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Seismic design of friction damped braced steel plane frames by energy methods Filiatrault, Andre 1988

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SEISMIC DESIGN OF FRICTION DAMPED BRACED STEEL PLANE FRAMES BY ENERGY METHODS by ANDRE FILIATRAULT B.Ing., Universite de Sherbrooke, 1983 M.A.Sc., The University of B r i t i s h Columbia, 1985 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1988 © Andre F i l i a t r a u l t , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l E n g i n e e r i n g The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e O c t o b e r 15 , 1988 DE-6(3/81) ABSTRACT The i n v e s t i g a t i o n described i n t h i s thesis represents the f i r s t known attempt to develop a s i m p l i f i e d method for the seismic design of structures equipped with a novel f r i c t i o n damping system. The system has been shown experimentally to perform very well and i s an e x c i t i n g development i n earthquake r e s i s t a n t design. The design of a b u i l d i n g equipped with the f r i c t i o n damping system i s achieved by determining the optimum s l i p load d i s t r i b u t i o n to minimize s t r u c t u r a l response. A new e f f i c i e n t numerical modelling approach for the analysis and design of F r i c t i o n Damped Braced Frames (FDBF) i s presented. The h y s t e r e t i c properties of the f r i c t i o n devices are derived t h e o r e t i c a l l y and included i n a F r i c t i o n Damped Braced Frame Analysis Program (FDBFAP), which i s adaptable to a microcomputer environment. The optimum s l i p load d i s t r i b u t i o n i s determined by minimizing a Relative Performance Index (RPI) derived from energy concepts. The steady-state response of a si n g l e storey f r i c t i o n damped structure subjected to sinusoidal ground motion i s investigated a n a l y t i c a l l y . Basic design information on the optimum s l i p load for the f r i c t i o n device i s obtained. The parameters governing the optimum s l i p load, which minimizes the amplitude for any forcing frequency, are derived. The study indicates that the optimum s l i p load depends on the c h a r a c t e r i s t i c s of the ground motion and of the structure. Using v a r i a t i o n a l p r i n c i p l e s on a shear beam analogy, an optimum s l i p load d i s t r i b u t i o n along the height of the structure i s derived when the t o t a l amount of s l i p load i s s p e c i f i e d . It i s shown that the optimum s l i p load i s proportional to the slope of the deflected shape of the structure. The r e s u l t s of the study reveal that only a small improvement i n the response i s obtained by using t h i s optimum d i s t r i b u -t i o n compared to the response obtained with a uniform d i s t r i b u t i o n . Therefore the use of an optimum uniform d i s t r i b u t i o n seems adequate for the design of f r i c t i o n damped structures. Taking into account the a n a l y t i c a l r e s u l t s obtained, FDBFAP i s then used i n a parametric study which leads to the construction of a design s l i p load spectrum. The spectrum depends on the properties of the structure and ground motion an t i c i p a t e d at the construction s i t e . I t i s believed that the a v a i l a b i l i t y of t h i s design s l i p load spectrum w i l l lead to a greater acceptance by the engineering profession of t h i s new and innovative s t r u c t u r a l concept. TABLE OF CONTENTS Page Abstract i i Table of Contents i v L i s t of Tables i x L i s t of Figures x i Acknowledgements x v i i i Dedication x i x 1. INTRODUCTION 1 1.1 Some Innovative Earthquake Resistant Steel Systems 2 1.1.1 Base Isolated Structures 4 1.1.2 Eccentric Braced Frames (EBF) 5 1.1.3 Braced Frames With Y i e l d i n g Devices 6 1.1.4 F r i c t i o n Damped Braced Frames (FDBF) 7 1.2 Description of F r i c t i o n Damping System 8 1.2.1 Physical Description 8 1.2.2 Current A n a l y t i c a l Models 10 1.2.3 Concept of Optimum S l i p Load 18 1.3 Motivation 20 1.4 Objectives 21 2. THEORY AND PROGRAMMING CONSIDERATIONS FOR FRICTION DAMPED BRACED FRAME ANALYSIS PROGRAM (FDBFAP) 23 2.1 General 24 2.2 Global S t i f f n e s s Matrix Formulation 24 2.3 Global Linear S t i f f n e s s Matrix 26 2.3.1 Beam Element S t i f f n e s s Matrix 27 2.3.2 Column Element S t i f f n e s s Matrix 28 2.4 Nonlinear Global S t i f f n e s s Matrix 29 2.4.1 Formulation for Slender Braces 30 2.4.2 Equilibrium V e r i f i c a t i o n at F u l l Slippage 46 2.4.3 Formulation for Stubby Braces 47 2.4.4 Summary 50 2.5 Global Mass Matrix Formulation 50 2.6 Global Damping Matrix Formulation 52 2.7 Undamped Natural Frequency Calculations 54 2.8 Nonlinear Dynamic Analysis 59 2.8.1 Incremental Equations of Motion 60 2.8.2 Average Acceleration Method 61 2.8.3 Time Step Integration Procedure 63 2.8.4 Summary of Integration Procedure 64 - i v -TABLE OF CONTENTS (Continued) Page 2.9 Energy Calculations 65 2.9.1 Continuous Energy Expressions 65 2.9.2 Discrete Energy Expressions 68 2.9.3 Remark on Energy Calculations 69 2.10 S l i p Load Optimization 70 3. STUDY OF FDBFAP PERFORMANCE THROUGH RESPONSE OF FRICTION DAMPED STRUCTURES 75 3.1 General 76 3.2 Numerical Example #1: Ten Storey Single Bay Frame 77 3.3 Numerical Example #2: Low Rise Frame 82 3.A Numerical Example #3: Three Storey Test Frame 88 3.5 Solution Times 93 4. STUDY OF AN ANALOGOUS NONLINEAR MECHANICAL SYSTEM 95 4.1 Motivation 96 4.2 Formulation 96 4.3 Solution 99 4.4 Analysis 103 4.5 Numerical V e r i f i c a t i o n 107 4.6 Remarks I l l 5. SIMULATION OF EARTHQUAKE ACCELEROGRAMS 113 5.1 Motivation 114 5.2 Strong Motion Duration of Earthquake 115 5.2.1 Concept of Strong Motion Duration 115 5.2.2 Local S t a t i o n a r i t y of Strong Motion Segment 117 5.2.3 D e f i n i t i o n of Strong Motion Duration 119 5.3 Power Spectral Density Functions of Earthquake Accelerograms 122 5.3.1 A n a l y t i c a l Presentation of Power Spectral Density 122 5.4 Summary of Governing Ground Motion Parameters 126 5.5 Simulation of Earthquake Accelerograms 128 5.5.1 Simulation Procedure 128 5.5.2 Simulation Example 132 - v -TABLE OF CONTENTS (Continued) Page 6. DIMENSIONAL ANALYSIS 141 6.1 General 142 6.2 Governing Parameters 142 6.2.1 Characterization of the Structure 142 6.2.2 Characterization of the Ground Motion 144 6.2.3 Summary of Governing Parameters 144 6.3 Ap p l i c a t i o n of Buckingham's Theorem 144 7. PARAMETRIC STUDY FOR ONE STOREY STRUCTURES 148 7.1 General Approach 149 7.2 S e n s i t i v i t y Study 151 7.2.1 D e f i n i t i o n of a Standard Earthquake 151 7.2.2 D e f i n i t i o n of a Standard Structure 156 7.2.3 S e n s i t i v i t y to Viscous Damping (£..) 158 7.2.4 S e n s i t i v i t y to C h a r a c t e r i s t i c Buckling Load ( P c r ) 160 7.2.5 S e n s i t i v i t y to Braced Period (T b) 162 7.2.6 S e n s i t i v i t y to Predominant Ground Period (T_) .... 162 7.2.7 S e n s i t i v i t y to Ground Damping (h„) 165 7.2.8 S e n s i t i v i t y to Peak Ground Acceleration (a ) 165 7.2.9 Discussion 168 7.3 Parametric Study 168 7.3.1 Strategy 168 7.3.2 Results for Least Square Slopes, M 172 7.3.3 C o r r e l a t i o n Between M and T b / T u 181 7.3.4 C o r r e l a t i o n Between Least Square Slope Parameters and T g / T u 181 7.4 Proposed Design Equation 185 7.4.1 Design S l i p Load Spectrum 186 7.5 Appraisal of Proposed Design Equation 187 7.6 Physical Limitations on Design Equation 193 7.6.1 Lower Bound 194 7.6.2 Upper Bound 195 7.6.3 Remarks 195 7.7 Choice of Diagonal Cross-Braces 196 7.8 Proposed Design Procedure 198 7.9 Design Examples 200 - v i -TABLE OF CONTENTS (Continued) Page 7.9.1 Conventional Design 201 7.9.2 Design With F r i c t i o n Damping System 20A 7.9.3 Comparison of Performance 208 8. OPTIMIZATION OF DISTRIBUTION OF FRICTION DEVICES IN MULTI-STOREY STRUCTURES 212 8.1 General 213 8.2 The Shear Beam Analogy 213 8.2.1 D e f i n i t i o n of a Shear Building 214 8.2.2 Governing Equations for a F r i c t i o n Damped Shear Beam 214 8.2.3 Optimum S l i p Load D i s t r i b u t i o n for Resonant Si t u a t i o n 220 8.2.4 Numerical V e r i f i c a t i o n 226 8.3 The "Weak-Girder-Strong Column" Building 231 8.3.1 D e f i n i t i o n of a "Weak-Girder-Strong Column" Building 232 8.3.2 Optimum S l i p Load D i s t r i b u t i o n for Resonant S i t u a t i o n 232 8.3.3 Numerical V e r i f i c a t i o n 233 8.4 Optimum S l i p Load D i s t r i b u t i o n for General Structure Under Earthquake Load 238 8.4.1 Optimum S l i p Load D i s t r i b u t i o n 238 8.4.2 Numerical V e r i f i c a t i o n 240 8.5 Discussion 243 9. PARAMETRIC STUDY FOR MULTISTOREY STRUCTURES 246 9.1 Strategy 247 9.1.1 Str u c t u r a l Model 248 9.1.2 Range of Values Considered 248 9.2 Results 250 9.3 Proposed Design Equation 252 9.4 Design S l i p Load Spectrum 254 9.5 Proposed Design Procedure 257 9.6 Design Example 261 10. CONCLUSIONS 266 10.1 Summary and Conclusions 267 10.2 Future Research 269 - v i i -TABLE OF CONTENTS (Continued) Page BIBLIOGRAPHY . 271 APPENDIX A: FDBFAP USER'S GUIDE 276 APPENDIX B: OVERVIEW OF RANDOM PROCESSES AND SPECTRAL ANALYSIS ... 286 APPENDIX C: CONTRIBUTION OF STRONG MOTION SEGMENT OF EARTHQUAKE ON STRUCTURAL RESPONSE 291 APPENDIX D: GESER USER'S GUIDE 299 APPENDIX E: EFFECT OF STRONG MOTION SEGMENT OF NONSTATIONARY EARTHQUAKE ON STRUCTURAL RESPONSE 302 APPENDIX F: THE SPECTRAL MOMENTS METHOD FOR ESTIMATION OF KANAI-TAJIMI PARAMETERS FROM REAL EARTHQUAKE ACCELEROGRAMS 309 APPENDIX G: SIMEA USER'S GUIDE 312 APPENDIX H: BILINEAR LEAST SQUARE FIT FOR LEAST SQUARE SLOPE PARAMETERS 315 APPENDIX I: EXPRESSIONS FOR FAMILY OF BILINEAR CURVES OF LEAST SQUARE SLOPE PARAMETERS 318 BIOGRAPHICAL INFORMATION 321 - v i i i -LIST OF TABLES Page TABLE 2.1 C o e f f i c i e n t s C1 and C 2, Slender Braces (Pfi > P c r) 51 2.2 C o e f f i c i e n t s Cj and C 2, Stubby Braces (Pj £ P c r) 52 3.1 S t r u c t u r a l Properties for Numerical Example #1 78 3.2 Undamped Fundamental Natural Frequencies for Numerical Example #1 79 3.3 Undamped Fundamental Natural Frequencies for Numerical Example #2 85 3.4 Natural Frequencies of Test Frame 91 3.5 Solution Times for Low Rise Frame on Amdahl V8 93 3.6 Comparison of CPU Times for FDBFAP 94 4.1 Values of S l i p Load Considered i n Numerical Investigation 108 6.1 Physical Quantities Considered i n the Dimensional Analysis 145 7.1 Properties of Simulated Earthquakes Used i n S e n s i t i v i t y Study 155 7.2 Nondimensional Parameters for Standard Models 156 7.3 Versions of Basic Structural Configuration 171 7.4 Values of Parameters Used i n Parametric Study 171 7.5 Least Square Slopes 172 7.6 Estimation of Least Square Slope Parameters a,b 184 7.7 C h a r a c t e r i s t i c s of Seismic Events 193 7.8 Appraisal of Proposed Design Equation 193 8.1 RPI for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n 231 - i x -LIST OF TABLES (Continued) Page 8.2 RPI for Uniform F r i c t i o n Damped Flexural Building Subjected to Sinusoidal Base E x c i t a t i o n 236 8.3 Results for Optimum S l i p Load D i s t r i b u t i o n 245 9.1 Versions of Basic Structural Configuration 249 9.2 Values of Parameters Used i n Parametric Study 250 9.3 Least Square Slopes M for NS = 3 250 9.4 Least Square Slopes M for NS = 5 251 9.5 Least Square Slopes M for NS = 10 251 9.6 Ordinates a and B for Construction of Design S l i p Load Spectrum 256 - x -LIST OF FIGURES P a g e Figure 1.1 Earthquake Representation by Energy Concept 2 1.2 Typical Locations of F r i c t i o n Device 9 1.3 F r i c t i o n Device 9 1.4 Hysteresis Loop from F r i c t i o n Device 10 1.5 Hysteresis Loop of a F r i c t i o n Joint with Braces Designed i n Tension Only 12 1.6 Hysteretic Behaviour of a Simple F r i c t i o n Damped Braced Frame During One Cycle of Loading 13 1.7 S i m p l i f i e d Model of a F r i c t i o n Damped Braced Frame ... 15 1.8 Unstable Mode of a F r i c t i o n Damped Braced Frame Modelled Only with Truss Elements 16 1.9 Refined Model of a F r i c t i o n Damped Braced Frame 18 1.10 Free Body Diagram of a F r i c t i o n Device When Slippage Occurs 20 2.1 Global S t i f f n e s s Matrix Formulation for F r i c t i o n Damped Braced Frame 25 2.2 Degrees of Freedom Considered i n Global Linear S t i f f n e s s Matrix Formulation 26 2.3 Forces and Deformations of Beam Element 27 2.4 Forces and Deformations of Column Element 28 2.5 F r i c t i o n Device Element 29 2.6 F r i c t i o n Device Element with Geometric Constraints ... 30 2.7 Hysteresis Loop of F r i c t i o n Device Element With Slender Braces 31 2.8 Free Body Diagram of F r i c t i o n Device Element During State #1 32 2.9 F r i c t i o n Device Element During State #2 36 - x i -LIST OF FIGURES (Continued) Page 2.10 Free Body Diagram of F r i c t i o n Device Element During State #2 ,.. 36 2.11 F r i c t i o n Pad A x i a l Load Ratio During State #2 39 2.12 F r i c t i o n Device Element During State #3 42 2.13 Hysteresis Loop of F r i c t i o n Device Element With Non-Slender Braces 48 2.14 E f f e c t of Mass Dependent Damping on Modal Damping Ratios 54 2.15 Average Acceleration Method 61 2.16 S t r a i n Energy Time-History for F r i c t i o n Damped Braced Frame with a P a r t i c u l a r S l i p Load D i s t r i b u t i o n 72 3.1 Str u c t u r a l Layout for Numerical Example #1 77 3.2 S l i p Load Optimization for Numerical Example #1 80 3.3 Energy Time-Histories for Numerical Example #1 81 3.4 Top Floor Relative Displacement Time-Histories for Numerical Example #1 83 3.5 Str u c t u r a l Damage Predicted by DRAIN-2D for Numerical Example #1 84 3.6 Str u c t u r a l Configuration for Numerical Example #2 84 3.7 S l i p Load Optimization for Numerical Example #2 86 3.8 Energy Time-Histories for Numerical Example #2 87 3.9 Top Floor Relative Displacement Time-Histories for Numerical Example #2 89 3.10 S t r u c t u r a l Damage Predicted by DRAIN-2D for Numerical Example #2 90 3.11 General Arrangement of Test Frame 90 3.12 Time-Histories of Third Floor L a t e r a l Displacements for Numerical Example #3 92 - x i i -LIST OF FIGURES (Continued) Page 4.1 Single Storey F r i c t i o n Damped Structure 97 4.2 B i l i n e a r Hysteresis 97 4.3 Frequency Response Functions for F r i c t i o n Damped Structure 110 4.4 Optimum S l i p Load for Harmonic Base Exci t a t i o n s with Various Amplitudes I l l 5.1 Accelerogram of the 1952 Taft Eathquake, S69E Component 116 5.2 Estimated Strong Motion Segment of the 1952 Taft Earthquake S69E Component 116 5.3 Scattergram of Peak Ground Acceleration vs Strong Motion Duration (after Vanmarcke and L a i , 1980) 121 5.4 Kanai-Tajimi Power Spectral Density Functions 124 5.5 Simulation of the 1952 Taft Earthquake, S69E Component 135 5.6 St r u c t u r a l Damage After Earthquakes, Taft vs Simulated Earthquakes 136 5.7 Envelope of Late r a l Deflections, East Side, Taft vs Simulated Earthquakes 137 5.8 Envelopes of Beam Bending Moments, East Bay, Taft vs Simulated Earthquakes 138 5.9 Envelopes of Column Bending Moments, East Side, Taft vs Simulated Earthquakes 139 5.10 Envelopes of Column Shear Forces, East Side, Taft vs Simulated Earthquakes 140 7.1 Histograms of Earthquake Parameters 153 7.2 Sample A r t i f i c i a l Accelerograms for Standard Earthquakes 154 7.3 Physical Properties of Standard Structures 157 - x i i i -LIST OF FIGURES (Continued) Page 7.4 S e n s i t i v i t y to Viscous Damping (E.) 159 7.5 S e n s i t i v i t y to C h a r a c t e r i s t i c Buckling Load ( P c r ) 161 7.6 S e n s i t i v i t y to Braced Period (T b) 163 7.7 S e n s i t i v i t y to Predominant Ground Period (T_) 164 7.8 S e n s i t i v i t y to Ground Damping (hg) 166 7.9 S e n s i t i v i t y to Peak Ground Acceleration (a_) 167 7.10 Basic S t r u c t u r a l Configuration Used i n Parametric Study 170 7.11 Least Square Slopes, T g / T u = 0.0512 173 7.12 Least Square Slopes, T g / T u = 0.0998 173 7.13 Least Square Slopes, T g / T u = 0.2657 174 7.14 Least Square Slopes, T g / T u = 0.3585 174 7.15 Least Square Slopes, T g / T u = 0.6989 175 7.16 Least Square Slopes, T g / T u = 0.7170 175 7.17 Least Square Slopes, T g / T u = 0.8045 176 7.18 Least Square Slopes, T g / T u = 1.0243 176 7.19 Least Square Slopes, T g / T u = 1.3978 177 7.20 Least Square Slopes, T g/T„ = 1.8597 177 7.21 Least Square Slopes, T g / T u = 1.9968 178 7.22 Least Square Slopes, T g / T u = 3.7195 178 7.23 Least Square Slopes, T g / T u = 5.3135 179 7.24 Least Square Slopes, T g / T u = 5.6315 179 7.25 Least Square Slopes, T g / T u = 11.2631 180 7.26 Least Square Slopes, T g / T u = 16.0901 180 7.27 Co r r e l a t i o n Between M and T b / T u 182 - x i v -LIST OF FIGURES (Continued) Page 7.28 Co r r e l a t i o n Between Least Square Slope Parameters and T g / T u 183 7.29 Design S l i p Load Spectrum for Single Storey F r i c t i o n Damped Structure 188 7.30 Earthquake Accelerograms Used for the Appraisal of the Design Equation 189 7.31 Appraisal of Proposed Design Equation, Structure #1 .. 190 7.32 Appraisal of Proposed Design Equation, Structure #2 ... 191 7.33 Appraisal of Proposed Design Equation, Structure #3 ... 192 7.34 Average Values of RPI at Optimum S l i p Load 197 7.35 Layout for Design Examples 201 7.36 Conventional Design Examples 205 7.37 St r u c t u r a l Damage for Design Examples, Eureka Earthquake 209 7.38 La t e r a l Deflections, Eureka Earthquake 210 7.39 Optimum S l i p Load Study for Design Example 211 8.1 Dynamic Equilibrium of a Shear Beam 215 8.2 Ty p i c a l Storey Displacement of Shear Building 216 8.3 Hypothetical Uniform Shear Building 227 8.4 Fundamental Unbraced Mode Shape of Hypothetical Uniform Shear Building 228 8.5 Maximum Floor Deflections for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n 229 8.6 Energy Time-Histories for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n 230 8.7 Slope of the Fundamental Mode Shape for a Flexural Beam 233 - xv -LIST OF FIGURES (Continued) Page 8.8 Fundamental Unbraced Mode Shape of Hypothetical Uniform Flexural Building 235 8.9 Maximum Floor D e f l e c t i o n for Uniform F r i c t i o n Damped Flexural Building Subjected to Sinusoidal Base E x c i t a t i o n 236 8.10 Energy Time-Histories for Uniform F r i c t i o n Damped Flex u r a l Building Subjected to Sinusoidal Base E x c i t a t i o n 237 8.11 Six Storey Structure Used for S l i p Load Optimization 241 8.12 Ten Storey Structure Used for S l i p Load Optimization 242 8.13 Top Floor L a t e r a l D e f l e c t i o n Time-Histories for Six Storey Structure, Long Beach Earthquake 244 9.1 Basic S t r u c t u r a l Configuration Used i n Parametric Study 249 9.2 Least Square Slope Parameters a and b 253 9.3 Construction of Design S l i p Load Spectrum 255 9.4 Design S l i p Load Spectrum for Design Example 263 9.5 Optimum S l i p Load Study for Design Example 265 C.l S l i p Load Optimization for F r i c t i o n Damped Structure Under Taft Earthquake 292 C.2 St r u c t u r a l Damage After Earthquake, Ta f t (0.54g) 294 C.3 Envelopes of L a t e r a l Deflections, East Side, Taft (0.54g) 295 C.4 Envelopes of Beam Bending Moments, East Bay, Taft (0.54g) 296 C.5 Envelopes of Column Bending Moments, East Side, Taft (0.54g) 297 C.6 Envelopes of Column Shear Forces, East Side, T a f t (0.54g) 298 - xv i -LIST OF FIGURES (Continued) Page E . l Equivalent Stationary Records for the Taft Earthquake (3-14 seconds) 303 E.2 St r u c t u r a l Damage After Earthquakes, Taft vs Stationary Records 304 E.3 Envelopes of Late r a l D eflections, East side, Taft vs Stationary Records 305 E.4 Envelopes of Beam Bending Moments, East Bay, Taf t vs Stationary Records 306 E.5 Envelopes of Column Bending Moments, East Side, Taft vs Stationary Records 307 E.6 Envelopes of Column Shear Forces, East Side, Taft vs Stationary Records 308 1.1 Ordinates of a Values for T g / T u = 1 and 15 319 1.2 Ordinates of b Values for T /T u = 1 and 15 320 - x v i i -ACKNOWLEDGEMENTS This research was c a r r i e d out under the supervision of Dr. S. Cherry, whose encouragement and guidance i s g r a t e f u l l y acknowledged. I would l i k e to express my appreciation to Professor R.F. Hooley, M.D. Olson and T.J. U l r i c h for the stimulating discussions that I had with them during various stages of the work presented i n t h i s t h e s i s . Funding for t h i s project was provided by the Natural Sciences and Engineering Research Council of Canada. I t i s a pleasure to record t h i s assistance. November 1988 Vancouver, B r i t i s h Columbia - x v i i i -DEDICATED TO MY WIFE LOUISE, WHO HAS SACRIFICED MUCH AND LOVED ME MORE DURING THESE DIFFICULT TIMES - x i x -1 1. INTRODUCTION "The Universe i s f u l l of magical things p a t i e n t l y waiting for our wits to grow sharper." - Eden P h i l l p o t t s (1862-1955), English Novelist 2 CHAPTER 1 INTRODUCTION 1.1 Some Innovative Earthquake Resistant Steel Systems Seismic loadings on buildings complicate the task of the s t r u c t u r a l engineer because of the p r o b a b i l i s t i c and random nature of earthquakes. However, i n c e r t a i n regions, seismic e f f e c t s cannot be ignored because load combinations including earthquake e f f e c t s may lead to greater stresses and deformations i n various c r i t i c a l components of a structure than dead, l i v e , wind, and temperature loads combined. Earthquake e x c i t a t i o n can be represented by a f i n i t e quantity of energy that i s f i l t e r e d through a structure. F i g . 1.1 i l l u s t r a t e s the manner i n which the energy i s d i s t r i b u t e d at the conclusion of an earth-quake, when the structure i s at r e s t . Energy Lost i n Heat Energy Input Structure Energy Radiated Back Into the Ground Hysteretic Energy Absorbed by the Structure Figure 1.1 Earthquake Representation by Energy Concept The amount of energy delivered to a structure greatly depends on the r a t i o of the fundamental period of the structure to the predominant period of the ground motion. The damage experienced by a structure as a r e s u l t of an earthquake i s d i r e c t l y r e l a t e d to the amount of h y s t e r e t i c energy absorbed by the various elements of the structure. 3 It has long been recognized that i t i s not economical to design usual structures to remain free of damage during a major earthquake. The philosophy of most modern bu i l d i n g codes, including the National Building Code of Canada, involves the concept of d u c t i l i t y balance: a structure must be designed to ensure that the d u c t i l i t y demands devel-oped i n i t s members as a r e s u l t of the energy which i t absorbs during an earthquake are balanced by the ( d u c t i l i t y ) capacity of the members to s a f e l y absorb t h i s energy while undergoing large i n e l a s t i c deformations ( d u c t i l i t y requirements). In t h i s context, many bu i l d i n g codes adopt the following c r i t e r i a : 1. Minor earthquakes at the b u i l d i n g s i t e s h a l l not cause any struc-t u r a l or non-structural damage. 2. Moderate earthquakes s h a l l be the basis for the design. The b u i l d -ing s h a l l be proportioned to r e s i s t moderate ground motions, which can be expected during the l i f e t i m e of the b u i l d i n g , without s i g n i -f i c a n t damage to the basic structure. 3. Severe earthquakes may cause s i g n i f i c a n t s t r u c t u r a l damage; however, collapse of the structure and loss of human l i f e must be  avoided. This philosophy of earthquake protection of buildings concentrates e s s e n t i a l l y on preventing collapse of the main structure during a major seismic event. While t h i s approach has been considered to be adequate for many types of structures, safer approaches are desirable, and c e r t a i n l y the avoidance of s t r u c t u r a l collapse i s not s u f f i c i e n t for buildings whose functions are c r u c i a l i n the immediate aftermath of a severe earthquake (hospitals, p o l i c e s t a t i o n s , communication buil d i n g s , e t c . ) . A In recent years, many researchers have been concerned with the development of new aseismic s t r u c t u r a l systems. Among these innovative systems, the following have been shown to possess considerable merit for the earthquake r e s i s t a n t design of s t e e l b u i l d i n g s : • Base Isolated Structures • Eccentric Braced Frames (EBF) • Braced Frames with Y i e l d i n g Devices • F r i c t i o n Damped Braced Frames (FDBF) These systems are discussed b r i e f l y below. The present i n v e s t i g a t i o n i s intended as a contribution toward an understanding of the seismic performance and design of the FDBF system. 1.1.1 Base Isolated Structures The concept of base i s o l a t i o n of structures can be interpreted through a consideration of energy balance. In conventional b u i l d i n g systems, the earthquake normally introduces energy into the structure. Base i s o l a t o r s l i m i t t h i s energy tra n s f e r ; i f there i s no energy transmitted to the superstructure during a seismic event, i t i s t h e o r e t i c a l l y unnecessary to design the structure to absorb energy. Although i s o l a t o r s vary i n design, the p r i n c i p l e of a l l base i s o l a t i o n schemes i s quite s i m i l a r . B a s i c a l l y , a base i s o l a t i o n system consists of an i s o l a t o r , such as s t e e l r o l l e r bearings or rubber pads, placed between the b u i l d i n g found-ati o n and the b u i l d i n g columns (Kelly, 1979) , and an energy d i s s i p a t i n g mechanism. The i s o l a t o r reduces the o v e r a l l l a t e r a l s t i f f n e s s of the 5 structure and, as a r e s u l t , i t s fundamental frequency i s s h i f t e d below (outside) the frequency range associated with the high energy content of most (typical) earthquakes; and the energy d i s s i p a t i n g mechanism diss i p a t e s energy h y s t e r e t i c a l l y , which l i m i t s the forces transmitted to the superstructure. The idea of protection through base i s o l a t i o n has long appealed to engineers and inventors, but i t s a p p l i c a t i o n has been intermittent and, u n t i l recently, e s s e n t i a l l y unsuccessful. The concept has been given a fresh and serious impetus over the past decade, p r i m a r i l y due to the innovative work of Skinner and K e l l y (1979) , such that a number of modern structures i n various parts of the world have been designed recently incorporating the concept of base i s o l a t i o n . Although t h i s approach leads to b e n e f i c i a l r e s u l t s under t y p i c a l s i t u a t i o n s , some recent studies (Eisenberger and Rutenberg, 1986; F i l i a t r a u l t and Cherry, 1988) have shown that base i s o l a t e d structures can behave far worse than conventional structures when excited by earthquakes whose power e s s e n t i -a l l y i s concentrated at the low frequency end of the earthquake energy spectra (e.g. 1977 Romania, and 1985 Mexico earthquakes). For t h i s type of e x c i t a t i o n the fundamental frequency of a base i s o l a t e d struc-ture may f a l l within the frequency range associated with the main energy content of the earthquake, causing a quasi-resonant s i t u a t i o n . 1.1.2 Ecce n t r i c Braced Frames (EBF) Popov and Roeder (1978) have proposed the use of Eccentric Braced Frames (EBF) i n the aseismic design of s t e e l structures. The main c h a r a c t e r i s t i c of such systems, i n which moment r e s i s t i n g frames are combined with eccentric bracing, i s that the a x i a l forces induced i n the 6 braces are p r i n c i p a l l y transmitted either to a column, or to another brace, through shear and bending i n a f l o o r beam segment c a l l e d a l i n k . During a severe earthquake the l i n k s y i e l d and e f f e c t i v e l y d i s s i p a t e the input energy. Since the ultimate capacity of the l i n k s can be estimated accurately, the braces can be designed not to buckle, thereby eliminat-ing the p r i n c i p a l disadvantage encountered with c o n c e n t r i c a l l y braced frames. The e c c e n t r i c a l l y braced frame system i s an innovative earthquake r e s i s t a n t approach to s t e e l design. Considerable progress i n the development of t h i s concept has been made over the l a s t few years. A preliminary design procedure for hand-calculation of EBFs ex i s t s and some suggestions for brace connection d e t a i l s have been advanced (Popov et a l . , 1987). As a r e s u l t , the e c c e n t r i c a l l y braced frame system has been adopted for many projects, p a r t i c u l a r l y i n C a l i f o r n i a . L i t t l e d i f f i c u l t y has been reported i n the f a b r i c a t i o n and erection of t h i s system. Although the e c c e n t r i c a l l y braced frame seems to be a vi a b l e earth-quake r e s i s t a n t system, a substantial part of the structure, the beams, are s a c r i f i c e d during a major earthquake. 1.1.3 Braced Frames with Y i e l d i n g Devices The idea of using disc r e t e y i e l d i n g elements, that are implemented at the i n t e r s e c t i o n of the diagonal braces i n a frame, was f i r s t suggested by consulting engineers David Smith and Robert Henry of Auckland, New Zealand. These devices serve to absorb energy during severe earthquakes. Smith and Henry c a r r i e d out s t a t i c t e s t s on two 7 prototype devices, one of which was used subsequently i n a bu i l d i n g construction i n Auckland (Tyler, 1983). The r e s u l t s of t h e i r i n v e s t i g a -t i o n showed that, i n order to achieve optimum energy absorption, the devices should be geometrically s i m i l a r to the framework i n which they are incorporated. The tests also showed that the use of such devices appears to be a r e l i a b l e way of ensuring continuing frame action under earthquake attack. The reported drawback i s the heavy damage observed i n the y i e l d i n g devices a f t e r a severe earthquake. 1.1.4 F r i c t i o n Damped Braced Frames (FDBF) Recently an idea s i m i l a r to the Smith and Henry suggestion has been proposed by Canadian researchers ( P a l l and Marsh, 1982). The system i s b a s i c a l l y the same as described i n Section 1.1.3; however, the y i e l d i n g element i s replaced by a simple mechanism containing f r i c t i o n brake l i n i n g pads, introduced at the i n t e r s e c t i o n of frame cross-braces. The device i s designed not to s l i p under normal service loads and moderate earthquakes. During severe seismic e x c i t a t i o n s , the device s l i p s at a predetermined load, before any y i e l d i n g of the main members has occurred. Slipping of a device changes the natural frequency of the structure and allows the structure to a l t e r i t s dynamic c h a r a c t e r i s t i c s during a severe earthquake. The phenomenon of quasi-resonance between the structure and the earthquake e x c i t a t i o n i s reduced because of t h i s de-tuning c a p a b i l i t y of the structure. Large scale model frames containing these f r i c t i o n devices have been performance-tested recently at the U n i v e r s i t y of B r i t i s h Columbia ( F i l i a t r a u l t and Cherry, 1985,1987). Seismic t e s t s of a three storey 8 F r i c t i o n Damped Braced Frame (FDBF) model were performed on a 3m x 3m earthquake simulator. The proposed f r i c t i o n damping system was shown experimentally to be r e l i a b l e and to perform favourably; i t s use s i g n i f i c a n t l y reduced the seismic response of conventional structures. As a r e s u l t , these damping devices represent an e x c i t i n g development which may o f f e r new opportunities i n earthquake r e s i s t a n t design. The research described herein i s concerned with the development of a simple design procedure for t h i s new f r i c t i o n device, which can be incorporated i n s t e e l structures i n order to increase t h e i r earthquake-r e s i s t a n t performance. This study i s needed i f t h i s new design concept i s to be accepted by the engineering profession. Some of these devices are being i n s t a l l e d now i n structures i n Canada, Mexico and the U.S.A. (P a l l et a l . , 1987). However, the techniques used for t h e i r design are expensive and tedious and not r e a d i l y p r a c t i c a l for most design o f f i c e s . 1.2 Description of F r i c t i o n Damping System 1.2.1 Physical Description The system b a s i c a l l y consists of a simple mechanism containing f r i c t i o n brake l i n i n g pads introduced at the i n t e r s e c t i o n of frame cross-braces. F i g . 1.2 shows representative locations of the f r i c t i o n device i n a t y p i c a l s t e e l frame. The general arrangement of an actual f r i c t i o n damping device i s presented i n F i g . 1.3. A t y p i c a l hysteresis loop developed with prototypes of these f r i c t i o n devices i s shown i n F i g . l.A ( F i l i a t r a u l t and Cherry, 1985,1987). The hysteresis loop i s very nearly a perfect rectangle and e x h i b i t s n e g l i g i b l e fade even after 50 c y c l e s . Figure 1.2. Ty p i c a l Locations of F r i c t i o n Device. Moment Connections to Braces Slotted Slip Joints With Brake Lining Pads Figure 1.3. F r i c t i o n Device. 10 Figure 1.4. Hysteresis Loop from F r i c t i o n Device. 1.2.2 Current A n a l y t i c a l Models Two d i f f e r e n t computer models have been used with the program DRAIN-2D (Kannan and Powell, 1973) to p r e d i c t a n a l y t i c a l l y the seismic response of a FDBF. The f i r s t one, o r i g i n a l l y proposed by P a l l and Marsh (1982), i s based on an equivalent h y s t e r e t i c model and i s approxi-mate since i t does not take i n t o account the complete behaviour of the 11 f r i c t i o n devices. As a r e s u l t , a new r e f i n e d computer model was developed and the response obtained from analyses using both models was compared for a three storey model frame ( F i l i a t r a u l t and Cherry, 1985, 1987). I t was found that the r e f i n e d model gives higher member forces and d e f l e c t i o n s than the s i m p l i f i e d model. The differences i n the response were s i g n i f i c a n t ; the response predictions using the r e f i n e d model exceeded by as much as 30% the corresponding predictions obtained from the s i m p l i f i e d model. However, the seismic response predicted by the r e f i n e d model was s t i l l s i g n i f i c a n t l y superior to the corresponding behaviour of conventional b u i l d i n g systems (Moment Resisting Frames and Braced Moment Resisting Frames), so that the use of the f r i c t i o n devices was shown t h e o r e t i c a l l y to provide an e f f e c t i v e aseismic system. a) S i m p l i f i e d Model If the diagonal braces of an ordinary braced frame structure were designed not to buckle i n compression, a simple s l o t t e d f r i c t i o n j o i n t could be inserted i n each diagonal. In t h i s case each s l i p j o i n t would act independently of the other. However, i t i s frequently not econo-mical to design the braces i n compression and, more often, since the braces are quite slender, they are designed to be e f f e c t i v e i n tension only. In such cases, a simple f r i c t i o n j o i n t would s l i p i n tension but would not s l i p back during reversal of the tension load or i n the compression (buckled) regime. The energy absorption therefore would be r e l a t i v e l y poor, since the brace would not s l i p again u n t i l i t i s stretched beyond the previous elongated length, as shown i n F i g . 1.5. It i s possible to improve the energy absorption c a p a b i l i t y of the struc-ture through the use of the f r i c t i o n damping mechanism, as described below. 12 A Slippage Buckling T Figure 1.5. Hysteresis Loop of a F r i c t i o n Joint with Braces Designed i n Tension Only. Consider the h y s t e r e t i c behaviour of a simple FDBF under seismic load as shown i n F i g . 1.6. Let V = L a t e r a l load at the girder l e v e l representing the seismic load. A. = Relative displacement of node C with respect to node A (defined p o s i t i v e i n figure) A. = Relative displacement of node B with respect to node D (defined p o s i t i v e i n figure) P. = Load i n brace 1 (positive i n tension) P. = Load i n brace 2 (positive i n tension) 13 Figure 1.6. Hysteretic Behaviour of a Simple F r i c t i o n Damped Braced Frame During One Cycle of Loading. Figure 1.6 also i l l u s t r a t e s f i v e stages during a t y p i c a l load cycle. The load-deformation curves of both braces and the associated deformed shape of the frame are shown for each stage. The following points should be noted during the cycle: 1) In the very e a r l y stages of loading both braces are active and behave e l a s t i c a l l y i n tension and compression. 2) At very low load the compression brace buckles while the tension brace continues to st r e t c h e l a s t i c a l l y . 3) The device i s set to s l i p before y i e l d i n g occurs i n the tension brace. When slippage occurs, the four l i n k s of the s p e c i a l mechanism are activated and deform i n t o a rhomboid shape; i n the 1A s i m p l i f i e d model t h i s deformation pattern i s assumed to eliminate the buckled shape of the compression brace. Thus, at the end of the slippage, P 2 i s s t i l l the buckling load but now the compression brace i s s t r a i g h t . It should be recognized that t h i s approach neglects the p o t e n t i a l for l o c a l buckling and for permanent deformation of the compression brace, which i s a function of i t s sl e n d e r n e s s r a t i o . A) When the load i s reversed t h i s straightened brace can immediately absorb energy i n tension. 5) A f t e r the completion of one cycle, the r e s u l t i n g areas of the hysteresis loops are i d e n t i c a l for both braces. In t h i s way, the energy d i s s i p a t i o n i s comparable with that of a simple f r i c t i o n j o i n t when used with braces which are designed not to buckle i n compression. In other words, the energy dis s i p a t e d i n each cycle i s e s s e n t i a l l y doubled through the use of the f r i c t i o n devices i n combination with slender diagonal braces. The program DRAIN-2D (Kannan and Powell, 1973) contains truss elements which may be orientated a r b i t r a r i l y i n the x-y plane, but which can transmit a x i a l load only. Two a l t e r n a t i v e modes of i n e l a s t i c behaviour may be s p e c i f i e d , namely (a) y i e l d i n g i n both tension and compression, and (b) y i e l d i n g i n tension with e l a s t i c buckling i n compression DRAIN-2D can be used i n a very simple way to model a FDBF. For t h i s purpose we need only replace the f r i c t i o n device by two normal diagonal braces which can y i e l d both i n tension and compression as shown i n F i g . 