Applied Science, Faculty of
Civil Engineering, Department of
DSpace
UBCV
Murphy, Michael
Master of Applied Science - MASc
University of British Columbia
Structural Steel is one of the most important building materials used worldwide; special Seismic Force Resisting Systems (SFRS) have been developed to use steel to resist seismic forces in earthquake prone regions. In Canada, several steel SFRS have been adopted in the code, these include Moment Resisting Frames, Concentrically and Eccentrically Braced Frames, Buckling Restrained Braced Frames and Steel Plate Shear Walls; conventional construction frames may also be designed which have no seismic detailing. The design of these systems is covered comprehensively in literature; however no guidance has been provided regarding the selection of the best system for a project. In this thesis, the relative merits of each of the prequalified systems have been studied. A five story office building located in Vancouver, British Columbia, was redesigned nine times implementing each of the clauses for seismic design in CSA S16-09. The relative performance of each are compared using the Performance Based Earthquake Engineering (PBEE) method. PBEE accounts for the uncertainties in the seismic hazard, structural response and structural damage and their effect on the building performance during an earthquake. The relative merits of these systems were evaluated in terms of material usage and financial loss of the structure after a seismic event. The conclusion is that although the Moment Resisting Frame carries the lowest repair costs, it uses 20% more steel than the Eccentrically Braced Frame. The optimum systems in terms of material usage and repair costs were the Steel Plate Shear Wall (type ductile) and the Eccentric Braced Frame. The worst performing were the buildings designed with low ductility; both the conventional construction and limited ductility Concentrically Braced Frame structures performed poorly. Analysis shows that under the conditions of this thesis, most of the repair costs are related to the acceleration sensitive nonstructural components. Systems designed with higher ductility experienced lower accelerations and therefore lower costs. The PBEE methodology is an effective approach for evaluating different structures and comparing how they perform dynamically in an earthquake. Using PBEE, this thesis shows the advantages of frames designed in Canada for high ductility in economic terms.
Performance based evaluation of prequalified steel seismic force resisting structures in Canada by Michael Murphy B.S., The University of Washington, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2012 © Michael Murphy, 2012 ii Abstract Structural Steel is one of the most important building materials used worldwide; special Seismic Force Resisting Systems (SFRS) have been developed to use steel to resist seismic forces in earthquake prone regions. In Canada, several steel SFRS have been adopted in the code, these include Moment Resisting Frames, Concentrically and Eccentrically Braced Frames, Buckling Restrained Braced Frames and Steel Plate Shear Walls; conventional construction frames may also be designed which have no seismic detailing. The design of these systems is covered comprehensively in literature; however no guidance has been provided regarding the selection of the best system for a project. In this thesis, the relative merits of each of the prequalified systems have been studied. A five story office building located in Vancouver, British Columbia, was redesigned nine times implementing each of the clauses for seismic design in CSA S16-09. The relative performance of each are compared using the Performance Based Earthquake Engineering (PBEE) method. PBEE accounts for the uncertainties in the seismic hazard, structural response and structural damage and their effect on the building performance during an earthquake. The relative merits of these systems were evaluated in terms of material usage and financial loss of the structure after a seismic event. The conclusion is that although the Moment Resisting Frame carries the lowest repair costs, it uses 20% more steel than the Eccentrically Braced Frame. The optimum systems in terms of material usage and repair costs were the Steel Plate Shear Wall (type ductile) and the Eccentric Braced Frame. The worst performing were the buildings designed with low ductility; both the conventional construction and limited ductility Concentrically Braced Frame structures performed poorly. Analysis shows that under the iii conditions of this thesis, most of the repair costs are related to the acceleration sensitive nonstructural components. Systems designed with higher ductility experienced lower accelerations and therefore lower costs. The PBEE methodology is an effective approach for evaluating different structures and comparing how they perform dynamically in an earthquake. Using PBEE, this thesis shows the advantages of frames designed in Canada for high ductility in economic terms. iv Table of Contents Abstract .......................................................................................................................................... ii Table of Contents ......................................................................................................................... iv List of Tables............................................................................................................................... viii List of Figures .............................................................................................................................. xii List of Symbols ......................................................................................................................... xviii List of Abbreviations ................................................................................................................... xx Acknowledgements ..................................................................................................................... xxi Chapter 1: Introduction .............................................................................................................. 1 1.1 Literature Review ............................................................................................................ 4 1.2 Objectives and Scope ...................................................................................................... 7 1.3 Organization and Outline ................................................................................................ 8 Chapter 2: Seismic Hazard Analysis ........................................................................................ 10 2.1 Seismicity in British Columbia ..................................................................................... 10 2.2 Site Description ............................................................................................................. 18 2.3 Probabilistic Seismic Hazard Analysis ......................................................................... 20 2.4 Ground Motion Selection .............................................................................................. 27 Chapter 3: Structural Design .................................................................................................... 33 3.1 Description of the Prototype Building .......................................................................... 33 3.2 Type D Moment Resisting Frame (MRF) S16-09 [27.2] Rd=5.0 Ro=1.5 ..................... 38 v 3.3 Type MD Concentric Braced Frame (CBF) S16-09 [27.5] Rd=3.0 Ro=1.3 .................. 40 3.4 Type LD Concentric Braced Frame (CBFLD) S16-09 [27.6] Rd=2.0 Ro=1.3 .............. 42 3.5 Type D Eccentrically Braced Frame (EBF) S16-09 [27.7] Rd=4.0 Ro=1.5 .................. 43 3.6 Type D Buckling Restrained Braced Frame (BRBF) S16-09 [27.8] Rd=4.0 Ro=1.2 .... 45 3.7 Type D Steel Plate Shear Wall (SPSW) S16-09 [27.9] Rd=5.0 Ro=1.6 ........................ 47 3.8 Type LD Steel Plate Shear Wall (SPSWLD) S16-09 [27.10] Rd=2.0 Ro=1.5 .............. 50 3.9 X-Braced Frame (XBF) S16-09 [27.11] Rd=1.5 Ro=1.3 ............................................... 52 3.10 Suspended Zipper Braced Frame (SZBF) Rd=3.0 Ro=1.9 ............................................. 53 Chapter 4: Analytical Modeling Approach ............................................................................. 55 4.1 Gravity System .............................................................................................................. 56 4.2 Concentrically Braced Frames ...................................................................................... 57 4.3 Eccentrically Braced Frames ......................................................................................... 60 4.4 Moment Resisting Frames ............................................................................................. 62 4.5 Buckling Restrained Braced Frames ............................................................................. 64 4.6 Steel Plate Shear Walls ................................................................................................. 66 Chapter 5: Seismic Performance Evaluation Procedure ........................................................ 68 5.1 The Performance Based Earthquake Engineering Method ........................................... 69 5.1.1 Performance Assessment Framework ....................................................................... 69 5.1.2 Evaluate Structural Response .................................................................................... 73 5.1.3 Performance Groups and Fragility Relations ............................................................ 74 5.1.4 Compute the Repair Cost .......................................................................................... 77 5.2 Performance Group Data for a 5 Story Vancouver Office Building ............................. 78 5.2.1 Structural Performance Group Data .......................................................................... 86 vi Chapter 6: Comparison of the Structural Performance......................................................... 92 6.1 OpenSees Model Data ................................................................................................... 92 6.2 Peak Story Drift and Acceleration ................................................................................ 93 6.3 Peak Story Shear ........................................................................................................... 99 6.4 Element Response ....................................................................................................... 105 6.5 Cost Analysis............................................................................................................... 112 6.6 Repair Cost Distribution by PG .................................................................................. 117 Chapter 7: Summary and Conclusions .................................................................................. 129 7.1 Steps for Performance Based Earthquake Engineering (PBEE) ................................. 130 7.1.1 Seismic Hazard Analysis ......................................................................................... 131 7.1.2 Structural Design ..................................................................................................... 131 7.1.3 Modeling ................................................................................................................. 133 7.1.4 Definition of Performance Groups .......................................................................... 133 7.1.5 Time History Analysis ............................................................................................ 134 7.1.6 Loss Analysis .......................................................................................................... 134 7.2 Performance Based Design and Evaluation of New Structural Systems .................... 134 7.3 Concluding Remarks ................................................................................................... 136 Bibliography .............................................................................................................................. 137 Appendices ................................................................................................................................. 142 Appendix A Structural Design ................................................................................................ 142 A.1 Gravity System ........................................................................................................ 142 A.2 Type D Moment Resisting Frame (MRF) S16-09 [27.2] Rd=5.0 Ro=1.5 ............... 152 A.3 Type MD Moment Resisting Frame (MRF) S16-09 [27.3] Rd=3.5 Ro=1.5 ............ 165 vii A.4 Type LD Moment Resisting Frame (MRF) S16-09 [27.4] Rd=2.0 Ro=1.3 ............. 166 A.5 Type MD Concentrically Braced Frame (CBF) S16-09 [27.5] Rd=3.0 Ro=1.3 ...... 166 A.6 Type LD Concentrically Braced Frame (CBFLD) S16-09 [27.6] Rd=2.0 Ro=1.3 .. 174 A.7 Type D Eccentrically Braced Frame (EBF) S16-09 [27.7] Rd=4.0 Ro=1.5 ............ 174 A.8 Type D Buckling Restrained Braced Frame (BRBF) S16-09 [27.8] Rd=4.0 Ro=1.2 187 A.9 Type D Steel Plate Shear Wall (SPSW) S16-09 [27.9] Rd=5.0 Ro=1.6 .................. 196 A.10 Type LD Steel Plate Shear Wall (SPSW) S16-09 [27.10] Rd=2.0 Ro=1.5 .............. 216 A.11 X-Braced Frame (XBF) S16-09 [27.11] Rd=1.5 Ro=1.3 ......................................... 217 A.12 Suspended Zipper Braced Frame (SZBF) Rd=3.0 Ro=1.9 ....................................... 219 Appendix B Modeling Parameter Details ............................................................................... 225 B.1 Steel01 Material ...................................................................................................... 225 B.2 Steel02 Material ...................................................................................................... 226 B.3 Hysteretic Material .................................................................................................. 227 B.4 Gravity System ........................................................................................................ 228 B.5 Concentrically Braced Frames ................................................................................ 228 B.6 Eccentrically Braced Frames ................................................................................... 229 B.7 Moment Resisting Frames ....................................................................................... 230 B.8 Buckling Restrained Braced Frames ....................................................................... 235 B.9 Steel Plate Shear Walls ........................................................................................... 236 Appendix C Development of Cost Data .................................................................................. 238 Appendix D Pushover Curves ................................................................................................. 245 viii List of Tables Table 2.1 Modified Mercalli Scale ................................................................................................ 17 Table 2.2 Historical Seismic Events in the Lower Mainland, British Columbia .......................... 17 Table 2.3 Seismic Sources within 300 km of Vancouver B.C. ..................................................... 19 Table 2.4 Criteria used to select Ground Motions ........................................................................ 28 Table 2.5 Selected Ground Motions for 2% in 50 Years Hazard .................................................. 30 Table 2.6 Selected Ground Motions for 10% in 50 Years Hazard ................................................ 30 Table 2.7 Selected Ground Motions for 10% in 50 Years Hazard ................................................ 31 Table 5.1 Summary of Performance Groups ................................................................................. 79 Table 5.2 CW Performance Group ................................................................................................ 81 Table 5.3 INTD Performance Group............................................................................................. 82 Table 5.4 Contents Performance Group ........................................................................................ 83 Table 5.5 Ceiling Performance Group .......................................................................................... 84 Table 5.6 Elevator Performance Group ......................................................................................... 85 Table 5.7 Structural BRBF PG ...................................................................................................... 87 Table 5.8 Structural CBF PG ........................................................................................................ 88 Table 5.9 Structural XBF PG ........................................................................................................ 88 Table 5.10 Structural MRF PG ..................................................................................................... 89 Table 5.11 Structural EBF PG....................................................................................................... 91 Table 5.12 Structural SPSW PG ................................................................................................... 91 Table 6.1 Structural Weight .......................................................................................................... 92 ix Table 6.2 Model Periods ............................................................................................................... 92 Table 6.3 Results for 2% in 50 Years Hazard Level ..................................................................... 95 Table 6.4 Results for 10% in 50 Years Hazard Level ................................................................... 95 Table 6.5 Results for 50% in 50 Years Hazard Level ................................................................... 96 Table 6.6 Maximum Response Quantities for 2% in 50 Years Hazard Level .............................. 96 Table 6.7 Percent Difference Between Median 2/50 EDP and Design Values ............................. 99 Table 6.8 Shear Results for 2% in 50 Years Hazard Level ......................................................... 102 Table 6.9 Shear Results for 10% in 50 Years Hazard Level ....................................................... 102 Table 6.10 Shear Results for 50% in 50 Years Hazard Level ..................................................... 102 Table 6.11 Maximum Response Quantities for 2% in 50 Years Hazard Level .......................... 103 Table 6.12 Percentage Difference Between NBCC and 2/50 Median Shear .............................. 103 Table 6.13 Pushover Analysis Nonlinearity Observed at ISD% ................................................. 106 Table 6.14 Yielding of Elements Under GM#302 ...................................................................... 107 Table 6.15 Total Repair Costs Associated with Different prequalified SFRS ............................ 115 Table A.1 Calculation of Additional Dead Load ........................................................................ 143 Table A.2 Load Combinations per NBCC 2005 Table 4.1.3.2 .................................................. 143 Table A.3 Members Satisfying Sx and Ix Limits Sorted by Mass ............................................... 147 Table A.4 Gravity Load Summary .............................................................................................. 150 Table A.5 Column Loading for Gravity System (interior columns) ........................................... 150 Table A.6 Vancouver Spectral Data ............................................................................................ 152 Table A.7 Design Parameters ...................................................................................................... 153 Table A.8 Building Component Weights .................................................................................... 154 x Table A.9 Sum of Building Element Weights per Floor ............................................................. 154 Table A.10 Calculation of Floor Shears ...................................................................................... 156 Table A.11 Deflections of Type D MRF..................................................................................... 160 Table A.12 Deflections of Type MD MRF ................................................................................. 165 Table A.13 Dr. Frame Deflections of Type LD MRF ................................................................. 166 Table A.14 Lateral Seismic Forces on CBF ................................................................................ 167 Table A.15 Axial Brace Loads from Elastic Model .................................................................... 168 Table A.16 Deflections of Type MD CBF .................................................................................. 173 Table A.17 Deflections of Type LD CBF ................................................................................... 174 Table A.18 Seismic Lateral Loads on EBF ................................................................................. 177 Table A.19 Comparison of Link Shear Calculations .................................................................. 178 Table A.20 Percent Differences the Two Force Estimation Methods......................................... 182 Table A.21 Link Properties ......................................................................................................... 183 Table A.22 Deflections of Type D EBF ...................................................................................... 185 Table A.23 Link Rotations .......................................................................................................... 186 Table A.24 Seismic Shear Forces for BRBF............................................................................... 188 Table A.25 Brace Core Area Design ........................................................................................... 189 Table A.26 Calculation of BRB Tmax and Cmax Forces ............................................................... 192 Table A.27 Deflections of Type D BRBF ................................................................................... 196 Table A.28 Member and Plate Sizes Ductile Plate Wall ............................................................. 197 Table A.29 RBS Connection Details........................................................................................... 197 Table A.30 Common A36 Plate Sizes ......................................................................................... 198 Table A.31 Pre-Qualified Connection Limitations ..................................................................... 211 xi Table A.32 Lateral Seismic Forces on SPSW ............................................................................. 213 Table A.33 Infill Plate Thickness ................................................................................................ 214 Table A.34 Check that MRF System Carries at Least 25% of Shear .......................................... 215 Table A.35 Deflections of SPSW ................................................................................................ 216 Table A.36 Deflections of Type LD SPSW ................................................................................ 216 Table A.37 Calculation of Seismic Lateral Shears ..................................................................... 218 Table A.38 Deflections of X-Braced Frame ............................................................................... 219 Table A.39 Deflections of SZBF................................................................................................. 224 Table B.1 Steel01 Material Parameters ....................................................................................... 225 Table B.2 Steel02 Parameters ..................................................................................................... 226 Table B.3 Hysteretic Material Parameters .................................................................................. 227 Table B.4 Lumped Hinge Behavior Properties ........................................................................... 228 Table B.5 Material Steel02 Parameters ....................................................................................... 229 Table B.6 Hinge Steel01 Parameters........................................................................................... 229 Table B.7 Steel02 Parameters ..................................................................................................... 230 Table B.8 Steel02 Hinge Parameters........................................................................................... 231 Table B.9 Steel02 Hinge Parameters for RBS ............................................................................ 231 Table B.10 Parameters for Hysteretic Material ........................................................................... 235 Table B.11 Clough Material Parameters ..................................................................................... 237 xii List of Figures Figure 2.1 Tectonic Plates and the “Ring of Fire” of Seismic Activity ........................................ 11 Figure 2.2 Plate Activity off the Coast of Vancouver Island ........................................................ 12 Figure 2.3 PGA Plot for 2% in 50 Years Seismic Hazard Level in Canada ................................. 12 Figure 2.4 Map of Historical Seismic Events in Canada 1627-2010 ............................................ 14 Figure 2.5 Modified Mercalli Intensity Scale for Historical Earthquakes .................................... 15 Figure 2.6 Significant Historical Earthquakes .............................................................................. 16 Figure 2.7 Building Site Location in Downtown Vancouver ....................................................... 19 Figure 2.8 Significant Seismic Sources Within 300 km of Site Location..................................... 20 Figure 2.9 Hazard Spectra for Downtown Vancouver, 5% Damped on Soil Class C .................. 23 Figure 2.10 Total Seismic Hazard Curve for PGA ....................................................................... 24 Figure 2.11 Deaggragation of seismic risk for PGA for different sources ................................... 25 Figure 2.12 Deaggragation of seismic risk by magnitude ............................................................. 26 Figure 2.13 Deaggregation of seismic risk by distance ................................................................ 27 Figure 2.14 Locations of Selected 2/50 Ground Motions ............................................................. 29 Figure 2.15 Selected ground motions scaled to target spectra 2/50 .............................................. 31 Figure 2.16 Selected ground motions scaled to target spectra 10/50 ............................................ 32 Figure 2.17 Selected ground motions scaled to target spectra 10/50 ............................................ 32 Figure 3.1 Isometric View of Building ......................................................................................... 34 Figure 3.2 Floor Plan of Building ................................................................................................. 35 Figure 3.3 Typical East-West Gravity Frame Bay – MRF System Only ...................................... 36 xiii Figure 3.4 Typical North-South Gravity Frame Bay – MRF System Only .................................. 36 Figure 3.5 Typical East-West Gravity Frame Bay ........................................................................ 37 Figure 3.6 Typical North-South Gravity Frame Bay .................................................................... 37 Figure 3.7 Plan Layout for Moment Resisting Frame Building .................................................... 39 Figure 3.8 Design For Moment Resisting Frame Design (MRF) ................................................. 39 Figure 3.9 Design for Concentric Braced Frame (CBF) ............................................................... 41 Figure 3.10 Design for Concentric Braced Frame (CBFLD) ........................................................ 42 Figure 3.11 Design for the Eccentrically Braced Frame ............................................................... 44 Figure 3.12 Design for Buckling Restrained Braced Frame (BRBF) ........................................... 46 Figure 3.13 Floor Plan of Steel Plate Shear Wall Building .......................................................... 48 Figure 3.14 Detail of bay DE-6 from the floor plan...................................................................... 48 Figure 3.15 Design for the Steel Plate Shear Wall (SPSW).......................................................... 49 Figure 3.16 Plan View of building for Type LD SPSW ............................................................... 51 Figure 3.17 Design for Type LD Steel Plate Wall (SPSWLD) ..................................................... 51 Figure 3.18 Design for X-Braced Conventional Construction (XBF) .......................................... 52 Figure 3.19 Design for Suspended Zipper Braced Frame (SZBF) ................................................ 54 Figure 4.1 Arrangement of Frames to Model ½ of the Building .................................................. 56 Figure 4.2 Typical Gravity Frame ................................................................................................. 57 Figure 4.3 Element Model for Buckling Brace ............................................................................. 59 Figure 4.4 Hysteresis Curve from OpenSees ................................................................................ 59 Figure 4.5 Schematic of Eccentric Braced Frame ......................................................................... 61 Figure 4.6 OpenSees Material Response for Shear Link .............................................................. 61 Figure 4.7 Zerolength Hinge Idealization for Moment Connection Behavior .............................. 63 xiv Figure 4.8 Overlay of SAC Test Data and OpenSees Simulation Result ..................................... 63 Figure 4.9 Hysteretic Response of Brace Material Model ............................................................ 65 Figure 4.10 Steel Plate Wall Test Specimen and Model ............................................................... 67 Figure 4.11 Opensees Load Deformation Response for Steel Plate Wall ..................................... 67 Figure 5.1 Steps for the PBEE Methodology ................................................................................ 71 Figure 5.2 Fragility Relation for a Typical PG ............................................................................. 75 Figure 5.3 Example of Unit Repair Cost Function........................................................................ 76 Figure 5.4 Fragility Curves for the Curtain Wall PG .................................................................... 80 Figure 5.5 Fragility Curves for INTD PG ..................................................................................... 82 Figure 5.6 Fragility Curve for Contents PG .................................................................................. 83 Figure 5.7 Fragility Curves for Ceiling PG ................................................................................... 84 Figure 5.8 Fragility Curve for Elevator PG .................................................................................. 85 Figure 5.9 Fragility Curve for BRBF Structural PG ..................................................................... 87 Figure 5.10 Fragility Curves for CBF ........................................................................................... 88 Figure 5.11 Fragility Curves for MRF Structural PG ................................................................... 89 Figure 5.12 Fragility Curves for EBF Structural PG .................................................................... 90 Figure 5.13 Fragility Curves for SPSW Structural PG ................................................................. 91 Figure 6.1 Median ISD Vs. Height from 2/50 Hazard .................................................................. 97 Figure 6.2 Median ISD Vs. Height from 10/50 Hazard ................................................................ 97 Figure 6.3 Median ISD Vs. Height from 50/50 Hazard ................................................................ 98 Figure 6.4 Median Shear Vs. Height from 2/50 Hazard ............................................................. 104 Figure 6.5 Median Shear Vs. Height from 10/50 Hazard ........................................................... 104 Figure 6.6 Median Shear Vs. Height from 50/50 Hazard ........................................................... 105 xv Figure 6.7 Hysteresis for First Floor Beam Hinge on MRF........................................................ 109 Figure 6.8 Hysteresis Curve for First Story Eccentric Link from EBF ....................................... 110 Figure 6.9 Hysteresis Curve for First Story Brace from BRBF .................................................. 110 Figure 6.10 Hysteresis Curve for a Typical Top Story Strip of SPSW ....................................... 111 Figure 6.11 Brace Hysteresis Curve for CBF Brace on Third Floor........................................... 111 Figure 6.12 CDF of the Repair Cost for Structures 2/50 Hazard Level ...................................... 112 Figure 6.13 CDF of the Repair Cost for Structures 10/50 Hazard Level .................................... 113 Figure 6.14 CDF of the Repair Cost for Structures 50/50 Hazard Level .................................... 114 Figure 6.15 Loss Curve for CBF ................................................................................................. 116 Figure 6.16 Repair Costs According to Performance Group for the BRBF ............................... 120 Figure 6.17 Repair Costs According to Performance Group for the CBF .................................. 121 Figure 6.18 Repair Costs According to Performance Group for the CBFLD ............................. 122 Figure 6.19 Repair Costs According to Performance Group for the SZBF ................................ 123 Figure 6.20 Repair Costs According to Performance Group for the XBF .................................. 124 Figure 6.21 Repair Costs According to Performance Group for the EBF .................................. 125 Figure 6.22 Repair Costs According to Performance Group for the MRF ................................. 126 Figure 6.23 Repair Costs According to Performance Group for the SPSW ............................... 127 Figure 6.24 Repair Costs According to Performance Group for the SPSWLD .......................... 128 Figure 7.1 Linked Column Frame ............................................................................................... 135 Figure A.1 Tributary Area for OWSJ.......................................................................................... 144 Figure A.2 Tributary Area for Beam ........................................................................................... 145 Figure A.3 Design Spectrum for Downtown Vancouver BC, Soil Class C ................................ 153 xvi Figure A.4 Eccentrically Applied Seismic Shear Forces ............................................................ 156 Figure A.5 Elastic Model of Moment Frame .............................................................................. 159 Figure A.6 Free Body Diagram of a Moment Resisting Frame .................................................. 161 Figure A.7 Free Body Diagram of Exterior Columns ................................................................. 162 Figure A.8 Bolted Stiffened End Plate Connection .................................................................... 162 Figure A.9 Elastic Model of the CBF ......................................................................................... 168 Figure A.10 Free Body Diagram of the Beam ............................................................................ 170 Figure A.11 Free Body Diagram Showing Column Loads ......................................................... 173 Figure A.12 Free Body Diagram for Link and Frame Shear ...................................................... 177 Figure A.13 Eccentric Braced Frame Elastic Model................................................................... 178 Figure A.14 Free Body Diagram of One Half of a Brace/Beam Assembly ................................ 179 Figure A.15 Free Body Diagram of EBF Frame ......................................................................... 184 Figure A.16 Allowable Drift Rotation as a Function of Link Length ......................................... 185 Figure A.17 EBF Link Detail ...................................................................................................... 186 Figure A.18 SAP2000 Model ...................................................................................................... 