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Seismic performance evaluation of concentrically braced and fiction damped braced frames through full-scale… Kullmann, Harald Georg 1995

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SEISMIC P E R F O R M A N C E E V A L U A T I O N O F C O N C E N T R I C A L L Y B R A C E D A N D F R I C T I O N D A M P E D B R A C E D F R A M E S T H R O U G H F U L L - S C A L E T E S T I N G by H A R A L D G E O R G K U L L M A N N B.A-Sc , The University of British Columbia, 1989 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Civil Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Apr i l 1995 @ Harald Georg Kullmann, 1995 In p resen t ing this thesis in partial f u l f i lmen t of t h e r e q u i r e m e n t s f o r an advanced d e g r e e at the Univers i ty o f Brit ish C o l u m b i a , I agree that t h e Library shall make it f reely available f o r re ference and s tudy. I fu r ther agree that pe rm iss ion f o r ex tens ive c o p y i n g of th is thesis f o r scholar ly p u r p o s e s may b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on o f this thesis fo r f inancial gain shall n o t be a l l o w e d w i t h o u t m y w r i t t e n pe rmiss ion . D e p a r t m e n t o f tZ-\\}[u- £ N j < ^ , i K l s s ^ g - ^ r - ^ The Univers i ty o f Brit ish C o l u m b i a Vancouver , Canada Da te c ? 5 " - o 4 - 2-1 DE-6 (2/88) ABSTRACT The study describes the analyses of a six storey concentrically braced frame (CBF) and an equivalent frame retrofitted with friction-damped bracing (FDBF). The CBF was designed according to the ductile braced frame requirements of the Canadian steel building standard CAN/CSA-S16.1-M89 (S16.1) and the National Building Code of Canada 1990. The optimum slip load for the friction dampers in the FDBF was determined on the basis of energy principles and its distribution was assumed to be uniform throughout the structure. Full-scale quasi-static cyclic and earthquake simulation testing was conducted on the storey predicted with the most severe damage. The analytical predictions of these two systems show that their overall response, in terms of peak interstorey drift, is similar. Full-scale tests conducted on the CBF using hollow steel structural sections for the cross-bracing show that the bracing may develop twice the energy dissipation ability predicted by computer analysis. This appears to be due to the coplanar interaction of the braces. Significant additional stresses are induced in the bracing due to this interaction, which the computer cannot model. Hollow structural section cross-bracing appears to provide significantly better energy dissipating properties than HSS single diagonal bracing or multilevel cross-bracing; however, this may be at the expense of early fatigue failure. The tests show that, when compared to other bracing configurations, significantly fewer load cycles can be carried by such braces prior to fatigue fracture. The test results indicate that the slenderness requirements set out in S16.1 appear to be adequate for the design of cross-bracing if based on the half length, as opposed to the full length of the brace diagonal. The width-to-thickness ratio requirements were inadequate to effectively delay fatigue fracture of the braces. The optimum slip-load study using the optimizing program FDBFAP for the six storey FDBF under investiga-tion shows that the best response is obtainable with a frame shear resistance of 80 kN. A more detailed analysis using the program DRAIN-2D showed that a local optimum existed at 80kN, but that the global optimum was 320kN. The testing of the FDBF showed that the friction damper can be a reliable method of dissipating energy. The locally installed friction pads used in the damper were, however, unreliable for the slip-load level required. Sig-nificant fade in the slip-load occurred during cyclic loading. The friction pads failed on two occasions, and at a panel shear slip-load (base shear) of approximately 130 kN. ii The tests show that the compression brace remains in compression during an entire half cycle of loading. This implies that a buckled brace is not restraightened with the initiation of slip, but rather with load reversal, result-ing in a slack in the hysteresis during loading and unloading, and therefore a reduction in the energy dissipation. Out-of-plane vibrations occurred during the testing. These vibrations appeared to be the result of a prestress which is induced in the system in the deformed position. The tests also indicated that significant brace bending occurred. This bending was found to be a function of the ratio of the frame panel size to damper size. It is expected that the combination of bending and tensile forces due to the prestressing effect can lead to brace yielding. i i i TABLE OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS 1 iv LIST OF TABLES x LIST OF FIGURES xii ACKNOWLEDGEMENT xxi CHAPTER 1 INTRODUCTION 1 1.1 Overview of Seismic Structural Systems 1 1.1.1 Conventional Structural Systems 2 1.1.2 Innovative Structural Systems 5 1.2 Motivation of the Study 7 1.3 Object and Scope of the Study 8 CHAPTER 2 BACKGROUND OF STRUCTURAL SYSTEMS INVESTIGATED 9 2.1 Performance of Braced Steel Structures 9 2.1.1 Behaviour of Braces Under Cyclic Loading 9 2.1.2 Effect of Brace Connections 15 2.1.3 New Design Techniques 16 2.1.3.1 Capacity Design Philosophy 16 2.1.3.2 CAN/CSA-S16.1-M89 and NBCC 1990 Seismic Requirements 16 2.1.3.3 Comments on CAN/CSA-S16.1-M89 20 2.2 Performance of Friction Damped Structures 21 2.2.1 Friction Damping 21 iv 2.2.2 Concept of an Optimum Slip-Load 24 2.2.3 The Pall System 27 CHAPTER 3 INCORPORATION OF E N E R G Y PRINCIPLES INTO DRAIN-2D COMPUTER C O D E 29 3.1 Review of Energy Principles 29 3.2 Application to DRAIN-2D Code 31 3.2.1 DRAIN-2D Basics 31 3.2.2 Energy Balance in DRAIN-2D 33 3.2.3 Single Degree of Freedom Example 35 CHAPTER 4 DESIGN OF A TYPICAL MULTISTOREY BRACED STRUCTURE 41 4.1 Design of the Conventional Braced Structure 41 4.1.1 Design Parameters 41 4.1.2 Selection of Brace Sections 43 4.1.3 Beam, Column, and Connection Design 47 4.1.4 Deflection Checks '. 48 4.2 Design of an Equivalent Friction Damped Structure 48 4.2.1 Optimum Slip-Load Study 50 4.2.2 Effect of Brace Section on the Optimum Slip-Load 51 4.2.3 Deflection Checks 53 CHAPTER 5 ANALYTICAL STUDY OF T H E BRACED STEEL STRUCTURE 54 5.1 Modelling of a Frame Within The Structure 55 5.2 Modelling of an Inelastic-Buckling Brace 56 v 5.3 Response Under the Design Earthquake .... 57 5.4 Energy Values 60 5.5 Response Under a Megathrust Event 63 CHAPTER 6 ANALYTICAL STUDY OF T H E FRICTION-DAMPED BRACED STRUC-T U R E 68 6.1 Theoretical Formulation 68 6.2 Effect of Damper Aspect Ratio and Size 77 6.3 Modelling of the Friction Damper 79 6.4 Response Under a Design Earthquake 82 6.5 RPI and DRAIN-2D 87 6.6 Effect of Inelastic Brace Buckling 90 CHAPTER 7 COMPARATIVE PERFORMANCE OF CBF AND FDBF 97 7.1 Performance Comparison Under Design Excitation 97 7.2 Performance Comparison Under Megathrust Excitation 101 7.3 Concluding Remarks 102 CHAPTER 8 EXPERIMENTAL PROGRAM 103 8.1 Testing Frame Design 103 8.1.1 Frame Constraints 104 8.1.2 Frame Design 105 8.1.3 Frame Fabrication and Assembly 109 8.1.4 Frame Tests and Performance 115 8.2 Testing Program 116 vi 8.2.1 Conventional Cross-Brace Tests 116 8.2.1.1 Fabrication and Assembly 117 8.2.1.2 Instrumentation : 121 8.2.1.3 Braced Frame Tests 121 8.2.2 Friction-Damped Frame Tests 124 8.2.2.1 Description of the Friction Damper 125 8.2.2.2 Instrumentation 128 8.2.2.3 Friction-Damped Frame Tests 129 CHAPTER 9 CBF TEST RESULTS 132 9.1 Coupon Tests 132 9.2 Frame Test Results 133 9.2.1 Evaluating the Test Frame Friction 133 9.2.2 CBF Cyclic Tests 134 9.2.3 Panel Shear Capacity 147 9.2.4 CBF Earthquake Simulation Test 154 9.2.5 Performance Evaluation 159 9.2.5.1 Shear Capacity Degradation 159 9.2.5.2 Gusset Performance 160 9.2.5.3 Failure Mode 160 9.2.5.4 Out-of-plane Deflections 161 9.2.5.5 Effect of Coplanar Members 162 9.2.5.6 Energy Comparisons 163 vii 9.2.5.7 Comparison with Past Research 167 CHAPTER 10 FDBF TEST RESULTS 169 10.1 Calibration Results 169 10.2 Frame Test Results 170 10.2.1 FDBF Cyclic Tests 170 10.2.2 Earthquake Simulation Tests 181 10.2.3 Damper Performance 183 10.2.4 Brace Performance 185 10.2.5 Energy Comparisons 189 CHAPTER 11 CONCLUSIONS 192 11.1 Conventional Concentric Cross-Bracing 192 11.2 Summary Discussion of Friction-Damped Bracing 193 11.3 Comparison of the Two Systems Under a Design Excitation 195 11.4 Recommendations for Additional Research 195 REFERENCES 197 APPENDIX A BRACING SCHEMES 199 A . l Typical Brace Types and Connections 200 A.2 Multi-Storey X-Braced Frame 203 A. 3 Single-Storey X-Braced Frame 205 APPENDIX B Energy Results 207 B. l DRAIN-2D 'ENERGY' Subroutine Coding 208 B.2 Example #1 Energy Results 211 viii APPENDIX C Six Storey CBF Mode Shapes and Frequencies 213 APPENDIX D 215 D . l Six Storey FDBF Mode Shapes and Frequencies (Damper Modelled as Actual-Sized) 215 D.2 Six Storey FDBF Mode Shapes and Frequencies (Damper Modelled as Over-Sized) 217 D.3 Six Storey FDBF Mode Shapes and Frequencies (No Damper Modelled - Zero Slip-Load) 219 APPENDIX E Testing Frame and Specimen Drawings 221 APPENDIX F CBF Coupon Test Results 230 APPENDIX G FDBF Test Results 245 ix LIST OF TABLES Table 2.1 - NBCC 1990 Force Modification Factors 18 Table 3.1 - Typical DRAIN-2D Energy Output File 34 Table 3.2 - Single Degree Of Freedom Parameters 36 Table 3.3 - Damping and Yielding of Examples 1,2,3 36 Table 3.4 - Energy Results for Example 2 37 Table 3.5 - Energy Results for Example 3 38 Table 4.1 - Structure Design Loads 43e Table 4.2 - Schedule of Braces Sections 44 Table 4.3 - Schedule of Braced Frame Members 47 Table 4.4 - FDBF Brace Properties 53 Table 5.1 - CBF Peak Interstorey Drifts 57 Table 5.2 - CBF Eigenvalues 58 Table 5.3 - CBF Energy Results 61 Table 6.1 - Influence of Compression Brace 75 Table 6.2 - Influence of Aspect Ratio 78 Table 6.3 - FDBF Eigenvalues 82 Table 6.4a - DRAIN-2D Energy Results of FDBF 86 Table 6.4b - FDBFAP Energy Results For FDBF 86 Table 6.5 - Damper Slippage 87 Table 6.6 - DRAIN-2D Strain Energy Quantities 88 Table 6.7 - Drift Quantities for FDBF 89 x Table 6.8 - Energy Results for EDBF400 92 Table 6.9 - Energy Results for FDBF600 93 Table 6.10 - Friction Dissipated Energy 94 Table 8.1 - Gusset Capacities 108 Table 9.1 - Energy Comparisons for CB2 and CB3 165 Table 10.1 - FDBF Cyclic Tests 174 Table 10.2 - Energy Comparison 191 xi L I S T O F F I G U R E S Figure 1.1 - Typical Braced Frame Configurations 3 Figure 1.2 - Typical EBFs, MRFs, and Shear Walls 4 Figure 1.3 - Innovative Structural Systems 6 Figure 2.1 - Strut 17 Hysteresis 10 Figure 2.2 - Typical W-shape Brace Connection 11 Figure 2.3 - Performance of a Braced Frame 12 Figure 2.4 - Strut 5 and Strut 19 Hysteresis 13 Figure 2.5 - Hysteresis with Bolt Slip 15 Figure 2.6 - Concept of Ductility Design 19 Figure 2.7 - Slip Characteristics of Different Surfaces 21 Figure 2.8 - Hysteresis of Steel on Brass 22 Figure 2.9 - Hysteresis of Cobalt on Steel 23 Figure 2.10 - Friction Damper within a Frame 23 Figure 2.11 - Friction Damper Hysteresis 24 Figure 2.12 - RPI and Structure 26 Figure 3.1 - Unbalanced Force Due to Member Yield 33 Figure 3.2 - Single Degree of Freedom Model 35 Figure 3.3 - Energy Balance of Examples 1, 2, and 3 39 Figure 3.4 - Energy Balance Error of Example 3 40 Figure 4.1 - Six Storey Office Building 42 Figure 4.2 - Design and Actual Shear Resistance 45 xii Figure 4.3 - Typical HSS Cross-Brace Details 46 Figure 4.4 - Design Seismic and Wind Deflections 48 Figure 4.5 - Original Damper 49 Figure 4.6 - New Damper 50 Figure 4.7 - RPI of the Six Storey FDBF 51 Figure 4.8 - Effect of Brace Area on the RPI Function 52 Figure 5.1 - Vancouver Design and Megathrust Records 54 Figure 5.2 - Design and Actual Response Spectra 55 Figure 5.3 - DRAIN-2D Model 56 Figure 5.4 - Predicted Drift Time-History for CBF 58 Figure 5.5 - Brace and Panel Hysteresis for CBF 59 Figure 5.6 - Damage Due to Design Excitation 60 Figure 5.7 - Energy Results of CBF Under Design Event 62 Figure 5.8 - CBF Deflections and Drifts under a Megathrust 63 Figure 5.9 - CBF Energy Results under a Megathrust Event 64 Figure 5.10 - CBF Damage Due to Megathrust Event 65 Figure 5.11 - Megathrust Hysteresis of 1st & 5th Storey 65 Figure 5.12 - Megathrust Column Forces 66 Figure 6.1 - Damper in the Deformed Position 69 Figure 6.2 - Theoretical Comparison with Filiatrault-Cherry 73 Figure 6.3 - Slip-Load of New and Conventional Damper 74 Figure 6.4 - Influence of Compression Brace Contribution 75 xiii Figure 6.5 - Frame and Damper in the Deformed Position 76 Figure 6.6 - Influence of Damper Aspect Ratio 78 Figure 6.7 - Complex Damper Model and Hysteresis 80 Figure 6.8 - FDBF100 Drift Time-History Under Design Event 83 Figure 6.9 - Comparison of DRAIN-2D and FDBFAP 84 Figure 6.10 - Energy Comparison for FDBF100 85 Figure 6.11 - Comparison of DRAIN-2D and FDBFAP Damper Slip 87 Figure 6.12 - RPI Comparison of DRAIN-2D and FDBFAP 88 Figure 6.13 - First Storey Drift Time-History of FDBF100,400 90 Figure 6.14 - Effect of Inelastic Brace Buckling on the RPI 94 Figure 6.15 - Brace Hysteresis for FDBF400,600 95 Figure 6.16 - Comparison of Elastic and Inelastic Response 96 Figure 7.1 - Peak Drift Comparison of CBF, FDBF100,400 97 Figure 7.2 - Drift Area Comparison of CBF, FDBF100,400 98 Figure 7.3-Comparison of Energies for CBF, FDBF100,400 99 Figure 7.4 - Comparisons of Bending Moments and Axials 100 Figure 7.5 - FDBFAP RPI Function for a Megathrust Event 101 Figure 7.6 - Envelope Displacements for CBF, FDBF100,400,800 102 Figure 8.1 - Proposed Testing Arrangement 104 Figure 8.2 - Force Diagram of Test Frame 106 Figure 8.3 - The Test Frame 106 Figure 8.4 - The Gusset Connections 109 xiv Figure 8.5 - a) Reaction base, b) Cross-beam 110 Figure 8.6 - Completed Test Frame Components I l l Figure 8.7 - a) Anchor installation, b) Installed Base 113 Figure 8.8 - a) Assembled Test Frame, b) 400kip Actuator 114 Figure 8.9 - Unloaded Test Frame Friction 115 Figure 8.10 - Brace End Connection 118 Figure 8.11 - Centre Connection: a) tongue, b) cover plate 119 Figure 8.12 - Prepared Specimen CB3 120 Figure 8.13 - CBF Specimen 122 Figure 8.14 - Cyclic Test Displacements 123 Figure 8.15 - CBF Earthquake Simulation Drift Time-History 123 Figure 8.16 - FDBF Double-Angle Brace 124 Figure 8.17 - The Friction Damper 125 Figure 8.18 - a) Dissassembled Damper, b) Centre Plate 126 Figure 8.19 - SP Belleville Spring Washer No. 680-245 127 Figure 8.20 - Friction Damper Assembled 127 Figure 8.21 - Damper in the Calibration Setup 128 Figure 8.22 - Brace Deformation Measurements 129 Figure 8.23 - FDBF Specimen 130 Figure 8.24 - FDBF Cyclic Test Displacements 131 Figure 8.25 - FDBF 1st Storey Drift Time-History 131 Figure 9.1 - Typical CBF Coupon Stress-Strain Curve 132 xv Figure 9.2 - Test Friction for CB2 and CB3 133 Figure 9.3 - Friction Release During Unloading 134 Figure 9.4 - CB1 Drift Time-History and Hysteresis 135 Figure 9.5 - CB1 Continuous Brace, Cycle 4 136 Figure 9.6 - CB1 Brace Deflected Shape (Compression Phase) 137 Figure 9.7 - CB1: Transverse crack at centre connection, a) Cycle 3, b) Cycle 4 138 Figure 9.8 - CB1 Failure Mode 139 Figure 9.9 - CB1 After Test 140 Figure 9.10 - CB1: a) Gusset at Cycle 3, b) Gusset Yielding 141 Figure 9.11 - CB2 Centre Connection Yielding After Test 143 Figure 9.12 - CB2 Drift Time-History and Hysteresis 144 Figure 9.13 - CB2 Brace Bending: a) Cycle 4, b) Cycle 5 145 Figure 9.14 - CB2 Cycle 5: Brace Buckling 146 Figure 9.15 - CB2 Brace Deflected Shape 147 Figure 9.16 - CB2 Local Buckling: a) Cycle 5, b) Cycle 6 148 Figure 9.17 - CB2 Rupture Sequence: a) to f) 149 Figure 9.18 - CB2: a) Failed Section, b) Failed Specimen 152 Figure 9.19 - CB3 Drift Time-History and Hysteresis 155 Figure 9.20 - CB3 Cycle 19: First Local Buckling 156 Figure 9.21 - CB3 Cycle 21: Brace Buckling 157 Figure 9.22 - CB3 Section Tearing: a) Cycle 22, b) Cycle 24 158 Figure 9.23 - Shear Capacity Degradation (at 20mm Frame Deformation) 159 xvi Figure 9.24 - Predicted and Actual CB3 Hysteresis 164 Figure 9.25 - Energy Comparison of CB3 with the Predicted 165 Figure 9.26 - Megathrust Input and 5th Storey Energy 166 Figure 9.27 - Strut 17 Hysteresis 167 Figure 10.1 - Friction Slip Test Results 169 Figure 10.2 - Friction Damper Calibration Tests '•• 170 Figure 10.3 - Full-scale FDBF Testing Arrangement 171 Figure 10.4 - Damper Support 173 Figure 10.5 - Typical Cyclic Test Hysteresis 174 Figure 10.6 - FDBF1.3 Hysteresis 175 Figure 10.7 - FDBF1.6 and FDBF1.7 Hysteresis 176 Figure 10.8 - FDBF1.8 Hysteresis 177 Figure 10.9 - a) Damper with Cracked Pads, b) Cracked Pad 178 Figure 10.10 - Link Plates: a) Local Yielding, b) Pad Scour 179 Figure 10.11 - FDBF2.5 and 2.6 180 Figure 10.12 - FDBF100 Drift Time-History and Hysteresis 182 Figure 10.13 - Calibration and Theoretical Hysteresis 183 Figure 10.14 - Comparison of FDBF and Theoretical Hysteresis 184 Figure 10.15 - Typical Brace Force Time-History 185 Figure 10.16 - Brace Bending 186 Figure 10.17 - Brace Bending Displacements from Cyclic Tests 186 Figure 10.18 - FDBF100 Upper and Lower Brace Bending 187 xvii Figure 10.19 - Predicted and Experimental Seismic Hysteresis 190 Figure 10.20 - Energy Time-History of the Seismic Response 191 Figure A . l - W-shape with web and flange connections 200 Figure A.2 - HSS with tongue plate and cover plates 201 Figure A.3 - HSS X-bracing with cover plate connections 202 Figure A.4 - Square X-bracing with cover plate connection 203 Figure A.5 - HSS X-bracing end connection and masonry wall 204 Figure A.6 - HSS X-bracing without connection plate 205 Figure A.l - Centre connection with missing weld 206 Figure C. l - CBF Model 213 Figure C.2 - Mode #1 213 Figure C.3 - Mode #2 213 Figure C.4 - Mode #3 214 Figure C.5 - Mode #4 214 Figure D . l - FDBF Model (Actual-Sized Damper) 215 Figure D.2 - Mode #1 215 Figure D.3 - Mode #2 215 Figure D.4 - Mode #3 216 Figure D.5 - Mode #4 • 216 Figure D.6 - FDBF Model (Oversized Damper) 217 Figure D.7 - Mode #1 217 Figure D.8 - Mode #2 217 xviii Figure D.9 - Mode #3 218 Figure D.10 - Mode #4 218 Figure D . l l - Unbraced Model 219 Figure D.12 - Mode #1 219 Figure D.13 - Mode #2 219 Figure D.14 - Mode #3 220 Figure D.15 - Mode #4 220 Figure E . l - DWG. 1992-001 - General Arrangement of Testing Frame 222 Figure E.2 - DWG. 1992-002 - Cross-Beam 223 Figure E.3 - DWG. 1992-003 - Column 224 Figure E.4 - DWG. 1992-004 - Reaction Base 225 Figure E.5 - DWG. 1992-005 - CBF General Arrangement 226 Figure E.6 - DWG. 1992-006 - FDBF General Arrangement 227 Figure E.7 - DWG. 1992-007 - Friction Damping Device 228 Figure E.8 - DWG. 1992-008 - Top and Bottom Gusset Plates 229 Figure F . l - Coupon Test Specimens 231 Figure F.2 - Tension Coupon Test Setup 232 Figure F.3 - Compression Coupon Test Setup 233 Figure F.4 - Tension Coupon Test #1 234 Figure F.5 - Tension Coupon Test #2 234 Figure F.6 - Tension Coupon Test #3 235 Figure F.7 - Tension Coupon Test #4 235 xix Figure F.8 - Compression Coupon Test 236 Figure F.9 - CBF Specimen Calibration Test Setup 237 Figure F.10 - CB1 Brace Deflection (Neutral Position) 238 Figure F . l l - CB1 Brace Deflection (Tension Phase) 238 Figure F.12 - CB1 Brace Deflection (Compression Phase) 239 Figure F.13 - CB2 Brace Deflection (Tension Phase) 240 Figure F.14 - CB2 Brace Deflection (Compression Phase) 240 Figure F.15 - CB2 Brace Force Results from Strain Gauge Data 241 Figure F.16 - CB2 Continuous Brace Hysteresis 241 Figure F.17 - CB2 Spliced Brace Hysteresis 242 Figure F.18 - CB3 Brace Deflection (Tension Phase) 242 Figure F.19 - CB3 Brace Deflection (Compression Phase) 243 Figure F.20 - CB3 Brace Force Results from Strain Gauge Data 243 Figure F.21 - CB3 Continuous Brace Hysteresis 244 Figure F.22 - CB3 Spliced Brace Hysteresis 244 Figure G . l - Friction Slip Test Setup 246 Figure G.2 - Slip Surfaces of Slip Test 247 Figure G.3 - Typical Friction Slip Test (3SP Washers) 248 xx ACKNOWLEDGEMENT Financial support for this study was provided by the Science Council of British Columbia, programs STDF/A-GAR #322(SA-1) and STDF/AGAR #74(SA-3). This support is gratefully acknowledged. The friction damping device was supplied by Pall Dynamics Ltd. of Montreal, and the friction pads were installed by Industrial and Automotive Frictions Ltd. of Vancouver. The author would like to thank his supervisors, Professor Sheldon Cherry of the University of British Columbia and Professor Andre Filiatrault of Ecole Polytechnique, for their guidance throughout the project. Thanks are also extended to the technical support staff of the Department of Civil Engineering, especially Dick Postgate, Howard Nichol, and Paul Symons. Special thanks are due to Robert Tremblay for the many stimulating discussions and for his assistance in the design of the full-scale testing frame. The author also thanks Marnie Cockburn for editing this manuscript. The author also acknowledges the assistance of Norm Low and Robert Muller for their help in developing the AutoCAD drawings. xxi 1 INTRODUCTION 1.1 Overview of Seismic Structural Systems Earthquake induced ground motions produce base shears in buildings and cause the buildings to vibrate and absorb energy. These vibrations in turn create lateral inertia forces throughout the structure. In zones of high seismic activity these lateral forces can become significant, and must be resisted. In most cases, how-ever, it is considered uneconomical to resist these high forces elastically, so allowance is made for some yielding of the structural members. Since the seismic forces are transient and oscillate, the yielding is expected to be limited and reversing. The yielding process dissipates energy thereby reducing the vibrations. The structural members which are expected to yield must be designed to be able to sustain large inelastic deformations without failure; the ability to sustain repetitive yielding without failure is called the member's ductility capacity. Proper design involves a fine balance between the induced forces, the strength, and the ductility capacity of the structure. Provision is made in most codes to allow for yielding in the structure. Also, modern building codes typically state the following objectives for seismic design: - for minor earthquakes, the structure should not sustain any structural or non-structural damage (i.e. require stiffness), - for moderate earthquakes, the structure can sustain minor inelastic deformations and non-structural damage (i.e. require strength), and - for severe earthquakes, the structure can sustain major damage but must avoid collapse (i.e. require ductility). Post-disaster structures, such as hospitals, exhibit stricter requirements since they must remain operational immediately after the seismic event. The National Building Code of Canada (NBCC 1990) attempts to meet this requirement by increasing the base shear of such structures by 30% over what is specified for typical buildings. Several structural forms are available to resist seismic forces. These can be divided into two groups: conven-tional and innovative systems. Conventional systems are merely adaptations of older structural forms used for resisting 'static' lateral loads, such as wind, but modified and detailed to provide ductility without collapse. Some examples of conventional systems are Concentrically Braced Frames (CBFs), Moment Resisting Frames (MRFs), Eccentrically Braced Frames (EBFs), and Shear Walls. Although it can be 1 argued that EBFs are not conventional systems, they use concepts that are similar to those employed in con-ventional systems. Innovative systems generally alter and/or reduce the vibrations induced in a building, whereas conventional systems simply resist the induced vibrational forces. Some examples of innovative systems include Base Isolated Frames (BIFs), Friction Damped Frames (FDFs), and Active or Semi-Active Controlled Frames (ACFs). Some aspects of these seismic resisting systems are briefly discussed below and illustrated in Figures 1.1 to 1.3. 