1.7. A f i c t i t i o u s y i e l d stress i n tension i s defined for the brace material, which corresponds to the stress i n the tension brace when the 15 Ty=Fictitious Tensile Yield Load Cy=Fictitious Compressive Yield Load Figure 1.7. S i m p l i f i e d Model of a F r i c t i o n Damped Braced Frame. device s l i p s . A very low f i c t i t i o u s compression y i e l d stress i s also assigned to the brace material, corresponding to the buckling stress of the compression brace. The s i m p l i f i e d model described by Figs. 1.6 and 1.7 was o r i g i n a l l y proposed and used by P a l l and Marsh (1982). The Pall-Marsh model i s only approximate since i t does not ade-quately account for the complete deformation pattern of the f r i c t i o n device. Their assumed hysteresis behaviour i s accurate only i f the device s l i p s and the buckled brace i s f u l l y straightened at every cycle, which i s not the case i n an actual earthquake. In many cycles the ten-sion brace w i l l not s l i p but the compression brace w i l l buckle. Under such conditions the Pall-Marsh model w i l l no longer be v a l i d , since the mechanism w i l l not be activated to p u l l back the buckled brace. 16 Therefore t h i s s i m p l i f i e d model overestimates the energy absorption of the f r i c t i o n device. For t h i s reason a more accurate or r e f i n e d model was developed to evaluate the s i g n i f i c a n c e of these s i m p l i f y i n g assumptions on the o v e r a l l response of a structure under a major earthquake ( F i l i a t r a u l t and Cherry, 1985,1987). b) Refined Model To eliminate the assumptions of the Pall-Marsh model, each member of the f r i c t i o n mechanism inserted at a brace i n t e r s e c t i o n i s modelled as an i n d i v i d u a l element. Truss elements, each with t h e i r i n d i v i d u a l load-deformation curves, were i n i t i a l l y used i n t h i s new model. How-ever, assuming pinned connections at the four corners of the mechanism leads to an unstable condition, as shown i n F i g . 1.8. Therefore the V Figure 1.8. Unstable Mode of a F r i c t i o n Damped Braced Frame Modelled Only with Truss Elements. 17 mechanism connections must be capable of carrying a small amount of bending to ensure s t a b i l i t y ; as explained below, the arrangement shown i n F i g . 1.3 s a t i s f i e s t h i s requirement. The beam-column elements i n the program DRAIN-2D (Kannan and Powell, 1973) possess f l e x u r a l and a x i a l s t i f f n e s s . In the program, y i e l d i n g may take place only i n concentrated p l a s t i c hinges at the element ends. D i f f e r e n t y i e l d moments may be s p e c i f i e d at these loca-tions for p o s i t i v e and negative bending. The i n t e r a c t i o n between a x i a l force and moment i n producing y i e l d i s taken into account approximately by i n t e r a c t i o n surfaces. However, i n the program, i n e l a s t i c a x i a l deformations are assumed not to occur i n beam-column elements, because of the d i f f i c u l t y of considering the i n t e r a c t i o n between a x i a l and f l e x u r a l deformations a f t e r y i e l d i n g commences. Hence, t h i s type of element cannot be used alone to model the f r i c t i o n device, since we would lose track of the i n e l a s t i c a x i a l deformations during slippage. A r e f i n e d model which r e f l e c t s the r e a l behaviour of the f r i c t i o n device can be developed by superimposing truss and beam-column elements for the diagonal braces. The r e f i n e d model i s i l l u s t r a t e d i n F i g . 1.9. F i r s t , a l l of the members are modelled by truss elements with t h e i r own load-deformation curves. The four outside diagonal braces are allowed to buckle e l a s t i c a l l y i n compression and the two insi d e diagonals of the f r i c t i o n device s l i p i n tension and compression. Beam-column elements with zero cross-sectional area are superimposed on the diagonal brace truss elements; these beam-column members can carry bending moments only. This simple approach i s introduced to ensure the f r i c t i o n devices remain stable. To represent the pinned connections at 18 Truss Elements Beam Column Elements (1) Diagonal Braces (2) Links (3) Friction Pads Figure 1.9. Refined Model of a F r i c t i o n Damped Braced Frame. the four corners of the frame, zero p l a s t i c moments are s p e c i f i e d at these locations i n the beam-column elements. The r e a l behaviour of a FDBF can be represented accurately by t h i s r e f i n e d model. However, t h i s r e f i n e d model requires many more elements and degrees of freedom than the s i m p l i f i e d model and i n any analysis the computer time therefore i s increased s i g n i f i c a n t l y . 1.2.3 Concept of Optimum S l i p Load The energy d i s s i p a t i o n i n a FDBF i s equal to the product of s l i p load by the t o t a l s l i p t r a v e l summed over a l l devices. For very high s l i p loads the energy d i s s i p a t i o n i n f r i c t i o n w i l l be zero, as there 19 w i l l be no slippage. In t h i s s i t u a t i o n the structure w i l l behave exactly as a normal braced moment r e s i s t i n g frame. If the s l i p loads are very low, large s l i p t r a v e l s w i l l occur but the amount of energy d i s s i p a t i o n again w i l l be n e g l i g i b l e . In t h i s case the structure w i l l behave exactly as an unbraced moment r e s i s t i n g frame. Between these extremes, there i s an intermediate s l i p load d i s t r i b u t i o n which r e s u l t s i n optimum energy d i s s i p a t i o n . This intermediate d i s t r i b u t i o n i s defined as the "Optimum S l i p Load D i s t r i b u t i o n " . As noted i n F i g . 1.6, when the tension i n one of the braces causes the f r i c t i o n j o i n t to s l i p the four l i n k s i n the device are activated; t h i s forces the j o i n t i n the other brace to s l i p simultaneously. F i g . 1.10 shows a free body diagram of a f r i c t i o n device when slippage o c c u r s . The " G l o b a l S l i p Load (P )" i s d e f i n e d as the l o a d i n the g t e n s i o n brace when slippage occurs, while the "Local S l i p Load (Pg)" - s defined as the corresponding load i n each f r i c t i o n pad. As can be seen from equilibrium requirements i n F i g . 1.10, the r e l a t i o n between the Global S l i p Load and the Local S l i p Load i s given by P = 2P n - P g -where Pc__ i s the c r i t i c a l buckling load of the compression brace. For slender braces the buckling load i s very small and can be neglected. In t h i s case we obtain: P = 2P„ g -. (1.2) 20 Figure 1.10. Free Body Diagram of a F r i c t i o n Device When Slippage Occurs. These two d e f i n i t i o n s of s l i p load must be d i f f e r e n t i a t e d since they control the d i f f e r e n t computer models used. The s i m p l i f i e d model proposed by P a l l and Marsh i s based on an equivalent h y s t e r e t i c behaviour and the Global S l i p Load i s the governing parameter. The re f i n e d model considers the actual material properties of each member separately and the l o c a l s l i p load governs i n t h i s case. 1.3 Motivation At t h i s stage of development of t h i s new damping system, i t i s considered e s s e n t i a l to develop a s i m p l i f i e d design method for evaluat-21 ing the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure; i n e l a s t i c time-history dynamic computer analyses are too expensive to be used for c a l c u l a t i n g the optimum s l i p load d i s t r i b u t i o n on a regular basis i n design o f f i c e s . E a r l i e r i n v e s t i g a t i o n s of FDBF ( P a l l et a l . , 1982,1987; F i l i a t r a u l t and Cherry, 1985,1987,1988) defined a uniform optimum s l i p load d i s t r i -bution (same s l i p load for a l l devices) of a f r i c t i o n damped structure as the uniform s l i p load d i s t r i b u t i o n minimizing a p a r t i c u l a r response parameter ( d e f l e c t i o n , shear, moment, e t c . ) . However, the primary func-t i o n of the f r i c t i o n damping system i s to d i s s i p a t e energy and therefore i t i s appropriate that the optimum s l i p load d i s t r i b u t i o n of a structure should be based on energy d i s s i p a t i o n as the prime design parameter. The research presented i n t h i s study i s concerned with the devel-opment of such an optimizing procedure, for the purpose of providing the p r a c t i c i n g engineer with a simple and d i r e c t approach to the seismic design of F r i c t i o n Damped Braced Frames. 1.4 Objectives The objectives of the research are: a) To create a computer program which w i l l evaluate the optimum s l i p load d i s t r i b u t i o n of a given FDBF structure based on energy concepts. Approximate analyses w i l l be used i n order to reduce the computer costs. Some e f f o r t also w i l l be invested to adapt the program to the microcomputer environment. b) For t y p i c a l f u l l scale b u i l d i n g s , to study the performance of the program and make corrections i n the program's assumptions i f necessary. 22 c) To determine a n a l y t i c a l l y the steady state response of simple sin g l e storey f r i c t i o n damped structures under harmonic base e x c i -t a t i o n i n order to obtain fundamental information on the parameters influ e n c i n g the optimum s l i p load. d) Taking into consideration the r e s u l t s of (c) , to undertake a dimensional analysis i n order to determine the non-dimensional parameters which govern the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure. e) Using v a r i a t i o n a l p r i n c i p l e s on a shear beam analogy, to deduce an optimum s l i p load d i s t r i b u t i o n along the height of the structure when the t o t a l amount of s l i p load i s f i x e d . f) Based on the s l i p load d i s t r i b u t i o n deduced i n (e) , to perform a parametric study to optimize the t o t a l s l i p load using the computer program created i n (a) and to express the r e s u l t s i n terms of non-dimensional parameters. g) To analyse and i n t e r p r e t the r e s u l t s of (f) i n order to deduce design curves and/or equations. h) To evaluate the design equations by case studies using r e a l (existing) b u i l d i n g s . 23 2. THEORY AND PROGRAMMING CONSIDERATIONS FOR FRICTION DAMPED BRACED FRAME ANALYSIS PROGRAM (FDBFAP) "Everything should be made as simple as possib l e , but not simpler." - Albert E i n s t e i n (1879-1955), German S c i e n t i s t 24 CHAPTER 2 THEORY AND PROGRAMMING CONSIDERATIONS FOR FRICTION DAMPED BRACED FRAME ANALYSIS PROGRAM (FDBFAP) 2.1 General The computer program "FDBFAP" created for the dynamic analysis of f r i c t i o n damped plane frame structures i s described i n t h i s chapter. FDBFAP performs a serie s of dynamic response analyses of F r i c t i o n Damped Braced Plane Frames (FDBF) of a r b i t r a r y configuration for d i f f e r e n t d i s t r i b u t i o n s of s l i p load. The program consists of a serie s of sub-routines carrying out step-by-step dynamic analyses. Energy c a l c u l a -tions are made at the end of each time step and an optimizing parameter i s calculated a f t e r each a n a l y s i s . The optimum s l i p load d i s t r i b u t i o n of the structure i s the d i s t r i b u t i o n which minimizes t h i s parameter. The computer program user's guide and a l i s t i n g of the input data for an example structure are included i n Appendix A. 2.2 Global S t i f f n e s s Matrix Formulation Laboratory t e s t i n g ( F i l i a t r a u l t and Cherry, 1985,1987) and computer studies ( P a l l et a l . , 1982,1987; F i l i a t r a u l t and Cherry, 1988) have demonstrated that the main s t r u c t u r a l elements of F r i c t i o n Damped Braced Frames remain e l a s t i c and free from damage under severe earthquakes; y i e l d i n g i s i n i t i a t e d during catastrophic conditions. Therefore, i n the program i t i s assumed that the main s t r u c t u r a l elements (beams and columns) remain e l a s t i c and that the i n e l a s t i c deformations are l i m i t e d to two sources: s l i p p i n g of the f r i c t i o n pads, and buckling i n compres-sion of the diagonal braces. It should be noted that under catastrophic conditions, when large displacement d u c t i l i t i e s may be involved i n the 25 FDBF, the v a l i d i t y of the assumption that the main s t r u c t u r a l elements remain e l a s t i c w i l l no longer be v a l i d . In order to reduce computing time on element s t i f f n e s s matrix assemblages and step-by-step dynamic analyses, the global tangent s t i f f n e s s matrix of the structure i s divided into a l i n e a r component and a nonlinear component as shown i n F i g . 2.1. The global tangent s t i f f n e s s matrix of the structure ([K(t)]) i s written as [K(t)] = [K-] + C K ^ U ) ] (2.1) The l i n e a r p o r t i o n of the s t i f f n e s s matrix ([K^]) represents the global s t i f f n e s s matrix of the frame alone (beams and columns) and i s calculated only once before the s t a r t of the dynamic analyses. [K(t)l IKJ Figure 2.1 Global S t i f f n e s s Matrix Braced Frame Formulation for F r i c t i o n Damped 26 The n o n l i n e a r portion of the s t i f f n e s s matrix ([Kj-^(t)]) i s calcu-lated at each time-step and i s formed from an assemblage of f r i c t i o n device element s t i f f n e s s matrices. 2.3 Global Linear S t i f f n e s s Matrix The d i r e c t s t i f f n e s s method (Laursen, 1978) i s used to form the g l o b a l l i n e a r s t i f f n e s s m a t r i x ([K - J) from an assemblage of beams and columns s t i f f n e s s matrices. The a x i a l deformations of the hori z o n t a l beams are neglected so that a single h o r i z o n t a l degree of freedom i s s u f f i c i e n t to represent the ho r i z o n t a l displacement of each f l o o r . The a x i a l deformations of the columns are also neglected, and therefore only an a d d i t i o n a l r o t a t i o n a l degree of freedom i s needed at each node. The e f f e c t of a x i a l deformation of the columns i s not important for low-rise structures and probably for medium-rise buildings that are not slender; Before Condensat ion After Condensat ion Figure 2.2 Degrees of Freedom Considered i n Global Linear S t i f f n e s s Matrix Formulation. 27 studies on t a l l FDBF structures have not been considered i n t h i s i n v e s t i g a t i o n . Second order e f f e c t s are ignored. F i g . 2.2 presents the s t r u c t u r a l degrees of freedom considered for the frame. A s t a t i c condensation procedure (Cook, 1981) i s applied to the global l i n e a r s t i f f n e s s matrix such that only one hor i z o n t a l degree of freedom per f l o o r i s kept. The s t r u c t u r a l degrees of freedom remaining a f t e r s t a t i c condensation are also shown i n F i g . 2.2. 2.3.1 Beam Element S t i f f n e s s Matrix Beam elements are assumed to be h o r i z o n t a l and possess f i n i t e f l e x u r a l s t i f f n e s s and i n f i n i t e a x i a l s t i f f n e s s . The element forces (S 1,S 2) and deformations ( s 1 , s J ) are shown i n F i g . 2.3. El, i y Figure 2.3 Forces and Deformations of Beam Element The f l e x u r a l s t i f f n e s s of a beam element i s given by (Laursen, 1978) 5 1 I _ S r4 2, l S l 5 2 b s 2 (2.2) where E = Young 1s modulus 1^ = Moment of i n e r t i a of beam element about bending axis L, = Span of beam element 28 2.3.2 Column Element S t i f f n e s s Matrix Column elements are assumed to be v e r t i c a l and possess f i n i t e f l e x u r a l s t i f f n e s s and i n f i n i t e a x i a l s t i f f n e s s . The element forces (S 1,S 2,S 3,S I i) and deformations ( S J . S J . S ^ S , ) are shown i n F i g . 2.A. The f l e x u r a l s t i f f n e s s of a column element i s given by (Laursen, 1978) — r ~\ s, 12 -6L -12 -6L s. 1 c c 1 s, EI -6L AL J 6L 2L 2 s, 2 c c c c c 2 < > = = < > s, L 3 -12 6L 12 6L s, 3 c c c 3 s. -6L 2L J 6L AL 2 s. 4 c c c c <— v J (2.3) where E = Young's modulus I = Moment of i n e r t i a of column element about bending axis L = Length of column element s3 Figure 2.A Forces and Deformations of Column Element 29 2.4 Global Non-linear S t i f f n e s s Matrix The d i r e c t s t i f f n e s s method i s used at the end of each time-step to form the g l o b a l non-linear s t i f f n e s s matrix ([K ^ C t ) ) ] from an assemb-lage of f r i c t i o n device element s t i f f n e s s matrices. The tangent element s t i f f n e s s matrix of a f r i c t i o n device depends on the deformed configura-t i o n of the f r i c t i o n device at the end of the time-step. To save computing time, a l l the possible deformation states of a f r i c t i o n device are considered and the tangent s t i f f n e s s matrix for each state i s derived and stored i n the program. In t h i s s e c t i o n , an element s t i f f n e s s matrix [K^] i s derived for each of the i possible states of deformation of a f r i c t i o n device, i n terms of the condensed global degrees of freedom,-as shown i n F i g . 2.5. The assemblage of members included inside the dashed l i n e s i n F i g . 2.5 forms a " f r i c t i o n device element", which includes the f r i c t i o n mechanism i t s e l f along with the four external diagonal braces. R r r i Figure 2.5 F r i c t i o n Device Element 30 To determine a l l the possible deformation states of such a f r i c t i o n device element, consider a t y p i c a l element connected to a r i g i d beam at a t r a n s l a t i n g upper f l o o r and restrained from t r a n s l a t i o n a l motion at the adjacent lower f l o o r , as shown i n F i g . 2.6. We are intere s t e d i n the load-deformation curve of the element i f a c y c l i c l a t e r a l load (R.) i s applied at the upper f l o o r l e v e l , causing t r a n s l a t i o n (r.) at the upper f l o o r . The load-deformation curve of the element depends on the behaviour of the diagonal braces, and two cases must be distinguished: 1) The compression brace buckles before the device s l i p s . 2) The device s l i p s before the compression brace buckles. 2.4.1 Formulation for Slender Braces F i r s t consider the case of slender braces, where the compression brace buckles before s l i p p i n g of the device occurs. R V r 1 Figure 2.6 F r i c t i o n Device Element with Geometric Constraints 31 I f a c y c l i c t e s t were performed on the f r i c t i o n device element of F i g . 2.6, the hysteresis loop shown i n F i g . 2.7 would be obtained. It can be seen that the f r i c t i o n device element undergoes four d i f f e r e n t states during a t y p i c a l load cycle. Each state i s a function of the deformed configuration of the f r i c t i o n device and the brace condition. The d i f f e r e n t states may be defined as follows: State #1 - E l a s t i c State ( S t i f f n e s s = Kj) - Both diagonal braces are active i n tension and compression. - The f r i c t i o n pads are not s l i p p i n g . State #2 - Buckling State ( S t i f f n e s s = K 2) - The compression brace has buckled and the tension brace s t i l l stretches e l a s t i c a l l y i n tension. - The f r i c t i o n pads are not s l i p p i n g . Figure 2.7 Hysteresis Loop of F r i c t i o n Device Element With Slender A R 1 Braces 32 State #3 - P a r t i a l Slipping State ( S t i f f n e s s = K 3) - The compression brace has buckled. - The tension pad i s s l i p p i n g but the compression pad i s not s l i p p i n g . State ilk - F u l l Slipping State ( S t i f f n e s s = K A = 0) - Both f r i c t i o n pads are s l i p p i n g . - The compression brace i s being p u l l e d s t r a i g h t by the action of the mechanism. We can c a l c u l a t e an element s t i f f n e s s matrix and determine the deformation l i m i t s for which the s t i f f n e s s matrix applies for each state of the f r i c t i o n device element. State ill - E l a s t i c State By symmetry, the load i n each diagonal brace ( F ^ ^ ) i s the same and the l o a d i n each f r i c t i o n pad ( F ^ ^ ) i s the same, as shown i n F i g . 2.8. The superscript r e f e r s to the state condition number. Figure 2.8 Free Body Diagram of F r i c t i o n Device Element During State #1 33 From equilibrium of Loop A i n the hori z o n t a l d i r e c t i o n we have: 2F 15 1 )cosa - 2 F ( 1 ) c o s a = 0 b p = F ( 1 ) (2.4) b p where F^ and F^ r e f e r to the f o r c e s i n the braces and f r i c t i o n pads re s p e c t i v e l y . From hori z o n t a l equilibrium of the upper f l o o r we have: -(1) (1) 1 K } = , (2.5) b 2 cosa By co m p a t i b i l i t y of deformations we can write: 2Ail) + A ( 1 ) = r . ( 1 ) c o s a (2.6) b p 1 where = elongation of a diagonal brace during state #1 = elongation of a f r i c t i o n pad during state #1 From Hooke's law we have: A b (1) (2.7) m f ( 1 ) l p E A (2.8) where L b = length of diagonal brace as shown i n F i g . 2.6 = cross-sectional area of a diagonal brace Lp = length of a diagonal pad as shown i n F i g . 2.6 Ap = cross-sectional area of a diagonal pad Substituting Equations (2.7) and (2.8) into Equation (2.6) leads to: 2 F ' \ F ( 1 » L ... Substituting Equations (2.4) and (2.5) into Equation (2.9) y i e l d s : ^ £ + - r i ( 1 ) or r. L * • O A J cosa E cosa A^ 2A 1 (1) = E cos*a (1) ( 2 > 1 Q ) 1 L, L I [— + — E ] *b 2 A P Hence \ 2 A P The element s t i f f n e s s matrix for state #1 ( [ K j ) i s then: R ( i ) (*2 J r i J 2 , (2.12) where 35 The f r i c t i o n device element remains i n the e l a s t i c state u n t i l the a x i a l load i n the compression braces reach the c r i t i c a l Euler buckling load ( P c r ) ; when t h i s occurs Equation (2.5) y i e l d s : R {P . = 2 P cosa (2.13) l ( c r ) cr Substituting Equation (2.13) into Equation (2.10) leads to: r. (} } . = A. = + ^E-] (2.14) l ( c r ) 1 E cosa A^ 2A^ In general the f r i c t i o n device element remains i n the e l a s t i c state i f : where -A. _ (r.-r.-A s) _ A. (2.15) 2 P L, L A = cr , b p -1 E cosa lAh 2A J P A g = permanent l a t e r a l displacement r e s u l t i n g from slippage of f r i c t i o n device element (see F i g . 2.7) State #2 - Buckling State During the buckling state the compression brace has buckled and only the tension brace i s active as shown i n F i g . 2.9. To determine the load i n each f r i c t i o n pad due to an increment (2) (F^ ) i n the tension brace load during state #2, we introduce a cut i n the tension pad to make the structure s t a t i c a l l y determinate as shown i n Fi g . 2.10. From requirements of j o i n t equilibrium we obtain: Figure 2.10 Free Body Diagram of F r i c t i o n Device Element During State #2 37 - - F ( 2 ) ) sina P (2.16) - «l" - F ( 2 ) ) cosa P (2.17) = F<2> P (2.18) The increment i n s t r a i n energy of the whole system (U ) produced during state #2 i s given by (Laursen, 1978): U ( 2 ) = , ] _ [( 2 ( F . 2 ) ) 2 L.) + ( ( F ( 2 ) ) 2 L )] (2.19) p i = l 1 1 P P Substituting Equations (2.16), (2.17) and (2.18) into Equation (2.19) y i e l d s : T T(2) 1 r o T r T r(2) _ ( 2 k 2 .2 . „ ,_(2) p ( 2 ) . 2 2 U = 0 _ . [2L (F, -F ) s i n a + 2L,(F, -F ) cos a 2 E A v b p h b p + ( F ( 2 ) - F , ! 2 ) ) 2 L + ( F ( 2 ) ) 2 L ] (2.20) p b p p p where L, , L and L are defined i n F i g . 2.6. h v p 6 Using Castigliano's second theorem (Laursen, 1978) we can write: ^^VrT = 0 ( 2 - 2 1 ) 3F^ ; Substituting Equation (2.20) into (2.21) y i e l d s : 2 2 L s i n a + L, cos a + 0.5 L .(2) = f_Y h E p(2) { 2 2 2 ) P 2 2 b L s i n a + L,cos a + L v h p 38 By geometry: L, = L cosa (2.23) h p and L = L sina (2.24) v p The a x i a l load i n the tension pad i s obtained by s u b s t i t u t i n g Equations (2.23) and (2.24) into Equation (2.22) to y i e l d : p(2) = s i n 3 a + c o s 3 a + 0.5 (2) ( , p s i n 3 a + c o s 3 a + 1 b The a x i a l load i n the compression pad can be found by s u b s t i t u t i n g Equation (2.25) in t o Equation (2.18): F c 2 ) = U , • . — ~ . x 1 J F 1 ? 2 ) (2.26) 5 2(sxn 3a + c o s 3 a +1) b Dividing Equation (2.26) by Equation (2.25) and taking the absolute value y i e l d s the r a t i o of a x i a l loads (ALR) i n the f r i c t i o n pads: ALR = F(2) F5 F(2) P 2 ( s i n3 a + co s 3 a + 0.5) (2.27) F i g . 2.11 shows the v a r i a t i o n of the a x i a l load r a t i o (ALR) with the angle of i n c l i n a t i o n of the f r i c t i o n pads (a). I t can be seen that the value of t h i s r a t i o i s always les s than unity, i n d i c a t i n g that the a x i a l load i n the tension pad i s always higher than the load i n the compres-sion pad and therefore the tension pad w i l l always s l i p f i r s t . 39 o O o .2 x 0.5 0.4 0.3 0.2 0.1 0.0 0 10 20 30 40 50 60 70 80 90 Angle of Inclination [Degrees] Figure 2.11 F r i c t i o n Pad A x i a l Load Ratio During State #2 From F i g . 2.9 and F i g . 2.10, for ho r i z o n t a l equilibrium at the upper f l o o r l e v e l we can write: .(2) _ R l (2) 'b cosa (2.28) Using c o m p a t i b i l i t y of deformations. Equations (2.6 and (2.9) are s t i l l v a l i d during state #2: 0 A ( D A A ( D (1) 2A-^  + A = r 1 cosa (2.6) 2 F < 2 ) L h F ( 2 ) L „ . b b , _p_ p (2) — _ r - r + r-- , r = r. cosa E A^ E A 1 (2.9) AO Substituting Equations (2.25) and (2.28) into Equation (2.9) y i e l d s : ,(2) _ E cos 2 a  * 2^b , s i n 3 a + cos 3 a + 0.5. ^p, s i n 3 a + cos 3 a + 1 .(2) (2.29) [ Hence: K, = -= r - (2.30) Z L b - J-'- ' ' " r r__  + ^ s i n 3 a + cos 3 a + 0.5^ p-j ^b s i n 3 a + c o s 3 a + 1 A p The element s t i f f n e s s matrix for state #2 ([K 2]) i s then: R < 2 ) , 2 , r > r ( 2 ) (2.31) where [ K J = E cos'a 2L L b + ^sin 3ct + c o s 3 a + 0.5^ p-j s i n 3 a + c o s 3 a + 1 A p [ 1 _ 1 ] L - l 1 J *b The f r i c t i o n device element remains i n the buckling state u n t i l the t o t a l a x i a l load i n the tension pad reaches the l o c a l s l i p load, P^: F = F ( 2 ) , + P = P n (2.32) p(t o t a l ) p(cr) cr J (2) where F , % i s the c r i t i c a l increment of load i n the tension pad i n p ( c r ) r state #2 to produce i n c i p i e n t slippage of that pad. Al Substituting Equations (2.25), (2.28) and (2.29) into Equation (2.32): (P -P ) L L ,.(2) _ A - i cr r„,sin 3a + cos 3 a + 1 N b , _p, , 0 r l ( c r ) " A> ~ E cosa [2i~~Z T T T T K * s m 3 a + c o s 3 a + 0.5 D p In general, the system remains i n the buckling state i f : -(A :+A 2) £ ( r r r 2 - A ) < -A, (2.3A) or Ax < t r r r r A ) £ A x + A s where 2 P L, L 1 E cosa ^ 2A J (P« - P ) • , j. . j. i L, L A = 8 cr s i n 3 a + cos 3 a + 1 , b _p_, 3 E cosa L ^ s i n 3 a + c o s 3 a + 0.5 A p J A g = permanent l a t e r a l displacement r e s u l t i n g from slippage of f r i c t i o n device element (see F i g . 2.7) At the end of the buckled state the load i n the compression pad i s : - F ( 2 ) Fc , = £ ^ P (2.35) 5(total) . , , , _,_ „ c v cr 2 ( s m 3 o + c o s 3 a + 0.5) 42 Substituting Equation (2.32) into Equation (2.35) y i e l d s : ( P c r " V F . , . -. = — P (2.36) 5(total) », . , , c r 2 ( s m 3 a + c o s 3 a + 0.5) State #3 - P a r t i a l S l i pping State During the p a r t i a l slippage state the compression brace has buckled and the tension pad i s s l i p p i n g as shown i n F i g . 2.12. From j o i n t equilibrium we get Figure 2.12 F r i c t i o n Device Element During State #3 A3 For h o r i z o n t a l equilibrium at the upper f l o o r l e v e l , Equation (2.28) i s s t i l l v a l i d : R ( 3 ) F ^ 3 ) = (2.28) b cosa By c o m p a t i b i l i t y of deformations we can write: 2A^ 3 ) - A p 3 ) + 2A^ 3 ) cosa + 2A^ 3 ) sina = r [ 3 ) cosa (2.AO) elongation of hor i z o n t a l l i n k during state #3 elongation of v e r t i c a l l i n k during state #3 Using Hooke's law and s u b s t i t u t i n g Equations (2.37), (2.38) and (2.39) int o Equation (2.40) y i e l d s : (3) , 0 x F' 2L, L 2L, c o s J a 2L„ s i n J a r » > c o s a , _ | _ t _ _ + J + _____ + _ _ _ _ _ _ D P P P Substituting Equations (2.23), (2.2A) and (2.28) into Equation (2.A1) leads to: where 43> ,(3) E c o s 2 a .(3) 2L L "1 t i — + (1 + 2cos 3a + 2sin 3a)] *b A P (2.A2) 44 Hence: K3 = E c o s 2 a 2L L [-.— + (1 + 2cos 3a + 2sin 3a)] *b AP (2.43) The element s t i f f n e s s matrix for state #3 ([K 3]) i s then: R i ,(3) > = [K 3] J .(3) > (2.44) where [K,] = E c o s2 a 2L L [ j — - + (1 + 2cos 3a + 2sin 3a)] D v [ 1 - 1] L - l 1 J The f r i c t i o n device element remains i n the p a r t i a l slippage state u n t i l the t o t a l a x i a l load i n the compression pad reaches the l o c a l s l i p load: = F (3) ( P c r " V L _ ,. , _ P = -P p ( t o t a l ) p(cr) 2 ( s i n 3 a + c o s 3 a + 0.5) cr 8 (2.45) (3) where F p ( c r ) i s the c r i t i c a l increment of load i n the compression pad i n state #3 to produce i n c i p i e n t slippage of that pad. Substituting Equations (2.39), (2.28) and (2.42) into Equation (2.45) y i e l d s : 45 CP -P 1 (3) = A - i cr r 2 _ , s i n 3 a + cos 3 a + 1 ., r l ( c r ) 3 E cosa L l s i n 3 a + cos 3 a + 0.5 J 2L, L [ T - ^ + (1 + 2 c o s 3 a + 2sin 3a)] (2.46) *b p In general the f r i c t i o n device element i s i n the p a r t i a l slippage state i f : -(A 1+A 2+A 3) <; (rj-r a-A ) < -(A,+A2) (2.47) or (A,+A2) < ( r r r 2 - A ) £ A ^ + A , where 2 P L, L 1 E cosa lkb 2k 1 x cr r_ , s m 3 a + cos 3 a + 1 . b _p_, 2 E cosa L l s i n 3 a + c o s 3 a + 0.5; A J (P -P ) A = 8 cr _ , s i n 3 a + c o s 3 a + 1  3 E cosa s i n 3 a + c o s 3 a + 0.5 2L, L [ T — + (1 + 2cos 3a + 2sin 3a)] % p A s = Permanent l a t e r a l displacement r e s u l t i n g from slippage of f r i c t i o n device element (see F i g . 2.7) 46 State #4 - F u l l Slipping State During the f u l l slippage state, both f r i c t i o n pads are s l i p p i n g and therefore the f r i c t i o n device element possess no s t i f f n e s s . The element s t i f f n e s s matrix for state #4 ([K A]) i s c l e a r l y : R ( 4 )  R2 = [KJ .(4) (2.48) where [KJ - [° °] In general the f r i c t i o n device element i s i n the f u l l slippage state i f : or (r.-r.-A s) > (A.+A.+A.) ( r , - r , - A ) < -(A 1+A 8+A,) (2.49) where A 1,A J,A 3 and A g are defined i n Equation (2.47). 2.4.2 Equilibrium V e r i f i c a t i o n at F u l l Slippage As a check on the previous d e r i v a t i o n for state #4, l e t us calcu-l a t e the t o t a l a x i a l load i n the tension brace when f u l l s l i p p i n g occurs. From Equations (2.5), (2.13), (2.25), (2.32), (2.39) and (2.45) we can write: 47 F b ( t o t a l ) = P + <V Pcr> [ s i n 3 a + c o s 3 a + 1 cr s i n 3 a + c o s 3 a + 0.5 2 ( s i n 3 a + c o s 3 a + 0.5) (2.50) or b( t o t a l ) cr = P g (2.51) Equation (2.51) i s i d e n t i c a l to Equation (1.1), which was obtained when the g l o b a l s l i p l o a d (P ) was derived by considering the e q u i l i -CP brium configuration of the f r i c t i o n device at s l i p p i n g . 2.4.3 Derivation for Stubby Braces Now consider the case where s l i p p i n g of the device occurs before the compression brace buckles. I f a c y c l i c t e s t i s performed on the f r i c t i o n device element, the hysteresis loop shown i n F i g . 2.13 would be obtained. It can be observed that the f r i c t i o n device element undergoes only two d i f f e r e n t states i n t h i s case. State / / l - E l a s t i c State ( S t i f f n e s s = K a) In t h i s state both diagonal braces are active i n tension and compression and the f r i c t i o n pads are not s l i p p i n g . From previous r e s u l t s (see Equation (2.12)), the element s t i f f n e s s matrix for state #1 UKJ) is > (2.52) 48 where [K.] = E c o sa a K L \ V [ 1 _ 1 ] L - l 1 J The f r i c t i o n device element remains i n the e l a s t i c state u n t i l the a x i a l loads i n the diagonal braces reach the l o c a l s l i p load (P^) ; for t h i s case we have from Equation (2.14): R U ) _ A _ _ _ _ _ _ _ r _ + ____] r l ( c r ) ° i E cosa ^ 2A 1 (2.53) In general the f r i c t i o n device element remains i n the e l a s t i c state 49 -A. _ ( r r r 2 - A s ) _ A, (2.54) where 2 P0 LK L L _ r_b + E cosa 2A A_ = Permanent l a t e r a l displacement r e s u l t i n g from slippage of f r i c t i o n device element (see F i g . 2.13). State #2 - F u l l S l i p p i n g State ( S t i f f n e s s = K 2 = 0) As i n the case of slender braces, the element s t i f f n e s s matrix during the f u l l slippage state ([K.]) i s c l e a r l y : (2.55) where k(2) • = [K_] « 1 R<2> r(2) 0 z V. J I 2 J = to O ] The f r i c t i o n device element remains i n the f u l l s l i p p i n g state i f : or (r.-r.-A.) > A, ( r . - r 2 - A s ) < -Aj (2.56) where 2 P n L P E cosa A^ 2A 50 A g = Permanent l a t e r a l displacement r e s u l t i n g from slippage of f r i c t i o n device element (see F i g . 2.13) 2.A.A Summary The program calculates the displacement of each f r i c t i o n device ( r j - T j - A g ) at the end of each time-step, and then decides which element s t i f f n e s s matrix i s needed for the next time-step. The element s t i f f n e s s m a t r i x [ K ^ ^ ( t ) ] can be expressed as a r e l a t i o n between i n c r e m e n t a l f o r c e s (ARj, AR 2) and the a s s o c i a t e d i n c r e m e n t a l displacements ( A r a , A r 2 ) for a f r i c t i o n device element: or (2.57) The values of Cj and C 2 depend on the brace slenderness and on the deformation state of the f r i c t i o n device element, as shown i n Tables 2.1 and 2.2. 2.5 Global Mass Matrix Formulation The t o t a l mass of the structure i s concentrated at the f l o o r s with v e r t i c a l and r o t a t i o n a l i n e r t i a neglected. The global mass matrix i s therefore diagonal with the only non-zero masses associated with the ho r i z o n t a l degrees of freedom at each f l o o r . 51 Table 2.1 C o e f f i c i e n t s C. and C,, Slender Braces (P„>P ) Deformation State Deformation Intervals E l a s t i c 1 0.50 Buckling 2 sin3a+cos 3a+0.5 A s-A 1-A a_r l-r a<A s-A 1 or A i + A s < r 1 - r ^ A s + A i + A » sin 3a+cos 3a+l P a r t i a l S l i p p i n g 2 l+2(cos 3a+sin 3a) A s-A 1-A.-A 3_r 1-r.<A s-A 1-A 2 or A s+A 1+A.<r 1-r J_A s+A 1+A-+A 3 F u l l S l i p p i n g CO 00 r . - r 2 < V A i " A ' ~ A 3 or r 1-r->A s+A 1+A 1+A 3 2P L, L » Ecosa l A b 2A p J , J. cr . s i n 3 a + c o s 3 a + 1 - b j p, 2 Ecosa s i n 3 a + c o s 3 a + 0.5 ^ A p (P -P ) , i cr r- f s i n 3 a + c o s 3 a + 1 3 Ecosa s i n 3 a + c o s 3 a + 0.5 2L, L [—- + ^  (1 + 2(cos 3a + s i n 3 a ) ] o p A_ = Permanent l a t e r a l deformation r e s u l t i n g from slippage 52 Table 2.2 C o e f f i c i e n t s C1 and C 2, Stubby Braces (P {£P ) Deformation State c 2 Deformation Intervals E l a s t i c 1 0.50 A -A.sr.-r.sA.+A s 1 1 2 1 s F u l l S l i p p i n g CO CO r.-r.<A -A. 1 2 s 1 or r.-r,>A.+A 1 2 1 S 1 Ecosa ^ 2A p J A = Permanent l a t e r a l deformation s r e s u l t i n g from slippage 2.6 Global Damping Matrix Formulation The main source of energy d i s s i p a t i o n i n a f r i c t i o n damped struc-ture r e s u l t s from the action of the f r i c t i o n devices which d i s s i p a t e the energy mechanically by s l i p p i n g during an earthquake. A secondary source of energy d i s s i p a t i o n i s the basic energy-loss mechanism inherent i n a non-yielding s t r u c t u r a l system undergoing small v i b r a t i o n s . In p r a c t i c a l structures t h i s energy-loss mechanism i s seldom f u l l y under-stood and i s u s u a l l y represented by an equivalent viscous damping, for which the damping c o e f f i c i e n t s must be evaluated by experimental methods. To study the influence of viscous damping on the response of f r i c t i o n damped structures, i t i s assumed for s i m p l i c i t y that the 53 viscous damping can be represented as a mass - dependent e f f e c t only. The global damping matrix of the system for t h i s s i t u a t i o n i s expressed as: [c] = o[M] (2.58) where [c] = Global damping matrix [M] = Global mass matrix a = Proportional constant The value of the proportional constant (a) i s determined by specifying a damping r a t i o i n the f i r s t mode of v i b r a t i o n (E 1) of the unbraced structure with frequency (u ) (Paz, 1985): (2.59) Using automatically defines the maximum value of viscous damping i n any mode; the viscous damping value corresponding to then follows from the a p p l i c a t i o n of Equation (2.59). Figure 2.1 A shows the influence of mass dependent damping on the modal damping r a t i o s . It can be seen that mass dependent damping implies that the high frequency modes of the structure provide very l i t t l e viscous damping. It should be noted that more sophisticated forms of equivalent viscous damping have been proposed by several investigators (Paz, 1985). The reason for choosing the expression given by Equation (2.58) i s the 54 1 2 3 4 5 6 7 8 9 10 Mode # Figure 2.14 E f f e c t of Mass Dependent Damping on Modal Damping Ratios computational s i m p l i c i t y involved, p a r t i c u l a r l y when a diagonal global mass matrix i s used. As observed i n previous a n a l y t i c a l i n vestigations ( F i l i a t r a u l t and Cherry, 1985,1987), the e f f e c t of viscous damping i n a f r i c t i o n damped s t e e l structure i s very small compared to the damping induced by the e f f e c t of the f r i c t i o n devices. 2.7 Undamped Natural Frequency Calculations At the beginning of each analysis the computer program determines the fundamental natural frequency range of the f r i c t i o n damped structure by c a l c u l a t i n g two fundamental natural frequencies: 55 • The fundamental n a t u r a l frequency of the unbraced frame alone (uiu) , corresponding to the fundamental frequency when a l l the devices are s l i p p i n g . • The fundamental n a t u r a l frequency of the f u l l y braced frame (ui^) , corresponding to the i n i t i a l natural frequency when none of the devices are s l i p p i n g and the diagonal braces remain e l a s t i c i n tension and compression. The Vianello-Stodola method (Adey and Brebbia, 1983) i s used for the c a l c u l a t i o n s of these two fundamental undamped natural frequencies. The basic d i f f e r e n t i a l equation for the undamped free v i b r a t i o n s of a l i n e a r e l a s t i c system i s given by: [M] {v} + [K] {v} = {0} (2.60) where [M] = Global mass matrix [K] = Global s t i f f n e s s matrix {v}, {v} = V e c t o r o f mass d i s p l a c e m e n t s and a c c e l e r a t i o n s r e s p e c t i v e l y In free v i b r a t i o n conditions, the following s o l u t i o n i s assumed: {v} = {A} sin(u)t+<|>) Substituting into Equation (2.60) y i e l d s : [[K] - w» [M] ] {A} s i n (ut + <J>) = {0} 56 This equation must be s a t i s f i e d for every value of t, therefore: [CK] - w2 [M] ] {A} = {0} (2.61) or [K] [I] {A} = w2 [M] {A} (2.62) where [I] = Identity matrix Since the global mass matrix i s diagonal we can write: [M] = [M] 1' 2 [M] 1' 2 (2.63) [I] = [M]- 1' 2 [MP' 2 (2.64) Substituting Equation (2.63) and (2.64) into Equation (2.62) leads to: [K] [M] " 1 ' 2 [M] 1 ' 2 {A} = u)2 [M] 1 ' 2 [M] 1 ' 2 {A} Pre-raultiplying by [M]" 1' 2 y i e l d s : [ [M] - i ' 2 [K] [M] " 1 ' 2 ] {[M] 1 ' 2 {A}} = U)2 {[M] 1 ' 2 {A}) (2.65) Equation (2.65) i s a c l a s s i c a l eigenvalue problem of the form [A]{X} = X{X), which leads to a set of n eigenvalues (natural frequen-cies) and n eigenvectors (mode shapes). 5 7 Equation ( 2 . 6 5 ) may be written as: [A] {X} = X{X} ( 2 . 6 6 ) where [A] = [[M]" 1' 2 [K] [M]-i' 2] = Real symmetric matrix of order n {X} = {[M] 1' 2 {A}} = Vector of unknown variables (eigenvector) X = u)2 = Eigenvalue Pre-multiplying Equation ( 2 . 6 6 ) by [A ] " 1 we get: [ A ] " i {X} = \ {X} ( 2 . 6 7 ) Select a " t r i a l " vector { X ^ } , such that: [A] " 1 {X ( 1 )} = {Y ( 1 )} ( 2 . 6 8 ) If the vector { X ^ } i s a true eigenvector then: {Y ( 1 )} = ^  {X ( 1 )} ( 2 . 6 9 ) for which a constant r a t i o should e x i s t between the c o e f f i c i e n t s of {Y ( 1 )} and {X ( 1 )}, such that: Y . ( 1 ) l — Q Y = ^ , i = 1 , 2 , . . . , n ( 2 . 