey (Figure 5.17(c)) structures. Notably, the dissipated energy was not equal to zero even

for building models that did not engage the lockup in any of the ten records. For example, the 6-

storey structure with � = 1.0 and ✓lock = 4.0% did not engage the lockup at all under DBE, but the

dissipated energy in the braces was still almost 50% of the fixed base value. This demand comes

from higher mode effects, and reflects why higher mode mitigation such as the systems described

in Section 2.2.5 are desirable in rocking systems where structural elements are designed to remain


0.5 DBEmedian percent DBEfixed base absorbed

energy (%)

DBEmedian percent DBEfixed base absorbed

energy (%)

MCEmedian percent DBEfixed base absorbed

energy (%)


energy (%)


energy (%)


energy (%)


energy (%)


energy (%)


energy (%)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

Figure 5.17: Median total energy dissipated by braces results: (a) 2-storey structure; (b) 4-storeystructure; (c) 6-storey structure


essentially elastic. As well, this demand further justifies the benefits of combining rocking with

ductile braces such as BRBs.

TheMCE level results revealed an important conclusion regarding frame collapse performance.

For the 2-storey structure (Figure 5.17(a)) the median dissipated energy dropped from over 200%

the DBE fixed base value to 100% and less for � = 0, � = 0.5, and � = 1.0, and ✓lock � 1.0%.

The results were similar for the 4 and 6-storey frames. While the largest reduction always occurred

at ✓lock = 4.0%, Figure 5.17 shows that there was still a large benefit in terms of dissipated energy

for smaller values of ✓lock. This results suggests that if a conventional fixed base structure was

able to survive a single MCE level event without collapsing (which is the inherent goal of conven-

tional seismic resistant systems), it could withstand two such events if hybrid ductile-rocking was

incorporated.

Maximum foundation tension

Figure 5.18 shows the median and median plus standard deviation of the peak foundation ten-

sion for all the building models. For the models with foundation uplift, this value was calculated as

the sum of the tension induced by the energy dissipation and lockup when engaged. Considering

the principles of capacity design, the force in the lockup device should be no greater than the tensile

force expected in the fixed base structures. In fact if supplemental energy dissipation elements are

provided, the ED elements will reduce the load on the lockup as it pulls down on the frame once it

has yielded in the tension. Consequently, the lockup force should be less than the foundation tension

in the fixed base structure since the braces are not expected to undergo as large nonlinear defor-

mations when rocking is permitted, and thus the overstrength forces applied to the system ought to

be smaller than under fixed-base conditions. The parametric study results, however, show that the

lockup force can exceed that of the fixed-base structure. This is evident from Figure 5.18 as the

foundation tension was amplified in many models compared to the fixed base structure. The forces

were reduced for larger values of ✓lock primarily because not every record engaged the lockup. The

most important amplification of the foundation tension was in the 2-storey frame (Figure 5.18 (a)).


0.5 DBEmedian peak

lockup tension (kN)

DBEmedian peak

lockup tension (kN)

MCEmedian peak

lockup tension (kN)

0.5 DBEmedian peak

lockup tension (kN)

DBEmedian peak

lockup tension (kN)

MCEmedian peak

lockup tension (kN)

0.5 DBEmedian peak

lockup tension (kN)

DBEmedian peak

lockup tension (kN)

MCEmedian peak

lockup tension (kN)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

Figure 5.18: Median peak foundation tension results: (a) 2-storey structure; (b) 4-storey structure;(c) 6-storey structure


For example, under MCE, the median lockup tension for the � = 0.5model with ✓lock = 2.0%was

more than 3 times as large as the fixed base alternative. This significant force amplification was

due to the excitation of vertical mass which is described in more detail in Section 5.4.

Maximum column compressive force

Figure 5.19 shows the median and median plus standard deviation peak column compressive

forces for all the building models. As is discussed in more detail in Section 5.4, excitation of

vertical mass had a large effect on column forces, especially in for the 2-storey frame, as illustrated

in Figure 5.19. For the 2-storey frame, allowing for uplift caused the first storey median peak

column force to exceed its ASCE capacity for � = 0, 0.5, and 1.0 for all earthquake levels and

� = 1.2 under the MCE suite of records. The column force amplification was apparent in the 4

and 6-storey frames, but not nearly as severe. It should be noted that the results for this parametric

study reflect a modeling decision where vertical mass was lumped at the beam column connections

(see Figure 5.2). Section 5.4 discusses how this modeling decision can affect the column force

results, and notes other possible vertical mass modeling options.

Maximum base overturning moment

Figure 5.20 shows the median and median plus standard deviation peak base overturning mo-

ments for all the building models. Base overturning moments were calculated by summing the

moments about the frame centerline at the base of the frame, including contributions from the first

storey axial forces, first storey column bending moments, and the contributions from the first storey

brace forces. In frame structures, storey overturning moments are resisted primarily by column ax-

ial force couples, and so just as the column forces were affected by the excitation of vertical mass,

so to were the overturning moments, especially for the 2-storey frame. In order to better under-

stand this phenomena, Figure 5.21 shows the base moment response for three three building sizes,

� = 1.0, and ✓lock = 1.0%. Figure 5.21(a) shows the base moment versus base rotation plots. The

6-storey plot demonstrates the flag-shape that was expected. The spike in base moment that oc-


0.5 DBEmedian peak column

compression (kN)

DBEmedian peak column

compression (kN)

MCEmedian peak column

compression (kN)


compression (kN)


compression (kN)


compression (kN)


compression (kN)


compression (kN)


compression (kN)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

0

5000

Pny

= 1809 kN

Pny

= 1809 kN

Pny

= 1809 kNP

ny= first storey

column capacity(note: capacitiesfor 4 and 6 storeyframes are beyondthe chart axis)

Figure 5.19: Median peak first storey column compression results: (a) 2-storey structure; (b)4-storey structure; (c) 6-storey structure


0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0

5x 10

4

0.5 DBEmedian peak base

overturningmoment (kN)

DBEmedian peak base


MCEmedian peak base




DBEmedian peak base


MCEmedian peak base




DBEmedian peak base


MCEmedian peak base

aoverturningmoment (kN)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

My =

7770 kN-m

My =

7770 kN-m

My =

7770 kN-m

My =

22900 kN-m

My =

22900 kN-m

My =

22900 kN-m

My =

28300 kN-m

My =

28300 kN-m

My =

28300 kN-m

My= yield moment

considering nominalbrace strengths

Figure 5.20: Median peak base overturning moment results: (a) 2-storey structure; (b) 4-storeystructure; (c) 6-storey structure


−2 0 2−4

−2

0

2

4x 10

4

−2 0 2−4

−2

0

2

4x 10

4

−2 0 2−4

−2

0

2

4x 10

4

−2 0 2−4

−2

0

2

4x 10

4

−2 0 2−4

−2

0

2

4x 10

4

−2 0 2−4

−2

0

2

4x 10

4


Base moment(kN-m)

Base moment(kN-m)

Roof displacement (% structure height)

(a)

(b)fixed baseHDR

base rotation (%) base rotation (%) base rotation (%)



Figure 5.21: Base moment example results for 2, 4, and 6-storey frames with � = 1.0 and ✓lock =1.0%, record ID#1 scaled to DBE: (a) base moment versus base rotation; (b) base moment versusroof drift with comparison to fixed base response

curred around 1.0% base rotation was due to the engagement of the left lockup device. Similarly,

the 4-storey frame had a similar response, although it was oscillating more, demonstrating that a

higher mode response interacted with the base moment-rotation response. Finally, the 2-storey

frame demonstrates a very large higher mode response. Higher lateral modes cannot influence the

response of the base rotation joint, rather it was the higher mode response from the excitation of

vertical mass that was affecting this structure. Near zero base rotation, there were very large spikes

in base moment due to large spikes in column forces.

Figure 5.21(b) shows the base moment versus roof drift response for all three HDR building

sizes, along with the corresponding fixed base responses. The flag-shape response was clear in the

6-storey structure (and somewhat clear in the 4-storey structure), and can be compared to the more

typical plastic response of the fixed-base structure.

Maximum base shear

Figure 5.22 shows the median and median plus standard deviation peak base shear forces for all

the building models. The base shears were calculated at each timestep by summing the horizontal


0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0

2000

4000

0.5 DBEmedian peak

base shear (kN)

DBEmedian peak

base shear (kN)

MCEmedian peak

base shear (kN)

0.5 DBEmedian peak

base shear (kN)

DBEmedian peak

base shear (kN)

MCEmedian peak

base shear (kN)

0.5 DBEmedian peak

base shear (kN)

DBEmedian peak

base shear (kN)

MCEmedian peak

base shear (kN)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

Figure 5.22: Median peak base shear results: (a) 2-storey structure; (b) 4-storey structure; (c)6-storey structure


component of the first storey brace forces and the first storey column shears. The base shears were

similar between the fixed base and HDR structures. This is explained by the fact that in both types

of frames the brace strengths limited the force experienced by the systems. Some reduction in base

shear can be observed in Figure 5.22 for lower � values in all three structures. Even though base

shear was limited by the brace strengths in both the fixed and HDR structures, the HDR structures

reduced the deformation demand on the braces, and thus the base shear was less since the braces did

not necessarily deform plastically as much as in the fixed-base structure, and thus they experienced

less overstrength. This is most apparent during the MCE suite of records where concentrations of

drift in the lower stories were most severe.

Maximum storey accelerations

Figure 5.23 shows the median and median plus standard deviation peak absolute storey accel-

erations for all the building models. The results are very similar for all of the different building

models as the beneficial acceleration control characteristics of the distributed ductility BRBFs were

maintained no matter how much allowable rocking or energy dissipation was provided.

Summary of parametric study conclusions

Considering the response parameters presented in this section, the following general conclu-

sions were drawn.

For models with a rocking strength lower than the superstructure strength (� = 0, 0.5, and 1.0

and � = 1.2 for the 4-storey frame):

• Peak interstorey drifts were similar for all values of ✓lock, although the shorter period 2-

storey frame experienced an increase in peak drifts when there was little or no supplemental

energy dissipation at the level of the rocking joint. This result demonstrates that seismic

displacement demand can be shared between a conventional plastic ductile mechanism and

a rocking mechanism with little or no penalty to peak drifts.

• For the 6-storey frame, peak drifts were generally reduced with increasing ✓lock.


0.5 DBEmedian peak storey

acceleration (g)

DBEmedian peak storey

acceleration (g)

MCEmedian peak storey

acceleration (g)


acceleration (g)


acceleration (g)


acceleration (g)


acceleration (g)


acceleration (g)


acceleration (g)

(a) 2 storey

(b) 4 storey

(c) 6 storey

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

β=0 β=0.5 β=1.0 β=1.2 β=1.4 β=1.6

0.125%

0.25%

0.5%

1.0%

2.0%

4.0%

st. dev

fixed

θlock

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

Figure 5.23: Median peak storey acceleration results: (a) 2-storey structure; (b) 4-storey structure;(c) 6-storey structure


• Peak base rotations were limited by the lockup device as well as increased energy dissipation

strength.

• The drift demand on the frame itself (calculated by subtracting the base rotation from the

interstorey drifts), was reduced with increasing ✓lock, although it did not become zero for

large ✓lock due to higher mode demands on the frame and elastic frame deformations.

• Residual drifts were reduced for lower level earthquakes, but not necessarily reduced for

larger earthquakes due to the sensitivity of residual drifts to the characteristics of individual

earthquake records. For many models the residual drifts were increased.

• The residual base rotations were all zero except for a few cases (primarily in the 6-storey

frame under the MCE suite), when the frame became stuck against the lockup.

• The energy dissipated by the BRBs was greatly reduced with increased ✓lock. Under theMCE

suite, the braces dissipated half or less energy than the corresponding fixed base structure for

✓lock � 1.0.

• The foundation tension was amplified compared to the tensile foundation force experienced

by the fixed base structure due to excitation of vertical mass in the models, especially in the

2-storey frame

• The base overturning moment experienced some amplification, especially in the 2-storey

structure, due to the excitation of vertical mass. For the 4 and 6-storey structures with larger

values of ✓lock, the base moment was sometimes reduced.

• The base shear was similar between the fixed and HDR structures, although some reduction

in base shear was noted since the brace’s experience less nonlinear deformations and thus

less overstrength.

• The storey accelerations were similar between the fixed and HDR structures.

5.4. INVESTIGATION OF VERTICAL MASS MODELING ON ANALYSIS RESULTS 97

For models with a rocking strength larger than the superstructure strength, the responses were very

similar to the fixed base structures.

5.4 Investigation of vertical mass modeling on analysis results

It was observed in the analyses that certain response parameters were sensitive to vertical mass

modeling choices. This sensitivity was most pronounced in the 2-storey structure and seemed

to affect the column forces and foundation tension more than other response parameters. Since

rocking includes a vertical displacement along with the horizontal displacement, and since this

thesis considers the dead load of the building as the primary restoring moment after uplift, it is

crucial to discuss if and how vertical mass should be modeled. For the parametric study models,

vertical mass was lumped at beam-column connections. This is in contrast to conventional analysis

practices where mass is typically modeled as a lumped horizontal point mass representing the entire

tributary area of the building, and no vertical mass is considered.