1.1.1 Conventional Structural Systems Concentrically Braced Frames (Figure 1.1) CBFs are very popular simply because they efficiently resist lateral loads with small deflections. CBFs are essentially vertical trusses. The main problem with this system is that the energy dissipation capacity is quite limited, especially if the bracing buckles under low loads. Under tensile yield, the brace becomes progressively longer from cycle to cycle. Thus, a pinched hysteresis loop develops where a region of pro-gressively lower lateral stiffness exists between peak deflections. As the building deflects through this soft region it gains momentum and impacts when the slack in the brace is used up. The pinching can result in a significant drop in energy dissipation ability. New design codes attempt to minimize this pinching behaviour by imposing higher compressive capacities of the braces. Thus, energy is dissipated through both inelastic buckling and tensile yielding. CBFs are discussed in more detail in Chapter 2. Moment Resisting Frames (Figure 1.2a) MRFs are among the most common framing systems used in low-rise seismic design. The main reason for this is that they have a very consistent and reliable way of dissipating energy. The energy is dissipated through flexural yielding of the beams and columns, which can lead to a relatively high ductility capacity of these members. MRFs also generally have a high structural redundancy and thus failure in one location can result in a redistribution of forces. Problems can occur with MRFs as a result of excessive deflections. Since lateral loads are resisted by bending, the resulting deflections are generally high and this can lead significant non-structural damage. MRFs are, therefore, generally restricted to low-rise buildings. 2 a) X—Bracing b) Chevron Bracing c) Diagonal Bracing d) K - B r a c i n g Figure 1.1 - Typical Braced Frame Configurations. The 1994 Northridge earthquake in California has prompted some investigation into the welding proce-dures of moment connected frames. Welded MRF's were found to develop significant cracks in the moment joints. Shear Walls (Figure 1.2b) Shear walls are simply vertical deep beams. The large depth allows for high lateral stiffness. Attention to proper detailing can provide good reliable energy dissipation. However, shear wall structures tend to have little redundancy. Also, due to the high lateral stiffness, higher mode effects tend to induce high seismic forces in the upper storeys. 3 a) M o m e n t F r a m e b) Shear Wall link c) D i a g o n a l E c c e n t r i c B r a c i n g link d) C h e v r o n E c c e n t r i c B r ac ing Figure 1.2 - Typical EBFs, MRFs, and Shear Walls. Shear walls are generally constructed of reinforced concrete, but research, Elgaaly and Caccese, 1990, shows that steel plate shear walls can provide excellent ductility and offer an economical alternative to concrete. The steel plate shear wall acts as a vertical plate girder and can be designed in accordance with plate girder principles. Design procedures for steel plate shear walls are expected to be incorporated into the next revision of the Canadian steel design code. Eccentrically Braced Frames (Figure 1.2c and d) EBFs combine the benefits of the MRFs and the CBFs, in that they possess the ductility of the M R F and the lateral stiffness of the CBF. Although the EBF is a relatively new concept, it has been incorporated into Canadian, American, and New Zealand design codes. The concept and design procedures for this system were developed at the University of California, Berkeley (Roeder and Popov, 1978). 4 The main features of the EBF system are the braces and the link beam. The braces are designed not to buckle under ultimate lateral forces. Instead, an eccentricity is introduced into the brace-beam and beam-column connection. This link beam between the connections is forced to be the weak link in the lateral capacity of the frame. Attention to detail in this link allows for reliable energy dissipation from cycle to cycle. 1.1.2 Innovative Structural Systems Under the design earthquake, conventional earthquake resistant building systems are expected to sustain structural damage resulting in the need for their repair. Engineers are concerned with the development of new structural systems which sustain less damage under a major earthquake than conventional systems and are safer and more reliable. Base-Isolated Frames (Figure 1.3a) Base isolation attempts to reduce the energy which is fed into the structure. The structure is isolated from the ground motion through the use of foundation bearings with low shear stiffness. The result is a reduc-tion in the overall lateral stiffness, which increases the natural period of the structure. The natural period of the structure is then shifted out of the range of periods where most energy is contained in typical earthquakes. Typical base isolation systems incorporate lead-rubber hysteretic bearings which are fastened between the foundation and the superstructure. The elastic shear stiffness of the bearings is typically about 10 times their post-elastic stiffness. This limits the base shear induced by the ground motion and the resulting energy transmitted to the structure; as a consequence, the structural frame is required to dissipate less energy. The post-elastic stiffness is governed by the yield force of the lead plug. At large ground motion the lead yields, dissipating energy and reducing the shear stiffness of the bearing. This system has been employed in a number of structures over the past decade. Its use has been restricted, in part due to the high associated engineering and construction costs. Friction Damped Frames (Figure 1.3b) FDFs are typically braced frames with slotted friction connections which dissipate energy through slip-page. The connections can be preset so that slippage occurs under a specific lateral seismic load. This load can be optimized to produce the best (minimum) structural response. 5 The optimum slip-loads for typical buildings is relatively low compared to conventional lateral seismic design forces; thus, the seismic forces within a friction damped structure are generally low. Analytical and experimental studies of FDF systems have shown that they out-perform conventional frame systems, and they do so with cost savings (Pall and Pall, 1993) which depend on the level of design seismic forces for a region. This system is discussed in more detail in Chapter 2. Rubber Bearing 1 / \ a) B a s e - I s o l a t e d F r a m e b) F r i c t i o n - D a m p e d c) A c t i v e l y - C o n t r o l l e d F r a m e F r a m e Figure 1.3 - Innovative Structural Systems. Actively and Semi-Actively Controlled Frames (Figure 1.3c) ACFs involve the application of externally applied forces to limit the inertial forces developed from vibra-tions. Through the use of electronic sensors, which monitor building vibrations, vibrations can be damp-ened with devices such as mass dampers or jet-propelled engines. Semi-ACFs use external energy to 6 modify the characteristics of the damping devices. These systems possess the best ability to minimize building vibrations and damage. However, due to very high engineering and construction costs, these sys-tems are not commonly used. 1.2 Motivation of the Study In 1990, the Canadian Standards Association (CSA) issued the latest revision of "Limit States Design of Steel Structures," CAN/CSA-S16.1-M89. This revision contains a new clause, Clause 27, which defines seismic design requirements. Past Canadian steel codes did not contain such a provision. CAN/CSA-S16.1-M89, Clause 27, "Seismic Design Requirements" is hereafter referred to as S16.1. Clause 27 is designed to be used in conjunction with the new format of the equivalent lateral seismic force design procedure defined in the National Building Code of Canada, 1990. This code is hereafter referred to as NBCC. The clause outlines design procedures for moment resisting frames as well as braced frames. The design requirements for braced frames exhibit very strict detailing procedures, since these frames can behave poorly under seismic loading. Essentially, no full-scale tests have been conducted on realistically sized braced frames, especially cross-braced frames. The new design requirements are generally based on scaled tests, as will be discussed in Chapter 2. It is then of particular interest to examine the performance of structures designed according to these new requirements under full-scale conditions. Recognizing that conventional building frames are only fairly reliable under seismic loading, designers and researchers have developed some innovative ways to improve the performance reliability of such structures. The friction-damped-braced system represents an example of this new technology. Although this system has been proven to have merit, both analytically and experimentally, very little testing has been conducted on full-scale models of this type of structure. This system is discussed in greater detail in Chapter 2. This thesis is concerned with: i) the seismic time-step computer analysis and full-scale tests of a conventional concentrically brace frame designed by the new Canadian steel code provisions, and ii) the seismic time-step computer analysis and full-scale tests on a friction-damped-concentrically-braced-frame. The behaviour of the two systems is also compared. 7 1.3 Object and Scope of the Study The objective of the study is to evaluate the performance of conventional concentrically-braced frames and of friction-damped frames under simulated seismic forces. These objectives were accomplished by using the following methodology. First, a typical office building with normal cross-bracing was designed in accordance with the new S16.1 design provisions for the ductile frames. The same design was used for the friction-damped building, except that the cross-braces were fitted with friction-dampers. Second, dynamic analyses were performed on both building frame forms using the design earthquake antici-pated at the building site. The program DRAIN-2D, with complex buckling elements (EL9 and EL10) was used in the analyses. The performance of the friction-damped frame was optimized by minimizing the damaging energy. Since DRAIN-2D does not calculate energy quantities, DRAIN-2D was first repro-grammed to give the necessary information for the slip-load optimization in the FDF. These energy quanti-ties also allow for more detailed insight into the performance of the frames. Third, full-scale panel tests on the braces of each frame type were performed under harmonic cyclic tests and under quasi-static seismic deformation conditions. These latter deformations corresponded to the inter-storey drift-time response histories of the most seriously deformed storey as predicted by computer analyses. The results of the tests are then discussed and the relative performance of the panels is evaluated. 8 2 BACKGROUND OF STRUCTURAL SYSTEMS INVESTIGATED The research presented and discussed in this thesis involves the computer analysis and testing of both conven-tional concentric and friction damped bracing systems. An evaluation of the results of the computer analysis and experimental testing require a prior understanding of the performance and the current state of research of these systems. 2.1 Performance of Braced Steel Structures 2.1.1 Behaviour of Braces Under Cyclic Loading CBF's have been a popular choice as a structural framing system in steel buildings. They resist lateral loads through axial shortening and elongation of the braced frame members and thus efficiently limit lat-eral deflections. Nevertheless, CBFs possess some undesirable characteristics. Since the forces are resisted axially by the frame members, under ultimate conditions the instability of the structure can be jeopardized if it is not properly designed. CBFs also have an inherent low redundancy. Although this is an attractive feature from an analytical and cost point of view, under ultimate conditions it is not. During a seismic event, if a member in the CBF fails, such as a brace fracturing, there may be insufficient capacity in the alternate load paths to take up the lateral force. Instability can quickly arise from the soft storey created by the brace fracture. Despite these problems, experience has shown that CBFs can perform to an acceptable level during strong seismic events if proper detailing is used. To better understand braced frames and how they perform under seismic loading, it is necessary to under-stand the behaviour of the braces. Figure 2.1 is a graph of a hysteresis curve from a single brace reverse load cyclic test by Black et al., 1980. During the compression excursion, the brace reaches the critical compressive load and buckles. For most practical brace sizes (slenderness ratio of less than 200) yielding generally accompanies buckling in the form of a hinge. During the tension excursion the hinge developed must yield again as the brace restraightens. This results in brace elongation even before the full tensile force is developed. The effect is indicated in Figure 2.1. Repeated cycling results in a continuous drop in the compressive resistance. After the hinge undergoes several cycles of inelastic deformation local buck-ling of the member elements, webs and flanges, etc., occurs. The number of cycles to the initiation of local buckling is dependent on the type of structural section used for the brace. Failure of the brace is generally a result of fatigue tearing of the section near the location of local buckling. 9 1.0 if i 3 . a UJ N - I < z <£ o z -1.0 - 8 -4 0 4 8 NORMALIZED DISPLACEMENT, (8/8,) Figure 2.1 - Strut 17 Hysteresis (from Black et al, 1980). The behaviour of a braced frame is dictated for the most part by the performance of the braces. Since the stiffness of the frame alone is much lower than the brace stiffness, the lateral loads on the braced frame system are carried almost entirely by the bracing. Although the frame connections are in many cases intended to be pinned, the brace connections will usually allow significant moments to be transferred from the beam to the column. Figure 2.2 is a typical connection of this type. Significant shear forces in the columns do not develop until large lateral deflections are reached. As an example, a single bay cross braced frame is used to illustrate the behaviour of a CBF under cyclic load reversals. This frame is shown in Figure 2.3. As the frame deforms, the compression brace buckles, usually inelastically, resulting in a loss of stiffness. As the tension brace undergoes yield, the stiffness drops further as the compression brace continues to deform inelastically. Upon reversal of the load, the buckled brace with the hinge goes into tension, the hinge begins to straighten and the previous tension brace begins to buckle. Since neither brace can plastically shorten during its compression state, the braces simply grow in length from cycle to cycle. The result is a slackening of the braces which, as seen in the figure, produces a region of low stiffness between peak deflections. This allows the frame to gain momen-tum as it sways, followed by high impact forces as the tension brace begins to become taught and takes load. As a result of this region of low stiffness, the frame must undergo large deflections before significant amounts of energy can be dissipated by the braces. Thus, overall stability can become a prob-lem after just a few severe excursions. 10 J I I I 1 I I L Figure 2.2 - Typical W-shape Brace Connection (Courtesy CANRON Inc). 11 Figure 2.3 - Behaviour of a Braced Frame (from Popov et al, 1976). Research conducted by Black et al, 1980, indicates that the slenderness ratio of the bracing is the single most important parameter which affects its performance under cyclic load reversals. Slender braces dissi-pate little energy during compressive buckling because there is little inelastic action as the buckled brace bends. This leads to the situation described above. Stocky braces can dissipate more energy since significant yielding occurs during brace buckling. Figure 2.4, from Black et al, 1980, illustrates the com-parison between braces having different slenderness ratios. Clearly, the brace with the lower slenderness ratio possesses a greater ability to dissipate energy. The stocky brace maintains a higher stiffness during tensile loading. Black et al, 1980, have ranked sections in order of their ability to dissipate energy. Hol-low sections provide the best performance and double-angles provide the worst. Local buckling within the brace section also effects the performance of the brace. Local buckling occurs when the bending strains at the hinge location become too great for the section webs, flanges, or legs. The result is a local instability; the section element buckles and the compressive resistance drops further. This local buckling relieves, to some degree, the strains on the element. Repeated buckling and straightening of the elements results in excessive straining. A fatigue rupture of the brace occurs soon after. In most cases this is the mode of failure of a brace under cyclic loading. A relationship between local buckling and overall buckling is given below. Local buckling is governed by the plate buckling equation (see, for example, Galambos, 1988,) 12 1.0 or a. a 4 O < s tr. o --• J#trl / ' ^zsrf/f J 11 -STRUT 9 1 1 .1 t/r '10 i i i 1 1 1 -1.0 - 8 -4 0 4 NORMALIZED DISPLACEMENT, (8/8, ) -4 0 4 NORMALIZED DISPLACEMENT, ii/t,) Figure 2.4 - Strut 5 and Strut 19 Hysteresis (from Black et al, 1980). = k-12(i-v z)( Overall brace buckling is governed by the Euler formula, n2E "brace f KL\2. [2.1] [2.2] 13 where, \ = width-to-thickness ratio of the plate element, k = factor accounting for the support conditions of the plate, v = Poisson's ratio, E = Young's modulus, ~ = effective overall brace slenderness ratio, K = effective brace length factor, L = brace length, and r = brace section radius of gyration. If the critical buckling stresses of two equations are equated, we see that a decrease in the overall slender-ness should be associated with a equivalent decrease in the width-to-thickness ratio. This is consistent with the results of research by Black et al, 1980, which has shown that local buckling tends to be more significant in members with lower slenderness. The width-to-thickness ratio of the brace section elements is a measure of their susceptibility to local buckling. A high ratio leads to early buckling; a low ratio produces a stiff element which has a greater resistance to buckling. While brace slenderness is a measure of the bending stiffness of the brace, width-to-thickness ratio can be considered to be a measure of the slenderness of the cross-sectional elements of the brace. Overall brace buckling is similar to local buckling in that slender braces, or elements, undergo less strain. With less strain the potential for a fatigue failure is reduced. Therefore, slender elements will buckle easily, but they are less susceptible to fatigue rupture than thick elements. An attempt was made by Tang and Goel, 1987, to relate brace slenderness and brace element width-to-thickness ratio to the brace cycle life in terms of 'standard cycles'. They established that cycle life was inversely proportional to the square of the width-to-thickness ratio and proportional to the slenderness ratio. This relationship was empirically calibrated for the rectangular tube Chevron bracing of a six-storey braced frame tested at the University of California, Berkeley and was incorporated into their dynamic analyses to predict a more accurate structural response after brace failure occurred. The calibration fac-tor, unfortunately, had a variation of over 50%, but it did improve the analytical response. 14 2.1.2 Effect of Brace Connections During a severe seismic event, braces are likely to experience their ultimate limit state. The connections must be designed to withstand these ultimate forces. The maximum connection force would be the speci-fied resistance (unfactored) of the brace. However, the probable resistance of the brace is higher than this value, and the connection may have sufficient additional overstrength resistance. Since the brace resistance is based on the 5th percentile of the resistance distribution for the brace section, it would seem more reasonable to use the average brace resistance. The New Zealand Steel Code, NZS 3404:1989, requires that overstrength factors be used for determining the connection design force. In research conducted by Astaneh-Asl et al, 1982, tests were conducted on several double angle braces. Their results show that, for out-of-plane buckling, the gusset plate end connections had premature failures if the gussets were restrained from rotations caused by brace buckling. Further, if too much of the gusset was able to rotate, the gusset itself would buckle. This would create a mechanism, and the brace would no longer buckle. As a result, the energy to be dissipated would then be concentrated in the gussets only, which generally possess little energy dissipation ability in comparison with the brace. Figure 2.5 - Hysteresis Curve with Bolt Slip (from Astaneh-Asl et al, 1987). In the same research, these authors found that bolted, bearing type connections would slip during loading. This slip contributed significantly to the overall pinched behaviour of the brace, as shown in Figure 2.5. The pinching contribution by bolt slip in this type of specimen was similar to the contribution to pinching 15 by the deformation to first yield in the first cycle, and over two times that amount by the eighth cycle. Over stretching of end bolt holes increased the pinching during loading. Slip-critical connections did not slip when designed for the ultimate limit state of the brace. 