7 0 ) X * l 58 In general, however, Equation (2.70) i s only approximately s a t i s f i e d . Then for a second " t r i a l " vector we take: { x ( 2 ) } = { Y ( 1 ) } (2.71) and proceed to compute: {Y ( 2 )} = [A]"i {X ( 2 )} (2.72) This process i s repeated, so that {x ( r )} -• { Y ( r _ 1 ) } {Y ( r )} = = [A]"* {X ( r )} (2.73) The i t e r a t i o n stops when a l l the corresponding c o e f f i c i e n t s of {Y^r^} and { X ^ } are i n a constant r a t i o within the desired p r e c i s i o n . A tolerance of 0.001 and a maximum of 100 i t e r a t i o n s are used i n the program for performing the computations. P r a c t i c a l l y , the number of i t e r a t i o n s to achieve convergence for a plane frame i s very much less than t h i s maximum. It can be shown (Adey and Brebbia, 1983) that when [A] i s a r e a l symmetric matrix t h i s i t e r a t i v e procedure converges to the largest 1 A value (as long as the t r i a l vector does not represent an exact higher mode shape), thus providing the inverse of the smallest eigenvalue (fundamental undamped natural frequency), which i s computed from: 59 Y i ( r ) 1 — r ~ r = — for any i (2.74) x (r) X1 i (r) The vector {X } i s the corresponding eigenvector, 2.8 Non-Linear Dynamic Analysis The governing dynamic equilibrium equations for a base excited f r i c t i o n damped structure whose main s t r u c t u r a l elements (beams and columns) remain e l a s t i c at a l l times can be written as: [M]{u(t)J + [c]{u(t)) + [K T]{u(t)} + {FJt)} = -[M]{I}x (t) L s g (2.75) where [M] = Global mass matrix [c] = Mass proportional global damping matrix [KjJ = Linear portion of the s t i f f n e s s matrix (beams and columns) {F (t)} = Non-linear force vector from f r i c t i o n device elements s (u(t)},(u(t)},(u(t)} = V e c t o r s of mass displacements, v e l o c i t i e s and accelerations r e s p e c t i v e l y r e l a t i v e to the moving base x (t) = Ground ac c e l e r a t i o n g {1} = Influence vector coupling the input ground motion to each degree of freedom (= i d e n t i t y vector i n t h i s case) The average a c c e l e r a t i o n time-step scheme i s used to integrate the dynamic equilibrium equations for the F r i c t i o n Damped Braced Frame. In a d d i t i o n to providing an unconditionally stable s o l u t i o n for any 60 time-step, t h i s method has the advantage of not introducing numerical damping into the c a l c u l a t i o n s , so that the higher frequency response components are retained i n the s o l u t i o n (Owen and Hinton, 1983). 2.8.1 Incremental Equations of Motion In incremental form Equation (2.75) becomes: [M]{Au(t)) + [c]{Au(t)) + [K.]{Au(t)} + {AF (t)} = -[M] {I}Ax„(t) U.76) L s g But (AF s(t)} = [ K ^ U ) ] (Au(t)} (2.77) where [Kjjj^(t) ] = N o n l i n e a r p o r t i o n of the s t i f f n e s s m a t r i x ( f r i c t i o n devices) Substituting Equation (2.77) into Equation (2.76) leads to: [M]{Au(t)}+[c]{Au(t)}+[[K L] [ K ^ t t ) ] ] (Au(t)} = -[M]{I}Ax g(t) (2.78) which i s the incremental equation of motion of a f r i c t i o n damped structure with e l a s t i c beams and columns. 61 2.8.2 Average Acceleration Method The basic assumption of t h i s method i s that the acceleration of each degree of freedom i s constant during each time increment and that the properties of the system remain constant during t h i s i n t e r v a l . As shown i n F i g . 2.15, the acceleration during a time-step i s assumed to be: The v e l o c i t y during the i n t e r v a l i s then: U : Acceleration - t U ; i Veloctity - t U : A Displacement t t+At - t Figure 2.15 Average Acceleration Method 62 u. (T) = u. (t) + J" u. (x) dx t or 1 u i ( x ) = u i ( t ) + 2 ( u i ( t ) + ^ ( t + A t ) ) (x-t) for t . T . t+At For x = t+At we have: u.(t+At) = u.(t) + y- (u.(t)+u.(t+At)) = u.(t) + ^ (2u.(t)+Au.(t)) X X _  X X X _ X X or AuAt) = u^t+At) - uAt) = Y (2u ±(t) + A u i ( t ) ) For a l l degrees of freedom we can write: (2.79) From a s i m i l a r development we can write the vector of incremental displacements as: (Au(t)) = At {u(t)} + ( A t ) 2 {\ (u(t)} + | (Au(t)}} (2.80) Equations (2.79) and (2.80) can be re-written as: (Au(t)} = At (u(t)} + r At (Au(t)} (2.81) (Au(t)} = At (u(t)} + ( A t ) 2 (u(t)} + B {Au(t)}} (2.82) 63 where r = 1/2 B = 1/4 2.8.3 Time-Step Integration Procedure To integrate the equations of motion we keep (Au(t)} as the basic v a r i a b l e of the an a l y s i s . From Equation (2.82) we have: {A;(t)} = m)>{Au(t)} - p i t { ; ( t ) ) - k {^t)] (2.83) Substituting Equation (2.83) into Equation (2.81) leads to: (Au(t)} = At (u(t)} + ^ (Au(t)} - * (u(t)} - ^ (u(t)} (2.84) Substituting Equations (2.83) and (2.84) into Equation (2.78) leads to a system of l i n e a r equations from which the vector of incremental displacements can be solved: [K?t)] (Au(t)} = (Aptt)} (2.85) where lUt)l = [K L] + [ ^ ( t ) ] + [M] + ^ [c] (Ap(t)} = [M] (u(t)} + |p (u(t)} - {I}x (t)} + [c] {u(t)J + (Jp - 1) At (u(t)}} 64 The vector of incremental v e l o c i t i e s can then be solved using Equation (2.84). The i n i t i a l displacements, v e l o c i t i e s and f r i c t i o n force vectors for the next time-step are then: (u(t+At)} = tu(t)} + (Au(t)} (u(t+At)} = (u(t)} + (Au(t)} (F s(t+At)} = (F s(t)} + {AF g(t)} (2.86) The accelerations for the next time step are calculated by imposing dynamic equilibrium: (u(t+At)} = [M]-i {-[MJ{I}x (t+At) - [c] (u(t+At)} g - [K T] (u(t+At)} - {F (t+At)}} L s (2.87) 2.8.4 Summary of Integration Procedure The following sequence i s used i n the program to integrate the governing equations of motion: 1) O b t a i n ( u ( t ) } , (u(t)} and {F g(t)} from i n i t i a l conditions or from preceeding time-step. 2) C a l c u l a t e [ K ^ L ( t ) ] for the deformed configuration of each f r i c t i o n device element from Equation (2.57). 3) Calculate (u(t)} from Equation (2.87). 4) Solve for (Au(t)} using Equation (2.85). 5) Solve for (Au(t)} using Equation (2.84). 65 6) Calculate new i n i t i a l conditions using Equation (2.86). 7) Repeat the sequence u n t i l end of int e g r a t i o n time. 2.9 Energy Calculations Energy c a l c u l a t i o n s are made at the end of each time-step with the following energies being calculated i n d i v i d u a l l y . 1) K i n e t i c Energy Energy stored by the masses of the structure; calculated at the end of each time-step. 2) S t r a i n Energy Recoverable energy stored i n the members; cal c u l a t e d at the end of each time-step. 3) Energy Dissipated i n Viscous Damping Calculated as a step function for each time-step; being continu-ously increased during the duration of motion. 4) Energy Dissipated by F r i c t i o n Product of the s l i p load by the t o t a l s l i p t r a v e l for each device; being summed during the duration of motion. These energies are summed to produce the t o t a l energy mobilized by the system at the end of each time-step. The energy input to the structure i s calculated separately at the end of each time-step by inte g r a t i n g the seismic forces through the r e l a t i v e displacement of each f l o o r . The accuracy of ca l c u l a t i o n s i s checked by comparing the sum of the energies mobilized with the input energy. 2.9.1 Continuous Energy Expressions To determine the energy expressions to be calculated at the end of each time-step l e t us consider the governing dynamic equations of motion given by Equation (2.75): 66 [M]{u(t)} + [c]{u(t)} + [K.]{u(t)} + {F (t)} = -[M]{I}x (t) (2.75) L s g T Pre-multiplying Equation (2.75) by (u(t)} and integ r a t i n g over the time domain y i e l d s : J {u(t)} T [M]{u(t)}dt + {u(t)} T [c] {u(t)}dt + / {u(t)} T [K L]{u(t)}dt + / {u(t)} T {F s(t)}dt o o t . T j" { u ( t ) K [M] {I}x (t)dt (2.88) But Substituting into Equation (2.88) u(t) . . u(t) . J" ( u ( t ) } 1 [M] (du(t)} + J ( u ( t ) } 1 [c] (du(t)} u(t) u(t) + J (du(t)} 1 [K L] (u(t)} + j (du(t)} 1 {F (t)} o o u(t) - - ; {du(t)} 1 [M] {1} x (t) o g i n t e g r a t i n g : 1 . T . u(t) . _ \ ( u ( t ) } 1 [M] {u(t)J + J {u(t)} 1 [c] (du(t)} o -, —, U(t) _ + ± {u(t)} 1 [K-] (u(t)} + J (du(t)} 1 {F g(t)} u(t) rp = - J (du(t)} 1 [M] {1} x (t) o 8 or where ,K(t) + D(t) + U(t) + F(t) = I(t) (2.89) K(t) = -| {u ( t ) } T [M] {u(t)J = K i n e t i c energy u(t) . D(t) = J" (u(t)} [c] (du(t)} = Energy d i s s i p a t e d i n viscous o damping 1 T U(t) = 2 (u(t)} [K-] (u(t)} = S t r a i n energy stored i n the main members (beams and columns) u(t) F(t) = J (du(t)} {F (t)} = Energy absorbed by the f r i c t i o n o device elements u(t) I(t) = - J (du(t)} [M] {1} x (t) = Energy input to the o g structure 68 The energy absorbed by the f r i c t i o n device elements (F(t)) i s the sum of the energy diss i p a t e d by f r i c t i o n (EF(t)) and the recoverable s t r a i n energy stored i n the members of the elements ( i . e . diagonal braces, pads and l i n k s ) , ( S ( t ) ) . F(t) = EF(t) + S(t) (2.90) The energy d i s s i p a t e d by f r i c t i o n (EF(t)) can be calculated d i r e c t l y at the end of each time-step as the product of s l i p load by t o t a l s l i p t r a v e l . NF EF(t) = 2 1 P .s.(t) i = l X 1 1 (2.91) where P ^ = Local s l i p load of the f r i c t i o n device #i s^(t) = Total slippage of the f r i c t i o n device #i at time t NF = Number of f r i c t i o n devices. The recoverable s t r a i n energy (S(t)) can be calculated at the end of each time-step by solving Equation (2.90), 2.9.2 Discrete Energy Expressions The following expressions are used i n the program for the c a l c u l a -tions of the d i f f e r e n t energy components at the end of each time-step: 69 K(t) = | {u( t ) } T [M] {u(t)} = K i n e t i c energy D(t) = D(t-At) + | (u(t-At) + u ( t ) } T [c] (u(t) - u(t-At)} = Energy dis s i p a t e d by viscous damping U(t) = | {u( t ) } T [K L] {u(t)} T = S t r a i n energy stored i n the main members (beams and columns) F(t) = F(t-At) + ^ (u(t) - u ( t - A t ) } T {F (t-At) + F (t)} 2 s s = Energy absorbed by the f r i c t i o n device elements NF EF(t) = 2 I ? ± s i ( t ) i = l = Energy d i s s i p a t e d by f r i c t i o n S(t) = F(t) - EF(t) = S t r a i n energy stored i n the members of the f r i c t i o n device elements I(t) = I(t-At) - (u(t) - u ( t - A t ) } T [M] {I}(x (t-At) + x_(t)) 2 g g = Input energy (2.92) 2.9.3 Remark on Energy Calculations The energy c a l c u l a t i o n s presented i n Sections 2.9.1 and 2.9.2 are based on equivalent l a t e r a l seismic forces applied to a r i g i d base f r i c t i o n damped structure. This approach eliminates consideration of the r i g i d body t r a n s l a t i o n of the structure. This formulation has the advantage of not i n t r o d u c i n g the ground displacement i n the 70 c a l c u l a t i o n s ; an "absolute" energy formulation requires the ground d i s -placement (energy input = base shear * ground displacement).The values of energy input are e s s e n t i a l l y i d e n t i c a l from both formulations i n the range of s t r u c t u r a l periods between 0.1 and 5 seconds. However, for very short fundamental periods the two ca l c u l a t i o n s diverge s i g n i f i -cantly and the absolute formulation should be used. 2.10 S l i p Load Optimization As discussed i n Chapter 1, the s l i p load d i s t r i b u t i o n r e s u l t i n g i n the best (smallest) response of a structure i s defined as the "Optimum S l i p Load D i s t r i b u t i o n " of the structure. The main question to be resolved i n the s l i p load optimization pertains to the choice of the "best" parameter to be optimized. O r i g i n a l l y , various parameters were considered i n the optimization of the l o c a l s l i p load d i s t r i b u t i o n : 1) Maximum top f l o o r d e f l e c t i o n 2) Maximum base shear 3) Maximum s t r a i n energy A) S t r a i n energy area (area under the s t r a i n energy time-history) 5) Percentage of energy dis s i p a t e d by f r i c t i o n 6) Maximum i n t e r s t o r e y d r i f t If a displacement c r i t e r i o n i s to be used the maximum in t e r s t o r e y d r i f t should be preferred to the maximum top f l o o r d e f l e c t i o n . The maximum i n t e r s t o r e y d r i f t i s more s i g n i f i c a n t when considering the t o t a l response from a l l contributing modes; the maximum top f l o o r d e f l e c t i o n may not r e f l e c t the response l e v e l of the structure i n the case where higher modes are contributing. 71 The percentage of energy dissipated by f r i c t i o n i s not a very mean-i n g f u l parameter because i t does not take into account the absolute value of the energy stored by the system; also the energy input i s not the same for d i f f e r e n t values of the s l i p load. For a given s l i p load the percentage of energy dissipated by f r i c t i o n can be large, but the remaining energy l e f t i n the system can be s u f f i c i e n t to cause a large s t r u c t u r a l response. On the basis of these observations, four parameters should be considered more c l o s e l y : • Maximum i n t e r s t o r e y d r i f t • Maximum base shear • Maximum s t r a i n energy • S t r a i n energy area The f i r s t three parameters involve an instantaneous maximum response l e v e l to assess the performance of the structure while the fourth involves the t o t a l time-history response. A v a r i e t y of proposed c r i t e r i a for evaluating damage due to non-l i n e a r s t r u c t u r a l response to earthquake ground motion were reviewed by McCabe and H a l l (1987). For nonlinear h y s t e r e t i c structures, they expressed the opinion that a performance (or damage) c r i t e r i o n based on maximum response i s not ne c e s s a r i l y adequate by i t s e l f ; knowledge of the number of cycles of v i b r a t i o n at a lower l e v e l should also be considered. In t h e i r study, McCabe and H a l l proposed a damage c r i t e r i o n for y i e l d i n g structures based on low cycle fatigue concepts. A philosophy s i m i l a r to that of McCabe and H a l l should be followed i n the choice of the parameter r e f l e c t i n g the performance l e v e l of the 72 structure. It can be argued that the o v e r a l l performance of a f r i c t i o n damped braced frame can be rela t e d to a combination of the instantaneous maximum response and the e f f e c t of the enti r e time-history at a lower response l e v e l . These two e f f e c t s can be related to the amount of e l a s t i c s t r a i n energy fed into the members of the b u i l d i n g . For each s l i p load d i s t r i -bution, the time-history of the e l a s t i c s t r a i n energy stored i n the structure can be calculated. Figure 2.16 shows an example of t h i s time-h i s t o r y for a given structure and a p a r t i c u l a r s l i p load d i s t r i b u t i o n . The best (smallest) response of the structure i s obtained when the s t r a i n energy i s a minimum at every instant of time. The i d e a l optimum performance of the structure would occur when the shaded area i n F i g . Time Figure 2.16 St r a i n Energy Time-History for F r i c t i o n Damped Braced Frame With a P a r t i c u l a r S l i p Load D i s t r i b u t i o n ' 73 2.16 i s zero, corresponding to a condition when a l l the members are s t r e s s l e s s during the e n t i r e duration of the earthquake. On the basis of t h i s reasoning, the s l i p load was optimized i n terms of a Relative Performance Index (RPI), defined as: R P I = I rSEA R P I 2 SEA U + max j (o) U (2.93) max(o) where SEA = S t r a i n Energy Area = Area under the s t r a i n energy time-h i s t o r y for a f r i c t i o n damped structure SEA. s = S t r a i n E n e rgy Area f o r the i d e n t i c a l s t r u c t u r e , but (o) b J without bracing ( s l i p load = 0) TJmax = Maximum s t r a i n energy for a f r i c t i o n damped structure U , v = Maximum s t r a i n energy f o r the i d e n t i c a l structure, but max(o) b J ' without bracing ( s l i p load = 0) Values of the Relative Performance Index (RPI) are such that RPI = 1 the response corresponds to the behaviour of an unbraced structure ( s l i p load = 0) RPI < 1 the response of the f r i c t i o n damped structure i s "smaller" than the response of the unbraced structure RPI > 1 the response of the f r i c t i o n damped structure i s " l a r g e r " than the response of the unbraced structure. 74 For each s l i p load d i s t r i b u t i o n the program calculates the value of the RPI and the optimum s l i p load d i s t r i b u t i o n of the structure i s defined to be the d i s t r i b u t i o n s l i p load for which the RPI i s a minimum. Note that Equation (2.93) i s based on the e l a s t i c s t r a i n energy and i s not an exact representation of the performance of the structure for small (approaching unbraced condition) and large (approaching ordinary braced condition) values of s l i p load, during which y i e l d i n g may take place i n the beams and columns. However, past investigations ( F i l i a t r a u l t and Cherry, 1985,1987,1988; P a l l and Marsh, 1982,1987) based on earthquakes having d i f f e r e n t i n t e n s i t i e s of ground motion and d i f f e r e n t predominant periods have shown that a FDBF tuned to i t s optimum s l i p load d i s t r i b u t i o n remains e l a s t i c and free from permanent damage when excited by severe ground motions. Therefore, to save computing time, h y s t e r e t i c energy d i s s i p a t i o n has not been included i n FDBFAP since only the optimum s l i p load response i s of i n t e r e s t . . STUDY OF FDBFAP PERFORMANCE THROUGH RESPONSE OF FRICTION DAMPED STRUCTURES "A theory i s no more l i k e a fac t than a photograph i s l i k e a person." - Ed Howe (1853-1937), American Author 76 CHAPTER 3 STUDY OF FDBFAP THROUGH RESPONSE OF FRICTION DAMPED STRUCTURES 3.1 General In t h i s chapter, three example structures are analyzed using FDBFAP. For the f i r s t two examples the r e s u l t s of FDBFAP are compared with those obtained from the well-known computer program DRAIN-2D, developed by Kannan and Powell (1973). In the DRAIN-2D analyses, two s t r u c t u r a l models are considered. The f i r s t one i s an "exact" model of the example structures; i t assumes 3 degrees of freedom per nodes, non-linear beam-column elements with a x i a l load-moment i n t e r a c t i o n surfaces and second order load-sway e f f e c t s . The second model analyzed by t h i s program incorporates the same assumptions used i n FDBFAP: (1) The t o t a l mass of the structure i s concentrated at the f l o o r s with v e r t i c a l and r o t a t i o n a l i n e r t i a neglected. (2) The a x i a l deformations of the beams and columns are neglected. (3) The main s t r u c t u r a l elements (beams and columns) of the structure remain e l a s t i c at a l l time. (4) The r e f i n e d model of a FDBF i s used (see section 1.2.2). For t h i s second case, the dynamic response r e s u l t s obtained using the DRAIN-2D program should be i d e n t i c a l to the ones obtained from FDBFAP. In the t h i r d example, FDBFAP i s used to model a three-storey s t e e l F r i c t i o n Damped Plane Braced Frame, which had previously been i n v e s t i g a -ted experimentally for seismic response on the earthquake simulator i n 77 the Earthquake Engineering Laboratory of the C i v i l Engineering Department at the U n i v e r s i t y of B r i t i s h Columbia ( F i l i a t r a u l t and Cherry, 1985,1987). 3.2 Numerical Example #1: Ten Storey Single Bay Frame The structure used i n the f i r s t numerical example consists of a ten storey s t e e l frame designed by Hanson and Fan (1969) according to the s p e c i f i c a t i o n s of the 1967 Uniform Building Code. This structure represents a good example of a s t e e l b u i l d i n g designed according to e a r l i e r code requirements and for which earthquake r e t r o f i t may be needed. The frame layout i s shown i n F i g . 3.1. Note that the frame i s not f u l l y braced; i t consists of alternate open storeys, which i s not a t y p i c a l of the arrangement used i n r e a l b u i l d i n g s . The s t r u c t u r a l properties of the frame are given i n Table 3.1. E h-CO @ o i Figure 3.1 Structural Layout for Numerical Example #1 78 Table 3.1 Str u c t u r a l Properties for Numerical Example #1 Floor Level Beams Braces Columns Section Ix(mm*) A(mm2) A(mm2) Section I x(mm») A(mm2) 10 18WF45 293xl0 6 8542 1858 14WF34 141x106 6452 9 18WF45 293xl0« 8542 14WF43 179x106 8161 8 18WF45 293x10* 8542 1858 14WF43 179x10s 8161 7 18WF45 293xl0 6 8542 14WF61 267x106 11574 6 18WF45 293xl0 6 8542 1858 14WF61 267x106 11574 5 18WF50 333xl0 6 9490 14WF74 332x106 14039 4 18WF50 333xl0 6 9490 1858 14WF74 332x106 14039 3 18WF50 333x106 9490 14WF87 402x106 16490 2 18WF50 333x10s 9490 1858 14WF87 402x106 16490 1 18WF50 333x10s 9490 14WF103 485x106 19523 Dead Load = 195.7 kN/floor (excluding roof) No load on the roof Design c r i t e r i a = 'Member stresses •Lateral d e f l e c t i o n l i m i t of 0.35% of the height •UBC 1967 The ground motion selected for the analyses was the f i r s t s i x seconds of the El-Centro earthquake, May 18, 1940, S00E component, with a peak acceleration of 0.34 g. No viscous damping was considered i n the analyses. Preliminary analyses were made to determine the proper inte 79 gration time-step to be used. An int e g r a t i o n time-step of 0.005 second was found to be s u f f i c i e n t and was used with DRAIN-2D and FDBFAP. Table 3.2 compares the undamped fundamental natural frequencies predicted by FDBFAP with the ones predicted by an e x i s t i n g standard program for the dynamic analysis of plane frame problems (program DYNA, Law and Grigg, 1978). Table 3.2 Undamped Fundamental Natural Frequencies for Numerical  Example #1 Program Natural Frequency (Hz) Unbraced Frame ( A l l Devices Slipping) F u l l y Braced Frame (No Slippage) DYNA 0.5208 0.8496 FDBFAP 0.5448 0.9708 It can be seen that the natural frequencies predicted by FDBFAP are s l i g h t l y higher than the ones predicted by the plane frame analysis program. This i s expected since, unlike the program DYNA, FDBFAP neglects the a x i a l deformations of the main members (beams and columns) and the masses are only associated with the hori z o n t a l displacements of the f l o o r s . Therefore the FDBFAP s t r u c t u r a l model i s " s t i f f e r " than the model analyzed by the standard plane frame program. The r e s u l t s of an optimization study performed by FDBFAP for a uniform s l i p load d i s t r i b u t i o n of the f r i c t i o n devices are presented i n F i g . 3.2 for 12 d i f f e r e n t values of the l o c a l s l i p load i n the e l a s t i c range of the cross-braces (0 - 220 kN). 80 Local Slip Load [kN] Figure 3.2 S l i p Load Optimization for Numerical Example #1. I t can be seen that with respect to the data points the optimum s l i p load i s 67 kN for each f r i c t i o n device; the corresponding Relative Performance Index (RPI), Equation (2.94), i s 0.426. Note that for values of l o c a l s l i p load larger than 175 kN the RPI i s larger than un i t y and the response of the FDBF i s worse than the response of the unbraced structure (RPI = 1). F i g . 3.3 presents the energy time-histories for the unbraced structure ( l o c a l s l i p load = 0) and for the f r i c t i o n damped structure tuned to i t s uniform optimum l o c a l s l i p load value of 67 kN. I t can be observed that the energy input time-history i s d r a s t i c a l l y a f f e c t e d by the introduction of f r i c t i o n devices i n the structure; at the end of the Local Slip Load=0 60-1 Time [sec] Local Slip Load=67kN 60-\ Time [sec] Figure 3.3 Energy Time-Histories for Numerical Example #1. 82 record (t = 6 seconds) 79% more energy has been transmitted to the FDBF than to the unbraced frame. The e f f e c t of the f r i c t i o n devices i n improving the seismic response of the structure i s c l e a r l y shown by the s t r a i n energy time - h i s t o r i e s ; the maximum s t r a i n energy induced i n the FDBF i s only 50% of the maximum s t r a i n energy induced i n the unbraced structure. Also, the area under the s t r a i n energy time-history (Strain Energy Area) of the FDBF i s only 39% of the Strain Energy Area of the unbraced frame. At the end of the earthquake, 80% of the energy input has been di s s i p a t e d by the f r i c t i o n devices. The top f l o o r r e l a t i v e displacement time-histories for the unbraced frame and FDBF, as determined by the FDBFAP and DRAIN-2D programs, are shown i n F i g . 3.A. As expected, the FDBFAP r e s u l t s are i d e n t i c a l to the r e s u l t s obtained from DRAIN-2D with the FDBFAP assumptions. The maximum d e f l e c t i o n of the unbraced structure predicted by FDBFAP i s 39% lower than the value predicted by DRAIN-2D when "exact" model assumptions are used. For the f r i c t i o n damped structure the difference i n predicted peak de f l e c t i o n s reduces to 22%. This i s explained by the fact that some y i e l d i n g occurs i n the beams of the unbraced frame, as shown i n Fi g . 3.5, causing s i g n i f i c a n t differences i n the response. Part of the difference also may be due to the fact that FDBFAP neglects the e f f e c t of a x i a l deformation i n the columns. For a slender structure t h i s e f f e c t can be important. The f r i c t i o n damped structure remains e l a s t i c at a l l time and the response predicted by FDBFAP i s much closer to the actual response predicted by DRAIN-2D. 3.3 Numerical Example #2: Low Rise Frame The b u i l d i n g configuration employed for the second example i s taken (with minor changes) d i r e c t l y from a paper by Montgomery and Ha l l (1979). The structure i s a representative low-rise b u i l d i n g , as shown i n F i g . 3.6. 83 Local Slip Load=0 Time [sec] Local Slip Load=67kN Time [sec] Figure 3.4 Top Floor Relative Displacement Time-Histories for Numerical Example #1. 84 Unbraced Frame Member Yielded FDBF Figure 3.5 S t r u c t u r a l Damage Predicted by DRAIN-2D for Numerical Example #1. Diagonal Braces: 1858 mm Friction Devices: 0.98mx0.34m All Columns: W 14x68 Floor Weight 910 kN (1st and 2nd Floor) 416 kN (3rd Floor) Figure 3.6 S t r u c t u r a l Configuration for Numerical Example #2. 85 The Newmark-Blume-Kapur a r t i f i c i a l earthquake (Newmark et a l . , 1973) scaled to a peak acceleration of 0.5 g was used i n the analyses. This record has a duration of 15 seconds and i s an average of many t y p i c a l earthquakes. Viscous damping was introduced by specifying 2% c r i t i c a l damping i n the f i r s t mode of v i b r a t i o n of the unbraced structure. The undamped fundamental natural frequencies predicted by FDBFAP and DYNA are shown i n Table 3.3. Again i t can be seen that the natural frequencies predicted by FDBFAP are s l i g h t l y higher than the ones predicted by the plane frame analysis program. Table 3.3 Undamped Fundamental Natural Frequencies for Numerical Example #2 Program Natural Frequency (Hz) Unbraced Frame ( A l l Devices Slipping) F u l l y Braced Frame (No Slippage) DYNA 1.3907 2.6529 FDBFAP 1.3931 2.7241 F i g . 3.7 presents the optimization r e s u l t s for a uniform s l i p load d i s t r i b u t i o n of the devices for t h i s low r i s e structure. The optimum s l i p load i s 134 kN for each device, corresponding to a RPI of 0.219. F i g . 3.7 also shows that there i s very l i t t l e v a r i a t i o n i n the Relative Performance Index for l o c a l s l i p loads between 90 kN and 220 kN. This suggests that the seismic response of t h i s structure i s not p a r t i c u l a r l y s e n s i t i v e to v a r i a t i o n s i n the optimum s l i p load which may occur due to environmental and construction f a c t o r s , such as temperature change and adjustment v a r i a b i l i t y . 86 I—I P-. PS 0.2-0.8-0.6-0.4-0 0 50 100 150 200 Local Slip Load [kN] Figure 3.7 S l i p Load Optimization for Numerical Example #2. The energy tim e - h i s t o r i e s for the unbraced structure ( l o c a l s l i p load = 0) and for the f r i c t i o n damped frame ( l o c a l s l i p load = 134 kN) are presented i n F i g . 3.8. Again the f r i c t i o n devices are very e f f i c i e n t i n reducing the amplitudes of the strain-energy time-history and therefore i n improving the response of the structure. The maximum s t r a i n energy induced i n the FDBF i s only 23% of the maximum s t r a i n energy induced i n the unbraced structure. The S t r a i n Energy Area for the FDBF i s only 21% of the St r a i n Energy Area for the unbraced frame. At the end of the record, 86% of the input energy has been diss i p a t e d by f r i c t i o n . 87 X E? 400 E 300-200-100-0 J 0 400 E 300-200-100-0J Local Slip Load=0 Energies: Klntllc Dltilpaltd In Vltcout Damping Strain Dl**lpet»d by Friction  Input I-.- : i •.' < WW ^Vy*.?.?,^'/. .5 77me [sec] Local Slip Load=134kN Energies: Klnttle Dlstlpot*d In Vlteaui Damping Slroln Dlttlpottd by rrletlen  input 6 9 Time [sec] 15 Figure 3.8 Energy Time-Histories for Numerical Example #2. 88 F i g . 3.9 presents the top f l o o r r e l a t i v e displacement time-h i s t o r i e s for the unbraced structure and for the FDBF. Again the FDBFAP r e s u l t s are i d e n t i c a l to the ones obtained with DRAIN-2D incorporating the FDBFAP assumptions. The predictions of DRAIN-2D and FDBFAP for zero l o c a l s l i p load are i n i t i a l l y close, but the amplitude agreement becomes poor near the end of the response record. This i s due to the severe member y i e l d which, as shown i n F i g . 3.10, was predicted by DRAIN-2D. The fa c t that the e l a s t i c displacements i n t h i s region of the record are larger than the i n e l a s t i c response values i s probably due to the earth-quake producing a quasi-resonance behaviour i n the e l a s t i c structure. In the case of the FDBF, the members remain e l a s t i c at a l l times. The e f f e c t of column a x i a l deformation i s small for t h i s three storey structure and the s t r u c t u r a l response predictions are very good for the e n t i r e duration of the earthquake. 3.A Numerical Example #3: Three Storey Test Frame FDBFAP has been used to analyze a 1/3 scale model of a 3 storey FDBF which had been tested on a shaking table at the U n i v e r s i t y of B r i t i s h Columbia ( F i l i a t r a u l t and Cherry, 1985,1987). The o v e r a l l dimensions of the model frame were 2.05x1.AO m i n plan and 3.53 m i n height. A l l the main beams and columns were made of the smallest struc-t u r a l shape (S75x8) a v a i l a b l e . The dead load was simulated by concrete blocks at each f l o o r . The general arrangement of the t e s t frame i s shown i n F i g . 3.11. The t e s t frame was equipped with f r i c t i o n devices at each f l o o r . On the basis of the r e s u l t s obtained by DRAIN-2D, a l o c a l s l i p load of 3.5 kN for each f r i c t i o n device at each f l o o r was used for the study of the FDBF model. 89 300-1 Local Slip Load=0 200-100-0 -100 -200 H - 3 0 0 -Progroms: DRAIN-2D OR AM-20 zFDBFAP_ Uodtl FDBFAP 3 0 0 200 100 H 0 700 -200-- 3 0 0 J 0 'i i 6 9 Time [sec] 12 -1 15 Local Slip Load=134 kN Programs: DRAIN-2D DRAIN-2D ^ FOBFAP^ Uodth-FDBFAP n — 1 — • — r 3 6 -I 1 1 1 1 r— 9 12 15 Time [sec] Figure 3.9 Top Floor Relative Displacement Time-Histories for Numerical Example #2. 90 1 Unbraced Frame FDBF -• Member Yielded Figure 3.10 Structural Damage Predicted by DRAIN-2D for Numerical Example #2. Figure 3.11 General Arrangement of Test Frame. 91 Some preliminary tests were performed on the model frame to deter-mine i t s fundamental dynamic properties. The natural frequencies and equivalent viscous damping of the structure were determined experiment-a l l y from free and forced v i b r a t i o n tests using the shake table as the ex c i t a t i o n source. Table 3.4 shows the measured and computed funda-mental natural frequencies of the structure. The measured f r a c t i o n of c r i t i c a l damping i n the fundamental mode was 0.28%. Table 3.4 Natural Frequencies of Test Frame Source Natural Frequency (Hz) Unbraced Frame ( A l l Devices Slipping) F u l l y Braced Frame (No Slippage) Experimental 2.86 7.03 FDBFAP 2.77 6.90 Several ground ( i . e . , shaking table) motions were used i n the experimental i n v e s t i g a t i o n . Of these, the Taft Earthquake, July 21, 1952, V e r t i c a l Component, scaled to a peak ground acceleration of 0.90 g, was selected for c o r r e l a t i o n studies with FDBFAP. This ground motion caused extensive i n e l a s t i c behaviour of the unbraced moment r e s i s t i n g frame and s i g n i f i c a n t slippage i n a l l 3 f r i c t i o n devices of the FDBF. Figure 3.12 shows time-histories of the measured t h i r d f l o o r r e l a -t i v e l a t e r a l displacements for the unbraced structure and the f r i c t i o n damped structure. It can be seen that the FDBF analysis shows close c o r r e l a t i o n i n the s t r u c t u r a l period, but overestimates the amplitudes of the unbraced frame. This i s due to the s i g n i f i c a n t i n e l a s t i c defor-mations of the structure and also to the viscous damping values used i n the a n a l y s i s . The experimental damping values used with FDBFAP were 92 0) O Unbraced Moment Resisting Frame 150 100-£ -100 -150 -50-Time [sec] Friction Damped Braced Frame 150 j , -100-n • • i • • i • ' 1 • 1 1 1 • f 0 3 6 9 12 15 Time [sec] Figure 3.12 Time-Histories of Third Floor L a t e r a l Displacements for Numerical Example #3. 93 measured at very low amplitude exc i t a t i o n s and are not representative of the frame behaviour under large i n e l a s t i c v i b r a t i o n s . Close c o r r e l a -tions i n amplitude and s t r u c t u r a l period are observed for the FDBF, whose s t r u c t u r a l members remained e l a s t i c during the duration of the earthquake. 3.5 Solution Times CPU times necessary to perform the low r i s e frame analyses on the Amdahl V8 main frame computer at the Un i v e r s i t y of B r i t i s h Columbia are shown i n Table 3.5. Table 3.5 Solution Times for Low Rise Frame on Amdahl V8 Program CPU Time (seconds) Relative Performance DRAIN-2D (2 dynamic analyses) 42.00 1.00 FDBFAP (11 dynamic analyses) 11.9 19.41 It took 42 seconds of CPU for DRAIN-2D to perform two dynamic analyses ( l o c a l s l i p loads of 0 and 134 kN) , while FDBFAP took only 11.9 seconds of CPU to perform the requested 11 dynamic analyses (Fig. 3.7). I t can be seen that FDBFAP i s much more economical to use than DRAIN-2D. A comparison of CPU times required by FDBFAP on various computers a v a i l -able at the Un i v e r s i t y of B r i t i s h Columbia i s presented i n Table 3.6. From the r e s u l t s of the three example structures investigated i n t h i s chapter, i t i s concluded that FDBFAP i s s u f f i c i e n t l y simple and 94 Table 3.6 Comparison of CPU Times for FDBFAP Computer CPU Time (Seconds) Relative Performance Micro, 4.77 mHz PC, (with 8087) 1254.6 1.00 Micro, 10 mHz AT, (with 80287) 418.5 3.00 Mini, Sun 3/260, (with ffpa) 50.9 24.65 Main Frame, Amdahl V8 11.9 105.43 Microcomputer r e s u l t s obtained using Microsoft FORTRAN V.4. s u f f i c i e n t l y accurate to be used for the p r a c t i c a l determination of the optimum s l i p load d i s t r i b u t i o n of structures equipped with f r i c t i o n damping devices. Thus, the objective of developing a simple and r e l i a b l e computer program, which models the major c h a r a c t e r i s t i c s of a f r i c t i o n damped braced frame with moderate computational e f f o r t , has been achieved. It should be noted again here that t h i s conclusion i s based on the assumption that the s t r u c t u r a l members of a FDBF and t h e i r connections remain e l a s t i c when the system i s adjusted to i t s optimum s l i p load d i s t r i b u t i o n , and that the e f f e c t of a x i a l deformation of the columns i s not important (for low-rise structures and perhaps even for medium-r i s e structures that are not too slender). 95 h. STUDY OF AN ANALOGOUS NONLINEAR MECHANICAL SYSTEM "Analogy, although i t i s not i n f a l l i b l e , i s yet that telescope of the mind by which i t i s marvelously a s s i s t e d i n the discovery of both phys i c a l and moral t r u t h . " - Caleb C. Colton (1780-1832), English Clergy 96 CHAPTER 4 STUDY OF AN ANALOGOUS NONLINEAR MECHANICAL SYSTEM 4.1 Motivation In t h i s chapter an analogy i s made between a one-storey f r i c t i o n damped structure and a simple nonlinear mechanical system. The steady state response of the mechanical system to sinusoidal base e x c i t a t i o n i s determined a n a l y t i c a l l y to provide basic information on the nature of the response of structures equipped with the new f r i c t i o n damping system. The a n a l y t i c a l solutions are compared with the predictions of FDBFAP. 4.2 Formulation Consider a sing l e storey f r i c t i o n damped structure which i s excited by a harmonic base ac c e l e r a t i o n , as shown i n F i g . 4.1. Provided that s l i p p i n g of the device occurs before the compression brace buckles (see section 2.4.3), t h i s structure can be represented by an equivalent system co n s i s t i n g of a mass m which i s acted upon by a p e r i o d i c d i s t u r b i n g force P C O S U J t and which i s suspended from a b i l i n e a r spring that exhibits a hysteresis curve of the type shown i n F i g . 4.2. The equation of motion for t h i s system i s : m x(t) + K^Ftx.u.t) = P cosw_t (4.1) Figure 4.2 B i l i n e a r Hysteresis 98 where F(x,u,t) i s the hysteresis restoring force per unit s t i f f n e s s , F(x,u,t) i s such that: F(x,u,t) = K i f 0 < x < x„ i f x > x„ (A.2) where x = Displacement of the mass r e l a t i v e to the moving base x 0 = L a t e r a l d e f l e c t i o n to cause slippage = K + K. u c K = "*"?F*C [oto»] = L a t e r a l s t i f f n e s s of the unbraced structure u h 3 2+30 2EA b = — — cos s a sina = L a t e r a l s t i f f n e s s of the diagonal braces P = I-ma | = A b solute amplitude of the equivalent disturbing force 6 " n c I, = Moment of i n e r t i a of the beam D I = Moment of i n e r t i a of one column c h = Storey height L = Bay width E = Young's modulus a = Angle of i n c l i n a t i o n of the diagonal cross-braces A^ = Cross-sectional area of each diagonal cross-brace The properties of t h e . b i l i n e a r hysteresis shown i n F i g . A.2 can be simply r e l a t e d to the physical properties of the r e a l structure. Thus, 99 K u = 1 - (^) x o = 2P^cosa K. (A.3) (4.A) Equation (4.1) can be written as: where x s m ID, t b p ma U) _ 7^ + F ( x , U , T ) " V 0 S ( ^ ) ^ 4.3 Solution The method of slowly varying parameters (Minorski, 1947) i s used to solve Equation (4.5). Applying the approach followed by Caughey (1960) we assume the s o l u t i o n to be of the form: 100 x(x) = R(x) cos(nx+<p(x)) (A.6) where R(x) and <£(x) are slow v a r y i n g amplitude and phase. D i f f e r e n t i a t i n g Equation (A.6): = x' = R'cosG - RnsinG - R^'sinG (A.7) where 9 = nx + <J>(T) (A.8) Now we can argue that we also expect the v e l o c i t y x' to look l i k e the l i n e a r case and therefore we set: R'cosG - R^'sinG = 0 (A.9) Hence from Equation (A.7) x" = -n JRcos6 - nR'sine - nR<f>'cos6 (A.10) Substituting Equation (A.10) into Equation (A.5): -nR'sine - nR^'cose - n JRcos6 + F,__ _ , = x cos(6-<j>) (A. 11) lKCOSo,U,X/ S M u l t i p l y i n g Equation (A.9) by nc° s6» Equation (A.11) by sinG, and subtracting y i e l d s : -nR'-n,2Rsin6cos6 + F,_ Q .sinG = x cos(9-<J>)sinG (A. 12) 101 M u l t i p l y i n g Equation (4.9) by nsin6, Equation (4.11) by cos6, and adding y i e l d s : -nR<r ' -n 2Rcos 2 e + F , _ a .cose = x cos ( e - * )cose (4.13) 1 R ( R C O S 6 , U , T ) s r Since R and $ are slow varying, i . e . they change l i t t l e over one cycle of main motion, we can integrate Equations (4.12) and (4.13) over one c y c l e and r e p l a c e R and <f> by t h e i r average values R and ^ , assumed constant. This leads to: (4.14) where -2nR' + S = x sintj» (R) S -2nRtj>' - n2R + C _ = x coscj) (R) S , 2TT c _ = - J" F _ cosede ( R ) V o ( R C O S 6 , U , T ) , 2TT S _ = - J F _ sin6d6 (R) V o (Rcose .u .r) The steady-state response i s obtained by se t t i n g R' and cj>' equal to zero i n Equation (4.14): (4.15) S ( R 0 ) = X s s i n * o (4.16) C ( R 0 ) ' n'R<> = X s C O S * < 102 where R 0 and <J>0 are steady state amplitude and phase. Solving for q i n Equation (4.16) y i e l d s : C ( R < > ) X e 2 S ( R 0 ) 2 1 ' 2 n' = — ^ - ± t ( ^ ) - (-^—) ] (4.17) Using F i g . 