This section presents a review of literature that focusses on rocking structure analyses and tests

that have relied on self-weight for the restoring moment. Examples of a column force amplification

are noted, and decisions and methods for modeling vertical mass are discussed.

A small parametric study is then performed by modeling the 2, 4, and 6, storey structures using

four different vertical mass options, including the lumped assumption from the parametric study.

A discussion of these results is presented.

5.4.1 Literature review of rocking studies relevant to vertical mass modeling

In their test of a 3-storeymoment resisting frame allowed to rock, Kelly and Tsztoo (1977) noted

that the compressive forces in the first storey columns were not critical, although it is apparent from

their column force time-histories that the column forces are greater in compression than tension for

the rocking configurations when dead load is not considered. The support under each column was

alternating layers of steel plate and neoprene rubber epoxied together, designed to cushion the


supports from impact following uplift. The stiffness of these composite pads was about 400 kips/in

(70 kN/mm). The axial stiffness of the first storey column, a W5X16 section 2060 mm long, was

295 kN/mm. Figure 5.24(a) notes a spike in column force upon impact for one test result for a

frame configuration that included a steel energy dissipation device.

Azuhata et al. (2003) studied numerical models of 5-storey moment and braced frames that

had yielding base plates that permitted column uplift. The mass of the structures was modeled by

lumping two thirds of the storey weight at the center of the beams, and one sixth of the weight

at each beam-column joint. It was observed in the numerical results that while tension force in

the columns was limited by the yielding base plates, the compression force was amplified due to

impact. These column spikes were as much as three times the value of the tension force when dead

load was subtracted. An example of these column force spikes can be seen in Figure 5.24(b).

Building on their earlier work, Azuhata et al. (2004) developed prediction models for their

yielding base plate system, and applied those models to the results of a 3-storey shake table test.

Expressions to determine column impact force were determined based on the velocity of the frame

at impact and by considering the excitation of the vertical mass mode along the height of a column.

These expressions conservatively predicted the column compressive forces. Figure 5.24(d) shows

the excited vertical mass mode described in their paper. The authors defined the compressive force

in the base of each column at a given floor, i, as:

N ic = �N i

T + IMPi (5.4.1)

where N iT is the variable and vertical tensile force at the column base on the i-th floor. If there

were no impact effects, thanN ic would be equal to �N i

T as the column forces would effectively be

limited by the rocking joint. In order to estimate IMP i the initial velocity shown in Figure 5.24(d)

is calculated by:

v0 =2⇡

Tuplift

�uplift,max (5.4.2)


(a)

(b)

(c)

(d)

Column force spike example

*

*

*

*

Figure 5.24: Examples of column force spikes in literature: (a) column force spike from threestorey shake table test (fromKelly and Tsztoo (1977)); (b) column force spike from five storeyshake table test (from Azuhata et al. (2003)); (c) comparison of rocking concrete wall model withand without vertical mass (from Kelley (2009)); (d) explanation of column force spike as excitationof vertical mass (from Azuhata et al. (2004))


where, Tuplift is the response vibration period of uplift which can be determined using the

equivalent linear stiffness of the whole system. �uplift,max is the maximum uplift. The maximum

value of IMP i can be evaluated using the maximum vertical deformation of the equivalent one-

mass system of the multi-mass-spring model shown in Figure 5.24(d):

�max =Tv

2⇡vo (5.4.3)

where Tvis the first natural period of the mass-spring model shown in Figure 5.24(d).

Midorikawa et al. (2006) presented the shake table results described by Azuhata et al. (2004),

along with results from a numerical model of these same tests. In their model, mass was placed

at 7 evenly spaced nodes distributed along each beam, and 4 along each column. The test results

again showed spikes in column forces of almost three times the tension force when dead load is not

considered. Their numerical model underestimated these column spikes by 20%.

Poirier (2008) presented a parametric study and shake table test of a viscously damped rocking

braced frame system. The parametric study considered 2, 4, and 6-storey frame heights in Van-

couver, Montreal, and Los Angeles. Vertical mass was not modeled for this study. However, for

the 2-storey shake table test, vertical mass was modeled by considering point masses at the beam-

column connections. Interestingly, while the numerical model predicted the drift response well,

it significantly underestimated the axial force response of the columns and braces as compared

to the experimental values, although the axial force response was predicted well under sinusoidal

and triangular input. The authors attributed this discrepancy to a high frequency response that was

not measured in the experimental program and concluded that the model could benefit from added

damping in the higher modes.

Kelley (2009) presented design guidelines for rocking walls and included the results of non-

linear time-history analysis on walls ranging from 1 to 6-storeys. These analyses did not consider

vertical mass, as per normal design office practice, but the effect of vertical mass was considered

on a subset of analyses. Figure 5.24(c) shows how including vertical mass caused a high frequency

response in the compressive reaction force, which was significantly larger than the maximum com-


pression without vertical mass. It was noted that the actual behavior of impact is more complex

than that which was captured by simply adding vertical mass to the model. Soil nonlinearity and

radiation damping, for example, would reduce the high frequency column resonance, and thus the

column force is likely to be less than what was predicted in the analyses.

The above references highlight how vertical mass can be excited by rocking motion, and that

different modeling choices will have different results for some response parameters, especially

column forces.

5.4.2 Parametric study on vertical mass modeling

A parametric study was performed on a few of the structures considered in this thesis in order to

investigate how vertical mass modeling choices influenced the dynamic analysis results. The 2, 4,

and 6-storey structures were modeled with four different vertical mass configurations. These mod-

els were allowed to uplift with a lockup at 1% allowable rotation, and no energy dissipation was

considered. The four different modeling choices are shown in Figure 5.25 for the 2-storey struc-

ture, and were determined based on the references noted above. The modeling choices are referred

to here as M1, M2, M3, and M4. An extra bay of gravity frame on each side of the braced frame

was modeled along with the lateral-force-resisting-system in order to more accurately capture how

the tributary vertical mass relates to the lateral system. In all scenarios the tributary gravity load

was modeled as point loads at the beam column connections. This decision was justified since the

load in the column (and thus the restoring moment after uplift) can relatively reliably be determined

from the tributary area of the columns no matter how much of that tributary mass is excited verti-

cally. Horizontal mass was modeled on the leaning columns, not shown in Figure 5.25. Note that

the gravity loads on the structures (in red in Figure 5.25) are larger than the weight of the associated

masses (in gray in Figure 5.25) since they were calculated considering 25% of the live load, while

the masses consider only the dead load, as per common analysis practice (although it is noted that

live load could potentially contribute an inertial response if excited vertically). All beam-column

connections were pins, and all columns were pinned at their base. Composite slab action was not


(a) M1: mass lumped at beam-column nodes

(b) M2: mass lumped at beam-column nodes (0.17M at beam ends, 0.67M at beam centers,where M is the tributary mass of a bay) (from Azuhata et al. (2003))

(c) M3: mass distributed at eighth points along beam (from Midorikawa et al. (2006))

(d) M4: no vertical mass modelled

90 kN

102.5 kN

90 kN 2160 kN

102.5 kN 2460 kN

180 kN

205 kN

180 kN

205 kN

90 kN

102.5 kN

90 kN 2160 kN

102.5 kN 2460 kN

180 kN

205 kN

180 kN

205 kN

90 kN

102.5 kN

90 kN 2160 kN

102.5 kN 2460 kN

180 kN

205 kN

180 kN

205 kN

90 kN

102.5 kN

90 kN 2160 kN

102.5 kN 2460 kN

180 kN

205 kN

180 kN

205 kN

85.1 kN/g 85.1 kN/g

90.1 kN/g

28.9 kN/g

30.6 kN/g

10.64 kN/g

22.5 kN/g typ.11.26 kN/g

21.3 kN/g typ.

28.9 kN/g

30.6 kN/g

114.0 kN/g

120.7 kN/g

57.9 kN/g

61.2 kN/g

57.9 kN/g

61.2 kN/g

114.0 kN/g

120.7 kN/g

114.0 kN/g

120.7 kN/g

170.1 kN/g

180.2 kN/g

170.1 kN/g

180.2 kN/g 90.1 kN/g

2040 kN/g

2164 kN/g

2040 kN/g

2164 kN/g

2040 kN/g

2164 kN/g

2550 kN/g

2710 kN/g

10.64 kN/g

11.26 kN/g

Figure 5.25: Schematic of vertical mass modeling assumptions


considered as a simplifying assumption, although it should be noted that the most accurate rep-

resentation of vertical mass excitation should consider the added stiffness and strength associated

with composite action, as well as the the stiffness of the slab across the the beam-column joints that

is not considered in these analyses since beams were assumed to be pinned. The structures were

subjected to ground motion ID#2 scaled to DBE (see Figure 4.4).

Responses considered in this analysis were the first storey left column axial force, the left lockup

device force, the first storey left brace force and deformation, and the first storey horizontal acceler-

ation recorded at the leaning column where the horizontal mass was placed. Figures 5.26, 5.27, and

5.28 show the time history results for the three structures for the first 20 seconds of the response.

It is clear from these responses that the vertical mass modeling seemed to have a large affect on

the 2-storey structure’s response, but minimal affect on the 4 and 6-storey responses. The column

forces in the two storey frame were amplified in compression, and that amplification is greatly de-

pendent on the vertical mass modeling choice. M1, the lumped mass assumption, yields the largest

amplification of column forces.ed

Figure 5.29 shows the peak interstorey drift profiles for the three structures with different mass

modeling assumptions. The drift profiles for the three structures were not greatly affected by the

different vertical mass modeling choices, although the 2-storey structure was more affected than

the 4 and 6-storey structures.

The peak results of the study can be compared in Table 5.6 . It can be seen from these results

that for the 2-storey structure that the M1 andM2 configurations caused the column force to exceed

the code capacity of 1809 kN. The peak compressive force in the M1 configuration was 4320 kN,

a greater than 500% increase over the peak compressive force for the M4 configuration of 797

kN, where no vertical mass was modeled. For the four storey structure the M1 compressive force

exceeded the M4 force by 26.6%, and for the six storey structure the M1 compressive force was

very similar to the M4 force.

Similarly to the column compressive force, the force in the lockup device was also greatly

influenced by themodeling of vertical mass. For the 2-storey structure themaximum tensile force in


1st storey column

force (kN)

1st storey brace force

(kN)

1st storey brace def.

(mm)

time (s)

1st storey accel.

(g)

1st storey left lockup

tension(kN)

M1M2M3M4

2 3 4 5 6 7 8 9 10

−4000

−2000

0

2000

2 3 4 5 6 7 8 9 10−2000

−1000

0

1000

2000

2 3 4 5 6 7 8 9 10

−50

0

50

2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

2 3 4 5 6 7 8 9 100

1000

2000

3000

4000

Figure 5.26: 2-storey response with different vertical mass modeling assumptions


1st storey column

force (kN)


(kN)


(mm)

time (s)

1st storey accel.

(g)


tension(kN)

M1M2M3M4

2 3 4 5 6 7 8 9 10

−4000

−2000

0

2000

2 3 4 5 6 7 8 9 10−2000

−1000

0

1000

2000

2 3 4 5 6 7 8 9 10

−20

0

20

2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

2 3 4 5 6 7 8 9 100

1000

2000

3000



1st storey column

force (kN)


(kN)


(mm)

time (s)

1st storey accel.

(g)


tension(kN)

M1M2M3M4

2 3 4 5 6 7 8 9 10−4000

−2000

0

2000

2 3 4 5 6 7 8 9 10−2000

−1000

0

1000

2000

2 3 4 5 6 7 8 9 10−50

0

50

2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

2 3 4 5 6 7 8 9 100

500

1000

1500

2000



Table 5.6: Summary of first storey response for vertical mass parametric study

peak leftcolumnforce(kN)

peak leftlockupforce(kN)

peak leftbraceforce(kN)

peak leftbracedef.(mm)

peakaccel.(g)

Peak In-terstoreydrift (%)

6 M1 3110 1946 1281 18.97 0.719 2.01Storey M2 2820 1980 1272 18.02 0.724 1.925

M3 2800 1843 1284 20.0 0.721 1.980M4 2920 1739 1229 20.9 0.640 1.892

ASCEcapacity

7797

4 M1 3014 2712 1416 21.7 0.932 2.21Storey M2 2540 2540 1502 35.9 0.898 2.50

M3 2529 2380 1471 31.5 0.906 2.26M4 2380 1973 1543 41.2 0.926 2.72

ASCEcapacity

5355

2 M1 4320 3520 985 44.3 0.745 2.81Storey M2 1890 1701 1017 52.8 0.638 2.67

M3 1140 1282 995 49.5 0.636 2.36M4 797 740 1067 63.2 0.532 3.13

ASCEcapacity

1809


0 2 40

1

2

0 2 40

1

2

3

4

0 2 40

1

2

3

4

5

6

Drift (%)

Storey

Storey

Storey

Drift (%) Drift (%)

M1

M2

M3

M4

Figure 5.29: Peak drift profiles for 2, 4, and 6-storey structures with different vertical mass mod-eling assumptions

the lockup in theM1 configurationwasmore than 4 times the force obtained in theM4 configuration

force. While the difference was not as severe for the taller structures, the modeling of vertical mass

actually affected the tensile force in the lockup for the six storey structure more than it affected the

column force. This suggests that the issue of vertical mass excitation must be carefully understood

if a lockup device is to be incorporated in conventionally designed frame structures.

