Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Performance evaluation of friction damped braced steel frames under simulated earthquake loads Filiatrault, André 1985

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1985_A7 F54.pdf [ 12.08MB ]
Metadata
JSON: 1.0062844.json
JSON-LD: 1.0062844+ld.json
RDF/XML (Pretty): 1.0062844.xml
RDF/JSON: 1.0062844+rdf.json
Turtle: 1.0062844+rdf-turtle.txt
N-Triples: 1.0062844+rdf-ntriples.txt
Original Record: 1.0062844 +original-record.json
Full Text
1.0062844.txt
Citation
1.0062844.ris

Full Text

PERFORMANCE EVALUATION OF FRICTION DAMPED BRACED STEEL FRAMES UNDER SIMULATED EARTHQUAKE LOADS  by ANDRE FILIATRAULT B.A.Sc, University of Sherbrooke, 1983  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1985  © Andre F i l i a t r a u l t , 1985  In presenting  this thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an  advanced degree at the University of B r i t i s h Columbia, I agree that Library s h a l l make i t freely available for reference and study.  the  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 rep-  resentatives.  It i s understood that copying or publication of this thesis  for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of  C i v i l Engineering  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  Date  October,  1985  ABSTRACT  This thesis presents the results obtained from q u a l i f i c a t i o n tests of a new f r i c t i o n damping system, which has been proposed i n order to improve the response of Moment Resisting Frames and Braced Moment Resisting Frames during severe earthquakes.  The system b a s i c a l l y consists of a s p e c i a l  inexpensive 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 main objective i s to study the  performance of a 3 storey F r i c t i o n Damped Braced Frame model under simulated earthquake loads. The main members of the test frame were chosen from available hotr o l l e d sections and the mass selected to provide the expected fundamental frequency of a three storey Moment Resisting Frame. was performed on an earthquake simulator table.  The seismic testing  The experimental  results  are compared with the findings of an i n e l a s t i c time-history dynamic analysis.  Two different computer models were used for this purpose.  The  f i r s t one i s based on an equivalent hysteretic model and i s only approximate, since i t  does not take into account the complete behaviour of the  f r i c t i o n devices.  A more refined computer model was then developed and the  results from the two models are compared.  It  i s found that the simpler  approximate model overestimates the energy dissipated by the devices, but the inaccuracy i s r e l a t i v e l y  small (10-20% i n resulting member f o r c e s ) .  To quantify the performance of the F r i c t i o n Damped Braced Frame r e l a tive to conventional aseismic systems, an equivalent viscous damping study i s made.  Viscous damping i s added to the Moment Resisting Frame and the  Braced Moment Resisting Frame u n t i l their responses become similar to the  - ii  -  response of the F r i c t i o n Damped Braced Frame.  The r e s u l t s show that f o r  this purpose 38% of c r i t i c a l damping must be added to the Moment Resisting Frame and 12% to the Braced Moment Resisting Frame.  The new system becomes  more e f f i c i e n t as the intensity of the earthquake increases. The economical p o t e n t i a l of the new damping system i s investigated by designing a reduced size F r i c t i o n Damped Braced Frame having response c h a r a c t e r i s t i c s which are similar to those of conventional s t r u c t u r a l systems with heavier members.  For the model frames studied, the results  show that i f the e f f e c t s of wind, l i v e and torsion loads are neglected, i t i s possible to reduce the members sizes of the F r i c t i o n Damped Braced Frame by 47% and s t i l l achieve a superior performance under strong earthquake, i n comparison to the seismic response of the two other conventional frames with t h e i r o r i g i n a l , heavier members. The r e s u l t s , both a n a l y t i c a l and experimental, c l e a r l y indicate the superior performance of the f r i c t i o n damped braced frame compared to conventional building systems.  Even an earthquake record with a peak  acceleration of 0.9 g causes no damage to the F r i c t i o n Damped Braced Frame, while the Moment Resisting Frame and the Braced Moment Resisting frame undergo large i n e l a s t i c deformations.  - iii  -  TABLE OF CONTENTS Page  ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF FIGURES  ..  viii  LIST OF TABLES  xiv  ACKNOWLEDGEMENTS  xvi  1.  2.  3.  4.  INTRODUCTION 1.1  Background  1.2  Object and Scope  1 .•  1 4  DAMPING SYSTEM  7  2.1  Description  7  2.2  Simplified Model  8  2.3  Refined Model  11  DESIGN OF TEST FRAMES  13  3.1  Dimensional Analysis  13  3.2  P r a c t i c a l Design  17  3.3  Remarks on F i n a l Design  19  ANALYTICAL STUDY  21  4.1  Concept of Optimum S l i p Load  21  4.2  Optimum S l i p Load Study  22  4.3  Comparison Between Approximate and Refined Models  25  4.4  Response Under Random White Noise Excitation  27  4.5  Equivalent Viscous Damping Study  30  - iv -  TABLE OF CONTENTS (Continued) Page  5.  6.  7.  8.  9.  CALIBRATION AND  QUALITY CONTROL OF FRICTION DEVICES  33  5.1  General  33  5.2  Experimental Set Up  34  5.3  S t a b i l i t y Tests  35  5.4  Tests of Complete Devices  36  5.5  C a l i b r a t i o n Curves  38  ESTIMATION OF MATERIALS PROPERTIES  39  6.1  General  39  6.2  Tests of Cross-Braces  40  6.3  Tests of Main Members  42  EXPERIMENTAL SET UP ON SHAKING TABLE  44  7.1  Shaking Table  44  7.2  Data Acquisition  45  7.3  Instrumentation  45  SYSTEM IDENTIFICATION THEORY  48  8.1  General  48  8.2  Undamped Free Vibration Analysis  49  8.3  Fourier Spectrum Analysis f o r Frequency Determination ....  51  8.4  Complex Frequency Response or Mobility Function  53  8.5  Frequency Assurance C r i t e r i a  57  8.6  Experimental Determination of Mode Shapes  58  8.7  Modal Assurance Matrix  60  8.8  Experimental Determination of Damping  61  SYSTEM IDENTIFICATION OF THE MOMENT RESISTING FRAME  68  9.1  68  Free Vibration Analysis  TABLE OF CONTENTS (Continued) Page  10.  11.  12.  13.  9.2  Harmonic Forced Vibration Test  69  9.3  Experimental Determination of Mode Shapes  70  9.4  Mobility Function  71  9.5  Experimental Determination of Damping  72  SYSTEM IDENTIFICATION OF THE BRACED MOMENT RESISTING FRAME  75  10.1 Free Vibration Analysis  75  10.2 Harmonic Forced Vibration Test  76  10.3 Experimental Determination of Mode Shapes  77  10.4 Experimental Determination of Damping  78  SYSTEM IDENTIFICATION OF THE FRICTION DAMPED BRACED FRAME UNDER LOW AMPLITUDE EXCITATIONS  80  11.1 Free Vibration Analysis  80  11.2 Harmonic Forced Vibration Test  82  11.3 Experimental Determination of Mode Shapes  83  11.4 Experimental Determination of Damping  84  SEISMIC TESTS ON SHAKING TABLE  86  12.1 Seismic Testing Program, Model Frame #1  86  12.2 Test Results, Model Frame #1  89  12.3 Supplementary Tests, Model Frame #1  93  12.4 Seismic Testing Program, Model Frame #2  96  12.5 Test Results, Model Frame #2  96  12.6 Energy Balance  98  ECONOMIC POTENTIAL OF FRICTION DAMPED BRACED FRAME  103  13.1 Introduction  103  13.2 Analysis of Different Section Sizes  103  - vi -  TABLE OF CONTENTS (Continued) Page  14.  CONCLUSIONS  106  13.1 Summary and Conclusions  106  13.2 Future Research  108  BIBLIOGRAPHY  110  FIGURES  112  TABLES  204  APPENDIX A:  LISTING OF PROGRAM "VIBRATION"  218  APPENDIX B:  LISTING OF PROGRAM "ENERGY"  220  - vii -  LIST OF FIGURES Page  1.1  Typical Hysteresis Loops of a Tension Brace (from Ref.:3)  112  1.2  Location of F r i c t i o n Device (from Ref.: 3)  112  2.1  Hysteresis Loops of Simple F r i c t i o n Joints (from Ref.:3)  113  2.2  Hysteresis Loop of a F r i c t i o n Joint Where the Braces are Designed i n Tension Only  114  2.3  Mechanism of F r i c t i o n Device (after Ref.: 3)  115  2.4  Possible Arrangements of F r i c t i o n Damped Braced Frames (from Ref. :3)  116  2.5  Hysteretic Behaviour of a Simple F r i c t i o n Damped Braced Frame  117  2.6 2.7  Simplified Model of a F r i c t i o n Damped Braced Frame Unstable Mode of a F r i c t i o n Damped Braced Frame Modelled With Truss Elements  118 119  2.8  Refined Model of a F r i c t i o n Damped Braced Frame  120  3.1  Dimensions and Member Sizes of Prototype Structure (from Ref:8) 121  3.2  General Arrangement of Model Frame  122  3.3  Details 1-2 of Model Frame  123  3.4  Details 3-4 of Model Frame  124  3.5  Details 5-6-7 of Model Frame  125  3.6  Model Frame Mounted on the Shaking Table  126  4.1  Concept of Optimum S l i p Load  127  4.2  Free Body Diagram of a F r i c t i o n Device at Slipping  128  4.3  Y i e l d Interaction Surfaces f o r Model Frame  129  4.4  Earthquakes Used for the Optimum Slip Load Study  130  - viii -  LIST OF FIGURES (Continued) P  a  S  e  4.5  Results of Optimum Slip Load Study, E l Centro Earthquake  131  4.6  Results of Optimum S l i p Load Study, P a r k f i e l d Earthquake  132  4.7  Results of Optimum Slip Load Study, A r t i f i c i a l Earthquake  133  4.8  Structural Damage After E l Centro Earthquake  134  4.9  Results Envelope for E l Centro Earthquake  135  4.10 Band Limited (0-25 Hz) White Noise Record  136  4.11 Structural Damage After White Noise Excitation  137  4.12 Results Envelope for White Noise E x c i t a t i o n  138  4.13 Time-Histories of Third Floor Deflection for White Noise Excitation  139  4.14 Equivalent Viscous Damping Study, Newmark-Blume-Kapur A r t i f i c i a l Earthquake  140  5.1  General Arrangement of F r i c t i o n Device  141  5.2  Details of F r i c t i o n Device  142  5.3  F r i c t i o n Surfaces of F r i c t i o n Device  143  5.4  F r i c t i o n Device on Model Frame  144  5.5  Experimental Set Up for Cyclic Tests of F r i c t i o n Devices  145  5.6  Experimental Set Up for S t a b i l i t y Tests  146  5.7  Typical Hysteresis Loop From S t a b i l i t y Tests  147  5.8  Results of S t a b i l i t y Tests for Various Frequencies  148  5.9  Hysteresis Loop From Original Device  149  5.10 Hysteresis Loop From Modified Device  150  5.11 C a l i b r a t i o n Curves of F r i c t i o n Devices  151  - ix -  LIST OF FIGURES (Continued) Page  6.1  D e t a i l of Brace Unit Used on the F r i c t i o n Damped Braced Frame  152  6.2  Permanent Deformed Shape of Brace Unit After U n i a x i a l Test #1  153  6.3  Load-Deformation Curve From Uniaxial Test #1 on Brace Unit .... 154  6.4  Load-Deformation Curve from Uniaxial Test #2 on Brace Unit .... 155  6.5  Load-Deformation Curve From Uniaxial Test on Main Members (S75x 8)  156  7.1  General Arrangement of Earthquake Simulator Table  157  7.2  Physical Arrangement of Data Acquisition System for Earthquake Simulator Table  158  S t r a i n Gage Accelerometer on F i r s t Floor Cross-Beam of Model Frame  159  Potentiometer Used to Measure Absolute Displacement of Model Frame  159  7.5  Strain Gage Unit on Base Column of Model Frame  160  7.6  Strain Gage Unit on F i r s t Floor Beam of Model Frame  161  8.1  Degrees of Freedom Considered to Determine the Experimental  7.3 7.4  Mode Shapes of the Model Frames  162  8.2  Bandwidth Method Applied to a Multi-Degree-of-Freedom System .. 163  8.3  Typical Acceleration Decay Record  163  9.1  Moment Resisting Frame on Shaking Table  164  9.2  Computer Model Used for the Free Vibration Analysis of the Moment Resisting Frame  165  Predicted Natural Frequencies and Mode Shapes of the Moment Resisting Frame  166  9.3  - x -  LIST OF FIGURES (Continued) P a  e  9.4  Fourier Spectrum Analysis of the Moment Resisting Frame  167  9.5  Measured vs. Predicted Natural Frequencies of the Moment Resisting Frame  168  9.6  Estimation of Mode Shapes for the Moment Resisting Frame  169  9.7  Measured vs. Predicted Mode Shapes of the Moment Resisting Frame  170  9.8  Experimental Mobility Functions f o r the Moment Resisting Frame 171  9.9  Bandwidth Method Applied to the F i r s t Floor Mobility Function of the Moment Resisting Frame Logarithmic Decrement Method Applied to the F i r s t Floor Time-Acceleration Decay of the Moment Resisting Frame  9.10 9.11  10.1 10.2  e  172 172  Predicted vs. Measured Mobility Functions of the Moment Resisting Frame  173  Computer Model Used f o r the Free Vibration Analysis of the Braced Moment Resisting Frame  174  Predicted Natural Frequencies and Mode Shapes of the Braced Moment Resisting Frame  175  10.3  Fourier Spectrum Analysis of the Braced Moment Resisting Frame 176  10.4  Estimation of Mode Shapes for the Braced Moment Resisting Frame  177  Logarithmic Decrement Method Applied to the F i r s t Floor TimeAcceleration Decay of the Braced Moment Resisting Frame  177  Predicted Mobility Functions of the Braced Moment Resisting Frame  178  11.1  F r i c t i o n Damped Braced Frame on Shaking Table  179  11.2  Computer Model Used f o r the Free Vibration Analysis of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  180  10.5 10.6  - xi -  LIST OF FIGURES (Continued) Page  11.3 11.4 11.5  11.6  Predicted Natural Frequencies and Mode Shapes of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitation, Modes #1-3  181  Predicted Natural Frequencies and Mode Shapes of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitation, Modes #4-9  182  Fourier Spectrum Analysis of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  183  Measured vs. Predicted Natural Frequencies of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  184  11.7  Estimation of Mode Shapes for the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations 185  11.8  Measured vs. Predicted Mode Shapes of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  186  Logarithmic-Decrement Method Applied to the F i r s t Floor TimeAcceleration Decay of the F r i c t i o n Damped Braced Frame  186  Predicted Structural Damage After A r t i f i c i a l Earthquake, Intensities 1, 2, 3  187  Shaking Table Performance, Newmark-Blume-Kapur A r t i f i c i a l Earthquake  188  11.9  12.1 12.2 12.3  Envelopes of Measured Horizontal Accelerations, Newmark-BlumeKapur A r t i f i c i a l Earthquake, I n t e n s i t i e s 1, 2, 3 189  12.4  Envelopes of L a t e r a l Deflections, Newmark-Blume-Kapur A r t i f i c i a l Earthquake, Intensities 1, 2, 3  190  12.5  Envelopes of Bending Moments i n the Beams, Newmark-Blume-Kapur A r t i f i c i a l Earthquake, I n t e n s i t i e s 1, 2, 3 191  12.6  Time-History of Third Floor Deflection, Newmark-Blume-Kapur A r t i f i c i a l Earthquake, Intensity 3  192  Slippage Time-History of Second Floor Device, Newmark-BlumeKapur A r t i f i c i a l Earthquake, Intensity 3  193  Recorded Time-Histories of Third Floor Deflection From Both Potentiometers of the F r i c t i o n Damped Braced Frame, NewmarkBlume-Kapur A r t i f i c i a l Earthquake, Intensity 3  193  12.7 12.8  - xii -  LIST OF FIGURES (Continued) Page  12.9  Shaking Table Performance, Taft Earthquake  12.10 Envelope of Measured Horizontal Accelerations, (0.60 g)  194 Taft Earthquake 195  12.11 Envelope of L a t e r a l Deflections, Taft Earthquake (0.60 g) .... 195 12.12 Envelope of Bending Moments i n the Beams, Taft Earthquake (0.60 g)  196  12.13 Time-History of Third Floor Deflection, Taft Earthquake (0.60 g)  197  12.14 Recorded Slippage Time-History of Second Floor Device, Taft Earthquake (0.60 g)  198  12.15 Recorded Time-Histories of Third Floor Deflection From Both Potentiometers of the F r i c t i o n Damped Braced Frame, Taft Earthquake (0.60 g)  198  12.16 Envelope of Measured Horizontal Accelerations, Taft Earthquake (0.90 g) 299  i  12.17 Time-History of Third Floor Accelerations, Taft Earthquake (0.90 g)  200  12.18 Slippage Time-Histories of F r i c t i o n Devices, Taft Earthquake (0.90 g)  201  12.19 Slippage Time-Histories of Both Second Floor Devices, Taft Earthquake (0.90 g)  202  12.20 Time-History of the Percentage of Energy Dissipated by the F r i c t i o n Devices, Taft Earthquake (0.90 g)  203  - xiii -  LIST OF TABLES Page  3.1  3.2  4.1  Dimensions of Governing Variables for Vibrations of E l a s t i c Structures (from ref:7)  204  Similitude Requirements for Vibrations of E l a s t i c Structures (from ref:7)  204  Comparison Between Simplified and Refined Models, E l Centro Earthquake  205  5.1  Result of Linear Regressions f o r C a l i b r a t i o n of F r i c t i o n Devices 206  9.1  Comparison of Measured and Predicted Natural Frequencies of the Moment Resisting Frame 207  9.2  Comparison of Measured and Predicted Mode Shapes of the Moment Resisting Frame  9.3  208  Measured Modal Damping Ratios of the Moment Resisting Frame .... 209  10.1 Measured Modal Damping Ratios of the Braced Moment Resisting Frame  210  11.1 Comparison Between Measured and Predicted Natural Frequencies of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations 211 11.2 Comparison of Measured vs. Predicted Mode Shapes of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations 211 11.3 Measured Modal Damping Ratios of the F r i c t i o n Damped Braced Frame  212  12.1 Envelopes of Bending Moments i n the Base Column at Location of S t r a i n Gages, Newmark-Blume-Kapur A r t i f i c i a l Earthquake, Intensities 1, 2, 3  213  12.2 Envelope of Bending Moments i n the Base Column at Location of S t r a i n Gages, Taft Earthquake (0.60 g)  214  - xiv -  LIST OF TABLES (Continued) Page  12.3 Comparison of Natural Frequencies for the Two Model Frames  215  12.4 Comparison of Damping Ratios f o r the Two Model Frames  216  13.1 Design of Reduced Size F r i c t i o n Damped Braced Frame, NewmarkBlume-Kapur A r t i f i c i a l Earthquake (0.30 g)  217  - xv -  ACKNOWLEDGEMENTS This research project was carried out under contract with the Department of Supply and Services Canada.  P a r t i a l funding was also  provided by the the Natural Sciences and Engineering Research Council of Canada, the Government of Quebec and the University of B r i t i s h Columbia. It i s a pleasure to record this assistance. The research was carried out under the supervision of Dr. S. Cherry, whose expert advise, guidance and confidence are g r a t e f u l l y acknowledged. Also the association with Dr. A.S. P a l l , Montreal Consulting Engineer, was very helpful i n the completion of the project. I would l i k e to thank Mr. M. Frank for his support and advise and CANRON Ltd. for the careful construction of the tests frames for the experimental  set up.  Also I wish to express my sincere thanks to Dr. R.F. Hooley for the stimulating discussions that I had with him throughout the project. The professional work of a l l members of the technical s t a f f of the Department of C i v i l Engineering, p a r t i c u l a r l y Max Nazar, Mark Abraham and Guy Kirsh, i s greatly appreciated.  Also, without a l l the friendly help  from fellow graduate students, p a r t i c u l a r l y Bryan Folz, James D. Dolan and C. Yao, the experimental  study could not have achieved the  required by the research schedule.  progress  The typing of this thesis was done by  Kelly Lamb and her professional work i s acknowledged. F i n a l l y , I wish to express the most sincere gratitude to my wife, Louise, for her endless support, patience and love. Andre F i l i a t r a u l t October 1985 Vancouver, B r i t i s h Columbia - xvi -  1. 1.  1.1  INTRODUCTION  Background Earthquake loading i s unique among the types of loads that a  structural engineer must consider because a great earthquake may  cause  greater stresses and deflections i n various c r i t i c a l components of a structure than a l l the other loadings combined.  From an energy point of  view, earthquakes induce k i n e t i c energy i n a structure.  The amount of  energy fed into the structure depends greatly on the natural frequency of the structure compared to the energy d i s t r i b u t i o n of the ground motion expressed i n terms of the frequency content of that motion.  The l e v e l of  damage i s determined by the manner i n which t h i s k i n e t i c energy i s absorbed by the structure. Generally i t i s not economically f e a s i b l e to design a structure to r e s i s t a major earthquake within the e l a s t i c range of the construction materials.  The philosophy of most building codes, including the National  Building Code of Canada, i s based on a c r i t e r i o n of d u c t i l i t y i n which the demand on a structure must be balanced by i t s d u c t i l i t y capacity, i . e . the capacity of the members to safely absorb the induced energy while undergoing i n e l a s t i c deformations.  In this context, the following design  c r i t e r i a are usually adopted:  1.  A minor earthquake at the building s i t e should not cause any  structural or non-structural damage. 2.  A moderate earthquake which may reasonably be expected at the building  s i t e during the l i f e of the structure should be taken as the basis of  2.  design.  The building should be proportioned  to r e s i s t this i n t e n s i t y of  ground motion without s i g n i f i c a n t damage to the basic structure. 3.  During the most severe earthquake that could possibly be expected to  occur at the building site during the l i f e of the structure, i t i s economically j u s t i f i e d to permit s i g n i f i c a n t s t r u c t u r a l damage.  However,  collapse and loss of l i f e must be avoided. To meet these design c r i t e r i a , t r a d i t i o n a l methods of aseismic place reliance on the d u c t i l i t y of the s t r u c t u r a l elements.  design  To optimize  the energy absorption without producing a p l a s t i c collapse mechanism, i t i s necessary to consider  the yielding pattern of a structure.  The "weak  beams-strong columns" design philosophy i s a t y p i c a l example of the control of this pattern: the beams are s a c r i f i c e d f i r s t ,  since l o c a l y i e l d i n g of  the beams does not seriously a f f e c t the v e r t i c a l - l o a d - c a r r y i n g capacity of the structure, whereas l o c a l column y i e l d i n g could e a s i l y lead to collapse. For s t e e l construction, current aseismic grouped into two  s t r u c t u r a l systems can  be  categories:  1)  Braced Moment Resisting Frames;  2)  Moment Resisting Frames. Braced Moment Resisting Frames, where the braces are designed i n  tension only, are known to be economical and are e f f e c t i v e i n c o n t r o l l i n g l a t e r a l deflections due to wind and moderate earthquakes; but extreme earthquakes these structures do not perform well.  during  Being s t i f f e r ,  they tend to a t t r a c t higher seismic forces, and t h e i r energy d i s s i p a t i o n i s very poor due  to the deteriorating hysteresis loops of the braces. (3)  Figure 1.1  shows a t y p i c a l hysteresis loop of a tension brace  This kind of hysteresis i s c a l l e d "pinched hysteresis". shock the tension brace f i r s t  .  During a severe  stretches and subsequently buckles i n  3. compression during the it  reversal of load.  same d i r e c t i o n , t h e e l o n g a t e d i s taut  this  kind  again.  Therefore,  brace  energy  of s t r u c t u r a l system  On t h e n e x t  a p p l i c a t i o n of load i n  i s not e f f e c t i v e i n tension  d i s s i p a t i o n degrades very  i s viewed  with  suspicion  until  q u i c k l y and  f o r earthquake  resistance. Moment capability loads.  due t o t h e i r  stable  These s t r u c t u r e s  forces. since  R e s i s t i n g F r a m e s a r e known f o r t h e i r  However,  their  interstorey drift  structural  damage.  structural  stability  In  recent  d u c t i l e behaviour  are very great  flexible  flexibility  under  and tend leads  resistance  repeated  reversing  t o induce lower  to economical  problems  i s often many  because  of their  jeopardized  authors  have  large  by t h e P-A  shown g r e a t  Frames and t h e s t i f f n e s s  Japan, designers braces  often  are designed  employ  to carry  of Braced  B r a c e d Moment only  a portion  Moment  non-  deflections,  factor. i n t e r e s t i n the devel-  opment o f s t r u c t u r a l s y s t e m s w h i c h c o m b i n e t h e d u c t i l e b e h a v i o u r Resisting  seismic  a n d d e f l e c t i o n s m u s t be c o n t r o l l e d t o p r e v e n t  Furthermore,  years,  earthquake  o f Moment  R e s i s t i n g Frames.  In  R e s i s t i n g Frames i n w h i c h t h e of the l a t e r a l  loads^^.  An  (2 ) . eccentric system yield  braced  frame  the diagonal  i s a further  brace  joints  a n d d i s s i p a t e more e n e r g y .  during  a major earthquake,  repairs  a r e needed.  step  i n this  direction  a r e made e c c e n t r i c The s t r u c t u r e  to force  i s saved  b u t t h e beams a r e s a c r i f i c e d  Another  *  In t h i s  t h e beams t o  from  collapse  and major,  a p p r o a c h makes u s e o f a b a s e  costly  isolation  (12) system  .  In t h i s  technique  the structure  quake e x c i t a t i o n , and i t s l a t e r a l its  first  typical  natural  frequency  earthquakes.  earthquakes within post-earthquake  stiffness  i s well  None o f t h e s e  the e l a s t i c  repairs.  limit  below  i s uncoupled i s thereby  reduced,  the frequency  systems a r e i n t e n d e d of the materials  from  content  the earthsuch  o f most  to r e s i s t  and w i l l  that  major  require  4. Recently a novel s t r u c t u r a l system for aseismic design of s t e e l framed (3) buildings has been proposed and patented by A.S. P a l l  .  The system  b a s i c a l l y consists of an inexpensive mechanism containing f r i c t i o n brake l i n i n g pads introduced  at the intersection of frame cross-braces.  Figure  1.2 shows the l o c a t i o n of the f r i c t i o n devices i n a t y p i c a l s t e e l frame. The device Is designed not to s l i p under normal service loads and moderate earthquakes.  During severe seismic excitations, the device s l i p s at a  predetermined load, before any y i e l d i n g and cracking 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 fundamental mode shape during a severe earthquake.  The phenomenon of quasi-resonance between the  structure and the earthquake excitation i s prevented because of this de-tuning c a p a b i l i t y of the structure. 1.2  Object and Scope Although the response of F r i c t i o n Damped Braced Frames has been (3)  studied a n a l y t i c a l l y ing.  , q u a l i f i c a t i o n tests of these structures are lack-  The main objective of the investigation reported  study experimentally  i n this thesis i s to  the seismic performance of F r i c t i o n Damped Steel  Braced Frame structures.  This was done by placing a 3 storey scale model  of the structure on a shaking table and subjecting i t to representative ground motion time-histories. A dimensional analysis was f i r s t performed to develop a 1/3 scale model of a Moment Resisting Frame.  The frame was then designed  according  to the National Building Code of Canada 1980 and the CAN3 S16.1-M78 "Limit State Design-Steel  Structures  for Building" Code.  taken into account using the s t a t i c method.  The seismic loads were  A s t a t i c , l i n e a r analysis was  5.  made and the members were chosen from available hot-rolled sections.  The  connections of the test frame were designed so that the model could e a s i l y be transformed into a Braced Moment Resisting Frame or a F r i c t i o n Damped Braced Frame as desired.  By a proper choice of the test sequence, i t was  possible to carry out a comparative study of the 3 model types using only a single frame. Two i d e n t i c a l frames were fabricated f o r t h i s study.  The second frame  was used as a backup for the primary tests and/or for undertaking further studies, depending on the results of the i n i t i a l experiments. A l l the seismic testing was carried out on the Earthquake Simulator Table i n the C i v i l Engineering Department of the University of B r i t i s h Columbia. The experimental results are compared to the predictions of an i n e l a s t i c time-history dynamic analysis using the computer program "DRAINZD", which was developed at the University of C a l i f o r n i a i n Berkeley.  This  program consists of a series of subroutines which carry out a step-by-step integration of the dynamic equilibrium equations using a constant acceleration algorithm within any time step. Two different computer models were used to predict a n a l y t i c a l l y the response of the F r i c t i o n Damped Braced Frame.  