4.2 and Equation (4.15) i t can be shown (Caughey, 1960) that: r - u R , s i n J 6 * Ro x o i f — > — x x s s R o x o i f — < — X X s s (A.18) '(R 0) r R 0 R 0 x 0 — [u9* + (I-U)TT - | sin29*] i f — > — s s R 0 x o i f — < — X X s s (A.19) where 2(x 0/x ) Substituting Equations (A.18) and (A.19) into Equation (A.17) leads to the amplitude equation: i- i R 0 x o I + — = — " (R 0/x ) i f X < — X 0 s s s n2 = i [U9* +(1-U)TT- I sin26*]±[ ) ,usin*9* N Jn i ' J > j r _ IT L F Ro X x o > — X o s s s (A.21) 103 4.4 Analysis F o r a p a r t i c u l a r v a l u e of u and x 0 / x g , n 2 can be s o l v e d f o r s p e c i f i e d v a l u e s of ( R 0 / x _ ) . The maximum amplitude (resonance) w i l l occur at the point where n has a double root, that i s at the point where: (4.22) ^usin28*^ 2 ( V x - ) 2 Substituting Equation (4.20) into Equation (4.22) y i e l d s the amplitude of the motion at resonance: R0 4 U / T T (x 0/x g) x ~ 4 U / T T - ( X /x 0) s s 0 (4.23) Since R 0 i s by d e f i n i t i o n p o s i t i v e , bounded response i s obtained provided that s 4u X . TT (4.24) Substituting Equations (4.3,4.4 and 4.5) into Equation (4.24) y i e l d s the following condition on the l o c a l s l i p load to provide bounded amplitudes at resonance: 2P„cosa a S TT _ W 4 g (4.25) From Equation (4.17) the resonant frequency r a t i o n i s given by C, n 2 = 'r '(R„) ~ RT (4.26) 104 The value of the l o c a l s l i p minimizing the amplitude of the motion at resonance can be obtained by: which leads to the condition: (4.28) Substituting Equations (4.3, 4.4 and 4.5) into Equation (4.28) y i e l d s * the l o c a l s l i p load P 0 which minimizes the resonant amplitude: * 2P ncosct a 2 TT W 2 g (4.29) The minimum resonant amplitude R0* can be obtained by s u b s t i t u t i n g Equation (4.28) into Equation (4.23): (4.30) Equation (4.30) also reveals that the f r i c t i o n device w i l l always be s l i p p i n g at resonance regardless of the value of the s l i p load used, s i n c e R 0*/x > x /x . The frequency r a t i o n* at which t h i s optimum s s 105 resonance occurs can be found by su b s t i t u t i n g Equations (4.28) and (4.30) into Equation (4.26): (4.31) where u K — and m b m The condition for which the amplitude of the motion i s a minimum f o r a p a r t i c u l a r e x c i t a t i o n frequency w (other than resonance) cannot be found a n a l y t i c a l l y , since Equation (4.21) i s transcendental and R 0/x g cannot be solved for every frequency r a t i o n. However, an analysis of the s t a b i l i t y of the steady-state s o l u t i o n (Caughey, 1960) shows that t h i s system i s always stable and that the family of frequency response curves that can be generated from Equation (4.21) are a l l single valued; hence a jump phenomenon, which i s c h a r a c t e r i s t i c of c e r t a i n nonlinear systems (Nayfeh and Mook, 1979) , i s not expected to occur. Formally we can invert Equation (4.21) as: — - G ( — , — , u) X W, X s b s (4.32) where G i s a sing l e valued function. The value of the l o c a l s l i p load minimizing the amplitude of the motion at any forcing frequency can be obtained by: 3G 3 ( x 0 / x s ) = 0 (4.33) 106 which w i l l y i e l d a condition such as: x 0 u — = H (-S s B , u) (4.34) where H i s an unknown function. Substituting Equations (4.3, 4.4 and 4.5) into (4.34) y i e l d s a r e l a t i o n s h i p for the optimum l o c a l s l i p load P 0: 2P 0cosa W = (1 K a u - -* H(-S V 8 K V 1 " ^ ) (4.35) or: 2P.cosa a T u 0 E W g Q ( ^ , T ; g u (4.36) where = Y = Natural period of the f u l l y braced structure b T = — = Natural period of the unbraced structure u U) u T_ = Period of the ground motion a = Peak ground ac c e l e r a t i o n g = Acceleration of g r a v i t y W = Weight of the structure a = Angle of i n c l i n a t i o n of diagonal cross-braces with the h o r i z o n t a l P 0 = Optimum l o c a l s l i p load Q = Unknown function 107 The s i g n i f i c a n c e of Equation (4.36) i s that i t reveals the non-dimensional parameters governing the optimum s l i p load of a one-storey FDBF under harmonic ground motion. I t can be expected that these parameters w i l l also be important i n the case of a FDBF excited by a general earthquake ground motion. An important conclusion that can be drawn from Equation (4.36) i s that the optimum s l i p load depends on the frequency and amplitude of the ground motion and i s not s t r i c t l y a s t r u c t u r a l property. Therefore, the earthquake ground motion expected at the construction s i t e w i l l have to be considered i n the design of structures equipped with t h i s new f r i c t i o n damping system. Also, i t i s i n t e r e s t i n g to note that the value of the optimum s l i p load i s l i n e a r l y proportional to the peak ground a c c e l e r a t i o n . 4.5 Numerical V e r i f i c a t i o n To further v e r i f y the accuracy of FDBFAP, the steady-state response of a one storey FDBF i s calculated by t h i s program and the r e s u l t s are checked against the a n a l y t i c a l s o l u t i o n derived above. The physical properties of the structure are: m = 45.34 N-sVmm I c = 9.03 x 10 6 mm h - CO a g = 0.05 g E = 200 000 MPa h = 4570 mm L 7600 mm 108 The properties of the analogous mechanical system are: m = 45.34 N-sVmm P = 22.24 kN = 32.25 mm2 u = 0.70 K u = 453.65 N/mm % =5.89 rad/s K d = 1067.68 N/mm X s = 14.14 mm «b = 1521.33 N/mm Table 4.1 p r e s e n t s the d i f f e r e n t v a l u e s of l o c a l s l i p load, P^, considered i n the numerical i n v e s t i g a t i o n along with the corresponding x /x„ values, s 0 Table 4.1 Values of S l i p Load Considered i n Numerical Investigation P a (kN) x s / x 0 10.1854* 0.8935 15.0000 0.6066 20.3708+ 0.4467 25.0000 0.3639 30.0000 0.3033 35.0000 0.2600 *Minimum s l i p load value to provide bounded amplitude at resonance + S l i p load value minimizing the resonant amplitude In the numerical i n v e s t i g a t i o n , the f r i c t i o n damped structure was excited by a serie s of harmonic ground ac c e l e r a t i o n time-histories 109 having a constant amplitude a and d i f f e r e n t values of forcing frequency w . An i n t e g r a t i o n time-step of 0.01 second was used i n a l l analyses and the amplitude at steady-state (R 0) was considered to be the amplitude of the motion a f t e r 20 seconds of e x c i t a t i o n . The comparison between the a n a l y t i c a l and numerical solutions i s presented i n F i g . A.3. I t can be seen that the numerical r e s u l t s are v i r t u a l l y i d e n t i c a l to the a n a l y t i c a l s olutions. The s l i g h t differences come from the fa c t that the steady-state condition may not be t o t a l l y achieved i n the numerical analyses a f t e r 20 seconds of v i b r a t i o n . From the fi g u r e i t i s apparent that the resonant frequency increases as the s l i p load increases. For very low s l i p loads the system behaves as an unbraced frame with a resonant frequency r a t i o n given by Wg/ui^ = i^/w^ = VK^/K^ = 0.55. For l a r g e s l i p loads, the system behaves as a f u l l y braced frame with n = 1. Note that the minimum resonant amplitude o c c u r s when a s l i p load value of 20.3708 kN i s used, as predicted by Equation (4.29). The frequency r a t i o n* at which t h i s optimum resonance occurs i s 0.8057, as predicted by Equation (4.31). These trends, which are i l l u s t r a t e d on the basis of t h e o r e t i c a l c a l c u l a t i o n s , are also r e f l e c t e d i n the experimental r e s u l t s reported by Baktash and Marsh (1986). Figure 4.4 presents the values of the s l i p load P 0 which minimizes the steady-state amplitude of the response when the structure i s excited by a s i n u s o i d a l ground motion at a p a r t i c u l a r frequency other than the resonance frequency (n = 0.90). The r e s u l t s are based on a ser i e s of n u m e r i c a l analyses. For each value of peak ground ac c e l e r a t i o n a , the s l i p load to ensure minimum steady state amplitude i s determined by evaluating the amplitude at s l i p load increments of 2 kN. I t can be 110 Analytical o Numerical Figure 4 . 3 Frequency Response Functions for F r i c t i o n Damped Structure I l l seen that the optimum s l i p load P. i s l i n e a r l y proportional to the peak ground a c c e l e r a t i o n a , as predicted by Equation (4.36). 8 4.6 Remarks Some valuable a n a l y t i c a l r e s u l t s have been obtained from the study of a simple analogous nonlinear mechanical system: 1) A lower bound value of the s l i p load has been established such that bounded amplitudes occur at resonance. 2) The value of the s l i p load minimizing the amplitudes at resonance has been determined along with the frequency at which t h i s optimum resonance occurs. I t has also been pointed out that the system i s always stable and a jump phenomenon should not occur. 112 The nondimensional r a t i o s governing the value of the l o c a l s l i p load minimizing the steady-state amplitude for any forcing frequency have been derived. I t can be expected that these r a t i o s w i l l also influence the optimum s l i p load of multi-storey f r i c t i o n damped structures under general earthquake loading. These r a t i o s c l e a r l y indicate that the optimum s l i p load i s a function of the amplitude and frequency of the ground motion and i s not s t r i c t l y a s t r u c t u r a l property. 113 5. SIMULATION OF EARTHQUAKE ACCELEROGRAMS "There i s a long and wearisome step between admiration and i m i t a t i o n . " - Jean Paul Richter (1763-1826), German Humorist 114 CHAPTER 5 SIMULATION OF EARTHQUAKE ACCELEROGRAMS 5.1 Motivation In Chapter 4 i t was shown that the c h a r a c t e r i s t i c s of a harmonic base e x c i t a t i o n have an important influence on the optimum s l i p load of a FDBF. By extension, i t may be expected that the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure w i l l be influenced by the c h a r a c t e r i s t i c of the earthquake ground motion anticipated at the con-s t r u c t i o n s i t e (see Equation (4.36)). Therefore, i t becomes necessary to consider a v a r i e t y of ground motions i n the parametric study invo l v -ing the optimum s l i p load d i s t r i b u t i o n . Since earthquakes are random i n nature, i t i s u n l i k e l y that the same earthquake ground motion w i l l be repeated at some future time at a given s i t e . The use of actual past earthquake records may not lead to meaningful r e s u l t s i n the sense that i n d i v i d u a l r e a l earthquake records are c o n d i t i o n a l on a sing l e r e a l i z a t i o n of a set of random parameters (magnitude, f o c a l depth, attenuation c h a r a c t e r i s t i c s , frequency content, duration, e t c . ) , a r e a l i z a t i o n that w i l l l i k e l y never occur again and that may not be s a t i s f a c t o r y for design purposes. This shortcoming can be avoided by the use of a r t i f i c i a l l y generated earthquakes of the same class as the past observed earthquakes. A stochastic representation of earthquake ground motion i s described i n t h i s chapter. In t h i s model, o r i g i n a l l y proposed by Vanmarcke (1983), the strong motion duration captures the e s s e n t i a l t r a n s i e n t c h a r a c t e r of earthquake ground motion w h i l e the 115 power spectral density function represents i t s "equivalent stationary" frequency content. This representation o f f e r s the advantage of completely describing the ground motion by seismic parameters that can be estimated at a given s i t e from geophysical information. The proced-ure used for the simulation of earthquake accelerograms based on t h i s stochastic ground motion representation i s described. A b r i e f review of random processes and spectral analysis i s presented i n Appendix B to complement t h i s chapter. 5.2 Strong Motion Duration of Earthquakes 5.2.1 Concept of Strong Motion Duration Based on past observations, and a n a l y t i c a l studies, i t i s generally agreed that most of the s t r u c t u r a l damage r e s u l t i n g from an earthquake i s caused by a s p e c i f i c segment of the motion, referred to as "strong ground shaking". These observations led to the concept of "strong-motion duration" to characterize the duration of an earthquake. To i l l u s t r a t e t h i s concept, l e t us consider the accelerogram of the 1952 Taft earthquake, S69E component, as shown i n F i g . 5.1. Although the t o t a l duration of the motion i s 54 seconds, most of the strong shaking occurs i n a l i m i t e d segment of the record. By simply observing the record, one can estimate the segment of strong motion to be roughly between 3 and 14 seconds, as shown i n F i g . 5.2. The contribution of t h i s strong shaking segment to the s t r u c t u r a l response of a t y p i c a l b u i l d i n g i s i l l u s t r a t e d i n Appendix C. The r e s u l t s of these analyses i l l u s t r a t e that the use of the "strong-motion duration" to characterize the duration of an earthquake leads to very good estimates of the envelopes of s t r u c t u r a l response. This i s i n agreement with the opinion of many researchers (Vanmarcke, 1977). Therefore i t would appear that the concept of strong-motion 116 0.20 -0.20 I 1 i 1 i 1 i 1 i 1 i 1 0 10 20 30 40 50 60 Time [sec] Figure 5.1 Accelerogram of the 1952 Taft Earthquake, S69E Component. 0.20-1 : : • : • , Time [sec] Figure 5.2 Estimated Strong Motion Segment of the 1952 Taft Earthquake, S69E Component. 117 duration can s a t i s f a c t o r i l y represent the earthquake duration i n a para-metric study re l a t e d to the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure. However, i n the case of harmonic-type ground motion, neglecting the i n i t i a l part of the record, even i f i t i s of low i n t e n s i t y , may lead to some s i g n i f i c a n t underestimation of the s t r u c t u r a l response. 5.2.2 Local S t a t i o n a r i t y of Strong Motion Segment A casual look at any earthquake accelerogram i s s u f f i c i e n t to con-vince one of the necessity of representing earthquakes as nonstationary random processes. The shape of the acceleration i n t e n s i t y envelope has very d i s t i n c t i v e r i s i n g and decaying portions, as i l l u s t r a t e d i n F i g . 5.1. While many studies have shown that the enti r e accelerogram i s non-stationary (see for example Moayyad and Mohraz, 1982; T i l l i o u i n e et a l . , 1984; Nau et a l . , 1980), i t has been suggested that the strong motion segment of an accelerogram may be considered as stationary (see for example Vanmarcke, 1983; Moayyad and Mohraz, 1982; Housner and Jennings, 1964). Moayyad and Mohraz (1982) used a d i s t r i b u t i o n - f r e e t e s t of hypo-the s i s (the runs test) to examine the stationary assumption of the strong motion segment of t y p i c a l earthquake accelerograms (including the Taft S69E earthquake) . The runs t e s t i s a technique for t e s t i n g the hypothesis that a var i a b l e i s random i n nature. The t h e o r e t i c a l basis of the runs te s t can be found i n many texts on s t a t i s t i c s (see for example Walpole and Myers, 1978; Eisenhart and Swed, 1943). Although the runs t e s t can be based on one of several s t a t i s t i c a l averages, Moayyad and Mohraz selected the mean square value as the basic random 118 v a r i a b l e for the t e s t ; i f the mean square value i s shown to be random, the strong motion segment of the accelerogram can be considered stationary. The r e s u l t s of the runs t e s t performed by Moayyad and Mohraz indicated that the strong motion segment of t y p i c a l accelerograms have very weak stationary c h a r a c t e r i s t i c s . The e f f e c t of t h i s weak l o c a l s t a t i o n a r i t y on s t r u c t u r a l response can be determined by replacing the strong motion segment of a r e a l accelerogram by an ensemble of strongly stationary records with equiva-lent energy (Power Spectral Density), frequency content and duration, and comparing the e f f e c t s of the r e a l accelerogram and replacement ensemble on the response of a t y p i c a l structure. For t h i s purpose the computer program "GESER" (Generation of Equivalent Stationary Earthquake Records) was created. The computer program user's guide i s presented i n Appendix D. Appendix E presents DRAIN-2D analyses of the same low r i s e b u i l d i n g described i n section 3.3 when excited by the strong motion segment of the r e a l Taft earthquake and also by an ensemble of equivalent stationary records. The r e s u l t s i l l u s t r a t e the fact that the nonstationarity or weak s t a t i o n a r i t y of the strong motion segment of an accelerogram i s not the main factor i n f l u e n c i n g the s t r u c t u r a l response. To obtain good e s t i -mates of s t r u c t u r a l response, one can r e p r e s e n t an earthquake accelerogram by a f i n i t e segment of a stationary process having the same energy and frequency content, i . e . power spectral density function, as the ground motion a n t i c i p a t e d at the b u i l d i n g s i t e . This representation of the ground motion, o r i g i n a l l y proposed by Vanmarcke (1983), w i l l be used i n the parametric study to determine simple design equations for the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure. 119 5.2.3 D e f i n i t i o n of Strong Motion Duration The concept of strong motion duration has been i l l u s t r a t e d i n Section 5.2.1 by q u a l i t a t i v e l y s e l e c t i n g a portion of the Taft accelero-gram as the strong motion segment. A more systematic procedure i s needed for estimating the strong motion duration of earthquake ground motion records. No s i n g l e quantitative measure of the duration of strong shaking i s i n common usage i n earthquake engineering. Studies of the dependence of duration on magnitude (Housner, 1965) and on distance and magnitude (Esteva, 1964) have been c a r r i e d out, but these are not based on formal, quantitative d e f i n i t i o n s of duration. Two crude but simple measures of strong motion duration appeared i n the earthquake engineering l i t e r a t u r e p r i o r to 1975. The f i r s t defines t h i s duration as the time i n t e r v a l between the f i r s t and l a s t peaks greater or equal to a given l e v e l , u s u a l l y 0.05g, on the accelerogram (Page, 1975). The second d e f i n i t i o n i s based on the concept of cumulative energy obtained by integrating squared accel e r a t i o n s : duration i s the time i n t e r v a l required to accu-mulate a prescribed f r a c t i o n of the t o t a l energy (Husid et a l . , 1969; Trifunac and Brady, 1975; and Bolt, 1975). An improved d e f i n i t i o n of strong motion duration, proposed more recently by Vanmarcke and L a i (1980,1982), i s used i n the present study. It i s based on the preservation of the t o t a l energy of the ground motion and on the existence of a consistent r e l a t i o n s h i p between the peak ground a c c e l e r a t i o n , the strong motion root mean square acceleration and the strong motion duration. The strong motion duration (s 0) i s obtained by solving a simple set of two equations with two unknowns. These equations can be written as: 120 where t 0 I 0 = o 0* s 0 - / aHt) dt (5.1) 0 a_ r [2 In (2s 0/T 0)]*'» s 0 _ 1 . 3 6 T 0 _ o L V2 (5.2) s 0 < 1.36 T 0 I 0 = Arias Intensity (Arias, 1970) o 0 = root mean square acceleration of strong motion segment s 0 = strong motion duration t 0 = length of the d i g i t i z e d accelerogram a(t) = ground a c c e l e r a t i o n a_ = peak ground a c c e l e r a t i o n T 0 = predominant period of strong motion segment In words, Equation (5.1) ensures that the t o t a l energy I 0 of the ground motion i s preserved and i s d i s t r i b u t e d uniformly, at constant average power o 0 2 , over the strong motion i n t e r v a l s 0 . Equation (5.2) i s based on the idea that a consistent r e l a t i o n s h i p e x i s t s between o 0 and a . S p e c i f i c a l l y , E q u a t i o n (5.2) i s t h e o r e t i c a l l y derived from a random process which i s both stationary and Gaussian and whose value of a /o 0 has a p r o b a b i l i t y of e" 1 of not being exceeded during s 0 . For a p a r t i c u l a r h i s t o r i c a l ground motion record, I 0 can r e a d i l y be computed, a can be measured d i r e c t l y , and the predominant period (T 0) can be estimated by counting the number of zero crossing within the a p p a r e n t s t r o n g shaking s e c t i o n of the r e c o r d ; the v a l u e of a /o 0 predicted by Equation (5.2) i s r e l a t i v e l y i n s e n s i t i v e to the choice of T 0 within the range of most common values (0.2 to 0.6 seconds). Then 121 the strong motion duration ( s 0 ) and the rms a c c e l e r a t i o n (o 0) can be solved from Equations (5.1) and (5.2) by a t r i a l - a n d - e r r o r procedure. Vanmarcke and L a i (1980) c a l c u l a t e d the strong motion duration ( s 0 ) and rms a c c e l e r a t i o n (o 0) associated w i t h the data obtained from 140 h o r i z o n t a l a c c e l e r a t i o n components of 70 western United States strong motion records. This data set was o r i g i n a l l y s e l e c t e d by McGuire and Barnhard (1977) so as to be r e p r e s e n t a t i v e of a broad range of earth-quake magnitudes, e p i c e n t r a l d i s t a n c e s , motion i n t e n s i t i e s , and s i t e c o n d i t i o n s . The scattergram of peak ground a c c e l e r a t i o n versus strong motion d u r a t i o n i s presented i n F i g . 5.3 f o r the 140 records analyzed by Vanmarcke and L a i (1980). Their r e s u l t s show that the strong motion duration ( s 0 ) i s i n v e r -s e l y c o r r e l a t e d w i t h the peak ground a c c e l e r a t i o n (a ). Based on Figure 5.3 Scattergram of Peak Ground A c c e l e r a t i o n vs Strong Motion Duration ( a f t e r Vanmarcke and L a i , 1980) 122 t h e i r data, they suggested the following regression equation for e s t i -mating strong motion duration (s 0) for a given peak ground acceleration s 0 = 30 exp (-3.254 a 0 , 3 S ) (5.3) where a = peak ground ac c e l e r a t i o n i n g s 0 = strong motion duration i n seconds This Vanmarcke and L a i formula i s also p l o t t e d i n F i g . 5.3. From Equation (5.3) i t follows that a and s 0 can be assumed to be dependent parameters and only one of the two can be used i n a dimen-si o n a l a n alysis. However, the large scatter noted i n F i g . 5.3 places the accuracy of Equation (5.3) i n question. For example, t h i s equation underestimates s i g n i f i c a n t l y the strong motion duration of the recorded motions during the 1979 Imperial V a l l e y earthquake and also the corres-ponding durations recorded during recent foreign earthquakes, such as the 1985 Mexico earthquake, the 1985 Chile earthquake and the 1978 Miyagi-Ken-Oki earthquake. 5.3 Power Spectral Density Functions of Earthquake Accelerograms 5.3.1 A n a l y t i c a l Presentation of Power Spectral Density The frequency content of the earthquake ground motion i s described by the ground ac c e l e r a t i o n power sp e c t r a l density function (S (u))) of an Si equivalent stationary random process. Based on s t a t i s t i c a l studies of h i s t o r i c a l earthquakes, several investigators have proposed expressions for the power spectral density function of the ground motion. These expressions are generally of the form (see for example Shinozuka, 1969): 123 c x + c 2 (w/u) ) 2 S (OJ) = 8 (5.A) a [l-(_/u ) 2 P + Ah2(w/_ ) 2 g g g where u is the dominant ground frequency and h is a parameter which indicates the sharpness of the unique peak spectrum (ground damping); c x and c. are constants. In the present study i t was decided to use a specific form of this expression, which was first suggested by Kanai (1957) and later was used by Tajimi (1960) to determine the maximum response of simple structures. Their expression has the form: S.(u) = 1 + Ah2(_/w ) 2 g _ [l-(w/ui ) 2 P + Ah2(_/ui ) 2 g g g S A = |H(uj) I2 S A (5.5) where |H(ui)|2 = transfer function of soil layer = power spectral density function at bedrock level. Equation (5.5) is the same as Equation (5. A) with c=S. and c =4h2 S.. A g A Physically, the Kanai-Tajimi ground power spectral density function is derived from an "ideal white noise" excitation (S.) at bedrock, which is then appropriately filtered through the overlaying soil deposits. Examples of Kanai-Tajimi power spectral density functions are illustra-ted in Fig. 5,A. When the transfer characteristics of the soil filter 124 0 1 2 3 Circular Frequency Ratio ^ w/wg7 Figure 5.4 Kanai-Tajimi Power Spectral Density Functions (|H(u)| 2) are properly selected, the r e s u l t i n g power sp e c t r a l density function provides a good representation of the r e a l ground motion. Due to the fa c t that most h i s t o r i c a l earthquake power spectral density functions are quite e r r a t i c , i t i s d i f f i c u l t to determine the parameters of the corresponding smooth Kanai-Tajimi power spectral density function. The "spectral moments" method has been proposed by Vanmarcke (1977) to estimate the Kanai-Tajimi parameters from r e a l accelerograms; t h i s method has been used by several investigators (see for example Binder, 1978; L a i , 1979). A b r i e f overview of the method i s presented Appendix F. 125 The method of spectral moments has been applied by Vanmarcke and Lai (1980) to the same data set selected by McGuire and Barnhard (1977). Based on the method of moving average s t a t i s t i c s , Vanmarcke and Lai have proposed several empirical equations for estimating the Kanai-Tajimi parameters (w ,h ) . These equations i n v o l v e geophysical information that may be av a i l a b l e at a given s i t e . Thus ui = 1.12 u - 5.15 g c (5.6) where Kanai-Tajimi predominant frequency (rad/sec) ce n t r a l frequency (rad/s) = 2TT/T0 . u = 27 - 0.09 R g 10 km £ R £ 160 km (5.7) where R = ep i c e n t r a l distance (km) where u = 65 - 7.5 MT 5 <; MT <. 1 g L L (5.8) MT = l o c a l Richter magnitude 126 h = 0.32 = constant g (5.9) Vanmarcke and L a i observed t h a t the K a n a i - T a j i m i damping (h ) i n c r e a s e s s l i g h t l y with the l o c a l Richter Magnitude (M^) but decided to hold t h i s value constant at 0.32. In the present study, however, i t was decided to l e t h^ vary as an independent parameter. The j o i n t e f f e c t s of M^ and R on the Kanai-Tajimi parameters were not considered by Vanmarcke and L a i ; therefore discrepancies may r e s u l t from the use of Equations (5.34) and (5.35). 5.4 Summary of Governing Ground Motion Parameters As discussed i n Sections 5.2 and 5.3, the s t r u c t u r a l response can be estimated with reasonable accuracy by representing the earthquake ground motion as a segment of a stationary random process; the duration of the segment i s the strong motion duration ( s 0 ) defined e a r l i e r . Four seismic parameters are needed to define completely the ground accelera-t i o n : i ) peak ground a c c e l e r a t i o n (a^) i i ) strong motion duration (s 0) i i i ) Kanai-Tajimi predominant or c h a r a c t e r i s t i c ground period (T = 2 T T / U ) g g iv) Kanai-Tajimi sharpness parameter or c h a r a c t e r i s t i c ground damping r a t i o (h ) g 127 We have seen that the f i r s t or i n v e r s e l y correlated (see F i g . defined by only three independent two seismic parameters are negatively 5.3). Hence the ground motion can be parameters: a_ = peak ground acceleration T = Kanai-Tajimi predominant period h = Kanai-Tajimi sharpness parameter As an example, consider that the design peak ground acceleration at a given s i t e i s 1/3 g and that the estimated design Kanai-Tajimi period and damping r a t i o i s 0.31 second and 0.32 r e s p e c t i v e l y . The correspond-ing strong motion duration (s 0) i s equal to 3.27 seconds, based on E q u a t i o n ( 5 . 3 ) . The c e n t r a l frequency (w_) i s estimated to be equal to 22.72 rad/s, based on Equation ( 5 . 6 ) . The predominant period (T 0) i s equal to 2TT/_c, the rms a c c e l e r a t i o n (o 0) i s estimated to be equal to 0.13 g (see E q u a t i o n 5.2). The white noise bedrock i n t e n s i t y (S^) can be found by i n t e g r a t i n g Equation ( 5 . 5 ) : a02 = J" S a (u) d_ = S. J |H(u) 12 du 0 a A 0 which y i e l d s : A o 0 2 h SA = TT. (l+4h?) ( 5 ' 1 0 ) g g 128 When the preceding quantities are substituted into Equation (5.10) a value of S. equal to 23157.56 mmVsec 3 i s obtained. S (w) can then be A 61 calc u l a t e d using Equation (5 .5) . Hence the design ground motion i s f u l l y described and an ensemble of records having these c h a r a c t e r i s t i c s can be simulated and used i n a parametric study. 5.5 Simulation of Earthquake Accelerograms 5.5.1 Simulation Procedure The simulation of earthquake accelerograms with a prescribed Kanai-Tajimi power sp e c t r a l density, and with strong motion duration ( s 0 ) , begins by considering a sub-ensemble of the universe of earthquake records. The power sp e c t r a l density function of each sample record S a^ n^ (ui) i n the sub-ensemble i s the same as the smooth power sp e c t r a l d e n s i t y o f t h e u n d e r l y i n g s t a t i o n a r y random pro c e s s S (w). From SL Equations (B.13) and (B.1A) we can write S (n) (ui) - Sa(u>) = f - |Fn(u.,s0)|» (5.11) a Substituting Equation (5.5) into Equation (5.11) y i e l d s : S (u) = |H(u) |» S = J - |F (ui.s ) | 2 3. A s rt n (5.12) Therefore, the desired earthquake accelerograms can be simulated by the generation of an ensemble of acce l e r a t i o n time-series from a s p e c i f i e d Fourier amplitude spectrum (|F ( u , s 0 ) | ) : 129 l F n ( u , s 0 ) S A s o i / j lH(u) I H j—] (5.13) The v a l u e of the white noise bedrock i n t e n s i t y (S^) can be calcu-l a t e d using Equation (5.10). However, for s i m p l i c i t y , i t was decided to normalize the Fourier amplitude spectrum i n the frequency domain and to scale the simulated acceleration record, determined by i n v e r t i n g t h i s spectrum, i n order to obtain the proper rms a c c e l e r a t i o n ( o 0 ) . The normalized Fourier amplitude spectrum i s : 1 + 4h*(_/i_ ) a | F n ( . , s 0 ) | = |H(_)| = [ « + 4g ] g g g 1 / 2 (5.14) A f i n i t e set of complex conjugate Fourier transforms with random phases i s s p e c i f i e d i n the frequency domain. i j = 1 |H(u.)|e J j = 2,3... . ,N/2 J |H(u.) 1 j = N/2 + 1 _ F n ( w ( N + 2 - j ) ' S o ) 3 = N/2 + 2, N/2 + 3....N (5.15) where N = number of data points s p e c i f i e d F* = complex conjugate of F n 130 i - 7-1 <J>j = random phase angle The reason for s e t t i n g the array of Fourier transforms i n the form given by Equation (5.15) i s that the d i s c r e t e fast Fourier transform of a r e a l time-series (earthquake accelerogram) has conjugate symmetry (Newland, 1975). For example, consider a r e a l time-series of length 10. The d i s c r e t e f a s t Fourier transform i s as follows: j 1 2 3 A 5 6 7 8 9 10 F n ( u j ' s » > A B C D E F E* D* C* B* where E*,D*,C* and B* are the complex conjugates of E,D,C and B, re s p e c t i v e l y . Also, the disc r e t e fast Fourier transform at j = 1 and j = (N/2)+l are both r e a l . A and F are therefore both r e a l and F occurs at the "foldover frequency" (or Nyquist frequency). The phase angle (<JK) i n Equation (5.15) i s a random var i a b l e having a uniform p r o b a b i l i t y d i s t r i b u t i o n over the i n t e r v a l (0-2TT) . r- 1/2TT 0 < <f>. < 2TT P(<f>.) = 3 (5.16) J -- 0 otherwise The s i m u l a t e d earthquake accelerogram ( x ^ ( t . ) ) i s obtained by g J taking the inverse d i s c r e t e f a s t Fourier transform of the F (u.,s„): 131 , . N iu, t . (5.17) In the frequency domain, the data must be given at frequencies w: u = 0, 2TT 4TT 2 (N-1) IT NAt ' N A t N A t (5.18) where N = number of data points At = time-increment of time seri e s The frequency sampling (Aw) i s then: Aw = 2TT NAt (5.19) In the time domain, the data must be taken at time (t) t = 0, At, 2At, .... (N-1)At (5.20) where At = N (5.21) The rms acceleration i s calculated by s u b s t i t u t i n g Equation (5.6) into Equation (5.2): 132 S [2 l n [0.28 s 0(6.28 + 5.15 T )/T 0 g g (5.22) The a c c e l e r a t i o n r e c o r d i s then s c a l e d to the proper rms acce l e r a t i o n : *(nht.) -~ l N > ' 5 ^ ^ g ( t J ) ) 2 (5.23) where a ^ n ^ ( t ^ ) = scaled earthquake accelerogram The computer program "SIMEA" (SIMulation of Earthquake Accelero-grams) was created for the simulation of earthquake accelerograms based on the above procedure. Appendix G contains a user's guide for the program. 5.5.2 Simulation Example To i l l u s t r a t e the use of the SIMEA program, l e t us generate an ensemble of a r t i f i c i a l accelerograms having the c h a r a c t e r i s t i c s of the 1952 Taft earthquake, S69E component, shown i n F i g . 5.1. Vanmarcke and L a i (1980,1982) estimated the following parameters for t h i s record: • a = 0.179 g • T = 0.334 second 133 • h = 0.35 g From Equation (5.3) the duration of the strong motion segment i s estimated as: • s 0 = 5.049 seconds From Equation (5.22) the strong motion rms acc e l e r a t i o n i s estimated as: • o0 - 0.067 g Vanmarcke and L a i computed 10.72 seconds and 0.060 g for the strong motion duration and rms acceleration r e s p e c t i v e l y for the r e a l Taft earthquake. The differences between the measured and estimated values can be explained by the scatter of the data i n F i g . 5.3. Figure 5.5 shows an ensemble of four a r t i f i c i a l accelerograms generated by SIMEA. The time increment used i n the simulation was 0.0024 sees and the frequency increment was 1.244 rad/sec. Each p l o t i s derived from 2048 data points. Figures 5.6 to 5.10 present the r e s u l t s obtained from DRAIN-2D analyses i n which the low r i s e b u i l d i n g considered previously was excited by these four simulated accelerograms. The r e s u l t s are also compared with the corresponding values obtained using the 1952 Taft earthquake as the e x c i t a t i o n source. A l l the ground acceleration values were scaled by a factor 3 to enhance s t r u c t u r a l damage. From these figures i t can be seen that the simulated accelerograms cause s t r u c t u r a l responses that compare well with the responses developed by the r e a l 1952 Taft earthquake, i n d i c a t i n g that the proposed 134 earthquake model seems adequate to predict s t r u c t u r a l response. Note that more dynamic analyses, using a v a r i e t y of s t r u c t u r a l configurations and earthquake ground motions, are needed to evaluate extensively the adequacy of the earthquake model proposed by Vanmarcke. This evaluation was not the subject of the present study and the Vanmarcke earthquake model was used here without further i n v e s t i g a t i o n . 135 0.3 - i -0.3-Simulated Accelerogram , 1 \ VV n*. 2 J r/me /secj 0 . 3 C D 0 . 0 -0.3 Simulated Accelerogram §2 CD 0 . 3 - 0 . 3 VI/ V vyv 2 3 77m e [sec] Simulated Accelerogram #Z \ A An. A h\. A A,A • • r yv 1  1  V ' \ 1  / 2 3 Time [sec] Simulated Accelerogram #4 Time [sec] Figure 5.5 Simulation of the 1952 Taft Earthquake, S69E Component 136 Unbraced X-Braced Friction Damped IS « s. S r CD F F R T3 3 S CM * . i 1 1 CO 1 & 11 o 2 •o 9 05 1? o u < • • Member Yielded Figure 5.6 St r u c t u r a l Damage a f t e r Earthquakes, Taft vs Simulated Earthquakes Q) Q) — J v.. O O C <D CD O O c CD _> CD •^J V. O O C 3 - i 2-1-Figure 5.7 Unbraced Structure 137 '' _#_>^ > • Ml Carlhauokt o Slmuloltd Aeetltrogrom fl a Slmuloltd Aeetltrogrom (2 • Slmuloltd Aeetltrogrom f3 * Slmuloltd Aeetltrogrom £4 T i i • Uton of Stoflonory Rteordt 50 100 150 Deflection [mm] X-Braced Structure 200 • Tofl Earlhquokt o Slmuloltd Aeetltrogrom fl o Slmuloltd Aeetltrogrom (2 * Slmuloltd Aeetltrogrom f$ " §L^Mi?tfP.Ap.?fJ*r?9.r.?m.j(*. • Uton of Stallonory Rteordt 50 100 150 Deflection [mm] Friction Damped Structure 72 200 • Tofl Eorthquokt o Slmuloltd Aeetltrogrom fl o Slmuloltd Aeetltrogrom fl • Slmuloltd Aeetltrogrom fi » Sbnulottd_Aeetltroanm_/4 • Uton of Stationary Rteords T 50 100 150 Deflection [mm] 200 Envelopes of Later a l Deflections, East Side, Taft vs Simulated Earthquakes Floor Level K> U Floor Level Kj U Floor Level C r 3 O CD a o c 0 0 Unbraced Structure 139 CD CD —J O o CD CD v.. o o CD CD o o Figure 5.9 100 200 300 400 500 Bending Moment [kN-m] X-Braced Structure 600 100 200 300 400 500 Bending Moment [kN-m] Friction Damped Structure 600 100 200 300 400 500 Bending Moment [kN-m] 600 Envelopes of Column Bending Moments, East Side, Taft vs Simulated Earthquakes CD CD O o CD 2» CD —J O O C CD > CD —I O o Unbraced Structure 140 2-S*mjlot9d Aee»l*r*grom ft  S*nMo*»d Aectltrognen / J <> d D» 50 700 /50 200 250 Shear Force [kN-m] X—Braced Structure 300 2-1-o i T • SlmiMottd Atctlfcgram fS • Wan of SknulaHd *c»«>f aygmt "~i ' 1 « 1 ' 1 ' r * * * 50 100 150 200 250 Shear Force [kN-m] Friction Damped Structure 300 1-•9-• Stw/W>rf AcctUroonm §3 • flfr^ifHAf^fffs^sm.if. IHm tH SlmuloHd Att*t*r»ar*mt —-E> I -e-50 100 150 200 Shear Force [kN-m] 250 300 Figure 5.10 Envelopes of Column Shear Forces, East Side, Taft vs Simulated Earthquakes 6. DIMENSIONAL ANALYSIS "We cannot impose our w i l l s on nature unless we f i r s t a s c e r t a i n what her w i l l i s . Working without regard to law brings nothing but f a i l u r e ; working with law enables us to do what seemed at f i r s t impossible." - Ralph Tyler Flewelling (1871-1967), American Philosopher 142 CHAPTER 6  DIMENSIONAL ANALYSIS 6.1 General Investigations of engineering phenomena are generally reported i n terms of dimensionless parameters obtained by dimensional analysis (Bridgman, 1922; Langhaar, 1951). Dimensional analysis reduces the number of variables to be studied i n a phenomenon, generally by the number of basic dimensions involved. For example, most engineering problems involve Force, Length and Time as the basic dimensions, and the number of dimensionless parameters i s three less than the number of variables r e l a t e d to the problem. In t h i s chapter a dimensional analysis i s performed i n order to determine the dimensionless parameters which may govern the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure. The a n a l y t i c a l r e s u l t s obtained i n Chapter 4 w i l l be used as guidelines to determine these dimensionless parameters. 