2.1.3 New Design Techniques 2.1.3.1 Capacity Design Philosophy The capacity design philosophy is similar in concept to plastic design except that it applies primarily to the earthquake load case. Plastic design applies to all load cases, dead, live, wind, etc. In capacity design, as in plastic design, the structure is reduced from its indeterminate form to a self-equilibrated determinate state (mechanism). In this form, no additional loads can be accepted. The structural elements are then designed according to the maximum forces which are transferred by hinged beams and columns and other yielded members. The critical areas are then detailed to withstand the yield levels expected. Thus, the ductility of the structure is insured. Analyses determine the deflection limits expected. The weak elements, those in which the designer accepts damage, undergo plastification during seismic loading. They reach peak force levels based on their ultimate strengths. These forces are transferred to the strong elements, which must be capable of accepting these force levels with minor or no yielding and with stability intact. In the case of moment frames, the weak elements are typically the beams and the strong elements are the columns. In braced frames, the braces are the weak elements and the beams and columns are the strong elements. In eccentrically braced frames, the shear links are the weak elements. 2.1.3.2 CAN/CSA-S16.1-M89 and NBCC 1990 Seismic Requirements Canadian design codes have recently revised the design requirements with respect to earthquake load-ing. Changes to the NBCC include a modification to the equivalent static seismic force equation to better reflect the effect of ductility capacity of the chosen structural form. The new equation is, Vt = vSIFW, V = Q>.6 — . [2.3] D L J 16 Where, V e = elastic seismic base shear, V = design seismic base shear, W = total weight of the structure, v = zonal velocity ratio, S = seismic response factor (reflects the peak acceleration), I = importance factor, F = foundation factor, and R = force modification factor. R reflects the relative ductility which the structure possesses. Table 2.1 lists the NBCC R values for various structural forms. In 1990 the Canadian Standards Association released the new steel design code, CAN/CSA-S16.1-M89 (S16.1). This revised code contains a new clause, Clause 27, "Seismic Design Requirements." This clause outlines the requirements for the design of moment resisting frames and braced frames. The research presented in this thesis is only concerned wiui braced frames, and only this type of frame is discussed in this study. S16.1 provides design requirements for two types of braced frames, ductile braced frames (DBF) and nominal ductile braced frames (NDBF) frames. The ductile braced frame has an R value of 3.0, whereas the nominal ductile frame has an R of 2.0. This assumes that the ductile frame is capable of undergoing the same deflections as the nominal ductile frame but at two-thirds of the strength level, as illustrated in Figure 2.6. S16.1 describes the detailing requirements needed to achieve these ductilities; the requirements are stricter for ductile braced frames than for nominal ductile braced frames. Some examples of new bracing and connection designs are shown in Appendix A. Ductile Braced Frames S16.1 has restrictions on the bracing configurations which are acceptable for a ductile system. Chevron, V, and K bracing configurations are not considered ductile frames. Although these frame configura-tions are very efficient for resisting lateral static forces, Tremblay et al, 1991, they exhibit poor response under reverse cyclic loading in the post yield range, Roeder, 1989. Ductile bracing requirements state that the compression braces shall resist alone at least 30% of the total lateral shear force taken by the braces. This means that the compressive resistance of the brace should be almost half of the tensile 17 Table 2.1 - NBCC 1990 Force Modification Factors Steel Structures ductile moment resisting frame 4.0 ductile eccentrically brace frame 3.5 ductile braced frame 3.0 nominal ductile moment frame 3.0 nominal ductile braced frame 2.0 systems not defined above 1.5 Reinforced Concrete Structures ductile moment frame 4.0 ductile flexural wall 3.5 nominal ductile moment frame 2.0 nominal ductile wall 2.0 systems not defined above 1.5 Timber Structures nailed plywood shear panel 3.0 ductile concentrically braced frame 2.0 ductile moment frame 2.0 systems not defined above 1.5 Masonry Structures reinforced masonry 1.5 unreinforced masonry 1.0 Systems not defined above 1.0 resistance, if there is no overstrength provided as a result of the brace selected. Therefore, tension-only bracing is not considered to be ductile; this is due to the increased pinching behaviour associated with bracing having low compressive resistance. The brace compressive resistance is calculated in the 18 usual manner, but factored down to account for the drop in buckling load during cyclic loading. This factor, however, need not be applied if the tension brace has sufficient reserve capacity resulting from the selection of an oversized brace. The slenderness ratio requirements for ductile frames are defined as, [2.4] - = 1 1 0 , for F=300MPa. r y For braces near the top of a braced frame, where design lateral loads are small, it is likely that the slenderness requirements will govern brace selection. If slenderness governs, the selected brace can be considerably larger in area than that required by the strength criteria. Width-to-thickness ratio requirements must be less than or equal to 80% of Class 1 sections. CD CO a CD si CO V, NDBF V, DBF - 6=Ductility 6=2 ' / 9=3 " m a x D i s p l a c e m e n t Figure 2.6 - Concept of Ductility Design. Welded brace connections are required to withstand the unfactored tensile yield strength of the brace. Bolted connections must resist the unfactored ultimate tensile strength. S16.1 also attempts to limit the connection size where the brace may be considerably oversized. Thus, the brace connections do not need to be designed for a force greater than twice the brace seismic load and the specified gravity loads. This is based on the assumption that the frame will not see a force greater than that based on an elastic analysis, and a minor amount of ductility is available in all steel structures resulting in an R factor of 1.5. Allowances must also be made for ductile rotation of the gusset during in or out-of-plane buckling. 19 The beams, columns and their connections are designed for loads similar to the brace connections. Nominal Ductile Braced Frames All bracing configurations are allowed as NDBF, including tension-only braces. In the case of Chevron and V braces, the intersected beam shall be continuous and be able to support its tributary gravity loads without considering the support provided by the braces. The braces are not limited in slenderness, and sections must be Class 1 or 2. All connections, beams and columns need not be designed for a force greater than 1.33 times the seis-mic member force (R= 1.5) plus gravity loads. 2.1.3.3 Comments on CAN/CSA-S16.1-M89 No provision is made for material overstrength in the braces when considering the forces induced in the connections, beams, and columns. Since the strength considered is the 5th percentile of resistance distribution, the actual strength of the brace is likely to be much higher. The New Zealand steel code uses material overstrength factors ranging from 1.35 to 1.7, depending on the steel grade and the cate-gory (ductility) of the member. The reduction factor for the brace compressive resistance is dependent on slenderness. The maximum reduction occurs with maximum slenderness. This maximum reduction is 33% for 300W steel. Exper-imental research indicates that the brace compressive resistance drops to 50% within a couple of cycles under cyclic loading, Black et al, 1980, and Astaneh-Asl et al, 1982. However, considering the variability in the cyclic deflections under seismic loading, 33% seems reasonable. This same research also indicates that the brace compressive reduction is inversely proportional to the brace slenderness. This seems reasonable when one considers that an elastically buckled brace, with very high slenderness, has no drop in compressive resistance. Inelastic action during buckling is what leads to a drop in com-pressive resistance. Only at very low slenderness, where little buckling occurs, does the compressive resistance drop become small. Results from Black et al, 1980, show that a compressive resistance drop at slenderness ratios of 40 is still quite high. Tests by Astaneh-Asl et al, 1982, show that in bolted connections, net section failures occur prema-turely. Reinforced net sections exhibit superior performance. Although the tests by these researchers 20 were conducted only on double-angle connections, it seems reasonable to extend this observation to connections with other brace sections. No provision is made in S16.1 to reinforce the net sections of bolted brace connections. 2.2 Performance of Friction Damped Structures 2.2.1 Friction Damping Using friction to dampen the response of buildings under earthquake loading is a relatively new concept. Pall et al, 1980, proposed a method to dampen the response of large panel structures through the use of Limited Slip Bolted Joints (LSBJ). Figure 2.7 shows the hysteretic behaviour of several different frictional surfaces tested during this research. *<•) Mill wait o) Sand blattttf Figure 2.7 - Slip Characteristics of Different Surfaces (from Pall et al, 1980). 21 Preliminary tests were conducted by Grigorian et al, 1992 on simple slotted bolted connections for diago-nal bracing. These studies show the benefits of using simple friction connections to dampen the response of structures. Figure 2.8 illustrates the hysteretic response developed by brass on steel slip joints. The tests, which were conducted at slip forces as high as 270 kN, proved to be very repeatable and reliable. Although ductilities in these systems appear to be extremely high, it is likely that slip forces required in practical structures would be much higher than those used in the tests. STEEL ON BRASS HYSTERESIS DIAGRAM 40 -2 -1.5 -1 -05 0 1 1.5 2 DISPLACEMENT (INCHES) Figure 2.8 - Hysteresis of Steel on Brass (from Grigorian et al, 1992). Similar tests were carried out by Tremblay and Stiemer, 1993, on single diagonal braces with bolted slip joints. These tests were conducted on near full-scale members and at 750 kN slip loads which were much higher than those used by Grigorian. Extensive studies were performed on simple and inexpensive slip surfaces to establish materials leading to the most stable and repeatable hysteresis. A combination of mill-scale surfaced steel and cobalt plate slip surfaces were found to provide these desirable characteris-tics. Near full-scale tests were conducted on single diagonal braces with 16 clamping bolts in a 4-slot gusset plate. Some increase in slip force occurred due to the galling of the steel and cobalt; however, after a few cycles, the slip force gradually dropped due mostly to the wearing of the plates. Figure 2.9 shows the hysteresis developed. After retorquing the clamping bolts, the joint was able to sustain a further 30 cycles with similar hysteresis. Pall and Marsh, 1982, developed a new friction damping device for application in a cross-braced configu-ration. Figure 2.10 shows the damper in its location in a single portal frame. 22 Figure 2.9 - Hysteresis of Cobalt on Steel (from Tremblay and Stiemer, 1993). Figure 2.10 - Friction Damper and Friction Damper within a Frame (from Filiatrault and Cherry, 1990). Figure 2.10 also describes how the damper operates within the frame. As the frame deforms, the tension brace force increases and the compression brace buckles. The damper slips at a pre-set slip load and the link mechanism acts to pull the buckled brace taught. When the deformation reverses, the compression brace is ready to undergo tension and slip the damper in the opposite direction. The use of brake lining pads as the wear material has been shown to lead to extremely reliable and repeatable hysteresis curves, 23 as shown in Figure 2.11. Earthquake simulator tests have been conducted on a scaled three storey build-ing fitted with these devices, Filiatrault and Cherry, 1987. Simulator tests involving the Pall dampers were also conducted on a scaled nine storey building, Aiken et al, 1988. ( l b s ) 4 <*>N) [ « c l U l l o n Fr«qu«ncy • O . J H i _ l s.eo 1000 •0.40 -0.90 . 0.30 0.40 •10.li -H .oa • i o . l i - .o  IOOO -4— -*•** -3000 —4— -g.oo Figure 2.11 - Friction Damper Hysteresis (from Filiatrault and Cherry, 1987). These friction dampers have been employed in a new library building in Montreal, Quebec, Pall et al, 1987, in the rehabilitation program for a school in Sorel, Quebec, following the earthquake damage aris-ing from the 1988, Saguenay earthquake, Pall et al, 1991, as well as some other recent buildings in Can-ada, Pall et al, 1993,1994. The Pall System is discussed in more detail in Section 2.2.3. 2.2.2 Concept of an Optimum Slip-Load In a friction damped structure, there is essentially one dominant variable which affects the performance of the structure. This variable is the load at which the damping system slips. Analysis of friction damped structures shows that there is a single slip-load which produces the best minimum response. Analytically, one can imagine that at a very low slip load the friction joint slips constantly. Since the energy dissipated is the product of the slip force and the total slip travel, a very low slip load dissipates little energy and the structure performs as an unbraced frame. If the slip load is set very high, the joint will rarely or never slip, thereby dissipating little or no energy and essentially acting as a moment resisting braced frame. The actual optimum slip-load is the load dissipating the maximum amount of energy in the system. This load 24 is developed when the friction damped structure lies between the two extremes noted above. The opti-mum slip-load is a function of the structural characteristics and the ground motion characteristics, Filia-trault and Cherry, 1990. The task of optimizing the slip load begins with selecting the quantity which is to be optimized. The slip load can be optimized based on many parameters: member forces, deflections, velocities, accelerations and energy quantities. Filiatrault and Cherry, 1990, optimized a performance index expressed as a func-tion of the strain energy within a structure. The structure they investigated was equipped with the Pall System. The optimization parameter, referred to as the Relative Performance Index (RPI), minimizes a combination of the total strain energy and peak strain energy in the friction damped structure relative to the same structure with the slip load set to zero. 2 SEA0 UmaXo_ ' ^ where, SEA = area under the strain energy curve of the friction damped structure, SE A 0 = area under the strain energy curve of the identical unbraced structure (slip load = 0), Umax = peak strain energy of the friction damped structure during the response, U m a x o = peak strain energy of the identical unbraced structure. For values of the RPI: = 1 - the response of the friction damped system corresponds to the behaviour of an unbraced structure, < 1 - the response of the friction damped system is less than the response of the unbraced struc-ture, > 1 - the response of the friction damped system is greater than the response of the unbraced structure. Figure 2.12 shows a graph of the RPI versus slip load for the structure shown. Friction Damped Braced Frame Analysis Program (FDBFAP) is a computer program developed by Filiatrault, Filiatrault and Cherry, 1988a, to optimize the slip load for structures with the Pall System. The program optimizes the RPI and other parameters based on an elastic response of the structure. 25 3 It 0.6m Typical Friction Device pi O.J -r 50 100 J50 200 Local Slip Load [kN] Figure 2.12 - RPI and Structure (from Filiatrault and Cherry, 1988a). A parametric study conducted by Filiatrault and Cherry, 1990, on a single storey friction damped braced frame structure subjected to a sinusoidal ground motion led to the equation below. 2P 0 cosa ma. • = F TU'TU [2.6] where, T D = natural period of the braced structure, T u = natural period of the unbraced structure, Tjj = period of the harmonic ground motion, ag = peak ground acceleration, m = mass of the structure, a = angle of the brace to the horizontal, P G = optimum local slip-load (slip-load of the sliding plates), and F = unknown function. This indicates that the response of a FDBF is dependent on the amplitude and the frequency content of the ground motion. It would seem reasonable that this observation can be extended to most structures with friction damping. It should be noted that the local slip-load refers to the load developed at the slid-ing plates during slip. Global slip-load refers to the sum of the tension and compression brace force dur-ing slip, and the panel shear resistance refers to the frame shear capacity during slip. Filiatrault and Cherry, 1988a, also performed parametric studies to investigate the optimum slip-load dis-tribution (the slip-load for each damper throughout the structure) for a given structure equipped with the 26 Pall System. Their results showed that the optimum slip-load distribution for the structure has the same shape as the dominant mode of vibration of the structure. The study also suggests that the optimum uni-form distribution is a reasonable approximation to the actual optimum distribution. In research conducted by Akbay and Aktan, 1991, a semi-active control approach was applied to a friction damped system. In these analytical studies, the slip load was controlled so that the joint slipped continu-ously during a seismic event. In the control algorithm, a feedback system is used to check if the joint is slipping; if not, the clamping force is dropped until slip occurs. The results indicate that the performance of the simulated structure with variable slip force is superior to the performance of the same structure under constant slip force. The analytical model, however, used brace stiffnesses and storey masses which are unrealistically low. The reason for this is that their model cannot accurately follow the response for a more realistic situation. Research is presently underway at the University of British Columbia by Dowdell and Cherry on the topic of semi-active control of friction damped systems. 2.2.3 The Pall System In tests conducted by Filiatrault and Cherry, 1987, a friction damper, designed by Pall Dynamics, Ltd. of Montreal was used in a one-third scale model of a three storey steel frame structure. Shake table tests were conducted on this friction-damped moment resisting braced frame, an equivalent moment resisting braced frame and an equivalent moment resisting frame. The performance of the FDBF was superior to the performance of the other frames; no yield damage occurred in the beams and columns of the FDBF. Equivalent viscous damping studies by these authors also showed that under a peak ground acceleration of 0.3g the MRF required 71% critical viscous damping and the braced frame 12% critical viscous damp-ing to attain the same response as the FDBF. It was also noted that the energy dissipated in the FDBF increased during more severe ground accelerations. This is due to the fact that the dampers slip more often with stronger ground motion. As the friction damped frame undergoes lateral deflections due to ground motion some dampers, usually those near the ground storeys, begin to slip. This creates a more flexible storey response and tends to isolate the rest of the structure from the ground motion. However, since damping is high when the damp-ers slip, deflections are small and stability is not jeopardized. 27 Another beneficial characteristic of the FDBF is that the natural frequencies of the structure are con-stantly changing, increasing and decreasing, as the dampers start and stop slipping, thus avoiding the development of a quasi-resonance response. In conventional structures the frequencies only change due to damage and the original frequencies are not recoverable. The frequency of a conventional structure may coincide with the predominant frequency of the ground motion. Under such conditions a quasi-resonance response may be induced in the structure and its stability may be jeopardized. In tests conducted by Filiatrault and Cherry, 1987, and Aiken et al, 1988, significant out-of-plane vibra-tions were observed in the friction devices. This was attributed to the out-of-plane eccentricities within the dampers. The damper has since been redesigned to eliminate these eccentricities. Many research papers, such as Pall et al, 1982, Baktash et al, 1986, and Filiatrault and Cherry, 1988b, have been published which compare the response of the FDBF with conventional systems and other innovative systems. The FDBF was shown to out-perform the other systems in most cases. No full-scale tests have been conducted on friction dampers fitted to realistically sized braces. In real structures, brace buckling is possible, depending on the brace size selected, and it is likely that braces will buckle inelastically. As described in Section 2.1, inelastic buckling can lead to elongation of the brace upon restraightening. The effect of brace elongation on the performance of frames fitted with the Pall friction damped system is presently unclear. This question is examined in Chapter 6. 