191 Figure A.19 Backbone Curve for the PowerCAT Brace ............................................................. 192 Figure A.20 Free Body Diagram of Diagonal Brace Bay ........................................................... 193 Figure A.21 Column Capacity Forces from Braces Loaded in Tension ..................................... 194 Figure A.22 Elastic Model with Bending Moment Diagram ...................................................... 195 Figure A.23 Strip Model Idealizations for Analysis of a Steel Plate Shear Wall ....................... 199 Figure A.24 Typical HBE with an Idealized Infill Plate Tension Strip ...................................... 202 Figure A.25 Axial Load Free Body Diagram for HBE ............................................................... 204 Figure A.26 Shear Load on HBE from Infill Plate Loading ....................................................... 206 xvii Figure A.27 Limited VBE Model for Estimating Moment and Shear on VBE .......................... 209 Figure A.28 Schematic of a Typical RBS Connection................................................................ 211 Figure A.29 Elastic Strip Model of Steel Plate Shear ................................................................. 215 Figure A.30 Elastic Model of Tension-Only System .................................................................. 218 Figure A.31 Free Body Diagram of Top Story Braces ............................................................... 221 Figure A.32 Free Body Diagram of 4th Level Beam ................................................................... 223 Figure B.1 Backbone Curves for Steel01 .................................................................................... 226 Figure B.2 Backbone Curves for Hysteretic Material ................................................................. 227 Figure B.3 Overlay of Test Data with the Model Results for RBS ............................................. 232 Figure B.4 Diagram of Fiber Model for RBS Plastic Hinge ....................................................... 233 Figure B.5 Fiber P-M Model Vs. Steel02 Model for Zerolength Element ................................. 233 Figure B.6 Moment-Rotation Behavior under Axial Load ......................................................... 234 Figure D.1 Pushover Curve for Moment Resisting Frame.......................................................... 246 Figure D.2 Pushover Curve for Eccentric Braced Frame............................................................ 246 Figure D.3 Pushover Curve for Buckling Restrained Braced Frame .......................................... 247 Figure D.4 Pushover Curve for Steel Plate Shear Wall .............................................................. 247 xviii List of Symbols Ag Gross area of section (mm2) Aw Area of the web, equal to (d-2tf)tw (mm2) Cy "squash load" equal to AgFy (kN) dz Panel Zone thickness (mm) e Link beam length (mm) (EBF system) E Modulus of elasticity (Mpa) (200 Gpa for steel) Fu Fracture strength of material (Mpa) Fy Yield strength of material (Mpa) Fysc Yield strength of buckling restrained brace material (Mpa) (BRBF system) G shear modulus (Mpa) (77,000 Mpa for steel) h Story height (mm) or wideflange web height (mm) hx Height of floor x (mm) Ix Moment of inertia for section (mm4) L Span length (mm) Lwall Length of steel plate wall (mm) (SPSW system) Mf Moment demand (kN-m) Mp Plastic modulus of section Mp = ZxFy (kN-m) Mv higher mode factor N number of inclined strips per direction (SPSW system) Pf Axial demand (kN) Ry Material overstrength factor xix Sa Spectral acceleration (units of g) Sx Section modulus Ta structural period (seconds) tp Thickness of infill plate (mm) (SPSW system) tz Panel Zone thickness (mm) Vf Shear demand (kN) W Structural weight (kN) (uppercase W) wg Distributed load (kN/m) wz Panel zone width (mm) x Distance to plastic hinge from column face (mm) (lowercase x) Zx Plastic modulus of section (mm4) α angle of inclination of strip elements per S16-09 [20.4.1] (radians) (SPSW system) γ Inelastic link rotation (radians) (EBF system) ∆ story deflection (mm) θp Column rotation (radians) ø Material resistance factor (0.9 for steel) Note: subscripts “b” and “c” refer to beam and column properties, respectively xx List of Abbreviations BRBF Buckling Restrained Braced Frame CBF Concentrically Braced Frame CBFLD Concentrically Braced Frame Limited Ductility DM Damage Measure DS Damage State DV Decision Variable EBF Eccentrically Braced Frame EDP Engineering Demand Parameter, interstory drift (ISD) or story acceleration IM Earthquake intensity measure, for example the peak ground acceleration. ISD Interstory Drift MRF Moment Resisting Frame NBCC National Building Code of Canada, 2005 Edition PBEE Performance Based Earthquake Engineering PG Performance Group SFRS Seismic Force Resisting System SPSW Steel Plate Shear Wall SPSWLD Steel Plate Shear Wall Limited Ductility SZBF Suspended Zipper Braced Frame XBF X-Braced Frame xxi Acknowledgements I would like to first thank Dr. Tony Yang for all of his help as my adviser on this project. I would like to particularly thank him for all the hours spent in his office talking over the finer points of design, and answering all of my many questions over the last couple of years. I want to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Steel Structures Education Foundation (SSEF) for all of their financial support via. my Industrial Postgraduate Scholarship (IPS) award. Particularly I want to thank David MacKinnon of the SSEF for helping me secure this award, and for his interest in my research. I would also like to thank Dr. Didier Pettinga, P.Eng and Rob Simpson, P.Eng of Glotman Simpson consulting engineers for their help in the design and modeling phases of this project. I would like to thank my mother, for all her years of moral and financial support. I want to also thank my father, who inspired my passion for engineering from an early age. 1 Chapter 1: Introduction Historically, structures were designed based on a set of assumptions. The most important of these are made regarding material behavior (from the material codes) and the distribution and intensity of loading (based on the building code). Codes based on this sort of approach are termed “prescriptive”, because they prescribe the way in which the structure is to be designed. A building so designed is considered “up to code”, or acceptable; if the structure is not designed in this way it is not in compliance with the code and therefore unacceptable. Essentially, code acceptance is binary, either acceptable or unacceptable. This is in contrast with other engineering disciplines which are often performance based. For example, a car may be designed with the performance limit of 35 miles to the gallon, or the requirement that it be able to go from 0-60 mph in 8 seconds. The reason for this disparity in engineering disciplines is obvious: it is a simple matter to test and redesign a car in order to get the 0-60 time up to the desired performance limit (8 seconds). During the design cycle of a car several prototypes may be built, tested and redesigned in order to meet performance criteria. Structures are far too large, and building a “prototype” structure to test the performance is totally uneconomical. Furthermore, in contrast to cars only one structure of a given design will usually be built, therefore it is impossible to spread testing costs out over multiple produced units. New and advanced computer modeling technologies have given engineers new options in recent years. It is now possible to use a computer to test a structural model and, if necessary, redesign it in order to meet specific performance targets. In light of this, building codes are slowly changing 2 from “prescriptive” design approaches to performance based ones. Ultimately, performance based design criteria are more useful to society. What is the probability of collapse of a structure during a once in a hundred years earthquake? Will the bracing withstand the loading of a rare hurricane force wind? What kind of damage will be sustained during these events? These are issues performance based design aims to address. This sort of performance based design methodology has only become practical recently due to advances in computer technology and developments in the field of structural engineering. Factors which make this approach possible include: • The increasing power of computer hardware: computer modeling of structural dynamics is very computationally intensive. As computers become more powerful, performing complicated structural analysis has become faster and computers are able to handle more complex models. • Development of more advanced computer analysis software has made it easier for engineers to model and analyze structures. • Due to the efforts of seismologists, an increasingly large and diverse body of earthquake ground motion accelerometer data is available. In the past there was a considerable dearth of data, and this made it difficult to conduct realistic computer simulations. • Advancements in the fields of seismology and geology have allowed engineers to better understand what causes earthquakes and how to predict future events. This is an area of ongoing research, and it is probably the single biggest source of uncertainty in earthquake engineering to date. 3 • A growing body of test data has enabled engineers to better model the full range of nonlinear behavior of structural elements. • In recent years there has been increased interest in full scale testing. The Hyogo Earthquake Engineering Research Center in Japan, also known as the “E-Defense” facility, is capable of full scale earthquake testing of structures with up to 10 stories. Data from these facilities has enabled engineers to better understand the structural demands resulting from strong ground motions. The contribution to Earthquake Engineering of advancements in computer hardware and software cannot be understated. Considerable uncertainty exists regarding seismicity and structural response. Probabilistic methods which can only be practicably implemented on a computer can be used to deal with these uncertainties. Increased testing in labs around the world is improving the knowledge about the complicated nonlinear response of beams, columns, slabs and other structural components. Details of this structural behavior can be input into the computer model in order to ensure that the global building response is correct. In parallel, seismologists are advancing the field of seismic geology in order to better understand how and why ground motions occur. Considerable uncertainties exist regarding the seismicity; things such as fault locations, plate slip rates and seismic sources affecting regional seismicity is an area of ongoing and intensive research. The Canadian Steel Code CSA S16-09 contains several pre-qualified steel Seismic Force Resisting Systems (SFRS), all of which are equally acceptable for use in steel buildings. These systems include moment resisting frames, several different classes of braced frames as well as 4 steel plate shear walls. The code includes design requirements but gives zero guidance as to the relative merits of these systems. In this thesis the Performance Based Earthquake Engineering (PBEE) method will be used to select the optimum prequalified system for a 5 story office building in downtown Vancouver BC. Because one building will be designed for each of the different seismic systems, this study will draw conclusions as to which prequalified systems are the best, and which perform the worst for the particular geometry selected. The design will conform to the 2005 National Building Code of Canada (NBCC 2005) as well as CSA S16-09. The primary performance criteria will be financial loss due to earthquake damage; consideration will also be made of the material usage of the different systems. This methodology, once developed, will be applicable to other building types and configurations not included in this study. 1.1 Literature Review The structural engineering community has been slowly moving towards performance based design criteria for several years. Superseding previous documents such as FEMA 273, 274 and FEMA 356, ASCE 41-06 “Seismic Rehabilitation of Existing Buildings” published by the American Society of Civil Engineers lays out performance criteria, modeling and analysis guidelines as well as seismic rehabilitation procedures. This document establishes performance levels for structural elements such as “immediate occupancy”, “life safety” and “collapse prevention”. Although this document provides performance specifications, it is primarily geared towards evaluation and modeling of existing structures. It falls short of providing a design 5 approach for new buildings, and fails to provide a complete methodology for new structural design. The Applied Technology Council (ATC) is an American nonprofit organization devoted to reducing the hazards of nature, particularly from earthquakes, on the human built environment. Project ATC-58 is an initiative of the Applied Technology Council whose purpose is to develop a procedure to evaluate the damage and associated repair costs incurred on a structure resulting from an earthquake. The ATC-58 project has funded many testing programs at various universities and research centers to gather data on building structural and nonstructural elements. For this project, data developed for the ATC-58 project will be used to evaluate the damage sustained by buildings during a seismic event. In the paper “Seismic Performance Evaluation of facilities: Methodology and Implementation” by Yang et al. (2009a) the theoretical background for the Performance Based Earthquake Engineering, or PBEE methodology is laid out. This paper concludes with an example application of the methodology to a three story steel building in California. This building is subjected to a series of ground motions, and statistics of the structural response are compiled (mean and standard deviation of the story drifts and story accelerations). Based on this information, the damage sustained by the building is estimated using probability distributions based on the structural response. Finally, repair cost data is used to estimate the cumulative distribution function of the repair cost for the structure. This procedure is carried out at three different seismic hazard levels (2%, 10% and 50% probability of exceedance in 50 years); this information is then used to calculate the mean cumulative annual total repair cost for the 6 structure. Stated in another way, the mean cumulative annual cost is the amount of money which would need to be set aside annually in order to cover repair costs for the building if an earthquake were to occur. This is the essence of the PBEE methodology, in this way total lifecycle costs to the building owner can be quantified. Construction costs can be estimated based on manpower and material usage; the costs incurred due to earthquake damage can be estimated using PBEE. The Pacific Earthquake Engineering Research Center (PEER) is an organization headquartered at the University of California at Berkeley which conducts research on earthquake engineering in general and performance based design in particular. This organization regularly publishes PEER reports, which focus on research to advance the state of the art of this field. The PEER report titled “Performance Evaluation of Innovative Steel Braced Frames” by Yang, Moehle and Stojadinovic (2009b) is a case study in which the PBEE methodology is applied to two different three story braced frame structures. The PEER report by Moehle et al. (2011) is a similar application of the methodology to a highrise office tower. In both of these reports the local seismicity, structural response and repair cost data is used to evaluate the lifecycle costs to the building. Probably the most critical step is defining the probabilistic damage relations, known as fragility relations which link the structural response with the probability of the structure being in various damage states. These relationships are an ongoing area of research, and several papers have been published containing fragility relations for various different structural components. The reports by Yang et al. (2009b) and Moehle et al. (2011) contains fragility data for several nonstructural components (ceiling lights, partition walls.ect) and structural components (SFRS 7 and gravity). A paper published by Baldvins et al. (2012) has developed fragility relations for the steel plate shear wall, and a report by Gulec et al. (2010) has developed fragility relations for shear links used in eccentric braced frames. Once the damage state has been determined, it is necessary to associate the damaged elements with repair costs. The reports by Yang et al. (2009a and 2009b) and Moehle et al. (2011) contain cost data for several of the structural and nonstructural building components. Cost data is also an area of ongoing research, where cost numbers are not available the repair cost can be estimated based on logical repair actions (cutting concrete slabs, welding steel shapes.ect). 1.2 Objectives and Scope There are four objectives of this thesis: 1. Use the PBEE procedure for steel buildings in Canada to minimize lifecycle costs to the building owners. In order to simplify this study, torsional effects on the structure will be ignored. 2. Compare the strengths and weaknesses of the prequalified structural SFRS systems in S16-09. Using the results of the methodology, the lifecycle costs and taking into account structural steel usage (considered a good indicator of initial construction cost), determine which prequalified systems are the most economical. 3. Provide modeling guidelines for each of the different structural systems. Detailed nonlinear finite element models will be created for each of the prequalified systems in S16-09. A guide will be created in order to show future engineers and researchers how to 8 best model the necessary parts of the frame with nonlinear elements. Appendix B summarizes the detailed modeling approach for this report. 4. By designing one steel building in Canada using the same overall layout but each of the prequalified systems in S16-09, an overview of the design approach for each of these systems will be given. Engineers will be able to use this guide as a reference for the design of the various SFRS systems. This design guide is included in Appendix A. 1.3 Organization and Outline This report is divided up into seven chapters, each one illustrating a critical step of the PBEE procedure. 1. Chapter 2 Seismic Hazard Analysis: Using historical and geological information about the seismicity of Vancouver British Columbia, design spectra are developed for 2%, 10% and 50% probability of exceedance in 50 years for the building site downtown. Using these hazard spectra, appropriate ground motions are selected and scaled to each of the three hazard levels. These ground motions may then be used for time history analysis of the finite element models. 2. Chapter 3 Structural Design: Using the NBCC 2005 as well as the steel code requirements of CSA S16-09 one five story office building is designed for each of the prequalified steel SFRS systems. 3. Chapter 4 Analytical Modeling Approach: Nonlinear finite element models are created for each of the different SFRS, and a detailed explanation is given for the different 9 nonlinear element types used for braces, moment connections, shear walls.ect. The energy dissipating elements (for example, the braces in braced frames) are calibrated to experimental data in order to make the numerical models match the behavior of test specimens as closely as possible. Modeling is done in the software OpenSees, developed at the University of California at Berkeley. 4. Chapter 5 Seismic Performance Evaluation: An overview of the PBEE method is given. Critical components of the building are grouped into structural and nonstructural performance groups. Performance groups are defined and explained, and fragility relations are given for each. Cost data for each of the different performance groups is presented, based on the latest developments in this field. 5. Chapter 6 Results: Time history analysis of each of the finite element models is performed using the amplitude scaled ground motions from the seismic hazard analysis. The results of the time history analysis are presented. The results of the PBEE procedure are shown in order that the lifecycle costs of the different frames may be compared. The structural steel usage of each frame system is also shown in order to give an idea of the costs associated with construction of each of the different SFRS. 6. Chapter 7 Summary and Conclusions: A summary of the performance based design procedure is given, and conclusions are drawn as to which systems prequalified in S16-09 are the most economical. Recommendations are given for the analysis of new structural systems. 10 Chapter 2: Seismic Hazard Analysis The first step of the PBEE procedure is to quantify the seismic hazard of the site. In this study, the prototype building is located in downtown Vancouver, British Columbia, Canada. The seismic hazard for Vancouver is described in detail in this chapter and is broken down into the following sections: 1. A discussion of historical seismicity in British Columbia. 2. A description of the site and seismic sources which contribute to the overall hazard. 3. Probabilistic Seismic Hazard Analysis (PSHA) for the building site. 4. Selection and scaling of the ground motions. 2.1 Seismicity in British Columbia The West Coast of British Columbia is part of the “Pacific ring of fire”, a belt of seismically active zones surrounding the Pacific Ocean. Figure 2.1a shows the worldwide tectonic plates; Figure 2.1b shows the Pacific ring of fire. These plates are constantly moving relative to one another. Strain energy builds up over time, periodically the energy is released through slippage; these releases are felt as earthquakes. Figure 2.2 shows a cross section of three plates off the coast of Vancouver Island. The Pacific Plate, Juan de Fuca Plate and the North American plate all intersect out in the ocean near the British Columbia coast; the interactions and relative motions of these plates are the major cause of earthquakes in this region. Figure 2.3 shows the seismic map of Canada showing the Peak Ground Acceleration (PGA) with a 2% probability of 11 exceedance in 50 years. This plot shows that the high seismic hazard is concentrated along both coast lines of Canada; the interior provinces have low or negligible seismic activity. Figure 2.1 Tectonic Plates and the “Ring of Fire” of Seismic Activity Source: US Geological Survey (a) and the Wikimedia commons (en.wikipedia.org) (b) 12 Figure 2.2 Plate Activity off the Coast of Vancouver Island Source: Geological Survey of Canada Figure 2.3 PGA Plot for 2% in 50 Years Seismic Hazard Level in Canada Source: Geological Survey of Canada 13 Figure 2.4 shows the historical earthquakes in Canada from 1627 to 2010, with seismic events shown as dots of various sizes based on their intensity. As indicated by the Seismicity map of Figure 2.3, Figure 2.4 shows that historically earthquakes in Canada have been concentrated along the East and West coastlines, with the west coast of BC by far showing the most activity. Figure 2.5 shows the effects of two seismic events using the Modified Mercalli Intensity Scale (MMI); this scale provides a qualitative rather than a quantitative measure of an earthquake’s effects. Table 2.1 provides a guide to the qualitative meaning attached to each of the MMI scale intensities. The map of Figure 2.5, together with the legend for the scale, gives an idea of what people would have felt depending on how far they were from the epicenter. The MMI scale gives an account of the earthquake which can be related to more easily than, say, a seismicity map such as that of Figure 2.3. Looking at Figure 2.5, it can be seen that the 1949 Queen Charlotte Islands earthquake could be felt 850 km away in Whitehorse and nearly 930 km away in Seattle, Washington. Similarly, the MMI recorded level V shaking for the 1872 Lake Chelan quake was felt nearly 280 km away from the epicenter in Portland, Oregon. Figure 2.6 shows the locations of some other large recorded historic earthquakes in Western Canada and Washington State. Table 2.2 lists several of these events. Of particular concern to seismologists and engineers alike is the Cascadia Subduction Zone magnitude 9 event in 1700. Subduction zones can produce what are referred to as “megathrust earthquakes”, and are some of the most powerful quakes in the world. It should be noted that this hazard source is not included in the probabilistic seismic hazard analysis used to calculate the spectral accelerations presented in the National Building Code of Canada. 14 Figure 2.4 Map of Historical Seismic Events in Canada 1627-2010 Source: Geological Survey of Canada 15 Figure 2.5 Modified Mercalli Intensity Scale for Historical Earthquakes Source: Geological Survey of Canada 16 Figure 2.6 Significant Historical Earthquakes Source: Geological Survey of Canada The Modified Mercalli scale is a measure of the intensity of shaking felt in a particular area. It measures the effect of the earthquake on humans and structures and cannot be compared to a magnitude scale. A summary of the scale is presented in Table 2.1. 17 Table 2.1 Modified Mercalli Scale Intensity Description I Not felt except by a very few under especially favorable conditions. II Felt only by a few persons at rest, especially on upper floors of buildings. III Felt quite noticeably by persons indoors, especially on upper floors of buildings. Many people do not recognize it as an earthquake. Standing motor cars may rock slightly. Vibrations similar to the passing of a truck. Duration estimated. IV Felt indoors by many, outdoors by few during the day. At night, some awakened. Dishes, windows, doors disturbed; walls make cracking sound. Sensation like heavy truck striking building. Standing motor cars rocked noticeably. V Felt by nearly everyone; many awakened. Some dishes, windows broken. Unstable objects overturned. Pendulum clocks may stop. VI Felt by all, many frightened. Some heavy furniture moved; a few instances of fallen plaster. Damage slight. VII Damage negligible in buildings of good design and construction; slight to moderate in well- built ordinary structures; considerable damage in poorly built or badly designed structures; some chimneys broken. VIII Damage slight in specially designed structures; considerable damage in ordinary substantial buildings with partial collapse. Damage great in poorly built structures. Fall of chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. IX Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb. Damage great in substantial buildings, with partial collapse. Buildings shifted off foundations. X Some well-built wooden structures destroyed; most masonry and frame structures destroyed with foundations. Rails bent. XI Few, if any (masonry) structures remain standing. Bridges destroyed. Rails bent greatly. XII Damage total. Lines of sight and level are distorted. Objects thrown into the air. Source: U.S. Geological Survey - The Modified Mercalli Scale Table 2.2 Historical Seismic Events in the Lower Mainland, British Columbia Year Location Magnitude 1700 Cascadia Subduction Zone 9 1872 Lake Chelan, Washington 7.4 1920 Vancouver Gulf Islands 5.5 1946 Vancouver Island 7.3 1965 Puget Sound, Washington 6.5 18 2.2 Site Description The proposed prototype building is located in downtown Vancouver B.C, at the intersection of West Georgia and Howe Street as shown in Figure 2.7. The soil in this area is known to be very dense, and is designated Soil Class C per the 2005 National Building Code of Canada (NBCC) definition. The program EZ-FRISK (Risk Engineering 2011) was used to perform a probabilistic seismic hazard analysis (PSHA) of the proposed site. Using this software, seismic sources within 300 km of Vancouver were identified. Unlike regions such as Southern California which has well defined fault lines between plates, seismic sources in British Columbia are less well defined. In BC the seismic sources are defined as “area sources”, meaning that the sources are considered to be distributed over an area. The exact definition of an area source is beyond the scope of this study, but is explained in the text “Geotechnical Earthquake Engineering” by Kramer (1996). Figure 2.8 is a map of all area sources within 300 km of the building site; Table 2.3 is a key listing the names of all these area sources. The attenuation relation is an equation used to measure the effect of the area source on the building site location. Two equations were used; BjfWall and YisWall, and they take into account source geometry and the distance to the site location of interest which is to be affected by the source. Attenuation equations are discussed in the next section. 19 Figure 2.7 Building Site Location in Downtown Vancouver Source: Google maps and EZ-Frisk Table 2.3 Seismic Sources within 300 km of Vancouver B.C. Source Name Attenuation relation Map Code Brooks Peninsula BjfWall BPEN Cascade Mountains BjfWall CASM Coastal BjfWall CO Explorer Plate Bending BjfWall EPB Georgia Straight/Puget Sound YisWall GSPS Juan de Fuca Plate Bending, Offshore YisWall JDFOF Juan de Fuca Plate Bending, Onshore BjfWall JDFON Nootka Fault BjfWall NF Northern BC BjfWall NBC Offshore BjfWall OFF Southern BC BjfWall SBC 20 Figure 2.8 Significant Seismic Sources Within 300 km of Site Location Source: EZ-Frisk software 2.3 Probabilistic Seismic Hazard Analysis Engineers, geologists and planners are interested in the hazard posed by sources and faults. This is complicated by the uncertainty in magnitude and timing of earthquakes. There are two different approaches to seismic hazard analysis, Deterministic and Probabilistic Seismic Hazard Analysis, or DSHA and PSHA, respectively. DSHA is used to gauge the effect of a “design” earthquake, without considering the probability of it happening. On the other hand, PSHA incorporates the probabilities of the different seismic sources being active as well as the hazards posed by them; for this study PSHA will be used. A detailed explanation of PSHA is beyond the scope of this study; interested readers are referred to the text “Geotechnical Earthquake Engineering” by Kramer (1996). In general, this approach includes the following steps: 21 1. Identification and characterization of seismic sources. The probability distribution of potential rupture locations within the source must be characterized, as well as the source geometry (point, area, line or volumetric source). 2. The time distribution of earthquake recurrence must be characterized for each source. A recurrence relationship is defined in order to specify the average rate at which an earthquake of a given magnitude will be exceeded. 3. The ground motions produced at the site of interest generated by earthquakes within the source are determined using predictive relationships. 4. The uncertainties in earthquake location, magnitude and ground motion parameter predictions are combined to calculate the probability that the parameter of interest will be exceeded during a given time period. All seismic sources near Vancouver are considered area sources. In Canada, there are two primary models used to characterize the seismic hazard, the R and the H model. The H model uses small source zones drawn around historic seismicity clusters, while the R model establishes larger regional zones. In the west, the R model collects the crustal earthquakes around Vancouver, Seattle, central Vancouver Island and Cascade Mountains into one earthquakes zone. In general, the R model is accepted as the more appropriate model for Western Canada. Once the model has been selected, it is necessary to determine how the sources affect sites far away from them. For this purpose, attenuation relations are used. For this study, two attenuation relationships were used, these are BfjWall and YisWall. BjfWall is an equation used for shallow crustal sources such as the Cascade Mountains and the Juan de Fuca plate; it has been adapted from the model proposed by Boore et al. (1997). YisWall is used for deeper earthquakes in sub 22 crustal regions such as the Georgia Straight/Puget sound region; this equation was proposed by Youngs et al. (1997). Further details regarding the West R Model and the attenuation equations may be obtained from OpenFile 4459 published by the Geological Survey of Canada. Table 2.3 lists which equations are used for attenuation of the considered sources. The NBCC spectral acceleration values were developed using a software package known as GSC Frisk developed by Risk Engineering, which implements the PSHA methodology. A commercially available software known as EZ-Frisk (Risk Engineering 2011) has been shown to reproduce results within 2-5% of values obtained from GSC Frisk (Geological Survey of Canada). In order to properly quantify seismic risk, the building will be subjected to ground motions scaled to a range of intensities. Earthquake records were scaled to intensities of 2%, 10% and 50% probability of exceedance in 50 years. In order to do this, it was necessary to obtain design spectra for these hazards. The probabilities of 2%, 10% and 50% correspond to return periods of 2475, 475 and 72 years. This indicates that, for example, an event with a 2% probability of exceedance in 50 years (referred to henceforth as a 2/50 event) will occur on average every 2475 years. The return period is then said to be 2475 years. Equation 2.1 shows how the return period is calculated. In this equation RT is the Return Period in years, X is the probability of the event being exceeded in Y years. Equation 2.1 yx RT /)1log(101 1 − − = 23 Figure 2.9 shows a plot of the hazard spectra obtained from EZ-Frisk, as well as the NBCC 2005 2/50 design spectrum. As shown by the figure, the 2/50 spectrum from NBCC matches up very well with the spectrum that obtained from the EZ-Frisk software, except the NBCC code is conservative in the short periods. The NBCC hazard spectrum does not consider the effect of the Cascadia Subduction Zone source; the 2/50 hazard level curve is also shown with consideration of this source. Inclusion of the Subduction zone increases the spectral accelerations; this curve was not used in this study, and is only shown for comparison. Figure 2.9 Hazard Spectra for Downtown Vancouver, 5% Damped on Soil Class C *2/50 hazard includes Cascadia subduction zone source, YisWall attenuation equation used Figure 2.10 shows the annual frequancy of exceedance of the Peak Ground Acceleration (PGA) generated by the PSHA using the West R model in EZ-Frisk. A 2% probability of exceedance in 50 years has an annual probability of exceedance of 0.0004. This is calculated using Equation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 S p e ct ra l A cc e le ra ti o n ( u n it s o f g ) Periods (seconds) EZ-Frisk 2/50* EZ-Frisk 2/50 EZ-Frisk 10/50 EZ-Frisk 50/50 NBCC2005 2/50 24 2.1, where the RT=2475 years and the annual probability of exceedance equals 1/2475. Using an annual frequency of exceedance of 0.0004 with Figure 2.10, this indicates a corresponding PGA of 0.468g. This is very close to the value of 0.46g contained in the NBCC 2005 for Vancouver. Since EZ-Frisk is predicting a PGA which is less than 2% different than that given by the building code, this indicates that the West R model is a good match for the NBCC hazard spectrum. Figure 2.11 shows a breakdown of the contribution of the different seismic sources to the overall hazard at the building site. From this figure, it can be seen that the two largest contributors to the hazard are the Cascade Mountain and the Georgia Straight/Puget Sound sources. Figure 2.10 Total Seismic Hazard Curve for PGA Source: EZ-Frisk Software 25 Figure 2.11 Deaggragation of seismic risk for PGA for different sources Source: EZ-Frisk Software Deaggragation is a powerful tool to separate the contributions to the total seismic hazard by magnitude and distance. Once these values are found, they can be used to select apropriate ground motions which fit the seismicity of the region. Since the majority of the structural systems explored in this report are braced frames with a fundamental period of ~0.5 seconds, this was the period used for the deaggragation. Figure 2.12 shows the deaggragation of the site using a period of 0.5 seconds and the associated NBCC PGA amplitude of 0.64g. This plot shows that the mean magnitude is 6.7 with a maximum of 7.7. This means that if the acceleration exceeds 26 0.64g, the magnitudes likely to have caused it are as shown. According to this figure, earthquakes greater than 7.7 need not be considered. Figure 2.13 shows the distance deaggragation. Although there is some activity up to about 130 km away, this plot suggests that an apropriate range is to consider earthquakes from 0-80 km away. Figure 2.12 Deaggregation of seismic risk by magnitude Source: EZ-Frisk Software 27 Figure 2.13 Deaggregation of seismic risk by distance Source: EZ-Frisk Software 2.4 Ground Motion Selection Using the hazard spectra and deaggregation information developed from the PSHA, ground motions were selected from the Pacific Earthquake Engineering Research (PEER) center Next- Generation Attenuation (NGA) database (PEER 2012). Table 2.4 summarizes the search criteria used for ground motion selection. Although the PSHA indicated that a 0-80 km search range and a maximum magnitude of 7.7 was appropriate, it was necessary to expand the criteria in order to find a sufficient number of records. Although the database contains many records, a significant 28 number were repeated from the same event. In order to prevent bias, a maximum of two records were selected from each event. Furthermore, the ground motions were amplitude scaled with a scaling factor between 0.2-6.8. These records were scaled such that the average of the motions did not fall below 10% of the target spectrum. Motions were scaled over the interval of 0.2 Tmin to 1.5 Tmax, where Tmin is the shortest fundamental period of the set of building models and Tmax is the longest; this works out to a scaling interval of 0.1-1.9 seconds. Both fault normal and fault parallel motions were taken. Overall a total of 17, 18 and 16 pairs of records were selected for the 2/50, 10/50 and 50/50 hazard levels, respectively. Figure 2.14 shows the geographic locations from which the 2/50 hazard level ground motion records were taken from. Table 2.5-Table 2.7 summarizes the ground motions used in this study. Figure 2.15, 2.16 and 2.17 shows the response spectra of the selected ground motion records, along with the target spectrum and the average of the spectra. Note that most of the records (with the exception of ground motion #1812) contained both fault normal and fault parallel ground acceleration records, this brings the number of ground motions to a total of 99 for all three hazard levels. Table 2.4 Criteria used to select Ground Motions Magnitude 5.5-8 Fault Type all types Distance (km) 0-100 Vs30 (m/s) 360-760 Pulse no pulse Scale Factor Range 0.2-6.8 Period Range (sec) 0.1-1.9 29 Figure 2.14 Locations of Selected 2/50 Ground Motions Source: google maps 30 Table 2.5 Selected Ground Motions for 2% in 50 Years Hazard NGA# Year Event Station Scaling Factor Mag Mechanism 1521 1999 Chi-Chi-Taiwan TCU089 1.14 7.62 Reverse-Oblique 762 1989 LomaPrieta Fremont-MissionSanJose 2.91 6.93 Reverse-Oblique 139 1978 Tabas-Iran Dayhook 1.12 7.35 Reverse 2113 2002 Denali-Alaska TAPSPumpStation#09 5.89 7.9 Strike-Slip 1812 1999 HectorMine MillCreekRangerStation 6.45 7.13 Strike-Slip 975 1994 Northridge-01 Glendora-NOakbank 6.05 6.69 Reverse 1512 1999 Chi-Chi-Taiwan TCU078 0.80 7.62 Reverse-Oblique 357 1983 Coalinga-01 Parkfield-StoneCorral3E 3.64 6.36 Reverse 126 1976 Gazli-USSR Karakyr 0.61 6.8 Unkown 164 1979 ImperialValley-06 CerroPrieto 1.79 6.53 Strike-Slip 291 1980 Irpinia-Italy-01 RioneroInVulture 3.54 6.9 Normal 942 1994 Northridge-01 Alhambra-FremontSchool 3.43 6.69 Reverse 302 1980 Irpinia-Italy-02 RioneroInVulture 2.95 6.2 Normal 769 1989 LomaPrieta GilroyArray#6 2.14 6.93 Reverse-Oblique 265 1980 Victoria-Mexico CerroPrieto 1.05 6.33 Strike-Slip 1162 1999 Kocaeli-Turkey Goynuk 2.79 7.51 Strike-Slip 1795 1999 HectorMine JoshuaTreeN.M.