6.2 Governing Parameters In the f i r s t step of a dimensional an a l y s i s , the variables which may e f f e c t s i g n i f i c a n t l y the phenomenon under consideration must be determined from experience and engineering judgement. Based on the r e s u l t s of Chapter 4, the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure depends on two fundamental sets of v a r i a b l e s : 1) C h a r a c t e r i s t i c s of the structure 2) C h a r a c t e r i s t i c s of the ground motion 6.2.1 Characterization of the Structure The dynamic properties of a f r i c t i o n damped structure are i n f l u -enced by the primary moment-resistant structure (beams and columns) and 143 also by the secondary bracing system ( f r i c t i o n devices and diagonal braces). The dynamic properties of a f r i c t i o n damped structure can be represented by the following independent v a r i a b l e s : w = Weight of the bu i l d i n g T u = Undamped fundamental period of the unbraced structure; corresponds to a l l the devices s l i p p i n g . T b = Undamped fundamental period of the f u l l y braced frame; corresponds to no slippage and a l l the braces behaving e l a s t i c a l l y i n tension and. compression. 5i = C h a r a c t e r i s t i c damping r a t i o . P cr = C h a r a c t e r i s t i c buckling load. NS = Number of storeys. g = Acceleration of gravi t y . The c h a r a c t e r i s t i c damping r a t i o (£j) i s defined to be the damping r a t i o i n the f i r s t mode of v i b r a t i o n of the unbraced structure, according to the global damping matrix formulation i n the FDBFAP program (see Section 2.6). The c h a r a c t e r i s t i c buckling load (P ) i s defined as the sum of the cr Euler buckling loads for the diagonal braces of each f r i c t i o n device: cr TTj E NT i = l T 2 (6.1) 144 Young's Modulus. Moment of i n e r t i a of d i a g o n a l braces around weak bending axis for f r i c t i o n device element i . Length of diagonal braces for f r i c t i o n device element i (see F i g . 2.6). Number of f r i c t i o n devices i n the structure. 6.2.2 Characterization of the Ground Motion As discussed i n Chapter 5, the ground motion can be completely defined by three independent v a r i a b l e s : a_ = Peak ground ac c e l e r a t i o n . T = Kanai-Tajimi predominant period. h = Kanai-Tajimi sharpness parameter. 6.2.3 Summary of Governing Parameters Based on the material presented i n Sections 6.2.1 and 6.2.2, the p h y s i c a l quantities believed to be pertinent i n a dimensional analysis for the optimum s l i p load of f r i c t i o n damped structures are summarized with t h e i r dimensions i n Table 6.1. 6.3 A p p l i c a t i o n of Buckingham's Theorem The theory of dimensions can be summarized by a general theorem stated by Buckingham (1914): "Any dimensionally homogeneous equation involving c e r t a i n p h y s i c a l quantities can be reduced to an equivalent equation inv o l v i n g a complete set of dimensionless products." where E 145 Table 6.1 Physical Quantities Considered i n the Dimensional Analysis QUANTITY DESCRIPTION DIMENSIONS* V c Optimum s l i p shear** F NS Number of storeys -g Acceleration of gra v i t y LT" 2 W Weight of the bu i l d i n g F T u Fundamental period of the unbraced structure T T b Fundamental period of the f u l l y braced structure T l i C h a r a c t e r i s t i c damping r a t i o -P cr C h a r a c t e r i s t i c buckling load F a g Peak ground ac c e l e r a t i o n LT" 2 T g Kanai-Tajimi predominant period T h g Kanai-Tajimi sharpness parameter -*p L T = Force = Length = Time N . NS Di **V0 = _ I 2 P 0. .cosa. . ° i-1 j = l ^ where P.. . = Optimum l o c a l s l i p load for the j ^ f r i c t i o n device i n the a. . .th . l storey. th = Angle of i n c l i n a t i o n from h o r i z o n t a l of the j braces i n - V • t h 4-the l storey. N. .th Di = Number of f r i c t i o n devices i n the i storey 146 This theorem states that the so l u t i o n equation for some physical quantity of i n t e r e s t , i . e . , F ( x l t x a x ) = 0 (6.2) can be expressed i n the equivalent form G(TTA,TT. n ) = 0 (6.3) The TK terms are dimensionless products of the phys i c a l quantities x1,x1,...,x . G e n e r a l l y , i t can be s t a t e d t h a t t h e number of dimensionless product (m) i s equal to the differ e n c e between the number of ph y s i c a l variables (n) and the number of fundamental measures (r) that are involved. For the problem under consideration, the number of variables and dimensions are n = 1 1 p h y s i c a l v a r i a b l e s ( V 0 , NS, g, W, T , T b > P__, a g' T , h ) g g r =3 ph y s i c a l dimensions (F,L,T) By Buckingham's theorem, the number of TK terms (m) that can be found i s then: m = n - r = 8 TT. terms I (6.4) 147 To apply Buckingham's theorem we f i r s t choose any 3 variables that contain the 3 dimensions of the problem (F,L,T). • W (F) • g (LT~ 2) • T u (T) We then group the remaining 8 variables (V 0, T, , £ l t P , a , T , D cr g g hg, NS) w i t h the primary v a r i a b l e s (W.g.T^) such that a l l groups are dimensionless. Accordingly we get: V 0 T, P a T *<W~ ' T ' \f ' t ' f ' V N S ) = ° ( 6 ' 5 ) u B u B or W N . NS Di I 1 2P n..cosa. . i-1 1-1 1 3 1 J W = G(; cr a _ W * g T _ T ' u NS) (6.6) where F and G are unknown functions. The way i n which the optimum s l i p shear V 0 must be d i s t r i b u t e d along the height of the structure remains to be determined. Chapter 8 i s concerned with t h i s optimum d i s t r i b u t i o n for multi-storey f r i c t i o n damped structures. Based on Equation (6.6), a parametric study was undertaken using FDBFAP i n order to determine design equations for the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped structure. 148 7. PARAMETRIC STUDY FOR ONE STOREY STRUCTURES "Man must go back to nature for information." - Thomas Paine (1737-1809) , English/American P o l i t i c i a n 149 CHAPTER 7 PARAMETRIC STUDY FOR ONE STOREY STRUCTURES 7.1 General Approach This chapter i s concerned with the development of a s i m p l i f i e d design procedure for one storey structures equipped with f r i c t i o n damp-ing devices. This type of structure can be reduced to a single-degree-of-freedom system for which the optimum s l i p load i s given by: W 2cosa F [ — I u _2£ _g _ f i W * g ' T 6 u (7.1) where: W u P cr a g g T g h = Optimum l o c a l s l i p load = Weight of the structure = Angle of i n c l i n a t i o n of the cross-braces from the ho r i z o n t a l = Fundamental period of the f u l l y braced structure = Fundamental period of the unbraced structure = C h a r a c t e r i s t i c damping r a t i o = C h a r a c t e r i s t i c buckling load = Peak ground ac c e l e r a t i o n = Acceleration of gra v i t y = Kanai-Tajimi ground predominant period = Kanai-Tajimi ground damping = Unknown function to be estimated 150 Using the program FDBFAP as a t o o l , a parametric study i s performed on one storey, singled bay structures i n order to estimate the unknown function F which w i l l lead to a design equation for the optimum l o c a l s l i p load P.. Equation (7.1) can be modified to take into account structures with several bays containing f r i c t i o n devices: (7.2) where V 0 = Optimum s l i p shear o i N. BF .th = Optimum l o c a l s l i p load i n i bay with f r i c t i o n device = Angle of i n c l i n a t i o n of the cross-braces i n the i t b bay = Number of bays with f r i c t i o n devices The influence of each nondimensional parameter on the optimum l o c a l s l i p load P 0 i s evaluated by a s e n s i t i v i t y study. For purposes of t h i s study we define a standard earthquake as the earthquake whose character-i s t i c parameters (T ,h and a ) are equal to the corresponding average 6 6 6 values determined from an ensemble of h i s t o r i c a l accelerograms. We also define three standard structures. These structures exhibit d i f f e r e n t l a t e r a l s t i f f n e s s e s - from s t i f f structures (T < 0.25 s) to f l e x i b l e u s t r u c t u r e s (T > 0.50 s) . The combination of each structure with the standard earthquake i s defined as a standard model. The s e n s i t i v i t y study i s c a r r i e d out by independently varying each parameter over i t s range of values while a l l the other parameters are held constant at 151 t h e i r standard values, and observing the influence of t h i s v a r i a t i o n on the optimum l o c a l s l i p load P 0. If a parameter does not influence sub-s t a n t i a l l y the optimum s l i p load, i t can be discarded i n the subsequent parametric study; t h i s w i l l obviously s i m p l i f y the unknown function F. Afte r the important nondimensional parameters have been established from the s e n s i t i v i t y study, a complete parametric study i s performed during which these parameters are varied over t h e i r range of p r a c t i c a l values. Then, using the least square method, an estimate of the unknown function F i s obtained. An appraisal of the proposed design equation i s performed by comparing the optimum s l i p load predicted by the estimated function F with the actual optimum s l i p load calculated by FDBFAP for t y p i c a l one storey structures excited by r e a l h i s t o r i c a l earthquakes. Next, a general design procedure i s proposed i n which some physical constraints ( e f f e c t s of wind and yiel d i n g ) are applied to the estimation of the unknown function F. F i n a l l y a complete design example i s given based on the proposed procedure and the performance of the structure i s compared with the performance of a conventional structure designed by the requirements of the National Building code of Canada (1985). 7.2 S e n s i t i v i t y Study 7.2.1 D e f i n i t i o n of a Standard Earthquake The d e f i n i t i o n of a standard earthquake i s based on the mean values of T , h and a computed f o r the ensemble of h i s t o r i c a l accelerograms g g g considered by Vanmarcke and L a i (1982) i n Chapter 5. The histograms of the predominant ground p e r i o d T , the ground damping h , and the peak 152 ground a c c e l e r a t i o n a are presented i n F i g . 7.1. The mean values of these parameters are: Mean value of T g = T = g 0. ,3095 Mean value of h g = h = g 0. ,32 Mean value of a g = a = g 0. ,0883 It should be recognized that t h i s d e f i n i t i o n of a standard earthquake i s subject to the s t a t i s t i c a l l i m i t a t i o n s inherent i n the data considered by Vanmarcke and L a i . The program SIMEA was used to simulate an ensemble of 5 accelero-grams based on these mean values. The sample accelerograms are shown i n F i g . 7.2. The strong motion duration s 0 and strong-motion rms accelera-t i o n o 0 for t h i s ensemble were calculated from Equations 5.3 and 5.22: s 0 = 7.46 seconds a0 = 0.0313 g A t o t a l of 18 d i f f e r e n t accelerogram ensembles were generated by SIMEA for use i n the s e n s i t i v i t y study; each ensemble consisted of 5 samples. The earthquake parameters defining each ensemble were varied around t h e i r standard (mean) values: T = 0.1s, 0.1893s, 0.3095s, 0.5s, 1.0s, 1.25s, 1.50s, 2.00s § h = 0.05, 0.10, 0.32, 0.50, 0.7.5, 0.95 6 a = 0.005g, O.OlOg, 0.050g, 0.0883g, 0.2500g, 0.5000g *Standard values The range of values considered for each parameter i s consistent w i t h the histograms shown i n F i g . 7.1 except for the upper bound of T ; the upper l i m i t for t h i s parameter was increased to 2 seconds to include 153 0 . 2 5 - 1 0.20-0.15 0.10 0.05 0.00 0.20 0.16 0.12-0.08 0.04-0.00 0.25 0.20-0.15 0.10-0.05-0.00J TQ [sec] Figure 7.1 Histograms of Earthquake Parameters 154 0.12 O) 0.00 -0.12 0.12 O) 0.00 -0.12 H 0.12 D> 0.00 -0.12 0.12 D> 0.00 -0.12 0.12 Cr> 0.00 -0.12 - J Sample Accelerogram #\ Time [sec] Sample Accelerogram #2 2 4 6 Time [sec] Sample Accelerogram #3 77m e [sec] Sample Accelerogram 04 2 4 6 Time [sec] Sample Accelerogram / 5 6 8 Time [sec] Figure 7.2 Sample A r t i f i c i a l Accelerograms for Standard Earthquake. 155 the p o s s i b i l i t y of large period ground motions, such as those that have occurred r e c e n t l y i n Romania (1977) and Mexico C i t y (1985). The para-meters for the a r t i f i c i a l accelerograms generated i n the s e n s i t i v i t y study are presented i n Table 7.1. Table 7.1 Properties of Simulated Earthquakes Used i n S e n s i t i v i t y Study Earthquake T8 a g so ° 0 Ensemble* No. (s) (g) (s) (g) 0.3095 0.32 0.0883 7.4606 0.0313 2 0.1000 II H 0.0282 3 0.5000 II it 0.0329 4 1.0000 II II 0.0351 5 1.5000 ti ti 0.0362 6 2.0000 ii II 0.0370 7 1.2448 II II 0.0357 8 0.3095 0.05 II 0.0313 9 0.10 it it 10 0.50 it tt 11 0.75 it ti 12 0.95 tt it 13 0.32 0.0050 18.0257 0.0016 14 it 0.0100 15.6730 0.0033 15 II 0.0500 9.5907 0.0172 16 ti 0.2500 4.0476 0.0964 17 it 0.5000 2.3354 0.2109 18 0.1893 it 0.0883 7.4606 0.0298 *5 Sample Accelerograms Generated for Each Earthquake Ensemble. **Standard Earthquake 156 7,2.2 D e f i n i t i o n of Standard Structures Three d i f f e r e n t standard structures were used i n the s e n s i t i v i t y study. The phys i c a l properties of these structures are defined i n F i g . 7.3. S t r u c t u r e s #1 and #2 represent f l e x i b l e structures (T = 1,2448s and 0.9511s), w h i l e s t r u c t u r e #3 r e p r e s e n t s a s t i f f s t r u c t u r e (T = 0.1893s). The combination of the standard earthquake with a standard structure i s define as a standard model. The nondimensional parameters of the three standard models are presented i n Table 7.2. Table 7.2 Nondimensional Parameters for Standard Models Structure No. V T u 5i Pcr/W a g/g T /T 1 0.3147 0.05 0.0003 0.0883 0.2486 0.32 2 0.2955 0.05 0.0316 0.0883 0.3254 0.32 3 0.5381 0.05 0.1556 0.0883 1.6350 0.32 A s e n s i t i v i t y study was employed to examine the influence of each nondimensional parameter on the optimum s l i p load. FDBFAP was used for these analyses, i n which each nondimensional parameter was varied i n d i v i d u a l l y while holding a l l the other nondimensional parameters constant (at t h e i r standard values given i n Table 7.2). The r e s u l t s of the s e n s i t i v i t y study are discussed i n sections 7.2.3 through 7.2.8 and displayed g r a p h i c a l l y i n Figs. 7.4 to 7.9, which show p l o t s of P0/W against the p a r t i c u l a r parameter under consideration. A l l the calculated data are shown on each graph (5 samples per ensemble), which also displays the curve defining the mean of the data. The 95% confidence i n t e r v a l for the mean i s also given and i s based on w Properties Structure #1 Structure #2 Structure #3 H (mm) 3660 2540 3050 L (ram) 7620 5080 6100 h (mm) 370 250 310 S (mm) 760 510 610 W (kN) 167 1245 492 (mm*) 4 x 10 s 29 x 10* 551 x 106 h (mm*) CO 40 x 106 551 x 106 (mm2) 99 994 2790 h (mm*) 195 126 x 10 3 2 x 10 6 E (MPa) 200,000 200,000 200,000 0.05 0.05 0.05 T u (s) 1.2448 0.9511 0.1893 T P (s) 0.3917 0.2811 0.1019 (kN) 0.0552 39.3666 76.5575 Figure 7.3 Physical Properties of Standard Structures 158 small sample theory (number of samples l e s s than 30) for an approximate normal population (Walpole and Myers, 1978): - t0.025 S . - . t0.025 S , 7 x — < u < x + (7.3) Vn Vn where: x = Mean value of the sample S = Standard deviation of the sample n = Sample s i z e = 5 (5 sample accelerograms) t0 025 = ^ a ^ u e t d i s t r i b u t i o n w i t h 4 degrees-of-freedom having a p r o b a b i l i t y of exceedance of 0.025. Vi = Mean value of the population 7.2.3 S e n s i t i v i t y to Viscous Damping (£ :) The f i r s t parameter examined i n the s e n s i t i v i t y study was the c h a r a c t e r i s t i c viscous damping ( E j ) . The d i f f e r e n t values of considered were: l1 = 0.00, 0.02, 0.05, 0.10, 0.15, 0.20 The r e s u l t s of the study are presented i n F i g . 7.4, which shows the influence on P0/W of v a r i a t i o n s i n ^ for the three structures. I t can be seen that the optimum s l i p load i s e s s e n t i a l l y independent of the viscous damping for the range of values considered. This was expected, since the v i b r a t i o n a l properties of a structure are e s s e n t i a l l y unchanged for viscous damping r a t i o s up to 20% c r i t i c a l , and also since the energy d i s s i p a t e d by viscous damping i s n e g l i g i b l e compared to the Structure #1 0.05 0.04 0.03-\ 0.02 O . O M 0.00-i 95% Confidence Interval for the Mean • Sample Data Mean 0.00 0.05 0.10 0.15 Structure #2 0.15 0.12 A 0.09 0.06 0.03 95% Confidence Interval for the Mean • Sample Data Mean 0.00 0.00 0.05 0.10 0.15 Structure #3 0.25 0.20-0.15-0.10 95% Confidence Interval for the Mean • Sample Data Mean 0.20 0.20 ° ' 0 5 (j^iiti^irffiffiOPffi 5£vs.< >»:•»:•»:•:•»: >>zo>>>>>>z<»>i .C<<K<KK*&*A*S xxxxxxxxxxxxx^ 0.00^-0.00 0.05 0.10 0.15 0.20 Figure 7.4 S e n s i t i v i t y to Viscous Damping (Ej) 160 much higher values of energy dis s i p a t e d by f r i c t i o n ; t h i s l a t t e r fact has already been observed i n F i g . 3.8. Therefore, the c h a r a c t e r i s t i c viscous damping E- was discarded as a parameter inf l u e n c i n g the optimum s l i p load, and was set to a constant value of 0.05 for a l l analyses performed i n the parametric study. 7.2.A S e n s i t i v i t y to C h a r a c t e r i s t i c Buckling Load (P ) _ _ c r ' To e v a l u a t e the i n f l u e n c e of the c h a r a c t e r i s t i c buckling load P__ on the optimum s l i p load, P__ was v a r i e d over the e l a s t i c range of the c r o s s - b r a c e s of each s t r u c t u r e : 0 _ P <; A, o , where o , the y i e l d c r b y y stress of the cross-braces, was taken to be equal to 300 MPa. Structure #1 P /W 0.00, 0.05, 0.10, 0.15, 0.18 Structure #2 P c r / W = ° ' 0 0 ' ° * 0 4 , ° * 0 6 ' ° * 1 2 ' ° ' 1 8 , ° ' 2 A Structures #3 P /W = 0.00, 0.07, 0.15, 0.21, 0.28, 0.35 Figure 7.5 presents the r e s u l t s of t h i s s e n s i t i v i t y study. I t can be seen t h a t P0/W i s e s s e n t i a l l y independent of P__/W. Only a s l i g h t decrease of P0/W i s observed for very small values or P__/W. Physic-a l l y , because of the behaviour of the f r i c t i o n devices, the c r i t i c a l buckling load of the cross-braces should not be an important parameter inf l u e n c i n g the response of a f r i c t i o n damped structure; the l a t e r a l 161 Structure #1 0.05 0.04 95% Confidence Interval for the Mean • Sample Data Mean 0.00 0.00 0.06 0.12 Pcr/W Structure #2 0.18 0.J5-I 0.12-__: 0.09-<__ 0.06-0.03 A 0.00 • Sample Dafa 95% Confidence Interval for the Mean M e a n I A A A A X A A A A AA" 0.00 0.06 0.12 Pcr/W 0.18 0.24 Structure #3 0.25 0.20-^ 0.15 £ o.w 0.05 0.00-95% Confidence Interval for the Mean • Sample Data Mean Figure 7.5 S e n s i t i v i t y to C h a r a c t e r i s t i c Buckling Load (P ) 162 shear c a r r i e d by the cross-braces during s l i p p i n g i s not dependent on the buckling load. The buckling load of the cross-braces only i n f l u -ences the t r a n s i t i o n from a f u l l y braced state to a complete s l i p p i n g state, as was shown i n Figs. 2.7 and 2.13. Since t h i s t r a n s i t i o n most l i k e l y takes place i n a short period of time, i t should not a f f e c t s e r i o u s l y the o v e r a l l response of the structure. The c h a r a c t e r i s t i c buckling load therefore was discarded as a parameter infl u e n c i n g the parametric study; i t was set to a nominal value of 10% of the y i e l d load of the cross-braces i n a l l subsequent analyses. 7.2.5 S e n s i t i v i t y to Braced Period (T b) The r a t i o T,/T can be v a r i e d i n d e p e n d e n t l y of the other non-fa U f J dimensional parameters by changing the cross-sectional area of the d i a g o n a l b r a c e s . The p o s s i b l e range of values of T,/T i s O^T./T £l. D U D U T^/T^ = 0 occurs when the d i a g o n a l braces are made i n f i n i t e l y s t i f f against a x i a l deformation; T, /T = 1 for an unbraced structure. D u The r e s u l t s of the s e n s i t i v i t y study are presented i n F i g . 7.6. I t can be seen that the optimum s l i p load i s negatively correlated with T, /T r a t i o for the three standard structures. Hence T,/T was kept as b u b u e a varying parameter for the parametric study. 7.2.6 S e n s i t i v i t y to Predominant Ground Period (T^) As shown i n Table 7.1, 7 ensembles, each containing 5 sample accelerograms, were simulated with d i f f e r e n t values of T and constant s t a n d a r d v a l u e s of h and a . T h i s p e r m i t t e d the s e n s i t i v i t y of the optimum s l i p l o a d to the nondimensional parameter 1^/1^ to be studied. The r e s u l t s of t h i s s e n s i t i v i t y study are shown i n F i g . 7.7. The curves 163 C L Structure #1 0.05-0.04-_ 0.03-> • C L 0.02-0.01-0.00-95% C o n f i d e n c e Interval for the Mean • Sample Data Mean 0.2 0.4 0.6 Tb/Tu Structure #2 0.8 0.2 0.4 0.6 Tb/Tu Structure #3 0.8 0.25-0.20-^ 0.15-(£ 0.10-0.05-0.00-95% C o n f i d e n c e Interval for the Mean • Sample Data Mean 0.2 0.4 0.6 Tb/Tu Figure 7.6 S e n s i t i v i t y to Braced Period (T^) 0.15-0.10-• Sample Data 95% C o n f i d e n c e Interval for the Mean Mean 0.05-• 0.00-Structure #1 o.wo-0.075-95% C o n f i d e n c e Interval for the Mean • Sample Data Mean Tg/Tu Structure #2 0.15 0.12 ^ 0.09-f£ 0.06-0.03 H 0.00 95% C o n f i d e n c e Interval for the Mean • Sampl* Data Structure #3 0.25-0.20-0.15-> • 0.10-0.05-0.00-95% C o n f i d e n c e Interval for the Mean • Sample Data Uton 2.4 0 3 6 9 Tg/Tu Figure 7.7 S e n s i t i v i t y to Predominant Ground Period (T ) g 12 165 indi c a t e that the predominant ground period i s an important parameter i n f l u e n c i n g the optimum s l i p load. P0/W i s proportional to T^/T^ for T /T _ 1; f o r v a l u e s of 1 /T > 1, Pn/W remains almost c o n s t a n t , g u g u 0 T h e r e f o r e TJT^ i s kept as a varying parameter i n the parametric study. 7.2.7 S e n s i t i v i t y to Ground Damping (h ) In order to study the influence of h on the optimum s l i p load, 6 accelerogram ensembles, each consisting of 5 samples, were simulated by s p e c i f y i n g d i f f e r e n t v a l u e s of h while maintaining constant standard v a l u e s of T and a , as shown i n Table 7.1. Figure 7.8 presents the r e s u l t s of t h i s s e n s i t i v i t y study. It can be seen that P0/W i s not i n f l u e n c e d s i g n i f i c a n t l y by h ; the curves passing through the mean of the data are e s s e n t i a l l y h o r i z o n t a l . The ground damping h therefore was discarded as a varying parameter and was set to a constant value of 0.32, as proposed by Vanmarcke and L a i (1980,1982). 7.2.8 S e n s i t i v i t y to Peak Ground Acceleration (a ) 1 g Table 7.1 also shows that 6 ensembles, each having 5 sample accelerograms, were generated with d i f f e r e n t values of a and constant s t a n d a r d values of T and h i n order to study the influence of the peak ground ac c e l e r a t i o n on the optimum s l i p load. The r e s u l t s of the study are presented i n F i g . 7.9, where i t can be observed that P0/W i s s t r o n g l y c o r r e l a t e d with a /g. The r e l a t i o n between P0/W and a /g can be approximated reasonably well by a s t r a i g h t l i n e through the o r i g i n . This r e s u l t i s i d e n t i c a l to the one obtained a n a l y t i c a l l y for a harmonic ground excited one storey f r i c t i o n damped structure e x h i b i t i n g b i l i n e a r 166 Structure #1 0.05 0.04-^ 0.03-r£ 0.02-0.01-0.00 95% C o n f i d e n c e Interval for fhe Mean • Sompln Data Mean I 1 I 1 I 0.2 0.4 0.6 Structure #2 0.8 0.15 0.12 0.09 > 0.06 0.03 0.00 0.25 0.20 0.15 > 0.10 0.05 0.00 95% C o n f i d e n c e Interval for fhe Mean • Sample Data Mean 0.2 0.4 0.6 h9 Structure / J 0.8 95% C o n f i d e n c e Interval for the Mean • Sample Data Mean Figure 7.8 S e n s i t i v i t y to Ground Damping (h ) Structure #1 0.10 Structure #2 0.15 0.12-95X Confldonc* tnltrvol for th* Moon 0.25 0.20 Structure #3 9SX Confloonc* lnl«r»ol for th* Moan Figure 7.9 S e n s i t i v i t y to Peak Ground Acceleration (a 168 h y s t e r e t i c properties (see Equation 4.36). Therefore a /g was kept as a varying parameter i n the parametric study. 7.2.9 Discussion The r e s u l t s of the s e n s i t i v i t y study on one storey f r i c t i o n damped structures i n d i c a t e that the optimum s l i p load i s mainly a function of three nondimensional r a t i o s : P 0 . T T a — = — F T — -^1 (7 4) W 2cosa T ' T ' g K ' , H ) u u & Of these t h r e e r a t i o s , a /g i s the one for which P0/W i s the most s e n s i t i v e . The r e l a t i o n s h i p between P0/W and a /g i s p r a c t i c a l l y l i n e a r and has zero ordinate at the o r i g i n . An important conclusion from the s e n s i t i v i t y study i s that the optimum s l i p load depends on the c h a r a c t e r i s t i c s of the earthquake expected at the construction s i t e : on the peak a c c e l e r a t i o n amplitude a , and on the dominant ground period s T ; i t i s not s t r i c t l y a s t r u c t u r a l property. s 7.3 Parametric Study 7.3.1 Strategy From the r e s u l t s of the s e n s i t i v i t y study, i t i s proposed to approximate the unknown function F i n Equation (7.4) i n the following manner: 169 where M i s a function of T,/T and T /T : b u g u T, T M = F * [ ^ , ^ ] u u (7.6) For f i x e d v a l u e s of T, /T and T /.T , M can be estimated by the b u g u 3 l e a s t square method. The sum of the square of the errors I can be written as: a 2P 0cosa ( ) a 1 = ^  [(-Tr-)^ (7.7) where 2P 0cosa ( ^ ).. = True optimum s l i p load obtained with FDBFAP for i W ' i N analysis = Number of analyses with d i f f e r e n t a /g values g The l e a s t square method consists i n minimizing the square of the er r o r s : i = ° ( 7 - 8 ) which l e a d s to a r e l a t i o n for M obtained for p a r t i c u l a r values of T,/T r b u and T /T . g u (7.9) 170 An array of M values can then be generated for d i f f e r e n t values of T^/T^ and T /T , from which F* i n Equation (7.6) can be estimated, g u The basic s t r u c t u r a l configuration used i n the parametric study i s shown i n F i g . 7.10. The dimensions represent a t y p i c a l bay of an i n d u s t r i a l b u i l d i n g . Table 7.3 presents 4 d i f f e r e n t versions of the basic s t r u c t u r a l configuration used i n the parametric study. Each version has a d i f f e r e n t l a t e r a l s t i f f n e s s ; the corresponding structures range from a very s t i f f system (T = 0.1243s) to a very f l e x i b l e system (T = 1.9525s). The values of the parameters used i n the parametric study are given i n Table 7.4. W=444kN U »Ji Figure 7.10 Basic S t r u c t u r a l Configuration Used i n Parametric Study 171 Table 7.3 Versions of Basic S t r u c t u r a l Configuration Version # Columns Section (mm*) T u (s) 1 W30 x 124 2.23 x 10 9 0.1243 2 W16 x 45 243 x 106 0.3764 3 W8 x 24 34 x 10 6 1.0016 4 W6 x 12 9 x 10 6 1.9525 Table 7.4 Values of Parameters Used i n Parametric Study Parameter Values T u (s) 0.1243, 0.3764, 1.0016, 1.9525 V T u 0.20, 0.40, 0.60, 0.80 T g / T u 0.1/TU, 0.7/Tu, 1.4/TU, 2.0/T u a g/g 0.005, 0.05, 0.10, 0.15, 0.20, 0.30, 0.40 The a /g values are the nominal values of hori z o n t a l peak ground a c c e l -s e r a t i o n associated with the 7 seismic zones i n Canada (NBCC, 1985); each value has a p r o b a b i l i t y of exceedence of 10% i n 50 years. For each combination of the parameters given i n Table 7.4, 5 d i f f e r e n t sample accelerograms were simulated with SIMEA. The t o t a l number of cases subjected to FDBFAP analyses was 2240. A microcomputer version of FDBFAP was used on an IBM-PC to perform these analyses. For each case, the optimum v a l u e of the s l i p l o a d was e s t a b l i s h e d from a n a l y s e s i n which a Pg/W increment equal to 0.01 was used. The minimum number of increments was 25; each increment corresponds to one time-h i s t o r y dynamic an a l y s i s . 172 7.3.2 Results for Least Square Slopes, M The r e s u l t s of the parametric study are shown i n Figs. 7.11 to 7.26. Each figure presents the values of 2P 0 cosa/W obtained for a fi x e d value of T /T and for a l l the values of T./T and a /g considered g u b u g 6 i n Table 7.A. A l l the data are presented, along with the least square s l o p e s M o b t a i n e d from E q u a t i o n (7.9), for p a r t i c u l a r values of TJT^ and T^/T u» The c a l c u l a t e d l e a s t square slopes are presented i n Table 7.5. Table 7.5 Least Square Slopes T /T M T b / T u =0.20 T b / T u =0.40 T b / T u =0.60 T b / T u =0.80 0.0512 0.1597 0.0780 0.0403 0.0343 0.0998 0.3274 0.1396 0.0785 0.0396 0.2657 0.6208 0.3120 0.2019 0.0918 0.3585 0.5774 0.3087 0.1435 0.0599 0.6989 0.9221 0.6491 0.3174 0.1338 0.7170 0.9057 0.7109 0.2850 . 0.0841 0.8045 1.1185 0.9715 0.7172 0.3068 1.0243 0.9199 0.8460 0.4975 0.2000 1.3978 1.0532 0.7995 0.5466 0.2300 1.8597 1.2197 0.8983 0.5145 0.2164 1.9968 0.9889 0.7385 0.4740 0.2411 3.7195 1.1370 0.9163 0.6004 0.2288 5.3135 1.0557 0.8635 0.5833 0.2192 5.6315 1.1658 1.0391 0.7037 0.4099 11.2631 1.1497 0.9865 0.7661 0.4238 16.0901 1.0679 0.9257 0.7450 0.3910 173 CO O o o C L 0.8-0.6-0.4-0.2-T/T =0.0512 ° Tt/Tu=0.20 O Tb/7y=0.40 x Tk/ry=o.eo 0.00 Least Square Slopes: Mfl>.20.0.05l2>0.!3»7 UfO.40.0.0512]-0.0780 Mf0.60,0.0312]=0.0403 Uf0.8O,O.OS12j=O.0J4J I ' 0.48 0.60 Figure 7.11 Least Square Slopes, T /T = 0.0512 > to O o o C L CN 0.8-0.6-0.4-0.2-Tg/Tu=0.0998 • T^/Ty=0.40 • Tt/Tv«0.80 Least Square Slopes: o r— Uf0.20,0.0W8]=0.3274 -M[0 .40 ,0 .0»»B>0. t3»6 — M[0.60,0.0»»8]«0.078» _—Mf0,80.0.0»»81»0.03»8 0.45 0.60 Figure 7.12 Least Square Slopes, T /T = 0.0998 g u 174 > CO O O o C L C N I 1 0.8 0.6-0.4-TgAu=0.2657 O Tt/T¥B0.20 X Tk/Tys0.60 • Tt/TumO.BO 0.00 Least Square Slopes: o •1(0.20,0.26971*0.6208 MfQ.40.0.26S7]=0.3120 U(0.60,0.26S7]*0.20t9 M[0.60,0.26S7]a0.09t8 0.12 0.24 0.36 a a/9 —I— 0.48 Figure 7.13 Least Square Slopes, T g / T u = 0.2657 0.60 1 CO O ° o 0 8-,6-4-0.2-TgAu=0.3585 * Tk/Tu*o.eo • T^/T^O.BO 0.00 Least Square Slopes: Mf0.20.0.33B5]=0.377« M(D.40,0.3585]=0.SO87 Mjp.60.0.3S»S)»0.1433 M{0.8O.0.3SB5]B0.Oa99 1 0.48 0.60 Figure 7.14 Least Square Slopes, T g / T u = 0.3585 175 > O o O CN 1 0.8-0.6-TgAu=0.6989 O 7^=0.20 * T>/Tv=0.60 0.00 Least Square Slopes: U|b-20.0.SMS)-0.I221 U[0.40.0.«98«]-0.84S1 — UfO.S0.0.«889]-0.3174 MfD.B0.0.6MB]>0.13J8 0.12 0.24 0.36 <=>a/9 0.48 0.60 Figure 7.15 Least Square Slopes, T g / T u = 0.6989 > CO O O o a. CN / 0.8-0.6-0.4-0.2-0 Tg/T=0.7170 O T^/T^O.20 a T^/TU=O.40 X TK/TY=0.60 • T^/T^mO.eO 0.00 Least Square Slopes: 11(6.20,0.71701=0.9057 M[0.40.0.7t70]«0.7109 U{0.e0,0.7l70]a0.2830 Hf0.80.0.7t70}«0.0841 0.48 0.60 Figure 7.16 Least Square Slopes, T /T = 0.7170 g u 176 0.60 Figure 7.17 Least Square Slopes, T /T = 0.8045 Tg/Tu=1.0243 O Tt/7V"0.20 0 Q _jO Tt/T=0.40 * T^/T=0.60 • Tt/Tv=0.80 ^ 0.6H C/) o a. CN o 0.4-0.2-0.00 Least Square Slopes: — M(0.20,1.024J)-0.»19» — M[0.40,1.02*S]=0.8460 MtO.60.t0243J-0.4J7S W(D.BO.I.0243>0.2000 0.12 0.24 0.36 0.48 og/g Figure 7.18 Least Square Slopes, T g / T u = 1.0243 0.60 177 Tg/T =1.3978 ° rt/i=o.20 • Tb/Ty=0.40 0.00 Least Square Slopes: M[0.2O.L3»78]*1.O532 M[0.40,LS978]=0.7»9S M [0 .«0, I .S978) -0 .54«6 Figure 7.19 Least Square Slopes, T /T^ = 1.3978 0.60 j Tg/T=1.8597 O Tb/T„=0.20 0.8 A° T>/T<=0-40 • Tt/Tu=O.BO 0.00 Least Square Slopes: MfD.2CM.aSS7lBl.2rS7 - M f 0 . 4 0 . t - 8 S S 7 J m 0 . 8 S 6 3 M [ 0 . ( 0 . 1 . B S S 7 ] - 0 . 5 1 4 9 M [ 0 . 8 0 . 1 . 8 3 S 7 ] « 0 . 2 1 6 4 0.12 0.24 0.36 0.48 og/g Figure 7.20 Least Square Slopes, T g / T u = 1.8597 0.60 178 > CO O O o CL CN 1 0.8-0.6-T/T =1.9968 O Tt/Tv=0.20 O T . / T „ = 0 . 4 0 Least Square Slopes: Mp>.20,L9M8>0.»88« Mp.40,Lt*6«l>0.7M5 ti(o.«o.t»««B>o.mo Mf0.80.t»968)=0.2411 0.00 0.12 0.24 0.36 °a/9 0.48 0.60 Figure 7.21 Least Square Slopes, T g / T u = 1.9968 > CO O 1 0.8-0.6-CU CN b 0.4-O T^/T^O.20 • Tb/Ty=0.40 * Tt/Tv=O.BO O Least Square Slopes: • WfD.20.3.7t84]*1.1370 M(S.40.3.7t»4]*0.aie3 U(p.S0.3.7N4]=0.e004 M{0.80.3.71»4]=0.2288 0.00 0.12 0.24 0.36 0.48 og/g Figure 7.22 Least Square Slopes, T g / T u = 3.7195 0.60 179 to O O o CL 0.8-0.6-Tg/Tu=5.3135 ° Tt/T¥=0.20 a rb/r¥=o.40 X Tb/Ty=0.60 • Tb/l¥mO.B0 0.00 Least Square Slopes: M[0.20.S.313S]>1.0557 -f0.40,5.3133]=0.8633 0.12 0.24 0.36 aa/9 0.48 0.60 Figure 7.23 Least Square Slopes, T g / T u = 5.3135 TgAu=5.6315 O Tb/T=0.20 0.8H° r^1o=0*° • Tb/ls*O.BO ^ 0.6H (o O °o 0.4 A CL CN 0.2 A O Least Square Slopes: W(D.20,S.63lS)=1.165B W(0.40.S.631S]=t.039t M[0.eO,5.63lS)=0.7037 M[0.60,S.S31S]-0.40»9 0.00 0.12 a a/9 0.24 0.36 0.48 0.60 Figure 7.24 Least Square Slopes, T /T =5.6315 g u 180 CO o o o a. CN 1 0.8-0.6-T/T =11.2631 ° T*AUB0.20 D Tb/l=0.40 * T,/rv=o.60 * T>AU=0.80 O Least Square Slopes: • Hfl>.20.H.tt3fl«t.14»7 W[0.40,t1.26Sfj°0.986S MfO.«0.n.M3()»0 7MI 0.00 0.12 Figure 7.25 Least Square Slopes, T g / T u = 11.2631 0.8-co O °o 0.4 CL CN 0.2 A Tg/T =16.0901 o rD/r,,=o.20 X Tk/Ty=0.60 • r»/T,=o.so 0.00 Least Square Slopes: Mf0.20.t6.090l)=t0e79 Mfp.60.16.090<|=0.7450 0./2 0.24 0.36 0.45 Figure 7.26 Least Square Slopes, T g / T u = 16.0901 0.60 181 7.3.3 Correlation Between M and T, /T b u The data contained in Table 7.5 is shown in graphical form in Fig. 7.27, which illustrates the relations obtained between the least square slopes M and T, /T for different values of T /T . It can be seen that M ^ b u g u is negatively correlated with T^/T ; the relationship is very nearly linear, such that: (7.10) where a and b are functions of T /T . Table 7.6 lists the values of a g u and b calculated from a least square analysis of the graphs in Fig. 7.27. As shown below in section 7.3.A, analytical expressions for the parameters a and b can be determined by the least square method. 7.3.A Correlation Between Least Square Slope Parameters and T^/T^ The a and b parameters corresponding to the Tg/Tu values shown in Table 7.6 are plotted in Fig. 7.28. A bilinear least square f i t was used to approximate the data, resulting in the following bilinear expressions for the least square slope parameters a and b in terms of T /T (see Appendix H): g u T T -1.4697 (^) 0 < ^  £ 1 u u a = T T 0.0232 (=£) - 1.4929 -S > T 1 u A. u (7.11) T T 1.4639 (^) u T 0 < ^  £ u T 1 b = -0.0069 (^) + 1.4708 _ u -£ > T 1 U Q) a o D Cr 0 —I 00 Figure 7.27 Correlation Between M and T, /T b u 183 Figure 7.28 C o r r e l a t i o n Between Least Square Slope Parameters and T /T 184 Table 7.6 Estimation of Least Square Slope Parameters a,b T /T a b 0.0512 - 0.2070 0.1816 0.0998 - 0.4622 0.3775 0.2657 - 0.8487 0.7309 0.3585 - 0.8590 0.7019 0.6989 - 1.3485 1.1799 0.7170 - 1.4452 1.2190 0.8045 - 1.3447 1.4509 1.0243 - 1.2542 1.2429 1.3978 - 1.3614 1.3380 1.8597 - 1.6968 1.5606 1.9968 - 1.2538 1.2375 3.7195 - 1.5203 1.4809 5.3135 - 1.3950 1.3780 5.6315 - 1.3017 1.4805 11.2631 - 1.1991 1.4312 16.0901 - 1.1058 1.3353 Substituting Equations (7.11) and (7.12) into Equation (7.10) leads to the following r e l a t i o n for the le a s t square slopes: T T, T, T M = K 4 ^ f + K 2 f + K 3 f + K, u u u u (7.13) 185 where K, -1.4697 0.0232 0 < T /T £ I g u T /T > 1 g u 0 0 < T / T ' i l g u -1.4929 T /T > 1 g u K, 1.4639 -0.0069 0 < T /T * 1 g u T /T > 1 g u K, 0 1.4708 0 < T /T £ 1 g u T /T > 1 g u 7.4 Proposed Design Equation Based on the r e s u l t s of the parametric study, the following design equation i s proposed for estimating the optimum s l i p shear of the f r i c t i o n devices for one storey structures: NBF V A = 2 1 P„. cosa. = i-1 T T, T, T a I K i f f + K 2 f + K 3 ^ + K J -S W u u u u B (7.14) where -1.4697 0 < T /T <; 1 g u K, 0.0232 T /T > 1 g u 186 K, 0 0 < T /T £ 1 g u -1.4929 T /T > 1 g u K, 1.4639 0 < T /T £ 1 g u -0.0069 T /T > 1 g u K, 0 0 < T /T £ 1 g u 1.4708 T /T > 1 g u a. l N BF u V 0 = Optimum s l i p shear P 0^ = Optimum l o c a l s l i p l o a d of the f r i c t i o n device i n the i bay = Angle of i n c l i n a t i o n of the cross-braces i n the i ^ bay = Number of bays with f r i c t i o n devices = Kanai-Tajimi predominant period = Fundamental period of the f u l l y braced structure = Fundamental period of the unbraced structure a^ = Peak ground acceleration g = Acceleration of g r a v i t y W = Weight of the structure .th 7.4.1 Design S l i p Load Spectrum A design s l i p load spectrum can be constructed by rewriting Equation (7.14) as v„ T, T T, 0 ma [K, ^ + Ks] T [Ka Y + KJ g u u u (7.15) 187 where m i s the t o t a l mass of the structure and V 0 i s the optimum t o t a l s l i p shear. F i g u r e 7.29 p r e s e n t s curves of V n/ma vs T /T f o r p a r t i c u l a r 6 r 0 g g u r v a l u e s of T, /T based on Equation (7.15). It should be noted that for b u 1/1 > 1 the curves are not too se n s i t i v e to the v a r i a t i o n i n T . This g u g i s a f o r t u n a t e s i t u a t i o n s i n c e T v a l u e s are not commonly known with g ' accuracy or provided i n codes at present. 7.5 Appraisal of Proposed Design Equation The design equation proposed i n Section 7.4 was evaluated by sub-j e c t i n g the three standard structures used i n the s e n s i t i v i t y study (see Section 7.2.2) to r e a l h i s t o r i c a l earthquakes, and comparing the optimum s l i p load predicted by Equation (7.14) with the value of the optimum s l i p load determined by FDBFAP. Three earthquake records having widely varying c h a r a c t e r i s t i c s were chosen for t h i s comparative t e s t : • P a r k f i e l d , C a l i f o r n i a Earthquake, June 27, 1966, COMP N40W • Long Beach, C a l i f o r n i a Earthquake, May 10, 1933, COMP N51W • Eureka, C a l i f o r n i a Earthquake, December 21, 1954, COMP N46W The accelerograms of these three earthquakes are shown i n F i g . 7.30 and t h e i r c h a r a c t e r i s t i c s are given i n Table 7.7. The r e s u l t s of the appraisal study are presented i n Figs. 7.31 to 7.33. I t can be seen that the proposed design equation predicts the optimum s l i p load reasonably w e l l . The differences i n RPI values associated with s l i p loads determined by the design equation and by the ap p l i c a t i o n of FDBFAP generally are small, as shown i n Table 7.8. 15 o 0.5 : T b / T u = 0.40 T b / T u = 0 . 5 0 T b / T u = 0.60 T b / T u = 0.70 , . r ... T k / T „ = 0.80 72 15 Figure 7.29 Design S l i p Load Spectrum for Single Storey F r i c t i o n Damped Structure. 189 0.3 D> 0.0 Park field Earthquakejune 27,1966,N40W -0.3 ItftlrJ/JlUM 0.1-i 0 5 10 15 20 25 Time [sec] Long Beach Earthquake,May 10,1933,N51W 30 D> 0.0 Mill i ^A A / \ 7 * 10 15 20 25 30 0.3n Time [sec] Eureka Earthquake,December 21,1954,N46W D> 0.0 -0.3 0 5 10 15 20 25 30 Time [sec] Figure 7.30 Earthquake Accelerograms Used for the Appraisal of the Design Equation 190 0.00 0.00 Parkfield Earthquake,June 27,1966,N40W 0.02 0.04 0.06 P/W 0.08 0.10 Long Beach Earthquake.May 10,1933,N51W P / W - D e s i g n Equation P 0 / W - D a t a 0.02 0.04 0.06 P,/W 0.08 0.10 Eureka Earthquake.December 21,1954,N46W 0.00 0.02 0.04 0.06 P,/W 0.08 0.10 Figure 7.31 Appraisal of Proposed Design Equation, Structure #1 191 Parkfield Earthquake,June 27,1966.N40W 0.00 0.02 0.04 0.06 0.08 0.10 p,/w Figure 7.32 Appraisal of Proposed Design Equation, Structure #2 192 Parkffeld Earthquake.June 27,1966,N40W 7 i 0.8- P 0 / W - D e s l g n Equat ion—v H 0.6-p> 0.4-P 0 / W - D a t a — N 0.2-0- i. 1 0 . 