28 3 INCORPORATION OF ENERGY PRINCIPLES INTO DRAIN-2D COMPUTER CODE Dynamic analyses of the structures investigated in this study were performed using DRAIN-2D, a General Pur-pose Computer Program for Dynamic Analysis of Inelastic Plane Structures, developed by Kanaan and Powell, 1973. DRAIN-2D is particularly useful due to its ability to incorporate the inelastic brace buckling elements, EL9 and EL10, developed for DRAIN-2D by Jain and Goel, 1978. This allows the performance of frames with inelastic buckled braces to be predicted. Although the program FDBFAP was employed to evaluate the opti-mum slip-load, it does not model inelastic response. DRAIN-2D is used to observe the effect of brace buckling on the optimum slip load. Since DRAIN-2D does not yield energy quantities, the program was modified to account for these parameters. Chapter 3 is devoted to a review of the evaluation of the energy quantities used in the analysis and the manner in which DRAIN-2D was programmed to incorporate these quantities. Examples are also given showing exact energy calculations versus the output from DRAIN-2D. 3.1 Review of Energy Principles In developing the energy formulation for a structure subjected to ground motion, one starts with the basic differential equation which governs the motion of a structure. For a single degree of freedom system, the basic differential equation is, mx + cx + kx = - m x g , [3.1] where, m = mass of the system, c = viscous damping in the system, k = stiffness of the system, x = relative acceleration of the system, x = relative velocity of the system, x = relative displacement of the system, and x, = ground acceleration imposed on the system. For a multi degree of freedom system, the equation becomes, [m]{x} + [c]{x> + [fc]{x> = -[m]{/>x f f. [3.2] 29 where, [m] = mass matrix of the system, [c] = viscous damping matrix in the system, [k] = stiffness matrix of the system, {x} = relative acceleration vector of the system, <x> = relative velocity vector of the system, {x> = relative displacement vector of the system, {1} = identity vector, and {xB} = ground acceleration imposed on the system. These equations hold true for linear elastic systems. For nonlinear systems, however, the stiffness matrix, [k], is dependent on the displacement, x. The revised equation of motion is then, [m]{x> + [c]{x> + [fc(x)]<x> = -[m]{/>x0. [3.3] When the equation is multiplied by the nodal velocity vector and then integrated over time, the desired energy quantities are obtained since, ENERGY = J FORCE • vdt. [3 .4] Equation 3.3 then becomes, J{x}T[m]{x}df + J{x} T[c]{x}di + f { x } 7 [ f c ( x ) ] { x } d « = - J {x}T\_mUI}xgdt. [3.5] t i t t However, dx .. dx x = ——, x = ——. [3.6] dt dt L J Substituting equations 3.6 into 3.5 the energy balance equation becomes, [m] J{x} T {dx} + [c] J {x}T{dx}+ J[fcO)]{x}{dx} T = -[m] J {I}xg{dx}T . [3.7] X X X X Smplifying gives, 30 [3.8] or, E K + E D + E u E f [3.9] where, ER; = kinetic energy = :;{x}7 [m]{x}, Ej) = viscous damped energy = [c]/ 3 C{x} 7 {dx}, ETJ = strain energy = / x[fc(x)]{x}{dx} T, and ET = input energy = - [m]/ x {7>x 9{dx> T. The formulation presented above is for motion of the structure relative to the ground. In other words, the formulation does not consider rigid body translation of the structure due to ground displacements. An abso-lute energy formulation can be derived but requires the introduction of the ground displacement in the calcu-lations. For practical ranges of structural periods, 0.1 to 5 seconds, the relative energy input and absolute input are the same. For periods outside this range, an absolute formulation should be used. Before discussing how DRAIN-2D is programmed to give energy results, it is necessary to understand how DRAIN-2D solves the basic equilibrium equation. DRAIN-2D uses the constant acceleration method, Clough and Penzien, 1975, to integrate the equations of motion of a system. This method provides stable solutions for all structural periods and time steps used. The incremental equation of dynamic equilibrium can be written as, {Ax}, {Ax}, {Ax}, and { A P } are the incremental kinematic vectors and load vector, respectively, at time, t, and 3.2 Application to DRAIN-2D Code 3.2.1 DRAIN-2D Basics [m]{Ax} + [c T]{Ax} + [ K T ] { A x } = {AP}, [3.10] where, 31 [ C T ] and [ K T ] are the tangential damping and stiffness matrices at time, t. The change in acceleration and velocity over a time increment can be expressed in terms of the corre-sponding change in displacement, and the initial conditions of the time increment. Substituting these rela-tionships into the incremental equation of motion leads to an expression involving the unknown, Ax, namely, ^ [ M ] + |^[C T]+[^ 7]j<Ax} = {P> + [ M ] ^ 2 { x 0 } + { x 0 > ^ j + [C 7]{2x 0> [3.11] where, { x 0 } and { x 0 > are the nodal acceleration and nodal velocity vectors at the start of the time-step, respectively. The vector of change in nodal displacements, { A x>, is calculated from this equation. From the relation-ships between this vector and {Ax}and { A x}, all kinematic quantities can be calculated and the status of the structure is then known at the end of the time increment. The kinematic results are then used as initial conditions for the next time increment, and so on until the specified time limit is reached. During each increment, DRAIN-2D calculates the forces and displacements of each element and transfers the quantities defined by the user to an output file. For an inelastic analysis program like DRAIN-2D, changes in the structure stiffness may occur as a result of yielding. After evaluating the kinematic quantities, DRAIN-2D checks to determine whether yielding has occurred in any element of the structure during that time increment. If yielding has occurred, equilib-rium is not satisfied because the element force is based on the stiffness of the previous increment. Figure 3.1 illustrates this problem. Although, an iterative procedure can be used at this point to re-evaluate displacements based on the true stiffness, this is costly in terms of processing time. To minimize this error, DRAIN-2D calculates an unbalanced force, F U D . This force is the difference between the force which satisfies equilibrium and the actual force based on the element's force-displacement relationship for the displacement calculated. This force is added to the incremental load vector {A P}, for the next time step. 32 A t A t Deflec t ion Figure 3.1 - Unbalanced Force Due to Member Yield. 3.2.2 Energy Balance in DRAIN-2D As discussed above, DRAIN-2D, as do most time dependent finite element programs, solves the basic equilibrium equation at discrete time intervals defined in the data input file. Thus, the incorporation of energy balance in DRAIN-2D requires formulation that must also be discrete. The energy balance equation discussed earlier is, E! E K + E D+ E v . [3.9] For a discrete calculation, however, f ;(f) = f , a - A 0 + - { A x ( 0 } T [ m ] { / } ( x 8 ( ( - A 0 + x „ ( f ) ) , ^^(0 = 2 < ^ ( 0 } 7 [ m ] { x ( 0 > , ED(0 = ED(t-L\0 + -{x(t-L\n+x(0}T[CTUAx(0}, and Eu(t) = Eu(t-L\t}+ -{x(t- L\t} + x(t)}T[KT]{Ax(0} [3.12] [3.13] [3.14] [3.15] where, A x ( 0 = x ( Q - x(t - At). 33 The potential presence of the unbalanced forces (UBF) described earlier must also be considered. Since these forces are applied as external loads, they must be included as energy input. The energy they impart to the structure is the product of these forces by the displacements through which they move. Therefore, the discrete energy input becomes, £ ; ( 0 = f / ( ' - A O - | { A x } T [ m ] { / > ( x f f ( 0 + x f , ( « - A O ) + { f ^ > { A x ( 0 > r . [3.16] The strain energy can be calculated by using the tangent stiffness matrix. This approach, however, does not give strain energy results for different elements or groups of elements. The strain energy is calculated, at the element level, by multiplying the end forces by the end displacements (or rotations). At this point it is also possible to distinguish between elastic (recoverable) energy and plastic (non-viscous) damped energy. Table 3.1 - DRAIN-2D Energy Output File SDOF - Example 1 : U n i t s : kN-mm-sec TIME-STEP=0.0200 ALPHA= 0.000 BETA= 0.000 BETA0= 0.000 DELTA= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.000 % PERCENT KINETIC ENERGY AREA 54.954 % PERCENT VISCOUS ENERGY AREA 0.000 % PERCENT STRAIN ENERGY AREA 45.046 % PERCENT OF STRAIN ENERGY, EL1 100.000 % PERCENT OF STRAIN ENERGY, EL2 0.000 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 0.000 % MAXIMUM KINETIC ENERGY 6071.86 ROOT-MEAN-SQUARE KINETIC ENERGY 1173.82 KINETIC ENERGY AREA 34147.67 STRAIN ENERGY AREA 27991.32 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 40.000 1431.56 24.33 0.00 1407.23 1431.56 0.00 The discrete energy formulation described in Equation 3.16 has been programmed into DRAIN-2D. Appendix B . l contains the various coding additions to DRAIN-2D which are required for calculating the energy quantities. 34 Table 3.1 is an example of the energy output file. The total energy column is the sum of the kinetic, viscous damped, and strain energies. The energy results for each time-step interval are not shown in Table 3.1. The first line in Table 3.1 is the title used in the DRATN-2D data file. ALPHA, BETA, BETAO, and D E L T A are the damping coefficients specified in the DRAIN-2D data file. The percent error in energy is the difference between the input and the total energy divided by the input energy calculated at the end of the analysis. The kinetic energy area is the area under the time-history kinetic energy curve. The percent kinetic energy area is the percentage of the input energy area contributed by the kinetic energy area. The viscous damped and strain energy areas are similarly defined. The percent strain energy, ELi, is the per-cent of the strain energy area contributed by element 'i'. The input energy includes the U.B.F. (unbal-anced force) energy term. 3.2.3 Single Degree of Freedom Example Three examples involving an arbitrary single degree of freedom (SDOF) model were used to test the accuracy of the energy calculations. The choice of a SDOF allows for easy calculation of strain energy. Figure 3.2 illustrates the SDOF model, in which the mass is linked by a spring (two force member) to the excited base. The structural characteristics of the model are described below in Table 3.2; the values are not intended to represent realistic sizes and materials. The horizontal base excitation is provided by 0 to 20 seconds of the El Centro NS, 1940, acceleration record, factored to O.Olg. The three examples, listed in Table 3.3, use different arrangements of damping and yielding to observe their effect on the energy quan-tities. a g / A,E,L ' / s / \ / / M Figure 3.2 - Single Degree of Freedom Model. 35 Table 3.2 - SDOF Parameters Mass 100 kNs2/mm Area 1000 mm2 Young's Modulus 100 kN/mm2 Length 1000 mm Natural Freq. 0.16 Hz Table 3.3 - Damping and Yielding of Examples 1,2,3 Example 1 Example 2 Example 3 Damping None Mass & Stiffness None Yielding None None Elastic-Plastic DRAIN-2D output and energy results for Example 1 are presented in Appendix B-2. Some energy results for the three examples are listed in Tables 3.1, 3.4, and 3.5. A sample calculation is provided for Example 1 as a check strain energy calculations. For a linear elastic response, the strain energy is calculated as, * « , ( 0 = | < * ( 0 } 7 [ * . . „ « , c K * ( 0 } . t 3 - 1 7 ] where, K = — [3.18] elastic £ ' In this structure, with the El Centro excitation, a peak displacement of 7.83540 mm occurs at time t= 18.14 seconds. Then, 36 ^(18.14) =-{7.8354} (100)(1000)' (1000) {7.8354} = 3069.67kN- mm. This agrees identically with the strain energy given by DRAIN-2D in Appendix B.2 at time t= 18.14 sec-onds. Table 3.4 - Energy Results For Example 2 SDOF - Example 2 (damping): U n i t s : kN-mm-sec TIME-STEP=0.0200 ALPHA= 0.010 BETA= 0.010 BETA0= ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.000 % PERCENT KINETIC ENERGY AREA 36.986 % PERCENT VISCOUS ENERGY AREA 34.552 % PERCENT STRAIN ENERGY AREA 28.463 % PERCENT OF STRAIN ENERGY, EL1 100.000 % PERCENT OF STRAIN ENERGY, EL2 0.000 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 0.000 % MAXIMUM KINETIC ENERGY 6008.43 ROOT-MEAN-SQUARE KINETIC ENERGY 985.88 KINETIC ENERGY AREA 26468.59 STRAIN ENERGY AREA 20369.07 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN ENERGY ENERGY ENERGY ENERGY 40.000 1742.97 13.91 1057.12 671.93 0.000 DELTA= 0.000 TOTAL U.B.F. ENERGY ENERGY 1742.97 0.00 Also, the output listed in Table 3.1 for Example 1 shows that the input energy is identically equal to the sum of kinetic and strain energies. The Example 2 results also show that the input energy is identically equal to the total energy, the sum of kinetic, viscous, and strain energy. Since the ground motion is stopped after 20 seconds, the response after this time takes the form of free vibrations. This is evident from Figure 3.3, where input energy is constant after 20 seconds, during which time it is merely transferred between strain and kinetic energies at the natural frequency of the system. In the damped system represented by Example 2, the kinetic and strain energy responses can be seen to decay after 20 seconds due to the presence of viscous damping. 37 Table 3.5 - Energy Results For Example 3 SDOF - Example 3 ( y i e l d i n g ) : U n i t s : kN -mm-sec TIME-STEP=0.0200 ALPHA= 0.000 BETA ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY PERCENT KINETIC ENERGY AREA PERCENT VISCOUS ENERGY AREA PERCENT STRAIN ENERGY AREA PERCENT OF STRAIN ENERGY, EL1 PERCENT OF STRAIN ENERGY, EL2 PERCENT OF STRAIN ENERGY, EL9 PERCENT OF STRAIN ENERGY, EL10 MAXIMUM KINETIC ENERGY ROOT-MEAN-SQUARE KINETIC ENERGY KINETIC ENERGY AREA STRAIN ENERGY AREA FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS ENERGY ENERGY ENERGY 40.000 2414.44 2.50 0.00 0.000 BETA0= 0.000 DELTA= 0.000 0.006 % 10.964 % 0.000 % 89.030 % 100.000 % 0.000 % 0.000 % 0.000 % 6370.79 642.71 9109.80 73973.70 STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY 2411.81 2414.31 5.24 The third example is used to illustrate the effect of member yielding on the energies of the system. As discussed earlier, yielding introduces error into the calculations. The unbalanced forces used to minimize this error impart energy into the structure which appears as a term in the input energy calculations. In Example 3, the yield force of the member was set at about one sixth of the peak elastic force. The results indicate that the unbalanced force energy accounts for about 0.2% of the total energy but minimizes the total energy error to only 0.006%. Figure 3.4 is a plot of error in total energy for the time-history response of Example 3. No error exists in the elastic response of Example 1 or 2. 38 E x a m p l e 1 10 20 Time (s) E x a m p l e 2 Time (s) Figure 3.3 - Energy Balance of Examples 1,2, and 3. 39 Example 3 0.14 j 1 0.13 -0.12 -0.11 -0.1 -0.09 -Time (s) Figure 3.4 - Energy Balance Error of Example 3. The error in the energy calculations is dependent on the time-step chosen for time-integration. It should be noted, however, that this error is not necessarily an indication of the accuracy of the response. Rather, the accuracy of the time-history response is also dependent on how well the ground motion is represented, which is dependent on the time step chosen for integration. In Example 1, the error in energy calculations is zero, but the ground motion for the El Centro 1920 NS accelerogram is poorly represented by DRAIN-2D with a time step of 0.02 seconds. Since DRAIN-2D assumes a constant acceleration over each time-step, the ground motion at each time-step would be better represented with a smaller time-step. 40 4 DESIGN OF A TYPICAL MULTISTOREY BRACED STRUCTURE This chapter discusses the design of the cross-braced structure and the structure retrofitted with friction damp-ers. The design parameters affecting these structures are discussed, as well as code requirements and testing constraints. The design procedure is presented in some detail to illustrate the capacity design approach. 4.1 Design of the Conventional Braced Structure 4.1.1 Design Parameters The structure investigated is a typical medium-rise office building in the downtown core of Vancouver, Canada. The elevation and floor plan of the structure is illustrated in Figure 4.1. The number of storeys chosen is dependent in the dynamic performance of the structure, since deflection limitations exist for the experimental study. Firm ground soil conditions were assumed at the building site. For the full-scale experimental part of the study, seismic storey drifts cannot exceed 75 mm. This is con-trolled by the stroke limitation of the actuator. Since the performance of the bracing was being evaluated, the most severely damaged braces in the structures, as predicted by dynamic analyses, were the elements isolated for the experimental study. The most severely damaged braces were defined as those which expe-rienced the peak drift within the structure. Again, as a result of experimental limitations, the most severely damaged storey was limited by a shear strength which was not to exceed about 2600 kN. Several structures were designed to establish rough member sizes. The number of storeys ranged from two to ten. Dynamic analyses were performed under design earthquakes to obtain rough deflection limits which satis-fied the experimental constraints. The preliminary results indicated that the most suitable response was obtained with a six storey structure, in which the majority of damage was concentrated at the fifth storey. The steel structure shown in Figure 4.1 is designed with composite floors having a 76 x 65 steel deck. The roof consists of open web steel joist, girder and steel deck construction. Frame Y was chosen for the investigation, since the effect of gravity loads is significantly less in this frame than in Frame X. Table 4.1 lists the loads used in the design procedure as well as the parameters applied in calculating the equivalent lateral seismic force. 41 2 @ 7 2 0 0 5 0 0 0 2 @ 7 2 0 0 ® U3 Elevation (Line 1,6) Figure 4.1 - Six Storey Office Building. 42 Table 4.1 - Structure Design Loads Gravity loads Dead loads = 4.5 kPa Occupancy loads = 3.0 kPa Snow loads = 2.0 kPa Total weight = 38740 kN Seismic loads Parameters: v = 0.2 S = 1.5 (based on 1.2 TNBCC) Base shear: V e = 11622 kN V = 2300 kN Since the examined frames were situated at the external walls of the building, they were assumed to take all the torsion induced by seismic excitation. The centre-of-mass and the centre-of-stiffness coincide; thus, only accidental torsion was considered in the design. This torsion was accounted for by increasing the storey masses by 10%, as required by NBCC 1990. This additional 10% was not considered as a gravity load, but only as a horizontal inertial contribution for the purpose of lateral force design. The lateral resistance provided by the beams, columns, and non-structural components outside of the braced bays of the building frame were taken to be negligible; the beams and columns outside of the braced frame are typically 'pin-connected' by clip-angle connections and thus provide little lateral stiffness in comparison to the braced frame. 4.1.2 Selection of Brace Sections This section discusses the brace design procedure and the selected braces. Three different brace sections were considered. The most economical arrangement was selected for the design of the frame and for testing purposes. The brace size was based on the design requirements of S16.1. For design purposes, the compressive resistance, C R , of any brace is governed by the equation, 43 where, Vf = design shear force at the storey in question, and 9 = angle of the brace from the horizontal. This procedure satisfies the requirement that the compression brace take at least 30% of the horizontal design shear force. Table 4.2 lists the brace schedule of the three different section arrangements (rectangular hollow sections, double angle sections, and wide flange sections) chosen for the cross-braced design. Table 4.2 - Schedule of Brace Sections Storey Vf (kN) HSS Double Angle W-flange 6 337 127x76.2x6.35 op 125x90x16 op W100xl9 op 5 611 127x76.2x6.35 op 125x90x16 op W100xl9 op 4 832 127x76.2x9.53 op 125x90x16 op W150x24 ip 3 1000 152x101.6x6.35 op 125x90x16 op W150x24 ip 2 1115 152x101.6x7.95 op 125x90x16 op W130x28 ip 1 1177 177.8x127x6.35 op 125x90x16 op W200x31 ip Notes: op - represents out-of-plane buckling predicted ip - represents in-plane buckling predicted Design for out-of-plane buckling assumes that rotation of the end gussets is possible without developing significant bending moments in the braces. Also, the connection intersection of the braces was assumed to transfer the full tensile yield load produced by the brace. The HSS sections listed are Class C. When choosing the brace size, it became clear that the most efficient brace is one which is oriented so that the force which causes in-plane buckling is nearly equal to the force which causes out-of-plane buck-ling. Although this does not ensure that the best section size is chosen, it does provide a method of rank-ing eligible sections based on compressive resistance. 44 Figure 4.2 presents plots of the storey design shear force, the storey shear resistance of the three sections chosen, and the total mass associated with these brace schemes. Because the section sizes available increase discretely, it is unlikely that the shear resistance profile for the structure will match the design shear force. Slenderness and width-to-thickness ratio requirements further increase the spread between resistances of the acceptable sections. Lateral Resistance Double-angle W— shape I ! I ! + t i i . -U-p 2,000 „ 1,000 500 1,000 1,500 2,000 S h e a r E e s i s t a n c e ( k N ) HSS W-shape Double Angle Figure 4.2 - Design and Actual Shear Resistance. The double angle schedule suffers from the fact that few sections are available which satisfy the require-ments for slenderness and width-to-thickness ratios. Thus, a large overstrength exists for the double-angle sections. Clause 27 requires that both legs of the angles have a maximum b/t of 6.6, in an effort to limit local buckling. However, since out-of-plane buckling was expected, only the outstanding legs were largely susceptible to local buckling. It was therefore decided to waive this requirement for the back-to-back legs, but to ensure that their b/t is still reasonable. Although Class H HSS sections can be used for the design, the weight benefit would be nominal since the Class C schedule is already pushing the limits for slenderness. The weight savings that could be gained by using Class H sections could be offset by their increased cost. The wide flange and HSS schedules are considerably more economical than the double angle schedule, as seen by comparing the total mass. The wide flange option becomes less attractive when the connection design is considered. Since there are four free edges in a wide flange cross-section, shear lag becomes a 45 significant factor in the connection design. Although the code accounts for shear lag in the tension mem-ber by a single factor, this can significantly underestimate the length required to develop the full strength of the section. Appendix A . l shows some W-shape bracing connections. The HSS schedule represents the most desirable bracing on the basis of economics, ease of fabrication, and, as discussed in Chapter 2, provision of the best energy dissipation. Thus, this system was chosen for the design of the six storey office building. Figure 4.3 shows the details of an HSS cross-brace envisioned for the six storey building under investiga-tion. The cross-brace design is all-welded with one continuous and one spliced diagonal brace. The spliced brace is fully welded to the continuous brace at the centre, so that both diagonals act as continuous braces. A tongue plate is inserted through the centre of the continuous brace for this purpose and is welded to the spliced braces as shown in Figure 4.3. This plate serves to transfer the 'web' stresses across the continuous brace, since the web of the continuous brace alone cannot transfer transverse stresses. This arrangement allows the full tensile capacity of the spliced brace to be realized without significantly increasing its out-of-plane bending stiffness. The welded cross-brace unit is then bolted to the frame by the end gussets. The longitudinal weld lengths of the tongue plate and the end gussets are designed so that the full tensile capacity of the braces can be developed. Figure 4.3 - Typical HSS Cross-Brace Details. 46 The example shown in Appendix A.3 illustrates a poor bracing design, since it does not contain a centre gusset to develop the full member capacity. Clearly, there would be a significant difference between the shear capacities and the stiffnesses in the two directions of the brace panel. 