5.5 Chapter summary

This chapter presented a parametric study that investigated the affect of allowing limit column

uplift on conventionally designed BRB frames. The 2, 4, and 6-storey frames presented in Chapter

4 were modified with varying amounts of allowable uplift before lockup and supplemental energy

dissipation element strength.

Unique modeling assumptions for the rocking joints were presented, and the behaviour of an

example 6-storey HDR model was investigated by performing pushover and push-pull analyses.

The response of this example structure to one record was presented in order to demonstrate how

the rocking joints perform during dynamic loading and to show how allowing rocking can reduce

drift demands on ductile braces.

The cumulative results of the parametric study were presented, and it was concluded that allow-


ing rocking can be beneficial for reducing displacement demands and residual drifts on distributed

ductility structures. It was noted that residual drifts were increased in some of the models, espe-

cially the lower storey buildings.

Finally, a side study was performed in order to investigate how vertical mass modeling choices

can affect results. Significant excitation of vertical mass was observed in structures where that

mass was modeled as lumped at beam-column joints, especially in the 2-storey structure. This

dynamic effect was lessened by spreading the mass to nodes along the beams, and by not modeling

vertical mass at all. Further research is required to better understand under which scenarios vertical

mass excitation is an issue for rocking structures, and how vertical mass should be modeled during

numerical analysis.

Chapter 6

Example design of 6-storey BRB HDR

structure with cast steel energy dissipation

elements

This chapter presents the design and analysis of a 6-storey BRB HDR frame that was carried out

to further validate the HDR concept that was developed in this thesis. The frame superstructure

is the 6-storey reference frame first presented in Chapter 4. The column-foundation connections

were designed to incorporate the hybrid ductile-rocking modifications, based on the results of the

parametric study in Chapter 5. The energy dissipation elements were designed based on the cast

steel yielding brace system from Section 5.1. This chapter begins by presenting an overview of the

cast steel triangular yielding fingers, and demonstrates how such a fuse can be modeled numerically

in the context of an HDR system. Next, the design of the HDR rocking joint is outlined, and the

properties of the fuse elements that were selected are described. The results of nonlinear time-

history analyses of the six-storey frame with the cast steel fuse are then presented and compared to

the response of the fixed base BRB structure. Finally, the design of a column-foundation detail is

presented that includes the cast steel fuse and lockup device.

110

6.1. CAST STEEL YIELDING CONNECTOR USED AS ROCKING FUSE ELEMENT 111

6.1 Cast steel yielding connector used as rocking fuse element

Chapter 5 presented a brief overview of the yielding brace system, which was initially devel-

oped as an alternative to buckling restrained braces (Gray, 2012). The yielding brace system uses

cantilevered triangular-shaped yielding fingers made of cast steel that can accommodate large non-

linear cyclic displacement demands since the profile of the fingers matches the bending moment

diagram for a cantilever, and thus nonlinear strains are distributed along the entire length of the

finger, rather than at a concentrated location. While the YBS system was initially envisioned as an

alternative non-buckling brace system that offers similar, albeit improved (see Gray et al. (2014))

system performance to BRBs, the small size and large ductility of the components makes them an

attractive option for a rocking system fuse.

Yielding brace system properties

Gray (2012) showed that the yield strength for cast steel yielding brace system connector, Pp

is:

Pp =nboh

2

4LFy (6.1.1)

where n is the number of yielding fingers, bo is the width of the yielding finger at the widest

point, L is the length of each finger, h is the height of each finger, and Fy is the yield stress of

the connector material. This strength expression is based on the plastic section modulus of each

yielding finger. The elastic stiffness of the connection, Kp, is given by:

Kp =nboEh3

6L3(6.1.2)

and the monotonic backbone curve after yielding can be expressed considering second order

geometric deformations of the fingers by the following closed-form equation:

112 CHAPTER 6. EXAMPLE DESIGN OF 6-STOREY BRB HDR STRUCTURE

(a) (b)

−50 −40 −30 −20 −10 0 10 20 30 40 50−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Deformation (mm)

Force(kN)

testOpenSees

deformed cast steeltriangular fingers

undeformed connection

Figure 6.1: Cast steel yielding fuse numerical model calibration: (a) comparison of OpenSeeselement output and test results; (b) photo of fuse test, University of Toronto 2014

P =Pp

cos(2dL)

(6.1.3)

6.2 Validation of cast steel material in OpenSees

OpenSees includes a material option called “CastFuse” that was developed by Gray (2012) to

capture the unique hysteretic properties of a yielding brace system ductile connection including

a stiffening and strengthening effect at large displacements. The material behaves similarly to

the “Steel02” material while also capturing the post-yield increase in stiffness and strength that

are caused by second-order axial deformation of the yielding fingers. Figure 6.1 shows the cast

fuse material calibrated against a test on a yielding connector that was performed at the University

of Toronto in 2014 (Gray et al., 2014). The material captures the overall elastic and nonlinear

behaviour of the connector well. Notably, the elastic stiffness of the real connector is lower than

the idealized numerical model as the flexibility of the cast steel arms connecting the yielding fingers

to the brace was not accounted for in the model. The fuse geometric and material properties used

for this model calibration are shown in Table 6.1 and Table 6.2 , respectively.

6.2. VALIDATION OF CAST STEEL MATERIAL IN OPENSEES 113

Table 6.1: YBS fuse geometric parameters for OpenSees calibration

Geometricparameters

Property

n 8bo (mm) 146.1h (mm) 27.2L (mm) 146.1Fy (MPa) 300E (MPa) 200000Pp (kN) 444Kp

(kN/mm)251

Table 6.2: YBS fuse material modeling parameters for OpenSees calibration

OpenSees cyclicstrain hardeningand Bauschinger

parameters

Property

b 0.00Ro 20.0cR1 0.925cR2 0.1500a1 0.0350a2 0.700a3 0.0350a4 0.700


6.3 Design of 6-storey HDR frame

This section presents a single example design for the 6-storey BRB frame with hybrid ductile-

rocking. The frame has been modified to allow column uplift, lockup after a predetermined frame

rotation, and accommodate supplemental energy dissipation based on the cast steel yielding brace

system.

6.3.1 Superstructure design

This example design uses the 6-storey reference frame described in Chapter 4 as the superstruc-

ture frame. The structural members are given in Table 4.1, and the design procedure is given in

Appendix A.

6.3.2 Fuse and lockup properties

The results of the parametric study from Chapter 5 showed that for the 6-storey frame, a �

value of 1.0 allowed for a large supplemental energy dissipation capability without resulting in a

lateral rocking strength that was greater than the lateral strength of the fixed base BRB frame. For

this reason, � = 1.0 was targeted for this example design. The geometric properties of the YBS

connector that was chosen for this application are given in Table 6.3 . From Equation 5.1.1, � = 1.0

which confirms the design intent.

The parametric study results also showed that a value of ✓lock = 1.0% provided much of the

benefit of the HDR system without the risk of excessive rocking deformations under larger earth-

quake demands. Thus, ✓lock = 1.0% was chosen for this design. This value corresponds to a

maximum vertical deformation of 91.4 mm. While the slab-SFRS compatibility was not modeled

in this example, it was presumed that limiting the ✓lock value would help to minimize the vertical

deformation demands if such a compatibility system were to be designed.

Considering ✓lock = 1.0%, the geometric properties of the fuse were chosen in order to pro-

vide adequate ductility considering the maximum deformation demand of 91.4 mm. A personal

6.3. DESIGN OF 6-STOREY HDR FRAME 115

Table 6.3: Example design fuse properties

Geometricparameters

Fuse property

n 10bo (mm) 200h (mm) 49L (mm) 300Fy (MPa) 300E (MPa) 200000Pp (kN) 1201Kp

(kN/mm)290

communication with Cast Connex Corporation, the commercial providers of the YBS devices, was

engaged in order to determine a satisfactory fuse design that met this ductility requirement. It is

noted that since the rocking connection can only deform vertically in one direction, the cyclic de-

mands on such a fuse detail are effectively half of the demands on a conventional YBS fuse cycled

to the same deformation in both directions. This characteristic of the rocking joint is beneficial in

terms of the low-cycle fatigue life of the connection.

6.3.3 Modeling assumptions

The fixed base and HDR frames were modeled in a similar way as in Chapters 4 and 5 except

for a few notable differences. The columns were modeled using the “beamwithhinges” elements in

OpenSees in order to allow for plastic hinging under larger displacement demands. For the HDR

frame, the energy dissipation was modeled using the “CastFuse” material as described and cali-

brated above, rather than the elasto-plastic material that was used Chapter 5. An analysis timestep

of 0.0005 seconds was used. Vertical mass was lumped at the beam column connections. This

assumption corresponds to the M1 scenario described in Section 5.4, and likely represents an upper

bound on forces related to the excitation of vertical mass.


6.3.4 Pushover response

Figure 6.2 shows the pushover response of the example design with the YBS fuse. This analysis

was performed in the same manner as in Section 5.2.1. While the HDR frame in Section 5.2.1

had similar � and ✓lock as this example design, the frame from Section 5.2.1 was modeled with

elastoplastic energy dissipation while the YBS fuse had an increase in strength and stiffness after

yield which can be viewed in the force-deformation plot in Figure 5.3(b). Thus, the pushover results

in Figure 6.2 are similar but notably different from the results in Figure 5.3. While the post-rocking

stiffness from Figure 6.2 was negative before the lockup was engaged, the post-rocking stiffness

in Figure 6.2 started off negative after the fuses yielded, but became slightly positive before the

lockup was engaged.

6.3.5 Push-pull response

Figure 6.3 shows the cyclic response of the system. The cyclic analysis was performed in

the same manner as in Section 5.2.2. The base shear versus roof drift is shown in 6.3(a). While

similar to the model with a general elastoplastic fuse that was presented in Figure 5.4, this response

is more rounded since the YBS fuse was modeled with more representative strain hardening and

Bauschinger effects. As well, while the nominal yield load of 1201 kN gives a � value of just less

than 1.0, the actual response was not fully self-centering during smaller amplitude displacements

since the strength of the fuse increased after yield. This increase in strength in both tension and

compression can be observed in Figure 6.3(b), and was due to the characteristic second-order post-

yield behaviour of the YBS device as well as the material nonlinearity. The contact and lockup

hystereses are shown in Figures 6.3(c) and (d) respectively. Their behaviour is nearly identical

to the behaviour in Figure 5.4. Finally, the first storey left brace hysteresis can be seen in Figure

6.3(e). As expected, the peak displacements on the brace for a given system deformation cycle was

less for the HDR structure than for the fixed base design.



Bas

e Sh

ear

(kN

)

Pushover Response

Fixed BaseHDR structure with cast steelenergy dissipation1% lockup, ED=1201 kN

Fuse Force-Deformation

Def. (mm)

Forc

e (k

N)

Contact Force-Deformation

Def. (mm)

Fo

rce

(kN

)Lockup Force-Deformation

Def. (mm)

Fo

rce

(kN

)

Def. (mm)

Forc

e (k

N)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

2500

3000

−100 0 100

−1000

0

1000

−100 0 100

−1000

0

1000

−100 0 100

−2000

0

2000

0 50 100 1500

500

1000

1500

2000First Story Left Brace Force-Deformation

column Decompression

fuse yield lockup

(a)

(b)

(e)

(d)(c)

Figure 6.2: Pushover response of 6-storey fixed base frame and 6-storey frame with cast steelfuse and ✓lock = 1% and � = 1.0: (a) base shear versus roof drift; (b) fuse force-deformation;(c) contact element force-deformation; (d) lockup force-deformation; (e) first storey left braceforce-deformation



Bas

e Sh

ear

(kN

)

Push-pull Response

Fuse Hysteresis

Def. (mm)

Forc

e (k

N)

Contact Hysteresis

Def. (mm)

Forc

e (k

N)

Lockup Hysteresis

Def. (mm)

Forc

e (k

N)

First Storey Left Brace Hystereses

Def. (mm)

Forc

e (k

N)

(a)

(b)

(e)

(d)(c)

flag shaped duringsmaller amplitudedisplacements

−5 −4 −3 −2 −1 0 1 2 3 4 5

−3000

−2000

−1000

0

1000

2000

3000

−100 0 100−2000

0

2000

−100 0 100

−5000

0

5000

−100 0 100−5000

0

5000

−200 −100 0 100 200−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

Fixed Base

HDR structure with cast steelenergy dissipation1% lockup, ED=1201 kN

Figure 6.3: Push-pull response of 6-storey fixed base frame and 6-storey frame with cast steelfuse and ✓lock = 1% and � = 1.0: (a) base shear versus roof drift; (b) fuse hysteresis; (c) contactelement hysteresis; (d) lockup hysteresis; (e) first storey left brace hysteresis


6.3.6 Time-history results under a sample record

The response of the example design subjected to record ID#1 scaled to DBE is presented here.

This response is very similar to the HDR model from Section 5.2.3 since the fuses are almost the

same strength.