0 0 0 . 0 4 O . O S 0 . / 2 0.16 Long Beach Earthquake.May 10,1933,N51W 0.00 0.04 0.08 0.12 P,/W 0.16 Eureka Earthquake.December .21,1954,N46W 0.20 0.20 P 0 / W - D e s i g n Equation P c / W - D a t a 0.00 0.04 0.08 P,/W Figure 7.33 Appraisal of Proposed Design Equation, Structure #3 193 Table 7.7 Ch a r a c t e r i s t i c s of Seismic Events Earthquake Comp Local Magnitude Ep i c e n t r a l Distance (km) T g (I) (s) P a r k f i e l d N40W 5.3 5.3 0.275 0.1935 0.22 1.99 Long Beach N51W 6.3 59.0 0.097 0.9607 0.61 13.48 Eureka N46W 6.6 40.0 0.201 0.6912 0.36 3.52 Table 7.8 Appraisal of Proposed Design Equation Structure No. Earthquake FDBFAP Design Equation RPI RPI 1 P a r k f i e l d Long Beach Eureka 0.0458 0.0334 0.0708 0.1787 0.1297 0.1271 0.0296 0.0416 0.0620 0.2040 0.1596 0.1310 2 P a r k f i e l d Long Beach Eureka 0.0536 0.0393 0.0679 0.0514 0.0293 0.0475 0.0322 0.0558 0.0841 0.0696 0.0636 0.0598 3 P a r k f i e l d Long Beach Eureka 0.1750 0.0333 0.0875 0.0243 0.1264 0.1220 0.1035 0.0377 0.0773 0.1053 0.1400 0.1400 Therefore, the proposed design equation (Equation 7.14) seems adequate for p r e d i c t i n g the optimum s l i p load of sing l e storey f r i c t i o n damped structures. 7.6 Physical Limitations on Design Equation The proposed design equation for evaluating the optimum s l i p load of a sing l e storey f r i c t i o n damped structure has been constructed with-out considering the e f f e c t of wind on the structure. Also, i t has been assumed that the value of the optimum s l i p load given by Equation (7.14) 194 i s l e s s than the load which produces y i e l d i n g i n the t e n s i o n c r o s s -brace. These two phenomena (wind e f f e c t and y i e l d i n g ) w i l l impose c o n s t r a i n t s on equation (7.14). 7.6.1 Lower Bound The f r i c t i o n devices must be designed so that they do not s l i p under wind l o a d . T h e r e f o r e the g l o b a l s l i p load P (load i n t e n s i o n g brace to produce s l i p p i n g of the f r i c t i o n device) must be l a r g e r than the design load generated i n the t e n s i o n cross-braces by wind. For slender braces ( P n . > P . ) : °i c r i P . = 2 P B. - P . * P . g i °i c r i wi or P-. ^ P . + P . wi c r i (7.16) For stubby braces (P.. £ P .) J °i crx P.. * P . °i wi (7.17) where P 0^ = Optimum l o c a l s l i p load i n f r i c t i o n device of the i ^ bay P w^ = Load induced i n the t e n s i o n brace of the i ^ bay by wind P c r £ = C r i t i c a l b u c k l i n g load of the compression brace of the i t h bay 195 7.6.2 Upper Bound The value of the optimum s l i p predicted by Equation (7.14) must be such that the tension diagonal cross-brace of the structure remains e l a s t i c . For slender braces (P.. > P . ) : °i c r i (7.18) For stubby braces (P 0. £ P .) c r i £ A, .a . DI y i (7.19) where = Cross-sectional area of the tension brace i n i ^ bay o . = Y i e l d stress of the brace material i n i ^ bay y i 7.6.3 Remarks If the above equations are not s a t i s f i e d , the value of the optimum s l i p load predicted by Equation (7.14) cannot be achieved p h y s i c a l l y and therefore the design of the structure must be re-examined. If Equations (7.16) or (7.17) are not s a t i s f i e d , the lower bound value of the s l i p load given by one of these equations could be used but dynamic analyses should be performed to examine more c a r e f u l l y the behaviour of the structure under t y p i c a l earthquake records expected at 196 the construction s i t e . However, i t should be recognized that for most s i t e s there may not be s u f f i c i e n t data to make a r e l i a b l e p r e d i c t i o n of such records. Another a l t e r n a t i v e would be to increase the s i z e of the main members (beams and columns) so that the moment r e s i s t i n g frame can carry a larger portion of the wind load. If Equations (7.18) or (7.19) are not s a t i s f i e d , the upper bound value of the s l i p load given by one of these equations could be used but again dynamic analyses should be performed. A l t e r n a t i v e l y , the size of the cross-braces could be increased. 7.7 Choice of Diagonal Cross-Braces In a design s i t u a t i o n , d i f f e r e n t member sizes are av a i l a b l e for the diagonal cross-braces. S i m i l a r l y , i n the case of seismic r e t r o f i t of a structure, i t may be decided to replace completely the diagonal cross-braces. For a conventional design, the cross-braces normally would be chosen to carry a c e r t a i n portion of the l a t e r a l seismic force. How-ever, i n the case of the design or r e t r o f i t of a structure equipped with f r i c t i o n devices, the diagonal cross-braces should be chosen to optimize the response of the structure. The best choice of cross-braces i s the one for which the r e l a t i v e performance index RPI i s the o v e r a l l minimum among the family of RPI values evaluated at the optimum s l i p loads corresponding to the ava i l a b l e diagonal member choices. To determine that best choice of cross-braces, the r e s u l t s of the parametric study can be used to f i n d the value of T, /T which minimizes the RPI when the b u f r i c t i o n devices are set at t h e i r optimum s l i p loads. Figure 7.34 presents the average values of RPI at the optimum s l i p l o a d f o r d i f f e r e n t values of T,/T and T /T . Each point on the curve b u g u r i s an average of 35 RPI values at the optimum s l i p load taken across the a /g va l u e s (see Table 7.4). I t can be seen that the RPI i s propor-PM P4 0.8-1 0.6 0.4 0.2 0 •0.0512 A •0.0998 + •0.2657 X 0.3585 o UA«= •0.6989 V •0.7170 H 0.8045 X •1.0243 TVLu= U978 1.8597 & 1.9968 ffl 13 •5.3135 V V 5.6315 • 11.2631 • Ta/Tu= 16.0901 0.2 0.4 0.6 0.8 Figure 7.34 Average Values of RPI at Optimum S l i p Load ^1 198 t i o n a l to T, /T and the best response (minimum RPI) i s obtained for b u r small values of T^/T^ which corresponds to large diagonal cross-braces. Therefore the diagonal cross-braces should be chosen with the largest possible c r oss-sectional area within the l i m i t s of cost and a v a i l a b i l i t y of material. Most of the curves i n F i g . 7.34 e x h i b i t steeper slopes for v a l u e s of T^/T^ l a r g e r than 0.4. Therefore, preferably, the diagonal cross-braces should be chosen such that T, /T < 0.40. b u 7.8 Proposed Design Procedure Based on the r e s u l t s of the parametric study, the following design procedure i s proposed for the design of si n g l e storey f r i c t i o n damped structures. Step 1 • In the case of a new structure, design the unbraced moment r e s i s t i n g frame to carry s a f e l y the usual load combinations but without considering earthquake e f f e c t s ; the unbraced structure should also be designed to have a minimum l a t e r a l s t i f f n e s s , y i e l d i n g resistance, and d u c t i l i t y so as to ensure an optimum response of the whole structure. • In the s i t u a t i o n of r e t r o f i t of an e x i s t i n g b u i l d i n g , v e r i f y that the unbraced moment r e s i s t i n g frame can carry s a f e l y the usual load combinations but without considering earthquake e f f e c t s ; i f necessary, the unbraced structure should also be r e t r o f i t t e d to have a minimum l a t e r a l s t i f f n e s s , y i e l d i n g resistance, and d u c t i l i t y i n order to ensure an optimum response of the whole structure. For a more conservative approach, the designer should include earthquake loads i n the combination and design the moment r e s i s t i n g frame with f u l l conventional d u c t i l i t y to account for a p o t e n t i a l 199 malfunction of the f r i c t i o n devices, or for catastrophic conditions. Step 2 • Calculate the fundamental period of the unbraced structure, T . • Choose s e c t i o n s f o r the diagonal cross-braces such that T^/T^ < 0.40 i f economically p o s s i b l e . The code l i m i t a t i o n s on the slenderness of the braces should be s a t i s f i e d for these sections. • E s t i m a t e the peak ground a c c e l e r a t i o n a and the predominant ground p e r i o d T f o r the c o n s t r u c t i o n s i t e . In Canada, a can be taken g g d i r e c t l y from the National Building Code (NBCC, 1985); the values i n that reference are based on a p r o b a b i l i t y of exceedence of 10% i n 50 years. However, at present, no maps e x i s t i n Canada for determining T . I f p a s t accelerogram or s o i l b o r i n g r e c o r d s which a l l o w T g g estimates to be made for the construction s i t e are not a v a i l a b l e , i t i s recommended t h a t T be approximated by the e m p i r i c a l formulas proposed by Vanmarcke and L a i (Equations (5.7) and (5.8)). Step 3 • V e r i f y that the nondimensional r a t i o s f a l l within the following l i m i t s : 0.20 £ T, /I <. 0.80 b u 0.05 <. T /T £ 20 g u 0.005 <. a /g <. 0.40 g These bounds correspond to those used i n t h i s study and represent reasonable p r a c t i c a l l i m i t s . I f the i n e q u a l i t i e s are not v e r i f i e d , the optimum s l i p load should be determined from dynamic analyses (FDBFAP). 200 If the i n e q u a l i t i e s are v e r i f i e d , estimate the optimum s l i p shear from Equation (7.14) or from the design s l i p load spectrum (Fig. 7.29). Step 4 • Calculate the a x i a l load induced i n the cross-braces from wind e f f e c t and v e r i f y Equations (7.16) or (7.17). If these equations are not v e r i f i e d , choose one of the following two a l t e r n a t i v e s : 1) Use the value of the s l i p load s a t i s f y i n g the e q u a l i t y given by one of these equations and perform dynamic analyses (FDBFAP) to examine the response of the structure. 2) Modify the unbraced moment r e s i s t i n g frame to carry a larger p o r t i o n of the wind load and return to step 2. Step 5 • Estimate the t e n s i l e y i e l d load and the c r i t i c a l buckling load of the cross braces and v e r i f y Equations (7.18) or (7.19). If these equa-tions are not v e r i f i e d , choose one of the following two a l t e r n a t i v e s : 1) Use a value of the s l i p load s a t i s f y i n g the equality i n one of these equations and perform dynamic analyses (FDBFAP) to examine the response of the structure. 2) Increase the s i z e of the diagonal cross-braces and return to step 3. 7.9 Design Example To i l l u s t r a t e the use of the design procedure proposed i n section 7.8, a t y p i c a l structure i s designed with f r i c t i o n devices and i t s response i s compared with a structure designed by the conventional 201 requirements of the National Building Code of Canada (1985). The layout of the structure i s shown i n F i g . 7.35; i t represents a one storey f i r e s t a t i o n located on firm rock i n Vancouver. The structure i s composed of two p a r a l l e l plane frames consisting of three bays. It i s assumed that the bracing system i s incorporated i n each external bay. 3 @ 7. 6m Figure 7.35 Layout for Design Examples It i s also assumed that the f i r e h a l l o f f i c e space i s contained on a mezzanine f l o o r , which transmits i t s load to the roof l e v e l . 7.9.1 Conventional Design F i r s t the structure i s designed according to the requirements of the National Building Code of Canada (NBCC, 1985). A dead load D of 4.8 kN/m2 i s assumed to be uniformly d i s t r i b u t e d on the roof. S i m i l a r l y , i t i s assumed that the uniform l i v e load L due to occupancy i s 2.4 kN/m2; the l i v e load due to snow i s calculated based on the code equation: 202 S = S n C, C C C (7.20) 0 b w s a where S = Snow pressure on roof i n kPa S 0 = Snow pressure on the ground = 1.9 kPa for Vancouver = Snow load factor = 0.8 C w = Wind exposure factor = 1.0 C g = Slope factor = 1.0 for f l a t roof C & = Accumulation factor = 1.0 for uniformly d i s t r i b u t e d load. The e f f e c t of a l i v e load Q due to wind i s considered by calculat-ing the external wind pressure from the code formula: P = q C C C (7.21) e g p where P = Wind pressure acting perpendicular to the surface of the bu i l d i n g i n kPa. q = Reference dynamic wind pressure having a p r o b a b i l i t y of exceedence of 0.01/year = 0.67 kPa for Vancouver C g = Exposure factor = 0.9 since the t o t a l height of the bu i l d i n g i s l e s s than 6 m. C = Gust factor = 2.0 for the design of the main elements g 5 0^ = P r e s s u r e f a c t o r , which depends on the geometry of the bu i l d i n g 203 The l a t e r a l l i v e load Q due to earthquake i s calculated by the code formula: V = v S K I F W (7.22) where V = Seismic base shear v = Zonal h o r i z o n t a l ground v e l o c i t y r a t i o = 0.20 for Vancouver S = Seismic response c o e f f i c i e n t = 0.44 K = Building c o e f f i c i e n t = 0.70 for unbraced moment r e s i s t i n g frame design = 0.80 for the braced moment r e s i s t i n g frame design I = Importance factor = 1.3 for a c i v i l p r o tection b u i l d i n g F = Foundation factor = 1.0 for s o l i d rock W = Weight of the b u i l d i n g = dead load + 25% of the snow load The l i m i t state design c r i t e r i a i s used for the design of the structure: a QD + r^[a LL + a QQ] £ <J>R (7.23) where 1.25 a L =1.50 a Q = 1.50 r = 1.0 ,1.00 i f only one of L and Q are considered 0.70 i f both L and Q are present cj> = Resistance index R = Limit state resistance 204 Two d i f f e r e n t conventional l a t e r a l load r e s i s t i n g systems are considered: a) Unbraced moment r e s i s t i n g frame b) Braced moment r e s i s t i n g frame. Figure 7.36 shows the member sizes r e s u l t i n g from the design of these conventional b u i l d i n g systems. The bracing system for the braced moment r e s i s t i n g frame has been designed to carry 100% of the l a t e r a l load, independently of the moment r e s i s t i n g frame. 7.9.2 Design with F r i c t i o n Damping System The design procedure proposed i n section 7.8 i s used to design the structure with f r i c t i o n devices. Step 1 The unbraced moment r e s i s t i n g frame i s f i r s t designed to carry the usual load combination without considering seismic loads. The members chosen are i d e n t i c a l to the main members of the conventional braced moment r e s i s t i n g frame shown i n F i g . 7.36. Step 2 The fundamental period of the unbraced structure, T , i s calculated to be equal to 1.0891 second. The diagonal braces chosen are the same as those used i n the conventional braced moment r e s i s t i n g frame; the r a t i o T^/T u i s c a l c u l a t e d to be equal to 0.34, which i s less than 0.40 as suggested. 205 a) Unbraced Moment Resisting Frame W250X115 W250X115 W250X115 co x o tf) CM yyyyyyyyysyysyyyyyy Oi co X o LO OJ yyyyyyyysyyyyyyyy Oi co X o If) CM yyyyyyyysyy^yyyyyy CO X o If) CM yyyyyyyy^yyyyyy b) Braced Moment Resisting Frame W250X131 W250X115 W250X131 Diagonal Braces:2L25X25X3 Figure 7.36 Conventional Design Examples 206 Assume that the peak ground acceleration and the dominant ground period of the design earthquake expected at the construction s i t e are the same as the corresponding parameters associated with the Eureka earthquake, December 21, 1954, COMP N46W. For t h i s seismic event: a = 0.201 g (which i s equal to the NBCC (1985) ^ value assigned to Vancouver) T = 0,6912 sec g Step 3 V e r i f y that the nondimensional r a t i o s f a l l within the appropriate l i m i t s : 0.20 <; T,/T = 0.34 £ 0.80 b u 0.05 <; T /T = 0.64 <; 20 g u 0.005 <; a /g = 0.201 <; 0.40 Since the preceding constraints are s a t i s f i e d , the optimum l o c a l s l i p load can be estimated by Equation (7.14): T T, T, T a V = rK — + K — + K + K 1 W u u u u 5 which y i e l d s : P„ = 32.5 kN for each f r i c t i o n device. 207 Step 4 The c r i t i c a l buckling load of each diagonal cross-brace i s : T T 2 E T P = I L - ^ = 1.976 kN The a x i a l load induced i n the tension cross-braces by wind e f f e c t i s P = 10.94 kN w From Equation (7.16) P 0 = 32.5 kN P +P v „ C r = 6.46 kN P +P _w_cr r o * 2 Hence the f r i c t i o n devices w i l l not s l i p under wind e f f e c t . Step 5 Assuming a y i e l d s t r e s s i n t e n s i o n o^ equal to 300 MPa for the diagonal cross-braces, we can v e r i f y Equation (7.18): P 0 = 32.5 kN A, o + P -S- 2^ — = 43.29 kN 208 A, a + P ro s 2 Hence the f r i c t i o n devices w i l l s l i p before y i e l d i n g of the cross-braces occurs. 7.9.3 Comparison of Performance The computer program DRAIN-2D was used to cal c u l a t e the seismic response of the d i f f e r e n t s t r u c t u r a l configurations due to the 1954 Eureka earthquake (see F i g . 7.30), i n order to assess the adequacy of the design procedure. The damage induced i n the various members of the structures i s shown i n F i g . 7.37. Substantial y i e l d i n g occurs i n the members of the unbraced moment r e s i s t i n g frame and the braced moment r e s i s t i n g frame; a l l the members of the f r i c t i o n damped structure remain e l a s t i c . The time-history of the l a t e r a l d e f l e c t i o n of each b u i l d i n g system i s presented i n F i g . 7.38. Again, the superior performance of the structure equipped with f r i c t i o n devices i s evident. It may be noted that the periods of the unbraced and braced moment r e s i s t i n g frames are e s s e n t i a l l y equal a f t e r the i n i t i a l 6 seconds of v i b r a t i o n . This i s due to the fac t that the braces i n the l a t t e r system underwent severe y i e l d i n g a f t e r t h i s i n i t i a l time i n t e r v a l and the structure behaved as an unbraced frame. Figure 7.39 compares the optimum s l i p load obtained from the design s l i p load spectrum with the value obtained from FDBFAP. It can be seen that the design equation i s adequate for p r e d i c t i n g the optimum s l i p load of the structure. 209 a) Unbraced Moment Resisting Frame b) Braced Moment Resisting Frame c) F r i c t i o n Damped Braced Frame yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyy^yyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyy M e m b e r Y i e l d e d Figure 7.37 S t r u c t u r a l Damage for Design Examples, Eureka Earthquake 210 Unbraced Moment Resisting Frame E. C i [mm] 80--40-c o • -0--40-Q) -80-80-40-C .g 0-"o -40--80-25 10 15 20 Time [sec] Braced Moment Resisting Frame 15 20 Time [sec] Friction Damped Braced Frame 10 15 20 25 Time [sec] Figure 7.38 L a t e r a l Deflections, Eureka Earthquake J O Eureka Earthquake,December 21,1954 Figure 7.39 Optimum S l i p Load Study for Design Example 212 8. OPTIMIZATION OF DISTRIBUTION OF FRICTION DEVICES IN MULTI-STOREY STRUCTURES "Logical consequences are the scarecrows of fools and the beacons of wise men." - Thomas H. Huxley (1825-1895), English S c i e n t i s t 213 CHAPTER 8 OPTIMIZATION OF DISTRIBUTION OF FRICTION DEVICES  IN MULTI-STOREY STRUCTURES 8.1 General The new damping system i s added to a structure to reduce i t s dyna-mic response through energy d i s s i p a t i o n by f r i c t i o n . In a multi-storey b u i l d i n g , the s l i p load of each f r i c t i o n device can be adjusted and the energy d i s s i p a t i o n i s therefore c o n t r o l l a b l e . When incorporating f r i c -t i o n devices i n a multi-storey structure, the following question needs to be addressed. What i s the most e f f e c t i v e s l i p load d i s t r i b u t i o n throughout the b u i l d i n g : should the d i s t r i b u t i o n be uniform, with equal t o t a l s l i p load at each f l o o r , or i s there an optimum d i s t r i b u t i o n that w i l l minimize the response when the sum of s l i p loads i s f i x e d (a cost function)? This question w i l l be examined i n t h i s chapter by means of a n a l y t i c a l and numerical studies. An optimum s l i p load d i s t r i b u t i o n w i l l be deduced from the r e s u l t s of these investigations and w i l l be used i n a parametric study of general multi-storey damped structures. 8.2 The Shear Beam Analogy In many cases i t i s quite reasonable to model the dynamic behaviour of a multi-storey b u i l d i n g as an analogous c a n t i l e v e r shear beam. This analogy has been widely used i n the past and u s u a l l y leads to an accept-able degree of accuracy. In t h i s section the shear beam analogy i s employed to express the optimization of the d i s t r i b u t i o n of s l i p load a n a l y t i c a l l y , so that a closed form s o l u t i o n can be obtained. 214 8.2.1 D e f i n i t i o n of a Shear Building In many bui l d i n g s , the r i g i d i t i e s of the beams are s u f f i c i e n t l y large compared to the r i g i d i t i e s of the columns that i t can be assumed the l a t e r a l d e f l e c t i o n of the b u i l d i n g r e s u l t s from column flexure only and no r o t a t i o n takes place at the j o i n t s . This type of structure i s r e f e r r e d to as a "shear b u i l d i n g " and i t s deflected shape w i l l e x h i b it many of the features of a c a n t i l e v e r beam that i s deflected by shear forces only. To r e a l i z e such d e f l e c t i o n s , a b u i l d i n g must s a t i s f y the following requirements: • The t o t a l mass of the structure must be concentrated at the f l o o r • The beams must be i n f i n i t e l y r i g i d compared to the columns. • The deformation of the structure must be independent of the a x i a l forces present i n the columns (ignore a x i a l shortening). 8.2.2 Governing Equations for a F r i c t i o n Damped Shear Beam In t h i s section, a multi-storey f r i c t i o n damped structure i s modelled as an analogous " f r i c t i o n damped" shear beam. A shear beam i s a distributed-parameter system having an i n f i n i t e number of degrees-of-freedom. The equation of motion of such a system i s a p a r t i a l d i f f e r e n -t i a l equation. The d e f l e c t i o n of a shear beam y(x,t) i s due to the e f f e c t of shear forces only. The dynamic equilibrium of an i n f i n i t e s i -mal section of a shear beam whose supports are subjected to ground motion i s shown i n F i g . 8.1. For h o r i z o n t a l dynamic equilibrium: l e v e l s . av(x.t) 3x m(x) a*y(x,t) at 2 + m(x) y (t) (8.1) 2 V(x,t) + «5 [V(x,t) ] dx dx (5 [yg(t)+y(x,t)] V(x,t) m(x) dx fit y(x,t) Figure 8.1. Dynamic Equilibrium of a Shear Beam. where V(x,t) = t o t a l shear force at l o c a t i o n x and time t m(x) = mass per unit length of the beam y(x,t) = displacement of the beam r e l a t i v e to i t s moving base y (t) = ground ac c e l e r a t i o n g The t o t a l shear force can be written as the sum of two components V(x,t) = K(x) 3 y ^ > t } + V D(x,t) = V m(x,t) + V D(x,t) (8.2) where K(x) = d i s t r i b u t e d shear s t i f f n e s s of the beam Vp(x,t) = shear force c a r r i e d by the f r i c t i o n devices V (x,t) = shear force c a r r i e d by the continuum m 216 The d i s t r i b u t e d s t i f f n e s s i n shear K(x) of the analogous shear beam can e a s i l y be r e l a t e d to the properties of the shear b u i l d i n g which i t i s modelling. F i g . 8.2 shows the deformation of the columns i n a t y p i c a l storey of a shear b u i l d i n g . From elementary beam theory the shear f o r c e V^(x,t) developed i n the columns of the i t h storey i s given by: 12E A.(t) N c i V c(x.,t) = L ? 1 1 I ± j l j =l J (8.3) where E L. I. . i j N . ci A.(t) = Young's modulus = height of i t h storey = moment of i n e r t i a of the j t h column i n the i t h storey = number of columns i n the i t h storey = d r i f t of the i t h storey. The s t o r e y d r i f t A^(t) can be r e l a t e d to the slope of the e l a s t i c curve of the equivalent shear beam continuum by: L: x.A i y(x,.t) Figure 8.2. T y p i c a l Storey Displacement of Shear Building. 217 A i ( t ) wx±>v 1 l Substituting Equation (8.4) into Equation (8.3) y i e l d s : 12E 3y(x.,t) i j=i j i In general, the shear force c a r r i e d by the continuum i n the analogous shear beam at a distance x from i t s base i s : where vjx.t) - K(X) Srijbt). m (8.6) N 12E L? j-1 c l 1 lu for 0 <. x < h. K(x) 12E N c 2 2 j=l J for h : £ x < h 2 12E Ncm 2 1 . L J . , mi m j = l J for h , s x < h m-1 m Substituting Equation (8.2) into Equation (8.1) leads to the governing d i f f e r e n t i a l equation of motion for an analogous f r i c t i o n damped shear beam: 218 , , 3'y(x,t) 3 r v f . 3y(x.t), 8 V D ( X , t ) . , " ... m(x) lt2> - ^  [K(x) y 8 x' ] = -m(x) y g ( t ) (8.7) The general s o l u t i o n of Equation (8.7) can be expressed as the product of a function of x alone and a function of t alone: y(x,t) = <{>(x) q(t) (8.8) Substituting into Equation (8.7): 3V (x,t) m(x)cMx)q(t) " (K(x)4>' (x)) 'q(t) ^ r - = -m(x)y (t) (8.9) ox g Mult i p l y i n g Equation (8.9) by 3y/3t and double integ r a t i n g over length and time y i e l d s : £ t . 1 t J m(x)<f>2(x)dx J q ( t ) q ( t ) d t - J (K(x) <f> • (x)) ' 4> (x) dx J q ( t ) q ( t ) d t o o o o 4 t 3V (x,t) . 2 t .. . - J" J ^ cj>(x)q(t)dtdx = -J m(x)cj>(x)dx J" y ( t ) q ( t ) d t o o o o 6 (8.10) where fl i s the length of the beam. Assuming zero i n i t i a l conditions q(o) = q(o) = 0, we can write: t . . . q ( t ) . . . J q ( t ) q ( t ) d t = J q ( t ) d q ( t ) = j q * ( t ) (8.11) o o 219 t . q(t) J q(t)q(t)dt = J q(t)dq(t) = \ q 2 (t) o o (8.12) Some of the terras i n Equation (8.10) can be integrated by parts. When the boundary conditions for a ca n t i l e v e r shear beam with o r i g i n at i t s base are applied, these terms can be written as: -J" ( K W (x)) •*(x)dx = J K(x) (<*>' (x))2dx (8.13) o o s av (x.t) J -J" *(x)dx = J VD(x,t)*'(x)dx o 0 (8.14) Substituting Equations (8.11) to (8.14) into Equation (8.10) y i e l d s the energy balance equation for a f r i c t i o n damped c a n t i l e v e r shear beam: 4 J m(x)<f>2(x)dx qa(t) + \ J K(x)(f (x))'dx q'(t) x t . a t .. + J J VD(x,t)<J>' (x)q(t)dt dx = -J m(x)<j>(x)dx / y ( t ) q ( t ) d t o o o o (8.15) The various terms i n Equation (8.15) can e a s i l y be recognized as: -z J m(x)<f>2 (x)dx q 2 ( t ) = K i n e t i c energy 220 2 J K(x) (<j>' ( x ) ) 2 dx q 2 ( t ) = Recoverable s t r a i n energy o 8 t J J" V D(x,t) <f>' (x) q(t)dt dx = Energy dis s i p a t e d by f r i c t i o n o o £ t .. - J m(x)<f>(x)dx / y (t) q(t)dt = Energy input due to base motion o o 8 8.2.3 Optimum S l i p Load D i s t r i b u t i o n for Resonant S i t u a t i o n We w i l l f i r s t consider the optimum s l i p load d i s t r i b u t i o n for response i n a sin g l e dominant mode, i n order to provide some in s i g h t into the more general and complex problem of a general earthquake input, where many modes may contribute to the response. Assume that the analogous f r i c t i o n damped shear beam i s subjected to a sinusoidal ground motion described by: y (t) = a s i n u t (8.16) g g u where ui^ i s the undamped fundamental frequency of the unbraced struc-t u r e . Assume also that the peak ground ac c e l e r a t i o n a g i s large enough to cause a l l the f r i c t i o n devices to s l i p during every h a l f cycle of v i b r a t i o n . The e x c i t i n g frequency i s taken as rather than w ,^ since at resonance, when the devices s l i p , the system w i l l behave as an unbraced frame. Under such conditions the beam w i l l respond mainly i n i t s f u n d a m e n t a l mode cj> (x) . Hence the s t e a d y - s t a t e response can be approximated as: 221 y(x,t) = <f>(x)q(t) = <f> (x) sin(uM: + 6) (8.17) where 0 i s a phase angle. Also, the shear force c a r r i e d by the f r i c t i o n devices V^(x,t) can be approximated by: V D(x,t) = V s(x)sgn ( 8 y ^ ' t } ) = V g(x) sgn(cos(u ut+e)) (8.18) where V g ( x ) i s the shear f o r c e d i s t r i b u t i o n c a u s i n g slippage of a l l f r i c t i o n devices; V g(x) i s always p o s i t i v e . V ( x ) > 0 0 <; x £ x (8.19) s Substituting Equations (8.17) and (8.18) into Equation (8.15) leads to the energy balance equation for the resonant s i t u a t i o n : ui2 S. i J m(x)<|)2(x)dx cos 2(ui t+8) + \ J K(x) ( 6 ' (x)) 2dx sin 2(ui t+0) b XI XI \X XI O O £ t + ui f V (x)4' (x)dx f cos(u) t+6)sgn(cosw t+0)dt u J s r u J u 6 u o o S t = -a w f m(x)* (x) dx f sinu t cos(ui t+8)dt (8.20) P U J T u U U & o o The energy dis s i p a t e d by f r i c t i o n EF(t) i n Equation (8.20) can be written as: t EF(t) = Iui J cos(u t+0) sgn(cos(ui t+0))dt (8.21) o 222 where S I = J V g(x )4^(x)dx o The constants and 9 are independent of the s l i p load so that to maximize EF(t) i t i s only necessary to maximize the i n t e g r a l I. Such a problem i s c a l l e d functional optimization; some function V g(x) i s to be established to cause the i n t e g r a l I to reach an extreme (maximum or minimum) value. This i s a c l a s s i c a l calculus of v a r i a t i o n problem. ' To bound the problem some contraints should be applied i n terms of cost functions. However, because cost data on the f r i c t i o n damping system are not cur r e n t l y a v a i l a b l e , the t o t a l amount of s l i p shear V 0 i n the system can be considered as a constraint. The t o t a l s l i p shear V 0 i s expressed as an i n t e g r a l constraint. The so l u t i o n can be obtained by using Lagrange m u l t i p l i e r p r i n c i p l e s . The problem reduces to the determination of a function V g(x) which maximizes the i n t e g r a l I: i I = j V s(x) <f>^ (x) dx (8.22) o subject to the constraint: J V s(x)dx = V 0 (8.23) o 2 Since V (x) i s a p o s i t i v e function i t can be replaced by C (x): 2 V (x) = C (x) (8.24) 223 Substituting Equation (8.24) into Equations (8.22) and (8.23) * 2 I = J C (x)<j>' (x)dx (8.25) o with a 2 V 0 = J" C (x)dx (8.26) o o From the calculus of v a r i a t i o n s , the function C(x) must be a s o l u t i o n of the Euler-Lagrange Equation (Weinstock, 1974): where F(x) = C 2(x)(|. u(x) + XC 2(x) X = Lagrange m u l t i p l i e r Equation (8.27) becomes: C(x)(X + <f>u(x)) = 0 (8.28) M u l t i p l y i n g Equation (8.28) by C(x) y i e l d s : V g(x) (X + <fS^(x)) = 0 (8.29) Equation (8.29) i s an eigen problem where X i s the eigenvalue and V g ( x ) i s the e i g e n f u n c t i o n . Since X i s constant, and the i n t e g r a l of V (x) i s constant and has d e f i n i t e non-negative and non-zero value, i t s 224 can be concluded t h a t V (x) has the form of a Dirac delta function at s some point x along the length of the structure such that: x = -<j>;(x0) 0 <; x 0 s a (8.30) V g(x) = V 06(x-x 0) 0 <; x 0 <. I (8.31) Substituting Equation (8.31) into Equation (8.22) y i e l d s : I = V 0 J o(x-x 0)*V(x)dx or i = v 0 *;<x0> (8.32) To make I a maximum, 4^(x 0) should be a maximum. Therefore, x 0 i s the l o c a t i o n where the slope (or derivative) of the undamped fundamental mode shape i s a maximum. Therefore, to achieve maximum energy d i s s i p a -t i o n by f r i c t i o n , the t o t a l s l i p load V 0 must be lumped at the l o c a t i o n where the slope of t h i s mode shape i s a maximum. For a uniform shear beam (m(x) = m, K(x) = K), the undamped fundamental mode shape i s given by (Paz, 1985): 4 (x) = sin(g) 0 <; x <. 1 (8.33) then 4 u(x) = |j cos(g) 0 <; x <; I (8.34) 225 and - *i(x 0) = TT for x 0 = 0 maximum Hence, i n a f r i c t i o n damped shear beam v i b r a t i n g i n i t s fundamental unbraced mode, the energy dis s i p a t e d by f r i c t i o n w i l l be a maximum when a l l the supplemental f r i c t i o n damping i s i n s t a l l e d at the base of the beam (x=0). By analogy, t h i s r e s u l t indicates that for a r e a l multi-storey structure v i b r a t i n g i n i t s unbraced fundamental mode shape, the f r i c t i o n devices should be i n s t a l l e d at the lo c a t i o n where the slope of t h i s mode shape i s a maximum. For a uniform multi-storey shear b u i l d i n g t h i s optimum l o c a t i o n i s the f i r s t f l o o r . Although such a d i s t r i b u t i o n at f i r s t may seem to v i o l a t e engineering i n t u i t i o n , the r e s u l t given by Equation (8.32) can e a s i l y be v i s u a l i z e d p h y s i c a l l y as follows. A f r i c t i o n device i s connected between adjacent f l o o r s i n such a way that i t w i l l function only when the f l o o r s experience a f i n i t e r e l a t i v e displacement. Thus, the energy diss i p a t e d by f r i c t i o n i s proportional to the i n t e r - s t o r e y displacement Ay(x,t) (or d r i f t ) . For a uniform shear b u i l d i n g c o n s i s t i n g of many i d e n t i c a l storeys, the i n t e r -storey d r i f t Ay(x,t) can be approximated by: where h i s the height of each storey. Substituting Equation (8.17) into Equation (8.35) y i e l d s : Ay(x,t) = h (8.35) (8.36) 226 Hence the energy dissipated by f r i c t i o n i s proportional to the slope of the mode shape <j>^(x). 8.2.A Numerical V e r i f i c a t i o n Figure 8.3 shows a ten-storey hypothetical uniform shear b u i l d i n g . The s t i f f n e s s of each f l o o r i s constant and equal mass i s assigned at each f l o o r l e v e l . The fundamental undamped (unbraced) mode shape of the structure i s presented i n F i g . 8.A. The de r i v a t i v e of the f i r s t mode d>',.. has been obtained by numerical d i f f e r e n t i a t i o n : u ( i ) J r u ( i ) *u(i) " * u ( i - l ) (8.37) where i s the i t h storey height and B i s an a r b i t r a r y s c a l i n g constant taken equal to 0.025 i n F i g . 8.A. The structure was subjected to a sin u s o i d a l ground ac c e l e r a t i o n described by: y (t) = 0.1 g s i n u t (8.38) g u where the peak amplitude of the ground motion i s 10% of the acceleration of g r a v i t y and the e x c i t a t i o n frequency equals the natural frequency of the undamped (unbraced) fundamental mode of the structure. The response was c a l c u l a t e d using the program FDBFAP; two d i f f e r e n t s l i p load d i s t r i b u t i o n s were considered: 1) Uniform d i s t r i b u t i o n with a l o c a l s l i p load of 90 kN at each f l o o r . 2) A l o c a l s l i p load of 900 kN lumped at the base f l o o r . 227 Beams are I n f i n i t e l y R i g i d E = 200000 MPa Floor Level Column I n e r t i a (106mm*) Brace Area (mm2) Floor Weight (kN) 10 1030 6452 400 9 1030 6452 800 8 1030 6452 800 7 1030 6452 800 6 1030 6452 800 5 1030 6452 800 4 1030 6452 800 3 1030 6452 800 2 1030 6452 800 1 1030 6452 800 6.1 m Figure 8.3. Hypothetical Uniform Shear Building. 228 Fundamental Unbraced Natural Frequency ui =5.61 rad/s u 1 / I / / 1 / / 1 1 / / Floor Level T u T u 1.0000 10 0.00009 0.9877 9 0.00025 0.9511 8 0.00042 0.8910 7 0.00057 0.8090 6 0.00071 0.7071 5 0.00083 0.5878 4 0.00093 0.4540 3 0.00101 0.3090 2 0.00106 0.1564 1 0.00109 0 Figure 8.4. Fundamental Shear Build; Unbraced Mode ing. Shape of Hypothetical Uniform 229 The uniform s l i p load d i s t r i b u t i o n of 90 kN at each f l o o r was chosen so as to ensure that a l l the f r i c t i o n devices were s l i p p i n g i n each h a l f cycle of v i b r a t i o n when the b u i l d i n g was excited by the ground motion described i n Equation (8.38). The maximum f l o o r d e f l e c t i o n s given i n F i g . 8.5 show that the constant s l i p load d i s t r i b u t i o n i s less —I O O 10- 1 1 Q - Uniform Shear Building / 1 3 Harmonic Base Excitation 8- / / 7-6- / / 5-* 4-3- Local Slip Load Distribution: 2- $0 khJ Constant oi Each Floor 1-0-900 kN Lumptd of Base Door 200 400 600 Maximum Floor Deflection [mm] 800 Figure 8.5 Maximum Floor Deflections for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n . e f f i c i e n t i n reducing the response than the lumped d i s t r i b u t i o n ; the maximum d e f l e c t i o n r e s u l t i n g from a constant d i s t r i b u t i o n s i t u a t i o n i s reduced by about 10% when the s l i p load i s lumped at the base f l o o r . The energy time-histories for t h i s system are shown i n F i g . 8.6. It can be seen that lumping the t o t a l s l i p load at the base f l o o r r e s u l t s i n a reduction i n the maximum amplitude of the s t r a i n energy 230 Constant Slip Load Distribution 6,000 - i 1 1 Time [sec] Slip Load Lumped at Base 6,000 - i Time [sec] Figure 8.6. Energy Time-Histories for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n . 231 r e l a t i v e to the corresponding value developed for the uniform d i s t r i b u -t i o n case. A f t e r 10 seconds of v i b r a t i o n the base concentrated f r i c t i o n devices absorbed 48% of the energy input; for the s i t u a t i o n i n which the devices are uniformly d i s t r i b u t e d along the b u i l d i n g height, t h i s figure i s 37%. The Relative Performance Indices (RPI) are presented i n Table 8.1. This index also i l l u s t r a t e s that the lumped d i s t r i b u t i o n i s more e f f i -cient than the uniform d i s t r i b u t i o n . Table 8.1 RPI for Uniform F r i c t i o n Damped Shear Building Subjected to Sinusoidal Base E x c i t a t i o n D i s t r i b u t i o n RPI 90 kN constant at each f l o o r 0.4034 900 kN lumped at base f l o o r 0.3552 These numerical r e s u l t s v e r i f y the a n a l y t i c a l p r e d i c t i o n represen-ted by Equation (8.32): to achieve maximum energy d i s s i p a t i o n for a shear beam bu i l d i n g i n a resonant s i t u a t i o n , the t o t a l s l i p load must be lumped at the l o c a t i o n where the slope of the undamped fundamental mode shape i s a maximum. 8.3 The "Weak-Girder-Strong Column" Building In the design of an earthquake-resistant structure, a reasonable balance must be maintained between the strength of the columns and gir d e r s ; a weak-girder-strong column philosophy i s recommended. The deformations of buildings designed i n accordance with t h i s philosophy may be due mainly to flexure, i n which case the shear beam analogy becomes u n r e a l i s t i c . These buildings can be modelled as f l e x u r a l beams i n which the shear deformation i s neglected. In t h i s section an optimum 232 s l i p load d i s t r i b u t i o n i s determined for a weak-girder-strong column b u i l d i n g at resonance, based on the r e s u l t given by Equation (8.32). This optimum d i s t r i b u t i o n i s v e r i f i e d numerically with the program FDBFAP. 8.3.1 D e f i n i t i o n of a "Weak-Girder-Strong Column" Building The f l e x u r a l deformation of a multi-storey weak-girder-strong column b u i l d i n g for which the a x i a l deformations of the columns are not too important (low-rise to medium-rise structures which are not too slender) i s due mainly to the r o t a t i o n of the beam-column j o i n t s . To r e a l i z e such d e f l e c t i o n s , the r i g i d i t i e s of the beams must be very small compared to the r i g i d i t i e s of the columns. This s i t u a t i o n would represent a p a r t i c u l a r case of a f l e x u r a l b u i l d i n g . 8.3.2 Optimum S l i p Load D i s t r i b u t i o n for Resonant S i t u a t i o n Although Equation (8.32) has been derived for an analogous shear beam, p h y s i c a l l y i t s t i l l i s reasonable to apply t h i s r e s u l t to a weak-girder-strong column structure v i b r a t i n g i n a p a r t i c u l a r mode. For the case of a uniform f l e x u r a l beam, the undamped fundamental mode i s given by (Paz, 1985): . / v • /1.875x^ . , ,li875x, <J>u(x) = sin(— j j ) - smh(—j- ) - 1.362 ( c o s ( 1 - ^ 7 5 x ) - c o s h ( 1 , 8 7 5 x ) ) (8.39) Then: , i / v 1.875 . ,1.875xx ,1.875x, 4^(x) = — — [ c o s ( — ) - cosh(—£ ) + 1 . 