The f i r s t one, o r i g i n a l l y  (3)  proposed by A.S. P a l l  , i s based on an equivalent hysteretic model and i s  only approximate, since i t does not take into account the complete behaviour of the f r i c t i o n devices.  As a r e s u l t , a new, refined computer  model was developed and the results obtained from analyses using both models are compared. The f r i c t i o n devices were fabricated at a machine shop i n Montreal to develop the optimum s l i p load as determined from the above noted model  6.  analyses.  The optimum s l i p load i s the load which leads to maximum energy  d i s s i p a t i o n i n the f r i c t i o n device. Quality control testing under c y c l i c loading i s necessary to c a l i b r a t e the f r i c t i o n devices; these tests were carried out at the University of B r i t i s h Columbia. An equivalent viscous damping analysis i s used to quantify the performance of the F r i c t i o n Damped Braced Frame r e l a t i v e to the behaviour of conventional aseismic s t r u c t u r a l systems.  In this analysis, viscous  damping i s added to the Moment Resisting Frame and the Braced Moment Resisting u n t i l their responses become similar to the response of the F r i c t i o n Damped Braced Frame. F i n a l l y , the economical potential of the new damping system i s evaluated by designing a reduced size F r i c t i o n Damped Braced Frame having a response which i s similar to the response of conventional s t r u c t u r a l systems with heavier members.  7.  2.  2.1  DAMPING SYSTEM  Description The proposed damping system uses f r i c t i o n pads to develop additional  energy dissipating sources which can be marshalled to protect the main members from s t r u c t u r a l damage.  To be e f f e c t i v e the s l i p joints must  present very stable, non-deteriorating hysteresis loops. Several s t a t i c and dynamic f r i c t i o n tests on s l i p joints having different surface treatments have been conducted under repeated reversals of loads and are reported i n the l i t e r a t u r e (4,5,6).  Figure 2.1  reproduces  the hysteresis loops obtained from a variety of these s l i p joint surfaces clamped together by 12.7 mm diameter ASTM A325 high strength bolts.  Note  that the best results are obtained with heavy duty brake l i n i n g pads inserted between the s l i d i n g s t e e l surfaces. The performance  i s reliable  and repeatable, and the hysteresis loops are rectangular with negligible fade over many more cycles than are encountered i n successive earthquakes. For these reasons, this type of s l i p joint was used i n the present study. If the diagonal braces of an ordinary braced frame structure were designed not to buckle i n compression, a simple f r i c t i o n joint could be inserted i n each diagonal. independently of the other.  In this case each s l i p joint would act The s l i p load should be lower than the y i e l d  load of the braces so that the j o i n t can be activated before the member yields.  However, i t i s not economical to design the braces i n compression  and, more often, the braces are quite slender and 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 joint would  s l i p i n tension but would not s l i p back during reversal of the tension load and i n the compression  (buckled) regime.  The energy absorption would be  8.  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 was beyond the previous elongated  length, as shown i n Figure  stretched  2.2.  The braces can be made to s l i p both i n tension and compression by connecting a special mechanism at the i n t e r s e c t i o n of frame cross-braces indicated i n Figure 1.2.  as  The d e t a i l s of this mechanism are shown  schematically i n Figure 2.3.  When a seismic l a t e r a l load i s induced i n the  frame, one of the braces goes i n t o tension while the other brace buckles very early i n compression.  If the s l i p load of the f r i c t i o n  joint i s lower  than the y i e l d load of the brace, when the load i n the tension brace reaches the s l i p load i t forces the joint to s l i p and activates the four links.  This i n turn, forces the j o i n t i n the other brace to s l i p  simultaneously. each half cycle.  In this manner, energy i s dissipated i n both braces i n Moreover, i n each half cycle, the mechanism straightens  the buckled brace and makes i t ready to absorb energy immediately when the load i s reversed.  In this way  the energy d i s s i p a t i o n  comparable with that of a simple f r i c t i o n  of this system i s  joint used with braces which are  designed not to buckle i n compression. The f r i c t i o n devices can be used i n any configuration of the bracing system.  Some possible bracing arrangements are shown i n Figure 2.4.  These  devices also can be conveniently incorporated i n existing framed buildings to upgrade their earthquake resistance. 2.2  Simplified Model Consider the hysteretic behaviour of a simple F r i c t i o n Damped Braced  Frame under seismic load as shown on Figure 2.5. V  Let  = L a t e r a l load at the girder representing the seismic load = Relative displacement positive i n figure)  of node C with respect to node A (defined  9.  A  2  P  1  = Load i n brace 1 (positive i n tension)  P  2  = Load i n brace 2 (positive i n tension)  = Relative displacement of node B with respect to node D (defined p o s i t i v e i n figure)  Figure 2.5 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. For each stage the load-deformation curves of both braces and the associated deformed shape of the frame are shown.  The following points  should be noted during the cycle: 1)  In the early stages of loading both braces are active and behave e l a s t i c a l l y i n tension and  2)  compression.  At very low load the compression brace buckles while the tension brace s t i l l stretches e l a s t i c a l l y i n tension.  3)  The s l i p load i s reached before y i e l d i n g occurs i n the tension brace. As a r e s u l t , the four links of the special mechanism are activated and deform into the rhomboid form, which eliminates the buckled shape of the compressive brace under the same buckling load. slippage P  2  i s s t i l l the buckling load but now  At the end of the  the compression brace  is straight. 4)  When the load i s reversed the buckled brace i s straight and 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. The program "Drain-2D" contains truss elements which may be a r b i t r a r -  i l y oriented i n the x,y plane, but which can transmit a x i a l load only. 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)  Two  10.  yielding i n both tension and compression and (b) yielding 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 F r i c t i o n Damped Braced Frame.  For this prupose we need only replace the f r i c t i o n device by  two normal braces which can y i e l d both i n tension and compression as shown i n Figure 2.6.  We define a f i c t i t i o u s y i e l d stress i n tension, which  corresponds to the stress i n the tension brace when the device s l i p s .  We  also assign a very low f i c t i t i o u s compression y i e l d stress to the material, corresponding to the buckling stress of the compression brace.  This simple  (3)  model was o r i g i n a l l y proposed and used by A.S. P a l l The P a l l model i s only approximate since i t does not adequately account for the complete deformation pattern of the f r i c t i o n device.  The  assumed hysteresis behaviour i s accurate only i f the device s l i p s at every cycle, which i s not the case i n an actual earthquake.  In many cycles the  tension brace w i l l not s l i p but the compression brace w i l l buckle.  Under  such conditions, the assumed hysteresis behaviour i s no longer v a l i d , since the mechanism has not activated the l i n k s to p u l l back the buckled brace. The simple model also assumes that the slippage of the device i s large enough to straighten completely the buckled brace.  If i n a given cycle the  slippage i s not large enough to achieve this condition, parts of the energy stored i n the compression brace w i l l be restored to the structure (see Figure 2.2).  Therefore this simple model overestimates the energy absorp-  t i o n of the f r i c t i o n device.  For t h i s reason a more accurate or refined  model was developed to evaluate the significance of these simplifying assumptions on the o v e r a l l response of the structure under a major earthquake.  11. 2.3  Refined Model To eliminate the assumptions used with the simple model, each member  of the mechanism i n the F r i c t i o n Damped Braced Frame i s taken as a 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  s t r a i n curves, were i n i t i a l l y used i n this new model.  stress-  However, assuming  pinned connections at the four corners of the mechanism leads to an unstable condition, as shown on Figure 2.7.  Therefore these four mechanism  connections must be r i g i d and carry a small amount of bending to ensure stability. The program "Drain-2D" contains beam-column elements which may a r b i t r a r i l y oriented i n the xy plane. axial stiffness.  Yielding may  hinges at the element ends.  be  These elements possess flexural and  take place only i n concentrated p l a s t i c  The y i e l d moments may  be specified  to be  d i f f e r e n t at the two element ends, and f o r positive and negative bending. The interaction  between axial 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 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 after y i e l d .  Hence, we cannot use this type of  element alone to model the f r i c t i o n device, since we would loose track of the i n e l a s t i c axial deformations during slippage. A refined model which accurately r e f l e c t s the true behaviour of the f r i c t i o n device can be developed by superposing truss and beam-column elements f o r the diagonal braces. 2.8.  This proposed model i s shown i n Figure  We f i r s t model a l l the members by truss elements with their  s t r e s s - s t r a i n curves.  own  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 inside diagonal pads s l i p i n  12.  tension and compression.  For the diagonal braces we superpose beam-column  elements with a zero cross-sectional area such that they can carry bending moment only, which i s required to ensure s t a b i l i t y .  To represent the  pinned connections at the four corners of the frame we specify zero p l a s t i c moments capacity for the beam-column elements. With this refined model we can accurately represent the real behaviour of a F r i c t i o n Damped Braced Frame.  However, i t requires many more elements  and degrees of freedom than the s i m p l i f i e d model and the computer time i s increased s i g n i f i c a n t l y .  In the a n a l y t i c a l study the two models w i l l be  compared and a conclusion drawn on the v a l i d i t y of the approximate model.  13. 3.  3.1  DESIGN OF MODEL FRAME  Dimensional Analysis Any  s t r u c t u r a l model s h o u l d be d e s i g n e d , l o a d e d , and  interpreted  a c c o r d i n g to a set of s i m i l i t u d e requirements t h a t r e l a t e the model to the real structure.  These s i m i l i t u d e requirements a r e based upon the theory o f  modeling, which can be d e r i v e d from a d i m e n s i o n a l a n a l y s i s of the p h y s i c a l phenomena i n v o l v e d i n the behaviour of the s t r u c t u r e . The  three g e n e r a l c l a s s e s of p h y s i c a l problems, namely, mechanical  ( s t a t i c and dynamic), thermodynamic, and e l e c t i c a l , a r e c o n v e n i e n t l y described q u a l i t a t i v e l y 1.  Length  2.  F o r c e (or mass)  3.  Time  4. 5.  i n terms of the f o l l o w i n g fundamental  measures:  Temperature Electric  charge.  Most s t r u c t u r a l modeling problems  a r e m e c h a n i c a l ; thus the measures o f  l e n g t h , f o r c e , and time are most important i n s t r u c t u r a l work. Keeping the above d e f i n i t i o n s o f measure i n mind, the theory of dimensions  can be summarized by a g e n e r a l theorem  s t a t e d by Buckingham i n  1914:  "Any d i m e n s i o n a l l y homogeneous e q u a t i o n i n v o l v i n g c e r t a i n p h y s i c a l q u a n t i t i e s can be reduced to an e q u i v a l e n t equat i o n i n v o l v i n g a complete s e t o f d i m e n s i o n l e s s p r o d u c t s . "  T h i s theorem  s t a t e s t h a t the s o l u t i o n e q u a t i o n f o r some p h y s i c a l  q u a n t i t y of i n t e r e s t , i . e . ,  14.  ..., X ) n  =  0  (3.1)  =  0  (3.2)  can equivalently be expressed i n the form  G(ir ,Tr , ... ,ir ) m 1  2  The P i terms are dimensionless products of the physical quantities Xj,X ,...,X « 2  n  Generally, i t can be stated that the number of dimensionless  products (m) i s equal to the difference between the number of physical variables (n) and the number of fundamental measures (r) that are involved. The main question to be resolved i n applying the Buckingham P i theorem pertains to the formation of appropriate Pi terms.  The best method for  a r r i v i n g at the groupings of Pi terms i s open to personal preference; are a number of rather formal techniques which involve setting up appropriate dimensional equations.  One  there  the  less formal approach involves the  following steps: 1.  Choose r variables that embrace the r dimensions (fundamental measures) required to express a l l variables of the problem, and are dimensionally  independent.  that  This means that i f a problem involves  the dimensions of force F, length L, and time T, then the three variables chosen must c o l l e c t i v e l y have dimensions which include F, L and T, but no two variables can have the same dimensions.  Variables  that are i n themselves dimensionless ( s t r a i n , Poisson's r a t i o , angles) cannot be chosen i n the set of r variables. 2.  Form the m Pi terms by taking the remaining (n-r) variables and grouping them with the r variables i n such a fashion that a l l groups  15. are dimensionless.  This procedure w i l l guarantee a set of  independent, dimensionless terms.  A consideration of the variables that govern the behaviour of v i b r a t ing structures reveals that i n addition to length (L) and force (F), which are considered  i n s t a t i c loading situations, we must also include time (T)  as one of the fundamental quantities before we proceed with dimensional analysis. Consider an e l a s t i c structure made of a homogeneous i s o t r o p i c material whose vibration conditions are to be determined.  A t y p i c a l length i n the  structure i s designated by % and a t y p i c a l force by Q.  The materials of  both the model and prototype can be characterized by the material constants:  the modulus of e l a s t i c i t y E, the Poisson's r a t i o v, and the mass  density p. The important parameters to be determined from the s t r u c t u r a l vibration are the deflected shape <5, the natural frequency f , and the dynamic stresses a.  The acceleration due to gravity g must also be  included since i t i s common to both model and prototype structures. dimensions of the governing variables i n both absolute  The  and common engineer-  ing units are shown i n Table 3.1. For the problem of v i b r a t i o n of an e l a s t i c structure, the number of variables and dimensions are:  n  =  9 variables (£,Q,E,v,p,6,a,f,g)  r  =  3 dimensions (F,L,T)  The number of P i terms that can be formed i s then:  m  =  n-r = 6 P i terms  16. To apply the Buckingham P i theorem we choose 3 variables that contain the 3 dimensions of the problem (F,L,T).  For example we can choose: £(L),  E(FL" ) and g ( L T ~ ) . 2  2  We then group the remaining 6 variables (6,a,v,p,Q,f) with the primary variables (£,E,g) such that a l l groups are dimensionless.  For a true  model, dimensionless parameters that govern the behaviour w i l l then be:  E  E£  2  Equation (3.3) means that we can determine  the dynamic c h a r a c t e r i s t i c s  of the structure by means of a model test by forcing the dimensionless products of the left-hand side of Equation (3.3) to be i d e n t i c a l i n model and prototype.  To impose these r e s t r i c t i o n s on the model design we  normally choose the scaling factors for length and modulus of e l a s t i c i t y :  S £  E  =  length of model _ _m length of prototype £^  ^.  Modulus of E l a s t i c i t y of Model _ _m Modulus of E l a s t i c i t y of Prototype E^  (••->)  We can then express a l l the other scaling factors as a function of S E and S .  The implied scale factors that govern these relationships are  summarized i n Table 3.2. As we can see from this Table, the density scale i s equal to S /S. f o r E Z a true e l a s t i c model and i s usually d i f f e r e n t from one.  This means that  the model should be made of a d i f f e r e n t material than the prototype.  In  practice this i s usually very expensive and often the effect of gravity  17.  forces i s neglected.  Table 3.2 gives also the scaling factors f o r the case  when the gravity forces are neglected. In the Table note that the time scale i s equal to S ^ e l a s t i c model and to  - 1 / 2  i n the case of a model where gravity  e f f e c t s can be neglected.  for a true loading  This means that an actual earthquake record  should have a d i f f e r e n t time scale when used as the input f o r the test of a model.  The frequency of v i b r a t i o n of the model, which i s inversely  proportional to the period, w i l l be S ~ neglecting gravity, respectively. higher frequencies  3.2  1 / 2  and S  -  1  considering and  This means that the model w i l l have  than the prototype structure.  P r a c t i c a l Design The f i r s t step i n the design of the model frame was to choose a f u l l  scale prototype structure. the frame used by Workman  I t was decided to choose the f i r s t 3 floors of as the prototype structure.  The dimensions,  member sizes, and other properties of a l l the Moment Resisting Frame members and braces are shown i n Figure 3.1. The t o t a l height of the prototype structure i s 10.98 m.  Considering the v e r t i c a l clearance of the  Earthquake Simulator Room (4 m), a 1/3 scaling factor was selected for length.  Also, since available hot-rolled section were to be used for the  main members, the scaling factor for modulus of e l a s t i c i t y was fixed as unity.  Thus  S  0  = 1/3  (3.6)  - 1  (3.7)  To represent a true model the scaling factor for density should be:  18.  (3.8)  Thus t h e o r e t i c a l l y we should use a model material having 3 times the density of s t e e l .  P r a c t i c a l l y , t h i s could be achieved by attaching a large  number of lumped masses to the main steel members. would be very expensive and tedious.  However, this procedure  Furthermore hot-rolled sections  meeting the similitude requirements for cross-sectional area and moment of i n e r t i a (S£ and Si *) do not e x i s t . 2  1  For these reasons, i t was not possible  to follow the requirements of the dimensional analysis.  However i t i s not  necessary to represent an exact model since the study w i l l compare the response behaviour of the Moment Resisting Frame, the Braced Moment Resisting Frame and the F r i c t i o n Damped Braced Frame. decided to select the fundamental  Therefore i t was  frequency of the model frame on the basis  of the National Building Code suggestion:  T  =  Io  =  To  =  °*  3 0  s e c  *  ( o r  f  =  3 H z )  ( 3  *  9 )  where N = Number of storey T = Fundamental period i n seconds  The main members were chosen from available hot-rolled sections and the connections designed according to the CAN3 S16.1-M78 "Limit State Design-Steel Structures for Buildings" code. arrangement of the r e s u l t i n g model frame.  Figure 3.2 shows the general  19. The o v e r a l l dimensions of the model frame are 2.05 x 1.4 m i n plan and 3.53 m i n height.  The frame i s composed of several separate assemblages  which are bolted together f o r ease of handling.  The side frames, oriented  p a r a l l e l to the d i r e c t i o n of the excitation, represent a Moment Resisting Construction. are bolted.  The frames i n the d i r e c t i o n perpendicular to the excitation, The f i r s t two floors are loaded with two concrete blocks, each  weighing 1700 kg., while the t h i r d f l o o r i s loaded with a 1150 kg concrete block.  The frame was analyzed and i t s fundamental frequency was found to  be 2.83 Hz. The frame i s mounted on a r i g i d base beam to f a c i l i t a t e the i n s t a l l a tion on the shaking table.  The t o t a l weight of the model frame with the  base beam and the concrete blocks i s 6000 kg.  3.3  Remarks on F i n a l Design Figures 3.3, 3.4 and 3.5 show the d e t a i l s of the model frame.  A l l the  main beams and columns are made of the smallest S shapes (S75*8) a v a i l a b l e . The cross-section has a depth of 75 mm and flange width of 59 mm.  The  column cross-sections are reinforced at their bases by 2 plates, each of 6 mm thickness and over a length of 400 mm, i n order to delay the formation of p l a s t i c hinges at the column bases. The beam-column connections were designed such that the Moment Resisting Frame could be transformed easily into a Braced Moment Resisting Frame or a F r i c t i o n Damped Braced Frame as needed. The cross-section of the braces for the Braced Moment Resisting Frame and the F r i c t i o n Damped Braced Frame was designed such that the following similitude i s respected:  20.  Brace Cross-Section of Model _ Brace Cross-Section of Prototype Column Cross-Section of Model Column Cross-Section of Prototype  This led to the choice of a 6 mm  square bar (A = 36 mm ) 2  g>  for the bracing  members. In the d i r e c t i o n perpendicular to the excitation, heavy cross-braces were i n s t a l l e d to separate the t r a n s l a t i o n a l frequencies i n each d i r e c t i o n and thus reduce the problem of t o r s i o n a l resonance.  As noted above, the  fundamental frequency i n the d i r e c t i o n p a r a l l e l to the excitation was  found  by c a l c u l a t i o n to be 2.83 Hz; i n the d i r e c t i o n perpendicular to the excitation the analysis yielded a much higher fundamental frequency of 14.31  Hz. The dead load of each concrete block i s transmitted to i t s supporting  beams through the flanges of 6 channels welded to these main members (see Figure 3.2).  With t h i s system, representing point loads, the diaphragm  effect of the concrete blocks i s reduced. Figure 3.6 shows the actual model of the Moment Resisting Frame mounted on the shaking  table.  21. 4.  4.1  ANALYTICAL STUDY  Concept of Optimum S l i p Load The energy d i s s i p a t i o n of a f r i c t i o n device 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 .  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 w i l l be no slippage.  In  this situation the structure w i l l behave exactly as a normal Braced Moment Resisting Frame.  If the s l i p load i s very low, large s l i p travels w i l l  occur but the amount of energy dissipation again w i l l be n e g l i g i b l e . Between these extremes, there i s an intermediate value of the s l i p load which results i n the maximum energy d i s s i p a t i o n . i s defined as the "Optimum S l i p Load".  This intermediate  value  This concept of optimum s l i p load  i s i l l u s t r a t e d on Figure 4.1. As noted i n Chapter 2, when tension i n one of the braces forces the joint to s l i p , i t activates the four links which forces the joint i n the other brace to s l i p simultaneously.  Figure 4.2 shows a free body diagram  of a f r i c t i o n device when slippage occurs.  We define the "Global S l i p Load  (P )" as the load i n the tension brace when slippage occurs; also, the "Local S l i p Load (P^)" i s defined as the load i n each f r i c t i o n pad when slippage occurs. As can be seen from equilibrium i n Figure 4.2, the r e l a t i o n between the Global Slip Load and the Local S l i p Load i s given by:  P  where P^  r  e  = 2P - P % cr  (4.1) '  v  i s the c r i t i c a l buckling load of the compression brace.  Usually the buckling load i s very small and can be neglected; i n t h i s case we obtain:  22.  (4.2)  These two definitions 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 i s based  on an equivalent hysteretic behaviour and the Global S l i p Load i s the governing parameter. the  The refined model considers each j o i n t separately and  Local Slip Load governs i n this case.  4.2  Optimum S l i p Load Study To determine the optimum s l i p load of the model frame, i n e l a s t i c time  history dynamic analyses were performed for d i f f e r e n t values of the s l i p load.  The computer program "Drain-2D", was used f o r t h i s purpose.  This  program consists of series of subroutines which carry out a step-by-step integration of the dynamic equilibrium equations:  [M]{x> + [C]{x> + [K]{x>  =  ~[M]{I} x g  (4.3)  where [M]  = global mass matrix  [C]  = global damping matrix  [K]  = global s t i f f n e s s matrix  {x}  = vector of mass displacements r e l a t i v e to the moving base  x  = ground acceleration  g {1}  =  influence vector coupling the input ground motion to each degree of freedom  The constant acceleration method i s used within each time step and provides an unconditional stable solution.  The global mass matrix i s  assumed to remain constant during the earthquake.  However, the damping and  s t i f f n e s s matrices may change at the beginning of a time step, depending on the p l a s t i c deformation state of the structure at the end of the previous time step.  The global mass matrix and the global s t i f f n e s s matrix are  formed by the direct method. The damping matrix i s described by considering Rayleigh type damping:  [ C ] = o [ M ] + 0[K] + 3 [ K ] o  q  (4.4)  where: [K]  = Updated global s t i f f n e s s matrix  [K ]  = I n i t i a l e l a s t i c global s t i f f n e s s matrix  q  a,3,3  Q  = Damping c o e f f i c i e n t s  F l e x u r a l and a x i a l deformations are considered and the i n t e r a c t i o n between a x i a l forces and moments at y i e l d are taken into account by means of y i e l d interaction surfaces.  Figure 4.3 shows the y i e l d i n t e r a c t i o n  surfaces used i n the analyses.  The y i e l d stress of the s t e e l i s assumed to  be 300 MPa i n tension and compression.  The P-A effect i s considered  approximately by adding the global geometric s t i f f n e s s matrix from s t a t i c loading to the updated global s t i f f n e s s matrix.  No viscous damping i s  considered i n the optimum s l i p load study; the value of t h i s parameter i s e s s e n t i a l l y negligible compared to the very much greater f r i c t i o n damping mechanism.  Rigid foundations are assumed and s o i l - s t r u c t u r e i n t e r a c t i o n i s  neglected.  The s t a t i c dead loads are considered by means of i n i t i a l forces  on the elements. It i s known that d i f f e r e n t earthquake  records, even when normalized to  the same intensity, give widely varying s t r u c t u r a l response, and the  24.  results obtained using a single record may not be conclusive.  For t h i s  reason, the optimum s l i p load study was carried out for 3 d i f f e r e n t earthquake records as follows:  1)  El-Centro earthquake 1940, component S00E, 0-6 seconds, scaled to a peak acceleration of 0.52 g.  2)  P a r k f i e l d earthquake 1966, component N65E, 0-9 seconds, scaled to a peak acceleration of 0.52 g.  3)  Newmark-Blume-Kapur a r t i f i c i a l earthquake, 0-15 seconds, scaled to a peak acceleration of 0.30 g.  The acceleration records and the amplitudes of the Fourier spectra f o r these three earthquakes are shown on Figure  4.4  To save computation costs, the s i m p l i f i e d computer model described i n Chapter 2 was used for a l l the analyses.  Preliminary analyses were made t  determine the proper time step to be used.  An integration time step of  0.005 sec was found to be s u f f i c i e n t and was used i n a l l the analyses. The r e s u l t s of the optimum s l i p load study are given i n Figures 4.5, 4.6, and 4.7.  Deflection envelopes, maximum moments i n the beams and  columns and maximum shear forces i n the columns are plotted f o r d i f f e r e n t values of the global s l i p load.  The results are given for global s l i p  loads ranging from 0 to 10 kN, representing cross-braces.  