Figure 6.4 shows the behaviour of the frame’s foundation elements, including the elastic-no-

tension contact element deformation and force, the lockup force, and the energy dissipation element

force. Figure 6.4(a) shows the contact element deformation, which is equal to the deformation of

the other foundation elements since the elements act in parallel. The left side engaged the lockup

once, while the right side did not at all. Figure 6.4(b) shows the elastic-no-tension contact element

forces. These elements began in compression due to gravity loading and did not go into tension

at all during the record. Figure 6.4(c) shows the lockup force. Figure 6.4(d) shows the force in

the fuses. The forces began slightly negative since some gravity load flowed through the fuses

before the transient response began. After first yield, the force alternated between the positive and

negative yield force plus the increased strength due to strain hardening. Figure 6.4(e) shows the

corresponding ground motion.

The roof displacement, base shear, and base overturning moment responses are shown in Figure

6.5. Overall the response is very similar to the response that was presented in Figure 5.6 that

included the elastoplastic fuse from the parametric study. Figure 6.5(a) shows how the displacement

demand on the HDR frame was primarily contributed to from the base rotation, whereas the fixed

base roof displacement, which was similar in magnitude to the HDR structure, was primarily due

to brace axial deformations. Figure 6.5(b) shows how the base shear response was similar between

the two frames, and Figure 6.5(c) shows how the overturning moments in the HDR frame were less

than the corresponding fixed frame except when the lockup engaged, at which point the moments

were of similar magnitude, since after lockup the system moments were limited by the capacity of

the BRBs.

The hysteretic response of the braces and fuses is shown in Figure 6.6. As in Figure 5.7, the

addition of rocking significantly reduced the displacement demand on the BRBs, especially in the


0 2 4 6 8 10 12 14 16 18 20−50

0

50

100

0 2 4 6 8 10 12 14 16 18 20−4000

−3000

−2000

−1000

0

1000

0 2 4 6 8 10 12 14 16 18 200

500

1000

1500

2000

2500

time (s)

0 2 4 6 8 10 12 14 16 18 20−3000

−2000

−1000

0

1000

2000

3000

time (s)

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

0.624

Contact Deformation

(mm)

Contact Force(kN)

Lockup Force(kN)

Fuse Force(kN)

(a)

(b)

(c)

(d)

Groundacceleration

(g)

(e)

Left side

Right side

time (s)

Figure 6.4: Foundation element behaviour for model with YBS fuse: (a) elastic-no-tension contactelement deformation; (b) contact force; (c) lockup force; (d) Fuse force; (e) ground acceleration(ID#1)


0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time (s)

0 2 4 6 8 10 12 14 16 18 20−4000

−2000

0

2000

4000

0 2 4 6 8 10 12 14 16 18 20−6e4

−4e4

−2e4

0

2e4

4e4

6e4

RoofDisplacement

(%)

Base Shear(kN)

OverturningMoment(kN-m)

(a)

(b)

(c)

lockupengaged

lockupengaged

time (s)

Fixed based roof displacementHDR Roof displacementHDR Rigid body rotationHDR Roof disp. minus rigid body rotation

Fixed baseHDR structure

Fixed baseHDR structure

Figure 6.5: Roof displacement and system forces of HDR example design with cast steel fusecompared to fixed base response: (a) roof displacement and base rotation of fixed and HDRstructures; (b) base shear time-history for fixed base and HDR structures; (c) overturning momenttime-history for fixed base and HDR structures


−100 −50 0 50 100−2000

0

2000

−100 −50 0 50 100−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−2000

0

2000

−100 −50 0 50 100−2000

0

2000

−100 −50 0 50 100−2000

0

2000


(kN)


(kN)


(kN)

3rd StoreyBrace Force

(kN)

2nd StoreyBrace Force

(kN)

1st StoreyBrace Force

(kN)

Deformation (mm)

Fuse Force(kN)

Deformation (mm)

(a) Left Side of Frame Right Side of Frame

(b)


Figure 6.6: Hysteretic response of 6-storey fixed base structure and HDR structure with cast steelfuses: (a) BRB hystereses; (b) fuse hystereses


lower stories as can be clearly seen in Figure 6.6(a). Figure 6.6(b) shows the hysteretic response of

the uplift fuses. The characteristic post-yield increase stiffness and strength increases are clearly

visible in the fuse responses.

6.3.7 Response of HDR structure to suites of records

Table 6.4 shows a summary of the main statistical results comparing the fixed base structure to

the example HDR design with the YBS fuse when subjected to the three record suites described in

Chapter 4.

Peak interstorey drift and drift minus base rotation

Figure 6.7 shows the interstorey drift and interstorey drift minus base rotation (DMR) results

for the fixed base and HDR frames. Figure 6.7(a) shows the peak interstorey drift results. As can

be seen, the drift concentrations were much less for the HDR frame. From Table 6.4 the median

peak drift at DBE dropped from 3.3% for the fixed base frame to 1.99% for the rocking frame. This

decrease means that the HDR frame passed the 2% interstorey drift code requirement whereas the

fixed base frame did not.

For conventional structures, peak drifts are an important parameter for estimating low-cycle

fatigue (LCF) life on fuse elements as well as determining demands on non-structural (NS) ele-

ments. For an HDR structure, it is the interstorey drift minus the base rotations (DMR) that reflect

the peak demands contributed from the structural frame itself, and thus are important when con-

sidering cyclic damage and LCF life of HDR braces (peak interstorey drifts are still important for

NS elements in HDR frames). The peak DMR results are shown in Figure 6.7(b). Under the 0.5

DBE suite, the peak drift minus base rotation value of 0.72% reflects a greater than 50% reduction

from the fixed base median peak drift of 1.54%. Under the DBE suite, the value dropped 56% from

3.3% to 1.44%, and under MCE the value dropped 28% from 4.4% to 3.1%.

Figure 6.7(c) summarizes the important interstorey drift results by comparing the median and

median plus standard deviation peak interstorey drifts from the fixed base frame to the median and


Table 6.4: Summary of statistical results

0.5 DBE DBE MCE

Fixedmedian(med. +st. dev.)

HDRmedian(med. +st. dev.)





peak drift (%) 1.535(2.18)

1.069(1.522)

3.26(4.81)

1.992(2.80)

4.36(7.39)

3.85(6.39)

peak drift minusbase rotation (%)

- 0.715(0.895)

- 1.437(2.01)

- 3.13(5.39)

residual drift (%) 0.536(0.998)

0.1212(0.211)

1.650(2.82)

0.659(1.233)

2.96(5.57)

1.385(4.29)

peak baserotation (%)

- 0.641(0.876)

- 0.907(1.066)

- 1.022(1.084)

residual baserotation (%)

- 0.00263(0.00449)

0.01008(0.01629)

- 0.1816(0.405)

peak global uplift(mm)

- -0.0901(0.1944)

- 0.647(1.256)

- 0.791(1.553)

peak storeyacceleration (g)

0.521(0.657)

0.513(0.619)

0.961(1.151)

0.856(1.070)

1.409(1.687)

1.472(1.764)

energy dissipatedby BRBs (kN-m)

705(1170)

217(317)

2380(4210)

1043(1580)

4560(8350)

2470(4100)

peak foundationtension (kN)

2060(2120)

1446(1795)

2350(2650)

2290(3300)

2830(3360)

4060(5430)

peak base shear(kN)

2110(2270)

1883(1995)

2510(2850)

2290(2440)

2920(3500)

2760(3140)

peak baseoverturningmoment (kN-m)

33300(34000)

27900(30300)

36700(40000)

33200(38200)

41800(47600)

43500(51100)

peak columncompression(kN)

3690(3730)

3520(3660)

3870(4030)

4110(4390)

4070(4370)

4570(5020)


Peak interstorey drift (%)

Peak interstorey driftminus base rotation, DMR (%)

Peak interstorey drift, ID, or peak interstorey driftminus base rotation, DMR (%)

6 StoreyHDR withcast steel

fuse

6 Storeyfixed base

0.5 DBE DBE MCE(a)

(b)

(c)

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6


fuse

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

storey

storey

storey

median of 10 recordsmedian + st. dev.

Fixed base structure IDHDR structure DMR

median of 10 records

median + st. dev.

Figure 6.7: Peak interstorey drift results: (a) peak interstorey drift for fixed base and HDR frame;(b) peak interstorey drift minus base rotation for HDR frame; (c) comparison of fixed base medianpeak interstorey drifts to HDR median peak interstorey drifts minus base rotations


0 1 2 30

2

4

6

0 1 2 30

2

4

6

0 1 2 30

2

4

6

0 1 2 30

2

4

6

0 1 2 30

2

4

6

0 1 2 30

2

4

6


Peak storey height (% total structure height)

6 StoreyHDR withYBS fuse

6 Storeyfixed base

0.5 DBE DBE MCE

Figure 6.8: Peak floor displacement results for fixed base and HDR frames

median plus standard deviation peak DMR values for the HDR frame. This figure clearly shows

how the HDR system greatly reduces the displacement demand on the frame, especially at the lower

stories.

Peak floor displacements

Figure 6.8 shows the peak floor displacements. The reduction in drift concentrations in the

HDR frames is evident as the displacement profiles are much straighter than those for the fixed

base frame.

Base rotation

Table 6.4 shows the median peak and median plus standard deviation base rotation results and

residual base rotation results. The number of records that engaged the lockup for each record suite

is shown in Table 6.5 . Under the 0.5 DBE suite of records, only one record engaged the lockup,

and both the median and median plus standard deviation peak base rotation values were less than

✓lock = 1.0%. For the DBE suite of records, five out of ten records engaged the lockup, although the


Table 6.5: HDR structure lockup engagement

0.5 DBE DBE MCE

# of records that engaged thelockup

1 5 8

median peak base rotation was still less than ✓lock = 1.0%. For both the 0.5 DBE and DBE suites

the residual base rotation was effectively 0%. Under the MCE suite, eight of ten records engaged

the lockup, and the median and median plus standard deviation peak base rotation results were both

essentially equal to ✓lock = 1.0%. They were slightly greater since there was some elastic flexibility

in the lockup and the foundation. There were some residual base rotations under the MCE suite.

These residuals were possible because the post-yield strengthening of the HDR energy dissipation

meant that the moment-rotation joint at the base of the structure was not fully self-centering, even

if it was designed for � < 1 based on nominal properties.

Residual drifts

Figure 6.9 shows the median and median plus standard deviation residual drift results for the

fixed base and HDR frames. The addition of hybrid ductile-rocking effectively reduced residual

drifts under all the earthquake levels. Notably, while the median values for the fixed base structure

exceeded the 0.5% residual drift criterion described in Section 4.2.3, under the DBE suite in the first

four storeys the HDR frame was below this threshold at all levels except for the first storey. That

being said a substantial number of records still failed the criteria even with the HDRmodifications.

Under the 0.5 DBE suite, however, the HDR frame passed the criteria for all the records, while the

fixed base frame failed for six out of ten records.

Energy dissipated by buckling restrained braces

Table 6.4 shows the median and median plus standard deviation results for the total energy

dissipated by the buckling restrained braces. The reduction in dissipated energy during the 0.5

DBE, DBE, and MCE suites between the fixed and HDR frames was 69.2%, 56.2%, and 45.8%


0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

storey


Residual interstorey drift (%)


fuse

6 Storeyfixed base

0.5 DBE DBE MCE

Figure 6.9: Residual drifts results for fixed base and HDR structure

respectively. This large decrease in demand on the superstructure highlights how the HDR frame

has a reduced low-cycle fatigue (LCF) demand compared to the fixed base frame, which represents

an increase in the resilience of the system. After a DBE level seismic loading, the BRBs of the

conventional structure would likely require replacement as they would have depleted their LCF

life whereas the BRBs of the HDR structure would be capable of resisting another DBE level event

before depleting their LCF life.

Maximum foundation tension

Table 6.4 shows the median and median plus standard deviation peak foundation tension. For

the fixed base structure this value corresponds to the maximum tension force experienced by the

foundation, and for the HDR structure this value corresponds to the maximum tension force in the

lockup devices and energy dissipation. Under the 0.5 DBE suite of records, there was a signification

reduction in foundation tension, since only one record engaged the lockup device (in the absence

of lockup engagement, foundation tension is solely due to the yielding of the fuse). Under the DBE


0 2000 4000 60000

2

4

6

0 2000 4000 60000

2

4

6

0 2000 4000 60000

2

4

6

0 2000 4000 60000

2

4

6

0 2000 4000 60000

2

4

6

0 2000 4000 60000

2

4

6

storey


Peak column compression (kN)


fuse

6 Storeyfixed base

0.5 DBE DBE MCE

Figure 6.10: Peak column compression results for 6-storey fixed base and HDR frames

suite of records the HDR foundation tension was similar to the fixed base response, although the

standard deviation was greater. Half of the records engaged the lockup, and thus there was a large

spread of tension values since for the records that did not engage the lockup foundation tension

was limited by the overstrength of the energy dissipation elements, whereas for the records that did

engage the lockup this tension was determined by brace yielding and excitation of vertical mass.

Under the MCE suite of records the foundation tension was greater than the fixed base response.

Eight out of ten records engaged the lockup, and the force in the lockup was greater than the fixed

base tensile force due to the excitation of the vertical mass, as was highlighted in Section 5.4.