-KeysView 5.33 7.13 Strike-Slip Table 2.6 Selected Ground Motions for 10% in 50 Years Hazard NGA# Year Event Station Scaling Factor Mag Mechanism 3471 1999 Chi-Chi-Taiwan-06 TCU075 2.55 6.3 Reverse 2982 1999 Chi-Chi-Taiwan-05 CHY088 2.14 6.2 Reverse 139 1978 Tabas-Iran Dayhook 0.57 7.35 Reverse 762 1989 LomaPrieta Fremont-MissionSanJose 1.49 6.93 Reverse-Oblique 2113 2002 Denali-Alaska TAPSPumpStation#09 3.03 7.9 Strike-Slip 1812 1999 HectorMine MillCreekRangerStation 3.31 7.13 Strike-Slip 975 1994 Northridge-01 Glendora-NOakbank 3.11 6.69 Reverse 357 1983 Coalinga-01 Parkfield-StoneCorral3E 1.87 6.36 Reverse 126 1976 Gazli-USSR Karakyr 0.31 6.8 Unkown 291 1980 Irpinia-Italy-01 RioneroInVulture 1.81 6.9 Normal 164 1979 ImperialValley-06 CerroPrieto 0.92 6.53 Strike-Slip 299 1980 Irpinia-Italy-02 Brienza 4.45 6.2 Normal 926 1992 BigBear-01 RanchoCucamonga-FF 4.08 6.46 Strike-Slip 942 1994 Northridge-01 Alhambra-FremontSchool 1.76 6.69 Reverse 265 1980 Victoria-Mexico CerroPrieto 0.53 6.33 Strike-Slip 769 1989 LomaPrieta GilroyArray#6 1.10 6.93 Reverse-Oblique 684 1987 WhittierNarrows-01 PlayaDelRey-Saran 5.48 5.99 Reverse-Oblique 1162 1999 Kocaeli-Turkey Goynuk 1.43 7.51 Strike-Slip 31 Table 2.7 Selected Ground Motions for 10% in 50 Years Hazard NGA# Year Event Station Scaling Factor Mag Mechanism 2982 1999 Chi-Chi-Taiwan-05 CHY088 0.80 6.2 Reverse 2624 1999 Chi-Chi-Taiwan-03 TCU073 2.75 6.2 Reverse 1812 1999 HectorMine MillCreekRangerStation 1.23 7.13 Strike-Slip 139 1978 Tabas-Iran Dayhook 0.21 7.35 Reverse 762 1989 LomaPrieta Fremont-MissionSanJose 0.55 6.93 Reverse-Oblique 975 1994 Northridge-01 Glendora-NOakbank 1.16 6.69 Reverse 357 1983 Coalinga-01 Parkfield-StoneCorral3E 0.69 6.36 Reverse 2113 2002 Denali-Alaska TAPSPumpStation#09 1.13 7.9 Strike-Slip 1767 1999 HectorMine Banning-TwinPinesRoad 3.15 7.13 Strike-Slip 164 1979 ImperialValley-06 CerroPrieto 0.34 6.53 Strike-Slip 291 1980 Irpinia-Italy-01 RioneroInVulture 0.67 6.9 Normal 299 1980 Irpinia-Italy-02 Brienza 1.66 6.2 Normal 1015 1994 Northridge-01 LB-RanchoLosCerritos 0.87 6.69 Reverse 926 1992 BigBear-01 RanchoCucamonga-FF 1.52 6.46 Strike-Slip 265 1980 Victoria-Mexico CerroPrieto 0.20 6.33 Strike-Slip 734 1989 LomaPrieta APEEL3EHaywardCSUH 0.87 6.93 Reverse-Oblique Figure 2.15 Selected ground motions scaled to target spectra 2/50 32 Figure 2.16 Selected ground motions scaled to target spectra 10/50 Figure 2.17 Selected ground motions scaled to target spectra 10/50 33 Chapter 3: Structural Design The prototype building was a five story office building designed according to the 2005 National Building Code of Canada and the Canadian Steel Code CSA S16-09. The intent is to compare the performance of the prototype buildings utilizing each of the steel Seismic Force Resisting Systems (SFRS) prequalified by the Code. Appendix A includes a detailed design summary for each frame system. 3.1 Description of the Prototype Building This section provides a summary of the gravity system design for the prototype building. The dimensions of this building are 45 meters by 63 meters by 18.85 meters high, with five stories. Structural steel is ASTM A992 for wideflange sections and A36 for plates. Hollow Structural Steel (HSS) sections are G40.21 grade 350W. Note that in all drawings, a dot at the end of the member represents an idealized pin connection, whereas a triangle represents an idealized fixed connection. All structural members were selected using the available CISC standard wide flange or HSS shapes. All units presented in the drawings are in meters unless otherwise noted. Figure 3.1 shows an isometric view of the prototype building. Figure 3.2 shows a typical floor plan of the structure. In Figure 3.2 the locations of the SFRS are indicated by triangles at either end of the bay. 34 For the braced frames there are a total of 6 braced bays in each of the two orthogonal directions. For the Moment Resisting Frame and the Steel Plate Shear Wall structural designs the plan is different owing to the significantly different capacities of these systems. The gravity columns for the Moment Resisting Frame are heavier members than for the other systems because it is the only system considered a “sway” frame, and must be designed with consideration of increased P- delta effects. Practically speaking this is the result of designing the MRF gravity columns for a column buckling coefficient k=1.2 rather than k=1.0 for the braced and shear wall frames. Figure 3.3 and 3.4 show the gravity members for the moment resisting frame, whilst Figures 3.5 and 3.6 show the gravity members for the other frames. Gravity members were designed accounting for live load reductions as allowed by NBCC 2005. Figure 3.1 Isometric View of Building 5 bays @ 9 7 bays @ 9 4 @ 3. 65 4. 25 N 35 Figure 3.2 Floor Plan of Building 7 bays @ 9 5 b ay s @ 9 N A B C D E F G H 1 2 3 4 5 6 36 Figure 3.3 Typical East-West Gravity Frame Bay – MRF System Only 3. 65 3. 65 3. 65 3. 65 4. 25 9 W690x125 W690x125 W690x125 W690x125 W690x125 W 31 0x 79 W 31 0x 17 9 Figure 3.4 Typical North-South Gravity Frame Bay – MRF System Only 3. 65 3. 65 3. 65 3. 65 4. 25 9 W460x60 W460x60 W460x60 W460x60 W460x60 W 31 0x 79 W 31 0x 17 9 37 Figure 3.5 Typical East-West Gravity Frame Bay 3. 65 3. 65 3. 65 3. 65 4. 25 9 W690x125 W690x125 W690x125 W690x125 W690x125 W 25 0x 67 W 25 0x 10 1 Figure 3.6 Typical North-South Gravity Frame Bay 3. 65 3. 65 3. 65 3. 65 4. 25 9 W460x60 W460x60 W460x60 W460x60 W460x60 W 25 0x 67 W 25 0x 10 1 38 3.2 Type D Moment Resisting Frame (MRF) S16-09 [27.2] Rd=5.0 Ro=1.5 The design of the Moment Resisting Frame (MRF) is typically controlled by the code deflection limit. Member sizes in a MRF that satisfy drift limits often automatically satisfy the strength limit; regardless member strength limits must be checked. Members selected for strength are usually smaller than those selected to satisfy the limit state of 2.5% interstory drift (ISD) per floor. Figure 3.7 shows the plan view of this system; ten moment resisting bays per direction were required in order to limit drifts. Figure 3.8 shows the member sizes used. The design approach for a MRF system involves the following steps: 1. Calculate the NBCC lateral forces 2. Design the beams and columns so the ISD at every level is 2.5% or less 3. Increase the column sizes so that they can withstand the forces caused by the beams forming plastic hinges. This is called the “Strong column – Weak beam” criteria 4. Check the columns for the shear forces resulting from the beams forming plastic hinges 5. Check the frame members against the NBCC equivalent static forces The biggest difference with the less ductile moment resisting frame systems (S16-09 clause 27.3 and 27.4) is that the RdR0 factor is smaller, and there are less detailing requirements. Because this frame is controlled by story drifts rather than member strength, there is an interesting phenomenon when the frame is designed with smaller reduction factors. Although design forces on the less ductile frames are higher, the story drifts are multiplied by a smaller value of RdRo to obtain realistic deflections. Since the frame is fundamentally controlled by drift limits, member sizes for the Type MD and LD are the same as the Type D MRF. 39 Figure 3.7 Plan Layout for Moment Resisting Frame Building A B C D E F G H 6 5 4 3 2 1 7 bays @ 9 5 b ay s @ 9 Figure 3.8 Design For Moment Resisting Frame Design (MRF) 3. 65 3. 65 3. 65 3. 65 4. 25 9 999 9 W690x125 W 36 0x 38 2 (W 36 0x 59 2 in te rio r) W690x125 W840x193 W840x193 W840x193 W 36 0x 34 7 (W 36 0x 59 2 in te rio r) W690x125 W690x125 W840x193 W840x193 W840x193 W690x125 W690x125 W840x193 W840x193 W840x193 W690x125 W840x193 W690x125 W690x125 W840x193 W840x193 W840x193 W840x193 W690x125 W840x193 40 3.3 Type MD Concentric Braced Frame (CBF) S16-09 [27.5] Rd=3.0 Ro=1.3 The design philosophy for the Type MD Concentrically Braced Frame (CBF) is to design the braces for the elastic axial loads, and then design the beams such that they can carry the gravity loads in the event the compression brace buckles. Because the beams must be designed to resist the considerable unbalanced load resulting when the compression brace buckles, this system uses the second most structural steel in this study. Although there are many possible configurations for CBF frames, the chevron configuration has been chosen because it is very common. CSA S16-09 clause [27.5.2.4] is specifically written for this brace configuration. The final design is shown in Figure 3.9. The steps for the design of a CBF are as follows: 1. Calculate the NBCC lateral forces 2. Design the braces for the story shears 3. Design the beams so that they can withstand the unbalanced force resulting from compression brace buckling 4. Design the columns so they can resist the capacity forces transmitted by the braces 5. Check the frame members against the NBCC equivalent static forces 6. Check the lateral story drifts against the 2.5% ISD limit 41 Figure 3.9 Design for Concentric Braced Frame (CBF) 9 W760x350 W 31 0x 34 2 Braces are HSS178x178x9.5 W760x350 W760x350 W760x389 W760x484 W 31 0x 14 3 Braces are HSS178x178x9.5 Braces are HSS178x178x9.5 Braces are HSS203x203x9.5 Braces are HSS203x203x13 42 3.4 Type LD Concentric Braced Frame (CBFLD) S16-09 [27.6] Rd=2.0 Ro=1.3 The primary difference between the Type LD CBF (CBFLD) and type MD CBF is that this system need not consider the unbalanced load on the beam resulting from compression brace buckling. Otherwise, the design of the Type LD frame is the same. The higher design forces result in larger braces and columns, but the beam sizes are significantly smaller. Although this system uses less structural steel than the type MD CBF, it is still relatively heavy compared to the other frames in this study. In this study, the Type LD SFRS uses 6% less structural steel than the Type MD system. The final design is shown in Figure 3.10. Figure 3.10 Design for Concentric Braced Frame (CBFLD) 3. 65 Braces are HSS203x203x13 4. 25 Braces are HSS203x203x16 3. 65 3. 65 3. 65 Braces are HSS203x203x9.5 Braces are HSS203x203x9.5 Braces are HSS203x203x13 W 31 0x 41 5 W690x265 W 31 0x 14 3 W690x265 W690x265 W690x265 W690x265 9 43 3.5 Type D Eccentrically Braced Frame (EBF) S16-09 [27.7] Rd=4.0 Ro=1.5 The Eccentric Braced Frame (EBF) is one of the most ductile systems. Depending on the length of the “link beam”, the link is classified as a shear or flexural controlled link. Shear links are generally accepted to behave in a more ductile manner and are preferred over flexural links. A shear controlled link dissipates energy by shearing of the web of the link; a flexural link dissipates energy by bending moments developed in the flanges. In this study, only shear links will be used. One particularly attractive aspect of this system is that it is very lightweight; the EBF in this study uses the least structural steel, and it is 20% lighter than the MRF. Although there are different configurations possible for EBF, the most common configuration is the eccentric chevron configuration. The final design is shown in Figure 3.11. The steps for the design of a EBF are as follows: 1. Calculate the NBCC lateral forces. 2. Design the link beams to resist the lateral shear; although the links need not be the same members as the beams, for economic reasons they usually are. 3. Using a rational method, determine the loads transmitted from the link to the beam and braces. Capacity design these elements to withstand the maximum forces generated by the shear link. 4. Design the columns so they can resist the capacity forces transmitted by the links. 5. Check the frame members against the NBCC equivalent static forces 6. Check the lateral story drifts against the 2.5% ISD limit. 44 Figure 3.11 Design for the Eccentrically Braced Frame W360x79 W 31 0x 25 3 W360x79 W360x79 W360x79 W360x79 W 31 0x 17 9 9 1.0 Link TYP. 3. 65 3. 65 3. 65 3. 65 4. 25 Braces are W250x131 Braces are W250x131 Braces are W250x131 Braces are W250x131 Braces are W250x131 45 3.6 Type D Buckling Restrained Braced Frame (BRBF) S16-09 [27.8] Rd=4.0 Ro=1.2 The Buckling Restrained Braced Frame (BRBF) is one of the newest systems in the code. In this study a cross diagonal configuration was used, although chevron systems are also common. To help control drifts and to provide a backup system, the beams and columns of this system are designed as a moment resisting frame per S16-09 clause [27.3]. Although not required by the code, this is a dual system. For this study, the force-deformation response of a PowerCAT brace was used. This product is manufactured by Star Seismic, a structural engineering company in the state of Utah in the United States. As the third lightest frame, the BRBF is very efficient with material. The final design is shown in Figure 3.12, note that brace areas given are for the steel core section of the BRB only. The steps for the design of a BRBF are as follows: 1. Calculate the NBCC lateral forces. 2. Design the braces for 100% of the story shear. 3. Design the beams and columns to withstand the capacity forces of the brace. 4. Design the surrounding moment resisting frame, increase column sizes as necessary to satisfy the strong column-weak beam criteria. 5. Check the frame members against the NBCC equivalent static forces 6. Check the lateral story drifts against the 2.5% ISD limit. 46 Figure 3.12 Design for Buckling Restrained Braced Frame (BRBF) W530x165 9 W530x165 W530x165 W530x165 W530x165 W 31 0x 28 3 W 31 0x 41 5 Brace Area 7000 mm square Brace Area 7000 mm square Brace Area 5200 mm square Brace Area 3700 mm square Brace Area 3700 mm square3. 65 3. 65 3. 65 3. 65 4. 25 47 3.7 Type D Steel Plate Shear Wall (SPSW) S16-09 [27.9] Rd=5.0 Ro=1.6 The Type D Steel Plate Shear Wall (SPSW) is the most ductile system in the code. There are two types of pre-qualified steel plate shear walls in CSA S16-09. The fundamental difference between the Type D and Type LD SPSW is the Type D wall requires the beams to be moment connected to the columns. Testing shows that steel plate shear walls with simply supported beams have “pinched” hysteretic curves, and therefore these systems exhibit significantly less ductility. Because of the considerable capacity of the SPSW system, only two walls were needed per each principal direction of the building. Figure 3.13 shows the plan view for this system. During the design, it was found that a 9 meter long wall was uneconomical so a 6 meter wall was centered in the 9 meter spans in order to maintain the bay length. A detail of this arrangement is shown in Figure 3.14. The final member sizes for the wall are shown in Figure 3.15. Reduced Beam Section (RBS) moment connections were used for the beams, a summary of the section cut information is given in Appendix A. The steps for the design of a SPSW are as follows: 1. Calculate the NBCC lateral forces. 2. Design the infill plates for 100% of the story shear. 3. Design the beams and columns to withstand the capacity forces of the infill plates. 4. Design the moment resisting frame so that it can resist 25% of the lateral story shear. 5. Check beams and columns against the stiffness requirements of S16-09. 6. Check all members for elastic loads from the NBCC forces. 7. Check the lateral story drifts against the 2.5% ISD limit. 48 Figure 3.13 Floor Plan of Steel Plate Shear Wall Building 7 bays @ 9 5 b ay s @ 9 A B C D E F G H 1 2 3 4 5 6 Figure 3.14 Detail of bay DE-6 from the floor plan SPSW-2 9 9 6 1.5 49 Figure 3.15 Design for the Steel Plate Shear Wall (SPSW) 3. 65 3. 65 3. 65 3. 65 4. 25 6 1.59 mm thk. A36 Plate W690x265 W690x265 W690x323 W690x323 W690x323 W920x449 W 36 0x 10 86 W 36 0x 67 7 2.66 mm thk. A36 Plate 4.76 mm thk. A36 Plate 6.35 mm thk. A36 Plate 6.35 mm thk. A36 Plate 50 3.8 Type LD Steel Plate Shear Wall (SPSWLD) S16-09 [27.10] Rd=2.0 Ro=1.5 Type LD or Limited Ductility Steel Plate Shear Walls (SPSWLD) are similar to the Type D walls except for several relaxations given in the code. The beams need not be rigidly connected to the columns, and the beams can be class 1 or 2 members. Since the code assigns a much lower RdRo factor, the walls are also subject to considerably higher design forces. Even with higher story shears, thin plates are still capable of resisting very high lateral loads. The downside with the less ductile plate wall is that the beams and columns are required to be larger sections in order to withstand the capacity forces of the thicker infill plates. For this reason, the design of the type LD system uses a 9 meter wall. The plan view of this structure is as shown in Figure 3.16. The longer bay span allows the infill plates to have a greater shear capacity, but also results in bigger bending moments in the beams – thus the beams are large sections. The columns are also large, owing to the vertical compressive loads from the yielded and strain hardened infill plates. The final design is shown in Figure 3.17. 51 Figure 3.16 Plan View of building for Type LD SPSW 7 bays @ 9 5 b ay s @ 9 N A B C D E F G H 1 2 3 4 5 6 Figure 3.17 Design for Type LD Steel Plate Wall (SPSWLD) 3. 65 3. 65 3. 65 3. 65 4. 25 9 2.66 mm thk. A36 Plate 4.76 mm thk. A36 Plate 7.94 mm thk. A36 Plate 9.53 mm thk. A36 Plate 9.53 mm thk. A36 Plate W920x1191 W690x265 W690x384 W690x548 W690x384 W690x419 W 36 0x 59 2 W 36 0x 10 86 52 3.9 X-Braced Frame (XBF) S16-09 [27.11] Rd=1.5 Ro=1.3 The conventional construction clause allows a structure to be designed without having to take into account any seismic detailing. There are several restrictions to this system, and the static lateral loads are higher than the other SFRS. This frame will be designed as “Tension-Only”, meaning that the contribution of the compression braces will be neglected in the design. Designing the frame with this assumption is conservative. The design is given in Figure 3.18. The steps for the design of a conventional construction frame are as follows: 1. Calculate the NBCC forces; they are increased 2% for every meter above 15 meters. 2. Design the braces for 100% of the story shear. 3. Design the beams and columns to withstand the capacity forces of the braces, 4. Check the lateral story drifts against the 2.5% ISD limit. Figure 3.18 Design for X-Braced Conventional Construction (XBF) 3. 65 3. 65 3. 65 3. 65 4. 25 9 W530x182 W530x182 W530x182 W530x182 W530x182 W 31 0x 34 2 W 31 0x 17 9 Braces are HSS178x178x6.4 Braces are HSS203x203x9.5 Braces are HSS203x203x16 Braces are HSS254x254x13 Braces are HSS254x254x16 53 3.10 Suspended Zipper Braced Frame (SZBF) Rd=3.0 Ro=1.9 The Suspended Zipper Braced Frame (SZBF) is not codified in S16-09; it is however essentially a modified version of the chevron concentrically braced frame. Rather than designing the beams to be strong to resist the unbalanced forces when the brace buckles, a zipper column is installed in order to resist this unbalanced force. When the brace at a level buckles, the zipper column is engaged, which then pulls on the story above. In this way damage spreads across the structure “zipping up” as compression braces successively buckle. Normal Zipper frames become unstable once the zipper columns have engaged all the way up to the top story; the Suspended Zipper Braced Frame differs because the top story braces are designed not to buckle. This system was originally proposed in the PEER report by Yang et al. (2009b). Because this system is a modified version of the chevron concentrically braced frame it will be designed assuming Rd=3.0 and Ro=1.3. The final design is given in Figure 3.19. The steps for the design of a SZBF are as follows: 1. Calculate the NBCC lateral forces 2. Design the braces for 100% of the story shear 3. Design the zipper columns for the unbalanced force when the compression brace buckles 4. Design the top level braces to resist the capacity forces of the zipper columns 5. Design columns for the capacity forces of the braces, including the top level braces 6. Check the lateral story drifts against the 2.5% ISD limit 54 Figure 3.19 Design for Suspended Zipper Braced Frame (SZBF) 9 W460x128 W460x235 W460x128 W460x128 W460x128 W 36 0x 21 6 W 36 0x 34 7 3. 65 3. 65 3. 65 3. 65 4. 25 HSS203x203x13 Brace HSS152x152x13 Zipper Column HSS203x203x9.5 Brace HSS203x203x16 Zipper Column HSS178x178x9.5 Brace HSS254x254x16 Zipper Column HSS178x178x9.5 brace HSS305x305x16 Zipper Column W310x202 Brace 55 Chapter 4: Analytical Modeling Approach Finite element models of the prototype building were created for each of the nine structural systems using the analysis program known as the Open System for Earthquake Engineering Simulation (OpenSees) (PEER 1997). Due to the symmetric nature of the building, only half of the structure was modeled. For simplicity, only the response in the East-West direction was studied. A 2D model consisting of the frames in lines 1, 2 and 3 is placed in series and tied together using multi-point constraints at each floor. Figure 4.1 shows the arrangement of the frames as modeled. The first frame contains the seismic force resisting system, whilst frames 2 and 3 carry the majority of the building mass. Members expected to undergo inelastic deformation were modeled using nonlinear elements calibrated to experimental test data. Rotational hinges were used to model connection behavior, and beam columns were modeled using the fiber based flexibility formulated beam column elements in OpenSees (de Souza 2000). Masses and gravity loads were lumped at the nodes according to the tributary area. The P- ∆ effect was accounted for using the corotational transformation in OpenSees (Filippou and Fenves 2004). Rayleigh damping of 2.5% was assigned to the first and third vibrational modes of each building. 56 Figure 4.1 Arrangement of Frames to Model ½ of the Building 4.1 Gravity System Figure 4.2 shows a typical frame with gravity resisting members. The beam columns were modeled using the fiber based beam column element (de Souza 2000). At the ends of the members, zerolength rotational hinges were added to represent the effect of connection stiffness. In Figure 4.2 the red dots are beam hinges and the green dots indicate column base hinges. The rotational stiffness of these elements are modeled using the Steel01 material model in OpenSees, this is a uniaxial bilinear material model with kinematic hardening and optional isotropic hardening. The beam column elements themselves are 57 assigned a Steel01 material with A992 material properties; elastic modulus of 200 Gpa and a yield strength of 345 Mpa, material hardening is set at 2%. Beam hinges (red dots in Figure 4.2) are modeled based on parameters proposed by Astaneh-Asl (2005). The green column hinges were modeled with stiffness relationships proposed in a paper by Fahmy et al. (1998); in this paper behavior is proposed for idealized “fixed” and “pinned” column base connections. Details for the Steel01 material as well as the parameters used to model beam and column connections are given in Appendix B. Figure 4.2 Typical Gravity Frame 4.2 Concentrically Braced Frames A special element was developed to model the concentric braces in the Chevron Concentrically Braced Frames (CBF, CBFLD), the Suspended Zipper Braced Frame (SZBF) and the Conventional Construction X-Braced Frame (XBF). This element is based off of a model originally proposed in a report by Yang et al. (2009b). Figure 4.3 shows a schematic of the brace model; the brace has been split with a node halfway along its length, 58 this node is offset by 0.1% of the braces length. At either end of the brace, a zerolength element rotational hinge is located which models the gusset plate connection. The brace member is a fiber beam column element (de Souza 2000); the rotational hinges at the ends are modeled with zerolength elements. The beam column fibers are assigned to a Steel02 material with A992 steel material properties. Steel02 is a Giuffre Menegotto-Pinto material model contained in OpenSees. In order to determine the Steel02 properties, the hinge stiffness and the offset of the midpoint node, this model was calibrated to test data of a HSS steel brace. In a report by Black et al. (1980), an HSS102x102x13 brace was tested cyclically until failure; data from this test was used to calibrate the behavior of the OpenSees model. Parameters for this model are given in Appendix B. Figure 4.4 shows a comparison of the experimental data with that given by the numerical model. In Figure 4.3, the end of the brace was moved with a displacement signal identical to the one used by Black et al. for testing of the real brace; the response of the model was used to create Figure 4.4. Figure 4.4(a) shows a comparison of the test data with the numerical model, Figure 4.4(b) shows the numerical model normalized for Euler buckling load and percent elongation. Although the numerical model buckles in plane, the resulting behavior matches up well with the data from the tested brace which buckles out of plane. The beams and columns in the braced frame surrounding the braces are modeled similar to the gravity elements. Column baseplates and beam connections are modeled with the same zerolength end hinges as described for the gravity system. See Appendix B for details. 59 Figure 4.3 Element Model for Buckling Brace Figure 4.4 Hysteresis Curve from OpenSees Source: Black et al. (1980) 60 4.3 Eccentrically Braced Frames A shear hinge was used to model the behavior of the shear link in the Eccentrically Braced Frame (EBF). The modeling approach is illustrated in Figure 4.5; midway through the link beam a zerolength element with rigid axial and rotational properties has a vertical degree of freedom (DOF). The shear DOF has a force deformation behavior modeled using a Steel02 material. This model is appropriate for shear links which have no axial load. The behavior of the shear link was calibrated to match data from a W250x49 section tested in a report by Gulec et al. (2010). The elastic stiffness and yield load of the link is based off parameters defined in ASCE41, a document developed by the American Society of Civil Engineers. Details of this model is given in Appendix B. Figure 4.6 shows a comparison of the experimental data from the test by Gulec et al. (2010) and the numerical model. Figure 4.6 (a) is a superposition of the numerical model for the same section tested, and Figure 4.6(b) is a plot of the numerical model data by itself. Other than the shear hinge, all other elements in the frame are modeled by rules already established for the gravity system. The braces are rigidly connected to the link, on the opposite side where the brace connects to the beam and column a zerolength rotational hinge is located to model the gusset connection stiffness. The brace itself does not have the ability to buckle, and the zerolength element behavior is modeled according to the rules established for the concentric brace hinge. 61 Figure 4.5 Schematic of Eccentric Braced Frame Figure 4.6 OpenSees Material Response for Shear Link Source: Gulec et al. (2010) 62 4.4 Moment Resisting Frames For the Moment Resisting Frame (MRF), as well as the moment connections found in the Buckling Restrained Braced Frame (BRBF) system, a model similar to that of the gravity connections was developed. Figure 4.7 shows the location of the zerolength hinges used to model the connection stiffness. The Steel02 material was used to model the moment rotation behavior of the connections; the hinges themselves were located infinitely close to the column face and the plastic hinge length was neglected. The initial stiffness and yield strength of the hinges are as proposed by Astaneh-Asl (2005). The behavior of the model was calibrated to match test data from the SAC phase 2 project; data from testing of a W760x147 member was used. Figure 4.8 shows a comparison of the test data taken from the SAC project with the numerical model results. Further details for the model are given in Appendix B. Note that this element model is only valid for beams with negligible axial load. The other members: beams, columns, and column base connections were modeled per the rules already established for gravity members. Because the Steel Plate Shear Wall (SPSW) uses moment connections which will experience significant axial load, the model developed for the MRF is unsuitable. For the moment connection in the SPSW, a fiber section located within a zerolength element was used to model the rotational stiffness of the reduced beam section (RBS) moment connection. This allows the beam to yield at different levels of moment depending on the axial load, taking into account P-M behavior. This model was calibrated to test data of a 63 RBS connection taken from the SAC project. For the RBS connection model the location of the plastic hinge (center of the reduced beam cut) was not neglected. Further detail for this element model is provided in Appendix B. Figure 4.7 Zerolength Hinge Idealization for Moment Connection Behavior Figure 4.8 Overlay of SAC Test Data and OpenSees Simulation Result Source: SAC project 64 4.5 Buckling Restrained Braced Frames The braces for the Buckling Restrained Braced Frame (BRB) are modeled as truss elements in OpenSees. Truss elements are unable to buckle, have no rotational restraint at the ends, and experience axial deformation and force response only. Truss elements are one of the simplest element types, and are defined only by length, cross sectional area and material type. The brace cross sectional areas were set per the design, and the Hysteretic material in OpenSees was used to describe the stress-strain behavior of the brace core material. The brace material was calibrated to match data for a Star Seismic PowerCAT buckling restrained braced frame. The backbone curve in the BRBF section of Appendix A was used to calibrate hysteretic material. The hysteric degradation parameters of this material model were calibrated to data obtained for a PowerCAT brace tested in a report by Merritt et al. (2003). Further information regarding this model is provided in Appendix B. Figure 4.9 shows a comparison of the numerical and test data from the report; the model agrees well with the experimental data. Other members in the frame are modeled according to the rules established for the gravity members, ie. beams and columns are modeled with fiber cross sections and have zerolength hinges to model the connection behavior. The moment connections in the dual system are per the rules established for the MRF system. 65 Figure 4.9 Hysteretic Response of Brace Material Model Source: ©University of California San Diego (2011), by permission 66 4.6 Steel Plate Shear Walls A model for the steel infill plate was developed for the Steel Plate Shear Wall (SPSW) as well as the limited ductility wall (SPSWLD). This is based on the inclined strip model given in the Canadian steel code S16-09 and explained in AISC Design Guide 20. Ten strips in each direction with an angle of inclination of α=40 degrees are used. The strips were modeled as truss elements in OpenSees with each strip having an area, As, given by Equation A.72. The stress strain behavior of the strips was assigned to the Clough material model in OpenSees. Figure 4.10(a) shows a one story steel plate shear wall tested for a Ph.D dissertation by Rezai (1999). This wall was recreated in OpenSees, shown in Figure 4.10(b). Using the data from Rezai, the Clough material was calibrated so that the numerical response of the model matched the test data from the one story wall. Figure 4.11 shows a comparison of the wall test data with the results of the numerical model given in Figure 4.10. Details for the numerical model are given in Appendix B. Other elements of the steel plate wall are as established for the gravity system. For the reduced beam section connections in the ductile plate wall, zerolength elements located at the center of the beam cut with fiber sections are used. The fiber sections represent the minimum section of the reduced beam flange cut. Details for this model are provided in the section on MRF modeling and in Appendix B. 67 Figure 4.10 Steel Plate Wall Test Specimen and Model Source: ©Rezai (1999), by permission Figure 4.11 Opensees Load Deformation Response for Steel Plate Wall Source: ©Rezai (1999), by permission 68 Chapter 5: Seismic Performance Evaluation Procedure The purpose of this thesis is to compare the performance of the different structural systems to select the most efficient one for the final design. Two quantities, material usage and the repair costs associated with fixing the structure after an earthquake, will be used to compare the relative performance of the systems. In this study, the configuration (number of floors, plan layout.ect) has already been determined, the final choice of the SFRS (concentric braced frame, moment resisting frame.ect) will be made based on whichever system has the best performance when subjected to different levels of earthquake shaking. A paper written by Yang et al. titled “Seismic Performance Evaluation of Facilities: Methodology and Implementation” (2009a) introduces the framework of the Performance Based Earthquake Engineering (PBEE) method. PBEE uses probabilistic data to link the seismic response of a structure to possible damage states. Once the damage to the building has been established, data is used to calculate the expected repair cost. This procedure accounts for the uncertainty in the seismic hazards, structural response and in the repair cost using probabilistic methods. In the this paper a framework is laid out in which the probable repair costs for a structure can be determined based on the structural design, the local seismic hazard and a list of building components likely to be damaged. 69 5.1 The Performance Based Earthquake Engineering Method The steps of the PBEE method can be summarized as follows: 1. Determine the seismic hazard of the building site. 2. Determine the structural demands based on the site seismicity. 3. Determine the damage based on the structural demands. 4. Determine the repair costs based on the damage sustained. Due to the uncertain nature of earthquakes, many quantities must be dealt with probabilistically. The seismic hazard is probably the most uncertain factor. Based on the analysis, there will be a statistical distribution for the structural demand as well as the damage state of the structure after the seismic event. As a result, ultimately the repair costs are also probabilistic quantities, and the costs can be expressed several different ways. The theoretical explanation of the PBEE method is given in the paper by Yang et al. (2009a). 5.1.1 Performance Assessment Framework In order to describe the methodology the following terminology is defined: • IM – earthquake intensity measure, for example the peak ground acceleration. • EDP – Engineering Demand Parameter, for example the interstory drift (ISD) or story acceleration. These are the response parameters taken from the analyses. 