Results  the e l a s t i c region of the  for zero global s l i p load represent  the response of  a Moment Resisting Frame. The figures c l e a r l y show the effectiveness of the f r i c t i o n devices i n improving the seismic response of the frame.  As we increase the global  s l i p load the deflections, moments and shear forces decrease steadily up t a global s l i p load of 6 kN.  For a global s l i p load between 6 kN and 10 kN  25. there i s very l i t t l e v a r i a t i o n i n the response.  The r e s u l t s indicate that  during a major earthquake, the devices extract enough energy to prevent y i e l d i n g i n s t r u c t u r a l members. This lower bound value of 6 kN f o r the global s l i p load i s observed for the 3 different earthquakes studied.  This suggests that the optimum  value of the global s l i p load may be independent of the ground motion input.  If this i s found to hold true for a l l cases, the optimum s l i p load  can be considered as a s t r u c t u r a l property.  This observation may  greatly  simplify the development of a design procedure for the f r i c t i o n devices. On the basis of the results obtained, an optimum global s l i p load value of 7 kN was subsequently used for the study of the F r i c t i o n Damped Braced Frame model.  From equation 4.2,  the corresponding l o c a l s l i p load  w i l l be 3.5 kN i f the buckling load of the brace Is neglected.  It i s  i n t e r e s t i n g to note that t h i s global s l i p load i s 20% of the weight of the model.  It might be expected that the same percentage would also apply to  the prototype structure.  4.3  Comparison Between Simplified and Refined Models The s i m p l i f i e d computer model i s based on an equivalent hysteretic  model and i s only approximate, since i t does not take into account the complete behaviour of the f r i c t i o n devices.  The refined model considers  each element of the f r i c t i o n device as an individual member with i t s own material (stress-strain) properties.  It can therefore represent the  complete behaviour of the f r i c t i o n devices but requires many more degrees of freedom and i s much more expansive to run than the s i m p l i f i e d model. For  this reason, the results of the two models are compared for the  El-Centro 1940 earthquake only. An optimum global s l i p load of 7 kN was used with the s i m p l i f i e d model  26. and an optimum l o c a l s l i p load of 3.5 kN was used with the refined model. Some t r i a l analyses were made to determine the proper integration time step to be used with the refined model.  I t was found that a much lower time  step than used with the simplified model was needed i n order to obtain accurate r e s u l t s ; a time step of 0.0015 second was used i n the analysis. The response parameters obtained using the two models of the F r i c t i o n Damped Braced Frame are also compared to the corresponding responses of the Moment Resisting Frame and the Braced Moment Resisting Frame. 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 at the end of the El-Centro (0-6 sec.) earthquake i s i l l u s t r a t e d i n Figure 4.8.  S i g n i f i c a n t damage occurs i n the Moment Resisting Frame, i n which the  f i r s t and second floor beams reach their p l a s t i c moment capacity.  A l l the  cross-braces of the Braced Moment Resisting Frame have yielded i n tension and w i l l need replacement.  Furthermore, the l e v e l of damage i n the Braced  Moment Resisting Frame i s a lower bound, since the computer program neglects the effect of the degrading s t i f f n e s s of the braces (pinched hysteresis) .  A l l the members remain e l a s t i c f o r the s i m p l i f i e d and refined  models of the F r i c t i o n Damped Braced Frame. The envelopes of the response parameters f o r the d i f f e r e n t frames are given i n Figure 4.9.  I t can be seen that the deflection at the top of the  F r i c t i o n Damped Braced Frame i s about 20% of the equivalent d e f l e c t i o n i n the Moment Resisting Frame and about 55% of the d e f l e c t i o n i n the normal Braced Moment Resisting Frame.  A l l the beams, except those i n the top  storey, have yielded i n the Moment Resisting Frame, and a l l the braces have yielded i n the Braced Moment Resisting Frame; but none of the beams or braces have yielded i n the F r i c t i o n Damped Braced Frame.  The maximum shear  at the base of the F r i c t i o n Damped Braced Frame i s only 55% and 32% of the corresponding shear In the Braced Moment Resisting Frame and the Moment Resisting Frame respectively.  27.  The maximum moment at the base of the F r i c t i o n Damped Braced Frame i s 26% and 51% of the corresponding moment i n the Moment Resisting Frame and the Braced Moment Resisting Frame, respectively. The results of the s i m p l i f i e d and refined models of the F r i c t i o n Damped Braced Frame are compared i n Table 4.1. The refined model gives higher member forces and deflections than the simplified model since the refined model takes i n t o account the r e a l deformation pattern of the f r i c t i o n devices; as discussed i n Chapter 2, the approximate model overestimates the energy dissipated by the f r i c t i o n devices.  However the  results are reasonably close f o r p r a c t i c a l application. The s i m p l i f i e d model w i l l give exact r e s u l t s under extreme e x c i t a tions, when the devices are slipping at every cycle.  This means that the  two models w i l l converge to the same response as the ground motion gets more severe. From the above comparison i t can be concluded that the s i m p l i f i e d model i s simpler and cheaper to use than the refined model.  Its a p p l i c a -  t i o n i n the study of F r i c t i o n Damped Braced Frames i s s a t i s f a c t o r y f o r a l l p r a c t i c a l purposes, leading to results which are within the variations t y p i c a l l y expected i n earthquake analysis.  4.4  Response Under Random White Noise E x c i t a t i o n When using a p a r t i c u l a r earthquake record as ground motion input, the  energy transmitted to the structure i s d i f f e r e n t for each type of frame (F.D.B.F., B.M.R.F., M.R.F.), since the natural frequencies of these frames are  a l l different.  However the energy input f o r the three different model  frames can be made i d e n t i c a l by using band limited white noise as the excitation source.  28. Band limited white noise i s defined as a random variable having a constant Power Spectral Density Function over a certain frequency  range.  It i s possible to generate a band limited white noise s i g n a l by the technique of summation of sinusoids (16).  In our case, to make the energy  input i d e n t i c a l for the three test frames, the frequency band of the white noise signal must span the estimated natural frequency range of the three d i f f e r e n t model frames.  The computer program SIMU.S, developed at the  University of B r i t i s h Columbia, was used to generate a band limited (0-25 Hz) white noise s i g n a l .  The acceleration record and i t s normalized Power  Spectral Density Function i s presented i n Figure 4.10.  The record has a  duration of 20 seconds with a peak acceleration of 1 g (9810 mm/s ). 2  The  amplitude of the Power Spectral Density Function i s constant for frequencies from 0 to 25 Hz. When t h i s record i s used as a ground motion input to the model frames, a l l the frequencies within the range of 0 to 25 Hz are excited with equal i n t e n s i t y . This band limited white noise record, scaled to a peak ground acceleration of 0.50 g, was used to study the performance of the F r i c t i o n Damped Braced Frame. kN.  The s i m p l i f i e d model was used with a global s l i p load of 7  The response parameters obtained with the F r i c t i o n Damped Braced Frame  are compared to the corresponding responses of the Moment Resisting Frame and the Braced Moment Resisting Frame. The s t r u c t u r a l damage i n the various members of the three model frames after the end of the white noise record ( i . e . resulting from the 20 seconds excitation) i s i l l u s t r a t e d i n Figure 4.11.  Serious damage occurs i n the  Moment Resisting Frame, i n which the f i r s t and second floor beams and both base columns reach their p l a s t i c moment capacity.  Significant structural  damage also occurs i n the Braced Moment Resisting Frame, where a l l the i  29.  cross-braces have yielded i n tension.  Notice that a l l the s t r u c t u r a l  members of the F r i c t i o n Damped Braced Frame remain e l a s t i c . The envelopes of the response parameters for the d i f f e r e n t model frames are presented i n Figure 4.12.  The deflection at the top of the  F r i c t i o n Damped Braced Frame i s 12% of the equivalent deflection i n the Moment Resisting Frame and 25% of the deflection i n the Braced Moment Resisting Frame.  The maximum shear at the base of the F r i c t i o n Damped  Braced Frame i s only 21% and 45% of the corresponding shear i n the Moment Resisting Frame and the Braced Moment Resisting Frame respectively.  The  maximum moment at the base of the F r i c t i o n Damped Braced Frame i s 20% and 43% of the corresponding values i n the Moment Resisting Frame and the Braced Moment Resisting Frame respectively. Time-histories of the d e f l e c t i o n at the top of building f o r the three frames are shown i n Figure 4.13.  The peak amplitude of the F r i c t i o n Damped  Braced Frame i s f a r less than the corresponding amplitude of the two other model frames.  Notice that the vibrations at the end of the excitation are  almost n e g l i g i b l e f o r the F r i c t i o n Damped Braced Frame compared with the vibrations of the other two frames.  This indicates that the building  recovers with almost no permanent deformation.  As suggested by A.S. P a l l  (3), i f the building i s s l i g h t l y out of alignment, i t can be corrected by loosening the bolts i n the device and then retightening. The results of this investigation c l e a r l y indicate the superior performance of the F r i c t i o n Damped Braced Frame compared to conventional aseismic building systems.  By performing this study under conditions which  ensure that the energy induced i n the structures i s the same for a l l frames, i t has been shown that the frequency content of a particular earthquake, i n r e l a t i o n to the d i f f e r e n t frame frequencies, i s not the underlying source of this difference i n structural response.  30. 4.5  Equivalent Viscous Damping Study The results of the previous sections have shown that the use of  inexpensive f r i c t i o n devices i n the bracings of s t e e l framed buildings can s i g n i f i c a n t l y enhance their earthquake resistance.  One method to quantify  the performance of the F r i c t i o n Damped Braced Frame r e l a t i v e to conventional aseismic systems i s by means of an equivalent viscous damping study.  Viscous damping was added to the Moment Resisting Frame and the  Braced Moment Resisting Frame u n t i l their responses become similar to the response of the F r i c t i o n Damped Braced Frame.  The maximum deflection at  the top of the building was choosen as the comparative response parameter. The program "Drain-2D" assumes that the viscous damping results from a combination 3  0  of mass-dependent and stiffness-dependent effects, so that i f  = 0 (see Equation (4.4)): [C] = a[M] + 3[K]  (4.5)  i n which a and 3 are constants to be s p e c i f i e d by the program user.  To  ensure the existence of c l a s s i c a l normal modes, the viscous damping ratios must be defined i n the following way  (14);  3OJ  r  2u) r  2  x  where: th ?  r  = viscous damping ratio of the r  mode t tl  OJ^  = undamped natural frequency of the r  a,3  - damping c o e f f i c i e n t s  mode (rad/s)  Considering only the f i r s t two modes of v i b r a t i o n s , the damping c o e f f i c i e n t s a,3 can be determined:  '  31. 4TT(T C; 2  a  -  2  T ^ ) (4.7a)  =  T  - T  2 2  T T (T q 1  B  L  -  2  2  2  T l  C ) 2  (4.7b)  = X  2  l  i  where: T|,T  = Undamped p e r i o d s of the f i r s t  2  and second modes.  To make t h e e q u i v a l e n t v i s c o u s damping study dependent on a s i n g l e variable,  i t was assumed that the v i s c o u s damping r a t i o s a r e the same f o r  the  and second modes:  first  C,  -  -  C  (4.8)  Hence: 4TT a  ?(T - T ) 2  (4.9a)  T  T  3  =  the  - T j  2 2  1 2 T  T  The f i r s t  x  =  C  (  2  T  2 _  ~ • T  2  T  2  l  )  (4.9b)  two n a t u r a l p e r i o d s of both the Moment R e s i s t i n g Frame and  Braced Moment R e s i s t i n g Frame were c a l c u l a t e d and found t o be ( s e e  S e c t i o n s 9.1 and 10.1):  1)  Moment R e s i s t i n g Frame T  1  T  2)  2  =  0.3532 s e c .  =  0.1090 s e c .  Braced Moment R e s i s t i n g Frame T T  l  2  =  0.1966 s e c .  =  0.0686 s e c .  32. Therefore: 1)  2)  Moment Resisting Frame a  =  27.1882 ?  6  =  0.0833 e  Braced Moment Resisting Frame a  -  47.3845 c,  B  =  0.0509 5  Several i n e l a s t i c time-history  dynamic analyses were performed f o r the  Moment Resisting Frame and the Braced Moment Resisting Frame with d i f f e r e n t values of the viscous damping r a t i o u n t i l their responses become similar to the response of the F r i c t i o n Damped Braced Frame.  The Newmark-Blume-Kapur  a r t i f i c i a l earthquake scaled to peak accelerations of 0.05, 0.10 and 0.30 g was used i n the analyses. The results of the analyses are shown i n Figure 4.14.  The equivalent  viscous damping r a t i o necessary to match the response of the F r i c t i o n Damped Braced Frame i s plotted vs. the ground peak acceleration.  As  expected, less viscous damping i s needed for the Braced Moment Resisting Frame than the Moment Resisting Frame.  Notice that the equivalent  damping r a t i o increases with the ground peak acceleration.  viscous  This indicates  that the F r i c t i o n Damped Braced Frame becomes more e f f i c i e n t as the intensity of the ground motion increases;  this i s due to the fact that the  energy i s dissipated mechanically throughout the height of the building rather than by l o c a l i z e d i n e l a s t i c action of the main structural members. For a peak ground acceleration of 0.30 g, 38% of c r i t i c a l damping i s needed for  the Moment Resisting Frame and 12% for the Braced Moment Resisting  Frame.  5.  5.1  CALIBRATION AND QUALITY CONTROL OF FRICTION DEVICES  General In the l a s t chapter, the optimum global s l i p load of the F r i c t i o n  Damped Braced Frame was found to be 7 kN.  Based on this value, seven  f r i c t i o n devices were fabricated i n a machine shop i n Montreal. This chapter i s concerned with the c y c l i c load tests performed at d i f f e r e n t frequencies on the f r i c t i o n devices i n order to c a l i b r a t e these units for the optimum s l i p load.  The tests also v e r i f y the r e l i a b i l i t y of  the hysteresis loops a f t e r many cycles. The general arrangement of an actual f r i c t i o n device i s presented i n Figure 5.1. Note that the overall dimensions of the mechanism are 355x200 mm.  The four connections between the pads and the braces are made with two  20 mm diameter ASTM A325 high strength bolts, representing moment connections. For research purposes, each f r i c t i o n device i s provided with a compression  spring (Korfund & Simpson Ltd., Type WSCB K = 1950 l b s / i n )  which allows adjustments to be made to the clamping force and therefore the s l i p load.  The free length of the compression  spring i s 146 mm.  Springs  w i l l not be used i n the prototype structure; the clamping force w i l l be developed  by a bolt torqued to the proper value.  Figure 5.2 shows the d e t a i l s of a f r i c t i o n device. made from 50x8 mm mild steel plates.  The f r i c t i o n  The mechanism i s  surfaces are 3 mm thick  heavy duty brake l i n i n g pads (No. 55B by "ASBESTONOS"), which are glued to the steel plates with plasti-lock glue.  Each device i s provided with four  f r i c t i o n surfaces: one f r i c t i o n surface i n each j o i n t and one common f r i c tion surface between the joints i n the form of a washer, to which brake  34. l i n i n g pads are glued on each side. surfaces i s shown i n Figure 5.3.  A view of the d i f f e r e n t f r i c t i o n  The fabrication tolerances were s p e c i f i e d  according to the CSA standard CAN3-S16.1-M78.  The nominal diameter of the  holes were d r i l l e d 2 mm greater than the nominal bolt s i z e s . Figure 5.4 shows an actual f r i c t i o n device mounted on the model frame.  5.2  Experimental Set Up The c y c l i c load tests on the f r i c t i o n devices were carried out i n the  Structural Laboratory of the C i v i l Engineering Department at the University of B r i t i s h Columbia. shown i n Figure  The general experimental set up f o r the tests i s  5.5.  One end of a diagonal l i n k of the f r i c t i o n device was bolted to a r i g i d testing bench while the opposite end was attached to a 50 kN capacity v e r t i c a l hydraulic actuator having a piston area of 2400 mm  2  stroke of 250 mm.  and a maximum  The servo valve of the actuator has a flow rate of 20  gallons/minute. The movement of the piston was controlled by an MTS  console which  contains an o s c i l l a t o r that allows controlled c y c l i c displacement be performed with variable amplitudes and frequencies. the piston was monitored  tests to  The displacement of  by a LVDT (Linear Variable Displacement  Transducer) mounted inside the actuator.  The load was measured by a 5  metric ton capacity load c e l l . The analog signals from the LVDT and the load c e l l were fed into a P h i l i p s analog-analog  tape recorder.  The data were stored d i r e c t l y on tape  for subsequent reduction and manipulation. was connected  A Hewlett-Packard  X-Y  Plotter  i n p a r a l l e l with the tape recorder so that a permanent record  of the hysteresis loops could be obtained on paper as the tests were  performed.  A storage oscilloscope was also connected i n p a r a l l e l with the  tape recorder i n order to v i s u a l i z e immediately  the hysteresis loops on the  screen; this served to monitor the plots desired f o r the permanent record.  5.3  S t a b i l i t y Tests The f i r s t series of tests on the f r i c t i o n devices were performed  with  a simple f r i c t i o n j o i n t (2 f r i c t i o n surfaces) i n order to v e r i f y the s t a b i l i t y and r e l i a b i l i t y of the brake l i n i n g pads under repetitive c y c l i c loads.  The experimental set up f o r t h i s series of tests i s presented i n  Figure  5.6.  The tests were conducted with d i f f e r e n t values of the s l i p load and also at various frequencies. Hz to 4.0 Hz.  The range of frequencies covered was  from  0.2  Each test was subjected to 50 cycles of loading.  Figure 5.7 shows a t y p i c a l load-displacement curve obtained from this series of tests.  The performance of the brake l i n i n g pads i s seen to be  r e l i a b l e and repeatable.  The hysteresis loop i s very nearly a perfect  rectangle and exhibits n e g l i g i b l e fade even a f t e r 50 cycles. It may  be seen that two d i s t i n c t plateaus occur at two opposite  corners of the hysteresis loop, where the t r a n s i t i o n between tensioncompression  and compression-tension  takes places.  The f i r s t plateau i s due  to the e c c e n t r i c i t y of the f r i c t i o n j o i n t which r e s u l t s i n a small out-ofplane movement of the central bolt and spring unit when the load i s applied.  The second plateau i s due to the clearance provided between the  central bolt and the washer onto which the brake l i n i n g pads are glued. The 2 mm  tolerance on the hole size was found to be too large and, as a  r e s u l t , the central bolt I n i t i a l l y moved without contacting the washer and thus f r i c t i o n resistance was not f u l l y developed i n the early stage of the loading.  36.  Figure 5.8 shows t y p i c a l hysteresis loops obtained at various frequencies of loading.  For the range of frequencies studied (0.2-4.0 Hz), i t  seems that the s l i p load remains constant.  Because of the physical l i m i t a -  tions of the actuator, the total movement of the piston i s reduced as the e x c i t a t i o n frequency  increases.  This leads to an apparent reduction i n the  e f f i c i e n c y of the f r i c t i o n joint at high frequency  loading since, although  the t o t a l movement of the piston i s reduced, the two plateau remain constant.  To be f u l l y e f f e c t i v e , the f r i c t i o n joint must s l i p more  than a c e r t a i n threshold necessary size.  displacements  This problem may  by a high frequency  to overcome the tolerance of the hole  be encountered during an earthquake characterized  input, since the resulting s t r u c t u r a l  displacements  w i l l usually remain small. Note that f o r a frequency  of 4 Hz the s l i p load i s not reached.  This  i s due to the fact that the slippage i s too small to overcome the tolerance of the hole size and not because the f r i c t i o n properties of the pads are altered.  As noted below, this problem was  solved by adjusting the hole  size of the washer.  5.4  Tests of Complete Device The tests described i n Section 5.3 were used to v e r i f y the r e l i a b i l i t y  of the brake l i n i n g pads; i t was demonstrated that the hysteresis loops developed by these pads do not degrade and remain stable even a f t e r 50 cycles.  This section deals with a series of tests which were performed on  the complete f r i c t i o n devices. As mentioned e a r l i e r , the diameter of a l l the b o l t holes were specified 2 mm tolerance was  greater than the nominal bolt s i z e .  This f a b r i c a t i o n  found to be too large and, as a r e s u l t , a rectangular  37.  hysteresis loop was not obtained, as shown i n Figure 5.9.  This i s due to  the fact that, when the loaded link s l i p s , the 4 corner bolts of the mechanism (see Figure 5.1)  f i r s t s l i d e i n t h e i r holes without developing  any bearing resistance. During this i n t e r v a l , the mechanism i s not activated and energy i s not absorbed i n the f r i c t i o n j o i n t of the other link.  This problem occurs i n both tension and compression.  The displace-  ment required to get both joints to s l i p i s so large that the value of the global s l i p load cannot be reached, as shown i n Figure  5.9.  The r e s u l t s of these tests c l e a r l y indicate that a rectangular loaddeformation curve can only be obtained i f the f a b r i c a t i o n tolerances of the f r i c t i o n devices are minimized.  This was achieved by inserting s t e e l bush-  ings i n the 4 corner holes of the mechanism and also i n the center slots of the f r i c t i o n pads. 2 mm  to 0.25  mm.  As a r e s u l t , the f a b r i c a t i o n tolerance was reduced from These steel bushings were fabricated i n the C i v i l  Engineering Workshop at the University of B r i t i s h Columbia. A t y p i c a l hysteresis loop developed with the modified device i s shown i n Figure 5.10.  Note that the incorporation of the s t e e l bushings  dramatically improves the performance of the f r i c t i o n devices.  Because the  f a b r i c a t i o n tolerances are minimized, both f r i c t i o n j o i n t s s l i p simultaneously; this results i n an almost perfectly rectangular hysteresis loop. Notice, however, that some imperfections s t i l l remain at two opposite corners of the hysteresis loop.  A perfectly rectangular hysteresis loop  presumably could be obtained by completely eliminating the f a b r i c a t i o n tolerance;  however, i t was decided that the results obtained with these  modified devices are satisfactory f o r p r a c t i c a l applications. Again, the hysteresis loop associated with the complete device i s very stable and displays negligable fade.  For high s l i p loads (> 5 kN)  the  38.  pads and the central portion of the mechanism become very hot a f t e r 50 cycles.  However, the s l i p load i s not influenced by the elevation i n  temperature.  A test was  performed when the pads were very hot, and another  at room temperature with the same spring load; no difference was  observed  i n the load-deformation curves.  5.5  C a l i b r a t i o n Curves In order to c a l i b r a t e the f r i c t i o n devices f o r the desired global s l i p  load, a l l 7 devices were tested under repeated c y c l i c loads with d i f f e r e n t values of the spring load. Before each test, the length of the compression spring was measured with a precision vernier.  It was  then possible to establish a c o r r e l a t i o n  between the global s l i p load and the length of the compression spring. each device a linear regression analysis was  For  performed to obtain an equa-  t i o n r e l a t i n g the global s l i p load and the length of the spring. The  resulting equations represent  devices. 5.11. 5.1.  The  c a l i b r a t i o n curves for the  friction  These curves are plotted with the experimental data i n Figure results of a l l the linear regression analyses are shown i n Table  Note that the c o r r e l a t i o n c o e f f i c i e n t s are very close to unity; t h i s  i s to be expected since the r e l a t i o n between the global s l i p load spring length should be l i n e a r according  to the conventional  and  friction  law.  Once these c a l i b r a t i o n curves are available, i t i s very easy to set each device for the optimum s l i p load of 7 kN  (see Chapter 4) by simply  adjusting the length of the spring with a precision vernier.  39. 6.  6.1  DETERMINATION OF  MATERIAL PROPERTIES  General T h i s chapter  describes  the procedures f o l l o w e d  p h y s i c a l p r o p e r t i e s of the m a t e r i a l s used i n the frames.  to determine  the  f a b r i c a t i o n of the model  These v a l u e s are needed i n the n o n - l i n e a r ,  t i m e - h i s t o r y dynamic  a n a l y s i s program, from which a n a l y t i c a l s o l u t i o n s were generated f o r comparison w i t h the experimental For  results.  the main members used i n the c o n s t r u c t i o n of the model frame  (S75*8), the  f o l l o w i n g p r o p e r t i e s must be  determined:  Young's Modulus, E Tangent Modulus, E ' P l a s t i c Moment, M  P For  the c r o s s - b r a c e s  used i n the Braced Moment R e s i s t i n g Frame and  in  the F r i c t i o n Damped Braced Frame, the f o l l o w i n g p r o p e r t i e s must be determined:  Young's Modulus, E Tangent Modulus, E ' Yield Stress, a Buckling  To  estimate  y Stress,  these m a t e r i a l p r o p e r t i e s , a s e r i e s of u n i a x i a l t e s t s were  c a r r i e d out on s t e e l specimens i n the S t r u c t u r a l L a b o r a t o r y Engineering  Department at the U n i v e r s i t y of B r i t i s h  of the  Columbia.  Civil  40. 6.2  Tests of Cross-Braces For  the cross-braces, these tests were conducted on a complete brace  unit as used with the F r i c t i o n Damped Braced Frame.  The unit consists of a  6 mm square bar made of AINSI C-1018 cold formed s t e e l , which was welded to 2 plates of mild s t e e l , as shown i n Figure 6.1. Two u n i a x i a l tests were conducted on 2 d i f f e r e n t specimens.  In the  f i r s t test, the specimen was loaded through one cycle of tension and compression without rupture.  Figure 6.2 shows the permanent deformation of  the specimen at the end of this test.  The load deformation curve obtained  from this test i s shown i n Figure 6.3.  The slope (m) of the i n i t i a l  e l a s t i c part of the curve can be measured from the graph and i s equal to:  AE  (6.1)  where: A  cross-sectional area of the specimen  E  Young's Modulus  L  Length of the specimen  Young's Modulus can then be evaluated from:  E  mL A~~  (6.2)  Similarly the tangent modulus can be estimated by:  (6.3)  where m' i s the slope of the i n e l a s t i c part of the curve determined from a l i n e a r regression analysis. The y i e l d load C^y) i y i e l d stress ( ) a  y  c a n  s  determined d i r e c t l y from the curve and the  then be found since:  P a y  =  The c r i t i c a l buckling load (^ )  (6.4)  -r%-  A  c a n  cr  ke determined d i r e c t l y from the  load-deformation curve and i s equal to:  P  r  - ifSL(KL)  C r  (6  .5)  2  from which the effective length factor can be estimated:  v K  =  —  (6.6)  L 'p cr  where: K = 1  for a perfectly pin-pin brace  K = 0.5  for a perfectly f i x - f i x brace.  