Maximum column compressive force

Figure 6.10 shows the peak column compression for the fixed base and HDR frames. The force

profiles are very similar between the fixed base and HDR structures, which is reasonable since the

column forces were limited by the capacity of the braces in both frames. The HDR frame column

force profiles had a larger standard deviation which is due to the added variability in column forces


0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

0 2 4 60

2

4

6

storeymedian of 10 recordsmedian + st. dev.nominal overturning momentoverstrength overturning moment

Peak overturning moment (kN-m)

6 Storeyfixed base

0.5 DBE DBE MCE


fuse

Figure 6.11: Peak storey overturning moment results for fixed base and HDR frames

from the excitation of vertical mass, as discussed in Section 5.4.

Maximum storey overturning moment

Figure 6.11 shows the peak storey overturningmoment for the fixed base andHDR frames. Like

the column forces, the overturning moments were similar between the fixed and HDR structures.

Similar to the column forces, the HDR structure had a larger standard deviation for overturning

moments due to the variability in column forces from excitation of vertical mass (see Section 5.4).

Maximum storey shear

Figure 6.12 shows the storey shear results for the fixed base and HDR frames. The results are

very similar between the two structures. This is reasonable since the storey shears were limited

by the brace strengths in both frames. That being said, there is a clear increase in storey shear

forces at the lower stories of the fixed base frame due to concentrations of drift (and thus increased

brace deformations and brace forces). While the HDR frame also has this increase in forces at the

6.4. PRELIMINARY DETAIL DESIGN 131

storey

Peak storey shear (kN)


fuse

6 Storeyfixed base

0.5 DBE DBE MCE

median of 10 recordsmedian + st. dev.nominal storey shearoverstrength storey shear

0 2000 40000

2

4

6

0 2000 40000

2

4

6

0 2000 40000

2

4

6

0 2000 40000

2

4

6

0 2000 40000

2

4

6

0 2000 40000

2

4

6

Figure 6.12: Peak storey shear results for fixed base and HDR frames

lower stories, it is notably smaller since the braces of the HDR frame deformed less. This result is

reflected in the median peak base shear values from Table 6.4. The median peak base shear values

for the HDR frame were less than for the fixed base frame at all earthquake levels.

Maximum storey accelerations

Figure 6.13 shows the storey acceleration results for the fixed base and HDR frames. Both

frames have very similar acceleration profiles that reflect the beneficial storey acceleration response

typical of BRB frames as previously described.

6.4 Preliminary Detail Design

This section presents a preliminary detail design for the 6-storey BRBF column-foundation

connection in order demonstrate how the HDR system can be practically implemented in a typical

BRBF frame.



Peak acceleration (g)


fuse

6 Storeyfixed base

0.5 DBE DBE MCE

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

6

Figure 6.13: Peak storey acceleration results for fixed base and HDR structures

6.4.1 Overview of detail

The main requirements of the HDR column-foundation detail are to (1) resist the seismic base

shear; (2) resist foundation tension after lockup; (3) accommodate the supplemental energy dis-

sipation elements; and (4) accommodate the brace connection. An overview of the proposed de-

tail is shown in Figure 6.14. Figure 6.14(a) shows an isometric view of the completed detail and

highlights the primary features of the concept. Figure 6.14(b) shows the column assembly which

includes two cast steel fuses that are bolted to the column web. As well, gusset plates with slotted

holes are welded to the column flanges and allow for foundation tension to be resisted after an uplift

displacement predetermined by the slotted hole length. One of the gusset plates also accommodates

the BRB connection, which is a true pin connection in this example. Finally, Figure 6.14(c) shows

the base plate assembly which includes the energy dissipation reaction collar and lockup reaction

plates, as well as holes for foundation anchor rods. For this preliminary design all welds were

designed as complete joint penetration welds.

Figure 6.15 shows the proposed construction sequence for the HDR column-foundation detail.


(a) (b)

(c)

(d)

gusset plate& lockup

cast steel EDbolted tocolumn web

anchor rods transmitbase shear and resistfoundation tensionafter lockup

slotted holesin gusset platespermit uplift& lockup, & resistbase shear

ED reaction“collar”

lockup reactionplates

base plate

W14X132column

buckling-restrainedbrace

ion gusset plate& lockup

buckling-restrai

ods transmitbase shear and rfoundation tension

in gusset platespermit uplift& lockupbase shear

Figure 6.14: Overview of HDR column-foundation detail: (a) main features of detail; (b) columnassembly including energy dissipation and gusset plates; (c) base plate assembly including energydissipation reaction collar and lockup reaction plates; (d) top and side views of fully assembleddetail with brace removed for clarity


Off-site fabrication of base plate assembly:

Off-site fabrication of column assembly:

On site assembly of detail:

ED reaction collarfabricated fromwelded plates

lockup reactionplates and ED reactioncollar welded to base plate

base plate assembly

Note: fasteners are notshown for clarity

Column assembly

cast steel energydissipation bolts tocolumn web

gusset plates weldto column flanges

column assemblyis lowered intoplace

brace is pinned togusset plate

ED is bolted to ED reaction collarand gusset plates are bolted to lockupreaction plates

completed HDR column-foundation detail

Figure 6.15: Overview of proposed construction sequence for HDR column-foundation detail


Table 6.6: Connection design forces

Force Value (kN)

Lockup tension 4540Foundation compression 8230Base shear 1878ED strength 1201Design ED strength 2402

The base plate assembly and column assembly can be assembled separately in the shop. The base

plate assembly is fabricated fromwelded plates, and includes the energy dissipation reaction collars,

lockup reaction plates, and the base plate. The column assembly includes the gusset plates which

are welded to the column flanges, and the cast steel fuse units which are through-bolted to the

columnweb. Once the base plate assembly is fastened to the foundation of the structure, the column

assembly can be dropped into place, and the buckling restrained brace installed on-site.

6.4.2 Design forces and material properties

As with all braced frame column-foundation details, the HDR detail was capacity designed to

resist shear, compression, and tension forces. Additionally, this detail included special considera-

tion for the forces from the energy dissipation elements. A summary of the design forces is shown

in Table 6.6. The seismic shear force was determined using capacity design principals, consider-

ing full compressive and tensile overstrength in the first storey BRBs. Earthquake loads governed

the compression and tension forces and were determined using (1.2 + 0.2SDS)D + ⇢QE + L and

(0.9 � 0.2SDS)D + ⇢QE respectively. Note that the maximum tension force in the lockup was

lessened by the strength of the energy dissipation, since at full lockup the yielded energy dissi-

pation elements pull down on the column. The energy dissipation strength was determined using

Equation 6.1.1 and the material and geometric properties from Table 6.1. For the design of the

energy dissipation and ED connecting elements, an overstrength factor of 2.0 was applied in order

to conservatively take into consideration the overstrength due to strain hardening and geometric


Table 6.7: Material properties for detail design

Elementtype

ASTM standardFy

(Mpa)Fu

(Mpa)

Nominaltensilestrength,Fnt (Mpa)

Nominal shear strengthin bearing-typeconnections, Fnv

(Mpa)

column andplate

A572 Gr. 50 345 448

anchor rods F1554 Gr. 55 379 7517cast steelenergydissipation

- 300 450

bolts A490-X 780 579

nonlinearity.

Table 6.7 shows the material specifications and properties assumed for all components of this

detailed design.

6.4.3 Energy dissipation elements

The energy dissipation elements were designed based on the YBS concentric braced frame sys-

tem as previously described. Two separate cast units were designed to bolt to the web of the column.

One of these fuses is shown in Figure 6.16. Figure 6.16 shows an isometric view of the unit and

notes important features including the ductile triangular fingers and holes for connecting the fuse

to the column web and reaction collar. The backing plate was designed to resist the combined axial

and bending moments associated with the overstrength yield force in the fuse. The forces in the

backing plate were conservatively determined by assuming the plate acted as a pin-supported beam-

column (with supports occurring at the centroid of each bolt group), and loading corresponding to

the applied bending moment and shear from each yielding finger. Ten 1 inch bolts were chosen

to fasten the energy dissipation to the column webs. The bolts were designed to be through-bolts

across both energy dissipation units and the column web. The bolts were checked for combined


876

76.20

127

190.5

300

250

132

200

148 67.8 33.9

50.8 TYP

50.80 TYP

25.40 TYP

backing plate

bolt holes for connectionto column web

bolt holes for connectionto reaction collar

ductile triangular yieldingfingers

Figure 6.16: Cast steel energy dissipating supplemental fuse for HDR column-foundation connec-tion

shear and tension as per the AISC manual Equation J3-2 (AISC (2007)).

6.4.4 Gusset plates

The gusset plates were designed to accommodate the lockup forces and shear transfer described

in previous chapters. Figure 6.17 shows the two gusset plates and their connection to the column.

The right plate includes a pin to connect a BRB brace. It is noted that a different brace connection,

such as a bolted connection, could be used instead of the pin configuration shown. The gusset plates

were designed to be welded to the flanges of the column. The primary loads resisted by this weld

are the lockup tension after uplift and the maximum brace compressive force. A complete joint

penetration weld was used for this connection, and thus it was the backing material that controlled

its design. Thus the gusset plate was checked so that there was no yielding under shear or bending

moment. For the lockup, ten 1-1/8 inch A-490 bolts were chosen, and were considered to act in

double shear in the X configuration (threads excluded from the shear planes). The gusset plates

were checked for bolt tear-out, bearing failure, and shear block failure as per the AISC manual

(AISC (2007)). The slotted holes were designed to carry the full base shear and checked using


266

91.4

76.2 TYP

88.9 TYP

240

346

206.5

258

508

PYT 1.51R 186.7

373

38.1

373

508

CJP

TYP

Figure 6.17: Gusset plate and column detail


1389

895

52.4 TYP

540

115

346

191 127

19

CJP

TYP

CJP

TYP

Figure 6.18: Base plate assembly

AISC manual Equation J3-6c (AISC (2007)).

6.4.5 Base plate assembly

The base plate assembly consists of the energy dissipation reaction collar, lockup reaction

plates, and base plate. An overview of this assembly is shown in Figure 6.18.

The energy dissipation reaction collar was designed to connect the energy dissipation elements

to the foundation and is shown in Figure 6.19 (a). The collar consists of a welded plate assembly

that is in turn welded to the base plate. Slotted holes allow for the fuses to react without causing

excessive tension forces in the yielding fingers due to second order geometric effects (see Gray

(2012)). Since the energy dissipation elements act eccentrically to the centroid of this built up

section, each half of the reaction collar was checked to ensure that the section did not yield under


(a) (b)

(c)

411

889

12.7 TYP

180

CJP

TYP

114

216

132

384

191 T

YP

1

27 T

YP

R13

63 96 193

457

30.2 88.9 50.8

50.8

19.0

5

1389

895

191

318

540

850

1072

1199

38.1

115

346

549

780

38

Figure 6.19: Components of base plate assembly drawings: (a) energy dissipation reaction collar;(b) lockup reaction plate; (c) base plate


combined axial forces and bending moments. Complete joint penetration welds were specified to

fabricate the collar and connect it to the base plate. A 1/4 inch gap was left between the column

and the reaction collar in order to allow for some rotation of the column after uplift. It is noted,

however, that the collar could serve as a secondary base shear transfer if the gusset plate bolts were

damaged during an earthquake.

The lockup reaction plates were designed to weld to the base plate on each side of the gusset

plate. 1-1/8 inch bolts are fastened through all three plates and ensure that the seismic base shear

is transferred perpendicular to the direction of the slotted holes in the gusset plates, and that the

lockup tension is transferred vertically. A detail drawing of one lockup reaction plate is shown in

Figure 6.19(b).

Finally, the base plate was designed to transfer the tension, compression, and shear force to the

foundation via a series of 50.8 mm diameter anchor rods. A 38.1 mm thick base plate was chosen,

which satisfies the minimum thickness requirement from AISC manual section 14-4.

6.5 Chapter summary

This chapter presented the design of a six storey buckling restrained braced frame that was mod-

ified with hybrid ductile-rocking (HDR) and included supplemental energy dissipation elements.

Section 6.1 described the yielding brace system, and how that system could be adapted as a sup-

plemental damper in a rocking structure. Equations for strength, stiffness, and post-yield stiffness

of a YBS element were presented. Section 6.2 showed how a YBS connector could be modeled in

OpenSees using a special hysteretic element that considers the unique strengthening and stiffening

properties associated with the ductile yielding fingers. A numerical model of a YBS element was

compared to real test results from tests performed at the University of Toronto and shown to be in

good agreement.

Section 6.3 presented an example HDR design based on the 6-storey reference structure. ✓lock =

1.0% was chosen based on the results from Chapter 5, as this amount of allowable lockup in the


6-storey frame provided improved performance over the fixed base BRB while limiting the possi-

bility of excessive uplift and rocking drift under larger earthquake demands. Fuse properties were

chosen with a target energy dissipation parameter of � = 1.0. Pushover and push-pull results

were presented in order to show how this more realistic fuse model changed the system behaviour

compared to the idealized elastoplastic fuses that were presented in Chapter 5. Finally, results of a

nonlinear time-history analysis were presented to highlight the improved system performance over

the conventional fixed base BRBF.