70 • PG – Performance Group. Structural and non-structural components where response is affected by the same EDP are grouped into PGs. • DM – Damage measure. • DS – Damage state. This is the state defined by the engineer to quantify the degree of damage for each PG. • DV – Decision variable. This could be monetary loss, down time, casualty count or whatever other variable the engineer is interested in. • ))Pr()|( yyXxyxG =<= Denotes the conditional complementary cumulative distribution function (CCCDF) of a random variable X given a particular outcome Y=y of a random variable Y. • )(imλ - seismic hazard curve which represents the annual probability of exceedance of intensity measure. For example, the annual rate at which the PGA exceeds 1.0 g. • )|( imedpG - the response analysis where the outcome is usually expressed in terms of the CCCDF of an edp given the IM has reached a threshold value. • )|( edpdmG - the response analysis where the outcome is usually expressed in terms of CCCDF of a dm given that the EDP has reached a threshold value. • )|( dmdvG - the loss analysis where the outcome is usually expressed in terms of the CCCDF of a dv given that the DM has reached a threshold value. The method is summarized in Figure 5.1. The steps are compartmentalized and are summarized as follows: 71 1. Seismic Hazard Analysis – In this first phase, seismologists, geologists and other experts assess the seismic hazard of the site. In this report, this step was carried out in Chapter 2, the outcome of the seismic hazard analysis are the uniform hazard spectra for the 2/50, 10/50 and 50/50 hazard levels for the downtown Vancouver. Based on these spectra, representative ground motions were selected and scaled. 2. Response Analysis – Computer models are created to characterize the behavior of the structure and are subjected to the selected ground motions. Based on this analysis, the EDP are determined. 3. Damage Analysis – Using the EDP from part 2, the damage to building PGs is assessed. The EDP are used to determine what damage state each PG is in. 4. Loss Analysis – Based on the damage state, associated repair actions and their costs are used to gauge the repair cost of the building. At every step along the way, uncertainties are associated with the different quantities and so the results of the Loss Analyses are themselves a probabilistic quantity. In the most general case, the output of the loss analysis can be the mean rate of exceeding a certain threshold repair cost for a set period of time, the cumulative expected monetary loss for repair of the facility or the total monetary loss for a specific hazard level. Figure 5.1 Steps for the PBEE Methodology 72 Equation 5.1 shows the relation for the mean annual rate of the decision variable (total dollar loss or down time) exceeding a threshold value. A full derivation is given by Yang et al. (2009a). From this equation it is seen that the four stages of the methodology are present. The term )(imλ requires hazard analysis, )|( imedpG requires response analysis, )|( edpdmG requires damage analysis and )|( dmdvG requires loss analysis. The assumption that the structure is completely repaired after each seismic event is implicit, since there is no way to keep track of deterioration with subsequent ground motions. Equation 5.1 ∫ ∫ ∫=< im dmedp imdimedpdGedpdmdGdmdvGDVdv |)(|)|()|()|()( λλ Due to its complexity, Equation 5.1 will not be evaluated as a closed form integral; it is however an efficient way to summarize the methodology. For this project, a MATLAB script was used to carry out the analysis. Carrying out the evaluation of the quantities in Equation 5.1 using the MATLAB script involves the following practical steps: 1. Define structural and nonstructural component PGs. 2. Conduct seismic hazard analysis and ground motion selection. 3. Evaluate the response of the building, this involves time history analysis. 73 4. Generate additional correlated EDP vectors. This allows the Loss analysis to be run with the equivalent of many thousands of time history analysis results without actually having to run more than a few actual computer analysis cases. 5. Computer the total repair cost. 6. Generate different expressions of the total repair cost data. 5.1.2 Evaluate Structural Response Once the seismic hazard analysis has been performed and ground motions have been selected, it is necessary to evaluate the response of the building. Nonlinear time history analysis is conducted in OpenSees to determine the peak EDP resulting from each ground motion record. For each record, the EDP matrix consists of the peak interstory drift ratio (ISD) at each floor together with the peak story acceleration. A total of 17, 18 and 16 ground motions were run for each of the 2/50, 10/50 and 50/50 hazard levels, respectively. For each of these records the analysis was run for the fault normal and fault parallel acceleration file (GM#1812 only had one); this results in a total of 99 ground motion records per structural model. Studies by the authors of the Yang et al. paper (2009a) have shown that in order to generate good results from the loss analysis, a minimum of about 200 EDP vectors at each hazard level are required. This is difficult to achieve by direct time history analysis, both because of the computational cost and because of the limited number of appropriate ground motion files available. An alternative is to generate a large number of additional EDP vectors which have the same statistical properties as the original smaller set obtained from direct time history analysis. Detailed explanation is beyond the scope of this 74 thesis, but a procedure to do this is explained in detail by Yang et al. (2009a). The procedure is therefore to generate a small set of EDP vectors directly from the nonlinear time history analysis, using the statistical methods presented in Yang et al. (2009a) to generate a much larger set of data which is correlated to the original set, and then to use this larger set of data for Loss Analysis. Although 200 EDP vectors are sufficient, using this procedure it is easy to generate thousands of vectors without much computational effort. For this report, the EDP vectors at each hazard level were extrapolated to a matrix of 100,000 vectors. 5.1.3 Performance Groups and Fragility Relations Performance groups are used to quantify the performance of building components. One example could be the interior nonstructural interstory drift sensitive building components on the first floor. This PG would have all appropriate building components assigned to it; these could include partition walls, door frames, bathroom fixtures and anything else damaged by interstory drift at that level. This PG is defined by fragility relations which model how damage is sustained to it. Figure 5.2 shows an example fragility curve for this PG; it has three damage states, NO DAMAGE, MEDIUM DAMAGE and HEAVY DAMAGE. Each of the two blue lines in Figure 5.2 defines the transition between two states; these are themselves defined by the median EDP (the EDP at a probability of 0.5) and a beta value which defines the curvature. The fragility relation defines the probability of the PG being in a damage state depending on the value of the EDP. For the relation of 75 Figure 5.2, suppose the ISD at the first floor is 1%. If this were true, this PG would have a 75% probability of being in DS3, a 25% probability of being in DS2 and essentially a 0% probability of being in DS1. A random number generator which varies between 0 and 1 is used to pick a value, and depending on where it lands along a vertical line at the appropriate EDP the damage state is determined. Once the damage state has been determined, repair actions together with the associated cost are used to gauge the loss associated with this PG. Figure 5.2 Fragility Relation for a Typical PG Source: ©Yang et al. (2009a), by permission Development of appropriate PGs, Fragility Relations and repair costs for different elements is an area of intensive research as of the writing of this report. Fragility relations are usually based on extensive test programs of the building element, and repair costs are based on industry practices. Once the damage state is determined, the repair cost is also a function of how many units are affected. For example, if the interior nonstructural interstory drift sensitive PG on the first floor is found to be in Damage State 3, the repair cost will depend 76 on the quantity and cost of the assigned components. Let’s say this PG is associated with partition walls (repair cost is $10/sq. meter at DS3), door frames (repair cost is $100 per frame at DS3) and bathroom fixtures (altogether $500 per bathroom at DS3). The first floor has 100 square feet, 3 door frames and 1 bathroom. The repair costs incurred at the first floor for this PG are then as follows: 100*$10 (partition walls) + 3*$100 (door frames) + $500 (1 bathroom) = $1800 Often a unit repair cost function will be defined in order to give a discount depending on the repair quantities. A lower and upper cost threshold will be given depending on the repair quantities. If only 100 square feet worth of building needs to have the partition walls repaired, the cost will be $10/sq. meter; but if the repair quantity is 500 square meters the cost will drop to only $5/sq. meter. This reflects the discount contractors give for more work. Figure 5.3 shows a graphical representation of this concept. Figure 5.3 Example of Unit Repair Cost Function Source: ©Yang et al. (2009a), by permission 77 5.1.4 Compute the Repair Cost Once the repair cost for each PG has been determined, the costs for all PG’s are summed. For each PG, the damage state is determined probabilistically using a random number generator, the EDP for that particular “earthquake” and the fragility curve. The process of calculating the repair costs for each PG and summing them up is completed for each EDP vector at the hazard level of interest. The result is a statistical distribution of repair costs at each hazard. The probabilistic variation in the damage state is accounted for via. fragility curves and the probabilistic nature of the earthquakes are taken into account via. the large number of statistically correlated EDP vectors representing several thousand earthquakes. In this report, two different representations of the results will be presented, the cumulative distribution of the total repair cost and the mean annual rate of the repair cost exceeding a threshold value. These will be discussed in the next chapter. For more detail refer to Yang et al. (2009a) and Yang et al. (2009b). 78 5.2 Performance Group Data for a 5 Story Vancouver Office Building Components of the building were assigned to 26 PGs. These include: one structural PG at every floor level (1-5), one exterior (6-10) and one interior (11-15) drift sensitive nonstructural component PG at each floor, two interior acceleration sensitive nonstructural component PGs at each floor (16-25) and one elevator PG (PG26). PG data is based on the reports by Moehle et al. (2011) and Yang et al. (2009b). PGs constitute building elements likely to be damaged and have costly repair actions associated with them after a seismic event. The structural PG at each level is based on the SFRS of the system. All chevron concentric braced frames share the same structural PGs, both SPSW models share the same PGs, the MRF, XBF, EBF and BRBF have different structural PG’s based on the behavior of their elements. All other nonstructural PG’s are identical for every building. Although the Moehle et al. (2009) report did include a PG for the structural gravity system, the interstory drifts which triggered damage were much higher than those experienced by the models during the time analyses and were not included. PGs whose median EDP required to trigger damage which were much higher than the maximum EDP sustained by the building models were excluded because they would not contribute significantly to repair cost. Table 5.1 gives a summary of the PG’s used for this report. 79 Table 5.1 Summary of Performance Groups PG Name Location EDP Description 1 SH12 between levels 1 and 2 du1 structural system specific: based on replacement of braces, moment connections.ect 2 SH23 between levels 2 and 3 du2 3 SH34 between levels 3 and 4 du3 4 SH45 between levels 4 and 5 du4 5 SH5R between levels 5 and R du5 6 CW12 between levels 1 and 2 du1 Curtain Wall: based on repair/replacement of 5'x6' exterior panels 7 CW23 between levels 2 and 3 du2 8 CW34 between levels 3 and 4 du3 9 CW45 between levels 4 and 5 du4 10 CW5R between levels 5 and R du5 11 INTD12 between levels 1 and 2 du1 Interior nonstructural drift sensitive: partitions, doors, glazing,etc 12 INTD23 between levels 2 and 3 du2 13 INTD34 between levels 3 and 4 du3 14 INTD45 between levels 4 and 5 du4 15 INTD5R between levels 5 and R du5 16 Contents2 below level 2 a2 Interior contents: bookshelves, computers, furniture.ect 17 Contents3 below level 3 a3 18 Contents4 below level 4 a4 19 Contents5 below level 5 a5 20 ContentsR below level R aR 21 Ceiling2 below level 2 a2 Interior nonstructural acceleration sensitive: ceilings, lights, sprinkler heads, etc 22 Ceiling3 below level 3 a3 23 Ceiling4 below level 4 a4 24 Ceiling5 below level 5 a5 25 CeilingR below level R aR 26 Elevator at level R aR 4 elevators in building The Curtain Wall (CW) PG was used in the report by Moehle et al. (2011) to describe the damage to the exterior nonstructural panels after an earthquake. In this report, the quantity is based on damage sustained to 5’x6’ rectangular panels. Two damage states are defined for this PG. The first state, DS2 is reached at a median EDP of 2.34% ISD. A parameter, Beta = 0.3 is given; this defines the shape of the fragility curve. A linear cost function similar to that of Figure 5.3 is defined in order to decrease unit repair costs when more panels are damaged. If only 20 panels are damaged, then the cost is $2960 per panel; if 100 80 panels are damaged, the repair cost goes down to $1580 per panel with linear interpolation in between. Since the first floor is 4.25 meters (13.94 ft) high, and 63 meters (206.7 ft) long going in the East-West direction, the total area is 206.7’x13.94’=2881.4 square feet. Since each panel is 5’x6’=30 sq/ft, 2881.4/30=96 panels per exterior wall. Since there are two exterior walls in the East-West direction, the number of panels on the first story is 96x2=192 panels. A similar calculation will show that the other floors (only 3.65 meters in height) have 165 panels. What this means, is that if the first and second floor are damaged to DS2, then the repair quantity for this PG is 192+165=357 panels. Since the cost equation gives the lowest cost at 100 panels, this will cost $1580 per panel to repair/replace. Table 5.2 summarizes the PG data for the CW Performance Group. The fragility curves obtained from this data are plotted in Figure 5.4. Costs are in US Dollars. Figure 5.4 Fragility Curves for the Curtain Wall PG 81 Table 5.2 CW Performance Group DS2 DS3 EDP ISD (%) ISD (%) Qty Min 20 20 Qty Max 100 100 Cost Min $2,960 $3,848 Cost Max $1,580 $2,054 Median EDP 2.34 3.1 Beta 0.3 0.3 Quantity 165 panels; (192 lvl1 only) Reference: Moehle et al. (2011) The Moehle et al. (2011) report includes a Performance Group used to determine the damage sustained by the interior nonstructural drift sensitive components; this PG is called INTD and is summarized in Table 5.3. Figure 5.5 shows the fragility curves obtained from using these median EDP and beta values. This PG is intended to account for partition walls, doors, glazing and any other items which are damaged when the ceiling and floor are displaced relatively. The repair quantity is defined in terms of linear feet per floor; the equation to estimate the length of partition/doors/framing is 0.001x(Floor Area ft2). The building of this project is 63 meters by 45 meters which is 2835 m2. Using the conversion factor of 3.281 ft/m this results in 30,518 ft2. Using the equation 0.001x30518=30.52 linear feet; since this analysis only considers motion in the East-West direction this number should be divided by two to yield a repair quantity of 15 feet per floor. 82 Figure 5.5 Fragility Curves for INTD PG Table 5.3 INTD Performance Group DS2 EDP ISD (%) Qty Min 0.56 Qty Max 5.56 Cost Min $3,510 Cost Max $2,160 Median EDP 0.24 Beta 0.65 Quantity 15 ft /floor Reference: Moehle et al. (2011) So far we have not considered the contents of the building, these include computer stations, servers, office furniture, bookshelves, and other nonstructural components which are not tied down to the structure. During an earthquake these things may be heavily damaged, as bookshelves crash down and smash computers, digital data is lost and furniture is ruined. This PG is difficult to quantify, so a value of $10 per square foot is used. The cost can be 83 calculated as (30519/ft2)*($10/ft2)=$305,190 per floor. The details are summarized in Table 5.4; Figure 5.6 shows the Fragility curves defined by the data in this table. Since building contents are damaged when the building suddenly moves, this PG is acceleration sensitive. Figure 5.6 Fragility Curve for Contents PG Table 5.4 Contents Performance Group DS2 EDP Story Acceleration Qty Min 1 Qty Max 10000 Cost Min $305,190 Cost Max $305,190 Median EDP 0.25 Beta 0.5 Quantity 1/floor Reference: Moehle et al. (2011) 84 A PG is defined to gauge the damage sustained by the ceiling. In most office buildings every floor has a suspended ceiling system. This includes ceiling panels, lights, sprinklers and heating vents. The Ceiling PG is intended to model the damage at each floor sustained by this system. Table 5.5 summarizes this PG. The repair quantity is determined by dividing by 250 ft2, therefore 30519/250=122 per floor. There are no units to this number, but it is linked to the repair costs. That is to say, if the first floor is at DS2, since 122>10 this will cost $326*122=$39,772. The resulting fragility curves are plotted in Figure 5.7. Figure 5.7 Fragility Curves for Ceiling PG Table 5.5 Ceiling Performance Group DS2 DS3 DS4 EDP Story Acceleration Qty Min 1 1 1 Qty Max 10 10 10 Cost Min $471 $3,770 $7,830 Cost Max $326 $3,610 $5,420 Median EDP 1 1.8 2.4 Beta 0.4 0.4 0.4 Quantity 122 per floor Reference: Moehle et al. (2011) 85 The elevator system has its own PG, and it depends on the maximum acceleration at any floor. Since the minimum and maximum costs are the same, the quantity min/max is essentially irrelevant. Nevertheless, something must be input for both, so Qty Min=1 (a small number) and Qty Max=10,000 (a large number). The data for the elevator PG is summarized in Table 5.6 and the fragility curve is plotted in Figure 5.8. Figure 5.8 Fragility Curve for Elevator PG Table 5.6 Elevator Performance Group DS2 EDP Max Acceleration Qty Min 1 Qty Max 10000 Cost Min $7,851 Cost Max $7,851 Median EDP 0.35 Beta 0.4 Quantity 4 elevators Reference: Moehle et al. (2011) 86 5.2.1 Structural Performance Group Data So far all of the PG’s have been for nonstructural components; these are common to all 9 models. Since each model in this report utilizes a different SFRS, the seismic PG’s will be different. Some simplifying assumptions were made. All frames utilizing chevron concentric braces used the same SFRS PG (CBF, CBFLD and SZBF), the steel plate walls used the same PG even though one utilizes moment connected beams (SPSW and SPSWLD). All other models have their own unique structural seismic PG (BRBF, EBF, XBF, MRF). Table 5.7 summarizes the PG for the BRBF, the fragility curve is shown in Figure 5.9. This PG only has two states, corresponding to undamaged braces Vs. fractured braces. Although the repair costs are relatively high for a fractured brace, the Median EDP indicates that they can experience up to 2% drift without any damage (50% of the time). This PG was not specified as being for cross braced or chevron braced BRB members. It is possible that in the future, more refined data may indicate that these values can be improved upon. Table 5.8 summarizes the data for the concentric braced frame and Table 5.9 is for the X-Braced conventional construction frame, fragility curves for the CBF are plotted in Figure 5.10. Note that the Median EDP are the same for Table 5.8 and Table 5.9, but the shape of the fragility relations (beta) and the costs differ for these two bracing configurations. Although not shown, the fragility curves for the XBF DS3 and DS4 would look slightly steeper. Table 5.10 summarizes the data for the MRF with fragility curves plotted in Figure 5.11. 87 Figure 5.9 Fragility Curve for BRBF Structural PG Table 5.7 Structural BRBF PG DS2 EDP ISD (%) Qty Min 1 Qty Max 10000 Cost Min $49,100 Cost Max $49,100 Median EDP 2 Beta 0.4 Quantity 6 braces/floor Reference: Moehle et al. (2011) 88 Figure 5.10 Fragility Curves for CBF Table 5.8 Structural CBF PG DS2 DS3 DS4 EDP ISD (%) Qty Min 1 1 1 Qty Max 10000 10000 10000 Cost Min $30,000 $36,500 $37,000 Cost Max $30,000 $36,500 $37,000 Median EDP 0.92 1.67 2.23 Beta 0.3 0.15 0.15 Quantity 6 braced bays/floor Reference: provided via. email from the ATC to Dr. Tony Yang Table 5.9 Structural XBF PG DS2 DS3 DS4 EDP ISD (%) Qty Min 1 1 1 Qty Max 10000 10000 10000 Cost Min $30,900 $51,200 $54,900 Cost Max $30,900 $51,200 $54,900 Median EDP 0.92 1.67 2.23 Beta 0.3 0.1 0.1 Quantity 6 braced bays/floor Reference: provided via. email from the ATC to Dr. Tony Yang 89 Figure 5.11 Fragility Curves for MRF Structural PG Table 5.10 Structural MRF PG DS2 DS3 DS4 EDP ISD (%) Qty Min 1 1 1 Qty Max 10000 10000 10000 Cost Min $30,400 $52,400 $52,400 Cost Max $30,400 $52,400 $52,400 Median EDP 3 4 5 Beta 0.3 0.3 0.3 Quantity 10 bays/floor Reference: Moehle et al. (2011) The PG data for the EBF was more difficult to obtain. Fragility curves are given in a report by Gulec et al. (2010); the curves are plotted in Figure 5.12. This report did not include costs associated with the damage states. One approach could be to average the unit repair costs of the MRF and the CBF, which would be logical since the EBF is a structural system in between these other two. A more rigorous approach is to break down the repair costs into 90 their constituent actions and sum them up. In the report by Yang et al. (2009b), repair costs for several common tasks are given. Together with the definitions of the damage states given by Gulec et al. (2010) these repair action costs are used to estimate the cost of each damage state. Calculation of the cost values are given in Appendix C. Cost and fragility data is summarized in Table 5.11. The fragility data for the steel plate wall was obtained from a report by Baldvins et al. (2012). Costs associated with each damage state are calculated in a manner similar to those for the EBF, these are explained in Appendix C. Fragility curves are plotted in Figure 5.13, and cost data is given in Table 5.12. Figure 5.12 Fragility Curves for EBF Structural PG 91 Table 5.11 Structural EBF PG DS2 DS3 DS4 EDP ISD (%) Qty Min 1 1 1 Qty Max 10000 10000 10000 Cost Min $33,965 $37,037 $48,806 Cost Max $33,965 $37,037 $48,806 Median EDP 0.44 0.66 0.88 Beta 0.3 0.3 0.32 Quantity 6 bays/floor Reference Gulec et al. (2010), Yang et al. (2009b) Figure 5.13 Fragility Curves for SPSW Structural PG Table 5.12 Structural SPSW PG DS2 DS3 DS4 EDP ISD (%) Qty Min 1 1 1 Qty Max 10000 10000 10000 Cost Min $24,756 $134,101 $193,875 Cost Max $24,756 $134,101 $193,875 Median EDP 0.6 1.5 2.75 Beta 0.3 0.3 0.3 Quantity 2 walls/floor Reference Baldvins et al. (2012), Yang et al. (2009b) 92 Chapter 6: Comparison of the Structural Performance 6.1 OpenSees Model Data Table 6.1 is a summary of the structural weight used in the design to calculate the NBCC equivalent static lateral forces; these were also used to assign mass to the nodes in the dynamic model. Note that because of the heavier gravity column sizes, the MRF floor weights are slightly heavier. Table 6.2 shows the five structural periods of the models. Table 6.1 Structural Weight FLOOR MRF Weight (kN) Weight of other Frames (kN) 5 7319 7299 4 11816 11795 3 11987 11853 2 11987 11853 1 12038 11882 Total 55147 54682 Table 6.2 Model Periods MRF EBF BRBF CBF CBFLD SZBF SPSW SPSWLD XBF T1 (sec) 1.28 0.66 0.90 0.54 0.49 0.57 0.58 0.37 0.45 T2 (sec) 0.42 0.22 0.33 0.19 0.18 0.19 0.22 0.16 0.18 T3 (sec) 0.22 0.14 0.19 0.12 0.11 0.11 0.14 0.10 0.12 T4 (sec) 0.13 0.10 0.14 0.09 0.09 0.08 0.11 0.08 0.09 T5 (sec) 0.10 0.09 0.10 0.08 0.07 0.06 0.08 0.06 0.07 93 6.2 Peak Story Drift and Acceleration The OpenSees models were subjected to each of the ground motions selected in Chapter 2; 99 ground motions were analyzed for each model. The maximum structural response was obtained from each and used to evaluate the performance. Tables 6.3-6.5 show the statistics of the peak structural response at each of the three hazard levels considered, this data is shown graphically in Figures 6.1-6.3. For each record, the peak response was obtained; the tables show the median of these peak responses for each hazard level. The standard deviation of the responses is in parentheses. Table 6.6 shows the maximum peak responses for the 2/50 hazard level. As shown in in Table 6.6, the maximum interstory drift of 2.5% occurs on the first story of the MRF, this happens during the 1994 Northridge ground motion (ground motion #942). The maximum story acceleration of 1.9g happens at the roof level of the XBF model, this peak value occurs during the 1980 Irpina Italy ground motion (ground motion #302). It is easier to pick trends out of Figures 6.1-6.3 than from the tables. Ranking the frames from greatest to least ISD at the first level for the 2/50 hazard the systems are: MRF, EBF, BRBF, SZBF, CBF, CBFLD, XBF, SPSW, SPSWLD. With slight variation, this trend holds across each of the hazard levels. The MRF, without any braces or infill plates to stiffen it, is naturally the system with the greatest drift. The SPSWLD system has the least ISD at the first story, the SPSW also tends to have low story drifts at the first story in comparison to the others; clearly the stiffness of the infill plate is successful at restricting 94 drifts. An interesting trend is that for some of the frames the ISD increases for the higher stories. The MRF is the only system for which the ISD decreases for successively higher stories, for every hazard. In particular, the SPSW drifts dramatically increase at the 5th floor, and then reduce a little at the roof; this trend is apparent at all hazards. The BRBF has decreasing ISD at the 3rd floor but by the 5th floor it increases to be about the same as at the ground level; drifts reduce considerably at the roof; this trend holds across all hazards. Two of the least ductile systems, the CBFLD and the XBF have the highest story accelerations; accelerations for the SPSWLD are also very high. Since these systems are designed to have little ductility, they are expected to remain largely elastic. Because these frames are so stiff, they experience high accelerations. The lowest accelerations were experienced by the EBF, which is one of the most ductile systems. It can be concluded that structures designed with low values of RdRo will experience much higher story accelerations than those structures designed with larger force reduction factors. The average standard deviation for drift is 0.06 (%), for acceleration the average is 0.08 (g). For ISD, the average standard deviation is 0.10, 0.06, and 0.03 for the 2/50, 10/50 and 50/50 hazards, respectively. For acceleration the average deviations are 0.11, 0.08 and 0.04 for the three decreasing hazard levels. This shows that the variation in response tends to decrease with the decreasing hazard. Since all ground motions were amplitude scaled together in order to match the design spectra as closely as possible, it is not surprising that the variance in structural response is small. 95 Table 6.3 Results for 2% in 50 Years Hazard Level EDP [units] EBF CBF CBFLD SZBF BRBF XBF SPSW SPSWLD MRF du2 [%] 0.88(0.22) 0.48(0.12) 0.44(0.18) 0.55(0.13) 0.63(0.14) 0.33(0.07) 0.28(0.05) 0.27(0.1) 1.43(0.45) du3 [%] 0.35(0.13) 0.46(0.12) 0.53(0.16) 0.49(0.10) 0.49(0.14) 0.39(0.09) 0.30(0.04) 0.20(0.03) 1.11(0.38) du4 [%] 0.21(0.05) 0.51(0.11) 0.28(0.07) 0.44(0.08) 0.59(0.12) 0.33(0.06) 0.40(0.07) 0.20(0.01) 0.93(0.22) du5 [%] 0.18(0.02) 0.23(0.04) 0.26(0.04) 0.23(0.03) 0.6(0.11) 0.39(0.11) 0.63(0.12) 0.30(0.05) 0.84(0.15) duR [%] 0.10(0.02) 0.12(0.02) 0.13(0.02) 0.04(0.01) 0.34(0.06) 0.32(0.05) 0.52(0.15) 0.20(0.02) 0.72(0.11) ag [g] 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) 0.36(0.10) a2 [g] 0.49(0.10) 0.61(0.14) 0.69(0.17) 0.67(0.12) 0.56(0.13) 0.61(0.14) 0.51(0.14) 0.63(0.11) 0.53(0.09) a3 [g] 0.42(0.09) 0.65(0.11) 0.77(0.11) 0.67(0.11) 0.56(0.11) 0.81(0.16) 0.53(0.12) 0.69(0.10) 0.51(0.10) a4 [g] 0.36(0.07) 0.70(0.09) 0.78(0.15) 0.62(0.11) 0.57(0.10) 0.96(0.16) 0.54(0.10) 0.86(0.11) 0.51(0.11) a5 [g] 0.41(0.06) 0.65(0.08) 0.90(0.13) 0.65(0.07) 0.55(0.07) 1.1(0.20) 0.58(0.07) 1.01(0.09) 0.50(0.08) aR [g] 0.60(0.10) 0.92(0.14) 1.14(0.21) 0.76(0.11) 0.82(0.13) 1.56(0.21) 0.82(0.10) 1.33(0.07) 0.80(0.12) Note: median of maximum values shown (standard deviation in parentheses) Table 6.4 Results for 10% in 50 Years Hazard Level EDP [units] EBF CBF CBF LD SZBF BRBF XBF SPSW SPSW LD MRF du2 [%] 0.4(0.1) 0.25(0.08) 0.23(0.06) 0.26(0.08) 0.33(0.08) 0.17(0.05) 0.21(0.03) 0.16(0.03) 0.74(0.23) du3 [%] 0.19(0.02) 0.27(0.05) 0.23(0.09) 0.26(0.06) 0.31(0.06) 0.2(0.06) 0.24(0.03) 0.13(0.03) 0.57(0.19) du4 [%] 0.17(0.02) 0.26(0.11) 0.19(0.05) 0.25(0.08) 0.34(0.07) 0.17(0.05) 0.28(0.04) 0.13(0.03) 0.48(0.15) du5 [%] 0.13(0.02) 0.18(0.02) 0.17(0.05) 0.17(0.03) 0.34(0.08) 0.2(0.06) 0.36(0.06) 0.16(0.04) 0.44(0.12) duR [%] 0.07(0.01) 0.08(0.01) 0.08(0.02) 0.03(0.01) 0.21(0.04) 0.17(0.05) 0.32(0.04) 0.14(0.03) 0.38(0.09) ag [g] 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) 0.19(0.05) a2 [g] 0.32(0.07) 0.35(0.07) 0.34(0.07) 0.36(0.07) 0.32(0.07) 0.34(0.07) 0.33(0.08) 0.36(0.06) 0.29(0.05) a3 [g] 0.27(0.07) 0.48(0.07) 0.44(0.1) 0.45(0.08) 0.32(0.07) 0.48(0.09) 0.39(0.08) 0.47(0.08) 0.28(0.05) a4 [g] 0.27(0.04) 0.5(0.13) 0.49(0.13) 0.43(0.1) 0.3(0.07) 0.51(0.12) 0.42(0.06) 0.56(0.1) 0.27(0.06) a5 [g] 0.31(0.04) 0.53(0.08) 0.52(0.15) 0.5(0.1) 0.33(0.06) 0.58(0.15) 0.45(0.07) 0.63(0.13) 0.27(0.06) aR [g] 0.42(0.08) 0.64(0.1) 0.65(0.18) 0.53(0.1) 0.51(0.1) 0.83(0.22) 0.67(0.07) 0.93(0.17) 0.42(0.08) Note: median of maximum values shown (standard deviation in parentheses) 96 Table 6.5 Results for 50% in 50 Years Hazard Level EDP [units] EBF CBF CBF LD SZBF BRBF XBF SPSW SPSWLD MRF du2 [%] 0.14(0.05) 0.08(0.03) 0.07(0.02) 0.09(0.03) 0.13(0.03) 0.07(0.02) 0.10(0.02) 0.06(0.01) 0.28(0.09) du3 [%] 0.12(0.03) 0.09(0.03) 0.08(0.02) 0.10(0.03) 0.12(0.03) 0.08(0.02) 0.12(0.02) 0.05(0.01) 0.21(0.07) du4 [%] 0.10(0.02) 0.09(0.03) 0.07(0.02) 0.09(0.02) 0.13(0.03) 0.07(0.02) 0.14(0.02) 0.05(0.01) 0.18(0.05) du5 [%] 0.08(0.02) 0.07(0.02) 0.06(0.02) 0.06(0.01) 0.13(0.03) 0.08(0.02) 0.19(0.03) 0.06(0.01) 0.17(0.05) duR [%] 0.04(0.01) 0.03(0.01) 0.03(0.01) 0.01(0.01) 0.08(0.02) 0.07(0.02) 0.17(0.02) 0.05(0.01) 0.14(0.04) ag [g] 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) 0.07(0.02) a2 [g] 0.13(0.03) 0.13(0.03) 0.13(0.03) 0.13(0.02) 0.13(0.03) 0.13(0.02) 0.12(0.03) 0.14(0.03) 0.10(0.02) a3 [g] 0.14(0.03) 0.17(0.04) 0.16(0.04) 0.16(0.03) 0.12(0.03) 0.18(0.04) 0.16(0.03) 0.18(0.04) 0.11(0.02) a4 [g] 0.14(0.03) 0.17(0.05) 0.18(0.05) 0.15(0.04) 0.12(0.03) 0.20(0.05) 0.16(0.03) 0.22(0.04) 0.10(0.02) a5 [g] 0.17(0.04) 0.19(0.05) 0.19(0.05) 0.18(0.04) 0.13(0.03) 0.22(0.06) 0.19(0.04) 0.24(0.05) 0.10(0.02) aR [g] 0.21(0.04) 0.24(0.05) 0.25(0.06) 0.19(0.04) 0.19(0.04) 0.31(0.08) 0.27(0.06) 0.35(0.06) 0.16(0.03) Note: median of maximum values shown (standard deviation in parentheses) Table 6.6 Maximum Response Quantities for 2% in 50 Years Hazard Level EDP [units] EBF CBF CBFLD SZBF BRBF XBF SPSW SPSWLD MRF du2 [%] 1.49 0.74 0.91 0.93 0.97 0.50 0.42 0.58 2.50 du3 [%] 0.82 0.74 0.90 0.65 0.91 0.59 0.39 0.29 2.13 du4 [%] 0.39 0.81 0.63 0.61 0.86 0.43 0.56 0.24 1.48 du5 [%] 0.24 0.43 0.38 0.35 0.96 0.66 0.90 0.45 1.12 duR [%] 0.15 0.16 0.19 0.06 0.49 0.40 0.91 0.25 0.91 ag [g] 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 a2 [g] 0.78 0.98 1.14 0.97 0.84 0.91 0.86 0.85 0.75 a3 [g] 0.65 0.87 1.01 0.98 0.78 1.17 0.86 0.96 0.75 a4 [g] 0.55 1.01 1.26 1.02 0.94 1.23 0.83 1.25 0.94 a5 [g] 0.65 0.96 1.19 0.83 0.75 1.50 0.86 1.22 0.77 aR [g] 0.91 1.31 1.65 1.06 1.14 1.89 1.11 1.45 1.06 Note: maximum of maximum values shown 97 Figure 6.1 Median ISD Vs. Height from 2/50 Hazard Figure 6.2 Median ISD Vs. Height from 10/50 Hazard 98 Figure 6.3 Median ISD Vs. Height from 50/50 Hazard Table 6.7 shows the percentage difference between the design interstory drifts and the median EDP. The design ISD is from the elastic frame model when the NBCC forces are applied to check the 2.5% drift limit (See Appendix A). Because the MRF design is governed by the ISD from an elastic frame analysis and the NBCC forces, it is not surprising that the differences are smallest for this system. Note that a positive value of percentage difference indicates that the “design” ISD was greater than the median response from the analysis. This also makes sense, since the frame analysis used for design did not include the contribution of the gravity system; in the OpenSees model the gravity elements clearly contribute significantly to lateral resistance via. frame action. 99 Table 6.7 Percent Difference Between Median 2/50 EDP and Design Values EDP [units] EBF CBF CBFLD SZBF BRBF XBF SPSW SPSWLD MRF du2 [%] 2% 4% 12% 8% 30% 59% 77% 78% 11% du3 [%] 56% 34% 12% 39% 62% 72% 77% 71% 45% du4 [%] 70% 27% 44% 45% 58% 75% 69% 71% 48% du5 [%] 70% 62% 48% 62% 50% 70% 63% 70% 48% duR [%] 75% 60% 57% 80% 51% 68% 60% 80% 40% Note: design values from NBCC forces and the elastic model 6.3 Peak Story Shear Tables 6.8-6.10 show the median of the peak responses for the story shear from the 2/50, 10/50 and 50/50 hazard level ground motions, respectively. The story shear values are expressed as percentages of the weight of the building. The data from Tables 6.8-6.10 is represented graphically in Figures 6.4-6.6 for the 2/50, 10/50 and 50/50 hazard levels, respectively. Table 6.11 shows the maximum of the maxima of the shears from the 2/50 hazard. The maximum shear of 26,100 kN was experienced by the SPSWLD during ground motion #265, the 1980 Victoria-Mexico record at the 2/50 hazard. Trends were generally consistent across the hazard levels. For every frame at every hazard, shear decreased at successively higher floor levels. The systems with the highest shears were the SPSWLD, SPSW, CBFLD and the XBF; likewise the lowest shears were experienced by the MRF, BRBF and the EBF. At the 2/50 hazard level, the CBF and the 100 SZBF had nearly identical shears at every story. At the 10/50 hazard level, the CBF and the CBFLD had nearly identical story shears; likewise at this hazard level the BRBF and the EBF experienced nearly identical shears at every level above ground. At the 50/50 hazard the CBFLD matched the CBF shears at lower levels; at the top two floors it experienced the same forces as the XBF. Overall the trend was for systems designed with higher reduction factors to experience lower story shears, the system with the highest reduction factor however, the SPSW, experienced some of the highest shears. Generally, the less ductile the system, the greater the shear forces it will experience. The fact that the SPSW experiences such high shear forces and yet is designed with the highest RdRo reduction factor may be explained by conservative clauses in S16-09. The steel code requires the SPSW be designed such that the infill plate has the capacity to take 100% of the NBCC lateral forces – but it also requires that the moment frame be able to withstand 25% of the story shear. This results in a conservative design that is also very stiff; thus this system attracted more force than might have been expected. It is possible that the requirement for the moment frame to carry 25% of the load is unnecessarily conservative. Code stiffness clauses require the beams and columns to be very large sizes, especially the beams at the top and bottom of the wall. This causes members to be larger than required for strength, for more detail refer to Appendix A. The average standard deviations for the 2/50, 10/50 and 50/50 hazard level data is 2%, 2% and 1%, respectively. This trend is similar as for the ISD and story acceleration, where the deviation decreases with hazard level. 101 Table 6.12 shows the percentage difference between the median responses of the models at the 2/50 hazard with the NBCC shear forces. Looking at this table, V1 represents the base shear. Interestingly, the percentage difference for the EBF, CBFLD, BRBF and XBF are small. This indicates that the code reduction factors are probably appropriate for these frames. The SPSW’s large deviation from the code shears may be explained by the conservative code clauses and the additional stiffness of the included moment frame. The limited ductility SPSWLD also was designed with several very conservative code clauses, which may account for some of its excessive base shear. Surprisingly, the MRF also shows a base shear much higher than that predicted by the NBCC. This may be a result of the increased stiffness provided by the gravity system. The stiffness provided by the gravity members was insignificant compared to the stiffness of the SFRS for the braced frames and shear walls, but it is significant compared with the moment frame lateral stiffness. As a result, the MRF was stiffer than predicted by the code and had a higher base shear than expected. 