The e f f e c t i v e length factor can be expected to l i e between these two values, since the end conditions of the brace unit are not c l e a r l y defined. The second u n i a x i a l t e n s i l e test on a brace unit was carried out up to rupture of the specimen. obtained from this test.  Figure 6.4 shows the load-deformation curve  The ultimate load (P ) can be determined from the u curve; the resulting ultimate stress i s given by:  42.  (6.7)  Also, the ultimate strain ( ) can be calcualted from: e  u  A*  e  u  •  -r  (6  -> 8  where hi  u  = ultimate elongation of the brace,  Based on these uniaxial tests, the following materials properties were obtained f o r the braces:  Young's Modulus, E = 180,000 MPa Tangent Modulus, E' = 58,000 MPa Yield Stress, a = 495 MPa y Ultimate Stress, a = 720 MPa u Ultimate Strain, e = 0.0074 u E f f e c t i v e Length Factor K = 0.65 C r i t i c a l Buckling Stress a =74 MPa cr a = 4 MPa cr 6.3  f o r F.D.B.F. for B.M.R.F.  Tests of Main Members The material properties of the main members (S75x8) were evaluated by  performing a uniaxial compression test on a complete S75x8 section 300 mm long.  The load-deformation  Figure 6.5.  curve obtained from t h i s test i s shown i n  43. Following the same procedure described i n Section 6.2, the material properties of the main members were determined as follows:  Young's Modulus, E  = 170,000 MPa  Tangent Modulus, E' = 0 Yield Stress, a  = 400 MPa  y Yield Strain, e y  = 0.0024  The p l a s t i c moment i s given by:  M  P  =  Z oy  (6.9)  where Z  =  Therefore, M  p l a s t i c section modulus = 31.9xl0  P  = 12.76  kN-m.  3  mm  3  f o r S75x8  44.  7.  7.1  EXPERIMENTAL SET UP ON SHAKING TABLE  Shaking Table A l l the seismic tests reported i n t h i s thesis were carried out on the  Earthquake Simulator i n the Earthquake Engineering Engineering  Laboratory of the C i v i l  Department at the University of B r i t i s h Columbia.  The shake  table i s 3 m x 3 m i n plan and i s fabricated as a c e l l u l a r box from welded aluminum plate.  The table i s mounted on four pedestal legs with universal  joints at each end to ensure linear horizontal motion.  Lateral movement of  the table (or yaw) i s restrained by three hydrostatic bearings. specimens are attached  Test  to the table by steel bolts which thread into s t e e l  i n s e r t s arranged i n a g r i d .  Figure 7.1 shows the general arrangement of  the shaking table. The shaking table i s activated by an MTS hydraulic power supply which provides regulated hydraulic pressure and flow to the servo valve and actuator.  The hydraulic f l u i d i s supplied at a pressure of 3000 p s i and at  a flow rate of up to 70 gpm. l i n e a r actuator.  The MTS hydraulic jack i s a double acting  A manifold supplies surges i n hydraulic f l u i d demand and  reduces fluctuations In l i n e pressure during dynamic conditions.  A three  stage servo valve i s used to control e l e c t r i c a l l y the d i r e c t i o n and magnitude of the hydraulic flow.  The third stage of the servo valve i s position  controlled by e l e c t r i c feedback from a spool l i n e a r variable displacement transducer (LVDT) and a valve c o n t r o l l e r . The simulator can be programmed by a D i g i t a l PDP 11/04 mini-computer to excite the test specimens with very generalized seismic ground motions.  functions, including  Some of the system s p e c i f i c a t i o n s are as follows:  maximum specimen weight equals 16000 kg; maximum horizontal acceleration equals 2.5 g; and maximum peak-to-peak horizontal displacement i s 150 mm. The mass of the empty table i s 2090 kg.  45. 7.2  Data Acquisition The data a c q u i s i t i o n system of the Earthquake Engineering Laboratory  i s controlled by a D i g i t a l PDP11/04 mini-computer having 56K bytes of memory, a RT11 operating software system and two RX01 disk drives.  The  physical arrangement of the data acquisition system i s shown i n Figure 7.2. During an experiment,  the analog signals coming from the d i f f e r e n t  sensors mounted on the test structure are fed into a 16 channel multiplexed Analog-Digital converter.  Data i s stored d i r e c t l y onto floppy disks  for l a t e r reduction and manipulation. A d i r e c t l i n e l i n k s the mini-computer to the University's main (Amdahl) computer, thereby assuring powerful data processing capability at the convenience of a main frame system.  F a c i l i t i e s also e x i s t to display  test data on storage oscilloscopes, on chart recorders and X-Y plotters, and on a Calcomp plotter and Tektronix graphics terminals equipped with hard copy units. The Earthquake Simulator at the University of B r i t i s h Columbia i s probably the most sophisticated test f a c i l i t y of i t s kind i n Canada. I t s a b i l i t y to command the table and to handle test data by means of a minicomputer allows for accuracy and e f f i c i e n c y when dealing with  earthquake  generated excitations.  7.3  Instrumentation A variety of sensors were mounted on the structures and on the shaking  table i n order to measure the response of the model frames due to earthquake ground motion excitations.  46. Four accelerometers were used to measure the horizontal acceleration at various levels of the structure.  A K i s t l e r servo accelerometer  was  clamped under the shaking table i n order to monitor the table acceleration. Two Statham model A514TC s t r a i n gage accelerometers, with a working  range  of ±2.5 g, were i n s t a l l e d on the f i r s t and second f l o o r cross-beams of the model frame.  Figure 7.3 shows the f i r s t floor accelerometer bolted on the  cross-beam.  To monitor the t h i r d f l o o r horizontal acceleration, a Statham  model 1525 strain gage accelerometer, with a working range of ±20 g, was mounted on the t h i r d f l o o r cross-beam.  The analog signals from these four  accelerometers were fed into amplifiers and to the multi-plexed AnalogD i g i t a l converter. Six  l i n e a r potentiometers were connected to a fixed panel situated  beyond the table; these were used to measure the absolute displacements of the frame.  Two potentiometers were attached to each f l o o r , one to each  side frame p a r a l l e l to the d i r e c t i o n of the excitation.  By analysing the  phase s h i f t between the two displacement records at each f l o o r , i t was possible to measure and detect any torsional motion developed i n the structure.  Figure 7.4 shows one potentiometer mounted on the fixed panel  located outside the table.  A l l the potentiometers were excited by one  power supply, whose excitation voltage was 9 v o l t s .  D.C.  The analog signals  from the potentiometers were fed d i r e c t l y into the multi-plexed AnalogD i g i t a l converter. The absolute displacement of the table was measured by a Linear Variable Displacement Transducer (LVDT) mounted inside the table actuator. The r e l a t i v e displacement of each f l o o r of the structure with respect to i t s moving base ( i . e . , the table) was obtained by subtracting the d i g i t i z e d record of the table displacement from each corresponding potentiometer record.  47. Four sets of s t r a i n gages were i n s t a l l e d to monitor the strain-time history at various locations i n the model.  Each set consisted of two  s t r a i n gages, which were glued to the outer faces of each flange of the main members. were recorded.  Half bridge c i r c u i t s were used, so that only bending strains One set of s t r a i n gages was mounted just above the r e i n -  forcing plate of one of the base colums, as shown i n Figure 7.5.  The  remaining three pairs of s t r a i n gages were mounted at one end of each f l o o r beam, so that the strains i n the beams could be monitored.  Figure 7.6  shows the set of s t r a i n gages mounted on the f i r s t f l o o r beam.  The analog  signals from the s t r a i n gages were fed through amplifiers to the multiplexed Analog-Digital converter.  A l l the s t r a i n gages were calibrated by  connecting a c a l i b r a t i o n resistance simulating a known change of s t r a i n i n p a r a l l e l with the bridge. Linear Variable Displacement Transducers  (LVDT) were used to record  the slippage-time history of the f r i c t i o n pads.  In the f i r s t series of  tests only one of the second f l o o r f r i c t i o n devices was instrumented f o r slippage, because the 15 other recording channels of the data a c q u i s i t i o n system were otherwise occupied. of  the s i x potentiometers  However, i n the second test series three  were disconnected, since I t was found that  t o r s i o n a l motion was n e g l i g i b l e ; t h i s allowed a t o t a l of four f r i c t i o n devices to be  instrumented.  A l l the LVDT's on the f r i c t i o n devices were excited by a single D.C. power supply having an excitation voltage of 6 v o l t s . F i n a l l y , Tens-Lac b r i t t l e lacquer was sprayed on various parts of the frame to detect the s t r a i n (stress) patterns.  The b r i t t l e lacquer cracks  at a c e r t a i n s t r a i n l e v e l ; the cracks occur perpendicular to the maximum tensile strains and are present at locations where the s t r a i n l e v e l has exceeded a certain threshold s t r a i n .  For the b r i t t l e lacker used, this  threshold was nominally 500 microstrain (ue).  48. 8.  8.1  SYSTEM IDENTIFICATION THEORY  General Dynamic tests of the model frames were divided into two parts:  mination tests, and evaluation tests.  deter-  The purpose of the determination  tests was to obtain the dynamic c h a r a c t e r i s t i c s of the model frames. Natural frequencies, damping parameters, mode shapes, and frequencyresponse functions of each model frame were obtained from the determination tests.  Determination tests were conducted at low excitation l e v e l s .  Evaluation tests were l a t e r used to study the performance of the model frames under s p e c i f i c ground motion time-histories.  Evaluation tests were  conducted at r e l a t i v e l y high e x c i t a t i o n l e v e l s . Determination tests are based upon system i d e n t i f i c a t i o n theory. Two basic approaches can be used to estimate the dynamic c h a r a c t e r i s t i c s of a test  structure: 1)  Time-response method  2)  Frequency response method.  The choice of the method used to estimate the dynamic c h a r a c t e r i s t i c s of a model structure depends on many factors: capability of the shaking table and the data acquisition system, instrumentation available, degree of non-linearity  of the test structure,  etc.  Often, a combination of the two  methods i s used i n a determination test program. In this chapter the parameter estimation theory of linear e l a s t i c systems i s reviewed and the methods necessary to determine the dynamic characteristics of the model frames are developed taking into account the Instrumentation described i n Chapter 7 and the c a p a b i l i t i e s of the Earthquake Simulator Table.  49.  8.2  Undamped Free Vibration Analysis The basic d i f f e r e n t i a l equation for the undamped free vibrations of a  linear e l a s t i c system i s given by:  [M]{x} + [K]{x}  In steady-state  =  {0}  (8.1)  conditions, the following solution i s assumed:  {x} = {A}  sin(o)t + <(,)  Substituting into (8.1) y i e l d s :  8 i n ( u t + <f>) =  [[K] - u [M]] {A} 2  {0}  This equation must be s a t i s f i e d for every value of t, therefore:  [[K] - u> [Ml] {A}  =  2  {0}  (8.2)  or [K][I]{A}  OJ [M]{A} 2  =  Assuming a diagonal mass matrix we can  [M]  =  [M]  (8.2a)  write:  1 / 2  [M]  1 / 2  (8.2b) [I]  =  Substituting into (8.2a) leads to:  [M]~  1 / 2  [M]  1 / 2  50.  [K][M]~  Pre-multiplying by [ M ]  [[M]  _ 1 / 2  [M]  1/2  {A} =  1 / 2  u> [M] 2  [M]  1/2  1 / 2  {A}  yields:  - 1 / 2  [K][M]  _ 1 / 2  ] {[M]  {A}}  1 / 2  =  a  {[M]  2  1 / 2  {A}}  Equation (8.3) i s a c l a s s i c a l eigenvalue problem of the form: X{y},  (8.3)  [B]{y} =  which leads to a set of n eigenvalues (natural frequencies) and n  eigenvectors (mode shapes).  r-  2  The frequency matrix i s defined as:  J  (8.4)  -  i n  where [~J  represents a diagonal matrix  By substituting each value of I D solve for the corresponding  set of  2  i n turn into Equation (8.3), we i.e.  {A},  ({A^^},  { A ^ ^ },...  can  {A^ ^ }). N  These corresponding { A } can be arranged i n a modal matrix:  [A]  =  [ { A  (  1  )  } , { A  (  2  )  } , . . . { A  (  N  )  } ]  (8.5)  Since the { A } vectors are eigenvectors, they do not have an absolute value; only their shapes are determined.  If the normal vectors are  normalized such that:  { A  (  R  )  }  T  [M]  {A  (  R  )  }  -  1  (8.6)  The following orthogonality properties of the normal modes can be shown (14) to exist:  51. [A]  [A]  T  [M][A]  T  [K][A]  (8.7)  - [I]  =  [L^J  (8.8)  The undamped natural frequencies and mode shapes of the d i f f e r e n t model frames were predicted using the computer program "DYNA" from the University of B r i t i s h Columbia C i v i l  Engineering Program Library.  "DYNA"  i s an interactive graphics program which performs l i n e a r e l a s t i c small deformation  dynamic analysis of plane frame problems.  The structure may be  formed from pinned-pinned, fixed-fixed and fixed-pinned members. "DYNA" reads the structural data from a data f i l e and assembles the global s t i f f n e s s and mass matrices.  The global s t i f f n e s s matrix i s  constructed from standard plane frame beam s t i f f n e s s matrices as described i n any introductory structural analysis reference.  For modelling the mass  d i s t r i b u t i o n , half of the weight of each member may be concentrated at each end of the member, and point masses may be superimposed at any node.  The  resulting global mass matrix i s diagonal. The undamped natural frequencies and mode shapes are found by solving Equation  (8.3).  A modified i t e r a t i v e (power) method i s used.  The s o l u t i o n  i s performed i n double precision.  8.3  Fourier Spectrum Analysis For Frequency  Determination  The f i r s t step i n the parameter estimation of a test structure i s to measure experimentally the natural frequencies of the system. method of exhibiting the frequency  One standard  content of a time-history record  involves the use of a Fourier Amplitude Spectrum.  52. The i  th  coordinate acceleration response of a l i g h t l y damped multi-  degree of freedom system subjected to i n i t i a l conditions {x } n  and (x } i s n  given by (14):  X i  where  L  (t)  =  (r)  and N  I [A r (r)  ( r ) i  e  (l>  cos u t + N  r )  C r )  s i n a t)]  (8.9)  are modal constants which depend on the i n i t i a l  conditions and on the modal natural frequency and damping value. Note that x^(t) i s a sum of simple harmonic signals whose frequencies correspond to the natural frequencies of the system.  The Fourier Amplitude  Spectrum of this signal, which i l l u s t r a t e s i t s frequency content, therefore w i l l exhibit a "spike" at each of the frequency components of the signal (the  natural frequencies of the system).  The amplitudes of the spikes w i l l  depend on the damping c h a r a c t e r i s t i c of the system and also on the i n i t i a l conditions. A very simple method of measuring experimentally the natural frequencies of the three different model frames was adopted using the Fourier Amplitude Spectrum.  The model frames were excited under a harmonic base  motion having a frequency well separated from the calculated resonant frequencies.  The ground motion was then stopped suddenly and the t h i r d  floor horizontal acceleration-time decay was recorded for a s u f f i c i e n t period of time.  The Fourier Amplitude Spectrum of t h i s recorded motion was  calculated using the Earthquake Simulator Laboratory software program "EDSPEC".  "EDSPEC" allows the Fourier Amplitude  from formatted data f i l e s .  Generated  Spectrum to be plotted  spectral values may  be Hanned (18),  and can be stored i n a print f i l e f o r l a t e r hard copy printout. The natural frequencies of each model frame correspond to the frequencies defining the maximum spectral values (the 'spikes') of the Fourier Amplitude  Spectrum.  53. 8.4  Complex Frequency Response or Mobility Function One convenient way to determine the dynamic c h a r a c t e r i s t i c s of a test  structure i s to find the magnitude of i t s complex frequency response function (or mobility function |H(io)|) under harmonic base motion.  The  governing d i f f e r e n t i a l equation f o r a base motion problem can be written i n the following form:  [M]{x} + [C]{x} + [K]{x}  =  -[M]{I} x  '  (8.10)  where [M]  =  global mass matrix  [C]  =  global damping matrix  [K]  =  global s t i f f n e s s matrix  {x}  =  vector of displacements r e l a t i v e to the moving base  {1}  =  influence vector coupling the input ground motion to each degree of freedom  x  8  =  ground acceleration  Assume that the ground acceleration i s a known harmonic function:  x g  =  a s i n tot  (8.11)  Under steady state conditions, the following solution of Equation (8.10) can be assumed:  {x}  -  {P} cos wt + {Q} s i n tot  Substituting into (8.10) y i e l d s :  (8.12)  54. [([K]  - u [M]){P} + u[C]{Q}] cos wt +  [([K]  - u [M]){P} - u[C]{Q}] s i n ut  2  = ~[M]{I} a s i n ut  2  This equation must be s a t i s f i e d for any time t . Therefore:  ([K] - w [M]){P} + u[C]{Q}  =  {0}  (8.13a)  ([K] - w [M]){P} - w[C]{Q}  -  ~[M]{I} a  (8.13b)  2  2  By using the orthogonality properties of the mode shapes as given by Equations (8.7) and (8.8), we may write:  [M]  =  [A]  [I] [ A ] "  [K]  =  [A]"  T  T  (8.14)  1  f ^ i ] [A]  (8.15)  - 1  Assuming modal damping, the damping matrix i s also orthogonal (14) to the modal matrix:  [C]  -  [A]"  r2c ft» J  T  ±  [A]"  ±  (8.16)  1  Substituting Equations (8.14-8.16)into Equations (8.13a) and (8.13b):  [A]"  T  [A]"  T  ["^-^JtA]  - 1  ^ } + [A]  T  r2? w wJ[A]" {Q} = {0}  (8.17a)  1  1  i  [u^-uZjtAr^P} - [ A ] " r Z C ^ J f A l - ^ Q } = - [ A l " [ I ] [ A ] " {l}a T  T  1  (8.17b)  55.  From which {P} and {Q} are solved:  {P} = [A] r 2 C ^ " / ( ( ^ - w ) 2  i  {Q} = -[A]  i  2  i  2  + (2^0)^)2)] [A]  + (2C u) w) )J  [tu -u )/((u -u ) 2  2  2  ±  2  2  2  ±  For the experimental  i  ±  set-up on the shaking  were measured r e l a t i v e to a fixed reference.  - 1  {I}a  [A]  - 1  {l}a  (8.18)  (8.19)  table, the displacements  The vector of t o t a l displace-  ment from the fixed reference, {x^,}, i s expressed as  {x }  =  {x} + {1} x  where x^ i s the harmonic displacement of the table.  (8.20)  Then  (L> = {x> + {1} x  {x^J = (a{l) - uJ {H}) s i n ait - ai {P} cos uit 2  2  (8.21)  The vector of the mobility functions at any forcing frequency of the table can be defined as the r a t i o of the r e s u l t i n g vector of absolute maximum acceleration amplitudes to the absolute maximum amplitude of the ground motion acceleration ( i . e . the magnification factors or maximum r a t i o of response to excitation for any forcing frequency):  ,  - ,  {|H(u)|} =  The mobility function f o r the r  { | X  TI  „  }  (8.22)  degree of freedom of the system i s :  ::co  (r) |H (w)|  Substituting Equation  (8.21) yields:  |H^ (S)| r;  (8.23)  P  ( r )  ) 2 + (a-w Q 2  ( r )  )2]  1 / 2  -  (8.24)  Define:  Substituting into Equation  |H (S)| (r)  -  (r)  p(r)  _  p(D  = 2  Q  (8.25a)  (r) (8.25 b)  (8.24) y i e l d s :  (1 - ll  2  Q  ( r )  + u>[(P  ( r )  )  2  + (Q  ( r >  ) ]) 2  1 / 2  (8.26)  Knowing the natural frequencies and mode shapes given by the program "DYNA" (see Section 8.2) and using the measured damping r a t i o s (see Section 8.8),  the vector of mobility functions can be calculated for various forc-  ing frequencies. created.  For t h i s purpose, the computer program "VIBRATION" was  A l i s t i n g of the program i s given i n Appendix A.  The program  "VIBRATION" reads the natural frequencies, mode shapes and damping r a t i o s of the system i n a data f i l e and uses Equation  (8.26) to generate the  mobility functions (magnification factors) i n a given range of forcing frequencies for a l l the degrees of freedom considered.  57. The magnitudes of the mobility functions are expressed i n decibles (db); this i s accomplished by multiplying the l ° g factor 2 0 .  10  (magnitude) by the  This f a c i l i t a t e s the use of the Half-Power method with the  mobility functions for estimating damping ratios (see Section 8 . 8 ) . The mobility functions for each f l o o r of the model frames can be generated experimentally  by exciting the structure with various harmonic  base motions and recording, at steady state, the acceleration amplification factors between the table and each floor (harmonic excitation method). experimental  The  mobility functions are generated by p l o t t i n g these magnifica-  tion factors ( i n db) as a function of the forcing frequencies.  8.5  Frequency Assurance C r i t e r i a The natural frequencies of the d i f f e r e n t model frames can be predicted  by the computer program "DYNA" which solves the c l a s s i c a l problem (Equation ( 8 . 3 ) ) . experimentally  eigenvalue  The natural frequencies can also be measured  by performing a Fourier Spectrum Analysis on the third f l o o r  time-acceleration decay record.  It i s necessary to e s t a b l i s h a systematic  method of comparing these two results i n order to judge the accuracy  of the  a n a l y t i c a l model. The most obvious comparison of the measured and predicted natural frequencies can be made by a simple tabulation of the two sets of r e s u l t s . A more useful format involves p l o t t i n g the experimental  value against the  predicted one for each of the modes included i n the comparison. way,  In this  i t i s possible to see not only the degree of c o r r e l a t i o n between the  two sets of results, but also the nature (and possible cause) of any discrepancies which do e x i s t .  I d e a l l y , the points plotted should l i e on or  close to a straight l i n e of slope 1.  If they l i e close to a line of  58. d i f f e r e n t slope then almost c e r t a i n l y the cause of the discrepancy i s an erroneous material property used i n the a n a l y t i c a l model.  If the points  l i e scattered widely about a straight l i n e then the a n a l y t i c a l model seriously f a i l s to represent evaluation i s c a l l e d f o r .  the test structure and a fundamental r e -  For a s a t i s f a c t o r y model, i t may be expected  that the scatter w i l l be small and randomly d i s t r i b u t e d about a 45° l i n e . If a l i n e a r regression analysis i s performed on the plotted points, the correlation c o e f f i c i e n t i s expected to be close to unity for a suitable model.  The c o r r e l a t i o n c o e f f i c i e n t i s defined as the "Frequency Assurance  C r i t e r i a " (F.A.C.) and can be used to indicate the degree of agreement between the measured natural frequencies and the a n a l y t i c a l natural frequencies.  8.6  Determination of Mode Shapes Consider a base motion problem i n which the system i s excited at i t s  r  th  . ,. natural frequency: x (t) g  =  a sin  t r  OJ  The steady state solution for the floor displacements r e l a t i v e to the moving base has been derived i n Section 8.4 and i s given by:  {x}  =  {G} cos a> t + {H} s i n m^t r  (8.12)  where {G} and {H} are defined by Equations (8.18) and (8.19).  ttl Since the system i s excited at i t s r t tl w i l l be primarily i n i t s r  natural frequency, the response  mode and therefore we can neglect the  influence of the other modes (for £  < 0.20):  59.  { P }  =  _2  [{0},{0},...,{A },...,{0}][A](r)  1  {1}  (8.27)  r r {Q}  =  {0}  {D}  =  [A]  (8.28)  Let - 1  {1} = {D ,D ,...,D 1  2  r  D >  (8.29)  T  n  Then setting (8.27) and (8.28) into (8.12) y i e l d s :  {x}  =  a D — 2C 0) r r  (  {A  U ;  } cos u t  (8.30)  2  Similarly the accelerations r e l a t i v e to the moving base are given by:  {x}  =  a D - •=—r 1  {A^ } X)  C  cos OJ t  (8.31)  r  Notice that {x} i s proportional to the r*"* mode shape {A^ ^}; t h i s means 1  r  that system accelerations can be used to define i t s r*"* mode shape when a 1  system i s excited by a harmonic base motion at a frequency which equals i t s r*"* natural frequency. 1  The mode shapes of the model frames were measured experimentally by exciting the structure at i t s different natural frequencies the absolute h o r i z o n t a l acceleration at each f l o o r {x^,}. considered  and recording  The structure was  as a three degree of freedom system, as shown i n Figure 8.1.  The r e l a t i v e acceleration at each f l o o r of the structure with respect to i t s moving base ( i . e . the table) {x}, was obtained by subtracting the d i g i t i z e d record of the table acceleration from each corresponding accelerometer record.  The mode shapes were then obtained by normalizing  to a unit  length the vector containing the amplitudes of the r e l a t i v e acceleration at each f l o o r .  60. 8.7  Modal Assurance Matrix The a n a l y t i c a l mode shapes of the d i f f e r e n t model frames can be  determined  by using the computer program "DYNA" which solves the c l a s s i c a l  eigenvalue problem (Equation (8.3)).  The experimental mode shapes can be  found by exciting the frames at their various natural frequencies and recording the horizontal acceleration at each f l o o r .  It i s also necessary  to establish a systematic way of comparing these two results i n order to judge the accuracy of the a n a l y t i c a l model. th ttl F i r s t consider a plot of the r experimental mode shape vs. the s a n a l y t i c a l mode shape.  If a l i n e a r regression analysis i s performed  on the  data points, the resulting correlation c o e f f i c i e n t i s defined as the "Modal Assurance C r i t e r i a " (M.A.C.).  The expected values of the M.A.C. are:  M.A.C. = 1  for two corresponding modes (r = s)  M.A.C. = 0  for two uncorrelated modes (r * s)  Having m^ experimental mode shapes and m^ a n a l y t i c a l mode shapes, i t Is possible to construct a matrix m A.  correlation c o e f f i c i e n t s . Matrix" [M.A.M.].  xm  which contains a l l the possible A  This matrix i s defined as the "Modal Assurance  For perfect correlation,  the [M.A.M.] should be equal to  the identity matrix, i . e . i d e a l l y :  [M.A.M.] =  1 0 0  0 1 0  0 0 1  Some authors (15) suggest that the a n a l y t i c a l model i s accurate enough if:  61. M.A.C.  > 0.9  f o r corresponding modes (r = s)  M.A.C.  < 0.05  for uncorrelated modes (r t s)  The Model Assurance Matrix was used to compare the measured and predicted mode shapes of the three d i f f e r e n t model frames.  8.8  Experimental Determination of Damping In the program "Drain-2D", the damping matrix i s described by  considering Rayleigh type damping:  [CJ  =  o[M] + 6[K] + 0 [ K 1 O  (8.32)  O  where:  [M]  =  Global mass matrix  [K]  =  Updated global s t i f f n e s s matrix  [K ]  =  I n i t i a l e l a s t i c global s t i f f n e s s matrix  =  Damping  Q  01J$J8Q  coefficients  The f i r s t term on the right-hand side of Equation (8.36) i s known as the i n e r t i a l damping matrix.  The corresponding damping force on each  concentrated mass i s proportional to i t s momentum.  I t represents the  energy loss associated with change i n momentum ( f o r example, during an impact). matrices.  The second and third terms are known as the s t i f f n e s s damping The corresponding damping force i s proportional to the rate of  change of the deformation forces at the j o i n t s .  62. If the damping matrix i s orthogonal to the model matrix [A], as expressed by Equation (8.16), i t follows that the damped motion can be uncoupled into i t s individual modal responses.  This means that the damped  system (as well as the undamped system) possesses c l a s s i c a l normal modes. Assuming $  0  = 0, i t can be shown (14) that, i n the case of Rayleigh  type damping, the damping matrix i s orthogonal to the modal matrix i f the damping r a t i o s are defined such that:  Cr  So) = 0 ^ - + - ^  2oi r  2  v  (8.33) '  where: ?  r  =  Damping ratio of the r  mode til  a>  =  Undamped natural frequency of the r  a,8  =  Damping  r  mode  coefficients  Knowing the damping ratios i n the f i r s t two modes  (£^,£2)»  t  n  e  damping  c o e f f i c i e n t s (a,B) can be determined from:  4ir(T C 2  a  =  2  T  - Tjtj)  2  T  _  2  l  l  L  T T (T C 1  B  2  2  = T  2  l  (8.34a)  2  1  2  _  - T  1  T  l  l  l C 2  )  2  where: T^ ,T  2  =  Undamped periods of the f i r s t and second modes.  (8.34b)  63. Two d i f f e r e n t methods were used to measure the damping r a t i o s of the d i f f e r e n t model frames:  1)  1)  Half-Power (Bandwidth) Method  2)  Logarithmic Decrement Method  Half-Power (Bandwidth) Method F i r s t consider the mobility function of a l i g h t l y damped (5 < 0.2)  single-degree-of-freedom-system.  |H(u»)|  Using the result of Equation (8.26):  O)  -  J(u -w ) 2  2  2  (8.35) +  2  (2£U)U))  2  The peak amplification occurs when the denominator of Equation (8.35) i s a minimum:  [(a> -u> ) + (2co)ui) ] = 0 2  2  2  a  =  (8.36)  2  dui  from which i t follows that  o V l ^ 2c*  (8.37)  Substituting (8.37) into Equation (8.35), the peak magnification can be found as  |H(w)l max 1  =  Q  1  =  (8.38) 1  25/I-2c  For  a l i g h t l y damped system (£ < 0.2)  2  64.  Q  The b a n d w i d t h ( h a l f - p o w e r ) c u r v e when t h e m a g n i t u d e i s is  d e n o t e d by AOJ = u)  2  -  15  ( 8  i s d e f i n e d as  l / f l times  the  the w i d t h of  peak v a l u e  Aw-«) + 2  2 2  (2ca>o))  Expanding Equation (8.40) y i e l d s  U  2  (Q).  This bandwidth of  1_ /  (8.40)  2  f o r a l i g h t l y damped  system  < 0.2):  c  T h i s m e t h o d c a n be e x t e n d e d widely  spaced  resonance  damping r a t i o i s g i v e n  -  2.7  F o r t h e r "* mode o f 1  1  systems  -  4 1 )  having  v i b r a t i o n the  by:  Au)  1  If  ( 8  to m u l t i - d e g r e e - o f - f r e e d o m  frequencies.  C  r  the m o b i l i t y f u n c t i o n i s  =  f 2  — to r  (8.42) '  plotted i n decibels  (see  Section 8.4),  bandwidth corresponding to a resonance  i s g i v e n by t h e w i d t h o f  t u d e p l o t a t 3 db b e l o w t h a t  peak,  2)  3 9 )  the m o b i l i t y  ®i» w h e r e I D ^ , ^ a r e g i v e n by t h e r o o t s  1_  '  resonance  as  the  the magni2.  shown i n F i g u r e 8 . $ .  L o g a r i t h m i c Decrement Method C o n s i d e r a v i s c o u s l y damped s i n g l e - d e g r e e - o f - f r e e d o m  some i n i t i a l  condition.  form of a time  decay:  The d i s p l a c e m e n t  response  of  this  system e x c i t e d  by  system takes  the  65. =  x (t) T  -tut  x(t) =  XQ  e  (8.43)  s i n OJ^ t  where:  o> - oj/l _. _ - rz,  = =  2  The  damped natural frequency  (8.44)  acceleration response of the system i s given by:  (t) =  x  Q  e ^  U ) t  [(C OJ -OJ ) 2  2  2  d  s i n a> t - 2COJU> cos a^t]  (8.45)  d  If the acceleration response at t = t^ i s denoted by x^ and the response at t = t.+2irr/uJ i s denoted by x,, then i t can be s-hown: l D i+r r  - 5 2-  — — X  =  2irr  e  (8.46)  i  If x^ corresponds to a peak point on the acceleration decay record with magnitude A^, then x ^ i n the acceleration-time  + r  corresponds to the peak point r cycles l a t e r  history and i t s magnitude i s denoted by A^ ^ as +  shown i n Figure 8.4. It follows  that  2irr i+r  =  e  (8.47)  Substituting Equation (8.44) and eliminating the exponential leads to  66.  A  *n[-p] A  i  =  2nr  (8.48)  For a l i g h t l y damped system (? < 0.2)  A 2nr A  (8.49)  i  from which (8.50)  For multi-degree-of-freedom systems, the modal damping r a t i o for each mode can be determined using t h i s method i f the i n i t i a l e x c i t a t i o n i s such that the decay takes place primarily i n one mode of v i b r a t i o n . There are l i m i t a t i o n s ( i n accuracy) of damping values that are experimentally determined by these two methods.  In the logarithmic  decrement  method, the procedure i s f i r s t to excite the model frame at the desired resonant frequency and then to cease suddenly the excitation; i t i s t a c i t l y assumed that the model frame can be excited i n a single mode. ing transient vibrations are invariably influenced  The r e s u l t -  by modal interaction.  This introduces a c e r t a i n amount of error into the measured damping values. In the half-power (bandwidth) method, the accuracy of the damping ratio becomes poor at very low damping ratios (<1%). this i s the d i f f i c u l t y i n obtaining data points on the mobility of a l i g h t l y damped system.  The main reason for  a s u f f i c i e n t number of experimental  curve i n the v i c i n i t y of a resonance  frequency  As a result, the mobility curve i s poorly  defined i n the neighbourhood of a weakly damped resonance peak.  To reduce  this problem, the mobility functions of the model frames were determined experimentally with the best frequency resolution that could be achieved with the o s c i l l a t o r of the Earthquake Simulator ( i . e . , 0.01 Hz increments) . The expressions used i n computing the damping parameters of the model frames are based on linear system theory.  However, during the evaluation  tests the Moment Resisting Frame and the Braced Moment Resisting Frame w i l l undergo i n e l a s t i c deformations and w i l l exhibit some non-linear behaviour. If the degree of non-linearity i s high, the measured damping values w i l l not  be representative of the actual damping values.  Measured viscous  damping i n the model frames should increase with the amplitude of motion (13).  68. 9.  9.1  SYSTEM IDENTIFICATION OF THE MOMENT RESISTING FRAME  Free Vibration Analysis The f i r s t model frame considered represents  the standard  construction  of a Moment Resisting Frame, as shown i n Figure 9.1. The computer program "DYNA" (see Chapter 8) was used to predict the natural frequencies and mode shapes of this frame.  A refined lumped mass system was developed to model  the structure as i l l u s t r a t e d i n Figure 9.2.  The mass matrix was formed by  lumping the mass of the concrete blocks and the structure at the indicated nodes.  The lumped masses were assumed to be active i n the x and y d i r e c -  tions, but rotational masses were neglected.  The material properties  determined from uniaxial tests on s t e e l specimens (see Chapter 6) were used in the program.  Shear deformations were not considered  i n the analysis.  The f i r s t s i x natural frequencies and mode shapes r e s u l t i n g from the analysis are shown i n Figure 9.3. Notice that the t h i r d , fourth and f i f t h modes of v i b r a t i o n correspond to v e r t i c a l deformations of the beams. In the test program, v e r t i c a l excitation of the base i s needed i n order to measure these three mode shapes.  Since the shaking table input i s only  quasi-horizontal, these three mode shapes do not contribute to the h o r i zontal response of the structure and therefore they were discarded. For purposes of comparison of the predicted and measured natural frequencies, the predicted horizontal natural frequencies of the Moment Resisting Frame are:  f  L  =  2.831 Hz  o)  f  2  =  9.174 Hz  OJ  f  3  = 15.930 Hz  o)  1  3  2  =  17.788 rad/s  =  57.642 rad/s  = 100.091 rad/s  6 9 .  Considering only 3 degrees-of-freedom (see Figure 8.1), the predicted mode shapes of the Moment Resisting Frame are:  {0.21, 0.57, 0.80} (A (A  where {A  9.2  }  (2) T  {0.57, 0.46, -0.68}  }  (3) ,T  {0.75, -0.58, 0.32}  th represents the transposed matrix of the i mode shape.  Harmonic Forced V i b r a t i o n Test The natural frequencies of the Moment Resisting Frame were measured  experimentally by f i r s t exciting the structure with harmonic base motions having frequencies which d i f f e r e d s i g n i f i c a n t l y from the natural frequencies of the structure and then suddenly  stopping the base excitations. The  t h i r d f l o o r time-acceleration decays were recorded at t h i s stage and a standard Fast Fourier Transform  program was used to convert these records  into Fourier Amplitude Spectra, from which the natural frequencies of the Moment Resisting were determined. Three d i f f e r e n t tests were made with base harmonic motions having frequencies of 5, 11, and 20 Hz (31.42, 69.12, 125.66 rad/s). ing Fourier Amplitude Spectra are shown i n Figure 9.4.  The r e s u l t -  From these Spectra,  the measured natural frequencies of the Moment Resisting Frame were found to be:  f  l  2.86 Hz  =  17.97  rad/s  f  2  9.08 Hz  =  57.05  rad/s  f  3  =  89.22  rad/s  =  14.4 Hz  u  3  70. The predicted and measured natural frequencies of the Moment Resisting Frame are tabulated i n Table 9.1.  Figure 9.5 shows a plot of the measured  vs. predicted natural frequencies of the Moment Resisting Frame. that the scatter of the points i s small. to the a n a l y t i c a l model.  Notice  As a r e s u l t , no change was made  The Frequency Assurance C r i t e r i a (F.A.C.) was  found to be equal to 0.9980, which i s very close to the expected value of 1.  9.3  Experimental Determination of Mode Shapes The horizontal mode shapes of the Moment Resisting Frame were measured  experimentally  by exciting the structure at the desired natural  frequencies  and recording the acceleration of each f l o o r r e l a t i v e to the moving base (I.e.  table). Figure 9.6 shows the steady-state,  time-acceleration records of each  floor f o r the f i r s t three horizontal modes considered. ing  Notice the follow-  c h a r a c t e r i s t i c s of c l a s s i c a l normal modes:  Mode 1 (ujj = 17.97 rad/s): A l l the floor records are i n phase. Mode 2 (OJ = 57.05 rad/s): The f i r s t and second f l o o r records are i n phase while the t h i r d f l o o r record i s out-of-phase by 180°. 2  Mode 3 ( U K = 89.22 rad/s): The f i r s t and t h i r d f l o o r records are i n phase while the second f l o o r record i s out-of-phase by 180°.  The measured mode shapes of the Moment Resisting Frame were normalized by equating the vector containing the amplitudes of the r e l a t i v e acceleration at each f l o o r to a unit length; t h i s leads to the following r e s u l t :  71. {A  ( 1 )  }  T  =  (0.22, 0.56, 0.80}  {A  ( 2 )  }  T  =  {0.55, 0.41, -0.72}  {A  ( 3 )  }  T  =  {0.86, -0.47, 0.21}  Table 9.2 shows the comparison between the measured and predicted mode shapes of the Moment Resisting Frame.  Notice that the Modal Assurance  C r i t e r i a (M.A.C.) are very close to the expected value of 1 for corresponding modes. Figure 9.7.  The measured vs. predicted mode shapes are plotted i n  The scatter of the points i s small and randomly distributed  about the expected 45° l i n e .  The r e s u l t i n g Modal Assurance Matrix (M.A.M.)  was found to be:  [M.A.M] =  0.9996 -0.4947 -0.1632  -0.5251 0.9998 -0.0889  -0.2634 0.0813 0.9823  As mentioned e a r l i e r (see Section 8.7), some authors suggest that the off-diagonal terms of the Modal Assurance Matrix should be smaller than 0.05 for satisfactory results.  ^  However this was not obtained for the  M.A.M. of the Moment Resisting Frame since only 3 degrees of freedom were considered for defining the mode shapes (see Figure 8.2).  If more degrees  of freedom are used to determine the mode shapes, the off-diagonal terms of the M.A.M. should converge to zero.  9.4  Mobility Function The mobility functions f o r each f l o o r of the Moment Resisting Frame  were determined experimentally by exciting the structure with various base harmonic motions and recording, at steady state, the amplifications of the acceleration responses between the table and each f l o o r .  The resulting  72. experimental Mobility Functions ( i n db) are shown i n Figure 9.8. two natural frequencies  The f i r s t  of the moment Resisting Frame (17.97 and 57.05  rad/s) are e a s i l y recognizable and correlate very well with the r e s u l t s of the Fourier Spectrum Analysis (Section 9.2).  However the third natural  frequency i s not clearly defined, as two peaks a r i s e around 90 rad/s. observation  By  of the frame behaviour, i t was found that the f i r s t peak i s the  result of an i n t e r a c t i o n between horizontal and v e r t i c a l modes due to the rocking of the table.  Since the shaking table i s mounted on four v e r t i c a l  legs with universal j o i n t s at each end, a small v e r t i c a l excitation i s transmitted  to the structure due to the arcing motion of the table as i t i s  excited h o r i z o n t a l l y .  This motion causes the table to rock s l i g h t l y , and  because the v e r t i c a l natural frequencies  of the Moment Resisting Frame are  very close to the t h i r d c l a s s i c a l horizontal natural frequency, an interaction occurs i n the Mobility Functions.  This e f f e c t can be elimina-  ted by i n s t a l l i n g two accelerometers ( i n opposite directions) at each f l o o r and subtracting the records to cancel the v e r t i c a l component.  The second  peak at 89.22 rad/s corresponds to the t h i r d c l a s s i c a l horizontal natural frequency of the Moment Resisting Frame.  9.5  Experimental Determination of Damping As mentioned e a r l i e r (see Section 8.8),  two d i f f e r e n t methods were  used to estimate the modal damping ratios of the Moment Resisting Frame: 1)  Half-Power (Bandwidth) Method  2)  Logarithmic Decrement Method  The Half-Power Method was used d i r e c t l y on the experimental mobility functions (see Figure 9.8).  Figure 9.9 i l l u s t r a t e s the method on a  73.  t y p i c a l d e t a i l of the f i r s t f l o o r mobility function of the Moment Resisting Frame. The damping r a t i o s were also estimated by the Logarithmic Decrement Method after exciting the structure i n each mode and recording the timeacceleration decays at each f l o o r .  Figure 9.10 i l l u s t r a t e s the method on a  t y p i c a l f i r s t floor time-acceleration decay i n the f i r s t mode of v i b r a t i o n . The damping r a t i o s determined from the two methods are compared i n Table 9.3.  The two methods give damping ratios which are of the same order  of magnitude.  Notice that the values obtained by the Bandwidth Method are  generally higher than those obtained by the Logarithmic Decrement Method. The Bandwidth Method tends to overestimate damping because of the d i f f i c u l t y i n obtaining a sharp experimental peak at resonance.  Based on  these r e s u l t s , the following damping r a t i o s were used f o r the Moment Resisting Frame:  5  X  =  0.0028  (0.28%)  C  2  =  0.0021  (0.21%)  C  3  -  0.0074  (0.74%)  By considering only the above f i r s t two modal damping ratios (Cj» C ), 2  the damping c o e f f i c i e n t s f o r Rayleigh type damping, as obtained from  Equations (8.34a) and (8.34b) are: a  =  0.0853  3  =  0.00015  These two damping c o e f f i c i e n t s were used i n the non-linear, timehistory dynamic analysis program to generate a n a l y t i c a l solutions for the Moment Resisting Frame.  74. The modal damping r a t i o s and the r e s u l t s of the Free Vibration Analysis  (see Section 9.1) were used i n the program "Vibration" (see  Section 8.4) to predict the mobility functions of the Moment Resisting Frame.  The predicted and measured mobility functions of the Moment Resist-  ing Frame are plotted i n Figure 9.11.  The two curves correlate very well,  especially for the f i r s t two modes of vibrations.  The amplitudes of the  resonant peaks are closely matched, i n d i c a t i n g a good estimation of the modal damping r a t i o s .  75. 10.  SYSTEM IDENTIFICATION OF THE BRACED MOMENT RESISTING FRAME  10.1 Free Vibration Analysis The second type of frame considered represents a Braced Moment Resisting  construction for which the braces were designed to be e f f e c t i v e i n  tension only.  The braces were made of 6 mm square bars (AINSI C-1018 cold  formed s t e e l ) .  They were not connected together at mid-length, so that  their e f f e c t i v e length i s essentially the t o t a l length of a brace.  This  reduces the buckling load of the braces and allows the structure to be modelled as a l i n e a r system i n which the braces act i n tension only. The computer program "DYNA" was used to predict the natural frequencies and mode shapes of the Braced Moment Resisting Frame.  Figure 10.1  i l l u s t r a t e s the computer model employed for the free vibration analysis. Only one brace was considered at each storey, since the compression braces were neglected. The f i r s t s i x natural frequencies and mode shapes r e s u l t i n g from the analysis are shown i n Figure 10.2.  As expected, the f i r s t natural  frequency of the Braced Moment Resisting Frame i s higher than the corresponding frequency of the Moment Resisting Frame (see Figure 9.3), since the braces have s t i f f e n e d the structure.  Notice that the same three v e r t i c a l  modes encountered with the Moment Resisting Frame are also present with the Braced Moment Resisting Frame; the incorporation of the braces does not affect the bending s t i f f n e s s of the beams.  The three natural frequencies  corresponding to the v e r t i c a l modes (87.336, 95.756 and 99.965 rad/s) are very close to the c l a s s i c a l horizontal second natural frequency (91.61 rad/s) and therefore we expect some d i f f i c u l t y i n c l e a r l y d i f f e r e n t i a t i n g  76. the c l a s s i c a l second mode i n the Fourier Spectrum Analysis (see Section 10.2). For purposes of comparison, the predicted horizontal natural frequencies of the Braced Moment Resisting Frame are:  Considering  fj  -  5.087 Hz  OJ -  31.962 rad/s  f  2  =  14.581 Hz  u>  91.617 rad/s  f  3  =  22.025 Hz  u)  1  2  3  =  = 138.389 rad/s  only 3 degrees-of-freedom (see Figure 8.1), the predicted  mode shapes of the Braced Moment Resisting Frame are:  {A  ( 1 )  }  T  = (0.27, 0.59, 0.76}  {A  ( 2 )  }  T  = {0.63, 0.32, -0.71}  {A  ( 3 )  }  T  = {0.65, -0.64, 0.40}  10.2 Harmonic Forced Vibration Test Two tests were performed i n order to measure the natural  frequencies  of the Braced Moment Resisting Frame using the Fourier Amplitude Spectrum method.  These were carried out with harmonic base motions having e x c i t a -  tion frequencies  of 8 and 18 Hz. The r e s u l t i n g Fourier Amplitude  Spectra  are presented i n Figure 10.3. The  fundamental natural frequency of the Braced Moment Resisting  Frame, which can easily be established from the Fourier Spectrum of the f i r s t test, was found to be:  f  L  =  5.29 Hz  UK  =  33.24 rad/s  77.  However the second and t h i r d natural frequencies are not easily differentiable.  The Fourier Amplitude Spectrum of the second test shows  many peaks between 13 and 20 Hz (81.68 - 125.66 rad/s). correspond to the closely spaced natural frequencies  These peaks  found through the Free  Vibration Analysis (see Figure 10.2) and are present i n the Spectrum because of the interaction between v e r t i c a l and horizontal modes due to the rocking of the table. The investigation of the natural frequencies of the Braced Moment Resisting Frame was not pursued further, since the contributions of the second and higher modes to the response of the structure w i l l be n e g l i gible.  10.3  Experimental Determination of Mode Shapes Since i t was d i f f i c u l t to separate the higher modes of the Braced  Moment Resisting Frame, only the f i r s t mode shape was measured experimenta l l y . The structure was excited at i t s f i r s t natural frequency (5.29 Hz) and the steady-state  r e l a t i v e accelerations at each floor were recorded.  Figure 10.4 shows the time-acceleration records for the three f l o o r s . Again, the c h a r a c t e r i s t i c of a c l a s s i c a l f i r s t normal mode can be observed, since a l l the f l o o r records are i n phase.  The measured f i r s t mode shape of  the Braced Moment Resisting Frame was found to be  (A  ( 1 )  }  T  =  {0.34, 0.60, 0.72}  A comparison of the measured and predicted f i r s t mode shape of the Braced Moment Resisting Frame (see Section 10.1) led to a Modal Assurance C r i t e r i o n equal to 0.9994.  78. 10.4 Experimental Determination of Damping In the l a s t chapter, the Bandwidth method and the Logarithmic Decrement method were used to estimate the modal damping r a t i o s of the Moment Resisting Frame.  The two methods gave results which correlated very well.  Therefore, i t was decided to use only the Logarithmic Decrement method to estimate the damping c h a r a c t e r i s t i c s of the Braced Moment Resisting Frame. This avoided the need to generate experimental mobility functions, which require repetitive testing. Since i t was d i f f i c u l t to separate the higher modes of the Braced Moment Resisting Frame (see Section 10.2), only the f i r s t modal damping r a t i o was measured.  To determine the values of the damping c o e f f i c i e n t s  used i n the Rayleigh type damping (Equations (8.34a) and (8.34b)), i t was assumed that the two lower modal damping r a t i o s were equal (^ = ? ) ' 2  Figure 10.5 i l l u s t r a t e s the Logarithmic Decrement method as applied to the f i r s t f l o o r acceleration-time decay i n the f i r s t mode of v i b r a t i o n . Table 10.1 shows the different damping ratios measured.  Based on these  t e s t s , the following (average) damping r a t i o s were used f o r the Braced Moment Resisting Frame:  ^1  =  ?  2  =  °-  0 1 2 7  (1-27%)  Notice that the incorporation of the braces changes the damping c h a r a c t e r i s t i c s of the structure:  r,l (Braced Moment Resisting Frame) = (Moment Resisting Frame)  = 0.0028  4.54  79. The damping c o e f f i c i e n t s defining the Rayleigh type damping and obtained by Equations (8.34a) and (8.34b) were:  a  =  0.6195  0  =  0.00064  These two damping c o e f f i c i e n t s were used i n the non-linear,  time-history  dynamic analysis to generate a n a l y t i c a l solutions f o r the Braced Moment Resisting Frame. Assuming the same damping r a t i o s i n the f i r s t three horizontal modes of vibrations (^ = ?  = 2  S3)  an<  * using the results of the free v i b r a t i o n  analysis (see Section 10.1), the predicted mobility functions of the Braced Moment Resisting Frame were generated by the program "Vibration" (see Section 8.4).  The predicted mobility functions are shown i n Figure 10.6.  80. 11.  SYSTEM IDENTIFICATION OF THE FRICTION DAMPED BRACED FRAME UNDER LOW AMPLITUDE EXCITATIONS  11.1 Free Vibration Analysis The t h i r d type of frame considered i n t h i s investigation represents a F r i c t i o n Damped Braced Frame as shown i n Figure 11.1.  Because of the  incorporation of the f r i c t i o n devices, the structure can no longer be modelled as a linear system.  The non-linearity of the structure b a s i c a l l y  results from two sources: 1)  Slipping of the F r i c t i o n Devices  2)  Buckling of the Compression  Braces  The dynamic c h a r a c t e r i s t i c s of the F r i c t i o n Damped Braced Frame (natural frequencies, damping parameters, mode shapes, mobility functions) therefore are a function of the amplitude of the e x c i t a t i o n .  Two extreme  fundamental frequencies can be obtained for a F r i c t i o n Damped Braced Frame as follows: the lowest natural frequency w i l l happen when a l l the devices are  slipping and w i l l be i d e n t i c a l to the fundamental frequency of the  Moment Resisting Frame (2.86 Hz); the highest natural frequency w i l l occur under very low amplitude excitations when none of the devices s l i p and a l l the braces behave e l a s t i c a l l y i n tension and compression.  For any combina-  tion of slippage of the devices and buckling of the compression braces, the fundamental natural frequency of the F r i c t i o n Damped Braced Frame w i l l l i e between these two extreme values. S i m i l a r l y , the viscous damping c h a r a c t e r i s t i c s of the structure w i l l vary with the amplitude of the excitation.  However, the viscous damping  w i l l be very small compared to the hysteretic damping dissipated by the devices and therefore can be assumed constant.  For this reason, i t was  81.  decided to estimate the dynamic parameters of the F r i c t i o n Damped Braced Frame under low amplitude excitations, during which none of the devices s l i p and a l l the braces behave e l a s t i c a l l y i n tension and  compression.  Under such conditions, the structure can be modelled as a linear  system.  Figure 11.2 i l l u s t r a t e s the computer model developed f o r the free v i b r a tions analysis using the computer program "DYNA".  Since the compression  braces were assumed not to buckle under low amplitude excitations, two e l a s t i c braces were considered at each storey.  The f r i c t i o n pads were  modelled by e l a s t i c bars since i t was assumed that no slippage takes place. The mass of the devices were considered and lumped at the four corner nodes of  the mechanism. The f i r s t nine natural frequencies and mode shapes resulting from the  analysis are shown i n Figures 11.3 and 11.4.  The f i r s t three natural  frequencies are closely spaced (24.448, 24.746 and 24.829 rad/s) and correspond to rotational modes of the devices due to the bending of the braces.  A rotational e x c i t a t i o n i s needed to be able to measure these  three modes and since the motion of the table i s quasi-horizontal, they w i l l not contribute to the horizontal response of the frame and therefore they were considered as higher modes and neglected.  In a real building  these three modes of vibrations w i l l occur at much higher frequencies, since the mass of the devices would be n e g l i g i b l e compared to the mass of the structure. The f i r s t c l a s s i c a l horizontal mode of the structure occurs at a frequency of 44.059 rad/s; this i s higher than the corresponding frequency for  the normal Braced Moment Resisting Frame (see Figure 10.2), since the  braces are assumed to be e f f e c t i v e both i n tension and compression under low amplitude excitations.  Notice, that the devices also rotate at this  82. same frequency; t h i s deformation i s caused by the bending of the braces. The next three modes are v e r t i c a l modes and they are i d e n t i c a l to the ones obtained with the Moment Resisting Frame (see Figure 9.3) and the Braced Moment Resisting Frame (see Figure 10.2); incorporation of the f r i c t i o n devices does not a l t e r the bending s t i f f n e s s of the beams.  Again,  these three modes were considered as higher modes and neglected. F i n a l l y , the second and t h i r d horizontal modes were obtained at frequencies of 122.348 and 177.919 rad/s. For purposes of comparison with subsequent experimental r e s u l t s , the predicted horizontal natural frequencies of the F r i c t i o n Damped Braced Frame under low amplitude excitations can be summarized as follows:  f  1  =  7.012  Hz  f  2  =  19.472  Hz  w  f  3  =  28.316  Hz  a>  2  3  Considering only 3 degrees-of-freedom  =  44.059  rad/s  =  122.348  rad/s  =  177.919  rad/s  (see Figure 8.1), the predicted  mode shapes of the F r i c t i o n Damped Braced Frame under low amplitude e x c i t a tions are:  {A  ( 1  V  - {0.29, 0.60, 0.75}  {A< >} - {0.66, 0.27, -0.70} 2  T  { A < V = {0.60, -0.67, 0.44} 3  11.2 Harmonic Forced V i b r a t i o n Test To measure the natural frequencies of the F r i c t i o n Damped Braced Frame under low amplitude excitations, two harmonic tests were performed  with  83. base motions having frequencies of 8 and 35 Hz. The r e s u l t i n g Fourier Amplitude Spectra are shown i n Figure 11.5.  The f i r s t two horizontal  natural frequencies of the F r i c t i o n Damped Braced Frame under low amplitude excitations correspond  to the frequencies at which the main spikes occur on  the spectra; these take place at:  7.03 Hz =  =  44.17 rad/s  oo, = 115.6  18.4 Hz  rad/s  The investigation of the third horizontal natural frequency becomes very d i f f i c u l t i n terms of the table c a p a b i l i t i e s because of the high value of this frequency  (« 28 Hz) r e l a t i v e to the frequency  range of the table.  Since only 2 modal damping r a t i o s are needed to determine the damping c o e f f i c i e n t s used i n the Rayleigh type damping, no experimental  attempt was  made to v e r i f y this t h i r d natural frequency. The predicted and measured natural frequencies of the F r i c t i o n Damped Braced Frame under low amplitude excitations are tabulated i n Table 11.1. Figure 11.6 shows a plot of the measured vs. predicted natural frequencies. The data points are very close to the expected 45° l i n e and therefore no change was made to the a n a l y t i c a l model.  