4.1.3 Beam, Column, and Connection Design Table 4.3 lists the member sections used for the beams, columns, and braces in the braced frame of the six storey building; the load case which governs the brace selection is also shown in the table. As discussed in Chapter 2, S16.1 provides an upper limit for beam, column, and connection resistance. This upper limit is the forces generated by gravity plus twice the seismic force, D2Q. Although not stated in S16.1, the com-mentary to this Standard, provided in the Handbook of Steel Construction, 1990, suggests that this upper limit be ignored for the upper storeys of buildings. This comment is in recognition of the fact that the higher modes contribute to higher shear forces near the top of a building. Because of this recommenda-tion, the beam, column, and connection resistances were based on the gross yield capacity of the brace. Table 4.3 - Schedule of Braced Frame Members Storey Load Case Beams Columns Braces 6 D2Q W310x67 W310x60 HSS 127x76.2x6.35 5 GBF W310x67 W310x60 HSS 127x76.2x6.35 4 GBF W310x67 W310xll8 HSS 127x76.2x7.95 3 GBF W310x86 W310xll8 HSS 152x101.6x6.35 2 GBF W310x86 W310x226 HSS 152x101.6x7.95 1 GBF W310x86 W310x226 HSS 177x76.2x6.35 Notes: GBF - gross brace force D2Q - gravity plus 2 x seismic force P-A effects were ignored in the design. An evaluation of the influence of the P-A on the design forces of this type of structure, provided by NEHRP, 1991, indicates that the P-A effects can be neglected. 47 4.1.4 Deflection Checks Since the building is located in the downtown core of Vancouver, the dynamic approach for calculating wind pressures was used. This approach, which is outlined in the Supplement to the NBCC 1990, allows for buildings in urban areas to be designed for lower wind pressures. Figure 4.4 shows the building deflec-tions under the design wind loads; the interstorey drifts are less than the maximum allowable limit of 1/500 of the storey height for wind loading. This figure also shows the building deflections expected during a design seismic event. These displacements are determined by multiplying the elastic deflections under the lateral seismic design force by the force modification factor for the structural form. The force modification factor for the ductile braced frame (DBF) is equal to 3. The interstorey deflections are well within the code requirement of 1/50 of the storey height (72mm) for seismic loading. <L> > i-l >> o 6 r-e 3 k U 0 * 0 0 CBF-W i n d W i n d ^ S t d . ) | 50 100 150 200 Deflection (mm) 250 300 Figure 4.4 - Design Seismic and Wind Deflections. 4.2 Design of an Equivalent Friction Damped Structure The friction damped structure used in this study represents a retrofit of the conventional cross braced build-ing discussed in Section 4.1. Friction damped bracing is fitted in the frame in place of the cross bracing used in the conventional system. The optimum slip load of the damper was evaluated by using the analysis 48 program FDBFAP, Filiatrault and Cherry, 1988a. The braces were then selected so that tensile yielding did not occur before the friction elements slipped. A deflection check was made to ensure that the axial stiffness of the braces was adequate to limit the deflections under wind conditions. It should be noted that although the building was designed as a concentric braced frame, the actual brace connections would allow for the transfer of bending moments between the beams and the columns. As a result, in the dynamic analyses, the frame was modelled as moment resisting. As discussed in Chapter 2, the original friction damper had inherent out-of-plane eccentricities. These eccentricities led to significant out-of-plane vibrations; in the latest model of the Pall and Marsh damper these eccentricities have been eliminated. Figure 4.5 shows the original damper and Figure 4.6 shows the redesigned (new) damper. Figure 4.5 - Original Damper. Both damper models function in essentially the same manner; they use a the four-link mechanism to allow 'brace' slip in both directions as the frame deforms. In the original damper, the diagonals contain the fric-tion pads, while the new damper has a centre plate containing the friction pads. The link mechanism in the new damper serves to transfer the slip force to the braces, whereas in the original damper the slip force is transferred directly to the braces. 49 Brace connections Sec t ion A—A Figure 4.6 - New Damper. 4.2.1 Optimum Slip-Load Study The results of the FDBFAP optimum slip load study for the six storey building under consideration are expressed in terms of a Relative Performance Index (RPI) versus the global slip-load; this is illustrated in Figure 4.7. The optimum global slip-load predicted by FDBFAP is 100 kN. This translates to a storey shear resistance of 80 kN. The simplicity of the ductility design concept of the NBCC is based on the principle of equal displace-ments. This principle allows us to lower the strength level of a structure so that even though yielding occurs it is possible to estimate reasonably the resulting ultimate deflections without conducting a non-linear time-step analysis. This principle appears to hold true for the friction damped frame; this is reflected in the RPI being relatively constant over a large range of shear strengths (slip-loads), as shown in Figure 4.7. This result is consistent with research conducted by Filiatrault and Cherry, 1990 and 1987. 50 0 200 400 600 Global S l ip-Load Figure 4.7 - RPI of the Six Storey FDBF. 4.2.2 Effect of Brace Section on the Optimum Slip-Load A parametric study was conducted to examine the effect of brace buckling and brace stiffness on the opti-mum slip load. Brace buckling effectively reduces the elastic stiffness, which is similar to a reduced brace area. An increase in the brace section leads to a reduction in the RPI (improved performance), although not a significant reduction. Therefore, in the absence of brace buckling the RPI value is reduced. Figure 4.8 illustrates how the RPI varies with the brace section area. Since FDBFAP incorporates only elastic brace buckling, it cannot be used to investigate the effect of inelastic buckling on the RPI. This effect can be examined using DRAIN-2D, as is discussed in Chapter 6. It may be seen that brace area has some effect on the optimum slip-load. The larger brace results in a optimum slip-load of 200kN. However, this larger brace, which has three times the weight of the lighter brace, shows only a marginal improvement in the response. For economic reasons, the smaller brace with the lower slip-load was chosen for the six storey building. The brace section recommended by Pall Dynamics, Ltd. (private communication) is a built-up symmetric section, such as double-angles or double-channels. These sections allow for easy installation and avoid out-of-plane eccentricities. Other sections are acceptable as well, provided that the axial force transfer through the brace remains concentric out-of-plane. 51 X v -d ti <u o ti o SH PH 4) 0.9 O.B 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1060 mm? I 1620 mm 2 1 3240 mraf l 200 400 Global Slip-Load 600 Figure 4.8 - Effect of Brace Area on the RPI Function. As mentioned earlier, the brace section must be chosen to avoid yielding before slip occurs. The total brace force at the optimum condition was shown to be 100 kN. If buckling is not allowed the brace force is equally distributed between both braces and the maximum force in each brace is 50 kN. If buckling is allowed, the maximum brace force is dependent on the minimum compressive force in the buckled brace. Filiatrault and Cherry, 1990, recommend that the brace section chosen should minimize the ratio of the braced structural period, T D , to the unbraced period, T u . They show that by limiting <0.4 [ 4 . 2 ] the Relative Performance Index at the optimum slip load is itself a global minimum. Based on the above constraints, double angle sections, 2L-65x50x5mm, were selected as the brace mem-bers. Smaller and cheaper sections are available, but they would be impractical in terms of the connection fabrication. Table 4.4 lists the properties of the braces. The tensile resistance, T r , assumes that the net section is reinforced. FDBFAP yields a T D / T U ratio of 0.26 when using these braces. 52 Table 4.4 - F D B F Brace Properties 2L-65x50x5 Area 1100 mm 2 T r 297 kN Q 118 kN Ix 0.243xl06 mm 4 where, Tr = factored tensile resistance, Cr = factored compressive resistance, Ix = moment of inertia. 4.2.3 Deflection Checks Under design wind loads, deflections are well within code requirements for the selected braces. The top-storey deflection under the design wind load is 8mm. However, the design wind loads result in slippage of the damper at the first storey. The global slip-load required to avoid slippage under wind is 110 kN, instead of 100 kN. As indicated in Figure 4.7, the Relative Performance Index is not very sensitive to slip-loads higher than 100 kN. A global slip-load as high as 297 kN is acceptable for the braces chosen in terms of their yield load, and the minimum RPI value is not significantly altered by this load. In open rural locations, the design wind load can be much larger. In the case of this particular structure, the slip-load must be set high enough to guard against slip during design wind loads. The seismic per-formance must then be evaluated at this higher-than-optimum slip-load. If the slip-load is well outside the optimum performance range it may be necessary to select a different form of seismic resistant system. This requires some investigation but is not considered in this research. The top-storey deflection predicted by FDBFAP under the seismic design load is 110mm. The maximum interstorey drift occurs at the second storey and is approximately 35mm. The NBCC 1990 sets the inter-storey deflection limit of the design structure at 72mm when applying the equivalent static design approach. 53 5 ANALYTICAL STUDY OF THE BRACED STEEL STRUCTURE Dynamic analyses of the braced frame were performed using the general purpose computer program DRAIN-2D. The ground motion selected for the analyses was the first 20seconds of the N-S component of the El Centro 1940 earthquake. This acceleration record was chosen since it reflects the acceleration/velocity ratio for the Vancouver region. The accelerations were factored to meet the peak ground acceleration of 0.21g, which is the NBCC specified amplitude of the Vancouver design earthquake. Several San Fernando, 1971, records which meet the Vancouver design earthquake criteria, were also considered. However, since these events resulted in less predicted damage in the structure than would occur with the E l Centro event, they were subsequently discarded. The use of this more severe ground motion to test the design requirements leads to an upper bound of the damage expected under a design seismic event. Time (s) Figure 5.1 - Vancouver Design and Megathrust Records. An artificially generated megathrust acceleration record was also considered in this study to predict the building response under a very large ampUtude, long duration, ground motion which is feasible for the Vancouver area. The computer program SIMEA was used to generate this artificial record, Filiatrault and Cherry, 1988a. It is noted that SIMEA generates only the strong motion portion of the record based on a stationary random pro-cess. The peak ground acceleration selected was 0.3g; the dominant period was 0.8seconds; the ground damp-ing ratio was 0.32. 54 The ground acceleration time-histories of the El Centro, San Fernando, and the megathrust events are shown in Figure 5.1. Figure 5.2 shows the elastic and design spectra along with response spectra for the three earth-quakes. 0.2 -0.1 -0 E l Centro NS B San Fernando 247 | Megathrust £ Elastic Design Design (R = 3) H A A . 5* Damping —A 0 2 4 Structure Period (s) Figure 5.2 - Design and Actual Response Spectra. 5.1 Modelling of a Frame Within The Structure The modelled frame consists of the braced portion of the frame shown in Figure 4.1. The beam-column assembly outside of the braced frame was considered to have no effect on the structural response. Since the brace connections of a ductile braced frame tend to be very large, the beam-to-column connections of the braced frame were assumed to be fully fixed due to the influence of the brace connections and to be capable of transferring their plastic moment capacity. The beams and columns were modelled using the usual beam-column elements provided with DRAIN-2D. The braces were designed to buckle out-of-plane. The end gussets allow rotation for this to happen and the braces were, therefore, assumed to be pinned to the beam-column assembly. Because the cross-braces are fixed at their intersection point, some interaction between the braces is expected. Since DRAIN-2D can only model in-plane effects (and these effects are expected to be minor) the braces are not connected together at the intersection in the model, thus reducing the number of elements required. The true effective buckling length of the braces is dependent on the tension stiffening contribution of the brace under tension; to 55 account for this interaction an estimated effective length of 0.7 was used in the analyses. The tensile strengths of the braces were modified to reflect the coupon test results and the shear capacity of a full-scale braced panel test. The masses were lumped at the storey levels at each node. To be consistent with the design lateral loads (NBCC), an additional 10% of the storey mass was added to account for accidental torsional amplification. Geometric effects were included in the response prediction. P-A effects were ignored since, as discussed in Chapter 4, they are negligible for this particular structure. For valid comparisons between the CBF and the FDBF, the same damping matrix was used for both analy-ses. The damping matrix was assigned only a mass proportional component of 5% critical damping in the fundamental mode of the unbraced model. Figure 5.3 shows the structural model which was used in DRAIN-2D. Six Storey Braced Frame Lines(l)and(2) (see Figure 4.1) Floor Mass = 6700 kN Floor Gravity Load = 320 kN 5000 Figure 5.3 - DRAIN-2D Model. 5.2 Modell ing of an Inelastic-Buckling Brace The braces were modelled using the inelastic buckling element, EL9, developed by Jain and Goel, 1978, for use in DRAIN-2D. The use of this element has been shown to significantly improve the predicted response of structures with inelastic buckling components. 56 The EL9 element accounts for the drop in buckling load from cycle to cycle by dropping the compressive capacity only after the first buckling load is reached. The buckling load after this remains constant at a lower load. An effective length factor and radius of gyration must be specified. These are used to calculate the overall slenderness and reflects the fullness of the hysteresis loops. 5.3 Response Under the Design Earthquake Table 5.1 lists the calculated peak interstorey drifts for each storey of the six storey structure as a result of the El Centro NS 1940 earthquake excitation scaled to 0.21g. The storey corresponding to the highest drift generally experiences the most damage. In this case, the fifth storey has a substantially higher drift than the other storeys. This would suggest higher modes of vibration are predominant. Recall that the design inter-storey deflection limit set by NBCC 1990 is about 72mm; the actual deflections are well within this specified limit. Table 5.1 - C B F Peak Interstorey Drifts (El Centro NS 1940) Storey Level Drift (mm) 6 29.1 5 43.9 4 24.9 3 24.2 2 20.4 1 20.1 Table 5.2 lists the first four eigenvalues (mode frequencies and periods) determined for the structure. The corresponding mode shapes are plotted in Appendix C. Since the structure investigated is modelled as a plane frame, the torsional modes cannot be isolated by the analysis. 57 Table 5.2 - CBF Eigenvalues Mode Frequency Period No. (Hz) (s) 1 0.570 1.755 2 1.682 0.595 3 3.013 0.332 4 3.119 0.321 E l Cen t ro NS (0.21g) Time (s) Figure 5.4 - Predicted Drift Time-History for CBF. From the response spectra in Figure 5.2, it is seen that the majority of the energy fed into the structure is concentrated at periods which coincide with the higher mode periods of the structure, rather than with its fundamental period. As a result, it is anticipated that the most significant structural damage will result from the higher mode response involving the upper levels of the building. Based on the interstorey drifts listed in Table 5.1, it would appear that the fifth storey suffered the most damage. The drift time-history for the fifth storey, predicted by the computer analysis, was subsequently used in the earthquake simulation tests on the full-scale braced panels for this storey. The results of this test are presented in Chapter 9. The interstorey drift of the fifth storey predicted by the computer analysis is shown in Figure 5.4. The panel and brace hys-58 Br ac e 1 /J •0.7 I 1 1 1 1 1 1 1 ' ' -15 -5 5 15 25 A x i a l D e f o r m a t i o n ( m m ) Brace 2 j> 0.2 £ o.i •a o 3-0.1 : yf/, i i i i i i A x i a l D e f o r m a t i o n ( m m ) P a n e l 1 * 1 1 Floor Displacement (mm) jure 5.5 - 5th Storey Brace and Panel Hysteresis for CBF. 59 teresis curves are plotted in Figure 5.5. As can be seen, only one excursion under gross yield occurred for Brace 1, early during the event. This initial deformation resulted in the time-history response, shown in Figure 5.4, being one-sided (the structure deforms mostly to one side). Following this excursion of yield, all energy dissipation was through inelastic buckling and restraightening. Brace 2 did not reach gross section yield. Figure 5.6 illustrates the extent of damage experienced by the analysed frame. The only strong element pre-dicted to undergo plastic deformation is a fifth storey column. The DRAIN-2D analysis predicted a peak compressive force of 1600kN occurred in this member. Although buckling was not a factor, the compressive force was sufficient for the column to reach the yield interaction surface and initiate hinging. This is attrib-uted to a failure to account for the overstrength of the braces when determining the peak axial forces devel-oped in the strong elements. Nevertheless, stability of the structure in this case was not jeopardized. • •- Buckled and Yielded Brace •€> ©- Buckled Brace -• • Yielded Column Figure 5.6 - Damage Due to Design Excitation. 5.4 Energy Values Table 5.3 lists the various energy quantities after 20 seconds of the E l Centro 0.21g seismic event. Figure 5.7 shows the energy balance diagram, the strain energy (cumulative) breakdown for the individual storey 60 braces, and the total error in energy. It can be seen that the majority of the strain energy is concentrated in the fifth storey and that essentially zero strain energy was dissipated in the sixth storey. It is noted that the total error in energy is insignificant. The peak error for a particular time-step is 0.3%. Table 5.3 -C B F Energy Results (El Centro NS 1940 0.21g) Six S to rey Braced Frame - UNITS :kN-mm-sec TIME-STEP=0.0040 ALPHA= 0.073 BETA= 0.000 BETA0= 0.000 DELTA= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.1279 % PERCENT KINETIC ENERGY AREA 7.538 % PERCENT VISCOUS ENERGY AREA 12.034 % PERCENT STRAIN ENERGY AREA 80.301 % PERCENT OF STRAIN ENERGY, EL2 5.030 % PERCENT OF STRAIN ENERGY, EL9 94.970 % MAXIMUM KINETIC ENERGY 75248.77 ROOT-MEAN-SQUARE KINETIC ENERGY 23586.61 KINETIC ENERGY AREA 343996.63 STRAIN ENERGY AREA 3664744.33 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 20.000 336036.51 3659.86 50209.28 281688.90 335558.04 284.56 Notes: Element EL2 - beams and columns, Element EL9 - braces. 61 E n e r g y B a l a n c e 400 , 1 Time (s) Input Energy E r ro r 8 10 12 14 16 18 20 T i m e (s) Figure 5.7 - Energy Results of CBF under a Design Seismic Event. 62 5.5 Response Under a Megathrust Event The deflection envelopes and peak storey drifts of the frame resulting from the Vancouver megathrust event are shown in Figure 5.8. From the deflected shape, it is concluded that the response is basically in the funda-mental mode. As seen in Figure 5.2, the megathrust energy is concentrated in the vicinity of the fundamental mode of the structure. 0 100 200 300 400 500 600 700 800 Displacement (mm) Figure 5.8 - CBF Deflections and Drifts under a Megathrust Event. Figure 5.9 shows the energy time-history results. The megathrust energy input is ten times that of the design event. This energy must be dissipated in the braces. The strain energies illustrates that the first storey dissi-pates almost half of the input energy. Since the level of inelastic action is significantly higher than for the design event, the total energy error is much higher. As shown in Figure 5.10, most of the beams and columns have yielded; however, a significant amount of energy has not been dissipated by the beams and columns. 63 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 E n e r g y B a l a n c e -Inpu t Energy / \ -A/ /-^  ^ — St ra in Energy -. . Kinet ic Energy A A . A . ^ . . - , < \ , „. / W T T A * Viscous Energy 6 8 10 12 14 16 IB 20 T i m e (s) B r a c e S t r a i n E n e r g y 4.5 4.0 o § 3.5 X a 3-° i a.25 Q 2.0 CJ IB 1.5 1.0 0.5 0.0 I npu t E n e r g y Input Energy E r r o r Figure 5.9 - CBF Energy Results under a Megathrust Event. 6 4 Buckled and Yielded Brace O- Buckled Brace Yielded Column or Beam Figure 5.10 - CBF Damage Due to Megathrust Event. Figure 5.11 shows the hysteresis of the first and fifth storey bracing. Significant brace yielding can be seen in the braces of both storeys. 1st Storey 5th Storey -200 0 Floor Displacement (mm) 120 -100 -80 -60 -40 -20 0 20 40 Floor Displacement (mm) Figure 5.11 - Megathrust Brace Hysteresis of 1st and 5th Storey. Figure 5.12 shows the envelopes of the axial forces and bending moments for the columns. All of the column axial resistances were exceeded during this event. This is attributed to the mainly fundamental mode response. In this mode, the columns experience the maximum axial load due to the fact that all of the braces 65 are contributing to a particular column's compressive force at the same time increment. Thus, the influence of the brace overstrength on the column load was significant, leading to stability problems, which could cause collapse. Buckling elements for the beams and columns were used in an attempt to improve the response predictions; however, it was found that this led to a dramatic increase in the energy error. Column Moments 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 Bending Moment (kN—m) Column Axial Force 0 2 , 0 0 0 4 , 0 0 0 6 , 0 0 0 8 , 0 0 0 Axial Force (kN) Figure 5.12 - Megathrust Column Forces. 66 Since the interstorey deflections, seen in Figure 5.11 are beyond the deformation range of the available test-ing equipment, laboratory tests corresponding to the megathrust response were not conducted. Further-more, as discussed in the previous paragraph the recognized inadequacy of the analytical results would not justify an experimental comparison. 67 6 ANALYTICAL STUDY OF THE FRICTION-DAMPED BRACED STRUCTURE As discussed in Chapter 4, the friction damped frame is being treated as a retrofit of the conventional frame; thus, the beams and columns remain the same. The earthquakes used with the braced steel structure were also apphed in the analyses of the friction damped frame, FDBF. 6.1 Theoretical Formulation Damper Performance The damper slip-load is expected to vary in a non-linear manner with the damper deformation. In the case of the original damper shown in Figure 4.5, this results from the compression slip joint sliding less than the tension joint. The theoretical deformation limit of the new damper corresponds to 0= 90°, as seen in Figure 6.1; the actual limit is, however, dependent on the length of the slipping slot. At ©=90°, the tension slip joint has slipped a total of 230 + 305-(2302 + 305 2)V 2= 153mm, whereas the compression joint has slipped twice this amount [(2302 + 3052)!