102 Table 6.8 Shear Results for 2% in 50 Years Hazard Level Story Shear EBF CBF CBF LD SZBF BRBF XBF SPSW SPSW LD MRF V1 [%] 10(1) 20(1) 26(1) 19(1) 13(1) 37(7) 26(1) 45(2) 15(3) V2 [%] 9(1) 18(1) 23(1) 18(1) 11(1) 34(6) 21(1) 40(2) 12(3) V3 [%] 9(1) 15(1) 20(2) 15(1) 11(1) 27(4) 17(1) 34(2) 11(2) V4 [%] 8(1) 12(1) 15(2) 12(1) 9(1) 21(3) 12(0) 24(1) 9(1) V5 [%] 4(1) 6(1) 8(1) 5(1) 5(1) 10(1) 7(1) 12(0) 5(1) Note: median of maximum values shown (standard deviation in parentheses) Table 6.9 Shear Results for 10% in 50 Years Hazard Level Story Shear EBF CBF CBF LD SZBF BRBF XBF SPSW SPSW LD MRF V1 [%] 8(0) 19(3) 19(5) 16(3) 10(2) 20(5) 23(2) 31(6) 8(2) V2 [%] 8(0) 16(3) 17(4) 14(3) 8(2) 18(5) 18(2) 27(6) 6(2) V3 [%] 7(1) 14(2) 14(4) 12(2) 7(1) 14(4) 15(2) 22(5) 6(2) V4 [%] 6(1) 10(1) 10(3) 8(2) 6(1) 11(3) 11(1) 17(3) 5(1) V5 [%] 3(1) 4(1) 4(1) 3(1) 3(1) 6(1) 6(1) 9(2) 3(1) Note: median of maximum values shown (standard deviation in parentheses) Table 6.10 Shear Results for 50% in 50 Years Hazard Level Story Shear EBF CBF CBF LD SZBF BRBF XBF SPSW SPSW LD MRF V1 [%] 6(1) 6(2) 6(2) 6(2) 4(1) 8(2) 10(2) 12(3) 3(1) V2 [%] 5(1) 6(2) 6(2) 5(2) 3(1) 7(2) 7(2) 10(2) 2(1) V3 [%] 4(1) 5(1) 5(1) 4(1) 3(1) 5(2) 6(1) 9(2) 2(1) V4 [%] 3(1) 3(1) 4(1) 3(1) 2(1) 4(1) 5(1) 6(1) 2(0) V5 [%] 1(0) 2(0) 2(0) 1(0) 1(0) 2(1) 2(0) 3(1) 1(0) Note: median of maximum values shown (standard deviation in parentheses) 103 Table 6.11 Maximum Response Quantities for 2% in 50 Years Hazard Level Story Shear EBF CBF CBF LD SZBF BRBF XBF SPSW SPSW LD MRF V1 [%] 11% 21% 28% 20% 16% 45% 28% 48% 20% V2 [%] 11% 19% 24% 19% 15% 40% 22% 42% 17% V3 [%] 10% 17% 23% 17% 12% 33% 18% 36% 15% V4 [%] 9% 15% 21% 15% 12% 23% 13% 25% 12% V5 [%] 6% 8% 11% 7% 7% 12% 9% 12% 7% Note: maximum of maximum values shown Table 6.12 Percentage Difference Between NBCC and 2/50 Median Shear % Diff. EBF CBF CBF LD SZBF BRBF XBF SPSW SPSWLD MRF V1 [%] 4% -26% -9% -21% -1% -8% -234% -114% -130% V2 [%] 8% -24% -5% -22% 5% -8% -191% -111% -114% V3 [%] -9% -20% -10% -20% -7% -4% -186% -112% -118% V4 [%] -37% -39% -19% -38% -31% -15% -201% -113% -152% V5 [%] -75% -65% -37% -37% -80% -30% -268% -143% -211% Note: a negative % difference implies the median value is greater 104 Figure 6.4 Median Shear Vs. Height from 2/50 Hazard Figure 6.5 Median Shear Vs. Height from 10/50 Hazard 105 Figure 6.6 Median Shear Vs. Height from 50/50 Hazard 6.4 Element Response In order to get an idea of how far the structure must be deformed to behave nonlinearly, a pushover analysis was conducted on the MRF, EBF, BRBF, CBF and SPSW models. Pushover curves are presented in Appendix D. Table 6.13 shows the approximate interstory drift of each model when nonlinear behavior is observed. 106 Table 6.13 Pushover Analysis Nonlinearity Observed at ISD% Level MRF EBF BRBF CBF SPSW du2 [%] 1.0% 0.1% 0.2% 0.1% 0.2% du3 [%] 1.0% 0.1% 0.4% 0.2% 0.2% du4 [%] 1.0% 0.2% 0.4% 0.3% 0.2% du5 [%] 1.0% 0.2% 0.4% 0.3% 0.2% duR [%] 1.0% 0.2% 0.4% 0.3% 0.2% Comparison of the values in Table 6.13 with the median interstory drifts at the 2/50 hazard level in Table 6.3 enables the engineer to get an idea of whether or not the structure is expected to behave nonlinearly. This assumption is based on the fact that the pushover curve was done using the NBCC lateral seismic force distribution; this is a 1st mode force pattern, and it is assumed that for this building the response will be primarily in the 1st mode. For example, the MRF goes nonlinear at 1% interstory drift according to the pushover curve; the median EDP for ISD is 1.43% (1st floor) and 1.11% (2nd floor), but less than 1% at the top three levels. It is expected then that the MRF will behave nonlinearly under most of the ground motions at the first two levels, but not as much on the top three floors. A similar comparison suggests that the SPSW and BRBF are expected to yield at every level, whereas the EBF and CBF are expected to be somewhat elastic on the top two stories. Although 99 ground motions were run per model, one earthquake record was chosen in order to examine element behaviors. Since the peak roof acceleration of 1.9g of the XBF was achieved under the 1980 Irpinia Italy ground motion (ground motion #302), this record was chosen. The structural response of each model to ground motion #302 was studied, and a typical yielding element at each floor was examined. Table 6.14 represents a summary of this analysis. 107 Table 6.14 Yielding of Elements Under GM#302 Level MRF EBF BRBF CBF CBFLD SZBF SPSW SPSWLD XBF Element M. Conn. Link Brace/ M.Conn. Brace Brace Brace Plate/ M.Conn. Plate Brace 1 yes yes yes/no yes no yes 40%/no 100% no 2 yes yes yes/no no yes yes 35%/no 50% no 3 yes limited yes/no yes no yes 55%/no 40% no 4 no limited yes/no no no no 60%/no 95% no 5 no no no/no no no no 60%/no 0% no The row labeled “element” indicates the designated yielding component in each system. In the MRF, the yielding component is the beam, and in particular at the zerolength hinge element at the ends of the beam elements. In the EBF, the yielding component is the eccentric link beam. In the concentrically braced frames, the yielding component is the brace. In the steel plate walls, the yielding components are the strips. Some of the systems, for example the BRBF have two designated yielding components, the brace and the moment connected beam; in this case, two elements are separated by a slash. Based on an analysis of the element forces during GM# 302, the CBF experiences yielding/buckling on the first and third stories while the CBFLD only buckles on the second story; this result seems to suggest that once a brace buckles drift demand and nonlinearity concentrates there. The SZBF experienced considerable yielding on the first three stories, which is largely a result of the top level truss which remains elastic per the design intent. Because the top level braces cannot buckle, demand concentrates at the lower levels. The data for the steel plate walls is expressed as a percentage, this represents the quantity of strip elements at each floor that yield (100% is all strips yield, 0% is all strips behave elastically). All 108 of the reduced beam section connections behaved elastically, but somewhat surprisingly there was more yielding at upper floors of the SPSW than at the lower floors. This can be explained by the fact that lower floors had much greater overstrength in the design calculations than the upper floors, this was necessary in order to use commonly available infill plate sizes. The SPSWLD generally experienced more yielding than its more ductile counterpart, which is likely a result of not having a moment resisting frame to help resist seismic forces. The XBF model remained totally elastic, which helps to explain why its story accelerations were so much higher. The XBF was considerably overdesigned because of the extremely conservative provisions of S16-09 regarding conventional construction buildings. Figures 6.7-6.11 are hysteresis curves for some typical elements from the time history analysis of the models under ground motion #302. At the beginning of this section, pushover drifts in Table 6.13 were compared with median ISD response values in Table 6.3 to predict whether or not the frame would go nonlinear. This hypothesis can be checked for ground motion #302 by looking at Table 6.14. A comparison of Table 6.13 with Table 6.3 indicated that the first three levels of the MRF ought to go nonlinear, referring to Table 6.14 this assumption was correct (for GM#302). Comparing the 2/50 median peak drift to the yield drift from Table 6.13 predicted the element yielding at the appropriate floor levels for the MRF, EBF, BRBF, and SPSW – these were also models for which pushover curves were generated. For the CBF, CBFLD and SPSWLD a comparison of Table 6.13 to Table 6.3 did not correctly predict element yielding. It is worthwhile to note that numerical convergence problems were encountered during the pushover analysis of the CBF when the brace buckled on the 3rd floor. 109 The XBF and SZBF are different enough that separate pushover curves would need to be done for them to establish yield drifts. It is impossible to establish a “rule” for predicting yield based on a comparison of pushover results to median 2/50 response because of lack of sufficient data. More comprehensive pushover curves would be required, and Table 6.14 would need to be expanded to include all the earthquake records. The limited comparison here does suggest that this is a worthwhile direction for future research. Figure 6.7 Hysteresis for First Floor Beam Hinge on MRF -3000 -2000 -1000 0 1000 2000 3000 -1 -0.5 0 0.5 M o m en t ( kN - m ) Hinge Rotation (degrees) 110 Figure 6.8 Hysteresis Curve for First Story Eccentric Link from EBF Figure 6.9 Hysteresis Curve for First Story Brace from BRBF -800 -600 -400 -200 0 200 400 600 800 -50 0 50 100 Li n k Sh ea r (kN ) Link Shear Deformation (mm) -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 -30 -20 -10 0 10 20 30 Br ac e Fo rc e (kN ) Axial Deformation (mm) 111 Figure 6.10 Hysteresis Curve for a Typical Top Story Strip of SPSW Figure 6.11 Brace Hysteresis Curve for CBF Brace on Third Floor -400 -300 -200 -100 0 100 200 300 400 -3 -1 1 3 5 St rip A x ia l L o ad (k N ) Strip deformation (mm) -2000 -1500 -1000 -500 0 500 1000 1500 2000 -15 -10 -5 0 5 10 Br ac e Fo rc e (kN ) Brace Axial Deformation (mm) 112 6.5 Cost Analysis Using the PBEE methodology presented in Chapter 5, the structural response information together with the fragility data was used to assess the performance of the building in terms of repair costs. The cumulative distribution function (CDF) of the repair cost for the 2/50, 10/50 and 50/50 hazard levels is shown in Figure 6.12, 6.13 and 6.14, respectively. The performance groups are identical for each model, except that the structural performance group depends on the system. Looking at the figures, it is clear that at the 2/50 hazard level, most of the models perform similarly. It is at the lower hazard levels that the differences arise. Figure 6.12 CDF of the Repair Cost for Structures 2/50 Hazard Level 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500000 1500000 2500000 3500000 4500000 Pr o ba bi lit y o f T o ta l R ep ai r Co st be in g le ss th an $ Cost ($) BRBF CBF CBFLD SZBF XBF EBF MRF SPSW SPSWLD 113 Figure 6.13 CDF of the Repair Cost for Structures 10/50 Hazard Level 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100000 600000 1100000 1600000 2100000 2600000 3100000 Pr o ba bi lit y o f T o ta l R ep ai r Co st be in g le ss th an $ Cost ($) BRBF CBF CBFLD SZBF XBF EBF MRF SPSW SPSWLD 114 Figure 6.14 CDF of the Repair Cost for Structures 50/50 Hazard Level In the paper by Yang et al. (2009a) an approach is given to determine the mean annual rate of the repair cost exceeding a threshold value. This result is obtained by first computing the complement of the cumulative distribution function as shown in Figures 6.12, 6.13 and 6.14. The complement is then multiplied by the slope of the hazard curve at the corresponding ground motion intensity level and finally the resulting curves are integrated across the seismic hazard intervals considered in the hazard analysis. Repeating this process for all repair cost values produces a loss curve that represents the mean annual rate of the repair cost exceeding a threshold value. Figure 6.15 shows the loss curve which is generated for the CBF data. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 250000 500000 750000 1000000 1250000 Pr o ba bi lit y o f T o ta l R ep ai r Co st be in g le ss th an $ Cost ($) BRBF CBF CBFLD SZBF XBF EBF MRF SPSW SPSWLD 115 The expected cumulative value of a cost X resulting from all earthquakes over a period of one year is given by Equation 6.1, where the last expression is obtained from integration by parts. Thus, the area under the loss curve represents the mean cumulative total repair cost for all earthquake events in one year’s time. This value has relevance to the premium that could be charged for building insurance against seismic damage. Curves similar to Figure 6.15 were generated for each of the nine structural models of this study, and the mean cumulative annual repair costs were calculated for each. Table 6.15 shows a summary of these repair costs, along with the median value of the CDF curves at each of the considered hazard levels. Finally, the structural steel usage for each frame is given, in kilograms; this gives an idea of which structures will cost more to build. For convenience, a comparison is given of each frame with the heaviest model (the MRF) and the most costly to repair (the XBF). Equation 6.1  =  | ()| =   ()   Table 6.15 Total Repair Costs Associated with Different prequalified SFRS Steel Usage. (kg) Compare to MRF Median Repair Cost from CDF Mean Cum. Annual repair Cost Compare to XBF 2/50 10/50 50/50 MRF 702897 100% $1,720,000 $1,080,000 $97,200 $ 19,800 42% CBF 682403 97% $1,690,000 $1,560,000 $338,000 $ 39,300 83% SPSW LD 656177 93% $1,730,000 $1,590,000 $610,000 $ 39,900 84% CBF LD 642654 91% $1,760,000 $1,560,000 $337,000 $ 44,500 93% SZBF 621532 88% $1,690,000 $1,360,000 $305,000 $ 37,600 79% SPSW 601896 86% $1,740,000 $1,380,000 $370,000 $ 31,300 66% BRBF 604979 86% $1,720,000 $1,320,000 $305,000 $ 35,400 74% XBF 591510 84% $1,850,000 $1,590,000 $610,000 $ 47,600 100% EBF 561057 80% $1,830,000 $1,220,000 $305,000 $ 33,600 71% Note: costs are in US Dollars 116 Figure 6.15 Loss Curve for CBF Costs in US Dollars Based on the results presented in Table 6.15, the MRF has the lowest annual repair cost, although it uses the most material. The EBF uses 20% less steel than the MRF, this makes it the lightest system. The XBF is the second most lightweight system, but carries the highest annual repair costs. Although most of the systems have relatively similar median CDF costs at the 2/50 hazard level, the differences begin to be considerable at the 10/50 and the 50/50 hazard levels. It is clear why the MRF carries such a low mean cumulative annual repair cost; at the 10/50 hazard level the median CDF is much lower than the others, and at the 50/50 hazard level the median CDF value is less than $100,000; less than 1/6th that of the XBF at this hazard level. 117 6.6 Repair Cost Distribution by PG Figures 6.16-6.24 show the breakdown of the total repair costs for each of the prequalified SFRS at each of the hazard levels. The vertical axis shows the probability associated with the costs. The horizontal axes show the 26 PGs included in the study. For example, the top chart of Figure 6.17 shows CBF at the 2/50 hazard level. A quick look at these figures shows that by far the greatest contributor to costs are the “contents” PGs, intended to model the costs associated with damaged furniture, computer equipment, and other things used by building occupants. This PG only has one damage state, and it is acceleration dependent. The ceiling performance groups, also acceleration dependent, contributed significantly to the repair cost of structures which have high story accelerations such as the XBF. Looking at Figure 6.1, the MRF had the highest ISD at every level, and the XBF had the highest story accelerations at every level except the 2nd and 4th levels; the CBFLD had the highest story accelerations for these two EDP. It is interesting to note that the MRF, with one of the highest RdRo reduction factors experienced the greatest drifts (Ref. Figure 6.1-6.3), while the least ductile structures experienced the highest accelerations. As it turns out, high story accelerations result in high costs from the ceiling and contents performance groups. As far as the MRF goes, damage is only sustained to the moment connections at 3% ISD, and the most costly damage state, DS4 only occurs at 5% ISD; therefore this system is easily able to accommodate the higher drifts without sustaining damage (recall the maximum ISD experienced by the MRF is 2.5%). Although the nonstructural drift sensitive PG’s do contribute to the costs for the MRF, these are not very significant. 118 In Figure 6.14 for the 50/50 hazard, the MRF system carries the lowest median repair cost; whereas the SPSWLD and the XBF are tied for highest median cost. Looking at Figures 6.16 through 6.24, it is clear that the only PG that significantly contributes to cost at the 50/50 hazard level is the contents group. This PG has a median EDP of 0.25g required to trigger damage. Looking at Table 6.5, the median peak response for the MRF is about 0.1g at every floor; it should not be surprising then that the MRF carries the lowest cost. With a median EDP required to trigger damage of 0.25g for the contents PG, and the structural response being on the median so much lower, this is not going to contribute much to the cost. On the other hand, the SPSWLD median EDP ranges from 0.14g to 0.35g at the roof; since these numbers are much closer or above the EDP required to cause damage, this is why the repair costs at the 50/50 hazard level are higher. The most expensive system at the 50/50 hazard level is determined by which one has the highest median acceleration response; in this case it was the SPSWLD. The MRF is the cheapest system to repair after a seismic event and the XBF is the most costly. The EBF uses the least structural steel. Note that the increased weight of the MRF is in part due to the larger gravity column sizes. The difference between the MRF and the EBF is 141,840 kg of steel, at a steel price of $794 per tonne this is worth $112,621. The mean cumulative annual repair cost of the EBF is $13,800 more than the MRF; it will thus take $112,621/$13,800=8.16 or about 8 years before the reduced annual financial risk of the MRF pays off the increased investment in steel. Whether or not this timeframe is significant, or if this increase in construction material and cost is important will depend on building owner preferences and steel prices. 119 As a general rule, the less ductile systems (structures with lower RdRo) tended to do badly and have higher repair costs. This is because they are designed to remain more or less elastic, or in the case of the XBF entirely elastic, and this results in very high accelerations which drive up acceleration sensitive PG related costs. The four lowest median cumulative annual repair costs are for the MRF, SPSW, EBF and the BRBF, in that order. The systems which use the least material are the EBF, XBF, SPSW and the BRBF, in that order. Note that the MRF, EBF and BRBF also had the highest drifts at the 2/50 hazard; it seems that high ductility and drifts are preferable over high accelerations. Therefore, this analysis tends to favor the EBF and the SPSW. Additionally, the SPSW only requires four walls to be built, compared to the 12 braced frames required in the EBF; it is thus likely that SPSW connection costs will be low compared to the other frames. The CBFLD is probably the worst structural system, using 91% as much steel as the MRF and having the second highest median repair cost. 120 Figure 6.16 Repair Costs According to Performance Group for the BRBF 121 Figure 6.17 Repair Costs According to Performance Group for the CBF 122 Figure 6.18 Repair Costs According to Performance Group for the CBFLD 123 Figure 6.19 Repair Costs According to Performance Group for the SZBF 124 Figure 6.20 Repair Costs According to Performance Group for the XBF 125 Figure 6.21 Repair Costs According to Performance Group for the EBF 126 Figure 6.22 Repair Costs According to Performance Group for the MRF 127 Figure 6.23 Repair Costs According to Performance Group for the SPSW 128 Figure 6.24 Repair Costs According to Performance Group for the SPSWLD 129 Chapter 7: Summary and Conclusions With the advancement of the computer hardware and software, simulating the full range of structural response under a large range of earthquake shaking intensities has become possible. This thesis utilizes a state-of-the art modeling approach and a performance-based assessment procedure to quantify the performance of the prequalified steel SFRS included in the Canadian steel code. Based on the results of this study, the optimum design of the five story office building for Vancouver BC involves use of either EBF or a ductile SPSW. This is based on the criteria of minimizing material usage, but also seeking the lowest post-earthquake repair costs. If steel prices are low, a MRF may also be desirable; this system carries by far the lowest expected repair costs at the expense of using more structural steel than the other system. Furthermore, the analysis of this report shows the drawbacks of the limited ductility systems. It is important to bear in mind that these conclusions are limited to the design approach presented here. Different assumptions about the building site, configuration and orientation of the SFRS may lead to different conclusions. Based on this study, it appears that structures designed for more ductility tend to perform better than those with very low RdR0 reduction factors. It is important to realize that these results are taken in the context of the 2005 National Building Code of Canada and the Canadian Steel Code. The NBCC 2005 and S16-09 provisions design the systems to have limited deformation at the 2/50 hazard level. This leads to structures which have limited drift related damage. Using the PBEE methodology on structures designed per the International Building 130 Code (USA), EUROCODE (European Union) or other nationally adopted code may yield different results. Using the NBCC, this analysis showed that the acceleration sensitive “contents” PGs were the biggest contributors to repair costs; in light of this proactive steps could be taken, such as to secure furniture, bookshelves and computer equipment in order to limit damage. On the other hand, designs which maximize ductility and minimize story accelerations can also reduce the contribution of these PGs to costs. 7.1 Steps for Performance Based Earthquake Engineering (PBEE) The PBEE methodology developed in this thesis was used to evaluate the prequalified systems in CSA S16-09 using one five-story office building in downtown Vancouver BC. This methodology may be applied to the design of other structures. The PBEE method can be broken down into several steps, and applied to any structure. The method can be summarized as follows: 1. Seismic Hazard Analysis 2. Structural Design 3. Modeling 4. Definition of Performance Groups 5. Time History Analysis 6. Loss Analysis 131 7.1.1 Seismic Hazard Analysis The seismic hazard analysis of the building site is critical in order to quantify the seismicity of the region. This step was performed for this report in Chapter 2. Depending on the building site, a Probabilistic or a Deterministic Seismic Hazard Analysis may be more appropriate. For this step, expert geotechnical engineers or seismologists may be brought in to provide expertise and improve the accuracy of this step. In this study, ground motions were scaled to three hazard levels, 2/50, 10/50 and 50/50. There is no correct number of hazard levels which must be used, however in Yang et al. (2009a) and (2009b) three hazard levels were used. It is likely that using ground motions scaled to four or more hazards would produce more refined results. In this report, a total of 99 ground motions were used. Statistically, it will be best to use as many ground motion records as possible to improve the accuracy of the simulation. 7.1.2 Structural Design The structural design is discussed in great length in Appendix A. For this thesis, capacity design principles were used in to design each system to withstand the maximum forces developed by the energy dissipating element. For example, in the BRBF system, the beams and columns were designed in order to withstand the forces generated by the buckling restrained braces when they developed their maximum forces. For the design of this system, the columns were designed assuming every brace at each floor developed its maximum force simultaneously. This is a very 132 conservative approach, and leads to a certain amount of overdesign. Each of the different prequalified SFRS has energy dissipating elements as follows: • Moment Resisting Frame – the beam dissipates energy through flexural bending. • Concentric Braced Frame – the brace dissipates energy through inelastic buckling. • Eccentric Braced Frame – the link beam dissipates energy either through shear deformations (shear link) or flexural bending (flexural link). • Buckling Restrained Braced Frame – the brace dissipates energy through axial deformations. • Steel Plate Shear Wall – the infill plate buckles in compression and yields in tension to dissipate energy. Appendix A explains how each of these energy dissipating elements may be designed, and how to properly capacity design the remaining elements. It is also possible to use alternative design approaches, such as using modal analysis to estimate the maximum forces in structural elements; this is however much more complicated and is probably only justified for taller buildings. New structural systems may be devised in which these energy dissipating elements are combined. In this study, the BRBF system includes a moment resisting frame as a backup system, as did the SPSW. One new and innovative system being developed is known as the Linked Column Frame system (LCF). This system combines a moment resisting frame with shear links from the eccentric braced frame system. The design of any SFRS can be accomplished by designing the individual components per the approach in Appendix A. Design the remaining elements to withstand the maximum forces developed by the energy dissipating 133 element, and then check members against the elastic forces. and the ISD limit specified in the NBCC. 7.1.3 Modeling The modeling phase is often the most time consuming. Chapter 4 of this thesis provides a guide to modeling the yielding elements. These elements were calibrated to match test data. A similar approach can be followed for other designated yielding elements. Elements not expected to deform nonlinearly can be modeled using elastic elements. 7.1.4 Definition of Performance Groups Performance group definition, specifically fragility curve development and associated cost data for the damage states is an area of ongoing research. An engineer interested in using the PBEE evaluation procedure is recommended to check for the latest papers and reports. The data used in this study and in the references is obtained from the most cutting edge research literature at the time of writing. 134 7.1.5 Time History Analysis Peak structural response should be obtained for a range of earthquakes. This can be accomplished through the use of nonlinear dynamic analyses. In this study, OpenSees was used for analysis of 2D models of the buildings. Although torsion was ignored for this thesis, it can be included by using a 3D model. 7.1.6 Loss Analysis The loss analysis is used to quantify the structural performance. The final outcome can be used to make an informed risk-based decision. In the methodology presented in Chapter 5, the distribution of the repair costs at each hazard level and the mean annual repair costs are calculated. Useful charts can also be created in to visualize the repair cost distribution across the PGs. 7.2 Performance Based Design and Evaluation of New Structural Systems Using the PBEE methodology presented so far, new structural systems can be evaluated using the same approach. The Linked Column Frame (LCF) System is a new steel structural system (Lopes, Dusicka and Berman (2012), Yang and Tung (2012a) and (2012b)). It is characterized by columns connected by a series of shear links, similar in behavior to those found in an eccentric braced frame. These “linked columns” are linked by means of moment resisting frames; in this way the LCF is a hybrid system combining elements of the EBF and the MRF. Figure 7.1 shows 135 a basic representation of the LCF system showing the linked columns and the moment connected members. In order to compare this system and demonstrate how well it compares with the other systems in this study (the MRF, EBF, SPSW and others), the following steps may be followed. Figure 7.1 Linked Column Frame 1. Using the methods of Appendix A, calculate the NBCC forces and design the frame system. Design the members per the elastic equivalent forces, and per capacity design principles as demonstrated in Appendix A. The NBCC interstory drift limit of 2.5% must also be met. The reduction factors for the EBF may be used conservatively. 2. Once the design has been completed, the frame can be modeled. The gravity system, moment resisting frame and shear links may be modeled according to the rules established in this Chapter 4 of this thesis and in Appendix B. 3. The structural and nonstructural elements will then need to be defined, and PGs compiled in order to characterize them. Using the methods presented in Chapter 5, the performance group data for all elements likely to result in costly repair actions should be assembled. 136 4. A hazard analysis similar to that of Chapter 2 should be conducted, and appropriate ground motions selected and scaled to 3 hazard levels. 5. The analysis should be run; ideally as many unique ground motions as possible ought to be used. Once completed, the post processing phase can extract the EDP. 6. The loss analysis as outlined in Chapter 5 and 6 can then be conducted in order to determine the probabilistic repair costs, and the steel tonnage used can be calculated. The CDF curves at the various hazard levels, steel usage and the mean cumulative annual repair cost may be compared with that of the other frames in this thesis. 7.3 Concluding Remarks Overall, the median cumulative annual repair costs, the CDF curves, the ISD and story accelerations as well as the story shears are performance criteria can be used to compare seismic performance of structures. Using the results of this report, this data can be used as a benchmark for design of similar low to midrise structures. The PBEE methodology may be applied to new buildings to determine the optimal structural system or configuration; it may also be applied to existing buildings to estimate the potential monetary losses in the event of an earthquake. This method is very versatile, and can be employed by researchers and engineering professionals alike. The results of this thesis indicate that structures designed per CSA S16-09 for higher ductility (larger RdRo factors) tend to have lower post-earthquake repair costs. Structures designed with lower reduction factors, while easier to design (Ref. Appendix A), have higher story accelerations and therefore greater repair costs after a seismic event. 137 Bibliography ASCE-41, (2007) “Seismic Rehabilitation of Existing Buildings”, American Society of Civil Engineers Astaneh-Asl, A. (2005) “Design of shear tab connections for gravity and seismic loads,” SteelTips, Structural Steel Educational Council. ATC. 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University of California at Berkeley http://peer.berkeley.edu/peer_ground_motion_database (accessed February 2012) Rezai (1999), “Seismic behaviour of steel plate shear walls by shake table testing,” doctoral dissertation at the University of British Columbia R. R. Youngs, S. -J. Chiou, W. J. Silva and J. R. Humphrey, (1997), Strong Ground Motion Attenuation SAC steel project website, (1994) (accessed February 2012). http://nees.org UCB (1997). “Open system for earthquake engineering simulation (OpenSees) framework,” Pacific Earthquake Engineering Research Center, University of California, Berkeley, http://opensees.berkeley.edu/. U.S. Geological Survey, (2012) www.usgs.gov (accessed January 2012) VicWest (2012) company website http://vicwest.com/ (accessed January 2011) Yang (2011) notes for Civil 511 “Advanced Design of Steel Structures”, September 2011, University of British Columbia Yang, T.Y., Bozidar Stojadinovic and Jack Moehle. (accepted for publication) “Demonstration of a practical method for seismic performance assessment of structural systems,” Earthquake Spectra, EE-RI. Yang, T.Y., Moehle, J., Stojadinovic, B. and Der Kiureghian, A. (2009a). “Performance evaluation of structural systems: theory and implementation,” Journal of Structural Engineering, ASCE. Vol. 135, No. 10, pg. 1146 – pg 1154 141 Yang, T.Y., Moehle. J. and Stojadinovic, B. (2009b). “Performance evaluation of innovative Steel Braced Frames,” PEER report, Pacific Earthquake Engineering Research Center, University of California, Berkeley. Report number 2009/103 Yang T.Y. and Tung D.P. (2009a), “Performance-based plastic design of linked column frames”, the 4th Asia- Pacific Young Researchers and Graduates Symposium (YRGS2012)), Hong Kong, China, 3-4 December 2012 Yang T.Y. and Tung D.P. (2009b), “Performance evaluation of linked Column Frame Systems” Proceedings, First International Conference on Performance-based and Life-cycle Structural Engineering (PLSE 2012), Hong Kong, China, 5-7 December 2012 142 Appendices Appendix A Structural Design The buildings are designed per the principles of Capacity Design, and the individual SFRS are designed per best practices as referenced in the design texts: American Institute of Steel Construction design guides, Steel Tips documents and the book “Ductile Design of Steel Structures” by Bruneau et al. (2011). This section will layout the design approach as explained in these documents, suggest alternative options and direct the reader to references where necessary. Every system was based on a simplified code based equivalent static force design procedure; a more complicated dynamic analysis based approach may lead to a more efficient design in some cases, but it is the opinion of the author that this would be unnecessarily complicated. A.1 Gravity System Referring to Table A.1, the additional dead load (ADL) is calculated to be 1.67 kPa. In the NBCC 2005, Table [4.1.5.3] under “Office Areas” and then under “Floors above the first story” the minimum specified live load is 2.4 kPa. NBCC 2005 Table [4.1.3.2] specifies the load combinations which must be considered, they are shown in Table A.2. 