11.3 Experimental Determination  of Mode Shapes -  The f i r s t two horizontal mode shapes of the F r i c t i o n Damped Braced Frame under low amplitude excitations were measured experimentally.  The  structure was excited at the desired natural frequencies and the horizontal accelerations of each floor ( r e l a t i v e to the table) were recorded under steady-state conditions.  84. Figure 11.7 shows the r e l a t i v e time-acceleration records f o r the three floors of the model frame.  Again notice that the results exhibit the same  c h a r a c t e r i s t i c s of c l a s s i c a l normal modes as were discussed e a r l i e r i n Section 9.3.  From these records, the measured f i r s t two horizontal mode  shapes of the F r i c t i o n Damped Braced Frame under low amplitude excitations are:  Table 11.2 mode shapes.  {A  ( 1 )  }  T  = {0.40, 0.50,  0.77}  {A  ( 2 )  }  T  = {0.58, 0.19, -0.79}  shows the comparison between the measured and predicted  The Modal Assurance C r i t e r i a (M.A.C.) are very close to the  expected value of 1 for the f i r s t two modes, hence confirming the v a l i d i t y of the a n a l y t i c a l model. plotted i n Figure 11.8.  The measured vs. predicted mode shapes are Notice that the scatter of the data points i s  small and randomly distributed about the expected 45° l i n e .  The Modal  Assurance Matrix (M.A.M.) was calculated to be:  [M.A.M.]  =  0.9661 -0.4301  -0.5235 0.9972  11.4 Experimental Determination of Damping The f i r s t two modal damping r a t i o s of the F r i c t i o n Damped Braced Frame under low amplitude excitations were estimated by the Logarithmic Decrement Method.  The structure was excited i n turn i n i t s f i r s t two modes and the  floor time-acceleration decays were recorded.  Figure 11.9  i l l u s t r a t e s the  results obtained f o r the f i r s t f l o o r time-acceleration decay corresponding to the f i r s t mode of v i b r a t i o n .  Table 11.3 summarizes the d i f f e r e n t  85. damping r a t i o s measured.  Note that a damping r e s u l t i s missing f o r the  second mode; this i s due to the fact that a malfunction of the second f l o o r amplifier was experienced during the test. Based on these results, the following modal damping ratios were assigned to the F r i c t i o n Damped Braced Frame:  r,  1  C  2  = 0.0060  (0.60%)  = 0.0039  (0.39%)  The damping c o e f f i c i e n t s defining calculated  the Rayleigh type damping were  from Equations (8.34a) and (8.34b) to y i e l d :  a  =  0.4667  6  =  0.0001  These two damping c o e f f i c i e n t s were used i n the non-linear, timehistory dynamic analysis Damped Braced Frame.  to generate a n a l y t i c a l solutions  f o r the F r i c t i o n  86. 12.  SEISMIC TESTS ON SHAKING TABLE  12.1 Seismic Testing Program, Model Frame #1 It was decided to use the Newmark-Blume-Kapur a r t i f i c i a l earthquake described i n Chapter 4, for the test of the f i r s t model frame on the shaking  table.  This earthquake record provides a response spectrum which  matches the Newmark-Blume-Kapur design spectrum and represents an average of many earthquake records. As mentioned e a r l i e r , the connections of the test frame were designed so  that the model could easily be transformed Into any of the three  s t r u c t u r a l configurations to be investigated (M.R.F., B.M.R.F., F.D.B.F.). By a proper choice of the intensity of the ground motion and the test sequence, i t was possible to carry out a comparative study of the three model types using only a single frame. Three d i f f e r e n t earthquake i n t e n s i t i e s , expressed i n terms of the peak acceleration of the ground motion, were used to study the performance of the  model frames i n both the e l a s t i c and i n e l a s t i c ranges.  In order to  accommodate the testing of the three s t r u c t u r a l configurations with a single frame, the three i n t e n s i t i e s of the ground motion were chosen to meet the following requirements:  Intensity 1 • The maximum stress i n the Moment Resisting Frame should be less than 50% of the y i e l d stress. • A l l members remain e l a s t i c for the Braced Moment Resisting Frame and the  F r i c t i o n Damped Braced Frame.  87. Intensity 2 • The maximum stress i n the Moment Resisting Frame should be between 50 and 75% of the y i e l d stress. • A l l members remain e l a s t i c for the Braced Moment Resisting Frame and the  F r i c t i o n Damped Braced Frame.  Intensity 3 • Some yielding should occur i n the Moment Resisting Frame. • Some of the cross-braces of the Braced Moment Resisting Frame should y i e l d but a l l the other structural elements should remain e l a s t i c ; a l l members remain e l a s t i c i n the F r i c t i o n Damped Braced Frame.  For  any given earthquake, the maximum ground motion intensity which  can be delivered by the shaking table i s l i m i t e d .  Considering the maximum  possible displacement of the table (±75 mm), i t was found that the peak acceleration that can be developed by the table with the Newmark-BlumeKapur a r t i f i c i a l earthquake i s 0.34 g. To determine the appropriate earthquake accelerations corresponding to the  i n t e n s i t i e s noted above, several non-linear time-history dynamic  analyses were performed with d i f f e r e n t peak ground accelerations.  The  computer model used for this purpose was similar to the one described i n Chapter 4.  Viscous damping was considered using the measured damping  c o e f f i c i e n t s determined experimentally (see Chapter 9 and 10).  From the  results of these analyses the following ground motion accelerations were found to s a t i s f y the intensity requirements stated above and were used f o r the  seismic tests of the f i r s t model frame:  88.  Intensity  The  Peak. Ground Acceleration (g)  1  0.05  2  0.10  3  0.30  predicted s t r u c t u r a l damage r e s u l t i n g from the Newmark-Blume-Kapur  a r t i f i c i a l earthquake i s shown i n Figure 12.1 f o r the Moment Resisting Frame and the Braced Moment Resisting Frame under these i n t e n s i t i e s . Notice that these three i n t e n s i t i e s meet the requirements formulated above. The following testing sequence was adopted i n order to test a l l three structural configurations using only a single frame.  Test Sequence  Frame Type  Intensity  Structural Damage  1  F.D.B.F.  1  None  2  F.D.B.F.  2  None  3  B.M.R.F.  1  None  4  B.M.R.F.  2  None  5  M.R.F.  1  None  6  M.R.F.  2  None  7  F.D.B.F.  3  None  8  B.M.R.F.  3  9  M.R.F.  3  Some preliminary  Only cross-braces  yield  Main members y i e l d  tests were performed at low acceleration amplitudes  to v e r i f y the table performance i n reproducing the Newmark-Blume-Kapur a r t i f i c i a l earthquake.  Figure 12.2 compares the acceleration records and  89. the Fourier Amplitude Spectra of the actual earthquake and the shaking table motions.  Notice that the Fourier Amplitude Spectrum of the actual  earthquake contains a large peak at very low frequency (»0.2 Hz) which i s not reproduced i n the table record.  For mechanical reasons, the integrat-  ing c i r c u i t for evaluating the table displacement corresponding to the acceleration input has a r o l l off at frequencies fore f i l t e r s the low frequencies;  12.2  lower than 1 Hz and there-  the table i s displacement controlled.  Test Results, Model Frame it I The experimental results presented i n t h i s section are compared to the  predictions of the i n e l a s t i c time-history dynamic analysis.  The material  properties determined from the u n i a x i a l tests on s t e e l specimens (see Chapter 6) were used i n the a n a l y t i c a l model.  Also, the damping values  measured at low amplitude vibrations (see chapters 9-11) were used to predict the responses of the frames. The envelopes of the measured horizontal accelerations f o r the three i n t e n s i t i e s of the ground motion are shown i n Figure 12.3.  (The program  Drain-2D does not provides r e s u l t s i n terms of acceleration; the a n a l y t i c a l values of acceleration are therefore not shown here).  For the Intensity 3  earthquake (peak acceleration = 0.30 g), a peak horizontal acceleration of 2.00 g and 1.51 g was experienced at the top of the Moment Resisting Frame and the Braced Moment Resisting Frame respectively; the measured peak horizontal acceleration at the top of the F r i c t i o n Damped Braced Frame was only 0.59 g. The influence of the new damping system i n reducing  seismic  response can be visualized by comparing the maximum acceleration amplifications experienced by the three frames under the Intensity 3 earthquake:  90.  Maximum Amplification  Frame M.R.F.  6.63  B.M.R.F.  5.08  F.D.B.F.  1.98  Figure 12.4 shows the envelopes of l a t e r a l deflections for the three i n t e n s i t i e s of the ground motion.  Good agreement i s observed between the  experimental results and the predictions of the i n e l a s t i c dynamic analysis.  time-history  Notice that the smallest deflections were always  obtained with the F r i c t i o n Damped Braced Frame.  The responses of the  F r i c t i o n Damped Braced Frame and the Braced Moment Resisting Frame are very similar for low intensity earthquakes, when there i s no slippage of the f r i c t i o n devices.  As the i n t e n s i t y of the ground motion increases, the  f r i c t i o n devices become active and improve the performance of the F r i c t i o n Damped Braced Frame compared to the two other frames. The a n a l y t i c a l model underestimates the deflections of the Braced Moment Resisting Frame for the Intensity 3 earthquake. expected since the a n a l y t i c a l model does not consider  This r e s u l t was the s t i f f n e s s degra-  dation of the braces when they undergo several i n e l a s t i c loops (pinched hysteresis).  The predictions of the Moment Resisting Frame deflections are  not very accurate for the Intensity 3 earthquake since the damping values, measured at very low amplitude excitations, and used i n the analysis, are not representative The  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 .  influence of damping i s i l l u s t r a t e d i n Figure 12.4 by showing the  change i n the deflections r e s u l t i n g from the use of a damping value which i s 5 times the actual damping measured at low i n t e n s i t i e s . No c e r t a i n explanation  can be offered f o r the apparent excessive  91. d i f f e r e n c e between t h e measured t h i r d f l o o r d e f l e c t i o n and i t s p r e d i c t e d value. for  I t i s p o s s i b l e that the c a l i b r a t i o n of the a n a l o g - d i g i t a l c o n v e r t e r  t h a t r e c o r d i n g channel was s e t i n e r r o r i n t h e t e s t ; t h e d i s p l a c e m e n t  appears  to be e x c e s s i v e by a f a c t o r of two.  The envelopes o f the bending moments i n t h e beams a r e p r e s e n t e d i n F i g u r e 12.5. of  The e x p e r i m e n t a l bending moments were o b t a i n e d from r e a d i n g s  t h e s t r a i n gages.  predicted values.  Good agreement i s observed between t h e measured and  However, the a n a l y t i c a l p r e d i c t i o n o v e r e s t i m a t e s the  a c t u a l damage i n t h e beams o f the Moment R e s i s t i n g Frame under the I n t e n s i t y 3 earthquake.  Only s l i g h t  y i e l d i n g was measured i n the f i r s t  f l o o r beam w h i l e the second f l o o r beam remained e l a s t i c . related of ing  to the f a c t  This i s again  that the measured damping v a l u e s a r e not r e p r e s e n t a t i v e  t h e a c t u a l frame behaviour under l a r g e amplitude v i b r a t i o n s . v a l u e s of the model frames  The damp-  should i n c r e a s e w i t h the amplitude o f  motion. T a b l e 12.1 compares t h e measured and p r e d i c t e d maximum bending moments in  the base column a t the l o c a t i o n of the s t r a i n gages.  Again good agree-  ment i s observed except f o r the Moment R e s i s t i n g Frame under I n t e n s i t y 3 earthquake, where the a n a l y t i c a l model o v e r e s t i m a t e s the bending moments i n the  base column. The t i m e - h i s t o r i e s o f t h e d e f l e c t i o n s a t t h e top of t h e frames f o r t h e  I n t e n s i t y 3 earthquake a r e presented i n F i g u r e 12.6. p r e d i c t s reasonably w e l l t h e responses o f t h e frames.  The a n a l y t i c a l model Notice that the  amplitudes of the v i b r a t i o n s f o r the F r i c t i o n Damped Braced Frame are f a r l e s s than t h e c o r r e s p o n d i n g v i b r a t i o n s o f t h e two o t h e r model frames. F i g u r e 12.7 shows the measured and p r e d i c t e d of  slippage  time-histories  t h e second f l o o r d e v i c e d u r i n g t h e f i r s t 9 seconds o f t h e I n t e n s i t y 3  92.  earthquake.  The slippage predicted by the refined a n a l y t i c a l model i s of  the same order of magnitude as the measured values.  However, notice that  the s i g n a l from the measured slippage i s very noisy.  This was due to the  l a t e r a l vibrations of the devices, which influenced the readings of the L.V.D.T.'s.  These l a t e r a l vibrations mainly occur because of the weight of  the devices compared to the weight of the model frame; i n a r e a l building these l a t e r a l vibrations w i l l be p r a c t i c a l l y non-existent. The torsional motion developed i n the structure was measured by analyzing the phase s h i f t between the displacement records given by the two potentiometers at each f l o o r .  Figure 12.8 shows the time-history of the  t h i r d f l o o r d e f l e c t i o n recorded from both potentiometers f o r the F r i c t i o n Damped Braced Frame during the Intensity 3 earthquake.  The two signals are  exactly i n phase with similar amplitudes and therefore no s i g n i f i c a n t torsional stresses were induced i n the structure. The r e s u l t s of t h i s f i r s t series of tests c l e a r l y indicate the superior performance  of the F r i c t i o n Damped Braced Frame compared to the  two frames, which represent a class of conventional building systems.  From  these results the following conclusions can be drawn: 1.  The amplitudes of the displacements and accelerations are considerably reduced for the F r i c t i o n Damped Braced Frame r e l a t i v e to the corresponding responses of the two other building systems; thus nons t r u c t u r a l damaged i s minimized.  2.  When subjected to the Intensity 3 earthquake many cross-braces of the Braced Moment Resisting Frame yielded i n tension and the f i r s t beams of the Moment Resisting Frame reached t h e i r y i e l d moment.  floor No  material yielding was involved i n the process of energy dissipation i n the F r i c t i o n Damped Braced Frame;  since t h i s frame was not damaged  93. d u r i n g the earthquake,  i t was  a b l e to f a c e f u t u r e earthquakes w i t h the  same e f f i c i e n c y . 3.  No s i g n i f i c a n t t o r s i o n a l motion o f the F r i c t i o n Damped Braced Frame was  4.  observed.  The deformations of the F r i c t i o n Damped Braced Frame a t the end o f the earthquakes  were n e g l i g i b l e ,  i n d i c a t i n g t h a t the s t r u c t u r e r e c o v e r e d  from these shocks without any permanent s e t .  12.3  Supplementary At  T e s t s , Model Frame #1  the end of the f i r s t  s e r i e s o f t e s t s , the f i r s t model frame had  s l i g h t l y y i e l d e d i n the f i r s t  f l o o r beams.  n o t been formed and the permanent s e t was  But a f u l l  very small.  p l a s t i c hinge  had  T h e r e f o r e i t was  decided to t e s t again the Moment R e s i s t i n g Frame and the F r i c t i o n Damped Braced Frame w i t h a d i f f e r e n t earthquake deformation which was  present.  r e c o r d d e s p i t e the s m a l l p l a s t i c  The Braced Moment R e s i s t i n g Frame was  not  r e - t e s t e d , s i n c e many of i t s c r o s s - b r a c e s had y i e l d e d i n t e n s i o n i n t h e first  s e r i e s of t e s t s , and c o u l d not be used a g a i n . The c h o i c e o f the new  earthquake  r e c o r d was  based on the f o l l o w i n g  requirements:  1.  I t s frequency content must be h i g h e r than t h a t o f the earthquake  so t h a t a h i g h e r peak a c c e l e r a t i o n can be a c h i e v e d w i t h the  same displacement of the shaking 2.  artificial  I t s F o u r i e r amplitude spectrum  table.  s h o u l d be r e l a t i v e l y  range of n a t u r a l f r e q u e n c i e s of the model frames  c o n s t a n t over the  so that the  i n p u t i s almost c o n s t a n t f o r t h e t h r e e t y p e s o f c o n s t r u c t i o n .  energy  94. Based on these requirements the following earthquake record was chosen: Kern County C a l i f o r n i a Earthquake (Taft Lincoln School Tunnel), July 21, 1952, Comp. Vert., 0-25 sec. This earthquake does not completely s a t i s f y the second requirement; however, an a n a l y t i c a l study using 'white noise' as the excitation source was also performed (see Section 4.4) to f u l f i l l that requirement. Some preliminary tests were performed at low intensity to v e r i f y the a b i l i t y of the shaking table to reproduce the Taft earthquake.  The accel-  eration record and the Fourier Amplitude Spectrum of the actual earthquake and the shaking table are presented i n Figure 12.9.  On the basis of these  tests, i t was found that the peak acceleration which can be developed on the shaking table with the Taft earthquake i s 0.90 g.  Notice t h i s value i s  much higher than the 0.34 g obtained with the Newmark-Blume-Kapur A r t i f i c i a l Earthquake.  For the actual tests, i t was decided to scale the  earthquake record to a peak acceleration of 0.60  g.  Again the experimental  results were compared with the predictions of the i n e l a s t i c time-history dynamic analysis.  The slight damage which had been induced i n the Moment  Resisting Frame by the e a r l i e r test was ignored when making t h i s comparison. The envelope of the measured horizontal accelerations are shown i n Figure 12.10.  As expected, smaller accelerations were induced i n the  F r i c t i o n Damped Braced Frame where, at the top storey, an acceleration of 1.10  g was experienced compared to 2.23 g for the corresponding value i n  the Moment Resisting Frame. Figure 12.11 shows the envelope of l a t e r a l deflections.  It can be  seen that the measured deflection at the top of the F r i c t i o n Damped Braced  95. Frame i s only 31% of t h e e q u i v a l e n t d e f l e c t i o n i n t h e Moment R e s i s t i n g Frame.  The a n a l y t i c a l p r e d i c t i o n s overestimate  the d e f l e c t i o n s of the  Moment R e s i s t i n g Frame; a g a i n i t i s b e l i e v e d t h a t t h e damping v a l u e s , determined a t low amplitude e x c i t a t i o n s , are s m a l l e r than the a c t u a l v a l u e s for  h i g h amplitude v i b r a t i o n s .  the i n i t i a l  An a n a l y t i c a l s o l u t i o n i n c l u d i n g 8 times  v i s c o u s damping i s a l s o shown i n F i g u r e 12.11 f o r the Moment  R e s i s t i n g Frame. The Notice  envelopes o f beam bending moments a r e shown i n F i g u r e 12.12.  that the f i r s t  f l o o r beam of the Moment R e s i s t i n g Frame reaches i t s  p l a s t i c moment c a p a c i t y under t h i s earthquake, whereas t h e e q u i v a l e n t moment In the F r i c t i o n Damped Braced Frame i s only 39% of the p l a s t i c moment. T a b l e 12.2 compares t h e measured and p r e d i c t e d maximum bending moments in  the base columns of the three frames a t the l o c a t i o n of the s t r a i n  gages.  The measured maximum moment i n the F r i c t i o n Damped Braced Frame i s  only 28% of the value i n the Moment R e s i s t i n g Frame. The  t i m e - h i s t o r i e s o f t h e d e f l e c t i o n s a t t h e t o p o f t h e frames a r e  presented  i n F i g u r e 12.13.  The measured amplitudes of the v i b r a t i o n s o f  the F r i c t i o n Damped Braced Frame a r e f a r l e s s than t h e v a l u e s o b t a i n e d  with  the Moment R e s i s t i n g Frame. F i g u r e 12.14 p r e s e n t s floor  f r i c t i o n device.  maximum recorded  t h e measured s l i p p a g e t i m e - h i s t o r y o f the second  Significant  v a l u e i s 7.64 mm.  s l i p p a g e occurs  I n the d e v i c e ;  the  The maximum s l i p p a g e o c c u r r e d a t t h e  same time as the maximum ground a c c e l e r a t i o n (see F i g u r e 12.9) and hence as the maximum i n e r t i a f o r c e s developed i n t h e s t r u c t u r e . The again.  t o r s i o n a l motion of the F r i c t i o n Damped Braced Frame was checked F i g u r e 12.5 shows t h e t i m e - h i s t o r y o f t h e t h i r d f l o o r d e f l e c t i o n  96. recorded from both potentiometers f o r the F r i c t i o n Damped Braced Frame. The  two signals are exactly in phase with similar amplitudes and,  there-  fore, i t can be concluded again that no s i g n i f i c a n t t o r s i o n a l stresses were induced i n the structure. 12.4  Seismic Testing Program, Model Frame #2 The results presented i n the l a s t section c l e a r l y show the superior  performance of the F r i c t i o n Damped Braced Frame compared to the convent i o n a l seismic s t r u c t u r a l systems. acceleration of 0.6  Even the Taft record scaled to a peak  g caused no damage i n the F r i c t i o n Damped Braced Frame,  while the Moment Resisting Frame underwent large i n e l a s t i c deformations. For the series of tests on the second model frame, i t was  decided to  study the performance of the three s t r u c t u r a l configurations when subjected to the Taft Earthquake scaled to the maximum intensity that can be a l l y r e a l i z e d by the shaking table (peak acceleration = 0.9  physic-  g) for t h i s  p a r t i c u l a r excitation. I n i t i a l l y the second model frame was mounted on the shaking table and some preliminary properties.  tests were performed to v e r i f y i t s fundamental dynamic  The natural frequencies and damping c h a r a c t e r i s t i c s of the  three d i f f e r e n t types of construction are compared i n Tables 12.3 for both model frames.  It can be seen that good agreement was  and  12.4  obtained f o r  the two model frames. From the results' of the f i r s t tests s e r i e s , i t was torsional motion was  found that  n e g l i g i b l e ; therefore, i n the second test series three  of the s i x potentiometers were disconnected and three more f r i c t i o n  devices  were instrumented for slippage. 12.5  Test Results, Model Frame #2 The Moment Resisting Frame and the Braced Moment Resisting Frame did  not perform well during the tests.  Very large strains occurred  i n the base  column, and i n the f i r s t and second f l o o r beams of the Moment Resisting Frame, indicating that the f u l l p l a s t i c moment capacity was these locations.  reached at  Although the main s t r u c t u r a l members of the Braced Moment  Resisting Frame remained e l a s t i c , many cross-braces yielded i n tension. The elongation of the braces was  very large and they buckled s i g n i f i c a n t l y  in the compression regime; this indicates that heavy non-structural damage would have occurred i n a r e a l building (cracks i n walls, broken glass, etc.).  However, the F r i c t i o n Damped Braced Frame performed very well; no  damage occurred i n any member and the deflections and accelerations were far  less than the values measured i n the two other types of Figure 12.16  construction.  i l l u s t r a t e s the superior performance of the F r i c t i o n  Damped Braced, expressed i n terms of the envelope of the measured horizontal accelerations.  A peak acceleration of 1.42  g was measured at  the top of the F r i c t i o n Damped Braced Frame compared to peak acceleration values of 2.24  g and 2.67  g for the Braced Moment Frame and the Moment  Resisting Frame respectively.  Notice that some variations occurred i n the  peak table acceleration (input) to which the frames were subjected. Although the intent was  to apply the same base motion i n t e n s i t y to a l l  three frames, i t i s believed that a frame-table i n t e r a c t i o n occurred as a result of the very large base shears which were developed at this strong l e v e l of e x c i t a t i o n .  However, the input variations were small and  the  results are s t i l l comparable. Figure 12.17 accelerations.  shows the time-histories of the measured t h i r d f l o o r The  trends noted i n the f i r s t series of tests are also  evident here, alhtough i n a more exaggerated sense. Since the excitations developed were extremely severe, i t was  possible  to measure s i g n i f i c a n t slippage i n the f r i c t i o n devices of the F r i c t i o n Damped Braced Frame. these devices.  Figure 12.18  shows the slippage time-histories of  Peak slippages of 5.89  mm,  10.16  mm  and 4.91  mm were  98. recorded with the f i r s t ,  second and t h i r d f l o o r devices r e s p e c t i v e l y .  Notice that a l l the f r i c t i o n devices experienced  peak slippage at a time  which coincided with time at which the peak ground acceleration occurred (see Figure 12.9). In this test series the two second f l o o r f r i c t i o n devices were instrumented;  Figure 12.19 compares their slippage time-histories.  It can  be seen that the signals are exactly i n phase and have the same amplitude. This confirms that the devices dissipated energy simultaneously and, therefore, that no s i g n i f i c a n t t o r s i o n a l stresses were induced i n the structure even under the extreme ground excitation used i n this experiment.  12.6 Energy Balance As mentioned i n Section 5.4, some imperfections were noted i n the hysteresis loops of even the modified devices.  If Figure 5.10 i s examined,  i t can be seen that a minimum slippage of about 6 mm i s required to develop the f u l l s l i p load of the f r i c t i o n devices.  Since at many times during the  test the slippage of the devices was less than 6 mm, i t i s of i n t e r e s t to evaluate the equivalent e f f e c t i v e constant s l i p load of the f r i c t i o n devices developed during the l a s t test on the F r i c t i o n Damped Braced Frame. This can be achieved by considering an energy balance of the system. From Newton's second law, the energy input (E^ ) to the structure in during a ground motion of duration t^ i s the product of the base shear and the ground displacement  and can be expressed as  99.  where N.D.O.F.  =  number of degrees of freedom considered  m^  =  lumped mass at the i " *  =  t o t a l acceleration of the i  =  displacement of the ground.  Xg  1  1  degree of freedom degree of freedom  The increment of energy input (AE^jp) during a time step of f i n i t e length can be calculated on the basis of the average t o t a l acceleration during the time step f o r each degree of freedom:  (j)  N.D.O.F. (xj = [-< — ., 1  m  j )  +  ... , (x ~ x ^ ) ] ^ g g  ~ , .  ( j )  (12.2)  n  Having N points i n each time-history, the t o t a l energy input can be written as:  E  = i n  N N.D.O.F. I I j=2 i - l  ( x ^+ x^ m, l i i 2 1  - 1 )  ) , -(x  ( j )  , ... - x^" ^ ^ 1  (12.3)  The number of degrees of freedom can be reduced to three by considering only the l a t e r a l movement of each f l o o r of the F r i c t i o n Damped Braced Frame (see Figure 8.1). written as:  3=2  Therefore the t o t a l energy input can be  100. where: m^,m,m  =  mass of the concrete blocks  x ,x ,x  =  absolute acceleration at each f l o o r  =  ground displacement  2  1  2  3  3  x^  Note that when the i n e r t i a forces and ground displacement are of opposite sign energy i s being radiated from the structure back into the ground.  Hence at the end of the earthquake the t o t a l energy input to the  system represents  the net energy that the system must dissipate.  Since a l l the floors of the F r i c t i o n Damped Braced Frame were i n s t r u mented for t o t a l acceleration, and the shaking table for ground displacement, i t i s possible to calculate the energy input during an actual test on the shaking table by simple numerical integration of the measured records. For this purpose, the computer program "Energy" was t h i s program i s provided  created; a l i s t i n g of  i n Appendix B.  The energy absorbed through f r i c t i o n by a l l the f r i c t i o n devices  (E ) Q  can also be calculated by a numerical integration of the slippage timehistories:  E  D  =  4P  J j=2  [(  ( j ) S l  - s ^ )  +  (sp>  - s^-V)  +  (sp>  -  sp" *)] 1  (12.5) where P  = l o c a l s l i p load  JO s ,s ,s 1  2  3  = slippage time-history at each f l o o r .  