3 6 2 ( s i n ( 1 ^ 7 5 x ) + s i n h ( 1 , 8 7 5 x ) ) ] (8.40) 233 The v a r i a t i o n of the slope of the unbraced fundamental mode shape Figure 8.7. Slope of the Fundamental Mode Shape for a Flexural Beam. From Equation (8.32), the t o t a l s l i p load must be lumped at the top f l o o r of the f l e x u r a l b u i l d i n g i f the energy dissipated by f r i c t i o n i s to be maximized when the structure vibrates i n i t s fundamental undamped mode. It should be noted that Equation (8.32) should not be applied to a slender f l e x u r a l b u i l d i n g i n which a x i a l deformations of the columns are important. For t h i s case the t o t a l s l i p load should be placed at the lo c a t i o n where the maximum tangential i n t e r s t o r e y d r i f t (the maximum deviation between the tangents of adjacent floors) occurs. 8.3,3 Numerical V e r i f i c a t i o n The structure shown i n F i g . 8.3 was modified to represent a uniform f l e x u r a l structure i n which the a x i a l deformation of the columns i s 234 neglected. The s t i f f n e s s of the columns was m u l t i p l i e d by a factor of 100 while the s t i f f n e s s of the beams was reduced to zero. The fundamen-t a l unbraced mode shape of the structure i s presented i n F i g . 8.8. Although t h i s leads to a structure whose period i s lower than that norm-a l l y associated with a ten storey b u i l d i n g , t h i s does not influence the comparative r e s u l t s sought. The structure was excited by the ground motion described i n Equation (8.38) i n order to produce a resonance s i t u a t i o n i n the fundamental undamped (unbraced) mode. The response was calc u l a t e d by FDBFAP considering three d i f f e r e n t s l i p load d i s t r i b u -t i o n s : 1) A uniform d i s t r i b u t i o n with a l o c a l s l i p load of 90 kN at each f l o o r . 2) A l o c a l s l i p load of 900 kN lumped at the top f l o o r . 3) A l o c a l s l i p load of 900 kN lumped at the base f l o o r . The maximum f l o o r d e f l e c t i o n s presented i n F i g . 8.9 show that a s l i p load of 900 kN lumped at the top f l o o r i s the most e f f i c i e n t d i s -t r i b u t i o n for reducing the response. Concentrating the devices i n the top storey r e s u l t s i n a d e f l e c t i o n at that l e v e l which i s 28% smaller than the corresponding d e f l e c t i o n i n the structure with uniform s l i p load d i s t r i b u t i o n . The energy time-histories are shown i n F i g . 8.10. The minimum s t r a i n energy amplitudes are obtained when a l l the s l i p load i s lumped at the top of the b u i l d i n g . Note that lumping the s l i p load at the base f l o o r i s i n e f f i c i e n t for t h i s type of b u i l d i n g . A f t e r 10 seconds of v i b r a t i o n , 73% of the input energy was di s s i p a t e d by the top f l o o r concentrated f r i c t i o n devices; for the uniform d i s t r i b u t i o n case, the comparative figure was 62%. Table 8.2 presents the Relative Performance Indices (RPI) obtained for these analyses. 235 Fundamental Unbraced Natural Frequency ID =3.62 rad/s Floor Level T u T u 1.0000 10 0.00096 0.8621 9 0.00095 0.7249 8 0.00093 0.5903 7 0.00090 0.4605 6 0.00084 0.3389 5 0.00076 0.2294 4 0.00065 0.1362 3 0.00050 0.0638 2 0.00033 0.0168 1 0.00012 0 Figure 8.8. Fundamental Unbraced Mode Flexural B u i l d i n g . Shape of Hypothetical Uniform 236 Maximum Floor Deflection [mm] Figure 8.9. Maximum Floor D e f l e c t i o n for Uniform F r i c t i o n Damped Flex u r a l Building Subjected to Sinusoidal Base E x c i t a t i o n . Table 8.2 RPI for Uniform F r i c t i o n Damped Flexural Building Subjected  to Sinusoidal Base E x c i t a t i o n D i s t r i b u t i o n RPI 90 kN constant at each f l o o r 0.1538 900 kN lumped at top f l o o r 0.1031 900 kN lumped at base f l o o r 0.9342 It can be seen that the smallest RPI i s obtained when a l l the s l i p load i s lumped at the top f l o o r of the b u i l d i n g . 237 E i CD 0 6,000 I CD 5,000-4,000-3,000-2,000-1,000-0-I CD C U l 6,000 5,000-4,000-3,000-2,000-1,000-0-Slip Load Lumped at Base 6,000-5,000- Energies: 4,000-Kinetic Streln 3,000- Dlstipgted by Friction 2,000-Inpul 1,000-T 2 4 6 8 Time [sec] Constant Slip Load Distribution w Energies: Kinetic Sfroln OlttlpoHd by Friction  Input 8 4 6 Time [sec] Slip Load Lumped at Top 10 Energies: Klnellc Strain _ Ollllpoltd by Frlcllon  Input 4 6 8 Time [sec] 10 Figure 8.10. Energy Time-Histories for Uniform F r i c t i o n Damped Flexural Building Subjected to Sinusoidal Base E x c i t a t i o n . 238 The numerical r e s u l t s tend to confirm the adequacy of Equation (8.32) for p r e d i c t i n g the optimum s l i p load d i s t r i b u t i o n for a general structure v i b r a t i n g i n a p a r t i c u l a r mode as long as the a x i a l deforma-tions of i t s columns can be ignored. Any c l a s s i c a l method can be used to determine the mode shape a n a l y t i c a l l y (e.g. Rayleigh-Ritz, matrix i t e r a t i o n ) . Then the deri v a t i v e of the mode shape can be obtained by numerical d i f f e r e n t i a t i o n (Equation (8.37)), 8.A Optimum S l i p Load D i s t r i b u t i o n for General Structure Under  Earthquake Load When a r e a l multi-storey f r i c t i o n damped b u i l d i n g i s excited by an actual earthquake more than one mode w i l l p a r t i c i p a t e i n the response, since an accelerogram normally has a wide frequency spectrum. Neverthe-l e s s , the optimum s l i p load d i s t r i b u t i o n given by Equation (8.32) i s s t i l l r e a s o n a b l e p r o v i d e d <f>u(x0) i s r e p l a c e d by the ins t a n t a n e o u s maximum slope of the deflected shape of the bu i l d i n g 8 y ( x 0 , t ) / 3 t . In p r i n c i p l e , at any instant of time the t o t a l s l i p load should be lumped at the l o c a t i o n where the in t e r s t o r e y d r i f t i s a maximum. This optimum s l i p load d i s t r i b u t i o n would be very d i f f i c u l t to achieve i n pr a c t i c e since an active control system would be necessary to continuously adapt the s l i p load of each f r i c t i o n device i n response to the monitored deflected shape of the b u i l d i n g . From a p r a c t i c a l point of view, an optimum s l i p load d i s t r i b u t i o n which i s independent of time must be used. 8.4.1 Optimum S l i p Load D i s t r i b u t i o n To account for a l l contributing modes, the t h e o r e t i c a l r e s u l t (Equation 8.32) obtained for any sing l e mode should be corrected. This co r r e c t i o n i s r e f l e c t e d i n the following expression proposed (on the 239 basis of i n t u i t i v e reasoning) for the optimum s l i p shear d i s t r i b u t i o n V (x) s V (x) = K(<f>' ( x ) ) r (8.41) where y and K are constants and <J>(x) i s the mode shape dominating the response of the structure. It i s obvious that i f y=0 the s l i p load d i s t r i b u t i o n i s uniform. In the l i m i t , when y approaches i n f i n i t y , the t o t a l s l i p load i s lumped at the l o c a t i o n where the slope of the assumed mode shape i s a maximum. For a di s c r e t e multi-storey f r i c t i o n damped structure we can rewrite Equation (8.41) as: s ( i ) N D i = X 2P„. . cosa. . = NS (8.42) where s ( i ) (i) N Di NS = optimum s l i p shear i n i t h storey N NS Di X X 2P. cosa. . = a s p e c i f i e d t o t a l optimum s l i p shear i = l j = l i j 1 J = optimum l o c a l s l i p l o a d f o r j t h f r i c t i o n d e v i c e i n i t h storey = angle of i n c l i n a t i o n from ho r i z o n t a l of j t h braces i n the i t h storey = value of the slope of the assumed mode shape i n i t h storey = number of f r i c t i o n devices i n i t h storey = number of storeys 240 8.4.2 Numerical V e r i f i c a t i o n Three representative multi-storey structures, i n which a x i a l defor-mations of the columns are neglected, were used i n an a n a l y t i c a l study to examine the f e a s i b i l i t y of applying Equation (8.42) to define the optimum s l i p load d i s t r i b u t i o n under earthquake conditions: • The three storey low r i s e b u i l d i n g described i n section 3.3 (Montgomery and H a l l , 1979). • The s i x storey b u i l d i n g shown i n F i g . 8.11 (Uang and Bertero, 1986). • The ten storey structure shown i n F i g . 8.12 (Workman, 1969). The three earthquake records described i n section 7.5 were used i n the study undertaken to optimize the s l i p load d i s t r i b u t i o n based on Equation (8.42). The t o t a l s l i p shear V 0, previously s p e c i f i e d as a constrained value, was now allowed to vary and was optimized i n terms of the minimum Relative Performance Index RPI, which was determined for the d i f f e r e n t patterns of s l i p load defined by Equation (8.42). The sel e c -ted values of y were 0, 1, 2, 4, 10, 15 and ». For each value of y, two d i f f e r e n t mode shapes were considered i n the s l i p load d i s t r i b u t i o n c a l c u l a t i o n s : • The unbraced fundamental mode shape, cj>u. • The braced fundamental mode shape, cf^. These two mode shapes were assumed to have the most influences on the s t r u c t u r a l response; i t i s expected t h a t <j>^  w i l l predominate for an earthquake causing moderate slippage while <J>u w i l l dominate the response for an earthquake causing large slippage. The minimum Relative Performance Indices obtained from FDBFAP analyses are presented i n Table 8.3 for a l l the d i f f e r e n t cases 2 @ 7.5 m Floor Level Beams Braces Columns Floor Weight (kN) 6 18W35 HSSA x A x .1875 10W33 10W33 10WA9 200 5 18W35 HSS5 x 5 x .1875 10W33 10W33 10WA9 2A0 A 18W35 HSS6 x 6 x 1/4 10W39 10W60 12W65 2A0 3 18W35 HSS6 x 6 x 1/4 10W39 10W60 12W65 2A0 2 18W35 HSS6 x 6 x 1/4 12W50 12W79 12W79 2A0 1 18W35 HSS6 x 6 x 1/4 12W65 12W106 12W87 2A0 Figure 8.11. Six Storey Structure Used for S l i p Load Optimization. 242 Beams Braces Columns Floor Level Section Ix(mm*) Adnra1) Section Ix(mm*) 10 W18x50 333x10 s 1858 W14x34 141x10s 9 W18x50 333x10 s 1858 W14x53 226x10 s 8 W18x50 333x10 s 1858 W14x53 226x10 s 7 W18x50 333x10 s 1858 W14x78 354x10 s 6 W18x50 333x10* 1858 W14x78 354x10 s 5 W18x60 410x10 s 1858 W14xl03 485x10 s 4 W18x60 410x10 s 1857 W14xl03 485x10 s 3 W18x60 410x10 s 1857 W14xll9 572x10 s 2 W18x60 410x10 s 2181 W14xll9 572x10 s 1 W18x77 535x10 s 2181 W14xl84 947x10 s Dead Load - 587.7 kN/floor Figure 8.12. Ten Storey Structure Used for S l i p Load Optimization. 243 considered; the corresponding V0/W r a t i o s are also tabulated. It can be seen that the lumped d i s t r i b u t i o n (y=°°) i s the least e f f i c i e n t for a l l the s i t u a t i o n s examined. The optimum response (minimum RPI) of the structures i s obtained for values of y ranging from 0 to 4. 8.5 Discussion The numerical c a l c u l a t i o n s suggest that the procedure proposed for optimizing the d i s t r i b u t i o n of s l i p load i n multi-storey structures, which involves the a p p l i c a t i o n of Equation (8.42), i s f e a s i b l e provided that the most dominant mode shape i s used i n t h i s expression. The choice of t h i s shape must be made by the designer. A close look at the data i n Table 8.3 reveals only small differences between the RPI values r e s u l t i n g from the optimum s l i p load d i s t r i b u t i o n s (*) and the optimum uniform d i s t r i b u t i o n (f=0). Thus, i n ordinary design s i t u a t i o n s i t may not be necessary to optimize y; i t may be expedient to simply use the optimum uniform d i s t r i b u t i o n (7=0) as the optimum s l i p load d i s t r i b u -t i o n . Figure 8.13 presents the top f l o o r l a t e r a l d e f l e c t i o n time-h i s t o r i e s for the s i x storey b u i l d i n g excited by the Long Beach earth-quake. It can be seen that very l i t t l e reduction i n the response i s obtained by using the optimum value of y=4. Note that the deflec t i o n s of the unbraced structure also are shown i n the top diagram. The b e n e f i c i a l e f f e c t of introducing f r i c t i o n dampers i s very apparent. As a r e s u l t of the outcome of the studies reported i n t h i s chapter, for design purposes i t was decided to use a uniform s l i p shear d i s t r i b u -t i o n (y=0) when performing the parametric study for multi-storey structures. This approach s i m p l i f i e s the design procedure and has the added advantage of reducing the r i s k during construction of improperly d i s t r i b u t i n g the f r i c t i o n dampers s p e c i f i e d for a structure. 2AA 100 -T 50-o CD - 5 0 -Unbraced Structure 7=0 V„/W=0.40 ViiJtZ'y.e/ZZh??.. T 4 6 Time [sec] 8 10 10 c o o CD CD C J - 5 --10 Enlargement of FDBF Responses Above t . n „ n n A A A lifl / in. A l l ' l l I I i i <, \ .-. ,vYl< -»f*i i> t.M .y=* y.c/yff.i-AP.. / % ii V < \ ' " 1 v\ A1'/ i 1 A V W U if \ y l 0 4 6 Time [sec] 8 10 Figure 8.13 Top Floor Lat e r a l D e f l e c t i o n Time-Histories for Six Storey Structure, Long Beach Earthquake. Table 8.3 Results for Optimum S l i p Load D i s t r i b u t i o n Structure Earthquake Minimum RPI (V0/W) r=o r = r u b r=i r=2 T=4 r=io r=15 r=- r=l T=2 r=4 r=io r=l5 7"=» Three Storey P a r k f i e l d 0.1305 (0.24) 0.1330 (0.18) 0.1340 (0.18) 0.1425 (0.14) 0.1423 (0.10) 0.1624 (0.11) 0.1721 (0.10) 0.1252* (0.20) 0.1277 (0.16) 0.1600 (0.13) 0.1739 (0.14) 0.1737 (0.14) 0.2116 (0.10) Long Beach 0.0401 (0.12) 0.0419 (0.10) 0.0448 (0.09) 0.0563 (0.08) 0.1095 (0.04) 0.1168 (0.03) 0.1365 (0.03) 0.0378 (0.08) 0.0324* (0.09) 0.0424 (0.09) 0.0503 (0.07) 0.0575 (0.07) 0.1584 (0.03) Eureka 0.0394 (0.23) 0.0372 (0.21) 0.0387 (0.22) 0.0557 (0.20) 0.1073 (0.12) 0.1229 (0.10) 0.1209 (0.10) 0.0365 (0.20) 0.0327* (0.20) 0.0369 (0.22) 0.0625 (0.22) 0.0706 (0.24) 0.1209 (0.10) Six Storey P a r k f i e l d 0.0430* (1.10) 0.0612 (1.80) 0.0591 (2.30) 0.0995 (0.70) 0.1403 (0.80) 0.2046 (0.10) 0.3326 (0.10) 0.0690 (1.20) 0.0884 (0.30) 0.0985 (0.40) 0.1691 (0.20) 0.1941 (2.40) 0.2824 (0.10) Long Beach 0.0257 (0.40) 0.0261 (0.40) 0.0288 (0.60) 0.0177* (1.30) 0.1596 (0.40) 0.1775 (0.90) 0.4597 (0.10) 0.0296 (0.60) 0.0296 (0.80) 0.0295 (2.20) 0.1789 (0.30) 0.1851 (0.80) 0.5411 (0.20) Eureka 0.0077 (0.90) 0.0092 (1.20) 0.0088 (1.40) 0.0069* (2.40) 0.0901 (1.10) 0.1058 (2.40) 0.3334 (0.20) 0.0090 (1.40) 0.0090 (2.10) 0.035 (2.40) 0.0950 (0.70) 0.1201 (1.20) 0.2654 (0.20) Ten Storey P a r k f i e l d 0.0476* (0.24) 0.0594 (0.20) 0.0650 (0.16) 0.0671 (0.12) 0.1160 (0.16) 0.1729 (0.12) 0.4284 (0.04) 0.0487 (0.24) 0.0515 (0.16) 0.0890 (0.16) 0.1623 (0.16) 0.1975 (0.20) 0.5412 (0.04) Long Beach 0.1731 (0.24) 0.1872 (0.24) 0.1951 (0.20) 0.1944 (0.24) 0.3513 (0.28) 0.4490 (0.08) 0.5356 (0.08) 0.1559 (0.20) 0.1499* (0.24) 0.2186 (0.32) 0.4476 (0.08) 0.4768 (0.08) 0.5923 (0.04) Eureka 0.1984* (0.12) 0.2133 (0.12) 0.2416 (0.12) 0.2834 (0.12) 0.3862 (0.08) 0.4534 (0.08) 0.8440 (0.04) 0.2072 (0.16) 0.2649 (0.16) 0.3174 (0.08) 0.4181 (0.08) 0.4758 (0.04) 0.7389 (0.08) *0ptimum S l i p Load D i s t r i b u t i o n 246 9. PARAMETRIC STUDY FOR MULTI-STOREY STRUCTURES "Method i s l i k e packing things i n a box; a good packer w i l l get i n h a l f as much again as a bad one." - Richard C e c i l (1748-1777) , English Divine 247 CHAPTER 9 PARAMETRIC STUDY FOR MULTI-STOREY STRUCTURES 9.1 Strategy As discussed i n Chapter 8, a uniform s l i p shear d i s t r i b u t i o n can be used for the p r a c t i c a l design of multi-storey f r i c t i o n damped struc-tures. A parametric study i s performed i n t h i s chapter i n order to determine an approximate design equation for the t o t a l optimum s l i p shear V 0. Using the notation given i n Chapter 7, the proposed design equation may be expressed as: with and (9.1) M - a ( ^ ) + b u (9.2) a = Fj (T /T u, NS) (9.3) b = F 2 (T /T u, NS) (9.4) where ¥ t and F 2 are unknown functions to be estimated. 248 9.1.1 St r u c t u r a l Model The basic s t r u c t u r a l model used i n the parametric study of mu l t i -storey f r i c t i o n damped structures i s shown i n F i g . 9.1; 3, 5 and 10 storey versions were investigated. Each version has a uniform mass d i s t r i b u t i o n and a mass proportional unbraced l a t e r a l s t i f f n e s s d i s t r i b u t i o n . NS X w. I i = l f V ^ I i (9.5) where I. = moment of i n e r t i a of the i t h f l o o r columns a. Wj = weight of the j t h f l o o r W = t o t a l weight of the structure The same diagonal braces were used throughout a structure. The proper-t i e s of the d i f f e r e n t structures are l i s t e d i n Table 9.1; i t may be seen that a t o t a l of 33 d i f f e r e n t structures was considered i n t h i s parametric study. 9.1.2 Range of Values Considered The values of the d i f f e r e n t v a r i a b l e s used i n the parametric study of multi-storey f r i c t i o n damped structures are given i n Table 9.2. For each combination of parametric values, 5 d i f f e r e n t sample accelerograms were simulated by SIMEA. The t o t a l number of FDBFAP analyses performed was 2640; the ca l c u l a t i o n s were made with a SUN 3/260 minicomputer. 249 Table 9.1 Versions of Basic Structural Configuration NS Version No. W T u A b (mmJ) (kN) (mm*) (s) T b/T u=0.20 T b/T u=0.50 T b/T u=0.80 1 42x109 0.1104 370990 41938 6452 3 2 1110 2xl0 9 0.4939 18550 2097 323 3 416xl0 6 1.1045 3710 419 65 4 125xl0 6 2.0165 1113 126 19 1 125x10s 0.1187 967800 96780 16130 5 2 2000 6xl0 9 0.5307 48390 4839 807 3 l x l O 9 1.1867 9678 968 161 4 416xl0 6 2.0555 3226 323 54 1 31xl0 9 0.5209 180656 22582 3226 10 2 4225 9x10" 0.9617 52992 6624 946 3 2xl0 9 2.0173 12044 1505 215 250 Table 9.2 Values of Parameters Used i n Parametric Study Parameter Values V T u 0.20, 0.50, 0.80 V T u 0.1/TU, 0.7/Tu, 1.4/TU, 2.0/T u a g/g 0.05, 0.10, 0.20, 0.40 NS 3, 5, 10 9.2 Results The procedure described i n section 7.3.1 was used to estimate the value of M i n Equation (9.1). A l l the calculated l e a s t square slopes are presented i n Tables 9.3, 9.4 and 9.5. Table 9.3 Least Square Slopes M for NS = 3 T /T g' u M T b/T u=0.20 T b/T u=0.50 T b/T u=0.80 0.0496 0.3929 0.1506 0.1365 0.0905 0.7976 0.2141 0.1294 0.2025 1.0471 0.3929 0.1600 0.3471 1.0565 0.3506 0.1482 0.6338 2.0494 0.4447 0.1906 0.6943 2.0424 0.4776 0.1012 0.9058 3.9812 1.2329 0.6871 0.9918 2.3294 1.2776 0.2800 1.2675 2.5012 1.5082 0.3341 1.4173 2.6259 1.1153 0.3082 1.8108 2.4306 1.0729 0.3365 2.8346 2.8118 1.3671 0.4047 4.0494 2.7624 1.2471 0.2682 6.3406 3.1765 1.8400 0.8376 12.6812 3.0096 1.8259 0.9412 18.1159 2.9812 1.5859 1.0165 Table 9.4 Least Square Slopes M for NS = 5 T /T g' u M T b/T u=0.20 T b/T u=0.50 T b/T u=0.80 0.0486 0.7671 0.3059 0.2165 0.0843 1.3788 0.3859 0.2447 0.1884 1.7271 0.7671 0.3671 0.3405 2.0894 0.5600 0.2494 0.5899 3.3835 0.8047 0.2918 0.6811 3.5294 1.0306 0.2071 0.8425 4.8000 2.5082 0.8612 0.9730 3.6282 1.8965 0.3718 1.1797 3.9529 1.6141 0.6447 1.3190 4.0094 1.7694 0.6024 1.6853 3.8824 1.8541 0.6024 2.6380 4.2447 2.3953 0.6259 3.7686 4.2212 2.0376 0.5459 5.8972 4.6000 2.6212 0.9365 11.7944 4.5500 2.7482 1.1341 16.8492 4.3000 2.6447 1.0776 Table 9.5 Least Square Slopes M for NS = 10 T /T g' u M T b/T u=0.20 T b/T u=0.50 T b/T u=0.80 0.0496 1.6659 0.6588 0.4235 0.1040 2.5318 1.3082 0.5271 0.1920 3.3412 1.5718 0.6776 0.3470 4.5929 1.1953 0.5082 0.6940 6.9082 2.0988 0.3294 0.7279 7.7271 2.6165 0.7435 0.9914 7.2565 3.9059 0.7435 1.3438 7.8024 4.6118 1.2894 1.4558 7.3600 3.9153 0.8612 2.0801 8.3859 4.1318 1.1388 2.6877 8.3953 5.6188 1.3647 3.8395 8.4612 5.3553 1.2612 252 Figure 9.2 presents the estimated values of the parameters a and b (Equation 9.2) along with b i l i n e a r l e a s t square f i t s through the data for d i f f e r e n t values of NS. The procedure follows the presentation i n 7.3.3. The one storey r e s u l t s also are shown on the f i g u r e . The family of b i l i n e a r curves i s approximated by the following r e l a t i o n s (see Appendix I ) : a = (-1.2378NS - 0.3072) T /T 0 < T /T <. 1 g u g u (0.0117NS + 0.0148) T /T - 1.2495NS - 0.3220 T /T > 1 g u g u (9.6) b = (1.0373NS + 0.4259) T /T 0 < T /T £ 1 g u g u (-0.0022NS + 0.0016) T /T + 1.0395NS + 0.4243 T /T > 1 g u g u (9.7) 9.3 Proposed Design Equation An approximate design equation for the t o t a l optimum -slip shear V 0 i s obtained by s u b s t i t u t i n g Equations (9.2), (9.6) and (9.7) into Equation (9.1). "((-1.2378NS -- 0.3072)T,/T + 1.0373NS + 0.4259)T /T b u g u 0 £ T /T < g u 1 ma g ((0.0117NS + + (0.0016 -0.0148)T /T - 1.2495NS - 0.3220)T,/T g u b u 0.0022NS)T /T + 1.0395NS + 0.4243 g u - T /T > g u 1 (9.8) 253 -Q 20 18 16 14 12 10 8 6 4 0 • NS=1 O NS=3 m NS=5 • NS=IO Bi-Lln»ar Least Sguor* Fits n r i H • m l n> ' o mmwm w—m- > i ; » < 0 8 10 12 14 J9/Ju 16 18 20 Figure 9.2 Least Square Slope Parameters a and b 254 where N NS Di V„ = Z 1 2P„. .cosa.. = Total optimum s l i p shear i = l j = l J J m = Total mass of the structure a^ = Peak ground acceleration T^ = Fundamental period of the f u l l y braced structure T = Fundamental period of the unbraced structure u c NS = Number of storeys T = Kanai-Tajimi predominant period s 9.4 Design S l i p Load Spectrum Equation (9.8) can be used d i r e c t l y to estimate the t o t a l optimum s l i p shear V 0. However, a graphical representation of t h i s equation i n the form of a design s l i p load spectrum provides a more convenient and s i m p l i f i e d method for e s t a b l i s h i n g V 0. I f a p l o t of V0/ma vs T /T i s made f o r p a r t i c u l a r v a l u e s of T^/T^ and NS, u s i n g Equation (9.8), a b i l i n e a r curve w i l l be obtained. This curve represents a general design s l i p load spectrum for multi-storey f r i c t i o n damped structures. This spectrum i s completely described by specifying the ordinate a, representing the upper bound of the f i r s t branch of the b i l i n e a r curve, and c o r r e s p o n d i n g t o Tg/T u = 1 for a l l NS, and any other ordinate data compatible with the second l i n e a r branch and taken here as the ordinate value at T /T =15. g u a 255 1 15 Figure 9.3 Construction of Design Slip Load Spectrum The ordinates a and B of these points are given by Equation (9.8) o = (-1.2378NS - 0.3072) T./T + 1.0373NS + 0.4259 D U (9 .9) B = (-11.0740NS - 0.1000) T./T + 1.0065NS + 0.4483 D U (9 .10) The values of a and B have been calculated for different values of T./T and NS and are presented in Table 9.6. With the use of this b u table, a design slip load spectrum can quickly be constructed for a particular design or retrofit situation. Table 9.6 Ordinates a and B for Construction of Design S l i p Load NS = • 1 NS = •- 2 NS = 3 NS = 4 NS = = 5 NS = •- 6 NS = •- 1 NS = = 8 NS = = 9 NS = = 10 T./T b ti a B a P a P a P a P a P a P a P a P a P 0.20 1.15 1.22 1.94 2.01 2.73 2.80 3.52 3.60 4.31 4.39 5.10 5.18 5.89 5.97 6.68 6.76 7.47 7.55 8.26 8.35 0.25 1.08 1.16 1.80 1.90 2.53 2.64 3.26 3.38 3.99 4.11 4.72 4.85 5.44 5.59 6.17 6.33 6.90 7.07 7.63 7.80 0.30 1.00 1.10 1.67 1.79 2.33 2.47 3.00 3.16 3.66 3.84 4.33 4.52 5.00 5.21 5.66 5.89 6.33 6.58 6.99 7.26 0.35 0.92 1.04 1.53 1.67 2.13 2.31 2.73 2.94 3.34 3.57 3.94 4.20 4.55 4.83 5.15 5.46 5.76 6.09 6.36 6.72 0.40 0.85 0.99 1.39 1.56 1.93 2.14 2.47 2.72 3.01 3.29 3.56 3.87 4.10 4.45 4.64 5.02 5.18 5.60 5.72 6.18 0.45 0.77 0.93 1.25 1.45 1.73 1.97 2.21 2.50 2.69 3.02 3.17 3.54 3.65 4.07 4.13 4.59 4.61 5.11 5.09 5.64 0.50 0.69 0.87 1.11 1.34 1.53 1.81 1.95 2.28 2.36 2.75 2.78 3.22 3.20 3.68 3.62 4.15 4.04 4.62 4.46 5.09 0.55 0.61 0.81 0.97 1.22 1.33 1.64 1.68 2.06 2.04 2.47 2.40 2.89 2.75 3.30 3.11 3.72 3.47 4.14 3.82 4.55 0.60 0.54 0.75 0.83 1.11 1.13 1.47 1.42 1.84 1.71 2.20 2.01 2.56 2.30 2.92 2.60 3.29 2.89 3.65 3.19 4.01 0.65 0.46 0.69 0.69 1.00 0.92 1.31 1.16 1.62 1.39 1.93 1.62 2.23 1.86 2.54 2.09 2.85 2.32 3.16 2.55 3.47 0.70 0.38 0.63 0.55 0.89 0.72 1.14 0.89 1.40 1.07 1.65 1.24 1.91 1.41 2.16 1.58 2.42 1.75 2.67 1.92 2.93 0.75 0.30 0.57 0.41 0.78 0.52 0.98 0.63 1.18 0.74 1.38 0.85 1.58 0.96 1.78 1.07 1.98 1.18 2.18 1.29 2.38 0.80 0.23 0.52 0.27 0.66 0.32 0.81 0.37 0.96 0.42 1.10 0.46 1.25 0.51 1.40 0.56 1.55 0.60 1.69 0.65 1.84 Ln 257 9.5 Proposed Design Procedure The following procedure i s proposed for the design of multi-storey f r i c t i o n damped structures. Step 1 • In the case of a new structure, design the unbraced moment r e s i s t i n g frame to carry s a f e l y the usual load combinations but without con-si d e r i n g earthquake e f f e c t s ; the unbraced structure should also be designed to have a minimum l a t e r a l s t i f f n e s s , y i e l d i n g resistance, and d u c t i l i t y so as to ensure an optimum response of the whole structure. Also see note i n step 1 of section 7.8. Step 2 • Calculate the fundamental period of the unbraced structure, T . u • Choose s e c t i o n s f o r the diagonal cross-braces such that T./T < 0.40 ° b u i f economically possible; i f necessary, the unbraced structure should also be r e t r o f i t t e d to have a minimum l a t e r a l s t i f f n e s s , y i e l d i n g resistance, and d u c t i l i t y i n order to ensure an optimum response of the whole structure. Step 3 • V e r i f y that the nondimensional r a t i o s f a l l w ithin the following l i m i t s : 0.20 ^ T,/T £ 0.80 b u 258 0.05 £ T /T £ 20 g u 0.005 £ a /g ^ 0.40 g NS £ 10 These bounds correspond to those used i n t h i s study and represent reasonable p r a c t i c a l l i m i t s . I f the i n e q u a l i t i e s are not v e r i f i e d , the optimum s l i p load should be determined from dynamic analyses (FDBFAP). Step A • Determine the c o e f f i c i e n t s a and B from Table 9.6 and construct the appropriate design s l i p load spectrum as indicated i n F i g . 9.3. Step 5 • Use the constructed design s l i p load spectrum to estimate the t o t a l s l i p shear V 0. D i s t r i b u t e t h i s t o t a l s l i p shear uniformly among the f l o o r s of the structure. V s ( i ) = NS (9.11) where V (i) i s the s l i p shear i n the i t h f l o o r . 259 Step 6 • D i s t r i b u t e the s l i p shear of each f l o o r V g ( i ) among the number of f r i c t i o n devices inserted i n each f l o o r . N Di I 2P, j=l cosa. 1J V (i) s (9.12) where N_. . i s the number of f r i c t i o n d e v i c e s i n s e r t e d i n the i t h D i f l o o r . Step 7 • Calculate the a x i a l loads induced i n the cross-braces from wind e f f e c t and v e r i f y that the f r i c t i o n devices are not s l i p p i n g under these loads (see section 7.6.1). For slender braces (P... > P/ s . . ) : ° i j (cr)ij P + P wij ( c r ) i j (9.13) For stubby braces (P n. . i P. .. .) 3 °ij ( c r ) i j ' P.. . ^ P . . (9.14) 260 where P .. i s the a x i a l wind l o a d induced i n the t e n s i l e diagonal wij 5 b r a c e o f the i t h braced bay of the i t h f l o o r and P, ... i s the J 3 ( c r ) i j corresponding buckling load of that brace. I f Equations (9.13) or (9.14) are not v e r i f i e d , choose one of the following two a l t e r n a t i v e s : 1) Use the value of the s l i p load s a t i s f y i n g the e q u a l i t y i n one of these equations and perform dynamic analyses (FDBFAP) to examine the response of the structure. 2) Modify the unbraced moment r e s i s t i n g frame to carry a larger portion of the wind load and return to step 2. Step 8 • Estimate the t e n s i l e y i e l d load and v e r i f y that the cross-braces do not y i e l d before s l i p p i n g occurs (see section 7.6.2). For slender braces (P.. . > P, ...) °ij ( c r ) i j ' A,, .o . . + P, . . . °ij 2 (9.15) For stubby braces (P 0 . . <, P ,,) i j ( c r ) i j P.. . £ A, . .o °1J Dl] ' IJ yij (9.16) 261 where A, . . and a . . are the c r o s s - s e c t i o n a l area and t e n s i l e y i e l d s tress for the j t h t e n s i l e brace i n the i t h f l o o r . I f Equations (9.15) or (9.16) are not v e r i f i e d , choose one of the following two a l t e r n a t i v e s : 1) Use a value of the s l i p load s a t i s f y i n g the e q u a l i t y i n one of these equations and perform dynamic analyses (FDBFAP) to examine the response of the structure. 2) Increase the s i z e of the diagonal cross-braces and return to step 3. 9.6 Design Example To i l l u s t r a t e the use of the design procedure proposed i n section 9.5, assume that the low r i s e frame described i n section 3.3 i s to be r e t r o f i t t e d with f r i c t i o n devices. Step 1 This structure was designed o r i g i n a l l y by Montgomery and H a l l (1979). Assume that the unbraced structure can carry s a f e l y the usual load combinations without considering earthquake e f f e c c t . Step 2 The fundamental periods of the structure have been calculated i n section 3.3 and are presented i n Table 3,3. T = 0.72 sec. u T, = 0.38 sec. 262 Let the Eureka earthquake, December 21, 1954, COMP N46W be the design earthquake s p e c i f i e d for the contruction s i t e . The parameters for t h i s seismic event are: a = 0.20 g T = 0.69 sec g Step 3 V e r i f y that the nondimensional r a t i o s are within the appropriate l i m i t s : 0.20 <; T./T - 0.53 <. 0.80 b u 0.50 <; T /T = 0.96 £ 20 g u 0.005 <. a /g = 0.201 £ 0.40 g NS = 3 S 10 Step 4 The c o e f f i c i e n t s a and B are estimated from Table 9.6. a = 1.43 B = 1.73 Step 5 The design s l i p load spectrum i s constructed as shown i n F i g . 9.4. From t h i s spectrum the t o t a l optimum s l i p shear i s estimated as v 0 = 1.14 ma g 263 1 15 Figure 9.4 Design S l i p Load Spectrum for Design Example V 0 = 514.44 kN This s l i p shear i s d i s t r i b u t e d uniformly among the f l o o r s . V ... = 171.48 kN i = 1,2,3 s ( i ) Step 6 The optimum l o c a l s l i p load P 0 for each f r i c t i o n device i s : V s ( i ) P. = T T ^ 2 - = 90.75 kN 0 2cosa 264 Step 7 I t i s assumed that t h i s s t r u c t u r e i s lo c a t e d i n Vancouver. The design wind pressure a c t i n g on the b u i l d i n g i s c a l c u l a t e d by the Na t i o n a l B u i l d i n g Code of Canada (NBCC, 1985): P = q C C C e q p where the symbols are as defined i n s e c t i o n 7.9.1. A s t a t i c a n a l y s i s of the s t r u c t u r e r e v e a l s that the maximum a x i a l load induced by the wind i n the diagonal cross braces i s 11.48 kN. The b u c k l i n g load f o r a cross brace member (2L75 x 75 x 6) i s estimated to be 97.10 kN: the braces w i l l not buckle under the design wind load. From Equation (9.14): P o i j. = 90.75 kN i = 1,2,3; j = l P .. = 11.48 kN i = 1,2,3; j = l w i j ••• P . . . * p . . -°ij wij and the f r i c t i o n devices w i l l not s l i p under the design wind load. Step 8 Assuming a t e n s i l e y i e l d s t r e s s °y^j e q u a l t o 300 MPa f o r the diagonal cross-braces, v e r i f y Equation (9.16). P o i j = 90.75 kN i = 1,2,3; j = l ^ . . o ^ . = 557.40 kN i = 1,2,3; j = l ••• P 0 . . £ A, . .o . . °ij T>ij y i j 265 Hence the f r i c t i o n devices w i l l s l i p occurs. The optimum s l i p load obtained i s compared with the value obtained r e s u l t s agree very c l o s e l y . before y i e l d i n g of the cross-braces from the design s l i p load spectrum from FDBFAP i n F i g . 9.5. The two Eureka Earthquake,December 21,1954 n , V,/ma Figure 9.5 Optimum S l i p Load Study for Design Example 10. CONCLUSION "In everything we ought to look to the end." - Jean de l a Fontaine (1621-1695), French Poet 267 CHAPTER 10  CONCLUSIONS 10.1 Summary and Conclusions 1) A simple and e f f i c i e n t numerical modelling procedure for structures equipped with a new f r i c t i o n damping system has been presented. The i n c l u -sion of a l l possible tangent s t i f f n e s s matrices of a f r i c t i o n device element i n a time-step i n t e g r a t i o n procedure leads to a s o l u t i o n algorithm (FDBFAP) which i s adaptable to a microcomputer environment. The optimum s l i p load d i s t r i b u t i o n for the f r i c t i o n devices i s determined by minimizing a Relative Performance Index (RPI) derived from energy concepts. From the r e s u l t s of example structures investigated, i t would appear that FDBFAP i s s u f f i c i e n t l y simple and accurate to be of value for the p r a c t i c a l determination of the optimum s l i p load d i s t r i b u t i o n of structures equipped with f r i c t i o n devices. It should be noted that FDBFAP neglects the e f f e c t of column a x i a l deforma-t i o n and therefore i t s use for p r e d i c t i n g the optimum s l i p load d i s t r i b u t i o n of very slender braced frames may lead to important inaccuracies. 2) The following valuable a n a l y t i c a l r e s u l t s have been obtained from the study of a simple f r i c t i o n damped structure excited by sinusoidal ground motion: a) A lower bound value of the s l i p load has been established (Equation A.25) such that bounded amplitudes occur at resonance. b) The value of the s l i p load which minimizes the amplitude at resonance has been determined (Equation A.29) along with the frequency at which t h i s optimum resonance occurs (Equation (A.31). c) The nondimensional r a t i o s governing the value of the s l i p load which minimizes the steady-state amplitude for any f o r c i n g frequency have been 268 derived (Equation (A.36). These r a t i o s c l e a r l y indicate that the o p t i -mum s l i p load i s a function of the amplitude and frequency of the ground motion and i s not s t r i c t l y a s t r u c t u r a l property. Furthermore, for harmonic e x c i t a t i o n the optimum s l i p load i s l i n e a r l y proportional to the amplitude of the ground motion. 3) Taking i n t o consideration the r e s u l t s of item 2, and using a stochastic earthquake model, a dimensional analysis was performed and a formal design equation for the optimum s l i p load d i s t r i b u t i o n of a f r i c t i o n damped braced frame was deduced (Equation 6.6). A s e n s i t i v i t y study performed on a one storey FDBF i d e n t i f i e d the most important parameters c o n t r o l l i n g the optimum s l i p load. It was found that the optimum s l i p load i s almost l i n e a r l y proportional to the peak value of the earthquake ground acceleration. 4) By applying v a r i a t i o n a l p r i n c i p l e s to an harmonically excited f r i c t i o n damped shear beam, which i s analogous to a f r i c t i o n damped shear b u i l d i n g , i t was shown that to achieve maximum energy d i s s i p a t i o n by f r i c t i o n i n a structure v i b r a t i n g i n a p a r t i c u l a r mode the t o t a l s l i p load should be lumped at the l o c a t i o n i n the structure where the slope of that mode shape i s a maximum. For a general structure subjected to earthquake e x c i t a t i o n , numerical analyses support the f e a s i b i l i t y of using an optimum s l i p shear d i s t r i b u t i o n which i s proportional to the slope of the most dominant mode shape of the structure. However, the use of t h i s optimum d i s t r i b u t i o n o f f e r s very l i t t l e advantage over the use of the simpler uniform d i s t r i b u t i o n . Therefore, for design purposes, i t seems adequate to use a uniform s l i p shear d i s t r i b u t i o n . 5) The r e s u l t s of a parametric study of multi-storey f r i c t i o n damped braced frames led to the construction of a design s l i p load spectrum (Fig. 9.3) for a quick evaluation of the t o t a l optimum s l i p shear. The spectrum takes into 269 account the properties of the structure and of the ground motion anticipated at the construction s i t e . The a v a i l a b i l i t y of t h i s design s l i p load spectrum should lead to a greater acceptance by the engineering profession of t h i s innovative design concept. 10.2 Future Research The i n v e s t i g a t i o n described i n t h i s thesis represents the f i r s t known attempt to develop a s i m p l i f i e d method for the seismic design of f r i c t i o n damped structures. The proposed method seems s a t i s f a c t o r y for ordinary design s i t u a t i o n s and should provide s t r u c t u r a l engineers with the basic p r i n c i p l e s needed for a r a t i o n a l seismic design using t h i s new s t r u c t u r a l concept. The e f f e c t of column a x i a l deformation on the optimum s l i p load d i s t r i -bution for very slender braced frames should be investigated. The proposed d e s i g n method contains T (predominant ground period of a b u i l d i n g s i t e ) as a parameter. It would be very desirable for b u i l d i n g codes to s p e c i f y T v a l u e s i n a manner analogous to the peak ground acceleration values now incorporated i n such documents; i t i s recognized that t h i s kind of information i s not widely a v a i l a b l e at present. In order to gain confidence i n , and p o s s i b l y to improve, the proposed method, i t would be desirable to t e s t model or f u l l scale structures designed by t h i s approach. Some c y c l i c t e s t s should be performed on a complete brace device u n i t i n order to v e r i f y the t h e o r e t i c a l h y s t e r e t i c behaviour incorporated i n FDBFAP. The r e s u l t s of these t e s t s may suggest possible improvements to the program. To gain wide acceptance by the profession of t h i s new system, serious studies should be undertaken to assess the environmental e f f e c t s on the f r i c t i o n devices; maintenance methods may have to be developed to ensure the long-term o p e r a b i l i t y of the devices. 270 F i n a l l y , the performance of a three-dimensional model of a f r i c t i o n damped braced frame should be examined i n order to obtain a more r e a l i s t i c assessment of i t s behaviour. 271 BIBLIOGRAPHY Adey, R.A. and Brebbia, C.A., "Basic Computational Techniques for Engineers", Wiley Interscience P u b l i c a t i o n , New York, 1983. A r i a s , A., "A Measure of Earthquake Intensity," Seismic Design of Nuclear Power Plants, R. Hansen Edi t o r , M.I.T. Press, Cambridge, 1970. Baktash, P. and Marsh, C , "Seismic Behaviour of F r i c t i o n Damped Braced Frames", Proceedings of the Third U.S. National Conference on Earthquake Engineering, Charleston, SC, Vol. I I , 1986, pp. 1099-1105. 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Mex. da Ing., Sismica, 3, 1964. 272 F i l i a t r a u l t , A. and Cherry, S., "Performance Evaluation of F r i c t i o n Damped Braced Steel Frames Under Simulated Earthquake Loads", Earthquake Engineering Laboratory Report, Dept. of C i v i l Engineering, Un i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C., November 1985. F i l i a t r a u l t , A. and Cherry, S., "Performance Evaluation of F r i c t i o n Damped Braced Steel Frames Under Simulated Earthquake Loads", Earthquake Spectra, Vol. 3:1, February 1987, pp. 57-78. F i l i a t r a u l t , A. and Cherry, S., "Comparative Seismic Performance of F r i c t i o n Damping Devices and Lead-Rubber Hysteretic Bearings for Earthquake R e t r o f i t and Design", Earthquake Engineering and S t r u c t u r a l Dynamics, V o l . 16, No. 3, A p r i l 1988, pp. 389-416. 