143 Table A.1 Calculation of Additional Dead Load Element PSF kPa Partitions 20 0.96 Floor Finish 0.8 0.04 Fireproofing 3 0.14 Mechanical/Electrical/Plumbing 3 0.14 Celling/Lights 4 0.19 Miscellaneous 4 0.19 Σ=ADL 35 1.67 Table A.2 Load Combinations per NBCC 2005 Table 4.1.3.2 CASE Load Combination Principal Loads Companion Loads 1 1.4D 2 (1.25D or 0.9D) + 1.5L 0.5S or 0.4W 3 (1.25D or 0.9D) +1.5S 0.5L or 0.4W 4 (1.25D or 0.9D) + 1.4W 0.5L or 0.5S 5 1.0D + 1.0E 0.5L + 0.25S The gravity beams are simply supported, and do not resist lateral (wind or earthquake) loading. The applicable load combinations for vertical loading are cases 1 and 2. They are: 1.4 D = 2.34 kPa 1.25 D + 1.5 L = 5.69 kPa  Governs Based on this loading a concrete slab on metal deck is selected from a VicWest product catalog, design tables available at http://vicwest.com/. A 141 mm thick slab with 0.91 mm thick steel deck, a load carrying capacity of 6.6 kPa and a self-weight of 2.14 kPa is chosen. An Open Web Steel Joist (OWSJ) is added in the North-South direction to limit the deflections in the slab. Including the self-weight of the slab, the load combinations are: 1.4DL = 5.33 kPa 144 1.25DL + 1.5LL = 8.36 kPa  Governs Service Loading is DL + LL = 6.21 kPa. Using a tributary width of 2250 mm (See Figure A.1) this gives a line load of 18.81 kN/m and a service line load of 14 kN/m. The service loading is used for calculation of deflections. Based on this loading, an OWSJ is chosen from a VicWest catalog. A joist 750 mm deep with a load carrying capacity of 21 kN/m and a specified service load of 14 kN/m is chosen. At 92% of this service load the deflection will be L/360, or 25mm. This OWSJ has a mass of 22.5 kg/m, which translates to an additional dead load contribution of 0.07194 kPa. Calculating the deflection of this joist shows that it will be 27mm at the full service load, which is over the limit of L/360=25mm, but it will be hidden from view underneath ceiling tiles so it is acceptable. Figure A.1 Tributary Area for OWSJ The East-West beam will now be designed. Recalculating the load combinations taking into account the joist dead load gives: 145 1.4DL = 5.43 kPa 1.25DL + 1.5LL = 8.45 kPa Governs Service load is DL + LL = 6.28 kPa (used for deflection calculations) Using a tributary area of 9 meters (Figure A.2), it is assumed that the slab acts with perfect 1- way action and that the entire load is carried into the beam. This beam will be designed according to two criteria; the first being strength to resist the gravity loading. The second criteria will be serviceability; deflections must be within code limits. Table D.1 of S16-09 recommends a deflection limit of L/360 for “Members of floors and roofs supporting construction and finishes susceptible to cracking”. The maximum deflection at midspan of a simply supported beam is given in Equation A.1. Since the beam will be continuously restrained across the top (compression) flange by the slab it is assumed that the member can develop the yield moment, therefore the strength of the beam is taken as Equation A.2. Although the code allows the beam to be designed for the full plastic moment, it is conservatively designed for the yield moment. Additionally the design moment of a simply supported beam is given as Equation A.3. Figure A.2 Tributary Area for Beam 146 Equation A.1 EI wL 384 5 4 =∆ Equation A.2 yxy FSM φ= Equation A.3 8 2wLM f = If Equation A.1 is combined with the deflection limit of L/360, and combined with Equation A.3, the following criteria can be developed for member selection: Equation A.4 E wLIrequired 16 75 3 = Equation A.5 y required F wLS φ8 2 = Equation A.4 is a serviceability limit, and therefore should be used with a distributed load of mkNmkPaw /52.56)9)(28.6( == . Likewise, Equation A.5 is a strength limit and should be used with a distributed load of mkNmkPaw /76)9)(45.8( == . Carrying these calculations out yields the following values: 46 3 10*966)9^10*200(16 )9)(/1000*52.56(75 mm Pa mmNIrequired == 33 2 10*2478)6^10*345)(9.0(8 )9)(/1000*76( mm Pa mmNSrequired == 147 Taking a list of all CISC wideflange sections, all members with Ix less than the limit above are removed; the remaining sections are found to automatically satisfy the yield modulus Sx requirement. The members are then sorted by mass. Table A.3 below shows the eight lightest members; a W690x125 and a W610x125 are both found to be acceptable. Table A.3 Members Satisfying Sx and Ix Limits Sorted by Mass Section (kg/m) Ix (106 mm4) Sx (103 mm3) Zx (103 mm3) W690x152 152 1510 4380 5000 W530x150 150 1010 3710 4150 W760x147 147 1660 4410 5100 W690x140 140 1360 3980 4550 W610x140 140 1120 3630 4150 W760x134 134 1500 4010 4630 W690x125 125 1190 3500 4010 W610x125 125 985 3220 3670 Because it has greater strength and the same mass a W690x125 member is chosen as the East- West gravity load carrying beam. This member has a midspan deflection of 21 mm at the service loading. The North-South beam will be designed. This section will summarize the design for the North- South beam. Since 1-way slab action is assumed, this beam will only need to carry loads to the columns across a tributary width of 2.25 meters. The criterion for selecting this beam is that the deflections are continuous with the joists. A beam will be selected with a similar equivalent moment of inertia. According to the VicWest table, 92% of the service load of 14 kN/m gives a deflection of 25mm, therefore: 148 46 33 10*220)9^10*200(16 )9)(/1000*14*92.0(75 16 75 mm Pa mmN E wIeffective === l A W460x60 has Ix=255*106 mm4. This is the lightest member with Ix>Ieffective, so it is acceptable. For the Roof an additional dead load of 1.67 kPa will be used, conservatively the same ADL as the floor. According to the NBCC 2005 section [4.1.6] the snow load can be estimated from values contained in the BC Building Code (given below). kPaSCCCCSIS raSWbSS 64.1]2.0)8.0(8.1[0.1])([ =+=+= Is=1.0 (Normal importance building) Sr=0.2 Ss=1.8 Cb=0.8 (basic roof snow load factor, standard value is 0.8) Cw=1.0 (Conservative, wind exposure factor) Cs=1.0 (roof is flat, roof slope factor) Therefore the loads are as follows: D = 1.67 + 0.0718 (joists) = 1.74 kPa L = 1.0 kPa (NBCC table 4.1.5.3) S = 1.64 kPa Relevant Load Combinations 1.4 D = 2.45 kPa 1.25 D + 1.55 L + 0.5 S = 4.5 kPa 149 1.25 D + 0.5 L + 1.5 S = 5.135 kPa  Governs The roof is designed using a corrugated metal deck supplied by the VicWest corporation. Using design tables provided by VicWest, a roof deck is selected as 0.91 mm thick metal, with 8.9 kPa factored load carrying capacity, 76 mm deep corrugations and a mass per area of 14.68 kg/m2. This will result in an area dead load of 0.144 kPa in addition to the other loads. Including this load, the factored roof load is 5.32 kPa, which is less than the floor factored load of 8.45 kPa. Since the floor beam system will be stronger, the same beams/joists will be used on the roof level as on the floor levels. The Column will be designed. Table A.4 is a summary of the gravity loads on the building. Per NBCC 2005 clause [4.1.5.9] part C the equation for the live load reduction is B/8.93.0 + . The columns have a tributary length of 9 meters in both directions for interior columns, so a tributary area of 81 m2 applies. The column loads for three load combinations in the NBCC are calculated in Table A.5. The greatest column loads are found to result from Combination #2; the lower column will be designed for a load of 2582 kN whilst the upper column will be designed for a load of 1512 kN. Because the Moment Frame is considered to be a “sway” frame, the lower column in this system will be designed with a value of K=2.0, the upper column will be designed for K=1.2. All of the braced frames will have columns designed for K=1.0 (upper and lower), because these systems are considered “non-sway” frames. As a result, the columns for the moment frame will be larger than the braced frame columns. Columns are otherwise designed for the loads shown in Table A.5. 150 Table A.4 Gravity Load Summary ROOF FLOOR type Load (kPa) type Load (kPa) DL 1.88 DL 3.88 LL 1 LL 2.4 SL 1.64 SL 0 Table A.5 Column Loading for Gravity System (interior columns) Story Tributary Area (m^2) B B/8.93.0 + Comb#1 (kN) 1.25DL+1.5SL+0.5LL Comb#2 (kN) 1.25DL+0.5SL+1.5LL Comb#3(kN) 1.4DL 5 81 0 1 430 378 213 4 81 81 0.648 886 960 653 3 81 162 0.546 1332 1512 1093 2 81 243 0.501 1773 2051 1533 1 81 324 0.474 2212 2582 1973 Summary of loads: Roof: • Dead Load = 1.67 kPa (ADL) + 0.072 kPa (joist load) + 0.14 kPa (roof weight) = 1.89 kPa • Roof Dead Load = 1.88 kPa • Roof Live Load = 1.0 kPa • Roof Snow Load = 1.64 kPa Floor: • Dead Load = 1.67 kPa (ADL) + 0.072 kPa (joist equivalent area load) + 2.14 kPa (slab self-weight) = 3.88 kPa • Floor Dead Load = 3.88 kPa 151 • Floor Live Load = 2.4 kPa Summary of gravity design: Roof Deck: • 0.91 mm thick steel corrugated deck, 0.14 kPa self-weight • 14.68 kg/m^2 • 76 mm deep Floor Slab: • 141 mm thick concrete, 2.14 kPa self-weight • 0.91 mm thick steel sheet HB308 Z275 galvanized steel Beams: • W690x125 beams spanning the East-West, 1-way slab carried the load to these beams • W460x60 beams spanning the short direction (North-South). This beam was chosen based on having a similar Ix as the Ieffective of the joists. Open Web Steel Joist: • 750 mm deep, 22.5 kg/m2 • 3 per bay spaced at 2250 mm • joists span into W690x125 beams Columns: • MRF Upper Column is a W310x79 section; MRF Lower Column is a W310x179 section • For other frames the Upper Column is a W250x67 section; the Lower Column is a W250x101 section columns are spliced at the 3rd floor, mid height 152 A.2 Type D Moment Resisting Frame (MRF) S16-09 [27.2] Rd=5.0 Ro=1.5 The Moment Resisting Frame will be designed according to the provisions of S16-09 section [27.2]; the Ductile System has the parameters Rd=5.0 and Ro=1.5. The design of the moment frame will follow these steps: 1. Define the Loading on the MRF 2. Design beams and columns to satisfy code drift requirements 3. Check and enforce the Strong-Column Weak-Beam condition to promote beam hinging instead of column hinge formation 4. Check members for strength The first step in the design is to determine the code required Seismic loading. The Type D Moment Resisting Frame (MRF) is designed according to the CSA S16-09 and NBCC 2005 equivalent static load procedure. Evaluation of the Seismic Loading requires an estimate of the fundamental building period. In the NBCC 2005 section [4.1.8.11 3] i) gives an estimate of the fundamental period of the MRF: Equation A.6 77.0)85.18(085.0)(085.0 4/34/3 === na hT seconds Table A.6 Vancouver Spectral Data aS (0.2) 0.94 aS (0.5) 0.64 aS (1.0) 0.33 aS (2.0) 0.17 153 Table A.7 Design Parameters Ta (sec) 0.77 RdR0 7.5 Mv 1 IE 1 Soil Class C Fv 1.0 Fa 1.0 Building Wt. (kN) 55147 Figure A.3 Design Spectrum for Downtown Vancouver BC, Soil Class C The spectral acceleration information for downtown Vancouver is given by Table A.6, the design spectrum is shown in Figure A.3. Other design parameters are given in Table A.7. Higher mode factor Mv is taken from NBCC 2005 table [4.1.8.11]. For Soil Class C, Fv= Fa = 1.0 for all values of Sa. The details of the building weight calculation are given in Table A.8 and Table A.9. According to NBCC 2005, the Seismic Base Shear Vbase is given an upper and lower limit; maxmin VVV Base << 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 0 1 2 3 4 5 S a [g ] Period [sec] 154 Table A.8 Building Component Weights Element wt/m (kN/m) wt/m2 (kPa) Qty/floor Area (m2) W690x125 1.23 42 W460x60 0.59 40 W310x179 1.75 48 W310x79 0.77 48 Floor Slab 2.14 2835 Roof Deck 0.144 2835 ADL Floor 1.67 2835 ADL Roof 1.67 2835 OWSJ 0.0719 2835 SNOW 1.64 2835 Beam Length (m) 9 Column Length (m) 3.65 Floor 1 Column Length (m) 4.25 Table A.9 Sum of Building Element Weights per Floor FLOOR BEAM WEIGHT (kN) Column Weight (kN) ADL Weight (kN) Slab/Deck Weight (kN) 25% Snow Weight (kN) Total Weight (kN) 5 674.73 136 4938 408 1162 7319 4 674.73 136 4938 6067 11816 3 674.73 307 4938 6067 11987 2 674.73 307 4938 6067 11987 1 674.73 358 4938 6067 12038 SUM-> 55147 Steel Wt. 4617 Steel % 8.4% Before the base shear can be calculated, the spectral acceleration at the building period, 0.77 seconds, needs to be determined. S(T=0.5) = 0.64 since Fv= Fa = 1.0 S(T=1.0)=0.33 since Fv= 1.0 155 Therefore, interpolating between these two values gives S(0.77) = 0.47. The base shear can be calculated using the equations of NBCC 2005 clause [4.1.5.11]. Note that in this case “W” is the structural weight, 4617 kN. Equation A.7 kN RR WIMSF RR WIMSV d EvaV d Ev 1250 5.7 )55147)(17.0()2.0()0.2( 00 min ==== Equation A.8 kN RR WIMTSV d Eva 3480 5.7 )55147)(47.0()( 0 === Equation A.9 kN RR WIMSV d Ev 4608 5.7 )55147)(94.0)(3/2()2.0()3/2( 0 max === Therefore, Vbase=3480 kN The shear is distributed to each floor according to Equation A.10. The variable hx is the floor height, Wx is the seismic weight associated with the floor and Ft is equal to 0.07TaV but need not exceed 0.025V and may be considered as zero when the fundamental lateral period does not exceed 0.7 seconds; for this frame Ft=187 kN. Table A.10 shows the result of carrying out the distribution of shear to each level. Note that the column “shear per node” is simply the nodal lateral shear force applied to the elastic frame model, accounting for eccentricity of the frame. Equation A.10 ∑ = − = n i ii xxt x hW hWFVF 1 )( 156 Table A.10 Calculation of Floor Shears Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shear (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.1*Vn 5 18.85 7319 601877 454251828 942 94 4 15.2 11816 601877 591312048 982 98 3 11.55 11987 601877 455848570 757 75 2 7.9 11987 601877 311792528 518 51 1 4.25 12038 601877 168443422 279 27 NBCC 2005 [4.1.8.11] requires torsional effects to be accounted for. According to this clause if B < 1.7 forces may be applied with a 10% eccentricity, otherwise dynamic analysis is required. Bx is defined as the ratio of the maximum story displacement at the extreme diaphragm point to the average story displacement of the diaphragm at a given floor; B is the maximum value of Bx across all floors. However, since the structure is symmetrical with a perimeter moment frame there should be no issues with amplifications of diaphragm displacements. Using the principle that in this case the diaphragm deflections are directly related to the force in the moment resisting bays, B can be expressed as the ratio of the maximum force in the bay with torsion considered to the force in the brace bay without torsion considered. Figure A.4 Eccentrically Applied Seismic Shear Forces 157 Consider Figure A.4, in this figure the 10% eccentricity is accounted for by simply increasing the seismic forces on one side. This can be proven to be statically correct; summing moments: 0 0)1.0()5.0(4.0)5.0(6.0 =−− DVDVDV The force on one moment resisting frame without eccentricity is Vbase/2=0.5Vbase; the force in one bay with eccentricity is 0.6*Vbase. Thus the Ratio B is: Equation A.11 7.12.1 5.0 6.0 <== V VB Since B<1.7 this approach is acceptable. Therefore accidental eccentricity may be accounted for in the elastic model by simply increasing the forces on the frame by 10%, as shown in Figure A.4. Now that the Seismic lateral loads have been calculated, the factored gravity loads must be calculated. Beams in seismic force resisting bays on the perimeter of the building have a tributary length of 4.5 meters. For the seismic loading case, the load combination is: 1.0DL + 1.0 E + 0.5LL + 0.25 SL Roof Loading: DL = 1.67 (ADL) + 0.0718 (joists) +0.144 (deck) = 1.89 kPa 158 LL = 1.0 kPa SL = 1.64 kPa Factoring these loads according to the load combination yields 2.8 kPa; using a tributary length of 4.5 meters gives 2.8*4.5=12.6 kN/m Floor Loading DL = 1.67 (ADL) + 0.0718 (joists) + 2.14 (slab) = 3.88 kPa LL = 2.4 kPa Factoring these loads and multiplying by a tributary length of 4.5 meters gives 22.86 kN/m In summary, the moment frame beams on the roof will be subjected to a distributed gravity load of 12.6 kN/m, while the beams at other floors will have a gravity load of 22.86 kN/m. An elastic model of the moment frame was created in Dr. Frame. The Dr. Frame software package, developed at the University of Washington, is a 2D structural analysis program. Figure A.5 is from Dr. Frame, with the Seismic and Gravity loads applied. This model includes 2nd Order effects; the elastic model is used to check deflections and determine forces for the member strength check. 159 Figure A.5 Elastic Model of Moment Frame NBCC 2005 specifies that the maximum Inter Story Drift ratio (ISD) permissible is 2.5%. Interstory Drift is defined as the story drift divided by the story height. Members were first selected for strength, and then increased in size through several iterations until deflections were within limits. Equation A.12 may be used with a spreadsheet to quickly calculate story deflections. Since the applied seismic shear forces were reduced by RdRo the drifts should be multiplied by RdR0 in order to get realistic values. Story drifts calculated using the elastic model and the Equation A.12 (under ‘Hand calcs’) are shown in Table A.11. Equation A.12             Σ + Σ =∆ b b i c ii i L I h IE hV 11 12 160 Table A.11 Deflections of Type D MRF Dr. Frame Story Drifts RdR0 7.5 'Hand Calc' Comparison Story Story Deflection (mm) Drift*RdR0 (mm) ISD (%) ISD (%) %Diff 5 40.76 42 1.2% 1.2% -1% 4 35.16 57.375 1.6% 2.4% -51% 3 27.51 64.2 1.8% 1.7% 2% 2 18.95 72.525 2.0% 2.0% -3% 1 9.28 69.6 1.6% 2.3% -40% The MRF design is usually governed by the 2.5% ISD limit, but elastic loads should still be checked against member strengths. Columns are also checked against capacity loads which arise in the event that all the beams form plastic hinges simultaneously (worst case scenario). Plastic shears from the beams were added up at each story, together with the gravity shear, which resulted in considerably higher column axial loads in the exterior columns than given by the elastic analysis. Panel zones are checked for strength, and if necessary doubler plates may be installed. At each beam-column intersection the strong column weak beam requirement is checked and if necessary enforced by increasing the column size. Figure A.6 is a free body diagram of the frame, Figure A.7 is a Free Body Diagram of the columns. Note that the plastic shear resulting from hinge formation in the beams adds up to create a tensile load in one of the exterior columns, a compression load in the other exterior columns, and cancels in the interior columns. The material overstrength factor Ry is taken as 385/345=1.11594 for A992 per S16-09 clause [27.1.7] since Fy=345 Mpa. Equations A-13 through A-17 define some of the terms shown in the figures. 161 Equation A.13 2 2 ' '1 wL L M V pbb −= Equation A.14 2 2 ' '2 wL L M V pbb += Equation A.15 22 cb ddx += Equation A.16 xLL 2' −= Equation A.17 yxbypb FZRM 1.1= Figure A.6 Free Body Diagram of a Moment Resisting Frame Source: ©Yang (2011), by permission 162 Figure A.7 Free Body Diagram of Exterior Columns Source: ©Yang (2011), by permission A W840x193 wideflange member was used for the first level beam. Note that this section was chosen because it is acceptable for use with the bolted stiffened end plate connection, which is pre-qualified for use in a Type D MRF. A typical connection of this type is shown in Figure A.8. It is preferable to use a prequalified connection because otherwise testing is required to prove acceptable performance. Because all of the pre-qualified connection types have requirements, the engineer must be mindful of the types that can be used with selected members. Figure A.8 Bolted Stiffened End Plate Connection 163 The columns are designed for the axial loads assuming that all beams have formed plastic hinges at either end. From Figure A.7 it can be seen that the shears resulting from plastic hinge formation cancel for interior columns and add for exterior columns. Depending on the direction of the plastic shear one exterior column will experience upward plastic shears loading it in tension, and the other will experience downward plastic shears loading it in compression. The column axial loads are summarized in Equation A.18 to A20. Columns have considerable overstrength based on these loads and size is ultimately driven by strong-column weak-beam requirements. Equation A.18 [ ]2/2 LwVC iibf +∑= (exterior column, compression) Equation A.19 [ ]2/1 LwVT iibf −∑= (exterior column, tension) Equation A.20 LwC if ∑= (interior column) Clause [27.2.3.1] (b) of S16-09 requires that if IEFaSa(1.0)>0.3, then the following limit of Equation A.21 shall apply to the column axial load. Since IEFaSa(1.0)=0.33, this clause applies. This is intended to preserve column ductility by limiting the axial load; this requirement is based on tests that show member ductility is decreased by significant axial loading. Equation A.21 ygf FAC 3.0< 164 Strong Column-Weak Beam philosophy is mandatory per Canadian code, in order to promote beam hinging in preference over column hinging. Beam plastic hinges are assumed to form at half of the beam depth away from the column face. CSA S16-09 Clause [27.2.3.2] provides equations to ensure strong-column weak-beam behavior, given as Equation A.22 and Equation A.23. Both equations are derived by summing moments about a free body diagram of a panel zone and accounting for the reduced moment capacity of the column based on its axial load. Equation A.22 ∑∑             ++≥ 2 1.1' chpbyrc d xVMRM Equation A.23 pc y f pcrc MC c MM φφφ ≤      −= 118.1' Panel Zones must be checked for strength, and if inadequate a doubler plate thickness must be specified. Because steel detailing is outside of the scope of this report, this topic will not be elaborated on further. Clause [27.2.4.3] requires that when 55.0)2.0( ≥aaE SFI (in this case 94.0)2.0( =aaE SFI ) the panel zone must conform to the ratios of Equation A.24. Equation A.24 90≤+ Z ZZ t wd 165 A.3 Type MD Moment Resisting Frame (MRF) S16-09 [27.3] Rd=3.5 Ro=1.5 The Type MD or Moderately Ductile Moment Resisting Frame is similar to the Type Ductile MRF except for a few relaxations of the code, these clauses do not affect the Type D MRF. In addition, the RdRo factor is smaller. This has an interesting effect, since although the design forces on the frame are higher, the story drifts are then multiplied by the correspondingly (smaller) value of RdRo. Since the Moment Resisting Frame is fundamentally controlled by drift limits and NOT by strength, the same members from the Ductile MRF satisfied the criteria for the Moderately Ductile MRF. The NBCC allows the drifts to be recalculated based on the period of the computer model; it is possible that using this code provision iteration of the design may potentially have yielded different member sizes for the less ductile moment resisting frame. This clause was not used, and all code lateral forces were calculated using Equation A.6 from the NBCC. A check of member strength against the elastic loads showed that all members were acceptable. Although the overstrength of the beams and columns was somewhat less, the story drifts were still relatively close to the code limits. Table A.12 shows the drifts obtained from the elastic model analysis. Because the interstory drifts are within the code tolerance of 2.5%, the Type D frame also satisfies the clause for a Moderately Ductile MRF. Table A.12 Deflections of Type MD MRF Frame Model Story Drifts RdR0 5.25 Story Story Deflection (mm) Drift*RdR0 (mm) ISD (%) 5 58.31 42.3675 1.2% 4 50.24 57.4875 1.6% 3 39.29 64.155 1.8% 2 27.07 72.66 2.0% 1 13.23 69.4575 1.6% 166 A.4 Type LD Moment Resisting Frame (MRF) S16-09 [27.4] Rd=2.0 Ro=1.3 The Type LD or Limited Ductility Moment Resisting Frame is the same as the Type MD MRF except for a few minor relaxations of some code clauses which do not concern the design in this report. Similar to the Type MD MRF, the RdRo is lower which results in higher design forces; it is also a smaller multiplier for the story drifts. Table A.13 includes the story deflections from the elastic Type LD MRF Model, they are found to be within code limits. A check of member strength shows that all members are acceptable. Because the deflections are within the code tolerance, the type Ductile MRF also satisfies the clauses of the Type LD MRF. Table A.13 Dr. Frame Deflections of Type LD MRF Dr. Frame Story Drifts RdR0 2.60 Story Story Deflection (mm) Drift*RdR0 (mm) ISD (%) 5 117.99 42.848 1.2% 4 101.51 57.564 1.6% 3 79.37 64.194 1.8% 2 54.68 72.774 2.0% 1 26.69 69.394 1.6% A.5 Type MD Concentrically Braced Frame (CBF) S16-09 [27.5] Rd=3.0 Ro=1.3 The design of the CBF is the most straightforward of the braced frames, but the price paid is that it is the least ductile. Because of the requirement to design the beams for the most severe combination of buckled and unbuckled brace forces, they are subject to a considerably severe bending moment as well as a high axial load. The result is relatively large beam sections. Braces will be square Hollow Structural Sections, or HSS. The Design strategy will involve the following Steps: 167 1. Determine the Seismic and Gravity Loads 2. Create an elastic model 3. Size braces for strength using the axial loading from the elastic model 4. Size beams for gravity loads and the most severe combination of brace forces 5. Size the column for the capacity loading based on the brace forces 6. Check drifts to make sure they are within code limits The period is estimated according to the Equation for braced frames given in NBCC 2005, shown in Equation A.25. The details of calculation of the seismic shear forces will not be repeated, but the summary is presented in Table A.14. Again, note that the column labeled “shear per node” contains the forces applied to each node of the elastic frame model accounting for 10% eccentricity. Equation A.25 47.085.18*025.0025.0 === na hT seconds Table A.14 Lateral Seismic Forces on CBF Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shea (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.1*Vn 5 18.85 7299 597908 1208844838 2021 202 4 15.2 11795 597908 1575276851 2634 263 3 11.55 11853 597908 1202927449 2011 201 2 7.9 11853 597908 822781545 1376 137 1 4.25 11882 597908 443700142 742 74 Figure A.9 shows the elastic model of the CBF together with the moment and shear diagrams. Beams are simply supported, and the column bases do not resist rotation. Table A.15 contains the 168 brace tensile and compression loads taken from the elastic model; these are used to design the HSS braces. Braces are considered pinned on both ends with K=1.0. Figure A.9 Elastic Model of the CBF Table A.15 Axial Brace Loads from Elastic Model Story Tu (Tensile) (kN) Cu (compression) (kN) 5 207 321 4 500 697 3 757 959 2 928 1129 1 1108 1332 CSA S16-09 clause [27.5.2.4] specifies that the beam must be designed for gravity loads assuming there is no vertical support provided by the braces, and also for the most severe loading case in the event that the compression brace buckles. Figure A.10 is a free body diagram of the beam, applied distributed gravity loading as well as brace forces are shown. Brace forces Tu and 169 Cu (from tension and compression braces, respectively) exert a net vertical downward force Qb and net axial loads +/- Paxial on the beam; half of the beam is loaded in compression and the other is loaded in tension. In the event of brace buckling, the force Tu will be much larger than Cu; this will give rise to an unbalanced force Qb. The force Qb will then result in a bending moment midspan of the beam. The beams will be designed based on the capacity loads from the braces, these will be much larger than the elastic loads. Per S16-09 clause [27.5.3.4] the probable brace tensile resistance shall be taken as Equation A.26. The probable (not buckled) compression brace resistance shall be taken as Equation A.27; Cr in this equation shall be calculated with RyFy. The post-buckling compression resistance of the brace shall be taken as Equation A.28; rC Cr calculated with RyFy. Cr is the compression strength of members subjected to elastic local buckling and is given in S16-09 clause [13.3.5]. Equation A.26 yygu FRAT = Equation A.27 )/2.1,min( φryygu CFRAC = Equation A.28 )/,2.0min( ' φryygu CFRAC = 170 Figure A.10 Free Body Diagram of the Beam The vertical component of the brace forces (when the brace has buckled) can be found by using trigonometry: Equation A.29 )sin()sin( ' θθ uub CTQ −= The applied moment on the beam will be the sum of the gravity moment assuming simple supports and the effect of Qb at midspan, this is given by Equation A.30: Equation A.30 48 2 LQwLM bf += The shear that the beam must be able to resist can be found from mechanics as Equation A.31: 171 Equation A.31 22 b f QwLV += This approach assumes that both sides of the beam (to the left and right of the brace connection) take an equal share of the force; this leads to a beam of which half is in tension and half in compression. The axial load in either half of the beam when the brace buckles is given by Equation A.32: Equation A.32 2 1)cos(][ ' θuuf CTP += There are essentially two cases which now must be considered. When the brace buckles, the bending moment in the beam is as given in Equation A.30 and the axial load is as given in Equation A.32. When the brace has not buckled but develops its maximum compressive strength as given in Equation A.27, the resultant axial load must be combined with the bending moment arising from gravity loads only. Therefore, the interaction of loads given by Equation A.30 and A-32 must be checked. The interaction of loads given by Equation A.33 and Equation A.34 must also be checked separately for the case of the compression brace not buckled. Equation A.33 2 1)cos(][ θuuf CTP += Equation A.34 8 2wLM f = 172 The columns will be designed for capacity forces assuming that the most severe loads are developed in the braces at every level simultaneously, even though this is unlikely. Code clause [27.5.3.4] gives probable tension forces for the tension brace, and probable forces for the compression brace in both its unbuckled and buckled condition. In order to provide a conservative design, the vertical component of the maximum unbuckled compression brace forces will be added at each level to determine the column compression load. Likewise, the brace tension forces will be added up level by level to determine the column tensile load. Figure A.11 is a free body diagram illustrating the capacity column loads. Equation A.35 may be used to calculate the column axial compression load; Equation A.36 is a similar equation for the column capacity tensile load. Note that in these equations the gravity load wgL includes the load from the adjacent gravity bay. S16-09 clause [27.5.5.2] specifies that columns should be designed for an additional moment in the direction of the braced bay of 0.2ZxFy in combination with other loads. The columns are checked for axial loads in addition to the bending moment of 0.2ZxFy. Equation A.35 [ ]LwCC guif +∑= θsin Equation A.36 [ ]LwTT guif −∑= θsin 173 Figure A.11 Free Body Diagram Showing Column Loads Interstory drift limits are not expected to govern, because this is a braced frame. Drifts are checked in Table A.16, and indeed they do not govern. Table A.16 Deflections of Type MD CBF Frame Story Drifts RdR0 3.90 Story Story Deflection (mm) Drift*RdR0 (mm) ISD Ratio (%) 5 27.36 12.44 0.3% 4 24.17 20.28 0.6% 3 18.97 25.62 0.7% 2 12.4 25.08 0.7% 1 5.97 23.28 0.5% 174 A.6 Type LD Concentrically Braced Frame (CBFLD) S16-09 [27.6] Rd=2.0 Ro=1.3 The main difference with the Type LD Concentrically braced frame is that the beams need not be designed in order to support the buckled compression brace. Otherwise, the design of this frame is a simplification of the Type MD Concentrically braced frame. Drifts are shown in Table A.17, but are well within limits. Table A.17 Deflections of Type LD CBF Frame Story Drifts RdR0 2.60 Story Story Deflection (mm) Drift*RdR0 (mm) ISD Ratio (%) 5 34.03 10.97 0.3% 4 29.81 17.81 0.5% 3 22.96 18.82 0.5% 2 15.72 20.41 0.6% 1 7.87 20.46 0.5% A.7 Type D Eccentrically Braced Frame (EBF) S16-09 [27.7] Rd=4.0 Ro=1.5 The Eccentrically Braced Frame is considerably more difficult to design than the previous systems. This is mainly as a result of having to determine the forces transmitted from the link to the beams, braces and columns. The design approach will be as follows: 1. Define Lateral seismic forces 2. Choose a link length. For this design a shear link will be used. 3. Calculate link shears. This can be done using a relationship derived from a Free Body Diagram or from an elastic model. 4. Based on the link shear, choose a link member. 175 5. Using capacity design, determine the forces transmitted by the link to the brace, beam outside the link and the column. Use these loads to design the surrounding frame. 6. From the elastic model check the drifts and the link rotations to make sure they are within code limits. Depending on the link length, the inelastic yielding of the link will be governed by shear or flexural behavior. The code does not express preference for one over the other, however literature strongly recommends that shear links be used because of their overall superior ductility. The shear resistance of the link is taken nominally as Vp=0.55wdFy, note that this is different and smaller than the static shear strength of the link given by clause [13.4]. Literature recommends that the link length be under the limit given in Equation A.37 in order to guarantee good inelastic shear behavior (e is the link length and Mp=ZxFy is the plastic moment of the section). This limit is in order to ensure good performance of a shear link. Equation A.37 P P V M e 3.1≤ Bruneau et al. (2011) recommends an upper limit for shear behavior, above which the link will undergo flexural and shear behavior, this is given in Equation A.38. Code provision [27.7.4] (limits on Link Length) is essentially identical to this limit except that it is adjusted for axial and shear interaction effects. 176 Equation A.38 P P V M e 6.1≤ Clause [27.7.4] requires that the link length shall not be less than the depth of the link beam. This clause requires that when 15.0)/( >yf AFP (where Pf is the axial load in tension or compression) the link length shall be according to Equation A.39. Since the axial load is very small for the chevron configuration, the limits of this clause do not apply. Equation A.39 (a) if f f g w P V A A 3.0≥ then                 −≤ p p g w f f V M A A V P e 6.15.015.1 (b) if f fw P V A A 3.0< then p p V M e 6.1≤ The design of this system was iterated several times, and the link length was set as 1 meter long on every story. This link length was found to satisfy Equation A.37 in every case; therefore all of the link beams for this design will undergo shear straining during an earthquake. The NBCC building period equation gives Ta=0.47 seconds. Table A.18 summarizes the calculation of the seismic lateral shear forces, with RdRo=6.0. 177 Table A.18 Seismic Lateral Loads on EBF Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shear (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.1*Vn 5 18.85 7299 597908 785749144 1314 131 4 15.2 11795 597908 1023929953 1712 171 3 11.55 11853 597908 781902842 1307 130 2 7.9 11853 597908 534808004 894 89 1 4.25 11882 597908 288405092 482 48 The link shears may be obtained two ways. For preliminary design they may be calculated according to a simple relationship given by Equation A.40. Figure A.12 is a free body diagram of half of a segment of the frame. Assuming the link resists 100% of the story shear and summing moments about Figure A.12 gives Equation A.40. In this equation Vi is the total story shear, h is the story height and L=9 meters. Alternatively the shear can be taken directly from the elastic model, this approach was used for this report. Figure A.13 is the elastic model with the loads applied. Table A.19 shows the result of a comparison of either method for calculation of link shear force. The differences are small. Figure A.12 Free Body Diagram for Link and Frame Shear 178 Equation A.40 L hVV iu = Table A.19 Comparison of Link Shear Calculations Floor h (m) Vi (kN) Vi*h/L (kN) Vi from Elastic Model % Difference 5 3.65 263 107 116 -9% 4 3.65 605 245 255 -4% 3 3.65 867 352 365 -4% 2 3.65 1046 424 436 -3% 1 4.25 1142 539 554 -3% Figure A.13 Eccentric Braced Frame Elastic Model A wideflange member is used as the link beam and the beam/link is one continuous member over the entire span. Clause [27.7.3] of S16-09 states that the resistance of the link shall be taken as Equation A.41. Equation A.42 is the nominal shear resistance of a wideflange, and Equation A.43 is a reduction of shear resistance taking into account axial/shear interaction effects. Equation A.44 is the reduced moment capacity of the link in the presence of an axial load. 179 Because of the configuration of this frame, the axial load will be negligible. In these equations Pf is taken as the greater of the tensile/compressive axial load. Equation A.41 )2,min( '' pp MV φφ Equation A.42 yp wdFV 55.0= Equation A.43 2 ' 1         −= y f pp AF P VV Equation A.44 p y f pp MAF P MM ≤         −= 118.1' There are two approaches to estimate the forces in the brace and in the beam outside of the link when the link has developed its maximum shear force. The first and most straightforward approach is to draw a free body diagram and estimate the forces from equations derived from mechanics. The second approach is the elastic force amplification method. Figure A.14 Free Body Diagram of One Half of a Brace/Beam Assembly 180 Since the link bends in reverse curvature and assuming the inflection point is midspan of the link, the moment in the link can be calculated per Equation A.45. The gravity loading added to the link moment is based on a beam fully fixed at either end. Assuming that the beam and brace are fixed against rotations at either end, the moment transferred from the link to either member can be estimated based on their relative stiffness’s. Equation A.46 is the moment in the beam transferred from the link plus the component contributed from gravity loading. The gravity moment contribution is based on the fixed end moment of a beam fixed at one end and simply supported at the other. Similarly, Equation A.47 is the moment transferred to the brace. Half of the beam end moment is carried over to the other end, thus beam shear can be calculated by Equation A.48. Consider the free body diagram of the joint between the link, beam and brace in Figure A.14. The link and beam shear is primarily balanced by the vertical component of the brace axial force. The brace shear force is very small and can be neglected. With these considerations, the brace axial load can be calculated per Equation A.49. Finally, the horizontal component of the brace axial force needs to be balanced by the axial force in the beam; therefore the beam axial load can be calculated per Equation A.50. The loads in the brace and beam when the link is fully loaded can be estimated using Equation A.45 through Equation A.50 when the link shear is as given by Equation A.42. Equation A.45 122 2ew VeM glinklink +      = Equation A.46 8// / 2beamg link bracebracebeambeam beambeam beam Lw M LILI LIM +      + = Equation A.47 link bracebracebeambeam bracebrace brace MLILI LIM       + = // / 181 Equation A.48 2 5.1 beamg beam beam beam Lw L MV += Equation A.49 θsin beamlink brace VVP += Equation A.50 θ θ tan cos beamlinkbracebeam VVPP +== The elastic force amplification method is slightly more complicated and computationally involved, but it accounts for member end connectivity and removes some of the assumptions implicit to the free body diagram. Table A.20 shows the comparison between brace and beam forces computed via. Equation A.45 – Equation A.50 and forces from the elastic force amplification method. In this table, positive percent differences indicate that the elastic force amplification method gives larger load values; in general this approach is conservative. Essentially the engineer wants to know how much moment, shear and axial load is in the brace and beam when the link is fully loaded. An elastic structural model such as the one shown in Figure A.13 automatically distributes the loads appropriately based on member stiffness and 2nd order effects. In order to determine the capacity loads in the beam and brace, the elastic loads are simply multiplied by the overstrength of the link. The overstrength of the link is calculated according to Equation A.51. In this Equation Vr is calculated per Equation A.41 and Vlink is the link shear from the elastic model. In this way the member loads are calculated exactly. For example, the link shear in the first story taken from the elastic model is 554 kN, the brace compression load is 827 kN. If the overstrength of the link per Equation A.51 is 1.14, then the 182 brace compression load when the link is fully loaded will be 1.14*827=943 kN. Since the loads scale linearly, this approach is correct. Equation A.51 link r V V φϕ /= Clause [27.7.9.2] of S16-09 requires that the beam outside the link be designed for forces corresponding to 1.30Ry times the design loading. When subject to these forces, the beam resistance may be multiplied by φ/yR . This is in recognition that since the beam is being loaded by forces arising from the link (in most cases the same member as the beam) it is unnecessary to include the effects of material overstrength in the loading. Clause [27.7.10.2] requires that the brace design forces also be amplified by a factor of 1.30Ry. In summary, the design forces for the brace and beam outside the link are taken from the elastic model and multiplied by ϕyR30.1 . Table A.21 is a summary of the link loads and their overstrength. Table A.20 Percent Differences the Two Force Estimation Methods Floor 1 2 3 4 5 Average Link Moment 8.5 9.5 8.6 14.9 19.3 12 Beam Moment 1.9 2.7 6.9 16.0 26.1 11 Beam Shear -3.6 9.1 18.5 37.6 56.2 24 Brace Moment 7.8 13.0 12.7 11.6 9.7 11 Brace Axial 1.9 4.2 6.2 11.1 17.7 8 Beam Axial 5.2 6.1 7.8 4.8 -8.0 3 Average +11% 183 Table A.21 Link Properties Link Vf (kN) Vp (kN) e (mm) Overstrength L1 554 568 1000 1.14 L2 436 568 1000 1.45 L3 364 568 1000 1.73 L4 256 568 1000 2.47 L5 116 568 1000 5.44 Figure A.15 is a free body diagram of half of the frame illustrating the forces on the column. The column demand will be calculated by summing the gravity loads at each story, in addition to the force resulting from the strain hardened link. S16-09 clause [27.7.12] states: “Columns shall be designed to resist the cumulative effect of yielding links together with the gravity loads. The link forces for links with wide-flange cross sections shall be taken as 1.15Ry times the nominal strength of the link, except that in the top two stories the force shall be taken as 1.30Ry times the nominal value. Column sections in braced bays shall be Class 1 or 2. Column resistances in the braced bays shall satisfy the requirements of Clause 13.8 assuming an additional bending moment in the direction of the braced bay of 0.2ZFy in combination with the computed bending moments and axial loads. In the top two stories, the additional bending moment shall be taken as equal to 0.4ZFy.” The column loads (tensile as well as compressive) are calculated per the Free Body Diagram of Figure A.15. Equation A.52 and Equation A.53 summarize the column compression and tensile loading calculations. The columns are designed for these axial loads, together with the moments from the elastic model and the additional bending moment required per S16-09. Note that Famplification=1.3 on the top two floors and 1.15 elsewhere, Vr is the link shear strength. 