It i s assumed that the l o c a l s l i p load (P£) i s constant and i s the same for a l l the f r i c t i o n  devices.  101. N e g l e c t i n g v i s c o u s damping, i t value for energy  the e q u i v a l e n t e f f e c t i v e  i n p u t and the e n e r g y  i s p o s s i b l e t o d e t e r m i n e an upper bound l o c a l s l i p l o a d ( P ^ ) by c o m p a r i n g  d i s s i p a t e d through f r i c t i o n  at  the  t h e end o f  each  time s t e p and i m p o s i n g :  In other words,  t h e maximum p e r c e n t a g e o f e n e r g y  to a unit value.  The p r o g r a m " E n e r g y " c a r r i e s o u t t h e n e c e s s a r y  determine the e q u i v a l e n t e f f e c t i v e the  shaking  ( 0 . 9 0 g))  were u s e d w i t h t h e  effective  l o c a l s l i p load for  l o a d , which corresponds  load i n s t e a d of v a l u e of  the  l o c a l s l i p l o a d (PJJ,) d u r i n g a t e s t  the l a s t  that  test.  to a value  l o a d was a l s o  t i m e when s l i p p a g e  percentage of energy Figure  (Taft  to on  Earthquake  program "Energy" to determine the  the c a l i b r a t e d value of  slip  test series  A value of of  1.7  7 kN (see  the g l o b a l  Section 5.5).  f o u n d by c o n s i d e r i n g t h e  occurs.  equivalent  k N was f o u n d  o n l y 3 . 4 kN f o r  h o r i z o n t a l dynamic e q u i l i b r i u m a t each l e v e l of particular  steps  table.  The e x p e r i m e n t a l r e s u l t s o f  this  d i s s i p a t e d was n o r m a l i z e d  This  requirements  t h e frame and a t  is  slip same for  a  The r e s u l t i n g t i m e - h i s t o r y o f  a b s o r b e d by t h e f r i c t i o n d e v i c e s  for  presented  the  in  12.20.  These r e s u l t s optimum s l i p  confirm that  l o a d of  the  the f r i c t i o n d e v i c e s were not a c t i n g a t  structure  (7 k N ) , a p p a r e n t l y due t o  t h e c o n s t r u c t i o n t o l e r a n c e was s t i l l effective,  the  friction  devices  the  I n o r d e r t o be  However,  frame  s t r u c t u r a l response  construction.  fully  e v e n though t h e optimum s l i p of  t h e F r i c t i o n Damped  B r a c e d Frame was d e m o n s t r a t i v e l y s u p e r i o r t o t h e r e s p o n s e s types of  that  s h o u l d be f a b r i c a t e d i n a p r e c i s i o n s h o p  w i t h minimum c o n s t r u c t i o n t o l e r a n c e . l o a d was n o t o b t a i n e d ,  too l a r g e .  the f a c t  the  o f t h e two  other  102  The results of the experimental investigations involving seismic tes on the shaking table c l e a r l y demonstrate that the new damping system has the a b i l i t y to dramatically  increase the earthquake resistance and damage  control capability of conventional framed buildings.  103. 13.  ECONOMIC POTENTIAL OF FRICTION DAMPED BRACED FRAME  13.1 Introduction The economical benefits achieved through the use of the new damping system can be evaluated by comparing the design requirements of a F r i c t i o n Damped Braced Frame to those of conventional s t r u c t u r a l systems conditions of similar response behaviour.  under  We want to know what savings of  materials can be derived by designing a reduced size frame equipped with f r i c t i o n devices which w i l l approach the responses of the two other types of aseismic s t r u c t u r a l systems.  In the analysis used f o r this purpose i t  was assumed that the observed behaviour of the Moment Resisting Frame and of the Braced Moment Resisting Frame i s acceptable during a severe earthquake.  13.2 Analysis f o r Different Section Sizes Several i n e l a s t i c time-history dynamic analyses were performed with three d i f f e r e n t t r i a l member sections.  The dead load of the concrete  blocks and the s l i p load of the devices were assumed similar f o r a l l frames. Table 13.1 presents the r e s u l t s of the analyses f o r the Newmark-BlumeKapur A r t i f i c a l Earthquake scaled to a peak acceleration of 0.30 g. By examining the M^/M^  ratio, i t may be seen that there i s a l i m i t to which  the cross-section can be reduced because of the requirement that the dead load be carried safely.  Notice that the r a t i o of the bending moment  induced by the i n e r t i a forces to the f u l l y p l a s t i c bending moment i n any reduced size F r i c t i o n Damped Braced Frame i s always smaller than the corresponding r a t i o of the two other conventional building systems.  This  104. result i s very important because i t seems to indicate that, i f earthquake loading i s neglected  i n the design of a structure equipped with f r i c t i o n  devices, i t s seismic response w i l l always be superior to a conventional structure for which earthquake loading has been taken into account i n the design.  Assuming this to be true, a possible design procedure f o r a  structure incorporating these f r i c t i o n devices i s : 1)  Design the structure so that i t s main members can carry safely a l l the possible load combinations but ignoring earthquake loading.  2)  Determine the optimum s l i p load of the devices.  3)  Provide the structure with f r i c t i o n devices at the optimum s l i p load.  If the preceding observation i s v a l i d for a l l cases, this approach should guarantee that the dynamic response of the designed structure with reduced member sizes would be less than the response of a conventional building system (with larger member sizes) designed by the code and including earthquake loading. For the model frame (and member sizes) tested, the material  savings  possible using this design approach was examined by designing the structure to carry the dead load only according to the National Building Code of Canada.  This resulted i n the following cross-section being selected for  the main members: Special Light P r o f i l e SLP-3" (from Coekerill-Belgium) A I M  = 565 mm  2  x P  = 0.56xl0  6  =4.42 kN-m  mm* 1  105. From Table 13.1,  I t can be seen that the performance of t h i s reduced size  F r i c t i o n Damped Braced Frame under a strong earthquake i s superior to the performance of the conventional building systems with heavier members. The material savings i s proportional to the reduction i n crosss e c t i o n a l area:  (reduced size)  c  ,  c  5 6 5  A, ^ (actual)  1070  = 0.528  In this example this represents a 47% savings i n material cost.  The  proposed damping system therefore appears to o f f e r savings i n material costs while assuring added security against collapse.  However i t should  be  noted that the e f f e c t s of wind loads, l i v e loads and t o r s i o n have been neglected i n this example and therefore the reduction i n member sizes w i l l be less f o r a r e a l b u i l d i n g .  106.  14.  CONCLUSIONS  14.1 Summary and Conclusions 1.  A refined computer model of a F r i c t i o n Damped Braced Frame was developed to eliminate the non-conservative assumptions used with the simplified model o r i g i n a l l y proposed by A.S. P a l l (3).  It was  found  that t h i s refined model can accurately represent the r e a l behaviour of a F r i c t i o n Damped Braced Frame.  However, i t requires many more  elements and degrees of freedom than the s i m p l i f i e d model and i t s use in analysis increases the computer time s i g n i f i c a n t l y .  By  comparing  these two models i t was concluded that the s i m p l i f i e d model i s simpler and cheaper to use than the refined model and yields results which s a t i s f y the accuracy normally associated with earthquake analysis. Furthermore, the two models w i l l provide results which converge to the same response as the ground motion becomes more severe.  2.  To study the performance of the F r i c t i o n Damped Braced Frame, a model of a 3-storey frame was designed.  Two model frames were subsequently  fabricated for use i n the experimental program. 3.  An optimum s l i p load study was performed to determine the value of the s l i p load which optimized the energy d i s s i p a t i o n of the devices.  friction  The results seem to indicate that the optimum s l i p load i s  independent of the ground motion time-history and i s rather a s t r u c t u r a l property.  The global s l i p load f o r the fabricated model  frames was found to be 7 kN.  107. 4.  Seven f r i c t i o n devices were fabricated f o r use with the model frames. Each device was provided with a compression spring i n order to adjust the desired s l i p load  The devices were f i r s t tested under c y c l i c  loads i n order to study the s t a b i l i t y of the brake l i n i n g pads and to calibrate their slipping loads.  The results of the tests c l e a r l y  indicate that the behaviour of the pads i s very stable even after 50 cycles.  However, a rectangular load-deformation  curve can only be  obtained i f the fabrication tolerances of the f r i c t i o n devices are minimized.  This problem was partly solved by i n s e r t i n g s t e e l bushings  i n the 4 corner holes of the mechanism and also i n the centre slots of the f r i c t i o n pads.  However, an energy balance c a l c u l a t i o n showed that  during an actual seismic test on the shaking table the f r i c t i o n devices were s t i l l not operating at their optimum s l i p load.  5.  Different a n a l y t i c a l studies were made as follows to quantify the performance of the F r i c t i o n Damped Braced Frame r e l a t i v e to conventional aseismic s t r u c t u r a l systems.  ( i ) In an equivalent  viscous damping study, viscous damping was added to the Moment Resisting Frame and to the Braced Moment Resisting Frame u n t i l t h e i r dynamic responses became similar to the response of the F r i c t i o n damped Braced Frame.  This equality was achieved by introducing 38%  c r i t i c a l viscous damping to the Moment Resisting Frame and 12% to the Braced Moment Resisting Frame.  It was found that the new damping  system becomes more e f f i c i e n t as the intensity of the earthquake increases,  ( i i ) The economical p o t e n t i a l of the new damping system  was evaluated by designing a reduced size F r i c t i o n Damped Braced Frame whose seismic response i s at l e a s t as good as the response of a conventional building system with heavier members.  I t was found that  108. a saving of 47% i n material cost could be achieved with the actual model frames considered.  Furthermore, the performance of this  reduced  size F r i c t i o n Damped Braced Frame was demonstrably f a r superior to the performances of conventional aseismic building systems with heavier members.  However the e f f e c t s of wind loads and torsion were neglected  i n this analysis and i t i s expected  that savings of material w i l l be  less i n a r e a l b u i l d i n g . 6.  F i n a l l y , seismic testing of the model frames on the shaking table under simulated earthquake loads confirmed the superior performance of the F r i c t i o n Damped Braced Frame compared to conventional aseismic building systems. of 0.90  Even an earthquake record with a peak acceleration  g did not cause any damage to the F r i c t i o n Damped Braced  Frame, while the Moment Resisting Frame and the Braced Moment Resisting Frame underwent large i n e l a s t i c deformations. earthquake (0.90 g), a peak acceleration of 2.67  Under that same g and 2.24  g were  measured at the top of the Moment Resisting Frame and the Braced Moment Resisting Frame respectively, while only a peak acceleration of 1.42  g was  experienced by the F r i c t i o n Damped Braced Frame at the same  location.  14.2  Future Research The work reported i n t h i s thesis should be viewed as as preliminary  study.  It i s presented as a f i r s t step i n examining the potential of  structures equipped with f r i c t i o n devices. system was  Although the proposed damping  experimentally shown to be r e l i a b l e and to perform  satisfac-  t o r i l y , and as such might lead to a breakthrough i n earthquake resistant design, extensive future research i s needed before these f r i c t i o n devices can be safely used.  109.  A simple design method for evaluating the optimum s l i p load must be developed;  i n e l a s t i c time-history dynamic computer analyses are much too  expensive to be used for c a l c u l a t i n g the s l i p load on a regular basis i n design o f f i c e s .  Therefore, a detailed parametric study should be under-  taken i n order to determine the parameters which govern the optimum s l i p load of a structure and to develop a simplified design method to calculate this load.  The f i r s t step would be to v e r i f y whether the optimum s l i p load  i s indeed a structural property and i s independent of the ground motion time-history.  If this i s found to be true, the development of a design  procedure could possibly be based on a fundamental parameter of the structure, such as i t s fundamental mode shape, f o r example. Each f r i c t i o n device used i n this research project was provided with a compression  spring which allows adjustments to be made to the clamping  force and therefore the s l i p load.  However, springs w i l l not be used i n a  r e a l structure; the clamping force w i l l be developed by a bolt torqued to the proper value.  Therefore, more experimental work i s needed to develop a  r a t i o n a l method of c a l i b r a t i o n f o r the f r i c t i o n devices.  Also, long term  studies should be undertaken to v e r i f y i f the devices creep and are s t i l l i n working condition a f t e r many years of service; maintenance methods should be developed  to ensure that there i s no deterioration i n the long-  term operability of the devices. 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 to provide a more r e a l i s t i c ment of i t s behaviour.  assess-  The computer program "Drain-Tabs", developed at the  University of C a l i f o r n i a , Berkeley, could be used for t h i s purpose.  110.  BIBLIOGRAPHY 1.  POPOV, E.P., TAKANASHI, K. and ROEDER, C.W., "Structural Steel Bracing Systems: Behaviour Under C y c l i c Loading," E.E.R.C. Report 76-17, Earthquake Engineering Research Center, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , June, 1976.  2.  POPOV, E.P. and ROEDER, C.W., "Eccentrically Braced Steel Frames for Earthquakes," ASCE, Journal of Structural D i v i s i o n , March 1978.  3.  PALL, A.S. and MARSH, C , "Response of F r i c t i o n Damped Braced Frames," ASCE, Journal of Structural Division, June 1982.  4.  PALL, A.S., "Limited S l i p Bolted Joints - A Device to Control the Seismic Response of Large Panel Structures," Ph.D. Thesis presented to the Centre for Building Studies, Concordia University, Montreal, Canada, 1979.  5.  PALL, A.S., MARSH, C , and FAZIO, P., " F r i c t i o n Joints for Seismic Control of Large Panel Structures," Journal of the Prestressed Concrete Institute, Nov./Dec, 1980, Vol. 25, No. 6, pp. 38-61.  6.  PALL, A.S., and MARSH, C , "Optimum Seismic Response of Large Panel Structures Using Limited S l i p Bolted Joints," Proceedings, Seventh World Conference on Earthquake Engineering, Istanbul, Turkey, Sept., 1980, Vol. 4, pp. 177-184.  7.  SABNIS, G.M., HARRIS, H.G., WHITE, R.N., and MIRZA, M.S., "Structural Modeling and Experimental Techniques", Prentice-Hall C i v i l Engineering and Engineering Mechanics Series, 1983.  8.  WORKMAN, G.H., "The I n e l a s t i c Behavior of Multistory Braced Frame Structures Subjected to Earthquake E x c i t a t i o n " , Research Report, University of Michigan, Ann Arbor, Michigan, September, 1969.  9.  PICARD, A. and BEAULIEU, D., "Calcul aux Etats Limites des Charpentes d'Acier", I n s t i t u t Canadien de l a Construction d'Acier, 1981.  10.  CANADIAN INSTITUTE OF STEEL CONSTRUCTION, "Handbook of Steel Construction", Willowdale, Ontario, December, 1980.  11.  NATIONAL RESEARCH COUNCIL OF CANADA, "National Building Code of Canada", Ottawa, Ontario, 1980.  12.  LEE, D.M. and MEDLAND, I.C., "Estimation of Base Isolated Structure Responses" B u l l e t i n of the New Zealand National Society f o r Earthquake Engineering, No. 4, December 1978.  13.  DE SILVA, C.W., "Dynamic Testing and Seismic Q u a l i f i c a t i o n Practice", Lexington Books, D.C. Heath and Company, Lexington, Massachusetts, 1983.  111. 14.  CLOUGH, R.W. and PENZIEN, J . , "Dynamics of Structures", McGraw-Hill Book Company, 1975.  15.  EWINS, D.J., "Model Testing: Theory and Practice", Research Studies Press, John Wiley & Sons, Inc., 1984.  16.  LAM, C.F., "Analytical and Experimental Studies of the Behaviour of Equipment V i b r a t i o n Isolators Under Seismic Conditions", M.A.Sc. Thesis submitted to the Department of C i v i l Engineering and the Faculty of Graduate Studies, University of B r i t i s h Columbia, Vancouver, Canada, A p r i l , 1985.  17.  KANNAN, A.E. and POWELL, G.H., "Drain-2D, A General Purpose Computer Program for Dynamic Analysis of I n e l a s t i c Plane Structures", A computer program distributed by NISEE/Computer Applications, College of Engineering, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , August 1975.  18.  KANASEWICH, E.R., "Time Sequence Analysis i n Geophysics", The University of Alberta Press, 1975.  P (kips)  .0  Figure 1.1 Typical Hysteresis Loops of Ref.:3)  a  Tension  Brace  Figure 1.2 Location of F r i c t i o n Device (from Ref.:3)  113.  Figure 2.1 Re f.:3 )  Hysteresis  Loops  of  Simple F r i c t i o n Joints (from  114.  p 11  >• A  Figure 2.2 Hysteresis Loop of a F r i c t i o n Joint where the are Designed in Tension Only  Braces  115.  Figure  2.3 Mechanism  of F r i c t i o n  Device  (after  Ref.:3)  116.  F i g u r e 2.4 Possible Frames (from R e f . : 3 )  Arrangements  of  Friction  Damped  Braced  117.  WW 1) E l a s t i c  P  stage  ^  2  L f  2) B u c k l I n g o f compression  .buck 1ed  i.  brace  \\\\\  < ii  3) S I I p p l n g o f tension  brace  J> WW 4) R e v e r s i n g o f  load  i  r  S) C o m p l e t i o n one  Figure Braced  of  J  cycle  2.5 Frame  t  Hysteretic  Behaviour  of  a Simple  Friction  Damped  Fcr=Fictitlous yield stress In compression Corresponds to the buckling stress of the compression brace  Brace 1  Brace 2  Figure 2.6 Simplified Model of a F r i c t i o n Damped Braced Frame  00  119.  F i g u r e 2.7 U n s t a b l e Mode of modeled w i t h T r u s s E l e m e n t s  a  Friction  Damped  Braced  Frame  Truss  Figure  2.8  120.  Elements  R e f i n e d Model of a F r i c t i o n  Damped B r a c e d  Frame  121.  in c E 3  O  o  in c E 3 0  o  (6 100mm) (a)  (b)  Moment  Braced  Resisting  Moment  Figure 3.1 D i m e n s i o n s (from r e f . : 8 )  Frame  (MRF)  Resisting  Frame  and Member  (BMRF)  Sizes  of Prototype  Structure  L I S T  O F  M A T E R I A L S  DESCRIPTIONS  S7SH9 3S13„ 9M.of  36  1121  ie lit s. 07  4C  5a M.t. C 140*1*01 (. 7a At. i  i 100*  fc  11 11  10.  Li..SI  11 22 ll  12-  13. 14.  9£1 7*  15-  lit.7»  16. 17.  M i . *  /IS X H If J2 /»w/m  18 19 20O  11-70  noio SO.it 121 « r  4  21/ 22  -M.i. IE i 7 0 X /VO X  2a  C 200X17  2 0O«<Mlt<lt  24 F 4 T O T A L WIIOUT:  626V. 61  NOTES: UKOIII »  uiu  MJUiHITU*  '  REFERENCE  SAAXM ro* i r t u : M M *  GENERAL I U T I : >/•*/»! AM* 4 l « t a ( > M M i  All IS  Figure 3.2 General Arrangement  of Model Frame  FRAME  ARRANQEMENT  NOTES  REFERENCE  FRAME  LENGTHS IN M I L L I M E T R E S WELDING DSINQ ELECTRODES  E«80XX  DETAILS  A L L H O L E S : l l / l B " DI A . A L L BOLTS : I T E M 18  j\ *? ? < 5  Figure 3.3 Details 1-2 of Model Frame  r»o» • or *  . ..  /-<£>  \  * i  ~7  DETAIL LAT KB A L  (*)  BBACIMO  Tl f ICAL CUBJMSB  —tt>I  DETAIL TiriCAL  I  -  &OMMBCT10M  0 •COLUMN  CONHBCT10N  J.  REFERENCE I  1 lIVlBlQJl-Ll.  DETAILS  NOTES:  SCALE-  LENGTH*  OBAWM  IN  BATS:  M1LLIMITSKS  WSLD1HU US1XO K L K C T m O D B I  Figure  3.4  IWIII  Details  3-4  of  Model  Frame  »••  tj*w«f B*  L O . * '  FRAME  REFERENCE NOTES:  DETAILS  LKN0T1I JM M l LLIMKTHCJS  •CAlJB »• • BATS : • / > « < * Ml AMI M : LD, AP  IVIJOIINO UMIHO Kl-LCTMUOUa ALL U O l t J '  FRAME  «»«yi*io«»»  '  n/*«"Oi*.  ALL. • O L T M ' IT KM (•  Figure  3.5 D e t a i l s 5-6-7 o f M o d e l  Frame  Ni  126.  Figure  3.6  M o d e l Frame mounted on t h e S h a k i n g  Table  (a)  High  ( b ) Low  slip  slip  ( c ) Optimum  load  load  slip  load  Figure 4.1 Concept of Optimum S l i p Load  128.  Figure  4.2  F r e e Body D i a g r a m o f a F r i c t i o n  Device at  Slipoi  129.  (a) Beams and columns  S75X8  9.57kN-m  -9.57kN-m  (b) R e i n f o r c e d base columns  S75X8  483kN  •16.02kN-m  16.02kN-m  483kN  Figure  4.3 Y i e l d  Interaction Surfaces  f o r Model  Frame  (a) El Centro (SOOE) (0-6sec.)  o  a E < O  10  2  (b) P a r k f l o l d  12 14 Time(sec . )  0.0  6.0  -1— 9.0  12.0  FREOUCNCY  (N65E) (0-9sec.)  (c) Newmark-Blume-Kapur a r t i f i c i a l  Accelerograms  earthquake  ~]  Fourier  Spectra  Figure 4.4 Earthquakes used for the Optimum S l i p Load Study  15.fi  (i',7)  (a)  Envelope  of bending  moments  (c)  In t h e beams  1st •+• 2nd 3rd  Envelope  of l a t e r a l  -©- 1 s t f l o o r  floor  •+  floor floor  -e—©—o 0.0  2.0  GLOBAL  (b) Envelope  4.0  6.0  10.0  8.0  «  0.0  moments  Figure 4.5 Earthquake  2.0  (d) Envelope  i n the columns  Results  I 4.0  GLOBAL  S L I P LOAD (KN)  of bending  deflections  of  Optimum  Slip  ©  ©  1 6.0  S L I P LOAD  of shear  Load  ©  forces  Study.El  2nd  floor  3rd  floor  e  1 8.0  ©-  (KN)  In t h e columns  Centro  I0.Q  (a)  E n v e l o p e o f b e n d i n g moments  (c)  I n t h e beams  GLOBAL SLIP LOAD (KN)  Figure 4.6 earthquake  Results  of  Envelope of l a t e r a l  deflections  GLOBAL SLIP LOAD (KN)  Optimum  Slip  Load  Study,Parkfield  (c) Envelope o f l a t e r a l  (a) E n v e l o p e o f b a n d i n g moments I n t h e beams  ~\ 0.0  i  I  I  I  I  I  I  L  1  1 2.0  1  1 4.0  1  1 6.0  1  I 8.0  GLOBRL  SLIP  LOAD  1  T 10.0  -©-  Mp-16.02kN-m +  floors  I  I  I  I  I  I  1  L  1  I 2.0  I  I 4.0  I  1 6.0  I  I 8.0  (KN)  GLOBAL  ( b ) E n v e l o p e o f b e n d i n g moments I n t h e c o l u m n s  base  0.0  SLIP  LOAD  (d) Envelope o f shear f o r c e s  T  10.0  (KN)  In t h e columns  base  Q  1st s t o r e y  -J- 2 n d s t o r e y  1st f l o o r 2nd  Mp»9.57kN-m  deflections  -6- 3 r d s t o r e y  floor f loor  -© 0.0  2.0 GLOBAL  I 4.0 SLIP  Figure 4.7 Earthquake  ©  * ft • *  6.0 LOAD  8.0  41  10.0  (KN)  Results  -e—&-—©—n  © (T  o.o  1  2.0 GLOBAL  of Optimum  Slip  Load  4.0 SLIP  6.0 LOAD  I 8.0 (KN)  Study A r t i f i c i a l  10.0  member  (c)  F r i c t i o n Damped B r a c e d simplified  Figure  4.8  Frame  yielded  ( ) F r i c t i o n Damped B r a c e d a  model  Structural  refined  Damage a f t e r  Frame  model  E l Centro  Earthquake  (a)  Envelope  of bending  moments  i n t h e beams  ( c ) Envelope  of l a t e r a l  deflections  LO  MOMENT (KN-M) (b) Envelope  of bending  moments  D E F L E C T I O N (MM) i n the columns  MOMENT (KN-M)  Figure  4.9 R e s u l t s  (d) Envelope  of shear  SHEAR  Envelope  for E l Centro  forces  FORCE  Earthquake  i n the columns  (KN)  (a)  Acceleration  Record  I SV  (b)  Power  l  Spectral  I  l  Density  I  Function  J  I  I  J  L  I  I  1  L  T3 0) N  o  z  0.0  i  i 5.0  r  -i 10.0  1—i 15.0  1  1 r 25.0 20.0 Frequency  Figure 4.10 Band Limited  i  1 30.0 (Hz)  r  i — i — i — i — i — r 3S.0  40.0  45.0  (0-25Hz) White Noise Record  50.0  (c) (a)  Moment R e s i s t i n g Frame  <) b  F r i c t i o n Damped B r a c e d Frame  Braced Moment R e s i s t i n g Frame  Figure 4.1 1 Structural Damage after White Noise Excitation  (a)  Envelope of Bending Moments I  I  I  I  I  I  in the Beams  I  (c)  Envelope of J  I L <B M . R . F .  I  Lateral I  Deflections  I  I  I  I  -f B . M . R . F . F.D.B.F. oj > OJ  -r  *(sImpIi fled) -i)  o o if-  0.0  1  ~i  "~I  2.0  1  4.0  1  r  6.0  10.0  8.0  Moment (kN-m). (b)  Envelope of Bending Moments J  I  I  J  I  100.0  60.0  D e f l e c t i o n (mm) (d)  in the Columns I  I  Envelope of I  L  I  Shear Forces in the Columns I  I  I  I  I I •+- M . R . F . O B.M.R.F. F.D.B.F. (a Imp I 1 f l e d )  QJ > QJ  O O  12.0  .Moment (kN-m) Figure  4.12  20.0  0.0  — i  4.0  8.0  r  12.0  Shear Force (kN)  R e s u l t s Envelope f o r White N o i s e  Excitation  16.0  20.0  139.  F i g u r e 4.13 Time H i s t o r i e s of Noise E x c i t a t i o n  Third  Floor  Deflection  for  White  140.  F i g u r e 4.14 E q u i v a l e n t A r t i f i c i a l Earthquake  Viscous  Damping  Study,Newmark-Blume-Kapur  141.  Figure 5.1 General Arrangement of F r i c t i o n Device  "t—» o P-1  1  ©  o  •<«o < so.a  l;lrtac^«y  -12^  h—>m  f  .  •t e l . . * I  E D I  •f~- MA  "<0 bet. AA  R]A'(fwfc b~k<./»xJO . p i .  i i i  C o m p r e s s i o n 5pr>'"3 7 S ^ " * ' t  o .  KOKRMO  4oJ.JO.I  U,  P-7.  1 •<  I .  P-3  Dr»M f  ;  "—•  P-4 (StJ. n.lU/')  P-9  *>  J  > P-IC  F2.IA  i *  P-2 P-5 P-4 P-5  It IC  -+  ^  * H  Friction  Devices  *!*•«.  r  no 16 10  I • . A L L ^ D i M l N H O H l IN m « t  . n i  SCHEDULE OF QUANTITIES. Mi. 0 ttitu H p-l 1  •-NPTES-  o tlo> so* ft  S A - I P J O M * t i n . (*»;«»<-Wit  MILD  a^tkL..  NAIL.0 l l t U - ^ K U k * ! * *  P7  8  P'  36 10 A&  A-525  t>OL1$  46  ft#ION  LlMlA  Rio PII P-IZ  f o r Model  Test  LOt*W  Frame  'AiMnB-os'C*'*!* *1  Ckwi*  Figure 5.2 Details of F r i c t i o n Device  TO  f u i r  i—  1  Link  without  Brake  Lining  Pad  Surface  #3  Link  without  Washer Friction  Glued  with on  Brake Brake  Each  Lining Lining  Pad Pads  Sides Central  Friction  Surface  Bolt  #4  Spring  Friction  Link  with Brake  Lining  Pad  \  ^v.L1nk  Surface  with  Friction  F i g u r e 5.3 F r i c t i o n  S u r f a c e s of F r i c t i o n  Device  Brake  Surface  #1  Lining  #2  Pad  144.  F i g u r e 5.4 F r i c t i o n D e v i c e on M o d e l Frame  Figure 5.5 Experimental Set Up Devices  for  Cyclic  Tests  of  Friction  Figure 5.6  Experimental  S e t Up  for S t a b i l i t y  Tests  6  6  7  E x c i t a t i o n Frequency = 0.2 Hz  Plateau#2  PJateau#1  Figure 5.7 Typical Hysteresis Loop From S t a b i l i t y Tests  148.  j  |  '  -17  I  TEST H22 FREQUENCY-I.QHZ  TEST t»21 FREQ.UENCY-a.20HZ i 1 I  I  I  I  .  ]  I  L .  ~i—i—i—i—i-*-!—i—i—t-i—i—r - u i - t a -tu I M 121 OISPLflCEItNTS O N  UI  TEST H24 FREOUENCY=3.0HZ j  i  i  i  Ui  -ta  - u i us  .  T "  ta  OISPUKEfSKTS (M  TEST H25 FREQUENCY-4.0HZ i—i— DEVICE «1 ^max=0.051n  T  1  -IM  I  OlSPLflCErtHTS ON)  I  U l  U i  OISPLACEflEKTS (DO  Figure  5.8 R e s u l t s o f S t a b i l i t y T e s t s f o r V a r i o u s F r e q u e n c i e s  E x c i t a t i o n Frequency = 0.2 Hz  (mm  ( in  Figure  5.9  H y s t e r e s i s L o o p from O r i g i n a l D e v i c e  Figure 5.10 Hysteresis Loop from Modified Device  Global S l i p Load (kN)  Figure 5.11 Calibration Curves of F r i c t i o n Devices  1  item # ? §"  (Inscription >  G R - f c e s  FoR  P06F-  quantityi 3o SCale< / r S date> Z5fo3ies~ drawn by Id, af Paffe 1 of c?. IAJPICATCO  F i g u r e 6.1 D e t a i l o f B r a c e U n i t B r a c e d Frame  Used  on  the  Friction  Damped  Ul  -  M  153.  Figure 6.2 Permanent Deformed Shape of Brace Unit after Uniaxial Test #1  (Lbs)  eooo  45O0  3000  1500  1.52  2.03  2.54  3.05  B u c k l I n g Regime  Unit  r e  6  '  3  L o a d  "  D e f o r m a t i o n  Curve from Uniaxial Test #1 on Brace  (Lbs)  •  6000  Rupture  of  Unit  450O  30O0  1500  .51  1 .02  1 .52  2 .03  2.54  3 . 05  Figure 6.4 Load-Deformation Curve from Uniaxial Test #2 on Brace Unit  ( ram)  Ui  p (kips)  A  UN)  x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\V CEILI NG 3045  P  1  <n* ro  67  •  439  01  U>  83  •  o  N  •  529  69  •  Oi  OI  • 530  Oi  Oi  • • •  530  m  444  OI Oi  85  •  • 533  • OI Oi  o  417  •  530  65  • ro  • <r> O  •  •  cn  • •  535  •  535  O  Oi  • 335  • 75  m  u> Oi  o  • 347  • 65  Ul  u> ro  397» III  Ul  530  Oi  • 3 4 7 • 67  Ul  m  o  o  •  Ul  ro  m  Oi  533  •  cn ro  Oi Oi  • 3 3 3 • 83 •b  o  on oi  en  Oi  Ul  53 5  Oi  440 •  OI  •  o  Oi  Oi Oi  535  • 65  IM  Ul  Oi  o  533  535  Ul  535  533  Oi O  Oi  * OI  A  437  •  e m  o>  o  •&  540  • 345  Oi Oi  o  •  533  530  Oi  Ul oi Ul  535  •  <n  Oi 01 OI  Ul  Oi Oi  OI Oi  OI OI  528  <J»  . •524 • •  530  o  •  542  Oi OI  Ul  440  •  cn  Ul  OI  OI OI  65  530  W  m  65  •  cn  OI OI  m  ro  •  OI  OI  445  528  cn IM  ro  OI  m  OI OI  542  cn  OI  <j» o*  425  •  1o•  Ul  O  532  o  • 350  cn  • 70  ro  o> ro  •  •  3045  V7777A  1  I  ,  180 —  *  1  1 1 1 1  SHAKE  1 1 1 1  TABLE  1— 1 1 •  o  — " 180  V  N0TE ALL  MEASUREMENTS ARE IN mm  Figure 7.1 General Arrangement of Earthquake Simulator Tabl  158.  