Section 6.4 presented a preliminary detail design for the column-foundation connection from

the example design in Section 6.3. The detail used two cast steel supplemental hysteretic dampers

in order to achieve the desired energy dissipating characteristics of the rocking joint. The fuses were

designed to bolt to the column web, and connect to the base plate by bolting to an assembly made of

welded plates. Gusset plates with ten slotted holes were designed to transfer base shear and allow

for uplift and lockup after 91.4mmof vertical deformation. TheBRBbracewas designed to connect

to one of these gusset plates. This detail design demonstrated that by making relatively simple

modifications to the column-foundation connections of conventional BRB frames, the performance

gains highlighted in Section 6.3 can be achieved. This detail design further highlighted the fact that

the HDR concept could be readily applied in the design and construction of steel buildings without

major modifications or particular challenges in the field.

Chapter 7

Summary and conclusions

This chapter presents a summary of the main aspects of this thesis. It also provides a discus-

sion on the viability of the proposed HDR system in the context of current building codes, and

recommendations for future research.

7.1 Background and literature review

Chapter 2 provided an overview of the mechanics and performance issues related to buckling

restrained brace frames and rocking frames. BRBFs are an attractive seismic-resistant system since

their high levels of ductility and low post-yield stiffness allow for low system forces and thus

economical designs. Despite this advantage, the low post-yield stiffness the system exhibits can

also lead to drifts accumulating in a few stories, excessive cyclic and residual damage, and in turn

necessitate the expensive repair or even demolition of the entire building.

On the other hand, rocking structures are usually not associated with large drift concentra-

tions or residual drifts as most of the inelastic action is accommodated at the base of the rocking

structure. Researchers have developed these systems in recent years so that the SFRS structural el-

ements are designed to remain essentially elastic, and system ductility occurs as a geometric frame

rotation that is not associated with damage, except possibly in supplemental energy dissipation

elements which can be replaced after a seismic event. While these systems are highly promising

143

144 CHAPTER 7. SUMMARY AND CONCLUSIONS

low-damage alternatives to conventional damage-based seismic systems, they too have drawbacks.

As shown byWiebe (2013), rocking structures do not limit the shear at the base of the structure and

overturning moments along the frame’s height, and thus higher mode demands can severely limit

the force-reduction capability of these systems. While Wiebe (2013) developed and tested highly

effective options for mitigating these higher-mode effects, these solutions represent an increased

design complexity and detailing cost that reduces the desirability of such controlled rocking frames.

If higher modes are not mitigated, large frame members are required to ensure an elastic response,

and even still these members could experience non-ductile damage if they are overloaded. As well,

the post-tensioning elements and slab-frame compatibility required for controlled rocking struc-

tures represent an increased complexity and cost that are not features of conventionally designed

BRBFs.

Thus, the hybrid ductile-rocking system that was proposed in this thesis allows some rocking

in a conventionally designed code-compliant BRB frame. The large force reduction, energy dis-

sipation, and acceleration control characteristics of the BRBs are maintained, while the tendency

to accumulate excessive cyclic and residual damage is reduced since a portion of the system dis-

placements is contributed by a base-rocking connection. A lockup device is provided at the base

in order to allow for the BRBF to develop its code-prescribed resistance under a larger earthquake

demand. The lockup also serves to limit the vertical displacement of the system as well as the

possible excessive deformation on supplemental energy dissipation elements.

7.2 Mechanics of the HDR system

Chapter 3 derived the basic mechanics of the HDR system and highlighted important charac-

teristics of the proposed system related to the P-Δ effect and residual deformations. Equations that

describe a base-rocking joint were presented based on the mechanics derivation for controlled rock-

ing structures from Wiebe (2013). A series of free body diagrams of a one-storey HDR structure

were presented in order to demonstrate the distinction between smaller system deformations that do

7.3. PARAMETRIC STUDY ON HYBRID DUCTILE-ROCKING BRBFS 145

not engage the lockup, and larger deformations that do engage it. Considering a single-degree-of-

freedom response, such smaller deformations are not associated with residual drifts in the braces

whereas if the lockup is engaged, residual deformations in the braces are possible. It was noted

that in a real multi-storey structure, higher mode demands can cause cyclic inelastic loading and

residual deformations even if the lockup is not engaged.

The pushover and push-pull behaviour of the HDR system was investigated. During a mono-

tonic pushover loading, since brace yielding is expected at a larger system displacement than for

a conventional fixed base structure, a reduction in system strength is expected because of the P-Δ

effects. During cyclic loading, however, that reduction depends on the specific loading history

since brace yielding can occur at any system displacement. Finally, in this chapter it was noted that

the residual drifts in HDR structures are sensitive to the loading history.

7.3 Parametric study on hybrid ductile-rocking BRBFs

Chapter 4 presented the reference buckling restrained braced frames used in this thesis: 2, 4,

and 6-storey frames were designed for Los Angeles, California (as outlined in Appendix A), and

models of these frames were created using OpenSees. Nonlinear elements were used to model

the buckling restrained braces and were calibrated to BRB tests by Black et al. (2002). Three

record suites of 10 records representing 0.5 DBE, DBE, and MCE were selected and scaled, and

the building models were analyzed using nonlinear time-history analysis. In general, the BRBs

accomplished their design intent of limiting system forces, including storey shears, overturning

moments, and column forces. That being said, cyclic and permanent deformations were excessive

especially at the lower stories of all three frames. The reference frame results suggested that these

buildings would require expensive repairs or, more likely, demolition after a design-level seismic

event due to excessive damage and permanent deformations concentrated at the first few floors.

In Chapter 5, a parametric study was performed on a total of 111 different building models

incorporating the HDR concept that investigated how the energy dissipation parameter, � (which is


directly related to the supplemental energy dissipation strength) and allowable base rotation before

lockup, θlock, affected the response of the reference frames. The results demonstrated a number

of different trends. While peak drifts were relatively similar or smaller for the HDR system as

compared to the fixed base frames for the 4 and 6-storey structures, allowing rocking generally

caused an increase in peak drifts for the 2-storey frame when there was little or no supplemental

energy dissipation. This result is in line with previous observations on deformations in flag-shaped

systems with short periods (see Christopoulos et al. (2002) and Wiebe and Christopoulos (2014)).

It was noted that while peak drifts are related to the low-cycle fatigue life of conventional fixed base

structures, it is the peak drift minus the base rotation (DMR) that helps predict LCF life in HDR

structures. Thus, the DMR values were calculated, and were shown to decrease with increased

θlock. Residual drifts were reduced for the 0.5 DBE suite, under which the lockup devices were

less frequently engaged, but were not necessarily reduced for the larger earthquake levels in the 2

and 4-storey frames. There was a notable reduction in residual deformations in the 6-storey frame.

The energy dissipated by the braces was shown to be significantly reduced with increased θlock.

System forces were shown to be similar between the HDR structures and the fixed base response

since in both cases forces were limited by the capacity of the braces. That being said, the storey

shears were somewhat reduced in many of the HDR frames since the overstrength demand on

the braces was lessened. An increase in overturning moment and column forces was observed,

especially for the shorter-period 2-storey frame, and was determined to be related to a dynamic

excitation of the lumped vertical masses.

Considering the observed excitation of vertical mass, a small study was performed in order to

investigate how vertical mass modeling choices affected the analysis results. A review of previous

experimental and numerical rocking investigations showed that while excitation of vertical mass is

a physical phenomena that can cause spikes in column forces, numerical techniques for approximat-

ing this effect are varied across different historical studies. A study was performed to investigate

how four different vertical mass modeling techniques observed in the literature affect the results.

It was determined that lumping the vertical mass at the beam-column connections likely represents

7.4. DETAIL DESIGN OF 6-STOREY BRB HDR FRAME WITH CAST STEEL FUSE 147

an upper bound on the column force amplification, and that this effect is much more pronounced

in shorter period structures. Further, even if the column forces were amplified, the brace force and

drift results were not greatly affected by the different vertical mass modeling techniques.

7.4 Detail design of 6-storey BRB HDR frame with cast steel fuse

In order to more clearly demonstrate the performance benefits of the proposed HDR system,

an example design and analysis was carried out. The 6-storey reference frame was modified to

incorporate the hybrid ductile-rocking system and HDR parameters were chosen based on the the

results of the parametric study in Chapter 5. A cast steel supplemental fuse based on the yielding

brace system (see Gray (2012)) was designed in order to provide adequate ductility at an expected

maximum deformation corresponding to θlock = 1.0%, and a numerical model of this fuse was

calibrated based on laboratory tests. This model explicitly captured the characteristic post-yield

stiffening and strengthening associated with the ductile yielding fingers of the YBS devices.

A nonlinear time-history analysis was carried out on this modified 6-storey frame as well as the

fixed base frame, and the results were compared. The HDR frame demonstrated a greatly improved

seismic performance across a number of response parameters. The peak drift profile of the HDR

frame did not feature large drift concentrations, and the DMRprofile demonstrated a large reduction

in seismic demand on the BRBs. This result was further emphasized by the roughly 50% reduction

in energy dissipated by the braces across all three earthquake intensities. The HDR frame’s residual

drifts were significantly less than the fixed base frame at all earthquake levels.

TheHDR systemmaximum seismic force values were very similar to the fixed base values since

in both frames the forces were limited by the capacities of the BRBs. Even still, the storey shears in

the lower stories of the HDR frame were lower than for the fixed frame since the HDR frame was

not associated with large deformation concentrations, and thus the BRBs did not experience as large

nonlinear overstrength forces. Furthermore, both the HDR and fixed frames similarly experienced

the beneficial storey acceleration characteristics of BRB frames.


Finally, a column-foundation detail concept was developed in order to demonstrate how the

HDR system could be practically implemented. Gusset plates with slotted holes were designed

to transfer the base shear and permit uplift and lockup. Cast steel energy dissipation elements

were designed to be through-bolted across the column web and connect to the base plate via a

collar fabricated from welded plates. The buckling restrained brace was designed to connect to

one of the gusset plates and was designed as a true-pin connection for this example. A proposed

construction sequence was developed to demonstrate how much of the fabrication of this detail can

be accomplished off-site so that no welding is required on-site.

This example design demonstrated that by modifying the column-foundation connections of

a conventionally designed BRBF to allow for a controlled uplift, the seismic performance and

resilience of the frame can be greatly improved.

7.5 Framework for application of HDR system and discussion of

future research

While controlled rocking structures (CRS), such as those proposed by Eatherton et al. (2008),

Roke et al. (2006), andWiebe (2013), are not yet codified seismic systems, theHDR system features

a code-compliant ductile BRB frame as its superstructure, and a lockup device that ensures, after a

predetermined amount of allowable base rocking, the development of the full superstructure codi-

fied specified lateral resistance and ductility. Thus, the HDR system is effectively a code-compliant

with respect to the strength requirements of the code. With respect to the global deformations of

the structure (which are linked to the global stability of the system), the parametric study in Chapter

5 and the analysis of the 6-storey example design in Chapter 6 showed that the peak drifts in the

HDR systems were typically similar if not smaller when compared to the conventional fixed-base

frames. Based on these results, it could then be inferred that the HDR structures considered in this

thesis do therefore meet the intent of the current seismic design codes with respect to the global

deformations of the system. In fact, the results of this thesis suggest that the collapse performance

7.5. FRAMEWORK FOR APPLICATION 149

of the HDR system should be improved over conventional fixed-base structures since the structure

can have similar or smaller peak drifts, interstorey drifts which are more evenly distributed along

the building height, and less inelastic deformation demand on the BRBs. Future research is recom-

mended in order to further confirm that the collapse performance of the HDR system is similar if

not improved over conventional fixed base structures.

It should also be noted that the idea of allowing small amounts of foundation uplift is already

recognized in current codes, even if rocking is not explicitly intended as an approved seismic sys-

tem. For example, AISC (2010) allows for column forces in an SFRS to be limited by the forces

that are associated with “the resistance of the foundation to overturning uplift”. Additionally, the

National Building Code of Canada (NBCC 2010) addresses foundation rocking in two clauses.

Clause 4.1.8.15(8) states that the design forces for the SFRS can be limited to the values associ-

ated with foundation rocking as long as the force modification factors correspond to those for the

chosen SFRS, and the foundations are designed according to clause 4.1.8.16(1). Clause 4.1.8.16(1)

states that when foundations are allowed to rock, the design forces for the foundation can be lim-

ited to those determined using an RdRo equal to 2.0. In the context of these code clauses, the HDR

column-foundation detail could be considered a form of foundation rocking in a conventional SFRS

that is already recognized by the building code.

While the HDR system requires modifications to a frame’s column-foundation detail, and thus

additional design complexity compared to conventional fixed base frame, the HDR system is not

as complex to design and detail as a controlled rocking structure. Controlled rocking structures

often use post-tensioning tendons to control rocking strength and post-uplift stiffness, while the

HDR systems studied in this thesis relied on gravity loads to supply the post-uplift restoring mo-

ment. As well, the higher-mode mitigation techniques presented byWiebe (2013), while achieving

significant response improvements, are in fact associated with increased design complexity and

construction cost. In the HDR system, the control of higher mode effects is achieved through the

ductile superstructure that limits forces. The HDR system does, however, require a slab-SFRS de-

tail that can accommodate column uplift while transferring dead load and inertial forces. Further


research is required in order to study details that can accommodate this displacement compatibility,

including how a conventional detail would perform under the uplift demands of the HDR system. It

is noted that the lockup device used in the HDR details in this thesis can limit the amount of vertical

uplift that is expected, and thus the amount of vertical slab deformation required is ultimately up

to the design engineer. The parametric study demonstrated that allowing 1.0% base rotation can

achieve significant performance benefits, and so research should be performed to determine sever-

ity of slab damage under roughly 90 mm of vertical uplift, depending on the frame proportions. As

well, the effect of the excitation of vertical mass studied in Section 5.4 should be further character-

ized considering how different slab-SFRS details may affect how gravity load is transferred to the

HDR frame and how the vertical loads are amplified as a result of this vertical vibration.