184 Equation A.52       +∑= r y ionamplificatgf V R FLwC φ Equation A.53       +−∑= r y ionamplificatgf V R FLwT φ Figure A.15 Free Body Diagram of EBF Frame Story drifts and link rotations must be checked. S16-09 clause [27.7.5] limits link rotations based on the link length. Figure A.16 illustrates the link rotation requirement; for this design the rotation must satisfy γp<0.08. The S16-09 commentary includes figures explaining the calculation of estimated drift rotation from inter story drift ∆ , the results of this will be explained in this section. Table A.22 shows a summary of the story drifts obtained from the elastic model, they are found to be within limits. 185 Figure A.16 Allowable Drift Rotation as a Function of Link Length Source: ©Yang (2011), by permission Table A.22 Deflections of Type D EBF Frame Story Drifts RdR0 6.00 Story Story Deflection (mm) Drift*RdR0 (mm) ISD Ratio (%) 5 21.2 13.02 0.4% 4 19.03 21.12 0.6% 3 15.51 27 0.7% 2 11.01 29.64 0.8% 1 6.07 36.42 0.9% Equation A.54 gives an estimation of the maximum link rotation based on the plastic rotation of the column θp. The plastic rotation θp is given by Equation A.55 and is based on 3 ∆; the elastic story drifts are multiplied by 3 in order to estimate the maximum inelastic rotation. When these equations are combined, Equation A.56 gives the expected link rotation as a function of the elastic story drifts. This can be used with the results of the elastic analysis in Table A.22 to determine if the link rotations are within the limits of S16-09 clause [27.7.5]. Note that this analysis is only valid for a chevron configuration of the EBF frame. Using these relations the link rotations are calculated in Table A.23. Link rotations are found to be within limits. Equation A.54 p e Lθγ = 186 Equation A.55 s p h ∆ = 3θ Equation A.56       ∆ = she L 3γ Table A.23 Link Rotations Story Story height (mm) Rotation γ (radians) Rotation Limit (radians) 5 3650 0.016 0.08 4 3650 0.026 0.08 3 3650 0.033 0.08 2 3650 0.037 0.08 1 4250 0.039 0.08 Link stiffeners are required in order to increase the ductility of the link in the presence of large inelastic strains. S16-09 contains several requirements for the link beam stiffeners. Because detailing is beyond the scope of this report, these details are omitted. Figure A.17 shows a typical detail showing link stiffeners at the ends and in the mid-section of the link. Figure A.17 EBF Link Detail 187 A.8 Type D Buckling Restrained Braced Frame (BRBF) S16-09 [27.8] Rd=4.0 Ro=1.2 The first step is to contact a manufacturer and obtain product specifications for the Buckling Restrained Braces. This frame will be designed for a Star Seismic PowerCAT brace. These braces are pinned at the ends. This design closely follows the Steel Tips article on design of buckling restrained braced frames by López, et al (2004). Specifications for the brace members were obtained from the report by Merrit et al. (2003), the backbone curve used in design was supplied by Star Seismic. The brace core material was taken to be A36 steel. The nominal yield stress is 250 Mpa, and the overstrength factor is Ry=1.16. The nominal brace force is a function of the area and the nominal yield stress. The design of the BRBF will follow these steps: 1. Define the Seismic and Gravity Loading on the Frame 2. Size braces assuming that they resist 100% of the lateral shear 3. Create a 3D elastic model and apply seismic lateral loads with 10% eccentricity in the North-South as well as the East-West directions; from this analysis determine the maximum expected loads in the braces 4. Using the maximum expected brace loads calculate the maximum expected brace axial deformations 5. Calculate the brace strains that correspond to 2 times the maximum expected brace axial deformations 6. Using an appropriate backbone curve, estimate the strain hardening factors for the buckling restrained braces. Using the strain hardening factors, calculate the maximum possible tensile and compression loads in the braces. 188 7. Using the maximum brace forces, design the beams and columns to resist these loads 8. Check drifts to ensure they are within the code limits The fundamental period is the same as the other braced frames per the code procedure. The only difference is that RdRo=4.8. Table A.24 summarizes the equivalent static loads. Table A.24 Seismic Shear Forces for BRBF Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shear (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.1*Vn Roof 18.85 7299 597908 982186431 1642 164 5 15.2 11795 597908 1279912441 2140 214 4 11.55 11853 597908 977378552 1634 163 3 7.9 11853 597908 668510005 1118 111 2 4.25 11882 597908 360506365 602 60 The braces will be sized based on the assumption that they carry 100% of the lateral shear. According to S16-09 Clause [27.8.3] the nominal resistance of the braces shall be taken Equation A.57. If the brace is oriented θ below the horizontal then the force in the brace can be expressed as Equation A.58. Equation A.57 yscscrr FACT φ== Equation A.58 θcos i u VP = Combining these two equations the nominal required area of the brace can be calculated with Equation A.59. Table A.25 carries out this calculation to determine the minimum allowable 189 brace areas; the chosen design areas are also included. Gravity forces are typically neglected in the design of the braces according to Bruneau et al. (2011). Equation A.59 θφ cosysc i sci F VA = Table A.25 Brace Core Area Design Floor Fx (kN) (0.6/3)*Fx (kN) Story Shear (kN) Angle Theta (Deg) Angle Theta (rad) Brace Force (kN) Required Brace Area (mm^2) Provided Brace Area (mm^2) % Difference 5 1643 329 329 22.08 0.385 355 1576 3700 -134.8% 4 2141 428 757 22.08 0.385 817 3629 3700 -2.0% 3 1635 327 1084 22.08 0.385 1169 5197 5200 -0.1% 2 1118 224 1307 22.08 0.385 1411 6269 7000 -11.7% 1 603 121 1428 25.28 0.441 1579 7018 7000 0.3% The capacity design will govern the selection of beam and column members. The next step is to determine the maximum brace forces accounting for strain hardened steel in the brace core. In order to find the maximum forces in the brace, Tmax and Cmax, the strain hardening factors must be determined. CSA S16-09 clause [27.8.3.4] states that the probable tensile Tysc and compressive Cysc shall be taken as follows: Equation A.60 yscyscysc FRAT ω= Equation A.61 yscyscysc FRAC βω= Where ω is defined in the code as a strain hardening adjustment factor obtained by dividing the maximum force developed in the buckling restrained brace in the qualification testing in clause 190 [27.8.6], up to a deformation corresponding to 2.0 times the seismic design story drift, by yscysc FRA . The factor β is defined as a friction adjustment factor obtained by dividing the maximum force developed in the buckling restrained brace in the qualification testing specified in clause [27.8.6], up to a deformation corresponding to 2.0 times the seismic design story drifty, by yscysc FRAω . The strain hardening factors ω and β were determined by creating a 3D model of the building in SAP2000, shown Figure A.18. The Seismic story shear forces listed in Table A.24 were applied to the rigid floor diaphragms from the North-South and East-West directions with a 10% eccentricity. The maximum brace forces were then extracted from the 3D model. The brace deformation can then be determined by Equation A.62. In this equation, Pmax is the force from the SAP2000 analysis. The deformation is multiplied by the RdRo factor to get a realistic value of maximum deformation. Because BRB members are typically necked down in the middle, but have larger ends, the actual stiffness of the members is about 1.4x that indicated by the core area; this is reflected in the axial model (stiffness can be increased by 40% by increasing the area by 40%). To reflect the fact that *real* braces are necked down in the middle, but the ends are much thicker steel sections, the effective length of the brace Lysc is taken as 0.66L, or 2/3 the nominal length. Once the axial brace deformation ∆b has been determined, the strain in the brace is calculated according to Equation A.63. Once the brace strains have been calculated, the strain hardening factors can be read off of a backbone curve. Figure A.19 is the backbone curve provided by Star Seismic used to determine the strain hardening factors. Equation A.62 EA RRLP ysc dysc b 0max =∆ 191 Equation A.63 ysc b b L ∆ = 0.2 ε Figure A.18 SAP2000 Model Table A.26 shows the calculations for the braces. Brace forces were extracted from SAP2000 and then used to calculate the brace deformations. These were then used with Equation A.63 to calculate maximum brace strains. Brace strains were used with Equation A.64 and Equation A.65 to calculate the strain hardening factors ω and ω β. Equation A.60 and Equation A.61 were then used to calculate the maximum expected brace tension and compression forces. 192 Figure A.19 Backbone Curve for the PowerCAT Brace Source: ©Star Seismic (2011), by permission Equation A.64 9103.02449.0 += εω Equation A.65 8687.04234.0 −= εωβ Table A.26 Calculation of BRB Tmax and Cmax Forces Floor Aysc (mm^2) Pmax (SAP2000) (kN) Lysc=0.66L (m) ∆b (mm) ysc b b L ∆ = 0.2 ε ω ωβ Tmax (kN) Cmax (kN) 5 3700 309 6.41 12.84 0.40% 1.01 1.04 1082 1114 4 3700 721 6.41 30.00 0.94% 1.14 1.26 1223 1357 3 5200 1019 6.41 30.15 0.94% 1.14 1.27 1720 1911 2 7000 1221 6.41 26.83 0.84% 1.12 1.22 2264 2483 1 7000 1326 6.57 29.85 0.91% 1.13 1.25 2300 2545 When the brace develops its maximum tensile force, the beam will be loaded in compression; the beam must be designed to withstand these forces. Figure A.20 is a free body diagram of a beam- column-brace intersection point where the brace has developed Tmax, its maximum inelastic tensile force. 193 Figure A.20 Free Body Diagram of Diagonal Brace Bay To simplify the analysis, the column shears will be neglected, and Fi will be taken as the sum of story collector forces corresponding to the mechanism under consideration. The force Fi is assumed equal on either side of the frame. From the free body diagram of Figure A.20 Equations A-66 and A-67 may be written. The beams are designed for the axial forces resulting from Equation A.67, as well as the bending moment from gravity loads. Because the beams are moment connected to the columns in this design, the moment resisting frame part of this system was designed to comply with the provisions for a Type MD MRF. Equation A.66 )sin()sin( 11maxmax ++−= iiii TTF ϕϕ Equation A.67 2 )sin( 11max iiiui FTP += ++ ϕ Column axial loads must consider the loads transferred from the braces and the plastic shears resulting from beam hinge formation. A capacity load approach will be used assuming all braces develop their maximum loads together with all beams simultaneously forming plastic hinges. 194 Consider Figure A.21, the maximum column compression load is given by Equation A.68. The maximum column tensile load is given by Equation A.70. These forces are used for the column design. A similar scenario can be considered when the braces are in compression; it was found that this case did not govern. Equation A.68 [ ]2/cosmax2 LwTVC gibAf ++Σ= θ Equation A.69 [ ]2/cosmax1 LwTVT gibAf −+Σ= θ Figure A.21 Column Capacity Forces from Braces Loaded in Tension 195 Figure A.22 is the elastic model, 2nd Order effects included. Clause [27.8.5.2] of S16-09 requires that an additional column moment of 0.2ZxFy is applied in the direction of the braced bay together with computed elastic bending loads. The columns were designed for the axial loads calculated above together with the bending moment 0.2ZxFy and the bending moment from the elastic model. Figure A.22 Elastic Model with Bending Moment Diagram Table A.27 below shows that the deflections for this frame are within code limits. Note that without the moment connected beams (and without the column bases fixed against rotation) this frame still meets code drift limits. The purpose of moment connecting the beams is to provide a backup system in the event of brace fracture, to provide system lateral stiffness while braces are replaced (braces damaged during a seismic event can be easily replaced) and to provide lateral stiffness after a strong seismic event when braces have large residual strains. 196 Table A.27 Deflections of Type D BRBF Frame Story Drifts RdR0 4.80 Story Story Deflection (mm) Drift*RdR0 (mm) ISD Ratio (%) 5 42.67 25.632 0.7% 4 37.33 45.024 1.2% 3 27.95 50.256 1.4% 2 17.48 47.424 1.3% 1 7.6 36.48 0.9% A.9 Type D Steel Plate Shear Wall (SPSW) S16-09 [27.9] Rd=5.0 Ro=1.6 For this design, the vertical columns will be referred to as VBE, or Vertical Bounding Elements. Likewise, the beams will be referred to as HBE, or Horizontal Bounding Elements. This system was designed based on the chapter on steel plate walls in the text by Bruneau et al. (2011) and the AISC Design Guide 20 on steel plate walls. A summary of the member sizes is given in Table A.28. Because the beams are connected to the columns with RBS connections, a summary of the RBS cut details is given in Table A.29. The Design strategy will involve the following Steps: 1. Determine the Seismic and Gravity Loading 2. Design infill plates for 100% of the story shear 3. Do a preliminary analysis to determine initial HBE and VBE members, sizes based on stiffness requirements 4. Capacity design to determine updated HBE members 5. Capacity design to determine updated VBE members 6. Create an elastic model representing the infill plates as a series of tension-only strips 7. Using the elastic model, check the HBE and VBE elements against the elastic loads 197 8. Check deflections 9. Check that the Moment Resisting Frame resists at least 25% of the story shear Table A.28 Member and Plate Sizes Ductile Plate Wall CISC Member Type (A992 Fy=345 Mpa, Ry=1.116) Infill Plate (A36 Fy=248 Mpa, Ry=1.3) Infill Plate thk. (mm) Effective Strip Area (mm2) HBE0 W920x449 First Story Infill 6.350 68 HBE1 W690x323 Second Story Infill 6.350 66 HBE2 W690x323 Third Story Infill 4.763 58 HBE3 W690x323 Fourth Story Infill 2.657 43 HBE4 W690x265 Fifth Story Infill 1.588 33 HBE5 W690x265 Lower VBE W360x1086 Upper VBE W360x667 Table A.29 RBS Connection Details Member a (mm) b (mm) c (mm) %Reduction Flange %Reduction Zx Radius of Cut (mm) Plastic Hinge Length from column centerline (Sh) W690x323 180 470 72 40% 32% 420 657 W690x265 179 459 72 40% 32% 402 650 Note: a, b and c are section cut parameters, these are explained later in this section Once the lateral seismic shears have been calculated, the first and most important step in the design of this system is to determine the infill plate thickness. Clause [27.9.2.1] of S16-09 requires that infill plates be designed for 100% of the applied factored shear force. This clause then gives Equation A.70 for the strength of the plate. This equation may be re-arranged to give the specified plate thickness in terms of the story shear. Derivation of this equation is beyond the scope of this report and is therefore omitted. 198 Equation A.70 )2sin(4.0 αφ wallpyr LtFV = Table A.30 is a list of common thicknesses for A36 steel; for this material, Ry=1.3. One common challenge faced by the designer of a steel plate wall is selection of plates from standard sizes. Using Equation A.70 an “ideal” thickness can be calculated to resist 100% of the story shear, it is up to the engineer to choose the closest plate size from those available from suppliers. Table A.30 Common A36 Plate Sizes in mm 0.0625 1.588 0.0747 1.897 0.1046 2.657 0.125 3.175 0.1345 3.416 0.1875 4.763 0.25 6.350 0.3125 7.938 0.375 9.525 0.4375 11.113 0.5 12.700 0.625 15.875 Although stiffened plate walls have been built, unstiffened plate walls are more economical. When an unstiffened steel plate wall is loaded in shear; the plate yields in one direction, and buckles (in compression) along a perpendicular direction. This led to the development of the strip model for analysis. Figure A.23 shows two strip models, one for cyclic and the other for static analysis. For Figure A.23 a), the strips are given zero compression strength. Other models for behavior of the steel plate have been proposed, but that shown in Figure A.23 is the most commonly used and is the one included in CSA S16-09. The angle α is the angle of inclination of 199 the tension strips, and is given by S16-09 clause [20.4.1]. This clause contains Equation A.71. In Figure A.23 b) note how the angle α is taken relative to the strips. Figure A.23 Strip Model Idealizations for Analysis of a Steel Plate Shear Wall Equation A.71       ++ + = LI h A wh A wL cb c 360 11 2 1 tan 3 4 α The angle of the tension strips is given by the relation in Equation A.71, it is very complicated and tedious to evaluate. Clause [20.4.2] of S16-09 allows a single angle of inclination taken as the average of that given by Equation A.71 at each floor to be used for the entire system when constructing a strip model of a multistory wall. Clause [20.4.2] allows α to be taken conservatively as 40 degrees if the height/length ratio is within the following range: 5.2/6.0 ≤≤ hL The commentary for S16-09 section [20.3] reports that 10 strips per panel have generally been found to be adequate, however depending on the ratio of the height and length of the wall more strips may be needed. When creating the finite element model, the area of the strips can be taken 200 as Equation A.72. Table A.31 includes the “Equivalent Strip Areas” column which is calculated using Equation A.72 and assuming 10 strips per panel. Equation A.72 N whLAS )]sin()cos([ αα + = Particularly at the upper floors, the plate thickness given by Equation A.70 will often be significantly smaller than any readily available size. It is also important to realize that very thin plates will be difficult to weld to the surrounding frame. One solution is to cut circular perforations out of the plate. The result of this is to reduce the “effective” thickness of the plate. Perforated infill plates are covered in clause [27.9.2.3] of S16-09. Another interesting strategy that has been developed to control excessive infill plate overstrength and to mitigate the problems caused by very thin plates is low yield stress steel. This approach has been pioneered in Japan, and Japanese steel mills have developed several different low yield stress steels. If the steel can be counted on to yield at a lower strength, correspondingly thicker plates (which are easier to weld) can be used. Essentially the design approach for the steel plate shear wall is to design the infill plate to resist the lateral shear, and then to design the HBE and VBE to resist the capacity of the infill plates. Testing has shown that design of the frame for strength alone is not sufficient however. Fully strained plates exert considerable “pull-in” on the surrounding beams and columns, and improperly designed HBE and VBE members take on an “hourglass” shape when the plate pulls them in. In order to avoid this, S16-09 provides flexibility limits for the VBE and HBE, particularly severe limits apply to the boundary HBE at the top and bottom of the wall. Clause 201 [20.5] of S16-09 contains these flexibility limits. Clause [20.5.1] requires that the column flexibility parameter hω not exceed 2.5. Equation A.73 25.0 2 7.0       = c p h LI t hω Equation A.74 25.044 4 7.0             += L t I L I h p bc Lω Clause [20.5.2] requires that the beam flexibility parameter Lω : • Not exceed 2.5 at the top panel of the plate wall • Not exceed 2.0 at the bottom panel of the plate wall • Be greater than hω85.0 Clause [27.9.3.2] of S16-09 requires that beams at every story shall have sufficient flexural resistance such that at least 25% of the applied factored shear force is resisted by the HBE and VBE forming a moment resisting frame. Axial loads in beams and gravity load effects on beams may be neglected in this calculation. In lieu of a more complicated analysis, the S16-09 code commentary recommends that the resistance of the moment resisting frame formed by the VBE and HBE at every story be taken as Equation A.75: Equation A.75 h M V pbrMRF 2 = 202 The HBE are loaded heavily in shear, axial compression and in bending. Clause [27.9.3.1] requires all beams to be Class 1.The loading on the HBE comes from the infill plates above and below, from gravity forces and from the surrounding VBE. Figure A.24 shows a typical HBE with an idealized tension strip (enlarged for clarity) acting on it. Referring to the figure, and making use of the trigonometric substitution ααα cossin22sin = , Equation A.76 can be derived for the horizontal distributed load on the HBE. Similarly, Equation A.77 can be derived for the vertical component of the infill plate load on the HBE. Recognizing that the capacity load on the HBE will be the resultant difference between the pull upwards from the plate above and the pull down from the plate below, the distributed loads for the HBE are given by Equation A.78 and Equation A.79. Equation A.76 )2sin( 2 1sinsin/ ασασαωω w dx wds dx PdxHxbH ===== Equation A.77 )(coscoscos/ 2 ασασαωω w dx wds dx PdxVybV ===== Figure A.24 Typical HBE with an Idealized Infill Plate Tension Strip Reference: Bruneau et al. (2011) 203 Equation A.78 αω 21 cos)( +−= pipiyyybi ttFR (Vertical load on HBE) Equation A.79 )2sin()(2 1 1 αω +−= pipiyyxbi ttFR (Horizontal load on HBE) A similar analysis can be done on the VBE. Since the VBE are only subject to an infill plate on one side, the distributed load only depends on the thickness of the adjacent plate. Equation A.80 and Equation A.81 are the loads which the VBE are subject to. Equation A.80 )2sin(2 1 αω piyyyci tFR= (Vertical load on VBE) Equation A.81 αω 2sin2 1 piyyxci tFR= (Horizontal load on VBE) The HBE may be designed for a moment given by Equation A.82. The text by Bruneau et al. (2011) contains a proof showing that if the HBE are designed for twice the moment of Equation A.82, in-span plastic hinges will not form in the beams. If a HBE is designed for twice this moment, plastic hinges will only form at the ends of the HBE. Analysis shows that in-span plastic hinging results in detrimental performance, therefore it is the recommendation of the authors of Bruneau et al. (2011) that the HBE be designed for the moment of Equation A.83. Equation A.82 8 ][ Lw M ybigHBEi ω+ = Equation A.83 4 ][ Lw M ybigHBEi ω+ = 204 Figure A.25 Axial Load Free Body Diagram for HBE Reference: Bruneau et al. (2011) The axial load in the HBE comes from two sources. PHBEVBE is the axial load from the VBE; when the infill plate pulls on both VBE, the HBE acts as “shoring” keeping them apart. Basically, the HBE is pinched between the two VBE, these forces tend to load the HBE in compression. PHBEWEB is the axial load resulting from the horizontal component of the infill plate, this force loads the HBE in tension. Assuming that the VBE delivers forces equally to the top and bottom of each story, the force of the VBE on the HBE can be calculated as Equation A.84; the summation in this equation indicates that the force from the plate above and below the HBE should be added together. The axial load on the HBE from the infill plate is calculated assuming equal collector conditions on either side of the wall. PHBEWEB is calculated per Equation A.85. The force PHBEVBE will usually be the larger of the two forces, and will load the HBE in compression, PHBEWEB loads the HBE in tension and relieves some of the compressive load. In Figure A.25, the HBE is split in half, and the maximum forces are solved via. the free body 205 diagram. The force FiL is in compression, whereas FiR may be a tensile or a compression force depending on the magnitudes of PHBEWEB and PHBEVBE. The design axial loads for the HBE are summarized in Equation A.86 and Equation A.87. Usually for the topmost and bottommost HBE FiR will be a tensile force (indicated mathematically by a negative sign). Equation A.84 α 2sin 2 1 piyyHBEVBEi tFhRP Σ= Equation A.85 xbiHBEWEBi LP ω2 1 = Equation A.86 ][ HBEWEBHBEVBEiL PPF += Equation A.87 ]HBEWEBHBEVBEiR PPF −= Finally, the HBE is heavily loaded in shear. In addition to the shear caused by gravity loading and by plastic hinge formation at the ends of the beam, both components of infill plate loading give rise to shear load. The vertical resultant load from the infill plate gives rise to shear in the HBE; however because the HBE has a finite depth, the axial component of infill shear also gives rise to a shear load on the member. The horizontal infill shear from the plate above and below creates a moment about the HBE centerline, which gives rise to a shear force; essentially the deeper the wideflange member used the more severe this contribution will be. Figure A.26 illustrates the shear loading on the HBE from both components of the infill plate; Equation A.88 is the distributed load used to calculate the HBE shear forces. These forces are given in Equation A.89 and Equation A.90. 206 Equation A.88 )2sin( 2 1* αω piyyybi tFR= Equation A.89 2 L V ybiVi ω = Equation A.90 2 ][ 1** bybiybi hi dV ++= ωω Figure A.26 Shear Load on HBE from Infill Plate Loading Reference: Bruneau et al. (2011) 207 The total shear on the HBE is calculated as Equation A.91 accounting for gravity, both components of infill plate tension and the shear from when a plastic hinge forms in the beam. Note that Sh is the distance from the column centerline to the expected plastic hinge location on the HBE. The HBE are very heavily loaded in shear and this may govern. It is best to avoid very deep wideflange members in order to minimize Vh. Equation A.91 hv gravity h pb HBE VV L SL M V +++ − = 22 2 ω Design of the VBE is somewhat more complicated than the HBE. A literature review will recommend several approaches for calculating the moment and shear demand for the VBE. The most straightforward method is to calculate axial loads with a capacity approach. VBE moments and shears are calculated based on a limited VBE elastic model. The axial loads are calculated per Equation A.92 and Equation A.93 for the compressive and tensile loads, respectively. Clause [27.9.4.1] requires all columns to be Class 1. These equations are derived by simply adding up the contribution of the infill plates, the shear forces as well as the gravity loads at every floor. Obviously this approach is very conservative, since it is unlikely that the full shear forces and infill plate forces will act at every floor simultaneously. There are other approaches for estimating the VBE loads, such as to use a dynamic model, but this is more complicated. Equation A.92       +++ − +Σ= 22 2 Lw VV SL M hC givihi h pbi iyciVBE ω 208 Equation A.93       −−+ − +Σ= 22 2 Lw VV SL M hT givihi h pbi iyciVBE ω The limited VBE model consists of an elastic model like that shown in Figure A.27. The horizontal components of the infill load are applied to each member, ωxci, together with the moments caused by plastic hinge formation at the end of each HBE and the code equivalent lateral forces at each story. Clause [27.9.3.4] allows the moments caused by plastic hinge formation in the beams to be taken as Equation A.94 or Equation A.95, whichever is smaller. In the limited VBE model, the beams are represented by springs with a stiffness given by Equation A.96. Once the model is created, the maximum moments and shears are combined with the axial loads calculated per Equation A.92 and Equation A.93 to check the VBE strength. Equation A.94 yxypb FZRM 1.1= Equation A.95         −= yb f yxypb FA C FZRM φ1)1.1(18.1 Equation A.96 2/L EAk bHBE = 209 Figure A.27 Limited VBE Model for Estimating Moment and Shear on VBE It is clear that this approach will lead to very high axial loads on the VBE, especially as the number of stories increases. In many cases, the axial demand on the VBE’s will be too large for a practical wideflange section to be used. Overall, Steel Plate Shear Walls require very heavy sections for the VBE and HBE. For taller buildings, this approach will become impractical, at which point a dynamic analysis should be employed in order to determine design forces. When large member sizes must be used, built up plate girders are an option. One approach that has 210 been successfully used in the past is to use concrete filled steel tubes for the VBE, rather than to use rolled wideflange sections. In a steel plate wall, the HBE are heavily loaded in bending, compression and in shear; depending on the situation any of these may govern. Due to the heavy loading the engineer will be tempted to use a very large wideflange section for the HBE. Unfortunately, all of the pre- qualified moment connection types have depth, mass and flange width limitations imposed on them. Table A.31 lists the prequalified moment connection types ( S16-09 and AISC 358-10). Flange thickness requirements severely restrict moment connected beams to the smaller members. The Reduced Beam Section (RBS) may be used with members nearly twice as large as any other pre-qualified connection. There are particular advantages for using a RBS connection, some of the advantages are as follows: • The RBS results in smaller end moments, so smaller forces will be transferred to the VBE • The RBS connection accommodates larger wideflange sections. Because the flange thickness limit is much larger, thick and compact sections can be used. • Most of the deep sections with thin flanges (tf<25mm) do not satisfy compression slenderness limits, such members are not ideal for carrying high axial loads as they are vulnerable to local buckling. 211 Table A.31 Pre-Qualified Connection Limitations Maximum Depth Max Flange Thk. (mm) Other Minimum Span/Depth Ratio Prequalified? Bolted Unstiffened W610 or (W760 for Type LD MRF) 19 Type D/MD=7 Type LD=5 CISC Bolted Stiffened W920 25 Type D/MD=7 Type LD=5 CISC Reduced Beam Section (RBS) W920 (max 446 kg/m) 44 b/2t<7.3 with b taken at 2/3 of RBS cut 7 CISC Bolted Flange Plate (BFP) W920 (max 224 kg/m) 25 SMF=9 IMF=7 AISC only Welded Unreinforced Flange Welded Web (WUF-W) W920 (224 kg/m) 25 SMF=7 IMF=5 AISC only Kaiser Bolted Bracket (KBB) W840 (195 kg/m) 25 flange width min 152/250 mm for W/B series SMF/IMF=9 AISC only Figure A.28 Schematic of a Typical RBS Connection 212 A schematic diagram of the RBS connection type is shown in Figure A.28, this figure shows the three different cut parameters a, b and c. The radius of the “dogbone” cut is given by Equation A.97. The plastic hinge is located a distance Sh from the column centerline as given by Equation A.98. The reduced plastic modulus, ZRBS is given by Equation A.99. In these equations subscript c is for column, b is for beam and tbf is the flange thickness. Equation A.100 is the distance between plastic hinges. Equation A.97 c bcR 8 4 22 + = Equation A.98 2/2/ badS ch ++= Equation A.99 ][2 bfbbfxbRBS tdctZZ −−= Equation A.100 hh SLL 2−= Because the reduction in beam strength affects the flexibility of the frame, literature suggests that a 50% reduction in flange area corresponds to a 10% increase in Moment Resisting Frame drift calculated from an elastic analysis. Flange reductions less than 50% may be interpolated to use a proportionately smaller increase in MRF drift. Because a steel plate wall is being designed rather than a Moment Resisting Frame, drifts are not expected to govern. Conservatively, drifts may be multiplied by a factor of 1.1 (and flange reductions kept to less than 50%). For further information regarding details of the RBS prequalified connection design refer to the CISC (2008) document on prequalified connection design in Canada. 213 Because of the extremely high resistance of the infill plates, it was decided to use only two plate walls per direction. From the NBCC Section [4.1.8.11] code equation, the period for shear walls is given as Equation A.101, or 0.45 seconds. From S16-09, RdR0=8.0. A summary of the equivalent static lateral load calculation is given in Table A.32. Equation A.101 45.0)85.18(05.0)(05.0 4/34/3 === na hT seconds Table A.32 Lateral Seismic Forces on SPSW Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shear (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.3*Vn 5 18.85 7299 597908 589311858 985 295 4 15.2 11795 597908 767947465 1284 385 3 11.55 11853 597908 586427131 980 294 2 7.9 11853 597908 401106003 670 201 1 4.25 11882 597908 216303819 361 108 Section [20.4] of S16-09 allows the angle of inclination of the tension strips to be taken as 40 degrees when the ratio of bay length to height is between 0.6 and 2.5, or 0.6<L/h<2.5. Since 9/4.25=2.11 and 9/3.65=2.46, the angle of inclination of strips can be taken as 40 degrees at every story. All design calculations were done in MathCAD, which allowed for easy iteration of member sizes for VBE and HBE. The equation for shear strength of the plate can be re-arranged in order to find the *optimum* plate thickness. Table A.33 shows a summary of the infill thickness calculation. Note that Vi are the cumulative story shears; ie. V1 is the total base shear. Although the overstrength on the fourth floor is slightly less than 1, the designer is limited by the available plate sizes, and the desire to keep overstrength to a minimum so as not to drive up the 214 capacity loads on the HBE and VBE members. The plate material was chosen as A36 steel, both because of its widespread availability but also because of the relatively thin plate sizes it is produced in. For A36 the yield strength is specified as 36 ksi or about 248 Mpa, the overstrength factor Ry=1.3. Table A.33 Infill Plate Thickness Floor Vi (kN) Calculated tp (Eq. A-70) (mm) tprovided (mm) tprovided/tp 5 591 1.22 1.587 1.30 4 1362 2.81 2.657 0.95 3 1950 4.02 4.763 1.19 2 2353 4.29 6.35 1.48 1 2570 5.38 6.35 1.18 Preliminary design of VBE and HBE members was done using the flexibility limits of S16-09 section [20.5]. Members were chosen based on meeting the requirements of Equation A.73 and Equation A.74. The design was iterated several times, final member sizes are shown in Table A.28. The “Strip Area” is calculated per Equation A.72. HBE1-5 are connected using Reduced Beam Section (RBS) connections. The ground level anchor beam (HBE0) is simply supported. Table A.29 contains the parameters for the RBS connection and the associated flange cuts. Elastic loads from the Finite Element Model were found not to govern. An elastic model was created, shown in Figure A.29. This model was also used to check deflections. 