/2-(305-230) = 307mm]. As discussed in Chapter 2, the operation of the new damper is somewhat different than the operation of the original model. It has two slip joints which slide relative to the links. By equating energy, it is possible to determine the relationship between the brace force required to cause slip, the friction force at the slip joint, and the damper deformation. Figure 6.1 shows the damper illustrated in Figure 4.6 in its deformed shape. For simplicity the compression brace force is initially assumed to be equal to zero: the formulation with a compression brace contribution is discussed later in this section. The external work, W e , of the system is, [6.1] Ft externally apphed tension force causing slip, and A s diagonal damper slip coincident with F(. However, F^ is expected to vary with L \ SO the work expression becomes, [6.2] A 68 The internal work, Wi, of the system is, Vl = 4-Ffbf, [6.3] where, Ff = dynamic friction force at the slotted joint, and 6 / = relative displacement in the slotted joint. Since there are four slip surfaces, one on each side at the top and bottom of the centre plate, the equation is multiplied by four. A / s^' Slip Surface Figure 6.1 - Damper in the Deformed Position. In Figure 6.1, the relative movement between the centre plate and the upper and lower link plates is 6f. Thus, where, 69 = aG then [6.4] This is a simplified formulation since the centre plate actually rotates relative to the link plate about a radius of half the damper height. For large damper deformations, 9 approaching 90°, this formulation can lead to significant error. For the damper deformation limits imposed by the slots, this formulation should be valid. Then, A + A = o — S a2 + b 2 -2abcos | - + 9 2 and [6.5] A. ( a 2 + b 2 ) 2 . [6.6] Substituting Equation 6.4 into 6.6 gives, A - a 2 + b 2 - 2abcos( -+ - 6 , 2 a 1 ( a2+ b 2 ) 2 . [6.7] Rearranging, cos i Az + 2 ( a 2 + b 2 ) 2 A ; V -2ab J [6.8] Substituting Equation 6.8 into Equation 6.3 leads to the following expression for the internal work, W, = 2Ffa . - i A2 + 2 ( a 2 + b 2 ) 2 A -2ab [6.9] Equating W e =Wj yields, F , d A s = 2F fa : o s - W A2 + 2 ( a 2 + b 2 ) 2 A s V -2ab [6.10] Differentiating both sides with respect to Ag, gives 70 F , - 2 F , a -A. cos A 2 + 2 ( a z + b 2 ) 2 A s -2ab 7 2 [6.11] Using the trigonometric identity, d 1 du. -cos u = d x s J l - U Z D X then, Ff 2 A s + 2 (a 2 + b 2 ) 2 1 — I A 2 + 2 C a 2 + b 2 ) A [6.12] From Equation 6.12, it can be seen that F( is nonlinear with respect to the damper deformation, Ag. As a comparison, a similar formulation is presented for the original damper, Figure 4.6. Equating work, f F,dA, = FfA. + F,i [6.13] where, A s = tension slip joint deformation and A s . = compression slip joint deformation. Differentiating gives, [6.14] where, similar to Equation 6.6, a 2 + b 2 - 2abcos| ~^-Q ( a 2 + b 2 ) \ [6.15] and, 71 Substituting Equation 6.15 into Equation 6.14 gives, A s , = ( a 2 + b 2 ) 2 - ( a 2 + b 2 _ A 2 _ 2 ( a 2 + b 2 ) i A J . [6.17] Differentiating yields, A s + ( a 2 + b 2 ) 2 [6.18] A 2 - 2 ( a 2 + b 2 Substituting into Equation 6.14 leads to, f A s + ( a 2 + b 2 ) 2 1 + [6.19] V ( A 2 - 2 ( a 2 + b 2 ) i A j J The theoretical performance of the original damper used by Filiatrault and Cherry, 1987, is shown in Figure 6.2a. The actual experimental results obtained by these authors in their study are illustrated in Figure 6.2b. The hysteresis losses evident at the corners of the experimental hysteresis loop are due to fabrication toler-ances in the corner bolt holes of the damper link plates. It may be seen that the theory and experiment are in excellent agreement. Equation 6.12 and 6.19 are plotted in Figure 6.3. The new damper shows very slightly more nonlinear beha-viour than the original damper. Of particular note, for a given clamping force, the new damper produces a brace force which is 25% higher than that of the original damper. 72 O r i g i n a l Damper (200x355) (a) Theoretical. 12 _10 z M 8 -1$ _l I I 1 I L re Ol I .&2 i-H -12 -8 -10 -12 _i i i i i i i_ 4 8 12 Elongation (mm) 16 (lbs) <i (kN) E x c i t a t i o n F r e q u e n c y • O.i H i (b) Experimental. Figure 6.2 - Theoretical Comparison with Filiatrault and Cherry, 1987. It should be noted that the formulation assumes that the brace force, F ,^ is independent of the rate of slip. This may not be true, and is dependent on the characteristics of the friction materials. Also, the formulation assumes that there is no compression brace influencing the performance. In the presence of a compression brace, the external work on the system becomes, 73 Figure 6.3 - Slip-Load of New and Original Damper. W.= f F,dAs+ f FcdAs,, [6.20] where, F c = compression brace force, and As. = diagonal damper slip coincident with F c . This expression for work is the same for both damper types. Obviously this equation is much more difficult to evaluate. However, the variation in total brace force for the new damper can be approximated by sum-ming Equation 6.12, representing the tension brace, and the same equation applied to a brace under com-pression. The compression brace equation is simply the negative of the tension case, but 180° out of phase. Each brace force equation can be factored to represent its percentage of contribution. Figure 6.4 illustrates the resulting total brace force variation for several combinations. The combination of 50/50 represents the case where no brace buckling occurs and axial stiffness in tension is identical to compression. The 100/0 combination represents the case where the braces can resist zero compression force. Table 6.1 lists six brace combinations with the variation in brace force. The force variation is measured between the slip-load at zero damper deformation and the slip-load at a 40mm diagonal elongation of the damper. The 40mm elongation was chosen since this was the maximum diagonal deformation calculated in 74 the dampers used in the six storey FDBF. An equal force contribution from each brace (no brace buckling) results in the most constant brace force. The 100/0 combination represents the largest deviation from the constant slip-load approximation. Figure 6.4 - Influence of Compression Brace Contribution. Table 6.1 - Influence of Compression Brace Combination Force Variation (tens./comp.) (@ As = 40mm) 100/0 22% 90/10 18% 80/20 13% 70/30 8.8% 60/40 5.2% 50/50 2.5% Damper Response within a Frame Panel It can be shown that a secondary tension force is induced in both braces within a braced panel as the panel 75 deforms. Thus, consider a panel of the six storey frame in Figure 4.1 in its deformed state as a result of a lateral panel displacement of 100mm, as shown in Figure 6.5. The resultant diagonal elongation, A bfl00t of the frame can be calculated as follows: - { 100 \ sin 1 = 1.59175°, 1,3600,1 V36002+ 50002 = 6161.17mm, V36002 + 5000 2-2- 3600 • 5000 • cos(90+ 1.592) = 6241.79mm, 6241.79-6161.17 = 80.62mm. where, 6 { 0 = the original diagonal length of the undeformed frame panel, bfioo= the diagonal length of the frame due to a lateral panel displacement of 100mm, and A 6 / J 0 0 = the elongation of the frame diagonal due to a lateral panel displacement of 100mm. Figure 6.5 - Frame and Damper in the Deformed Position. Assuming that the frame and bracing system is rigid in comparison to the damper under slip, the diagonal elongation of 80.62mm occurs entirely within the damper. Thus, the diagonal elongation of the damper cor-responding to a lateral frame displacement of 100mm, A6 d, 0 0must also be 80.62mm. Based on this, the diagonal shortening in the opposite direction is similarly calculated as, 76 _ 1 ( (382 + 80 .62 ) 2 -230 2 -305 2 , 9rf, = cos - =60.965 c 2-230-305 6 d y o o , = V230 2 +305 2 -2- 230- 305- cos(60.965) = 278.98mm, A 6 d , 0 0 . = 382-278.98= 103.01mm. where, A6 d ; 0 0 . = the shortening of the damper diagonal due to a lateral panel displacement of 100mm. Yet, the frame diagonal deformation is, 6 / / 0 0 , = V3600 2+ 5000 2 -2 - 3600- 5000 • cos(90- 1.592) = 6079.47mm, .-. A 6 / J 0 0 , = 6161.17-6079.47 = 81.70mm. where, A 6 / J 0 0 . = the shortening of the frame diagonal due to a lateral panel displacement of 100mm. The difference in diagonal deformation, A 6, is 103.01 - 81.70 mm = 21 mm. Thus, an increase in tension must occur in both braces as the frame deflects from the zero position. 6.2 Effect of Damper Aspect Ratio and Size From Equations 6.12 and 6.19, which define the slip load variation of the dampers, it is clear that the aspect ratio influences the nonlinearity of the slip load. The friction dampers were designed to maintain the same aspect ratio as the frame panel; therefore, it is of interest to investigate the effect of the aspect ratio on the slip-load variation. Figure 6.6 shows the results of such an investigation. Three different aspect ratios are used and compared with the damper under investigation. Table 6.2 lists the characteristics of the dampers. The base dimension each damper is 305mm. Again, the variation refers to the difference in brace force between zero elongation and 40 mm elongation. The brace force factor is the ratio of the brace force over the friction force at the slip joint. 77 ase Length, b = 305 m m -20 0 20 40 Diagonal Damper Elongation (mm) Figure 6.6 - Influence of Damper Aspect Ratio. TABLE 6.2 - Influence of Aspect Ratio Height Base Ratio Variation Brace Force (mm) (mm) (a to b) (40mm) Factor 153 305 l to2 26% 2.24 230 305 1 to 1.3 22% 2.51 305 305 I t o l 19% 2.83 610 305 2 t o l 12% 4.47 Based on the results shown in Figure 6.6 and Table 6.2, it may be seen that there can be significant deviation from the assumed constant slip-load with a change in the damper aspect ratio. Also, it is noted that the larger the aspect ratio, the lower is the required clamping force to generate a particular slip-load. This fea-ture can be important when selecting the frictional surfaces. The dampers in the superstructure of the Concordia Library, Pall et al, 1987, are similar in design to the new damper and have aspect ratios of 1 to 2.2. The resultant slip load variation is in excess of 30%. This, of course, assumes that the braces do not transfer compressive forces. 78 The relative size of the damper compared to the frame panel also influences the variation in slip-load. The larger the damper, the lower the slip-load variation, since under a certain frame deformation the damper angle change is reduced. The basis for selecting the size of the damper relative to the panel size would seem to be dependent on the maximum elongation required of the damper. Physical limits exist on the length of the damper slots. Larger slips would require longer slots and, therefore, a larger damper. The smallest possible damper would be the most economical. 6.3 Modelling of the Friction Damper A single bay FDBF was analysed using DRAIN-2D. The analysis was conducted to investigate the mathe-matical performance predicted above. The damper was modelled as accurately as possible with twelve beam-column elements, EL2, (four for the centre plate and eight for the link plates) and two truss elements, ELI (used for the yielding elements which connect the centre plate to the link plates at the slots), as indi-cated in Figure 6.7. The braces, beams and columns of the frame were modelled with buckling elements, EL10. The beam and columns are not shown in Figure 6.7 for clarity. The hysteresis curve shown in Figure 6.7 confirms the brace force based on a 50 kN local slip-load and the characteristic variation in slip load which was predicted mathematically above; however, the predicted nonlinear behaviour was not observed, possibly because it represents a second order effect. To reduce the computing time when analysing the six storey structure, the damper was modelled in its sim-pler original form involving only six truss elements. The difference in damper performance between the original and the new damper is considered negligible, since the energy dissipated in a single cycle is essentially the same for the two models. It is important to recognize that the brace force imparted to the beam-column frame is the governing crite-rion for optimum performance. Representing the new damper by the simpler, original model leads to one deficiency. Specifically, the local slip-load in the original damper leads to a global slip-load which is higher than the global slip-load created by the new damper with the same local slip-load. Thus, it is important to relate performance to global slip-load and not to local slip-load. The damper elements were modelled as pinned truss elements. The braces were modelled with special beam-column elements similar to EL9 as described in Chapter 5.2. The brace elements were taken as pinned to the damper and fixed to the beam-column frame. Previous modelling techniques by Filiatrault and 79 C o m p l e x D a m p e r M o d e l Figure 6.7 - Complex Damper Model and Hysteresis. Cherry, 1987, used a combination of bending and axial members to model the braces to avoid the unstable rigid body rotation of the damper. By fixing the braces to the beam-column frame this instability is avoided. This is realistic, since the brace to beam-column connection has two bolts, moments can be transferred across this joint. Initial attempts to analyse the six storey FDBF led to some interesting observations. The damper underwent rigid body rotations, by as much as 40 degrees, even though the braces were not allowed to yield in bending and the frame deflections were reasonable. Since the brace bending resistance is still present, it would appear that the damper rotation was not associated with an instability; although the damper rotational stiff-ness was very low, the analysis was still numerically stable. Decreasing the time-step in the analysis has no 80 effect on this anomaly. This apparent rigid rotation may be related to the observation, noted previously, that in the deformed state of the frame, secondary forces are induced in the damper; hence, the damping unit wants to undergo rotation to redistribute these force evenly into the braces. Such damper rotations were not reported in the experiments described by Filiatrault and Cherry, 1987; however, the braces used in their experiments were fixed to both the beam-column frame and the damper, providing a stiffer assembly. In an attempt to minimize this problem, the size of the damper in the computer model was increased to one 1/5 of the frame panel size from the actual 1/16 size. The damper aspect ratio was kept the same. The large rigid rotations of the damper disappeared when the damper size was altered. Increasing the damper size has two effects. It decreases the secondary forces induced in the deformed shape of the frame and increases the bending stiffness of the braces. This serves to minimize the tendency for the damper to rotate. The question then arises, is this a valid alternative to the actual damper size? To answer this question, the effect of the brace force on the frame must be examined. Under the same local slip-load the brace force will remain the same. The difference in the analysis arises because of the nonlinear nature of the brace force. By increasing the damper size its distortion during slip is decreased for a given panel deformation and, as a result, the degree of nonlinearity of the brace force is decreased. Since this tends to be relatively minor, the panel with the increased damper size should lead to only negligible change in the analytical results. An analysis using FDBFAP with the oversized damper resulted in a 0.1% and 0.5% decrease in the energy input and the energy dissipated by friction, respectively. The RPI value for a 50kN slip-load was decreased by 0.5%. Thus, it was concluded that the oversized damper would be acceptable. The natural frequencies of the structure were dramatically changed under elastic conditions by altering the damper size. Under the slip condition, the frequencies were independent of the damper size. Table 6.3 lists the predicted frequencies of the FDBF with the damper modelled as actual-sized and with the over-sized damper. Appendix D shows the mode shapes corresponding to these frequencies. Since the response is controlled primarily by the performance under the slip condition, the overall response should not be greatly affected by damper size. 81 Table 6.3 - FDBF Natural Frequencies (Hz) Mode Actual-sized Over-sized Unbraced Damper Damper Structure 1 0.202 0.417 0.113 2 0.525 1.196 0.276 3 0.920 2.111 0.520 4 1.154 2.828 0.656 6.4 Response Under a Design Earthquake Analysis using DRAIN-2D with FDBF100 (lOOkN global slip-load), under E l Centro NS 1940 scaled to 0.21g, indicates that the first storey damper underwent significantly more slippage than dampers in the other storeys. The drift of this storey was therefore used in the full-scale testing program. The drift time-history of this storey is plotted in Figure 6.8; also shown is the first storey hysteresis curve. The results of the DRAIN-2D analysis indicate that the compression braces remained in compression during their entire 'compression' excursion. This would suggest that if the brace buckled, it would not be straightened until the frame displacement reversed. This is contrary to the assumption, illustrated in Figure 2.10, that the buckled brace is pulled straight when the damper begins to slip. Such behaviour could dramati-cally drop the stiffness of the system during load reversal, especially if the brace is buckled. Figure 6.9 is a comparison of the DRAIN-2D top storey deflection time-history with the corresponding deflection obtained from the FDBFAP analysis under the same lOOkN optimum global slip-load. DRAIN-2D predicted an entirely elastic response and, since the FDBFAP response is elastic as well, a good agreement is observed. The largest discrepancy arises during times of high slippage. This is likely the result of FDBFAP's greater cumulative error during the computations. DRAIN-2D minimizes the effect of equi-librium unbalance during the overshoot of the yield plateau, as discussed in Chapter 3, by adding unbalanced forces. FDBFAP does not contain this feature. It should be stated that the overshoot of the yield plateau arises from the slipping elements, which are simply yielding elements. This cumulative error is reflected in the energy balance error calculation in the output of each program. The final energy quantities and the error in energy balance for FDBFAP and DRAIN-2D are presented in Table 6.4a) and b). The FDBFAP energy 82 Dri f t -T ime History I i i i i i i i i i i i i i i i i i 0 2 4 6 8 10 12 14 16 18 20 Time (s) • Hysteresis Curve 100, ^ 1 Panel Displacement (mm) Figure 6.8 - FDBF100 First Storey Response Under Design Seismic Event. error is six times that of the DRAIN-2D response (0.1785% versus 0.0270%, respectively), although both are relatively small. For the purpose of comparing the energy dissipated by friction, in the DRAIN-2D response this energy is the strain energy from element, ELI. However, it should be noted that the ELI energy consists of both elastic and plastic energy; the plastic energy is equivalent to the energy dissipated by friction. 8 3 FDBF100 80 , • —130 I I i i i i i i i i i i i i i i i i i i i i _ 0 2 4 6 8 10 12 14 16 18 20 Time (s) Figure 6.9 - Comparison of Top Storey Deflection of DRAIN-2D and FDBFAP. The differences in the energy input and the energy dissipated by friction are likely a result of the FDBFAP's simplified representation of the structure. FDBFAP assumes the beams and columns have an infinite axial stiffness. This results in a stiffer structure and, therefore, based on the excitation response spectra, the energy fed into the structure would be slightly higher. Figure 6.10 shows the energy balance resulting from the design excitation for each of the models studied. Both methods of analysis yield characteristically similar results. It should be noted that the strain energy is almost entirely due to energy dissipated by friction. A comparison of the damper slippage envelope of each model is shown in Figure 6.11 and Table 6.5. As can be seen, the DRAIN-2D slip profile shows a more fundamental mode response, while the FDBFAP slip pro-file can be viewed as containing higher mode contributions. 84 DRAIN-2D 400 | 1 350 -300 -250 -200 -150 -Time (s) FDBFAP 400 | 1 350 -300 -250 -200 -150 \-Time (s) Figure 6.10 - Energy Balance Comparison for FDBF100. 85 Table 6.4a - DRAIN-2D Energy Output For FDBF100 F667 - Local SL=50kN : kN-mm-sec TIME-STEP=0.0020 ALPHA= 0.073 BETA= 0.000 BETA0= 0.000 DELTA= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.0270 % PERCENT KINETIC ENERGY AREA 3.803 % PERCENT VISCOUS ENERGY AREA 13.063 % PERCENT STRAIN ENERGY AREA 83.107 % PERCENT OF STRAIN ENERGY, EL1 93.909 % PERCENT OF STRAIN ENERGY, EL2 5.795 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 0.297 % MAXIMUM KINETIC ENERGY 51672.25 ROOT-MEAN-SQUARE KINETIC ENERGY 6311.24 KINETIC ENERGY AREA 77517.97 STRAIN ENERGY AREA 1693795.30 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U .B .F . ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 20.000 82264.94 1766.91 10970.40 69503.64 82240.65 15.9 Notes: ELI elements - friction damper EL2 elements - beams and columns EL10 elements - braces Table 6.4b - FDBFAP Energy Output For FDBF100 FRICTION - UNITS: kN-mm-sec DEVICE* LOCAL SLIP LOAD TOTAL SLIPPAGE 1 50.00000 597.46770 2 50.00000 680.20920 3 50.00000 481.32350 4 50.00000 195.37350 5 50.00000 113.64660 6 50.00000 6.80353 ENERGY BALANCE AT TIME= 20.0000 ENERGY INPUT= 118519.30000 ENERGY DISSIPATED BY FRICTION= 103741.20000 ENERGY DISSIPATED BY VISCOUS DAMPING= 13448.90000 RESIDUAL KINETIC ENERGY= 725.90380 RESIDUAL STRAIN ENERGY= 391.68530 ERROR IN ENERGY BALANCE= 1785% 86 0 200 400 600 800 1,000 Damper Slippage (mm) Figure 6.11 - DRAIN-2D and FDBFAP Damper Slippage. 