This thesis only considered the two-dimensional response of the seismic force resisting-systems

that were studied. Since the HDR system behaves similarly to a conventional frame, it should be

able to accommodate three dimensional effects in the same manner as a conventional frame. In that

respect, further analyses should be performed that confirm the column-foundation detail’s ability

to accommodate out of plane rotations due to building deformations that are perpendicular to the

plane of the lateral frame.

Further research is recommended to expand the HDR system to include other SFRSs besides

BRBFs that are prone to excessive damage, including conventional concentric braced frames (CBF),

moment resisting frames (MRF), or eccentrically braced frames (EBF). Additionally, the HDR con-

cept could be used to increase the seismic force reduction factor of a lower-ductility system. For

example, a special concentrically braced frame (SCBF), which has an R factor of 6 (ASCE, 2010a)

could be designed with a larger R factor if it was designed as an HDR system.

7.6 Low-damage, economical seismic design

This thesis presented a new hybrid ductile-rocking steel frame seismic system. It was shown

that while conventional codified structures may be relatively simple to design and inexpensive to

7.6. LOW-DAMAGE, ECONOMICAL SEISMIC DESIGN 151

construct, excessive damage can result in their total loss even after experiencing a single design-

level earthquake. While controlled rocking structures are promising high-performance systems

since they can be designed to withstand multiple earthquakes while sustaining little or no dam-

age, higher mode demands and increased detailing requirements make them more complex and

expensive options as compared to conventional ductile systems. Thus, the HDR system combines

the positive aspects of conventional ductility and rocking frames in order to achieve a low dam-

age, cost effective seismic system that offers improved performance and resilience over current

damage-based systems. The result from the example structure that was presented in Chapter 6 in-

dicated that with the HDR detail the cyclic inelastic demands were reduced in the BRBs to such an

extent during even an MCE event that the structure could potentially resist a second MCE event (as

long as the residual interstorey drifts were not excessive after the first MCE record), thus greatly

enhancing the resilience of this highly ductile codified steel BRBF.

While this thesis focused on BRBFs as an example of conventional systems, the HDR concept

can be extended to other damage-based, distributed-ductility, codified seismic systems such as

CBFs, MRFs, and/or EBFs and be further developed as a general strategy to enhance the resilience

of current codified ductile structures.

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Appendix A

Design of reference structures

This section describes the design process for the reference structures. The structures used in this

study were designed based on similar structures from Choi et al. (2008). The frames were designed

using the loading defined by ASCE 7-10 Minimum Design Loads for Buildings and Other Struc-

tures ASCE (2010a). The steel members were designed according to the provisions of AISC 360-10

Load and Resistance Factor Design Specification for Structural Steel Buildings ASCE (2010b), and

ANSI/AISC 341-10 Seismic Provisions for Structural Steel Buildings (including Supplement No.

1) (AISC, 2010).

A.1 Description of reference structures and preliminary design

A typical steel frame office building located in downtown Los Angeles, California was chosen.

The plan of the office building consisted of three 9.14 m bays in the north-south direction, and

five 9.14 m bays in the east-west direction. The lateral force-resisting system in the north-south

direction consisted of two BRB braced frames, along the east and west edges of the building. The

planwas symmetrical in both the north-south and east-west direction, and thus there was no inherent

torsion in this structure. The design of the lateral system in the east-west direction is not described

in this thesis. Figure 4.1shows the building plan and elevation. Table A.1 shows the steel material

properties used in the design. Lateral load resisting beams and columns were designed with ASTM

159

160 APPENDIX A. DESIGN OF REFERENCE STRUCTURES

Table A.1: Material properties

Property ASTM A992 A36

Modulus of Elasticity, E (Mpa) 200000 200000Shear Modulus, G (Mpa) 78000 78000Specified Minimum Yield Stress, Fy (Mpa) 345 345Specified Minimum Tensile Strength, Fu (Mpa) 448 400Ratio of the Expected Yield Stress to the SpecifiedMinimum Yield Stress, Ry

1.1 1.5(1.3)*

Ratio of the Expected Tensile Strength of the SpecifiedMinimum Tensile Strength, Rt

1.1 1.2(1.2)*

*Values for A36 plate are given in brackets (as opposed toA36 rolled sections)

A992 steel. The beams and columns that were only part of the gravity frame were presumed to use

A36 steel, although the design of these members is not included in this report.

The building is located in Los Angeles, California. The soil was assumed to correspond to Site

Class D – stiff soil. The design response spectrum was determined using the USGS online tool

(http://earthquake.usgs.gov/designmaps). The mapped acceleration parameters were Ss = 2.434

g and S1 = 0.853 g. The site coefficients, Fa and Fv, were 1.0 and 1.5 respectively. Using

these values, the maximum considered earthquake spectral response acceleration parameters were

determined using for short periods as SMS = FaSs = 1.0(2.434) = 2.343 g (ASCE 7-10 Equation

11.4-1), and at a period of 1 s as SM1 = FvS1 = 1.5(0.853) = 1.280 g (ASCE 7-10 Equation

11.4-2).

The design spectral acceleration parameters were equal to 2/3 of the maximum considered

earthquake spectral response acceleration parameters, giving SDS = 1.631g and SD1 = 0.858g.

The fundamental period parameters for the design response spectrum were To = 0.2( SD1SDS

) =

0.1052s and TS = SD1

SDS

= 0.526s. The long-period transition period was determined from the

map given in ASCE 7-10 Figure 22-12 as TL = 8.Using these parameters the design spectrum was

produced using ASCE 7-10 section 11.4.5 (ASCE, 2010a).

A.1. DESCRIPTION OF REFERENCE STRUCTURES AND PRELIMINARY DESIGN 161

Table A.2: Seismic loading parameters

Seismic Force-Resisting System BRBF

ASCE 7 section where detaling requirements are specified 14.1 and 12.2.5.5Response Modification Factor, R 8Overstrength Factor, Ωo 2.5Deflection Amplification Factor, Cd 5

Table A.3: Gravity loading

Roof Load (kPa) Floor Load (kPa)

Roofing 0.24 Floor Covering 0.10Insulation 0.10 Partitions 0.48Concrete fill on metal deck 2.20 Concrete fill on metal deck 2.20Fireproofing 0.10 Fireproofing 0.10Ceiling 0.24 Ceiling 0.24Mechanical/electrical 0.48 Mechanical/electrical 0.48Steel framing 0.72 Steel framing 0.72

Total 4.07 Total 4.31

The building model was assumed to be an office building, and thus the occupancy category was

II according the ASCE 7-10 table 1.5-1. This meant that the importance factor according to ASCE

7-10 table 1.5-2 was I = 1. Since SDS > 1.5, the seismic design category was E for this system,

as per ASCE 7-10 section 11.6. The seismic loading parameters are given in Table A.2 .

The dead load gravity loadings that were assumed at a typical floor level and roof level are

given in Table A.3 . For office buildings the minimum uniformly distributed live load on a typical

floor is 2.4 kPa (50 psf) and 0.96 kPa (20 psf) on the flat roof according to ASCE 7-10 Table 4-1.

Live load reduction factors were applied as prescribed by ASCE 7-10 section 4.8 and section 4.9

for the column and beam design loadings based on their tributary areas. The final live loads for the

exterior beams and columns are given in Table A.4 . In the table, Lo is the unreduced design live

load, L is the reduced live load, KLL is the live load element factor for interior floors, R1 is the


Table A.4: Live loads

Storey Live Load,Lo (kPa)

Column Beam

KLL or R1 At (m^2) L (kPa) KLL or R1 At (m^2) L (kPa)

6 0.96 0.725 43.2 0.696 0.740 41.8 0.715 2.4 4 86.4 1.19 2 41.8 1.84 2.4 4 129.5 0.96 2 41.8 1.83 2.4 4 259 0.96 2 41.8 1.82 2.4 4 518 0.96 2 41.8 1.81 2.4 4 1036 0.96 2 41.8 1.8

reduction factor for roof live loads, and At is the tributary area on the column or beam.

The effective seismic weight considered the dead load for each storey. These values are given

in Table A.5 .

Wind loading did not control the design of this structure, as determined from checks not shown

here, based on Choi et al. (2008).

To begin the design of the model structure, the equivalent lateral force procedure was performed

in order to obtain lateral loads that led to a first-iteration design of the structural members. This pro-

cedure did not depend on the stiffness properties of the structure. Since modal response spectrum

was subsequently performed, the ASCE 7-10 (ASCE, 2010a) upper limit on fundamental period,

Tu = 1.4Ta, was used for the initial design.

The approximate fundamental period for the BRBFs was determined using ASCE 7-10 section

12.8.2.1 Equation (12.8-7). The approximate fundamental period was given by Ta = Cthxn. Table

A.6 shows the fundamental periods for the frames.

Once the approximate fundamental period was calculated, the seismic base shear was deter-

mined using ASCE 7-10 section 12.8.1, where V = CsW . Cs is the seismic response coefficient,

given by ASCE 7-10 Equation (12.8-2) as Cs = SDS

(RI

)subject to ASCE 7-10 Equation (12.8-3)

to (12.8-5), Cs SD1

T (RI

)for T > TL, Cs S

D1TL

T 2(RI

)for T > TL, and Cs = 0.044SDSIe � 0.01.

Since S1 > 0.6, the the lower limit for Cs was determined by ASCE 7-10 Equation (12.8-6) as


Table A.5: Effective seismic weight

Storey hx(m) wx(kN)

6 storey frame (k=1.268) 6 21.9 51005 18.3 54104 14.6 54103 11 54102 7.3 54101 3.7 5410

Total 32150

4 Storey frame (k=1.133) 4 14.6 51003 11 54102 7.3 54101 3.7 5410

Total 21330

2 Storey frame (k=1.0) 2 7.3 51001 3.7 5410

Total 10510

Table A.6: Determination of approximate fundamental period


Ct 0.0731 0.0731 0.0731

x 0.75 0.75 0.75

Ta (s) 0.74 0.547 0.323

Tu (s) 1.036 0.766 0.452


Table A.7: Calculation of seismic base shear


W (kN) (forentire floor)

32150 21330 10510

R 8 8 8Ta (s) 0.74 0.547 0.323Cs 0.204 0.204 0.204Cs lower limit 0.01 0.01 0.01Cs upper limit 0.1036 0.1400 0.237V (kN) 3328 2990 2144

Cs � 0.5S1

(RI

). Figure A.7 shows the parameters involved in the calculation of the seismic base shear

for the frames.

Subsequently, the vertical distribution of seismic force was determined using ASCE 7-10 sec-

tion 12.8.3. The lateral seismic force at level x is given by Fx = CvxV , where Cvx is the vertical

distribution factor given by Cvx = wx

hk

xPn

i=1 wi

hk

i

, where wx is the portion of the total effective weight

of the structure located or assigned to level i or x, hx is the height from the base to level i or x, and k

is an exponent that takes into account higher mode effects. k is equal to 1.0 if T 0.5s and is equal

to 2.0 if T � 2.5s. For intermediate periods k can be determined as a linear interpolation. Figure

A.8 shows the lateral seismic force calculations for the BRBFs. Storey weights and subsequent

forces are given for half of the total base shear, given that there are two lateral resistant frames in

each orthogonal direction.

Once the storey shear was determined, preliminary design forces were obtained for each frame

using simplifying assumptions that did not rely on member stiffnesses. Statics was used to find

brace forces by presuming that all of the storey shear was to be resisted by the horizontal component

of force in each brace at a given level. Once the design brace force was determined for each brace,

brace core areas were calculated using the steel minimum specified yield strength, rounding up to

the nears 0.5 square inch. Finally, the beam and column forces were determined with the principles


Table A.8: Lateral seismic force calculations

Storey wx(kN) hx(m) Fx(kN) Vx(kN)

6 storey frame (k=1.268) 6 5100 21.9 1008 10085 5410 18.29 854 18624 5410 14.62 642 25043 5410 10.97 446 29502 5410 7.32 267 32171 5410 3.66 110.7 3328

Total 32150

4 Storey frame (k=1.133) 4 5100 14.62 1211 12113 5410 10.97 927 21382 5410 7.32 586 27241 5410 3.66 267 2991

Total 21330

2 Storey frame (k=1.0) 2 5100 7.32 1401 14011 5410 3.66 743 2144

Total 10510


of capacity design.

In order to account for the effect of accidental torsion, the storey shear values determined from

the equivalent lateral force procedure were multiplied by 1.1, representing a 10 per cent increase

in the forces in each frame due to the offset of the centre of mass by 5 per cent from the center in

each orthogonal direction.