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Law, D. and Grigg, R., "Dynamic Analysis of Plane Structures, Graphics", C i v i l Engineering Program Library, U n i v e r s i t y of B r i t i s h Columbia, 1978. L i n , Y.K., " P r o b a b i l i t y Theory of St r u c t u r a l Dynamics", McGraw H i l l , New York, 1967. McCabe, S.L. and H a l l , W.J., "Evaluation of St r u c t u r a l Response and Damage Resulting From Earthquake Ground Motion", Report No. SRS538, Dept. of C i v i l Engineering, U n i v e r s i t y of I l l i n o i s , Urbana, I I . , 1987. McGuire, R.K. and Barnhard, J.A., "Magnitude, Distance and Intensity Data for C.I.T. Strong Motion Records", U.S. Geological Survey Research Journal, 5, No. 4, 1977. Minorski, N. , "Nonlinear Mechanics", J.W. Edwards, Ann Arbor, Michigan, 1947. Moayyad, P. and Mohraz, B. , "A Study of Power Spectral Density of Earthquake Accelerograms", C i v i l and Mechanical Engineering Dept., Southern Methodist Un i v e r s i t y , D a l l a s , Texas, Report PFR 8004824, 1982. Montgomery, C.J. and H a l l , W.J., "Seismic Design of Low-Rise Steel Buildings", ASCE, Journal of the St r u c t u r a l D i v i s i o n , V o l. 105, ST10, October 1979. National Research Council of Canada, "National Building Code of Canada 1985", NRC No. 23174F, Ottawa, Ontario, 1985. Nau, R.F., O l i v e r , R.M. and P i s t e r , K.C, "Simulating and Analyzing A r t i f i c i a l Nonstationary Earthquake Ground Motions", UCB/EERC-80/ 36, Earthquake Engineering Research Center, Uni v e r s i t y of C a l i f o r n i a , Berkeley, 1980. Nayfeh, A.H. and Mook, P.T., "Nonlinear O s c i l l a t i o n s " , John Wiley, New York, 1979. Newmark, N.M., Blume, J.A. and Kapur, K.K., "Seismic Design Spectra for Nuclear Power Plants", ASCE, Journal of the Power D i v i s i o n , V o l. 99, No. P02, November 1973. 274 Newland, D.E., "Random Vi b r a t i o n and Spectral Analysis", McGraw H i l l , New York, 1975. Owen, D.R.J, and Hinton, E., " F i n i t e Elements i n P l a s t i c i t y : Theory and Pr a c t i c e " , Pineridge Press Limited, Swansea, U.K., 1980. Page, R.A., Boore, D.M. and D i e t r i c h , J.H., "Estimation of Bedrock Motion at the Ground Surface", Profess. Paper 941-A, 1975. P a l l , A.S. and Marsh, C. , "Response of F r i c t i o n Damped Braced Frames", ASCE, Journal of Str u c t u r a l D i v i s i o n , V o l . 108, June 1982, pp. 1313-1323. P a l l , A.S., Verganelakis, V. and Marsh, C , " F r i c t i o n Dampers for Seismic Control of Concordia Un i v e r s i t y Library Building", Proceedings of the 5th Canadian Conference on Earthquake Engineering, Ottawa, Canada, July 1987, pp. 191-200. Papoulis, A., " P r o b a b i l i t y , Random Variables and Stochastic Processes", Second E d i t i o n , McGraw H i l l , New York, 1984. Paz, M., "S t r u c t u r a l Dynamics, Theory and Computation", Second E d i t i o n , Van Nostrand Reinhold Company, New York, 1985. Popov, E.P. and Roeder, C.W., " E c c e n t r i c a l l y Braced Steel Frames for Earthquakes", ASCE, Journal of Str u c t u r a l D i v i s i o n , ST3, March 1978, pp. 391-410. Popov, E.P., Kasai, K. and Engelhardt, M.D., "Advances i n Design of E c c e n t r i c a l l y Braced Frames", Earthquake Spectra, Vol. 3, No. 1, February 1987, pp. 43-55. Shinozuka, M., "Methods of Safety and R e l i a b i l i t y Analysis", Proceedings of the International Conference on Str u c t u r a l Safety and R e l i a b i l i t y , 1969. Taj i m i , H., "A S t a t i s t i c a l Method of Determining the Maximum Response of Building Structures During an Earthquake", Proceedings of the Second World Conference on Earthquake Engineering, Tokyo and Kyoto, Vol. 2, 1960, pp. 781-797. T i l l i o u i n e , B. , Azevedo, J. and Shah, H. , "A Computer Program for Nonstationary Analysis and Simulation of Strong Motion Earthquake Records", Report No. 63, The John A. Blume Earthquake Engineering Center, Dept. of C i v i l Engineering, Stanford University, Stanford, CA, 1984. Trifu n a c , M.D. and Brady, A.G., "A Study of the Duration of Strong Earthquake Ground Motion", B u l l e t i n of the Seismological Society of America, Vol. 65, 1975, pp. 581-626. Ty l e r , R.G., "Preliminary Tests on an Energy Absorbing Element for Braced Structures Under Earthquake Loading", B u l l e t i n of the New Zealand Society for Earthquake Engineering, Vol. 16, No. 3, September 1983. 275 Uang, CM. and Bertero, V.V., "Earthquake Simulation Tests and Associated Studies of a 0.3-Scale Model of a Six-Storey Concentrically Braced Steel Structure", Report No. UCB/EERC-86/10, Univ e r s i t y of C a l i f o r n i a , Berkeley, 1986. Vanmarcke, E.H., "Properties of Spectral Moments with Applications to Random Vi b r a t i o n " , ASCE, Journal of the Engineering Mechanics D i v i s i o n , V ol. 98, No. EM2, A p r i l 1972, pp. A25-AA6. Vanmarcke, E.H., "St r u c t u r a l Response to Earthquakes", Chapter 8, Seismic Risk and Engineering Decisions, C. Lomnitz and E. Rosenblueth E d i t o r s , E l s e v i e r Publishing Co., New York, 1977. Vanmarcke, E.H. and L a i , S.P., "Strong Motion Duration and RMS Amplitude of Earthquake Records", B u l l e t i n of the Seismological Society of America, V o l . 70, No. A, August 1980, pp. 1293-1307. Vanmarcke, E.H., " E f f i c i e n t Stochastic Representation of Earthquake Ground Motion", Proceedings of the Fourth Canadian Conference on Earthquake Engineering, Vancouver, Canada, pp. K1-K1A, 1983. Walpole, R.E. and Myers, R.H., " P r o b a b i l i t y and S t a t i s t i c s for Engineers and S c i e n t i s t s " , Macmillan Publishing Co., Inc., New York, 1978. Weinstock, R., "Calculus of V a r i a t i o n s " , Dover Publications, Inc., New York, 197A. Workman, G.H., "The I n e l a s t i c Behaviour of Multistorey Braced Frame Structures Subjected to Earthquake E x c i t a t i o n " , Research Report, U n i v e r s i t y of Michigan, Ann Arbor, September 1969. 276 APPENDIX A FDBFAP USER'S GUIDE I d e n t i f i c a t i o n FDBFAP.S: F r i c t i o n Damped Braced Frame Analysis Program Programmer: A. F i l i a t r a u l t , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1987. Purpose To determine the optimum s l i p load of F r i c t i o n Damped Braced Plane Frames of a r b i t r a r y configuration. The program consists of a seri e s of subroutines for carrying out step-by-step earthquake dynamic analyses for d i f f e r e n t values of s l i p load within the e l a s t i c range of the cross-braces. Energy c a l c u l a t i o n s are made at the end of each time step and a Relative Performance Index (RPI) i s calculated a f t e r each a n a l y s i s . The optimum s l i p load for the structure i s considered to be the value of the s l i p load which minimizes t h i s Relative Performance Index (RPI). The RPI i s defined as: « » - * < & 7 T + ! K T ] (o) max(o) where SEA = S t r a i n Energy Area = Area under the s t r a i n energy time-h i s t o r y . SEA. . = S t r a i n Energy Area for the unbraced structure ( s l i p load = 0). U = Maximum s t r a i n energy max b J U , . = Maximum s t r a i n energy for the unbraced structure ( s l i p max(o) r load = 0). 277 I d e a l i z a t i o n The structure i s i d e a l i z e d as a number of nodes (joints), connected by elements (members). The nodes must be numbered sequentially s t a r t i n g with the f i r s t f l o o r nodes. The structure i s assumed to have a r i g i d foundation and therefore the base nodes must not be numbered. Three types of elements are incorporated into the program: 1) Beam elements ( h o r i z o n t a l ) ; 2) Column elements ( v e r t i c a l ) ; and 3) F r i c t i o n device elements. The main s t r u c t u r a l elements (beams and columns) remain e l a s t i c at a l l time. The i n e l a s t i c deformations of the structure are s t r i c t l y due to s l i p p i n g of the f r i c t i o n pads, and to buckling i n compression of the diagonal braces. The a x i a l deformations of the beams and columns are neglected and the t o t a l mass of the structure i s concentrated at the f l o o r s ; v e r t i c a l and r o t a t i o n a l i n e r t i a are neglected. Capacity Limitations Some capacity l i m i t a t i o n s r e s u l t i n g from the use of fi x e d dimensions are associated with the input data. Maximum Number of Degrees of Freedom Before Condensation = 175 Maximum Number of Columns = 150 Maximum Number of Beams = 125 Maximum Number of F r i c t i o n Devices = 125 Maximum Number of Dynamic Analyses = 25 Maximum Number of Integration Time Steps i n Each Dynamic Analysis = 3000 Maximum Number of Time-Acceleration Pairs Defining the Ground Motion = 1500 278 These l i m i t a t i o n s can be relaxed by changing a number of dimension statements i n the program. A microcomputer version of the FDBFAP program was also created. The program was compiled with the Microsoft Fortran Optimizing Compiler (Version 4.0) using an overlay structure to accommodate the RAM memory of DOS. Because of t h i s RAM memory capacity (640 kbytes), some l i m i t a t i o n s r e s u l t i n g from the use of fi x e d dimensions are associated with the input data. Maximum Number of Degrees of Freedom Before Condensation = 30 Maximum Number of Columns = 20 Maximum Number of Beams = 12 Maximum Number of F r i c t i o n Devices = 10 Maximum Number of Integration Time Steps i n Each Dynamic Analysis = 2000 Maximum Number of Dynamic Analyses = 25 Maximum Number of Time-Acceleration Pairs Defining the Ground Motion = 1500 These l i m i t a t i o n s can be relaxed to some extent by optimizing the source code of the program. The same source code implemented on the Amdahl V8 main frame computer was used for t h i s microcomputer version of the program. Input Data The following l i n e s define the problem to be solved. Consistent u n i t s must be used throughout. 279 I . General Information 1 l i n e @ 214, F15.5, 414, 2F15.5, 214, F15.5 NUMSTR NFLOOR HEIGHT MODES NUMCOL NUMBEA NUMFRI EYOUNG ACCGRA NSLIPS IDFLAG ZETA NUMSTR: Structure number. NFLOOR: Number of f l o o r s . HEIGHT: Height of the structure. NNODES: Number of nodes. NUMCOL: Number of v e r t i c a l columns. NUMBEA: Number of hori z o n t a l beams. NUMFRI: Number of f r i c t i o n devices. EYOUNG: Young's modulus for a l l members. ACCGRA: Acceleration of g r a v i t y i n proper u n i t s . NSLIPS: Number of dynamic analyses to be performed. IDFLAG: Data checking code. Punch 1 i f only a data checking run i s required. Punch 2 i f a data checking run with c a l c u l a t i o n of the nondimensional parameters of the problem are required. Punch 0 i f the problem i s to be executed. ZETA: Modal damping r a t i o i n f i r s t mode of v i b r a t i o n of the undamped structure. 280 I I . Node Information NNODES l i n e s 0 215 NODNUM LFLOOR NODNUM: Node number. LFLOOR: Floor l e v e l . I I I . Column Information NUMCOL l i n e s @ 314, 2F15.5 ICOLNU NODEJl N0DEJ2 HEIGHT CINERT ICOLNU NODEJl NODEJ2 HEIGHT CINERT Column number. F i r s t node number (punch 0 i f base column) Second node number. Height of the column. Column moment of i n e r t i a . IV. Beam Information NUMBEA l i n e s @ 314, 2F15.5 IBEANU NODEJl NODEJ2 BLENGT BINERT IBEANU: Beam number. NODEJl: F i r s t node number. -N0DEJ2: Second node number. BLENGT: Beam length. BINERT: Beam moment of i n e r t i a . 281 V. Floor Mass Information NFLOOR l i n e s @ 14, F15.5 IFLOOR FLMASS IFLOOR: Floor number. FLMASS: Floor mass. VI. F r i c t i o n Device Information NUMFRI l i n e s @ 214, 6F15.5, II NDEVIC LFLOOR BRACEL PADLEN ABRACE ALINKS ALPHAS PCRITI NDEVIC: Device number. LFLOOR: Floor l o c a t i o n . BRACEL: Length of diagonal brace. PADLEN: Length of diagonal pad. ABRACE: Cross-sectional area of diagonal brace. ALINKS: Cross-sectional area of diagonal pad. ALPHAS: Angle of i n c l i n a t i o n of f r i c t i o n pad (radians) PCRITI: Buckling load of diagonal brace. 282 VII. S l i p Load Information 1 l i n e @ F15.5, II IANALY IWRITE IANALY: Analysis number. IWRITE: Energy time-histories p r i n t i n g code. Punch 1 i f energy time-histories are to be printed. Punch 0 i f energy time-history are not to be printed. Repeat NUMFRI l i n e s @ 13, F15.5, II NSLIPS Times NDEVIC PSLIPS IPRINT NDEVIC: Device number. PSLIPS: Local s l i p load. IPRINT: Slippage time-history p r i n t i n g code. Punch 1 i f slippage time-history i s to be printed for t h i s device i n t h i s a n a l y s i s . Punch 0 i f slippage time-history i s not to be pri n t e d for t h i s device i n t h i s a n a l y s i s . VIII. Time-Step Integration Control Information 1 l i n e @ 16, F15.9 NSTEPS DELTAT NSTEPS: DELTAT: Number of in t e g r a t i o n time-steps. Integration time-step. 283 IX. Earthquake Record Information 1 l i n e @ 14, 2F15.5, 15 NUMQUA ACCMAX TQUAKE NPDATA NUMQUA: Earthquake record number. ACCMAX: Peak ground acc e l e r a t i o n . TQUAKE: Period of peak ground spectral a c c e l e r a t i o n . NPDATA: Number of time-acceleration p a i r s defining the ground motion. X. Ground Acceleration Time-History As many l i n e s as needed to sp e c i f y NPDATA time-acceleration p a i r s , 6 p a i r s to a l i n e (12F6.0). Can be entered i n any u n i t s , the record i s scaled such that the peak ground ac c e l e r a t i o n equals ACCMAX. Input F i l e for a Sample Problem Based on the information determined above, the input f i l e for the second example structure (low r i s e frame) studied i n Chapter 3 (section 3.2) i s now s p e c i f i e d . Data may be entered following the format s p e c i f i c a t i o n s given above, or a l t e r n a t i v e l y i t may be entered with commas separating each data entry. The l a t t e r option i s simpler, and i s thus preferred. This method i s employed below for t h i s example's input f i l e . I.1.10300.,11.II,t.1.100000..9110..11,0.0.01. 1.1. 1.1. I. l. *.l. S.2. (.1. 7.1. 9.7. *.), 10.1. U.J. II. J. 1.0.1.HOC.,3013313)1.1, I, 0.3.3*00. .)013S145!.l. 3.0.).1*00.,3011)1)11.1. *.o.*.)*oo..joi3)i)S2.i. ).i.).iioo..)oi3si))i.i. t.j.i.jtoe. .loiiiDai.i. 7.3.7,3*00. .3013)1)31.1. *.*.*.3*00. .301)3)332.1, 9.S,9.3*00..301)31331.1. 10.4.10.3*00. .301J*-13)2.1. II. 7.11.J400.,1013)1)52.1. 11.1.11.3*00..3013513)1.1. 1.1.1.9100.,111316331-7, 1.3.1.9800. .4493)5)07.3. 3.3.4,9*00..213211331.7. 4.3.4.9*00..1*4071)11.9. 1,4.7,9100..1**011411.*, 4,7,1.9800..1I107B)1*.4. 7.9.10.MOO..130711081.4. 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(1 1117. 13.47 -24*. 13.7) 179. 13.7* 1704. 13.4) -172. 1).41-24*1. 13.97 0003. 14.0) -*4|. 14.0* 177*. 14.13 til. 14.21 -7. 14.27 -914. 14. ))-l«03. 14) 9-1049. 14,45 -34. 14.31 14. 14.37 449. 14.(1 132. 14.49 -11* 14.73 -243. 14.11 -543. 14.47 -490. 14.9) -540. 14.t* -310. 286 APPENDIX B OVERVIEW OF RANDOM PROCESSES AND SPECTRAL ANALYSIS This appendix presents a b r i e f review of random processes and sum-marizes some of the concepts of s p e c t r a l analysis used i n t h i s study. The material can be found i n a number of references (see for example, Clough and Penzien, 1975; L i n , 1967; Bendat and P i e r s o l , 1971; Newland, 1975; Jenkin and Watts, 1968; and Popoulis, 198A). B.1 Random Process A time-series or a time-history i s the c o l l e c t i o n of observations i n time. I f the observations can be predicted p r e c i s e l y , the process i s c a l l e d d e t e r m i n i s t i c . I f , however, the observations can only be defined i n terms of p r o b a b i l i t y statements, the process i s r e f e r r e d to as stochastic (or random). Earthquake ground motions obviously f a l l under the c l a s s i f i c a t i o n of a random process. A random process can formally be defined as an i n f i n i t e set or ensemble of " e q u a l l y - l i k e l y " sample time-histories with some s t a t i s t i c a l or p r o b a b i l i s t i c information about the samples. A random process can best be described by i t s f i r s t and second order moment, namely: +» E [ x ( t j ) ] = u ( t j ) = J x p (x.tj) dx (B.l) X X —oo +00 +00 E W t j ) x ( t 2 ) ] = R x ( t 1 ( t a ) = J" J " x i x 2 P x tei.Xj^i.t,) dx.dx, —00 —00 (B.2) 287 where t x and t 2 are arbitrary times, E[] is the ensemble average or expected value of [] , pCx.tj) is the probability density function of process {x(t)), where {} denotes the ensemble of sample records x(t) , and p ( x t , x 2 ; t t , t 2 ) is the joint probability density function. The first and second order moments are of special interest and are referred to as the mean value u (t :) and the autocorrelation function R ( t 1 , t 2 ) , respectively. If the mean value and autocorrelation function in Equations (B.l) and (B.2) vary with time, then the random process (x(t)} is said to be nonstationary. For the special case where the mean is constant and the autocorrelation function is a function of the time lag T only (x = t 2 - t 1 ) , the process {x(t)} is said to be stationary. Two other quantities of interest are the mean square value ^ ( t ^ ) and the variance o^Ctj) defined as: *»(tj) = Etx'ttj)] (B.3) o» ( t j ) = EUxCt , ) - px(t1))»] (B.4) For a random process with zero mean, the mean square value equals the variance of the process. B.2 Finite Fourier Transform and Power Spectral Density Let a^ n^(t) be a sample record ground acceleration of an ensemble of earthquakes {a(t)J having similar characteristics. If 288 J | a W (t) |dt < ~ (B.5) then the F o u r i e r t r a n s f o r m of a ^ n ^ ( t ) e x i s t s , a c o n d i t i o n which i s u s u a l l y s a t i s f i e d i n p r a c t i c e . The sample r e c o r d a ^ n ^ ( t ) uniquely corresponds to i t s Fourier transform and conversely, the Fourier (n) transform uniquely defines the sample record a (t) through: +00 t F (u,t 0) = J a (t)e dt = J a (t)e dt (B.6) -oo 0 +oo a ( n ) ( t ) = ^ J* F n ( u ) , t 0 ) e i u t du (B.7) —00 where t 0 i s the duration of the record. For a sample accelerogram i n the f i n i t e i n t e r v a l 0 £ t £ t 0 , the power s p e c t r a l density function of the sample can be defined as (Bendat and P i e r s o l , 1971): G (n,u>,t0) = f - F*(w,t 0) F (w,t 0) (B.8) a t Q n n * where F n ( i u , t 0 ) i s the complex conjugate of F n ( u ) , t 0 ) . The power s p e c t r a l density function of the underlying stationary random process i s defined as: G (to) = lira E [G (n.w,t 0)] (B.9) where E[G (n,w,t 0)] i s the expected value of the power s p e c t r a l density Si over the ensemble of records. Substituting Equation (B.8) i n t o Equation (B.9) y i e l d s : 289 1 G„(u) = lira 7— E [|F„(u),t0) | 2] (B.10) a t . n t -*<*> 0 S i n c e a ^ n ^ ( t ) i s r e a l , G (ui) i s an even f u n c t i o n of frequency 3L (Bendat and P i e r s o l , 1971): G (-ui) = G (ui) ( B . l l ) a a For the one-sided power spectral density function (S (ui)) S (ui) = 2 lira 7- E[|F ( u i . t j l 2 ] 0 <; u < » (B.12) a t n n 0 to-co 0 A s t a t i o n a r y random process contains an i n f i n i t e number of sample records with i n f i n i t e duration, whereas records of p h y s i c a l phenomena (such as earthquakes) are few i n number and have f i n i t e durations. Consequently, an estimate of the power sp e c t r a l density function of the ensemble {S (ui)} can be obtained by f i r s t computing the power spectral Si d e n s i t y f u n c t i o n S^ n^(u) of each sample record (n) and then averaging the ensemble of s p e c t r a l d e n s i t y components at each frequency S (ui) Si (Newland, 1975). The averaging i s intended to approximate the expected value i n Equation (B.12), which can be replaced by the following two equations: S a ( n ) ( u i ) = \- | F n ( u i , t 0 ) | 2 (B.13) N S (u) = 5 - I S a ( n ) ( u i ) (B.1A) a N , a s n=l where N i s the sample s i z e , s ^ 290 A basic property of the power sp e c t r a l density function i s that the mean square a c c e l e r a t i o n a 2 i s obtained by integr a t i n g S (ui) over a l l di frequencies: 00 a 2 = J" S (ui) du (B.15) 0 a 291 APPENDIX C CONTRIBUTION OF STRONG MOTION SEGMENT OF EARTHQUAKE  ON STRUCTURAL RESPONSE The contribution of the strong shaking segment of an earthquake on the s t r u c t u r a l response can be evaluated by computing the response of a structure when excited by the strong shaking segment alone and comparing the r e s u l t s with those obtained when the structure i s excited by the complete earthquake record. For t h i s purpose the low-rise b u i l d i n g configuration described i n Section 3.3 was analyzed by the program Drain-2D (Kannan and Powell, 1973). A family of three frames was considered for a n a l y s i s . Included were (1) an unbraced structure; (2) an x-braced structure with the diagonal braces allowed to y i e l d i n tension and buckle e l a s t i c a l l y i n compression and (3) a f r i c t i o n damped structure. To demonstrate the superior performance of the f r i c t i o n damped structure over conventional b u i l d i n g systems, i n e l a s t i c time-history dynamic analyses were c a r r i e d out using the Taft earthquake with the a c c e l e r a t i o n values, scaled to three times t h e i r actual values (peak ground a c c e l e r a t i o n = 0.54 g) to enhance s t r u c t u r a l damage. The FDBFAP program was i n i t i a l l y used to determine a uniform optimum s l i p load d i s t r i b u t i o n of the f r i c t i o n damped structure under the Taft earthquake. The r e s u l t s of the optimum s l i p load study are shown i n F i g . C . l . It can be seen that the optimum s l i p load i s 107 kN for each f r i c -t i o n device; the associated Relative Performance Index (RPI), Equation (2.93), i s 0.240. Note that t h i s value of the optimum s l i p load i s d i f f e r e n t than the value (134 kN) obtained i n Section 3.3, which was concerned with the analysis of the same structure excited by the 292 0 50 100 150 200 Local Slip Load [kN] Figure C . l S l i p Load Optimization for F r i c t i o n Damped Structure Under Taft Earthquake. Newmark-Blume-Kapur a r t i f i c i a l earthquake. This suggests that the earthquake record influences the optimum s l i p load of a f r i c t i o n damped structure. The s t r u c t u r a l damage i n the various members of the d i f f e r e n t frames a f t e r the end of the complete Ta f t record and a f t e r the end of the strong motion segment of the Taft record i s i l l u s t r a t e d i n F i g . C.2 I t can be seen that the damage pattern caused by the strong motion segment i s i d e n t i c a l to the damage pattern caused by the complete earth-quake record. Also, the superior performance of the f r i c t i o n damped structure i s apparent. The analysis indicated that only s l i g h t y i e l d i n g occurred i n the base columns of the f r i c t i o n damped frame compared to 293 serious y i e l d i n g i n the beams, columns, and braces of the two conven-t i o n a l s t r u c t u r a l systems. Quantitative r e s u l t s of the analyses are presented i n Figs. C.3 to C.6 i n terms of envelopes of l a t e r a l d e f l e c t i o n s , beam and column bend-ing moments and column shear forces. For each of these parameters, the values obtained when considering only the strong motion portion of the earthquake are very s i m i l a r to the values obtained when the complete earthquake record i s used. 294 Complete Record Strong Motion Segment Unbraced Unbraced < Unbraced » < ) < > < > • • a • i > 4 • 4 • ft Unbraced • i »_ j • L 1 [ i J • X-Braced X-Braced X-Braced X-Braced Friction Damped Friction Damped 4 Friction Damped Friction Damped i * i 1 • • Member Yielded Figure C.2 Str u c t u r a l Damage Afte r Earthquake, Taft (0.54g) 295 Unbraced Structure CD *> o o 0) .^ 0) o o 0) 0) —J o o 2-1-2-2-1-• Complete Record o Strong Motion Segment 50 100 150 Deflection [mm] X-Braced Structure 200 • Complete Record o Strong Motion Segment 50 100 150 Deflection [mm] Friction Damped Structure 200 • Complete Record o Strong Motion Segment 50 100 150 Deflection [mm] 200 Figure C.3 Envelopes of La t e r a l Deflections, East Side, Taft (0.54g) 296 Unbraced Structure CD Co o CO CO k. o o 4) - J o o 100 200 300 400 Bending Moment [kN-m] X-Braced Structure 500 100 200 300 400 Bending Moment [kN-m] Friction Damped Structure 500 100 200 300 400 Bending Moment [kN-m] 500 Figure C.4 Envelopes of Beam Bending Moments, East Bay, Taft (0.54g) 297 3-1 <D 2 O O O o u: 3 i i-Unbraced Structure — " i • Complete Record o Strong Motion Segment f \ ^ - C ^ ^ . Plostfc 100 200 300 400 500 Bending Moment [kN-m] X-Braced Structure 600 • Complete Record ° Strong Motion Segment ! Plastic Moment ^ 100 200 300 400 500 Bending Moment [kN-m] 600 0) - J o o 2-1-Friction Damped Structure • Complete Record o Strong Motion Segment Plostic Moment \ 100 200 300 400 500 Bending Moment [kN-m] 600 Figure C.5 Envelopes of Column Bending Moments, East Side, Taft (0.54g) 298 > —I o o c: Q) > — J O O CD — J O o 2-1-0 Unbraced Structure — T t — • Comphlt Rmcord  o Strong Uoflon S»gm»nt T o 50 100 150 200 250 Shear Force [kN] X-Braced Structure 2-1-o 3 2-1-300 • Comptott Record © Strong Motion Stgmont A_, ? 50 100 150 200 250 300 Shear Force [kN] Friction Damped Structure • Compitt* Record o Strong Motion Segment I 50 100 150 200 250 Shear Force [kN] 300 Figure C.6 Envelopes of Column Shear Forces, East Side, Ta f t (0.54g) 299 APPENDIX D GESER USER'S GUIDE I d e n t i f i c a t i o n GESER.S: Generation of Equivalent Stationary Earthquake Records Programmer: A. F i l i a t r a u l t , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1987. Purpose To generate an ensemble of stationary and Gaussian time-series having the same power, frequency content and duration as a s p e c i f i e d time-series. Each sample time-series has the same Fourier amplitude spectrum as the s p e c i f i e d time-series but the phase spectrum i s d i f f e r e n t . Equation (B.13) i s used by the program to c a l c u l a t e the power spe c t r a l density of the strong segment of the r e a l record. The Fourier t r a n s f o r m of the r e a l record ( F n ( u , s 0 ) ) i s obtained by using a Discrete Fast Fourier Transform (DFFT) subroutine: F n ( V 8 « ) (D.l) S (n) | F n ( V S o > l ' (D.2) a where N 'k number of a c c e l e r a t i o n data points k ^ d i s c r e t e frequency i n the frequency range of i n t e r e s t d i s c r e t e value of acceleration at time t . J 300 i = V - l s 0 = strong motion duration The F o u r i e r t r a n s f o r m of the e q u i v a l e n t s t a t i o n a r y r e c o r d (s) (F ( u , s 0 ) ) i s such t h a t i t s amplitude i s the same as the amplitude of the Fourier transform of the r e a l record but i t s phase i s modified randomly: ~ (n ) • i ' » (D.3) where i s a random v a r i a b l e having a uniform p r o b a b i l i t y d i s t r i b u t i o n over the i n t e r v a l [0,2T T]. This procedure ensures that the corresponding a c c e l e r a t i o n record i s stationary and Gaussian (Papoulis, 1984). The (s) e q u i v a l e n t s t a t i o n a r y a c c e l e r a t i o n record (a (t)) i s obtained by an Inverse Discrete Fast Fourier Transform (IDFFT): ( } , N , » iw,t • 3 s ° k-1 n Capacity Limitations The maximum number of a c c e l e r a t i o n data points i s 5000. This l i m i t a t i o n can be relaxed by changing a few dimension statements at the beginning of the main subroutine. Input Data I. General Information 1 l i n e § 14, F4.2, 12, F15.5 301 NDATA DELTA NREC SEED NDATA: Number of data points i n s p e c i f i e d time-series, required to be even. DELTA: Time increment. NREC: Number of equivalent stationary time-series required. SEED: Seed for random number generator. S p e c i f i e d Time-Series As many l i n e s as needed to sp e c i f y NDATA values, 8 values to a l i n e (8F10.0) 302 APPENDIX E EFFECT OF STRONG MOTION SEGMENT OF NONSTATIONARY EARTHQUAKE ON STRUCTURAL RESPONSE The estimated strong motion segment (3 to 14 seconds), of the 1952 Taft S69E earthquake, F i g . 5.2, was used with GESER to generate an ensemble of 4 equivalent stationary accelerograms. F i g . E . l presents these 4 stationary accelerograms along with the strong motion segment of the r e a l Taft record. The computer program DRAIN-2D was used to analyze the same low r i s e b u i l d i n g described i n Section 3.3. This structure was excited by the ensemble of 4 equivalent stationary records with a l l a c c e l e r a t i o n values m u l t i p l i e d by a factor of 3. Figures E.2 to E.6 compare the r e s u l t s of these analyses with the ones obtained when the structure i s excited by the strong motion segment of the r e a l Taft accelerogram (also scaled by a factor of 3 to 0.54g). I t can be seen that the mean response values determined from the ensemble compare reasonably well with the values based on an analysis using the strong motion segment . of the r e a l earthquake. The v a r i a t i o n s noted are of an order normally expected and accepted i n earthquake engineering p r a c t i c e . 303 0 . 2 - i D J 0.0 Strong Motion Segment of Real Taft S69E Accelerogram -0.2--0.2-0.2 D > 0.0 -0.2 0.2-i CD 0.0 -0.2 4 6 8 Time [sec] Equivalent Stationary Accelerogram #1 2 4 6 8 Time [sec] Equivalent Stationary Accelerogram #2 4 6 8 Time [sec] Equivalent Stationary Accelerogram / J Time [sec] Equivalent Stationary Accelerogram #4 4 6 8 Time [sec] J L/v vvy i r 10 A r\ h\ \ / A ffl A A nflA-Yl/l Ar V /l/i ^ 1 • • Figure E . l Equivalent Stationary Records for the Taft Earthquake (3-14 seconds) 304 Unbraced X-Braced Friction Damped c g 1 E D) D) « 2 co 55 i. a o CO DC 1 — 1 co c o CO CO CM # O i i i r >, CO S 8 co ac CO c .o +5 CO o o o DC Member Yielded Figure E .2 S t r u c t u r a l Damage Afte r Earthquakes, Taft vs Stationary Records 305 Unbraced Structure O o 0) —J O o 0) — J o o 2-1-2-1-* * >* w • Strong Uotlon S*gm*nl o Stotlonory Rteord fl o Stotlonory Rteord §2 • Stotlonory Rteord /3 • Stotlonory Record _/4 • Uton of Stotlonory Rteordt T ' 1 ' 1 ' 1 ' 50 100 150 Deflection [mm] X-Braced Structure 200 WJ(rt ... T • Strong Uotlon Stgmtnt • Stationary Rteord tt o Stotlonory Rteord /2 t Stotlonory Rteord fi <• Stotlonory Rteord jf4 • Uton of Stotlonory Rteordt 50 100 150 Deflection [mm] Friction Damped Structure 200 • Strong Motion Sogmont o Stotlonory focord jl o Stationary Record §2 • Stotlonory ftoeord / J " Stotlonory theord jf4 • Uton of Stotlonory Rocordt 50 100 150 Deflection [mm] 200 Figure E.3 Envelopes of La t e r a l D eflections, East Side, Taft vs Stationary Records OP n (D tn j> in r t „ p> W rt 3 O fl> S o i-l *o srf CD W 2? o ° o i-t CO W CD (D P-OQ O g ro 3 en W Pi Floor Level _ M u Floor Level — M u ^ CD o aaoaig m m » x x x x J F/oor Leve/ K ) CM 307 —I O O —I O O — J o o c: 2-1-Unbraced Structure 100 200 300 400 500 Bending Moment [kN-m] X-Braced Structure 600 • Strong Mo/ton S»gmont • ^ W t e f t o r i / » » c o n t f / I ° $*o*ta**rf *fCO>* i2 " S^tonorf Record 03 _ « Stationary kecori jf4 • ftfeon of Stationary Hoeord* ^ ^ ^ k ^ i ^ ^ ^ - M o m t n t 100 200 300 400 500 Bending Moment [kN-m] Friction Damped Structure 600 100 200 300 400 500 Bending Moment [kN-m] 600 Figure E.5 Envelopes of Column Bending Moments, East Side, Taft vs Stationary Records 308 <D CD O O CD CD —J 4* O o CD CD - J O o 3 - T 2-1-0-2-1-Unbraced Structure • Strong Uotlon S+gm+nt o S W t e M r y t » c t v t f / /  " r. ItfF^cP. jfA • K a o n of Slottonorr mmoord% rr -*-9 5 0 700 750 200 250 Shear Force [kN-m] 300 X—Braced Structure f .1 • Strong Motion S o y m o n f • 8*Ql(ei%ory »»contf / ! _ o SToHo/iorr Eocortf /7 • ^ ' k ^ ^ r . 50 100 150 200 250 Shear Force [kN-m] Friction Damped Structure 300 50 100 150 200 Shear Force [kN-m] 250 300 Figure E.6 Envelopes of Column Shear Forces, East Side, Taft vs Stationary Records 309 APPENDIX F THE SPECTRAL MOMENTS METHOD FOR ESTIMATION OF KANAI-TAJIMI PARAMETERS FROM REAL EARTHQUAKE ACCELEROGRAMS The s p e c t r a l moments of a power spectral density function are key parameters i n the analysis of earthquake accelerograms i n the frequency domain. The i * " * 1 s p e c t r a l moment (X^) i s defined as (Vanmarcke, 1977): X. = J" w1 S (u) du i = 0,1,2,... (F.l) 1 0 a A measure of where the spe c t r a l values are concentrated along the frequency axis i s given by the cent r a l frequency ( u £ ) : ^ 2 1 ^ 2 w = (r-) (F.2) c Xo which i s analogous to the root mean square value of a random va r i a b l e (Vanmarcke, 1977). A convenient measure of the spread of the dispersion of the power sp e c t r a l density function about i t s c e n t r a l frequency i s given by the shape factor (6): V 1 , 2 x 0 x 2 6 i s dimensionless, always l i e s between 0 and 1, and increases with increasing bandwith (Vanmarcke, 1972,1977). The method of spectral 310 moments computes the Kanai-Tajimi parameters i n such a way that the zero s p e c t r a l moment ( X 0 ) , the shape factor (6) and the ce n t r a l frequency (w ) o f t h e a c t u a l e a r t h q u a k e s p e c t r u m and those of the f i t t e d c Kanai-Tajimi power sp e c t r a l density function are the same. Substituting Equation (5.5) into Equation ( F . l ) , and using a f i n i t e frequency range of 0~wmax. the shape factor and central frequency of the f i t t e d Kanai-Tajimi power sp e c t r a l density function are obtained from: 6 = [1 - < T ^ - ) ] ( F . A ) J 0 J 2 J 2 1 ' 2 where and c J 0 8 u /ui max g J. = J" a 1 S(a) da (F.6) 1 0 1 + Ah Ja 2 § ( a ) = (1-a.). S Ah*a* = | H ( W ) | 2 ( F ' 7 ) g ui a = — ui g The procedure used i n the method of spectral moments can be summarized as follows: (1) Compute X0,X1 and X2 from Equation (F.l) using the actual earth-quake accelerogram power s p e c t r a l density with ^> m a x as the upper l i m i t of the i n t e g r a l . 311 (2) U s i n g X Q, X x and X z computed i n (1) c a l c u l a t e wc and o from Equations (F.2) and (F.3). (3) Assume v a l u e s of h and ui and use them i n Equations (F.A) and g g (F.5) to compute 6 and u>c. Compare 6 and ui c with the corresponding values computed i n (2). Repeat the procedure u n t i l s a t i s f a c t o r y agreement i s reached between the assumed and computed values. (A) Compute the bedrock power sp e c t r a l density function by equating the zero s p e c t r a l moments from the raw and smooth power spectral d e n s i t i e s : S A - 3 7 (F.8) 312 APPENDIX G SIMEA USER'S GUIDE I d e n t i f i c a t i o n SIMEA.S: SIMulation of Earthquake Accelerogram. Programmer: A. F i l i a t r a u l t , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1987. Purpose To simulate an ensemble of earthquake accelerograms having the c h a r a c t e r i s t i c s of a target accelerogram. The earthquake accelerograms are simulated based on the stochastic model proposed by Vanmarcke and L a i (1980,1982). The earthquake accelerogram i s represented by an ensemble of segments of a stationary stochastic process; the segments have l i m i t e d duration. The duration (s 0) i s calculated according to a regression formula: s 0 = 30 exp (-3.254 a 0.35 g ) where a g = Peak ground ac c e l e r a t i o n i n g. = Strong motion duration i n seconds. The frequency content of the ground motion i s described i n terms of the Kanai-Tajimi power spectral density functions (Kanai (1957) ; Tajimi (I960)): 313 where S a(w) = Power sp e c t r a l density function. h = Sharpness parameter or ground damping r a t i o . 6 u = Predominant ground frequency. S^ = White noise bedrock spectral i n t e n s i t y . The simulation procedure consists of specifying a Fourier amplitude spectrum and generating a random Fourier phase spectrum, each phase value having a uniform p r o b a b i l i t y d i s t r i b u t i o n over the i n t e r v a l [0,2T T]. The simulated accelerogram i s generated by an Inverse Fast Fourier Transform (IFFT). The rms ac c e l e r a t i o n i s calculated according to the Vanmarcke and L a i formula: a o = 8 : 0 [2 Sn[0.295 s0(6.28+5.15 T )/T ]]*'» 0 g g rms acc e l e r a t i o n . Peak ground acc e l e r a t i o n . Strong motion duration. Predominant ground period (T = 2u/u) ). g g The simulated accelerogram i s scaled i n the time domain to y i e l d the proper rms a c c e l e r a t i o n . where °o = a T g 314 Capacity Limitions The Fast Fourier Transform (FFT) subroutine used accepts data whose dimensions are a power of 2 only ( i . e . , 2, 4, 8, 16, e t c . ) . The maximum number of data points that can be s p e c i f i e d i s 2048 ( 2 1 1 ) . Input Data The input data are supplied i n t e r a c t i v e l y to the program i n free format. The following input data must be s p e c i f i e d : SEED: Seed for random phase generator. IP: Power spe c i f y i n g the number of points i n time seri e s ( l s l P s i l l ) . AG Peak ground ac c e l e r a t i o n (g). TG Ground predominant period (seconds). HG Ground damping r a t i o . NREC: Number of simulated accelerograms required. 315 APPENDIX H BILINEAR LEAST SQUARE FIT FOR LEAST SQUARE SLOPE PARAMETERS The parameters a and b i n F i g . 7.29 are approximated by b i l i n e a r curves with the two l i n e a r branches j o i n i n g at T /T =1. g u a = m, • (T /T ) i a g u m. • (T /T ) + C J a g u  £ 0 £ T /T £ 1 g u T./T > 1 b u (H.l) b = m 3 b • ( T / T u ) + C b 0 £ T /T £ 1 g u T /T > 1 g u (H.2) From con t i n u i t y at T /T = 1 we get: g u m, = m. _ C 2 a  la a (H.3) m 2 b = raib " c b (H.4) Substituting Equations (H.3) and (H.4) into (H.l) and (H.2) y i e l d s : a = m. • (T /T ) i a g u (m. -C ) • (T /T ) + C 1a a g u £ 0 £ T /T s i g u T /T > 1 g u (H.5) b = mib ' W ( m i b - C b ) ' ( Y V + Cb O ^ T / T S i g u T /T > 1 g u (H.6) 316 The siims of the square of the erros I and 1^ can be written as: N ( 1 ) (1) (1) * I = X [ar ; - m, • (T /T ) K }] a i = 1 l ia g u i N<2> (2) (2) 1 + X [ a ^ ; - (ml -C ) • (T /T ) - C ] (H.7) . , l J a a g u . a i = l 6 l N ( 1 ) (1} f l ) J I, = 2 [b)i} - m 1 K • (T /T ) ] b . , I xb g u . i= l 6 l N ( 2 ) (2) (2) 2 + X [ b | Z ; - (m 1 K-C K) • (T /T ) - C,] (H.8) . , l *b b g u . b i = l 6 l where the superscripts (1) and (2) r e f e r to the f i r s t l i n e a r branch (T /T < 1) and the second l i n e a r branch (T /T > 1) r e s p e c t i v e l y . g u g u r J (2) and N are the number of data points i n the corresponding branches. The l e a s t square method consists i n minimizing the square of the er r o r s : 31 a 8m. 1a = 0 (H.9) 31 a ac a = 0 (H.10) 3 m i b = 0 (H.ll) ac, b = 0 (H.12) 317 which leads to a p a i r of systems of l i n e a r equations: and r "> m. I a. • (T /T ). 1a i-1 1 g U 1 [A] > N ( 2 ) 1 [a< 2 ) - a f 2 ) • ( T / T ) f 2 ) ] C a L > Li-i 1 1 g u l . (H.13) f N ( 1 ) + N ( 2 ) ^ mib . l = 1 b i ' ( T g / T u > i [A] i N ( 2 ) X [bf2) -bf2) • ( T / T ) f 2 ) ] > b Li-1 1 1 8 « i (H.1A) where [A] = A J 1 A 2 2 and N ( 1 V 2 ) 2 i-1 (T /T ). g u ' i N (2) I [T /T ) f 2 ) - ( ( T /T ) f 2 ) ) 2 ] i = 1 g u l g u 1 N ( 2 ) ^22 = tl [ ( ( T g / T u ) { 2 ) ) S - 2 ( T g / T u ) { 2 ) + 1] from which the c o e f f i c i e n t s m. , m., , C and C, can be evaluated. l a lb a b 318 APPENDIX I EXPRESSIONS FOR FAMILY OF BILINEAR CURVES  OF LEAST SQUARE SLOPE PARAMETERS The ordinates of the b i l i n e a r curves shown i n F i g . 9.2 are c a l c u l a t e d f o r two p a r t i c u l a r values of T /T : T /T = 1 and T /T =15. g u g u g u These ordinates are presented i n F i g s . E . l and E.2 along w i t h l e a s t square r e g r e s s i o n l i n e s . These ordinates are approximated by: a(T /T = 1) = -1.2378NS - 0.3072 g u a(T /T = 15) = -1.0740NS - 0.1000 g u b(T II = 1) = 1.0373NS + 0.4259 g u b(T /T = 15) = 1.0065NS + 0.4483 g u (I.D (1.2) (1.3) (1.4) These four equations describe completely the f a m i l y of b i l i n e a r curves f o r a and b: a = (-1.2378NS - 0.3072) T /T g u 0 < T /T S i g u (0.0117NS + 0.0148) T /T - 1.2495NS - 0.3220 T /T > 1 g. u g u (1.5) b = " (1.0373NS + 0.4259) T /T g u 0 < T /T S g u 1 (-0.0022NS + 0.0016) T /T + 1.0395NS + 0.4243 g u T /T > g u 1 (1.6) 319 Figure E . l Ordinates of a Values for T /T = 1 and 15 g u 320 Figure E.2 Ordinates of b Values for T /T = 1 and 15 g u 

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