6.5 RPI and DRAIN-2D The Relative Performance Index (RPI) function introduced by Filiatrault and Cherry, 1990, was developed in the present study through analysis using DRAIN-2D. It should be noted that the computational effort required to develop these RPI values by DRAIN-2D is extremely tedious, since it involves numerous time consuming dynamic analyses. Figure 6.12 presents the RPI function developed for the 6-storey friction damped frame from the DRAIN-2D analyses; the function established by the program FDBFAP is also pro-vided for comparative purposes. Table 6.6 lists the energy quantities used to calculate the DRAIN-2D RPI values. Table 6.5 - Damper Slip page Damper Storey DRAIN-2D FDBFAP 6 13 mm 7 mm 5 36 114 4 71 195 3 157 481 2 302 680 1 768 597 Total 1347 mm 2074 mm 87 Global Slip-Load (kN) Figure 6.12 - RPI Comparison of DRAIN-2D and FDBFAP. Table 6.6 - DRAIN-2D Strain Energy Quantities Global Slip-Load Strain Energy Area (kN-mm-s) Peak Strain Energy (kN-mm) 0 124697 13731 30 63194 8335 50 55356 8571 100 51338 9236 150 50299 9570 200 50659 9126 250 51964 8549 300 53759 7837 400 59686 6332 500 63619 7296 600 69552 9626 800 75744 15111 88 The DRAIN-2D RPI function defines the optimum slip-load for the system as 400kN, although a local mini-mum is observed around lOOkN. A lOOkN optimum value was obtained by FDBFAP approach. A better optimum slip-load is expected from the more accurate representation offered by the DRAIN-2D analysis. An examination of the interstorey drift provides some insight into the reasons for the difference in the slip-loads by the two analysis methods. Interstorey drift is of particular interest since it is the underlying cause of damage to structural and nonstructural components of buildings. Table 6.7 lists various drift values for the structure from the DRAIN-2D analyses. The peak drifts listed in Table 6.7 all occur in the first storey. The drift area, the area under the drift time-history graph, represents the cumulative drift experienced during the event. The drift quantities listed for each storey are also drift areas. Table 6.7 - Drift Quantities for FDBF (DRAIN-2D) Drift Global Slip-Load 100 200 300 400 500 600 800 Peak (mm) 67 67 59 47 38 38 50 RMS (mm) 9.3 8.7 8.3 8.0 9.0 8.9 9.3 Drift Area 662 614 631 691 830 797 816 Storey - 6 25 53 70 85 91 86 93 5 51 96 89 100 170 182 153 4 53 64 75 136 158 123 151 3 90 65 67 90 115 111 117 2 136 100 104 93 105 105 110 1 307 233 224 189 189 188 192 Under the lOOkN slip-load it is clear that the majority of the deformation occurs in the first storey. This is consistent with the DRAIN-2D results of Table 6.5 listing the storey slippage quantities. From the results in Table 6.7 a more uniform distribution of drift area is seen with a slip-load of 400kN. In addition, the peak and RMS drifts at 400kN slip-load are much improved over the lOOkN slip-load. 89 Since DRAIN-2D predicted a different slip-load than FDBFAP, it was intended that both slip-loads would be used in the full-scale testing program; however, as noted later in Chapter 10, the damper could not sustain a 400kN slip-load. Figure 6.13 shows the first storey drift time-history of the two slip conditions. B R A I N - 2 B a Time (s) Figure 6.13 - First Storey Drift Time-History Comparison of FDBF100,400. 6.6 Effect of Inelastic Brace Buckling In practice, braces which buckle usually buckle inelastically. This is because the slenderness of typical braces is low enough that some inelastic action will occur. Braces with slenderness ratios above 300 generally have negligible yielding during buckling; braces with slenderness ratios below 200 generally have significant yield-ing. As discussed in Chapter 2, this inelastic action substantially accelerates fatigue failures relative to such failures under elastic deformation conditions. In addition, cyclic inelastic brace buckling causes cyclic elongation of the braces due to tensile inelastic restraightening. This elongation results in increased axial compressive deformations. The buckling of braces attached to friction dampers can be expected to experience similar problems, although such problems are not as severe due to the onset of slip. Also, the secondary forces induced in the system in the deformed position result in an increase in tension, and, if the axial stiffness is high enough, this tensile increase can lead to stresses that are higher than those for which the section is designed. Although 90 the computer model results do not indicate that this occurred, as discussed in Chapter 10, test results con-firm this behaviour. It is possible to model the inelastic buckling of the braces by introducing the end-moment buckling element, EL10, provided by DRAIN-2D. Dynamic analyses were performed on the six storey structure for two global slip-loads, 400kN and 600kN. To observe the effect of brace buckling on the dynamic response, the braces sizes were chosen so that inelas-tic buckling occurred prior to damper slip. The brace slenderness in the direction of buckling was estimated to be 120. The energy results of these analyses are presented in Tables 6.8 and 6.9. These tables also provide the results for the elastic, no buckling, case for comparative purposes. For the 400kN slip-load analysis there is a negligible increase in input energy when comparing the elastic (179626kN-mm) and inelastic (184028kN-mm) buckling cases. The total kinetic and strain energies also remain essentially the same. The energy dissipated in the two cases is, however, significantly different. Under elastic conditions the braces account for 1% of the strain energy, but once inelastic buckling is allowed the braces consume almost 26% of the strain energy. For the 600kN case, the corresponding brace strain energy changes from 2% to 36%. Table 6.10 lists the energies dissipated by friction for the analysed cases. It is seen that inelastic buckling can substantially reduce the effectiveness of the damper. In the 400kN case, some of the energy dissipated by friction is simply transferred to the braces without much effect on the performance of the structure (see Table 6.8). In the 600kN case inelastic brace buckling did affect the performance of the structure (see Table 6.9). For this latter case, the results showed that the drift area was increased, although the overall peak drift was not affected. The effect of inelastic buckling on the RPI function is shown in Figure 6.14. The increase in the RPI with inelastic buckling is a reflection of the poorer response which accompanies inelastic brace buckling. 91 Table 6.8 - Energy Results for FDBF400 FDBF - LSL=200 kN ( e l a s t i c ) : kN -mm-sec TIME-STEP=0.0020 ALPHA= 0.073 BETA= 0.000 BETA0= 0 000 DELTA= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.0279 % PERCENT KINETIC ENERGY AREA 2.730 % PERCENT VISCOUS ENERGY AREA 8.776 % PERCENT STRAIN ENERGY AREA 88.466 % PERCENT OF STRAIN ENERGY, EL1 96.834 % PERCENT OF STRAIN ENERGY, EL2 2.112 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 1.054 % MAXIMUM KINETIC ENERGY 59738.95 ROOT-MEAN-SQUARE KINETIC ENERGY 8226.83 KINETIC ENERGY AREA 118854.75 STRAIN ENERGY AREA 3851847.96 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 30.000 179625.84 396.23 17351.68 161826.85 179574.76 33.40 FDBF - LSL=200 kN ( i n e l a s t i c ) : kN-mm-sec TIME-STEP=0.0020 ALPHA= 0.073 BETA= 0.000 BETA0= 0 000 DELTA= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 1.3685 % PERCENT KINETIC ENERGY AREA 2.721 % PERCENT VISCOUS ENERGY AREA 8.920 % PERCENT STRAIN ENERGY AREA 86.990 % PERCENT OF STRAIN ENERGY, EL1 71.935 % PERCENT OF STRAIN ENERGY, EL2 2.352 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 25.713 % MAXIMUM KINETIC ENERGY 64571.73 ROOT-MEAN-SQUARE KINETIC ENERGY 8707.40 KINETIC ENERGY AREA 121205.57 STRAIN ENERGY AREA 3874756.23 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 30.000 184028.45 446.94 17695.04 163111.65 181253.63 2979.20 Notes: ELI elements - friction damper EL2 elements - beams and columns EL10 elements - braces 92 Table 6.9 - Energy Results for FDBF600 FDBF - LSL=300 kN ( e l a s t i c ) : kN -mm-sec TIME-STEP=0.0020 ALPHA= 0.073 BETA= 0.000 BETA0= 0 000 DELTA-= 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 0.0226 % PERCENT KINETIC ENERGY AREA 3.233 % PERCENT VISCOUS ENERGY AREA 9.895 % PERCENT STRAIN ENERGY AREA 86.850 % PERCENT OF STRAIN ENERGY, EL1 96.230 % PERCENT OF STRAIN ENERGY, EL2 2.082 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 1.688 % MAXIMUM KINETIC ENERGY 75836.85 ROOT-MEAN-SQUARE KINETIC ENERGY 10857.58 KINETIC ENERGY AREA 167158.30 STRAIN ENERGY AREA 4490488.11 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 30.000 209300.79 1878.94 24403.68 182965.67 209248.29 34.31 FDBF - LSL=300 kN ( i n e l a s t i c ) : kN-mm-sec TIME-STEP=0.0020 ALPHA= 0.073 BETA= 0.000 BETA0= 0 000 DELTA= = 0.000 ENERGY ENVELOPE VALUES PERCENT ERROR IN ENERGY 1.2185 % PERCENT KINETIC ENERGY AREA 3.117 % PERCENT VISCOUS ENERGY AREA 9.949 % PERCENT STRAIN ENERGY AREA 85.715 % PERCENT OF STRAIN ENERGY, EL1 61.234 % PERCENT OF STRAIN ENERGY, EL2 2.329 % PERCENT OF STRAIN ENERGY, EL9 0.000 % PERCENT OF STRAIN ENERGY, EL10 36.437 % MAXIMUM KINETIC ENERGY 86834.52 ROOT-MEAN-SQUARE KINETIC ENERGY 11841.61 KINETIC ENERGY AREA 173257.35 STRAIN ENERGY AREA 4764216.62 FINAL ENERGY RESULTS TIME INPUT KINETIC VISCOUS STRAIN TOTAL U.B.F. ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY 30.000 227750.28 986.13 25294.36 198284.82 224565.31 3334.42 Notes: ELI elements - friction damper EL2 elements - beams and columns EL10 elements - braces 93 Table 6.10 - Friction Dissipated Energy (kN mm) 400 kN 600 kN No Buckling 159000 180000 Inelastic Buckling 113200 119400 %Drop 29% 34% 1 0 . 1 h 0 I 1 i i i i i i i i 0 200 400 600 800 Global Sl ip-Load (kN) Figure 6.14 - Effect of Inelastic Brace Buckling on the RPI. Figure 6.15 shows the hysteresis of the braces predicted by DRAIN-2D. Although the modelling of the braces is approximate, the brace hysteresis curves are very 'full' suggesting that they are undergoing signifi-cant inelastic action. The inelastic behaviour is expected to be capable of fatigue failure under long duration cyclic loading, such as during a megathrust event. Although the initiation of damper slip minimizes the compressive forces within the braces, it does appear that significant buckling can occur. Some residual elongation in all of the braces can be seen in Figure 6.15. Depending on the extent of inelastic damage experienced by the braces, it seems possible that an incorrectly chosen brace section, although it would satisfy the design criteria, could lead to a significant deterioration in the performance of a FBDF, and 94 F D B F 4 0 0 F D B F 4 0 0 150 100 50 0 - 5 0 -100 -150 -200 -250 -300 -350 ( Left Brace ) - 7 - 5 - 3 - 1 1 3 Axial Deformation (mm, -ve = shortening) 150 r 1 100 -o ° 50 -II Si 0 -+ 1" - 5 0 -- 1 0 0 -4) u o - 1 5 0 -- 2 0 0 -- 2 5 0 -- 3 0 0 -- 3 5 0 -( Right Brace ) - 4 - 2 0 2 4 Axial Deformation (mm, -ve = shortening) F D B F 6 0 0 F D B F 6 0 0 200 100 0 - 1 0 0 - 2 0 0 - 3 0 0 - 4 0 0 - 5 0 0 • ^ ( Left Brace ) - 6 - 4 - 2 0 2 4 f Axial Deformation (mm, -ve = shortening) g -100 V £ - 2 0 0 3 Ex 3-300 3 ( Right Brace ) 1 1 • \ - 1 0 - 8 - 6 - 4 - 2 0 2 4 Axial Deformation (mm, —ve = shortening) Figure 6.15 - Brace Hysteresis for FDBF400,600. in the extreme case, failure of the brace sections. Friction damped structures with high unbraced fundamen-tal frequencies, namely those with few storeys, would be more susceptible to brace fatigue, since they would undergo more vibration cycles. Figure 6.16 presents a comparison of the elastic and inelastic first storey response of FDBF400. Clearly, from an analytical point of view, the displacement time-history is not significantly affected. The hysteresis curves indicate that only a slight drop in stiffness occurs in the inelastic case, relative to the elastic case, dur-ing the loading and unloading cycles. Based on these analyses, it can be concluded that for the system under consideration brace buckling does not significantly affect the structural performance. 95 Inelastic B u c k l i n g No B u c k l i n g -10 10 30 Panel Displacement (mm) -10 10 Panel Displacement (mm) F i r s t Storey Figure 6.16 - Comparison of Elastic and Inelastic Response of FDBF400. 96 7 COMPARATIVE PERFORMANCE OF CBF AND FDBF This chapter provides a comparison of the behaviour of the conventional concentrically braced frame and the innovative friction damped braced frame systems. However, it must be kept in mind that the two structures are designed to perform differently. For instance, the strain energy within each system is substantially different since the weak links in the two systems lead to the development of vastly different loads in the structure. 7.1 Performance Comparison Under Design Excitation A comparison of drift is of prime importance since it is the factor controlling damage in the structure, as previously discussed. Figure 7.1 shows a graph of peak drifts for each storey for the CBF and FDBF with lOOkN and 400kN global slip loads for the design (El Centro) excitation; Figure 7.2 provides a comparison of their drift areas. > a> u O -»-> -H-0 $ A <t)A S> B ED <SJa—a A IB-10 20 30 40 50 60 70 Figure 7.1 - Peak Drift Comparison of CBF, FDBF100, and FDBF400. The drift areas of the friction damped structure, both for the 100 and 400kN global slip-loads, are generally significantly smaller than the corresponding areas of the concentric braced structure. However, the peak interstorey drift experienced by the structures occurred in the first storey of the FDBF, suggesting the poten-tial for non-structural damage at this level. 97 6 5 4 3 3 2 1 0 50 100 150 200 250 300 350 Drift Area ( m m - s ) Figure 7.2 - Drift Area Comparison of CBF, FDBF100, and FDBF400. It was shown that the RPI for the CBF is 1.43; this is significantly higher than the 0.47 value associated with the FDBF for the optimum slip-load provided in Figure 6.11. This high RPI value is a result of the high brace force imparted to the beam-column assembly. Figure 7.3 compares the strain energy time-history of the strong elements, (beams and columns) of the three frames under consideration. The high brace forces in the CBF account for the high strong element strain energy. Figure 7.3 also shows the kinetic energy time-histories of the two systems. The kinetic energy devel-oped in the CBF is significantly higher than the corresponding energy in the FDBF systems; this is due to the greater energy contribution arising from the higher mode response of the CBF. Higher modes are generally associated with higher velocities and lower displacements. Figure 7.4 shows the envelopes of the beam moments, column moments, and column axial forces of the three different frames. The FDBFs develop very high beam and column bending moments at the first storey. This results from the FDBF operating as a moment resisting frame during slip. Since the FDBF first storey peak drifts are higher than those in the CBF, it seems reasonable that the FDBF moments are higher at this loca-tion. The column axial forces are, however, higher in the CBF; this is due to the high stiffness and capacity of the CBF braces. 98 St r a in Energy 36 i -34 \-Q I I I 1 I I I I I I I I I 1 I I I I I I I L_ 0 2 4 6 8 10 12 14 16 18 20 Time (s) Kine t i c Energy 80 I 1 Time (s) Figure 7.3 - Comparison of Energies of CBF, FDBF100, and FDBF400. 99 Beam Moments B e n d i n g M o m e n t ( k N — m ) C o l u m n Moments 400 B e n d i n g M o m e n t ( k N — m ) C o l u m n Axia l s 6,000 A x i a l F o r c e ( k N ) Figure 7.4 - Comparisons of Bending Moments and Axials. 100 7.2 Performance Comparison Under Megathrust Excitation Analysis of the FDBF under a megathrust excitation illustrates the strong dependency of the optimum slip-load on the characteristics of the ground motion. This is shown by the RPI function resulting from the megathrust excitation. The RPI function, which was established by FDBFAP, is plotted in Figure 7.5. The optimum slip-load indicated in Figure 7.5 is in the range of 4000 to 8000kN. However, slip-loads of this magnitude in a single damper would be impractical and, therefore, it would be necessary to introduce addi-tional dampers to develop this shear capacity in the frame. 2,000 4,000 6,000 Global Slip-Load 8,000 (kN) 10,000 Figure 7.5 - F D B F A P RPI Function for a Megathrust Event. Figure 7.6 shows the envelope of displacements for various slip-load cases and for the CBF due to the mega-thrust event. The deflections predicted by FDBFAP resulted in significant inelastic bending of the beams and columns. Thus, the assumption that the beams and columns remain elastic, which is assumed in FDBFAP, is not valid and the results illustrated in Figure 7.6 are suspect. An analysis of FDBF400 using DRAIN-2D indicated that collapse would occur due to excessive deflections. 101 0 1,000 2,000 3,000 4,000 5,000 D e f l e c t i o n ( m m ) Figure 7.6 - Envelope Displacements for CBF, FDBF100,400,8000 for a Megathrust Event. 7.3 Concluding Remarks Comparisons between the response of the friction damped structure and the conventional braced frame structure under the design event indicate that, for the six storey building under consideration, little improve-ment in deflections is realized by using friction damping. Strong element axial forces in the FDBF cases are, however, significantly lower and yielding occurs in the braces of the CBF. The FDBF response is expected to be superior to the CBF response if both systems have similar modal contributions. The computer analysis predicts that both the friction damped frame and the conventional braced frame would collapse under megathrust conditions. It is likely that the conventional frame would have survived had it been designed by a true capacity design method. It should be noted that survival is also dependent on the fatigue life of the braces, which can not be modelled by analysis. The optimum global slip-load demanded by the megathrust excitation, as compared to the design event (see Figure 6.12), illustrates the strong dependency of the optimum slip-load on the characteristics of the ground motion. 102 8 EXPERIMENTAL PROGRAM The majority of experimental research conducted to date on braced frames and friction damped braced frames has been performed under small-scale conditions. Doubt is frequently cast on the validity of applying the results of these tests to full-scale conditions. In the experimental program undertaken in this investigation an attempt was made to eliminate this uncertainty by employing full-scale testing. The program seeks to verify various aspects of the analytical research and to examine the failure criteria of the bracing systems. The experimental program is divided into two parts. The first part involves the full-scale experimental testing of the concentric cross-bracing described in Chapter 5. The second part involves the full-scale testing of the friction-damped braced panel described in Chapter 6. Three tests were performed on single cross-braced panels: two reverse cyclic tests under increasing displace-ments and one seismic simulation test. Displacements for the seismic test were derived from the computer simulations analysed in Chapter 5. Coupon tests were also conducted to establish the yield stress of the brace section. The friction damped brace experiments consisted of several tests on single braced panels. One friction damper was used for all the experiments. These experiments included several reverse cyclic tests under varying slip-loads and displacements and involved several earthquake simulation tests. Calibration tests were initially con-ducted on the damper to verify the mathematical model assumed for the device and to set its desired slip-loads. Calibration tests were also conducted on a friction pad to evaluate the clamping technique for the damper and the friction coefficient. A frame of suitable size and strength was required to conduct the full-scale panel tests. The design and fabrica-tion of this testing frame is discussed in the following section. 8.1 Testing Frame Design A general schematic of the test frame is shown in Figure 8.1. It is basically a four-pinned portal frame which is driven by an hydraulic actuator located at the cross-beam. 103 Figure 8.1 - Proposed Testing Arrangement. 8.1.1 Frame Constraints The design constraints for the system were dictated by the maximum displacement and the maximum force developed in the critical panel of the building under consideration, as defined by the computer anal-The maximum displacements developed in the panel during the earthquake simulation studies were + 45mm and -35mm for the CBF and + 70mm and -30mm for the FDBF. The maximum total required displacement was therefore 100mm. The maximum total test frame displacement that could be developed was limited by the stroke of the available actuators, which is 150mm. Reverse cyclic tests were therefore limited to ±75mm displacement. The required shear capacity of the test frame was dictated by the plastic shear capacity of the CBF cross braces. This was evaluated from the unfactored code capacity multiplied by an overstrength factor of 1.3, resulting in a shear capacity of about 1300kN. ysis. V= 1.3(812 + 419)-0.8 V= \280kN. 104 where, T r = peak brace tension force, C r = peak brace compression force, and V = required actuator force (after frictional losses). The largest actuator available in Structures Laboratory is a double acting model built by Team Corpora-tion, which possesses a 2000kN push capacity and a 1500kN pull capacity under 3000psi hydraulic operat-ing pressure. This allows for a 13% load reserve capacity. To minimize the test frame cost, the actuator and test frame reactions were accommodated by the strong floor of the Structures Laboratory. The anchor bolt layout for these reactions was dictated by a 24 inch spacing of bolt holes in the floor. For simplicity and symmetry, the test frame size was governed by the floor-bolt layout. As a result, the base length of the frame panel was set at 4876.8mm (192in.), which is 2.5% smaller than the 5000mm used for the design and analysis of the six storey braced frame. Similarly, the panel height was set at 3657.6mm (144in.), or 1.6% larger than the 3600mm used in the analysis. The frame aspect ratio is therefore 0.75. The actual design was based on an aspect ratio of 0.72, a 4% differ-ence. This increase in aspect ratio results in a 1.5% drop in the horizontal shear capacity, which is consid-ered to be negligible in terms of the present study. 8.1.2 Frame Design General The testing frame was designed as a truly pinned portal frame and is similar in concept to an existing testing frame at the University of Michigan, Astaneh-Asl, 1982. For the purpose of versatility, the frame was designed to accept two actuators operating in tandem. The combined capacity of the actuators, 1780kN (400kips) pull and lOOOkN (225kips) push, selected was 2780kN (625kips) in each direction. The test frame was designed to accommodate the above specified actuators and a specimen which would induce the highest forces in the main frame members, namely a diagonal brace. The resulting force dia-gram is shown in Figure 8.2. The frame member sizes were chosen based on these forces. 105 Figure 8.2 - Force Diagram of Test Frame. Figure 8.3 provides some details of the final test frame design and Appendix E shows the frame compo-nents. The end plate for attaching the second actuator is not shown since it was not installed. The end plate was not needed for this study but can be attached at any time with relative ease. COLUMN + + + + + + W + + + + + W310xl5S + + + 17 + + + + 4+ + + + + + + + W310xl58 + + + + + + + + Figure 8.3 - The Test Frame (Plan View). 