A.2 Dynamic Analysis

2D numerical models were constructed using SAP2000 in order to perform a modal analysis

for each frame. Once the preliminary design was determined using the equivalent lateral force

procedure, a modal analysis was performed in order to accurately determine earthquake loads and

drifts. The structures were then redesigned to reflect the new design forces, and then reanalyzed.

This process was repeated until the design converged. Only the final design is described in this

thesis.

In order to accurately approximate the increased stiffness of the BRBs due to the stiff connec-

tions regions and transitions of the braces, the elastic modulus of the braces was multiplied by 2.0,

as explained in the OpenSees modeling description in Section 4.1. Centerline beam-column ele-

ments were used to model the beams and columns. Columns were continuous between floors and

column splices were modeled as fixed moment connections. Beam-column connections and brace

connections were modeled as pins. Masses were modeled as lumped horizontal mass at each storey

corresponding to half the seismic weight from Table A.5 since only one frame was modeled which

carries half of the tributary weight of the entire structure.

Response spectrum analysis was performed in SAP2000. For the four and six storey structures

three modes were considered, while twomodes were considered for the two storey structure. Modal

analysis results for both structural systems, including � are listed in Table A.9 , and themode shapes

are shown in Figure A.1 .

Table A.10 shows mass participation factors for the structures, and indicates that a sufficient

A.2. DYNAMIC ANALYSIS 167

Table A.9: Modal analysis results

Mode 1 Mode 2 Mode 3

6 storey frame Tm (s) 1.016 0.369 0.219Sam(g) 0.844 1.631 1.631�6m 1.000 1.000 1.000�5m 0.835 -0.135 -0.945�4m 0.656 -0.530 -0.093�3m 0.470 -0.859 0.145�2m 0.293 -0.082 0.952�1m 0.134 -0.482 0.841

4 Storey frame Tm (s) 0.615 0.236 0.1449Sam(g) 1.395 1.631 1.631�4m 1.000 1.000 1.000�3m 0.747 -0.402 -2.078�2m 0.481 -0.981 0.422�1m 0.224 -0.075 1.809

2 Storey frame Tm (s) 0.384 0.1588Sam(g) 1.631 1.631�2m 1.000 1.000�1m 0.505 -1.867


(a)

T=1.016 s T=0.369 s T=0.219 s

T=0.615 s T=0.236 s T=0.1449 s

T=0.384 T=0.1588 s

(b)

(c)

T=0.369 s

Figure A.1: Mode shapes from SAP2000 modal analysis (a) 6 storey structure; (b) 4 storey struc-ture; (c) 2 storey structure

A.2. DYNAMIC ANALYSIS 169

Table A.10: Response spectrum analysis parameters

Mode 1 Mode 2 Mode 3

6 storey frame Mass Participation Factor 0.750 0.150 0.0452Seismic Base Shear, Vt (kN) 2824Scaling Factor 1.0

4 Storey frame Mass Participation Factor 0.817 0.1371 0.0350Seismic Base Shear, Vt (kN) 3220Scaling Factor 1.0

2 Storey frame Mass Participation Factor 0.900 0.0989Seismic Base Shear, Vt (kN) 1944Scaling Factor 1.0

Table A.11: Storey deflection and drift calculations

6 Storey 4 Storey 2 Storey�6(%) 0.805�5(%) 0.931�4(%) 1.002 0.788�3(%) 1.079 0.887�2(%) 1.092 0.919 0.691�1(%) 1.069 0.913 0.681

number of modes was considered during the design since the sum of the mass participation factors

were greater than 0.9 as per ASCE 7-10 section 12.9.1. The updated base shear, Vt, is also included

in this table, as it was determined with SAP2000 using the SRSS combination rule. Since Vt >

0.85V for all the the structures, no scaling factor was necessary.

A storey drift limit is imposed by ASCE 7-10 section 12.12.1 where�a is the storey drift limit

governed by �a = 0.025hsx for occupancy category II, where hsx is the storey height, and the

structure is four storeys or less, and �a = 0.02h for all other structures, including the six storey

BRBF design. The storey drifts � were determined using the SRSS rule. Table A.11 shows the

results of the drift calculations.

The influence of P-Δ effects must be checked according to ASCE 7-10 section 12.8.7. These


effects must be considered only if ✓ = Px

�x

Vx

hsx

Cd

0.10 is not true, where ✓ is the stability coefficient

according to ASCE 7-10 Eq. (12.8-16). Px is the total vertical design load at and above level x,

using the load combination 1.0D+ 0.5L,� is the design storey drift, Vx is the seismic shear force

acting between levels x and x� 1 as determined from the modal analysis, hsx is the storey height

below level x, andCd is the deflection amplification factor as previously defined. For all the frames,

✓ 0.10 and thus P-Delta effects were not an issue for this structure.

A.3 Design of Structural Members

The structural members were selected using capacity design principles. While all ASCE7-10

load combinations were be considered in design, it was determined that combinations including

seismic loads governed all aspects of the design. These load combinations are defined in ASCE 7-

10 section 2.3 and 12.4.2.3 as (1.2+0.2SDS)D+⇢QE+L+0.2S and (0.9�0.2SDS)D+⇢QE+1.6H ,

where SDS is the design spectrum acceleration parameter previously defined,D is the dead load, ⇢

is the redundancy factor, QEa is the effect of the horizontal seismic forces including consideration

of capacity design principles, L is the live load, S is the snow load, andH is the load due to lateral

earth pressure, groundwater pressure or pressure of bulkmaterials. ⇢ is the redundancy factor factor

as defined by ASCE 7-10 section 12.3.4.2. This factor was assumed to be 1.0 as per ASCE 7-10

section 12.3.4.2. All beams and columns were selected to satisfy the requirements for sectional

ductility as defined by ANSI/AISC 341-10 Table I-8-1.

Before the elastic elements (beams, columns), were designed, the BRBswere designed based on

the results of the response spectrum analysis. The buckling restrained brace design axial strength is

given by ANSI/AISC 341-10 section F4 as �Pysc = �FyscAsc, where � = 0.9,Asc is the area of the

BRB steel core, and Fysc is the specified minimum yield stress of the steel core, or the actual yield

stress of the steel core. The overstrength brace strength is given as Pn0 = �!RyPysc for compres-

sion, and Tn0 = !RyPysc for tension. In these expressions, the overstrength factor is included in

the brace axial strength, Pysc, and is not explicitly considered as Ry. � is the compression strength

A.3. DESIGN OF STRUCTURAL MEMBERS 171

Table A.12: Buckling restrained brace parameters

BRB Property Fy

, Mpa Fymax

, Mpa ! � Ry

248 323 1.5 1.1 1.0

Table A.13: BRB design parameters

Storey Angle (deg.) P1 (kN) Asc (in^2) �Pysc(kN)

6 storey frame 6 38.7 345 2.5 3605 38.7 550 4 5764 38.7 696 5 7203 38.7 820 6 8642 38.7 925 6.5 9361 38.7 1005 7 1008

4 Storey frame 4 38.7 470 3.5 5043 38.7 791 6 8642 38.7 1010 7.5 10801 38.7 1135 8 1152

2 Storey frame 2 38.7 449 3.5 5041 38.7 684 5 720

adjustment factor, or the ratio of the maximum compressive force to the maximum tensile force in

the BRB, and was taken as 1.1. Additionally, ! is the strain hardening adjustment factor, which

is the ratio of the maximum tension force to the specified minimum yield force, and was taken as

1.5. The design values are given for the BRB frames in Table A.13, where P1 is the design axial

load. Fymax = 323Mpa was used for these calculations. Buckling-restrained brace parameters are

given in Table A.12 , and a summary of the brace design calculations is given in Table A.16.

Column forces were determined considering the applied overstrength brace forces and the cor-

responding free body diagram, as in Choi et al. (2008). Column sections, design forces, and resis-

tances are shown in Table A.14 , where Pf is the factored compressive load and Pny is the nominal

axial strength considering weak axis buckling.


Table A.14: Column sections, design forces, and resistances

Storey Column Size Pf (kN) �Pny (kN)

6 Storey frame 6th and 5th W12X35 1059 10883rd and 4th W12X96 3465 48191st and 2nd W14X132 6608 6982

4 Storey frame 3rd and 4th W10X45 1197 18301st and 2nd W12X96 4375 4819

2 Storey frame 1st and 2nd W8X40 1171 1628

Figure A.2: Unbalanced force on beams in BRBFs (from Choi et al. (2008))

The result of the unbalanced vertical load due to the differing tension and compression strengths

of BRBs, as shown in Figure A.2 , was included in the design of the beams and columns. The

vertical force on the beam was found as the difference between the adjusted brace strengths P 0n

and T 0n multiplied by the brace angle, Pu�vert = (P 0

n � T 0n)sin✓. Because of this vertical force,

there was an additional moment that was added to moments from dead and live loads, and was

determined using the load combination, M1 = (1.2 + 0.2SDS)MD + ⇢Mvert + 0.5ML, where

MD is the moment cause by dead load, ML is the moment caused by live load, and Mvert is the

moment caused by the unbalanced force. In order to determine the compressive force in the beams,

the horizontal seismic force must be inferred from the brace capacities. It was assumed that the

seismic force entered the SFRS from the diaphragm, with half of the total force acting on each

side of the frame. In this way, the seismic force, Fhalf was determined as Fhalf = Pu�hor/2, from


Table A.15: Forces and resistances of beams

Storey T 0n

(kN)P 0n

(kN)Pu�vert

(kN)Pcomp�total

(kN)M1

(kN-m)M2

(kN-m)

6 storey frame 6 781 860 -48.8 641 207 3045 1250 1375 -78.1 1361 191 4234 1563 1719 -97.6 1477 146 4233 1875 2063 -117.2 1721 101 4232 2032 2235 -126.9 1715 79 4231 2188 2407 -136.7 1837 57 423

4 Storey frame 4 1094 1203 -68.3 897 163 3043 1875 2063 -117.2 2105 101 4232 2344 2579 -146.5 2215 34 4231 2501 2751 -156.2 2081 12 423

2 Storey frame 2 1094 1203 -68.3 897 163 3041 1563 1719 -97.6 1605 146 423

the difference between the horizontal forces of the brace capacity above and below a given storey,

Pu�hor = (T 0n + P 0

n)icos✓ � (T 0n + P 0

n)i+1cos✓i+1, where i is the storey level below the beam and

i + 1 is the storey level above the beam. Finally, from statics the compressive force in half of

the beam, Pcomp�total, is the sum of the seismic force and the horizontal component of the tensile

capacity of the brace in the storey above, and is given as Pcomp�total = Fhalf + T0

n(i+1) ⇤ cos✓.

The interaction between themoments caused by gravity loads and the unbalanced vertical loads,

and the compressive force in half the beam was taken into consideration when beam sections were

chosen. The forces and resistances of the beams are given in Table A.15 . The design forces

were similar for all stories of each structure, and so for simplicity only one section was chosen

for each structure based on the worse case. Table A.15 includes the maximum compressive and

tensile overstrength force in each brace, P 0n and T 0

n, the maximum compressive force in the beam

Pcom , which occurs on one side of the braces while the other side is in tension, and the maximum

bending moment in the beam, M1, determined considering the BRB’s were not designed to carry

gravity load as per AISC 345-10. It should be noted that the unbalanced force due to the higher


compressive overstrength axial force in the brace led to an unequal net force that the braces applied

to the beam. The vertical component of this force acts upwards on the beam, and tends to counteract

the effect of gravity. Considering this, the non-seismic load combinations, especially 1.2D+1.6L,

was considered for beam design, and is given as M2 in Table A.15. However, in this case the

lower seismic bending moment acts with a high compressive force in the beam, and this combined

axial-bending loading condition controlled the design of all beams.

Axial-bending interaction can be verified using AISC 360-10 equations (H1-1a), and (H1-1b)

Pr

Pc

+8

9

✓Mrx

Mcx

+Mry

Mcy

◆ 1.0For

Pr

Pc

� 0.2 (A.3.1)

Pr

2Pc

+

✓Mrx

Mcx

+Mry

Mcy

◆ 1.0For

Pr

Pc

where Pr and Pc are the required and available axial compressive strength, respectively, and

Mrand Mc are the required available flexural strength. As shown in Table A.16, the interaction

ratios are all less than 1.0, and the flexural strength of the beams is greater than the bending moment

from gravity load.


Table A.16: BRB frame forces and beam calculations

Storey Section M2�b

Mn

(kN-m) Ratio from eq. A.3.1

6 storey frame 6 W18X55 0.534 0.6435 W18X55 0.742 0.9774 W18X55 0.742 0.9523 W18X55 0.742 0.9872 W18X55 0.742 0.9411 W18X55 0.742 0.955

4 Storey frame 4 W18X65 0.450 0.5653 W18X65 0.625 0.9432 W18X65 0.625 0.8741 W18X65 0.625 0.787

2 Storey frame 2 W18X60 0.486 0.6231 W18X60 0.676 0.908

Appendix B

Push-pull analysis overview of building models

for parametric study

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Figure B.1: Push-pull response of 2 storey structures for � = 0, � = 0.5, and � = 1.0

178 APPENDIX B. PUSH-PULL ANALYSIS OVERVIEW OF BUILDING MODELS

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Documents

Seismic performance of new hybrid ductile-rocking braced