Figure 7.2 Physical Arrangement of Data Acquisition Earthquake Simulator Table  System  for  159.  Figure of  Model  7.3  Strain  Gage  Accelerometer  on  First  Floor  Cross-Beam  Frame  F i g u r e 7.4 Potentiometer of Model Frame  used  to  Measure  Absolute  Displacement  160.  F i g u r e 7.5 S t r a i n Gages U n i t on Base Column o f M o d e l Frame  161.  Figure  7.6  Strain  Gages  Unit  on  First  Floor  Beam  of  Model  Frame  162.  Figure 8.1 Degrees of Freedom Considered Experimental Mode Shapes of the Model Frames  o  to  Determine the  163.  k Magnltude  (db)  Figure 8.3  Tv  Pical-Acceleration-Decay Record  164.  Figure  9.1  Moment R e s i s t i n g  Frame on  Shaking  Table  165.  J ^ l - f t  "  ^  f  t  ^  r  1 175  b+X^ ht 323.5  *\t—*p203  4»r-  3  g  203  7  397  »H 203  »f* »> 323.5  1175  ^ t ^ t ^ D.O.F.  InX.Y.e  Lumped Mass All  Dimensions  4  A  Directions  in X.Y  I  775  Directions  1n mm 400  \\\W  Figure 9.2 Computer Model Used for the Free of the Moment Resisting Frame  Vibration  Analysis  166.  Mode#1.  17.788  rad/s  Mode#3.  37.336  rad/s  Mode*5.  99.965  rad/s  Figure 9.3 P r e d i c t e d N a t u r a l Moment R e s i s t i n g Frame  Mode/?2.  Mode/y4,  Mode*6.  Frequencies  57.642  95.756  rad/s  rad/s  lOO.091rad/s  a n d Mode Shapes o f t h e  167.  f ^ Y S I S OF TKIRO FLOOR I IME-fiCCCLERRT ION-OECRT MPMFNT RESISTING FRRHE FREQUENCY OF i N i r i n i onsE MOTION INPUT.SMZ  FOURIER srCCTRUM  FOURIER SPECTRUM RNfiLYSIS OF THIRD FLOOR TIME-nCCCLERRTION-OECHT MOMENT RESISTING FRRME OF IN ITlfiL Of!SE MOTION INPUTi20KZ  FREQUENCY  1/1 l  01  3  o.  E <  IU.0 FREQUENCY  FOURIER SPECTRUM RNfiLYSIS OF THIRO FLOOR TIME-flCCELERATION-OECflr MOMENT RESISTING FRBME FREQUENCY OF INITIAL SflSE MOTION i N P U T i I I K Z  «  in i 01  Figure 9.4 Fourier Spectrum Analysis Frame  of  the Moment  Resisting  168.  ANALYTICAL  F i g u r e 9 . 5 Measured vs Moment R e s i s t i n g Frame  (RfiD/S)  Predicted  Natural  Frequencies  of  the  169.  Hode Shape Tests,Mode 81,Frequency=2.86Hz  Time (sec.)  node Shape Tests.Mode tt2,Frequency=9.08Hz i  i  i 0.0  i  i  i  i  i  i  i  i  i  i  i  i  i  i—i  1 i 1 1 1 i i i i i i i ir 0.2  0.4  0.6  0.8  1.0  1.2  1.4  Time (sec.)  Mode Shape Tests.Mode H3,Frequency=14.4Hz  Tine (sec.)  Figure Frame  9.6 E s t i m a t i o n  o f Mode Shapes f o r  t h e Moment  Resisting  170.  flNfOTICRL  Figure 9.7 Measured R e s i s t i n g Frame  vs  Predicted  Mode  Shapes  o f t h e Moment  171. EXPERinENTRL  Floor  1st  - i — i — i — i — i — i — i — i — i — i — i — i — i — r 24j <aj MJ Tzi au ao o u  FREQUENCY U (RflO/SI -J  EXPERlnENTRL  I  I  2nd  I  i_  Floor  T — i — i — i — i — i — i — i — i — i — i — i — i — r 24J  4U  SU  t»a  au  72J  FREQUENCY U (RflO/S) I  i  EXPERIMENT RL  I  I  I  3rd  I  I  c u  i_  Floor  •4J  tZU  FREOUENCY U (RflO/S)  Figure 9.8 Experimental R e s i s t i n g Frame  Mobility  Functions  for  the  Moment  172.  17.84  17.88  17.92  17.36  18.0  18.04  18.08  18.12  FREQUENCY U (RAD/S)  Figure 9.9 Bandwidth Method Applied to the F i r s t Floor Function of the Moment Resisting Frame  Mobility  Damping Test,Moment R e s i s t i n g Frame 1st F l o o r T1me-Accelerat1on-Decay 1st Mode Frequency of I n i t i a l Base Motion Input=2.86hz  67 .5  Figure 9.10 Logarithmic Decrement Method Applied to the F i r s t Floor Time-Acceleration-Decay of the Moment Resisting Frame  173. ' -  '  L_  Floor  1st  flNPLYTICPL •  EXPERIMENTS!.  — i — i — i — i — i — i — i — i — i — i — i — 310 1U US 801 9U UM  FREQUENCY IKRflO/S)  RNPOTICRL +  2nd  + EXPERIfiENTRL  »  Floor  *A  V / •  1 33J  +  :  <** t»a eu FREQUENCY UIRPO/S)  RNRLYTICRL EXPERIMENTAL  3rd  Floor  • •  -  J  f1 / \ i \ ' J \. - 7 X .y V  x  + i  JA I T f \ J \ /  * Il ++ 11 +* l\ V +  7  \  -  32J  4U  SiJ nj  FREQUENCY UlRRO/S)  F i g u r e 9.11 P r e d i c t e d vs Moment R e s i s t i n g Frame  Measured  Mobility  Functions  of  the  A l l 01mens Ions i n mm  Figure 10.1 Computer Model Used for the Free Vibrations Analysis of the Braced Moment Resisting Frame  175.  Mode *5. 99.956  Figure Braced  rad/s  Mode #6,  138.389  10.2 Predicted Natural Frequencies Moment R e s i s t i n g Frame  rad/s  and Mode Shapes o f t h e  176. F2UR1FR SPECTRUM RNfiLYSIS CF THIRD FLOCR TIME-RCCELFRflTIGN-QECflY BRflCFD MfiMENT RESISTING FRAME FREQUENCY SF INITIfiL BASE MOTION INPUT:18HZ  Figure 10.3 Fourier Resisting Frame  Spectrum  Analysis  of  the Braced  177.  Mode Shape Tests.Node  tt1.Frequency=5.29Hz  Time (sec.)  Figure 10.4 Estimation of Mode Resisting Frame  D a m p i n g T e s t , B r a c e d Moment R e s i s t i n g 1st F l o o r T1me-Accelerat1on-Decay 1 s t Mode £  Frequency  of  Initial  Base Motion  Shapes  for the Braced  Moment  Frame  Input=5.29Hz  Figure 10.5 Logarithmic Decrement Method Applied on the F i r s t Floor Time-Acceleration-Decay of the Braced Moment Resisting Frame  178. ANALYTICAL  10  40.0  1st  80.0  120.0  Floor  160.0  2O0.0  FREQUENCY U (RflO/S) J_ 2nd  RNRLYTICRL  Floor  —i—i—i—r 120J  200.0  160.0  FREQUENCY U (RflD/S)  ANALYTICAL  3rd  Floor  9 <H  40.0  i  80.0  i  1 1—i 120.0 160 0  r  200.0  FREQUENCY U (RAD/S)  Figure 10.6 Predicted Mobility Functions of Resisting Frame  the  Braced  Moment  179.  11.1  Friction  Damped B r a c e d Frame on  Shaking  Figure 11.2 Computer Model Used for the Free Vibrations Analysis of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  181.  Mode/M . 24.448 rad/s  Mode/f2, 24.746 rad/s  Mode/»3, 24.829 rad/s  Figure 11.3 Predicted Natural Frequencies and Mode Shapes of the Friction Damped Braced Frame Under Low Amplitude Excitation,Modes#1-#3  182.  Mode#4,  44.059  rad/s  95.732  rad/s  Mode#6,  Mode#8.  122.348  rad/s  Models.  87.332  rad/s  Mode*7,  33.956  rad/s  Mode#9.  177.919  rad/s  Figure 11.4 Predicted Natural Frequencies and Mode Shapes of the Friction Damped Braced Frame Under Low Amplitude Excitation,Modes#4-#9  183. FOUR'FR SPECTRUM <~,NRL SIS OF THIRD FLOOR T IME-RCCELF.RR Ii3N-OECfi FRICTION DfiMPED BRRCFO FRAME FREQUENCY OF INITIAL 5RSF MOT I ON SHZ V  T  y  FOURIER SPECTRUM RNRLYS15 OF THIRD FLOOR T IME-RCCELF.RRT ION-DECRY FRICTION DfiMFED BRRCED FRRME FREQUENCY OF IN ITI RL 5R5E MOTION; 3'JHZ T o  in  Figure 11.5 Fourier Spectrum Analysis of the Braced Frame Under Low Amplitude Excitations  Friction  Damped  184.  Figure 1 1 . 6 Measured vs Predicted Natural Frequencies of the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  185.  Mode Shape Tests,node tf1,Frequency=7.03Hz — i — i — i — i — i — i — i  i  i  i  i  i  i  i  i  i  i  3rd 2nd  1st  I  0-0  T  I  II  T  I  0-2  1  1  0.3  1  1  1  1  0.5  0.4  1  1  0.6  1  1  0.7  1  1  0.8  1 0.9  1t.O  Time (sec.)  node Shape Tests,node 82,Frequency=18.4Hz  J — i — i  i  i  i  i  i  i  i  i  i  i  i  i  2nd  Time (sec.)  Figure 11.7 Estimation of Mode Shapes for the F r i c t i o n Damped Braced Frame Under Low Amplitude Excitations  186.  2  1  i -IJJ  1  -gj  1—i  -u  1  i  i  i — i — i — i  -u -u ANRLYTICBL  1  U  i  i  0.4  1 — i — i — i — r 03 u  IJO  Figure 11.8 Measured vs Predicted Mode Shapes of the Damped Braced Frame Under Low Amplitude Excitations  Friction  D a m p i n g T e s t , F r l e t 1 o n Damped B r a c e d Frame 1st F l o o r T 1 m e - A c c e 1 e r a t 1 o n - O e c a y 1 s t Mode Frequency of I n i t i a l Base Motion Input=7.03Hz  A =o.0495g  r = 27  Figure 1 1 . 9 Logarithmic Decrement Method Applied to the F i r s t Floor Time-Acceleration-Decay of the F r i c t i o n Damped Braced Frame  187. (a)  j  ^ m a x - 2 0 . 447mm  Intensity  Moment R e s i s t i n g  j  1  A |  >  m a x = 40 . 923mm  Intensity  Mmax/Mp=0.34  Frame  |  2  B r a c e d Moment R e s i s t i n g  1  m a x = 102 . 017mm  3  Mmax/Mp=1.00 Frame  max=3.198mm  Intens1ty  A  Intensity  Mmax/Mp=0.57 (b)  Mmax/Mp»0.16  >  max=19.472mm  Intensity  2  Mmax/Mp»0.20  Figure 12.1 Predicted Structural Earthquake,Intensities 1 2,3  Intensity  3  Mmax/Mp=0.36  Damage  After  Artificial  A c c e l e r a t ion  Acceleration  > o o  ro o o  a  a  T  a o a  -> DI  O  o>  cr  3  73  ro o o 1 a  >  o n  r r v Q  •-• C  CD  l-h i-l H" fD  >  <D T 01  O  wCU  0  C 01  I— —• 1  ro  0)  n • Ctf  -J  D C  to  0)  rt  IS  C OJ  Cu  fD  D  -3 0)  cr fD  Ampli tude fD  •1 i-M  o rt  3 cu 3 O fD  Z fD *  3 cu I  w  \->  c 3  fD I  Pi  cu  TJ C 1  '881  Ampli tude  189.  J  J  L  I n t e n s i t y 1 (O.OSg)  L  I  I  0 M.R.F. + B.M.R.F. <> F . D . B . F .  I  I  0.4  J  L  —[  I  0.8  1  T~  1.2 Acceleration (g)  J  I n t e n s l t y 2 (0.10g)  1.6  2.0  L 0 M.R.F. + B.M.R.F. <* F . D . B . F .  I n t e n s i t y 3 (0.30g)  QJ  t3 M.R.F. -+• B.M.R.F. * F.D.B.F.  SR.  1 CM  •  0.0  0.4  0.8  1.2  Acceleration  i r~ (g)  1.6  2.0  Figure 12.3 Envelopes of Measured Horizontal Accelerations, Newmark-Blume-Kapur A r t i f i c i a l Earthquake,Intensities 1,2,3  190.  I n t e n s i t y 1 (0.05g)  30.0  <S M.R.F.  60.0  90.0  120.0  150.0  D e f l e c t i o n (mm) i  i  i  i  i  i  I n t e n s i t y 2 (0.10g)  — i  0 M.R.F.  1 1 1 1 1 1 1 1 r  0.0  30.0  60.0  90.0  120.0  150.0  D e f l e c t i o n (mm) i  i  i  i  i  i  i  i  I n t e n s i t y 3 (0.30g)  D e f l e c t i o n (mm)  Figure 12.4 Envelopes of Lateral Deflections,Newmark-Blume-Kapur A r t i f i c i a l Earthquake,Intensities 1,2,3  191. J_ 1 (0.05g)  Intensity  0 A +  QJ > QJ  M.R.F  8 M.R.F. F.D.B. F .  2  —  a o  — Measured  10  Predlcted'o  -  a  —-r a . a "  1^ 9.0  i  0.0  6.0  3.0  12.0  15.0  Moment (kN-m)  Intensity  2  (0.10g)  0 M.R.F. A B.M.R.F. +  F.D.B.F.  aj >  z  z  ID  ID  _ ——Measured a o  — —  Predicted  a  K  1  0.0  1 3.0  I  I 6.0  I  I 3.0  12.0  15.0  Moment (kN-m) 1  I  Intensity  I 3  I  (0.30g)  C3  cn"  0  QJ > QJ  M.R.F.  A  B.M.R.F.  +  F.D.B.F.  Measured  a o  Predicted  311  o.o  ~1~ 3.0  T  6.0  9.0  12.0  15.0  Moment (kN-m)  Figure 12.5 Envelopes of Bending Moments i n the Beams,NewmarkBlume-Kapur A r t i f i c i a l Earthquake,Intensities 1,2 ,3  192. (a) Moment R e s i s t i n g Frame  1  i  i  i  0.0  i  2.0  I  i  4.0  i  i  6.0  i  i  8.0  i  i  10.0  i  r~  12.0  14.0  Time (sec.) (b) Braced Moment R e s i s t i n g Frame I .—.  I  I  I  L _  i  Measured  CD  e <=J_ e cn  • Predicted Peak  a  u cn. QJ  I  QJ Q  cr, CD  in  2.0  0.0-  1  1  6.0  4.0  8.0  r  ~ i  12.0  10.0  14.0  Time (sec.) (c) F r i c t i o n Damped Braced Frame 1 ,—  CD CD'  E= cn  I I Measured  I  I  J  I  L  Predicted  _  —  Peak  a  •—i •M (_>  QJ  CD CD  ro _ i  .—1  U_  QJ Q  CD CD  in  i 0.0  i 2.0  i  i 4.0  i  i  i  6.0  i 8.0  i  i 10.0  r  12.0  i  r  14.0  Time (sec.) Figure 12.6 Time-Histories of Third Floor Deflection,NewmarkBlume-Kapur A r t i f i c i a l Earthquake,Intensity 3  193.  4.0  6.0  Tine (sec.) Figure 12.7 Slippage Time-Histories of Second Floor Device,Newmark-Blume-Kapur Artificial Earthquake,Intensity 3  C3  o" • CZ  a  V  Ay\  fl  »i  J"  L/\  A  .»«  ,  Aw  . - / v . .  u QJ I OJ  a C3  a n . i  0.0  2.0  4.0  6.0  8.0  10.0  12.0  14.0  Time (sec.) Figure 12.8 Recorded Time Histories of Third Floor Deflection From Both Potentiometers of the F r i c t i o n Damped Braced Frame,Newmark-Blume-Kapur. A r t i f i c i a l Earthquake, Intensity 3  A c c e l e r a t i o n Record.Actua1  Earthquake  F o u r i e r Amplitude Spectrum,Actua1  Earthquake  FREQUENCY  Figure 12.9 Shaking Table Performance,Taft Earthquake  iKZl  195.  0.0  0.34  0.68  1.02  1.37  1.71  Acceleration (g)  Figure 12.10 Envelope of Measured Horizontal Earthquake (0.60g)  Accelerations,Taft  J  1  L  0 M.R.F. A B.M.R.F. -f- F.D.B.F.  O  60.0  90.0  Deflection (mm) (0 60*g) 9  1 2 , 1 1  E n V e l  °P  e  o f  L  a  t  e  r  a  l  M.R.F.  ( 8 X Damping)  —  Measured  —•  Predicted  n  120.0  r  Deflections,Taft  150.0  Earthquake  196.  00  QJ > QJ  #  z  ® Aa.  ID  R  -G  I CM  a a LJL  ^ 3 — — — Measured Predicted  T 0.0  3.0  6.0  T 9.0  12.0  15.0  Moment (kN-m) Figure 12.12 Envelope Earthquake (0.60g)  of  Bending  Moments  in the Beams,Taft  197.  (a) Moment R e s i s t i n g Frame  a CD CM  n 0.0  i  1  1  3.84  1  1  i  i  7.69  i  11.53  i  i  15.38  i  19.23  i  i  23.07  Time (sec.)  (b)  F r i c t i o n Damped Braced Frame  CD CD*  .1  Measured  I  Predicted  CD  _  a />^^li')V>  •5w>i&>y*fiH . Jvf  U CD CLI —I  U_  d  "=T _| I  Q CD CM  o.o  i  r  3.84  i  7.69  r  11.53  15.38  19.23  23.07  Time (sec.)  Figure 12.13 Time-Histories Earthquake (0.60g)  of  Third  Floor  Deflection,Taft  198.  o  |  1  co :  !  |  |  |  1  I  1  C3  l.i!  1= ^»|^Nni^|W^j 71 CO i  o 03..  ni  0.0  3.84  i  7.69  i  i  11.53  i  I r  15.38  19.23  23.07  Time (sec.) Figure 12.14 Recorded Slippage Device,Taft Earthquake (0.60g)  Time-History of Second Floor  co  °.  3  SiWlJislili 111  c o u QJ  C D  I  QJ  / r W| i|lYr  CD CD CD  ro. i  0.0  3.84  7.69  11.53  15.38  19.23  23.07  Time (sec.) Figure 12.15 Recorded Time-Histories of Third Floor Deflection From Both Potentiometers of the F r i c t i o n Damped Braced Frame ,Taft Earthquake (0.60g)  3.0  Acceleration (g) Figure 12.16 Envelope of Measured Horizontal Earthquake (0.9g)  Accelerations,Taft  vO VO  (a)  Moment R e s i s t i n g  200.  Frame  J  L  a  cn (_  QJ  CD  QJ U C_)  I  CC  n 0.0  2.0  r 4.0  i  r  T  6.0  8.0  10.0  12.0  14.0  Time (sec.) (b) Braced Moment R e s i s t i n g Frame  6.0  8.0  10.0  Time (sec.) ( c ) F r i c t i o n Dampea Braced Frame  4.0  6.0  8.0  10.0  12.0  Time (sec.) Figure 1 2 . 1 7 Time-Histories Earthquake (0.9g.)  of Third Floor  Accelerations,Taft  (a)  First  1  Floor  1  201.  Device  1  1  1  1  1  1  1  1  1  to  QJ cn  JUjU , h M « U W\J«v\|  \I\AL  ro o  CL . CL I  o d l  0.0  1  1 2.0  1  1 4.0  1  1 6.0  1  1 1  8.0  i  10.0  1 14.0  1  i  12.0  Time (sec.) (b)  Second F l o o r  Device  _]  L  2.0  6.0  8.0  10.0  Time (sec.) (c)  Third  1  Floor  Device  1  1  1  1  1  1  1  1  1  1  1  1  CD  cri"  QJ cn  ro <=> a. CN .  - •_  "  r.  - A . f-A  1  v"-^-/ 1  ^Arfi  r  „-,(—i  1  1  n A  CL I  CD CD*  I  1  1  1  1  1  1  1  1  1  1  1  1  Time (sec.)  Figure 12.18 Slippage Time-Histories Earthquake (0.9g)  of  Friction  Devices,Taft  Figure 12.19 Slippage Devices,Taft Earthquake  Time-Histories (0.9g)  of  Both  Second  Floor  N3 O  0.0  2.0  4.0  6.0  8.0  10.0  12.0  Time (sec.)  Figure 12.20 Time-History of the Percentage of Energy Dissipated by the F r i c t i o n Devices,Taft Earthquake (0.90g)  14.0  204.  Dimensions Quantity  Absolute System  Engineering System  L MLT'  L F FL-i  Mass density, p Deflection, 5 Stress, a Frequency, f  ML'> L ML' T~-  FTIL-* L  r-i  LT-i  r-i  Acceleration, g  Length, /  1  Force, Q Modulus of elasticity, £ Poisson's ratio, v  1  Table 3.1 Dimensions of Governing Variables for Vibration of E l a s t i c Structure (from ref.:7)  Scale Factors  Group  Quantity  Dimension (engineering units)  (1)  (2)  (3)  Loading  Force, Q Gravitational acceleration, g Time, /  Geometry  Linear dimension, / Displacement, S Frequency,/  Material properties  Modulus, E Stress, a Poisson's ratio, v Density, p  Exact Scaling (4)  Gravity Forces Neglected (5)  F LT'  S Sf  s s?  1  1  T L L  S]' St Si sr s Se  E  1  l/1  r-i  FL' FL'  1  1  E  1  1  FL~  i  Table 3.2 Similitude Requirements Structure (from ref.:7)  s is, E  E  S, s, s,  sf S  E  SE 1 Neglected  for Vibration  of  Elastic  205.  Refined Response  parameter  S impl1f1ed  model  Ref1ned  mode 1  s 1 mp 1st  floor  max.  deflection  4.249mm  4.616mm  1 .09  2nd  floor  max.  deflection  8.909mm  1 1.497mm  1 . 29  3rd  floor  max.  deflection  10.559mm  13.434mm  1 . 27  1st  floor  beam max.  moment  3.634kN-m  4.234kN-m  1.17  2nd f l o o r  beam max.  moment  2.496kN-m  3.257kN-m  1 .30  3rd  beam max.  moment  1.354kN-m  1.521kN-m  1.12  floor 1st  storey max.  column  moment  2nd s t o r e y max. 3rd  1st  1st  column  moment  s t o r e y column max. moment  4.039kN-m  4.143kN-m  1 .03  2.360kN-m  3.055kN-m  1 . 29  1.354kN-m  1 . 52 1kN-m  1.12  storey  max.  shear  5.227kN  5.22<r N  1 .00  2nd s t o r e y  max.  shear  3.719kN  4 . 747kN  1 .28  3rd  max.  shear  2 . 263kN  2.356kN  1 .04  storey  storey  max.  axial  force  2 7 . 138kN  28.531kN  1 .05  2nd s t o r e y  max.  axial  force  1S.394kN  17.527kN  1 .07  3rd  max.  axial  force  6.580kN  7.790kN  1.18  292. 1 16sec.  3.59  storey  C . P .U.  T1 me  T a b l e 4.1 C o m p a r i s o n Centro Earthquake  81.293sec.  between S i m p l i f i e d  and  1 i f i ed  Refined  Models,El  206.  CorrelatIon Device  Linear  »  Regression Coeff tclent  1  -0.994 1  l = -3.12P+148.02  2  L=-2.63P+147.53  -0.9979  3  L=-3.09P+148.47  -0.9971  4  L=-2.67P+142.41  -0.999S  5  L=-3.36P+148.83  -0.9987  6  L=-3.10P+148.19  -0.9811  7  L=-3.68P+146.53  P=Global  Slip  Load  Table 5.1 Results F r i c t i o n Devices  (kN)  L » L e n g t h of  of Linear  -0.9977  Spring  Regressions  (mm)  for Calibration  of  207.  Natural  Frequencies  Natural  Frequencies  (rad/s)  (hz)  Difference  Mode Measured  Predicted  Measured  Predicted  1  2.86  2.831  17.97  17.788  -1 . 0 5  2  9.08  3  14.4  °  9 . 174  57.05  57.642  1 .04  15.930  89.22  100.091  10.63  Table 9.1 Comparison Between Measured Frequencies of the Moment Resisting Frame  (%)  and Predicted Natural  2 0 8 .  Mode  Shapes M.A.C.  Mode measured  . predicted  1  {0.22.0.56.0.80)  {0.21.0.57.0.80)  0.9996  2  {0.55,0.41,-0.72)  {0.57,0.46,-0.68)  0.9998  3  {0.86.-0.47,0.21}  {0.75.-0.58,0.32)  0.9823  Table 9.2 Comparison of Measured and Predicted the Moment Resisting Frame  Mode  Shapes of  209.  Mode  Table Frame  F 1 oor  Method  Modal Damping R a t i o  1  1  Bandw1dth  0.0029  1  1  Log-Decrement  0.0027  1  2  Bandwidth  0.0025  1  2  Log-Decrement  0.0030  1  3  '  Bandwidth  0.0025  3  Log-Decrement  0.0029  2  1  Bandwidth  0.0023  2  1  Log-Decrement  0.0017  2  2  Bandwidth  0.0025  2  2  Log-Decrement  0.OO17  2  3  Bandwidth  0.0024  2  3  3  1  Bandwidth  0.0109  3  1  Log-Decrement  0.004 1  3  2  Bandwidth  0.0103  3  2  Log-Decrement  0.004 1  3  3  Bandwidth  0.0076  Log-Decrement  0.0017  9.3 Measured Modal Damping Ratios of the Moment Resisting  210.  Mode  F1 oor  Method  Modal Damping R a t i o  1  1  Log-Decrement  0.0146  1  2  Log-Decrement  0.0125  1  3  Log-Decrement  0.0109  Table 10.1 Measured R e s i s t i n g Frame  Modal  Damping  Ratios  of the Braced  Moment  Natural  Frequencies Natural  (rad/s)  (hz)  D i f f e r e n c e {%)  Mode Measured  Predicted  1  7 .03  7.012  2  18.4  19.472  3  *  28.316  Measured  44 . 17  Predicted  44.059  -0. 26  115.6  122 . 348  5.83  *  177.919  TABLE 11.1 C o m p a r i s o n Between Frequencies of the F r i c t i o n Amplitude Excitations * Unable to V e r i f y  211.  Frequencies  Measured Damped  and P r e d i c t e d Natural Braced Frame Under Low  Experimentally  Mode Shapes M.A.C.  Mode predicted  measured  1  {0.29.0.60,0.75.)  (0.40,0.50.0.77)  0.9661  2  {0.66.0.27,-0.70)  (0.58.O.19,-0.79}  0.9972  T a b l e 1 1 . 2 C o m p a r i s o n of M e a s u r e d vs P r e d i c t e d Mode t h e F r i c t i o n Damped B r a c e d Frame Under Low A m p l i t u d e  Shapes of Excitations  212.  Floor  Method  Modal Damping R a t i o  1  1  Log-Decrement  0.0065  1  2  Log-Decrement  0.0060  1  3  Log-Oecrement  0.0056  2  1  Log-Decrement  0.0043  2  3  Log-Decrement  0.0034  Mode  T a b l e 11.3 M e a s u r e d B r a c e d Frame  Modal  Damping  Ratios  of the F r i c t i o n  Damped  213.  Bending Moment (kN-m) Intenslty  Type of Measured  1  2  3  Predicted  M.R F .  1 . 48  2.11  B.M.R . F .  0. 47  0. 46  F . D . BF .  0.22  0. 34  M.R. F .  4 .05  4.18  B.M.R F .  1 .06  0.96  F . D . BF .  0.45  0.70  M.R. F .  7.07  10.01  B.M.R F .  2.61  2.82  F . 0 . Br.  0.94  1 .47  T a b l e 12.1 E n v e l o p e s of B e n d i n g Moments i n t h e Base Location of Strain Gages,Newmark-Blume-Kapur E a r t h q u a k e , I n t e n s i t i e s 1,2,3  Column a t Artificial  214.  Bending Moment (kN-m) Frame Measured  Pred i c t e d  FD.B.F.  2.70  3 . 50  B.M.R.F.  6.50  M.R.F.  9.60  * Unable t o V e r i f y  *  10. 30  Experimentally  T a b l e 12.2 E n v e l o p e of B e n d i n g Moments i n the Base L o c a t i o n of S t r a i n G a g e s , T a f t E a r t h q u a k e (0.60g)  Column  215.  Natural  Frame  F r e q u e n c i e s (Hz)  Measured  Mode D r»aH \ f t o H r T S Q ' ^- lc<J  Frame* 1  M.R.F.  B.M.R.F.  1  2.831  2.86  2.81  2  9. 174  9.08  8.91  3  15.93  14.4  13.90  1  5 .087  5 . 29  6 . 10  2  14.581  *  *  22.025  *  *  3 1 F.D.B.F.  2 3  •Unable t o V e r i f y  T a b l e 12.3 Frames  Frame#2  7.012  7 .03  19.472  18.4  28.316  •  6.00  *  Experimentally  C o m p a r i s o n of N a t u r a l F r e q u e n c i e s  f o r the  Two  Model  216.  Modal Damping R a t i o Frame  Mode Frame*1  M.R.F.  B.M.R.F.  1  0.0028  0.0026  2  0.0021  0.0023  3  0.0074  0.0040  1  0.0127  0.0058 *  2  1  0.0060  2  0.0039  *  3  *  *  * Unable t o V e r i f y  Table  12.4  *  •  3  F.D.B.F.  Frame/f2  0.0058  Experimentally  C o m p a r i s o n of Damping  Ratios  f o r t h e Two  Model  Frames  217.  * M.R.F.  Mmax/Mp  Mp(kN-m)  ^max(mm)  Md/Mp  Ml/Mp  S75x8  9.57  102.02  0 . 16  0.84  1 .00  S75x8  9.57  19.47  0 . 16  0 . 32  0 . 48  0 . 16  0.17  0.33  C r o s s - S e c t Ion  Frame  CD N 1/1  «  B.M.R.F.  C  *  01 •  •  S75x8  F.D.B.F.  9.57  9.05  I 5.89  10.48  0 . 25  0 . 19  0 . 44  HSS 3 8 . 1x38. 1x4.78mm  2.76  13.87  0 . 54  0 . 10  0.64  HSS  1 .80  14.32  SLP-3"  tze  F.D.B.F.  educed  1/5  F.D.B.F.  1 1  F.D.B.F.  38.1x38.1x2.54mm  0.82  0 . 12  0 . 94  » T e s t e d on S h a k i n g Mp=Plast1c  Md=Max1mum Moment M1'Maximum  Table  Moment  Moment  From Dead  Load  From I n e r t i a  Mmax=Max1mum Moment  From I n e r t i a  ^max=Max1mum T h i r d  Floor  Load  Lateral  F o r c e s + Dead  Load  Deflection  T a b l e 13.1 D e s i g n of a Reduced Size Friction Damped Frame,Newmark-Blume-Kapur A r t i f i c i a l E a r t h q u a k e (0.30g)  Braced  218. Appendix  Q* **********  C C C C C C C C C C C C  *  A:  Listing  of  Program  "VIBRATION"  **********************************************************  PROGRAM VIBRATION BY ANDRE FILIATRAULT APRIL 1985 UNIVERSITY OF BRITISH-COLUMBIA GENERATION OF THE AMPLITUDE FREQUENCY RESPONSE FUNCTION FOR AN ELASTIC M.D.O.F.S. WITH MODAL VISCOUS DAMPING UNDER HARMONIC BASE EXCITATION IN PHASE LOGICAL UNIT 5:DATA-FILE CONTAINING THE NO. OF DEGREES OF FREEDOM,THE NATURAL FREQUENCIES ,MODE SHAPES,DAMPING RATIOS AND FREQUENCY RANGE LOGICAL UNIT 6:OUTPUT F I L E  Q*** *******************************************************  ************  IMPLICIT REAL*8(A-H,0-Z) DIMENSION A ( 3 , 3 ) , A I N V ( 3 , 3 ) , D l ( 3 , 3 ) , D 2 ( 3 , 3 ) , G ( 3 ) , H ( 3 ) , H W ( l 0 0 0 , 3 ) DIMENSION W(1000),WN<3),ZETA(3),DA(3 , 3) , ADA(3,3),IPERM(6) C C C  READ DATA FROM DATA-FILE  10  C  100 101 102 103 111  C  READ(5 , 100)N READ(5,101)(WN(I),I=1,N) DO 10 I=1,N READ(5,102)(A(J,I),J=1,N) CONTINUE READ(5,10 3 ) ( Z E T A ( I ) , I = 1 , N) READ(5,111) Wl,W2 FORMAT(15) FORMAT(3F7.2) FORMAT(3F7.2) FORMAT(3F6.4) FORMAT(2F7.2) WRITE DATA IN OUTPUT F I L E  C  C C C C  WRITE(6,104) WRITE(6,105) WRITE(6,106)N WRITE(6,107)(WN(I),1 = 1 ,N) WRITE(6,108) DO 20 1=1,N WRITE(6,109)(A(I,J),J=1,N) 20 CONTINUE WRITE(6,1 1 0 ) ( Z E T A ( I ) , I = 1 ,N) WRITE(6,116)W1,W2 WRITE(6,104) 104 FORMAT(//,80('*'),//) 105 FORMAT(T25,'PROGRAM VIBRATION') 106 FORMAT(T25,'NO. DEGREES OF FREEDOM CONSIDERED= ',15) 107 FORMAT(T25,'NATURAL FREQUENCIES (RAD/S)= ' ,3 (F7.2,1X)) 108 FORMAT(T25,'MODAL MATRIX') 109 FORMAT(T25,3(F4.2,IX)) 110 FORMAT(T25,'MODAL DAMPING RATIOS= ',3(F6.4,1X)) 116 FORMAT(T25,'FREQUENCY RANGE (RAD/S)= ' , 2X, F7.2,'TO '.F7.2) CALL UBC MATRIX SUBROUTINE TO INVERSE THE MODAL MATRIX CALL INV(N,N,A,I PERM,N,AINV,DET,JEXP,COND)  219. C C  SE? UP FREQUENCY  RANGE  DO 2 5 I=1,N DO 2 6 J=1,N DI (I ,J)=0.D0 D2(I,J)=0.D0 26 CONTINUE 25 CONTINUE DELW=(W2-W1)/!000.D0 WBAR=W1-DELW  C C C  STARTS LOOP THROUGH FREQUENCY RANGE DO 30 1=1,1000 WBAR=W3AR+DELW DO 40 J = 1 , N DENO=DABS((WN(J)**2.D0-WBAR**2.DO))**2.D0 DENO=DENO+(2.DO*2ETA(J)*WN(J)*WBAR)**2.DO D U J , J ) = (2* ZETA(J)*WN(J)*WBAR)/DENO D2(J,J)=(WN(J)**2.D0-WBAR**2.D0)/DENO 40 CONTINUE CALL DGMULT(D1,AINV,DA,N,N,N,N,N,N) CALL DGMULT(A,DA,ADA,N,N,N,N,N,N) DO 50 J=1,N G(J)=0.D0 DO 60 K=1,N G(J)=G(J)+ADA(J,K) 60 CONTINUE 50 CONTINUE CALL DGMULT(D2,AINV,DA,N,N,N,N,N,N) CALL DGMULT(A,DA,ADA,N,N,N,N,N,N) DO 70 J=1,N H(J)=0.D0 DO 80 K=1,N H(J)=H(J)+ADA(J,K) 80 CONTINUE H(J)=-H(J) 70 CONTINUE DO 90 J=1,N HW(I,J) = (DABS(G(J))**2.D0+DABS(H(J))**2.DO)*WBAR**4 .DO HW(I,J)=HW(I,J)+1.D0-(2.D0*H(J)*WBAR**2.D0) HW(I,J)=DSQRT(HW(I,J)) HW(I,J)=20.D0*DLOG10(HW(I,J)) 90 CONTINUE W(I)=WBAR 30 CONTINUE  C C C  WRITE RESULTS IN OUTPUT  97 95 112 113  FILE  WRITE(6,104) WRITE(6,112) DO 95 J=1 ,N WRITE(6,113) J WRITE(6,114) DO 97 K=1,1000 WRITE(6,115) W(K),HW(K,J) CONTINUE CONTINUE FORMAT(T25,'RESULTS') FORMAT(//,'DEGREE OF FREEDOM NO. ',12)  114 FORMAT(T15,'FREQUENCY W (RAD/S) ',5X,'H(W)*) 115 FORMAT(T15,F7.2,T40,F7.2) STOP END  Appendix  C C C C C C C C C C C C C C C C C C  C C C  B:  Listing  of  Program  "ENERGY  220.  THIS PROGRAM CALCULATE THE ENERGY DISSIPATED BY THE FRICTION DEVICES AND THE TOTAL ENERGY INPUTED IN THE FRICTION DAMPED BRACED FRAME DURING A SEISMIC TEST ON THE SHAKING TABLE UNITS USED MUST BE CONSISTENT UNIT 1=RECORDED FIRST FLOOR ACCELERATION UNIT 2=REC0RDED SECOND FLOOR ACCELERATION UNIT 3=RECORDED THIRD FLOOR ACCELERATION UNIT 4=RECORDED TABLE DISPLACEMENT UNIT 6=OUTPUT F I L E CONTAINING ENERGY INPUT UNIT 7=RECORDED SLIPPAGE OF F I R S T FLOOR DEVICE UNIT 8=RECORDED SLIPPAGE OF SECOND FLOOR DEVICE UNIT 9=RECORDED SLIPPAGE OF THIRD FLOOR DEVICE UNIT 10=OUTPUT F I L E CONTAINING ENERGY D I S S I P A T E D BY FRICTION UNIT 11=OUTPUT F I L E CONTAINING % ENERGY D I S S I P A T E D IMPLICIT REAL*8(A-H,0-Z) DIMENSION A 1 ( 5 0 0 0 ) , A 2 ( 5 0 0 0 ) ,A3 ( 5 0 0 0 ) , X G ( 5 0 0 0 ) , S 1 (5000) ,S2(5000),S3 *(5000),WM(3),EI(3),ED(3),P(3) READ DATA WRITE(5,100) 100 FORMAT('ENTER NUMBER OF DATA POINTS') READ(5,*) N WRITE(5,10l) 101 FORMAT('ENTER THE MASS OF EACH CONCRETE BLOCK') READ(5,*)(WM(I),1=1,3) WRITE(5,102) 102 FORMAT('ENTER LOCAL S L I P LOAD OF FRICTION DEVICES') READ(5,*) P S L I P WRITE(5,111) READ(5,*) DELTAT 111 FORMAT('ENTER SAMPLE INTERVAL ') AN=N/8 NLINE=DINT(AN)  C C C  DO 10 1=1,NLINE K=(8*I)-7 K7=K+7 READ(1,103)(Al(J),J=K,K7) READ(2,103)(A2(J),J=K,K7) R E A D O , 103) (A3 ( J ) ,J=K,K7) READ(4,103)(XG(J),J=K,K7) READ(7,103)(SI(J),J=K,K7) READ(8,103)(S2(J),J=K,K7) READ(9,103)(S3(J),J=K,K7) 10 CONTINUE 103 FORMAT(8F10.5) N=NLINE*8 CALCULATE ENERGY WRITE(5,104) WRITE(6,104) 104 FORMAT('***PROGRAM WRITE(5,105) N WRITE(6,105) N  ENERGY***')  221.  105 FORMAT{'NUMBER OF DATA POINTS CONSIDERED',15) DO 30 1=1,3 WRITE(5,106) I,WM(I) WRITE(6,106) I,WM(I) 30 CONTINUE 106 FORMAT('MASS OF FLOOR NO. ',12,'=',F15.5) WRITE(5,107) PSLIP WRITE(6,107) PSLIP 107 FORMAT('LOCAL SLIP LOAD OF FRICTION DEVICES=',F 1 5 . 5) WRITE(5,108)DELTAT WRITE(6,108)DELTAT 108 FORMAT('SAMPLE INTERVAL=',F15.5) WRITE(6,109) WRITE(10,115) WRITE(11,116) 115 FORMAT('TIME',5X,'ENERGY DISSIPATED') 116 FORMAT('TIME',5X,'% ENERGY DISSIPATED') 109 FORMAT('TIME',5X,'ENERGY INPUT') DO 15 1=1,3 EI (I)=0.D0 ED(I)=0.D0 15 CONTINUE EITOT=0.D0 EDTOT=0.D0 DO 20 I=2,N DELTAX=-(XG(I)-XG(l-1)) EI(1)=EI(1)+(WM(1)*(A1(I)+A1(1-1))*DELTAX) C MINUS SINCE TABLE LVDT CALIBRATED IN REVERSE WITH ACCELEROMETER EI(2)=EI(2)+(WM(2)*(A2(I)+A2(I-1))*DELTAX) EI(3)=EI(3)+(WM(3)*(A3(l)+A3(I-1))*DELTAX) ED(1)=ED(1)+DA3S(S1(I)-S1(1-1)) ED(2)=ED(2)+DABS(S2(I ) - S 2 ( I - 1 ) ) ED(3)=ED(3)+DABS(S3(I)-S3(l-1)) TIME=DELTAT*I EITOT=EITOT+((EI(1)+EI(2)+EI(3))/2.DO) EDTOT=EDTOT+((ED(1)+ED(2)+ED(3))*4.D0*PSLIP) PTOT=EDTOT/EITOT*100.DO WRITE(6,110) TIME,EITOT WRITE(10,110) TIME.EDTOT WRITE(11,110) TIME,PTOT 20 CONTINUE 110 FORMAT(F7.4,F15.5) WRITE(5,112) EITOT WRITE(6,112) EITOT WRITE(5,113) EDTOT WRITE(6,113) EDTOT WRITE(5,114) PTOT WRITE(6,114) PTOT 112 FORMAT('TOTAL ENERGY INDUCED IN THE STRUCTURE=*,F15.5) 113 FORMAT('TOTAL ENERGY DISSIPATED BY FRICTION=',F15.5) 114 FORMAT('% TOTAL ENERGY DISSIPATED BY FRICTION-',F15.5) STOP END  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 9 1
Sweden 5 0
Mexico 4 0
Canada 3 2
United Kingdom 3 4
China 3 23
France 2 3
India 1 1
Turkey 1 0
Poland 1 0
Iran 1 11
Algeria 1 1
Italy 1 0
City Views Downloads
Unknown 7 8
Stockholm 5 0
Mexico City 4 0
Beijing 3 0
Ashburn 3 0
Jacksonville 2 0
Richmond 2 2
Sheffield 2 3
Palatine 2 0
Gorizia 1 0
London 1 0
Algiers 1 1
Sunnyvale 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0062844/manifest

Comment

Related Items