106 Structural Sections and Grade The structural sections selected for the frame include a double-channel, C380x74, for the cross beam and wide flanges, W310x158, for the columns. A double-channel was chosen for the cross beam so that the web of the W-shape could be inserted between the channels at the pin locations. This provides a concen-tric path for force transfer. The W-shapes are made from CSA-G40.21 Grade 300W steel; the double-channel steel cross-beam conforms to ASTM A36 grade steel. The double-channel member was stitched together with a 60mm back-to-back spacing by twenty-one 100mm long welded stitch plates arranged in three rows. Two longitudinal stiffeners were attached to each channel to increase the overall buckling resistance of the cross-beam. Pins and Bushings Pins 'A' and 'D' are 140mm in diameter, and pins 'B' and ' C are 100mm in diameter. All pins were fabri-cated from AISI4140 grade steel, and were heat treated and stress relieved. The machined bushings were fabricated from solid AISI 4140 stock. The bushings for the 100mm and 140mm diameter pins have 140mm and 190mm outer diameters, respectively. The inside diameter of the bushings is approximately 0.5mm oversized relative to their respective pin diameters. The reinforcing plates on the web of the W-shapes at the pins are CSA-G40.21 Grade 700QT. The clearances of the plates for the pins are the same as for the bushings. The smaller pins are retained by using external snap rings. The larger pins are retained by using set screws and a top cap plate. Floor Anchorage The test frame was anchored to the strong floor by SAE Grade B-7 prestressed anchor bolts. Since the accuracy of the location of the strong floor bolt holes is poor (±10mm), it was necessary to use anchor bolts that were significantly smaller than the bolt holes. This introduced an opportunity for the frame bases to slip. It is particularly desirable to avoid slippage of the test frame reaction bases for the purpose of obtaining accurate frame displacement measurements. A similar anchorage system is used for a concrete beam element tester in the Structures Laboratory. Tests of this apparatus showed that bases clamped with six similarly prestressed anchor bolts slipped on the concrete strong floor at approximately 1200kN (200kN per bolt.) 107 To ensure that the reaction bases of the new test frame would not slip, eight anchor bolts per reaction base were used, resulting in a 1600kN capacity against slippage. Higher reaction resistances can be obtained by adding base plates and bolts. Although the test frame reaction bases must avoid slip, slippage of the actuator reaction base is not critical to accurate frame displacement measurements and, therefore, slip is allowed. The cross-beam is supported on slide bearings which are snug tightened to the floor. The slide bearings have an uplift capacity of 250kN each. Gusset Connections The details of the gusset connections are shown in Figure 8.4. These connections were designed to suit the research test requirements of this and other experimental programs planned for the equipment. One inch diameter bolts are used for the connections. Due to the eccentricity of the upper gusset connection, additional bolts are required to provide the same capacity as the lower gusset connection. The gusset capacities, as defined by S16.1 are listed in Table 8.1. Table 8.1 - Gusset Capacities Capacity (kN) Friction (A490) 2290 Friction (A325) 1900 Bearing 3660 The specimen gusset plates can be up to 19mm thick and can be either friction or bearing connected. The fraying surfaces of the gusset connections were clean mill scale. Although not recognized by the code, higher friction capacities can be expected if the specimen gusset plates are blast cleaned. It should also be noted that S16.1 does not allow the re-use of A490 bolts in friction connections. The use of a full bearing capacity connection was not selected since some inelastic deformation would be experienced around the bolt holes. This would make the specimen gusset plate difficult to remove. The gusset connection capacity can be increased by installing additional connection plates. 108 12 , £ 6 5 I9nn P u To 350^645 C2/CONNECTION) UPPER GUSSET 19nn P|764nn x 175i LDWER GUSSET Figure 8.4 - The Gusset Connections. Clearance The vertical distance between the test frame centreline and the floor was set at 370mm. For the CBF tests, the clear distance to the floor was 305mm. Additional clearance could be provided by shimming the reaction bases. 8.1.3 Frame Fabrication and Assembly Fabrication All of the test frame welding and machining was done in the Department's Machine Shop. Carbon diox-ide shielded metal arc welding was used; the weld wire was flux cored, E480XX. Figure 8.5 shows a reaction base after welding. The final machining of the bushing was completed after welding to minimize the distortion due to welding. Mild steel pins were placed in the bushings during the welding process to maintain alignment of the bushing plates. Figure 8.6 is a photograph of the completed test frame components. 109 Figure 8.5 - a) Reaction base after welding, b) Cross-beam during fabrication. 110 I l l Assembly 1 The reaction bases were anchored to the floor with 2" 0 x 36" long SAE Grade B-7 threaded anchor rods and high strength 2H-A192 nuts. The anchor bolts were pretensioned to lOOOkN. Figure 8.7a illustrates the arrangement of the jacking assembly, which consists of dual-hydraulic-cylinders and a 10,000psi Ener-pac hydraulic pump unit. The jacking assembly was attached to the anchor bolt by a 5" long coupling nut. A high strength 2H nut and bearing plate were attached to the anchor bolt on the underside of the strong floor. Figure 8.7b is a view of the installed reaction base. Figure 8.8 shows the assembled the testing frame with the 400kip Team double acting actuator. 112 Figure 8.8 - a) Assembled Test Frame, b) 400kip Double-acting Actuator. 114 8.1.4 Frame Tests and Performance Performance Tests Several tests were conducted on the 'bare' (without specimens) test frame. These initial tests were used to determine the no-load frictional force within the frame. For this purpose, the test frame was adapted to accommodate a small and more responsive actuator with a more sensitive load cell. Figure 8.9 provides a typical graph of the displacement versus the friction force of the test frame. A con-stant frictional force of 4.3kN was found to exist in the 'bare' test frame. The tests indicated that the frictional force within the test frame was relatively independent of the cycling frequency. No Sp e c i m e n ---1 • i i i i 1 I-i i -70 -50 -30 -10 10 " 30 50 70 F r a m e D i s p l a c e m e n t ( m m ) Figure 8.9 - Unloaded Test Frame Friction. Performance Aspects During the course of this research project the test frame underwent a total of 22 cycles during the three CBF tests, in which a peak actuator force of 1245kN was developed. The frame was subsequently sub-jected to a further 360 cycles at a peak actuator force of 670kN in a second, independent research pro-gram. Finally, the test frame experienced another 500 cycles, at a peak actuator force of 130kN, when examining the FDBF behaviour. 115 Some bearing problems occurred during the second (independent) test program. The large pins began to gall as a result of using a poor lubricant and of using similar materials for the pins and the bushings. The pins and bushings were re-machined and molybdenum grease was substituted for the original heavy oil. As a result, galling did not reoccur during subsequent tests. 8.2 Testing Program As discussed earlier, the testing program was divided into two parts: conventional bracing tests and friction bracing tests. The following sections describe the details of the test specimens, fabrication, and instrumenta-tion of the tests performed. 8.2.1 Conventional Cross-Brace Tests Class C, HSS 127x76.2x6.35 sections were used for the X-bracing in the conventional brace tests. This was the design brace section for the fifth storey of the CBF structure. The unfactored code tensile capacity of this section is 812kN. Its in-plane slenderness is estimated as: KL 0 . 3 x 4 5 6 0 — = ' 4 5 . r 3 0 . 1 The out-of-plane slenderness is estimated as: KL 0 . 7 x 4 5 6 0 V 4 5 7 i — " 7 1 -Based on the coupon tests and the first cyclic test, the strength values used in the dynamic analyses were modified to reflect the actual strength values. The brace-to-frame connection is a gusset plate welded through the centroid of the brace. The length of the weld was based on a shear lag calculation which indicated that the shear stresses in the weld decays essentially to zero within 450mm. Therefore, a weld of greater length offers no additional benefits. A stress check revealed that the yield stress of the gusset plate was slightly exceeded, indicating that some minor localized yielding would occur during the test. General practice in selecting a weld length for this type of connection is to use a length equal to the perimeter length of the HSS section. If a star-shaped gusset is used, the weld length would be half of the 116 HSS perimeter. For the in-plane gusset used in this design, the weld length should be (127+ 76.2)x2 = 406mm. This is close to the weld length obtained from the shear lag calculation. A weld length of 410mm was selected. Due to the 'X' configuration of the braces, at least one brace must be spliced. To ensure that the full tensile plastic force can be transferred between the spliced braces, a gusset plate or cover plates must be used at the intersection of the braces. Since some force can be transferred by butt welding the spliced braces to the continuous brace, a shorter gusset-to-braces weld length, and therefore a shorter gusset, can be used. A single 600mm gusset plate was used; it was welded to the spliced brace along slots cut into the brace centroid. The single plate was used since it contributes little to the bending stiffness of the brace. To ensure that movement in the gusset-to-frame connections did not influence the recorded data, slip-critical (friction type) connections were used. Code calculations indicated that ten 1" diameter A325 bolts were required. To maintain symmetry in the gusset connection bolt layout, 12 bolts are used. Four 6mm by 410mm long welds connect the brace to the gusset. 8.2.1.1 Fabrication and Assembly The braces were first slotted by flame cutting. Prior to welding the braces to the gussets, the gussets and braces were prefit in the test frame. This ensured that a 'perfect' fit was achieved and that minimal effort was required to install the braces. The brace-gusset connections were then tack welded and removed from the frame for final welding. Figure 8.10 shows the completed and installed brace end-connection. Again, to develop the full brace capacity, it was necessary to weld its slotted end. For ease of welding, the connections were welded outside of the test frame, except for the brace inter-section connection. Due to the design of the test frame gusset connections, the intersection connection was welded after the braces were installed in the test frame. To accommodate this, the slot in the continuous brace had to be lengthened by approximately 20mm. This oversized slot was fill welded to develop the full capacity of the brace. It was difficult to install a proper backing plate to provide a good closure weld, and the quality of this connection was therefore questionable. Because of this difficulty, only the first X-brace was fabricated in this manner. Subsequent specimens used cover plates, top and bottom, to avoid slotting the continuous brace and avoid the installation difficulties. Figure 8.11a and b show the centre connection with the tongue and cover plates, respectively. 117 118 A lime white-wash was apphed to the braces and the gussets to help detect the initiation and spread of yielding. When yielding takes place, the white-wash begins to flake off with the mill-scale, indicating yield locations. Although Class C hollow structural sections do not have mill-scale, as a result of cold-forming, some flaking off of the white-wash did occur. Figure 8.12 is a photograph of the third specimen, CB3, ready for testing. a * e 8 Figure 8.12 - Prepared Specimen CB3. 120 8.2.1.2 Instrumentation The displacement of the cross-beam relative to the floor was measured by a displacement transducer located at the mid length of the cross-beam. The load was measured in three ways: by the actuator load cell, by electronic hydraulic pressure trans-ducers, and by strain gauges attached to the braces. The strain gauges were located 400mm from the connection weld. This location was chosen to avoid the gauges being affected by stress concentrations caused by the connection, and to maintain adequate distance from the point of hinging due to overall brace buckling. A calibration specimen was used to correlate voltage output with load. The deflected shape of the continuous brace was monitored by measuring the out-of-plane deflection relative to a spring-loaded wire attached to the frame pins, 'B' and 'D' (see Figure 8.3.) 8.2.1.3 Braced Frame Tests Three tests were conducted on cross-braced specimens. Each specimen was constructed in an identical manner except for the centre connection, as discussed earlier. Figure 8.13 shows the CBF specimen used in the full-scale frame tests. The first two specimens, CB1 and CB2, were tested under controlled reverse cyclic displacements. The CB2 test was a repeat of CB1 due to problems experienced during the CB1 test; this is discussed in Chapter 9. The cyclic displacements were increased by multiples of the yield displacement. The yield displacement was reached when the braced panel underwent full plastic deformation, which occurred when the peak load was reached. The specimens were subjected to two cycles at each displacement ampUtude. An example of the cyclic displacement time-history is shown in Figure 8.14. Cyclic defor-mations were carried out until the failure of a brace occurred. 121 HSS 127x76.2x6.35mn Ff_ 19mm HSS 127x76.2x6.35mm TYPE T SECTION I B SCALE 1'5 V — TYPE '2' Figure 8.13 - C B F Specimen. The third test in this series, CB3, represented an earthquake simulation test. The apphed drift dis-placements in this test were derived from a computer analysis of the six storey building discussed in Chapter 5. The building base excitation corresponded to a Vancouver design earthquake. The analysis showed that the fifth storey bracing suffered the most damage during this earthquake and the drift displacement time-history of this storey was therefore used as the input parameter for the full-scale testing. 122 - 6 6 Cycle # 10 12 Figure 8.14 - CBF Cyclic Test Displacements. The actual fifth storey drift time-history is shown in Figure 8.15 along with a simplified version of this response parameter, which was applied in the test. It is noted that since the test was conducted in a quasi-static manner due to equipment limitations, the time axis for the test is not consistent with that of the predicted response. It was assumed that the small high frequency displacement cycles of the actual drift time-history would not significantly affect the test results and could therefore be neglected. A c t u a l S i m p l i f i e d Figure 8.15 - CBF Earthquake Simulation Drift Time-History (5th Storey). 123 8.2.2 Fr ic t ion-Damped Frame Tests Double-angle sections, 2L 65x50x5, were used for the friction damped braced frame tests. The short legs were stitched with a 16mm back-to-back spacing. One 50mm stitch was found to be sufficient for the brace length. Al l brace connections were slip-critical (friction type) and utilized two 1" 0 A325 bolts. The code capacity of the connection is 512kN. The brace-to-frame gussets were similar to those used in the CBF tests. The brace section chosen was impractical from the point of view of the connections used since the bolt size required a large brace. However, a small brace was desired to investigate brace buckling effects. Oversized plates were welded to the brace ends to allow for the bolt size. Figure 8.16 shows the double-angle brace section. Figure 8.16 - F D B F Double-Angle Brace. 124 8.2.2.1 Description of the Friction Damper Figure 8.17 shows the layout and dimensions of the friction damper. Figure 8.18a provides a disas-sembled view of the friction damper. All the damper plates are made from mild steel. The centre friction plate is fitted with a standard brake lining pad, as shown in Figure 8.18b. The brake lining pad is non-asbestos 1/8" thick Scan-Pac type RF-38 and is high temperature bonded to the steel plate. The sliding link plates were polished with a high speed grinder. Figure 8.17 - The Friction Damper. The clamping bolts are 1" 0 UNF SAE Grade 8. The six link bolts are 1" 0 UNC SAE Grade 5. The link bolts were machined to ensure a tight fit was achieved. Set screws were installed in the link-bolt nuts to avoid loosening. D E L R O N spacer washers, 3" diameter by 1/8" thick, were placed between the 125 126 link plates to stiffen the damper and to avoid plate galling during slipping. Belleville spring washers, Spae-naur SP-series No. 680-245, were used to maintain the clamping force on the sliding plates. The spring washer and its specifications are listed in Figure 8.19. Figure 8.20 shows the assembled damper. O.H. Bolt Size - 1" D = 2.756" d = 1.102" t = 0.275" D O.H.m„ = 0.362" O.H.^ = 0.317" Load @ O.H^ = 21 kips (94kN) Figure 8.19 - SP Belleville Spring Washer No. 680-245. Figure 8.20 - Friction Damper Assembled. 127 8.2.2.2 Instrumentation Calibration Tests Calibration tests of the damper were performed in the Earthquake Simulation Laboratory at U .B .C . The damper slip load was measured with a 20 kip load cell. Figure 8.21 shows the damper in the calibration setup. Figure 8.21 - Damper in the Calibration Setup. 128 The damper was fitted on all four arms with strain gauges; the arrangement functioned as a load cell. The strain gauges were calibrated with the 20 kip load cell noted above. Due to the design of the damper arms, calibration factors were required for tension as well as for compression. The damper was also fitted with two Linear Variable Displacement Transducers (LVDT) in both diag-onal directions. These LVDTs measured the damper deformation, which included the slack (from the clearance in the bolt holes) within the damper. Frame Tests Test frame displacements were measured using the method employed in the CBF tests. LVDTs were utilized to measure brace bending and brace axial deformation on two of the four braces. Bending measurements were made over a 1725 mm span. The L V D T arrangement for recording the brace bending and axial deformations are schematically shown in Figure 8.22. 8.2.2.3 Friction-Damped Frame Tests Slip tests were performed to investigate the friction coefficient of the pad material under various clamping loads. The tests were conducted on a plate which was similar to the centre friction plate of the damper. Appendix G lists the friction slip test results. A = Axial Deflection Measurement B = Bending Deflection Measurement Figure 8.22 - Brace Deformation Measurements 129 Calibration tests were performed in the Earthquake Simulation Laboratory to establish the slip beha-viour of the friction damper. Several full-scale frame tests were conducted on the friction damped braced frame under reverse cyclic loading. The effects of brace orientation, loading rate, and slip-load were investigated. All tests employed SP spring washers in combination with the clamping bolts. The clamping force was set by deforming the washers to their rated minimum overall height. The FDBF test specimen is shown in Figure 8.23. Figure 8.24 is the cyclic displacement time-history which was used in the full-scale reverse cyclic tests. Cycle Tests 0 4 .8 12 16 20 24 2B Cycle * Figure 8.24 - FDBF Cyclic Test Displacements. Earthquake simulation tests were conducted to compare the analytically predicted response with the actual measured response. Figure 8.25 provides a graph of the predicted and simplified drift time-histories of the first storey of the six storey structure under consideration. As in the CBF tests, the small high frequency cycles in the calculated drift were assumed to have negligible effect on the response, and the simplified drift time-history was used for the full-scale simulation test. Again, as in the CBF tests, the time axis for the FDBF tests is not consistent with that of the predicted response. Time (s) Time (s) Figure 8.25 - FDBF Earthquake Simulation Drift Time-History (1st Storey). 131 9 C B F T E S T R E S U L T S Chapter 9 presents the results obtained from the full-scale cross-brace tests and other associated tests. 9.1 Coupon Tests The data for coupon tests are presented in Appendix F. A typical stress-strain curve obtained from the tension tests on specimens prepared from the HSS section used for the braces is shown in Figure 9.1; the yield stress of the section is defined by the 2% offset line intersection with the stress-strain curve. Clearly, the 2% offset stress, 380MPa, is not representative of the actual yield stress for modelling purposes; by observation, 400MPa is a better representation of the yield stress. This translates into a brace yield force of 928kN. 500 0.2% o f f s e t 400 U n l o a d cn cn cu £ 200 100 0 0 0.04 0.08 0.12 0.16 0.2 0.24 Strain Figure 9.1 - Typical C B F Coupon Stress-Strain Curve. A compression coupon test yielded similar results. 132 9.2 Frame Test Results 9.2.1 Evaluating the Test Frame Friction The friction in the test frame was evaluated under the no-load condition, as discussed in Chapter 8. The results showed that the internal friction of the test frame was 4.3kN, which is attributed primarily to the slide bearing resistance. Friction is also expected within the pin joints of the frame under load conditions. To evaluate the pin friction, the apphed actuator force was compared with the sum of the brace force components at loads within the elastic range of the braces. The initial load cycles of tests CB2 and CB3 were used to evaluate the friction in the test frame. As shown in Figure 9.2, there was good agreement between the brace forces determined from the strain gauge data and the actuator force established by the load cell data. The average difference in these forces is about 10%, with a peak force difference of 45kN. C B 2 C B 3 20 40 T ime (s) Figure 9.2 - Test Friction for CB2 and CB3. The test frame friction can also be examined by analyzing the force-displacement data obtained during the unloading stage, immediately after the peak displacement was achieved. Figure 9.3 shows the frictional force in a typical unloading condition. The data indicate that the average frictional force during unloading is approximately 45kN. It should be noted that this force also includes the friction force derived from the slide bearings. 133 1.2 5 7 9 11 13 15 Frame Displacement (mm) Figure 9.3 - CB2 Friction Release During Unloading. The estimate of the frictional force in the test frame involves a high degree of uncertainty. However, based on the limited information available, it would seem reasonable to correct the load-cell force by a 10% reduction to a maximum limit of 45kN. 9.2.2 C B F Cycl ic Tests CBl CB1 was tested under reverse cyclic deformation. The cyclic displacement of the test frame is shown in Figure 9.4 together with the associated hysteresis curve of the bracing system. The yield displacement shown in Figure 9.4 is the displacement to cause full plastification of the braced panel. Due to an electrical power failure during the experiment, the computer recorded data of the first two cycles were lost. However, the actuator loads and displacements were automatically plotted in hard copy. Thus, the load and displacement data were recovered by digitizing this plotted record. This data, how-ever, includes the slippage of the actuator reaction bracket and must be adjusted to remove this effect. The hysteresis curve shown in Figure 9.4 was corrected for this actuator slip and for the test frame friction. 134 135 Hinging, due to buckling, occurred near the mid span of the lower braces, where the strain gauges were located. As a result of the local yielding that occurred in the walls of the HSS section, the strain gaug