215 Figure A.29 Elastic Strip Model of Steel Plate Shear The moment resisting frame must resist at least 25% of the lateral shear. The strength of this system is calculated using the reduced beam section plastic modulus of the HBE, ZRBS. Table A.34 below shows the percentage of the lateral story shear that the VBE and HBE are able to resist when forming a moment resisting frame according to Equation A.75. It is clear that at each story the MRF is capable of resisting greater than 25% of the applied story shear. Table A.34 Check that MRF System Carries at Least 25% of Shear Floor Vi (kN) VrMRF (VrMRF/Vi)*100% 5 591 1208 204% 4 1362 1208 89% 3 1950 1502 77% 2 2353 1502 64% 1 2570 1502 58% 216 Deflections are not expected to govern, but must still be checked. Story drifts shown in Table A.35 are found to be within code limits. Drifts are conservatively increased by 10% in order to account for the flange reductions. Predictably, the increased drifts are still well below the NBCC limit of 2.5%. Table A.35 Deflections of SPSW Frame Story Drifts RdR0 8.00 Story Story Deflection (mm) Drift*RdR0*1.1 (mm) ISD Ratio (%) 5 28.75 46.28 1.3% 4 23.49 60.80 1.7% 3 16.58 48.57 1.3% 2 11.06 47.34 1.3% 1 5.68 49.98 1.2% Note: Deflections increased by 10% to account for RBS connection A.10 Type LD Steel Plate Shear Wall (SPSW) S16-09 [27.10] Rd=2.0 Ro=1.5 The design of the Limited Ductility Steel Plate Shear Wall is essentially a simplification of the design for the Ductile wall, except that the HBE are not moment connected to the VBE. Story drift limits do not govern, they are however shown in Table A.36. Because the force reduction factors are considerably lower, the steel infill plates are much thicker. This results in extremely heavy member sizes. Table A.36 Deflections of Type LD SPSW Frame Story Drifts RdR0 3.00 Story Story Deflection (mm) Drift* RdR0 (mm) ISD Ratio (%) 5 57.77 36 1.0% 4 45.77 35.22 1.0% 3 34.03 26.07 0.7% 2 25.34 24.36 0.7% 1 17.22 51.66 1.2% 217 A.11 X-Braced Frame (XBF) S16-09 [27.11] Rd=1.5 Ro=1.3 For comparison with the other frames, a conventional construction braced frame is designed. The X-Bracing configuration is one of the most popular, and these structures are often designed in Vancouver per the conventional construction clause .The design will be as follows: 1. Calculate static seismic shears per the code procedure. Lateral forces must be increased 2% per meter of height above 15 meters. 2. Create a 2D elastic model with compression braces removed. Use the forces in the elastic model to design the tension braces from steel HSS sections. 3. Design beams to withstand the capacity of the maximum forces exerted by the braces. 4. The code requires the columns to be designed for the effect of gravity loads combined with 1.3 times the compressive loads imparted by the seismic shear forces. 5. Check deflections to ensure they are within limits. The structural period determined via. the NBCC is the same for all braced frames; 0.47 seconds. Clause [27.11.3] requires that the lateral forces be increased by 2% for every meter above 15 meters of the structure in height. Since the structure is 3.85 meters higher than 15 meters, the lateral forces must be increased by 2%*3.85=7.7%. The calculation is given in Table A.37. 218 Table A.37 Calculation of Seismic Lateral Shears Floor # xh (m) xW (kN) ∑ = n i ii hW 1 xxt hWFV )( − Story Shear (kN) ∑ = − = n i ii xxt x hW hWFVF 1 )( Shear per Node = 0.1*Vn Increase forces by 7.7% per S16-09 [27.11.3] (a) 5 18.85 7299 597908 2417689675 4043 404 435 4 15.2 11795 597908 3150553701 5269 526 567 3 11.55 11853 597908 2405854897 4023 402 433 2 7.9 11853 597908 1645563090 2752 275 296 1 4.25 11882 597908 887400284 1484 148 159 The loads from Table A.37 are applied to a tension-only elastic model, and the elastic brace loads are used to design the members. Figure A.30 shows the elastic model with member axial loads. Compression braces are assumed buckled and thus unable to provide resistance. Figure A.30 Elastic Model of Tension-Only System 219 The forces from the elastic model are used to design the braces. The beams are designed for the greater of the elastic forces or the capacity loads exerted by the braces. The free body diagram used to design the beam is identical to that derived in the section for the BRBF system and will not be repeated. Columns are designed based on the greater of the elastic loads or the capacity loads from the braces; the free body diagram for this is identical to that in the BRBF section. Since this is a braced frame it is expected that drifts will not govern, it is still important to check. Table A.38 shows the story deflections and drifts; they are within the NBCC limit. Table A.38 Deflections of X-Braced Frame Frame Story Drifts RdR0 1.95 Story Story Deflection (mm) Drift*RdR0 (mm) ISD Ratio (%) 5 112.15 37.6155 1.0% 4 92.86 48.1065 1.3% 3 68.19 48.0675 1.3% 2 43.54 50.1345 1.4% 1 17.83 34.7685 0.8% A.12 Suspended Zipper Braced Frame (SZBF) Rd=3.0 Ro=1.9 The design procedure will follow the steps as outlined in the PEER report by Yang et al. (2009a), except that the provisions of S16-09 clause [27.5] will also be applied. The Suspended Zipper Braced Frame is essentially a variation on the chevron concentric braced frame. The Design will involve the following Steps: 1. Design the frame for seismic loads from the NBCC and without zipper struts. 220 2. Zipper Columns are added and designed to resist the vertical unbalanced forces that result when the braces buckle. 3. Design the top story braces to elastically resist the vertical unbalanced forces that result when all the zipper columns have been simultaneously engaged. In order to ensure that the top story braces remain elastic, the top level seismic shear forces shall be multiplied by a factor of 1.7. 4. Design beams to resist the capacity axial load imparted on them by the braces in addition to gravity loads. It is appropriate to consider the beams as simply supported at either end in addition to a simple support midspan provided by the zipper column. 5. Design the second to top level beam to take the capacity forces of the top level braces (above) in addition to the capacity loads transmitted by the HSS braces below. 6. Design the columns for the capacity loads of the braces. 7. Check drifts to ensure they are within code limits. Since the ductility and overstrength factors for this structure are the same as that of S16-09 clause [27.5] for the Moderately Ductile Chevron Concentrically Braced Frame, the brace sizes will be identical. Seismic forces are as given in the section on the Type MD Concentric Braced Frame. The Zipper columns are to resist the unbalanced force in the beam that results when the compression brace buckles. Thus, the capacity of the zipper column is chosen to be as close as possible to Qb as shown in Figure A.10. The zipper column on the 2nd story is designed for Qb1, the unbalanced vertical force on the first story beam when the associated compression brace 221 buckles. The zipper column on the 3rd story is designed for Qb1+Qb2, or the cumulative force assuming both levels below have suffered compression brace buckling. In this way each successive zipper column going up the structure is designed assuming all floors below it have buckled compression braces. Equation A.102 gives the maximum tensile load that the zipper column at the top story will experience. Expected brace loads in tension, compression and buckled are as given in the section on the Type MD concentric braced frame. Equation A.102 biQT ∑=5 Figure A.31 is a Free Body Diagram of the top level braces. The lateral force V5=202.18 kN is from Table A.14 and is amplified by 1.7 to ensure that the braces remain elastic. The load T5 calculated per Equation A.102 is the accumulated zipper column force. Forces PA, given by Equation A.103, and PB, given by Equation A.104, are the tension and compression brace forces for the “Suspended” elastic braces which are designed not to buckle. It will turn out that both braces are in compression when the zipper columns have been engaged, thus PA is pointing in the wrong direction. The top level “suspended” braces are designed to resist PA and PB. Figure A.31 Free Body Diagram of Top Story Braces 222 Equation A.103 θθθ sin2sin4cos 7.1 55 TLwVP roofA −−= Equation A.104 θθθ sin2sin4cos 7.1 55 TLwVP roofB ++= Since the beams do not have to be designed for the unbalanced forces resulting from the braces buckling, the section sizes will be smaller than the regular chevron braced frame. The loading will assume that the beam is simply supported at the ends and with a simple support midspan. This allows the beam to be designed for a moment of wl2/32 rather than wl2/8. The beams will also be designed assuming maximum tension and compression (not-buckled) brace forces as given in S16-09 clause [27.5.3.4]. The beam design axial load is given by Equation A.33. The beam at the second to top story level must be designed to take the capacity loads from the top story braces (above) in addition to the loads transmitted from the HSS braces (below). The roof beam does not have braces to impart an axial force on it, so may be smaller. Figure A.32 shows a free body diagram of the beam at the 4th level; it is subject to capacity forces from both the braces above (the braces designed NOT to buckle) in addition to the capacity loads transmitted from the HSS braces below it. For the analysis of the Chevron Concentrically braced frame, the beam axial load was calculated from Tu4 and Cu4 (from the braces on the 4th level) and it was assumed that half of the beam took half of this force in compression, and the other half of the beam took it in tension. The braces in the top story develop compression forces PA and PB, which load the beam in Figure A.32 in tension; since these loads are applied at the ends of the beam. The axial (tensile) force from the top level braces will add to the beam axial loads from 223 the HSS braces below; thus both sides of the beam will be in tension but one side will have a higher tensile load. Equation A.105 is the axial load that the 4th level beam is designed for assuming that the top level braces have developed their maximum loads PA and PB, and the HSS brace below has buckled. Equation A.106 is the maximum possible compression load assuming that the HSS braces below have developed their maximum forces (compression brace not buckled) and the top level braces have not loaded the beam in tension at all. The design moment is simply wl2/32. Figure A.32 Free Body Diagram of 4th Level Beam Equation A.105 ( ) ( ) θθ cos 2 1 cos4 uubBAalthLevelAxi TCPPT +++= Equation A.106 ( ) θcos 2 1 4 uualthLevelAxi TCC += 224 The column design is similar to that of the Type MD Chevron Concentrically Braced Frame, except that the top story braces on both sides are in compression. The Free Body Diagram of Figure A.11 is essentially applicable except that the effect of the top level braces (those designed per the Free Body Diagram of Figure A.31) must be included. The column compression load is given by Equation A.107, the tensile load is given by Equation A.108. Since both of the top level braces are in compression, this effect reduces the tensile column load but also significantly increases the column compression load. Column bending moments are obtained in the same way as for the Concentrically Braced Frame. Table A.39 shows the story displacements calculated from the elastic model and the associated story drifts. As expected from a braced frame, the drifts are well under the NBCC limit of 2.5%. Equation A.107 [ ]LwCPC giuiBf +∑+= θsin Equation A.108 [ ]LwTPT giuiAf −∑+−= θsin Table A.39 Deflections of SZBF Frame Story Drifts RdRo 3.90 Story Story Deflection (mm) Drift* RdRo (mm) ISD Ratio (%) 5 28.39 7.917 0.2% 4 26.36 20.241 0.6% 3 21.17 27.651 0.8% 2 14.08 27.768 0.8% 1 6.96 27.144 0.6% 225 Appendix B Modeling Parameter Details In this section the numerical element models for structural components are presented. In all cases values presented are variables (ie. Mp=ZxFy) or nondimensional. Note that the material models used (Steel01, Steel02, Hysteretic, Clough) are unique to OpenSees; this being the case the parameters for these models in the tables of this section may not correspond to those found in other analysis programs. This guide is specifically geared towards the OpenSees user community. Further information regarding these material models, and the element types used in this project may be found in the OpenSees Command Language manual by Mazzoni et al. (2007). B.1 Steel01 Material Steel01 is a uniaxial bilinear steel material with kinematic hardening and optional isotropic hardening. The parameters for this material are given in Table B.1, Figure B.1 shows graphically what some of these variables control. Table B.1 Steel01 Material Parameters Fy yield strength E0 initial elastic tangent b strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent) a1 isotropic hardening parameter, increase of compression yield envelope as proportion of yield strength after a plastic strain of $a2*($Fy/E0). (optional) a2 isotropic hardening parameter (see explanation under $a1). (optional). a3 isotropic hardening parameter, increase of tension yield envelope as proportion of yield strength after a plastic strain of $a4*($Fy/E0). (optional) a4 isotropic hardening parameter (see explanation under $a3). (optional) Mazzoni et al. (2007) 226 Figure B.1 Backbone Curves for Steel01 Reference: Mazzoni et al. (2007) B.2 Steel02 Material Steel02 is a uniaxial Giuffre Menegotto-Pinto steel material with isotropic strain hardening. Table B.2 describes the material parameters. Table B.2 Steel02 Parameters Fy yield strength E0 initial elastic tangent b strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent) R0, CR1, CR2 parameters to control the transition from elastic to plastic branches. a1 isotropic hardening parameter, increase of compression yield envelope as proportion of yield strength after a plastic strain a2 isotropic hardening parameter a3 isotropic hardening parameter, increase of tension yield envelope as proportion of yield strength after a plastic strain a4 isotropic hardening parameter sigInit Initial Stress Value Source: Mazzoni et al. (2007) 227 B.3 Hysteretic Material The Hysteretic material is a uniaxial bilinear material with pinching of force and deformation, damage due to ductility and energy as well as degradation upon unloading stiffness based on ductility. Table B.3 describes the material parameters and Figure B.3 illustrates some of them. Table B.3 Hysteretic Material Parameters s1p, e1p 1st (+) stress and strain point s2p, e2p 2nd (+) stress and strain point s3p, e3p 3rd (+) stress and strain point (optional) s1n, e1n 1st (-) stress and strain point s2n, e2n 2nd (-) stress and strain point s3n, e3n 3rd (-) stress and strain point (optional) pinchX pinching factor for strain (or deformation) during reloading pinchY pinching factor for stress (or force) during reloading damage1 damage due to ductility damage2 damage due to energy beta power used to determine the degraded unloading stiffness based on ductility Source: Mazzoni et al. (2007) Figure B.2 Backbone Curves for Hysteretic Material Reference: Mazzoni et al. (2007) 228 B.4 Gravity System The shear connections at the ends of beams as well as the fixed/pinned column base connections have moment rotation behaviors described by the Steel01 material. The parameters used to define the Steel01 properties are based on the steel sections used. Zerolength hinges are given rigid axial and shear degrees of freedom, and the rotational degree is assigned to Steel01. Table B.4 describes the Steel01 material properties depending on what is being modeled. These element models do not model beam/column connection failure. Table B.4 Lumped Hinge Behavior Properties Model #1 Pin Connected Beam Model #2 Pinned Columns Model #3 Fixed Columns Material Model Steel01 Steel01 Steel01 My (yield moment) 0.25Mp 0.3Mp 0.9Mp k (initial slope) EIx/2L EIx/2L 2EIx/L b (post yield stiffness) 1.0% 1.0% 2% a1 0 0 0 a2 1 1 1 a3 0 0 0 a4 1 1 1 E is the elastic modulus of steel, Ix is the moment of inertia of the section and L is the bay width B.5 Concentrically Braced Frames The concentric braces were modeled using the force beam column element (de Souza 2000), but were assigned to a specially calibrated Steel02 material; the properties for this are given in Table B.5. At the ends of the braces zerolength hinges provided rotational flexibility; the moment rotation behavior was assigned to a Steel01 material with properties as defined in Table B.6. In the middle of the brace a node was located which was offset 0.1% of the brace length 229 perpendicular to the axis. All these were varied in order to calibrate the numerical model to match test data. The most critical factor is the offset of the middle node. The offset of 0.1% seems to make the brace buckle in compression correctly; however an offset of 1% results in a brace that buckles far too easily. The concentrically braced frame model does not model brace fracture but it does model brace buckling. Table B.5 Material Steel02 Parameters Fy (Mpa) 345 E (Gpa) 200 Hardening Ratio (b) 0.03 Transition Parameter (R0) 18.5 Transition Parameter (cR1) 0.925 Transition Parameter (cR2) 0.15 Iso Hardening Parameter (a1) 0 Iso Hardening Parameter (a2) 1 Iso Hardening Parameter (a3) 0 Iso Hardening Parameter (a4) 1 Initial Stress (sig0) 0 Table B.6 Hinge Steel01 Parameters Fy 0.25Mp E EIx/5Lb b 0.01 a1 0 a2 1 a3 0 a4 1 Lb is the brace length B.6 Eccentrically Braced Frames The link beam of the Eccentrically Braced Frame has a zerolength element located at its midpoint. This element is rigid in axial and shear degrees of freedom, but the shear DOF is assigned to a Steel02 material, with properties defined by Table B.7. The yield stress and slope, 230 Fy and E are as suggested by ASCE41; other parameters were calibrated to test data. This approach does not model shear link fracture. Table B.7 Steel02 Parameters Fy Vp=0.6FystAweb E Kv=GAweb/e b 0.002 R0 15 cR1 0.925 cR2 0.15 a1 0.0155 a2 1 a3 0.0155 a4 1 sig0 0 Aweb based on link section, e=link beam length, Fy is yield strength, G is shear modulus of steel B.7 Moment Resisting Frames A zerolength element with rigid axial and shear degrees of freedom was used to model connection stiffness; a Steel02 material was assigned to the rotational DOF. The parameters of the Steel02 material are given in Table B.8. Because a bolted stiffened connection was assumed in the design of the moment resisting frame, a SAC test specimen was chosen which matched this connection type. The SAC test specimen was a W760x147 (SAC reference ID M9904,M0005), which was chosen because it was similar in depth and connection type to the W840 section used in the MRF design in this report. This approach cannot account for the P-M, or axial moment interaction effect. This model also does not model connection failure. 231 Table B.8 Steel02 Hinge Parameters Fy Mp=ZxFy E 18EstIx/Lbeam b 0.025 R0 10 cR1 0.925 cR2 0.15 a1 0 a2 1 a3 0 a4 1 sig0 0 Ix, Zx based on section properties, Lbeam is beam length A model was created to simulate the behavior of a reduced beam section connection type. The Zerolength hinge model was calibrated to SAC phase 2 test data for a W760x147 (SAC reference ID LS-1) with a reduced beam section connection type, with a 67% strength reduction at the narrowest point of the flange. The parameters of the Steel02 material are given in Table B.9. Figure B.4 shows an overlay of the test and numerical data comparing the model and showing relatively good agreement. This approach cannot account for P-M interaction. Table B.9 Steel02 Hinge Parameters for RBS Fy Mp=ZRBSFy E 18EIx/L b 0.0008 R0 10 cR1 0.925 cR2 0.15 a1 0.007 a2 0.07 a3 0.005 a4 0.05 sig0 0 ZRBS based on the minimum necked down section 232 Figure B.3 Overlay of Test Data with the Model Results for RBS Source: SAC project website The steel plate shear wall uses RBS connections between the beam and column in the plate bays. Because the presence of the infill plate leads to potentially high beam axial loads, it is critical that the RBS model account for P-M interaction. To do this, a fiber section will be used to simulate the minimum flange section at the RBS. A Zerolength element will be placed at the location of the RBS connection (away from the face of the column), and a fiber section will be assigned to it. This fiber section will represent the minimum cross section of the RBS, therefore neglecting the gradual necking down of the cut. Figure B.5 illustrates this concept. The Zerolength fiber element model is calibrated with zero axial load to the test data of Figure B.4. The fiber model is given properties per Table B.5 except that the post yield stiffness is 1%. Figure B.6 shows a comparison of the fiber model for the Zerolength element and the model from Figure B.4. The output is almost identical, but the advantage of the fiber model is that it will yield at lower moments when subject to axial compressive load. 233 Figure B.4 Diagram of Fiber Model for RBS Plastic Hinge Figure B.5 Fiber P-M Model Vs. Steel02 Model for Zerolength Element The point of using the fiber model is to capture P-M interaction; one way to check this is to see if the hinge yields at lower moments in the presence of high axial load. Figure B.7 shows the moment-rotation behavior of three different cross sections of the fiber model with different axial -20000.00 -15000.00 -10000.00 -5000.00 0.00 5000.00 10000.00 15000.00 20000.00 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 M o m e n t (K ip -i n ) Rotation (radians) Fiber Model Steel02 Model 234 loads applied. This figure shows M-Rotation curves with zero axial load, half of the yield load Cy=AFy, and the full yield load. Horizontal lines are also plotted showing the expected reduced plastic moment capacity Mpr as given by the CSA/AISC code equation (Equation B.1) for reduced plastic moment capacity depending on the axial load. Yielding of the section under Pf=0.5Cy occurs slightly after the level predicted by Equation B.1, although because of the rounded edge it is difficult to pick the exact point. The section subject to Pf=1.0Cy is able to sustain a small range of elastic rotation before yielding. Equation B.1         −= y f Ppr C P MM 118.1 Mp is the plastic moment and Pf is the applied axial load Figure B.6 Moment-Rotation Behavior under Axial Load Mpr stands for reduced plastic moment (in the presence of axial load) 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0.00 0.01 0.02 0.03 0.04 M o m e n t (K ip -i n ) Rotation (radians) Test6 P=0 Test5 P=0.5Cy Test7 P=Cy Mpr P=0 Mpr P=0.5Cy 235 B.8 Buckling Restrained Braced Frames For the buckling restrained braces (BRB), the hysteretic material was calibrated in order to match the PowerCAT brace, a product of the Star Seismic Company. The points of the backbone curve (eXn, sXn) were calibrated to match the backbone curve given for this product in Figure A.19. The hysteresis parameters PinchX and PinchY were then calibrated to the test data given in the report by Merritt et al. (2003). The parameters for the brace core material are given in Table B.10. This data is in units of Pascals (stress), strain is dimensionless. The factors PinchX, PinchY, D1, D2 and beta govern how the hysteresis curve “loops” as the load is cycled, and were varied until numerical data agreed sufficiently with test data. It was found that variation of these factors dramatically affect the results. The ductility and energy damage parameters, as well as the Beta parameter, were not used. Note that this model does not capture the point at which the brace fractures at high drift levels. Table B.10 Parameters for Hysteretic Material e1p (strain) 0.0014427 s1p (stress)(Pa) 268381292 e2p (strain) 0.0250000 s2p (stress)(Pa) 440175899 e1n (strain) -0.0014386 s1n (stress)(Pa) -263780974 e2n (strain) -0.0249960 s2n (stress)(Pa) -563080604 PinchX 0.25 PinchY 0.8 Ductility Damage (D1) 0 Energy Damage (D2) 0 Beta 0 236 B.9 Steel Plate Shear Walls For the steel plate shear walls, truss elements modeled the “strips”, and the clough material model was assigned to the strips as the stress strain relationship. The clough model was originally developed to model reinforced concrete shear walls, it has been found to also be suitable for steel plate walls. Table B.11 shows the clough parameters which were used to model the steel infill plate behavior, stress units are Pascals. According to the literature, the strip elements are intended to have zero compression strength and only have tensile strength, however the Clough model as used here has both tensile and compression strength. This approach was found to empirically work much better as assigning strip elements zero compression strength resulted in convergence problems in OpenSees. The purpose behind modeling the plate wall with strips is to simulate the behavior of unstiffened plate walls – namely that the wall yields in tension and buckles in compression along easily visible lines. Although giving the strips compression strength does appear to give the wall the correct global behavior, it is possible that the forces at the local level are incorrect. This is due to the fact that the compression zones have strength, whereas in reality the plate wall simply buckles along the compression strip zones. The Clough model parameters are very complicated, and it was found that varying some of the values in B-11 had considerable effects, while variation of others had little effect. Although the Clough model was calibrated to test data from a steel plate wall which had beams fixed connected to the columns, the same model was used for the limited ductility steel plate wall 237 in which the beams are pin connected. This is potentially a source of inaccuracy, but because of the lack of available test data for limited ductility plate walls the Clough model was not recalibrated for this scenario. As more test data becomes available, it would be prudent to calibrate the model for this case. Table B.11 Clough Material Parameters Elastic Stiffness (Pa) 2E+11 Positive Yield Stress (Pa) 200000000 Negative Yield Stress (Pa) -200000000 Hardening Stiffness Ratio 0 Remaining Strength Ratio 0 Stiff Degradation Ratio -0.1 Dsp at Fc+ 10 Dsp at Fc- -10 Norm. Energy Capacity (LambdaS) 0 Norm. Energy Capacity (lambdaK) 0 Norm Energy Capacity (LambdaA) 0 Norm Energy Capacity (LambdaD) 0 Cyclic Deterioration Term (cS) 1 Cyclic Deterioration Term (cK) 1 Cyclic Deterioration Term (cA) 1 Cyclic Deterioration Term (cD) 1 238 Appendix C Development of Cost Data In this Appendix the costs associated with the Eccentric Braced Frame and the Steel Plate Shear Wall damage states are developed. Data from this section is used in Chapter 5, where the Performance Group Data is developed and explained. Eccentric Braced Frame Cost Values: Developments of costs associated with damage states 2 through 4 for the Eccentric Braced Frame are developed in this section. Damage states are defined in the report by Gulet et al. (2010), the costs associated with different repair actions are taken from the PEER report by Yang et al. (2009b). Damage State 2: according to Gulec et al. (2010) this state is defined as “damage to concrete slab above the link”. Therefore, repair costs would be associated with repair of the slab. The costs are summed as follows: Drywall removal $2.5/ft2  $27/m2 9m x 9m =81m2 (typical bay area) $27/m2 x 81m2=$2,187 Cut floor at damaged connection $200/ft2  $2153/m2 3m x 3m = 9m2 (assumed area of slab damaged around 1m long link) $2153/m2 x 9m2 = $19,377 Replace slab $20/ft2  $215/m2 $215m2 x 9m2 = $1,938 Wall re-framing and drywall $12/ft2  $130m2 81 m2 (typical bay area) $130/m2 x 81m2 = $10,463 239 Summing the costs for all operations gives $2187 + $19,377 + $1,938 + $10,463 = $33,965 Damage State 3: this state is defined as “web local buckling and/or flange local buckling”, the recommended repair method is heat straightening of the web or flange. This step involves using a torch to heat the section until it is malleable enough to bend it back into shape. It is assumed that if DS3 has been reached, then the concrete slab has also been damaged and will require all of the same steps as DS2. It is also likely that in order to properly perform heat straightening, the slab above the link would have to be removed in order to have access for the workmen to heat and straighten the link. Although no data is given for costs associated with using a heating torch, data is given for welding costs. Therefore, the heating operations will be based off of welding costs, this assumption seems rational since both require skilled workers, gas tanks and specific area setup procedures. Cost of all operations of DS2  $33,965 Welding protection $1.5/ft2  $16/m2 (it is assumed that the protection/setup costs for welding and those associated with use of a heating torch will carry similar costs) 81m2 (typical bay area) 81m2 x $16/m2 = $1,300 Carbon Welding $15/ft  $50/m 9m length of beam (link is 1m so this is conservative) 9m x $50/m = $443 Since welding will be considerably faster than heating, it is estimated that the costs for heating up the section will be ~3x more than the welding costs. Therefore $443x3= $1772 Summing costs gives $33,965 + $1,300 + $1,772 = $37,037 240 Damage State 4: This damage state is defined as “web and/or flange fracture”, the recommended repair procedure is replacement of the link section. Since the repair actions will involve access to the link beam and welding, it will be assumed that the link replacement costs will be added to the costs associated with DS2 and DS3. Cost of all operations of DS3  $37,037 Remove portion of damaged beam $80/ft2  $263/m2 3m x 3m = 9m2 (area around link) $263/m2 x 9m2 = $2362 Remove/Replace connection $6/lb  $13.23/kg (using 2.205 lb/kg) Costs are based on the link beam, a W360x79 member which is 9 meters long to be conservative. 9m x 79 kg/m x $13.23 kg = $9,407 Summing costs gives $37,037 + $2362 + $9,407 = $48,806 Note that in most cases, the costs associated with gaining access to the structural elements were more significant than the actual repair work. For DS4 for the EBF, removing and replacing the link ($9,407) and welding ($1772) only constitute $11,179, or 22% of the repair cost. This is expected, since removing the nonstructural elements and gaining access the building’s structural “skeleton” is typically very expensive. Note that these costs do not factor in downtime when the construction area is not available for the use of the occupants. 241 Steel Plate Shear Wall Cost Values: The fragility data for the steel plate wall was obtained from a report by Baldvins et al. (2012), costs associated with different repair actions are taken from the PEER report by Yang et al. (2009b). In this report by Baldvins et al. (2012), 12 damage states are defined and grouped into five repair categories: 1. Cosmetic Repair – this repair case carries trivial damage cost 2. Replace web plate – the costs will be associated with replacement of the steel plate 3. VBE Repair – replace the columns 4. HBE and connection repair – replacement of the beams and associated connections 5. Replace Boundary Elements – total replacement of the frame, VBE and HBE required Two general notes are worth mentioning. First, the replacement of the columns is typically expected to be much more expensive than beam replacement, because the beams must be shored against the gravity loading before they are removed. Secondly, the authors recommend merging the repair states 4 and 5 together. This is both because the median ISD triggering these damage states is relatively high and close together (DS4 and DS5 had median EDP of 2.75 and 3%, respectively), and because in tests DS4 only occurred when reduced beam section beam connections were used. It is the recommendation of the authors of this paper that DS5 and DS4 be merged into one damage state triggered at a median EDP of 2.75% ISD, to be conservative. Although the designs of this report utilize one wall with RBS connections (SPSW) and another without any moment resisting connection (SPSW_LD), the approach recommended by the authors of the Baldvins et al. paper will be used for both structures. This is done for simplicity, 242 and because this paper did not investigate steel plate walls without fixed connected beams. Because this report did not include repair cost data, a similar approach as that used for the EBF PG cost data will be taken. DS2 Replace web plate: the costs will be associated with replacement of the web infill plate. For length calculations, the bay length of 9 meters will be used; for area calculations it will be assumed that a 1 meter strip running along the beam is appropriate, or 1m x 9m = 9m2. Costs involving nonstructural element removal/replacement will be based on 9m x 9m = 91m2. Several costs will be repeated from the EBF cost calculations. Welding Protection $16/m2 x 81 m2 = $1,300 New Infill Plate costs based on a 6.35 mm (1/4” thick) steel plate, using prices from the McMaster-Carr website this costs $1.81/m2 9m x 4.25 m (1st story infill plate) = 38.25 m2 $1.81/m2 x 38.25 m2 = $70 Plate welding cost $50/m Plate perimeter is 9m x 2 + 4.25m x 2 = 26.5 meters 26.5m x $50/m = $1,325 Cutting out the old plate  assume this costs half as much as welding in the new plate $1,325/2= $663 Drywall removal  $2,187 (from EBF calculation) Wall re-framing and drywall  $10,463 (from EBF calculation) Ceiling system removal (for access to top of infill plate) $2/ft2  $22/m2 $22/m2 x 81 m2 = $1,782 243 Ceiling System Put-Back $8/ft2  $86/m2 $86/m2 x 81m2 = $6966 Summing costs $1,300 + $70 + $1,325 + $663 + $2,187 + $10,463 + $1,782 + $6966 = $24,756 DS3 VBE Repair: presumably if the VBE need to be replaced, this will also require replacement of the steel infill plate. This should be true both because the infill will have to be removed in order to replace the columns, and because damage that triggers column replacement will surely also have damaged the plate. Therefore, all costs from DS2 will be added to those associated with column replacement. Costs associated with DS2 $24, 756 Shoring up beams $2100 per location Remove portion of column $80/ft2  $262/m2 Using 9m2 from a 1 meter strip along the length of the beam 9m2 x $262/m2 = $2,358 Remove/replace connection $13.23/kg Based on the column member W360x1086 which is 3.65 m long x 2 columns gives 1086 kg/m x 3.65 m x 2 columns = 7928 kg $13.23/kg x 7928 kg = $104,887 Summing costs gives $24, 756 + $2100 + $2,358 + $104,887 = $134, 101 DS4 Replacement of frame: All the costs of DS3 will be added to the costs associated with replacement of the beam or HBE. DS3 costs $134, 101 244 Cut Slab for access (from EBF calculation) $19, 377 Replace beam $13.23/kg based on a W690x323 wideflange member 9m x 323 kg/m x $13.23/kg = $38,459 Replace slab (from EBF calculation) $1,938 Summing costs gives $134, 101 + $38,459 + $1,938 = $193, 875 245 Appendix D Pushover Curves The pushover curves in this section were used to determine the yield drift of the Moment Resisting Frame, the Eccentric Braced Frame, the Buckling Restrained Braced Frame and the Steel Plate Shear Wall. Figure D.1 through D.4 show the interstory drifts for all five floors of the MRF, EBF, BRBF and SPSW models, respectively. The OpenSees model was subjected to a load pattern proportioned per the NBCC 2005 equivalent static loads, this pattern is consistent with deforming the structure in its first mode shape. When subjecting the CBF model to a pushover analysis, as soon as the 3rd floor brace buckled (severely) the entire model became numerically unstable; therefore it was only possible to push this model up until this point. Because of this sudden buckling, the pushover curve was not shown. In the case of the CBF, the “yield drift” is taken when the model becomes unstable due to brace buckling on the 3rd floor. According to the fragility data of Chapter 5, the BRBF, CBF, MRF, EBF and SPSW structural systems fail (achieve maximum Damage State) at median interstory drifts of 2%, 2.23%, 5%, 0.88% and 2.75%, respectively. The pushover curves are therefore extended out to these limits, in order to establish at what point the system becomes nonlinear. Also note that because the structural element models do not model failure, these pushover curves will not be correct for large displacements at which point the SFRS should fail. Although the SPSW fragility data indicates that failure should occur at 2.75%, the pushover curve became erratic after roughly 1.5% ISD, therefore the figure is truncated at this value. Note that in Figure D.4 the pushover 246 curve initially has a negative slope; this is as a result of giving the strips which model the infill plate compression strength. Figure D.1 Pushover Curve for Moment Resisting Frame ISD# indicates the interstory drift at the #th floor level Figure D.2 Pushover Curve for Eccentric Braced Frame ISD# indicates the interstory drift at the #th floor level 0 2000 4000 6000 8000 10000 12000 14000 16000 0% 1% 2% 3% 4% 5% B as e Sh ea r (kN ) Inter Story Drift (%) ISD1 ISD2 ISD3 ISD4 ISD5 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 -0.1% 0.1% 0.3% 0.5% 0.7% 0.9% B as e Sh ea r (kN ) Inter Story Drift (%) ISD1 ISD2 ISD3 ISD4 ISD5 247 Figure D.3 Pushover Curve for Buckling Restrained Braced Frame ISD# indicates the interstory drift at the #th floor level Figure D.4 Pushover Curve for Steel Plate Shear Wall ISD# indicates the interstory drift at the #th floor level 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0.0% 0.5% 1.0% 1.5% 2.0% B as e Sh ea r (kN ) Inter Story Drift (%) ISD1 ISD2 ISD3 ISD4 ISD5 0 2000 4000 6000 8000 10000 12000 -0.5% 0.0% 0.5% 1.0% 1.5% B as e Sh ea r (kN ) Inter Story Drift (%) ISD1 ISD2 ISD3 ISD4 ISD5
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution 3.0 Unported
Graduate
Performance based evaluation of prequalified steel seismic force resisting structures in Canada
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